From 537a37e57f522f44a21864cd2cab66d9c5c0049c Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 1 Sep 2025 18:10:05 +0200 Subject: [PATCH 01/16] =?UTF-8?q?simple=20`Id`=20followed=20by=20`?= =?UTF-8?q?=E2=89=83`?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- src/graph-theory/equivalences-undirected-graphs.lagda.md | 4 ++-- src/graph-theory/morphisms-undirected-graphs.lagda.md | 2 +- src/group-theory/category-of-semigroups.lagda.md | 2 +- src/group-theory/homomorphisms-concrete-groups.lagda.md | 2 +- src/group-theory/homomorphisms-groups.lagda.md | 2 +- src/group-theory/isomorphisms-abelian-groups.lagda.md | 2 +- src/group-theory/isomorphisms-groups.lagda.md | 2 +- .../precategory-of-orbits-monoid-actions.lagda.md | 2 +- src/lists/lists.lagda.md | 2 +- src/lists/tuples.lagda.md | 2 +- src/order-theory/order-preserving-maps-posets.lagda.md | 2 +- src/order-theory/order-preserving-maps-preorders.lagda.md | 2 +- src/species/morphisms-finite-species.lagda.md | 2 +- .../infinite-cyclic-types.lagda.md | 2 +- src/synthetic-homotopy-theory/interval-type.lagda.md | 2 +- src/univalent-combinatorics/cyclic-finite-types.lagda.md | 2 +- src/univalent-combinatorics/finite-types.lagda.md | 6 +++--- src/univalent-combinatorics/necklaces.lagda.md | 2 +- src/univalent-combinatorics/partitions.lagda.md | 2 +- 19 files changed, 22 insertions(+), 22 deletions(-) diff --git a/src/graph-theory/equivalences-undirected-graphs.lagda.md b/src/graph-theory/equivalences-undirected-graphs.lagda.md index 4b98ef0d5e..8fef4d2b7f 100644 --- a/src/graph-theory/equivalences-undirected-graphs.lagda.md +++ b/src/graph-theory/equivalences-undirected-graphs.lagda.md @@ -231,7 +231,7 @@ module _ extensionality-equiv-Undirected-Graph : (f g : equiv-Undirected-Graph G H) → - Id f g ≃ htpy-equiv-Undirected-Graph f g + (f ＝ g) ≃ htpy-equiv-Undirected-Graph f g pr1 (extensionality-equiv-Undirected-Graph f g) = htpy-eq-equiv-Undirected-Graph f g pr2 (extensionality-equiv-Undirected-Graph f g) = @@ -272,7 +272,7 @@ module _ ( equiv-eq-Undirected-Graph) extensionality-Undirected-Graph : - (H : Undirected-Graph l1 l2) → Id G H ≃ equiv-Undirected-Graph G H + (H : Undirected-Graph l1 l2) → (G ＝ H) ≃ equiv-Undirected-Graph G H pr1 (extensionality-Undirected-Graph H) = equiv-eq-Undirected-Graph H pr2 (extensionality-Undirected-Graph H) = is-equiv-equiv-eq-Undirected-Graph H diff --git a/src/graph-theory/morphisms-undirected-graphs.lagda.md b/src/graph-theory/morphisms-undirected-graphs.lagda.md index aa80fba80a..79a56e28d8 100644 --- a/src/graph-theory/morphisms-undirected-graphs.lagda.md +++ b/src/graph-theory/morphisms-undirected-graphs.lagda.md @@ -173,7 +173,7 @@ module _ ( htpy-eq-hom-Undirected-Graph f) extensionality-hom-Undirected-Graph : - (f g : hom-Undirected-Graph G H) → Id f g ≃ htpy-hom-Undirected-Graph f g + (f g : hom-Undirected-Graph G H) → (f ＝ g) ≃ htpy-hom-Undirected-Graph f g pr1 (extensionality-hom-Undirected-Graph f g) = htpy-eq-hom-Undirected-Graph f g pr2 (extensionality-hom-Undirected-Graph f g) = diff --git a/src/group-theory/category-of-semigroups.lagda.md b/src/group-theory/category-of-semigroups.lagda.md index a610b13437..74c50be246 100644 --- a/src/group-theory/category-of-semigroups.lagda.md +++ b/src/group-theory/category-of-semigroups.lagda.md @@ -40,7 +40,7 @@ is-large-category-Semigroup G = fundamental-theorem-id (is-torsorial-iso-Semigroup G) (iso-eq-Semigroup G) extensionality-Semigroup : - {l : Level} (G H : Semigroup l) → Id G H ≃ iso-Semigroup G H + {l : Level} (G H : Semigroup l) → (G ＝ H) ≃ iso-Semigroup G H pr1 (extensionality-Semigroup G H) = iso-eq-Semigroup G H pr2 (extensionality-Semigroup G H) = is-large-category-Semigroup G H diff --git a/src/group-theory/homomorphisms-concrete-groups.lagda.md b/src/group-theory/homomorphisms-concrete-groups.lagda.md index d61de489f6..1f1c9f34c0 100644 --- a/src/group-theory/homomorphisms-concrete-groups.lagda.md +++ b/src/group-theory/homomorphisms-concrete-groups.lagda.md @@ -128,7 +128,7 @@ module _ ( f) extensionality-hom-Concrete-Group : - (g : hom-Concrete-Group G H) → Id f g ≃ htpy-hom-Concrete-Group g + (g : hom-Concrete-Group G H) → (f ＝ g) ≃ htpy-hom-Concrete-Group g extensionality-hom-Concrete-Group = extensionality-hom-∞-Group ( ∞-group-Concrete-Group G) diff --git a/src/group-theory/homomorphisms-groups.lagda.md b/src/group-theory/homomorphisms-groups.lagda.md index a4e286a07f..13d329ef42 100644 --- a/src/group-theory/homomorphisms-groups.lagda.md +++ b/src/group-theory/homomorphisms-groups.lagda.md @@ -157,7 +157,7 @@ module _ ( semigroup-Group H) extensionality-hom-Group : - (f g : hom-Group G H) → Id f g ≃ htpy-hom-Group f g + (f g : hom-Group G H) → (f ＝ g) ≃ htpy-hom-Group f g pr1 (extensionality-hom-Group f g) = htpy-eq-hom-Group f g pr2 (extensionality-hom-Group f g) = is-equiv-htpy-eq-hom-Group f g diff --git a/src/group-theory/isomorphisms-abelian-groups.lagda.md b/src/group-theory/isomorphisms-abelian-groups.lagda.md index 9d7b07d36e..2fe037a367 100644 --- a/src/group-theory/isomorphisms-abelian-groups.lagda.md +++ b/src/group-theory/isomorphisms-abelian-groups.lagda.md @@ -184,7 +184,7 @@ iso-eq-Ab A B p = iso-eq-Group (group-Ab A) (group-Ab B) (ap pr1 p) abstract equiv-iso-eq-Ab' : - {l : Level} (A B : Ab l) → Id A B ≃ iso-Ab A B + {l : Level} (A B : Ab l) → (A ＝ B) ≃ iso-Ab A B equiv-iso-eq-Ab' A B = ( extensionality-Group' (group-Ab A) (group-Ab B)) ∘e ( equiv-ap-inclusion-subtype is-abelian-prop-Group {A} {B}) diff --git a/src/group-theory/isomorphisms-groups.lagda.md b/src/group-theory/isomorphisms-groups.lagda.md index 0ab46ae767..4a65c7e7c5 100644 --- a/src/group-theory/isomorphisms-groups.lagda.md +++ b/src/group-theory/isomorphisms-groups.lagda.md @@ -231,7 +231,7 @@ module _ iso-eq-Group = iso-eq-Large-Precategory Group-Large-Precategory G abstract - extensionality-Group' : (H : Group l) → Id G H ≃ iso-Group G H + extensionality-Group' : (H : Group l) → (G ＝ H) ≃ iso-Group G H extensionality-Group' H = ( extensionality-Semigroup (semigroup-Group G) (semigroup-Group H)) ∘e ( equiv-ap-inclusion-subtype is-group-prop-Semigroup {s = G} {t = H}) diff --git a/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md b/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md index 5a51488a0e..037435ec5b 100644 --- a/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md +++ b/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md @@ -97,7 +97,7 @@ module _ extensionality-hom-orbit-action-Monoid : {x y : type-action-Monoid M X} (f g : hom-orbit-action-Monoid x y) → - Id f g ≃ htpy-hom-orbit-action-Monoid f g + (f ＝ g) ≃ htpy-hom-orbit-action-Monoid f g pr1 (extensionality-hom-orbit-action-Monoid f g) = htpy-eq-hom-orbit-action-Monoid f g pr2 (extensionality-hom-orbit-action-Monoid f g) = diff --git a/src/lists/lists.lagda.md b/src/lists/lists.lagda.md index dfa10f3084..2f0399cf9c 100644 --- a/src/lists/lists.lagda.md +++ b/src/lists/lists.lagda.md @@ -225,7 +225,7 @@ is-equiv-Eq-eq-list l l' = ( is-retraction-eq-Eq-list l l') equiv-Eq-list : - {l1 : Level} {A : UU l1} (l l' : list A) → Id l l' ≃ Eq-list l l' + {l1 : Level} {A : UU l1} (l l' : list A) → (l ＝ l') ≃ Eq-list l l' equiv-Eq-list l l' = pair (Eq-eq-list l l') (is-equiv-Eq-eq-list l l') diff --git a/src/lists/tuples.lagda.md b/src/lists/tuples.lagda.md index 5663fcc5b1..a7739f1bd6 100644 --- a/src/lists/tuples.lagda.md +++ b/src/lists/tuples.lagda.md @@ -154,7 +154,7 @@ module _ ( is-section-eq-Eq-tuple n u v) ( is-retraction-eq-Eq-tuple n u v) - extensionality-tuple : (n : ℕ) → (u v : tuple A n) → Id u v ≃ Eq-tuple n u v + extensionality-tuple : (n : ℕ) → (u v : tuple A n) → (u ＝ v) ≃ Eq-tuple n u v extensionality-tuple n u v = (Eq-eq-tuple n u v , is-equiv-Eq-eq-tuple n u v) ``` diff --git a/src/order-theory/order-preserving-maps-posets.lagda.md b/src/order-theory/order-preserving-maps-posets.lagda.md index 1638130acd..c1f944d16f 100644 --- a/src/order-theory/order-preserving-maps-posets.lagda.md +++ b/src/order-theory/order-preserving-maps-posets.lagda.md @@ -99,7 +99,7 @@ module _ is-equiv-htpy-eq-hom-Preorder (preorder-Poset P) (preorder-Poset Q) extensionality-hom-Poset : - (f g : hom-Poset P Q) → Id f g ≃ htpy-hom-Poset f g + (f g : hom-Poset P Q) → (f ＝ g) ≃ htpy-hom-Poset f g extensionality-hom-Poset = extensionality-hom-Preorder (preorder-Poset P) (preorder-Poset Q) diff --git a/src/order-theory/order-preserving-maps-preorders.lagda.md b/src/order-theory/order-preserving-maps-preorders.lagda.md index a45df09656..32c3406fb3 100644 --- a/src/order-theory/order-preserving-maps-preorders.lagda.md +++ b/src/order-theory/order-preserving-maps-preorders.lagda.md @@ -125,7 +125,7 @@ module _ ( htpy-eq-hom-Preorder f) extensionality-hom-Preorder : - (f g : hom-Preorder P Q) → Id f g ≃ htpy-hom-Preorder f g + (f g : hom-Preorder P Q) → (f ＝ g) ≃ htpy-hom-Preorder f g pr1 (extensionality-hom-Preorder f g) = htpy-eq-hom-Preorder f g pr2 (extensionality-hom-Preorder f g) = is-equiv-htpy-eq-hom-Preorder f g diff --git a/src/species/morphisms-finite-species.lagda.md b/src/species/morphisms-finite-species.lagda.md index fbf3a5e9aa..85b5ea4044 100644 --- a/src/species/morphisms-finite-species.lagda.md +++ b/src/species/morphisms-finite-species.lagda.md @@ -142,7 +142,7 @@ is-equiv-htpy-eq-hom-finite-species F G f = extensionality-hom-finite-species : {l1 l2 l3 : Level} (F : finite-species l1 l2) (G : finite-species l1 l3) (f g : hom-finite-species F G) → - Id f g ≃ htpy-hom-finite-species F G f g + (f ＝ g) ≃ htpy-hom-finite-species F G f g pr1 (extensionality-hom-finite-species F G f g) = htpy-eq-hom-finite-species F G f g pr2 (extensionality-hom-finite-species F G f g) = diff --git a/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md b/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md index bab47be956..4329545999 100644 --- a/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md +++ b/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md @@ -115,7 +115,7 @@ module _ is-equiv-equiv-eq-Cyclic-Type zero-ℕ X extensionality-Infinite-Cyclic-Type : - (Y : Infinite-Cyclic-Type l1) → Id X Y ≃ equiv-Infinite-Cyclic-Type Y + (Y : Infinite-Cyclic-Type l1) → (X ＝ Y) ≃ equiv-Infinite-Cyclic-Type Y extensionality-Infinite-Cyclic-Type = extensionality-Cyclic-Type zero-ℕ X module _ diff --git a/src/synthetic-homotopy-theory/interval-type.lagda.md b/src/synthetic-homotopy-theory/interval-type.lagda.md index eb421618ba..983521bbb9 100644 --- a/src/synthetic-homotopy-theory/interval-type.lagda.md +++ b/src/synthetic-homotopy-theory/interval-type.lagda.md @@ -94,7 +94,7 @@ module _ ( pr2 (pr2 y)) ( β))) - extensionality-Data-𝕀 : (x y : Data-𝕀 P) → Id x y ≃ Eq-Data-𝕀 x y + extensionality-Data-𝕀 : (x y : Data-𝕀 P) → (x ＝ y) ≃ Eq-Data-𝕀 x y extensionality-Data-𝕀 (pair u (pair v α)) = extensionality-Σ ( λ {u'} vα' p → diff --git a/src/univalent-combinatorics/cyclic-finite-types.lagda.md b/src/univalent-combinatorics/cyclic-finite-types.lagda.md index 160d23bc13..45597bd99a 100644 --- a/src/univalent-combinatorics/cyclic-finite-types.lagda.md +++ b/src/univalent-combinatorics/cyclic-finite-types.lagda.md @@ -211,7 +211,7 @@ module _ ( equiv-eq-Cyclic-Type k X) extensionality-Cyclic-Type : - (Y : Cyclic-Type l k) → Id X Y ≃ equiv-Cyclic-Type k X Y + (Y : Cyclic-Type l k) → (X ＝ Y) ≃ equiv-Cyclic-Type k X Y pr1 (extensionality-Cyclic-Type Y) = equiv-eq-Cyclic-Type k X Y pr2 (extensionality-Cyclic-Type Y) = is-equiv-equiv-eq-Cyclic-Type Y diff --git a/src/univalent-combinatorics/finite-types.lagda.md b/src/univalent-combinatorics/finite-types.lagda.md index ab9b151fe7..b89d762ec6 100644 --- a/src/univalent-combinatorics/finite-types.lagda.md +++ b/src/univalent-combinatorics/finite-types.lagda.md @@ -621,7 +621,7 @@ id-equiv-Finite-Type : {l : Level} (X : Finite-Type l) → equiv-Finite-Type X X id-equiv-Finite-Type X = id-equiv extensionality-Finite-Type : - {l : Level} (X Y : Finite-Type l) → Id X Y ≃ equiv-Finite-Type X Y + {l : Level} (X Y : Finite-Type l) → (X ＝ Y) ≃ equiv-Finite-Type X Y extensionality-Finite-Type = extensionality-subuniverse is-finite-Prop is-torsorial-equiv-Finite-Type : @@ -657,7 +657,7 @@ id-equiv-fam-Finite-Type Y x = id-equiv extensionality-fam-Finite-Type : {l1 l2 : Level} {X : UU l1} (Y Z : X → Finite-Type l2) → - Id Y Z ≃ equiv-fam-Finite-Type Y Z + (Y ＝ Z) ≃ equiv-fam-Finite-Type Y Z extensionality-fam-Finite-Type = extensionality-fam-subuniverse is-finite-Prop ``` @@ -706,7 +706,7 @@ eq-equiv-Type-With-Cardinality-ℕ k X Y = equiv-equiv-eq-Type-With-Cardinality-ℕ : {l : Level} (k : ℕ) (X Y : Type-With-Cardinality-ℕ l k) → - Id X Y ≃ equiv-Type-With-Cardinality-ℕ k X Y + (X ＝ Y) ≃ equiv-Type-With-Cardinality-ℕ k X Y pr1 (equiv-equiv-eq-Type-With-Cardinality-ℕ k X Y) = equiv-eq-Type-With-Cardinality-ℕ k pr2 (equiv-equiv-eq-Type-With-Cardinality-ℕ k X Y) = diff --git a/src/univalent-combinatorics/necklaces.lagda.md b/src/univalent-combinatorics/necklaces.lagda.md index 3c1af9b26c..a5cbb95309 100644 --- a/src/univalent-combinatorics/necklaces.lagda.md +++ b/src/univalent-combinatorics/necklaces.lagda.md @@ -111,7 +111,7 @@ module _ where extensionality-necklace : - (N1 N2 : necklace l m n) → Id N1 N2 ≃ equiv-necklace m n N1 N2 + (N1 N2 : necklace l m n) → (N1 ＝ N2) ≃ equiv-necklace m n N1 N2 extensionality-necklace N1 = extensionality-Σ ( λ {X} f e → diff --git a/src/univalent-combinatorics/partitions.lagda.md b/src/univalent-combinatorics/partitions.lagda.md index 1d697a9f42..3ecece3020 100644 --- a/src/univalent-combinatorics/partitions.lagda.md +++ b/src/univalent-combinatorics/partitions.lagda.md @@ -194,7 +194,7 @@ pr2 (pr2 (id-equiv-partition-Finite-Type X P)) = refl-htpy extensionality-partition-Finite-Type : {l1 l2 l3 : Level} (X : Finite-Type l1) (P Q : partition-Finite-Type l2 l3 X) → - Id P Q ≃ equiv-partition-Finite-Type X P Q + (P ＝ Q) ≃ equiv-partition-Finite-Type X P Q extensionality-partition-Finite-Type X P = extensionality-Σ ( λ {Y} Zf e → From ceae99b4145dc911be31f23f2841f5284a30ed57 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 1 Sep 2025 18:28:01 +0200 Subject: [PATCH 02/16] `Id` args without parentheses --- .../commutative-rings.lagda.md | 2 +- .../euclidean-domains.lagda.md | 2 +- .../integral-domains.lagda.md | 2 +- ...trict-inequality-rational-numbers.lagda.md | 2 +- .../universal-property-integers.lagda.md | 4 ++-- .../commutative-finite-rings.lagda.md | 2 +- src/finite-algebra/finite-fields.lagda.md | 2 +- src/finite-algebra/finite-rings.lagda.md | 4 ++-- .../abstract-quaternion-group.lagda.md | 4 ++-- .../delooping-sign-homomorphism.lagda.md | 4 ++-- .../groups-of-order-2.lagda.md | 4 ++-- src/finite-group-theory/permutations.lagda.md | 2 +- src/foundation-core/identity-types.lagda.md | 4 ++-- ...universal-property-identity-types.lagda.md | 2 +- .../equivalences-undirected-graphs.lagda.md | 8 ++++---- .../morphisms-undirected-graphs.lagda.md | 4 ++-- src/group-theory/category-of-groups.lagda.md | 2 +- .../category-of-semigroups.lagda.md | 2 +- src/group-theory/commutative-monoids.lagda.md | 2 +- .../equivalences-semigroups.lagda.md | 2 +- src/group-theory/groups.lagda.md | 2 +- .../homomorphisms-abelian-groups.lagda.md | 4 ++-- .../homomorphisms-concrete-groups.lagda.md | 2 +- .../homomorphisms-groups.lagda.md | 4 ++-- .../homomorphisms-semigroups.lagda.md | 4 ++-- .../isomorphisms-abelian-groups.lagda.md | 4 ++-- src/group-theory/isomorphisms-groups.lagda.md | 2 +- .../isomorphisms-monoids.lagda.md | 2 +- .../isomorphisms-semigroups.lagda.md | 2 +- src/group-theory/loop-groups-sets.lagda.md | 2 +- ...category-of-orbits-monoid-actions.lagda.md | 4 ++-- .../subgroups-abelian-groups.lagda.md | 2 +- src/linear-algebra/matrices.lagda.md | 4 ++-- src/lists/lists.lagda.md | 8 ++++---- src/lists/tuples.lagda.md | 6 +++--- ...-colimits-in-homotopy-type-theory.lagda.md | 2 +- src/order-theory/decidable-subposets.lagda.md | 2 +- .../decidable-subpreorders.lagda.md | 2 +- .../decidable-total-orders.lagda.md | 2 +- src/order-theory/finite-preorders.lagda.md | 2 +- .../finitely-graded-posets.lagda.md | 12 +++++------ .../order-preserving-maps-posets.lagda.md | 4 ++-- .../order-preserving-maps-preorders.lagda.md | 4 ++-- src/order-theory/subposets.lagda.md | 2 +- src/order-theory/subpreorders.lagda.md | 2 +- src/polytopes/abstract-polytopes.lagda.md | 8 ++++---- src/reflection/type-checking-monad.lagda.md | 4 ++-- .../invariant-basis-property-rings.lagda.md | 2 +- src/ring-theory/rings.lagda.md | 6 +++--- src/ring-theory/semirings.lagda.md | 4 ++-- src/species/morphisms-finite-species.lagda.md | 2 +- .../morphisms-species-of-types.lagda.md | 4 ++-- src/structured-types/h-spaces.lagda.md | 2 +- src/structured-types/wild-loops.lagda.md | 2 +- .../free-loops.lagda.md | 10 +++++----- .../infinite-cyclic-types.lagda.md | 2 +- .../interval-type.lagda.md | 10 +++++----- .../loop-spaces.lagda.md | 16 +++++++-------- .../triple-loop-spaces.lagda.md | 10 +++++----- .../universal-cover-circle.lagda.md | 18 ++++++++--------- src/trees/extensional-w-types.lagda.md | 2 +- .../dependent-type-theories.lagda.md | 20 +++++++++---------- .../fibered-dependent-type-theories.lagda.md | 8 ++++---- .../simple-type-theories.lagda.md | 14 ++++++------- .../2-element-decidable-subtypes.lagda.md | 2 +- .../classical-finite-types.lagda.md | 2 +- .../counting-dependent-pair-types.lagda.md | 2 +- .../cyclic-finite-types.lagda.md | 8 ++++---- .../embeddings-standard-finite-types.lagda.md | 2 +- .../equality-finite-types.lagda.md | 2 +- .../equality-standard-finite-types.lagda.md | 6 +++--- .../equivalences-cubes.lagda.md | 8 ++++---- .../ferrers-diagrams.lagda.md | 8 ++++---- .../fibers-of-maps.lagda.md | 2 +- .../finite-types.lagda.md | 10 +++++----- .../inequality-types-with-counting.lagda.md | 2 +- .../injective-maps.lagda.md | 2 +- .../repetitions-of-values.lagda.md | 2 +- 78 files changed, 174 insertions(+), 174 deletions(-) diff --git a/src/commutative-algebra/commutative-rings.lagda.md b/src/commutative-algebra/commutative-rings.lagda.md index 06b3e5d600..2f523ab320 100644 --- a/src/commutative-algebra/commutative-rings.lagda.md +++ b/src/commutative-algebra/commutative-rings.lagda.md @@ -335,7 +335,7 @@ module _ mul-Commutative-Ring' = mul-Ring' ring-Commutative-Ring ap-mul-Commutative-Ring : - {x x' y y' : type-Commutative-Ring} (p : Id x x') (q : Id y y') → + {x x' y y' : type-Commutative-Ring} (p : x ＝ x') (q : y ＝ y') → Id (mul-Commutative-Ring x y) (mul-Commutative-Ring x' y') ap-mul-Commutative-Ring p q = ap-binary mul-Commutative-Ring p q diff --git a/src/commutative-algebra/euclidean-domains.lagda.md b/src/commutative-algebra/euclidean-domains.lagda.md index abd586d5ef..1b375c384b 100644 --- a/src/commutative-algebra/euclidean-domains.lagda.md +++ b/src/commutative-algebra/euclidean-domains.lagda.md @@ -334,7 +334,7 @@ module _ mul-Integral-Domain' integral-domain-Euclidean-Domain ap-mul-Euclidean-Domain : - {x x' y y' : type-Euclidean-Domain} (p : Id x x') (q : Id y y') → + {x x' y y' : type-Euclidean-Domain} (p : x ＝ x') (q : y ＝ y') → Id (mul-Euclidean-Domain x y) (mul-Euclidean-Domain x' y') ap-mul-Euclidean-Domain p q = ap-binary mul-Euclidean-Domain p q diff --git a/src/commutative-algebra/integral-domains.lagda.md b/src/commutative-algebra/integral-domains.lagda.md index d960d94053..25b7728952 100644 --- a/src/commutative-algebra/integral-domains.lagda.md +++ b/src/commutative-algebra/integral-domains.lagda.md @@ -303,7 +303,7 @@ module _ mul-Commutative-Ring' commutative-ring-Integral-Domain ap-mul-Integral-Domain : - {x x' y y' : type-Integral-Domain} (p : Id x x') (q : Id y y') → + {x x' y y' : type-Integral-Domain} (p : x ＝ x') (q : y ＝ y') → Id (mul-Integral-Domain x y) (mul-Integral-Domain x' y') ap-mul-Integral-Domain p q = ap-binary mul-Integral-Domain p q diff --git a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md index e3986f030d..59ae9d47b5 100644 --- a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md +++ b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md @@ -451,7 +451,7 @@ abstract trichotomy-le-ℚ : {l : Level} {A : UU l} (x y : ℚ) → ( le-ℚ x y → A) → - ( Id x y → A) → + ( x ＝ y → A) → ( le-ℚ y x → A) → A trichotomy-le-ℚ x y left eq right diff --git a/src/elementary-number-theory/universal-property-integers.lagda.md b/src/elementary-number-theory/universal-property-integers.lagda.md index 2d36403b4e..123b49094e 100644 --- a/src/elementary-number-theory/universal-property-integers.lagda.md +++ b/src/elementary-number-theory/universal-property-integers.lagda.md @@ -143,7 +143,7 @@ pr2 (pr2 (reflexive-Eq-ELIM-ℤ P p0 pS (f , p , H))) = inv ∘ (right-inv ∘ H Eq-ELIM-ℤ-eq : { l1 : Level} (P : ℤ → UU l1) → ( p0 : P zero-ℤ) (pS : (k : ℤ) → (P k) ≃ (P (succ-ℤ k))) → - ( s t : ELIM-ℤ P p0 pS) → Id s t → Eq-ELIM-ℤ P p0 pS s t + ( s t : ELIM-ℤ P p0 pS) → s ＝ t → Eq-ELIM-ℤ P p0 pS s t Eq-ELIM-ℤ-eq P p0 pS s .s refl = reflexive-Eq-ELIM-ℤ P p0 pS s abstract @@ -185,7 +185,7 @@ abstract eq-Eq-ELIM-ℤ : { l1 : Level} (P : ℤ → UU l1) → ( p0 : P zero-ℤ) (pS : (k : ℤ) → (P k) ≃ (P (succ-ℤ k))) → - ( s t : ELIM-ℤ P p0 pS) → Eq-ELIM-ℤ P p0 pS s t → Id s t + ( s t : ELIM-ℤ P p0 pS) → Eq-ELIM-ℤ P p0 pS s t → s ＝ t eq-Eq-ELIM-ℤ P p0 pS s t = map-inv-is-equiv (is-equiv-Eq-ELIM-ℤ-eq P p0 pS s t) abstract diff --git a/src/finite-algebra/commutative-finite-rings.lagda.md b/src/finite-algebra/commutative-finite-rings.lagda.md index 035aeea3e7..0d1c9851e3 100644 --- a/src/finite-algebra/commutative-finite-rings.lagda.md +++ b/src/finite-algebra/commutative-finite-rings.lagda.md @@ -347,7 +347,7 @@ module _ mul-Finite-Ring' finite-ring-Finite-Commutative-Ring ap-mul-Finite-Commutative-Ring : - {x x' y y' : type-Finite-Commutative-Ring} (p : Id x x') (q : Id y y') → + {x x' y y' : type-Finite-Commutative-Ring} (p : x ＝ x') (q : y ＝ y') → Id (mul-Finite-Commutative-Ring x y) (mul-Finite-Commutative-Ring x' y') ap-mul-Finite-Commutative-Ring p q = ap-binary mul-Finite-Commutative-Ring p q diff --git a/src/finite-algebra/finite-fields.lagda.md b/src/finite-algebra/finite-fields.lagda.md index 27425b25c9..6468b8c8f6 100644 --- a/src/finite-algebra/finite-fields.lagda.md +++ b/src/finite-algebra/finite-fields.lagda.md @@ -282,7 +282,7 @@ module _ mul-Finite-Field' = mul-Finite-Ring' finite-ring-Finite-Field ap-mul-Finite-Field : - {x x' y y' : type-Finite-Field} (p : Id x x') (q : Id y y') → + {x x' y y' : type-Finite-Field} (p : x ＝ x') (q : y ＝ y') → Id (mul-Finite-Field x y) (mul-Finite-Field x' y') ap-mul-Finite-Field p q = ap-binary mul-Finite-Field p q diff --git a/src/finite-algebra/finite-rings.lagda.md b/src/finite-algebra/finite-rings.lagda.md index e78ca56b4d..7729cd4169 100644 --- a/src/finite-algebra/finite-rings.lagda.md +++ b/src/finite-algebra/finite-rings.lagda.md @@ -134,7 +134,7 @@ module _ ap-add-Finite-Ring : {x y x' y' : type-Finite-Ring R} → - Id x x' → Id y y' → Id (add-Finite-Ring x y) (add-Finite-Ring x' y') + x ＝ x' → y ＝ y' → Id (add-Finite-Ring x y) (add-Finite-Ring x' y') ap-add-Finite-Ring = ap-add-Ring (ring-Finite-Ring R) associative-add-Finite-Ring : @@ -296,7 +296,7 @@ module _ mul-Finite-Ring' = mul-Ring' (ring-Finite-Ring R) ap-mul-Finite-Ring : - {x x' y y' : type-Finite-Ring R} (p : Id x x') (q : Id y y') → + {x x' y y' : type-Finite-Ring R} (p : x ＝ x') (q : y ＝ y') → Id (mul-Finite-Ring x y) (mul-Finite-Ring x' y') ap-mul-Finite-Ring = ap-mul-Ring (ring-Finite-Ring R) diff --git a/src/finite-group-theory/abstract-quaternion-group.lagda.md b/src/finite-group-theory/abstract-quaternion-group.lagda.md index c8a8199ee7..c01ec2ead6 100644 --- a/src/finite-group-theory/abstract-quaternion-group.lagda.md +++ b/src/finite-group-theory/abstract-quaternion-group.lagda.md @@ -757,10 +757,10 @@ refl-Eq-Q8 -j-Q8 = star refl-Eq-Q8 k-Q8 = star refl-Eq-Q8 -k-Q8 = star -Eq-eq-Q8 : {x y : Q8} → Id x y → Eq-Q8 x y +Eq-eq-Q8 : {x y : Q8} → x ＝ y → Eq-Q8 x y Eq-eq-Q8 {x} refl = refl-Eq-Q8 x -eq-Eq-Q8 : (x y : Q8) → Eq-Q8 x y → Id x y +eq-Eq-Q8 : (x y : Q8) → Eq-Q8 x y → x ＝ y eq-Eq-Q8 e-Q8 e-Q8 e = refl eq-Eq-Q8 -e-Q8 -e-Q8 e = refl eq-Eq-Q8 i-Q8 i-Q8 e = refl diff --git a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md index cdb615bc50..3c3f8d641a 100644 --- a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md +++ b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md @@ -313,7 +313,7 @@ module _ ( X Y : UU l1) → ( eX : mere-equiv (Fin (n +ℕ 2)) X) → ( eY : mere-equiv (Fin (n +ℕ 2)) Y) → - Id X Y → + X ＝ Y → Id ( equivalence-class (R (n +ℕ 2) (X , eX))) ( equivalence-class (R (n +ℕ 2) (Y , eY))) @@ -365,7 +365,7 @@ module _ ( X Y : UU l1) ( eX : mere-equiv (Fin (n +ℕ 2)) X) ( eY : mere-equiv (Fin (n +ℕ 2)) Y) - ( p : Id X Y) → + ( p : X ＝ Y) → ( Id (tr (mere-equiv (Fin (n +ℕ 2))) p eX) eY) → ( sX : is-set X) ( sY : is-set Y) → diff --git a/src/finite-group-theory/groups-of-order-2.lagda.md b/src/finite-group-theory/groups-of-order-2.lagda.md index 05de139f31..6ec841ea0a 100644 --- a/src/finite-group-theory/groups-of-order-2.lagda.md +++ b/src/finite-group-theory/groups-of-order-2.lagda.md @@ -109,7 +109,7 @@ module _ where iso-eq-Group-of-Order-2 : - (H : Group-of-Order-2 l) → Id G H → iso-Group-of-Order-2 G H + (H : Group-of-Order-2 l) → G ＝ H → iso-Group-of-Order-2 G H iso-eq-Group-of-Order-2 H p = iso-eq-Group ( group-Group-of-Order-2 G) @@ -134,7 +134,7 @@ module _ ( iso-eq-Group-of-Order-2) eq-iso-Group-of-Order-2 : - (H : Group-of-Order-2 l) → iso-Group-of-Order-2 G H → Id G H + (H : Group-of-Order-2 l) → iso-Group-of-Order-2 G H → G ＝ H eq-iso-Group-of-Order-2 H = map-inv-is-equiv (is-equiv-iso-eq-Group-of-Order-2 H) ``` diff --git a/src/finite-group-theory/permutations.lagda.md b/src/finite-group-theory/permutations.lagda.md index 02097ffebf..1ca715c0af 100644 --- a/src/finite-group-theory/permutations.lagda.md +++ b/src/finite-group-theory/permutations.lagda.md @@ -314,7 +314,7 @@ module _ Id ( iterate (nat-Fin 2 k) (succ-Fin 2) x) ( iterate (nat-Fin 2 k') (succ-Fin 2) x) → - Id k k' + k ＝ k' is-injective-iterate-involution (inl (inr star)) (inl (inr star)) x p = refl diff --git a/src/foundation-core/identity-types.lagda.md b/src/foundation-core/identity-types.lagda.md index d1c48f2836..1477822e20 100644 --- a/src/foundation-core/identity-types.lagda.md +++ b/src/foundation-core/identity-types.lagda.md @@ -26,7 +26,7 @@ equipped with a {{#concept "reflexivity element" Disambiguation="identity type" Agda=refl}} ```text - refl : (x : A) → Id x x. + refl : (x : A) → x ＝ x. ``` In other words, the identity type is a reflexive @@ -79,7 +79,7 @@ introducing types equipped with induction principles. The only constructor of the identity type `Id x : A → 𝒰` is the reflexivity identification ```text - refl : Id x x. + refl : x ＝ x. ``` ```agda diff --git a/src/foundation/universal-property-identity-types.lagda.md b/src/foundation/universal-property-identity-types.lagda.md index bf40f8734e..7dfeb07d06 100644 --- a/src/foundation/universal-property-identity-types.lagda.md +++ b/src/foundation/universal-property-identity-types.lagda.md @@ -161,7 +161,7 @@ In this composite, the injectivity of `equiv-eq` is used in the third step. ```agda module _ {l : Level} (A : UU l) - (L : (a x y : A) → is-injective (equiv-eq {A = Id x y} {B = Id a y})) + (L : (a x y : A) → is-injective (equiv-eq {A = x ＝ y} {B = a ＝ y})) where injection-Id-is-injective-equiv-eq-Id : diff --git a/src/graph-theory/equivalences-undirected-graphs.lagda.md b/src/graph-theory/equivalences-undirected-graphs.lagda.md index 8fef4d2b7f..4b9800127d 100644 --- a/src/graph-theory/equivalences-undirected-graphs.lagda.md +++ b/src/graph-theory/equivalences-undirected-graphs.lagda.md @@ -187,7 +187,7 @@ module _ refl-htpy-hom-Undirected-Graph G H (hom-equiv-Undirected-Graph G H f) htpy-eq-equiv-Undirected-Graph : - (f g : equiv-Undirected-Graph G H) → Id f g → + (f g : equiv-Undirected-Graph G H) → f ＝ g → htpy-equiv-Undirected-Graph f g htpy-eq-equiv-Undirected-Graph f .f refl = refl-htpy-equiv-Undirected-Graph f @@ -239,7 +239,7 @@ module _ eq-htpy-equiv-Undirected-Graph : (f g : equiv-Undirected-Graph G H) → - htpy-equiv-Undirected-Graph f g → Id f g + htpy-equiv-Undirected-Graph f g → f ＝ g eq-htpy-equiv-Undirected-Graph f g = map-inv-is-equiv (is-equiv-htpy-eq-equiv-Undirected-Graph f g) ``` @@ -252,7 +252,7 @@ module _ where equiv-eq-Undirected-Graph : - (H : Undirected-Graph l1 l2) → Id G H → equiv-Undirected-Graph G H + (H : Undirected-Graph l1 l2) → G ＝ H → equiv-Undirected-Graph G H equiv-eq-Undirected-Graph .G refl = id-equiv-Undirected-Graph G is-torsorial-equiv-Undirected-Graph : @@ -277,7 +277,7 @@ module _ pr2 (extensionality-Undirected-Graph H) = is-equiv-equiv-eq-Undirected-Graph H eq-equiv-Undirected-Graph : - (H : Undirected-Graph l1 l2) → equiv-Undirected-Graph G H → Id G H + (H : Undirected-Graph l1 l2) → equiv-Undirected-Graph G H → G ＝ H eq-equiv-Undirected-Graph H = map-inv-is-equiv (is-equiv-equiv-eq-Undirected-Graph H) ``` diff --git a/src/graph-theory/morphisms-undirected-graphs.lagda.md b/src/graph-theory/morphisms-undirected-graphs.lagda.md index 79a56e28d8..eba9c66056 100644 --- a/src/graph-theory/morphisms-undirected-graphs.lagda.md +++ b/src/graph-theory/morphisms-undirected-graphs.lagda.md @@ -132,7 +132,7 @@ module _ ( vertex-hom-Undirected-Graph G H f) p) htpy-eq-hom-Undirected-Graph : - (f g : hom-Undirected-Graph G H) → Id f g → htpy-hom-Undirected-Graph f g + (f g : hom-Undirected-Graph G H) → f ＝ g → htpy-hom-Undirected-Graph f g htpy-eq-hom-Undirected-Graph f .f refl = refl-htpy-hom-Undirected-Graph f abstract @@ -180,7 +180,7 @@ module _ is-equiv-htpy-eq-hom-Undirected-Graph f g eq-htpy-hom-Undirected-Graph : - (f g : hom-Undirected-Graph G H) → htpy-hom-Undirected-Graph f g → Id f g + (f g : hom-Undirected-Graph G H) → htpy-hom-Undirected-Graph f g → f ＝ g eq-htpy-hom-Undirected-Graph f g = map-inv-is-equiv (is-equiv-htpy-eq-hom-Undirected-Graph f g) ``` diff --git a/src/group-theory/category-of-groups.lagda.md b/src/group-theory/category-of-groups.lagda.md index 5b7a7ec268..271b60bf50 100644 --- a/src/group-theory/category-of-groups.lagda.md +++ b/src/group-theory/category-of-groups.lagda.md @@ -31,7 +31,7 @@ is-large-category-Group : is-large-category-Group G = fundamental-theorem-id (is-torsorial-iso-Group G) (iso-eq-Group G) -eq-iso-Group : {l : Level} (G H : Group l) → iso-Group G H → Id G H +eq-iso-Group : {l : Level} (G H : Group l) → iso-Group G H → G ＝ H eq-iso-Group G H = map-inv-is-equiv (is-large-category-Group G H) Group-Large-Category : Large-Category lsuc (_⊔_) diff --git a/src/group-theory/category-of-semigroups.lagda.md b/src/group-theory/category-of-semigroups.lagda.md index 74c50be246..3e79172f68 100644 --- a/src/group-theory/category-of-semigroups.lagda.md +++ b/src/group-theory/category-of-semigroups.lagda.md @@ -45,7 +45,7 @@ pr1 (extensionality-Semigroup G H) = iso-eq-Semigroup G H pr2 (extensionality-Semigroup G H) = is-large-category-Semigroup G H eq-iso-Semigroup : - {l : Level} (G H : Semigroup l) → iso-Semigroup G H → Id G H + {l : Level} (G H : Semigroup l) → iso-Semigroup G H → G ＝ H eq-iso-Semigroup G H = map-inv-is-equiv (is-large-category-Semigroup G H) Semigroup-Large-Category : Large-Category lsuc (_⊔_) diff --git a/src/group-theory/commutative-monoids.lagda.md b/src/group-theory/commutative-monoids.lagda.md index 79b4faf392..2274f34fcc 100644 --- a/src/group-theory/commutative-monoids.lagda.md +++ b/src/group-theory/commutative-monoids.lagda.md @@ -176,7 +176,7 @@ module _ right-unit-law-mul-Monoid (monoid-Commutative-Monoid M) is-unit-Commutative-Monoid : type-Commutative-Monoid M → UU l - is-unit-Commutative-Monoid x = Id x unit-Commutative-Monoid + is-unit-Commutative-Monoid x = (x ＝ unit-Commutative-Monoid) is-unit-prop-Commutative-Monoid : type-Commutative-Monoid M → Prop l is-unit-prop-Commutative-Monoid x = diff --git a/src/group-theory/equivalences-semigroups.lagda.md b/src/group-theory/equivalences-semigroups.lagda.md index b0ec4a907d..c9148dd2f1 100644 --- a/src/group-theory/equivalences-semigroups.lagda.md +++ b/src/group-theory/equivalences-semigroups.lagda.md @@ -87,7 +87,7 @@ module _ ( t : Σ ( has-associative-mul (type-Semigroup G)) ( λ μ → preserves-mul-Semigroup G (pair (set-Semigroup G) μ) id)) → - Id center-total-preserves-mul-id-Semigroup t + center-total-preserves-mul-id-Semigroup ＝ t contraction-total-preserves-mul-id-Semigroup ( (μ-G' , associative-G') , μ-id) = eq-type-subtype diff --git a/src/group-theory/groups.lagda.md b/src/group-theory/groups.lagda.md index 7c99534d5c..44f5de6216 100644 --- a/src/group-theory/groups.lagda.md +++ b/src/group-theory/groups.lagda.md @@ -106,7 +106,7 @@ module _ mul-Group = pr1 has-associative-mul-Group ap-mul-Group : - {x x' y y' : type-Group} (p : Id x x') (q : Id y y') → + {x x' y y' : type-Group} (p : x ＝ x') (q : y ＝ y') → Id (mul-Group x y) (mul-Group x' y') ap-mul-Group p q = ap-binary mul-Group p q diff --git a/src/group-theory/homomorphisms-abelian-groups.lagda.md b/src/group-theory/homomorphisms-abelian-groups.lagda.md index 172b824ed0..628a5097b5 100644 --- a/src/group-theory/homomorphisms-abelian-groups.lagda.md +++ b/src/group-theory/homomorphisms-abelian-groups.lagda.md @@ -121,7 +121,7 @@ module _ refl-htpy-hom-Ab : (f : hom-Ab A B) → htpy-hom-Ab f f refl-htpy-hom-Ab f = refl-htpy-hom-Group (group-Ab A) (group-Ab B) f - htpy-eq-hom-Ab : (f g : hom-Ab A B) → Id f g → htpy-hom-Ab f g + htpy-eq-hom-Ab : (f g : hom-Ab A B) → f ＝ g → htpy-hom-Ab f g htpy-eq-hom-Ab f g = htpy-eq-hom-Group (group-Ab A) (group-Ab B) f g abstract @@ -136,7 +136,7 @@ module _ is-equiv-htpy-eq-hom-Ab f g = is-equiv-htpy-eq-hom-Group (group-Ab A) (group-Ab B) f g - eq-htpy-hom-Ab : {f g : hom-Ab A B} → htpy-hom-Ab f g → Id f g + eq-htpy-hom-Ab : {f g : hom-Ab A B} → htpy-hom-Ab f g → f ＝ g eq-htpy-hom-Ab = eq-htpy-hom-Group (group-Ab A) (group-Ab B) is-set-hom-Ab : is-set (hom-Ab A B) diff --git a/src/group-theory/homomorphisms-concrete-groups.lagda.md b/src/group-theory/homomorphisms-concrete-groups.lagda.md index 1f1c9f34c0..3136fa756e 100644 --- a/src/group-theory/homomorphisms-concrete-groups.lagda.md +++ b/src/group-theory/homomorphisms-concrete-groups.lagda.md @@ -136,7 +136,7 @@ module _ ( f) eq-htpy-hom-Concrete-Group : - (g : hom-Concrete-Group G H) → (htpy-hom-Concrete-Group g) → Id f g + (g : hom-Concrete-Group G H) → (htpy-hom-Concrete-Group g) → f ＝ g eq-htpy-hom-Concrete-Group g = map-inv-equiv (extensionality-hom-Concrete-Group g) ``` diff --git a/src/group-theory/homomorphisms-groups.lagda.md b/src/group-theory/homomorphisms-groups.lagda.md index 13d329ef42..08926c3883 100644 --- a/src/group-theory/homomorphisms-groups.lagda.md +++ b/src/group-theory/homomorphisms-groups.lagda.md @@ -134,7 +134,7 @@ module _ ( semigroup-Group G) ( semigroup-Group H) - htpy-eq-hom-Group : (f g : hom-Group G H) → Id f g → htpy-hom-Group f g + htpy-eq-hom-Group : (f g : hom-Group G H) → f ＝ g → htpy-hom-Group f g htpy-eq-hom-Group = htpy-eq-hom-Semigroup ( semigroup-Group G) @@ -161,7 +161,7 @@ module _ pr1 (extensionality-hom-Group f g) = htpy-eq-hom-Group f g pr2 (extensionality-hom-Group f g) = is-equiv-htpy-eq-hom-Group f g - eq-htpy-hom-Group : {f g : hom-Group G H} → htpy-hom-Group f g → Id f g + eq-htpy-hom-Group : {f g : hom-Group G H} → htpy-hom-Group f g → f ＝ g eq-htpy-hom-Group = eq-htpy-hom-Semigroup (semigroup-Group G) (semigroup-Group H) diff --git a/src/group-theory/homomorphisms-semigroups.lagda.md b/src/group-theory/homomorphisms-semigroups.lagda.md index b00a30a3fa..5bef1c713a 100644 --- a/src/group-theory/homomorphisms-semigroups.lagda.md +++ b/src/group-theory/homomorphisms-semigroups.lagda.md @@ -122,7 +122,7 @@ module _ refl-htpy-hom-Semigroup f = refl-htpy htpy-eq-hom-Semigroup : - (f g : hom-Semigroup) → Id f g → htpy-hom-Semigroup f g + (f g : hom-Semigroup) → f ＝ g → htpy-hom-Semigroup f g htpy-eq-hom-Semigroup f .f refl = refl-htpy-hom-Semigroup f abstract @@ -145,7 +145,7 @@ module _ ( htpy-eq-hom-Semigroup f) eq-htpy-hom-Semigroup : - {f g : hom-Semigroup} → htpy-hom-Semigroup f g → Id f g + {f g : hom-Semigroup} → htpy-hom-Semigroup f g → f ＝ g eq-htpy-hom-Semigroup {f} {g} = map-inv-is-equiv (is-equiv-htpy-eq-hom-Semigroup f g) diff --git a/src/group-theory/isomorphisms-abelian-groups.lagda.md b/src/group-theory/isomorphisms-abelian-groups.lagda.md index 2fe037a367..6270009d25 100644 --- a/src/group-theory/isomorphisms-abelian-groups.lagda.md +++ b/src/group-theory/isomorphisms-abelian-groups.lagda.md @@ -179,7 +179,7 @@ id-iso-Ab A = id-iso-Group (group-Ab A) ```agda iso-eq-Ab : - {l : Level} (A B : Ab l) → Id A B → iso-Ab A B + {l : Level} (A B : Ab l) → A ＝ B → iso-Ab A B iso-eq-Ab A B p = iso-eq-Group (group-Ab A) (group-Ab B) (ap pr1 p) abstract @@ -206,7 +206,7 @@ is-equiv-iso-eq-Ab A = ( iso-eq-Ab A) eq-iso-Ab : - {l : Level} (A B : Ab l) → iso-Ab A B → Id A B + {l : Level} (A B : Ab l) → iso-Ab A B → A ＝ B eq-iso-Ab A B = map-inv-is-equiv (is-equiv-iso-eq-Ab A B) ``` diff --git a/src/group-theory/isomorphisms-groups.lagda.md b/src/group-theory/isomorphisms-groups.lagda.md index 4a65c7e7c5..843c083f28 100644 --- a/src/group-theory/isomorphisms-groups.lagda.md +++ b/src/group-theory/isomorphisms-groups.lagda.md @@ -227,7 +227,7 @@ module _ {l : Level} (G : Group l) where - iso-eq-Group : (H : Group l) → Id G H → iso-Group G H + iso-eq-Group : (H : Group l) → G ＝ H → iso-Group G H iso-eq-Group = iso-eq-Large-Precategory Group-Large-Precategory G abstract diff --git a/src/group-theory/isomorphisms-monoids.lagda.md b/src/group-theory/isomorphisms-monoids.lagda.md index 5ed4766a19..dde9090b5a 100644 --- a/src/group-theory/isomorphisms-monoids.lagda.md +++ b/src/group-theory/isomorphisms-monoids.lagda.md @@ -191,7 +191,7 @@ module _ An equality between objects `x y : A` gives rise to an isomorphism between them. This is because by the J-rule, it is enough to construct an isomorphism given -`refl : Id x x`, from `x` to itself. We take the identity morphism as such an +`refl : x ＝ x`, from `x` to itself. We take the identity morphism as such an isomorphism. ```agda diff --git a/src/group-theory/isomorphisms-semigroups.lagda.md b/src/group-theory/isomorphisms-semigroups.lagda.md index 2c2bb15a9a..001bafb3e9 100644 --- a/src/group-theory/isomorphisms-semigroups.lagda.md +++ b/src/group-theory/isomorphisms-semigroups.lagda.md @@ -314,6 +314,6 @@ module _ id-iso-Semigroup = id-iso-Large-Precategory Semigroup-Large-Precategory {X = G} - iso-eq-Semigroup : (H : Semigroup l) → Id G H → iso-Semigroup G H + iso-eq-Semigroup : (H : Semigroup l) → G ＝ H → iso-Semigroup G H iso-eq-Semigroup = iso-eq-Large-Precategory Semigroup-Large-Precategory G ``` diff --git a/src/group-theory/loop-groups-sets.lagda.md b/src/group-theory/loop-groups-sets.lagda.md index fc595b0599..a10ea49025 100644 --- a/src/group-theory/loop-groups-sets.lagda.md +++ b/src/group-theory/loop-groups-sets.lagda.md @@ -102,7 +102,7 @@ module _ map-hom-inv-symmetric-group-loop-group-Set X Y f = inv (eq-equiv f) commutative-inv-map-hom-symmetric-group-loop-group-Set : - (X Y : UU l) (p : Id X Y) (sX : is-set X) (sY : is-set Y) → + (X Y : UU l) (p : X ＝ Y) (sX : is-set X) (sY : is-set Y) → Id ( map-hom-symmetric-group-loop-group-Set (Y , sY) (X , sX) (inv p)) ( inv-equiv diff --git a/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md b/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md index 037435ec5b..c6f7de583c 100644 --- a/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md +++ b/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md @@ -71,7 +71,7 @@ module _ htpy-eq-hom-orbit-action-Monoid : {x y : type-action-Monoid M X} (f g : hom-orbit-action-Monoid x y) → - Id f g → htpy-hom-orbit-action-Monoid f g + f ＝ g → htpy-hom-orbit-action-Monoid f g htpy-eq-hom-orbit-action-Monoid f .f refl = refl-htpy-hom-orbit-action-Monoid f @@ -105,7 +105,7 @@ module _ eq-htpy-hom-orbit-action-Monoid : {x y : type-action-Monoid M X} {f g : hom-orbit-action-Monoid x y} → - htpy-hom-orbit-action-Monoid f g → Id f g + htpy-hom-orbit-action-Monoid f g → f ＝ g eq-htpy-hom-orbit-action-Monoid {x} {y} {f} {g} = map-inv-is-equiv (is-equiv-htpy-eq-hom-orbit-action-Monoid f g) diff --git a/src/group-theory/subgroups-abelian-groups.lagda.md b/src/group-theory/subgroups-abelian-groups.lagda.md index 61238aa5c7..55bcda4a76 100644 --- a/src/group-theory/subgroups-abelian-groups.lagda.md +++ b/src/group-theory/subgroups-abelian-groups.lagda.md @@ -186,7 +186,7 @@ module _ eq-subgroup-ab-eq-ab : {x y : type-ab-Subgroup-Ab} → Id (map-inclusion-Subgroup-Ab x) (map-inclusion-Subgroup-Ab y) → - Id x y + x ＝ y eq-subgroup-ab-eq-ab = eq-subgroup-eq-group (group-Ab A) B set-ab-Subgroup-Ab : Set (l1 ⊔ l2) diff --git a/src/linear-algebra/matrices.lagda.md b/src/linear-algebra/matrices.lagda.md index b5ff40520f..7df740da58 100644 --- a/src/linear-algebra/matrices.lagda.md +++ b/src/linear-algebra/matrices.lagda.md @@ -78,7 +78,7 @@ vertically-empty-matrix = empty-tuple eq-vertically-empty-matrix : {l : Level} {n : ℕ} {A : UU l} - (x : matrix A 0 n) → Id vertically-empty-matrix x + (x : matrix A 0 n) → vertically-empty-matrix ＝ x eq-vertically-empty-matrix empty-tuple = refl is-contr-matrix-zero-ℕ : @@ -98,7 +98,7 @@ horizontally-empty-matrix {m = succ-ℕ m} = eq-horizontally-empty-matrix : {l : Level} {m : ℕ} {A : UU l} - (x : matrix A m 0) → Id horizontally-empty-matrix x + (x : matrix A m 0) → horizontally-empty-matrix ＝ x eq-horizontally-empty-matrix {m = zero-ℕ} empty-tuple = refl eq-horizontally-empty-matrix {m = succ-ℕ m} (empty-tuple ∷ M) = ap-binary _∷_ refl (eq-horizontally-empty-matrix M) diff --git a/src/lists/lists.lagda.md b/src/lists/lists.lagda.md index 2f0399cf9c..cbf193fa1b 100644 --- a/src/lists/lists.lagda.md +++ b/src/lists/lists.lagda.md @@ -175,11 +175,11 @@ refl-Eq-list nil = raise-star refl-Eq-list (cons x l) = pair refl (refl-Eq-list l) Eq-eq-list : - {l1 : Level} {A : UU l1} (l l' : list A) → Id l l' → Eq-list l l' + {l1 : Level} {A : UU l1} (l l' : list A) → l ＝ l' → Eq-list l l' Eq-eq-list l .l refl = refl-Eq-list l eq-Eq-list : - {l1 : Level} {A : UU l1} (l l' : list A) → Eq-list l l' → Id l l' + {l1 : Level} {A : UU l1} (l l' : list A) → Eq-list l l' → l ＝ l' eq-Eq-list nil nil (map-raise star) = refl eq-Eq-list nil (cons x l') (map-raise f) = ex-falso f eq-Eq-list (cons x l) nil (map-raise f) = ex-falso f @@ -187,7 +187,7 @@ eq-Eq-list (cons x l) (cons .x l') (pair refl e) = ap (cons x) (eq-Eq-list l l' e) square-eq-Eq-list : - {l1 : Level} {A : UU l1} {x : A} {l l' : list A} (p : Id l l') → + {l1 : Level} {A : UU l1} {x : A} {l l' : list A} (p : l ＝ l') → Id ( Eq-eq-list (cons x l) (cons x l') (ap (cons x) p)) ( pair refl (Eq-eq-list l l' p)) @@ -210,7 +210,7 @@ eq-Eq-refl-Eq-list nil = refl eq-Eq-refl-Eq-list (cons x l) = ap² (cons x) (eq-Eq-refl-Eq-list l) is-retraction-eq-Eq-list : - {l1 : Level} {A : UU l1} (l l' : list A) (p : Id l l') → + {l1 : Level} {A : UU l1} (l l' : list A) (p : l ＝ l') → Id (eq-Eq-list l l' (Eq-eq-list l l' p)) p is-retraction-eq-Eq-list nil .nil refl = refl is-retraction-eq-Eq-list (cons x l) .(cons x l) refl = diff --git a/src/lists/tuples.lagda.md b/src/lists/tuples.lagda.md index a7739f1bd6..d93ba4fa4f 100644 --- a/src/lists/tuples.lagda.md +++ b/src/lists/tuples.lagda.md @@ -116,10 +116,10 @@ module _ pr1 (refl-Eq-tuple (succ-ℕ n) (x ∷ xs)) = refl pr2 (refl-Eq-tuple (succ-ℕ n) (x ∷ xs)) = refl-Eq-tuple n xs - Eq-eq-tuple : (n : ℕ) → (u v : tuple A n) → Id u v → Eq-tuple n u v + Eq-eq-tuple : (n : ℕ) → (u v : tuple A n) → u ＝ v → Eq-tuple n u v Eq-eq-tuple n u .u refl = refl-Eq-tuple n u - eq-Eq-tuple : (n : ℕ) → (u v : tuple A n) → Eq-tuple n u v → Id u v + eq-Eq-tuple : (n : ℕ) → (u v : tuple A n) → Eq-tuple n u v → u ＝ v eq-Eq-tuple zero-ℕ empty-tuple empty-tuple eq-tuple = refl eq-Eq-tuple (succ-ℕ n) (x ∷ xs) (.x ∷ ys) (refl , eqs) = ap (x ∷_) (eq-Eq-tuple n xs ys eqs) @@ -132,7 +132,7 @@ module _ left-whisker-comp² (x ∷_) (is-retraction-eq-Eq-tuple n xs xs) refl square-Eq-eq-tuple : - (n : ℕ) (x : A) (u v : tuple A n) (p : Id u v) → + (n : ℕ) (x : A) (u v : tuple A n) (p : u ＝ v) → (Eq-eq-tuple _ (x ∷ u) (x ∷ v) (ap (x ∷_) p)) ＝ (refl , (Eq-eq-tuple n u v p)) square-Eq-eq-tuple zero-ℕ x empty-tuple empty-tuple refl = refl diff --git a/src/literature/sequential-colimits-in-homotopy-type-theory.lagda.md b/src/literature/sequential-colimits-in-homotopy-type-theory.lagda.md index 34e5e2e86f..af67a0c9e3 100644 --- a/src/literature/sequential-colimits-in-homotopy-type-theory.lagda.md +++ b/src/literature/sequential-colimits-in-homotopy-type-theory.lagda.md @@ -17,7 +17,7 @@ open import foundation.universe-levels using ( UU ) open import foundation.identity-types using - ( Id -- "path" + ( -- ＝ "path" ; refl -- "constant path" ; inv -- "inverse path" ; concat -- "concatenation of paths" diff --git a/src/order-theory/decidable-subposets.lagda.md b/src/order-theory/decidable-subposets.lagda.md index d57463aa0b..740b185f7b 100644 --- a/src/order-theory/decidable-subposets.lagda.md +++ b/src/order-theory/decidable-subposets.lagda.md @@ -41,7 +41,7 @@ module _ type-Subposet P (subtype-decidable-subtype S) eq-type-Decidable-Subposet : - (x y : type-Decidable-Subposet) → Id (pr1 x) (pr1 y) → Id x y + (x y : type-Decidable-Subposet) → Id (pr1 x) (pr1 y) → x ＝ y eq-type-Decidable-Subposet = eq-type-Subposet P (subtype-decidable-subtype S) diff --git a/src/order-theory/decidable-subpreorders.lagda.md b/src/order-theory/decidable-subpreorders.lagda.md index 57e290edc2..ff1398155c 100644 --- a/src/order-theory/decidable-subpreorders.lagda.md +++ b/src/order-theory/decidable-subpreorders.lagda.md @@ -41,7 +41,7 @@ module _ type-Subpreorder P (subtype-decidable-subtype S) eq-type-Decidable-Subpreorder : - (x y : type-Decidable-Subpreorder) → Id (pr1 x) (pr1 y) → Id x y + (x y : type-Decidable-Subpreorder) → Id (pr1 x) (pr1 y) → x ＝ y eq-type-Decidable-Subpreorder = eq-type-Subpreorder P (subtype-decidable-subtype S) diff --git a/src/order-theory/decidable-total-orders.lagda.md b/src/order-theory/decidable-total-orders.lagda.md index 5799377141..89e631f796 100644 --- a/src/order-theory/decidable-total-orders.lagda.md +++ b/src/order-theory/decidable-total-orders.lagda.md @@ -175,7 +175,7 @@ module _ antisymmetric-leq-Decidable-Total-Order : (x y : type-Decidable-Total-Order) → - leq-Decidable-Total-Order x y → leq-Decidable-Total-Order y x → Id x y + leq-Decidable-Total-Order x y → leq-Decidable-Total-Order y x → x ＝ y antisymmetric-leq-Decidable-Total-Order = antisymmetric-leq-Poset poset-Decidable-Total-Order diff --git a/src/order-theory/finite-preorders.lagda.md b/src/order-theory/finite-preorders.lagda.md index b7a2511b49..4ec0ab834a 100644 --- a/src/order-theory/finite-preorders.lagda.md +++ b/src/order-theory/finite-preorders.lagda.md @@ -190,7 +190,7 @@ module _ is-finite-type-decidable-subtype S (is-finite-type-Finite-Preorder P) eq-type-finite-Subpreorder : - (x y : type-finite-Subpreorder) → Id (pr1 x) (pr1 y) → Id x y + (x y : type-finite-Subpreorder) → Id (pr1 x) (pr1 y) → x ＝ y eq-type-finite-Subpreorder = eq-type-Decidable-Subpreorder (preorder-Finite-Preorder P) S diff --git a/src/order-theory/finitely-graded-posets.lagda.md b/src/order-theory/finitely-graded-posets.lagda.md index b4fb4a404b..762e20da05 100644 --- a/src/order-theory/finitely-graded-posets.lagda.md +++ b/src/order-theory/finitely-graded-posets.lagda.md @@ -138,7 +138,7 @@ If chains with jumps are never used, we'd like to call the following chains. path-faces-Finitely-Graded-Poset z tr-refl-path-faces-Finitely-Graded-Poset : - {i j : Fin (succ-ℕ k)} (p : Id j i) (x : face-Finitely-Graded-Poset j) → + {i j : Fin (succ-ℕ k)} (p : j ＝ i) (x : face-Finitely-Graded-Poset j) → path-faces-Finitely-Graded-Poset ( tr face-Finitely-Graded-Poset p x) ( x) @@ -204,7 +204,7 @@ eq-path-elements-Finitely-Graded-Poset : (x y : type-Finitely-Graded-Poset X) → (p : Id (shape-Finitely-Graded-Poset X x) (shape-Finitely-Graded-Poset X y)) → - path-elements-Finitely-Graded-Poset X x y → Id x y + path-elements-Finitely-Graded-Poset X x y → x ＝ y eq-path-elements-Finitely-Graded-Poset {k} X (pair i1 x) (pair .i1 .x) p refl-path-faces-Finitely-Graded-Poset = refl eq-path-elements-Finitely-Graded-Poset {k = succ-ℕ k} X (pair i1 x) @@ -243,7 +243,7 @@ module _ abstract eq-path-faces-Finitely-Graded-Poset : {i : Fin (succ-ℕ k)} (x y : face-Finitely-Graded-Poset X i) → - path-faces-Finitely-Graded-Poset X x y → Id x y + path-faces-Finitely-Graded-Poset X x y → x ＝ y eq-path-faces-Finitely-Graded-Poset {i} x y H = map-left-unit-law-Σ-is-contr ( is-proof-irrelevant-is-prop @@ -261,7 +261,7 @@ module _ (x y : type-Finitely-Graded-Poset X) → path-elements-Finitely-Graded-Poset X x y → path-elements-Finitely-Graded-Poset X y x → - Id x y + x ＝ y antisymmetric-path-elements-Finitely-Graded-Poset (pair i x) (pair j y) H K = eq-path-elements-Finitely-Graded-Poset X (pair i x) (pair j y) ( antisymmetric-leq-Fin (succ-ℕ k) @@ -490,7 +490,7 @@ module _ is-set-type-Set face-set-Finitely-Graded-Subposet eq-face-Finitely-Graded-Subposet : - (x y : face-Finitely-Graded-Subposet) → Id (pr1 x) (pr1 y) → Id x y + (x y : face-Finitely-Graded-Subposet) → Id (pr1 x) (pr1 y) → x ＝ y eq-face-Finitely-Graded-Subposet x y = eq-type-subtype S emb-face-Finitely-Graded-Subposet : @@ -594,7 +594,7 @@ module _ antisymmetric-leq-Finitely-Graded-Subposet : (x y : type-Finitely-Graded-Subposet) → - leq-Finitely-Graded-Subposet x y → leq-Finitely-Graded-Subposet y x → Id x y + leq-Finitely-Graded-Subposet x y → leq-Finitely-Graded-Subposet y x → x ＝ y antisymmetric-leq-Finitely-Graded-Subposet x y H K = is-injective-map-emb-type-Finitely-Graded-Subposet ( antisymmetric-leq-Finitely-Graded-Poset X diff --git a/src/order-theory/order-preserving-maps-posets.lagda.md b/src/order-theory/order-preserving-maps-posets.lagda.md index c1f944d16f..bc2be768b5 100644 --- a/src/order-theory/order-preserving-maps-posets.lagda.md +++ b/src/order-theory/order-preserving-maps-posets.lagda.md @@ -85,7 +85,7 @@ module _ refl-htpy-hom-Preorder (preorder-Poset P) (preorder-Poset Q) htpy-eq-hom-Poset : - (f g : hom-Poset P Q) → Id f g → htpy-hom-Poset f g + (f g : hom-Poset P Q) → f ＝ g → htpy-hom-Poset f g htpy-eq-hom-Poset = htpy-eq-hom-Preorder (preorder-Poset P) (preorder-Poset Q) is-torsorial-htpy-hom-Poset : @@ -104,7 +104,7 @@ module _ extensionality-hom-Preorder (preorder-Poset P) (preorder-Poset Q) eq-htpy-hom-Poset : - (f g : hom-Poset P Q) → htpy-hom-Poset f g → Id f g + (f g : hom-Poset P Q) → htpy-hom-Poset f g → f ＝ g eq-htpy-hom-Poset = eq-htpy-hom-Preorder (preorder-Poset P) (preorder-Poset Q) is-prop-htpy-hom-Poset : diff --git a/src/order-theory/order-preserving-maps-preorders.lagda.md b/src/order-theory/order-preserving-maps-preorders.lagda.md index 32c3406fb3..49e61f205f 100644 --- a/src/order-theory/order-preserving-maps-preorders.lagda.md +++ b/src/order-theory/order-preserving-maps-preorders.lagda.md @@ -104,7 +104,7 @@ module _ refl-htpy-hom-Preorder f = refl-htpy htpy-eq-hom-Preorder : - (f g : hom-Preorder P Q) → Id f g → htpy-hom-Preorder f g + (f g : hom-Preorder P Q) → f ＝ g → htpy-hom-Preorder f g htpy-eq-hom-Preorder f .f refl = refl-htpy-hom-Preorder f is-torsorial-htpy-hom-Preorder : @@ -130,7 +130,7 @@ module _ pr2 (extensionality-hom-Preorder f g) = is-equiv-htpy-eq-hom-Preorder f g eq-htpy-hom-Preorder : - (f g : hom-Preorder P Q) → htpy-hom-Preorder f g → Id f g + (f g : hom-Preorder P Q) → htpy-hom-Preorder f g → f ＝ g eq-htpy-hom-Preorder f g = map-inv-is-equiv (is-equiv-htpy-eq-hom-Preorder f g) ``` diff --git a/src/order-theory/subposets.lagda.md b/src/order-theory/subposets.lagda.md index d7cd52162e..9a09f651b4 100644 --- a/src/order-theory/subposets.lagda.md +++ b/src/order-theory/subposets.lagda.md @@ -42,7 +42,7 @@ module _ type-Subposet = type-Subpreorder (preorder-Poset X) S eq-type-Subposet : - (x y : type-Subposet) → Id (pr1 x) (pr1 y) → Id x y + (x y : type-Subposet) → Id (pr1 x) (pr1 y) → x ＝ y eq-type-Subposet = eq-type-Subpreorder (preorder-Poset X) S leq-Subposet-Prop : (x y : type-Subposet) → Prop l2 diff --git a/src/order-theory/subpreorders.lagda.md b/src/order-theory/subpreorders.lagda.md index 8ac0925ccd..1421f51657 100644 --- a/src/order-theory/subpreorders.lagda.md +++ b/src/order-theory/subpreorders.lagda.md @@ -42,7 +42,7 @@ module _ type-Subpreorder = type-subtype S eq-type-Subpreorder : - (x y : type-Subpreorder) → Id (pr1 x) (pr1 y) → Id x y + (x y : type-Subpreorder) → Id (pr1 x) (pr1 y) → x ＝ y eq-type-Subpreorder x y = eq-type-subtype S leq-Subpreorder-Prop : (x y : type-Subpreorder) → Prop l2 diff --git a/src/polytopes/abstract-polytopes.lagda.md b/src/polytopes/abstract-polytopes.lagda.md index 429d52d3ab..641ac47852 100644 --- a/src/polytopes/abstract-polytopes.lagda.md +++ b/src/polytopes/abstract-polytopes.lagda.md @@ -226,7 +226,7 @@ module _ cons-path-faces-Prepolytope a p = cons-path-faces-Finitely-Graded-Poset a p tr-refl-path-faces-Preposet : - {i j : Fin (succ-ℕ k)} (p : Id j i) (x : face-Prepolytope j) → + {i j : Fin (succ-ℕ k)} (p : j ＝ i) (x : face-Prepolytope j) → path-faces-Prepolytope (tr face-Prepolytope p x) x tr-refl-path-faces-Preposet = tr-refl-path-faces-Finitely-Graded-Poset finitely-graded-poset-Prepolytope @@ -265,19 +265,19 @@ module _ eq-path-elements-Prepolytope : (x y : type-Prepolytope) (p : Id (shape-Prepolytope x) (shape-Prepolytope y)) → - path-elements-Prepolytope x y → Id x y + path-elements-Prepolytope x y → x ＝ y eq-path-elements-Prepolytope = eq-path-elements-Finitely-Graded-Poset finitely-graded-poset-Prepolytope eq-path-faces-Prepolytope : {i : Fin (succ-ℕ k)} (x y : face-Prepolytope i) → - path-faces-Prepolytope x y → Id x y + path-faces-Prepolytope x y → x ＝ y eq-path-faces-Prepolytope = eq-path-faces-Finitely-Graded-Poset finitely-graded-poset-Prepolytope antisymmetric-path-elements-Prepolytope : (x y : type-Prepolytope) → path-elements-Prepolytope x y → - path-elements-Prepolytope y x → Id x y + path-elements-Prepolytope y x → x ＝ y antisymmetric-path-elements-Prepolytope = antisymmetric-path-elements-Finitely-Graded-Poset finitely-graded-poset-Prepolytope diff --git a/src/reflection/type-checking-monad.lagda.md b/src/reflection/type-checking-monad.lagda.md index 44a8550920..50a1e0f7bc 100644 --- a/src/reflection/type-checking-monad.lagda.md +++ b/src/reflection/type-checking-monad.lagda.md @@ -331,10 +331,10 @@ example was adapted from nil))) module _ (a b : ℕ) (p : a ＝ b) where - ex3 : Id a b + ex3 : a ＝ b ex3 = try-path! p - ex4 : Id b a + ex4 : b ＝ a ex4 = try-path! p ``` diff --git a/src/ring-theory/invariant-basis-property-rings.lagda.md b/src/ring-theory/invariant-basis-property-rings.lagda.md index ae9972819c..274f5069bf 100644 --- a/src/ring-theory/invariant-basis-property-rings.lagda.md +++ b/src/ring-theory/invariant-basis-property-rings.lagda.md @@ -34,5 +34,5 @@ invariant-basis-property-Ring : invariant-basis-property-Ring R = (m n : ℕ) → iso-Ring (Π-Ring (Fin m) (λ i → R)) (Π-Ring (Fin n) (λ i → R)) → - Id m n + m ＝ n ``` diff --git a/src/ring-theory/rings.lagda.md b/src/ring-theory/rings.lagda.md index 757f7994ee..af58d72d7d 100644 --- a/src/ring-theory/rings.lagda.md +++ b/src/ring-theory/rings.lagda.md @@ -115,7 +115,7 @@ module _ ap-add-Ring : {x y x' y' : type-Ring R} → - Id x x' → Id y y' → Id (add-Ring x y) (add-Ring x' y') + x ＝ x' → y ＝ y' → Id (add-Ring x y) (add-Ring x' y') ap-add-Ring = ap-add-Ab (ab-Ring R) associative-add-Ring : @@ -265,7 +265,7 @@ module _ zero-Ring = zero-Ab (ab-Ring R) is-zero-Ring : type-Ring R → UU l - is-zero-Ring x = Id x zero-Ring + is-zero-Ring x = x ＝ zero-Ring is-nonzero-Ring : type-Ring R → UU l is-nonzero-Ring x = ¬ (is-zero-Ring x) @@ -367,7 +367,7 @@ module _ mul-Ring' x y = mul-Ring y x ap-mul-Ring : - {x x' y y' : type-Ring R} (p : Id x x') (q : Id y y') → + {x x' y y' : type-Ring R} (p : x ＝ x') (q : y ＝ y') → Id (mul-Ring x y) (mul-Ring x' y') ap-mul-Ring p q = ap-binary mul-Ring p q diff --git a/src/ring-theory/semirings.lagda.md b/src/ring-theory/semirings.lagda.md index e8c7bc9900..97c595edf6 100644 --- a/src/ring-theory/semirings.lagda.md +++ b/src/ring-theory/semirings.lagda.md @@ -181,7 +181,7 @@ module _ unit-Commutative-Monoid (additive-commutative-monoid-Semiring R) is-zero-Semiring : type-Semiring R → UU l - is-zero-Semiring x = Id x zero-Semiring + is-zero-Semiring x = x ＝ zero-Semiring is-nonzero-Semiring : type-Semiring R → UU l is-nonzero-Semiring x = ¬ (is-zero-Semiring x) @@ -219,7 +219,7 @@ module _ mul-Semiring' x y = mul-Semiring y x ap-mul-Semiring : - {x x' y y' : type-Semiring R} (p : Id x x') (q : Id y y') → + {x x' y y' : type-Semiring R} (p : x ＝ x') (q : y ＝ y') → Id (mul-Semiring x y) (mul-Semiring x' y') ap-mul-Semiring p q = ap-binary mul-Semiring p q diff --git a/src/species/morphisms-finite-species.lagda.md b/src/species/morphisms-finite-species.lagda.md index 85b5ea4044..fc77041ff3 100644 --- a/src/species/morphisms-finite-species.lagda.md +++ b/src/species/morphisms-finite-species.lagda.md @@ -121,7 +121,7 @@ right-unit-law-comp-hom-finite-species F G f = refl htpy-eq-hom-finite-species : {l1 l2 l3 : Level} (F : finite-species l1 l2) (G : finite-species l1 l3) (f g : hom-finite-species F G) → - Id f g → htpy-hom-finite-species F G f g + f ＝ g → htpy-hom-finite-species F G f g htpy-eq-hom-finite-species F G f g refl X y = refl is-torsorial-htpy-hom-finite-species : diff --git a/src/species/morphisms-species-of-types.lagda.md b/src/species/morphisms-species-of-types.lagda.md index bd5a2617ff..da7e432629 100644 --- a/src/species/morphisms-species-of-types.lagda.md +++ b/src/species/morphisms-species-of-types.lagda.md @@ -74,7 +74,7 @@ refl-htpy-hom-species-types f X = refl-htpy htpy-eq-hom-species-types : {l1 l2 l3 : Level} {F : species-types l1 l2} {G : species-types l1 l3} {f g : hom-species-types F G} → - Id f g → htpy-hom-species-types f g + f ＝ g → htpy-hom-species-types f g htpy-eq-hom-species-types refl X y = refl is-torsorial-htpy-hom-species-types : @@ -95,7 +95,7 @@ is-equiv-htpy-eq-hom-species-types f = eq-htpy-hom-species-types : {l1 l2 l3 : Level} {F : species-types l1 l2} {G : species-types l1 l3} - {f g : hom-species-types F G} → htpy-hom-species-types f g → Id f g + {f g : hom-species-types F G} → htpy-hom-species-types f g → f ＝ g eq-htpy-hom-species-types {f = f} {g = g} = map-inv-is-equiv (is-equiv-htpy-eq-hom-species-types f g) ``` diff --git a/src/structured-types/h-spaces.lagda.md b/src/structured-types/h-spaces.lagda.md index be217b8539..72fb621b35 100644 --- a/src/structured-types/h-spaces.lagda.md +++ b/src/structured-types/h-spaces.lagda.md @@ -107,7 +107,7 @@ module _ mul-H-Space' x y = mul-H-Space y x ap-mul-H-Space : - {a b c d : type-H-Space} → Id a b → Id c d → + {a b c d : type-H-Space} → a ＝ b → c ＝ d → Id (mul-H-Space a c) (mul-H-Space b d) ap-mul-H-Space p q = ap-binary mul-H-Space p q diff --git a/src/structured-types/wild-loops.lagda.md b/src/structured-types/wild-loops.lagda.md index 0abc01f7c2..b86e858471 100644 --- a/src/structured-types/wild-loops.lagda.md +++ b/src/structured-types/wild-loops.lagda.md @@ -61,7 +61,7 @@ module _ mul-Wild-Loop' = mul-H-Space' h-space-Wild-Loop ap-mul-Wild-Loop : - {a b c d : type-Wild-Loop} → Id a b → Id c d → + {a b c d : type-Wild-Loop} → a ＝ b → c ＝ d → Id (mul-Wild-Loop a c) (mul-Wild-Loop b d) ap-mul-Wild-Loop = ap-mul-H-Space h-space-Wild-Loop diff --git a/src/synthetic-homotopy-theory/free-loops.lagda.md b/src/synthetic-homotopy-theory/free-loops.lagda.md index 2ca408cc36..3a7bdb47d6 100644 --- a/src/synthetic-homotopy-theory/free-loops.lagda.md +++ b/src/synthetic-homotopy-theory/free-loops.lagda.md @@ -87,7 +87,7 @@ module _ pr1 (refl-Eq-free-loop (pair x α)) = refl pr2 (refl-Eq-free-loop (pair x α)) = right-unit - Eq-eq-free-loop : (α α' : free-loop X) → Id α α' → Eq-free-loop α α' + Eq-eq-free-loop : (α α' : free-loop X) → α ＝ α' → Eq-free-loop α α' Eq-eq-free-loop α .α refl = refl-Eq-free-loop α abstract @@ -98,7 +98,7 @@ module _ ( is-torsorial-Id x) ( pair x refl) ( is-contr-is-equiv' - ( Σ (Id x x) (λ α' → Id α α')) + ( Σ (Id x x) (λ α' → α ＝ α')) ( tot (λ α' α → right-unit ∙ α)) ( is-equiv-tot-is-fiberwise-equiv ( λ α' → is-equiv-concat right-unit α')) @@ -134,7 +134,7 @@ module _ pr2 (refl-Eq-free-dependent-loop (pair y p)) = right-unit Eq-free-dependent-loop-eq : - ( p p' : free-dependent-loop α P) → Id p p' → Eq-free-dependent-loop p p' + ( p p' : free-dependent-loop α P) → p ＝ p' → Eq-free-dependent-loop p p' Eq-free-dependent-loop-eq p .p refl = refl-Eq-free-dependent-loop p abstract @@ -145,7 +145,7 @@ module _ ( is-torsorial-Id y) ( pair y refl) ( is-contr-is-equiv' - ( Σ (Id (tr P (loop-free-loop α) y) y) (λ p' → Id p p')) + ( Σ (Id (tr P (loop-free-loop α) y) y) (λ p' → p ＝ p')) ( tot (λ p' α → right-unit ∙ α)) ( is-equiv-tot-is-fiberwise-equiv ( λ p' → is-equiv-concat right-unit p')) @@ -162,7 +162,7 @@ module _ eq-Eq-free-dependent-loop : (p p' : free-dependent-loop α P) → - Eq-free-dependent-loop p p' → Id p p' + Eq-free-dependent-loop p p' → p ＝ p' eq-Eq-free-dependent-loop p p' = map-inv-is-equiv (is-equiv-Eq-free-dependent-loop-eq p p') ``` diff --git a/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md b/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md index 4329545999..1bd957651f 100644 --- a/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md +++ b/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md @@ -101,7 +101,7 @@ module _ id-equiv-Infinite-Cyclic-Type = id-equiv-Cyclic-Type zero-ℕ X equiv-eq-Infinite-Cyclic-Type : - (Y : Infinite-Cyclic-Type l1) → Id X Y → equiv-Infinite-Cyclic-Type Y + (Y : Infinite-Cyclic-Type l1) → X ＝ Y → equiv-Infinite-Cyclic-Type Y equiv-eq-Infinite-Cyclic-Type = equiv-eq-Cyclic-Type zero-ℕ X is-torsorial-equiv-Infinite-Cyclic-Type : diff --git a/src/synthetic-homotopy-theory/interval-type.lagda.md b/src/synthetic-homotopy-theory/interval-type.lagda.md index 983521bbb9..e4601a34f1 100644 --- a/src/synthetic-homotopy-theory/interval-type.lagda.md +++ b/src/synthetic-homotopy-theory/interval-type.lagda.md @@ -40,7 +40,7 @@ postulate target-𝕀 : 𝕀 - path-𝕀 : Id source-𝕀 target-𝕀 + path-𝕀 : source-𝕀 ＝ target-𝕀 ind-𝕀 : {l : Level} (P : 𝕀 → UU l) (u : P source-𝕀) (v : P target-𝕀) @@ -118,10 +118,10 @@ module _ refl-Eq-Data-𝕀 : (x : Data-𝕀 P) → Eq-Data-𝕀 x x refl-Eq-Data-𝕀 x = triple refl refl right-unit - Eq-eq-Data-𝕀 : {x y : Data-𝕀 P} → Id x y → Eq-Data-𝕀 x y + Eq-eq-Data-𝕀 : {x y : Data-𝕀 P} → x ＝ y → Eq-Data-𝕀 x y Eq-eq-Data-𝕀 {x = x} refl = refl-Eq-Data-𝕀 x - eq-Eq-Data-𝕀' : {x y : Data-𝕀 P} → Eq-Data-𝕀 x y → Id x y + eq-Eq-Data-𝕀' : {x y : Data-𝕀 P} → Eq-Data-𝕀 x y → x ＝ y eq-Eq-Data-𝕀' {x} {y} = map-inv-equiv (extensionality-Data-𝕀 x y) eq-Eq-Data-𝕀 : @@ -152,7 +152,7 @@ is-section-inv-ev-𝕀 (pair u (pair v q)) = tr-value : {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (f g : (x : A) → B x) {x y : A} - (p : Id x y) (q : Id (f x) (g x)) (r : Id (f y) (g y)) → + (p : x ＝ y) (q : Id (f x) (g x)) (r : Id (f y) (g y)) → Id (apd f p ∙ r) (ap (tr B p) q ∙ apd g p) → Id (tr (λ x → Id (f x) (g x)) p q) r tr-value f g refl q r s = (inv (ap-id q) ∙ inv right-unit) ∙ inv s @@ -179,7 +179,7 @@ abstract is-equiv-ev-𝕀 P = is-equiv-is-invertible inv-ev-𝕀 is-section-inv-ev-𝕀 is-retraction-inv-ev-𝕀 -contraction-𝕀 : (x : 𝕀) → Id source-𝕀 x +contraction-𝕀 : (x : 𝕀) → source-𝕀 ＝ x contraction-𝕀 = ind-𝕀 ( Id source-𝕀) diff --git a/src/synthetic-homotopy-theory/loop-spaces.lagda.md b/src/synthetic-homotopy-theory/loop-spaces.lagda.md index d150a83996..99790500d6 100644 --- a/src/synthetic-homotopy-theory/loop-spaces.lagda.md +++ b/src/synthetic-homotopy-theory/loop-spaces.lagda.md @@ -134,38 +134,38 @@ module _ {l1 : Level} {A : UU l1} {x y : A} where - equiv-tr-Ω : Id x y → Ω (pair A x) ≃∗ Ω (pair A y) + equiv-tr-Ω : x ＝ y → Ω (pair A x) ≃∗ Ω (pair A y) equiv-tr-Ω refl = pair id-equiv refl - equiv-tr-type-Ω : Id x y → type-Ω (pair A x) ≃ type-Ω (pair A y) + equiv-tr-type-Ω : x ＝ y → type-Ω (pair A x) ≃ type-Ω (pair A y) equiv-tr-type-Ω p = equiv-pointed-equiv (equiv-tr-Ω p) - tr-type-Ω : Id x y → type-Ω (pair A x) → type-Ω (pair A y) + tr-type-Ω : x ＝ y → type-Ω (pair A x) → type-Ω (pair A y) tr-type-Ω p = map-equiv (equiv-tr-type-Ω p) - is-equiv-tr-type-Ω : (p : Id x y) → is-equiv (tr-type-Ω p) + is-equiv-tr-type-Ω : (p : x ＝ y) → is-equiv (tr-type-Ω p) is-equiv-tr-type-Ω p = is-equiv-map-equiv (equiv-tr-type-Ω p) - preserves-refl-tr-Ω : (p : Id x y) → Id (tr-type-Ω p refl) refl + preserves-refl-tr-Ω : (p : x ＝ y) → Id (tr-type-Ω p refl) refl preserves-refl-tr-Ω refl = refl preserves-mul-tr-Ω : - (p : Id x y) (u v : type-Ω (pair A x)) → + (p : x ＝ y) (u v : type-Ω (pair A x)) → Id ( tr-type-Ω p (mul-Ω (pair A x) u v)) ( mul-Ω (pair A y) (tr-type-Ω p u) (tr-type-Ω p v)) preserves-mul-tr-Ω refl u v = refl preserves-inv-tr-Ω : - (p : Id x y) (u : type-Ω (pair A x)) → + (p : x ＝ y) (u : type-Ω (pair A x)) → Id ( tr-type-Ω p (inv-Ω (pair A x) u)) ( inv-Ω (pair A y) (tr-type-Ω p u)) preserves-inv-tr-Ω refl u = refl eq-tr-type-Ω : - (p : Id x y) (q : type-Ω (pair A x)) → + (p : x ＝ y) (q : type-Ω (pair A x)) → Id (tr-type-Ω p q) (inv p ∙ (q ∙ p)) eq-tr-type-Ω refl q = inv right-unit ``` diff --git a/src/synthetic-homotopy-theory/triple-loop-spaces.lagda.md b/src/synthetic-homotopy-theory/triple-loop-spaces.lagda.md index cdb70b73bf..5f2aaf3924 100644 --- a/src/synthetic-homotopy-theory/triple-loop-spaces.lagda.md +++ b/src/synthetic-homotopy-theory/triple-loop-spaces.lagda.md @@ -61,17 +61,17 @@ z-concat-Ω³ = z-concat-Id³ ap-x-concat-Ω³ : {l : Level} {A : UU l} {a : A} {α α' β β' : type-Ω³ a} - (s : Id α α') (t : Id β β') → Id (x-concat-Ω³ α β) (x-concat-Ω³ α' β') + (s : α ＝ α') (t : β ＝ β') → Id (x-concat-Ω³ α β) (x-concat-Ω³ α' β') ap-x-concat-Ω³ s t = ap-binary x-concat-Ω³ s t ap-y-concat-Ω³ : {l : Level} {A : UU l} {a : A} {α α' β β' : type-Ω³ a} - (s : Id α α') (t : Id β β') → Id (y-concat-Ω³ α β) (y-concat-Ω³ α' β') + (s : α ＝ α') (t : β ＝ β') → Id (y-concat-Ω³ α β) (y-concat-Ω³ α' β') ap-y-concat-Ω³ s t = j-concat-Id⁴ s t ap-z-concat-Ω³ : {l : Level} {A : UU l} {a : A} {α α' β β' : type-Ω³ a} - (s : Id α α') (t : Id β β') → Id (z-concat-Ω³ α β) (z-concat-Ω³ α' β') + (s : α ＝ α') (t : β ＝ β') → Id (z-concat-Ω³ α β) (z-concat-Ω³ α' β') ap-z-concat-Ω³ s t = k-concat-Id⁴ s t ``` @@ -113,8 +113,8 @@ left-unit-law-z-concat-Ω³ α = {- super-naturality-right-unit : - {l : Level} {A : UU l} {x y z : A} {p q : Id x y} {α β : Id p q} (γ : Id α β) - (u : Id y z) → + {l : Level} {A : UU l} {x y z : A} {p q : x ＝ y} {α β : p ＝ q} (γ : α ＝ β) + (u : y ＝ z) → Id (ap (λ ω → horizontal-concat-Id² ω (refl {x = u})) γ) {!!} super-naturality-right-unit α = {!!} -} diff --git a/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md b/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md index 2d69b1844c..efc341525b 100644 --- a/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md +++ b/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md @@ -69,7 +69,7 @@ functor-free-dependent-loop l {P} {Q} f = coherence-square-functor-free-dependent-loop : { l1 l2 l3 : Level} {X : UU l1} {P : X → UU l2} {Q : X → UU l3} - ( f : (x : X) → P x → Q x) {x y : X} (α : Id x y) + ( f : (x : X) → P x → Q x) {x y : X} (α : x ＝ y) ( h : (x : X) → P x) → Id ( inv ( preserves-tr f α (h x)) ∙ @@ -194,12 +194,12 @@ contraction-total-space {B = B} center x = path-total-path-fiber : { l1 l2 : Level} {A : UU l1} (B : A → UU l2) (x : A) → - { y y' : B x} (q : Id y' y) → Id {A = Σ A B} (pair x y) (pair x y') + { y y' : B x} (q : y' ＝ y) → Id {A = Σ A B} (pair x y) (pair x y') path-total-path-fiber B x q = eq-pair-eq-fiber (inv q) tr-path-total-path-fiber : { l1 l2 : Level} {A : UU l1} {B : A → UU l2} (c : Σ A B) (x : A) → - { y y' : B x} (q : Id y' y) (α : Id c (pair x y')) → + { y y' : B x} (q : y' ＝ y) (α : Id c (pair x y')) → Id ( tr (λ z → Id c (pair x z)) q α) ( α ∙ (inv (path-total-path-fiber B x q))) @@ -207,7 +207,7 @@ tr-path-total-path-fiber c x refl α = inv right-unit segment-Σ : { l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} → - { x x' : A} (p : Id x x') + { x x' : A} (p : x ＝ x') { F : UU l3} {F' : UU l4} (f : F ≃ F') ( e : F ≃ B x) (e' : F' ≃ B x') ( H : ((map-equiv e') ∘ (map-equiv f)) ~ ((tr B p) ∘ (map-equiv e))) (y : F) → Id (pair x (map-equiv e y)) (pair x' (map-equiv e' (map-equiv f y))) @@ -221,7 +221,7 @@ contraction-total-space' c x {F} e = equiv-tr-contraction-total-space' : { l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} (c : Σ A B) → - { x x' : A} (p : Id x x') → + { x x' : A} (p : x ＝ x') → { F : UU l3} {F' : UU l4} (f : F ≃ F') (e : F ≃ B x) (e' : F' ≃ B x') → ( H : ((map-equiv e') ∘ (map-equiv f)) ~ ((tr B p) ∘ (map-equiv e))) → ( contraction-total-space' c x' e') ≃ (contraction-total-space' c x e) @@ -250,7 +250,7 @@ tr-path-total-tr-coherence c x f e e' H y α = square-tr-contraction-total-space : { l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} (c : Σ A B) → - { x x' : A} (p : Id x x') + { x x' : A} (p : x ＝ x') { F : UU l3} {F' : UU l4} (f : F ≃ F') (e : F ≃ B x) (e' : F' ≃ B x') ( H : ((map-equiv e') ∘ (map-equiv f)) ~ ((tr B p) ∘ (map-equiv e))) (h : contraction-total-space c x) → @@ -267,7 +267,7 @@ square-tr-contraction-total-space c refl f e e' H h y = dependent-identification-contraction-total-space' : {l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} (c : Σ A B) → - {x x' : A} (p : Id x x') → + {x x' : A} (p : x ＝ x') → {F : UU l3} {F' : UU l4} (f : F ≃ F') ( e : F ≃ B x) (e' : F' ≃ B x') (H : ((map-equiv e') ∘ (map-equiv f)) ~ ((tr B p) ∘ (map-equiv e))) → (h : (y : F) → Id c (pair x (map-equiv e y))) → @@ -284,7 +284,7 @@ dependent-identification-contraction-total-space' map-dependent-identification-contraction-total-space' : { l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} (c : Σ A B) → - { x x' : A} (p : Id x x') → + { x x' : A} (p : x ＝ x') → { F : UU l3} {F' : UU l4} (f : F ≃ F') ( e : F ≃ B x) (e' : F' ≃ B x') ( H : ((map-equiv e') ∘ (map-equiv f)) ~ ((tr B p) ∘ (map-equiv e))) → ( h : contraction-total-space' c x e) → @@ -325,7 +325,7 @@ map-dependent-identification-contraction-total-space' equiv-dependent-identification-contraction-total-space' : { l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} (c : Σ A B) → - { x x' : A} (p : Id x x') → + { x x' : A} (p : x ＝ x') → { F : UU l3} {F' : UU l4} (f : F ≃ F') ( e : F ≃ B x) (e' : F' ≃ B x') ( H : ((map-equiv e') ∘ (map-equiv f)) ~ ((tr B p) ∘ (map-equiv e))) → ( h : contraction-total-space' c x e) → diff --git a/src/trees/extensional-w-types.lagda.md b/src/trees/extensional-w-types.lagda.md index 257d84375e..fd4b5cb8e3 100644 --- a/src/trees/extensional-w-types.lagda.md +++ b/src/trees/extensional-w-types.lagda.md @@ -41,7 +41,7 @@ A W-type `𝕎 A B` is said to be **extensional** if for any two elements `S T : 𝕎 A B` the induced map ```text - Id S T → ((U : 𝕎 A B) → (U ∈-𝕎 S) ≃ (U ∈-𝕎 T)) + S ＝ T → ((U : 𝕎 A B) → (U ∈-𝕎 S) ≃ (U ∈-𝕎 T)) ``` is an equivalence. diff --git a/src/type-theories/dependent-type-theories.lagda.md b/src/type-theories/dependent-type-theories.lagda.md index 91b1339c00..9dcb5a5ba3 100644 --- a/src/type-theories/dependent-type-theories.lagda.md +++ b/src/type-theories/dependent-type-theories.lagda.md @@ -87,7 +87,7 @@ homotopies of sections of fibered systems. ```agda tr-fibered-system-slice : {l1 l2 l3 l4 : Level} {A : system l1 l2} {B B' : fibered-system l3 l4 A} - (α : Id B B') (f : section-system B) (X : system.type A) → + (α : B ＝ B') (f : section-system B) (X : system.type A) → Id ( fibered-system.slice B (section-system.type f X)) ( fibered-system.slice B' @@ -96,7 +96,7 @@ homotopies of sections of fibered systems. Eq-fibered-system' : {l1 l2 l3 l4 : Level} {A : system l1 l2} {B B' : fibered-system l3 l4 A} - (α : Id B B') (f : section-system B) (g : section-system B') → + (α : B ＝ B') (f : section-system B) (g : section-system B') → fibered-system l3 l4 A fibered-system.type (Eq-fibered-system' {A = A} α f g) X = Id @@ -117,14 +117,14 @@ homotopies of sections of fibered systems. htpy-section-system' : {l1 l2 l3 l4 : Level} {A : system l1 l2} {B B' : fibered-system l3 l4 A} - (α : Id B B') (f : section-system B) (g : section-system B') → + (α : B ＝ B') (f : section-system B) (g : section-system B') → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) htpy-section-system' {A = A} α f g = section-system (Eq-fibered-system' α f g) concat-htpy-section-system' : {l1 l2 l3 l4 : Level} {A : system l1 l2} {B B' B'' : fibered-system l3 l4 A} - {α : Id B B'} {β : Id B' B''} (γ : Id B B'') (δ : Id (α ∙ β) γ) + {α : B ＝ B'} {β : B' ＝ B''} (γ : B ＝ B'') (δ : Id (α ∙ β) γ) {f : section-system B} {g : section-system B'} {h : section-system B''} (G : htpy-section-system' α f g) (H : htpy-section-system' β g h) → @@ -162,7 +162,7 @@ homotopies of sections of fibered systems. inv-htpy-section-system' : {l1 l2 l3 l4 : Level} {A : system l1 l2} {B B' : fibered-system l3 l4 A} - {α : Id B B'} (β : Id B' B) (γ : Id (inv α) β) + {α : B ＝ B'} (β : B' ＝ B) (γ : Id (inv α) β) {f : section-system B} {g : section-system B'} → htpy-section-system' α f g → htpy-section-system' β g f section-system.type (inv-htpy-section-system' {α = refl} .refl refl H) X = @@ -337,10 +337,10 @@ We show that systems form a category. left-whisker-comp-hom-system' : {l1 l2 l3 l4 l5 l6 : Level} {A : system l1 l2} {B B' : system l3 l4} {C C' : system l5 l6} {g : hom-system B C} {g' : hom-system B' C'} - (p : Id B B') + (p : B ＝ B') {p' : Id (constant-fibered-system A B) (constant-fibered-system A B')} (α : Id (ap (constant-fibered-system A) p) p') - (q : Id C C') + (q : C ＝ C') {q' : Id (constant-fibered-system A C) (constant-fibered-system A C')} (β : Id (ap (constant-fibered-system A) q) q') (r : Id (tr (λ t → t) (ap-binary hom-system p q) g) g') @@ -384,7 +384,7 @@ We show that systems form a category. ( γ (section-system.type H X)) ( section-system.slice H X) where - γ : {Y Y' : system.type B} (p : Id Y Y') → + γ : {Y Y' : system.type B} (p : Y ＝ Y') → Id ( tr ( λ t → t) @@ -405,7 +405,7 @@ We show that systems form a category. right-whisker-comp-hom-system' : {l1 l2 l3 l4 l5 l6 : Level} {A : system l1 l2} {B : system l3 l4} - {C C' : system l5 l6} (p : Id C C') {g : hom-system B C} + {C C' : system l5 l6} (p : C ＝ C') {g : hom-system B C} {g' : hom-system B C'} {p' : Id (constant-fibered-system B C) (constant-fibered-system B C')} (α : Id (ap (constant-fibered-system B) p) p') @@ -1449,7 +1449,7 @@ We define what it means for a dependent type theory to have Π-types. {- concat-htpy-hom-system' : {l1 l2 l3 l4 : Level} {A : system l1 l2} {B B' B'' : system l3 l4} - (p : Id B B') (q : Id B' B'') {f : hom-system A B} {g : hom-system A B'} + (p : B ＝ B') (q : B' ＝ B'') {f : hom-system A B} {g : hom-system A B'} {h : hom-system A B''} → htpy-hom-system' p f g → htpy-hom-system' q g h → htpy-hom-system' (p ∙ q) f h htpy-hom-system'.type (concat-htpy-hom-system' refl refl H K) = diff --git a/src/type-theories/fibered-dependent-type-theories.lagda.md b/src/type-theories/fibered-dependent-type-theories.lagda.md index a5da36d53a..0d1905bdbb 100644 --- a/src/type-theories/fibered-dependent-type-theories.lagda.md +++ b/src/type-theories/fibered-dependent-type-theories.lagda.md @@ -106,7 +106,7 @@ module fibered where ```agda double-tr : {l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} - (D : (x : A) → B x → C x → UU l4) {x y : A} (p : Id x y) + (D : (x : A) → B x → C x → UU l4) {x y : A} (p : x ＝ y) {u : B x} {u' : B y} (q : Id (tr B p u) u') {v : C x} {v' : C y} (r : Id (tr C p v) v') → D x u v → D y u' v' double-tr D refl refl refl d = d @@ -117,7 +117,7 @@ module fibered where (D : bifibered-system l7 l8 B C) {X : system.type A} (Y : fibered-system.type B X) {Z Z' : fibered-system.type C X} {d : bifibered-system.type D Y Z} {d' : bifibered-system.type D Y Z'} - (p : Id Z Z') (q : Id (tr (bifibered-system.type D Y) p d) d') → + (p : Z ＝ Z') (q : Id (tr (bifibered-system.type D Y) p d) d') → Id ( tr ( bifibered-system l7 l8 (fibered-system.slice B Y)) @@ -130,7 +130,7 @@ module fibered where {l1 l2 l3 l4 l5 l6 l7 l8 : Level} {A : system l1 l2} {B : fibered-system l3 l4 A} {C C' : fibered-system l5 l6 A} (D : bifibered-system l7 l8 B C) (D' : bifibered-system l7 l8 B C') - (α : Id C C') (β : Id (tr (bifibered-system l7 l8 B) α D) D') + (α : C ＝ C') (β : Id (tr (bifibered-system l7 l8 B) α D) D') (f : section-system C) (f' : section-system C') (g : section-fibered-system f D) (g' : section-fibered-system f' D') → bifibered-system l7 l8 B (Eq-fibered-system' α f f') @@ -167,7 +167,7 @@ module fibered where {B : fibered-system l3 l4 A} {C C' : fibered-system l5 l6 A} {D : bifibered-system l7 l8 B C} {D' : bifibered-system l7 l8 B C'} {f : section-system C} {f' : section-system C'} - {α : Id C C'} (β : Id (tr (bifibered-system l7 l8 B) α D) D') + {α : C ＝ C'} (β : Id (tr (bifibered-system l7 l8 B) α D) D') (H : htpy-section-system' α f f') (g : section-fibered-system f D) (h : section-fibered-system f' D') → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ l7 ⊔ l8) diff --git a/src/type-theories/simple-type-theories.lagda.md b/src/type-theories/simple-type-theories.lagda.md index b8e45f3112..1c1b961458 100644 --- a/src/type-theories/simple-type-theories.lagda.md +++ b/src/type-theories/simple-type-theories.lagda.md @@ -68,7 +68,7 @@ homotopies of sections of fibered systems. ```agda Eq-fibered-system' : {l1 l2 l3 l4 : Level} {T : UU l1} {A : system l2 T} {S : T → UU l3} - {B B' : fibered-system l4 S A} (α : Id B B') {f f' : (X : T) → S X} + {B B' : fibered-system l4 S A} (α : B ＝ B') {f f' : (X : T) → S X} (g : section-system B f) (g' : section-system B' f') → fibered-system l4 (λ t → Id (f t) (f' t)) A fibered-system.element (Eq-fibered-system' {B = B} refl {f} g g') {X} p x = @@ -86,7 +86,7 @@ homotopies of sections of fibered systems. htpy-section-system' : {l1 l2 l3 l4 : Level} {T : UU l1} {A : system l2 T} {S : T → UU l3} - {B B' : fibered-system l4 S A} (α : Id B B') {f f' : (X : T) → S X} + {B B' : fibered-system l4 S A} (α : B ＝ B') {f f' : (X : T) → S X} (H : f ~ f') (g : section-system B f) (g' : section-system B' f') → UU (l1 ⊔ l2 ⊔ l4) htpy-section-system' α H g g' = @@ -95,8 +95,8 @@ homotopies of sections of fibered systems. concat-htpy-section-system' : {l1 l2 l3 l4 : Level} {T : UU l1} {A : system l2 T} {S : T → UU l3} - {B B' B'' : fibered-system l4 S A} {α : Id B B'} {β : Id B' B''} - (γ : Id B B'') (δ : Id (α ∙ β) γ) {f f' f'' : (X : T) → S X} + {B B' B'' : fibered-system l4 S A} {α : B ＝ B'} {β : B' ＝ B''} + (γ : B ＝ B'') (δ : Id (α ∙ β) γ) {f f' f'' : (X : T) → S X} {H : f ~ f'} {H' : f' ~ f''} {g : section-system B f} {g' : section-system B' f'} {g'' : section-system B'' f''} (K : htpy-section-system' α H g g') @@ -129,7 +129,7 @@ homotopies of sections of fibered systems. {l1 l2 l3 l4 : Level} {T : UU l1} {A : system l2 T} {S : T → UU l3} {B B' : fibered-system l4 S A} - {α : Id B B'} (β : Id B' B) (γ : Id (inv α) β) + {α : B ＝ B'} (β : B' ＝ B) (γ : Id (inv α) β) {f f' : (X : T) → S X} {g : section-system B f} {g' : section-system B' f'} {H : f ~ f'} → htpy-section-system' α H g g' → htpy-section-system' β (inv-htpy H) g' g @@ -575,7 +575,7 @@ module dependent-simple Eq-fibered-system' : {l1 l2 l3 l4 : Level} {T : UU l1} {A : simple.system l2 T} {S : T → UU l3} - {B B' : simple.fibered-system l4 S A} (α : Id B B') + {B B' : simple.fibered-system l4 S A} (α : B ＝ B') {f f' : (X : T) → S X} (g : simple.section-system B f) (g' : simple.section-system B' f') → fibered.hom-fibered-system @@ -593,7 +593,7 @@ module dependent-simple htpy-section-system' : {l1 l2 l3 l4 : Level} {T : UU l1} {A : simple.system l2 T} {S : T → UU l3} - {B B' : simple.fibered-system l4 S A} (α : Id B B') {f f' : (X : T) → S X} + {B B' : simple.fibered-system l4 S A} (α : B ＝ B') {f f' : (X : T) → S X} {H : f ~ f'} {g : simple.section-system B f} {g' : simple.section-system B' f'} → simple.htpy-section-system' α H g g' → dependent.htpy-section-system' diff --git a/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md b/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md index f223fc9b23..892da4210f 100644 --- a/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md +++ b/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md @@ -94,7 +94,7 @@ module _ decidable-subtype-2-Element-Decidable-Subtype eq-is-in-2-Element-Decidable-Subtype : - {x : X} {y z : is-in-2-Element-Decidable-Subtype x} → Id y z + {x : X} {y z : is-in-2-Element-Decidable-Subtype x} → y ＝ z eq-is-in-2-Element-Decidable-Subtype {x} = eq-is-prop (is-prop-is-in-2-Element-Decidable-Subtype x) diff --git a/src/univalent-combinatorics/classical-finite-types.lagda.md b/src/univalent-combinatorics/classical-finite-types.lagda.md index 388ff0875d..6306188299 100644 --- a/src/univalent-combinatorics/classical-finite-types.lagda.md +++ b/src/univalent-combinatorics/classical-finite-types.lagda.md @@ -68,7 +68,7 @@ eq-succ-classical-Fin : eq-succ-classical-Fin k x .x refl = refl eq-Eq-classical-Fin : - (k : ℕ) (x y : classical-Fin k) → Eq-classical-Fin k x y → Id x y + (k : ℕ) (x y : classical-Fin k) → Eq-classical-Fin k x y → x ＝ y eq-Eq-classical-Fin (succ-ℕ k) (pair zero-ℕ _) (pair zero-ℕ _) e = refl eq-Eq-classical-Fin (succ-ℕ k) (pair (succ-ℕ x) p) (pair (succ-ℕ y) q) e = eq-succ-classical-Fin k diff --git a/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md b/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md index 398a1fe35d..077b75f365 100644 --- a/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md +++ b/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md @@ -149,7 +149,7 @@ count-fiber-map-section-family {l1} {l2} {A} {B} b e f (pair y z) = ( is-torsorial-Id' y) ( pair y refl)) ∘e ( inv-associative-Σ A - ( λ x → Id x y) + ( λ x → x ＝ y) ( λ t → Id (tr B (pr2 t) (b (pr1 t))) z))) ∘e ( equiv-tot (λ x → equiv-pair-eq-Σ (pair x (b x)) (pair y z)))) ( count-eq (has-decidable-equality-count (f y)) (b y) z) diff --git a/src/univalent-combinatorics/cyclic-finite-types.lagda.md b/src/univalent-combinatorics/cyclic-finite-types.lagda.md index 45597bd99a..a4393d21ea 100644 --- a/src/univalent-combinatorics/cyclic-finite-types.lagda.md +++ b/src/univalent-combinatorics/cyclic-finite-types.lagda.md @@ -185,7 +185,7 @@ module _ id-equiv-Cyclic-Type = id-equiv-Type-With-Endomorphism (endo-Cyclic-Type k X) equiv-eq-Cyclic-Type : - (Y : Cyclic-Type l k) → Id X Y → equiv-Cyclic-Type k X Y + (Y : Cyclic-Type l k) → X ＝ Y → equiv-Cyclic-Type k X Y equiv-eq-Cyclic-Type .X refl = id-equiv-Cyclic-Type is-torsorial-equiv-Cyclic-Type : @@ -216,7 +216,7 @@ module _ pr2 (extensionality-Cyclic-Type Y) = is-equiv-equiv-eq-Cyclic-Type Y eq-equiv-Cyclic-Type : - (Y : Cyclic-Type l k) → equiv-Cyclic-Type k X Y → Id X Y + (Y : Cyclic-Type l k) → equiv-Cyclic-Type k X Y → X ＝ Y eq-equiv-Cyclic-Type Y = map-inv-is-equiv (is-equiv-equiv-eq-Cyclic-Type Y) ``` @@ -236,7 +236,7 @@ module _ refl-htpy-equiv-Cyclic-Type e = refl-htpy htpy-eq-equiv-Cyclic-Type : - (e f : equiv-Cyclic-Type k X Y) → Id e f → htpy-equiv-Cyclic-Type e f + (e f : equiv-Cyclic-Type k X Y) → e ＝ f → htpy-equiv-Cyclic-Type e f htpy-eq-equiv-Cyclic-Type e .e refl = refl-htpy-equiv-Cyclic-Type e is-torsorial-htpy-equiv-Cyclic-Type : @@ -521,7 +521,7 @@ map-equiv-compute-Ω-Cyclic-Type : map-equiv-compute-Ω-Cyclic-Type k = map-equiv (equiv-compute-Ω-Cyclic-Type k) preserves-concat-equiv-eq-Cyclic-Type : - (k : ℕ) (X Y Z : Cyclic-Type lzero k) (p : Id X Y) (q : Id Y Z) → + (k : ℕ) (X Y Z : Cyclic-Type lzero k) (p : X ＝ Y) (q : Y ＝ Z) → Id ( equiv-eq-Cyclic-Type k X Z (p ∙ q)) ( comp-equiv-Cyclic-Type k X Y Z diff --git a/src/univalent-combinatorics/embeddings-standard-finite-types.lagda.md b/src/univalent-combinatorics/embeddings-standard-finite-types.lagda.md index 458ec1b433..3d9b26b588 100644 --- a/src/univalent-combinatorics/embeddings-standard-finite-types.lagda.md +++ b/src/univalent-combinatorics/embeddings-standard-finite-types.lagda.md @@ -76,7 +76,7 @@ abstract Id ( cases-map-reduce-emb-Fin k l f d x e) ( cases-map-reduce-emb-Fin k l f d x' e') → - Id x x' + x ＝ x' is-injective-cases-map-reduce-emb-Fin k l f (inl (pair t q)) x e x' e' p = is-injective-inl ( is-injective-is-emb diff --git a/src/univalent-combinatorics/equality-finite-types.lagda.md b/src/univalent-combinatorics/equality-finite-types.lagda.md index 220051d0d8..98592329a8 100644 --- a/src/univalent-combinatorics/equality-finite-types.lagda.md +++ b/src/univalent-combinatorics/equality-finite-types.lagda.md @@ -79,6 +79,6 @@ is-finite-eq-Finite-Type X = Id-Finite-Type : {l : Level} → (X : Finite-Type l) (x y : type-Finite-Type X) → Finite-Type l -pr1 (Id-Finite-Type X x y) = Id x y +pr1 (Id-Finite-Type X x y) = x ＝ y pr2 (Id-Finite-Type X x y) = is-finite-eq-Finite-Type X ``` diff --git a/src/univalent-combinatorics/equality-standard-finite-types.lagda.md b/src/univalent-combinatorics/equality-standard-finite-types.lagda.md index 637b3cf66a..3103dd72a7 100644 --- a/src/univalent-combinatorics/equality-standard-finite-types.lagda.md +++ b/src/univalent-combinatorics/equality-standard-finite-types.lagda.md @@ -64,11 +64,11 @@ refl-Eq-Fin : (k : ℕ) (x : Fin k) → Eq-Fin k x x refl-Eq-Fin (succ-ℕ k) (inl x) = refl-Eq-Fin k x refl-Eq-Fin (succ-ℕ k) (inr x) = star -Eq-Fin-eq : (k : ℕ) {x y : Fin k} → Id x y → Eq-Fin k x y +Eq-Fin-eq : (k : ℕ) {x y : Fin k} → x ＝ y → Eq-Fin k x y Eq-Fin-eq k refl = refl-Eq-Fin k _ eq-Eq-Fin : - (k : ℕ) {x y : Fin k} → Eq-Fin k x y → Id x y + (k : ℕ) {x y : Fin k} → Eq-Fin k x y → x ＝ y eq-Eq-Fin (succ-ℕ k) {inl x} {inl y} e = ap inl (eq-Eq-Fin k e) eq-Eq-Fin (succ-ℕ k) {inr star} {inr star} star = refl @@ -151,7 +151,7 @@ is-contr-is-zero-or-one-Fin-2 x = ```agda decidable-Eq-Fin : (n : ℕ) (i j : Fin n) → Decidable-Prop lzero -pr1 (decidable-Eq-Fin n i j) = Id i j +pr1 (decidable-Eq-Fin n i j) = i ＝ j pr1 (pr2 (decidable-Eq-Fin n i j)) = is-set-Fin n i j pr2 (pr2 (decidable-Eq-Fin n i j)) = has-decidable-equality-Fin n i j ``` diff --git a/src/univalent-combinatorics/equivalences-cubes.lagda.md b/src/univalent-combinatorics/equivalences-cubes.lagda.md index 4324b6570d..543b6c237c 100644 --- a/src/univalent-combinatorics/equivalences-cubes.lagda.md +++ b/src/univalent-combinatorics/equivalences-cubes.lagda.md @@ -63,7 +63,7 @@ id-equiv-cube : id-equiv-cube k X = pair id-equiv (λ x → id-equiv) equiv-eq-cube : - (k : ℕ) {X Y : cube k} → Id X Y → equiv-cube k X Y + (k : ℕ) {X Y : cube k} → X ＝ Y → equiv-cube k X Y equiv-eq-cube k {X} refl = id-equiv-cube k X is-torsorial-equiv-cube : @@ -92,7 +92,7 @@ is-equiv-equiv-eq-cube k X = ( λ Y → equiv-eq-cube k {X = X} {Y}) eq-equiv-cube : - (k : ℕ) (X Y : cube k) → equiv-cube k X Y → Id X Y + (k : ℕ) (X Y : cube k) → equiv-cube k X Y → X ＝ Y eq-equiv-cube k X Y = map-inv-is-equiv (is-equiv-equiv-eq-cube k X Y) @@ -117,7 +117,7 @@ refl-htpy-equiv-cube k X Y e = pair refl-htpy (λ d → refl-htpy) htpy-eq-equiv-cube : (k : ℕ) (X Y : cube k) (e f : equiv-cube k X Y) → - Id e f → htpy-equiv-cube k X Y e f + e ＝ f → htpy-equiv-cube k X Y e f htpy-eq-equiv-cube k X Y e .e refl = refl-htpy-equiv-cube k X Y e is-torsorial-htpy-equiv-cube : @@ -141,7 +141,7 @@ is-equiv-htpy-eq-equiv-cube k X Y e = eq-htpy-equiv-cube : (k : ℕ) (X Y : cube k) (e f : equiv-cube k X Y) → - htpy-equiv-cube k X Y e f → Id e f + htpy-equiv-cube k X Y e f → e ＝ f eq-htpy-equiv-cube k X Y e f = map-inv-is-equiv (is-equiv-htpy-eq-equiv-cube k X Y e f) ``` diff --git a/src/univalent-combinatorics/ferrers-diagrams.lagda.md b/src/univalent-combinatorics/ferrers-diagrams.lagda.md index 034b26e3ad..57d99eae92 100644 --- a/src/univalent-combinatorics/ferrers-diagrams.lagda.md +++ b/src/univalent-combinatorics/ferrers-diagrams.lagda.md @@ -168,7 +168,7 @@ module _ pr2 id-equiv-ferrers-diagram x = id-equiv equiv-eq-ferrers-diagram : - (E : ferrers-diagram l2 l3 A) → Id D E → equiv-ferrers-diagram E + (E : ferrers-diagram l2 l3 A) → D ＝ E → equiv-ferrers-diagram E equiv-eq-ferrers-diagram .D refl = id-equiv-ferrers-diagram is-torsorial-equiv-ferrers-diagram : @@ -197,7 +197,7 @@ module _ equiv-eq-ferrers-diagram eq-equiv-ferrers-diagram : - (E : ferrers-diagram l2 l3 A) → equiv-ferrers-diagram E → Id D E + (E : ferrers-diagram l2 l3 A) → equiv-ferrers-diagram E → D ＝ E eq-equiv-ferrers-diagram E = map-inv-is-equiv (is-equiv-equiv-eq-ferrers-diagram E) ``` @@ -224,7 +224,7 @@ module _ equiv-eq-ferrers-diagram-Finite-Type : (E : ferrers-diagram-Finite-Type l2 l3 A) → - Id D E → equiv-ferrers-diagram-Finite-Type E + D ＝ E → equiv-ferrers-diagram-Finite-Type E equiv-eq-ferrers-diagram-Finite-Type .D refl = id-equiv-ferrers-diagram-Finite-Type @@ -268,7 +268,7 @@ module _ eq-equiv-ferrers-diagram-Finite-Type : (E : ferrers-diagram-Finite-Type l2 l3 A) → - equiv-ferrers-diagram-Finite-Type E → Id D E + equiv-ferrers-diagram-Finite-Type E → D ＝ E eq-equiv-ferrers-diagram-Finite-Type E = map-inv-is-equiv (is-equiv-equiv-eq-ferrers-diagram-Finite-Type E) ``` diff --git a/src/univalent-combinatorics/fibers-of-maps.lagda.md b/src/univalent-combinatorics/fibers-of-maps.lagda.md index 2d9e458ea5..426d066387 100644 --- a/src/univalent-combinatorics/fibers-of-maps.lagda.md +++ b/src/univalent-combinatorics/fibers-of-maps.lagda.md @@ -122,7 +122,7 @@ abstract ( is-torsorial-Id' y) ( pair y refl)) ∘e ( inv-associative-Σ A - ( λ x → Id x y) + ( λ x → x ＝ y) ( λ t → Id (tr B (pr2 t) (b (pr1 t))) z))) ∘e ( equiv-tot (λ x → equiv-pair-eq-Σ (pair x (b x)) (pair y z)))) ( is-finite-eq (has-decidable-equality-is-finite (g y))) diff --git a/src/univalent-combinatorics/finite-types.lagda.md b/src/univalent-combinatorics/finite-types.lagda.md index b89d762ec6..e2dd89744d 100644 --- a/src/univalent-combinatorics/finite-types.lagda.md +++ b/src/univalent-combinatorics/finite-types.lagda.md @@ -419,7 +419,7 @@ number-of-elements-Finite-Type X = ```agda eq-cardinality : {l1 : Level} {k l : ℕ} {A : UU l1} → - has-cardinality-ℕ k A → has-cardinality-ℕ l A → Id k l + has-cardinality-ℕ k A → has-cardinality-ℕ l A → k ＝ l eq-cardinality H K = apply-universal-property-trunc-Prop H ( Id-Prop ℕ-Set _ _) @@ -634,11 +634,11 @@ is-torsorial-equiv-Finite-Type {l} X = ( is-torsorial-Id X) equiv-eq-Finite-Type : - {l : Level} → (X Y : Finite-Type l) → Id X Y → equiv-Finite-Type X Y + {l : Level} → (X Y : Finite-Type l) → X ＝ Y → equiv-Finite-Type X Y equiv-eq-Finite-Type X Y = map-equiv (extensionality-Finite-Type X Y) eq-equiv-Finite-Type : - {l : Level} → (X Y : Finite-Type l) → equiv-Finite-Type X Y → Id X Y + {l : Level} → (X Y : Finite-Type l) → equiv-Finite-Type X Y → X ＝ Y eq-equiv-Finite-Type X Y = map-inv-equiv (extensionality-Finite-Type X Y) ``` @@ -679,7 +679,7 @@ id-equiv-Type-With-Cardinality-ℕ X = id-equiv-component-UU-Level X equiv-eq-Type-With-Cardinality-ℕ : {l : Level} (k : ℕ) {X Y : Type-With-Cardinality-ℕ l k} → - Id X Y → equiv-Type-With-Cardinality-ℕ k X Y + X ＝ Y → equiv-Type-With-Cardinality-ℕ k X Y equiv-eq-Type-With-Cardinality-ℕ k p = equiv-eq-component-UU-Level p abstract @@ -700,7 +700,7 @@ abstract eq-equiv-Type-With-Cardinality-ℕ : {l : Level} (k : ℕ) (X Y : Type-With-Cardinality-ℕ l k) → - equiv-Type-With-Cardinality-ℕ k X Y → Id X Y + equiv-Type-With-Cardinality-ℕ k X Y → X ＝ Y eq-equiv-Type-With-Cardinality-ℕ k X Y = eq-equiv-component-UU-Level X Y diff --git a/src/univalent-combinatorics/inequality-types-with-counting.lagda.md b/src/univalent-combinatorics/inequality-types-with-counting.lagda.md index 137bb6bd70..7894a0ff69 100644 --- a/src/univalent-combinatorics/inequality-types-with-counting.lagda.md +++ b/src/univalent-combinatorics/inequality-types-with-counting.lagda.md @@ -48,7 +48,7 @@ refl-leq-count e x = antisymmetric-leq-count : {l : Level} {X : UU l} (e : count X) {x y : X} → - leq-count e x y → leq-count e y x → Id x y + leq-count e x y → leq-count e y x → x ＝ y antisymmetric-leq-count e H K = is-injective-map-inv-equiv ( equiv-count e) diff --git a/src/univalent-combinatorics/injective-maps.lagda.md b/src/univalent-combinatorics/injective-maps.lagda.md index b848851cf0..cb28686054 100644 --- a/src/univalent-combinatorics/injective-maps.lagda.md +++ b/src/univalent-combinatorics/injective-maps.lagda.md @@ -29,7 +29,7 @@ Injectiveness in the context of finite types enjoys further properties. ```agda is-decidable-is-injective-is-finite' : {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) → - is-finite A → is-finite B → is-decidable ((x y : A) → Id (f x) (f y) → Id x y) + is-finite A → is-finite B → is-decidable ((x y : A) → Id (f x) (f y) → x ＝ y) is-decidable-is-injective-is-finite' f HA HB = is-decidable-Π-is-finite HA ( λ x → diff --git a/src/univalent-combinatorics/repetitions-of-values.lagda.md b/src/univalent-combinatorics/repetitions-of-values.lagda.md index fe61953e54..cef067fbdd 100644 --- a/src/univalent-combinatorics/repetitions-of-values.lagda.md +++ b/src/univalent-combinatorics/repetitions-of-values.lagda.md @@ -80,7 +80,7 @@ repetition-of-values-is-not-injective-Fin k l f N = K = pr2 v w : (f x ＝ f y) × (x ≠ y) w = exists-not-not-for-all-count - ( λ _ → Id x y) + ( λ _ → x ＝ y) ( λ _ → has-decidable-equality-Fin k x y) ( count-is-decidable-is-prop From 7934dbd16089f1537b2aff9398e1ab009c8fe865 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 1 Sep 2025 18:31:25 +0200 Subject: [PATCH 03/16] =?UTF-8?q?another=20edge=20case:=20`=E2=89=83=20Id`?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../universal-property-integers.lagda.md | 2 +- src/foundation/effective-maps-equivalence-relations.lagda.md | 2 +- src/univalent-combinatorics/finite-types.lagda.md | 3 ++- 3 files changed, 4 insertions(+), 3 deletions(-) diff --git a/src/elementary-number-theory/universal-property-integers.lagda.md b/src/elementary-number-theory/universal-property-integers.lagda.md index 123b49094e..6b5827f733 100644 --- a/src/elementary-number-theory/universal-property-integers.lagda.md +++ b/src/elementary-number-theory/universal-property-integers.lagda.md @@ -99,7 +99,7 @@ equiv-comparison-map-Eq-ELIM-ℤ : { l1 : Level} (P : ℤ → UU l1) ( p0 : P zero-ℤ) (pS : (k : ℤ) → (P k) ≃ (P (succ-ℤ k))) → ( s t : ELIM-ℤ P p0 pS) (k : ℤ) → - Id ((pr1 s) k) ((pr1 t) k) ≃ Id ((pr1 s) (succ-ℤ k)) ((pr1 t) (succ-ℤ k)) + (pr1 s k ＝ pr1 t k) ≃ (pr1 s (succ-ℤ k) ＝ pr1 t (succ-ℤ k)) equiv-comparison-map-Eq-ELIM-ℤ P p0 pS s t k = ( ( equiv-concat (pr2 (pr2 s) k) (pr1 t (succ-ℤ k))) ∘e ( equiv-concat' (map-equiv (pS k) (pr1 s k)) (inv (pr2 (pr2 t) k)))) ∘e diff --git a/src/foundation/effective-maps-equivalence-relations.lagda.md b/src/foundation/effective-maps-equivalence-relations.lagda.md index 2e7cb51d1b..f229342fc7 100644 --- a/src/foundation/effective-maps-equivalence-relations.lagda.md +++ b/src/foundation/effective-maps-equivalence-relations.lagda.md @@ -21,7 +21,7 @@ open import foundation-core.identity-types ## Idea Consider a type `A` equipped with an equivalence relation `R`, and let -`f : A → X` be a map. Then `f` is effective if `R x y ≃ Id (f x) (f y)` for all +`f : A → X` be a map. Then `f` is effective if `R x y ≃ (f x ＝ f y)` for all `x y : A`. If `f` is both effective and surjective, then it follows that `X` satisfies the universal property of the quotient `A/R`. diff --git a/src/univalent-combinatorics/finite-types.lagda.md b/src/univalent-combinatorics/finite-types.lagda.md index e2dd89744d..f9d68827e6 100644 --- a/src/univalent-combinatorics/finite-types.lagda.md +++ b/src/univalent-combinatorics/finite-types.lagda.md @@ -756,7 +756,8 @@ abstract ```agda equiv-has-cardinality-id-number-of-elements-is-finite : {l : Level} (X : UU l) ( H : is-finite X) (n : ℕ) → - ( has-cardinality-ℕ n X ≃ Id (number-of-elements-is-finite H) n) + has-cardinality-ℕ n X ≃ + ( number-of-elements-is-finite H ＝ n) pr1 (equiv-has-cardinality-id-number-of-elements-is-finite X H n) Q = ap ( number-of-elements-has-finite-cardinality) From 26583b05488e6798c2821cbfcab6fd34633520e6 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 1 Sep 2025 18:33:08 +0200 Subject: [PATCH 04/16] oops --- .../sequential-colimits-in-homotopy-type-theory.lagda.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/literature/sequential-colimits-in-homotopy-type-theory.lagda.md b/src/literature/sequential-colimits-in-homotopy-type-theory.lagda.md index af67a0c9e3..34e5e2e86f 100644 --- a/src/literature/sequential-colimits-in-homotopy-type-theory.lagda.md +++ b/src/literature/sequential-colimits-in-homotopy-type-theory.lagda.md @@ -17,7 +17,7 @@ open import foundation.universe-levels using ( UU ) open import foundation.identity-types using - ( -- ＝ "path" + ( Id -- "path" ; refl -- "constant path" ; inv -- "inverse path" ; concat -- "concatenation of paths" From f89fefc755d42edf2ad13681e43ba4b1b885076d Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 1 Sep 2025 18:35:58 +0200 Subject: [PATCH 05/16] fix typecheck task --- .vscode/tasks.json | 18 ++++++++++++++++-- 1 file changed, 16 insertions(+), 2 deletions(-) diff --git a/.vscode/tasks.json b/.vscode/tasks.json index ac3e2254a1..9e33790ff3 100644 --- a/.vscode/tasks.json +++ b/.vscode/tasks.json @@ -9,7 +9,7 @@ "kind": "test", "isDefault": true }, - "dependsOn": ["pre-commit", "link-check", "typecheck"], + "dependsOn": ["link-check", "typecheck-after-pre-commit"], "dependsOrder": "parallel" }, { @@ -20,12 +20,26 @@ "args": ["check"], "problemMatcher": [], "group": { "kind": "build" }, - "dependsOn": "pre-commit", "presentation": { "panel": "dedicated", "clear": true } }, + { + "label": "typecheck-after-pre-commit", + "detail": "Typecheck the library after running pre-commit", + "type": "shell", + "command": "make", + "args": ["check"], + "problemMatcher": [], + "group": { "kind": "build" }, + "dependsOn": "pre-commit", + "presentation": { + "panel": "dedicated", + "clear": true + }, + "hide": true + }, { "label": "pre-commit", "detail": "Check and fix code conventions", From e04898043a7ed8c20e4434c46f9f14dad9d6bc7c Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 1 Sep 2025 18:36:27 +0200 Subject: [PATCH 06/16] `Id` with at most one parenthesis in either argument --- .../commutative-rings.lagda.md | 4 +-- .../euclidean-domains.lagda.md | 4 +-- .../integral-domains.lagda.md | 4 +-- .../products-commutative-rings.lagda.md | 6 ++-- .../pisano-periods.lagda.md | 2 +- .../universal-property-integers.lagda.md | 8 ++--- .../commutative-finite-rings.lagda.md | 4 +-- .../dependent-products-finite-rings.lagda.md | 10 +++---- src/finite-algebra/finite-fields.lagda.md | 4 +-- src/finite-algebra/finite-rings.lagda.md | 14 ++++----- .../products-finite-rings.lagda.md | 10 +++---- .../abstract-quaternion-group.lagda.md | 6 ++-- .../delooping-sign-homomorphism.lagda.md | 2 +- .../finite-abelian-groups.lagda.md | 2 +- .../orbits-permutations.lagda.md | 12 ++++---- ...ermutations-standard-finite-types.lagda.md | 2 +- src/finite-group-theory/permutations.lagda.md | 2 +- ...mpson-delooping-sign-homomorphism.lagda.md | 2 +- .../transpositions.lagda.md | 22 +++++++------- src/foundation/uniqueness-truncation.lagda.md | 2 +- .../morphisms-undirected-graphs.lagda.md | 2 +- src/graph-theory/vertex-covers.lagda.md | 2 +- src/group-theory/abelian-groups.lagda.md | 8 ++--- ...cartesian-products-abelian-groups.lagda.md | 6 ++-- ...sian-products-commutative-monoids.lagda.md | 4 +-- ...artesian-products-concrete-groups.lagda.md | 4 +-- .../cartesian-products-groups.lagda.md | 4 +-- .../cartesian-products-monoids.lagda.md | 4 +-- ...ushforward-concrete-group-actions.lagda.md | 2 +- src/group-theory/decidable-subgroups.lagda.md | 4 +-- src/group-theory/groups.lagda.md | 10 +++---- src/group-theory/loop-groups-sets.lagda.md | 6 ++-- ...category-of-orbits-monoid-actions.lagda.md | 2 +- src/group-theory/sheargroups.lagda.md | 4 +-- .../shriek-concrete-group-actions.lagda.md | 2 +- .../subgroups-abelian-groups.lagda.md | 8 ++--- src/group-theory/subgroups.lagda.md | 4 +-- ...ubstitution-functor-group-actions.lagda.md | 2 +- .../cartesian-products-higher-groups.lagda.md | 4 +-- .../higher-groups.lagda.md | 4 +-- src/linear-algebra/matrices-on-rings.lagda.md | 2 +- .../multiplication-matrices.lagda.md | 8 ++--- .../tuples-on-euclidean-domains.lagda.md | 2 +- src/linear-algebra/tuples-on-rings.lagda.md | 2 +- src/lists/concatenation-lists.lagda.md | 4 +-- ...ersal-property-lists-wild-monoids.lagda.md | 4 +-- src/order-theory/decidable-subposets.lagda.md | 2 +- .../decidable-subpreorders.lagda.md | 2 +- src/order-theory/finite-preorders.lagda.md | 2 +- .../finitely-graded-posets.lagda.md | 2 +- src/order-theory/subposets.lagda.md | 2 +- src/order-theory/subpreorders.lagda.md | 2 +- src/polytopes/abstract-polytopes.lagda.md | 2 +- .../dependent-products-rings.lagda.md | 10 +++---- src/ring-theory/products-rings.lagda.md | 10 +++---- src/ring-theory/rings.lagda.md | 14 ++++----- src/ring-theory/semirings.lagda.md | 10 +++---- src/structured-types/h-spaces.lagda.md | 6 ++-- src/structured-types/magmas.lagda.md | 2 +- src/structured-types/wild-loops.lagda.md | 6 ++-- .../cocones-under-spans.lagda.md | 2 +- .../free-loops.lagda.md | 2 +- .../infinite-cyclic-types.lagda.md | 2 +- .../interval-type.lagda.md | 10 +++---- .../loop-spaces.lagda.md | 4 +-- .../triple-loop-spaces.lagda.md | 30 +++++++++---------- .../universal-cover-circle.lagda.md | 22 +++++++------- .../universal-property-circle.lagda.md | 2 +- .../dependent-type-theories.lagda.md | 12 ++++---- .../fibered-dependent-type-theories.lagda.md | 4 +-- .../simple-type-theories.lagda.md | 6 ++-- .../2-element-decidable-subtypes.lagda.md | 2 +- .../2-element-subtypes.lagda.md | 6 ++-- .../cartesian-product-types.lagda.md | 4 +-- .../classical-finite-types.lagda.md | 2 +- .../counting-decidable-subtypes.lagda.md | 2 +- .../cyclic-finite-types.lagda.md | 2 +- .../dependent-pair-types.lagda.md | 2 +- .../double-counting.lagda.md | 4 +-- .../finite-types.lagda.md | 2 +- .../injective-maps.lagda.md | 2 +- ...tations-complete-undirected-graph.lagda.md | 8 ++--- .../ramsey-theory.lagda.md | 2 +- .../repetitions-of-values-sequences.lagda.md | 4 +-- 84 files changed, 220 insertions(+), 220 deletions(-) diff --git a/src/commutative-algebra/commutative-rings.lagda.md b/src/commutative-algebra/commutative-rings.lagda.md index 2f523ab320..7c21de00b1 100644 --- a/src/commutative-algebra/commutative-rings.lagda.md +++ b/src/commutative-algebra/commutative-rings.lagda.md @@ -142,7 +142,7 @@ module _ commutative-add-Commutative-Ring : (x y : type-Commutative-Ring) → - Id (add-Commutative-Ring x y) (add-Commutative-Ring y x) + add-Commutative-Ring x y ＝ add-Commutative-Ring y x commutative-add-Commutative-Ring = commutative-add-Ab ab-Commutative-Ring interchange-add-add-Commutative-Ring : @@ -336,7 +336,7 @@ module _ ap-mul-Commutative-Ring : {x x' y y' : type-Commutative-Ring} (p : x ＝ x') (q : y ＝ y') → - Id (mul-Commutative-Ring x y) (mul-Commutative-Ring x' y') + mul-Commutative-Ring x y ＝ mul-Commutative-Ring x' y' ap-mul-Commutative-Ring p q = ap-binary mul-Commutative-Ring p q associative-mul-Commutative-Ring : diff --git a/src/commutative-algebra/euclidean-domains.lagda.md b/src/commutative-algebra/euclidean-domains.lagda.md index 1b375c384b..ae580388d3 100644 --- a/src/commutative-algebra/euclidean-domains.lagda.md +++ b/src/commutative-algebra/euclidean-domains.lagda.md @@ -178,7 +178,7 @@ module _ commutative-add-Euclidean-Domain : (x y : type-Euclidean-Domain) → - Id (add-Euclidean-Domain x y) (add-Euclidean-Domain y x) + add-Euclidean-Domain x y ＝ add-Euclidean-Domain y x commutative-add-Euclidean-Domain = commutative-add-Ab ab-Euclidean-Domain interchange-add-add-Euclidean-Domain : @@ -335,7 +335,7 @@ module _ ap-mul-Euclidean-Domain : {x x' y y' : type-Euclidean-Domain} (p : x ＝ x') (q : y ＝ y') → - Id (mul-Euclidean-Domain x y) (mul-Euclidean-Domain x' y') + mul-Euclidean-Domain x y ＝ mul-Euclidean-Domain x' y' ap-mul-Euclidean-Domain p q = ap-binary mul-Euclidean-Domain p q associative-mul-Euclidean-Domain : diff --git a/src/commutative-algebra/integral-domains.lagda.md b/src/commutative-algebra/integral-domains.lagda.md index 25b7728952..a44507fc8b 100644 --- a/src/commutative-algebra/integral-domains.lagda.md +++ b/src/commutative-algebra/integral-domains.lagda.md @@ -147,7 +147,7 @@ module _ commutative-add-Integral-Domain : (x y : type-Integral-Domain) → - Id (add-Integral-Domain x y) (add-Integral-Domain y x) + add-Integral-Domain x y ＝ add-Integral-Domain y x commutative-add-Integral-Domain = commutative-add-Ab ab-Integral-Domain interchange-add-add-Integral-Domain : @@ -304,7 +304,7 @@ module _ ap-mul-Integral-Domain : {x x' y y' : type-Integral-Domain} (p : x ＝ x') (q : y ＝ y') → - Id (mul-Integral-Domain x y) (mul-Integral-Domain x' y') + mul-Integral-Domain x y ＝ mul-Integral-Domain x' y' ap-mul-Integral-Domain p q = ap-binary mul-Integral-Domain p q associative-mul-Integral-Domain : diff --git a/src/commutative-algebra/products-commutative-rings.lagda.md b/src/commutative-algebra/products-commutative-rings.lagda.md index dee1a7ee38..c0e1aa3cfd 100644 --- a/src/commutative-algebra/products-commutative-rings.lagda.md +++ b/src/commutative-algebra/products-commutative-rings.lagda.md @@ -117,7 +117,7 @@ module _ commutative-add-product-Commutative-Ring : (x y : type-product-Commutative-Ring) → - Id (add-product-Commutative-Ring x y) (add-product-Commutative-Ring y x) + add-product-Commutative-Ring x y ＝ add-product-Commutative-Ring y x commutative-add-product-Commutative-Ring = commutative-add-product-Ring ( ring-Commutative-Ring R1) @@ -150,7 +150,7 @@ module _ left-unit-law-mul-product-Commutative-Ring : (x : type-product-Commutative-Ring) → - Id (mul-product-Commutative-Ring one-product-Commutative-Ring x) x + mul-product-Commutative-Ring one-product-Commutative-Ring x ＝ x left-unit-law-mul-product-Commutative-Ring = left-unit-law-mul-product-Ring ( ring-Commutative-Ring R1) @@ -158,7 +158,7 @@ module _ right-unit-law-mul-product-Commutative-Ring : (x : type-product-Commutative-Ring) → - Id (mul-product-Commutative-Ring x one-product-Commutative-Ring) x + mul-product-Commutative-Ring x one-product-Commutative-Ring ＝ x right-unit-law-mul-product-Commutative-Ring = right-unit-law-mul-product-Ring ( ring-Commutative-Ring R1) diff --git a/src/elementary-number-theory/pisano-periods.lagda.md b/src/elementary-number-theory/pisano-periods.lagda.md index 231db7df28..8879673098 100644 --- a/src/elementary-number-theory/pisano-periods.lagda.md +++ b/src/elementary-number-theory/pisano-periods.lagda.md @@ -169,7 +169,7 @@ is-lower-bound-pisano-period k = cases-is-repetition-of-zero-pisano-period : (k x y : ℕ) → Id (pr1 (is-ordered-repetition-pisano-period k)) x → - Id (pisano-period k) y → is-zero-ℕ x + pisano-period k ＝ y → is-zero-ℕ x cases-is-repetition-of-zero-pisano-period k zero-ℕ y p q = refl cases-is-repetition-of-zero-pisano-period k (succ-ℕ x) zero-ℕ p q = ex-falso diff --git a/src/elementary-number-theory/universal-property-integers.lagda.md b/src/elementary-number-theory/universal-property-integers.lagda.md index 6b5827f733..598e3b8f54 100644 --- a/src/elementary-number-theory/universal-property-integers.lagda.md +++ b/src/elementary-number-theory/universal-property-integers.lagda.md @@ -59,7 +59,7 @@ abstract compute-zero-elim-ℤ : { l1 : Level} (P : ℤ → UU l1) ( p0 : P zero-ℤ) (pS : (k : ℤ) → (P k) ≃ (P (succ-ℤ k))) → - Id (elim-ℤ P p0 pS zero-ℤ) p0 + elim-ℤ P p0 pS zero-ℤ ＝ p0 compute-zero-elim-ℤ P p0 pS = refl compute-succ-elim-ℤ : @@ -85,7 +85,7 @@ ELIM-ℤ : ELIM-ℤ P p0 pS = Σ ( (k : ℤ) → P k) ( λ f → - ( ( Id (f zero-ℤ) p0) × + ( ( f zero-ℤ ＝ p0) × ( (k : ℤ) → Id (f (succ-ℤ k)) ((map-equiv (pS k)) (f k))))) Elim-ℤ : @@ -128,7 +128,7 @@ Eq-ELIM-ℤ : ( s t : ELIM-ℤ P p0 pS) → UU l1 Eq-ELIM-ℤ P p0 pS s t = ELIM-ℤ - ( λ k → Id (pr1 s k) (pr1 t k)) + ( λ k → pr1 s k ＝ pr1 t k) ( (pr1 (pr2 s)) ∙ (inv (pr1 (pr2 t)))) ( equiv-comparison-map-Eq-ELIM-ℤ P p0 pS s t) @@ -197,7 +197,7 @@ abstract is-prop-all-elements-equal ( λ s t → eq-Eq-ELIM-ℤ P p0 pS s t ( Elim-ℤ - ( λ k → Id (pr1 s k) (pr1 t k)) + ( λ k → pr1 s k ＝ pr1 t k) ( (pr1 (pr2 s)) ∙ (inv (pr1 (pr2 t)))) ( equiv-comparison-map-Eq-ELIM-ℤ P p0 pS s t))) ``` diff --git a/src/finite-algebra/commutative-finite-rings.lagda.md b/src/finite-algebra/commutative-finite-rings.lagda.md index 0d1c9851e3..475d5d1c43 100644 --- a/src/finite-algebra/commutative-finite-rings.lagda.md +++ b/src/finite-algebra/commutative-finite-rings.lagda.md @@ -168,7 +168,7 @@ module _ commutative-add-Finite-Commutative-Ring : (x y : type-Finite-Commutative-Ring) → - Id (add-Finite-Commutative-Ring x y) (add-Finite-Commutative-Ring y x) + add-Finite-Commutative-Ring x y ＝ add-Finite-Commutative-Ring y x commutative-add-Finite-Commutative-Ring = commutative-add-Ab ab-Finite-Commutative-Ring @@ -348,7 +348,7 @@ module _ ap-mul-Finite-Commutative-Ring : {x x' y y' : type-Finite-Commutative-Ring} (p : x ＝ x') (q : y ＝ y') → - Id (mul-Finite-Commutative-Ring x y) (mul-Finite-Commutative-Ring x' y') + mul-Finite-Commutative-Ring x y ＝ mul-Finite-Commutative-Ring x' y' ap-mul-Finite-Commutative-Ring p q = ap-binary mul-Finite-Commutative-Ring p q associative-mul-Finite-Commutative-Ring : diff --git a/src/finite-algebra/dependent-products-finite-rings.lagda.md b/src/finite-algebra/dependent-products-finite-rings.lagda.md index a1c2f98a52..712477404d 100644 --- a/src/finite-algebra/dependent-products-finite-rings.lagda.md +++ b/src/finite-algebra/dependent-products-finite-rings.lagda.md @@ -99,12 +99,12 @@ module _ associative-add-Π-Ring (type-Finite-Type I) (ring-Finite-Ring ∘ A) left-unit-law-add-Π-Finite-Ring : - (x : type-Π-Finite-Ring) → Id (add-Π-Finite-Ring zero-Π-Finite-Ring x) x + (x : type-Π-Finite-Ring) → add-Π-Finite-Ring zero-Π-Finite-Ring x ＝ x left-unit-law-add-Π-Finite-Ring = left-unit-law-add-Π-Ring (type-Finite-Type I) (ring-Finite-Ring ∘ A) right-unit-law-add-Π-Finite-Ring : - (x : type-Π-Finite-Ring) → Id (add-Π-Finite-Ring x zero-Π-Finite-Ring) x + (x : type-Π-Finite-Ring) → add-Π-Finite-Ring x zero-Π-Finite-Ring ＝ x right-unit-law-add-Π-Finite-Ring = right-unit-law-add-Π-Ring (type-Finite-Type I) (ring-Finite-Ring ∘ A) @@ -122,7 +122,7 @@ module _ commutative-add-Π-Finite-Ring : (x y : type-Π-Finite-Ring) → - Id (add-Π-Finite-Ring x y) (add-Π-Finite-Ring y x) + add-Π-Finite-Ring x y ＝ add-Π-Finite-Ring y x commutative-add-Π-Finite-Ring = commutative-add-Π-Ring (type-Finite-Type I) (ring-Finite-Ring ∘ A) @@ -142,12 +142,12 @@ module _ associative-mul-Π-Ring (type-Finite-Type I) (ring-Finite-Ring ∘ A) left-unit-law-mul-Π-Finite-Ring : - (x : type-Π-Finite-Ring) → Id (mul-Π-Finite-Ring one-Π-Finite-Ring x) x + (x : type-Π-Finite-Ring) → mul-Π-Finite-Ring one-Π-Finite-Ring x ＝ x left-unit-law-mul-Π-Finite-Ring = left-unit-law-mul-Π-Ring (type-Finite-Type I) (ring-Finite-Ring ∘ A) right-unit-law-mul-Π-Finite-Ring : - (x : type-Π-Finite-Ring) → Id (mul-Π-Finite-Ring x one-Π-Finite-Ring) x + (x : type-Π-Finite-Ring) → mul-Π-Finite-Ring x one-Π-Finite-Ring ＝ x right-unit-law-mul-Π-Finite-Ring = right-unit-law-mul-Π-Ring (type-Finite-Type I) (ring-Finite-Ring ∘ A) diff --git a/src/finite-algebra/finite-fields.lagda.md b/src/finite-algebra/finite-fields.lagda.md index 6468b8c8f6..baf4b87188 100644 --- a/src/finite-algebra/finite-fields.lagda.md +++ b/src/finite-algebra/finite-fields.lagda.md @@ -135,7 +135,7 @@ module _ commutative-add-Finite-Field : (x y : type-Finite-Field) → - Id (add-Finite-Field x y) (add-Finite-Field y x) + add-Finite-Field x y ＝ add-Finite-Field y x commutative-add-Finite-Field = commutative-add-Ab ab-Finite-Field interchange-add-add-Finite-Field : @@ -283,7 +283,7 @@ module _ ap-mul-Finite-Field : {x x' y y' : type-Finite-Field} (p : x ＝ x') (q : y ＝ y') → - Id (mul-Finite-Field x y) (mul-Finite-Field x' y') + mul-Finite-Field x y ＝ mul-Finite-Field x' y' ap-mul-Finite-Field p q = ap-binary mul-Finite-Field p q associative-mul-Finite-Field : diff --git a/src/finite-algebra/finite-rings.lagda.md b/src/finite-algebra/finite-rings.lagda.md index 7729cd4169..ba18843e7c 100644 --- a/src/finite-algebra/finite-rings.lagda.md +++ b/src/finite-algebra/finite-rings.lagda.md @@ -134,7 +134,7 @@ module _ ap-add-Finite-Ring : {x y x' y' : type-Finite-Ring R} → - x ＝ x' → y ＝ y' → Id (add-Finite-Ring x y) (add-Finite-Ring x' y') + x ＝ x' → y ＝ y' → add-Finite-Ring x y ＝ add-Finite-Ring x' y' ap-add-Finite-Ring = ap-add-Ring (ring-Finite-Ring R) associative-add-Finite-Ring : @@ -150,7 +150,7 @@ module _ is-group-additive-semigroup-Ring (ring-Finite-Ring R) commutative-add-Finite-Ring : - (x y : type-Finite-Ring R) → Id (add-Finite-Ring x y) (add-Finite-Ring y x) + (x y : type-Finite-Ring R) → add-Finite-Ring x y ＝ add-Finite-Ring y x commutative-add-Finite-Ring = commutative-add-Ring (ring-Finite-Ring R) interchange-add-add-Finite-Ring : @@ -229,11 +229,11 @@ module _ is-nonzero-finite-ring-Prop = is-nonzero-ring-Prop (ring-Finite-Ring R) left-unit-law-add-Finite-Ring : - (x : type-Finite-Ring R) → Id (add-Finite-Ring R zero-Finite-Ring x) x + (x : type-Finite-Ring R) → add-Finite-Ring R zero-Finite-Ring x ＝ x left-unit-law-add-Finite-Ring = left-unit-law-add-Ring (ring-Finite-Ring R) right-unit-law-add-Finite-Ring : - (x : type-Finite-Ring R) → Id (add-Finite-Ring R x zero-Finite-Ring) x + (x : type-Finite-Ring R) → add-Finite-Ring R x zero-Finite-Ring ＝ x right-unit-law-add-Finite-Ring = right-unit-law-add-Ring (ring-Finite-Ring R) ``` @@ -297,7 +297,7 @@ module _ ap-mul-Finite-Ring : {x x' y y' : type-Finite-Ring R} (p : x ＝ x') (q : y ＝ y') → - Id (mul-Finite-Ring x y) (mul-Finite-Ring x' y') + mul-Finite-Ring x y ＝ mul-Finite-Ring x' y' ap-mul-Finite-Ring = ap-mul-Ring (ring-Finite-Ring R) associative-mul-Finite-Ring : @@ -344,11 +344,11 @@ module _ one-Finite-Ring = one-Ring (ring-Finite-Ring R) left-unit-law-mul-Finite-Ring : - (x : type-Finite-Ring R) → Id (mul-Finite-Ring R one-Finite-Ring x) x + (x : type-Finite-Ring R) → mul-Finite-Ring R one-Finite-Ring x ＝ x left-unit-law-mul-Finite-Ring = left-unit-law-mul-Ring (ring-Finite-Ring R) right-unit-law-mul-Finite-Ring : - (x : type-Finite-Ring R) → Id (mul-Finite-Ring R x one-Finite-Ring) x + (x : type-Finite-Ring R) → mul-Finite-Ring R x one-Finite-Ring ＝ x right-unit-law-mul-Finite-Ring = right-unit-law-mul-Ring (ring-Finite-Ring R) ``` diff --git a/src/finite-algebra/products-finite-rings.lagda.md b/src/finite-algebra/products-finite-rings.lagda.md index 4555ad4b74..c87b8b2071 100644 --- a/src/finite-algebra/products-finite-rings.lagda.md +++ b/src/finite-algebra/products-finite-rings.lagda.md @@ -78,13 +78,13 @@ module _ left-unit-law-add-product-Finite-Ring : (x : type-product-Finite-Ring) → - Id (add-product-Finite-Ring zero-product-Finite-Ring x) x + add-product-Finite-Ring zero-product-Finite-Ring x ＝ x left-unit-law-add-product-Finite-Ring = left-unit-law-add-product-Ring (ring-Finite-Ring R1) (ring-Finite-Ring R2) right-unit-law-add-product-Finite-Ring : (x : type-product-Finite-Ring) → - Id (add-product-Finite-Ring x zero-product-Finite-Ring) x + add-product-Finite-Ring x zero-product-Finite-Ring ＝ x right-unit-law-add-product-Finite-Ring = right-unit-law-add-product-Ring (ring-Finite-Ring R1) (ring-Finite-Ring R2) @@ -118,7 +118,7 @@ module _ commutative-add-product-Finite-Ring : (x y : type-product-Finite-Ring) → - Id (add-product-Finite-Ring x y) (add-product-Finite-Ring y x) + add-product-Finite-Ring x y ＝ add-product-Finite-Ring y x commutative-add-product-Finite-Ring = commutative-add-product-Ring (ring-Finite-Ring R1) (ring-Finite-Ring R2) @@ -143,13 +143,13 @@ module _ left-unit-law-mul-product-Finite-Ring : (x : type-product-Finite-Ring) → - Id (mul-product-Finite-Ring one-product-Finite-Ring x) x + mul-product-Finite-Ring one-product-Finite-Ring x ＝ x left-unit-law-mul-product-Finite-Ring = left-unit-law-mul-product-Ring (ring-Finite-Ring R1) (ring-Finite-Ring R2) right-unit-law-mul-product-Finite-Ring : (x : type-product-Finite-Ring) → - Id (mul-product-Finite-Ring x one-product-Finite-Ring) x + mul-product-Finite-Ring x one-product-Finite-Ring ＝ x right-unit-law-mul-product-Finite-Ring = right-unit-law-mul-product-Ring (ring-Finite-Ring R1) (ring-Finite-Ring R2) diff --git a/src/finite-group-theory/abstract-quaternion-group.lagda.md b/src/finite-group-theory/abstract-quaternion-group.lagda.md index c01ec2ead6..91d626215e 100644 --- a/src/finite-group-theory/abstract-quaternion-group.lagda.md +++ b/src/finite-group-theory/abstract-quaternion-group.lagda.md @@ -126,7 +126,7 @@ inv-Q8 -j-Q8 = j-Q8 inv-Q8 k-Q8 = -k-Q8 inv-Q8 -k-Q8 = k-Q8 -left-unit-law-mul-Q8 : (x : Q8) → Id (mul-Q8 e-Q8 x) x +left-unit-law-mul-Q8 : (x : Q8) → mul-Q8 e-Q8 x ＝ x left-unit-law-mul-Q8 e-Q8 = refl left-unit-law-mul-Q8 -e-Q8 = refl left-unit-law-mul-Q8 i-Q8 = refl @@ -136,7 +136,7 @@ left-unit-law-mul-Q8 -j-Q8 = refl left-unit-law-mul-Q8 k-Q8 = refl left-unit-law-mul-Q8 -k-Q8 = refl -right-unit-law-mul-Q8 : (x : Q8) → Id (mul-Q8 x e-Q8) x +right-unit-law-mul-Q8 : (x : Q8) → mul-Q8 x e-Q8 ＝ x right-unit-law-mul-Q8 e-Q8 = refl right-unit-law-mul-Q8 -e-Q8 = refl right-unit-law-mul-Q8 i-Q8 = refl @@ -862,7 +862,7 @@ Q8-Group = ( pair inv-Q8 (pair left-inverse-law-mul-Q8 right-inverse-law-mul-Q8))) is-noncommutative-mul-Q8 : - ¬ ((x y : Q8) → Id (mul-Q8 x y) (mul-Q8 y x)) + ¬ ((x y : Q8) → mul-Q8 x y ＝ mul-Q8 y x) is-noncommutative-mul-Q8 f = Eq-eq-Q8 (f i-Q8 j-Q8) map-equiv-count-Q8 : Fin 8 → Q8 diff --git a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md index 3c3f8d641a..8bfee17597 100644 --- a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md +++ b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md @@ -174,7 +174,7 @@ module _ preserves-id-equiv-invertible-action-D-equiv : (n : ℕ) (X : Type-With-Cardinality-ℕ l1 n) → - Id (invertible-action-D-equiv n X X id-equiv) id-equiv + invertible-action-D-equiv n X X id-equiv ＝ id-equiv preserves-id-equiv-invertible-action-D-equiv n = compute-id-equiv-action-equiv-family-over-subuniverse ( mere-equiv-Prop (Fin n)) diff --git a/src/finite-group-theory/finite-abelian-groups.lagda.md b/src/finite-group-theory/finite-abelian-groups.lagda.md index 51bc616d82..e956bcfa34 100644 --- a/src/finite-group-theory/finite-abelian-groups.lagda.md +++ b/src/finite-group-theory/finite-abelian-groups.lagda.md @@ -98,7 +98,7 @@ module _ add-Finite-Ab' = mul-Group' group-Finite-Ab commutative-add-Finite-Ab : - (x y : type-Finite-Ab) → Id (add-Finite-Ab x y) (add-Finite-Ab y x) + (x y : type-Finite-Ab) → add-Finite-Ab x y ＝ add-Finite-Ab y x commutative-add-Finite-Ab = pr2 A ab-Finite-Ab : Ab l diff --git a/src/finite-group-theory/orbits-permutations.lagda.md b/src/finite-group-theory/orbits-permutations.lagda.md index 835bcc5323..604f62a17b 100644 --- a/src/finite-group-theory/orbits-permutations.lagda.md +++ b/src/finite-group-theory/orbits-permutations.lagda.md @@ -312,14 +312,14 @@ module _ le-min-reporting) pred-first : ℕ pred-first = pr1 is-successor-first-point-min-repeating - equality-pred-first : Id first-point-min-repeating (succ-ℕ pred-first) + equality-pred-first : first-point-min-repeating ＝ succ-ℕ pred-first equality-pred-first = pr2 is-successor-first-point-min-repeating is-successor-second-point-min-repeating : is-successor-ℕ second-point-min-repeating is-successor-second-point-min-repeating = is-successor-is-nonzero-ℕ np pred-second : ℕ pred-second = pr1 is-successor-second-point-min-repeating - equality-pred-second : Id second-point-min-repeating (succ-ℕ pred-second) + equality-pred-second : second-point-min-repeating ＝ succ-ℕ pred-second equality-pred-second = pr2 is-successor-second-point-min-repeating has-finite-orbits-permutation' : @@ -352,14 +352,14 @@ module _ le-min-reporting) pred-first : ℕ pred-first = pr1 is-successor-first-point-min-repeating - equality-pred-first : Id first-point-min-repeating (succ-ℕ pred-first) + equality-pred-first : first-point-min-repeating ＝ succ-ℕ pred-first equality-pred-first = pr2 is-successor-first-point-min-repeating is-successor-second-point-min-repeating : is-successor-ℕ second-point-min-repeating is-successor-second-point-min-repeating = is-successor-is-nonzero-ℕ np pred-second : ℕ pred-second = pr1 is-successor-second-point-min-repeating - equality-pred-second : Id second-point-min-repeating (succ-ℕ pred-second) + equality-pred-second : second-point-min-repeating ＝ succ-ℕ pred-second equality-pred-second = pr2 is-successor-second-point-min-repeating has-finite-orbits-permutation : @@ -621,7 +621,7 @@ module _ cases-decidable-equality : (T1 T2 : equivalence-class same-orbits-permutation) (t1 : type-Type-With-Cardinality-ℕ n X) → - Id T1 (class same-orbits-permutation t1) → + T1 ＝ class same-orbits-permutation t1 → is-decidable ( is-in-equivalence-class same-orbits-permutation T2 t1) → is-decidable (Id T1 T2) @@ -1409,7 +1409,7 @@ module _ h'-inl : ( k : Fin (number-of-elements-count h)) ( T : equivalence-class (same-orbits-permutation-count g)) → - Id (map-equiv-count h k) T → + map-equiv-count h k ＝ T → is-decidable ( is-in-equivalence-class (same-orbits-permutation-count g) T a) → is-decidable diff --git a/src/finite-group-theory/permutations-standard-finite-types.lagda.md b/src/finite-group-theory/permutations-standard-finite-types.lagda.md index d411bd518e..a50f9577df 100644 --- a/src/finite-group-theory/permutations-standard-finite-types.lagda.md +++ b/src/finite-group-theory/permutations-standard-finite-types.lagda.md @@ -127,7 +127,7 @@ abstract retraction-permutation-list-transpositions-Fin' : (n : ℕ) (f : Permutation (succ-ℕ n)) → (x : Fin (succ-ℕ n)) → Id (map-equiv f (inr star)) x → - (y z : Fin (succ-ℕ n)) → Id (map-equiv f y) z → + (y z : Fin (succ-ℕ n)) → map-equiv f y ＝ z → Id ( map-equiv ( permutation-list-transpositions diff --git a/src/finite-group-theory/permutations.lagda.md b/src/finite-group-theory/permutations.lagda.md index 1ca715c0af..8d38d934db 100644 --- a/src/finite-group-theory/permutations.lagda.md +++ b/src/finite-group-theory/permutations.lagda.md @@ -193,7 +193,7 @@ module _ ( type-Decidable-Prop ∘ P))))) ( λ li → Id k (mod-two-ℕ (length-list li)) × - Id f (permutation-list-transpositions li)))) + f ＝ permutation-list-transpositions li))) abstract is-contr-parity-transposition-permutation : diff --git a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md index 5e6d56f993..5a37978c7a 100644 --- a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md +++ b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md @@ -633,7 +633,7 @@ module _ abstract preserves-id-equiv-simpson-comp-equiv : (X : Type-With-Cardinality-ℕ l n) → - Id (simpson-comp-equiv X X id-equiv) id-equiv + simpson-comp-equiv X X id-equiv ＝ id-equiv preserves-id-equiv-simpson-comp-equiv X = eq-htpy-equiv left-unit-law-equiv diff --git a/src/finite-group-theory/transpositions.lagda.md b/src/finite-group-theory/transpositions.lagda.md index 1900b49eed..9b104bd41d 100644 --- a/src/finite-group-theory/transpositions.lagda.md +++ b/src/finite-group-theory/transpositions.lagda.md @@ -182,7 +182,7 @@ module _ abstract left-computation-standard-transposition : - Id (map-standard-transposition x) y + map-standard-transposition x ＝ y left-computation-standard-transposition with is-decidable-type-prop-standard-2-Element-Decidable-Subtype H p x ... | inl pp = @@ -197,7 +197,7 @@ module _ abstract right-computation-standard-transposition : - Id (map-standard-transposition y) x + map-standard-transposition y ＝ x right-computation-standard-transposition with is-decidable-type-prop-standard-2-Element-Decidable-Subtype H p y ... | inl pp = @@ -259,7 +259,7 @@ module _ where abstract - is-not-identity-map-transposition : Id (map-transposition P) id → empty + is-not-identity-map-transposition : map-transposition P ＝ id → empty is-not-identity-map-transposition f = is-not-identity-swap-2-Element-Type ( 2-element-type-2-Element-Decidable-Subtype P) @@ -291,7 +291,7 @@ module _ Σ X (λ x → map-transposition Y x ≠ x) element-is-not-identity-map-transposition = exists-not-not-for-all-count - ( λ z → Id (map-transposition Y z) z) + ( λ z → map-transposition Y z ＝ z) ( λ x → has-decidable-equality-count eX (map-transposition Y x) x) ( eX) ( λ H → is-not-identity-map-transposition Y (eq-htpy H)) @@ -370,7 +370,7 @@ module _ ( type-decidable-prop-pr1-two-elements-transposition) type-t-coproduct-id : (x : X) → - ( Id (pr1 two-elements-transposition) x) + + ( pr1 two-elements-transposition ＝ x) + ( Id (pr1 (pr2 two-elements-transposition)) x) → type-Decidable-Prop (pr1 Y x) type-t-coproduct-id x (inl Q) = @@ -400,7 +400,7 @@ module _ ( pr1 (pr2 two-elements-transposition)) ( type-decidable-prop-pr1-pr2-two-elements-transposition))) ( k3) → - ( Id (pr1 two-elements-transposition) x) + + ( pr1 two-elements-transposition ＝ x) + ( Id (pr1 (pr2 two-elements-transposition)) x) cases-coproduct-id-type-t x p h (inl (inr star)) (inl (inr star)) k3 K1 K2 K3 = @@ -429,7 +429,7 @@ module _ inl (ap pr1 (is-injective-equiv (inv-equiv h) (K2 ∙ inv K1))) coproduct-id-type-t : (x : X) → type-Decidable-Prop (pr1 Y x) → - ( Id (pr1 two-elements-transposition) x) + + ( pr1 two-elements-transposition ＝ x) + ( Id (pr1 (pr2 two-elements-transposition)) x) coproduct-id-type-t x p = apply-universal-property-trunc-Prop (pr2 Y) @@ -482,9 +482,9 @@ module _ is-decidable (Id (pr1 (pr2 two-elements-transposition)) x) → is-decidable (Id (pr1 two-elements-transposition) y) → is-decidable (Id (pr1 (pr2 two-elements-transposition)) y) → - ( ( Id (pr1 two-elements-transposition) x) × + ( ( pr1 two-elements-transposition ＝ x) × ( Id (pr1 (pr2 two-elements-transposition)) y)) + - ( ( Id (pr1 two-elements-transposition) y) × + ( ( pr1 two-elements-transposition ＝ y) × ( Id (pr1 (pr2 two-elements-transposition)) x)) cases-eq-two-elements-transposition x y np p1 p2 (inl q) r s (inl u) = inl (pair q u) @@ -560,9 +560,9 @@ module _ (x y : X) (np : x ≠ y) → type-Decidable-Prop (pr1 Y x) → type-Decidable-Prop (pr1 Y y) → - ( ( Id (pr1 two-elements-transposition) x) × + ( ( pr1 two-elements-transposition ＝ x) × ( Id (pr1 (pr2 two-elements-transposition)) y)) + - ( ( Id (pr1 two-elements-transposition) y) × + ( ( pr1 two-elements-transposition ＝ y) × ( Id (pr1 (pr2 two-elements-transposition)) x)) eq-two-elements-transposition x y np p1 p2 = cases-eq-two-elements-transposition x y np p1 p2 diff --git a/src/foundation/uniqueness-truncation.lagda.md b/src/foundation/uniqueness-truncation.lagda.md index 25971b85f3..47fe418314 100644 --- a/src/foundation/uniqueness-truncation.lagda.md +++ b/src/foundation/uniqueness-truncation.lagda.md @@ -74,7 +74,7 @@ module _ K = universal-property-set-quotient-is-set-quotient R C g Ug B f k : type-Set C → type-Set B k = pr1 (center K) - α : Id (precomp-Set-Quotient R C g B k) f + α : precomp-Set-Quotient R C g B k ＝ f α = eq-htpy-reflecting-map-equivalence-relation R B ( precomp-Set-Quotient R C g B k) ( f) diff --git a/src/graph-theory/morphisms-undirected-graphs.lagda.md b/src/graph-theory/morphisms-undirected-graphs.lagda.md index eba9c66056..b41b7b77af 100644 --- a/src/graph-theory/morphisms-undirected-graphs.lagda.md +++ b/src/graph-theory/morphisms-undirected-graphs.lagda.md @@ -151,7 +151,7 @@ module _ ( λ gE → (p : unordered-pair-vertices-Undirected-Graph G) → (e : edge-Undirected-Graph G p) → - Id (edge-hom-Undirected-Graph G H f p e) (gE p e))) + edge-hom-Undirected-Graph G H f p e ＝ gE p e)) ( equiv-tot ( λ gE → equiv-Π-equiv-family diff --git a/src/graph-theory/vertex-covers.lagda.md b/src/graph-theory/vertex-covers.lagda.md index 462cda70c8..4cd6e60929 100644 --- a/src/graph-theory/vertex-covers.lagda.md +++ b/src/graph-theory/vertex-covers.lagda.md @@ -40,7 +40,7 @@ vertex-cover G = edge-Undirected-Graph G p → type-trunc-Prop ( Σ (vertex-Undirected-Graph G) - ( λ x → is-in-unordered-pair p x × Id (c x) (inr star)))) + ( λ x → is-in-unordered-pair p x × c x ＝ inr star))) ``` ## External links diff --git a/src/group-theory/abelian-groups.lagda.md b/src/group-theory/abelian-groups.lagda.md index 43e415465c..8c2c72d7aa 100644 --- a/src/group-theory/abelian-groups.lagda.md +++ b/src/group-theory/abelian-groups.lagda.md @@ -168,7 +168,7 @@ module _ (x : type-Ab) → add-Ab x (neg-Ab x) ＝ zero-Ab right-inverse-law-add-Ab = right-inverse-law-mul-Group group-Ab - commutative-add-Ab : (x y : type-Ab) → Id (add-Ab x y) (add-Ab y x) + commutative-add-Ab : (x y : type-Ab) → add-Ab x y ＝ add-Ab y x commutative-add-Ab = pr2 A interchange-add-add-Ab : @@ -439,7 +439,7 @@ module _ transpose-eq-add-Ab : {x y z : type-Ab A} → - Id (add-Ab A x y) z → Id x (add-Ab A z (neg-Ab A y)) + add-Ab A x y ＝ z → Id x (add-Ab A z (neg-Ab A y)) transpose-eq-add-Ab = transpose-eq-mul-Group (group-Ab A) inv-transpose-eq-add-Ab : @@ -449,12 +449,12 @@ module _ transpose-eq-add-Ab' : {x y z : type-Ab A} → - Id (add-Ab A x y) z → Id y (add-Ab A (neg-Ab A x) z) + add-Ab A x y ＝ z → Id y (add-Ab A (neg-Ab A x) z) transpose-eq-add-Ab' = transpose-eq-mul-Group' (group-Ab A) inv-transpose-eq-add-Ab' : {x y z : type-Ab A} → - Id y (add-Ab A (neg-Ab A x) z) → Id (add-Ab A x y) z + Id y (add-Ab A (neg-Ab A x) z) → add-Ab A x y ＝ z inv-transpose-eq-add-Ab' = inv-transpose-eq-mul-Group' (group-Ab A) double-transpose-eq-add-Ab : diff --git a/src/group-theory/cartesian-products-abelian-groups.lagda.md b/src/group-theory/cartesian-products-abelian-groups.lagda.md index 8968d9ecc8..3116d3e5c3 100644 --- a/src/group-theory/cartesian-products-abelian-groups.lagda.md +++ b/src/group-theory/cartesian-products-abelian-groups.lagda.md @@ -77,11 +77,11 @@ module _ associative-add-product-Ab = associative-mul-Group group-product-Ab left-unit-law-add-product-Ab : - (x : type-product-Ab) → Id (add-product-Ab zero-product-Ab x) x + (x : type-product-Ab) → add-product-Ab zero-product-Ab x ＝ x left-unit-law-add-product-Ab = left-unit-law-mul-Group group-product-Ab right-unit-law-add-product-Ab : - (x : type-product-Ab) → Id (add-product-Ab x zero-product-Ab) x + (x : type-product-Ab) → add-product-Ab x zero-product-Ab ＝ x right-unit-law-add-product-Ab = right-unit-law-mul-Group group-product-Ab left-inverse-law-add-product-Ab : @@ -96,7 +96,7 @@ module _ right-inverse-law-mul-Group group-product-Ab commutative-add-product-Ab : - (x y : type-product-Ab) → Id (add-product-Ab x y) (add-product-Ab y x) + (x y : type-product-Ab) → add-product-Ab x y ＝ add-product-Ab y x commutative-add-product-Ab (pair x1 y1) (pair x2 y2) = eq-pair (commutative-add-Ab A x1 x2) (commutative-add-Ab B y1 y2) diff --git a/src/group-theory/cartesian-products-commutative-monoids.lagda.md b/src/group-theory/cartesian-products-commutative-monoids.lagda.md index 643ac5ce8a..9d8964b602 100644 --- a/src/group-theory/cartesian-products-commutative-monoids.lagda.md +++ b/src/group-theory/cartesian-products-commutative-monoids.lagda.md @@ -70,13 +70,13 @@ module _ left-unit-law-mul-product-Commutative-Monoid : (x : type-product-Commutative-Monoid) → - Id (mul-product-Commutative-Monoid unit-product-Commutative-Monoid x) x + mul-product-Commutative-Monoid unit-product-Commutative-Monoid x ＝ x left-unit-law-mul-product-Commutative-Monoid = left-unit-law-mul-Monoid monoid-product-Commutative-Monoid right-unit-law-mul-product-Commutative-Monoid : (x : type-product-Commutative-Monoid) → - Id (mul-product-Commutative-Monoid x unit-product-Commutative-Monoid) x + mul-product-Commutative-Monoid x unit-product-Commutative-Monoid ＝ x right-unit-law-mul-product-Commutative-Monoid = right-unit-law-mul-Monoid monoid-product-Commutative-Monoid diff --git a/src/group-theory/cartesian-products-concrete-groups.lagda.md b/src/group-theory/cartesian-products-concrete-groups.lagda.md index a60202281c..019271c1b6 100644 --- a/src/group-theory/cartesian-products-concrete-groups.lagda.md +++ b/src/group-theory/cartesian-products-concrete-groups.lagda.md @@ -151,13 +151,13 @@ module _ left-unit-law-mul-product-Concrete-Group : (x : type-product-Concrete-Group) → - Id (mul-product-Concrete-Group unit-product-Concrete-Group x) x + mul-product-Concrete-Group unit-product-Concrete-Group x ＝ x left-unit-law-mul-product-Concrete-Group = left-unit-law-mul-∞-Group ∞-group-product-Concrete-Group right-unit-law-mul-product-Concrete-Group : (y : type-product-Concrete-Group) → - Id (mul-product-Concrete-Group y unit-product-Concrete-Group) y + mul-product-Concrete-Group y unit-product-Concrete-Group ＝ y right-unit-law-mul-product-Concrete-Group = right-unit-law-mul-∞-Group ∞-group-product-Concrete-Group diff --git a/src/group-theory/cartesian-products-groups.lagda.md b/src/group-theory/cartesian-products-groups.lagda.md index 5d2fa94e90..13001b5809 100644 --- a/src/group-theory/cartesian-products-groups.lagda.md +++ b/src/group-theory/cartesian-products-groups.lagda.md @@ -66,12 +66,12 @@ module _ unit-product-Group = unit-Monoid monoid-product-Group left-unit-law-mul-product-Group : - (x : type-product-Group) → Id (mul-product-Group unit-product-Group x) x + (x : type-product-Group) → mul-product-Group unit-product-Group x ＝ x left-unit-law-mul-product-Group = left-unit-law-mul-Monoid monoid-product-Group right-unit-law-mul-product-Group : - (x : type-product-Group) → Id (mul-product-Group x unit-product-Group) x + (x : type-product-Group) → mul-product-Group x unit-product-Group ＝ x right-unit-law-mul-product-Group = right-unit-law-mul-Monoid monoid-product-Group diff --git a/src/group-theory/cartesian-products-monoids.lagda.md b/src/group-theory/cartesian-products-monoids.lagda.md index 69e92fb08f..3d3abec4b7 100644 --- a/src/group-theory/cartesian-products-monoids.lagda.md +++ b/src/group-theory/cartesian-products-monoids.lagda.md @@ -65,12 +65,12 @@ module _ pr2 unit-product-Monoid = unit-Monoid N left-unit-law-mul-product-Monoid : - (x : type-product-Monoid) → Id (mul-product-Monoid unit-product-Monoid x) x + (x : type-product-Monoid) → mul-product-Monoid unit-product-Monoid x ＝ x left-unit-law-mul-product-Monoid (pair x y) = eq-pair (left-unit-law-mul-Monoid M x) (left-unit-law-mul-Monoid N y) right-unit-law-mul-product-Monoid : - (x : type-product-Monoid) → Id (mul-product-Monoid x unit-product-Monoid) x + (x : type-product-Monoid) → mul-product-Monoid x unit-product-Monoid ＝ x right-unit-law-mul-product-Monoid (pair x y) = eq-pair (right-unit-law-mul-Monoid M x) (right-unit-law-mul-Monoid N y) diff --git a/src/group-theory/contravariant-pushforward-concrete-group-actions.lagda.md b/src/group-theory/contravariant-pushforward-concrete-group-actions.lagda.md index a9aa640c3c..24921c0bd0 100644 --- a/src/group-theory/contravariant-pushforward-concrete-group-actions.lagda.md +++ b/src/group-theory/contravariant-pushforward-concrete-group-actions.lagda.md @@ -30,6 +30,6 @@ module _ -- The following should be constructed as a set hom-action-Concrete-Group G X - ( subst-action-Concrete-Group G H f (λ y → Id (shape-Concrete-Group H) y)) + ( subst-action-Concrete-Group G H f (λ y → shape-Concrete-Group H ＝ y)) -} ``` diff --git a/src/group-theory/decidable-subgroups.lagda.md b/src/group-theory/decidable-subgroups.lagda.md index e523eb23cb..be44157b41 100644 --- a/src/group-theory/decidable-subgroups.lagda.md +++ b/src/group-theory/decidable-subgroups.lagda.md @@ -250,13 +250,13 @@ module _ left-unit-law-mul-Decidable-Subgroup : (x : type-group-Decidable-Subgroup) → - Id (mul-Decidable-Subgroup unit-Decidable-Subgroup x) x + mul-Decidable-Subgroup unit-Decidable-Subgroup x ＝ x left-unit-law-mul-Decidable-Subgroup = left-unit-law-mul-Subgroup G (subgroup-Decidable-Subgroup G H) right-unit-law-mul-Decidable-Subgroup : (x : type-group-Decidable-Subgroup) → - Id (mul-Decidable-Subgroup x unit-Decidable-Subgroup) x + mul-Decidable-Subgroup x unit-Decidable-Subgroup ＝ x right-unit-law-mul-Decidable-Subgroup = right-unit-law-mul-Subgroup G (subgroup-Decidable-Subgroup G H) diff --git a/src/group-theory/groups.lagda.md b/src/group-theory/groups.lagda.md index 44f5de6216..bc860de881 100644 --- a/src/group-theory/groups.lagda.md +++ b/src/group-theory/groups.lagda.md @@ -107,7 +107,7 @@ module _ ap-mul-Group : {x x' y y' : type-Group} (p : x ＝ x') (q : y ＝ y') → - Id (mul-Group x y) (mul-Group x' y') + mul-Group x y ＝ mul-Group x' y' ap-mul-Group p q = ap-binary mul-Group p q mul-Group' : type-Group → type-Group → type-Group @@ -152,11 +152,11 @@ module _ pr2 (is-unit-prop-Group' x) = is-prop-is-unit-Group' x left-unit-law-mul-Group : - (x : type-Group) → Id (mul-Group unit-Group x) x + (x : type-Group) → mul-Group unit-Group x ＝ x left-unit-law-mul-Group = pr1 (pr2 is-unital-Group) right-unit-law-mul-Group : - (x : type-Group) → Id (mul-Group x unit-Group) x + (x : type-Group) → mul-Group x unit-Group ＝ x right-unit-law-mul-Group = pr2 (pr2 is-unital-Group) coherence-unit-laws-mul-Group : @@ -197,7 +197,7 @@ module _ pr2 (pr2 (is-invertible-element-Group x)) = left-inverse-law-mul-Group x inv-unit-Group : - Id (inv-Group unit-Group) unit-Group + inv-Group unit-Group ＝ unit-Group inv-unit-Group = ( inv (left-unit-law-mul-Group (inv-Group unit-Group))) ∙ ( right-inverse-law-mul-Group unit-Group) @@ -643,7 +643,7 @@ module _ where is-idempotent-Group : type-Group G → UU l - is-idempotent-Group x = Id (mul-Group G x x) x + is-idempotent-Group x = mul-Group G x x ＝ x is-unit-is-idempotent-Group : {x : type-Group G} → is-idempotent-Group x → is-unit-Group G x diff --git a/src/group-theory/loop-groups-sets.lagda.md b/src/group-theory/loop-groups-sets.lagda.md index a10ea49025..35b72abd19 100644 --- a/src/group-theory/loop-groups-sets.lagda.md +++ b/src/group-theory/loop-groups-sets.lagda.md @@ -44,7 +44,7 @@ module _ where type-loop-Set : UU (lsuc l) - type-loop-Set = Id (type-Set X) (type-Set X) + type-loop-Set = type-Set X ＝ type-Set X is-set-type-loop-Set : is-set type-loop-Set is-set-type-loop-Set = @@ -93,12 +93,12 @@ module _ map-hom-symmetric-group-loop-group-Set : (X Y : Set l) → - Id (type-Set X) (type-Set Y) → (type-Set Y) ≃ (type-Set X) + type-Set X ＝ type-Set Y → (type-Set Y) ≃ (type-Set X) map-hom-symmetric-group-loop-group-Set X Y p = equiv-eq (inv p) map-hom-inv-symmetric-group-loop-group-Set : (X Y : Set l) → - (type-Set X) ≃ (type-Set Y) → Id (type-Set Y) (type-Set X) + (type-Set X) ≃ (type-Set Y) → type-Set Y ＝ type-Set X map-hom-inv-symmetric-group-loop-group-Set X Y f = inv (eq-equiv f) commutative-inv-map-hom-symmetric-group-loop-group-Set : diff --git a/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md b/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md index c6f7de583c..c5eb7fc864 100644 --- a/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md +++ b/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md @@ -45,7 +45,7 @@ module _ hom-orbit-action-Monoid : (x y : type-action-Monoid M X) → UU (l1 ⊔ l2) hom-orbit-action-Monoid x y = - Σ (type-Monoid M) ( λ m → Id (mul-action-Monoid M X m x) y) + Σ (type-Monoid M) ( λ m → mul-action-Monoid M X m x ＝ y) element-hom-orbit-action-Monoid : {x y : type-action-Monoid M X} → hom-orbit-action-Monoid x y → type-Monoid M diff --git a/src/group-theory/sheargroups.lagda.md b/src/group-theory/sheargroups.lagda.md index 42becd5b2b..7b76e4aae0 100644 --- a/src/group-theory/sheargroups.lagda.md +++ b/src/group-theory/sheargroups.lagda.md @@ -27,8 +27,8 @@ Sheargroup l = ( λ e → Σ (type-Set X → type-Set X → type-Set X) ( λ m → - ( (x : type-Set X) → Id (m e x) x) × - ( ( (x : type-Set X) → Id (m x x) e) × + ( (x : type-Set X) → m e x ＝ x) × + ( ( (x : type-Set X) → m x x ＝ e) × ( (x y z : type-Set X) → Id (m x (m y z)) (m (m (m x (m y e)) e) z)))))) ``` diff --git a/src/group-theory/shriek-concrete-group-actions.lagda.md b/src/group-theory/shriek-concrete-group-actions.lagda.md index 6e5364e97f..270643b890 100644 --- a/src/group-theory/shriek-concrete-group-actions.lagda.md +++ b/src/group-theory/shriek-concrete-group-actions.lagda.md @@ -38,5 +38,5 @@ module _ trunc-Set ( Σ ( classifying-type-Concrete-Group G) ( λ x → - type-Set (X x) × Id (classifying-map-hom-Concrete-Group G H f x) y)) + type-Set (X x) × classifying-map-hom-Concrete-Group G H f x ＝ y)) ``` diff --git a/src/group-theory/subgroups-abelian-groups.lagda.md b/src/group-theory/subgroups-abelian-groups.lagda.md index 55bcda4a76..2c1f5234a5 100644 --- a/src/group-theory/subgroups-abelian-groups.lagda.md +++ b/src/group-theory/subgroups-abelian-groups.lagda.md @@ -185,7 +185,7 @@ module _ eq-subgroup-ab-eq-ab : {x y : type-ab-Subgroup-Ab} → - Id (map-inclusion-Subgroup-Ab x) (map-inclusion-Subgroup-Ab y) → + map-inclusion-Subgroup-Ab x ＝ map-inclusion-Subgroup-Ab y → x ＝ y eq-subgroup-ab-eq-ab = eq-subgroup-eq-group (group-Ab A) B @@ -211,13 +211,13 @@ module _ left-unit-law-add-Subgroup-Ab : (x : type-ab-Subgroup-Ab) → - Id (add-ab-Subgroup-Ab zero-ab-Subgroup-Ab x) x + add-ab-Subgroup-Ab zero-ab-Subgroup-Ab x ＝ x left-unit-law-add-Subgroup-Ab = left-unit-law-mul-Subgroup (group-Ab A) B right-unit-law-add-Subgroup-Ab : (x : type-ab-Subgroup-Ab) → - Id (add-ab-Subgroup-Ab x zero-ab-Subgroup-Ab) x + add-ab-Subgroup-Ab x zero-ab-Subgroup-Ab ＝ x right-unit-law-add-Subgroup-Ab = right-unit-law-mul-Subgroup (group-Ab A) B @@ -239,7 +239,7 @@ module _ commutative-add-Subgroup-Ab : ( x y : type-ab-Subgroup-Ab) → - Id ( add-ab-Subgroup-Ab x y) (add-ab-Subgroup-Ab y x) + add-ab-Subgroup-Ab x y ＝ add-ab-Subgroup-Ab y x commutative-add-Subgroup-Ab x y = eq-subgroup-ab-eq-ab (commutative-add-Ab A (pr1 x) (pr1 y)) diff --git a/src/group-theory/subgroups.lagda.md b/src/group-theory/subgroups.lagda.md index 83b79f9284..c47c094a4b 100644 --- a/src/group-theory/subgroups.lagda.md +++ b/src/group-theory/subgroups.lagda.md @@ -283,12 +283,12 @@ module _ pr2 unit-Subgroup = contains-unit-Subgroup G H left-unit-law-mul-Subgroup : - (x : type-group-Subgroup) → Id (mul-Subgroup unit-Subgroup x) x + (x : type-group-Subgroup) → mul-Subgroup unit-Subgroup x ＝ x left-unit-law-mul-Subgroup x = eq-subgroup-eq-group (left-unit-law-mul-Group G (pr1 x)) right-unit-law-mul-Subgroup : - (x : type-group-Subgroup) → Id (mul-Subgroup x unit-Subgroup) x + (x : type-group-Subgroup) → mul-Subgroup x unit-Subgroup ＝ x right-unit-law-mul-Subgroup x = eq-subgroup-eq-group (right-unit-law-mul-Group G (pr1 x)) diff --git a/src/group-theory/substitution-functor-group-actions.lagda.md b/src/group-theory/substitution-functor-group-actions.lagda.md index d8d73e7e03..30c4e663e9 100644 --- a/src/group-theory/substitution-functor-group-actions.lagda.md +++ b/src/group-theory/substitution-functor-group-actions.lagda.md @@ -139,7 +139,7 @@ module _ ( type-Group G) ( λ g → ( Id (mul-Group H (map-hom-Group G H f g) h) h') × - ( Id (mul-action-Group G X g x) x')) + ( mul-action-Group G X g x ＝ x')) pr1 ( pr2 (equivalence-relation-obj-left-adjoint-subst-action-Group X)) ( h , x) = diff --git a/src/higher-group-theory/cartesian-products-higher-groups.lagda.md b/src/higher-group-theory/cartesian-products-higher-groups.lagda.md index 2183bb2862..56e64a9ddd 100644 --- a/src/higher-group-theory/cartesian-products-higher-groups.lagda.md +++ b/src/higher-group-theory/cartesian-products-higher-groups.lagda.md @@ -100,13 +100,13 @@ module _ left-unit-law-mul-product-∞-Group : (x : type-product-∞-Group) → - Id (mul-product-∞-Group unit-product-∞-Group x) x + mul-product-∞-Group unit-product-∞-Group x ＝ x left-unit-law-mul-product-∞-Group = left-unit-law-mul-Ω classifying-pointed-type-product-∞-Group right-unit-law-mul-product-∞-Group : (y : type-product-∞-Group) → - Id (mul-product-∞-Group y unit-product-∞-Group) y + mul-product-∞-Group y unit-product-∞-Group ＝ y right-unit-law-mul-product-∞-Group = right-unit-law-mul-Ω classifying-pointed-type-product-∞-Group diff --git a/src/higher-group-theory/higher-groups.lagda.md b/src/higher-group-theory/higher-groups.lagda.md index 6102d85536..188b007dbb 100644 --- a/src/higher-group-theory/higher-groups.lagda.md +++ b/src/higher-group-theory/higher-groups.lagda.md @@ -119,12 +119,12 @@ module _ associative-mul-∞-Group = associative-mul-Ω classifying-pointed-type-∞-Group left-unit-law-mul-∞-Group : - (x : type-∞-Group) → Id (mul-∞-Group unit-∞-Group x) x + (x : type-∞-Group) → mul-∞-Group unit-∞-Group x ＝ x left-unit-law-mul-∞-Group = left-unit-law-mul-Ω classifying-pointed-type-∞-Group right-unit-law-mul-∞-Group : - (y : type-∞-Group) → Id (mul-∞-Group y unit-∞-Group) y + (y : type-∞-Group) → mul-∞-Group y unit-∞-Group ＝ y right-unit-law-mul-∞-Group = right-unit-law-mul-Ω classifying-pointed-type-∞-Group diff --git a/src/linear-algebra/matrices-on-rings.lagda.md b/src/linear-algebra/matrices-on-rings.lagda.md index 35c07eaca1..63732106fb 100644 --- a/src/linear-algebra/matrices-on-rings.lagda.md +++ b/src/linear-algebra/matrices-on-rings.lagda.md @@ -93,7 +93,7 @@ module _ commutative-add-matrix-Ring : {m n : ℕ} (A B : matrix-Ring R m n) → - Id (add-matrix-Ring R A B) (add-matrix-Ring R B A) + add-matrix-Ring R A B ＝ add-matrix-Ring R B A commutative-add-matrix-Ring empty-tuple empty-tuple = refl commutative-add-matrix-Ring (v ∷ A) (w ∷ B) = ap-binary _∷_ diff --git a/src/linear-algebra/multiplication-matrices.lagda.md b/src/linear-algebra/multiplication-matrices.lagda.md index 361034808b..7a3d7ce025 100644 --- a/src/linear-algebra/multiplication-matrices.lagda.md +++ b/src/linear-algebra/multiplication-matrices.lagda.md @@ -43,7 +43,7 @@ mul-Mat mulK addK zero (v ∷ vs) m = mul-transpose : {l : Level} → {K : UU l} → {m n p : ℕ} → {addK : K → K → K} {mulK : K → K → K} {zero : K} → - ((x y : K) → Id (mulK x y) (mulK y x)) → + ((x y : K) → mulK x y ＝ mulK y x) → (a : Mat K m n) → (b : Mat K n p) → Id (transpose (mul-Mat mulK addK zero a b)) @@ -74,7 +74,7 @@ module _ ({l : ℕ} → Id (diagonal-product {n = l} zero) (add-tuple addK (diagonal-product zero) (diagonal-product zero))) → ((x y z : K) → (Id (mulK x (addK y z)) (addK (mulK x y) (mulK x z)))) → - ((x y : K) → Id (addK x y) (addK y x)) → + ((x y : K) → addK x y ＝ addK y x) → ((x y z : K) → Id (addK x (addK y z)) (addK (addK x y) z)) → (a : tuple K n) (b : Mat K n m) (c : Mat K n m) → Id (mul-tuple-matrix mulK addK zero a (add-Mat addK b c)) @@ -123,7 +123,7 @@ module _ (diagonal-product {n = l} zero) (add-tuple addK (diagonal-product zero) (diagonal-product zero))) → ((x y z : K) → (Id (mulK x (addK y z)) (addK (mulK x y) (mulK x z)))) → - ((x y : K) → Id (addK x y) (addK y x)) → + ((x y : K) → addK x y ＝ addK y x) → ((x y z : K) → Id (addK x (addK y z)) (addK (addK x y) z)) → (a : Mat K m n) (b : Mat K n p) (c : Mat K n p) → Id (mul-Mat mulK addK zero a (add-Mat addK b c)) @@ -148,7 +148,7 @@ module _ (diagonal-product {n = l} zero) (add-tuple addK (diagonal-product zero) (diagonal-product zero))) → ((x y z : K) → (Id (mulK (addK x y) z) (addK (mulK x z) (mulK y z)))) → - ((x y : K) → Id (addK x y) (addK y x)) → + ((x y : K) → addK x y ＝ addK y x) → ((x y z : K) → Id (addK x (addK y z)) (addK (addK x y) z)) → (b : Mat K n p) (c : Mat K n p) (d : Mat K p m) → Id (mul-Mat mulK addK zero (add-Mat addK b c) d) diff --git a/src/linear-algebra/tuples-on-euclidean-domains.lagda.md b/src/linear-algebra/tuples-on-euclidean-domains.lagda.md index 8b68b65aba..178f5e519a 100644 --- a/src/linear-algebra/tuples-on-euclidean-domains.lagda.md +++ b/src/linear-algebra/tuples-on-euclidean-domains.lagda.md @@ -148,7 +148,7 @@ module _ commutative-add-tuple-Euclidean-Domain : {n : ℕ} (v w : tuple-Euclidean-Domain R n) → - Id (add-tuple-Euclidean-Domain R v w) (add-tuple-Euclidean-Domain R w v) + add-tuple-Euclidean-Domain R v w ＝ add-tuple-Euclidean-Domain R w v commutative-add-tuple-Euclidean-Domain = commutative-add-tuple-Commutative-Ring ( commutative-ring-Euclidean-Domain R) diff --git a/src/linear-algebra/tuples-on-rings.lagda.md b/src/linear-algebra/tuples-on-rings.lagda.md index fa814cfa40..4c745aeefd 100644 --- a/src/linear-algebra/tuples-on-rings.lagda.md +++ b/src/linear-algebra/tuples-on-rings.lagda.md @@ -134,7 +134,7 @@ module _ commutative-add-tuple-Ring : {n : ℕ} (v w : tuple-Ring R n) → - Id (add-tuple-Ring R v w) (add-tuple-Ring R w v) + add-tuple-Ring R v w ＝ add-tuple-Ring R w v commutative-add-tuple-Ring empty-tuple empty-tuple = refl commutative-add-tuple-Ring (x ∷ v) (y ∷ w) = ap-binary _∷_ diff --git a/src/lists/concatenation-lists.lagda.md b/src/lists/concatenation-lists.lagda.md index 1face70497..246a58513c 100644 --- a/src/lists/concatenation-lists.lagda.md +++ b/src/lists/concatenation-lists.lagda.md @@ -52,12 +52,12 @@ associative-concat-list (cons a x) y z = left-unit-law-concat-list : {l1 : Level} {A : UU l1} (x : list A) → - Id (concat-list nil x) x + concat-list nil x ＝ x left-unit-law-concat-list x = refl right-unit-law-concat-list : {l1 : Level} {A : UU l1} (x : list A) → - Id (concat-list x nil) x + concat-list x nil ＝ x right-unit-law-concat-list nil = refl right-unit-law-concat-list (cons a x) = ap (cons a) (right-unit-law-concat-list x) diff --git a/src/lists/universal-property-lists-wild-monoids.lagda.md b/src/lists/universal-property-lists-wild-monoids.lagda.md index 67c4735fec..8dbc5be3be 100644 --- a/src/lists/universal-property-lists-wild-monoids.lagda.md +++ b/src/lists/universal-property-lists-wild-monoids.lagda.md @@ -144,7 +144,7 @@ module _ mul-Wild-Monoid M (f u) (map-elim-list-Wild-Monoid x) preserves-unit-map-elim-list-Wild-Monoid : - Id (map-elim-list-Wild-Monoid nil) (unit-Wild-Monoid M) + map-elim-list-Wild-Monoid nil ＝ unit-Wild-Monoid M preserves-unit-map-elim-list-Wild-Monoid = refl pointed-map-elim-list-Wild-Monoid : @@ -340,7 +340,7 @@ htpy-elim-list-Wild-Monoid {X = X} M g h H = ( α (concat-list x y) ∙ pr2 (pr1 h) x y) β nil y = {!!} β (cons x x₁) y = {!!} - γ : Id (pr2 g) (α nil ∙ pr2 h) + γ : pr2 g ＝ α nil ∙ pr2 h γ = ( inv right-unit) ∙ ( ( left-whisker-concat (pr2 g) (inv (left-inv (pr2 h)))) ∙ diff --git a/src/order-theory/decidable-subposets.lagda.md b/src/order-theory/decidable-subposets.lagda.md index 740b185f7b..3121a9fc93 100644 --- a/src/order-theory/decidable-subposets.lagda.md +++ b/src/order-theory/decidable-subposets.lagda.md @@ -41,7 +41,7 @@ module _ type-Subposet P (subtype-decidable-subtype S) eq-type-Decidable-Subposet : - (x y : type-Decidable-Subposet) → Id (pr1 x) (pr1 y) → x ＝ y + (x y : type-Decidable-Subposet) → pr1 x ＝ pr1 y → x ＝ y eq-type-Decidable-Subposet = eq-type-Subposet P (subtype-decidable-subtype S) diff --git a/src/order-theory/decidable-subpreorders.lagda.md b/src/order-theory/decidable-subpreorders.lagda.md index ff1398155c..f8776da04f 100644 --- a/src/order-theory/decidable-subpreorders.lagda.md +++ b/src/order-theory/decidable-subpreorders.lagda.md @@ -41,7 +41,7 @@ module _ type-Subpreorder P (subtype-decidable-subtype S) eq-type-Decidable-Subpreorder : - (x y : type-Decidable-Subpreorder) → Id (pr1 x) (pr1 y) → x ＝ y + (x y : type-Decidable-Subpreorder) → pr1 x ＝ pr1 y → x ＝ y eq-type-Decidable-Subpreorder = eq-type-Subpreorder P (subtype-decidable-subtype S) diff --git a/src/order-theory/finite-preorders.lagda.md b/src/order-theory/finite-preorders.lagda.md index 4ec0ab834a..2e56612100 100644 --- a/src/order-theory/finite-preorders.lagda.md +++ b/src/order-theory/finite-preorders.lagda.md @@ -190,7 +190,7 @@ module _ is-finite-type-decidable-subtype S (is-finite-type-Finite-Preorder P) eq-type-finite-Subpreorder : - (x y : type-finite-Subpreorder) → Id (pr1 x) (pr1 y) → x ＝ y + (x y : type-finite-Subpreorder) → pr1 x ＝ pr1 y → x ＝ y eq-type-finite-Subpreorder = eq-type-Decidable-Subpreorder (preorder-Finite-Preorder P) S diff --git a/src/order-theory/finitely-graded-posets.lagda.md b/src/order-theory/finitely-graded-posets.lagda.md index 762e20da05..9ad45ac90f 100644 --- a/src/order-theory/finitely-graded-posets.lagda.md +++ b/src/order-theory/finitely-graded-posets.lagda.md @@ -490,7 +490,7 @@ module _ is-set-type-Set face-set-Finitely-Graded-Subposet eq-face-Finitely-Graded-Subposet : - (x y : face-Finitely-Graded-Subposet) → Id (pr1 x) (pr1 y) → x ＝ y + (x y : face-Finitely-Graded-Subposet) → pr1 x ＝ pr1 y → x ＝ y eq-face-Finitely-Graded-Subposet x y = eq-type-subtype S emb-face-Finitely-Graded-Subposet : diff --git a/src/order-theory/subposets.lagda.md b/src/order-theory/subposets.lagda.md index 9a09f651b4..602d315d2d 100644 --- a/src/order-theory/subposets.lagda.md +++ b/src/order-theory/subposets.lagda.md @@ -42,7 +42,7 @@ module _ type-Subposet = type-Subpreorder (preorder-Poset X) S eq-type-Subposet : - (x y : type-Subposet) → Id (pr1 x) (pr1 y) → x ＝ y + (x y : type-Subposet) → pr1 x ＝ pr1 y → x ＝ y eq-type-Subposet = eq-type-Subpreorder (preorder-Poset X) S leq-Subposet-Prop : (x y : type-Subposet) → Prop l2 diff --git a/src/order-theory/subpreorders.lagda.md b/src/order-theory/subpreorders.lagda.md index 1421f51657..02eec90e6b 100644 --- a/src/order-theory/subpreorders.lagda.md +++ b/src/order-theory/subpreorders.lagda.md @@ -42,7 +42,7 @@ module _ type-Subpreorder = type-subtype S eq-type-Subpreorder : - (x y : type-Subpreorder) → Id (pr1 x) (pr1 y) → x ＝ y + (x y : type-Subpreorder) → pr1 x ＝ pr1 y → x ＝ y eq-type-Subpreorder x y = eq-type-subtype S leq-Subpreorder-Prop : (x y : type-Subpreorder) → Prop l2 diff --git a/src/polytopes/abstract-polytopes.lagda.md b/src/polytopes/abstract-polytopes.lagda.md index 641ac47852..a1ffd29517 100644 --- a/src/polytopes/abstract-polytopes.lagda.md +++ b/src/polytopes/abstract-polytopes.lagda.md @@ -264,7 +264,7 @@ module _ eq-path-elements-Prepolytope : (x y : type-Prepolytope) - (p : Id (shape-Prepolytope x) (shape-Prepolytope y)) → + (p : shape-Prepolytope x ＝ shape-Prepolytope y) → path-elements-Prepolytope x y → x ＝ y eq-path-elements-Prepolytope = eq-path-elements-Finitely-Graded-Poset finitely-graded-poset-Prepolytope diff --git a/src/ring-theory/dependent-products-rings.lagda.md b/src/ring-theory/dependent-products-rings.lagda.md index 4bb9add1af..cefc4eb518 100644 --- a/src/ring-theory/dependent-products-rings.lagda.md +++ b/src/ring-theory/dependent-products-rings.lagda.md @@ -83,11 +83,11 @@ module _ associative-add-Π-Ring = associative-add-Semiring semiring-Π-Ring left-unit-law-add-Π-Ring : - (x : type-Π-Ring) → Id (add-Π-Ring zero-Π-Ring x) x + (x : type-Π-Ring) → add-Π-Ring zero-Π-Ring x ＝ x left-unit-law-add-Π-Ring = left-unit-law-add-Semiring semiring-Π-Ring right-unit-law-add-Π-Ring : - (x : type-Π-Ring) → Id (add-Π-Ring x zero-Π-Ring) x + (x : type-Π-Ring) → add-Π-Ring x zero-Π-Ring ＝ x right-unit-law-add-Π-Ring = right-unit-law-add-Semiring semiring-Π-Ring left-inverse-law-add-Π-Ring : @@ -99,7 +99,7 @@ module _ right-inverse-law-add-Π-Ring = right-inverse-law-add-Ab ab-Π-Ring commutative-add-Π-Ring : - (x y : type-Π-Ring) → Id (add-Π-Ring x y) (add-Π-Ring y x) + (x y : type-Π-Ring) → add-Π-Ring x y ＝ add-Π-Ring y x commutative-add-Π-Ring = commutative-add-Semiring semiring-Π-Ring mul-Π-Ring : type-Π-Ring → type-Π-Ring → type-Π-Ring @@ -115,12 +115,12 @@ module _ associative-mul-Semiring semiring-Π-Ring left-unit-law-mul-Π-Ring : - (x : type-Π-Ring) → Id (mul-Π-Ring one-Π-Ring x) x + (x : type-Π-Ring) → mul-Π-Ring one-Π-Ring x ＝ x left-unit-law-mul-Π-Ring = left-unit-law-mul-Semiring semiring-Π-Ring right-unit-law-mul-Π-Ring : - (x : type-Π-Ring) → Id (mul-Π-Ring x one-Π-Ring) x + (x : type-Π-Ring) → mul-Π-Ring x one-Π-Ring ＝ x right-unit-law-mul-Π-Ring = right-unit-law-mul-Semiring semiring-Π-Ring diff --git a/src/ring-theory/products-rings.lagda.md b/src/ring-theory/products-rings.lagda.md index fcebad162a..86d85cd229 100644 --- a/src/ring-theory/products-rings.lagda.md +++ b/src/ring-theory/products-rings.lagda.md @@ -56,12 +56,12 @@ module _ pr2 (neg-product-Ring (x , y)) = neg-Ring R2 y left-unit-law-add-product-Ring : - (x : type-product-Ring) → Id (add-product-Ring zero-product-Ring x) x + (x : type-product-Ring) → add-product-Ring zero-product-Ring x ＝ x left-unit-law-add-product-Ring (x , y) = eq-pair (left-unit-law-add-Ring R1 x) (left-unit-law-add-Ring R2 y) right-unit-law-add-product-Ring : - (x : type-product-Ring) → Id (add-product-Ring x zero-product-Ring) x + (x : type-product-Ring) → add-product-Ring x zero-product-Ring ＝ x right-unit-law-add-product-Ring (x , y) = eq-pair (right-unit-law-add-Ring R1 x) (right-unit-law-add-Ring R2 y) @@ -88,7 +88,7 @@ module _ ( associative-add-Ring R2 y1 y2 y3) commutative-add-product-Ring : - (x y : type-product-Ring) → Id (add-product-Ring x y) (add-product-Ring y x) + (x y : type-product-Ring) → add-product-Ring x y ＝ add-product-Ring y x commutative-add-product-Ring (x1 , y1) (x2 , y2) = eq-pair ( commutative-add-Ring R1 x1 x2) @@ -113,12 +113,12 @@ module _ ( associative-mul-Ring R2 y1 y2 y3) left-unit-law-mul-product-Ring : - (x : type-product-Ring) → Id (mul-product-Ring one-product-Ring x) x + (x : type-product-Ring) → mul-product-Ring one-product-Ring x ＝ x left-unit-law-mul-product-Ring (x , y) = eq-pair (left-unit-law-mul-Ring R1 x) (left-unit-law-mul-Ring R2 y) right-unit-law-mul-product-Ring : - (x : type-product-Ring) → Id (mul-product-Ring x one-product-Ring) x + (x : type-product-Ring) → mul-product-Ring x one-product-Ring ＝ x right-unit-law-mul-product-Ring (x , y) = eq-pair (right-unit-law-mul-Ring R1 x) (right-unit-law-mul-Ring R2 y) diff --git a/src/ring-theory/rings.lagda.md b/src/ring-theory/rings.lagda.md index af58d72d7d..c639bb2552 100644 --- a/src/ring-theory/rings.lagda.md +++ b/src/ring-theory/rings.lagda.md @@ -115,7 +115,7 @@ module _ ap-add-Ring : {x y x' y' : type-Ring R} → - x ＝ x' → y ＝ y' → Id (add-Ring x y) (add-Ring x' y') + x ＝ x' → y ＝ y' → add-Ring x y ＝ add-Ring x' y' ap-add-Ring = ap-add-Ab (ab-Ring R) associative-add-Ring : @@ -127,7 +127,7 @@ module _ is-group-Semigroup (additive-semigroup-Ring R) is-group-additive-semigroup-Ring = is-group-Ab (ab-Ring R) - commutative-add-Ring : (x y : type-Ring R) → Id (add-Ring x y) (add-Ring y x) + commutative-add-Ring : (x y : type-Ring R) → add-Ring x y ＝ add-Ring y x commutative-add-Ring = commutative-add-Ab (ab-Ring R) interchange-add-add-Ring : @@ -276,10 +276,10 @@ module _ is-nonzero-ring-Prop : type-Ring R → Prop l is-nonzero-ring-Prop x = neg-Prop (is-zero-ring-Prop x) - left-unit-law-add-Ring : (x : type-Ring R) → Id (add-Ring R zero-Ring x) x + left-unit-law-add-Ring : (x : type-Ring R) → add-Ring R zero-Ring x ＝ x left-unit-law-add-Ring = left-unit-law-add-Ab (ab-Ring R) - right-unit-law-add-Ring : (x : type-Ring R) → Id (add-Ring R x zero-Ring) x + right-unit-law-add-Ring : (x : type-Ring R) → add-Ring R x zero-Ring ＝ x right-unit-law-add-Ring = right-unit-law-add-Ab (ab-Ring R) ``` @@ -368,7 +368,7 @@ module _ ap-mul-Ring : {x x' y y' : type-Ring R} (p : x ＝ x') (q : y ＝ y') → - Id (mul-Ring x y) (mul-Ring x' y') + mul-Ring x y ＝ mul-Ring x' y' ap-mul-Ring p q = ap-binary mul-Ring p q associative-mul-Ring : @@ -410,10 +410,10 @@ module _ one-Ring : type-Ring R one-Ring = unit-Monoid multiplicative-monoid-Ring - left-unit-law-mul-Ring : (x : type-Ring R) → Id (mul-Ring R one-Ring x) x + left-unit-law-mul-Ring : (x : type-Ring R) → mul-Ring R one-Ring x ＝ x left-unit-law-mul-Ring = left-unit-law-mul-Monoid multiplicative-monoid-Ring - right-unit-law-mul-Ring : (x : type-Ring R) → Id (mul-Ring R x one-Ring) x + right-unit-law-mul-Ring : (x : type-Ring R) → mul-Ring R x one-Ring ＝ x right-unit-law-mul-Ring = right-unit-law-mul-Monoid multiplicative-monoid-Ring ``` diff --git a/src/ring-theory/semirings.lagda.md b/src/ring-theory/semirings.lagda.md index 97c595edf6..8cd31fe350 100644 --- a/src/ring-theory/semirings.lagda.md +++ b/src/ring-theory/semirings.lagda.md @@ -190,13 +190,13 @@ module _ is-zero-semiring-Prop x = Id-Prop (set-Semiring R) x zero-Semiring left-unit-law-add-Semiring : - (x : type-Semiring R) → Id (add-Semiring R zero-Semiring x) x + (x : type-Semiring R) → add-Semiring R zero-Semiring x ＝ x left-unit-law-add-Semiring = left-unit-law-mul-Commutative-Monoid ( additive-commutative-monoid-Semiring R) right-unit-law-add-Semiring : - (x : type-Semiring R) → Id (add-Semiring R x zero-Semiring) x + (x : type-Semiring R) → add-Semiring R x zero-Semiring ＝ x right-unit-law-add-Semiring = right-unit-law-mul-Commutative-Monoid ( additive-commutative-monoid-Semiring R) @@ -220,7 +220,7 @@ module _ ap-mul-Semiring : {x x' y y' : type-Semiring R} (p : x ＝ x') (q : y ＝ y') → - Id (mul-Semiring x y) (mul-Semiring x' y') + mul-Semiring x y ＝ mul-Semiring x' y' ap-mul-Semiring p q = ap-binary mul-Semiring p q associative-mul-Semiring : @@ -285,12 +285,12 @@ module _ one-Semiring = unit-Monoid multiplicative-monoid-Semiring left-unit-law-mul-Semiring : - (x : type-Semiring R) → Id (mul-Semiring R one-Semiring x) x + (x : type-Semiring R) → mul-Semiring R one-Semiring x ＝ x left-unit-law-mul-Semiring = left-unit-law-mul-Monoid multiplicative-monoid-Semiring right-unit-law-mul-Semiring : - (x : type-Semiring R) → Id (mul-Semiring R x one-Semiring) x + (x : type-Semiring R) → mul-Semiring R x one-Semiring ＝ x right-unit-law-mul-Semiring = right-unit-law-mul-Monoid multiplicative-monoid-Semiring ``` diff --git a/src/structured-types/h-spaces.lagda.md b/src/structured-types/h-spaces.lagda.md index 72fb621b35..0e2199b8a0 100644 --- a/src/structured-types/h-spaces.lagda.md +++ b/src/structured-types/h-spaces.lagda.md @@ -108,7 +108,7 @@ module _ ap-mul-H-Space : {a b c d : type-H-Space} → a ＝ b → c ＝ d → - Id (mul-H-Space a c) (mul-H-Space b d) + mul-H-Space a c ＝ mul-H-Space b d ap-mul-H-Space p q = ap-binary mul-H-Space p q magma-H-Space : Magma l @@ -122,13 +122,13 @@ module _ left-unit-law-mul-H-Space : (x : type-H-Space) → - Id (mul-H-Space unit-H-Space x) x + mul-H-Space unit-H-Space x ＝ x left-unit-law-mul-H-Space = pr1 coherent-unit-laws-mul-H-Space right-unit-law-mul-H-Space : (x : type-H-Space) → - Id (mul-H-Space x unit-H-Space) x + mul-H-Space x unit-H-Space ＝ x right-unit-law-mul-H-Space = pr1 (pr2 coherent-unit-laws-mul-H-Space) diff --git a/src/structured-types/magmas.lagda.md b/src/structured-types/magmas.lagda.md index 13372059eb..de9500e352 100644 --- a/src/structured-types/magmas.lagda.md +++ b/src/structured-types/magmas.lagda.md @@ -75,7 +75,7 @@ is-semigroup-Magma M = ```agda is-commutative-Magma : {l : Level} → Magma l → UU l is-commutative-Magma M = - (x y : type-Magma M) → Id (mul-Magma M x y) (mul-Magma M y x) + (x y : type-Magma M) → mul-Magma M x y ＝ mul-Magma M y x ``` ### The structure of a commutative monoid on magmas diff --git a/src/structured-types/wild-loops.lagda.md b/src/structured-types/wild-loops.lagda.md index b86e858471..409b170d77 100644 --- a/src/structured-types/wild-loops.lagda.md +++ b/src/structured-types/wild-loops.lagda.md @@ -62,19 +62,19 @@ module _ ap-mul-Wild-Loop : {a b c d : type-Wild-Loop} → a ＝ b → c ＝ d → - Id (mul-Wild-Loop a c) (mul-Wild-Loop b d) + mul-Wild-Loop a c ＝ mul-Wild-Loop b d ap-mul-Wild-Loop = ap-mul-H-Space h-space-Wild-Loop magma-Wild-Loop : Magma l magma-Wild-Loop = magma-H-Space h-space-Wild-Loop left-unit-law-mul-Wild-Loop : - (x : type-Wild-Loop) → Id (mul-Wild-Loop unit-Wild-Loop x) x + (x : type-Wild-Loop) → mul-Wild-Loop unit-Wild-Loop x ＝ x left-unit-law-mul-Wild-Loop = left-unit-law-mul-H-Space h-space-Wild-Loop right-unit-law-mul-Wild-Loop : - (x : type-Wild-Loop) → Id (mul-Wild-Loop x unit-Wild-Loop) x + (x : type-Wild-Loop) → mul-Wild-Loop x unit-Wild-Loop ＝ x right-unit-law-mul-Wild-Loop = right-unit-law-mul-H-Space h-space-Wild-Loop diff --git a/src/synthetic-homotopy-theory/cocones-under-spans.lagda.md b/src/synthetic-homotopy-theory/cocones-under-spans.lagda.md index b3ac5c1716..e2d5bd126e 100644 --- a/src/synthetic-homotopy-theory/cocones-under-spans.lagda.md +++ b/src/synthetic-homotopy-theory/cocones-under-spans.lagda.md @@ -214,7 +214,7 @@ cocone-map-span-diagram {𝒮 = 𝒮} c = cocone-map-id : {l1 l2 l3 l4 : Level} {S : UU l1} {A : UU l2} {B : UU l3} (f : S → A) (g : S → B) {X : UU l4} (c : cocone f g X) → - Id (cocone-map f g c id) c + cocone-map f g c id ＝ c cocone-map-id f g c = eq-pair-eq-fiber ( eq-pair-eq-fiber (eq-htpy (ap-id ∘ coherence-square-cocone f g c))) diff --git a/src/synthetic-homotopy-theory/free-loops.lagda.md b/src/synthetic-homotopy-theory/free-loops.lagda.md index 3a7bdb47d6..bdd17f5d3d 100644 --- a/src/synthetic-homotopy-theory/free-loops.lagda.md +++ b/src/synthetic-homotopy-theory/free-loops.lagda.md @@ -122,7 +122,7 @@ module _ Eq-free-dependent-loop : (p p' : free-dependent-loop α P) → UU l2 Eq-free-dependent-loop (pair y p) p' = - Σ ( Id y (base-free-dependent-loop α P p')) + Σ ( y ＝ base-free-dependent-loop α P p') ( λ q → ( p ∙ q) ＝ ( ( ap (tr P (loop-free-loop α)) q) ∙ diff --git a/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md b/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md index 1bd957651f..da44084699 100644 --- a/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md +++ b/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md @@ -158,7 +158,7 @@ module _ ( ( equiv-right-swap-Σ) ∘e ( ( associative-Σ ( ℤ ≃ ℤ) - ( λ e → Id (map-equiv e zero-ℤ) zero-ℤ) + ( λ e → map-equiv e zero-ℤ ＝ zero-ℤ) ( λ e → ( map-equiv (pr1 e) ∘ succ-ℤ) ~ ( succ-ℤ ∘ map-equiv (pr1 e)))) ∘e diff --git a/src/synthetic-homotopy-theory/interval-type.lagda.md b/src/synthetic-homotopy-theory/interval-type.lagda.md index e4601a34f1..0bd6c8acef 100644 --- a/src/synthetic-homotopy-theory/interval-type.lagda.md +++ b/src/synthetic-homotopy-theory/interval-type.lagda.md @@ -48,11 +48,11 @@ postulate compute-source-𝕀 : {l : Level} {P : 𝕀 → UU l} (u : P source-𝕀) (v : P target-𝕀) - (q : dependent-identification P path-𝕀 u v) → Id (ind-𝕀 P u v q source-𝕀) u + (q : dependent-identification P path-𝕀 u v) → ind-𝕀 P u v q source-𝕀 ＝ u compute-target-𝕀 : {l : Level} {P : 𝕀 → UU l} (u : P source-𝕀) (v : P target-𝕀) - (q : dependent-identification P path-𝕀 u v) → Id (ind-𝕀 P u v q target-𝕀) v + (q : dependent-identification P path-𝕀 u v) → ind-𝕀 P u v q target-𝕀 ＝ v compute-path-𝕀 : {l : Level} {P : 𝕀 → UU l} (u : P source-𝕀) (v : P target-𝕀) @@ -109,7 +109,7 @@ module _ ( pair refl right-unit) ( λ u' → id-equiv) ( extensionality-Σ - ( λ {v'} α' q → Id (α ∙ q) α') + ( λ {v'} α' q → α ∙ q ＝ α') ( refl) ( right-unit) ( λ v' → id-equiv) @@ -152,9 +152,9 @@ is-section-inv-ev-𝕀 (pair u (pair v q)) = tr-value : {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (f g : (x : A) → B x) {x y : A} - (p : x ＝ y) (q : Id (f x) (g x)) (r : Id (f y) (g y)) → + (p : x ＝ y) (q : f x ＝ g x) (r : f y ＝ g y) → Id (apd f p ∙ r) (ap (tr B p) q ∙ apd g p) → - Id (tr (λ x → Id (f x) (g x)) p q) r + Id (tr (λ x → f x ＝ g x) p q) r tr-value f g refl q r s = (inv (ap-id q) ∙ inv right-unit) ∙ inv s is-retraction-inv-ev-𝕀 : diff --git a/src/synthetic-homotopy-theory/loop-spaces.lagda.md b/src/synthetic-homotopy-theory/loop-spaces.lagda.md index 99790500d6..2b28900ec2 100644 --- a/src/synthetic-homotopy-theory/loop-spaces.lagda.md +++ b/src/synthetic-homotopy-theory/loop-spaces.lagda.md @@ -42,7 +42,7 @@ module _ where type-Ω : UU l - type-Ω = Id (point-Pointed-Type A) (point-Pointed-Type A) + type-Ω = point-Pointed-Type A ＝ point-Pointed-Type A refl-Ω : type-Ω refl-Ω = refl @@ -147,7 +147,7 @@ module _ is-equiv-tr-type-Ω : (p : x ＝ y) → is-equiv (tr-type-Ω p) is-equiv-tr-type-Ω p = is-equiv-map-equiv (equiv-tr-type-Ω p) - preserves-refl-tr-Ω : (p : x ＝ y) → Id (tr-type-Ω p refl) refl + preserves-refl-tr-Ω : (p : x ＝ y) → tr-type-Ω p refl ＝ refl preserves-refl-tr-Ω refl = refl preserves-mul-tr-Ω : diff --git a/src/synthetic-homotopy-theory/triple-loop-spaces.lagda.md b/src/synthetic-homotopy-theory/triple-loop-spaces.lagda.md index 5f2aaf3924..cc980fb796 100644 --- a/src/synthetic-homotopy-theory/triple-loop-spaces.lagda.md +++ b/src/synthetic-homotopy-theory/triple-loop-spaces.lagda.md @@ -61,17 +61,17 @@ z-concat-Ω³ = z-concat-Id³ ap-x-concat-Ω³ : {l : Level} {A : UU l} {a : A} {α α' β β' : type-Ω³ a} - (s : α ＝ α') (t : β ＝ β') → Id (x-concat-Ω³ α β) (x-concat-Ω³ α' β') + (s : α ＝ α') (t : β ＝ β') → x-concat-Ω³ α β ＝ x-concat-Ω³ α' β' ap-x-concat-Ω³ s t = ap-binary x-concat-Ω³ s t ap-y-concat-Ω³ : {l : Level} {A : UU l} {a : A} {α α' β β' : type-Ω³ a} - (s : α ＝ α') (t : β ＝ β') → Id (y-concat-Ω³ α β) (y-concat-Ω³ α' β') + (s : α ＝ α') (t : β ＝ β') → y-concat-Ω³ α β ＝ y-concat-Ω³ α' β' ap-y-concat-Ω³ s t = j-concat-Id⁴ s t ap-z-concat-Ω³ : {l : Level} {A : UU l} {a : A} {α α' β β' : type-Ω³ a} - (s : α ＝ α') (t : β ＝ β') → Id (z-concat-Ω³ α β) (z-concat-Ω³ α' β') + (s : α ＝ α') (t : β ＝ β') → z-concat-Ω³ α β ＝ z-concat-Ω³ α' β' ap-z-concat-Ω³ s t = k-concat-Id⁴ s t ``` @@ -82,27 +82,27 @@ ap-z-concat-Ω³ s t = k-concat-Id⁴ s t ```agda left-unit-law-x-concat-Ω³ : {l : Level} {A : UU l} {a : A} (α : type-Ω³ a) → - Id (x-concat-Ω³ refl-Ω³ α) α + x-concat-Ω³ refl-Ω³ α ＝ α left-unit-law-x-concat-Ω³ α = left-unit right-unit-law-x-concat-Ω³ : {l : Level} {A : UU l} {a : A} (α : type-Ω³ a) → - Id (x-concat-Ω³ α refl-Ω³) α + x-concat-Ω³ α refl-Ω³ ＝ α right-unit-law-x-concat-Ω³ α = right-unit left-unit-law-y-concat-Ω³ : {l : Level} {A : UU l} {a : A} (α : type-Ω³ a) → - Id (y-concat-Ω³ refl-Ω³ α) α + y-concat-Ω³ refl-Ω³ α ＝ α left-unit-law-y-concat-Ω³ α = left-unit-law-horizontal-concat-Ω² right-unit-law-y-concat-Ω³ : {l : Level} {A : UU l} {a : A} (α : type-Ω³ a) → - Id (y-concat-Ω³ α refl-Ω³) α + y-concat-Ω³ α refl-Ω³ ＝ α right-unit-law-y-concat-Ω³ α = right-unit-law-horizontal-concat-Ω² left-unit-law-z-concat-Ω³ : {l : Level} {A : UU l} {a : A} (α : type-Ω³ a) → - Id (z-concat-Ω³ refl-Ω³ α) α + z-concat-Ω³ refl-Ω³ α ＝ α left-unit-law-z-concat-Ω³ α = ( left-unit-law-z-concat-Id³ α) ∙ ( ( inv right-unit) ∙ @@ -122,7 +122,7 @@ super-naturality-right-unit α = {!!} {- right-unit-law-z-concat-Ω³ : {l : Level} {A : UU l} {a : A} (α : type-Ω³ a) → - Id (z-concat-Ω³ α refl-Ω³) α + z-concat-Ω³ α refl-Ω³ ＝ α right-unit-law-z-concat-Ω³ α = ( right-unit-law-z-concat-Id³ α) ∙ {!!} @@ -171,7 +171,7 @@ interchange-y-z-concat-Ω³ α β γ δ = ```agda outer-eckmann-hilton-connection-x-y-concat-Ω³ : {l : Level} {A : UU l} {a : A} (α δ : type-Ω³ a) → - Id (y-concat-Ω³ α δ) (x-concat-Ω³ α δ) + y-concat-Ω³ α δ ＝ x-concat-Ω³ α δ outer-eckmann-hilton-connection-x-y-concat-Ω³ α δ = ( j-concat-Id⁴ ( inv (right-unit-law-x-concat-Ω³ α)) @@ -183,7 +183,7 @@ outer-eckmann-hilton-connection-x-y-concat-Ω³ α δ = inner-eckmann-hilton-connection-x-y-concat-Ω³ : {l : Level} {A : UU l} {a : A} (β γ : type-Ω³ a) → - Id (y-concat-Ω³ β γ) (x-concat-Ω³ γ β) + y-concat-Ω³ β γ ＝ x-concat-Ω³ γ β inner-eckmann-hilton-connection-x-y-concat-Ω³ β γ = ( j-concat-Id⁴ ( inv (left-unit-law-x-concat-Ω³ β)) @@ -196,7 +196,7 @@ inner-eckmann-hilton-connection-x-y-concat-Ω³ β γ = {- outer-eckmann-hilton-connection-x-z-concat-Ω³ : {l : Level} {A : UU l} {a : A} (α δ : type-Ω³ a) → - Id (z-concat-Ω³ α δ) (x-concat-Ω³ α δ) + z-concat-Ω³ α δ ＝ x-concat-Ω³ α δ outer-eckmann-hilton-connection-x-z-concat-Ω³ α δ = ( k-concat-Id⁴ ( inv (right-unit-law-x-concat-Ω³ α)) @@ -210,7 +210,7 @@ outer-eckmann-hilton-connection-x-z-concat-Ω³ α δ = {- inner-eckmann-hilton-connection-x-z-concat-Ω³ : {l : Level} {A : UU l} {a : A} (β γ : type-Ω³ a) → - Id (z-concat-Ω³ β γ) (x-concat-Ω³ γ β) + z-concat-Ω³ β γ ＝ x-concat-Ω³ γ β inner-eckmann-hilton-connection-x-z-concat-Ω³ β γ = ( k-concat-Id⁴ ( inv (left-unit-law-x-concat-Ω³ β)) @@ -224,7 +224,7 @@ inner-eckmann-hilton-connection-x-z-concat-Ω³ β γ = {- outer-eckmann-hilton-connection-y-z-concat-Ω³ : {l : Level} {A : UU l} {a : A} (α δ : type-Ω³ a) → - Id (z-concat-Ω³ α δ) (y-concat-Ω³ α δ) + z-concat-Ω³ α δ ＝ y-concat-Ω³ α δ outer-eckmann-hilton-connection-y-z-concat-Ω³ α δ = ( k-concat-Id⁴ ( inv (right-unit-law-y-concat-Ω³ α)) @@ -238,7 +238,7 @@ outer-eckmann-hilton-connection-y-z-concat-Ω³ α δ = {- inner-eckmann-hilton-connection-y-z-concat-Ω³ : {l : Level} {A : UU l} {a : A} (β γ : type-Ω³ a) → - Id (z-concat-Ω³ β γ) (y-concat-Ω³ γ β) + z-concat-Ω³ β γ ＝ y-concat-Ω³ γ β inner-eckmann-hilton-connection-y-z-concat-Ω³ β γ = ( k-concat-Id⁴ ( inv (left-unit-law-y-concat-Ω³ β)) diff --git a/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md b/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md index efc341525b..ac2ea7746f 100644 --- a/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md +++ b/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md @@ -99,7 +99,7 @@ abstract is-equiv-functor-free-dependent-loop-is-fiberwise-equiv (pair x l) {P} {Q} {f} is-equiv-f = is-equiv-map-Σ - ( λ q₀ → Id (tr Q l q₀) q₀) + ( λ q₀ → tr Q l q₀ ＝ q₀) ( is-equiv-f x) ( λ p₀ → is-equiv-comp @@ -190,7 +190,7 @@ contraction-total-space : { l1 l2 : Level} {A : UU l1} {B : A → UU l2} (center : Σ A B) → ( x : A) → UU (l1 ⊔ l2) contraction-total-space {B = B} center x = - ( y : B x) → Id center (pair x y) + ( y : B x) → center ＝ pair x y path-total-path-fiber : { l1 l2 : Level} {A : UU l1} (B : A → UU l2) (x : A) → @@ -199,9 +199,9 @@ path-total-path-fiber B x q = eq-pair-eq-fiber (inv q) tr-path-total-path-fiber : { l1 l2 : Level} {A : UU l1} {B : A → UU l2} (c : Σ A B) (x : A) → - { y y' : B x} (q : y' ＝ y) (α : Id c (pair x y')) → + { y y' : B x} (q : y' ＝ y) (α : c ＝ pair x y') → Id - ( tr (λ z → Id c (pair x z)) q α) + ( tr (λ z → c ＝ pair x z) q α) ( α ∙ (inv (path-total-path-fiber B x q))) tr-path-total-path-fiber c x refl α = inv right-unit @@ -235,7 +235,7 @@ equiv-contraction-total-space : ( x : A) → {F : UU l3} (e : F ≃ B x) → ( contraction-total-space c x) ≃ (contraction-total-space' c x e) equiv-contraction-total-space c x e = - equiv-precomp-Π e (λ y → Id c (pair x y)) + equiv-precomp-Π e (λ y → c ＝ pair x y) tr-path-total-tr-coherence : { l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} (c : Σ A B) (x : A) → @@ -243,7 +243,7 @@ tr-path-total-tr-coherence : ( H : ((map-equiv e') ∘ (map-equiv f)) ~ (map-equiv e)) → (y : F) (α : Id c (pair x (map-equiv e' (map-equiv f y)))) → Id - ( tr (λ z → Id c (pair x z)) (H y) α) + ( tr (λ z → c ＝ pair x z) (H y) α) ( α ∙ (inv (segment-Σ refl f e e' H y))) tr-path-total-tr-coherence c x f e e' H y α = tr-path-total-path-fiber c x (H y) α @@ -320,7 +320,7 @@ map-dependent-identification-contraction-total-space' ( map-equiv (equiv-tr-contraction-total-space' c refl f e e' H)) ( is-section-map-inv-is-equiv ( is-equiv-map-equiv - ( equiv-precomp-Π e' (λ y' → Id c (pair x y')))) + ( equiv-precomp-Π e' (λ y' → c ＝ pair x y'))) ( h')))))) equiv-dependent-identification-contraction-total-space' : @@ -349,7 +349,7 @@ equiv-dependent-identification-contraction-total-space' ( map-equiv (equiv-tr-contraction-total-space' c refl f e e' H)) ( is-section-map-inv-is-equiv ( is-equiv-map-equiv - ( equiv-precomp-Π e' (λ y' → Id c (pair x y')))) + ( equiv-precomp-Π e' (λ y' → c ＝ pair x y'))) ( h')))) ∘e ( ( equiv-concat ( inv @@ -449,7 +449,7 @@ contraction-total-universal-cover-circle-data : ( h) ( h)) → ( t : Σ X (universal-cover-circle l dup-circle)) → - Id (center-total-universal-cover-circle l dup-circle) t + center-total-universal-cover-circle l dup-circle ＝ t contraction-total-universal-cover-circle-data {l1} l dup-circle h p (pair x y) = map-inv-is-equiv @@ -577,7 +577,7 @@ point-universal-cover-circle l dup-circle = universal-cover-circle-eq : { l1 : Level} {X : UU l1} (l : free-loop X) → ( dup-circle : dependent-universal-property-circle l) → - ( x : X) → Id (base-free-loop l) x → universal-cover-circle l dup-circle x + ( x : X) → base-free-loop l ＝ x → universal-cover-circle l dup-circle x universal-cover-circle-eq l dup-circle .(base-free-loop l) refl = point-universal-cover-circle l dup-circle @@ -595,7 +595,7 @@ equiv-universal-cover-circle : { l1 : Level} {X : UU l1} (l : free-loop X) → ( dup-circle : dependent-universal-property-circle l) → ( x : X) → - ( Id (base-free-loop l) x) ≃ (universal-cover-circle l dup-circle x) + ( base-free-loop l ＝ x) ≃ (universal-cover-circle l dup-circle x) equiv-universal-cover-circle l dup-circle x = pair ( universal-cover-circle-eq l dup-circle x) diff --git a/src/synthetic-homotopy-theory/universal-property-circle.lagda.md b/src/synthetic-homotopy-theory/universal-property-circle.lagda.md index 9698cb554f..8b827dcb33 100644 --- a/src/synthetic-homotopy-theory/universal-property-circle.lagda.md +++ b/src/synthetic-homotopy-theory/universal-property-circle.lagda.md @@ -123,7 +123,7 @@ module _ ( Eq-free-dependent-loop α P ( ev-free-loop-Π α P f) ( ev-free-loop-Π α P g)) → - ( free-dependent-loop α (λ x → Id (f x) (g x))) + ( free-dependent-loop α (λ x → f x ＝ g x)) pr1 (free-dependent-loop-htpy {l2} {P} {f} {g} (p , q)) = p pr2 (free-dependent-loop-htpy {l2} {P} {f} {g} (p , q)) = map-compute-dependent-identification-eq-value f g (loop-free-loop α) p p q diff --git a/src/type-theories/dependent-type-theories.lagda.md b/src/type-theories/dependent-type-theories.lagda.md index 9dcb5a5ba3..9d8d82750e 100644 --- a/src/type-theories/dependent-type-theories.lagda.md +++ b/src/type-theories/dependent-type-theories.lagda.md @@ -124,7 +124,7 @@ homotopies of sections of fibered systems. concat-htpy-section-system' : {l1 l2 l3 l4 : Level} {A : system l1 l2} {B B' B'' : fibered-system l3 l4 A} - {α : B ＝ B'} {β : B' ＝ B''} (γ : B ＝ B'') (δ : Id (α ∙ β) γ) + {α : B ＝ B'} {β : B' ＝ B''} (γ : B ＝ B'') (δ : α ∙ β ＝ γ) {f : section-system B} {g : section-system B'} {h : section-system B''} (G : htpy-section-system' α f g) (H : htpy-section-system' β g h) → @@ -162,7 +162,7 @@ homotopies of sections of fibered systems. inv-htpy-section-system' : {l1 l2 l3 l4 : Level} {A : system l1 l2} {B B' : fibered-system l3 l4 A} - {α : B ＝ B'} (β : B' ＝ B) (γ : Id (inv α) β) + {α : B ＝ B'} (β : B' ＝ B) (γ : inv α ＝ β) {f : section-system B} {g : section-system B'} → htpy-section-system' α f g → htpy-section-system' β g f section-system.type (inv-htpy-section-system' {α = refl} .refl refl H) X = @@ -338,10 +338,10 @@ We show that systems form a category. {l1 l2 l3 l4 l5 l6 : Level} {A : system l1 l2} {B B' : system l3 l4} {C C' : system l5 l6} {g : hom-system B C} {g' : hom-system B' C'} (p : B ＝ B') - {p' : Id (constant-fibered-system A B) (constant-fibered-system A B')} + {p' : constant-fibered-system A B ＝ constant-fibered-system A B'} (α : Id (ap (constant-fibered-system A) p) p') (q : C ＝ C') - {q' : Id (constant-fibered-system A C) (constant-fibered-system A C')} + {q' : constant-fibered-system A C ＝ constant-fibered-system A C'} (β : Id (ap (constant-fibered-system A) q) q') (r : Id (tr (λ t → t) (ap-binary hom-system p q) g) g') {f : hom-system A B} {f' : hom-system A B'} → @@ -407,9 +407,9 @@ We show that systems form a category. {l1 l2 l3 l4 l5 l6 : Level} {A : system l1 l2} {B : system l3 l4} {C C' : system l5 l6} (p : C ＝ C') {g : hom-system B C} {g' : hom-system B C'} - {p' : Id (constant-fibered-system B C) (constant-fibered-system B C')} + {p' : constant-fibered-system B C ＝ constant-fibered-system B C'} (α : Id (ap (constant-fibered-system B) p) p') - {q' : Id (constant-fibered-system A C) (constant-fibered-system A C')} + {q' : constant-fibered-system A C ＝ constant-fibered-system A C'} (β : Id (ap (constant-fibered-system A) p) q') (H : htpy-section-system' p' g g') → (f : hom-system A B) → diff --git a/src/type-theories/fibered-dependent-type-theories.lagda.md b/src/type-theories/fibered-dependent-type-theories.lagda.md index 0d1905bdbb..ed84a99826 100644 --- a/src/type-theories/fibered-dependent-type-theories.lagda.md +++ b/src/type-theories/fibered-dependent-type-theories.lagda.md @@ -107,8 +107,8 @@ module fibered where double-tr : {l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} (D : (x : A) → B x → C x → UU l4) {x y : A} (p : x ＝ y) - {u : B x} {u' : B y} (q : Id (tr B p u) u') {v : C x} {v' : C y} - (r : Id (tr C p v) v') → D x u v → D y u' v' + {u : B x} {u' : B y} (q : tr B p u ＝ u') {v : C x} {v' : C y} + (r : tr C p v ＝ v') → D x u v → D y u' v' double-tr D refl refl refl d = d tr-bifibered-system-slice : diff --git a/src/type-theories/simple-type-theories.lagda.md b/src/type-theories/simple-type-theories.lagda.md index 1c1b961458..72974ca11d 100644 --- a/src/type-theories/simple-type-theories.lagda.md +++ b/src/type-theories/simple-type-theories.lagda.md @@ -70,7 +70,7 @@ homotopies of sections of fibered systems. {l1 l2 l3 l4 : Level} {T : UU l1} {A : system l2 T} {S : T → UU l3} {B B' : fibered-system l4 S A} (α : B ＝ B') {f f' : (X : T) → S X} (g : section-system B f) (g' : section-system B' f') → - fibered-system l4 (λ t → Id (f t) (f' t)) A + fibered-system l4 (λ t → f t ＝ f' t) A fibered-system.element (Eq-fibered-system' {B = B} refl {f} g g') {X} p x = Id ( tr @@ -96,7 +96,7 @@ homotopies of sections of fibered systems. {l1 l2 l3 l4 : Level} {T : UU l1} {A : system l2 T} {S : T → UU l3} {B B' B'' : fibered-system l4 S A} {α : B ＝ B'} {β : B' ＝ B''} - (γ : B ＝ B'') (δ : Id (α ∙ β) γ) {f f' f'' : (X : T) → S X} + (γ : B ＝ B'') (δ : α ∙ β ＝ γ) {f f' f'' : (X : T) → S X} {H : f ~ f'} {H' : f' ~ f''} {g : section-system B f} {g' : section-system B' f'} {g'' : section-system B'' f''} (K : htpy-section-system' α H g g') @@ -129,7 +129,7 @@ homotopies of sections of fibered systems. {l1 l2 l3 l4 : Level} {T : UU l1} {A : system l2 T} {S : T → UU l3} {B B' : fibered-system l4 S A} - {α : B ＝ B'} (β : B' ＝ B) (γ : Id (inv α) β) + {α : B ＝ B'} (β : B' ＝ B) (γ : inv α ＝ β) {f f' : (X : T) → S X} {g : section-system B f} {g' : section-system B' f'} {H : f ~ f'} → htpy-section-system' α H g g' → htpy-section-system' β (inv-htpy H) g' g diff --git a/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md b/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md index 892da4210f..58a5fc5eb3 100644 --- a/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md +++ b/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md @@ -294,7 +294,7 @@ module _ compute-swap-2-Element-Decidable-Subtype : (x y : type-2-Element-Decidable-Subtype P) → x ≠ y → - Id (map-swap-2-Element-Decidable-Subtype x) y + map-swap-2-Element-Decidable-Subtype x ＝ y compute-swap-2-Element-Decidable-Subtype = compute-swap-2-Element-Type (2-element-type-2-Element-Decidable-Subtype P) diff --git a/src/univalent-combinatorics/2-element-subtypes.lagda.md b/src/univalent-combinatorics/2-element-subtypes.lagda.md index 9c1bd2707d..c0ba7d21a6 100644 --- a/src/univalent-combinatorics/2-element-subtypes.lagda.md +++ b/src/univalent-combinatorics/2-element-subtypes.lagda.md @@ -242,7 +242,7 @@ module _ is-injective-element-unordered-pair : (p : unordered-pair A) → ¬ ( (x y : type-unordered-pair p) → - Id (element-unordered-pair p x) (element-unordered-pair p y)) → + element-unordered-pair p x ＝ element-unordered-pair p y) → is-injective (element-unordered-pair p) is-injective-element-unordered-pair (pair X f) H {x} {y} p = apply-universal-property-trunc-Prop @@ -252,10 +252,10 @@ module _ where first-element : (Fin 2 ≃ (type-2-Element-Type X)) → Σ ( type-2-Element-Type X) - ( λ x → ¬ ((y : type-2-Element-Type X) → Id (f x) (f y))) + ( λ x → ¬ ((y : type-2-Element-Type X) → f x ＝ f y)) first-element h = exists-not-not-for-all-count (λ z → (w : type-2-Element-Type X) → - Id (f z) (f w)) (λ z → {!!}) + f z ＝ f w) (λ z → {!!}) {!!} {!!} two-elements-different-image : Σ ( type-2-Element-Type X) diff --git a/src/univalent-combinatorics/cartesian-product-types.lagda.md b/src/univalent-combinatorics/cartesian-product-types.lagda.md index 43dee54458..c6ba5cccee 100644 --- a/src/univalent-combinatorics/cartesian-product-types.lagda.md +++ b/src/univalent-combinatorics/cartesian-product-types.lagda.md @@ -79,12 +79,12 @@ abstract equiv-left-factor : {l1 l2 : Level} {X : UU l1} {Y : UU l2} (y : Y) → - (Σ (X × Y) (λ t → Id (pr2 t) y)) ≃ X + (Σ (X × Y) (λ t → pr2 t ＝ y)) ≃ X equiv-left-factor {l1} {l2} {X} {Y} y = ( ( right-unit-law-product) ∘e ( equiv-tot ( λ x → equiv-is-contr (is-torsorial-Id' y) is-contr-unit))) ∘e - ( associative-Σ X (λ x → Y) (λ t → Id (pr2 t) y)) + ( associative-Σ X (λ x → Y) (λ t → pr2 t ＝ y)) count-left-factor : {l1 l2 : Level} {X : UU l1} {Y : UU l2} → count (X × Y) → Y → count X diff --git a/src/univalent-combinatorics/classical-finite-types.lagda.md b/src/univalent-combinatorics/classical-finite-types.lagda.md index 6306188299..86fd8c8636 100644 --- a/src/univalent-combinatorics/classical-finite-types.lagda.md +++ b/src/univalent-combinatorics/classical-finite-types.lagda.md @@ -57,7 +57,7 @@ nat-classical-Fin k = pr1 ```agda Eq-classical-Fin : (k : ℕ) (x y : classical-Fin k) → UU lzero -Eq-classical-Fin k x y = Id (nat-classical-Fin k x) (nat-classical-Fin k y) +Eq-classical-Fin k x y = nat-classical-Fin k x ＝ nat-classical-Fin k y eq-succ-classical-Fin : (k : ℕ) (x y : classical-Fin k) → Id {A = classical-Fin k} x y → diff --git a/src/univalent-combinatorics/counting-decidable-subtypes.lagda.md b/src/univalent-combinatorics/counting-decidable-subtypes.lagda.md index 6b06dd1332..5589ef86fe 100644 --- a/src/univalent-combinatorics/counting-decidable-subtypes.lagda.md +++ b/src/univalent-combinatorics/counting-decidable-subtypes.lagda.md @@ -122,7 +122,7 @@ is-decidable-count-subtype P e f x = ( count-decidable-subtype ( λ y → pair - ( Id (pr1 y) x) + ( pr1 y ＝ x) ( pair ( is-set-count e (pr1 y) x) ( has-decidable-equality-count e (pr1 y) x))) diff --git a/src/univalent-combinatorics/cyclic-finite-types.lagda.md b/src/univalent-combinatorics/cyclic-finite-types.lagda.md index a4393d21ea..cd6fbb7087 100644 --- a/src/univalent-combinatorics/cyclic-finite-types.lagda.md +++ b/src/univalent-combinatorics/cyclic-finite-types.lagda.md @@ -582,7 +582,7 @@ preserves-concat-equiv-compute-Ω-Cyclic-Type k {p} {q} = ( equiv-eq-Cyclic-Type k ( ℤ-Mod-Cyclic-Type k) ( ℤ-Mod-Cyclic-Type k) q)) type-Ω-Cyclic-Type : (k : ℕ) → UU (lsuc lzero) -type-Ω-Cyclic-Type k = Id (ℤ-Mod-Cyclic-Type k) (ℤ-Mod-Cyclic-Type k) +type-Ω-Cyclic-Type k = ℤ-Mod-Cyclic-Type k ＝ ℤ-Mod-Cyclic-Type k is-set-type-Ω-Cyclic-Type : (k : ℕ) → is-set (type-Ω-Cyclic-Type k) is-set-type-Ω-Cyclic-Type k = diff --git a/src/univalent-combinatorics/dependent-pair-types.lagda.md b/src/univalent-combinatorics/dependent-pair-types.lagda.md index bb19fd3a96..f67c29c4bf 100644 --- a/src/univalent-combinatorics/dependent-pair-types.lagda.md +++ b/src/univalent-combinatorics/dependent-pair-types.lagda.md @@ -109,7 +109,7 @@ abstract ( equiv-tot ( λ x → equiv-eq-pair-Σ (map-section-family b x) t)) ∘e ( ( associative-Σ A - ( λ (x : A) → Id x (pr1 t)) + ( λ (x : A) → x ＝ pr1 t) ( λ s → Id (tr B (pr2 s) (b (pr1 s))) (pr2 t))) ∘e ( inv-left-unit-law-Σ-is-contr ( is-torsorial-Id' (pr1 t)) diff --git a/src/univalent-combinatorics/double-counting.lagda.md b/src/univalent-combinatorics/double-counting.lagda.md index a6a55dcc85..0e7c84e5e9 100644 --- a/src/univalent-combinatorics/double-counting.lagda.md +++ b/src/univalent-combinatorics/double-counting.lagda.md @@ -29,14 +29,14 @@ abstract double-counting-equiv : {l1 l2 : Level} {A : UU l1} {B : UU l2} (count-A : count A) (count-B : count B) (e : A ≃ B) → - Id (number-of-elements-count count-A) (number-of-elements-count count-B) + number-of-elements-count count-A ＝ number-of-elements-count count-B double-counting-equiv (k , f) (l , g) e = is-equivalence-injective-Fin (inv-equiv g ∘e e ∘e f) abstract double-counting : {l : Level} {A : UU l} (count-A count-A' : count A) → - Id (number-of-elements-count count-A) (number-of-elements-count count-A') + number-of-elements-count count-A ＝ number-of-elements-count count-A' double-counting count-A count-A' = double-counting-equiv count-A count-A' id-equiv ``` diff --git a/src/univalent-combinatorics/finite-types.lagda.md b/src/univalent-combinatorics/finite-types.lagda.md index f9d68827e6..fbfca5b8be 100644 --- a/src/univalent-combinatorics/finite-types.lagda.md +++ b/src/univalent-combinatorics/finite-types.lagda.md @@ -388,7 +388,7 @@ module _ abstract compute-number-of-elements-is-finite : (e : count X) (f : is-finite X) → - Id (number-of-elements-count e) (number-of-elements-is-finite f) + number-of-elements-count e ＝ number-of-elements-is-finite f compute-number-of-elements-is-finite e f = ind-trunc-Prop ( λ g → diff --git a/src/univalent-combinatorics/injective-maps.lagda.md b/src/univalent-combinatorics/injective-maps.lagda.md index cb28686054..9c9ba4917b 100644 --- a/src/univalent-combinatorics/injective-maps.lagda.md +++ b/src/univalent-combinatorics/injective-maps.lagda.md @@ -29,7 +29,7 @@ Injectiveness in the context of finite types enjoys further properties. ```agda is-decidable-is-injective-is-finite' : {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) → - is-finite A → is-finite B → is-decidable ((x y : A) → Id (f x) (f y) → x ＝ y) + is-finite A → is-finite B → is-decidable ((x y : A) → f x ＝ f y → x ＝ y) is-decidable-is-injective-is-finite' f HA HB = is-decidable-Π-is-finite HA ( λ x → diff --git a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md index 6dd93e951d..a986699c5a 100644 --- a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md +++ b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md @@ -525,7 +525,7 @@ module _ abstract preserves-id-equiv-orientation-complete-undirected-graph-equiv : (X : Type-With-Cardinality-ℕ l n) → - Id (orientation-complete-undirected-graph-equiv X X id-equiv) id-equiv + orientation-complete-undirected-graph-equiv X X id-equiv ＝ id-equiv preserves-id-equiv-orientation-complete-undirected-graph-equiv X = eq-htpy-equiv ( λ d → @@ -1591,7 +1591,7 @@ module _ ( has-decidable-equality-count eX) ( np')) ( Y))))) → - Id (two-elements-transposition eX Y) two-elements → + two-elements-transposition eX Y ＝ two-elements → is-decidable (Id (pr1 two-elements) i) → is-decidable (Id (pr1 two-elements) j) → is-decidable (Id (pr1 (pr2 two-elements)) i) → @@ -1826,7 +1826,7 @@ module _ ( has-decidable-equality-count eX) ( np')) ( Y))))) → - Id (two-elements-transposition eX Y) two-elements → + two-elements-transposition eX Y ＝ two-elements → is-decidable (Id (pr1 two-elements) i) → is-decidable (Id (pr1 two-elements) j) → is-decidable (Id (pr1 (pr2 two-elements)) i) → @@ -2253,7 +2253,7 @@ module _ ( has-decidable-equality-count eX) ( np')) ( Y))))) → - Id (two-elements-transposition eX Y) two-elements → + two-elements-transposition eX Y ＝ two-elements → is-decidable (Id (pr1 two-elements) i) → is-decidable (Id (pr1 two-elements) j) → is-decidable (Id (pr1 (pr2 two-elements)) i) → diff --git a/src/univalent-combinatorics/ramsey-theory.lagda.md b/src/univalent-combinatorics/ramsey-theory.lagda.md index c5e0cc22ff..17087e6c27 100644 --- a/src/univalent-combinatorics/ramsey-theory.lagda.md +++ b/src/univalent-combinatorics/ramsey-theory.lagda.md @@ -47,7 +47,7 @@ is-ramsey-set {l} {k} q r A = ( (x : type-Finite-Type A) → type-Prop ((pr1 Q) x) → type-Prop ((pr1 P) x)) → - Id (c Q) i)) + c Q ＝ i)) {- is-ramsey-set-empty-coloring : (r : ℕ) → is-ramsey-set ex-falso r empty-Finite-Type is-ramsey-set-empty-coloring zero-ℕ c = {!!} diff --git a/src/univalent-combinatorics/repetitions-of-values-sequences.lagda.md b/src/univalent-combinatorics/repetitions-of-values-sequences.lagda.md index 41333f0272..15cf17c137 100644 --- a/src/univalent-combinatorics/repetitions-of-values-sequences.lagda.md +++ b/src/univalent-combinatorics/repetitions-of-values-sequences.lagda.md @@ -22,7 +22,7 @@ is-decidable-is-ordered-repetition-of-values-ℕ-Fin k f x = {!!} {- is-decidable-strictly-bounded-Σ-ℕ' x - ( λ y → Id (f y) (f x)) + ( λ y → f y ＝ f x) ( λ y → has-decidable-equality-Fin k (f y) (f x)) -} @@ -33,7 +33,7 @@ is-decidable-is-ordered-repetition-of-values-ℕ-count e f x = {!!} {- is-decidable-strictly-bounded-Σ-ℕ' x - ( λ y → Id (f y) (f x)) + ( λ y → f y ＝ f x) ( λ y → has-decidable-equality-count e (f y) (f x)) -} ``` From aadc54183afe6c0aca0b0fe713fc47684be233b3 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 1 Sep 2025 18:43:12 +0200 Subject: [PATCH 07/16] more replacement --- .../universal-property-integers.lagda.md | 4 ++-- .../orbits-permutations.lagda.md | 15 ++++++------ src/finite-group-theory/permutations.lagda.md | 4 ++-- .../transpositions.lagda.md | 4 ++-- ...universal-property-identity-types.lagda.md | 4 ++-- src/graph-theory/directed-graphs.lagda.md | 2 +- .../equivalences-undirected-graphs.lagda.md | 2 +- ...artesian-products-concrete-groups.lagda.md | 4 ++-- ...artesian-products-concrete-groups.lagda.md | 4 ++-- .../shriek-concrete-group-actions.lagda.md | 2 +- .../transposition-matrices.lagda.md | 2 +- src/lists/lists.lagda.md | 2 +- src/lists/tuples.lagda.md | 2 +- src/reflection/group-solver.lagda.md | 2 +- .../free-loops.lagda.md | 4 ++-- .../interval-type.lagda.md | 4 ++-- .../loop-spaces.lagda.md | 2 +- .../2-element-decidable-subtypes.lagda.md | 4 ++-- .../decidable-propositions.lagda.md | 8 +++---- .../equality-finite-types.lagda.md | 2 +- .../equality-standard-finite-types.lagda.md | 2 +- ...tations-complete-undirected-graph.lagda.md | 24 +++++++++---------- .../sequences-finite-types.lagda.md | 2 +- .../standard-finite-trees.lagda.md | 2 +- .../untruncated-pi-finite-types.lagda.md | 6 ++--- 25 files changed, 56 insertions(+), 57 deletions(-) diff --git a/src/elementary-number-theory/universal-property-integers.lagda.md b/src/elementary-number-theory/universal-property-integers.lagda.md index 598e3b8f54..e12b4b7012 100644 --- a/src/elementary-number-theory/universal-property-integers.lagda.md +++ b/src/elementary-number-theory/universal-property-integers.lagda.md @@ -110,7 +110,7 @@ zero-Eq-ELIM-ℤ : ( p0 : P zero-ℤ) (pS : (k : ℤ) → (P k) ≃ (P (succ-ℤ k))) → ( s t : ELIM-ℤ P p0 pS) (H : (pr1 s) ~ (pr1 t)) → UU l1 zero-Eq-ELIM-ℤ P p0 pS s t H = - Id (H zero-ℤ) ((pr1 (pr2 s)) ∙ (inv (pr1 (pr2 t)))) + (H zero-ℤ ＝ pr1 (pr2 s) ∙ inv (pr1 (pr2 t))) succ-Eq-ELIM-ℤ : { l1 : Level} (P : ℤ → UU l1) @@ -157,7 +157,7 @@ abstract ( pair (pr1 s) refl-htpy) ( is-torsorial-Eq-structure ( is-contr-is-equiv' - ( Σ (Id (pr1 s zero-ℤ) p0) (λ α → Id α (pr1 (pr2 s)))) + ( Σ (pr1 s zero-ℤ ＝ p0) (λ α → Id α (pr1 (pr2 s)))) ( tot (λ α → right-transpose-eq-concat refl α (pr1 (pr2 s)))) ( is-equiv-tot-is-fiberwise-equiv ( λ α → is-equiv-right-transpose-eq-concat refl α (pr1 (pr2 s)))) diff --git a/src/finite-group-theory/orbits-permutations.lagda.md b/src/finite-group-theory/orbits-permutations.lagda.md index 604f62a17b..9f4dc02c18 100644 --- a/src/finite-group-theory/orbits-permutations.lagda.md +++ b/src/finite-group-theory/orbits-permutations.lagda.md @@ -265,7 +265,7 @@ module _ abstract not-not-eq-second-point-zero-min-reporting : - ¬¬ (Id second-point-min-repeating zero-ℕ) + ¬¬ (second-point-min-repeating ＝ zero-ℕ) not-not-eq-second-point-zero-min-reporting np = contradiction-le-ℕ ( pred-first) @@ -323,7 +323,7 @@ module _ equality-pred-second = pr2 is-successor-second-point-min-repeating has-finite-orbits-permutation' : - is-decidable (Id second-point-min-repeating zero-ℕ) → + is-decidable (second-point-min-repeating ＝ zero-ℕ) → Σ ℕ (λ k → (is-nonzero-ℕ k) × Id (iterate k (map-equiv f) a) a) pr1 (has-finite-orbits-permutation' (inl p)) = first-point-min-repeating pr1 (pr2 (has-finite-orbits-permutation' (inl p))) = @@ -375,7 +375,7 @@ module _ ( has-decidable-equality-ℕ second-point-min-repeating zero-ℕ) where cases-second-point : - is-decidable (Id second-point-min-repeating zero-ℕ) → + is-decidable (second-point-min-repeating ＝ zero-ℕ) → (pr1 has-finite-orbits-permutation) ≤-ℕ (number-of-elements-count eX) cases-second-point (inl p) = tr @@ -458,8 +458,7 @@ module _ lemma : (h : Fin n ≃ type-Type-With-Cardinality-ℕ n X) (k : ℕ) → Σ ( ℕ) - ( λ j → - Id (j +ℕ k) (k *ℕ (pr1 (has-finite-orbits-permutation-a h)))) + ( λ j → j +ℕ k ＝ k *ℕ (pr1 (has-finite-orbits-permutation-a h))) lemma h k = subtraction-leq-ℕ ( k) @@ -624,7 +623,7 @@ module _ T1 ＝ class same-orbits-permutation t1 → is-decidable ( is-in-equivalence-class same-orbits-permutation T2 t1) → - is-decidable (Id T1 T2) + is-decidable (T1 ＝ T2) cases-decidable-equality T1 T2 t1 p1 (inl p) = inl ( ( p1) ∙ @@ -746,7 +745,7 @@ module _ ( b)) where induction-cases-equal-iterate-transposition : - is-decidable (Id k zero-ℕ) → + is-decidable (k ＝ zero-ℕ) → Id ( iterate k (map-equiv (composition-transposition-a-b g)) x) ( iterate k (map-equiv g) x) @@ -916,7 +915,7 @@ module _ Σ ( ℕ) ( λ l → is-nonzero-ℕ l × - Id (l +ℕ k) (pr1 (minimal-element-iterate g a b pa))) + (l +ℕ k ＝ pr1 (minimal-element-iterate g a b pa))) pair-k2 = (subtraction-le-ℕ k (pr1 (minimal-element-iterate g a b pa)) ineq) pr2 (neq-iterate-nonzero-le-minimal-element pa k (pair nz ineq)) r = diff --git a/src/finite-group-theory/permutations.lagda.md b/src/finite-group-theory/permutations.lagda.md index 8d38d934db..dc3ebdad92 100644 --- a/src/finite-group-theory/permutations.lagda.md +++ b/src/finite-group-theory/permutations.lagda.md @@ -192,8 +192,8 @@ module _ ( Σ ( type-Type-With-Cardinality-ℕ n X) ( type-Decidable-Prop ∘ P))))) ( λ li → - Id k (mod-two-ℕ (length-list li)) × - f ＝ permutation-list-transpositions li))) + ( k ＝ mod-two-ℕ (length-list li)) × + ( f ＝ permutation-list-transpositions li)))) abstract is-contr-parity-transposition-permutation : diff --git a/src/finite-group-theory/transpositions.lagda.md b/src/finite-group-theory/transpositions.lagda.md index 9b104bd41d..b644320bc3 100644 --- a/src/finite-group-theory/transpositions.lagda.md +++ b/src/finite-group-theory/transpositions.lagda.md @@ -478,9 +478,9 @@ module _ (x y : X) (np : x ≠ y) → type-Decidable-Prop (pr1 Y x) → type-Decidable-Prop (pr1 Y y) → - is-decidable (Id (pr1 two-elements-transposition) x) → + is-decidable (pr1 two-elements-transposition ＝ x) → is-decidable (Id (pr1 (pr2 two-elements-transposition)) x) → - is-decidable (Id (pr1 two-elements-transposition) y) → + is-decidable (pr1 two-elements-transposition ＝ y) → is-decidable (Id (pr1 (pr2 two-elements-transposition)) y) → ( ( pr1 two-elements-transposition ＝ x) × ( Id (pr1 (pr2 two-elements-transposition)) y)) + diff --git a/src/foundation/universal-property-identity-types.lagda.md b/src/foundation/universal-property-identity-types.lagda.md index 7dfeb07d06..f4aa575bf3 100644 --- a/src/foundation/universal-property-identity-types.lagda.md +++ b/src/foundation/universal-property-identity-types.lagda.md @@ -206,7 +206,7 @@ is a proper embedding. ```agda module _ {l : Level} (A : UU l) - (L : (a x y : A) → instance-preunivalence (Id x y) (Id a y)) + (L : (a x y : A) → instance-preunivalence (x ＝ y) (a ＝ y)) where emb-Id-is-injective-equiv-eq-Id : (a x : A) → (Id a ＝ Id x) ↪ (a ＝ x) @@ -235,7 +235,7 @@ module _ is-emb-Id-preunivalence-axiom : is-emb (Id {A = A}) is-emb-Id-preunivalence-axiom = is-emb-Id-is-injective-equiv-eq-Id A - ( λ a x y → is-injective-is-emb (L (Id x y) (Id a y))) + ( λ a x y → is-injective-is-emb (L (x ＝ y) (a ＝ y))) ``` #### `Id : A → (A → 𝒰)` is an embedding diff --git a/src/graph-theory/directed-graphs.lagda.md b/src/graph-theory/directed-graphs.lagda.md index 798b5afeb6..63c467e2c7 100644 --- a/src/graph-theory/directed-graphs.lagda.md +++ b/src/graph-theory/directed-graphs.lagda.md @@ -130,7 +130,7 @@ module equiv {l1 l2 : Level} where Directed-Graph' l1 l2 -> Directed-Graph l1 (l1 ⊔ l2) pr1 (Directed-Graph'-to-Directed-Graph (V , E , st , tg)) = V pr2 (Directed-Graph'-to-Directed-Graph (V , E , st , tg)) x y = - Σ E (λ e → (Id (st e) x) × (Id (tg e) y)) + Σ E (λ e → (st e ＝ x) × (tg e ＝ y)) ``` ## See also diff --git a/src/graph-theory/equivalences-undirected-graphs.lagda.md b/src/graph-theory/equivalences-undirected-graphs.lagda.md index 4b9800127d..ee6a634e8b 100644 --- a/src/graph-theory/equivalences-undirected-graphs.lagda.md +++ b/src/graph-theory/equivalences-undirected-graphs.lagda.md @@ -206,7 +206,7 @@ module _ ( λ gE → (p : unordered-pair-vertices-Undirected-Graph G) → (e : edge-Undirected-Graph G p) → - Id (edge-equiv-Undirected-Graph G H f p e) (map-equiv (gE p) e))) + edge-equiv-Undirected-Graph G H f p e ＝ map-equiv (gE p) e)) ( equiv-tot ( λ gE → equiv-Π-equiv-family diff --git a/src/group-theory/cartesian-products-concrete-groups.lagda.md b/src/group-theory/cartesian-products-concrete-groups.lagda.md index 019271c1b6..b7bb672221 100644 --- a/src/group-theory/cartesian-products-concrete-groups.lagda.md +++ b/src/group-theory/cartesian-products-concrete-groups.lagda.md @@ -110,14 +110,14 @@ module _ ( mere-eq-classifying-type-product-Concrete-Group shape-product-Concrete-Group X) - ( is-set-Prop (Id X Y)) + ( is-set-Prop (X ＝ Y)) ( λ where refl → apply-universal-property-trunc-Prop ( mere-eq-classifying-type-product-Concrete-Group shape-product-Concrete-Group Y) - ( is-set-Prop (Id shape-product-Concrete-Group Y)) + ( is-set-Prop (shape-product-Concrete-Group ＝ Y)) ( λ where refl → is-set-type-product-Concrete-Group)) classifying-1-type-product-Concrete-Group : Truncated-Type (l1 ⊔ l2) one-𝕋 diff --git a/src/group-theory/iterated-cartesian-products-concrete-groups.lagda.md b/src/group-theory/iterated-cartesian-products-concrete-groups.lagda.md index db2815fc12..75e7a17979 100644 --- a/src/group-theory/iterated-cartesian-products-concrete-groups.lagda.md +++ b/src/group-theory/iterated-cartesian-products-concrete-groups.lagda.md @@ -121,14 +121,14 @@ module _ ( mere-eq-classifying-type-iterated-product-Concrete-Group shape-iterated-product-Concrete-Group X) - ( is-set-Prop (Id X Y)) + ( is-set-Prop (X ＝ Y)) ( λ where refl → apply-universal-property-trunc-Prop ( mere-eq-classifying-type-iterated-product-Concrete-Group shape-iterated-product-Concrete-Group Y) - ( is-set-Prop (Id shape-iterated-product-Concrete-Group Y)) + ( is-set-Prop (shape-iterated-product-Concrete-Group ＝ Y)) ( λ where refl → is-set-type-iterated-product-Concrete-Group)) classifying-1-type-iterated-product-Concrete-Group : Truncated-Type l one-𝕋 diff --git a/src/group-theory/shriek-concrete-group-actions.lagda.md b/src/group-theory/shriek-concrete-group-actions.lagda.md index 270643b890..8166a6336f 100644 --- a/src/group-theory/shriek-concrete-group-actions.lagda.md +++ b/src/group-theory/shriek-concrete-group-actions.lagda.md @@ -38,5 +38,5 @@ module _ trunc-Set ( Σ ( classifying-type-Concrete-Group G) ( λ x → - type-Set (X x) × classifying-map-hom-Concrete-Group G H f x ＝ y)) + type-Set (X x) × (classifying-map-hom-Concrete-Group G H f x ＝ y))) ``` diff --git a/src/linear-algebra/transposition-matrices.lagda.md b/src/linear-algebra/transposition-matrices.lagda.md index 9ed858094f..b6d0964ec3 100644 --- a/src/linear-algebra/transposition-matrices.lagda.md +++ b/src/linear-algebra/transposition-matrices.lagda.md @@ -62,7 +62,7 @@ is-involution-transpose-matrix {m = succ-ℕ m} (r ∷ rs) = lemma-rest : {l : Level} → {A : UU l} → {m n : ℕ} → (x : tuple A n) → (xs : matrix A m n) → - Id (transpose-matrix xs) (map-tuple tail-tuple (transpose-matrix (x ∷ xs))) + transpose-matrix xs ＝ map-tuple tail-tuple (transpose-matrix (x ∷ xs)) lemma-rest {n = zero-ℕ} empty-tuple xs = refl lemma-rest {n = succ-ℕ n} (k ∷ ks) xs = ap diff --git a/src/lists/lists.lagda.md b/src/lists/lists.lagda.md index cbf193fa1b..c9ccea7272 100644 --- a/src/lists/lists.lagda.md +++ b/src/lists/lists.lagda.md @@ -168,7 +168,7 @@ Eq-list : {l1 : Level} {A : UU l1} → list A → list A → UU l1 Eq-list {l1} nil nil = raise-unit l1 Eq-list {l1} nil (cons x l') = raise-empty l1 Eq-list {l1} (cons x l) nil = raise-empty l1 -Eq-list {l1} (cons x l) (cons x' l') = (Id x x') × Eq-list l l' +Eq-list {l1} (cons x l) (cons x' l') = (x ＝ x') × Eq-list l l' refl-Eq-list : {l1 : Level} {A : UU l1} (l : list A) → Eq-list l l refl-Eq-list nil = raise-star diff --git a/src/lists/tuples.lagda.md b/src/lists/tuples.lagda.md index d93ba4fa4f..0350140c08 100644 --- a/src/lists/tuples.lagda.md +++ b/src/lists/tuples.lagda.md @@ -109,7 +109,7 @@ module _ Eq-tuple : (n : ℕ) → tuple A n → tuple A n → UU l Eq-tuple zero-ℕ empty-tuple empty-tuple = raise-unit l - Eq-tuple (succ-ℕ n) (x ∷ xs) (y ∷ ys) = (Id x y) × (Eq-tuple n xs ys) + Eq-tuple (succ-ℕ n) (x ∷ xs) (y ∷ ys) = (x ＝ y) × (Eq-tuple n xs ys) refl-Eq-tuple : (n : ℕ) → (u : tuple A n) → Eq-tuple n u u refl-Eq-tuple zero-ℕ empty-tuple = map-raise star diff --git a/src/reflection/group-solver.lagda.md b/src/reflection/group-solver.lagda.md index 5a1bd511ee..ebcf363baf 100644 --- a/src/reflection/group-solver.lagda.md +++ b/src/reflection/group-solver.lagda.md @@ -37,7 +37,7 @@ data Inductive-Fin : ℕ → UU lzero where zero-Inductive-Fin : {n : ℕ} → Inductive-Fin (succ-ℕ n) succ-Inductive-Fin : {n : ℕ} → Inductive-Fin n → Inductive-Fin (succ-ℕ n) -finEq : {n : ℕ} → (a b : Inductive-Fin n) → is-decidable (Id a b) +finEq : {n : ℕ} → (a b : Inductive-Fin n) → is-decidable (a ＝ b) finEq zero-Inductive-Fin zero-Inductive-Fin = inl refl finEq zero-Inductive-Fin (succ-Inductive-Fin b) = inr (λ ()) finEq (succ-Inductive-Fin a) zero-Inductive-Fin = inr (λ ()) diff --git a/src/synthetic-homotopy-theory/free-loops.lagda.md b/src/synthetic-homotopy-theory/free-loops.lagda.md index bdd17f5d3d..792a2cb3ba 100644 --- a/src/synthetic-homotopy-theory/free-loops.lagda.md +++ b/src/synthetic-homotopy-theory/free-loops.lagda.md @@ -81,7 +81,7 @@ module _ Eq-free-loop : (α α' : free-loop X) → UU l1 Eq-free-loop (pair x α) α' = - Σ (Id x (base-free-loop α')) (λ p → Id (α ∙ p) (p ∙ (loop-free-loop α'))) + Σ (x ＝ base-free-loop α') (λ p → α ∙ p ＝ p ∙ (loop-free-loop α')) refl-Eq-free-loop : (α : free-loop X) → Eq-free-loop α α pr1 (refl-Eq-free-loop (pair x α)) = refl @@ -98,7 +98,7 @@ module _ ( is-torsorial-Id x) ( pair x refl) ( is-contr-is-equiv' - ( Σ (Id x x) (λ α' → α ＝ α')) + ( Σ (x ＝ x) (λ α' → α ＝ α')) ( tot (λ α' α → right-unit ∙ α)) ( is-equiv-tot-is-fiberwise-equiv ( λ α' → is-equiv-concat right-unit α')) diff --git a/src/synthetic-homotopy-theory/interval-type.lagda.md b/src/synthetic-homotopy-theory/interval-type.lagda.md index 0bd6c8acef..55dd66d000 100644 --- a/src/synthetic-homotopy-theory/interval-type.lagda.md +++ b/src/synthetic-homotopy-theory/interval-type.lagda.md @@ -153,8 +153,8 @@ is-section-inv-ev-𝕀 (pair u (pair v q)) = tr-value : {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (f g : (x : A) → B x) {x y : A} (p : x ＝ y) (q : f x ＝ g x) (r : f y ＝ g y) → - Id (apd f p ∙ r) (ap (tr B p) q ∙ apd g p) → - Id (tr (λ x → f x ＝ g x) p q) r + apd f p ∙ r ＝ ap (tr B p) q ∙ apd g p → + tr (λ x → f x ＝ g x) p q ＝ r tr-value f g refl q r s = (inv (ap-id q) ∙ inv right-unit) ∙ inv s is-retraction-inv-ev-𝕀 : diff --git a/src/synthetic-homotopy-theory/loop-spaces.lagda.md b/src/synthetic-homotopy-theory/loop-spaces.lagda.md index 2b28900ec2..17bdfca373 100644 --- a/src/synthetic-homotopy-theory/loop-spaces.lagda.md +++ b/src/synthetic-homotopy-theory/loop-spaces.lagda.md @@ -166,7 +166,7 @@ module _ eq-tr-type-Ω : (p : x ＝ y) (q : type-Ω (pair A x)) → - Id (tr-type-Ω p q) (inv p ∙ (q ∙ p)) + tr-type-Ω p q ＝ inv p ∙ (q ∙ p) eq-tr-type-Ω refl q = inv right-unit ``` diff --git a/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md b/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md index 58a5fc5eb3..fbc6bca067 100644 --- a/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md +++ b/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md @@ -223,8 +223,8 @@ module _ eq-pair-Σ ( eq-equiv ( pair - ( map-commutative-coproduct (Id x z) (Id y z)) - ( is-equiv-map-commutative-coproduct (Id x z) (Id y z)))) + ( map-commutative-coproduct (x ＝ z) (y ＝ z)) + ( is-equiv-map-commutative-coproduct (x ＝ z) (y ＝ z)))) ( eq-pair-Σ ( eq-is-prop ( is-prop-is-prop diff --git a/src/univalent-combinatorics/decidable-propositions.lagda.md b/src/univalent-combinatorics/decidable-propositions.lagda.md index c825d693f8..c338b85b6b 100644 --- a/src/univalent-combinatorics/decidable-propositions.lagda.md +++ b/src/univalent-combinatorics/decidable-propositions.lagda.md @@ -55,7 +55,7 @@ count-type-Decidable-Prop P (inr f) = count-is-empty f ```agda cases-count-eq : {l : Level} {X : UU l} (d : has-decidable-equality X) {x y : X} - (e : is-decidable (Id x y)) → count (Id x y) + (e : is-decidable (x ＝ y)) → count (x ＝ y) cases-count-eq d {x} {y} (inl p) = count-is-contr ( is-proof-irrelevant-is-prop (is-set-has-decidable-equality d x y) p) @@ -63,12 +63,12 @@ cases-count-eq d (inr f) = count-is-empty f count-eq : - {l : Level} {X : UU l} → has-decidable-equality X → (x y : X) → count (Id x y) + {l : Level} {X : UU l} → has-decidable-equality X → (x y : X) → count (x ＝ y) count-eq d x y = cases-count-eq d (d x y) cases-number-of-elements-count-eq' : {l : Level} {X : UU l} {x y : X} → - is-decidable (Id x y) → ℕ + is-decidable (x ＝ y) → ℕ cases-number-of-elements-count-eq' (inl p) = 1 cases-number-of-elements-count-eq' (inr f) = 0 @@ -79,7 +79,7 @@ number-of-elements-count-eq' d x y = cases-number-of-elements-count-eq : {l : Level} {X : UU l} (d : has-decidable-equality X) {x y : X} - (e : is-decidable (Id x y)) → + (e : is-decidable (x ＝ y)) → Id ( number-of-elements-count (cases-count-eq d e)) ( cases-number-of-elements-count-eq' e) diff --git a/src/univalent-combinatorics/equality-finite-types.lagda.md b/src/univalent-combinatorics/equality-finite-types.lagda.md index 98592329a8..db146d3236 100644 --- a/src/univalent-combinatorics/equality-finite-types.lagda.md +++ b/src/univalent-combinatorics/equality-finite-types.lagda.md @@ -64,7 +64,7 @@ has-decidable-equality-has-cardinality-ℕ {l1} {X} k H = abstract is-finite-eq : {l : Level} {X : UU l} → - has-decidable-equality X → {x y : X} → is-finite (Id x y) + has-decidable-equality X → {x y : X} → is-finite (x ＝ y) is-finite-eq d {x} {y} = is-finite-count (count-eq d x y) is-finite-eq-is-finite : diff --git a/src/univalent-combinatorics/equality-standard-finite-types.lagda.md b/src/univalent-combinatorics/equality-standard-finite-types.lagda.md index 3103dd72a7..ab4d010816 100644 --- a/src/univalent-combinatorics/equality-standard-finite-types.lagda.md +++ b/src/univalent-combinatorics/equality-standard-finite-types.lagda.md @@ -90,7 +90,7 @@ is-decidable-Eq-Fin (succ-ℕ k) (inr x) (inl y) = is-decidable-empty is-decidable-Eq-Fin (succ-ℕ k) (inr x) (inr y) = is-decidable-unit has-decidable-equality-Fin : - (k : ℕ) (x y : Fin k) → is-decidable (Id x y) + (k : ℕ) (x y : Fin k) → is-decidable (x ＝ y) has-decidable-equality-Fin k x y = map-coproduct ( eq-Eq-Fin k) diff --git a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md index a986699c5a..b0fcabea91 100644 --- a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md +++ b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md @@ -776,8 +776,8 @@ module _ ( has-decidable-equality-count eX) ( np')) ( Y))))) → - is-decidable (Id (pr1 two-elements) i) → - is-decidable (Id (pr1 two-elements) j) → + is-decidable (pr1 two-elements ＝ i) → + is-decidable (pr1 two-elements ＝ j) → is-decidable (Id (pr1 (pr2 two-elements)) i) → Σ X (λ z → type-Decidable-Prop (pr1 Y z)) cases-orientation-two-elements-count @@ -1453,8 +1453,8 @@ module _ ( standard-2-Element-Decidable-Subtype ( has-decidable-equality-count eX) ( np)))))) → - (q : is-decidable (Id (pr1 two-elements) i)) → - (r : is-decidable (Id (pr1 two-elements) j)) → + (q : is-decidable (pr1 two-elements ＝ i)) → + (r : is-decidable (pr1 two-elements ＝ j)) → (s : is-decidable (Id (pr1 (pr2 two-elements)) i)) → (t : is-decidable (Id (pr1 (pr2 two-elements)) j)) → Id @@ -1592,8 +1592,8 @@ module _ ( np')) ( Y))))) → two-elements-transposition eX Y ＝ two-elements → - is-decidable (Id (pr1 two-elements) i) → - is-decidable (Id (pr1 two-elements) j) → + is-decidable (pr1 two-elements ＝ i) → + is-decidable (pr1 two-elements ＝ j) → is-decidable (Id (pr1 (pr2 two-elements)) i) → is-decidable (Id (pr1 (pr2 two-elements)) j) → Id @@ -1827,8 +1827,8 @@ module _ ( np')) ( Y))))) → two-elements-transposition eX Y ＝ two-elements → - is-decidable (Id (pr1 two-elements) i) → - is-decidable (Id (pr1 two-elements) j) → + is-decidable (pr1 two-elements ＝ i) → + is-decidable (pr1 two-elements ＝ j) → is-decidable (Id (pr1 (pr2 two-elements)) i) → is-decidable (Id (pr1 (pr2 two-elements)) j) → Id @@ -2254,8 +2254,8 @@ module _ ( np')) ( Y))))) → two-elements-transposition eX Y ＝ two-elements → - is-decidable (Id (pr1 two-elements) i) → - is-decidable (Id (pr1 two-elements) j) → + is-decidable (pr1 two-elements ＝ i) → + is-decidable (pr1 two-elements ＝ j) → is-decidable (Id (pr1 (pr2 two-elements)) i) → is-decidable (Id (pr1 (pr2 two-elements)) j) → Id @@ -2870,8 +2870,8 @@ module _ ( np')) ( pr1 T))))) → Id two-elements (two-elements-transposition eX (pr1 T)) → - is-decidable (Id (pr1 two-elements) i) → - is-decidable (Id (pr1 two-elements) j) → + is-decidable (pr1 two-elements ＝ i) → + is-decidable (pr1 two-elements ＝ j) → is-decidable (Id (pr1 (pr2 two-elements)) i) → is-decidable (Id (pr1 (pr2 two-elements)) j) → Id diff --git a/src/univalent-combinatorics/sequences-finite-types.lagda.md b/src/univalent-combinatorics/sequences-finite-types.lagda.md index aa9ed385fc..fb170408e4 100644 --- a/src/univalent-combinatorics/sequences-finite-types.lagda.md +++ b/src/univalent-combinatorics/sequences-finite-types.lagda.md @@ -130,7 +130,7 @@ minimal-element-repetition-of-values-sequence-Fin : minimal-element-ℕ (λ x → Σ ℕ (λ y → (le-ℕ y x) × (f y ＝ f x))) minimal-element-repetition-of-values-sequence-Fin k f = well-ordering-principle-ℕ - ( λ x → Σ ℕ (λ y → (le-ℕ y x) × (Id (f y) (f x)))) + ( λ x → Σ ℕ (λ y → (le-ℕ y x) × (f y ＝ f x))) ( λ x → is-decidable-strictly-bounded-Σ-ℕ' x ( λ y → f y ＝ f x) diff --git a/src/univalent-combinatorics/standard-finite-trees.lagda.md b/src/univalent-combinatorics/standard-finite-trees.lagda.md index 9ef39e079f..cc5b89ab82 100644 --- a/src/univalent-combinatorics/standard-finite-trees.lagda.md +++ b/src/univalent-combinatorics/standard-finite-trees.lagda.md @@ -59,5 +59,5 @@ is-leaf-Tree-Fin (tree-Fin (succ-ℕ n) _) = empty is-full-binary-Tree-Fin : Tree-Fin → UU lzero is-full-binary-Tree-Fin (tree-Fin zero-ℕ f) = unit is-full-binary-Tree-Fin (tree-Fin (succ-ℕ n) f) = - (Id 2 n) × ((k : Fin (succ-ℕ n)) → is-full-binary-Tree-Fin (f k)) + (2 ＝ n) × ((k : Fin (succ-ℕ n)) → is-full-binary-Tree-Fin (f k)) ``` diff --git a/src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md b/src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md index 716344eb21..378dfac342 100644 --- a/src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md +++ b/src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md @@ -497,7 +497,7 @@ has-finitely-many-connected-components-Σ-is-0-connected {A = A} {B} C H K = ( Prop-Set _) ( λ ω → trunc-Prop (dependent-identification B ω y y')) - P : type-trunc-Set (Id a a) → Prop _ + P : type-trunc-Set (a ＝ a) → Prop _ P = pr1 (center ℙ) compute-P : @@ -521,7 +521,7 @@ has-finitely-many-connected-components-Σ-is-0-connected {A = A} {B} C H K = ( unit-trunc-Set y')))) f : type-hom-Prop - ( trunc-Prop (Σ (type-trunc-Set (Id a a)) (type-Prop ∘ P))) + ( trunc-Prop (Σ (type-trunc-Set (a ＝ a)) (type-Prop ∘ P))) ( mere-eq-Prop {A = Σ A B} (a , y) (a , y')) f t = apply-universal-property-trunc-Prop t @@ -542,7 +542,7 @@ has-finitely-many-connected-components-Σ-is-0-connected {A = A} {B} C H K = ( v) e : mere-eq {A = Σ A B} (a , y) (a , y') ≃ - type-trunc-Prop (Σ (type-trunc-Set (Id a a)) (type-Prop ∘ P)) + type-trunc-Prop (Σ (type-trunc-Set (a ＝ a)) (type-Prop ∘ P)) e = equiv-iff ( mere-eq-Prop (a , y) (a , y')) From 1ca93aaef86bd5ccc56e3fda86ec13ad02ea2819 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 1 Sep 2025 18:55:31 +0200 Subject: [PATCH 08/16] Ids matching `\(Id ` and some fixes --- .../orbits-permutations.lagda.md | 27 ++++---- .../transpositions.lagda.md | 8 +-- .../equality-coproduct-types.lagda.md | 8 +-- src/graph-theory/vertex-covers.lagda.md | 2 +- .../multiplication-matrices.lagda.md | 64 +++++++++---------- .../free-loops.lagda.md | 2 +- ...tations-complete-undirected-graph.lagda.md | 22 +++---- 7 files changed, 65 insertions(+), 68 deletions(-) diff --git a/src/finite-group-theory/orbits-permutations.lagda.md b/src/finite-group-theory/orbits-permutations.lagda.md index 9f4dc02c18..ca95aa0d47 100644 --- a/src/finite-group-theory/orbits-permutations.lagda.md +++ b/src/finite-group-theory/orbits-permutations.lagda.md @@ -510,8 +510,8 @@ module _ ( h : Fin n ≃ type-Type-With-Cardinality-ℕ n X) (a b : type-Type-With-Cardinality-ℕ n X) → is-decidable - ( Σ ℕ (λ m → (m ≤-ℕ n) × (Id (iterate m (map-equiv f) a) b))) → - is-decidable (Σ ℕ (λ m → Id (iterate m (map-equiv f) a) b)) + ( Σ ℕ (λ m → (m ≤-ℕ n) × (iterate m (map-equiv f) a ＝ b))) → + is-decidable (Σ ℕ (λ m → iterate m (map-equiv f) a ＝ b)) is-decidable-iterate-is-decidable-bounded h a b (inl p) = inl (pair (pr1 p) (pr2 (pr2 p))) is-decidable-iterate-is-decidable-bounded h a b (inr np) = @@ -759,8 +759,8 @@ module _ induction-cases-equal-iterate-transposition (inr s) = equal-iterate-transposition x g C F Ind k (inr (Ind k p s)) cases-equal-iterate-transposition : - is-decidable (Id (iterate (succ-ℕ k) (map-equiv g) x) a) → - is-decidable (Id (iterate (succ-ℕ k) (map-equiv g) x) b) → + is-decidable (iterate (succ-ℕ k) (map-equiv g) x ＝ a) → + is-decidable (iterate (succ-ℕ k) (map-equiv g) x ＝ b) → Id ( iterate (succ-ℕ k) (map-equiv (composition-transposition-a-b g)) x) ( iterate (succ-ℕ k) (map-equiv g) x) @@ -952,14 +952,13 @@ module _ cases-equal-iterate-transposition-a (inl s) = inl s cases-equal-iterate-transposition-a (inr s) = inr (pair s ineq) lemma2 : - ( pa : Σ ℕ (λ k → Id (iterate k (map-equiv g) a) b)) → - is-decidable (Id (pr1 (minimal-element-iterate g a b pa)) zero-ℕ) → - Id - ( iterate - ( pr1 (minimal-element-iterate g a b pa)) - ( map-equiv (composition-transposition-a-b g)) - ( a)) - ( a) + ( pa : Σ ℕ (λ k → iterate k (map-equiv g) a ＝ b)) → + is-decidable (pr1 (minimal-element-iterate g a b pa) ＝ zero-ℕ) → + iterate + ( pr1 (minimal-element-iterate g a b pa)) + ( map-equiv (composition-transposition-a-b g)) + ( a) ＝ + a lemma2 pa (inl p) = ex-falso ( np @@ -1231,7 +1230,7 @@ module _ ( refl) where cases-lemma2 : - is-decidable (Id (pr1 (minimal-element-iterate-2-a-b g pa)) zero-ℕ) → + is-decidable (pr1 (minimal-element-iterate-2-a-b g pa) ＝ zero-ℕ) → (c : ( Id ( iterate @@ -1245,7 +1244,7 @@ module _ ( map-equiv g) ( x)) ( b))) → - Id c (pr1 (pr2 (minimal-element-iterate-2-a-b g pa))) → + ( c ＝ pr1 (pr2 (minimal-element-iterate-2-a-b g pa))) → ( sim-equivalence-relation ( same-orbits-permutation-count ( composition-transposition-a-b g)) diff --git a/src/finite-group-theory/transpositions.lagda.md b/src/finite-group-theory/transpositions.lagda.md index b644320bc3..7972e2e6c2 100644 --- a/src/finite-group-theory/transpositions.lagda.md +++ b/src/finite-group-theory/transpositions.lagda.md @@ -479,13 +479,13 @@ module _ type-Decidable-Prop (pr1 Y x) → type-Decidable-Prop (pr1 Y y) → is-decidable (pr1 two-elements-transposition ＝ x) → - is-decidable (Id (pr1 (pr2 two-elements-transposition)) x) → + is-decidable (pr1 (pr2 two-elements-transposition) ＝ x) → is-decidable (pr1 two-elements-transposition ＝ y) → - is-decidable (Id (pr1 (pr2 two-elements-transposition)) y) → + is-decidable (pr1 (pr2 two-elements-transposition) ＝ y) → ( ( pr1 two-elements-transposition ＝ x) × - ( Id (pr1 (pr2 two-elements-transposition)) y)) + + ( pr1 (pr2 two-elements-transposition) ＝ y)) + ( ( pr1 two-elements-transposition ＝ y) × - ( Id (pr1 (pr2 two-elements-transposition)) x)) + ( pr1 (pr2 two-elements-transposition) ＝ x)) cases-eq-two-elements-transposition x y np p1 p2 (inl q) r s (inl u) = inl (pair q u) cases-eq-two-elements-transposition x y np p1 p2 (inl q) r s (inr nu) = diff --git a/src/foundation/equality-coproduct-types.lagda.md b/src/foundation/equality-coproduct-types.lagda.md index 1728220e3f..591c95e1f6 100644 --- a/src/foundation/equality-coproduct-types.lagda.md +++ b/src/foundation/equality-coproduct-types.lagda.md @@ -161,11 +161,11 @@ module _ pr1 compute-Eq-coproduct-inl-inr = map-compute-Eq-coproduct-inl-inr pr2 compute-Eq-coproduct-inl-inr = is-equiv-map-compute-Eq-coproduct-inl-inr - compute-eq-coproduct-inl-inr : Id {A = A + B} (inl x) (inr y) ≃ empty + compute-eq-coproduct-inl-inr : (inl x ＝ inr y) ≃ empty compute-eq-coproduct-inl-inr = compute-Eq-coproduct-inl-inr ∘e extensionality-coproduct (inl x) (inr y) - is-empty-eq-coproduct-inl-inr : is-empty (Id {A = A + B} (inl x) (inr y)) + is-empty-eq-coproduct-inl-inr : is-empty (inl x ＝ inr y) is-empty-eq-coproduct-inl-inr = map-equiv compute-eq-coproduct-inl-inr module _ @@ -184,11 +184,11 @@ module _ pr1 compute-Eq-coproduct-inr-inl = map-compute-Eq-coproduct-inr-inl pr2 compute-Eq-coproduct-inr-inl = is-equiv-map-compute-Eq-coproduct-inr-inl - compute-eq-coproduct-inr-inl : Id {A = A + B} (inr x) (inl y) ≃ empty + compute-eq-coproduct-inr-inl : (inr x ＝ inl y) ≃ empty compute-eq-coproduct-inr-inl = compute-Eq-coproduct-inr-inl ∘e extensionality-coproduct (inr x) (inl y) - is-empty-eq-coproduct-inr-inl : is-empty (Id {A = A + B} (inr x) (inl y)) + is-empty-eq-coproduct-inr-inl : is-empty (inr x ＝ inl y) is-empty-eq-coproduct-inr-inl = map-equiv compute-eq-coproduct-inr-inl module _ diff --git a/src/graph-theory/vertex-covers.lagda.md b/src/graph-theory/vertex-covers.lagda.md index 4cd6e60929..f83c492312 100644 --- a/src/graph-theory/vertex-covers.lagda.md +++ b/src/graph-theory/vertex-covers.lagda.md @@ -40,7 +40,7 @@ vertex-cover G = edge-Undirected-Graph G p → type-trunc-Prop ( Σ (vertex-Undirected-Graph G) - ( λ x → is-in-unordered-pair p x × c x ＝ inr star))) + ( λ x → is-in-unordered-pair p x × (c x ＝ inr star)))) ``` ## External links diff --git a/src/linear-algebra/multiplication-matrices.lagda.md b/src/linear-algebra/multiplication-matrices.lagda.md index 7a3d7ce025..7d30a008b0 100644 --- a/src/linear-algebra/multiplication-matrices.lagda.md +++ b/src/linear-algebra/multiplication-matrices.lagda.md @@ -45,9 +45,8 @@ mul-transpose : {addK : K → K → K} {mulK : K → K → K} {zero : K} → ((x y : K) → mulK x y ＝ mulK y x) → (a : Mat K m n) → (b : Mat K n p) → - Id - (transpose (mul-Mat mulK addK zero a b)) - (mul-Mat mulK addK zero (transpose b) (transpose a)) + transpose (mul-Mat mulK addK zero a b) ＝ + mul-Mat mulK addK zero (transpose b) (transpose a) mul-transpose mulK-comm empty-tuple b = {!!} mul-transpose mulK-comm (a ∷ as) b = {!!} -} @@ -71,15 +70,18 @@ module _ left-distributive-tuple-matrix : {n m : ℕ} → - ({l : ℕ} → Id (diagonal-product {n = l} zero) - (add-tuple addK (diagonal-product zero) (diagonal-product zero))) → - ((x y z : K) → (Id (mulK x (addK y z)) (addK (mulK x y) (mulK x z)))) → + ( {l : ℕ} → + diagonal-product {n = l} zero ＝ + add-tuple addK (diagonal-product zero) (diagonal-product zero)) → + ((x y z : K) → (mulK x (addK y z) ＝ addK (mulK x y) (mulK x z))) → ((x y : K) → addK x y ＝ addK y x) → - ((x y z : K) → Id (addK x (addK y z)) (addK (addK x y) z)) → + ((x y z : K) → addK x (addK y z) ＝ addK (addK x y) z) → (a : tuple K n) (b : Mat K n m) (c : Mat K n m) → - Id (mul-tuple-matrix mulK addK zero a (add-Mat addK b c)) - (add-tuple addK (mul-tuple-matrix mulK addK zero a b) - (mul-tuple-matrix mulK addK zero a c)) + ( mul-tuple-matrix mulK addK zero a (add-Mat addK b c)) ＝ + ( add-tuple + ( addK) + ( mul-tuple-matrix mulK addK zero a b) + ( mul-tuple-matrix mulK addK zero a c)) left-distributive-tuple-matrix id-tuple _ _ _ empty-tuple empty-tuple empty-tuple = id-tuple left-distributive-tuple-matrix @@ -97,8 +99,8 @@ module _ where lemma-shuffle : {n : ℕ} → {x y z w : tuple K n} → - Id (add-tuple addK (add-tuple addK x y) (add-tuple addK z w)) - (add-tuple addK (add-tuple addK x z) (add-tuple addK y w)) + add-tuple addK (add-tuple addK x y) (add-tuple addK z w) ＝ + add-tuple addK (add-tuple addK x z) (add-tuple addK y w) lemma-shuffle {x = x} {y = y} {z = z} {w = w} = associative-add-tuples {zero = zero} addK-associative (add-tuple addK x y) z w ∙ (commutative-add-tuples @@ -119,16 +121,14 @@ module _ left-distributive-matrices : {n m p : ℕ} → ({l : ℕ} → - Id - (diagonal-product {n = l} zero) - (add-tuple addK (diagonal-product zero) (diagonal-product zero))) → - ((x y z : K) → (Id (mulK x (addK y z)) (addK (mulK x y) (mulK x z)))) → + diagonal-product {n = l} zero ＝ + add-tuple addK (diagonal-product zero) (diagonal-product zero)) → + ((x y z : K) → mulK x (addK y z) ＝ addK (mulK x y) (mulK x z)) → ((x y : K) → addK x y ＝ addK y x) → - ((x y z : K) → Id (addK x (addK y z)) (addK (addK x y) z)) → + ((x y z : K) → addK x (addK y z) = addK (addK x y) z) → (a : Mat K m n) (b : Mat K n p) (c : Mat K n p) → - Id (mul-Mat mulK addK zero a (add-Mat addK b c)) - (add-Mat addK (mul-Mat mulK addK zero a b) - (mul-Mat mulK addK zero a c)) + ( mul-Mat mulK addK zero a (add-Mat addK b c)) ＝ + ( add-Mat addK (mul-Mat mulK addK zero a b) (mul-Mat mulK addK zero a c)) left-distributive-matrices _ _ _ _ empty-tuple _ _ = refl left-distributive-matrices id-tuple k-distr addK-comm addK-associative (a ∷ as) b c = (ap (λ r → r ∷ mul-Mat mulK addK zero as (add-Mat addK b c)) @@ -144,16 +144,14 @@ module _ right-distributive-matrices : {n m p : ℕ} → ({l : ℕ} → - Id - (diagonal-product {n = l} zero) - (add-tuple addK (diagonal-product zero) (diagonal-product zero))) → - ((x y z : K) → (Id (mulK (addK x y) z) (addK (mulK x z) (mulK y z)))) → + diagonal-product {n = l} zero ＝ + add-tuple addK (diagonal-product zero) (diagonal-product zero)) → + ((x y z : K) → mulK (addK x y) z ＝ addK (mulK x z) (mulK y z)) → ((x y : K) → addK x y ＝ addK y x) → - ((x y z : K) → Id (addK x (addK y z)) (addK (addK x y) z)) → + ((x y z : K) → addK x (addK y z) ＝ addK (addK x y) z) → (b : Mat K n p) (c : Mat K n p) (d : Mat K p m) → - Id (mul-Mat mulK addK zero (add-Mat addK b c) d) - (add-Mat addK (mul-Mat mulK addK zero b d) - (mul-Mat mulK addK zero c d)) + mul-Mat mulK addK zero (add-Mat addK b c) d ＝ + add-Mat addK (mul-Mat mulK addK zero b d) (mul-Mat mulK addK zero c d) right-distributive-matrices _ _ _ _ empty-tuple empty-tuple _ = refl right-distributive-matrices {p = .zero-ℕ} @@ -166,11 +164,11 @@ module _ TODO: associativity associative-mul-matrices : - {l : Level} {K : UU l} {n m p q : ℕ} → - {addK : K → K → K} {mulK : K → K → K} {zero : K} → - (x : Mat K m n) → (y : Mat K n p) → (z : Mat K p q) → - Id (mul-Mat mulK addK zero x (mul-Mat mulK addK zero y z)) - (mul-Mat mulK addK zero (mul-Mat mulK addK zero x y) z) + {l : Level} {K : UU l} {n m p q : ℕ} → + {addK : K → K → K} {mulK : K → K → K} {zero : K} → + (x : Mat K m n) → (y : Mat K n p) → (z : Mat K p q) → + mul-Mat mulK addK zero x (mul-Mat mulK addK zero y z) ＝ + mul-Mat mulK addK zero (mul-Mat mulK addK zero x y) z associative-mul-matrices x y z = {!!} -} ``` diff --git a/src/synthetic-homotopy-theory/free-loops.lagda.md b/src/synthetic-homotopy-theory/free-loops.lagda.md index 792a2cb3ba..29f4024736 100644 --- a/src/synthetic-homotopy-theory/free-loops.lagda.md +++ b/src/synthetic-homotopy-theory/free-loops.lagda.md @@ -145,7 +145,7 @@ module _ ( is-torsorial-Id y) ( pair y refl) ( is-contr-is-equiv' - ( Σ (Id (tr P (loop-free-loop α) y) y) (λ p' → p ＝ p')) + ( Σ (tr P (loop-free-loop α) y ＝ y) (λ p' → p ＝ p')) ( tot (λ p' α → right-unit ∙ α)) ( is-equiv-tot-is-fiberwise-equiv ( λ p' → is-equiv-concat right-unit p')) diff --git a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md index b0fcabea91..83ce2b93e3 100644 --- a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md +++ b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md @@ -778,7 +778,7 @@ module _ ( Y))))) → is-decidable (pr1 two-elements ＝ i) → is-decidable (pr1 two-elements ＝ j) → - is-decidable (Id (pr1 (pr2 two-elements)) i) → + is-decidable (pr1 (pr2 two-elements) ＝ i) → Σ X (λ z → type-Decidable-Prop (pr1 Y z)) cases-orientation-two-elements-count i j Y (pair x (pair y (pair np' P))) (inl q) r s = @@ -1455,8 +1455,8 @@ module _ ( np)))))) → (q : is-decidable (pr1 two-elements ＝ i)) → (r : is-decidable (pr1 two-elements ＝ j)) → - (s : is-decidable (Id (pr1 (pr2 two-elements)) i)) → - (t : is-decidable (Id (pr1 (pr2 two-elements)) j)) → + (s : is-decidable (pr1 (pr2 two-elements) ＝ i)) → + (t : is-decidable (pr1 (pr2 two-elements) ＝ j)) → Id ( pr1 ( cases-orientation-two-elements-count i j @@ -1594,8 +1594,8 @@ module _ two-elements-transposition eX Y ＝ two-elements → is-decidable (pr1 two-elements ＝ i) → is-decidable (pr1 two-elements ＝ j) → - is-decidable (Id (pr1 (pr2 two-elements)) i) → - is-decidable (Id (pr1 (pr2 two-elements)) j) → + is-decidable (pr1 (pr2 two-elements) ＝ i) → + is-decidable (pr1 (pr2 two-elements) ＝ j) → Id ( pr1 ( orientation-aut-count @@ -1829,8 +1829,8 @@ module _ two-elements-transposition eX Y ＝ two-elements → is-decidable (pr1 two-elements ＝ i) → is-decidable (pr1 two-elements ＝ j) → - is-decidable (Id (pr1 (pr2 two-elements)) i) → - is-decidable (Id (pr1 (pr2 two-elements)) j) → + is-decidable (pr1 (pr2 two-elements) ＝ i) → + is-decidable (pr1 (pr2 two-elements) ＝ j) → Id ( pr1 ( orientation-aut-count @@ -2256,8 +2256,8 @@ module _ two-elements-transposition eX Y ＝ two-elements → is-decidable (pr1 two-elements ＝ i) → is-decidable (pr1 two-elements ＝ j) → - is-decidable (Id (pr1 (pr2 two-elements)) i) → - is-decidable (Id (pr1 (pr2 two-elements)) j) → + is-decidable (pr1 (pr2 two-elements) ＝ i) → + is-decidable (pr1 (pr2 two-elements) ＝ j) → Id ( pr1 ( map-orientation-complete-undirected-graph-equiv @@ -2872,8 +2872,8 @@ module _ Id two-elements (two-elements-transposition eX (pr1 T)) → is-decidable (pr1 two-elements ＝ i) → is-decidable (pr1 two-elements ＝ j) → - is-decidable (Id (pr1 (pr2 two-elements)) i) → - is-decidable (Id (pr1 (pr2 two-elements)) j) → + is-decidable (pr1 (pr2 two-elements) ＝ i) → + is-decidable (pr1 (pr2 two-elements) ＝ j) → Id ( standard-2-Element-Decidable-Subtype ( has-decidable-equality-count eX) From cc172dbd23aac996542229ff7604552dd541bc89 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 1 Sep 2025 19:10:48 +0200 Subject: [PATCH 09/16] another Id pattern --- src/commutative-algebra/euclidean-domains.lagda.md | 2 +- src/elementary-number-theory/pisano-periods.lagda.md | 8 +++++--- .../universal-property-integers.lagda.md | 10 +++++----- .../dependent-products-finite-rings.lagda.md | 4 ++-- src/finite-algebra/finite-rings.lagda.md | 8 ++++---- .../abstract-quaternion-group.lagda.md | 6 +++--- .../delooping-sign-homomorphism.lagda.md | 2 +- src/finite-group-theory/orbits-permutations.lagda.md | 8 ++++---- src/finite-group-theory/transpositions.lagda.md | 2 +- .../cartesian-products-abelian-groups.lagda.md | 4 ++-- src/group-theory/cartesian-products-groups.lagda.md | 4 ++-- src/group-theory/groups.lagda.md | 6 +++--- src/group-theory/homomorphisms-abelian-groups.lagda.md | 4 ++-- src/group-theory/homomorphisms-groups.lagda.md | 4 ++-- src/group-theory/homomorphisms-semigroups.lagda.md | 4 ++-- .../precategory-of-orbits-monoid-actions.lagda.md | 6 +++--- .../cartesian-products-higher-groups.lagda.md | 4 ++-- src/higher-group-theory/higher-groups.lagda.md | 4 ++-- src/linear-algebra/left-modules-rings.lagda.md | 6 +++--- src/linear-algebra/matrices-on-rings.lagda.md | 4 ++-- src/linear-algebra/right-modules-rings.lagda.md | 6 +++--- .../tuples-on-euclidean-domains.lagda.md | 4 ++-- src/linear-algebra/tuples-on-rings.lagda.md | 4 ++-- src/lists/concatenation-lists.lagda.md | 4 ++-- src/lists/flattening-lists.lagda.md | 2 +- src/lists/lists.lagda.md | 8 ++++---- src/lists/reversing-lists.lagda.md | 2 +- src/order-theory/order-preserving-maps-posets.lagda.md | 4 ++-- .../order-preserving-maps-preorders.lagda.md | 4 ++-- src/ring-theory/dependent-products-rings.lagda.md | 4 ++-- src/ring-theory/products-rings.lagda.md | 4 ++-- src/species/morphisms-finite-species.lagda.md | 4 ++-- src/species/morphisms-species-of-types.lagda.md | 4 ++-- src/synthetic-homotopy-theory/interval-type.lagda.md | 2 +- .../classical-finite-types.lagda.md | 4 ++-- .../cyclic-finite-types.lagda.md | 4 ++-- .../orientations-complete-undirected-graph.lagda.md | 2 +- 37 files changed, 84 insertions(+), 82 deletions(-) diff --git a/src/commutative-algebra/euclidean-domains.lagda.md b/src/commutative-algebra/euclidean-domains.lagda.md index ae580388d3..54b46a93f5 100644 --- a/src/commutative-algebra/euclidean-domains.lagda.md +++ b/src/commutative-algebra/euclidean-domains.lagda.md @@ -72,7 +72,7 @@ is-euclidean-valuation R v = ( is-nonzero-Integral-Domain R y) → Σ ( type-Integral-Domain R × type-Integral-Domain R) ( λ (q , r) → - ( Id x (add-Integral-Domain R (mul-Integral-Domain R q y) r)) × + ( x ＝ add-Integral-Domain R (mul-Integral-Domain R q y) r) × ( is-zero-Integral-Domain R r + ( v r <-ℕ v y))) ``` diff --git a/src/elementary-number-theory/pisano-periods.lagda.md b/src/elementary-number-theory/pisano-periods.lagda.md index 8879673098..dd93f4af79 100644 --- a/src/elementary-number-theory/pisano-periods.lagda.md +++ b/src/elementary-number-theory/pisano-periods.lagda.md @@ -168,8 +168,10 @@ is-lower-bound-pisano-period k = pr2 (pr2 (minimal-ordered-repetition-fibonacci-pair-Fin k)) cases-is-repetition-of-zero-pisano-period : - (k x y : ℕ) → Id (pr1 (is-ordered-repetition-pisano-period k)) x → - pisano-period k ＝ y → is-zero-ℕ x + (k x y : ℕ) → + pr1 (is-ordered-repetition-pisano-period k) ＝ x → + pisano-period k ＝ y → + is-zero-ℕ x cases-is-repetition-of-zero-pisano-period k zero-ℕ y p q = refl cases-is-repetition-of-zero-pisano-period k (succ-ℕ x) zero-ℕ p q = ex-falso @@ -205,7 +207,7 @@ is-repetition-of-zero-pisano-period k = compute-fibonacci-pair-Fin-pisano-period : (k : ℕ) → - Id (fibonacci-pair-Fin k (pisano-period k)) (fibonacci-pair-Fin k zero-ℕ) + fibonacci-pair-Fin k (pisano-period k) ＝ fibonacci-pair-Fin k zero-ℕ compute-fibonacci-pair-Fin-pisano-period k = ( inv (pr2 (pr2 (is-ordered-repetition-pisano-period k)))) ∙ ( ap (fibonacci-pair-Fin k) (is-repetition-of-zero-pisano-period k)) diff --git a/src/elementary-number-theory/universal-property-integers.lagda.md b/src/elementary-number-theory/universal-property-integers.lagda.md index e12b4b7012..d39b4c7a7e 100644 --- a/src/elementary-number-theory/universal-property-integers.lagda.md +++ b/src/elementary-number-theory/universal-property-integers.lagda.md @@ -65,7 +65,7 @@ abstract compute-succ-elim-ℤ : { l1 : Level} (P : ℤ → UU l1) ( p0 : P zero-ℤ) (pS : (k : ℤ) → (P k) ≃ (P (succ-ℤ k))) (k : ℤ) → - Id (elim-ℤ P p0 pS (succ-ℤ k)) (map-equiv (pS k) (elim-ℤ P p0 pS k)) + elim-ℤ P p0 pS (succ-ℤ k) ＝ map-equiv (pS k) (elim-ℤ P p0 pS k) compute-succ-elim-ℤ P p0 pS (inl zero-ℕ) = inv ( is-section-map-inv-is-equiv @@ -86,7 +86,7 @@ ELIM-ℤ P p0 pS = Σ ( (k : ℤ) → P k) ( λ f → ( ( f zero-ℤ ＝ p0) × - ( (k : ℤ) → Id (f (succ-ℤ k)) ((map-equiv (pS k)) (f k))))) + ( (k : ℤ) → f (succ-ℤ k) ＝ map-equiv (pS k) (f k)))) Elim-ℤ : { l1 : Level} (P : ℤ → UU l1) @@ -157,15 +157,15 @@ abstract ( pair (pr1 s) refl-htpy) ( is-torsorial-Eq-structure ( is-contr-is-equiv' - ( Σ (pr1 s zero-ℤ ＝ p0) (λ α → Id α (pr1 (pr2 s)))) + ( Σ (pr1 s zero-ℤ ＝ p0) (λ α → α ＝ pr1 (pr2 s))) ( tot (λ α → right-transpose-eq-concat refl α (pr1 (pr2 s)))) ( is-equiv-tot-is-fiberwise-equiv ( λ α → is-equiv-right-transpose-eq-concat refl α (pr1 (pr2 s)))) ( is-torsorial-Id' (pr1 (pr2 s)))) ( pair (pr1 (pr2 s)) (inv (right-inv (pr1 (pr2 s))))) ( is-contr-is-equiv' - ( Σ ( ( k : ℤ) → Id (pr1 s (succ-ℤ k)) (pr1 (pS k) (pr1 s k))) - ( λ β → β ~ (pr2 (pr2 s)))) + ( Σ ( ( k : ℤ) → pr1 s (succ-ℤ k) ＝ pr1 (pS k) (pr1 s k)) + ( λ β → β ~ pr2 (pr2 s))) ( tot (λ β → right-transpose-htpy-concat refl-htpy β (pr2 (pr2 s)))) ( is-equiv-tot-is-fiberwise-equiv ( λ β → diff --git a/src/finite-algebra/dependent-products-finite-rings.lagda.md b/src/finite-algebra/dependent-products-finite-rings.lagda.md index 712477404d..34992d7389 100644 --- a/src/finite-algebra/dependent-products-finite-rings.lagda.md +++ b/src/finite-algebra/dependent-products-finite-rings.lagda.md @@ -110,13 +110,13 @@ module _ left-inverse-law-add-Π-Finite-Ring : (x : type-Π-Finite-Ring) → - Id (add-Π-Finite-Ring (neg-Π-Finite-Ring x) x) zero-Π-Finite-Ring + add-Π-Finite-Ring (neg-Π-Finite-Ring x) x ＝ zero-Π-Finite-Ring left-inverse-law-add-Π-Finite-Ring = left-inverse-law-add-Π-Ring (type-Finite-Type I) (ring-Finite-Ring ∘ A) right-inverse-law-add-Π-Finite-Ring : (x : type-Π-Finite-Ring) → - Id (add-Π-Finite-Ring x (neg-Π-Finite-Ring x)) zero-Π-Finite-Ring + add-Π-Finite-Ring x (neg-Π-Finite-Ring x) ＝ zero-Π-Finite-Ring right-inverse-law-add-Π-Finite-Ring = right-inverse-law-add-Π-Ring (type-Finite-Type I) (ring-Finite-Ring ∘ A) diff --git a/src/finite-algebra/finite-rings.lagda.md b/src/finite-algebra/finite-rings.lagda.md index ba18843e7c..fb005ba1ea 100644 --- a/src/finite-algebra/finite-rings.lagda.md +++ b/src/finite-algebra/finite-rings.lagda.md @@ -255,13 +255,13 @@ module _ left-inverse-law-add-Finite-Ring : (x : type-Finite-Ring R) → - Id (add-Finite-Ring R (neg-Finite-Ring x) x) (zero-Finite-Ring R) + add-Finite-Ring R (neg-Finite-Ring x) x ＝ zero-Finite-Ring R left-inverse-law-add-Finite-Ring = left-inverse-law-add-Ring (ring-Finite-Ring R) right-inverse-law-add-Finite-Ring : (x : type-Finite-Ring R) → - Id (add-Finite-Ring R x (neg-Finite-Ring x)) (zero-Finite-Ring R) + add-Finite-Ring R x (neg-Finite-Ring x) ＝ zero-Finite-Ring R right-inverse-law-add-Finite-Ring = right-inverse-law-add-Ring (ring-Finite-Ring R) @@ -361,13 +361,13 @@ module _ left-zero-law-mul-Finite-Ring : (x : type-Finite-Ring R) → - Id (mul-Finite-Ring R (zero-Finite-Ring R) x) (zero-Finite-Ring R) + mul-Finite-Ring R (zero-Finite-Ring R) x ＝ zero-Finite-Ring R left-zero-law-mul-Finite-Ring = left-zero-law-mul-Ring (ring-Finite-Ring R) right-zero-law-mul-Finite-Ring : (x : type-Finite-Ring R) → - Id (mul-Finite-Ring R x (zero-Finite-Ring R)) (zero-Finite-Ring R) + mul-Finite-Ring R x (zero-Finite-Ring R) ＝ zero-Finite-Ring R right-zero-law-mul-Finite-Ring = right-zero-law-mul-Ring (ring-Finite-Ring R) ``` diff --git a/src/finite-group-theory/abstract-quaternion-group.lagda.md b/src/finite-group-theory/abstract-quaternion-group.lagda.md index 91d626215e..dba01d4504 100644 --- a/src/finite-group-theory/abstract-quaternion-group.lagda.md +++ b/src/finite-group-theory/abstract-quaternion-group.lagda.md @@ -147,7 +147,7 @@ right-unit-law-mul-Q8 k-Q8 = refl right-unit-law-mul-Q8 -k-Q8 = refl associative-mul-Q8 : - (x y z : Q8) → Id (mul-Q8 (mul-Q8 x y) z) (mul-Q8 x (mul-Q8 y z)) + (x y z : Q8) → mul-Q8 (mul-Q8 x y) z ＝ mul-Q8 x (mul-Q8 y z) associative-mul-Q8 e-Q8 e-Q8 e-Q8 = refl associative-mul-Q8 e-Q8 e-Q8 -e-Q8 = refl associative-mul-Q8 e-Q8 e-Q8 i-Q8 = refl @@ -661,7 +661,7 @@ associative-mul-Q8 -k-Q8 -k-Q8 -j-Q8 = refl associative-mul-Q8 -k-Q8 -k-Q8 k-Q8 = refl associative-mul-Q8 -k-Q8 -k-Q8 -k-Q8 = refl -left-inverse-law-mul-Q8 : (x : Q8) → Id (mul-Q8 (inv-Q8 x) x) e-Q8 +left-inverse-law-mul-Q8 : (x : Q8) → mul-Q8 (inv-Q8 x) x ＝ e-Q8 left-inverse-law-mul-Q8 e-Q8 = refl left-inverse-law-mul-Q8 -e-Q8 = refl left-inverse-law-mul-Q8 i-Q8 = refl @@ -671,7 +671,7 @@ left-inverse-law-mul-Q8 -j-Q8 = refl left-inverse-law-mul-Q8 k-Q8 = refl left-inverse-law-mul-Q8 -k-Q8 = refl -right-inverse-law-mul-Q8 : (x : Q8) → Id (mul-Q8 x (inv-Q8 x)) e-Q8 +right-inverse-law-mul-Q8 : (x : Q8) → mul-Q8 x (inv-Q8 x) ＝ e-Q8 right-inverse-law-mul-Q8 e-Q8 = refl right-inverse-law-mul-Q8 -e-Q8 = refl right-inverse-law-mul-Q8 i-Q8 = refl diff --git a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md index 8bfee17597..00e93b169c 100644 --- a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md +++ b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md @@ -366,7 +366,7 @@ module _ ( eX : mere-equiv (Fin (n +ℕ 2)) X) ( eY : mere-equiv (Fin (n +ℕ 2)) Y) ( p : X ＝ Y) → - ( Id (tr (mere-equiv (Fin (n +ℕ 2))) p eX) eY) → + ( tr (mere-equiv (Fin (n +ℕ 2))) p eX ＝ eY) → ( sX : is-set X) ( sY : is-set Y) → coherence-square-maps diff --git a/src/finite-group-theory/orbits-permutations.lagda.md b/src/finite-group-theory/orbits-permutations.lagda.md index ca95aa0d47..eaee934b81 100644 --- a/src/finite-group-theory/orbits-permutations.lagda.md +++ b/src/finite-group-theory/orbits-permutations.lagda.md @@ -86,7 +86,7 @@ module _ iso-iterative-groupoid-automorphism-Finite-Type : (x y : type-Finite-Type X) → UU l iso-iterative-groupoid-automorphism-Finite-Type x y = - Σ ℕ (λ n → Id (iterate n (map-equiv e) x) y) + Σ ℕ (λ n → iterate n (map-equiv e) x ＝ y) natural-isomorphism-iterative-groupoid-automorphism-Finite-Type : (x y : type-Finite-Type X) @@ -394,7 +394,7 @@ module _ mult-has-finite-orbits-permutation : (k : ℕ) → - Id (iterate (k *ℕ (pr1 has-finite-orbits-permutation)) (map-equiv f) a) a + iterate (k *ℕ (pr1 has-finite-orbits-permutation)) (map-equiv f) a ＝ a mult-has-finite-orbits-permutation zero-ℕ = refl mult-has-finite-orbits-permutation (succ-ℕ k) = ( iterate-add-ℕ @@ -1688,7 +1688,7 @@ module _ ( same-orbits-permutation-count (composition-transposition-a-b g)) ( T) ( b)) → - Id (h' (inv-h' T)) T + h' (inv-h' T) ＝ T retraction-h' T (inl Q) R = tr (λ w → @@ -2002,7 +2002,7 @@ module _ ( b)) ( refl))) section-h' : - (k : Fin (succ-ℕ (number-of-elements-count h))) → Id (inv-h' (h' k)) k + (k : Fin (succ-ℕ (number-of-elements-count h))) → inv-h' (h' k) ＝ k section-h' (inl k) = section-h'-inl k Q R ( is-decidable-is-in-equivalence-class-same-orbits-permutation diff --git a/src/finite-group-theory/transpositions.lagda.md b/src/finite-group-theory/transpositions.lagda.md index 7972e2e6c2..05d0d3f1f0 100644 --- a/src/finite-group-theory/transpositions.lagda.md +++ b/src/finite-group-theory/transpositions.lagda.md @@ -115,7 +115,7 @@ module _ (x : X) (d : is-decidable (is-in-2-Element-Decidable-Subtype P x)) (d' : is-decidable ( is-in-2-Element-Decidable-Subtype P (map-transposition' x d))) → - Id (map-transposition' (map-transposition' x d) d') x + map-transposition' (map-transposition' x d) d' ＝ x is-involution-map-transposition' x (inl p) (inl p') = ( ap ( λ y → map-transposition' (map-transposition' x (inl p)) (inl y)) diff --git a/src/group-theory/cartesian-products-abelian-groups.lagda.md b/src/group-theory/cartesian-products-abelian-groups.lagda.md index 3116d3e5c3..4d3e2dad34 100644 --- a/src/group-theory/cartesian-products-abelian-groups.lagda.md +++ b/src/group-theory/cartesian-products-abelian-groups.lagda.md @@ -86,12 +86,12 @@ module _ left-inverse-law-add-product-Ab : (x : type-product-Ab) → - Id (add-product-Ab (neg-product-Ab x) x) zero-product-Ab + add-product-Ab (neg-product-Ab x) x ＝ zero-product-Ab left-inverse-law-add-product-Ab = left-inverse-law-mul-Group group-product-Ab right-inverse-law-add-product-Ab : (x : type-product-Ab) → - Id (add-product-Ab x (neg-product-Ab x)) zero-product-Ab + add-product-Ab x (neg-product-Ab x) ＝ zero-product-Ab right-inverse-law-add-product-Ab = right-inverse-law-mul-Group group-product-Ab diff --git a/src/group-theory/cartesian-products-groups.lagda.md b/src/group-theory/cartesian-products-groups.lagda.md index 13001b5809..8077b4e5b9 100644 --- a/src/group-theory/cartesian-products-groups.lagda.md +++ b/src/group-theory/cartesian-products-groups.lagda.md @@ -81,13 +81,13 @@ module _ left-inverse-law-product-Group : (x : type-product-Group) → - Id (mul-product-Group (inv-product-Group x) x) unit-product-Group + mul-product-Group (inv-product-Group x) x ＝ unit-product-Group left-inverse-law-product-Group (pair x y) = eq-pair (left-inverse-law-mul-Group G x) (left-inverse-law-mul-Group H y) right-inverse-law-product-Group : (x : type-product-Group) → - Id (mul-product-Group x (inv-product-Group x)) unit-product-Group + mul-product-Group x (inv-product-Group x) ＝ unit-product-Group right-inverse-law-product-Group (pair x y) = eq-pair (right-inverse-law-mul-Group G x) (right-inverse-law-mul-Group H y) diff --git a/src/group-theory/groups.lagda.md b/src/group-theory/groups.lagda.md index bc860de881..1b52e49b88 100644 --- a/src/group-theory/groups.lagda.md +++ b/src/group-theory/groups.lagda.md @@ -183,11 +183,11 @@ module _ inv-Group = pr1 has-inverses-Group left-inverse-law-mul-Group : - (x : type-Group) → Id (mul-Group (inv-Group x) x) unit-Group + (x : type-Group) → mul-Group (inv-Group x) x ＝ unit-Group left-inverse-law-mul-Group = pr1 (pr2 has-inverses-Group) right-inverse-law-mul-Group : - (x : type-Group) → Id (mul-Group x (inv-Group x)) unit-Group + (x : type-Group) → mul-Group x (inv-Group x) ＝ unit-Group right-inverse-law-mul-Group = pr2 (pr2 has-inverses-Group) is-invertible-element-Group : @@ -439,7 +439,7 @@ module _ ```agda inv-inv-Group : - (x : type-Group G) → Id (inv-Group G (inv-Group G x)) x + (x : type-Group G) → inv-Group G (inv-Group G x) ＝ x inv-inv-Group x = is-injective-mul-Group ( inv-Group G x) diff --git a/src/group-theory/homomorphisms-abelian-groups.lagda.md b/src/group-theory/homomorphisms-abelian-groups.lagda.md index 628a5097b5..43fa154acc 100644 --- a/src/group-theory/homomorphisms-abelian-groups.lagda.md +++ b/src/group-theory/homomorphisms-abelian-groups.lagda.md @@ -185,13 +185,13 @@ associative-comp-hom-Ab A B C D = ```agda left-unit-law-comp-hom-Ab : { l1 l2 : Level} (A : Ab l1) (B : Ab l2) - ( f : hom-Ab A B) → Id (comp-hom-Ab A B B (id-hom-Ab B) f) f + ( f : hom-Ab A B) → comp-hom-Ab A B B (id-hom-Ab B) f ＝ f left-unit-law-comp-hom-Ab A B = left-unit-law-comp-hom-Semigroup (semigroup-Ab A) (semigroup-Ab B) right-unit-law-comp-hom-Ab : { l1 l2 : Level} (A : Ab l1) (B : Ab l2) - ( f : hom-Ab A B) → Id (comp-hom-Ab A A B f (id-hom-Ab A)) f + ( f : hom-Ab A B) → comp-hom-Ab A A B f (id-hom-Ab A) ＝ f right-unit-law-comp-hom-Ab A B = right-unit-law-comp-hom-Semigroup (semigroup-Ab A) (semigroup-Ab B) ``` diff --git a/src/group-theory/homomorphisms-groups.lagda.md b/src/group-theory/homomorphisms-groups.lagda.md index 08926c3883..8df523437e 100644 --- a/src/group-theory/homomorphisms-groups.lagda.md +++ b/src/group-theory/homomorphisms-groups.lagda.md @@ -199,7 +199,7 @@ module _ ```agda left-unit-law-comp-hom-Group : {l1 l2 : Level} (G : Group l1) (H : Group l2) (f : hom-Group G H) → - Id (comp-hom-Group G H H (id-hom-Group H) f) f + comp-hom-Group G H H (id-hom-Group H) f ＝ f left-unit-law-comp-hom-Group G H = left-unit-law-comp-hom-Semigroup ( semigroup-Group G) @@ -207,7 +207,7 @@ left-unit-law-comp-hom-Group G H = right-unit-law-comp-hom-Group : {l1 l2 : Level} (G : Group l1) (H : Group l2) (f : hom-Group G H) → - Id (comp-hom-Group G G H f (id-hom-Group G)) f + comp-hom-Group G G H f (id-hom-Group G) ＝ f right-unit-law-comp-hom-Group G H = right-unit-law-comp-hom-Semigroup ( semigroup-Group G) diff --git a/src/group-theory/homomorphisms-semigroups.lagda.md b/src/group-theory/homomorphisms-semigroups.lagda.md index 5bef1c713a..9f9bb485b1 100644 --- a/src/group-theory/homomorphisms-semigroups.lagda.md +++ b/src/group-theory/homomorphisms-semigroups.lagda.md @@ -224,7 +224,7 @@ module _ left-unit-law-comp-hom-Semigroup : { l1 l2 : Level} (G : Semigroup l1) (H : Semigroup l2) ( f : hom-Semigroup G H) → - Id ( comp-hom-Semigroup G H H (id-hom-Semigroup H) f) f + comp-hom-Semigroup G H H (id-hom-Semigroup H) f ＝ f left-unit-law-comp-hom-Semigroup G (pair (pair H is-set-H) (pair μ-H associative-H)) (pair f μ-f) = eq-htpy-hom-Semigroup G @@ -234,7 +234,7 @@ left-unit-law-comp-hom-Semigroup G right-unit-law-comp-hom-Semigroup : { l1 l2 : Level} (G : Semigroup l1) (H : Semigroup l2) ( f : hom-Semigroup G H) → - Id ( comp-hom-Semigroup G G H f (id-hom-Semigroup G)) f + comp-hom-Semigroup G G H f (id-hom-Semigroup G) ＝ f right-unit-law-comp-hom-Semigroup (pair (pair G is-set-G) (pair μ-G associative-G)) H (pair f μ-f) = eq-htpy-hom-Semigroup diff --git a/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md b/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md index c5eb7fc864..60b25923c5 100644 --- a/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md +++ b/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md @@ -53,7 +53,7 @@ module _ eq-hom-orbit-action-Monoid : {x y : type-action-Monoid M X} (f : hom-orbit-action-Monoid x y) → - Id (mul-action-Monoid M X (element-hom-orbit-action-Monoid f) x) y + mul-action-Monoid M X (element-hom-orbit-action-Monoid f) x ＝ y eq-hom-orbit-action-Monoid f = pr2 f htpy-hom-orbit-action-Monoid : @@ -165,14 +165,14 @@ module _ left-unit-law-comp-hom-orbit-action-Monoid : {x y : type-action-Monoid M X} (f : hom-orbit-action-Monoid x y) → - Id (comp-hom-orbit-action-Monoid (id-hom-orbit-action-Monoid y) f) f + comp-hom-orbit-action-Monoid (id-hom-orbit-action-Monoid y) f ＝ f left-unit-law-comp-hom-orbit-action-Monoid f = eq-htpy-hom-orbit-action-Monoid ( left-unit-law-mul-Monoid M (element-hom-orbit-action-Monoid f)) right-unit-law-comp-hom-orbit-action-Monoid : {x y : type-action-Monoid M X} (f : hom-orbit-action-Monoid x y) → - Id (comp-hom-orbit-action-Monoid f (id-hom-orbit-action-Monoid x)) f + comp-hom-orbit-action-Monoid f (id-hom-orbit-action-Monoid x) ＝ f right-unit-law-comp-hom-orbit-action-Monoid f = eq-htpy-hom-orbit-action-Monoid ( right-unit-law-mul-Monoid M (element-hom-orbit-action-Monoid f)) diff --git a/src/higher-group-theory/cartesian-products-higher-groups.lagda.md b/src/higher-group-theory/cartesian-products-higher-groups.lagda.md index 56e64a9ddd..920864f380 100644 --- a/src/higher-group-theory/cartesian-products-higher-groups.lagda.md +++ b/src/higher-group-theory/cartesian-products-higher-groups.lagda.md @@ -121,13 +121,13 @@ module _ left-inverse-law-mul-product-∞-Group : (x : type-product-∞-Group) → - Id (mul-product-∞-Group (inv-product-∞-Group x) x) unit-product-∞-Group + mul-product-∞-Group (inv-product-∞-Group x) x ＝ unit-product-∞-Group left-inverse-law-mul-product-∞-Group = left-inverse-law-mul-Ω classifying-pointed-type-product-∞-Group right-inverse-law-mul-product-∞-Group : (x : type-product-∞-Group) → - Id (mul-product-∞-Group x (inv-product-∞-Group x)) unit-product-∞-Group + mul-product-∞-Group x (inv-product-∞-Group x) ＝ unit-product-∞-Group right-inverse-law-mul-product-∞-Group = right-inverse-law-mul-Ω classifying-pointed-type-product-∞-Group ``` diff --git a/src/higher-group-theory/higher-groups.lagda.md b/src/higher-group-theory/higher-groups.lagda.md index 188b007dbb..0a27359d3b 100644 --- a/src/higher-group-theory/higher-groups.lagda.md +++ b/src/higher-group-theory/higher-groups.lagda.md @@ -138,12 +138,12 @@ module _ inv-∞-Group = inv-Ω classifying-pointed-type-∞-Group left-inverse-law-mul-∞-Group : - (x : type-∞-Group) → Id (mul-∞-Group (inv-∞-Group x) x) unit-∞-Group + (x : type-∞-Group) → mul-∞-Group (inv-∞-Group x) x ＝ unit-∞-Group left-inverse-law-mul-∞-Group = left-inverse-law-mul-Ω classifying-pointed-type-∞-Group right-inverse-law-mul-∞-Group : - (x : type-∞-Group) → Id (mul-∞-Group x (inv-∞-Group x)) unit-∞-Group + (x : type-∞-Group) → mul-∞-Group x (inv-∞-Group x) ＝ unit-∞-Group right-inverse-law-mul-∞-Group = right-inverse-law-mul-Ω classifying-pointed-type-∞-Group ``` diff --git a/src/linear-algebra/left-modules-rings.lagda.md b/src/linear-algebra/left-modules-rings.lagda.md index 82d977be40..2ac8a9a423 100644 --- a/src/linear-algebra/left-modules-rings.lagda.md +++ b/src/linear-algebra/left-modules-rings.lagda.md @@ -142,13 +142,13 @@ module _ left-unit-law-add-left-module-Ring : (x : type-left-module-Ring R M) → - Id (add-left-module-Ring R M (zero-left-module-Ring R M) x) x + add-left-module-Ring R M (zero-left-module-Ring R M) x ＝ x left-unit-law-add-left-module-Ring = left-unit-law-add-Ab (ab-left-module-Ring R M) right-unit-law-add-left-module-Ring : (x : type-left-module-Ring R M) → - Id (add-left-module-Ring R M x (zero-left-module-Ring R M)) x + add-left-module-Ring R M x (zero-left-module-Ring R M) ＝ x right-unit-law-add-left-module-Ring = right-unit-law-add-Ab (ab-left-module-Ring R M) ``` @@ -187,7 +187,7 @@ module _ abstract left-unit-law-mul-left-module-Ring : (x : type-left-module-Ring R M) → - Id ( mul-left-module-Ring R M (one-Ring R) x) x + mul-left-module-Ring R M (one-Ring R) x ＝ x left-unit-law-mul-left-module-Ring = htpy-eq-hom-Ab ( ab-left-module-Ring R M) diff --git a/src/linear-algebra/matrices-on-rings.lagda.md b/src/linear-algebra/matrices-on-rings.lagda.md index 63732106fb..30479f8f26 100644 --- a/src/linear-algebra/matrices-on-rings.lagda.md +++ b/src/linear-algebra/matrices-on-rings.lagda.md @@ -110,7 +110,7 @@ module _ left-unit-law-add-matrix-Ring : {m n : ℕ} (A : matrix-Ring R m n) → - Id (add-matrix-Ring R (zero-matrix-Ring R) A) A + add-matrix-Ring R (zero-matrix-Ring R) A ＝ A left-unit-law-add-matrix-Ring empty-tuple = refl left-unit-law-add-matrix-Ring (v ∷ A) = ap-binary _∷_ @@ -127,7 +127,7 @@ module _ right-unit-law-add-matrix-Ring : {m n : ℕ} (A : matrix-Ring R m n) → - Id (add-matrix-Ring R A (zero-matrix-Ring R)) A + add-matrix-Ring R A (zero-matrix-Ring R) ＝ A right-unit-law-add-matrix-Ring empty-tuple = refl right-unit-law-add-matrix-Ring (v ∷ A) = ap-binary _∷_ diff --git a/src/linear-algebra/right-modules-rings.lagda.md b/src/linear-algebra/right-modules-rings.lagda.md index 76b6e48da9..46e612a0ac 100644 --- a/src/linear-algebra/right-modules-rings.lagda.md +++ b/src/linear-algebra/right-modules-rings.lagda.md @@ -116,13 +116,13 @@ module _ left-unit-law-add-right-module-Ring : (x : type-right-module-Ring R M) → - Id (add-right-module-Ring R M (zero-right-module-Ring R M) x) x + add-right-module-Ring R M (zero-right-module-Ring R M) x ＝ x left-unit-law-add-right-module-Ring = left-unit-law-add-Ab (ab-right-module-Ring R M) right-unit-law-add-right-module-Ring : (x : type-right-module-Ring R M) → - Id (add-right-module-Ring R M x (zero-right-module-Ring R M)) x + add-right-module-Ring R M x (zero-right-module-Ring R M) ＝ x right-unit-law-add-right-module-Ring = right-unit-law-add-Ab (ab-right-module-Ring R M) ``` @@ -160,7 +160,7 @@ module _ left-unit-law-mul-right-module-Ring : (x : type-right-module-Ring R M) → - Id ( mul-right-module-Ring R M (one-Ring R) x) x + mul-right-module-Ring R M (one-Ring R) x ＝ x left-unit-law-mul-right-module-Ring = htpy-eq-hom-Ab ( ab-right-module-Ring R M) diff --git a/src/linear-algebra/tuples-on-euclidean-domains.lagda.md b/src/linear-algebra/tuples-on-euclidean-domains.lagda.md index 178f5e519a..ff3a592561 100644 --- a/src/linear-algebra/tuples-on-euclidean-domains.lagda.md +++ b/src/linear-algebra/tuples-on-euclidean-domains.lagda.md @@ -126,14 +126,14 @@ module _ left-unit-law-add-tuple-Euclidean-Domain : {n : ℕ} (v : tuple-Euclidean-Domain R n) → - Id (add-tuple-Euclidean-Domain R (zero-tuple-Euclidean-Domain R) v) v + add-tuple-Euclidean-Domain R (zero-tuple-Euclidean-Domain R) v ＝ v left-unit-law-add-tuple-Euclidean-Domain = left-unit-law-add-tuple-Commutative-Ring ( commutative-ring-Euclidean-Domain R) right-unit-law-add-tuple-Euclidean-Domain : {n : ℕ} (v : tuple-Euclidean-Domain R n) → - Id (add-tuple-Euclidean-Domain R v (zero-tuple-Euclidean-Domain R)) v + add-tuple-Euclidean-Domain R v (zero-tuple-Euclidean-Domain R) ＝ v right-unit-law-add-tuple-Euclidean-Domain = right-unit-law-add-tuple-Commutative-Ring ( commutative-ring-Euclidean-Domain R) diff --git a/src/linear-algebra/tuples-on-rings.lagda.md b/src/linear-algebra/tuples-on-rings.lagda.md index 4c745aeefd..177899f946 100644 --- a/src/linear-algebra/tuples-on-rings.lagda.md +++ b/src/linear-algebra/tuples-on-rings.lagda.md @@ -115,12 +115,12 @@ module _ where left-unit-law-add-tuple-Ring : - {n : ℕ} (v : tuple-Ring R n) → Id (add-tuple-Ring R (zero-tuple-Ring R) v) v + {n : ℕ} (v : tuple-Ring R n) → add-tuple-Ring R (zero-tuple-Ring R) v ＝ v left-unit-law-add-tuple-Ring = left-unit-law-add-tuple-Semiring (semiring-Ring R) right-unit-law-add-tuple-Ring : - {n : ℕ} (v : tuple-Ring R n) → Id (add-tuple-Ring R v (zero-tuple-Ring R)) v + {n : ℕ} (v : tuple-Ring R n) → add-tuple-Ring R v (zero-tuple-Ring R) ＝ v right-unit-law-add-tuple-Ring = right-unit-law-add-tuple-Semiring (semiring-Ring R) ``` diff --git a/src/lists/concatenation-lists.lagda.md b/src/lists/concatenation-lists.lagda.md index 246a58513c..19d14a5d5b 100644 --- a/src/lists/concatenation-lists.lagda.md +++ b/src/lists/concatenation-lists.lagda.md @@ -104,13 +104,13 @@ length-concat-list (cons a x) y = ```agda eta-list : {l1 : Level} {A : UU l1} (x : list A) → - Id (concat-list (head-list x) (tail-list x)) x + concat-list (head-list x) (tail-list x) ＝ x eta-list nil = refl eta-list (cons a x) = refl eta-list' : {l1 : Level} {A : UU l1} (x : list A) → - Id (concat-list (remove-last-element-list x) (last-element-list x)) x + concat-list (remove-last-element-list x) (last-element-list x) ＝ x eta-list' nil = refl eta-list' (cons a nil) = refl eta-list' (cons a (cons b x)) = ap (cons a) (eta-list' (cons b x)) diff --git a/src/lists/flattening-lists.lagda.md b/src/lists/flattening-lists.lagda.md index 3534e0b657..c06302a1bd 100644 --- a/src/lists/flattening-lists.lagda.md +++ b/src/lists/flattening-lists.lagda.md @@ -40,7 +40,7 @@ flatten-list = fold-list nil concat-list ```agda flatten-unit-list : {l1 : Level} {A : UU l1} (x : list A) → - Id (flatten-list (unit-list x)) x + flatten-list (unit-list x) ＝ x flatten-unit-list x = right-unit-law-concat-list x length-flatten-list : diff --git a/src/lists/lists.lagda.md b/src/lists/lists.lagda.md index c9ccea7272..d9a142c16b 100644 --- a/src/lists/lists.lagda.md +++ b/src/lists/lists.lagda.md @@ -195,7 +195,7 @@ square-eq-Eq-list refl = refl is-section-eq-Eq-list : {l1 : Level} {A : UU l1} (l l' : list A) (e : Eq-list l l') → - Id (Eq-eq-list l l' (eq-Eq-list l l' e)) e + Eq-eq-list l l' (eq-Eq-list l l' e) ＝ e is-section-eq-Eq-list nil nil e = eq-is-contr is-contr-raise-unit is-section-eq-Eq-list nil (cons x l') e = ex-falso (is-empty-raise-empty e) is-section-eq-Eq-list (cons x l) nil e = ex-falso (is-empty-raise-empty e) @@ -205,13 +205,13 @@ is-section-eq-Eq-list (cons x l) (cons .x l') (pair refl e) = eq-Eq-refl-Eq-list : {l1 : Level} {A : UU l1} (l : list A) → - Id (eq-Eq-list l l (refl-Eq-list l)) refl + eq-Eq-list l l (refl-Eq-list l) ＝ refl eq-Eq-refl-Eq-list nil = refl eq-Eq-refl-Eq-list (cons x l) = ap² (cons x) (eq-Eq-refl-Eq-list l) is-retraction-eq-Eq-list : {l1 : Level} {A : UU l1} (l l' : list A) (p : l ＝ l') → - Id (eq-Eq-list l l' (Eq-eq-list l l' p)) p + eq-Eq-list l l' (Eq-eq-list l l' p) ＝ p is-retraction-eq-Eq-list nil .nil refl = refl is-retraction-eq-Eq-list (cons x l) .(cons x l) refl = eq-Eq-refl-Eq-list (cons x l) @@ -273,7 +273,7 @@ list-Set A = pair (list (type-Set A)) (is-set-list (is-set-type-Set A)) ```agda length-nil : {l1 : Level} {A : UU l1} → - Id (length-list {A = A} nil) zero-ℕ + length-list {A = A} nil ＝ zero-ℕ length-nil = refl is-nil-is-zero-length-list : diff --git a/src/lists/reversing-lists.lagda.md b/src/lists/reversing-lists.lagda.md index 551bed29c5..cb92b6b36a 100644 --- a/src/lists/reversing-lists.lagda.md +++ b/src/lists/reversing-lists.lagda.md @@ -90,7 +90,7 @@ reverse-flatten-list (cons a x) = reverse-reverse-list : {l1 : Level} {A : UU l1} (x : list A) → - Id (reverse-list (reverse-list x)) x + reverse-list (reverse-list x) ＝ x reverse-reverse-list nil = refl reverse-reverse-list (cons a x) = ( reverse-snoc-list (reverse-list x) a) ∙ diff --git a/src/order-theory/order-preserving-maps-posets.lagda.md b/src/order-theory/order-preserving-maps-posets.lagda.md index bc2be768b5..cd0ab7058c 100644 --- a/src/order-theory/order-preserving-maps-posets.lagda.md +++ b/src/order-theory/order-preserving-maps-posets.lagda.md @@ -168,13 +168,13 @@ module _ left-unit-law-comp-hom-Poset : (f : hom-Poset P Q) → - Id ( comp-hom-Poset P Q Q (id-hom-Poset Q) f) f + comp-hom-Poset P Q Q (id-hom-Poset Q) f ＝ f left-unit-law-comp-hom-Poset = left-unit-law-comp-hom-Preorder (preorder-Poset P) (preorder-Poset Q) right-unit-law-comp-hom-Poset : (f : hom-Poset P Q) → - Id (comp-hom-Poset P P Q f (id-hom-Poset P)) f + comp-hom-Poset P P Q f (id-hom-Poset P) ＝ f right-unit-law-comp-hom-Poset = right-unit-law-comp-hom-Preorder (preorder-Poset P) (preorder-Poset Q) ``` diff --git a/src/order-theory/order-preserving-maps-preorders.lagda.md b/src/order-theory/order-preserving-maps-preorders.lagda.md index 49e61f205f..a809e54834 100644 --- a/src/order-theory/order-preserving-maps-preorders.lagda.md +++ b/src/order-theory/order-preserving-maps-preorders.lagda.md @@ -187,7 +187,7 @@ module _ left-unit-law-comp-hom-Preorder : (f : hom-Preorder P Q) → - Id ( comp-hom-Preorder P Q Q (id-hom-Preorder Q) f) f + comp-hom-Preorder P Q Q (id-hom-Preorder Q) f ＝ f left-unit-law-comp-hom-Preorder f = eq-htpy-hom-Preorder P Q ( comp-hom-Preorder P Q Q (id-hom-Preorder Q) f) @@ -196,7 +196,7 @@ module _ right-unit-law-comp-hom-Preorder : (f : hom-Preorder P Q) → - Id (comp-hom-Preorder P P Q f (id-hom-Preorder P)) f + comp-hom-Preorder P P Q f (id-hom-Preorder P) ＝ f right-unit-law-comp-hom-Preorder f = eq-htpy-hom-Preorder P Q ( comp-hom-Preorder P P Q f (id-hom-Preorder P)) diff --git a/src/ring-theory/dependent-products-rings.lagda.md b/src/ring-theory/dependent-products-rings.lagda.md index cefc4eb518..a73fae798a 100644 --- a/src/ring-theory/dependent-products-rings.lagda.md +++ b/src/ring-theory/dependent-products-rings.lagda.md @@ -91,11 +91,11 @@ module _ right-unit-law-add-Π-Ring = right-unit-law-add-Semiring semiring-Π-Ring left-inverse-law-add-Π-Ring : - (x : type-Π-Ring) → Id (add-Π-Ring (neg-Π-Ring x) x) zero-Π-Ring + (x : type-Π-Ring) → add-Π-Ring (neg-Π-Ring x) x ＝ zero-Π-Ring left-inverse-law-add-Π-Ring = left-inverse-law-add-Ab ab-Π-Ring right-inverse-law-add-Π-Ring : - (x : type-Π-Ring) → Id (add-Π-Ring x (neg-Π-Ring x)) zero-Π-Ring + (x : type-Π-Ring) → add-Π-Ring x (neg-Π-Ring x) ＝ zero-Π-Ring right-inverse-law-add-Π-Ring = right-inverse-law-add-Ab ab-Π-Ring commutative-add-Π-Ring : diff --git a/src/ring-theory/products-rings.lagda.md b/src/ring-theory/products-rings.lagda.md index 86d85cd229..acddf82bbc 100644 --- a/src/ring-theory/products-rings.lagda.md +++ b/src/ring-theory/products-rings.lagda.md @@ -67,13 +67,13 @@ module _ left-inverse-law-add-product-Ring : (x : type-product-Ring) → - Id (add-product-Ring (neg-product-Ring x) x) zero-product-Ring + add-product-Ring (neg-product-Ring x) x ＝ zero-product-Ring left-inverse-law-add-product-Ring (x , y) = eq-pair (left-inverse-law-add-Ring R1 x) (left-inverse-law-add-Ring R2 y) right-inverse-law-add-product-Ring : (x : type-product-Ring) → - Id (add-product-Ring x (neg-product-Ring x)) zero-product-Ring + add-product-Ring x (neg-product-Ring x) ＝ zero-product-Ring right-inverse-law-add-product-Ring (x , y) = eq-pair (right-inverse-law-add-Ring R1 x) (right-inverse-law-add-Ring R2 y) diff --git a/src/species/morphisms-finite-species.lagda.md b/src/species/morphisms-finite-species.lagda.md index fc77041ff3..10f2e12299 100644 --- a/src/species/morphisms-finite-species.lagda.md +++ b/src/species/morphisms-finite-species.lagda.md @@ -105,13 +105,13 @@ module _ left-unit-law-comp-hom-finite-species : {l1 l2 l3 : Level} (F : finite-species l1 l2) (G : finite-species l1 l3) (f : hom-finite-species F G) → - Id (comp-hom-finite-species F G G (id-hom-finite-species G) f) f + comp-hom-finite-species F G G (id-hom-finite-species G) f ＝ f left-unit-law-comp-hom-finite-species F G f = refl right-unit-law-comp-hom-finite-species : {l1 l2 l3 : Level} (F : finite-species l1 l2) (G : finite-species l1 l3) (f : hom-finite-species F G) → - Id (comp-hom-finite-species F F G f (id-hom-finite-species F)) f + comp-hom-finite-species F F G f (id-hom-finite-species F) ＝ f right-unit-law-comp-hom-finite-species F G f = refl ``` diff --git a/src/species/morphisms-species-of-types.lagda.md b/src/species/morphisms-species-of-types.lagda.md index da7e432629..0926e3fe12 100644 --- a/src/species/morphisms-species-of-types.lagda.md +++ b/src/species/morphisms-species-of-types.lagda.md @@ -124,12 +124,12 @@ module _ left-unit-law-comp-hom-species-types : {l1 l2 l3 : Level} {F : species-types l1 l2} {G : species-types l1 l3} (f : hom-species-types F G) → - Id (comp-hom-species-types (id-hom-species-types G) f) f + comp-hom-species-types (id-hom-species-types G) f ＝ f left-unit-law-comp-hom-species-types f = refl right-unit-law-comp-hom-species-types : {l1 l2 l3 : Level} {F : species-types l1 l2} {G : species-types l1 l3} (f : hom-species-types F G) → - Id (comp-hom-species-types f (id-hom-species-types F)) f + comp-hom-species-types f (id-hom-species-types F) ＝ f right-unit-law-comp-hom-species-types f = refl ``` diff --git a/src/synthetic-homotopy-theory/interval-type.lagda.md b/src/synthetic-homotopy-theory/interval-type.lagda.md index 55dd66d000..680dcc0126 100644 --- a/src/synthetic-homotopy-theory/interval-type.lagda.md +++ b/src/synthetic-homotopy-theory/interval-type.lagda.md @@ -158,7 +158,7 @@ tr-value : tr-value f g refl q r s = (inv (ap-id q) ∙ inv right-unit) ∙ inv s is-retraction-inv-ev-𝕀 : - {l : Level} {P : 𝕀 → UU l} (f : (x : 𝕀) → P x) → Id (inv-ev-𝕀 (ev-𝕀 f)) f + {l : Level} {P : 𝕀 → UU l} (f : (x : 𝕀) → P x) → inv-ev-𝕀 (ev-𝕀 f) ＝ f is-retraction-inv-ev-𝕀 {l} {P} f = eq-htpy ( ind-𝕀 diff --git a/src/univalent-combinatorics/classical-finite-types.lagda.md b/src/univalent-combinatorics/classical-finite-types.lagda.md index 86fd8c8636..6e5a8d7228 100644 --- a/src/univalent-combinatorics/classical-finite-types.lagda.md +++ b/src/univalent-combinatorics/classical-finite-types.lagda.md @@ -100,12 +100,12 @@ pr2 (classical-standard-Fin k x) = strict-upper-bound-nat-Fin k x ```agda is-section-classical-standard-Fin : {k : ℕ} (x : Fin k) → - Id (standard-classical-Fin k (classical-standard-Fin k x)) x + standard-classical-Fin k (classical-standard-Fin k x) ＝ x is-section-classical-standard-Fin {succ-ℕ k} x = is-section-nat-Fin k x is-retraction-classical-standard-Fin : {k : ℕ} (x : classical-Fin k) → - Id (classical-standard-Fin k (standard-classical-Fin k x)) x + classical-standard-Fin k (standard-classical-Fin k x) ＝ x is-retraction-classical-standard-Fin {succ-ℕ k} (pair x p) = eq-Eq-classical-Fin (succ-ℕ k) ( classical-standard-Fin diff --git a/src/univalent-combinatorics/cyclic-finite-types.lagda.md b/src/univalent-combinatorics/cyclic-finite-types.lagda.md index cd6fbb7087..a2f6f15d3b 100644 --- a/src/univalent-combinatorics/cyclic-finite-types.lagda.md +++ b/src/univalent-combinatorics/cyclic-finite-types.lagda.md @@ -328,7 +328,7 @@ module _ where left-unit-law-comp-equiv-Cyclic-Type : - Id (comp-equiv-Cyclic-Type k X Y Y (id-equiv-Cyclic-Type k Y) e) e + comp-equiv-Cyclic-Type k X Y Y (id-equiv-Cyclic-Type k Y) e ＝ e left-unit-law-comp-equiv-Cyclic-Type = eq-htpy-equiv-Cyclic-Type k X Y ( comp-equiv-Cyclic-Type k X Y Y (id-equiv-Cyclic-Type k Y) e) @@ -336,7 +336,7 @@ module _ ( refl-htpy) right-unit-law-comp-equiv-Cyclic-Type : - Id (comp-equiv-Cyclic-Type k X X Y e (id-equiv-Cyclic-Type k X)) e + comp-equiv-Cyclic-Type k X X Y e (id-equiv-Cyclic-Type k X) ＝ e right-unit-law-comp-equiv-Cyclic-Type = eq-htpy-equiv-Cyclic-Type k X Y ( comp-equiv-Cyclic-Type k X X Y e (id-equiv-Cyclic-Type k X)) diff --git a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md index 83ce2b93e3..a542795714 100644 --- a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md +++ b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md @@ -3451,7 +3451,7 @@ module _ ( pair X (unit-trunc-Prop (equiv-count eX)))) ( T) ( canonical-orientation-count))) → - Id (pr1 equiv-fin-2-quotient-sign-count (inv-orientation T H)) T + pr1 equiv-fin-2-quotient-sign-count (inv-orientation T H) ＝ T retraction-orientation T (inl H) = eq-effective-quotient' ( even-difference-orientation-Complete-Undirected-Graph From 74a7954d817829a9213cbe731e1fa088ce10de08 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 1 Sep 2025 19:16:28 +0200 Subject: [PATCH 10/16] another Id pattern --- .../orbits-permutations.lagda.md | 78 +++++++++---------- ...ermutations-standard-finite-types.lagda.md | 8 +- .../transpositions.lagda.md | 12 +-- src/graph-theory/matchings.lagda.md | 2 +- src/group-theory/abelian-groups.lagda.md | 8 +- src/group-theory/furstenberg-groups.lagda.md | 2 +- src/group-theory/groups.lagda.md | 6 +- .../homomorphisms-groups.lagda.md | 2 +- src/group-theory/inverse-semigroups.lagda.md | 4 +- ...ubstitution-functor-group-actions.lagda.md | 2 +- .../left-modules-rings.lagda.md | 2 +- .../right-modules-rings.lagda.md | 2 +- .../transposition-matrices.lagda.md | 2 +- src/linear-algebra/tuples-on-rings.lagda.md | 4 +- src/lists/concatenation-lists.lagda.md | 2 +- src/lists/functoriality-lists.lagda.md | 2 +- src/lists/lists.lagda.md | 2 +- src/lists/reversing-lists.lagda.md | 10 +-- .../dependent-products-rings.lagda.md | 4 +- src/ring-theory/rings.lagda.md | 12 +-- src/ring-theory/semirings.lagda.md | 2 +- src/structured-types/magmas.lagda.md | 2 +- .../morphisms-h-spaces.lagda.md | 2 +- .../pointed-dependent-functions.lagda.md | 2 +- src/structured-types/wild-semigroups.lagda.md | 2 +- .../infinite-cyclic-types.lagda.md | 2 +- .../loop-spaces.lagda.md | 6 +- .../universal-cover-circle.lagda.md | 12 +-- .../dependent-type-theories.lagda.md | 8 +- .../fibered-dependent-type-theories.lagda.md | 6 +- .../complements-isolated-elements.lagda.md | 2 +- .../necklaces.lagda.md | 2 +- ...tations-complete-undirected-graph.lagda.md | 20 ++--- 33 files changed, 117 insertions(+), 117 deletions(-) diff --git a/src/finite-group-theory/orbits-permutations.lagda.md b/src/finite-group-theory/orbits-permutations.lagda.md index eaee934b81..a3ef4b93f2 100644 --- a/src/finite-group-theory/orbits-permutations.lagda.md +++ b/src/finite-group-theory/orbits-permutations.lagda.md @@ -165,7 +165,7 @@ module _ Σ ( ℕ) ( λ m → ( le-ℕ m n) × - ( Id (iterate n (map-equiv f) a) (iterate m (map-equiv f) a)))) + ( iterate n (map-equiv f) a ＝ iterate m (map-equiv f) a))) pr1 (two-points-iterate-ordered-ℕ (inl p)) = point2-iterate-ℕ pr1 (pr2 (two-points-iterate-ordered-ℕ (inl p))) = point1-iterate-ℕ pr1 (pr2 (pr2 (two-points-iterate-ordered-ℕ (inl p)))) = @@ -200,17 +200,17 @@ module _ Σ ( ℕ) ( λ m → ( le-ℕ m n) × - ( Id (iterate n (map-equiv f) a) (iterate m (map-equiv f) a)))) + ( iterate n (map-equiv f) a ＝ iterate m (map-equiv f) a))) min-repeating = well-ordering-principle-ℕ ( λ n → Σ ( ℕ) ( λ m → ( le-ℕ m n) × - ( Id (iterate n (map-equiv f) a) (iterate m (map-equiv f) a)))) + ( iterate n (map-equiv f) a ＝ iterate m (map-equiv f) a))) ( λ n → is-decidable-bounded-Σ-ℕ n ( λ m → le-ℕ m n) - ( λ m → Id (iterate n (map-equiv f) a) (iterate m (map-equiv f) a)) + ( λ m → iterate n (map-equiv f) a ＝ iterate m (map-equiv f) a) ( λ m → is-decidable-le-ℕ m n) ( λ m → has-decidable-equality-count eX @@ -235,7 +235,7 @@ module _ Σ ( ℕ) ( λ m → ( le-ℕ m n) × - ( Id (iterate n (map-equiv f) a) (iterate m (map-equiv f) a)))) + ( iterate n (map-equiv f) a ＝ iterate m (map-equiv f) a))) ( first-point-min-repeating) is-lower-bound-min-reporting = pr2 (pr2 min-repeating) @@ -324,7 +324,7 @@ module _ has-finite-orbits-permutation' : is-decidable (second-point-min-repeating ＝ zero-ℕ) → - Σ ℕ (λ k → (is-nonzero-ℕ k) × Id (iterate k (map-equiv f) a) a) + Σ ℕ (λ k → (is-nonzero-ℕ k) × (iterate k (map-equiv f) a ＝ a)) pr1 (has-finite-orbits-permutation' (inl p)) = first-point-min-repeating pr1 (pr2 (has-finite-orbits-permutation' (inl p))) = is-nonzero-le-ℕ @@ -363,7 +363,7 @@ module _ equality-pred-second = pr2 is-successor-second-point-min-repeating has-finite-orbits-permutation : - Σ ℕ (λ k → (is-nonzero-ℕ k) × Id (iterate k (map-equiv f) a) a) + Σ ℕ (λ k → (is-nonzero-ℕ k) × (iterate k (map-equiv f) a ＝ a)) has-finite-orbits-permutation = has-finite-orbits-permutation' ( has-decidable-equality-ℕ second-point-min-repeating zero-ℕ) @@ -420,7 +420,7 @@ module _ same-orbits-permutation : equivalence-relation l (type-Type-With-Cardinality-ℕ n X) (pr1 same-orbits-permutation) a b = - trunc-Prop (Σ ℕ (λ k → Id (iterate k (map-equiv f) a) b)) + trunc-Prop (Σ ℕ (λ k → iterate k (map-equiv f) a ＝ b)) pr1 (pr2 same-orbits-permutation) _ = unit-trunc-Prop (0 , refl) pr1 (pr2 (pr2 same-orbits-permutation)) a b P = apply-universal-property-trunc-Prop @@ -448,7 +448,7 @@ module _ where has-finite-orbits-permutation-a : (h : Fin n ≃ type-Type-With-Cardinality-ℕ n X) → - Σ ℕ (λ l → (is-nonzero-ℕ l) × Id (iterate l (map-equiv f) a) a) + Σ ℕ (λ l → (is-nonzero-ℕ l) × (iterate l (map-equiv f) a ＝ a)) has-finite-orbits-permutation-a h = has-finite-orbits-permutation ( type-Type-With-Cardinality-ℕ n X) @@ -496,7 +496,7 @@ module _ ( is-decidable-iterate-is-decidable-bounded h a b ( is-decidable-bounded-Σ-ℕ n ( λ z → z ≤-ℕ n) - ( λ z → Id (iterate z (map-equiv f) a) b) + ( λ z → iterate z (map-equiv f) a ＝ b) ( λ z → is-decidable-leq-ℕ z n) ( λ z → has-decidable-equality-equiv @@ -676,22 +676,22 @@ module _ abstract minimal-element-iterate : - (g : X ≃ X) (x y : X) → Σ ℕ (λ k → Id (iterate k (map-equiv g) x) y) → - minimal-element-ℕ (λ k → Id (iterate k (map-equiv g) x) y) + (g : X ≃ X) (x y : X) → Σ ℕ (λ k → iterate k (map-equiv g) x ＝ y) → + minimal-element-ℕ (λ k → iterate k (map-equiv g) x ＝ y) minimal-element-iterate g x y = well-ordering-principle-ℕ - ( λ k → Id (iterate k (map-equiv g) x) y) + ( λ k → iterate k (map-equiv g) x ＝ y) ( λ k → has-decidable-equality-count eX (iterate k (map-equiv g) x) y) abstract minimal-element-iterate-nonzero : (g : X ≃ X) (x y : X) → - Σ ℕ (λ k → is-nonzero-ℕ k × Id (iterate k (map-equiv g) x) y) → + Σ ℕ (λ k → is-nonzero-ℕ k × (iterate k (map-equiv g) x ＝ y)) → minimal-element-ℕ - ( λ k → is-nonzero-ℕ k × Id (iterate k (map-equiv g) x) y) + ( λ k → is-nonzero-ℕ k × (iterate k (map-equiv g) x ＝ y)) minimal-element-iterate-nonzero g x y = well-ordering-principle-ℕ - ( λ k → is-nonzero-ℕ k × Id (iterate k (map-equiv g) x) y) + ( λ k → is-nonzero-ℕ k × (iterate k (map-equiv g) x ＝ y)) ( λ k → is-decidable-product ( is-decidable-neg (has-decidable-equality-ℕ k zero-ℕ)) @@ -702,17 +702,17 @@ module _ (g : X ≃ X) (x y z : X) → Σ ( ℕ) ( λ k → - ( Id (iterate k (map-equiv g) x) y) + - ( Id (iterate k (map-equiv g) x) z)) → + ( iterate k (map-equiv g) x ＝ y) + + ( iterate k (map-equiv g) x ＝ z)) → minimal-element-ℕ ( λ k → - ( Id (iterate k (map-equiv g) x) y) + - ( Id (iterate k (map-equiv g) x) z)) + ( iterate k (map-equiv g) x ＝ y) + + ( iterate k (map-equiv g) x ＝ z)) minimal-element-iterate-2 g x y z p = well-ordering-principle-ℕ ( λ k → - ( Id (iterate k (map-equiv g) x) y) + - ( Id (iterate k (map-equiv g) x) z)) + ( iterate k (map-equiv g) x ＝ y) + + ( iterate k (map-equiv g) x ＝ z)) ( λ k → is-decidable-coproduct ( has-decidable-equality-count eX (iterate k (map-equiv g) x) y) @@ -887,7 +887,7 @@ module _ ( λ p → lemma3 p k2 q))) where neq-iterate-nonzero-le-minimal-element : - ( pa : Σ ℕ (λ k → Id (iterate k (map-equiv g) a) b)) + ( pa : Σ ℕ (λ k → iterate k (map-equiv g) a ＝ b)) ( k : ℕ) → ( is-nonzero-ℕ k × le-ℕ k (pr1 (minimal-element-iterate g a b pa))) → ( iterate k (map-equiv g) a ≠ a) × (iterate k (map-equiv g) a ≠ b) @@ -923,7 +923,7 @@ module _ ( contradiction-le-ℕ k (pr1 (minimal-element-iterate g a b pa)) ineq (pr2 (pr2 (minimal-element-iterate g a b pa)) k r)) equal-iterate-transposition-a : - (pa : Σ ℕ (λ k → Id (iterate k (map-equiv g) a) b)) (k : ℕ) → + (pa : Σ ℕ (λ k → iterate k (map-equiv g) a ＝ b)) (k : ℕ) → le-ℕ k (pr1 (minimal-element-iterate g a b pa)) → ( Id ( iterate k (map-equiv (composition-transposition-a-b g)) a) @@ -963,7 +963,7 @@ module _ ex-falso ( np ( tr - ( λ v → Id (iterate v (map-equiv g) a) b) + ( λ v → iterate v (map-equiv g) a ＝ b) ( p) ( pr1 (pr2 (minimal-element-iterate g a b pa))))) lemma2 pa (inr p) = @@ -1001,7 +1001,7 @@ module _ is-successor-ℕ (pr1 (minimal-element-iterate g a b pa)) is-successor-k1 = is-successor-is-nonzero-ℕ p mult-lemma2 : - ( pa : Σ ℕ (λ k → Id (iterate k (map-equiv g) a) b)) (k : ℕ) → + ( pa : Σ ℕ (λ k → iterate k (map-equiv g) a ＝ b)) (k : ℕ) → Id ( iterate ( k *ℕ (pr1 (minimal-element-iterate g a b pa))) @@ -1025,7 +1025,7 @@ module _ ( zero-ℕ))) ∙ ( mult-lemma2 pa k)) lemma3 : - ( pa : Σ ℕ (λ k → Id (iterate k (map-equiv g) a) b)) (k : ℕ) → + ( pa : Σ ℕ (λ k → iterate k (map-equiv g) a ＝ b)) (k : ℕ) → iterate k (map-equiv (composition-transposition-a-b g)) a ≠ b lemma3 pa k q = contradiction-le-ℕ @@ -1076,7 +1076,7 @@ module _ ( λ p → np ( tr - ( λ v → Id (iterate v (map-equiv g) a) b) + ( λ v → iterate v (map-equiv g) a ＝ b) ( p) ( pr1 (pr2 (minimal-element-iterate g a b pa))))) @@ -1165,20 +1165,20 @@ module _ ( g : X ≃ X) → ( Σ ( ℕ) ( λ k → - ( Id (iterate k (map-equiv g) x) a) + - ( Id (iterate k (map-equiv g) x) b))) → + ( iterate k (map-equiv g) x ＝ a) + + ( iterate k (map-equiv g) x ＝ b))) → minimal-element-ℕ ( λ k → - ( Id (iterate k (map-equiv g) x) a) + - ( Id (iterate k (map-equiv g) x) b)) + ( iterate k (map-equiv g) x ＝ a) + + ( iterate k (map-equiv g) x ＝ b)) minimal-element-iterate-2-a-b g = minimal-element-iterate-2 g x a b equal-iterate-transposition-same-orbits : ( g : X ≃ X) ( pa : Σ ( ℕ) ( λ k → - ( Id (iterate k (map-equiv g) x) a) + - ( Id (iterate k (map-equiv g) x) b))) + ( iterate k (map-equiv g) x ＝ a) + + ( iterate k (map-equiv g) x ＝ b))) ( k : ℕ) → ( le-ℕ k (pr1 (minimal-element-iterate-2-a-b g pa))) → Id @@ -1211,8 +1211,8 @@ module _ ( pa : Σ ( ℕ) (λ k → - ( Id (iterate k (map-equiv g) x) a) + - ( Id (iterate k (map-equiv g) x) b))) → + ( iterate k (map-equiv g) x ＝ a) + + ( iterate k (map-equiv g) x ＝ b))) → ( sim-equivalence-relation ( same-orbits-permutation-count (composition-transposition-a-b g)) ( x) @@ -1257,11 +1257,11 @@ module _ cases-lemma2 (inl q) (inl c) r = inl ( unit-trunc-Prop - ( pair zero-ℕ (tr (λ z → Id (iterate z (map-equiv g) x) a) q c))) + ( pair zero-ℕ (tr (λ z → iterate z (map-equiv g) x ＝ a) q c))) cases-lemma2 (inl q) (inr c) r = inr ( unit-trunc-Prop - ( pair zero-ℕ (tr (λ z → Id (iterate z (map-equiv g) x) b) q c))) + ( pair zero-ℕ (tr (λ z → iterate z (map-equiv g) x ＝ b) q c))) cases-lemma2 (inr q) (inl c) r = inr (unit-trunc-Prop ( pair @@ -2190,7 +2190,7 @@ module _ where minimal-element-iterate-repeating : minimal-element-ℕ - ( λ k → is-nonzero-ℕ k × Id (iterate k (map-equiv g) a) a) + ( λ k → is-nonzero-ℕ k × (iterate k (map-equiv g) a ＝ a)) minimal-element-iterate-repeating = minimal-element-iterate-nonzero ( g) diff --git a/src/finite-group-theory/permutations-standard-finite-types.lagda.md b/src/finite-group-theory/permutations-standard-finite-types.lagda.md index a50f9577df..431c30d661 100644 --- a/src/finite-group-theory/permutations-standard-finite-types.lagda.md +++ b/src/finite-group-theory/permutations-standard-finite-types.lagda.md @@ -64,7 +64,7 @@ Permutation n = Aut (Fin n) ```agda list-transpositions-permutation-Fin' : (n : ℕ) (f : Permutation (succ-ℕ n)) → - (x : Fin (succ-ℕ n)) → Id (map-equiv f (inr star)) x → + (x : Fin (succ-ℕ n)) → map-equiv f (inr star) ＝ x → ( list ( Σ ( Fin (succ-ℕ n) → Decidable-Prop lzero) @@ -126,7 +126,7 @@ list-transpositions-permutation-Fin (succ-ℕ n) f = abstract retraction-permutation-list-transpositions-Fin' : (n : ℕ) (f : Permutation (succ-ℕ n)) → - (x : Fin (succ-ℕ n)) → Id (map-equiv f (inr star)) x → + (x : Fin (succ-ℕ n)) → map-equiv f (inr star) ＝ x → (y z : Fin (succ-ℕ n)) → map-equiv f y ＝ z → Id ( map-equiv @@ -190,7 +190,7 @@ abstract ( neq-inr-inl) P : Σ ( Permutation (succ-ℕ (succ-ℕ n))) - ( λ g → Id (map-equiv g (inr star)) (inr star)) + ( λ g → map-equiv g (inr star) ＝ inr star) P = pair ( transposition t ∘e f) @@ -284,7 +284,7 @@ abstract ( neq-inr-inl) P : Σ ( Permutation (succ-ℕ (succ-ℕ n))) - ( λ g → Id (map-equiv g (inr star)) (inr star)) + ( λ g → map-equiv g (inr star) ＝ inr star) P = pair ( transposition t ∘e f) ( ( ap (map-transposition t) p) ∙ diff --git a/src/finite-group-theory/transpositions.lagda.md b/src/finite-group-theory/transpositions.lagda.md index 05d0d3f1f0..5103dfc047 100644 --- a/src/finite-group-theory/transpositions.lagda.md +++ b/src/finite-group-theory/transpositions.lagda.md @@ -371,7 +371,7 @@ module _ type-t-coproduct-id : (x : X) → ( pr1 two-elements-transposition ＝ x) + - ( Id (pr1 (pr2 two-elements-transposition)) x) → + ( pr1 (pr2 two-elements-transposition) ＝ x) → type-Decidable-Prop (pr1 Y x) type-t-coproduct-id x (inl Q) = tr @@ -387,7 +387,7 @@ module _ (x : X) (p : is-in-2-Element-Decidable-Subtype Y x) → (h : Fin 2 ≃ type-2-Element-Decidable-Subtype Y) → (k1 k2 k3 : Fin 2) → - Id ( map-inv-equiv h (pair x p)) k1 → + map-inv-equiv h (pair x p) ＝ k1 → Id ( map-inv-equiv h ( pair @@ -401,7 +401,7 @@ module _ ( type-decidable-prop-pr1-pr2-two-elements-transposition))) ( k3) → ( pr1 two-elements-transposition ＝ x) + - ( Id (pr1 (pr2 two-elements-transposition)) x) + ( pr1 (pr2 two-elements-transposition) ＝ x) cases-coproduct-id-type-t x p h (inl (inr star)) (inl (inr star)) k3 K1 K2 K3 = inl (ap pr1 (is-injective-equiv (inv-equiv h) (K2 ∙ inv K1))) @@ -430,7 +430,7 @@ module _ coproduct-id-type-t : (x : X) → type-Decidable-Prop (pr1 Y x) → ( pr1 two-elements-transposition ＝ x) + - ( Id (pr1 (pr2 two-elements-transposition)) x) + ( pr1 (pr2 two-elements-transposition) ＝ x) coproduct-id-type-t x p = apply-universal-property-trunc-Prop (pr2 Y) ( coproduct-Prop @@ -561,9 +561,9 @@ module _ type-Decidable-Prop (pr1 Y x) → type-Decidable-Prop (pr1 Y y) → ( ( pr1 two-elements-transposition ＝ x) × - ( Id (pr1 (pr2 two-elements-transposition)) y)) + + ( pr1 (pr2 two-elements-transposition) ＝ y)) + ( ( pr1 two-elements-transposition ＝ y) × - ( Id (pr1 (pr2 two-elements-transposition)) x)) + ( pr1 (pr2 two-elements-transposition) ＝ x)) eq-two-elements-transposition x y np p1 p2 = cases-eq-two-elements-transposition x y np p1 p2 ( has-decidable-equality-count eX (pr1 two-elements-transposition) x) diff --git a/src/graph-theory/matchings.lagda.md b/src/graph-theory/matchings.lagda.md index 66d2d1cc2c..3ad3de0630 100644 --- a/src/graph-theory/matchings.lagda.md +++ b/src/graph-theory/matchings.lagda.md @@ -44,7 +44,7 @@ module _ Σ ( vertex-Undirected-Graph G) ( λ y → Σ ( edge-Undirected-Graph G (standard-unordered-pair x y)) - ( λ e → Id (c (standard-unordered-pair x y) e) (inr star))) + ( λ e → c (standard-unordered-pair x y) e ＝ inr star)) matching : Undirected-Graph l1 l2 → UU (lsuc lzero ⊔ l1 ⊔ l2) matching G = diff --git a/src/group-theory/abelian-groups.lagda.md b/src/group-theory/abelian-groups.lagda.md index 8c2c72d7aa..1bf0cfd7f1 100644 --- a/src/group-theory/abelian-groups.lagda.md +++ b/src/group-theory/abelian-groups.lagda.md @@ -439,22 +439,22 @@ module _ transpose-eq-add-Ab : {x y z : type-Ab A} → - add-Ab A x y ＝ z → Id x (add-Ab A z (neg-Ab A y)) + add-Ab A x y ＝ z → x ＝ add-Ab A z (neg-Ab A y) transpose-eq-add-Ab = transpose-eq-mul-Group (group-Ab A) inv-transpose-eq-add-Ab : {x y z : type-Ab A} → - Id x (add-Ab A z (neg-Ab A y)) → add-Ab A x y ＝ z + x ＝ add-Ab A z (neg-Ab A y) → add-Ab A x y ＝ z inv-transpose-eq-add-Ab = inv-transpose-eq-mul-Group (group-Ab A) transpose-eq-add-Ab' : {x y z : type-Ab A} → - add-Ab A x y ＝ z → Id y (add-Ab A (neg-Ab A x) z) + add-Ab A x y ＝ z → y ＝ add-Ab A (neg-Ab A x) z transpose-eq-add-Ab' = transpose-eq-mul-Group' (group-Ab A) inv-transpose-eq-add-Ab' : {x y z : type-Ab A} → - Id y (add-Ab A (neg-Ab A x) z) → add-Ab A x y ＝ z + y ＝ add-Ab A (neg-Ab A x) z → add-Ab A x y ＝ z inv-transpose-eq-add-Ab' = inv-transpose-eq-mul-Group' (group-Ab A) double-transpose-eq-add-Ab : diff --git a/src/group-theory/furstenberg-groups.lagda.md b/src/group-theory/furstenberg-groups.lagda.md index a9e760780e..aeeea480eb 100644 --- a/src/group-theory/furstenberg-groups.lagda.md +++ b/src/group-theory/furstenberg-groups.lagda.md @@ -29,5 +29,5 @@ Furstenberg-Group l = ( λ μ → ( (x y z : type-Set X) → Id (μ (μ x z) (μ y z)) (μ x y)) × ( Σ ( type-Set X → type-Set X → type-Set X) - ( λ δ → (x y : type-Set X) → Id (μ x (δ x y)) y))))) + ( λ δ → (x y : type-Set X) → μ x (δ x y) ＝ y))))) ``` diff --git a/src/group-theory/groups.lagda.md b/src/group-theory/groups.lagda.md index 1b52e49b88..a2f71e3c77 100644 --- a/src/group-theory/groups.lagda.md +++ b/src/group-theory/groups.lagda.md @@ -66,9 +66,9 @@ is-group-is-unital-Semigroup G is-unital-Semigroup-G = Σ ( type-Semigroup G → type-Semigroup G) ( λ i → ( (x : type-Semigroup G) → - Id (mul-Semigroup G (i x) x) (pr1 is-unital-Semigroup-G)) × + mul-Semigroup G (i x) x ＝ pr1 is-unital-Semigroup-G) × ( (x : type-Semigroup G) → - Id (mul-Semigroup G x (i x)) (pr1 is-unital-Semigroup-G))) + mul-Semigroup G x (i x) ＝ pr1 is-unital-Semigroup-G)) is-group-Semigroup : {l : Level} (G : Semigroup l) → UU l @@ -115,7 +115,7 @@ module _ associative-mul-Group : (x y z : type-Group) → - Id (mul-Group (mul-Group x y) z) (mul-Group x (mul-Group y z)) + mul-Group (mul-Group x y) z ＝ mul-Group x (mul-Group y z) associative-mul-Group = pr2 has-associative-mul-Group is-group-Group : is-group-Semigroup semigroup-Group diff --git a/src/group-theory/homomorphisms-groups.lagda.md b/src/group-theory/homomorphisms-groups.lagda.md index 8df523437e..e5cb8c5b7f 100644 --- a/src/group-theory/homomorphisms-groups.lagda.md +++ b/src/group-theory/homomorphisms-groups.lagda.md @@ -260,7 +260,7 @@ module _ preserves-inverses-Group : (type-Group G → type-Group H) → UU (l1 ⊔ l2) preserves-inverses-Group f = - {x : type-Group G} → Id (f (inv-Group G x)) (inv-Group H (f x)) + {x : type-Group G} → f (inv-Group G x) ＝ inv-Group H (f x) abstract preserves-inv-hom-Group : diff --git a/src/group-theory/inverse-semigroups.lagda.md b/src/group-theory/inverse-semigroups.lagda.md index 983cf5a04e..67ec5cbcc4 100644 --- a/src/group-theory/inverse-semigroups.lagda.md +++ b/src/group-theory/inverse-semigroups.lagda.md @@ -35,8 +35,8 @@ is-inverse-Semigroup S = is-contr ( Σ ( type-Semigroup S) ( λ y → - Id (mul-Semigroup S (mul-Semigroup S x y) x) x × - Id (mul-Semigroup S (mul-Semigroup S y x) y) y)) + ( mul-Semigroup S (mul-Semigroup S x y) x ＝ x) × + ( mul-Semigroup S (mul-Semigroup S y x) y ＝ y))) Inverse-Semigroup : (l : Level) → UU (lsuc l) Inverse-Semigroup l = Σ (Semigroup l) is-inverse-Semigroup diff --git a/src/group-theory/substitution-functor-group-actions.lagda.md b/src/group-theory/substitution-functor-group-actions.lagda.md index 30c4e663e9..f1820ab830 100644 --- a/src/group-theory/substitution-functor-group-actions.lagda.md +++ b/src/group-theory/substitution-functor-group-actions.lagda.md @@ -138,7 +138,7 @@ module _ exists-structure-Prop ( type-Group G) ( λ g → - ( Id (mul-Group H (map-hom-Group G H f g) h) h') × + ( mul-Group H (map-hom-Group G H f g) h ＝ h') × ( mul-action-Group G X g x ＝ x')) pr1 ( pr2 (equivalence-relation-obj-left-adjoint-subst-action-Group X)) diff --git a/src/linear-algebra/left-modules-rings.lagda.md b/src/linear-algebra/left-modules-rings.lagda.md index 2ac8a9a423..5527f53599 100644 --- a/src/linear-algebra/left-modules-rings.lagda.md +++ b/src/linear-algebra/left-modules-rings.lagda.md @@ -305,7 +305,7 @@ module _ abstract left-zero-law-mul-left-module-Ring : (x : type-left-module-Ring R M) → - Id ( mul-left-module-Ring R M (zero-Ring R) x) (zero-left-module-Ring R M) + mul-left-module-Ring R M (zero-Ring R) x ＝ zero-left-module-Ring R M left-zero-law-mul-left-module-Ring = htpy-eq-hom-Ab ( ab-left-module-Ring R M) diff --git a/src/linear-algebra/right-modules-rings.lagda.md b/src/linear-algebra/right-modules-rings.lagda.md index 46e612a0ac..537a9008fe 100644 --- a/src/linear-algebra/right-modules-rings.lagda.md +++ b/src/linear-algebra/right-modules-rings.lagda.md @@ -275,7 +275,7 @@ module _ left-zero-law-mul-right-module-Ring : (x : type-right-module-Ring R M) → - Id (mul-right-module-Ring R M (zero-Ring R) x) (zero-right-module-Ring R M) + mul-right-module-Ring R M (zero-Ring R) x ＝ zero-right-module-Ring R M left-zero-law-mul-right-module-Ring = htpy-eq-hom-Ab ( ab-right-module-Ring R M) diff --git a/src/linear-algebra/transposition-matrices.lagda.md b/src/linear-algebra/transposition-matrices.lagda.md index b6d0964ec3..13effe6619 100644 --- a/src/linear-algebra/transposition-matrices.lagda.md +++ b/src/linear-algebra/transposition-matrices.lagda.md @@ -44,7 +44,7 @@ transpose-matrix {n = succ-ℕ n} x = ```agda is-involution-transpose-matrix : {l : Level} → {A : UU l} → {m n : ℕ} → - (x : matrix A m n) → Id x (transpose-matrix (transpose-matrix x)) + (x : matrix A m n) → x ＝ transpose-matrix (transpose-matrix x) is-involution-transpose-matrix {m = zero-ℕ} empty-tuple = refl is-involution-transpose-matrix {m = succ-ℕ m} (r ∷ rs) = ( ap (_∷_ r) (is-involution-transpose-matrix rs)) ∙ diff --git a/src/linear-algebra/tuples-on-rings.lagda.md b/src/linear-algebra/tuples-on-rings.lagda.md index 177899f946..3a164d207a 100644 --- a/src/linear-algebra/tuples-on-rings.lagda.md +++ b/src/linear-algebra/tuples-on-rings.lagda.md @@ -151,7 +151,7 @@ module _ left-inverse-law-add-tuple-Ring : {n : ℕ} (v : tuple-Ring R n) → - Id (add-tuple-Ring R (neg-tuple-Ring R v) v) (zero-tuple-Ring R) + add-tuple-Ring R (neg-tuple-Ring R v) v ＝ zero-tuple-Ring R left-inverse-law-add-tuple-Ring empty-tuple = refl left-inverse-law-add-tuple-Ring (x ∷ v) = ap-binary _∷_ @@ -160,7 +160,7 @@ module _ right-inverse-law-add-tuple-Ring : {n : ℕ} (v : tuple-Ring R n) → - Id (add-tuple-Ring R v (neg-tuple-Ring R v)) (zero-tuple-Ring R) + add-tuple-Ring R v (neg-tuple-Ring R v) ＝ zero-tuple-Ring R right-inverse-law-add-tuple-Ring empty-tuple = refl right-inverse-law-add-tuple-Ring (x ∷ v) = ap-binary _∷_ diff --git a/src/lists/concatenation-lists.lagda.md b/src/lists/concatenation-lists.lagda.md index 19d14a5d5b..b70cbc2f90 100644 --- a/src/lists/concatenation-lists.lagda.md +++ b/src/lists/concatenation-lists.lagda.md @@ -45,7 +45,7 @@ concatenation. ```agda associative-concat-list : {l1 : Level} {A : UU l1} (x y z : list A) → - Id (concat-list (concat-list x y) z) (concat-list x (concat-list y z)) + concat-list (concat-list x y) z ＝ concat-list x (concat-list y z) associative-concat-list nil y z = refl associative-concat-list (cons a x) y z = ap (cons a) (associative-concat-list x y z) diff --git a/src/lists/functoriality-lists.lagda.md b/src/lists/functoriality-lists.lagda.md index eac0801600..cb283730ec 100644 --- a/src/lists/functoriality-lists.lagda.md +++ b/src/lists/functoriality-lists.lagda.md @@ -50,7 +50,7 @@ map-list f = fold-list nil (λ a → cons (f a)) ```agda length-map-list : {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) (l : list A) → - Id (length-list (map-list f l)) (length-list l) + length-list (map-list f l) ＝ length-list l length-map-list f nil = refl length-map-list f (cons x l) = ap succ-ℕ (length-map-list f l) diff --git a/src/lists/lists.lagda.md b/src/lists/lists.lagda.md index d9a142c16b..61636d25ed 100644 --- a/src/lists/lists.lagda.md +++ b/src/lists/lists.lagda.md @@ -363,7 +363,7 @@ tail-snoc-snoc-list (cons c x) a b = refl last-element-snoc : {l1 : Level} {A : UU l1} (x : list A) (a : A) → - Id (last-element-list (snoc x a)) (unit-list a) + last-element-list (snoc x a) ＝ unit-list a last-element-snoc nil a = refl last-element-snoc (cons b nil) a = refl last-element-snoc (cons b (cons c x)) a = diff --git a/src/lists/reversing-lists.lagda.md b/src/lists/reversing-lists.lagda.md index cb92b6b36a..59c919f3f3 100644 --- a/src/lists/reversing-lists.lagda.md +++ b/src/lists/reversing-lists.lagda.md @@ -40,7 +40,7 @@ reverse-list (cons a l) = snoc (reverse-list l) a ```agda reverse-unit-list : {l1 : Level} {A : UU l1} (a : A) → - Id (reverse-list (unit-list a)) (unit-list a) + reverse-list (unit-list a) ＝ unit-list a reverse-unit-list a = refl length-snoc-list : @@ -51,7 +51,7 @@ length-snoc-list (cons b x) a = ap succ-ℕ (length-snoc-list x a) length-reverse-list : {l1 : Level} {A : UU l1} (x : list A) → - Id (length-list (reverse-list x)) (length-list x) + length-list (reverse-list x) ＝ length-list x length-reverse-list nil = refl length-reverse-list (cons a x) = ( length-snoc-list (reverse-list x) a) ∙ @@ -109,14 +109,14 @@ head-reverse-list (cons a (cons b x)) = last-element-reverse-list : {l1 : Level} {A : UU l1} (x : list A) → - Id (last-element-list (reverse-list x)) (head-list x) + last-element-list (reverse-list x) ＝ head-list x last-element-reverse-list x = ( inv (head-reverse-list (reverse-list x))) ∙ ( ap head-list (reverse-reverse-list x)) tail-reverse-list : {l1 : Level} {A : UU l1} (x : list A) → - Id (tail-list (reverse-list x)) (reverse-list (remove-last-element-list x)) + tail-list (reverse-list x) ＝ reverse-list (remove-last-element-list x) tail-reverse-list nil = refl tail-reverse-list (cons a nil) = refl tail-reverse-list (cons a (cons b x)) = @@ -125,7 +125,7 @@ tail-reverse-list (cons a (cons b x)) = remove-last-element-reverse-list : {l1 : Level} {A : UU l1} (x : list A) → - Id (remove-last-element-list (reverse-list x)) (reverse-list (tail-list x)) + remove-last-element-list (reverse-list x) ＝ reverse-list (tail-list x) remove-last-element-reverse-list x = ( inv (reverse-reverse-list (remove-last-element-list (reverse-list x)))) ∙ ( ( inv (ap reverse-list (tail-reverse-list (reverse-list x)))) ∙ diff --git a/src/ring-theory/dependent-products-rings.lagda.md b/src/ring-theory/dependent-products-rings.lagda.md index a73fae798a..bbf0759490 100644 --- a/src/ring-theory/dependent-products-rings.lagda.md +++ b/src/ring-theory/dependent-products-rings.lagda.md @@ -79,7 +79,7 @@ module _ associative-add-Π-Ring : (x y z : type-Π-Ring) → - Id (add-Π-Ring (add-Π-Ring x y) z) (add-Π-Ring x (add-Π-Ring y z)) + add-Π-Ring (add-Π-Ring x y) z ＝ add-Π-Ring x (add-Π-Ring y z) associative-add-Π-Ring = associative-add-Semiring semiring-Π-Ring left-unit-law-add-Π-Ring : @@ -110,7 +110,7 @@ module _ associative-mul-Π-Ring : (x y z : type-Π-Ring) → - Id (mul-Π-Ring (mul-Π-Ring x y) z) (mul-Π-Ring x (mul-Π-Ring y z)) + mul-Π-Ring (mul-Π-Ring x y) z ＝ mul-Π-Ring x (mul-Π-Ring y z) associative-mul-Π-Ring = associative-mul-Semiring semiring-Π-Ring diff --git a/src/ring-theory/rings.lagda.md b/src/ring-theory/rings.lagda.md index c639bb2552..4bbf93fbc2 100644 --- a/src/ring-theory/rings.lagda.md +++ b/src/ring-theory/rings.lagda.md @@ -120,7 +120,7 @@ module _ associative-add-Ring : (x y z : type-Ring R) → - Id (add-Ring (add-Ring x y) z) (add-Ring x (add-Ring y z)) + add-Ring (add-Ring x y) z ＝ add-Ring x (add-Ring y z) associative-add-Ring = associative-add-Ab (ab-Ring R) is-group-additive-semigroup-Ring : @@ -298,11 +298,11 @@ module _ neg-Ring = neg-Ab (ab-Ring R) left-inverse-law-add-Ring : - (x : type-Ring R) → Id (add-Ring R (neg-Ring x) x) (zero-Ring R) + (x : type-Ring R) → add-Ring R (neg-Ring x) x ＝ zero-Ring R left-inverse-law-add-Ring = left-inverse-law-add-Ab (ab-Ring R) right-inverse-law-add-Ring : - (x : type-Ring R) → Id (add-Ring R x (neg-Ring x)) (zero-Ring R) + (x : type-Ring R) → add-Ring R x (neg-Ring x) ＝ zero-Ring R right-inverse-law-add-Ring = right-inverse-law-add-Ab (ab-Ring R) neg-neg-Ring : (x : type-Ring R) → neg-Ring (neg-Ring x) ＝ x @@ -373,7 +373,7 @@ module _ associative-mul-Ring : (x y z : type-Ring R) → - Id (mul-Ring (mul-Ring x y) z) (mul-Ring x (mul-Ring y z)) + mul-Ring (mul-Ring x y) z ＝ mul-Ring x (mul-Ring y z) associative-mul-Ring = pr2 has-associative-mul-Ring multiplicative-semigroup-Ring : Semigroup l @@ -425,7 +425,7 @@ module _ where left-zero-law-mul-Ring : - (x : type-Ring R) → Id (mul-Ring R (zero-Ring R) x) (zero-Ring R) + (x : type-Ring R) → mul-Ring R (zero-Ring R) x ＝ zero-Ring R left-zero-law-mul-Ring x = is-zero-is-idempotent-Ab ( ab-Ring R) @@ -434,7 +434,7 @@ module _ ( ap (mul-Ring' R x) (left-unit-law-add-Ring R (zero-Ring R)))) right-zero-law-mul-Ring : - (x : type-Ring R) → Id (mul-Ring R x (zero-Ring R)) (zero-Ring R) + (x : type-Ring R) → mul-Ring R x (zero-Ring R) ＝ zero-Ring R right-zero-law-mul-Ring x = is-zero-is-idempotent-Ab ( ab-Ring R) diff --git a/src/ring-theory/semirings.lagda.md b/src/ring-theory/semirings.lagda.md index 8cd31fe350..34ae774d3f 100644 --- a/src/ring-theory/semirings.lagda.md +++ b/src/ring-theory/semirings.lagda.md @@ -225,7 +225,7 @@ module _ associative-mul-Semiring : (x y z : type-Semiring R) → - Id (mul-Semiring (mul-Semiring x y) z) (mul-Semiring x (mul-Semiring y z)) + mul-Semiring (mul-Semiring x y) z ＝ mul-Semiring x (mul-Semiring y z) associative-mul-Semiring = pr2 has-associative-mul-Semiring multiplicative-semigroup-Semiring : Semigroup l diff --git a/src/structured-types/magmas.lagda.md b/src/structured-types/magmas.lagda.md index de9500e352..6d382f91b9 100644 --- a/src/structured-types/magmas.lagda.md +++ b/src/structured-types/magmas.lagda.md @@ -67,7 +67,7 @@ is-unital-magma-Unital-Magma M = pr2 M is-semigroup-Magma : {l : Level} → Magma l → UU l is-semigroup-Magma M = (x y z : type-Magma M) → - Id (mul-Magma M (mul-Magma M x y) z) (mul-Magma M x (mul-Magma M y z)) + mul-Magma M (mul-Magma M x y) z ＝ mul-Magma M x (mul-Magma M y z) ``` ### Commutative magmas diff --git a/src/structured-types/morphisms-h-spaces.lagda.md b/src/structured-types/morphisms-h-spaces.lagda.md index 970774ab0f..bdffcf852a 100644 --- a/src/structured-types/morphisms-h-spaces.lagda.md +++ b/src/structured-types/morphisms-h-spaces.lagda.md @@ -150,7 +150,7 @@ module _ preserves-left-unit-law-mul : ((x : type-Pointed-Type A) → μ (point-Pointed-Type A) x ＝ x) → - ((y : type-Pointed-Type B) → Id (ν (point-Pointed-Type B) y) y) → + ((y : type-Pointed-Type B) → ν (point-Pointed-Type B) y ＝ y) → UU (l1 ⊔ l2) preserves-left-unit-law-mul lA lB = (x : type-Pointed-Type A) → diff --git a/src/structured-types/pointed-dependent-functions.lagda.md b/src/structured-types/pointed-dependent-functions.lagda.md index 7df9e3085d..0de60179ff 100644 --- a/src/structured-types/pointed-dependent-functions.lagda.md +++ b/src/structured-types/pointed-dependent-functions.lagda.md @@ -55,6 +55,6 @@ module _ preserves-point-function-pointed-Π : (f : pointed-Π A B) → - Id (function-pointed-Π f (point-Pointed-Type A)) (point-Pointed-Fam A B) + function-pointed-Π f (point-Pointed-Type A) ＝ point-Pointed-Fam A B preserves-point-function-pointed-Π = pr2 ``` diff --git a/src/structured-types/wild-semigroups.lagda.md b/src/structured-types/wild-semigroups.lagda.md index 8791b8b74d..e9cd1ddadb 100644 --- a/src/structured-types/wild-semigroups.lagda.md +++ b/src/structured-types/wild-semigroups.lagda.md @@ -28,7 +28,7 @@ Wild-Semigroup l = Σ ( Magma l) ( λ M → (x y z : type-Magma M) → - Id (mul-Magma M (mul-Magma M x y) z) (mul-Magma M x (mul-Magma M y z))) + mul-Magma M (mul-Magma M x y) z ＝ mul-Magma M x (mul-Magma M y z)) module _ {l : Level} (G : Wild-Semigroup l) diff --git a/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md b/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md index da44084699..570be9e03b 100644 --- a/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md +++ b/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md @@ -164,7 +164,7 @@ module _ ( succ-ℤ ∘ map-equiv (pr1 e)))) ∘e ( ( equiv-right-swap-Σ) ∘e ( equiv-Σ - ( λ e → Id (map-equiv (pr1 e) zero-ℤ) zero-ℤ) + ( λ e → map-equiv (pr1 e) zero-ℤ ＝ zero-ℤ) ( equiv-Σ ( λ e → (map-equiv e ∘ succ-ℤ) ~ (succ-ℤ ∘ map-equiv e)) ( equiv-postcomp-equiv (equiv-left-add-ℤ (neg-ℤ x)) ℤ) diff --git a/src/synthetic-homotopy-theory/loop-spaces.lagda.md b/src/synthetic-homotopy-theory/loop-spaces.lagda.md index 17bdfca373..be414f65db 100644 --- a/src/synthetic-homotopy-theory/loop-spaces.lagda.md +++ b/src/synthetic-homotopy-theory/loop-spaces.lagda.md @@ -102,11 +102,11 @@ module _ inv-Ω = inv left-inverse-law-mul-Ω : - (x : type-Ω A) → Id (mul-Ω A (inv-Ω x) x) (refl-Ω A) + (x : type-Ω A) → mul-Ω A (inv-Ω x) x ＝ refl-Ω A left-inverse-law-mul-Ω x = left-inv x right-inverse-law-mul-Ω : - (x : type-Ω A) → Id (mul-Ω A x (inv-Ω x)) (refl-Ω A) + (x : type-Ω A) → mul-Ω A x (inv-Ω x) ＝ refl-Ω A right-inverse-law-mul-Ω x = right-inv x Ω-Wild-Quasigroup : Wild-Quasigroup l @@ -123,7 +123,7 @@ module _ associative-mul-Ω : (x y z : type-Ω A) → - Id (mul-Ω A (mul-Ω A x y) z) (mul-Ω A x (mul-Ω A y z)) + mul-Ω A (mul-Ω A x y) z ＝ mul-Ω A x (mul-Ω A y z) associative-mul-Ω x y z = assoc x y z ``` diff --git a/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md b/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md index ac2ea7746f..7712fb9d47 100644 --- a/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md +++ b/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md @@ -217,7 +217,7 @@ contraction-total-space' : { l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} (c : Σ A B) → ( x : A) → {F : UU l3} (e : F ≃ B x) → UU (l1 ⊔ l2 ⊔ l3) contraction-total-space' c x {F} e = - ( y : F) → Id c (pair x (map-equiv e y)) + ( y : F) → c ＝ pair x (map-equiv e y) equiv-tr-contraction-total-space' : { l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} (c : Σ A B) → @@ -270,8 +270,8 @@ dependent-identification-contraction-total-space' : {x x' : A} (p : x ＝ x') → {F : UU l3} {F' : UU l4} (f : F ≃ F') ( e : F ≃ B x) (e' : F' ≃ B x') (H : ((map-equiv e') ∘ (map-equiv f)) ~ ((tr B p) ∘ (map-equiv e))) → - (h : (y : F) → Id c (pair x (map-equiv e y))) → - (h' : (y' : F') → Id c (pair x' (map-equiv e' y'))) → + (h : (y : F) → c ＝ pair x (map-equiv e y)) → + (h' : (y' : F') → c ＝ pair x' (map-equiv e' y')) → UU (l1 ⊔ l2 ⊔ l3) dependent-identification-contraction-total-space' c {x} {x'} p {F} {F'} f e e' H h h' = @@ -279,7 +279,7 @@ dependent-identification-contraction-total-space' ( λ y → concat' c (segment-Σ p f e e' H y)) h) ~ ( precomp-Π ( map-equiv f) - ( λ y' → Id c (pair x' (map-equiv e' y'))) + ( λ y' → c ＝ pair x' (map-equiv e' y')) ( h')) map-dependent-identification-contraction-total-space' : @@ -312,7 +312,7 @@ map-dependent-identification-contraction-total-space' ( segment-Σ refl f e e' H) ( precomp-Π ( map-equiv f) - ( λ y' → Id c (pair x (map-equiv e' y'))) + ( λ y' → c ＝ pair x (map-equiv e' y')) ( h')) ( α))) ∙ ( inv @@ -341,7 +341,7 @@ equiv-dependent-identification-contraction-total-space' ( segment-Σ refl f e e' H) ( precomp-Π ( map-equiv f) - ( λ y' → Id c (pair x (map-equiv e' y'))) + ( λ y' → c ＝ pair x (map-equiv e' y')) ( h')))) ∘e ( ( equiv-funext) ∘e ( ( equiv-concat' h diff --git a/src/type-theories/dependent-type-theories.lagda.md b/src/type-theories/dependent-type-theories.lagda.md index 9d8d82750e..d60c0db666 100644 --- a/src/type-theories/dependent-type-theories.lagda.md +++ b/src/type-theories/dependent-type-theories.lagda.md @@ -339,10 +339,10 @@ We show that systems form a category. {C C' : system l5 l6} {g : hom-system B C} {g' : hom-system B' C'} (p : B ＝ B') {p' : constant-fibered-system A B ＝ constant-fibered-system A B'} - (α : Id (ap (constant-fibered-system A) p) p') + (α : ap (constant-fibered-system A) p ＝ p') (q : C ＝ C') {q' : constant-fibered-system A C ＝ constant-fibered-system A C'} - (β : Id (ap (constant-fibered-system A) q) q') + (β : ap (constant-fibered-system A) q ＝ q') (r : Id (tr (λ t → t) (ap-binary hom-system p q) g) g') {f : hom-system A B} {f' : hom-system A B'} → htpy-section-system' p' f f' → @@ -408,9 +408,9 @@ We show that systems form a category. {C C' : system l5 l6} (p : C ＝ C') {g : hom-system B C} {g' : hom-system B C'} {p' : constant-fibered-system B C ＝ constant-fibered-system B C'} - (α : Id (ap (constant-fibered-system B) p) p') + (α : ap (constant-fibered-system B) p ＝ p') {q' : constant-fibered-system A C ＝ constant-fibered-system A C'} - (β : Id (ap (constant-fibered-system A) p) q') + (β : ap (constant-fibered-system A) p ＝ q') (H : htpy-section-system' p' g g') → (f : hom-system A B) → htpy-section-system' q' (comp-hom-system g f) (comp-hom-system g' f) diff --git a/src/type-theories/fibered-dependent-type-theories.lagda.md b/src/type-theories/fibered-dependent-type-theories.lagda.md index ed84a99826..f4a605ae1d 100644 --- a/src/type-theories/fibered-dependent-type-theories.lagda.md +++ b/src/type-theories/fibered-dependent-type-theories.lagda.md @@ -117,7 +117,7 @@ module fibered where (D : bifibered-system l7 l8 B C) {X : system.type A} (Y : fibered-system.type B X) {Z Z' : fibered-system.type C X} {d : bifibered-system.type D Y Z} {d' : bifibered-system.type D Y Z'} - (p : Z ＝ Z') (q : Id (tr (bifibered-system.type D Y) p d) d') → + (p : Z ＝ Z') (q : tr (bifibered-system.type D Y) p d ＝ d') → Id ( tr ( bifibered-system l7 l8 (fibered-system.slice B Y)) @@ -130,7 +130,7 @@ module fibered where {l1 l2 l3 l4 l5 l6 l7 l8 : Level} {A : system l1 l2} {B : fibered-system l3 l4 A} {C C' : fibered-system l5 l6 A} (D : bifibered-system l7 l8 B C) (D' : bifibered-system l7 l8 B C') - (α : C ＝ C') (β : Id (tr (bifibered-system l7 l8 B) α D) D') + (α : C ＝ C') (β : tr (bifibered-system l7 l8 B) α D ＝ D') (f : section-system C) (f' : section-system C') (g : section-fibered-system f D) (g' : section-fibered-system f' D') → bifibered-system l7 l8 B (Eq-fibered-system' α f f') @@ -167,7 +167,7 @@ module fibered where {B : fibered-system l3 l4 A} {C C' : fibered-system l5 l6 A} {D : bifibered-system l7 l8 B C} {D' : bifibered-system l7 l8 B C'} {f : section-system C} {f' : section-system C'} - {α : C ＝ C'} (β : Id (tr (bifibered-system l7 l8 B) α D) D') + {α : C ＝ C'} (β : tr (bifibered-system l7 l8 B) α D ＝ D') (H : htpy-section-system' α f f') (g : section-fibered-system f D) (h : section-fibered-system f' D') → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ l7 ⊔ l8) diff --git a/src/univalent-combinatorics/complements-isolated-elements.lagda.md b/src/univalent-combinatorics/complements-isolated-elements.lagda.md index add3d8ff3f..c993815344 100644 --- a/src/univalent-combinatorics/complements-isolated-elements.lagda.md +++ b/src/univalent-combinatorics/complements-isolated-elements.lagda.md @@ -116,7 +116,7 @@ equiv-complement-element-Type-With-Cardinality-ℕ : Σ ( Type-With-Cardinality-ℕ l1 (succ-ℕ k)) ( type-Type-With-Cardinality-ℕ (succ-ℕ k))) → (e : equiv-Type-With-Cardinality-ℕ (succ-ℕ k) (pr1 X) (pr1 Y)) - (p : Id (map-equiv e (pr2 X)) (pr2 Y)) → + (p : map-equiv e (pr2 X) ＝ pr2 Y) → equiv-Type-With-Cardinality-ℕ k ( complement-element-Type-With-Cardinality-ℕ k X) ( complement-element-Type-With-Cardinality-ℕ k Y) diff --git a/src/univalent-combinatorics/necklaces.lagda.md b/src/univalent-combinatorics/necklaces.lagda.md index a5cbb95309..711246126a 100644 --- a/src/univalent-combinatorics/necklaces.lagda.md +++ b/src/univalent-combinatorics/necklaces.lagda.md @@ -124,7 +124,7 @@ module _ refl-extensionality-necklace : (N : necklace l m n) → - Id (map-equiv (extensionality-necklace N N) refl) (id-equiv-necklace m n N) + map-equiv (extensionality-necklace N N) refl ＝ id-equiv-necklace m n N refl-extensionality-necklace N = refl ``` diff --git a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md index a542795714..cdab2fa97c 100644 --- a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md +++ b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md @@ -734,7 +734,7 @@ module _ ( np)) ( Y))))) → is-decidable - ( Id (map-equiv e (pr1 two-elements)) (pr1 two-elements)) → + ( map-equiv e (pr1 two-elements) ＝ pr1 two-elements) → is-decidable ( Id (map-equiv e (pr1 (pr2 two-elements))) (pr1 (pr2 two-elements))) → Σ X (λ z → type-Decidable-Prop (pr1 Y z)) @@ -870,9 +870,9 @@ module _ (i j : X) (np : i ≠ j) (Y : 2-Element-Decidable-Subtype l X) (x : X) → x ≠ i → x ≠ j → - ( ( Id (pr1 (two-elements-transposition eX Y)) x) × + ( ( pr1 (two-elements-transposition eX Y) ＝ x) × ( Id (pr1 (pr2 (two-elements-transposition eX Y))) i)) + - ( ( Id (pr1 (two-elements-transposition eX Y)) i) × + ( ( pr1 (two-elements-transposition eX Y) ＝ i) × ( Id (pr1 (pr2 (two-elements-transposition eX Y))) x)) → Id ( pr1 (orientation-two-elements-count i j np Y)) @@ -953,9 +953,9 @@ module _ (i j : X) (np : i ≠ j) (Y : 2-Element-Decidable-Subtype l X) (x : X) → x ≠ i → x ≠ j → - ( ( Id (pr1 (two-elements-transposition eX Y)) x) × + ( ( pr1 (two-elements-transposition eX Y) ＝ x) × ( Id (pr1 (pr2 (two-elements-transposition eX Y))) i)) + - ( ( Id (pr1 (two-elements-transposition eX Y)) i) × + ( ( pr1 (two-elements-transposition eX Y) ＝ i) × ( Id (pr1 (pr2 (two-elements-transposition eX Y))) x)) → Id ( pr1 @@ -1144,9 +1144,9 @@ module _ (i j : X) (np : i ≠ j) (Y : 2-Element-Decidable-Subtype l X) (x : X) → x ≠ i → x ≠ j → - ( ( Id (pr1 (two-elements-transposition eX Y)) x) × + ( ( pr1 (two-elements-transposition eX Y) ＝ x) × ( Id (pr1 (pr2 (two-elements-transposition eX Y))) j)) + - ( ( Id (pr1 (two-elements-transposition eX Y)) j) × + ( ( pr1 (two-elements-transposition eX Y) ＝ j) × ( Id (pr1 (pr2 (two-elements-transposition eX Y))) x)) → Id ( pr1 (orientation-two-elements-count i j np Y)) @@ -1253,9 +1253,9 @@ module _ (i j : X) (np : i ≠ j) (Y : 2-Element-Decidable-Subtype l X) (x : X) → x ≠ i → x ≠ j → - ( ( Id (pr1 (two-elements-transposition eX Y)) x) × + ( ( pr1 (two-elements-transposition eX Y) ＝ x) × ( Id (pr1 (pr2 (two-elements-transposition eX Y))) j)) + - ( ( Id (pr1 (two-elements-transposition eX Y)) j) × + ( ( pr1 (two-elements-transposition eX Y) ＝ j) × ( Id (pr1 (pr2 (two-elements-transposition eX Y))) x)) → Id ( pr1 @@ -2869,7 +2869,7 @@ module _ ( has-decidable-equality-count eX) ( np')) ( pr1 T))))) → - Id two-elements (two-elements-transposition eX (pr1 T)) → + two-elements ＝ two-elements-transposition eX (pr1 T) → is-decidable (pr1 two-elements ＝ i) → is-decidable (pr1 two-elements ＝ j) → is-decidable (pr1 (pr2 two-elements) ＝ i) → From d330e69135936bfe07ae3a4efcc22005fa079f3f Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 1 Sep 2025 19:44:38 +0200 Subject: [PATCH 11/16] another Id pattern --- .../orbits-permutations.lagda.md | 12 ++++----- .../equality-coproduct-types.lagda.md | 8 +++--- src/group-theory/furstenberg-groups.lagda.md | 2 +- src/group-theory/sheargroups.lagda.md | 2 +- .../transposition-matrices.lagda.md | 2 +- src/lists/concatenation-lists.lagda.md | 2 +- ...ersal-property-lists-wild-monoids.lagda.md | 4 +-- .../finitely-graded-posets.lagda.md | 6 ++--- src/ring-theory/rings.lagda.md | 4 +-- .../morphisms-h-spaces.lagda.md | 2 +- .../morphisms-magmas.lagda.md | 2 +- .../triple-loop-spaces.lagda.md | 4 +-- .../universal-cover-circle.lagda.md | 27 ++++++++----------- .../dependent-type-theories.lagda.md | 21 ++++++++------- 14 files changed, 47 insertions(+), 51 deletions(-) diff --git a/src/finite-group-theory/orbits-permutations.lagda.md b/src/finite-group-theory/orbits-permutations.lagda.md index a3ef4b93f2..9864859945 100644 --- a/src/finite-group-theory/orbits-permutations.lagda.md +++ b/src/finite-group-theory/orbits-permutations.lagda.md @@ -240,8 +240,8 @@ module _ is-lower-bound-min-reporting = pr2 (pr2 min-repeating) same-image-iterate-min-reporting : - Id ( iterate first-point-min-repeating (map-equiv f) a) - ( iterate second-point-min-repeating (map-equiv f) a) + iterate first-point-min-repeating (map-equiv f) a ＝ + iterate second-point-min-repeating (map-equiv f) a same-image-iterate-min-reporting = pr2 (pr2 (pr1 (pr2 min-repeating))) leq-first-point-min-reporting-succ-number-elements : @@ -752,8 +752,8 @@ module _ induction-cases-equal-iterate-transposition (inl s) = tr ( λ k → - Id (iterate k (map-equiv (composition-transposition-a-b g)) x) - (iterate k (map-equiv g) x)) + iterate k (map-equiv (composition-transposition-a-b g)) x ＝ + iterate k (map-equiv g) x) ( inv s) ( refl) induction-cases-equal-iterate-transposition (inr s) = @@ -1805,7 +1805,7 @@ module _ ( Q)))) retraction-h' T (inr NQ) (inl R) = tr - (λ w → Id (h' (cases-inv-h' T (pr1 w) (pr2 w))) T) + (λ w → h' (cases-inv-h' T (pr1 w) (pr2 w)) ＝ T) {x = pair (inr NQ) (inl R)} {y = pair (is-decidable-is-in-equivalence-class-same-orbits-permutation @@ -1842,7 +1842,7 @@ module _ ( R)) retraction-h' T (inr NQ) (inr NR) = tr - (λ w → Id (h' (cases-inv-h' T (pr1 w) (pr2 w))) T) + (λ w → h' (cases-inv-h' T (pr1 w) (pr2 w)) ＝ T) {x = pair (inr NQ) (inr NR)} {y = pair (is-decidable-is-in-equivalence-class-same-orbits-permutation diff --git a/src/foundation/equality-coproduct-types.lagda.md b/src/foundation/equality-coproduct-types.lagda.md index 591c95e1f6..208bb98e5f 100644 --- a/src/foundation/equality-coproduct-types.lagda.md +++ b/src/foundation/equality-coproduct-types.lagda.md @@ -133,11 +133,11 @@ module _ pr1 compute-Eq-coproduct-inl-inl = map-compute-Eq-coproduct-inl-inl pr2 compute-Eq-coproduct-inl-inl = is-equiv-map-compute-Eq-coproduct-inl-inl - compute-eq-coproduct-inl-inl : Id {A = A + B} (inl x) (inl y) ≃ (x ＝ y) + compute-eq-coproduct-inl-inl : (inl x ＝ inl y) ≃ (x ＝ y) compute-eq-coproduct-inl-inl = compute-Eq-coproduct-inl-inl ∘e extensionality-coproduct (inl x) (inl y) - map-compute-eq-coproduct-inl-inl : Id {A = A + B} (inl x) (inl y) → x ＝ y + map-compute-eq-coproduct-inl-inl : (inl x ＝ inl y) → x ＝ y map-compute-eq-coproduct-inl-inl = map-equiv compute-eq-coproduct-inl-inl is-equiv-map-compute-eq-coproduct-inl-inl : @@ -219,11 +219,11 @@ module _ pr1 compute-Eq-coproduct-inr-inr = map-compute-Eq-coproduct-inr-inr pr2 compute-Eq-coproduct-inr-inr = is-equiv-map-compute-Eq-coproduct-inr-inr - compute-eq-coproduct-inr-inr : Id {A = A + B} (inr x) (inr y) ≃ (x ＝ y) + compute-eq-coproduct-inr-inr : (inr x ＝ inr y) ≃ (x ＝ y) compute-eq-coproduct-inr-inr = compute-Eq-coproduct-inr-inr ∘e extensionality-coproduct (inr x) (inr y) - map-compute-eq-coproduct-inr-inr : Id {A = A + B} (inr x) (inr y) → x ＝ y + map-compute-eq-coproduct-inr-inr : (inr x ＝ inr y) → x ＝ y map-compute-eq-coproduct-inr-inr = map-equiv compute-eq-coproduct-inr-inr is-equiv-map-compute-eq-coproduct-inr-inr : diff --git a/src/group-theory/furstenberg-groups.lagda.md b/src/group-theory/furstenberg-groups.lagda.md index aeeea480eb..0da3bff5fb 100644 --- a/src/group-theory/furstenberg-groups.lagda.md +++ b/src/group-theory/furstenberg-groups.lagda.md @@ -27,7 +27,7 @@ Furstenberg-Group l = ( type-trunc-Prop (type-Set X)) × ( Σ ( type-Set X → type-Set X → type-Set X) ( λ μ → - ( (x y z : type-Set X) → Id (μ (μ x z) (μ y z)) (μ x y)) × + ( (x y z : type-Set X) → μ (μ x z) (μ y z) ＝ μ x y) × ( Σ ( type-Set X → type-Set X → type-Set X) ( λ δ → (x y : type-Set X) → μ x (δ x y) ＝ y))))) ``` diff --git a/src/group-theory/sheargroups.lagda.md b/src/group-theory/sheargroups.lagda.md index 7b76e4aae0..d513a5332d 100644 --- a/src/group-theory/sheargroups.lagda.md +++ b/src/group-theory/sheargroups.lagda.md @@ -30,5 +30,5 @@ Sheargroup l = ( (x : type-Set X) → m e x ＝ x) × ( ( (x : type-Set X) → m x x ＝ e) × ( (x y z : type-Set X) → - Id (m x (m y z)) (m (m (m x (m y e)) e) z)))))) + m x (m y z) ＝ m (m (m x (m y e)) e) z))))) ``` diff --git a/src/linear-algebra/transposition-matrices.lagda.md b/src/linear-algebra/transposition-matrices.lagda.md index 13effe6619..72d36918e1 100644 --- a/src/linear-algebra/transposition-matrices.lagda.md +++ b/src/linear-algebra/transposition-matrices.lagda.md @@ -54,7 +54,7 @@ is-involution-transpose-matrix {m = succ-ℕ m} (r ∷ rs) = lemma-first-row : {l : Level} → {A : UU l} → {m n : ℕ} → (x : tuple A n) → (xs : matrix A m n) → - Id x (map-tuple head-tuple (transpose-matrix (x ∷ xs))) + x ＝ map-tuple head-tuple (transpose-matrix (x ∷ xs)) lemma-first-row {n = zero-ℕ} empty-tuple _ = refl lemma-first-row {n = succ-ℕ m} (k ∷ ks) xs = ap (_∷_ k) (lemma-first-row ks (map-tuple tail-tuple xs)) diff --git a/src/lists/concatenation-lists.lagda.md b/src/lists/concatenation-lists.lagda.md index b70cbc2f90..2e643dd79f 100644 --- a/src/lists/concatenation-lists.lagda.md +++ b/src/lists/concatenation-lists.lagda.md @@ -92,7 +92,7 @@ snoc-concat-unit (cons x xs) a = ap (cons x) (snoc-concat-unit xs a) ```agda length-concat-list : {l1 : Level} {A : UU l1} (x y : list A) → - Id (length-list (concat-list x y)) ((length-list x) +ℕ (length-list y)) + length-list (concat-list x y) ＝ length-list x +ℕ length-list y length-concat-list nil y = inv (left-unit-law-add-ℕ (length-list y)) length-concat-list (cons a x) y = ( ap succ-ℕ (length-concat-list x y)) ∙ diff --git a/src/lists/universal-property-lists-wild-monoids.lagda.md b/src/lists/universal-property-lists-wild-monoids.lagda.md index 8dbc5be3be..08d8f1c6bd 100644 --- a/src/lists/universal-property-lists-wild-monoids.lagda.md +++ b/src/lists/universal-property-lists-wild-monoids.lagda.md @@ -336,8 +336,8 @@ htpy-elim-list-Wild-Monoid {X = X} M g h H = ( unit-list x) ( l)))) β : (x y : pr1 (pr1 (list-Wild-Monoid X))) → - Id ( pr2 (pr1 g) x y ∙ ap-mul-Wild-Monoid M (α x) (α y)) - ( α (concat-list x y) ∙ pr2 (pr1 h) x y) + pr2 (pr1 g) x y ∙ ap-mul-Wild-Monoid M (α x) (α y) ＝ + α (concat-list x y) ∙ pr2 (pr1 h) x y β nil y = {!!} β (cons x x₁) y = {!!} γ : pr2 g ＝ α nil ∙ pr2 h diff --git a/src/order-theory/finitely-graded-posets.lagda.md b/src/order-theory/finitely-graded-posets.lagda.md index 9ad45ac90f..73f51fae4d 100644 --- a/src/order-theory/finitely-graded-posets.lagda.md +++ b/src/order-theory/finitely-graded-posets.lagda.md @@ -202,9 +202,9 @@ If chains with jumps are never used, we'd like to call the following chains. eq-path-elements-Finitely-Graded-Poset : {l1 l2 : Level} {k : ℕ} (X : Finitely-Graded-Poset l1 l2 k) (x y : type-Finitely-Graded-Poset X) → - (p : Id (shape-Finitely-Graded-Poset X x) - (shape-Finitely-Graded-Poset X y)) → - path-elements-Finitely-Graded-Poset X x y → x ＝ y + shape-Finitely-Graded-Poset X x ＝ shape-Finitely-Graded-Poset X y → + path-elements-Finitely-Graded-Poset X x y → + x ＝ y eq-path-elements-Finitely-Graded-Poset {k} X (pair i1 x) (pair .i1 .x) p refl-path-faces-Finitely-Graded-Poset = refl eq-path-elements-Finitely-Graded-Poset {k = succ-ℕ k} X (pair i1 x) diff --git a/src/ring-theory/rings.lagda.md b/src/ring-theory/rings.lagda.md index 4bbf93fbc2..e743cf7234 100644 --- a/src/ring-theory/rings.lagda.md +++ b/src/ring-theory/rings.lagda.md @@ -61,9 +61,9 @@ has-mul-Ab A = ( λ μ → ( is-unital (pr1 μ)) × ( ( (a b c : type-Ab A) → - Id (pr1 μ a (add-Ab A b c)) (add-Ab A (pr1 μ a b) (pr1 μ a c))) × + pr1 μ a (add-Ab A b c) ＝ add-Ab A (pr1 μ a b) (pr1 μ a c)) × ( (a b c : type-Ab A) → - Id (pr1 μ (add-Ab A a b) c) (add-Ab A (pr1 μ a c) (pr1 μ b c))))) + pr1 μ (add-Ab A a b) c ＝ add-Ab A (pr1 μ a c) (pr1 μ b c)))) Ring : (l1 : Level) → UU (lsuc l1) Ring l1 = Σ (Ab l1) has-mul-Ab diff --git a/src/structured-types/morphisms-h-spaces.lagda.md b/src/structured-types/morphisms-h-spaces.lagda.md index bdffcf852a..30381abe58 100644 --- a/src/structured-types/morphisms-h-spaces.lagda.md +++ b/src/structured-types/morphisms-h-spaces.lagda.md @@ -317,5 +317,5 @@ preserves-mul-htpy : {f g : A → B} (μf : preserves-mul μA μB f) (μg : preserves-mul μA μB g) → (f ~ g) → UU (l1 ⊔ l2) preserves-mul-htpy {A = A} μA μB μf μg H = - (a b : A) → Id (μf ∙ ap-binary μB (H a) (H b)) (H (μA a b) ∙ μg) + (a b : A) → μf ∙ ap-binary μB (H a) (H b) ＝ H (μA a b) ∙ μg ``` diff --git a/src/structured-types/morphisms-magmas.lagda.md b/src/structured-types/morphisms-magmas.lagda.md index e5d3920e3b..a627ee3b41 100644 --- a/src/structured-types/morphisms-magmas.lagda.md +++ b/src/structured-types/morphisms-magmas.lagda.md @@ -30,7 +30,7 @@ module _ preserves-mul-Magma : (type-Magma M → type-Magma N) → UU (l1 ⊔ l2) preserves-mul-Magma f = - (x y : type-Magma M) → Id (f (mul-Magma M x y)) (mul-Magma N (f x) (f y)) + (x y : type-Magma M) → f (mul-Magma M x y) ＝ mul-Magma N (f x) (f y) hom-Magma : UU (l1 ⊔ l2) hom-Magma = Σ (type-Magma M → type-Magma N) preserves-mul-Magma diff --git a/src/synthetic-homotopy-theory/triple-loop-spaces.lagda.md b/src/synthetic-homotopy-theory/triple-loop-spaces.lagda.md index cc980fb796..ab74b46e9b 100644 --- a/src/synthetic-homotopy-theory/triple-loop-spaces.lagda.md +++ b/src/synthetic-homotopy-theory/triple-loop-spaces.lagda.md @@ -38,7 +38,7 @@ module _ Ω³ A = iterated-loop-space 3 A type-Ω³ : {A : UU l} (a : A) → UU l - type-Ω³ a = Id (refl-Ω² {a = a}) (refl-Ω² {a = a}) + type-Ω³ a = (refl-Ω² {a = a} ＝ refl-Ω² {a = a}) refl-Ω³ : {A : UU l} {a : A} → type-Ω³ a refl-Ω³ = refl @@ -115,7 +115,7 @@ left-unit-law-z-concat-Ω³ α = super-naturality-right-unit : {l : Level} {A : UU l} {x y z : A} {p q : x ＝ y} {α β : p ＝ q} (γ : α ＝ β) (u : y ＝ z) → - Id (ap (λ ω → horizontal-concat-Id² ω (refl {x = u})) γ) {!!} + ap (λ ω → horizontal-concat-Id² ω (refl {x = u})) γ ＝ {!!} super-naturality-right-unit α = {!!} -} diff --git a/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md b/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md index 7712fb9d47..93ae7db590 100644 --- a/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md +++ b/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md @@ -200,17 +200,15 @@ path-total-path-fiber B x q = eq-pair-eq-fiber (inv q) tr-path-total-path-fiber : { l1 l2 : Level} {A : UU l1} {B : A → UU l2} (c : Σ A B) (x : A) → { y y' : B x} (q : y' ＝ y) (α : c ＝ pair x y') → - Id - ( tr (λ z → c ＝ pair x z) q α) - ( α ∙ (inv (path-total-path-fiber B x q))) + tr (λ z → c ＝ pair x z) q α ＝ α ∙ inv (path-total-path-fiber B x q) tr-path-total-path-fiber c x refl α = inv right-unit segment-Σ : { l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} → { x x' : A} (p : x ＝ x') { F : UU l3} {F' : UU l4} (f : F ≃ F') ( e : F ≃ B x) (e' : F' ≃ B x') - ( H : ((map-equiv e') ∘ (map-equiv f)) ~ ((tr B p) ∘ (map-equiv e))) (y : F) → - Id (pair x (map-equiv e y)) (pair x' (map-equiv e' (map-equiv f y))) + ( H : map-equiv e' ∘ map-equiv f ~ tr B p ∘ map-equiv e) (y : F) → + pair x (map-equiv e y) ＝ pair x' (map-equiv e' (map-equiv f y)) segment-Σ refl f e e' H y = path-total-path-fiber _ _ (H y) contraction-total-space' : @@ -242,9 +240,7 @@ tr-path-total-tr-coherence : { F : UU l3} {F' : UU l4} (f : F ≃ F') ( e : F ≃ B x) (e' : F' ≃ B x) ( H : ((map-equiv e') ∘ (map-equiv f)) ~ (map-equiv e)) → (y : F) (α : Id c (pair x (map-equiv e' (map-equiv f y)))) → - Id - ( tr (λ z → c ＝ pair x z) (H y) α) - ( α ∙ (inv (segment-Σ refl f e e' H y))) + tr (λ z → c ＝ pair x z) (H y) α ＝ α ∙ (inv (segment-Σ refl f e e' H y)) tr-path-total-tr-coherence c x f e e' H y α = tr-path-total-path-fiber c x (H y) α @@ -252,12 +248,12 @@ square-tr-contraction-total-space : { l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} (c : Σ A B) → { x x' : A} (p : x ＝ x') { F : UU l3} {F' : UU l4} (f : F ≃ F') (e : F ≃ B x) (e' : F' ≃ B x') - ( H : ((map-equiv e') ∘ (map-equiv f)) ~ ((tr B p) ∘ (map-equiv e))) + ( H : map-equiv e' ∘ map-equiv f ~ tr B p ∘ map-equiv e) (h : contraction-total-space c x) → ( map-equiv ( ( equiv-tr-contraction-total-space' c p f e e' H) ∘e - ( ( equiv-contraction-total-space c x' e') ∘e - ( equiv-tr (contraction-total-space c) p))) + ( equiv-contraction-total-space c x' e') ∘e + ( equiv-tr (contraction-total-space c) p)) ( h)) ~ ( map-equiv (equiv-contraction-total-space c x e) h) square-tr-contraction-total-space c refl f e e' H h y = @@ -269,14 +265,13 @@ dependent-identification-contraction-total-space' : {l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} (c : Σ A B) → {x x' : A} (p : x ＝ x') → {F : UU l3} {F' : UU l4} (f : F ≃ F') ( e : F ≃ B x) (e' : F' ≃ B x') - (H : ((map-equiv e') ∘ (map-equiv f)) ~ ((tr B p) ∘ (map-equiv e))) → + (H : map-equiv e' ∘ map-equiv f ~ tr B p ∘ map-equiv e) → (h : (y : F) → c ＝ pair x (map-equiv e y)) → (h' : (y' : F') → c ＝ pair x' (map-equiv e' y')) → UU (l1 ⊔ l2 ⊔ l3) dependent-identification-contraction-total-space' c {x} {x'} p {F} {F'} f e e' H h h' = - ( map-Π - ( λ y → concat' c (segment-Σ p f e e' H y)) h) ~ + ( map-Π (λ y → concat' c (segment-Σ p f e e' H y)) h) ~ ( precomp-Π ( map-equiv f) ( λ y' → c ＝ pair x' (map-equiv e' y')) @@ -286,7 +281,7 @@ map-dependent-identification-contraction-total-space' : { l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} (c : Σ A B) → { x x' : A} (p : x ＝ x') → { F : UU l3} {F' : UU l4} (f : F ≃ F') ( e : F ≃ B x) (e' : F' ≃ B x') - ( H : ((map-equiv e') ∘ (map-equiv f)) ~ ((tr B p) ∘ (map-equiv e))) → + ( H : map-equiv e' ∘ map-equiv f ~ tr B p ∘ map-equiv e) → ( h : contraction-total-space' c x e) → ( h' : contraction-total-space' c x' e') → ( dependent-identification-contraction-total-space' c p f e e' H h h') → @@ -327,7 +322,7 @@ equiv-dependent-identification-contraction-total-space' : { l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} (c : Σ A B) → { x x' : A} (p : x ＝ x') → { F : UU l3} {F' : UU l4} (f : F ≃ F') ( e : F ≃ B x) (e' : F' ≃ B x') - ( H : ((map-equiv e') ∘ (map-equiv f)) ~ ((tr B p) ∘ (map-equiv e))) → + ( H : map-equiv e' ∘ map-equiv f ~ tr B p ∘ map-equiv e) → ( h : contraction-total-space' c x e) → ( h' : contraction-total-space' c x' e') → ( dependent-identification (contraction-total-space c) p diff --git a/src/type-theories/dependent-type-theories.lagda.md b/src/type-theories/dependent-type-theories.lagda.md index d60c0db666..550570f8d9 100644 --- a/src/type-theories/dependent-type-theories.lagda.md +++ b/src/type-theories/dependent-type-theories.lagda.md @@ -343,7 +343,7 @@ We show that systems form a category. (q : C ＝ C') {q' : constant-fibered-system A C ＝ constant-fibered-system A C'} (β : ap (constant-fibered-system A) q ＝ q') - (r : Id (tr (λ t → t) (ap-binary hom-system p q) g) g') + (r : tr id (ap-binary hom-system p q) g ＝ g') {f : hom-system A B} {f' : hom-system A B'} → htpy-section-system' p' f f' → htpy-section-system' q' (comp-hom-system g f) (comp-hom-system g' f') @@ -571,15 +571,16 @@ weakening structure. field type : (X : system.type A) → - Id ( tr - ( system.element (system.slice B (section-system.type h X))) - ( section-system.type - ( preserves-weakening.type Wh X) - ( X)) - ( section-system.element - ( section-system.slice h X) - ( generic-element.type δA X))) - ( generic-element.type δB (section-system.type h X)) + Id + ( tr + ( system.element (system.slice B (section-system.type h X))) + ( section-system.type + ( preserves-weakening.type Wh X) + ( X)) + ( section-system.element + ( section-system.slice h X) + ( generic-element.type δA X))) + ( generic-element.type δB (section-system.type h X)) slice : (X : system.type A) → preserves-generic-element From 848484a1a69e02e7ced8e0185e571595479c0487 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 1 Sep 2025 19:51:02 +0200 Subject: [PATCH 12/16] another pattern --- .../dependent-type-theories.lagda.md | 33 ++++++++++--------- .../fibered-dependent-type-theories.lagda.md | 27 +++++++-------- .../simple-type-theories.lagda.md | 8 ++--- .../unityped-type-theories.lagda.md | 8 ++--- .../classical-finite-types.lagda.md | 3 +- .../counting-dependent-pair-types.lagda.md | 10 +++--- .../cyclic-finite-types.lagda.md | 4 +-- .../decidable-propositions.lagda.md | 4 +-- .../dependent-pair-types.lagda.md | 2 +- ...t-truncation-over-finite-products.lagda.md | 5 +-- .../fibers-of-maps.lagda.md | 2 +- ...tations-complete-undirected-graph.lagda.md | 18 +++++----- 12 files changed, 64 insertions(+), 60 deletions(-) diff --git a/src/type-theories/dependent-type-theories.lagda.md b/src/type-theories/dependent-type-theories.lagda.md index 550570f8d9..6d14984834 100644 --- a/src/type-theories/dependent-type-theories.lagda.md +++ b/src/type-theories/dependent-type-theories.lagda.md @@ -1416,22 +1416,23 @@ We define what it means for a dependent type theory to have Π-types. (preserves-weakening.type (hom-dtt.W (function-types.sys Π (natural-numbers.N N))) ?) ?)!} {- - Id ( section-system.type - ( weakening.type (type-theory.W A) X) - ( section-system.type - ( hom-dtt.sys (function-types.sys Π (natural-numbers.N N))) - ( section-system.type - ( weakening.type (type-theory.W A) (natural-numbers.N N)) - (natural-numbers.N N)))) - ( section-system.type - ( hom-dtt.sys - ( function-types.sys (function-types.slice Π X) - ( natural-numbers.N (natural-numbers-slice A Π N X)))) - ( section-system.type - ( weakening.type - ( type-theory.W (slice-dtt A X)) - ( natural-numbers.N (natural-numbers-slice A Π N X))) - ( natural-numbers.N (natural-numbers-slice A Π N X)))) + Id + ( section-system.type + ( weakening.type (type-theory.W A) X) + ( section-system.type + ( hom-dtt.sys (function-types.sys Π (natural-numbers.N N))) + ( section-system.type + ( weakening.type (type-theory.W A) (natural-numbers.N N)) + (natural-numbers.N N)))) + ( section-system.type + ( hom-dtt.sys + ( function-types.sys (function-types.slice Π X) + ( natural-numbers.N (natural-numbers-slice A Π N X)))) + ( section-system.type + ( weakening.type + ( type-theory.W (slice-dtt A X)) + ( natural-numbers.N (natural-numbers-slice A Π N X))) + ( natural-numbers.N (natural-numbers-slice A Π N X)))) -} ( section-system.element ( weakening.type (type-theory.W A) X) diff --git a/src/type-theories/fibered-dependent-type-theories.lagda.md b/src/type-theories/fibered-dependent-type-theories.lagda.md index f4a605ae1d..dfd266af16 100644 --- a/src/type-theories/fibered-dependent-type-theories.lagda.md +++ b/src/type-theories/fibered-dependent-type-theories.lagda.md @@ -619,19 +619,20 @@ generic elements to be equipped with generic elements. type : {X : system.type A} {Y : fibered-system.type B X} {x : system.element A X} (y : fibered-system.element B Y x) → - Id ( double-tr - ( λ α β γ → fibered-system.element B {X = α} β γ) - ( section-system.type - ( substitution-cancels-weakening.type S!WA x) - ( X)) - ( section-fibered-system.type - ( fibered-substitution-cancels-weakening.type S!WB y) - ( Y)) - ( generic-element-is-identity.type δidA x) - ( section-fibered-system.element - ( fibered-substitution.type SB y) - ( fibered-generic-element.type δB Y))) - ( y) + Id + ( double-tr + ( λ α β γ → fibered-system.element B {X = α} β γ) + ( section-system.type + ( substitution-cancels-weakening.type S!WA x) + ( X)) + ( section-fibered-system.type + ( fibered-substitution-cancels-weakening.type S!WB y) + ( Y)) + ( generic-element-is-identity.type δidA x) + ( section-fibered-system.element + ( fibered-substitution.type SB y) + ( fibered-generic-element.type δB Y))) + ( y) slice : {X : system.type A} (Y : fibered-system.type B X) → fibered-generic-element-is-identity diff --git a/src/type-theories/simple-type-theories.lagda.md b/src/type-theories/simple-type-theories.lagda.md index 72974ca11d..81f56fba52 100644 --- a/src/type-theories/simple-type-theories.lagda.md +++ b/src/type-theories/simple-type-theories.lagda.md @@ -452,10 +452,10 @@ We specialize the above definitions to nonhomogenous homotopies. field element : {X : T} (x : system.element A X) → - Id ( section-system.element - ( substitution.element S x) - ( generic-element.element δ X)) - ( x) + section-system.element + ( substitution.element S x) + ( generic-element.element δ X) ＝ + x slice : (X : T) → generic-element-is-identity diff --git a/src/type-theories/unityped-type-theories.lagda.md b/src/type-theories/unityped-type-theories.lagda.md index 99c7fb918f..b40d443dfc 100644 --- a/src/type-theories/unityped-type-theories.lagda.md +++ b/src/type-theories/unityped-type-theories.lagda.md @@ -283,10 +283,10 @@ module unityped where field element : (x : system.element σ) → - Id ( hom-system.element - ( substitution.element S x) - ( generic-element.element δ)) - ( x) + hom-system.element + ( substitution.element S x) + ( generic-element.element δ) ＝ + x slice : generic-element-is-identity ( substitution.slice S) diff --git a/src/univalent-combinatorics/classical-finite-types.lagda.md b/src/univalent-combinatorics/classical-finite-types.lagda.md index 6e5a8d7228..5a00365542 100644 --- a/src/univalent-combinatorics/classical-finite-types.lagda.md +++ b/src/univalent-combinatorics/classical-finite-types.lagda.md @@ -60,7 +60,8 @@ Eq-classical-Fin : (k : ℕ) (x y : classical-Fin k) → UU lzero Eq-classical-Fin k x y = nat-classical-Fin k x ＝ nat-classical-Fin k y eq-succ-classical-Fin : - (k : ℕ) (x y : classical-Fin k) → Id {A = classical-Fin k} x y → + (k : ℕ) (x y : classical-Fin k) → + x ＝ y → Id { A = classical-Fin (succ-ℕ k)} ( pair (succ-ℕ (pr1 x)) (pr2 x)) diff --git a/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md b/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md index 077b75f365..879eb9fefd 100644 --- a/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md +++ b/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md @@ -97,8 +97,8 @@ abstract number-of-elements-count-Σ' : {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (k : ℕ) (e : Fin k ≃ A) → (f : (x : A) → count (B x)) → - Id ( number-of-elements-count (count-Σ' k e f)) - ( sum-Fin-ℕ k (λ x → number-of-elements-count (f (map-equiv e x)))) + number-of-elements-count (count-Σ' k e f) ＝ + sum-Fin-ℕ k (λ x → number-of-elements-count (f (map-equiv e x))) number-of-elements-count-Σ' zero-ℕ e f = refl number-of-elements-count-Σ' (succ-ℕ k) e f = ( number-of-elements-count-coproduct @@ -112,8 +112,8 @@ abstract number-of-elements-count-Σ : {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (e : count A) (f : (x : A) → count (B x)) → - Id ( number-of-elements-count (count-Σ e f)) - ( sum-count-ℕ e (λ x → number-of-elements-count (f x))) + number-of-elements-count (count-Σ e f) ＝ + sum-count-ℕ e (λ x → number-of-elements-count (f x)) number-of-elements-count-Σ (pair k e) f = number-of-elements-count-Σ' k e f ``` @@ -150,7 +150,7 @@ count-fiber-map-section-family {l1} {l2} {A} {B} b e f (pair y z) = ( pair y refl)) ∘e ( inv-associative-Σ A ( λ x → x ＝ y) - ( λ t → Id (tr B (pr2 t) (b (pr1 t))) z))) ∘e + ( λ t → tr B (pr2 t) (b (pr1 t)) ＝ z))) ∘e ( equiv-tot (λ x → equiv-pair-eq-Σ (pair x (b x)) (pair y z)))) ( count-eq (has-decidable-equality-count (f y)) (b y) z) diff --git a/src/univalent-combinatorics/cyclic-finite-types.lagda.md b/src/univalent-combinatorics/cyclic-finite-types.lagda.md index a2f6f15d3b..bc93b6858d 100644 --- a/src/univalent-combinatorics/cyclic-finite-types.lagda.md +++ b/src/univalent-combinatorics/cyclic-finite-types.lagda.md @@ -427,7 +427,7 @@ preserves-pred-preserves-succ-map-ℤ-Mod k f H x = compute-map-preserves-succ-map-ℤ-Mod' : (k : ℕ) (f : ℤ-Mod k → ℤ-Mod k) → (f ∘ succ-ℤ-Mod k) ~ (succ-ℤ-Mod k ∘ f) → - (x : ℤ) → Id (add-ℤ-Mod k (mod-ℤ k x) (f (zero-ℤ-Mod k))) (f (mod-ℤ k x)) + (x : ℤ) → add-ℤ-Mod k (mod-ℤ k x) (f (zero-ℤ-Mod k)) ＝ f (mod-ℤ k x) compute-map-preserves-succ-map-ℤ-Mod' k f H (inl zero-ℕ) = ( ap (add-ℤ-Mod' k (f (zero-ℤ-Mod k))) (mod-neg-one-ℤ k)) ∙ ( ( inv (is-left-add-neg-one-pred-ℤ-Mod k (f (zero-ℤ-Mod k)))) ∙ @@ -470,7 +470,7 @@ compute-map-preserves-succ-map-ℤ-Mod' k f H (inr (inr (succ-ℕ x))) = compute-map-preserves-succ-map-ℤ-Mod : (k : ℕ) (f : ℤ-Mod k → ℤ-Mod k) (H : (f ∘ succ-ℤ-Mod k) ~ (succ-ℤ-Mod k ∘ f)) - (x : ℤ-Mod k) → Id (add-ℤ-Mod k x (f (zero-ℤ-Mod k))) (f x) + (x : ℤ-Mod k) → add-ℤ-Mod k x (f (zero-ℤ-Mod k)) ＝ f x compute-map-preserves-succ-map-ℤ-Mod k f H x = ( ap (add-ℤ-Mod' k (f (zero-ℤ-Mod k))) (inv (is-section-int-ℤ-Mod k x))) ∙ ( ( compute-map-preserves-succ-map-ℤ-Mod' k f H (int-ℤ-Mod k x)) ∙ diff --git a/src/univalent-combinatorics/decidable-propositions.lagda.md b/src/univalent-combinatorics/decidable-propositions.lagda.md index c338b85b6b..320887a5c0 100644 --- a/src/univalent-combinatorics/decidable-propositions.lagda.md +++ b/src/univalent-combinatorics/decidable-propositions.lagda.md @@ -89,8 +89,8 @@ cases-number-of-elements-count-eq d (inr f) = refl abstract number-of-elements-count-eq : {l : Level} {X : UU l} (d : has-decidable-equality X) (x y : X) → - Id ( number-of-elements-count (count-eq d x y)) - ( number-of-elements-count-eq' d x y) + number-of-elements-count (count-eq d x y) ＝ + number-of-elements-count-eq' d x y number-of-elements-count-eq d x y = cases-number-of-elements-count-eq d (d x y) ``` diff --git a/src/univalent-combinatorics/dependent-pair-types.lagda.md b/src/univalent-combinatorics/dependent-pair-types.lagda.md index f67c29c4bf..f1f8a7ba5f 100644 --- a/src/univalent-combinatorics/dependent-pair-types.lagda.md +++ b/src/univalent-combinatorics/dependent-pair-types.lagda.md @@ -110,7 +110,7 @@ abstract ( λ x → equiv-eq-pair-Σ (map-section-family b x) t)) ∘e ( ( associative-Σ A ( λ (x : A) → x ＝ pr1 t) - ( λ s → Id (tr B (pr2 s) (b (pr1 s))) (pr2 t))) ∘e + ( λ s → tr B (pr2 s) (b (pr1 s)) ＝ pr2 t)) ∘e ( inv-left-unit-law-Σ-is-contr ( is-torsorial-Id' (pr1 t)) ( pair (pr1 t) refl)))))) diff --git a/src/univalent-combinatorics/distributivity-of-set-truncation-over-finite-products.lagda.md b/src/univalent-combinatorics/distributivity-of-set-truncation-over-finite-products.lagda.md index 64609a61a2..8ef38ae7ee 100644 --- a/src/univalent-combinatorics/distributivity-of-set-truncation-over-finite-products.lagda.md +++ b/src/univalent-combinatorics/distributivity-of-set-truncation-over-finite-products.lagda.md @@ -170,8 +170,9 @@ module _ ( λ h → ( ( inv-equiv equiv-funext) ∘e ( equiv-precomp-Π e - ( λ x → Id ((map-equiv f ∘ unit-trunc-Set) h x) - ( map-Π (λ y → unit-trunc-Set) h x)))) ∘e + ( λ x → + map-equiv f (unit-trunc-Set h) x ＝ + map-Π (λ y → unit-trunc-Set) h x))) ∘e ( equiv-funext)))) ( is-contr-equiv' ( Σ ( ( type-trunc-Set ((x : Fin k) → B (map-equiv e x))) ≃ diff --git a/src/univalent-combinatorics/fibers-of-maps.lagda.md b/src/univalent-combinatorics/fibers-of-maps.lagda.md index 426d066387..17f21e3818 100644 --- a/src/univalent-combinatorics/fibers-of-maps.lagda.md +++ b/src/univalent-combinatorics/fibers-of-maps.lagda.md @@ -123,7 +123,7 @@ abstract ( pair y refl)) ∘e ( inv-associative-Σ A ( λ x → x ＝ y) - ( λ t → Id (tr B (pr2 t) (b (pr1 t))) z))) ∘e + ( λ t → tr B (pr2 t) (b (pr1 t)) ＝ z))) ∘e ( equiv-tot (λ x → equiv-pair-eq-Σ (pair x (b x)) (pair y z)))) ( is-finite-eq (has-decidable-equality-is-finite (g y))) ``` diff --git a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md index cdab2fa97c..0ed55c9e9a 100644 --- a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md +++ b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md @@ -736,7 +736,7 @@ module _ is-decidable ( map-equiv e (pr1 two-elements) ＝ pr1 two-elements) → is-decidable - ( Id (map-equiv e (pr1 (pr2 two-elements))) (pr1 (pr2 two-elements))) → + ( map-equiv e (pr1 (pr2 two-elements)) ＝ pr1 (pr2 two-elements)) → Σ X (λ z → type-Decidable-Prop (pr1 Y z)) cases-orientation-aut-count e Y (pair x (pair y (pair np P))) (inl q) r = @@ -871,9 +871,9 @@ module _ (Y : 2-Element-Decidable-Subtype l X) (x : X) → x ≠ i → x ≠ j → ( ( pr1 (two-elements-transposition eX Y) ＝ x) × - ( Id (pr1 (pr2 (two-elements-transposition eX Y))) i)) + + ( pr1 (pr2 (two-elements-transposition eX Y)) ＝ i)) + ( ( pr1 (two-elements-transposition eX Y) ＝ i) × - ( Id (pr1 (pr2 (two-elements-transposition eX Y))) x)) → + ( pr1 (pr2 (two-elements-transposition eX Y)) ＝ x)) → Id ( pr1 (orientation-two-elements-count i j np Y)) ( x) @@ -954,9 +954,9 @@ module _ (Y : 2-Element-Decidable-Subtype l X) (x : X) → x ≠ i → x ≠ j → ( ( pr1 (two-elements-transposition eX Y) ＝ x) × - ( Id (pr1 (pr2 (two-elements-transposition eX Y))) i)) + + ( pr1 (pr2 (two-elements-transposition eX Y)) ＝ i)) + ( ( pr1 (two-elements-transposition eX Y) ＝ i) × - ( Id (pr1 (pr2 (two-elements-transposition eX Y))) x)) → + ( pr1 (pr2 (two-elements-transposition eX Y)) ＝ x)) → Id ( pr1 ( orientation-aut-count @@ -1145,9 +1145,9 @@ module _ (Y : 2-Element-Decidable-Subtype l X) (x : X) → x ≠ i → x ≠ j → ( ( pr1 (two-elements-transposition eX Y) ＝ x) × - ( Id (pr1 (pr2 (two-elements-transposition eX Y))) j)) + + ( pr1 (pr2 (two-elements-transposition eX Y)) ＝ j)) + ( ( pr1 (two-elements-transposition eX Y) ＝ j) × - ( Id (pr1 (pr2 (two-elements-transposition eX Y))) x)) → + ( pr1 (pr2 (two-elements-transposition eX Y)) ＝ x)) → Id ( pr1 (orientation-two-elements-count i j np Y)) ( x) @@ -1254,9 +1254,9 @@ module _ (Y : 2-Element-Decidable-Subtype l X) (x : X) → x ≠ i → x ≠ j → ( ( pr1 (two-elements-transposition eX Y) ＝ x) × - ( Id (pr1 (pr2 (two-elements-transposition eX Y))) j)) + + ( pr1 (pr2 (two-elements-transposition eX Y)) ＝ j)) + ( ( pr1 (two-elements-transposition eX Y) ＝ j) × - ( Id (pr1 (pr2 (two-elements-transposition eX Y))) x)) → + ( pr1 (pr2 (two-elements-transposition eX Y)) ＝ x)) → Id ( pr1 ( orientation-aut-count From 250bff90597ead830c2ec4c9796175bc0801206b Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 1 Sep 2025 19:51:47 +0200 Subject: [PATCH 13/16] fix uneven indentation --- src/finite-group-theory/transpositions.lagda.md | 2 +- src/group-theory/homomorphisms-semigroups.lagda.md | 4 ++-- src/linear-algebra/left-modules-rings.lagda.md | 4 ++-- src/linear-algebra/right-modules-rings.lagda.md | 2 +- src/order-theory/order-preserving-maps-posets.lagda.md | 2 +- src/order-theory/order-preserving-maps-preorders.lagda.md | 2 +- 6 files changed, 8 insertions(+), 8 deletions(-) diff --git a/src/finite-group-theory/transpositions.lagda.md b/src/finite-group-theory/transpositions.lagda.md index 5103dfc047..ce3e270663 100644 --- a/src/finite-group-theory/transpositions.lagda.md +++ b/src/finite-group-theory/transpositions.lagda.md @@ -387,7 +387,7 @@ module _ (x : X) (p : is-in-2-Element-Decidable-Subtype Y x) → (h : Fin 2 ≃ type-2-Element-Decidable-Subtype Y) → (k1 k2 k3 : Fin 2) → - map-inv-equiv h (pair x p) ＝ k1 → + map-inv-equiv h (pair x p) ＝ k1 → Id ( map-inv-equiv h ( pair diff --git a/src/group-theory/homomorphisms-semigroups.lagda.md b/src/group-theory/homomorphisms-semigroups.lagda.md index 9f9bb485b1..2312dc77ab 100644 --- a/src/group-theory/homomorphisms-semigroups.lagda.md +++ b/src/group-theory/homomorphisms-semigroups.lagda.md @@ -224,7 +224,7 @@ module _ left-unit-law-comp-hom-Semigroup : { l1 l2 : Level} (G : Semigroup l1) (H : Semigroup l2) ( f : hom-Semigroup G H) → - comp-hom-Semigroup G H H (id-hom-Semigroup H) f ＝ f + comp-hom-Semigroup G H H (id-hom-Semigroup H) f ＝ f left-unit-law-comp-hom-Semigroup G (pair (pair H is-set-H) (pair μ-H associative-H)) (pair f μ-f) = eq-htpy-hom-Semigroup G @@ -234,7 +234,7 @@ left-unit-law-comp-hom-Semigroup G right-unit-law-comp-hom-Semigroup : { l1 l2 : Level} (G : Semigroup l1) (H : Semigroup l2) ( f : hom-Semigroup G H) → - comp-hom-Semigroup G G H f (id-hom-Semigroup G) ＝ f + comp-hom-Semigroup G G H f (id-hom-Semigroup G) ＝ f right-unit-law-comp-hom-Semigroup (pair (pair G is-set-G) (pair μ-G associative-G)) H (pair f μ-f) = eq-htpy-hom-Semigroup diff --git a/src/linear-algebra/left-modules-rings.lagda.md b/src/linear-algebra/left-modules-rings.lagda.md index 5527f53599..c1e3fd47bf 100644 --- a/src/linear-algebra/left-modules-rings.lagda.md +++ b/src/linear-algebra/left-modules-rings.lagda.md @@ -187,7 +187,7 @@ module _ abstract left-unit-law-mul-left-module-Ring : (x : type-left-module-Ring R M) → - mul-left-module-Ring R M (one-Ring R) x ＝ x + mul-left-module-Ring R M (one-Ring R) x ＝ x left-unit-law-mul-left-module-Ring = htpy-eq-hom-Ab ( ab-left-module-Ring R M) @@ -305,7 +305,7 @@ module _ abstract left-zero-law-mul-left-module-Ring : (x : type-left-module-Ring R M) → - mul-left-module-Ring R M (zero-Ring R) x ＝ zero-left-module-Ring R M + mul-left-module-Ring R M (zero-Ring R) x ＝ zero-left-module-Ring R M left-zero-law-mul-left-module-Ring = htpy-eq-hom-Ab ( ab-left-module-Ring R M) diff --git a/src/linear-algebra/right-modules-rings.lagda.md b/src/linear-algebra/right-modules-rings.lagda.md index 537a9008fe..a3f7cfe14e 100644 --- a/src/linear-algebra/right-modules-rings.lagda.md +++ b/src/linear-algebra/right-modules-rings.lagda.md @@ -160,7 +160,7 @@ module _ left-unit-law-mul-right-module-Ring : (x : type-right-module-Ring R M) → - mul-right-module-Ring R M (one-Ring R) x ＝ x + mul-right-module-Ring R M (one-Ring R) x ＝ x left-unit-law-mul-right-module-Ring = htpy-eq-hom-Ab ( ab-right-module-Ring R M) diff --git a/src/order-theory/order-preserving-maps-posets.lagda.md b/src/order-theory/order-preserving-maps-posets.lagda.md index cd0ab7058c..4505e38cb2 100644 --- a/src/order-theory/order-preserving-maps-posets.lagda.md +++ b/src/order-theory/order-preserving-maps-posets.lagda.md @@ -168,7 +168,7 @@ module _ left-unit-law-comp-hom-Poset : (f : hom-Poset P Q) → - comp-hom-Poset P Q Q (id-hom-Poset Q) f ＝ f + comp-hom-Poset P Q Q (id-hom-Poset Q) f ＝ f left-unit-law-comp-hom-Poset = left-unit-law-comp-hom-Preorder (preorder-Poset P) (preorder-Poset Q) diff --git a/src/order-theory/order-preserving-maps-preorders.lagda.md b/src/order-theory/order-preserving-maps-preorders.lagda.md index a809e54834..7338a27531 100644 --- a/src/order-theory/order-preserving-maps-preorders.lagda.md +++ b/src/order-theory/order-preserving-maps-preorders.lagda.md @@ -187,7 +187,7 @@ module _ left-unit-law-comp-hom-Preorder : (f : hom-Preorder P Q) → - comp-hom-Preorder P Q Q (id-hom-Preorder Q) f ＝ f + comp-hom-Preorder P Q Q (id-hom-Preorder Q) f ＝ f left-unit-law-comp-hom-Preorder f = eq-htpy-hom-Preorder P Q ( comp-hom-Preorder P Q Q (id-hom-Preorder Q) f) From 09cc9b4a8a32019c85b0038b3ce4905d2e766ef3 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 1 Sep 2025 20:54:35 +0200 Subject: [PATCH 14/16] manual pass --- .../commutative-rings.lagda.md | 9 +- .../euclidean-domains.lagda.md | 22 +++-- .../integral-domains.lagda.md | 9 +- .../products-commutative-rings.lagda.md | 38 ++++---- ...quences-positive-rational-numbers.lagda.md | 11 ++- ...tion-difference-integer-fractions.lagda.md | 28 +++--- ...quences-positive-rational-numbers.lagda.md | 13 ++- .../mediant-integer-fractions.lagda.md | 10 +-- .../pisano-periods.lagda.md | 19 ++-- .../universal-property-integers.lagda.md | 5 +- .../commutative-finite-rings.lagda.md | 9 +- .../dependent-products-finite-rings.lagda.md | 15 ++-- src/finite-algebra/finite-fields.lagda.md | 9 +- src/finite-algebra/finite-rings.lagda.md | 15 ++-- ...products-commutative-finite-rings.lagda.md | 87 +++++++++---------- .../products-finite-rings.lagda.md | 38 ++++---- .../delooping-sign-homomorphism.lagda.md | 5 +- .../orbits-permutations.lagda.md | 54 +++++------- src/finite-group-theory/permutations.lagda.md | 5 +- .../sign-homomorphism.lagda.md | 22 ++--- ...mpson-delooping-sign-homomorphism.lagda.md | 30 +++---- .../transpositions.lagda.md | 50 +++++------ src/foundation/truncations.lagda.md | 42 ++++----- ...e-arithmetic-dependent-pair-types.lagda.md | 12 +-- src/group-theory/abelian-groups.lagda.md | 5 +- ...cartesian-products-abelian-groups.lagda.md | 5 +- ...sian-products-commutative-monoids.lagda.md | 10 +-- ...artesian-products-concrete-groups.lagda.md | 20 ++--- .../cartesian-products-groups.lagda.md | 5 +- .../cartesian-products-monoids.lagda.md | 5 +- .../cartesian-products-semigroups.lagda.md | 5 +- src/group-theory/decidable-subgroups.lagda.md | 15 ++-- src/group-theory/groups.lagda.md | 5 +- .../homomorphisms-concrete-groups.lagda.md | 23 ++--- ...homomorphisms-generated-subgroups.lagda.md | 5 +- src/group-theory/inverse-semigroups.lagda.md | 23 +++-- ...artesian-products-concrete-groups.lagda.md | 57 +++++------- ...category-of-orbits-monoid-actions.lagda.md | 4 +- .../products-of-elements-monoids.lagda.md | 5 +- src/group-theory/semigroups.lagda.md | 5 +- .../subgroups-abelian-groups.lagda.md | 13 +-- ...roups-generated-by-subsets-groups.lagda.md | 16 ++-- src/group-theory/subgroups.lagda.md | 13 +-- ...ubstitution-functor-group-actions.lagda.md | 21 ++--- src/group-theory/symmetric-groups.lagda.md | 30 +++---- src/group-theory/torsors.lagda.md | 15 +--- .../cartesian-products-higher-groups.lagda.md | 10 +-- .../higher-groups.lagda.md | 5 +- ...-cartesian-products-higher-groups.lagda.md | 42 +++------ ...te-sequences-in-euclidean-domains.lagda.md | 19 ++-- .../left-modules-rings.lagda.md | 53 +++++------ src/linear-algebra/matrices-on-rings.lagda.md | 5 +- .../right-modules-rings.lagda.md | 53 +++++------ .../tuples-on-euclidean-domains.lagda.md | 15 ++-- src/linear-algebra/tuples-on-rings.lagda.md | 5 +- src/lists/concatenation-lists.lagda.md | 34 +++----- src/lists/flattening-lists.lagda.md | 12 +-- src/lists/functoriality-lists.lagda.md | 4 +- src/lists/lists.lagda.md | 5 +- src/lists/reversing-lists.lagda.md | 9 +- ...ersal-property-lists-wild-monoids.lagda.md | 21 ++--- .../dependent-products-rings.lagda.md | 4 +- src/ring-theory/localizations-rings.lagda.md | 36 ++++---- src/ring-theory/products-rings.lagda.md | 20 ++--- src/ring-theory/rings.lagda.md | 5 +- src/structured-types/h-spaces.lagda.md | 5 +- src/structured-types/wild-loops.lagda.md | 5 +- src/structured-types/wild-monoids.lagda.md | 43 +++------ src/structured-types/wild-semigroups.lagda.md | 5 +- .../dependent-type-theories.lagda.md | 6 +- .../2-element-decidable-subtypes.lagda.md | 26 +++--- .../cartesian-product-types.lagda.md | 7 +- .../coproduct-types.lagda.md | 17 ++-- .../counting-dependent-pair-types.lagda.md | 49 +++++------ .../cyclic-finite-types.lagda.md | 28 +++--- .../decidable-propositions.lagda.md | 5 +- .../embeddings-standard-finite-types.lagda.md | 5 +- .../fibers-of-maps.lagda.md | 12 ++- .../isotopies-latin-squares.lagda.md | 9 +- .../sums-of-natural-numbers.lagda.md | 5 +- 80 files changed, 581 insertions(+), 865 deletions(-) diff --git a/src/commutative-algebra/commutative-rings.lagda.md b/src/commutative-algebra/commutative-rings.lagda.md index 7c21de00b1..f970d153a0 100644 --- a/src/commutative-algebra/commutative-rings.lagda.md +++ b/src/commutative-algebra/commutative-rings.lagda.md @@ -656,11 +656,10 @@ module _ preserves-concat-add-list-Commutative-Ring : (l1 l2 : list type-Commutative-Ring) → - Id - ( add-list-Commutative-Ring (concat-list l1 l2)) - ( add-Commutative-Ring - ( add-list-Commutative-Ring l1) - ( add-list-Commutative-Ring l2)) + add-list-Commutative-Ring (concat-list l1 l2) ＝ + add-Commutative-Ring + (add-list-Commutative-Ring l1) + ( add-list-Commutative-Ring l2) preserves-concat-add-list-Commutative-Ring = preserves-concat-add-list-Ring ring-Commutative-Ring ``` diff --git a/src/commutative-algebra/euclidean-domains.lagda.md b/src/commutative-algebra/euclidean-domains.lagda.md index 54b46a93f5..3111ea0b91 100644 --- a/src/commutative-algebra/euclidean-domains.lagda.md +++ b/src/commutative-algebra/euclidean-domains.lagda.md @@ -635,11 +635,10 @@ module _ preserves-concat-add-list-Euclidean-Domain : (l1 l2 : list type-Euclidean-Domain) → - Id - ( add-list-Euclidean-Domain (concat-list l1 l2)) - ( add-Euclidean-Domain - ( add-list-Euclidean-Domain l1) - ( add-list-Euclidean-Domain l2)) + add-list-Euclidean-Domain (concat-list l1 l2) ＝ + add-Euclidean-Domain + ( add-list-Euclidean-Domain l1) + ( add-list-Euclidean-Domain l2) preserves-concat-add-list-Euclidean-Domain = preserves-concat-add-list-Integral-Domain integral-domain-Euclidean-Domain @@ -677,13 +676,12 @@ module _ equation-euclidean-division-Euclidean-Domain : ( x y : type-Euclidean-Domain) → ( p : is-nonzero-Euclidean-Domain y) → - ( Id - ( x) - ( add-Euclidean-Domain - ( mul-Euclidean-Domain - ( quotient-euclidean-division-Euclidean-Domain x y p) - ( y)) - ( remainder-euclidean-division-Euclidean-Domain x y p))) + x ＝ + add-Euclidean-Domain + ( mul-Euclidean-Domain + ( quotient-euclidean-division-Euclidean-Domain x y p) + ( y)) + ( remainder-euclidean-division-Euclidean-Domain x y p) equation-euclidean-division-Euclidean-Domain x y p = pr1 (pr2 (pr2 is-euclidean-domain-Euclidean-Domain x y p)) diff --git a/src/commutative-algebra/integral-domains.lagda.md b/src/commutative-algebra/integral-domains.lagda.md index a44507fc8b..5a5abcf20d 100644 --- a/src/commutative-algebra/integral-domains.lagda.md +++ b/src/commutative-algebra/integral-domains.lagda.md @@ -604,11 +604,10 @@ module _ preserves-concat-add-list-Integral-Domain : (l1 l2 : list type-Integral-Domain) → - Id - ( add-list-Integral-Domain (concat-list l1 l2)) - ( add-Integral-Domain - ( add-list-Integral-Domain l1) - ( add-list-Integral-Domain l2)) + add-list-Integral-Domain (concat-list l1 l2) ＝ + add-Integral-Domain + ( add-list-Integral-Domain l1) + ( add-list-Integral-Domain l2) preserves-concat-add-list-Integral-Domain = preserves-concat-add-list-Commutative-Ring commutative-ring-Integral-Domain diff --git a/src/commutative-algebra/products-commutative-rings.lagda.md b/src/commutative-algebra/products-commutative-rings.lagda.md index c0e1aa3cfd..916031cbac 100644 --- a/src/commutative-algebra/products-commutative-rings.lagda.md +++ b/src/commutative-algebra/products-commutative-rings.lagda.md @@ -87,9 +87,8 @@ module _ left-inverse-law-add-product-Commutative-Ring : (x : type-product-Commutative-Ring) → - Id - ( add-product-Commutative-Ring (neg-product-Commutative-Ring x) x) - zero-product-Commutative-Ring + add-product-Commutative-Ring (neg-product-Commutative-Ring x) x ＝ + zero-product-Commutative-Ring left-inverse-law-add-product-Commutative-Ring = left-inverse-law-add-product-Ring ( ring-Commutative-Ring R1) @@ -97,9 +96,8 @@ module _ right-inverse-law-add-product-Commutative-Ring : (x : type-product-Commutative-Ring) → - Id - ( add-product-Commutative-Ring x (neg-product-Commutative-Ring x)) - ( zero-product-Commutative-Ring) + add-product-Commutative-Ring x (neg-product-Commutative-Ring x) ＝ + zero-product-Commutative-Ring right-inverse-law-add-product-Commutative-Ring = right-inverse-law-add-product-Ring ( ring-Commutative-Ring R1) @@ -107,9 +105,8 @@ module _ associative-add-product-Commutative-Ring : (x y z : type-product-Commutative-Ring) → - Id - ( add-product-Commutative-Ring (add-product-Commutative-Ring x y) z) - ( add-product-Commutative-Ring x (add-product-Commutative-Ring y z)) + ( add-product-Commutative-Ring (add-product-Commutative-Ring x y) z) ＝ + ( add-product-Commutative-Ring x (add-product-Commutative-Ring y z)) associative-add-product-Commutative-Ring = associative-add-product-Ring ( ring-Commutative-Ring R1) @@ -140,9 +137,8 @@ module _ associative-mul-product-Commutative-Ring : (x y z : type-product-Commutative-Ring) → - Id - ( mul-product-Commutative-Ring (mul-product-Commutative-Ring x y) z) - ( mul-product-Commutative-Ring x (mul-product-Commutative-Ring y z)) + ( mul-product-Commutative-Ring (mul-product-Commutative-Ring x y) z) ＝ + ( mul-product-Commutative-Ring x (mul-product-Commutative-Ring y z)) associative-mul-product-Commutative-Ring = associative-mul-product-Ring ( ring-Commutative-Ring R1) @@ -166,11 +162,10 @@ module _ left-distributive-mul-add-product-Commutative-Ring : (x y z : type-product-Commutative-Ring) → - Id - ( mul-product-Commutative-Ring x (add-product-Commutative-Ring y z)) - ( add-product-Commutative-Ring - ( mul-product-Commutative-Ring x y) - ( mul-product-Commutative-Ring x z)) + ( mul-product-Commutative-Ring x (add-product-Commutative-Ring y z)) ＝ + ( add-product-Commutative-Ring + ( mul-product-Commutative-Ring x y) + ( mul-product-Commutative-Ring x z)) left-distributive-mul-add-product-Commutative-Ring = left-distributive-mul-add-product-Ring ( ring-Commutative-Ring R1) @@ -178,11 +173,10 @@ module _ right-distributive-mul-add-product-Commutative-Ring : (x y z : type-product-Commutative-Ring) → - Id - ( mul-product-Commutative-Ring (add-product-Commutative-Ring x y) z) - ( add-product-Commutative-Ring - ( mul-product-Commutative-Ring x z) - ( mul-product-Commutative-Ring y z)) + ( mul-product-Commutative-Ring (add-product-Commutative-Ring x y) z) ＝ + ( add-product-Commutative-Ring + ( mul-product-Commutative-Ring x z) + ( mul-product-Commutative-Ring y z)) right-distributive-mul-add-product-Commutative-Ring = right-distributive-mul-add-product-Ring ( ring-Commutative-Ring R1) diff --git a/src/elementary-number-theory/arithmetic-sequences-positive-rational-numbers.lagda.md b/src/elementary-number-theory/arithmetic-sequences-positive-rational-numbers.lagda.md index 0950117458..e32016c17b 100644 --- a/src/elementary-number-theory/arithmetic-sequences-positive-rational-numbers.lagda.md +++ b/src/elementary-number-theory/arithmetic-sequences-positive-rational-numbers.lagda.md @@ -171,12 +171,11 @@ module _ abstract compute-arithmetic-sequence-ℚ⁺ : ( n : ℕ) → - Id - ( add-ℚ - ( rational-ℚ⁺ (initial-term-arithmetic-sequence-ℚ⁺ u)) - ( mul-ℚ - ( rational-ℕ n) - ( rational-ℚ⁺ (common-difference-arithmetic-sequence-ℚ⁺ u)))) + ( add-ℚ + ( rational-ℚ⁺ (initial-term-arithmetic-sequence-ℚ⁺ u)) + ( mul-ℚ + ( rational-ℕ n) + ( rational-ℚ⁺ (common-difference-arithmetic-sequence-ℚ⁺ u)))) ＝ ( rational-ℚ⁺ (seq-arithmetic-sequence-ℚ⁺ u n)) compute-arithmetic-sequence-ℚ⁺ n = ( compute-standard-arithmetic-sequence-ℚ⁺ diff --git a/src/elementary-number-theory/cross-multiplication-difference-integer-fractions.lagda.md b/src/elementary-number-theory/cross-multiplication-difference-integer-fractions.lagda.md index 38bca51942..50d7b1c744 100644 --- a/src/elementary-number-theory/cross-multiplication-difference-integer-fractions.lagda.md +++ b/src/elementary-number-theory/cross-multiplication-difference-integer-fractions.lagda.md @@ -51,9 +51,7 @@ cross-mul-diff-fraction-ℤ x y = abstract skew-commutative-cross-mul-diff-fraction-ℤ : (x y : fraction-ℤ) → - Id - ( neg-ℤ (cross-mul-diff-fraction-ℤ x y)) - ( cross-mul-diff-fraction-ℤ y x) + neg-ℤ (cross-mul-diff-fraction-ℤ x y) ＝ cross-mul-diff-fraction-ℤ y x skew-commutative-cross-mul-diff-fraction-ℤ x y = distributive-neg-diff-ℤ ( numerator-fraction-ℤ y *ℤ denominator-fraction-ℤ x) @@ -84,9 +82,8 @@ module _ abstract cross-mul-diff-one-fraction-ℤ : - Id - ( cross-mul-diff-fraction-ℤ one-fraction-ℤ x) - ( (numerator-fraction-ℤ x) -ℤ (denominator-fraction-ℤ x)) + cross-mul-diff-fraction-ℤ one-fraction-ℤ x ＝ + (numerator-fraction-ℤ x) -ℤ (denominator-fraction-ℤ x) cross-mul-diff-one-fraction-ℤ = ap-diff-ℤ ( right-unit-law-mul-ℤ (numerator-fraction-ℤ x)) @@ -126,11 +123,10 @@ differences satisfy a transitive-additive property: abstract lemma-add-cross-mul-diff-fraction-ℤ : (p q r : fraction-ℤ) → - Id - ( add-ℤ - ( denominator-fraction-ℤ p *ℤ cross-mul-diff-fraction-ℤ q r) - ( denominator-fraction-ℤ r *ℤ cross-mul-diff-fraction-ℤ p q)) - ( denominator-fraction-ℤ q *ℤ cross-mul-diff-fraction-ℤ p r) + ( add-ℤ + ( denominator-fraction-ℤ p *ℤ cross-mul-diff-fraction-ℤ q r) + ( denominator-fraction-ℤ r *ℤ cross-mul-diff-fraction-ℤ p q)) ＝ + ( denominator-fraction-ℤ q *ℤ cross-mul-diff-fraction-ℤ p r) lemma-add-cross-mul-diff-fraction-ℤ (np , dp , hp) (nq , dq , hq) @@ -191,9 +187,8 @@ abstract lemma-left-sim-cross-mul-diff-fraction-ℤ : (a a' b : fraction-ℤ) → sim-fraction-ℤ a a' → - Id - ( denominator-fraction-ℤ a *ℤ cross-mul-diff-fraction-ℤ a' b) - ( denominator-fraction-ℤ a' *ℤ cross-mul-diff-fraction-ℤ a b) + denominator-fraction-ℤ a *ℤ cross-mul-diff-fraction-ℤ a' b ＝ + denominator-fraction-ℤ a' *ℤ cross-mul-diff-fraction-ℤ a b lemma-left-sim-cross-mul-diff-fraction-ℤ a a' b H = equational-reasoning ( denominator-fraction-ℤ a *ℤ cross-mul-diff-fraction-ℤ a' b) @@ -223,9 +218,8 @@ abstract lemma-right-sim-cross-mul-diff-fraction-ℤ : (a b b' : fraction-ℤ) → sim-fraction-ℤ b b' → - Id - ( denominator-fraction-ℤ b *ℤ cross-mul-diff-fraction-ℤ a b') - ( denominator-fraction-ℤ b' *ℤ cross-mul-diff-fraction-ℤ a b) + denominator-fraction-ℤ b *ℤ cross-mul-diff-fraction-ℤ a b' ＝ + denominator-fraction-ℤ b' *ℤ cross-mul-diff-fraction-ℤ a b lemma-right-sim-cross-mul-diff-fraction-ℤ a b b' H = equational-reasoning ( denominator-fraction-ℤ b *ℤ cross-mul-diff-fraction-ℤ a b') diff --git a/src/elementary-number-theory/geometric-sequences-positive-rational-numbers.lagda.md b/src/elementary-number-theory/geometric-sequences-positive-rational-numbers.lagda.md index 69f19c0b20..b0561a8ca6 100644 --- a/src/elementary-number-theory/geometric-sequences-positive-rational-numbers.lagda.md +++ b/src/elementary-number-theory/geometric-sequences-positive-rational-numbers.lagda.md @@ -190,13 +190,12 @@ module _ abstract compute-geometric-sequence-ℚ⁺ : (n : ℕ) → - Id - ( mul-ℚ⁺ - ( initial-term-geometric-sequence-ℚ⁺ u) - ( power-Monoid monoid-mul-ℚ⁺ - ( n) - ( common-ratio-geometric-sequence-ℚ⁺ u))) - ( seq-geometric-sequence-ℚ⁺ u n) + mul-ℚ⁺ + ( initial-term-geometric-sequence-ℚ⁺ u) + ( power-Monoid monoid-mul-ℚ⁺ + ( n) + ( common-ratio-geometric-sequence-ℚ⁺ u)) ＝ + seq-geometric-sequence-ℚ⁺ u n compute-geometric-sequence-ℚ⁺ n = ( compute-standard-geometric-sequence-ℚ⁺ ( initial-term-geometric-sequence-ℚ⁺ u) diff --git a/src/elementary-number-theory/mediant-integer-fractions.lagda.md b/src/elementary-number-theory/mediant-integer-fractions.lagda.md index d9e2828280..9300f0a752 100644 --- a/src/elementary-number-theory/mediant-integer-fractions.lagda.md +++ b/src/elementary-number-theory/mediant-integer-fractions.lagda.md @@ -52,9 +52,8 @@ mediant-fraction-ℤ (a , b) (c , d) = (add-ℤ a c , add-positive-ℤ b d) ```agda cross-mul-diff-left-mediant-fraction-ℤ : (x y : fraction-ℤ) → - Id - ( cross-mul-diff-fraction-ℤ x y) - ( cross-mul-diff-fraction-ℤ x ( mediant-fraction-ℤ x y)) + cross-mul-diff-fraction-ℤ x y ＝ + cross-mul-diff-fraction-ℤ x ( mediant-fraction-ℤ x y) cross-mul-diff-left-mediant-fraction-ℤ (nx , dx , px) (ny , dy , py) = equational-reasoning (ny *ℤ dx -ℤ nx *ℤ dy) @@ -71,9 +70,8 @@ cross-mul-diff-left-mediant-fraction-ℤ (nx , dx , px) (ny , dy , py) = cross-mul-diff-right-mediant-fraction-ℤ : (x y : fraction-ℤ) → - Id - ( cross-mul-diff-fraction-ℤ x y) - ( cross-mul-diff-fraction-ℤ (mediant-fraction-ℤ x y) y) + cross-mul-diff-fraction-ℤ x y ＝ + cross-mul-diff-fraction-ℤ (mediant-fraction-ℤ x y) y cross-mul-diff-right-mediant-fraction-ℤ (nx , dx , px) (ny , dy , py) = equational-reasoning (ny *ℤ dx -ℤ nx *ℤ dy) diff --git a/src/elementary-number-theory/pisano-periods.lagda.md b/src/elementary-number-theory/pisano-periods.lagda.md index dd93f4af79..d195b11890 100644 --- a/src/elementary-number-theory/pisano-periods.lagda.md +++ b/src/elementary-number-theory/pisano-periods.lagda.md @@ -60,10 +60,9 @@ inv-generating-map-fibonacci-pair-Fin k (pair x y) = is-section-inv-generating-map-fibonacci-pair-Fin : (k : ℕ) (p : Fin (succ-ℕ k) × Fin (succ-ℕ k)) → - Id - ( generating-map-fibonacci-pair-Fin k - ( inv-generating-map-fibonacci-pair-Fin k p)) - ( p) + generating-map-fibonacci-pair-Fin k + ( inv-generating-map-fibonacci-pair-Fin k p) ＝ + p is-section-inv-generating-map-fibonacci-pair-Fin k (pair x y) = ap-binary pair refl ( ( commutative-add-Fin @@ -76,10 +75,9 @@ is-section-inv-generating-map-fibonacci-pair-Fin k (pair x y) = is-retraction-inv-generating-map-fibonacci-pair-Fin : (k : ℕ) (p : Fin (succ-ℕ k) × Fin (succ-ℕ k)) → - Id - ( inv-generating-map-fibonacci-pair-Fin k - ( generating-map-fibonacci-pair-Fin k p)) - ( p) + inv-generating-map-fibonacci-pair-Fin k + ( generating-map-fibonacci-pair-Fin k p) ＝ + p is-retraction-inv-generating-map-fibonacci-pair-Fin k (pair x y) = ap-binary pair ( ( commutative-add-Fin @@ -107,9 +105,8 @@ fibonacci-pair-Fin k (succ-ℕ n) = compute-fibonacci-pair-Fin : (k : ℕ) (n : ℕ) → - Id - ( fibonacci-pair-Fin k n) - ( mod-succ-ℕ k (Fibonacci-ℕ n) , mod-succ-ℕ k (Fibonacci-ℕ (succ-ℕ n))) + fibonacci-pair-Fin k n ＝ + ( mod-succ-ℕ k (Fibonacci-ℕ n) , mod-succ-ℕ k (Fibonacci-ℕ (succ-ℕ n))) compute-fibonacci-pair-Fin k zero-ℕ = refl compute-fibonacci-pair-Fin k (succ-ℕ zero-ℕ) = ap-binary pair refl (right-unit-law-add-Fin k (one-Fin k)) diff --git a/src/elementary-number-theory/universal-property-integers.lagda.md b/src/elementary-number-theory/universal-property-integers.lagda.md index d39b4c7a7e..93f4616d70 100644 --- a/src/elementary-number-theory/universal-property-integers.lagda.md +++ b/src/elementary-number-theory/universal-property-integers.lagda.md @@ -118,9 +118,8 @@ succ-Eq-ELIM-ℤ : ( s t : ELIM-ℤ P p0 pS) (H : (pr1 s) ~ (pr1 t)) → UU l1 succ-Eq-ELIM-ℤ P p0 pS s t H = ( k : ℤ) → - Id - ( H (succ-ℤ k)) - ( map-equiv (equiv-comparison-map-Eq-ELIM-ℤ P p0 pS s t k) (H k)) + H (succ-ℤ k) ＝ + map-equiv (equiv-comparison-map-Eq-ELIM-ℤ P p0 pS s t k) (H k) Eq-ELIM-ℤ : { l1 : Level} (P : ℤ → UU l1) diff --git a/src/finite-algebra/commutative-finite-rings.lagda.md b/src/finite-algebra/commutative-finite-rings.lagda.md index 475d5d1c43..3a4e470b9c 100644 --- a/src/finite-algebra/commutative-finite-rings.lagda.md +++ b/src/finite-algebra/commutative-finite-rings.lagda.md @@ -632,11 +632,10 @@ module _ preserves-concat-add-list-Finite-Commutative-Ring : (l1 l2 : list type-Finite-Commutative-Ring) → - Id - ( add-list-Finite-Commutative-Ring (concat-list l1 l2)) - ( add-Finite-Commutative-Ring - ( add-list-Finite-Commutative-Ring l1) - ( add-list-Finite-Commutative-Ring l2)) + ( add-list-Finite-Commutative-Ring (concat-list l1 l2)) ＝ + ( add-Finite-Commutative-Ring + ( add-list-Finite-Commutative-Ring l1) + ( add-list-Finite-Commutative-Ring l2)) preserves-concat-add-list-Finite-Commutative-Ring = preserves-concat-add-list-Finite-Ring finite-ring-Finite-Commutative-Ring ``` diff --git a/src/finite-algebra/dependent-products-finite-rings.lagda.md b/src/finite-algebra/dependent-products-finite-rings.lagda.md index 34992d7389..4bbf10587c 100644 --- a/src/finite-algebra/dependent-products-finite-rings.lagda.md +++ b/src/finite-algebra/dependent-products-finite-rings.lagda.md @@ -92,9 +92,8 @@ module _ associative-add-Π-Finite-Ring : (x y z : type-Π-Finite-Ring) → - Id - ( add-Π-Finite-Ring (add-Π-Finite-Ring x y) z) - ( add-Π-Finite-Ring x (add-Π-Finite-Ring y z)) + ( add-Π-Finite-Ring (add-Π-Finite-Ring x y) z) ＝ + ( add-Π-Finite-Ring x (add-Π-Finite-Ring y z)) associative-add-Π-Finite-Ring = associative-add-Π-Ring (type-Finite-Type I) (ring-Finite-Ring ∘ A) @@ -135,9 +134,8 @@ module _ associative-mul-Π-Finite-Ring : (x y z : type-Π-Finite-Ring) → - Id - ( mul-Π-Finite-Ring (mul-Π-Finite-Ring x y) z) - ( mul-Π-Finite-Ring x (mul-Π-Finite-Ring y z)) + ( mul-Π-Finite-Ring (mul-Π-Finite-Ring x y) z) ＝ + ( mul-Π-Finite-Ring x (mul-Π-Finite-Ring y z)) associative-mul-Π-Finite-Ring = associative-mul-Π-Ring (type-Finite-Type I) (ring-Finite-Ring ∘ A) @@ -162,9 +160,8 @@ module _ right-distributive-mul-add-Π-Finite-Ring : (f g h : type-Π-Finite-Ring) → - Id - ( mul-Π-Finite-Ring (add-Π-Finite-Ring f g) h) - ( add-Π-Finite-Ring (mul-Π-Finite-Ring f h) (mul-Π-Finite-Ring g h)) + ( mul-Π-Finite-Ring (add-Π-Finite-Ring f g) h) ＝ + ( add-Π-Finite-Ring (mul-Π-Finite-Ring f h) (mul-Π-Finite-Ring g h)) right-distributive-mul-add-Π-Finite-Ring = right-distributive-mul-add-Π-Ring ( type-Finite-Type I) diff --git a/src/finite-algebra/finite-fields.lagda.md b/src/finite-algebra/finite-fields.lagda.md index baf4b87188..72efaff9f4 100644 --- a/src/finite-algebra/finite-fields.lagda.md +++ b/src/finite-algebra/finite-fields.lagda.md @@ -539,11 +539,10 @@ module _ preserves-concat-add-list-Finite-Field : (l1 l2 : list type-Finite-Field) → - Id - ( add-list-Finite-Field (concat-list l1 l2)) - ( add-Finite-Field - ( add-list-Finite-Field l1) - ( add-list-Finite-Field l2)) + ( add-list-Finite-Field (concat-list l1 l2)) ＝ + ( add-Finite-Field + ( add-list-Finite-Field l1) + ( add-list-Finite-Field l2)) preserves-concat-add-list-Finite-Field = preserves-concat-add-list-Finite-Ring finite-ring-Finite-Field ``` diff --git a/src/finite-algebra/finite-rings.lagda.md b/src/finite-algebra/finite-rings.lagda.md index fb005ba1ea..77f1e9a2cb 100644 --- a/src/finite-algebra/finite-rings.lagda.md +++ b/src/finite-algebra/finite-rings.lagda.md @@ -139,9 +139,8 @@ module _ associative-add-Finite-Ring : (x y z : type-Finite-Ring R) → - Id - ( add-Finite-Ring (add-Finite-Ring x y) z) - ( add-Finite-Ring x (add-Finite-Ring y z)) + ( add-Finite-Ring (add-Finite-Ring x y) z) ＝ + ( add-Finite-Ring x (add-Finite-Ring y z)) associative-add-Finite-Ring = associative-add-Ring (ring-Finite-Ring R) is-group-additive-semigroup-Finite-Ring : @@ -302,9 +301,8 @@ module _ associative-mul-Finite-Ring : (x y z : type-Finite-Ring R) → - Id - ( mul-Finite-Ring (mul-Finite-Ring x y) z) - ( mul-Finite-Ring x (mul-Finite-Ring y z)) + ( mul-Finite-Ring (mul-Finite-Ring x y) z) ＝ + ( mul-Finite-Ring x (mul-Finite-Ring y z)) associative-mul-Finite-Ring = associative-mul-Ring (ring-Finite-Ring R) multiplicative-semigroup-Finite-Ring : Semigroup l @@ -533,9 +531,8 @@ module _ preserves-concat-add-list-Finite-Ring : (l1 l2 : list (type-Finite-Ring R)) → - Id - ( add-list-Finite-Ring (concat-list l1 l2)) - ( add-Finite-Ring R (add-list-Finite-Ring l1) (add-list-Finite-Ring l2)) + ( add-list-Finite-Ring (concat-list l1 l2)) ＝ + ( add-Finite-Ring R (add-list-Finite-Ring l1) (add-list-Finite-Ring l2)) preserves-concat-add-list-Finite-Ring = preserves-concat-add-list-Ring (ring-Finite-Ring R) ``` diff --git a/src/finite-algebra/products-commutative-finite-rings.lagda.md b/src/finite-algebra/products-commutative-finite-rings.lagda.md index 46ad564c75..b2b7f718ca 100644 --- a/src/finite-algebra/products-commutative-finite-rings.lagda.md +++ b/src/finite-algebra/products-commutative-finite-rings.lagda.md @@ -96,11 +96,10 @@ module _ left-unit-law-add-product-Finite-Commutative-Ring : (x : type-product-Finite-Commutative-Ring) → - Id - ( add-product-Finite-Commutative-Ring - ( zero-product-Finite-Commutative-Ring) - ( x)) - ( x) + add-product-Finite-Commutative-Ring + ( zero-product-Finite-Commutative-Ring) + ( x) ＝ + x left-unit-law-add-product-Finite-Commutative-Ring = left-unit-law-add-product-Commutative-Ring ( commutative-ring-Finite-Commutative-Ring R1) @@ -108,11 +107,10 @@ module _ right-unit-law-add-product-Finite-Commutative-Ring : (x : type-product-Finite-Commutative-Ring) → - Id - ( add-product-Finite-Commutative-Ring - ( x) - ( zero-product-Finite-Commutative-Ring)) + add-product-Finite-Commutative-Ring ( x) + ( zero-product-Finite-Commutative-Ring) ＝ + x right-unit-law-add-product-Finite-Commutative-Ring = right-unit-law-add-product-Commutative-Ring ( commutative-ring-Finite-Commutative-Ring R1) @@ -120,10 +118,9 @@ module _ left-inverse-law-add-product-Finite-Commutative-Ring : (x : type-product-Finite-Commutative-Ring) → - Id - ( add-product-Finite-Commutative-Ring + add-product-Finite-Commutative-Ring ( neg-product-Finite-Commutative-Ring x) - ( x)) + ( x) ＝ zero-product-Finite-Commutative-Ring left-inverse-law-add-product-Finite-Commutative-Ring = left-inverse-law-add-product-Commutative-Ring @@ -132,11 +129,10 @@ module _ right-inverse-law-add-product-Finite-Commutative-Ring : (x : type-product-Finite-Commutative-Ring) → - Id - ( add-product-Finite-Commutative-Ring - ( x) - ( neg-product-Finite-Commutative-Ring x)) - ( zero-product-Finite-Commutative-Ring) + add-product-Finite-Commutative-Ring + ( x) + ( neg-product-Finite-Commutative-Ring x) ＝ + zero-product-Finite-Commutative-Ring right-inverse-law-add-product-Finite-Commutative-Ring = right-inverse-law-add-product-Commutative-Ring ( commutative-ring-Finite-Commutative-Ring R1) @@ -144,13 +140,12 @@ module _ associative-add-product-Finite-Commutative-Ring : (x y z : type-product-Finite-Commutative-Ring) → - Id - ( add-product-Finite-Commutative-Ring - ( add-product-Finite-Commutative-Ring x y) - ( z)) - ( add-product-Finite-Commutative-Ring - ( x) - ( add-product-Finite-Commutative-Ring y z)) + add-product-Finite-Commutative-Ring + ( add-product-Finite-Commutative-Ring x y) + ( z) ＝ + add-product-Finite-Commutative-Ring + ( x) + ( add-product-Finite-Commutative-Ring y z) associative-add-product-Finite-Commutative-Ring = associative-add-product-Commutative-Ring ( commutative-ring-Finite-Commutative-Ring R1) @@ -158,9 +153,8 @@ module _ commutative-add-product-Finite-Commutative-Ring : (x y : type-product-Finite-Commutative-Ring) → - Id - ( add-product-Finite-Commutative-Ring x y) - ( add-product-Finite-Commutative-Ring y x) + add-product-Finite-Commutative-Ring x y ＝ + add-product-Finite-Commutative-Ring y x commutative-add-product-Finite-Commutative-Ring = commutative-add-product-Commutative-Ring ( commutative-ring-Finite-Commutative-Ring R1) @@ -183,13 +177,12 @@ module _ associative-mul-product-Finite-Commutative-Ring : (x y z : type-product-Finite-Commutative-Ring) → - Id - ( mul-product-Finite-Commutative-Ring - ( mul-product-Finite-Commutative-Ring x y) - ( z)) - ( mul-product-Finite-Commutative-Ring - ( x) - ( mul-product-Finite-Commutative-Ring y z)) + mul-product-Finite-Commutative-Ring + ( mul-product-Finite-Commutative-Ring x y) + ( z) ＝ + mul-product-Finite-Commutative-Ring + ( x) + ( mul-product-Finite-Commutative-Ring y z) associative-mul-product-Finite-Commutative-Ring = associative-mul-product-Commutative-Ring ( commutative-ring-Finite-Commutative-Ring R1) @@ -215,13 +208,12 @@ module _ left-distributive-mul-add-product-Finite-Commutative-Ring : (x y z : type-product-Finite-Commutative-Ring) → - Id - ( mul-product-Finite-Commutative-Ring - ( x) - ( add-product-Finite-Commutative-Ring y z)) - ( add-product-Finite-Commutative-Ring - ( mul-product-Finite-Commutative-Ring x y) - ( mul-product-Finite-Commutative-Ring x z)) + mul-product-Finite-Commutative-Ring + ( x) + ( add-product-Finite-Commutative-Ring y z) ＝ + add-product-Finite-Commutative-Ring + ( mul-product-Finite-Commutative-Ring x y) + ( mul-product-Finite-Commutative-Ring x z) left-distributive-mul-add-product-Finite-Commutative-Ring = left-distributive-mul-add-product-Commutative-Ring ( commutative-ring-Finite-Commutative-Ring R1) @@ -229,13 +221,12 @@ module _ right-distributive-mul-add-product-Finite-Commutative-Ring : (x y z : type-product-Finite-Commutative-Ring) → - Id - ( mul-product-Finite-Commutative-Ring - ( add-product-Finite-Commutative-Ring x y) - ( z)) - ( add-product-Finite-Commutative-Ring - ( mul-product-Finite-Commutative-Ring x z) - ( mul-product-Finite-Commutative-Ring y z)) + mul-product-Finite-Commutative-Ring + ( add-product-Finite-Commutative-Ring x y) + ( z) ＝ + add-product-Finite-Commutative-Ring + ( mul-product-Finite-Commutative-Ring x z) + ( mul-product-Finite-Commutative-Ring y z) right-distributive-mul-add-product-Finite-Commutative-Ring = right-distributive-mul-add-product-Commutative-Ring ( commutative-ring-Finite-Commutative-Ring R1) diff --git a/src/finite-algebra/products-finite-rings.lagda.md b/src/finite-algebra/products-finite-rings.lagda.md index c87b8b2071..1f337cff1e 100644 --- a/src/finite-algebra/products-finite-rings.lagda.md +++ b/src/finite-algebra/products-finite-rings.lagda.md @@ -90,9 +90,8 @@ module _ left-inverse-law-add-product-Finite-Ring : (x : type-product-Finite-Ring) → - Id - ( add-product-Finite-Ring (neg-product-Finite-Ring x) x) - ( zero-product-Finite-Ring) + add-product-Finite-Ring (neg-product-Finite-Ring x) x ＝ + zero-product-Finite-Ring left-inverse-law-add-product-Finite-Ring = left-inverse-law-add-product-Ring ( ring-Finite-Ring R1) @@ -100,9 +99,8 @@ module _ right-inverse-law-add-product-Finite-Ring : (x : type-product-Finite-Ring) → - Id - ( add-product-Finite-Ring x (neg-product-Finite-Ring x)) - ( zero-product-Finite-Ring) + add-product-Finite-Ring x (neg-product-Finite-Ring x) ＝ + zero-product-Finite-Ring right-inverse-law-add-product-Finite-Ring = right-inverse-law-add-product-Ring ( ring-Finite-Ring R1) @@ -110,9 +108,8 @@ module _ associative-add-product-Finite-Ring : (x y z : type-product-Finite-Ring) → - Id - ( add-product-Finite-Ring (add-product-Finite-Ring x y) z) - ( add-product-Finite-Ring x (add-product-Finite-Ring y z)) + add-product-Finite-Ring (add-product-Finite-Ring x y) z ＝ + add-product-Finite-Ring x (add-product-Finite-Ring y z) associative-add-product-Finite-Ring = associative-add-product-Ring (ring-Finite-Ring R1) (ring-Finite-Ring R2) @@ -135,9 +132,8 @@ module _ associative-mul-product-Finite-Ring : (x y z : type-product-Finite-Ring) → - Id - ( mul-product-Finite-Ring (mul-product-Finite-Ring x y) z) - ( mul-product-Finite-Ring x (mul-product-Finite-Ring y z)) + mul-product-Finite-Ring (mul-product-Finite-Ring x y) z ＝ + mul-product-Finite-Ring x (mul-product-Finite-Ring y z) associative-mul-product-Finite-Ring = associative-mul-product-Ring (ring-Finite-Ring R1) (ring-Finite-Ring R2) @@ -155,11 +151,10 @@ module _ left-distributive-mul-add-product-Finite-Ring : (x y z : type-product-Finite-Ring) → - Id - ( mul-product-Finite-Ring x (add-product-Finite-Ring y z)) - ( add-product-Finite-Ring - ( mul-product-Finite-Ring x y) - ( mul-product-Finite-Ring x z)) + ( mul-product-Finite-Ring x (add-product-Finite-Ring y z)) ＝ + ( add-product-Finite-Ring + ( mul-product-Finite-Ring x y) + ( mul-product-Finite-Ring x z)) left-distributive-mul-add-product-Finite-Ring = left-distributive-mul-add-product-Ring ( ring-Finite-Ring R1) @@ -167,11 +162,10 @@ module _ right-distributive-mul-add-product-Finite-Ring : (x y z : type-product-Finite-Ring) → - Id - ( mul-product-Finite-Ring (add-product-Finite-Ring x y) z) - ( add-product-Finite-Ring - ( mul-product-Finite-Ring x z) - ( mul-product-Finite-Ring y z)) + ( mul-product-Finite-Ring (add-product-Finite-Ring x y) z) ＝ + ( add-product-Finite-Ring + ( mul-product-Finite-Ring x z) + ( mul-product-Finite-Ring y z)) right-distributive-mul-add-product-Finite-Ring = right-distributive-mul-add-product-Ring ( ring-Finite-Ring R1) diff --git a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md index 00e93b169c..4ae5ccaa9a 100644 --- a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md +++ b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md @@ -314,9 +314,8 @@ module _ ( eX : mere-equiv (Fin (n +ℕ 2)) X) → ( eY : mere-equiv (Fin (n +ℕ 2)) Y) → X ＝ Y → - Id - ( equivalence-class (R (n +ℕ 2) (X , eX))) - ( equivalence-class (R (n +ℕ 2) (Y , eY))) + equivalence-class (R (n +ℕ 2) (X , eX)) ＝ + equivalence-class (R (n +ℕ 2) (Y , eY)) map-quotient-delooping-sign-loop n X Y eX eY p = ap ( equivalence-class ∘ R (n +ℕ 2)) diff --git a/src/finite-group-theory/orbits-permutations.lagda.md b/src/finite-group-theory/orbits-permutations.lagda.md index 9864859945..6dcc6add5b 100644 --- a/src/finite-group-theory/orbits-permutations.lagda.md +++ b/src/finite-group-theory/orbits-permutations.lagda.md @@ -290,15 +290,13 @@ module _ ( f) ( tr ( λ x → - Id - ( iterate x (map-equiv f) a) - ( iterate (succ-ℕ pred-second) (map-equiv f) a)) + iterate x (map-equiv f) a ＝ + iterate (succ-ℕ pred-second) (map-equiv f) a) ( equality-pred-first) ( tr ( λ x → - Id - ( iterate first-point-min-repeating (map-equiv f) a) - ( iterate x (map-equiv f) a)) + iterate first-point-min-repeating (map-equiv f) a ＝ + iterate x (map-equiv f) a) ( equality-pred-second) ( same-image-iterate-min-reporting))))))) where @@ -334,9 +332,8 @@ module _ pr2 (pr2 (has-finite-orbits-permutation' (inl p))) = tr ( λ x → - Id - ( iterate first-point-min-repeating (map-equiv f) a) - ( iterate x (map-equiv f) a)) + iterate first-point-min-repeating (map-equiv f) a ＝ + iterate x (map-equiv f) a) ( p) ( same-image-iterate-min-reporting) has-finite-orbits-permutation' (inr np) = @@ -729,9 +726,8 @@ module _ ( Ind : (n : ℕ) → C (succ-ℕ n) → is-nonzero-ℕ n → C n) → (k : ℕ) → (is-zero-ℕ k + C k) → - Id - ( iterate k (map-equiv (composition-transposition-a-b g)) x) - ( iterate k (map-equiv g) x) + iterate k (map-equiv (composition-transposition-a-b g)) x ＝ + iterate k (map-equiv g) x equal-iterate-transposition x g C F Ind zero-ℕ p = refl equal-iterate-transposition x g C F Ind (succ-ℕ k) (inl p) = ex-falso (is-nonzero-succ-ℕ k p) @@ -746,9 +742,8 @@ module _ where induction-cases-equal-iterate-transposition : is-decidable (k ＝ zero-ℕ) → - Id - ( iterate k (map-equiv (composition-transposition-a-b g)) x) - ( iterate k (map-equiv g) x) + iterate k (map-equiv (composition-transposition-a-b g)) x ＝ + iterate k (map-equiv g) x induction-cases-equal-iterate-transposition (inl s) = tr ( λ k → @@ -761,9 +756,8 @@ module _ cases-equal-iterate-transposition : is-decidable (iterate (succ-ℕ k) (map-equiv g) x ＝ a) → is-decidable (iterate (succ-ℕ k) (map-equiv g) x ＝ b) → - Id - ( iterate (succ-ℕ k) (map-equiv (composition-transposition-a-b g)) x) - ( iterate (succ-ℕ k) (map-equiv g) x) + iterate (succ-ℕ k) (map-equiv (composition-transposition-a-b g)) x ＝ + iterate (succ-ℕ k) (map-equiv g) x cases-equal-iterate-transposition (inl q) r = ex-falso (pr1 (F (succ-ℕ k) p) q) cases-equal-iterate-transposition (inr q) (inl r) = @@ -824,9 +818,8 @@ module _ where equal-iterate-transposition-other-orbits : (k : ℕ) → - Id - ( iterate k (map-equiv (composition-transposition-a-b g)) x) - ( iterate k (map-equiv g) x) + iterate k (map-equiv (composition-transposition-a-b g)) x ＝ + iterate k (map-equiv g) x equal-iterate-transposition-other-orbits k = equal-iterate-transposition x g (λ k' → unit) (λ k' _ → @@ -925,9 +918,8 @@ module _ equal-iterate-transposition-a : (pa : Σ ℕ (λ k → iterate k (map-equiv g) a ＝ b)) (k : ℕ) → le-ℕ k (pr1 (minimal-element-iterate g a b pa)) → - ( Id - ( iterate k (map-equiv (composition-transposition-a-b g)) a) - ( iterate k (map-equiv g) a)) + iterate k (map-equiv (composition-transposition-a-b g)) a ＝ + iterate k (map-equiv g) a equal-iterate-transposition-a pa k ineq = equal-iterate-transposition a g ( λ k' → @@ -1181,9 +1173,8 @@ module _ ( iterate k (map-equiv g) x ＝ b))) ( k : ℕ) → ( le-ℕ k (pr1 (minimal-element-iterate-2-a-b g pa))) → - Id - ( iterate k (map-equiv (composition-transposition-a-b g)) x) - ( iterate k (map-equiv g) x) + iterate k (map-equiv (composition-transposition-a-b g)) x ＝ + iterate k (map-equiv g) x equal-iterate-transposition-same-orbits g pa k ineq = equal-iterate-transposition x g ( λ k' → le-ℕ k' (pr1 (minimal-element-iterate-2-a-b g pa))) @@ -1910,9 +1901,7 @@ module _ ( same-orbits-permutation-count (composition-transposition-a-b g)) ( h'-inl k (map-equiv-count h k) refl Q R) ( b))) → - Id - ( cases-inv-h' (h'-inl k (map-equiv-count h k) refl Q R) Q' R') - ( inl k) + cases-inv-h' (h'-inl k (map-equiv-count h k) refl Q R) Q' R' ＝ inl k section-h'-inl k (inl Q) R (inl Q') R' = ap inl ( is-injective-equiv (equiv-count h) @@ -2208,9 +2197,8 @@ module _ NP (unit-trunc-Prop (pair k R)) equal-iterate-transposition-a : (k : ℕ) → le-ℕ k (pr1 minimal-element-iterate-repeating) → - Id - ( iterate k (map-equiv (composition-transposition-a-b g)) a) - ( iterate k (map-equiv g) a) + iterate k (map-equiv (composition-transposition-a-b g)) a ＝ + iterate k (map-equiv g) a equal-iterate-transposition-a k ineq = equal-iterate-transposition a g ( λ k' → diff --git a/src/finite-group-theory/permutations.lagda.md b/src/finite-group-theory/permutations.lagda.md index dc3ebdad92..cd611b4f45 100644 --- a/src/finite-group-theory/permutations.lagda.md +++ b/src/finite-group-theory/permutations.lagda.md @@ -311,9 +311,8 @@ module _ ( f) is-injective-iterate-involution : (k k' x : Fin 2) → - Id - ( iterate (nat-Fin 2 k) (succ-Fin 2) x) - ( iterate (nat-Fin 2 k') (succ-Fin 2) x) → + ( iterate (nat-Fin 2 k) (succ-Fin 2) x ＝ + iterate (nat-Fin 2 k') (succ-Fin 2) x) → k ＝ k' is-injective-iterate-involution (inl (inr star)) (inl (inr star)) x p = diff --git a/src/finite-group-theory/sign-homomorphism.lagda.md b/src/finite-group-theory/sign-homomorphism.lagda.md index 877705c2d4..3dd0e7054e 100644 --- a/src/finite-group-theory/sign-homomorphism.lagda.md +++ b/src/finite-group-theory/sign-homomorphism.lagda.md @@ -67,9 +67,8 @@ module _ preserves-add-sign-homomorphism-Fin-2 : (f g : type-Type-With-Cardinality-ℕ n X ≃ type-Type-With-Cardinality-ℕ n X) → - Id - ( sign-homomorphism-Fin-2 (f ∘e g)) - ( add-Fin 2 (sign-homomorphism-Fin-2 f) (sign-homomorphism-Fin-2 g)) + sign-homomorphism-Fin-2 (f ∘e g) ＝ + add-Fin 2 (sign-homomorphism-Fin-2 f) (sign-homomorphism-Fin-2 g) preserves-add-sign-homomorphism-Fin-2 f g = apply-universal-property-trunc-Prop ( has-cardinality-type-Type-With-Cardinality-ℕ n X) @@ -166,10 +165,7 @@ module _ list-comp-f-g h = concat-list (list-trans f h) (list-trans g h) eq-list-comp-f-g : ( h : Fin n ≃ type-Type-With-Cardinality-ℕ n X) → - Id - ( f ∘e g) - ( permutation-list-transpositions - ( list-comp-f-g h)) + f ∘e g ＝ permutation-list-transpositions (list-comp-f-g h) eq-list-comp-f-g h = eq-htpy-equiv ( λ x → @@ -194,11 +190,8 @@ module _ ( list-trans g h)) eq-sign-homomorphism-Fin-2-transposition : - ( Y : - 2-Element-Decidable-Subtype l (type-Type-With-Cardinality-ℕ n X)) → - Id - ( sign-homomorphism-Fin-2 (transposition Y)) - ( inr star) + ( Y : 2-Element-Decidable-Subtype l (type-Type-With-Cardinality-ℕ n X)) → + sign-homomorphism-Fin-2 (transposition Y) ＝ inr star eq-sign-homomorphism-Fin-2-transposition Y = ap pr1 { x = @@ -222,9 +215,8 @@ module _ preserves-conjugation-sign-homomorphism-Fin-2 : ( f : type-Type-With-Cardinality-ℕ n X ≃ type-Type-With-Cardinality-ℕ n X) → ( g : type-Type-With-Cardinality-ℕ n X ≃ type-Type-With-Cardinality-ℕ n Y) → - Id - ( sign-homomorphism-Fin-2 n Y (g ∘e (f ∘e inv-equiv g))) - ( sign-homomorphism-Fin-2 n X f) + sign-homomorphism-Fin-2 n Y (g ∘e (f ∘e inv-equiv g)) ＝ + sign-homomorphism-Fin-2 n X f preserves-conjugation-sign-homomorphism-Fin-2 f g = apply-universal-property-trunc-Prop ( has-cardinality-type-Type-With-Cardinality-ℕ n X) diff --git a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md index 5a37978c7a..c593a2aaf3 100644 --- a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md +++ b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md @@ -202,12 +202,11 @@ module _ private abstract lemma : - Id - ( inr star) - ( sign-homomorphism-Fin-2 - ( number-of-elements-count eX) - ( Fin-Type-With-Cardinality-ℕ (number-of-elements-count eX)) - ( inv-equiv (equiv-count eX) ∘e (equiv-count eX ∘e transposition-eX))) + inr star ＝ + sign-homomorphism-Fin-2 + ( number-of-elements-count eX) + ( Fin-Type-With-Cardinality-ℕ (number-of-elements-count eX)) + ( inv-equiv (equiv-count eX) ∘e (equiv-count eX ∘e transposition-eX)) lemma = ( inv ( eq-sign-homomorphism-Fin-2-transposition @@ -463,10 +462,9 @@ module _ ( X , unit-trunc-Prop (equiv-count eX))) ( T) ( equiv-count eX))) → - Id - ( pr1 equiv-Fin-2-quotient-sign-comp-count - ( inv-Fin-2-quotient-sign-comp-count T H)) - ( T) + ( pr1 equiv-Fin-2-quotient-sign-comp-count + ( inv-Fin-2-quotient-sign-comp-count T H)) ＝ + ( T) retraction-Fin-2-quotient-sign-comp-count T (inl P) = eq-effective-quotient' ( sign-comp-equivalence-relation @@ -526,10 +524,9 @@ module _ ( X , unit-trunc-Prop (equiv-count eX))) ( pr1 equiv-Fin-2-quotient-sign-comp-count k) ( equiv-count eX))) → - Id - ( inv-Fin-2-quotient-sign-comp-count - (pr1 equiv-Fin-2-quotient-sign-comp-count k) (D)) - ( k) + ( inv-Fin-2-quotient-sign-comp-count + (pr1 equiv-Fin-2-quotient-sign-comp-count k) D) ＝ + ( k) section-Fin-2-quotient-sign-comp-count (inl (inr star)) (inl D) = refl section-Fin-2-quotient-sign-comp-count (inl (inr star)) (inr ND) = ex-falso @@ -643,9 +640,8 @@ module _ type-Type-With-Cardinality-ℕ n X ≃ type-Type-With-Cardinality-ℕ n Y) → ( f : type-Type-With-Cardinality-ℕ n Y ≃ type-Type-With-Cardinality-ℕ n Z) → - Id - ( simpson-comp-equiv X Z (f ∘e e)) - ( simpson-comp-equiv Y Z f ∘e simpson-comp-equiv X Y e) + simpson-comp-equiv X Z (f ∘e e) ＝ + simpson-comp-equiv Y Z f ∘e simpson-comp-equiv X Y e preserves-comp-simpson-comp-equiv X Y Z e f = eq-htpy-equiv ( λ h → associative-comp-equiv h e f) diff --git a/src/finite-group-theory/transpositions.lagda.md b/src/finite-group-theory/transpositions.lagda.md index ce3e270663..f0f270695a 100644 --- a/src/finite-group-theory/transpositions.lagda.md +++ b/src/finite-group-theory/transpositions.lagda.md @@ -236,10 +236,9 @@ module _ -- !! Why not a homotopy? eq-concat-permutation-list-transpositions : (l l' : list (2-Element-Decidable-Subtype l2 X)) → - Id - ( ( permutation-list-transpositions l) ∘e - ( permutation-list-transpositions l')) - ( permutation-list-transpositions (concat-list l l')) + ( permutation-list-transpositions l) ∘e + ( permutation-list-transpositions l') ＝ + ( permutation-list-transpositions (concat-list l l')) eq-concat-permutation-list-transpositions nil l' = eq-htpy-equiv refl-htpy eq-concat-permutation-list-transpositions (cons P l) l' = eq-htpy-equiv @@ -606,12 +605,7 @@ module _ Σ ( Fin n) ( λ y → Σ ( x ≠ y) - ( λ np → - Id - ( standard-2-Element-Decidable-Subtype - ( p) - ( np)) - ( Y))))) + ( λ np → standard-2-Element-Decidable-Subtype p np ＝ Y)))) ( eq-is-prop (is-prop-has-decidable-equality)) ( two-elements-transposition (count-Fin n) Y) @@ -1047,11 +1041,10 @@ eq-transposition-precomp-ineq-standard-2-Element-Decidable-Subtype : {l : Level} {X : UU l} (H : has-decidable-equality X) → {x y z w : X} (np : x ≠ y) (np' : z ≠ w) → x ≠ z → x ≠ w → y ≠ z → y ≠ w → - Id - ( precomp-equiv-2-Element-Decidable-Subtype - ( standard-transposition H np) - ( standard-2-Element-Decidable-Subtype H np')) - ( standard-2-Element-Decidable-Subtype H np') + precomp-equiv-2-Element-Decidable-Subtype + ( standard-transposition H np) + ( standard-2-Element-Decidable-Subtype H np') ＝ + standard-2-Element-Decidable-Subtype H np' eq-transposition-precomp-ineq-standard-2-Element-Decidable-Subtype {l} {X} H {x} {y} {z} {w} np np' nq1 nq2 nq3 nq4 = eq-pair-Σ @@ -1126,11 +1119,10 @@ module _ cases-eq-equiv-universes-transposition : ( P : 2-Element-Decidable-Subtype l X) (x : X) → ( d : is-decidable (is-in-2-Element-Decidable-Subtype P x)) → - Id - ( map-transposition' P x d) - ( map-transposition - ( map-equiv (equiv-universes-2-Element-Decidable-Subtype X l l') P) - ( x)) + map-transposition' P x d ＝ + map-transposition + ( map-equiv (equiv-universes-2-Element-Decidable-Subtype X l l') P) + ( x) cases-eq-equiv-universes-transposition P x (inl p) = ( ap pr1 ( inv @@ -1185,10 +1177,9 @@ module _ eq-equiv-universes-transposition : ( P : 2-Element-Decidable-Subtype l X) → - Id - ( transposition P) - ( transposition - ( map-equiv (equiv-universes-2-Element-Decidable-Subtype X l l') P)) + transposition P ＝ + transposition + ( map-equiv (equiv-universes-2-Element-Decidable-Subtype X l l') P) eq-equiv-universes-transposition P = eq-htpy-equiv ( λ x → @@ -1197,12 +1188,11 @@ module _ eq-equiv-universes-transposition-list : ( li : list (2-Element-Decidable-Subtype l X)) → - Id - ( permutation-list-transpositions li) - ( permutation-list-transpositions - ( map-list - ( map-equiv (equiv-universes-2-Element-Decidable-Subtype X l l')) - ( li))) + permutation-list-transpositions li ＝ + permutation-list-transpositions + ( map-list + ( map-equiv (equiv-universes-2-Element-Decidable-Subtype X l l')) + ( li)) eq-equiv-universes-transposition-list nil = refl eq-equiv-universes-transposition-list (cons P li) = ap-binary diff --git a/src/foundation/truncations.lagda.md b/src/foundation/truncations.lagda.md index 9523e78045..da13d2b08d 100644 --- a/src/foundation/truncations.lagda.md +++ b/src/foundation/truncations.lagda.md @@ -235,21 +235,19 @@ module _ ( Σ ( (t : total-truncated-fam-trunc B) → type-Truncated-Type (C t)) ( λ h → (x : A) (y : type-Truncated-Type (B x)) → - Id - ( h (unit-trunc x , map-compute-truncated-fam-trunc B x y)) - ( f x y))) + h (unit-trunc x , map-compute-truncated-fam-trunc B x y) ＝ + f x y)) dependent-universal-property-total-truncated-fam-trunc = is-contr-equiv _ ( equiv-Σ ( λ g → (x : A) → - Id - ( g (unit-trunc x)) - ( map-equiv-Π - ( λ u → type-Truncated-Type (C (unit-trunc x , u))) - ( compute-truncated-fam-trunc B x) - ( λ u → id-equiv) - ( f x))) + g (unit-trunc x) ＝ + map-equiv-Π + ( λ u → type-Truncated-Type (C (unit-trunc x , u))) + ( compute-truncated-fam-trunc B x) + ( λ u → id-equiv) + ( f x)) ( equiv-ev-pair) ( λ g → equiv-Π-equiv-family @@ -257,15 +255,14 @@ module _ ( inv-equiv equiv-funext) ∘e ( equiv-Π ( λ y → - Id - ( g (unit-trunc x , y)) - ( map-equiv-Π - ( λ u → - type-Truncated-Type (C (unit-trunc x , u))) - ( compute-truncated-fam-trunc B x) - ( λ u → id-equiv) - ( f x) - ( y))) + g (unit-trunc x , y) ＝ + map-equiv-Π + ( λ u → + type-Truncated-Type (C (unit-trunc x , u))) + ( compute-truncated-fam-trunc B x) + ( λ u → id-equiv) + ( f x) + ( y)) ( compute-truncated-fam-trunc B x) ( λ y → equiv-concat' @@ -297,10 +294,9 @@ module _ htpy-dependent-universal-property-total-truncated-fam-trunc : (x : A) (y : type-Truncated-Type (B x)) → - Id - ( function-dependent-universal-property-total-truncated-fam-trunc - ( unit-trunc x , map-compute-truncated-fam-trunc B x y)) - ( f x y) + function-dependent-universal-property-total-truncated-fam-trunc + ( unit-trunc x , map-compute-truncated-fam-trunc B x y) ＝ + f x y htpy-dependent-universal-property-total-truncated-fam-trunc = pr2 (center dependent-universal-property-total-truncated-fam-trunc) ``` diff --git a/src/foundation/type-arithmetic-dependent-pair-types.lagda.md b/src/foundation/type-arithmetic-dependent-pair-types.lagda.md index 022b2dc3a8..922b6cd1da 100644 --- a/src/foundation/type-arithmetic-dependent-pair-types.lagda.md +++ b/src/foundation/type-arithmetic-dependent-pair-types.lagda.md @@ -62,16 +62,12 @@ module _ ( ind-singleton a C ( λ x → ( y : B x) → - Id - ( ( map-inv-left-unit-law-Σ-is-contr ∘ - map-left-unit-law-Σ-is-contr) - ( x , y)) - ( x , y)) + map-inv-left-unit-law-Σ-is-contr + ( map-left-unit-law-Σ-is-contr (x , y)) ＝ + ( x , y)) ( λ y → ap ( map-inv-left-unit-law-Σ-is-contr) - ( ap - ( λ f → f y) - ( compute-ind-singleton a C (λ x → B x → B a) id)))) + ( ap (λ f → f y) ( compute-ind-singleton a C (λ x → B x → B a) id)))) is-equiv-map-left-unit-law-Σ-is-contr : is-equiv map-left-unit-law-Σ-is-contr diff --git a/src/group-theory/abelian-groups.lagda.md b/src/group-theory/abelian-groups.lagda.md index 1bf0cfd7f1..5b2831b180 100644 --- a/src/group-theory/abelian-groups.lagda.md +++ b/src/group-theory/abelian-groups.lagda.md @@ -753,9 +753,8 @@ module _ preserves-concat-add-list-Ab : (l1 l2 : list (type-Ab A)) → - Id - ( add-list-Ab (concat-list l1 l2)) - ( add-Ab A (add-list-Ab l1) (add-list-Ab l2)) + add-list-Ab (concat-list l1 l2) ＝ + add-Ab A (add-list-Ab l1) (add-list-Ab l2) preserves-concat-add-list-Ab = preserves-concat-mul-list-Group (group-Ab A) ``` diff --git a/src/group-theory/cartesian-products-abelian-groups.lagda.md b/src/group-theory/cartesian-products-abelian-groups.lagda.md index 4d3e2dad34..4a93240ebb 100644 --- a/src/group-theory/cartesian-products-abelian-groups.lagda.md +++ b/src/group-theory/cartesian-products-abelian-groups.lagda.md @@ -71,9 +71,8 @@ module _ associative-add-product-Ab : (x y z : type-product-Ab) → - Id - ( add-product-Ab (add-product-Ab x y) z) - ( add-product-Ab x (add-product-Ab y z)) + add-product-Ab (add-product-Ab x y) z ＝ + add-product-Ab x (add-product-Ab y z) associative-add-product-Ab = associative-mul-Group group-product-Ab left-unit-law-add-product-Ab : diff --git a/src/group-theory/cartesian-products-commutative-monoids.lagda.md b/src/group-theory/cartesian-products-commutative-monoids.lagda.md index 9d8964b602..bec9b949d0 100644 --- a/src/group-theory/cartesian-products-commutative-monoids.lagda.md +++ b/src/group-theory/cartesian-products-commutative-monoids.lagda.md @@ -58,9 +58,8 @@ module _ associative-mul-product-Commutative-Monoid : (x y z : type-product-Commutative-Monoid) → - Id - ( mul-product-Commutative-Monoid (mul-product-Commutative-Monoid x y) z) - ( mul-product-Commutative-Monoid x (mul-product-Commutative-Monoid y z)) + ( mul-product-Commutative-Monoid (mul-product-Commutative-Monoid x y) z) ＝ + ( mul-product-Commutative-Monoid x (mul-product-Commutative-Monoid y z)) associative-mul-product-Commutative-Monoid = associative-mul-Monoid monoid-product-Commutative-Monoid @@ -82,9 +81,8 @@ module _ commutative-mul-product-Commutative-Monoid : (x y : type-product-Commutative-Monoid) → - Id - ( mul-product-Commutative-Monoid x y) - ( mul-product-Commutative-Monoid y x) + mul-product-Commutative-Monoid x y ＝ + mul-product-Commutative-Monoid y x commutative-mul-product-Commutative-Monoid _ _ = eq-pair ( commutative-mul-Commutative-Monoid M _ _) diff --git a/src/group-theory/cartesian-products-concrete-groups.lagda.md b/src/group-theory/cartesian-products-concrete-groups.lagda.md index b7bb672221..b25c3b0fbf 100644 --- a/src/group-theory/cartesian-products-concrete-groups.lagda.md +++ b/src/group-theory/cartesian-products-concrete-groups.lagda.md @@ -143,9 +143,8 @@ module _ associative-mul-product-Concrete-Group : (x y z : type-product-Concrete-Group) → - Id - (mul-product-Concrete-Group (mul-product-Concrete-Group x y) z) - (mul-product-Concrete-Group x (mul-product-Concrete-Group y z)) + mul-product-Concrete-Group (mul-product-Concrete-Group x y) z ＝ + mul-product-Concrete-Group x (mul-product-Concrete-Group y z) associative-mul-product-Concrete-Group = associative-mul-∞-Group ∞-group-product-Concrete-Group @@ -162,9 +161,8 @@ module _ right-unit-law-mul-∞-Group ∞-group-product-Concrete-Group coherence-unit-laws-mul-product-Concrete-Group : - Id - ( left-unit-law-mul-product-Concrete-Group unit-product-Concrete-Group) - ( right-unit-law-mul-product-Concrete-Group unit-product-Concrete-Group) + left-unit-law-mul-product-Concrete-Group unit-product-Concrete-Group ＝ + right-unit-law-mul-product-Concrete-Group unit-product-Concrete-Group coherence-unit-laws-mul-product-Concrete-Group = coherence-unit-laws-mul-∞-Group ∞-group-product-Concrete-Group @@ -174,17 +172,15 @@ module _ left-inverse-law-mul-product-Concrete-Group : (x : type-product-Concrete-Group) → - Id - ( mul-product-Concrete-Group (inv-product-Concrete-Group x) x) - ( unit-product-Concrete-Group) + mul-product-Concrete-Group (inv-product-Concrete-Group x) x ＝ + unit-product-Concrete-Group left-inverse-law-mul-product-Concrete-Group = left-inverse-law-mul-∞-Group ∞-group-product-Concrete-Group right-inverse-law-mul-product-Concrete-Group : (x : type-product-Concrete-Group) → - Id - ( mul-product-Concrete-Group x (inv-product-Concrete-Group x)) - ( unit-product-Concrete-Group) + mul-product-Concrete-Group x (inv-product-Concrete-Group x) ＝ + unit-product-Concrete-Group right-inverse-law-mul-product-Concrete-Group = right-inverse-law-mul-∞-Group ∞-group-product-Concrete-Group diff --git a/src/group-theory/cartesian-products-groups.lagda.md b/src/group-theory/cartesian-products-groups.lagda.md index 8077b4e5b9..d914f3f536 100644 --- a/src/group-theory/cartesian-products-groups.lagda.md +++ b/src/group-theory/cartesian-products-groups.lagda.md @@ -56,9 +56,8 @@ module _ associative-mul-product-Group : (x y z : type-product-Group) → - Id - ( mul-product-Group (mul-product-Group x y) z) - ( mul-product-Group x (mul-product-Group y z)) + mul-product-Group (mul-product-Group x y) z ＝ + mul-product-Group x (mul-product-Group y z) associative-mul-product-Group = associative-mul-Semigroup semigroup-product-Group diff --git a/src/group-theory/cartesian-products-monoids.lagda.md b/src/group-theory/cartesian-products-monoids.lagda.md index 3d3abec4b7..47f0105340 100644 --- a/src/group-theory/cartesian-products-monoids.lagda.md +++ b/src/group-theory/cartesian-products-monoids.lagda.md @@ -54,9 +54,8 @@ module _ associative-mul-product-Monoid : (x y z : type-product-Monoid) → - Id - ( mul-product-Monoid (mul-product-Monoid x y) z) - ( mul-product-Monoid x (mul-product-Monoid y z)) + mul-product-Monoid (mul-product-Monoid x y) z ＝ + mul-product-Monoid x (mul-product-Monoid y z) associative-mul-product-Monoid = associative-mul-Semigroup semigroup-product-Monoid diff --git a/src/group-theory/cartesian-products-semigroups.lagda.md b/src/group-theory/cartesian-products-semigroups.lagda.md index 764cb0ae83..e93230d667 100644 --- a/src/group-theory/cartesian-products-semigroups.lagda.md +++ b/src/group-theory/cartesian-products-semigroups.lagda.md @@ -49,9 +49,8 @@ module _ associative-mul-product-Semigroup : (x y z : type-product-Semigroup) → - Id - ( mul-product-Semigroup (mul-product-Semigroup x y) z) - ( mul-product-Semigroup x (mul-product-Semigroup y z)) + mul-product-Semigroup (mul-product-Semigroup x y) z ＝ + mul-product-Semigroup x (mul-product-Semigroup y z) associative-mul-product-Semigroup (pair x1 y1) (pair x2 y2) (pair x3 y3) = eq-pair ( associative-mul-Semigroup A x1 x2 x3) diff --git a/src/group-theory/decidable-subgroups.lagda.md b/src/group-theory/decidable-subgroups.lagda.md index be44157b41..ef1abb5ec1 100644 --- a/src/group-theory/decidable-subgroups.lagda.md +++ b/src/group-theory/decidable-subgroups.lagda.md @@ -239,9 +239,8 @@ module _ associative-mul-Decidable-Subgroup : (x y z : type-group-Decidable-Subgroup) → - Id - ( mul-Decidable-Subgroup (mul-Decidable-Subgroup x y) z) - ( mul-Decidable-Subgroup x (mul-Decidable-Subgroup y z)) + mul-Decidable-Subgroup (mul-Decidable-Subgroup x y) z ＝ + mul-Decidable-Subgroup x (mul-Decidable-Subgroup y z) associative-mul-Decidable-Subgroup = associative-mul-Subgroup G (subgroup-Decidable-Subgroup G H) @@ -266,17 +265,15 @@ module _ left-inverse-law-mul-Decidable-Subgroup : ( x : type-group-Decidable-Subgroup) → - Id - ( mul-Decidable-Subgroup (inv-Decidable-Subgroup x) x) - ( unit-Decidable-Subgroup) + mul-Decidable-Subgroup (inv-Decidable-Subgroup x) x ＝ + unit-Decidable-Subgroup left-inverse-law-mul-Decidable-Subgroup = left-inverse-law-mul-Subgroup G (subgroup-Decidable-Subgroup G H) right-inverse-law-mul-Decidable-Subgroup : (x : type-group-Decidable-Subgroup) → - Id - ( mul-Decidable-Subgroup x (inv-Decidable-Subgroup x)) - ( unit-Decidable-Subgroup) + mul-Decidable-Subgroup x (inv-Decidable-Subgroup x) ＝ + unit-Decidable-Subgroup right-inverse-law-mul-Decidable-Subgroup = right-inverse-law-mul-Subgroup G (subgroup-Decidable-Subgroup G H) diff --git a/src/group-theory/groups.lagda.md b/src/group-theory/groups.lagda.md index a2f71e3c77..c6df41df88 100644 --- a/src/group-theory/groups.lagda.md +++ b/src/group-theory/groups.lagda.md @@ -663,9 +663,8 @@ module _ preserves-concat-mul-list-Group : (l1 l2 : list (type-Group G)) → - Id - ( mul-list-Group (concat-list l1 l2)) - ( mul-Group G (mul-list-Group l1) (mul-list-Group l2)) + mul-list-Group (concat-list l1 l2) ＝ + mul-Group G (mul-list-Group l1) (mul-list-Group l2) preserves-concat-mul-list-Group = distributive-mul-concat-list-Monoid (monoid-Group G) ``` diff --git a/src/group-theory/homomorphisms-concrete-groups.lagda.md b/src/group-theory/homomorphisms-concrete-groups.lagda.md index 3136fa756e..4b12ff6251 100644 --- a/src/group-theory/homomorphisms-concrete-groups.lagda.md +++ b/src/group-theory/homomorphisms-concrete-groups.lagda.md @@ -55,9 +55,8 @@ module _ preserves-point-classifying-map-hom-Concrete-Group : (f : hom-Concrete-Group) → - Id - ( classifying-map-hom-Concrete-Group f (shape-Concrete-Group G)) - ( shape-Concrete-Group H) + classifying-map-hom-Concrete-Group f (shape-Concrete-Group G) ＝ + shape-Concrete-Group H preserves-point-classifying-map-hom-Concrete-Group = preserves-point-classifying-map-hom-∞-Group ( ∞-group-Concrete-Group G) @@ -72,9 +71,7 @@ module _ preserves-unit-map-hom-Concrete-Group : (f : hom-Concrete-Group) → - Id - ( map-hom-Concrete-Group f (unit-Concrete-Group G)) - ( unit-Concrete-Group H) + map-hom-Concrete-Group f (unit-Concrete-Group G) ＝ unit-Concrete-Group H preserves-unit-map-hom-Concrete-Group = preserves-unit-map-hom-∞-Group ( ∞-group-Concrete-Group G) @@ -82,11 +79,10 @@ module _ preserves-mul-map-hom-Concrete-Group : (f : hom-Concrete-Group) {x y : type-Concrete-Group G} → - Id - ( map-hom-Concrete-Group f (mul-Concrete-Group G x y)) - ( mul-Concrete-Group H - ( map-hom-Concrete-Group f x) - ( map-hom-Concrete-Group f y)) + ( map-hom-Concrete-Group f (mul-Concrete-Group G x y)) ＝ + ( mul-Concrete-Group H + ( map-hom-Concrete-Group f x) + ( map-hom-Concrete-Group f y)) preserves-mul-map-hom-Concrete-Group = preserves-mul-map-hom-∞-Group ( ∞-group-Concrete-Group G) @@ -94,9 +90,8 @@ module _ preserves-inv-map-hom-Concrete-Group : (f : hom-Concrete-Group) (x : type-Concrete-Group G) → - Id - ( map-hom-Concrete-Group f (inv-Concrete-Group G x)) - ( inv-Concrete-Group H (map-hom-Concrete-Group f x)) + map-hom-Concrete-Group f (inv-Concrete-Group G x) ＝ + inv-Concrete-Group H (map-hom-Concrete-Group f x) preserves-inv-map-hom-Concrete-Group = preserves-inv-map-hom-∞-Group ( ∞-group-Concrete-Group G) diff --git a/src/group-theory/homomorphisms-generated-subgroups.lagda.md b/src/group-theory/homomorphisms-generated-subgroups.lagda.md index 3f3d1b219b..c196e22660 100644 --- a/src/group-theory/homomorphisms-generated-subgroups.lagda.md +++ b/src/group-theory/homomorphisms-generated-subgroups.lagda.md @@ -317,9 +317,8 @@ module _ eq-map-restriction-generating-subset-Group : ( f : hom-Group G G') (x : type-subtype S) → - Id - ( map-emb restriction-generating-subset-Group f x) - ( map-hom-Group G G' f (pr1 x)) + map-emb restriction-generating-subset-Group f x ＝ + map-hom-Group G G' f (pr1 x) eq-map-restriction-generating-subset-Group f x = ap ( map-hom-Group G G' f) diff --git a/src/group-theory/inverse-semigroups.lagda.md b/src/group-theory/inverse-semigroups.lagda.md index 67ec5cbcc4..1bba7e938f 100644 --- a/src/group-theory/inverse-semigroups.lagda.md +++ b/src/group-theory/inverse-semigroups.lagda.md @@ -68,9 +68,8 @@ module _ associative-mul-Inverse-Semigroup : (x y z : type-Inverse-Semigroup) → - Id - ( mul-Inverse-Semigroup (mul-Inverse-Semigroup x y) z) - ( mul-Inverse-Semigroup x (mul-Inverse-Semigroup y z)) + mul-Inverse-Semigroup (mul-Inverse-Semigroup x y) z ＝ + mul-Inverse-Semigroup x (mul-Inverse-Semigroup y z) associative-mul-Inverse-Semigroup = associative-mul-Semigroup semigroup-Inverse-Semigroup @@ -84,21 +83,19 @@ module _ inner-inverse-law-mul-Inverse-Semigroup : (x : type-Inverse-Semigroup) → - Id - ( mul-Inverse-Semigroup - ( mul-Inverse-Semigroup x (inv-Inverse-Semigroup x)) - ( x)) - ( x) + mul-Inverse-Semigroup + ( mul-Inverse-Semigroup x (inv-Inverse-Semigroup x)) + ( x) ＝ + x inner-inverse-law-mul-Inverse-Semigroup x = pr1 (pr2 (center (is-inverse-semigroup-Inverse-Semigroup x))) outer-inverse-law-mul-Inverse-Semigroup : (x : type-Inverse-Semigroup) → - Id - ( mul-Inverse-Semigroup - ( mul-Inverse-Semigroup (inv-Inverse-Semigroup x) x) - ( inv-Inverse-Semigroup x)) - ( inv-Inverse-Semigroup x) + mul-Inverse-Semigroup + ( mul-Inverse-Semigroup (inv-Inverse-Semigroup x) x) + ( inv-Inverse-Semigroup x) ＝ + inv-Inverse-Semigroup x outer-inverse-law-mul-Inverse-Semigroup x = pr2 (pr2 (center (is-inverse-semigroup-Inverse-Semigroup x))) ``` diff --git a/src/group-theory/iterated-cartesian-products-concrete-groups.lagda.md b/src/group-theory/iterated-cartesian-products-concrete-groups.lagda.md index 75e7a17979..8105aac994 100644 --- a/src/group-theory/iterated-cartesian-products-concrete-groups.lagda.md +++ b/src/group-theory/iterated-cartesian-products-concrete-groups.lagda.md @@ -160,42 +160,34 @@ module _ associative-mul-iterated-product-Concrete-Group : (x y z : type-iterated-product-Concrete-Group) → - Id - ( mul-iterated-product-Concrete-Group - ( mul-iterated-product-Concrete-Group x y) - ( z)) - ( mul-iterated-product-Concrete-Group - ( x) - ( mul-iterated-product-Concrete-Group y z)) + mul-iterated-product-Concrete-Group + ( mul-iterated-product-Concrete-Group x y) + ( z) ＝ + mul-iterated-product-Concrete-Group + ( x) + ( mul-iterated-product-Concrete-Group y z) associative-mul-iterated-product-Concrete-Group = associative-mul-∞-Group ∞-group-iterated-product-Concrete-Group left-unit-law-mul-iterated-product-Concrete-Group : (x : type-iterated-product-Concrete-Group) → - Id - ( mul-iterated-product-Concrete-Group - ( unit-iterated-product-Concrete-Group) - ( x)) - ( x) + mul-iterated-product-Concrete-Group unit-iterated-product-Concrete-Group x ＝ + x left-unit-law-mul-iterated-product-Concrete-Group = left-unit-law-mul-∞-Group ∞-group-iterated-product-Concrete-Group right-unit-law-mul-iterated-product-Concrete-Group : (y : type-iterated-product-Concrete-Group) → - Id - ( mul-iterated-product-Concrete-Group - ( y) - ( unit-iterated-product-Concrete-Group)) - ( y) + mul-iterated-product-Concrete-Group y unit-iterated-product-Concrete-Group ＝ + y right-unit-law-mul-iterated-product-Concrete-Group = right-unit-law-mul-∞-Group ∞-group-iterated-product-Concrete-Group coherence-unit-laws-mul-iterated-product-Concrete-Group : - Id - ( left-unit-law-mul-iterated-product-Concrete-Group - unit-iterated-product-Concrete-Group) - ( right-unit-law-mul-iterated-product-Concrete-Group - unit-iterated-product-Concrete-Group) + ( left-unit-law-mul-iterated-product-Concrete-Group + unit-iterated-product-Concrete-Group) ＝ + ( right-unit-law-mul-iterated-product-Concrete-Group + unit-iterated-product-Concrete-Group) coherence-unit-laws-mul-iterated-product-Concrete-Group = coherence-unit-laws-mul-∞-Group ∞-group-iterated-product-Concrete-Group @@ -206,21 +198,19 @@ module _ left-inverse-law-mul-iterated-product-Concrete-Group : (x : type-iterated-product-Concrete-Group) → - Id - ( mul-iterated-product-Concrete-Group - ( inv-iterated-product-Concrete-Group x) - ( x)) - ( unit-iterated-product-Concrete-Group) + mul-iterated-product-Concrete-Group + ( inv-iterated-product-Concrete-Group x) + ( x) ＝ + unit-iterated-product-Concrete-Group left-inverse-law-mul-iterated-product-Concrete-Group = left-inverse-law-mul-∞-Group ∞-group-iterated-product-Concrete-Group right-inverse-law-mul-iterated-product-Concrete-Group : (x : type-iterated-product-Concrete-Group) → - Id - ( mul-iterated-product-Concrete-Group - ( x) - ( inv-iterated-product-Concrete-Group x)) - ( unit-iterated-product-Concrete-Group) + mul-iterated-product-Concrete-Group + ( x) + ( inv-iterated-product-Concrete-Group x) ＝ + unit-iterated-product-Concrete-Group right-inverse-law-mul-iterated-product-Concrete-Group = right-inverse-law-mul-∞-Group ∞-group-iterated-product-Concrete-Group @@ -278,8 +268,7 @@ equiv-type-Concrete-group-iterated-product-Concrete-Group zero-ℕ G = ( is-set-is-contr is-contr-raise-unit raise-star raise-star) refl) is-contr-raise-unit equiv-type-Concrete-group-iterated-product-Concrete-Group (succ-ℕ n) G = - equiv-product - ( id-equiv) + equiv-product-right ( equiv-type-Concrete-group-iterated-product-Concrete-Group n (G ∘ inl)) ∘e equiv-type-Concrete-Group-product-Concrete-Group ( G (inr star)) diff --git a/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md b/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md index 60b25923c5..fe5e435588 100644 --- a/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md +++ b/src/group-theory/precategory-of-orbits-monoid-actions.lagda.md @@ -60,9 +60,7 @@ module _ {x y : type-action-Monoid M X} (f g : hom-orbit-action-Monoid x y) → UU l1 htpy-hom-orbit-action-Monoid {x} {y} f g = - Id - ( element-hom-orbit-action-Monoid f) - ( element-hom-orbit-action-Monoid g) + ( element-hom-orbit-action-Monoid f ＝ element-hom-orbit-action-Monoid g) refl-htpy-hom-orbit-action-Monoid : {x y : type-action-Monoid M X} (f : hom-orbit-action-Monoid x y) → diff --git a/src/group-theory/products-of-elements-monoids.lagda.md b/src/group-theory/products-of-elements-monoids.lagda.md index a4712a4f13..73285db9b2 100644 --- a/src/group-theory/products-of-elements-monoids.lagda.md +++ b/src/group-theory/products-of-elements-monoids.lagda.md @@ -47,9 +47,8 @@ module _ distributive-mul-concat-list-Monoid : (l1 l2 : list (type-Monoid M)) → - Id - ( mul-list-Monoid M (concat-list l1 l2)) - ( mul-Monoid M (mul-list-Monoid M l1) (mul-list-Monoid M l2)) + mul-list-Monoid M (concat-list l1 l2) ＝ + mul-Monoid M (mul-list-Monoid M l1) (mul-list-Monoid M l2) distributive-mul-concat-list-Monoid nil l2 = inv (left-unit-law-mul-Monoid M (mul-list-Monoid M l2)) distributive-mul-concat-list-Monoid (cons x l1) l2 = diff --git a/src/group-theory/semigroups.lagda.md b/src/group-theory/semigroups.lagda.md index 4709573fcf..80f0317523 100644 --- a/src/group-theory/semigroups.lagda.md +++ b/src/group-theory/semigroups.lagda.md @@ -68,9 +68,8 @@ module _ associative-mul-Semigroup : (x y z : type-Semigroup) → - Id - ( mul-Semigroup (mul-Semigroup x y) z) - ( mul-Semigroup x (mul-Semigroup y z)) + mul-Semigroup (mul-Semigroup x y) z ＝ + mul-Semigroup x (mul-Semigroup y z) associative-mul-Semigroup = pr2 has-associative-mul-Semigroup left-swap-mul-Semigroup : diff --git a/src/group-theory/subgroups-abelian-groups.lagda.md b/src/group-theory/subgroups-abelian-groups.lagda.md index 2c1f5234a5..15443acff8 100644 --- a/src/group-theory/subgroups-abelian-groups.lagda.md +++ b/src/group-theory/subgroups-abelian-groups.lagda.md @@ -203,9 +203,8 @@ module _ associative-add-Subgroup-Ab : ( x y z : type-ab-Subgroup-Ab) → - Id - ( add-ab-Subgroup-Ab (add-ab-Subgroup-Ab x y) z) - ( add-ab-Subgroup-Ab x (add-ab-Subgroup-Ab y z)) + add-ab-Subgroup-Ab (add-ab-Subgroup-Ab x y) z ＝ + add-ab-Subgroup-Ab x (add-ab-Subgroup-Ab y z) associative-add-Subgroup-Ab = associative-mul-Subgroup (group-Ab A) B @@ -223,17 +222,13 @@ module _ left-inverse-law-add-Subgroup-Ab : (x : type-ab-Subgroup-Ab) → - Id - ( add-ab-Subgroup-Ab (neg-ab-Subgroup-Ab x) x) - ( zero-ab-Subgroup-Ab) + add-ab-Subgroup-Ab (neg-ab-Subgroup-Ab x) x ＝ zero-ab-Subgroup-Ab left-inverse-law-add-Subgroup-Ab = left-inverse-law-mul-Subgroup (group-Ab A) B right-inverse-law-add-Subgroup-Ab : (x : type-ab-Subgroup-Ab) → - Id - ( add-ab-Subgroup-Ab x (neg-ab-Subgroup-Ab x)) - ( zero-ab-Subgroup-Ab) + add-ab-Subgroup-Ab x (neg-ab-Subgroup-Ab x) ＝ zero-ab-Subgroup-Ab right-inverse-law-add-Subgroup-Ab = right-inverse-law-mul-Subgroup (group-Ab A) B diff --git a/src/group-theory/subgroups-generated-by-subsets-groups.lagda.md b/src/group-theory/subgroups-generated-by-subsets-groups.lagda.md index bdcb3c854f..8627d87149 100644 --- a/src/group-theory/subgroups-generated-by-subsets-groups.lagda.md +++ b/src/group-theory/subgroups-generated-by-subsets-groups.lagda.md @@ -113,11 +113,10 @@ module _ preserves-concat-ev-formal-combination-subset-Group : (u v : formal-combination-subset-Group) → - Id - ( ev-formal-combination-subset-Group (concat-list u v)) - ( mul-Group G - ( ev-formal-combination-subset-Group u) - ( ev-formal-combination-subset-Group v)) + ( ev-formal-combination-subset-Group (concat-list u v)) ＝ + ( mul-Group G + ( ev-formal-combination-subset-Group u) + ( ev-formal-combination-subset-Group v)) preserves-concat-ev-formal-combination-subset-Group nil v = inv (left-unit-law-mul-Group G (ev-formal-combination-subset-Group v)) preserves-concat-ev-formal-combination-subset-Group @@ -153,10 +152,9 @@ module _ preserves-inv-ev-formal-combination-subset-Group : (u : formal-combination-subset-Group) → - Id - ( ev-formal-combination-subset-Group - ( inv-formal-combination-subset-Group u)) - ( inv-Group G (ev-formal-combination-subset-Group u)) + ev-formal-combination-subset-Group + ( inv-formal-combination-subset-Group u) ＝ + inv-Group G (ev-formal-combination-subset-Group u) preserves-inv-ev-formal-combination-subset-Group nil = inv (inv-unit-Group G) preserves-inv-ev-formal-combination-subset-Group diff --git a/src/group-theory/subgroups.lagda.md b/src/group-theory/subgroups.lagda.md index c47c094a4b..7cd7c96ecd 100644 --- a/src/group-theory/subgroups.lagda.md +++ b/src/group-theory/subgroups.lagda.md @@ -271,9 +271,8 @@ module _ associative-mul-Subgroup : (x y z : type-group-Subgroup) → - Id - ( mul-Subgroup (mul-Subgroup x y) z) - ( mul-Subgroup x (mul-Subgroup y z)) + mul-Subgroup (mul-Subgroup x y) z ＝ + mul-Subgroup x (mul-Subgroup y z) associative-mul-Subgroup x y z = eq-subgroup-eq-group ( associative-mul-Group G (pr1 x) (pr1 y) (pr1 z)) @@ -299,17 +298,13 @@ module _ left-inverse-law-mul-Subgroup : ( x : type-group-Subgroup) → - Id - ( mul-Subgroup (inv-Subgroup x) x) - ( unit-Subgroup) + mul-Subgroup (inv-Subgroup x) x ＝ unit-Subgroup left-inverse-law-mul-Subgroup x = eq-subgroup-eq-group (left-inverse-law-mul-Group G (pr1 x)) right-inverse-law-mul-Subgroup : (x : type-group-Subgroup) → - Id - ( mul-Subgroup x (inv-Subgroup x)) - ( unit-Subgroup) + mul-Subgroup x (inv-Subgroup x) ＝ unit-Subgroup right-inverse-law-mul-Subgroup x = eq-subgroup-eq-group (right-inverse-law-mul-Group G (pr1 x)) diff --git a/src/group-theory/substitution-functor-group-actions.lagda.md b/src/group-theory/substitution-functor-group-actions.lagda.md index f1820ab830..a49ac526a4 100644 --- a/src/group-theory/substitution-functor-group-actions.lagda.md +++ b/src/group-theory/substitution-functor-group-actions.lagda.md @@ -64,9 +64,8 @@ module _ preserves-id-subst-action-Group : {l3 : Level} (X : action-Group H l3) → - Id - ( hom-subst-action-Group X X (id-hom-action-Group H X)) - ( id-hom-action-Group G (obj-subst-action-Group X)) + ( hom-subst-action-Group X X (id-hom-action-Group H X)) ＝ + ( id-hom-action-Group G (obj-subst-action-Group X)) preserves-id-subst-action-Group X = refl preserves-comp-subst-action-Group : @@ -74,15 +73,13 @@ module _ (Y : action-Group H l4) (Z : action-Group H l5) (g : hom-action-Group H Y Z) (f : hom-action-Group H X Y) → - Id - ( hom-subst-action-Group X Z - ( comp-hom-action-Group H X Y Z g f)) - ( comp-hom-action-Group G - ( obj-subst-action-Group X) - ( obj-subst-action-Group Y) - ( obj-subst-action-Group Z) - ( hom-subst-action-Group Y Z g) - ( hom-subst-action-Group X Y f)) + hom-subst-action-Group X Z (comp-hom-action-Group H X Y Z g f) ＝ + comp-hom-action-Group G + ( obj-subst-action-Group X) + ( obj-subst-action-Group Y) + ( obj-subst-action-Group Z) + ( hom-subst-action-Group Y Z g) + ( hom-subst-action-Group X Y f) preserves-comp-subst-action-Group X Y Z g f = refl subst-action-Group : diff --git a/src/group-theory/symmetric-groups.lagda.md b/src/group-theory/symmetric-groups.lagda.md index bb5b38be9e..5d601670c4 100644 --- a/src/group-theory/symmetric-groups.lagda.md +++ b/src/group-theory/symmetric-groups.lagda.md @@ -105,14 +105,13 @@ module _ ( eq-equiv-eq-map-equiv refl) is-section-hom-inv-symmetric-group-equiv-Set : - Id - ( comp-hom-Group - ( symmetric-Group Y) - ( symmetric-Group X) - ( symmetric-Group Y) - ( hom-symmetric-group-equiv-Set) - ( hom-inv-symmetric-group-equiv-Set)) - ( id-hom-Group (symmetric-Group Y)) + comp-hom-Group + ( symmetric-Group Y) + ( symmetric-Group X) + ( symmetric-Group Y) + ( hom-symmetric-group-equiv-Set) + ( hom-inv-symmetric-group-equiv-Set) ＝ + id-hom-Group (symmetric-Group Y) is-section-hom-inv-symmetric-group-equiv-Set = eq-type-subtype ( preserves-mul-prop-Semigroup @@ -125,14 +124,13 @@ module _ ( eq-equiv-eq-map-equiv refl))) is-retraction-hom-inv-symmetric-group-equiv-Set : - Id - ( comp-hom-Group - ( symmetric-Group X) - ( symmetric-Group Y) - ( symmetric-Group X) - ( hom-inv-symmetric-group-equiv-Set) - ( hom-symmetric-group-equiv-Set)) - ( id-hom-Group (symmetric-Group X)) + comp-hom-Group + ( symmetric-Group X) + ( symmetric-Group Y) + ( symmetric-Group X) + ( hom-inv-symmetric-group-equiv-Set) + ( hom-symmetric-group-equiv-Set) ＝ + id-hom-Group (symmetric-Group X) is-retraction-hom-inv-symmetric-group-equiv-Set = eq-type-subtype ( preserves-mul-prop-Semigroup diff --git a/src/group-theory/torsors.lagda.md b/src/group-theory/torsors.lagda.md index 0e1b1d1375..c87c5d6c96 100644 --- a/src/group-theory/torsors.lagda.md +++ b/src/group-theory/torsors.lagda.md @@ -573,17 +573,10 @@ module _ preserves-mul-equiv-Eq-equiv-Torsor-Group : { p q : principal-Torsor-Group G ＝ principal-Torsor-Group G} → - Id - ( map-equiv - ( equiv-Eq-equiv-Torsor-Group (principal-Torsor-Group G)) - ( p ∙ q)) - ( mul-Group G - ( map-equiv - ( equiv-Eq-equiv-Torsor-Group (principal-Torsor-Group G)) - ( p)) - ( map-equiv - ( equiv-Eq-equiv-Torsor-Group (principal-Torsor-Group G)) - ( q))) + map-equiv (equiv-Eq-equiv-Torsor-Group (principal-Torsor-Group G)) (p ∙ q) ＝ + mul-Group G + ( map-equiv (equiv-Eq-equiv-Torsor-Group (principal-Torsor-Group G)) p) + ( map-equiv (equiv-Eq-equiv-Torsor-Group (principal-Torsor-Group G)) q) preserves-mul-equiv-Eq-equiv-Torsor-Group {p} {q} = ( ap ( Eq-equiv-Torsor-Group (principal-Torsor-Group G)) diff --git a/src/higher-group-theory/cartesian-products-higher-groups.lagda.md b/src/higher-group-theory/cartesian-products-higher-groups.lagda.md index 920864f380..0d0bd9c95a 100644 --- a/src/higher-group-theory/cartesian-products-higher-groups.lagda.md +++ b/src/higher-group-theory/cartesian-products-higher-groups.lagda.md @@ -92,9 +92,8 @@ module _ assoc-mul-product-∞-Group : (x y z : type-product-∞-Group) → - Id - ( mul-product-∞-Group (mul-product-∞-Group x y) z) - ( mul-product-∞-Group x (mul-product-∞-Group y z)) + mul-product-∞-Group (mul-product-∞-Group x y) z ＝ + mul-product-∞-Group x (mul-product-∞-Group y z) assoc-mul-product-∞-Group = associative-mul-Ω classifying-pointed-type-product-∞-Group @@ -111,9 +110,8 @@ module _ right-unit-law-mul-Ω classifying-pointed-type-product-∞-Group coherence-unit-laws-mul-product-∞-Group : - Id - ( left-unit-law-mul-product-∞-Group unit-product-∞-Group) - ( right-unit-law-mul-product-∞-Group unit-product-∞-Group) + left-unit-law-mul-product-∞-Group unit-product-∞-Group ＝ + right-unit-law-mul-product-∞-Group unit-product-∞-Group coherence-unit-laws-mul-product-∞-Group = refl inv-product-∞-Group : type-product-∞-Group → type-product-∞-Group diff --git a/src/higher-group-theory/higher-groups.lagda.md b/src/higher-group-theory/higher-groups.lagda.md index 0a27359d3b..783087da6b 100644 --- a/src/higher-group-theory/higher-groups.lagda.md +++ b/src/higher-group-theory/higher-groups.lagda.md @@ -113,9 +113,8 @@ module _ associative-mul-∞-Group : (x y z : type-∞-Group) → - Id - ( mul-∞-Group (mul-∞-Group x y) z) - ( mul-∞-Group x (mul-∞-Group y z)) + mul-∞-Group (mul-∞-Group x y) z ＝ + mul-∞-Group x (mul-∞-Group y z) associative-mul-∞-Group = associative-mul-Ω classifying-pointed-type-∞-Group left-unit-law-mul-∞-Group : diff --git a/src/higher-group-theory/iterated-cartesian-products-higher-groups.lagda.md b/src/higher-group-theory/iterated-cartesian-products-higher-groups.lagda.md index e9a5b7aaab..a10ad22a4b 100644 --- a/src/higher-group-theory/iterated-cartesian-products-higher-groups.lagda.md +++ b/src/higher-group-theory/iterated-cartesian-products-higher-groups.lagda.md @@ -113,45 +113,29 @@ module _ assoc-mul-iterated-product-∞-Group : (x y z : type-iterated-product-∞-Group) → - Id - ( mul-iterated-product-∞-Group - ( mul-iterated-product-∞-Group x y) - ( z)) - ( mul-iterated-product-∞-Group - ( x) - ( mul-iterated-product-∞-Group y z)) + mul-iterated-product-∞-Group (mul-iterated-product-∞-Group x y) z ＝ + mul-iterated-product-∞-Group x (mul-iterated-product-∞-Group y z) assoc-mul-iterated-product-∞-Group = associative-mul-Ω classifying-pointed-type-iterated-product-∞-Group left-unit-law-mul-iterated-product-∞-Group : (x : type-iterated-product-∞-Group) → - Id - ( mul-iterated-product-∞-Group - ( unit-iterated-product-∞-Group) - ( x)) - ( x) + mul-iterated-product-∞-Group unit-iterated-product-∞-Group x ＝ x left-unit-law-mul-iterated-product-∞-Group = left-unit-law-mul-Ω classifying-pointed-type-iterated-product-∞-Group right-unit-law-mul-iterated-product-∞-Group : (y : type-iterated-product-∞-Group) → - Id - ( mul-iterated-product-∞-Group - ( y) - ( unit-iterated-product-∞-Group)) - ( y) + mul-iterated-product-∞-Group y unit-iterated-product-∞-Group ＝ y right-unit-law-mul-iterated-product-∞-Group = right-unit-law-mul-Ω classifying-pointed-type-iterated-product-∞-Group coherence-unit-laws-mul-iterated-product-∞-Group : - Id - ( left-unit-law-mul-iterated-product-∞-Group - unit-iterated-product-∞-Group) - ( right-unit-law-mul-iterated-product-∞-Group - unit-iterated-product-∞-Group) + left-unit-law-mul-iterated-product-∞-Group unit-iterated-product-∞-Group ＝ + right-unit-law-mul-iterated-product-∞-Group unit-iterated-product-∞-Group coherence-unit-laws-mul-iterated-product-∞-Group = refl inv-iterated-product-∞-Group : @@ -161,22 +145,16 @@ module _ left-inverse-law-mul-iterated-product-∞-Group : (x : type-iterated-product-∞-Group) → - Id - ( mul-iterated-product-∞-Group - ( inv-iterated-product-∞-Group x) - ( x)) - ( unit-iterated-product-∞-Group) + mul-iterated-product-∞-Group (inv-iterated-product-∞-Group x) x ＝ + unit-iterated-product-∞-Group left-inverse-law-mul-iterated-product-∞-Group = left-inverse-law-mul-Ω classifying-pointed-type-iterated-product-∞-Group right-inverse-law-mul-iterated-product-∞-Group : (x : type-iterated-product-∞-Group) → - Id - ( mul-iterated-product-∞-Group - ( x) - ( inv-iterated-product-∞-Group x)) - ( unit-iterated-product-∞-Group) + mul-iterated-product-∞-Group x (inv-iterated-product-∞-Group x) ＝ + unit-iterated-product-∞-Group right-inverse-law-mul-iterated-product-∞-Group = right-inverse-law-mul-Ω classifying-pointed-type-iterated-product-∞-Group diff --git a/src/linear-algebra/finite-sequences-in-euclidean-domains.lagda.md b/src/linear-algebra/finite-sequences-in-euclidean-domains.lagda.md index 955193ee9e..fa50780b53 100644 --- a/src/linear-algebra/finite-sequences-in-euclidean-domains.lagda.md +++ b/src/linear-algebra/finite-sequences-in-euclidean-domains.lagda.md @@ -192,23 +192,20 @@ module _ left-inverse-law-add-fin-sequence-type-Euclidean-Domain : (n : ℕ) (v : fin-sequence-type-Euclidean-Domain R n) → - Id - ( add-fin-sequence-type-Euclidean-Domain - R n ( neg-fin-sequence-type-Euclidean-Domain R n v) v) - ( zero-fin-sequence-type-Euclidean-Domain R n) + add-fin-sequence-type-Euclidean-Domain R n + ( neg-fin-sequence-type-Euclidean-Domain R n v) + ( v) ＝ + zero-fin-sequence-type-Euclidean-Domain R n left-inverse-law-add-fin-sequence-type-Euclidean-Domain = left-inverse-law-add-fin-sequence-type-Commutative-Ring ( commutative-ring-Euclidean-Domain R) right-inverse-law-add-fin-sequence-type-Euclidean-Domain : (n : ℕ) (v : fin-sequence-type-Euclidean-Domain R n) → - Id - ( add-fin-sequence-type-Euclidean-Domain - ( R) - ( n) - ( v) - ( neg-fin-sequence-type-Euclidean-Domain R n v)) - ( zero-fin-sequence-type-Euclidean-Domain R n) + add-fin-sequence-type-Euclidean-Domain R n + ( v) + ( neg-fin-sequence-type-Euclidean-Domain R n v) ＝ + zero-fin-sequence-type-Euclidean-Domain R n right-inverse-law-add-fin-sequence-type-Euclidean-Domain = right-inverse-law-add-fin-sequence-type-Commutative-Ring ( commutative-ring-Euclidean-Domain R) diff --git a/src/linear-algebra/left-modules-rings.lagda.md b/src/linear-algebra/left-modules-rings.lagda.md index c1e3fd47bf..f884bf39f8 100644 --- a/src/linear-algebra/left-modules-rings.lagda.md +++ b/src/linear-algebra/left-modules-rings.lagda.md @@ -126,9 +126,8 @@ module _ ```agda associative-add-left-module-Ring : (x y z : type-left-module-Ring) → - Id - ( add-left-module-Ring (add-left-module-Ring x y) z) - ( add-left-module-Ring x (add-left-module-Ring y z)) + add-left-module-Ring (add-left-module-Ring x y) z ＝ + add-left-module-Ring x (add-left-module-Ring y z) associative-add-left-module-Ring = associative-add-Ab ab-left-module-Ring ``` @@ -162,17 +161,15 @@ module _ left-inverse-law-add-left-module-Ring : (x : type-left-module-Ring R M) → - Id - ( add-left-module-Ring R M (neg-left-module-Ring R M x) x) - ( zero-left-module-Ring R M) + add-left-module-Ring R M (neg-left-module-Ring R M x) x ＝ + zero-left-module-Ring R M left-inverse-law-add-left-module-Ring = left-inverse-law-add-Ab (ab-left-module-Ring R M) right-inverse-law-add-left-module-Ring : (x : type-left-module-Ring R M) → - Id - ( add-left-module-Ring R M x (neg-left-module-Ring R M x)) - ( zero-left-module-Ring R M) + add-left-module-Ring R M x (neg-left-module-Ring R M x) ＝ + zero-left-module-Ring R M right-inverse-law-add-left-module-Ring = right-inverse-law-add-Ab (ab-left-module-Ring R M) ``` @@ -212,11 +209,10 @@ module _ abstract left-distributive-mul-add-left-module-Ring : (r : type-Ring R) (x y : type-left-module-Ring R M) → - Id - ( mul-left-module-Ring R M r (add-left-module-Ring R M x y)) - ( add-left-module-Ring R M - ( mul-left-module-Ring R M r x) - ( mul-left-module-Ring R M r y)) + mul-left-module-Ring R M r (add-left-module-Ring R M x y) ＝ + add-left-module-Ring R M + ( mul-left-module-Ring R M r x) + ( mul-left-module-Ring R M r y) left-distributive-mul-add-left-module-Ring r x y = preserves-add-hom-Ab ( ab-left-module-Ring R M) @@ -228,11 +224,10 @@ module _ right-distributive-mul-add-left-module-Ring : (r s : type-Ring R) (x : type-left-module-Ring R M) → - Id - ( mul-left-module-Ring R M (add-Ring R r s) x) - ( add-left-module-Ring R M - ( mul-left-module-Ring R M r x) - ( mul-left-module-Ring R M s x)) + mul-left-module-Ring R M (add-Ring R r s) x ＝ + add-left-module-Ring R M + ( mul-left-module-Ring R M r x) + ( mul-left-module-Ring R M s x) right-distributive-mul-add-left-module-Ring r s = htpy-eq-hom-Ab ( ab-left-module-Ring R M) @@ -267,9 +262,8 @@ module _ abstract associative-mul-left-module-Ring : (r s : type-Ring R) (x : type-left-module-Ring R M) → - Id - ( mul-left-module-Ring R M (mul-Ring R r s) x) - ( mul-left-module-Ring R M r (mul-left-module-Ring R M s x)) + mul-left-module-Ring R M (mul-Ring R r s) x ＝ + mul-left-module-Ring R M r (mul-left-module-Ring R M s x) associative-mul-left-module-Ring r s = htpy-eq-hom-Ab ( ab-left-module-Ring R M) @@ -321,9 +315,8 @@ module _ right-zero-law-mul-left-module-Ring : (r : type-Ring R) → - Id - ( mul-left-module-Ring R M r (zero-left-module-Ring R M)) - ( zero-left-module-Ring R M) + mul-left-module-Ring R M r (zero-left-module-Ring R M) ＝ + zero-left-module-Ring R M right-zero-law-mul-left-module-Ring r = preserves-zero-hom-Ab ( ab-left-module-Ring R M) @@ -344,9 +337,8 @@ module _ abstract left-negative-law-mul-left-module-Ring : (r : type-Ring R) (x : type-left-module-Ring R M) → - Id - ( mul-left-module-Ring R M (neg-Ring R r) x) - ( neg-left-module-Ring R M (mul-left-module-Ring R M r x)) + mul-left-module-Ring R M (neg-Ring R r) x ＝ + neg-left-module-Ring R M (mul-left-module-Ring R M r x) left-negative-law-mul-left-module-Ring r = htpy-eq-hom-Ab ( ab-left-module-Ring R M) @@ -368,9 +360,8 @@ module _ right-negative-law-mul-left-module-Ring : (r : type-Ring R) (x : type-left-module-Ring R M) → - Id - ( mul-left-module-Ring R M r (neg-left-module-Ring R M x)) - ( neg-left-module-Ring R M (mul-left-module-Ring R M r x)) + mul-left-module-Ring R M r (neg-left-module-Ring R M x) ＝ + neg-left-module-Ring R M (mul-left-module-Ring R M r x) right-negative-law-mul-left-module-Ring r x = preserves-negatives-hom-Ab ( ab-left-module-Ring R M) diff --git a/src/linear-algebra/matrices-on-rings.lagda.md b/src/linear-algebra/matrices-on-rings.lagda.md index 30479f8f26..dc0b737311 100644 --- a/src/linear-algebra/matrices-on-rings.lagda.md +++ b/src/linear-algebra/matrices-on-rings.lagda.md @@ -74,9 +74,8 @@ module _ associative-add-matrix-Ring : {m n : ℕ} (A B C : matrix-Ring R m n) → - Id - ( add-matrix-Ring R (add-matrix-Ring R A B) C) - ( add-matrix-Ring R A (add-matrix-Ring R B C)) + add-matrix-Ring R (add-matrix-Ring R A B) C ＝ + add-matrix-Ring R A (add-matrix-Ring R B C) associative-add-matrix-Ring empty-tuple empty-tuple empty-tuple = refl associative-add-matrix-Ring (v ∷ A) (w ∷ B) (z ∷ C) = ap-binary _∷_ diff --git a/src/linear-algebra/right-modules-rings.lagda.md b/src/linear-algebra/right-modules-rings.lagda.md index a3f7cfe14e..41744df1db 100644 --- a/src/linear-algebra/right-modules-rings.lagda.md +++ b/src/linear-algebra/right-modules-rings.lagda.md @@ -100,9 +100,8 @@ module _ associative-add-right-module-Ring : (x y z : type-right-module-Ring R M) → - Id - ( add-right-module-Ring R M (add-right-module-Ring R M x y) z) - ( add-right-module-Ring R M x (add-right-module-Ring R M y z)) + add-right-module-Ring R M (add-right-module-Ring R M x y) z ＝ + add-right-module-Ring R M x (add-right-module-Ring R M y z) associative-add-right-module-Ring = associative-add-Ab (ab-right-module-Ring R M) ``` @@ -136,17 +135,15 @@ module _ left-inverse-law-add-right-module-Ring : (x : type-right-module-Ring R M) → - Id - ( add-right-module-Ring R M (neg-right-module-Ring R M x) x) - ( zero-right-module-Ring R M) + add-right-module-Ring R M (neg-right-module-Ring R M x) x ＝ + zero-right-module-Ring R M left-inverse-law-add-right-module-Ring = left-inverse-law-add-Ab (ab-right-module-Ring R M) right-inverse-law-add-right-module-Ring : (x : type-right-module-Ring R M) → - Id - ( add-right-module-Ring R M x (neg-right-module-Ring R M x)) - ( zero-right-module-Ring R M) + add-right-module-Ring R M x (neg-right-module-Ring R M x) ＝ + zero-right-module-Ring R M right-inverse-law-add-right-module-Ring = right-inverse-law-add-Ab (ab-right-module-Ring R M) ``` @@ -184,11 +181,10 @@ module _ left-distributive-mul-add-right-module-Ring : (r : type-Ring R) (x y : type-right-module-Ring R M) → - Id - ( mul-right-module-Ring R M r (add-right-module-Ring R M x y)) - ( add-right-module-Ring R M - ( mul-right-module-Ring R M r x) - ( mul-right-module-Ring R M r y)) + mul-right-module-Ring R M r (add-right-module-Ring R M x y) ＝ + add-right-module-Ring R M + ( mul-right-module-Ring R M r x) + ( mul-right-module-Ring R M r y) left-distributive-mul-add-right-module-Ring r x y = preserves-add-hom-Ab ( ab-right-module-Ring R M) @@ -200,11 +196,10 @@ module _ right-distributive-mul-add-right-module-Ring : (r s : type-Ring R) (x : type-right-module-Ring R M) → - Id - ( mul-right-module-Ring R M (add-Ring R r s) x) - ( add-right-module-Ring R M - ( mul-right-module-Ring R M r x) - ( mul-right-module-Ring R M s x)) + mul-right-module-Ring R M (add-Ring R r s) x ＝ + add-right-module-Ring R M + ( mul-right-module-Ring R M r x) + ( mul-right-module-Ring R M s x) right-distributive-mul-add-right-module-Ring r s = htpy-eq-hom-Ab ( ab-right-module-Ring R M) @@ -238,9 +233,8 @@ module _ associative-mul-right-module-Ring : (r s : type-Ring R) (x : type-right-module-Ring R M) → - Id - ( mul-right-module-Ring R M (mul-Ring R r s) x) - ( mul-right-module-Ring R M s (mul-right-module-Ring R M r x)) + mul-right-module-Ring R M (mul-Ring R r s) x ＝ + mul-right-module-Ring R M s (mul-right-module-Ring R M r x) associative-mul-right-module-Ring r s = htpy-eq-hom-Ab ( ab-right-module-Ring R M) @@ -291,9 +285,8 @@ module _ right-zero-law-mul-right-module-Ring : (r : type-Ring R) → - Id - ( mul-right-module-Ring R M r (zero-right-module-Ring R M)) - ( zero-right-module-Ring R M) + mul-right-module-Ring R M r (zero-right-module-Ring R M) ＝ + zero-right-module-Ring R M right-zero-law-mul-right-module-Ring r = preserves-zero-hom-Ab ( ab-right-module-Ring R M) @@ -313,9 +306,8 @@ module _ left-negative-law-mul-right-module-Ring : (r : type-Ring R) (x : type-right-module-Ring R M) → - Id - ( mul-right-module-Ring R M (neg-Ring R r) x) - ( neg-right-module-Ring R M (mul-right-module-Ring R M r x)) + mul-right-module-Ring R M (neg-Ring R r) x ＝ + neg-right-module-Ring R M (mul-right-module-Ring R M r x) left-negative-law-mul-right-module-Ring r = htpy-eq-hom-Ab ( ab-right-module-Ring R M) @@ -337,9 +329,8 @@ module _ right-negative-law-mul-right-module-Ring : (r : type-Ring R) (x : type-right-module-Ring R M) → - Id - ( mul-right-module-Ring R M r (neg-right-module-Ring R M x)) - ( neg-right-module-Ring R M (mul-right-module-Ring R M r x)) + mul-right-module-Ring R M r (neg-right-module-Ring R M x) ＝ + neg-right-module-Ring R M (mul-right-module-Ring R M r x) right-negative-law-mul-right-module-Ring r x = preserves-negatives-hom-Ab ( ab-right-module-Ring R M) diff --git a/src/linear-algebra/tuples-on-euclidean-domains.lagda.md b/src/linear-algebra/tuples-on-euclidean-domains.lagda.md index ff3a592561..50f4cb3853 100644 --- a/src/linear-algebra/tuples-on-euclidean-domains.lagda.md +++ b/src/linear-algebra/tuples-on-euclidean-domains.lagda.md @@ -109,9 +109,8 @@ module _ associative-add-tuple-Euclidean-Domain : {n : ℕ} (v1 v2 v3 : tuple-Euclidean-Domain R n) → - Id - ( add-tuple-Euclidean-Domain R (add-tuple-Euclidean-Domain R v1 v2) v3) - ( add-tuple-Euclidean-Domain R v1 (add-tuple-Euclidean-Domain R v2 v3)) + add-tuple-Euclidean-Domain R (add-tuple-Euclidean-Domain R v1 v2) v3 ＝ + add-tuple-Euclidean-Domain R v1 (add-tuple-Euclidean-Domain R v2 v3) associative-add-tuple-Euclidean-Domain = associative-add-tuple-Commutative-Ring ( commutative-ring-Euclidean-Domain R) @@ -163,18 +162,16 @@ module _ left-inverse-law-add-tuple-Euclidean-Domain : {n : ℕ} (v : tuple-Euclidean-Domain R n) → - Id - ( add-tuple-Euclidean-Domain R (neg-tuple-Euclidean-Domain R v) v) - ( zero-tuple-Euclidean-Domain R) + add-tuple-Euclidean-Domain R (neg-tuple-Euclidean-Domain R v) v ＝ + zero-tuple-Euclidean-Domain R left-inverse-law-add-tuple-Euclidean-Domain = left-inverse-law-add-tuple-Commutative-Ring ( commutative-ring-Euclidean-Domain R) right-inverse-law-add-tuple-Euclidean-Domain : {n : ℕ} (v : tuple-Euclidean-Domain R n) → - Id - ( add-tuple-Euclidean-Domain R v (neg-tuple-Euclidean-Domain R v)) - ( zero-tuple-Euclidean-Domain R) + add-tuple-Euclidean-Domain R v (neg-tuple-Euclidean-Domain R v) ＝ + zero-tuple-Euclidean-Domain R right-inverse-law-add-tuple-Euclidean-Domain = right-inverse-law-add-tuple-Commutative-Ring ( commutative-ring-Euclidean-Domain R) diff --git a/src/linear-algebra/tuples-on-rings.lagda.md b/src/linear-algebra/tuples-on-rings.lagda.md index 3a164d207a..047c896cea 100644 --- a/src/linear-algebra/tuples-on-rings.lagda.md +++ b/src/linear-algebra/tuples-on-rings.lagda.md @@ -101,9 +101,8 @@ module _ associative-add-tuple-Ring : {n : ℕ} (v1 v2 v3 : tuple-Ring R n) → - Id - ( add-tuple-Ring R (add-tuple-Ring R v1 v2) v3) - ( add-tuple-Ring R v1 (add-tuple-Ring R v2 v3)) + add-tuple-Ring R (add-tuple-Ring R v1 v2) v3 ＝ + add-tuple-Ring R v1 (add-tuple-Ring R v2 v3) associative-add-tuple-Ring = associative-add-tuple-Semiring (semiring-Ring R) ``` diff --git a/src/lists/concatenation-lists.lagda.md b/src/lists/concatenation-lists.lagda.md index 2e643dd79f..43c3f1b3a6 100644 --- a/src/lists/concatenation-lists.lagda.md +++ b/src/lists/concatenation-lists.lagda.md @@ -121,27 +121,23 @@ eta-list' (cons a (cons b x)) = ap (cons a) (eta-list' (cons b x)) ```agda head-concat-list : {l1 : Level} {A : UU l1} (x y : list A) → - Id - ( head-list (concat-list x y)) - ( head-list (concat-list (head-list x) (head-list y))) + head-list (concat-list x y) ＝ + head-list (concat-list (head-list x) (head-list y)) head-concat-list nil nil = refl head-concat-list nil (cons x y) = refl head-concat-list (cons a x) y = refl tail-concat-list : {l1 : Level} {A : UU l1} (x y : list A) → - Id - ( tail-list (concat-list x y)) - ( concat-list (tail-list x) (tail-list (concat-list (head-list x) y))) + tail-list (concat-list x y) ＝ + concat-list (tail-list x) (tail-list (concat-list (head-list x) y)) tail-concat-list nil y = refl tail-concat-list (cons a x) y = refl last-element-concat-list : {l1 : Level} {A : UU l1} (x y : list A) → - Id - ( last-element-list (concat-list x y)) - ( last-element-list - ( concat-list (last-element-list x) (last-element-list y))) + last-element-list (concat-list x y) ＝ + last-element-list (concat-list (last-element-list x) (last-element-list y)) last-element-concat-list nil nil = refl last-element-concat-list nil (cons b nil) = refl last-element-concat-list nil (cons b (cons c y)) = @@ -155,11 +151,10 @@ last-element-concat-list (cons a (cons b x)) y = remove-last-element-concat-list : {l1 : Level} {A : UU l1} (x y : list A) → - Id - ( remove-last-element-list (concat-list x y)) - ( concat-list - ( remove-last-element-list (concat-list x (head-list y))) - ( remove-last-element-list y)) + remove-last-element-list (concat-list x y) ＝ + concat-list + ( remove-last-element-list (concat-list x (head-list y))) + ( remove-last-element-list y) remove-last-element-concat-list nil nil = refl remove-last-element-concat-list nil (cons a nil) = refl remove-last-element-concat-list nil (cons a (cons b y)) = refl @@ -170,11 +165,10 @@ remove-last-element-concat-list (cons a (cons b x)) y = tail-concat-list' : {l1 : Level} {A : UU l1} (x y : list A) → - Id - ( tail-list (concat-list x y)) - ( concat-list - ( tail-list x) - ( tail-list (concat-list (last-element-list x) y))) + tail-list (concat-list x y) ＝ + concat-list + ( tail-list x) + ( tail-list (concat-list (last-element-list x) y)) tail-concat-list' nil y = refl tail-concat-list' (cons a nil) y = refl tail-concat-list' (cons a (cons b x)) y = diff --git a/src/lists/flattening-lists.lagda.md b/src/lists/flattening-lists.lagda.md index c06302a1bd..60423e48d9 100644 --- a/src/lists/flattening-lists.lagda.md +++ b/src/lists/flattening-lists.lagda.md @@ -45,9 +45,7 @@ flatten-unit-list x = right-unit-law-concat-list x length-flatten-list : {l1 : Level} {A : UU l1} (x : list (list A)) → - Id - ( length-list (flatten-list x)) - ( sum-list-ℕ (map-list length-list x)) + length-list (flatten-list x) ＝ sum-list-ℕ (map-list length-list x) length-flatten-list nil = refl length-flatten-list (cons a x) = ( length-concat-list a (flatten-list x)) ∙ @@ -55,9 +53,7 @@ length-flatten-list (cons a x) = flatten-concat-list : {l1 : Level} {A : UU l1} (x y : list (list A)) → - Id - ( flatten-list (concat-list x y)) - ( concat-list (flatten-list x) (flatten-list y)) + flatten-list (concat-list x y) ＝ concat-list (flatten-list x) (flatten-list y) flatten-concat-list nil y = refl flatten-concat-list (cons a x) y = ( ap (concat-list a) (flatten-concat-list x y)) ∙ @@ -65,9 +61,7 @@ flatten-concat-list (cons a x) y = flatten-flatten-list : {l1 : Level} {A : UU l1} (x : list (list (list A))) → - Id - ( flatten-list (flatten-list x)) - ( flatten-list (map-list flatten-list x)) + flatten-list (flatten-list x) ＝ flatten-list (map-list flatten-list x) flatten-flatten-list nil = refl flatten-flatten-list (cons a x) = ( flatten-concat-list a (flatten-list x)) ∙ diff --git a/src/lists/functoriality-lists.lagda.md b/src/lists/functoriality-lists.lagda.md index cb283730ec..c8cd460bb8 100644 --- a/src/lists/functoriality-lists.lagda.md +++ b/src/lists/functoriality-lists.lagda.md @@ -219,9 +219,7 @@ module _ preserves-concat-map-list : (l k : list A) → - Id - ( map-list f (concat-list l k)) - ( concat-list (map-list f l) (map-list f k)) + map-list f (concat-list l k) ＝ concat-list (map-list f l) (map-list f k) preserves-concat-map-list nil k = refl preserves-concat-map-list (cons x l) k = ap (cons (f x)) (preserves-concat-map-list l k) diff --git a/src/lists/lists.lagda.md b/src/lists/lists.lagda.md index 61636d25ed..f0d072dee6 100644 --- a/src/lists/lists.lagda.md +++ b/src/lists/lists.lagda.md @@ -188,9 +188,8 @@ eq-Eq-list (cons x l) (cons .x l') (pair refl e) = square-eq-Eq-list : {l1 : Level} {A : UU l1} {x : A} {l l' : list A} (p : l ＝ l') → - Id - ( Eq-eq-list (cons x l) (cons x l') (ap (cons x) p)) - ( pair refl (Eq-eq-list l l' p)) + Eq-eq-list (cons x l) (cons x l') (ap (cons x) p) ＝ + pair refl (Eq-eq-list l l' p) square-eq-Eq-list refl = refl is-section-eq-Eq-list : diff --git a/src/lists/reversing-lists.lagda.md b/src/lists/reversing-lists.lagda.md index 59c919f3f3..cd906c8a95 100644 --- a/src/lists/reversing-lists.lagda.md +++ b/src/lists/reversing-lists.lagda.md @@ -59,9 +59,7 @@ length-reverse-list (cons a x) = reverse-concat-list : {l1 : Level} {A : UU l1} (x y : list A) → - Id - ( reverse-list (concat-list x y)) - ( concat-list (reverse-list y) (reverse-list x)) + reverse-list (concat-list x y) ＝ concat-list (reverse-list y) (reverse-list x) reverse-concat-list nil y = inv (right-unit-law-concat-list (reverse-list y)) reverse-concat-list (cons a x) y = @@ -76,9 +74,8 @@ reverse-snoc-list (cons b x) a = ap (λ t → snoc t b) (reverse-snoc-list x a) reverse-flatten-list : {l1 : Level} {A : UU l1} (x : list (list A)) → - Id - ( reverse-list (flatten-list x)) - ( flatten-list (reverse-list (map-list reverse-list x))) + reverse-list (flatten-list x) ＝ + flatten-list (reverse-list (map-list reverse-list x)) reverse-flatten-list nil = refl reverse-flatten-list (cons a x) = ( reverse-concat-list a (flatten-list x)) ∙ diff --git a/src/lists/universal-property-lists-wild-monoids.lagda.md b/src/lists/universal-property-lists-wild-monoids.lagda.md index 08d8f1c6bd..ae5ec9d1c5 100644 --- a/src/lists/universal-property-lists-wild-monoids.lagda.md +++ b/src/lists/universal-property-lists-wild-monoids.lagda.md @@ -61,10 +61,9 @@ pr2 (pr2 (pr2 (pr2 (list-H-Space X)))) = refl ```agda unit-law-011-associative-concat-list : {l1 : Level} {X : UU l1} (y z : list X) → - Id - ( ( associative-concat-list nil y z) ∙ - ( left-unit-law-concat-list (concat-list y z))) - ( ap (λ t → concat-list t z) (left-unit-law-concat-list y)) + ( associative-concat-list nil y z) ∙ + ( left-unit-law-concat-list (concat-list y z)) ＝ + ( ap (λ t → concat-list t z) (left-unit-law-concat-list y)) unit-law-011-associative-concat-list y z = refl concat-list' : {l : Level} {A : UU l} → list A → list A → list A @@ -72,10 +71,9 @@ concat-list' x y = concat-list y x unit-law-101-associative-concat-list : {l1 : Level} {X : UU l1} (x z : list X) → - Id - ( ( associative-concat-list x nil z) ∙ - ( ap (concat-list x) (left-unit-law-concat-list z))) - ( ap (λ t → concat-list t z) (right-unit-law-concat-list x)) + ( associative-concat-list x nil z) ∙ + ( ap (concat-list x) (left-unit-law-concat-list z)) ＝ + ( ap (λ t → concat-list t z) (right-unit-law-concat-list x)) unit-law-101-associative-concat-list nil z = refl unit-law-101-associative-concat-list (cons x l) z = ( ( ( inv @@ -93,10 +91,9 @@ unit-law-101-associative-concat-list (cons x l) z = unit-law-110-associative-concat-list : {l1 : Level} {X : UU l1} (x y : list X) → - Id - ( ( associative-concat-list x y nil) ∙ - ( ap (concat-list x) (right-unit-law-concat-list y))) - ( right-unit-law-concat-list (concat-list x y)) + ( associative-concat-list x y nil) ∙ + ( ap (concat-list x) (right-unit-law-concat-list y)) ＝ + ( right-unit-law-concat-list (concat-list x y)) unit-law-110-associative-concat-list nil y = ap-id (right-unit-law-concat-list y) unit-law-110-associative-concat-list (cons a x) y = diff --git a/src/ring-theory/dependent-products-rings.lagda.md b/src/ring-theory/dependent-products-rings.lagda.md index bbf0759490..a8a9793828 100644 --- a/src/ring-theory/dependent-products-rings.lagda.md +++ b/src/ring-theory/dependent-products-rings.lagda.md @@ -133,9 +133,7 @@ module _ right-distributive-mul-add-Π-Ring : (f g h : type-Π-Ring) → - Id - ( mul-Π-Ring (add-Π-Ring f g) h) - ( add-Π-Ring (mul-Π-Ring f h) (mul-Π-Ring g h)) + mul-Π-Ring (add-Π-Ring f g) h ＝ add-Π-Ring (mul-Π-Ring f h) (mul-Π-Ring g h) right-distributive-mul-add-Π-Ring = right-distributive-mul-add-Semiring semiring-Π-Ring diff --git a/src/ring-theory/localizations-rings.lagda.md b/src/ring-theory/localizations-rings.lagda.md index 05d9c027cc..0c1a59aad1 100644 --- a/src/ring-theory/localizations-rings.lagda.md +++ b/src/ring-theory/localizations-rings.lagda.md @@ -57,21 +57,19 @@ inv-inverts-element-hom-Ring R S x f H = pr1 H is-left-inverse-inv-inverts-element-hom-Ring : {l1 l2 : Level} (R : Ring l1) (S : Ring l2) (x : type-Ring R) (f : hom-Ring R S) (H : inverts-element-hom-Ring R S x f) → - Id - ( mul-Ring S - ( inv-inverts-element-hom-Ring R S x f H) - ( map-hom-Ring R S f x)) - ( one-Ring S) + mul-Ring S + ( inv-inverts-element-hom-Ring R S x f H) + ( map-hom-Ring R S f x) ＝ + one-Ring S is-left-inverse-inv-inverts-element-hom-Ring R S x f H = pr2 (pr2 H) is-right-inverse-inv-inverts-element-hom-Ring : {l1 l2 : Level} (R : Ring l1) (S : Ring l2) (x : type-Ring R) (f : hom-Ring R S) (H : inverts-element-hom-Ring R S x f) → - Id - ( mul-Ring S - ( map-hom-Ring R S f x) - ( inv-inverts-element-hom-Ring R S x f H)) - ( one-Ring S) + mul-Ring S + ( map-hom-Ring R S f x) + ( inv-inverts-element-hom-Ring R S x f H) ＝ + one-Ring S is-right-inverse-inv-inverts-element-hom-Ring R S x f H = pr1 (pr2 H) ``` @@ -230,11 +228,10 @@ is-left-inverse-inv-inverts-subset-hom-Ring : {l1 l2 l3 : Level} (R : Ring l1) (S : Ring l2) (P : subset-Ring l3 R) (f : hom-Ring R S) (H : inverts-subset-hom-Ring R S P f) (x : type-Ring R) (p : type-Prop (P x)) → - Id - ( mul-Ring S - ( inv-inverts-subset-hom-Ring R S P f H x p) - ( map-hom-Ring R S f x)) - ( one-Ring S) + mul-Ring S + ( inv-inverts-subset-hom-Ring R S P f H x p) + ( map-hom-Ring R S f x) ＝ + one-Ring S is-left-inverse-inv-inverts-subset-hom-Ring R S P f H x p = is-left-inverse-inv-inverts-element-hom-Ring R S x f (H x p) @@ -242,11 +239,10 @@ is-right-inverse-inv-inverts-subset-hom-Ring : {l1 l2 l3 : Level} (R : Ring l1) (S : Ring l2) (P : subset-Ring l3 R) (f : hom-Ring R S) (H : inverts-subset-hom-Ring R S P f) (x : type-Ring R) (p : type-Prop (P x)) → - Id - ( mul-Ring S - ( map-hom-Ring R S f x) - ( inv-inverts-subset-hom-Ring R S P f H x p)) - ( one-Ring S) + mul-Ring S + ( map-hom-Ring R S f x) + ( inv-inverts-subset-hom-Ring R S P f H x p) ＝ + one-Ring S is-right-inverse-inv-inverts-subset-hom-Ring R S P f H x p = is-right-inverse-inv-inverts-element-hom-Ring R S x f (H x p) diff --git a/src/ring-theory/products-rings.lagda.md b/src/ring-theory/products-rings.lagda.md index acddf82bbc..e05f1cc8ba 100644 --- a/src/ring-theory/products-rings.lagda.md +++ b/src/ring-theory/products-rings.lagda.md @@ -79,9 +79,8 @@ module _ associative-add-product-Ring : (x y z : type-product-Ring) → - Id - ( add-product-Ring (add-product-Ring x y) z) - ( add-product-Ring x (add-product-Ring y z)) + add-product-Ring (add-product-Ring x y) z ＝ + add-product-Ring x (add-product-Ring y z) associative-add-product-Ring (x1 , y1) (x2 , y2) (x3 , y3) = eq-pair ( associative-add-Ring R1 x1 x2 x3) @@ -104,9 +103,8 @@ module _ associative-mul-product-Ring : (x y z : type-product-Ring) → - Id - ( mul-product-Ring (mul-product-Ring x y) z) - ( mul-product-Ring x (mul-product-Ring y z)) + mul-product-Ring (mul-product-Ring x y) z ＝ + mul-product-Ring x (mul-product-Ring y z) associative-mul-product-Ring (x1 , y1) (x2 , y2) (x3 , y3) = eq-pair ( associative-mul-Ring R1 x1 x2 x3) @@ -124,9 +122,8 @@ module _ left-distributive-mul-add-product-Ring : (x y z : type-product-Ring) → - Id - ( mul-product-Ring x (add-product-Ring y z)) - ( add-product-Ring (mul-product-Ring x y) (mul-product-Ring x z)) + mul-product-Ring x (add-product-Ring y z) ＝ + add-product-Ring (mul-product-Ring x y) (mul-product-Ring x z) left-distributive-mul-add-product-Ring (x1 , y1) (x2 , y2) (x3 , y3) = eq-pair ( left-distributive-mul-add-Ring R1 x1 x2 x3) @@ -134,9 +131,8 @@ module _ right-distributive-mul-add-product-Ring : (x y z : type-product-Ring) → - Id - ( mul-product-Ring (add-product-Ring x y) z) - ( add-product-Ring (mul-product-Ring x z) (mul-product-Ring y z)) + mul-product-Ring (add-product-Ring x y) z ＝ + add-product-Ring (mul-product-Ring x z) (mul-product-Ring y z) right-distributive-mul-add-product-Ring (x1 , y1) (x2 , y2) (x3 , y3) = eq-pair ( right-distributive-mul-add-Ring R1 x1 x2 x3) diff --git a/src/ring-theory/rings.lagda.md b/src/ring-theory/rings.lagda.md index e743cf7234..a78cbeea55 100644 --- a/src/ring-theory/rings.lagda.md +++ b/src/ring-theory/rings.lagda.md @@ -686,9 +686,8 @@ module _ preserves-concat-add-list-Ring : (l1 l2 : list (type-Ring R)) → - Id - ( add-list-Ring (concat-list l1 l2)) - ( add-Ring R (add-list-Ring l1) (add-list-Ring l2)) + add-list-Ring (concat-list l1 l2) ＝ + add-Ring R (add-list-Ring l1) (add-list-Ring l2) preserves-concat-add-list-Ring = preserves-concat-add-list-Ab (ab-Ring R) ``` diff --git a/src/structured-types/h-spaces.lagda.md b/src/structured-types/h-spaces.lagda.md index 0e2199b8a0..be995de740 100644 --- a/src/structured-types/h-spaces.lagda.md +++ b/src/structured-types/h-spaces.lagda.md @@ -133,9 +133,8 @@ module _ pr1 (pr2 coherent-unit-laws-mul-H-Space) coh-unit-laws-mul-H-Space : - Id - ( left-unit-law-mul-H-Space unit-H-Space) - ( right-unit-law-mul-H-Space unit-H-Space) + left-unit-law-mul-H-Space unit-H-Space ＝ + right-unit-law-mul-H-Space unit-H-Space coh-unit-laws-mul-H-Space = pr2 (pr2 coherent-unit-laws-mul-H-Space) diff --git a/src/structured-types/wild-loops.lagda.md b/src/structured-types/wild-loops.lagda.md index 409b170d77..25b46aed25 100644 --- a/src/structured-types/wild-loops.lagda.md +++ b/src/structured-types/wild-loops.lagda.md @@ -79,9 +79,8 @@ module _ right-unit-law-mul-H-Space h-space-Wild-Loop coh-unit-laws-mul-Wild-Loop : - Id - ( left-unit-law-mul-Wild-Loop unit-Wild-Loop) - ( right-unit-law-mul-Wild-Loop unit-Wild-Loop) + left-unit-law-mul-Wild-Loop unit-Wild-Loop ＝ + right-unit-law-mul-Wild-Loop unit-Wild-Loop coh-unit-laws-mul-Wild-Loop = coh-unit-laws-mul-H-Space h-space-Wild-Loop diff --git a/src/structured-types/wild-monoids.lagda.md b/src/structured-types/wild-monoids.lagda.md index 250cabb823..f91e23d40b 100644 --- a/src/structured-types/wild-monoids.lagda.md +++ b/src/structured-types/wild-monoids.lagda.md @@ -99,39 +99,25 @@ module _ associator-H-Space : UU l associator-H-Space = (x y z : type-H-Space M) → - Id - ( mul-H-Space M (mul-H-Space M x y) z) - ( mul-H-Space M x (mul-H-Space M y z)) + mul-H-Space M (mul-H-Space M x y) z ＝ + mul-H-Space M x (mul-H-Space M y z) is-unital-associator : (α : associator-H-Space) → UU l is-unital-associator α111 = Σ ( (y z : type-H-Space M) → - Id - ( ( α111 (unit-H-Space M) y z) ∙ - ( left-unit-law-mul-H-Space M - ( mul-H-Space M y z))) - ( ap - ( mul-H-Space' M z) - ( left-unit-law-mul-H-Space M y))) + ( α111 (unit-H-Space M) y z) ∙ + ( left-unit-law-mul-H-Space M (mul-H-Space M y z)) ＝ + ( ap (mul-H-Space' M z) (left-unit-law-mul-H-Space M y))) ( λ α011 → Σ ( (x z : type-H-Space M) → - Id - ( ( α111 x (unit-H-Space M) z) ∙ - ( ap - ( mul-H-Space M x) - ( left-unit-law-mul-H-Space M z))) - ( ap - ( mul-H-Space' M z) - ( right-unit-law-mul-H-Space M x))) + ( α111 x (unit-H-Space M) z) ∙ + ( ap (mul-H-Space M x) (left-unit-law-mul-H-Space M z)) ＝ + ( ap (mul-H-Space' M z) (right-unit-law-mul-H-Space M x))) ( λ α101 → Σ ( (x y : type-H-Space M) → - Id - ( ( α111 x y (unit-H-Space M)) ∙ - ( ap - ( mul-H-Space M x) - ( right-unit-law-mul-H-Space M y))) - ( right-unit-law-mul-H-Space M - ( mul-H-Space M x y))) + ( α111 x y (unit-H-Space M)) ∙ + ( ap (mul-H-Space M x) (right-unit-law-mul-H-Space M y)) ＝ + ( right-unit-law-mul-H-Space M (mul-H-Space M x y))) ( λ α110 → unit))) unital-associator : UU l @@ -210,9 +196,8 @@ module _ unit-law-110-associative-Wild-Monoid : (x y : type-Wild-Monoid) → - Id - ( ( associative-mul-Wild-Monoid x y unit-Wild-Monoid) ∙ - ( ap (mul-Wild-Monoid x) (right-unit-law-mul-Wild-Monoid y))) - ( right-unit-law-mul-Wild-Monoid (mul-Wild-Monoid x y)) + ( associative-mul-Wild-Monoid x y unit-Wild-Monoid) ∙ + ( ap (mul-Wild-Monoid x) (right-unit-law-mul-Wild-Monoid y)) ＝ + ( right-unit-law-mul-Wild-Monoid (mul-Wild-Monoid x y)) unit-law-110-associative-Wild-Monoid = pr1 (pr2 (pr2 (pr2 (pr2 M)))) ``` diff --git a/src/structured-types/wild-semigroups.lagda.md b/src/structured-types/wild-semigroups.lagda.md index e9cd1ddadb..81276f79f3 100644 --- a/src/structured-types/wild-semigroups.lagda.md +++ b/src/structured-types/wild-semigroups.lagda.md @@ -48,8 +48,7 @@ module _ associative-mul-Wild-Semigroup : (x y z : type-Wild-Semigroup) → - Id - ( mul-Wild-Semigroup (mul-Wild-Semigroup x y) z) - ( mul-Wild-Semigroup x (mul-Wild-Semigroup y z)) + mul-Wild-Semigroup (mul-Wild-Semigroup x y) z ＝ + mul-Wild-Semigroup x (mul-Wild-Semigroup y z) associative-mul-Wild-Semigroup = pr2 G ``` diff --git a/src/type-theories/dependent-type-theories.lagda.md b/src/type-theories/dependent-type-theories.lagda.md index 6d14984834..742c3dbcf2 100644 --- a/src/type-theories/dependent-type-theories.lagda.md +++ b/src/type-theories/dependent-type-theories.lagda.md @@ -88,10 +88,8 @@ homotopies of sections of fibered systems. tr-fibered-system-slice : {l1 l2 l3 l4 : Level} {A : system l1 l2} {B B' : fibered-system l3 l4 A} (α : B ＝ B') (f : section-system B) (X : system.type A) → - Id - ( fibered-system.slice B (section-system.type f X)) - ( fibered-system.slice B' - ( section-system.type (tr section-system α f) X)) + fibered-system.slice B (section-system.type f X) ＝ + fibered-system.slice B' (section-system.type (tr section-system α f) X) tr-fibered-system-slice refl f X = refl Eq-fibered-system' : diff --git a/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md b/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md index fbc6bca067..58c0c75ab7 100644 --- a/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md +++ b/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md @@ -213,9 +213,8 @@ module _ where is-commutative-standard-2-Element-Decidable-Subtype : - Id - ( standard-2-Element-Decidable-Subtype d np) - ( standard-2-Element-Decidable-Subtype d (λ p → np (inv p))) + standard-2-Element-Decidable-Subtype d np ＝ + standard-2-Element-Decidable-Subtype d (λ p → np (inv p)) is-commutative-standard-2-Element-Decidable-Subtype = eq-pair-Σ ( eq-htpy @@ -248,9 +247,8 @@ module _ where eq-equal-elements-standard-2-Element-Decidable-Subtype : - Id - ( standard-2-Element-Decidable-Subtype d np) - ( standard-2-Element-Decidable-Subtype d nq) + standard-2-Element-Decidable-Subtype d np ＝ + standard-2-Element-Decidable-Subtype d nq eq-equal-elements-standard-2-Element-Decidable-Subtype = eq-pair-Σ ( eq-htpy @@ -365,10 +363,9 @@ pr2 (precomp-equiv-2-Element-Decidable-Subtype e (pair P H)) = preserves-comp-precomp-equiv-2-Element-Decidable-Subtype : { l1 l2 l3 l4 : Level} {X : UU l1} {Y : UU l2} {Z : UU l3} (e : X ≃ Y) → ( f : Y ≃ Z) → - Id - ( precomp-equiv-2-Element-Decidable-Subtype {l3 = l4} (f ∘e e)) - ( ( precomp-equiv-2-Element-Decidable-Subtype e) ∘ - ( precomp-equiv-2-Element-Decidable-Subtype f)) + precomp-equiv-2-Element-Decidable-Subtype {l3 = l4} (f ∘e e) ＝ + precomp-equiv-2-Element-Decidable-Subtype e ∘ + precomp-equiv-2-Element-Decidable-Subtype f preserves-comp-precomp-equiv-2-Element-Decidable-Subtype e f = eq-htpy ( λ (pair P H) → @@ -420,11 +417,10 @@ module _ pr2 other-element-subtype-2-element-decidable-subtype-Fin abstract - unequal-elements-2-element-decidable-subtype-Fin : - ¬ ( Id - ( element-2-element-decidable-subtype-Fin) - ( other-element-2-element-decidable-subtype-Fin)) - unequal-elements-2-element-decidable-subtype-Fin p = + neq-elements-2-element-decidable-subtype-Fin : + element-2-element-decidable-subtype-Fin ≠ + other-element-2-element-decidable-subtype-Fin + neq-elements-2-element-decidable-subtype-Fin p = has-no-fixed-points-swap-2-Element-Type ( 2-element-type-2-Element-Decidable-Subtype P) { element-subtype-2-element-decidable-subtype-Fin} diff --git a/src/univalent-combinatorics/cartesian-product-types.lagda.md b/src/univalent-combinatorics/cartesian-product-types.lagda.md index c6ba5cccee..499521da4d 100644 --- a/src/univalent-combinatorics/cartesian-product-types.lagda.md +++ b/src/univalent-combinatorics/cartesian-product-types.lagda.md @@ -111,10 +111,9 @@ abstract product-number-of-elements-product : {l1 l2 : Level} {A : UU l1} {B : UU l2} (count-AB : count (A × B)) → (a : A) (b : B) → - Id - ( ( number-of-elements-count (count-left-factor count-AB b)) *ℕ - ( number-of-elements-count (count-right-factor count-AB a))) - ( number-of-elements-count count-AB) + ( number-of-elements-count (count-left-factor count-AB b)) *ℕ + ( number-of-elements-count (count-right-factor count-AB a)) ＝ + ( number-of-elements-count count-AB) product-number-of-elements-product count-AB a b = ( inv ( number-of-elements-count-product diff --git a/src/univalent-combinatorics/coproduct-types.lagda.md b/src/univalent-combinatorics/coproduct-types.lagda.md index ee2c6015ff..fbf9dccd07 100644 --- a/src/univalent-combinatorics/coproduct-types.lagda.md +++ b/src/univalent-combinatorics/coproduct-types.lagda.md @@ -230,9 +230,8 @@ abstract double-counting-coproduct : { l1 l2 : Level} {A : UU l1} {B : UU l2} ( count-A : count A) (count-B : count B) (count-C : count (A + B)) → - Id - ( number-of-elements-count count-C) - ( number-of-elements-count count-A +ℕ number-of-elements-count count-B) + number-of-elements-count count-C ＝ + number-of-elements-count count-A +ℕ number-of-elements-count count-B double-counting-coproduct count-A count-B count-C = ( double-counting count-C (count-coproduct count-A count-B)) ∙ ( number-of-elements-count-coproduct count-A count-B) @@ -240,10 +239,9 @@ abstract abstract sum-number-of-elements-coproduct : {l1 l2 : Level} {A : UU l1} {B : UU l2} (e : count (A + B)) → - Id - ( ( number-of-elements-count (count-left-summand e)) +ℕ - ( number-of-elements-count (count-right-summand e))) - ( number-of-elements-count e) + number-of-elements-count (count-left-summand e) +ℕ + number-of-elements-count (count-right-summand e) ＝ + number-of-elements-count e sum-number-of-elements-coproduct e = ( inv ( number-of-elements-count-coproduct @@ -310,9 +308,8 @@ pr2 (coproduct-Type-With-Cardinality-ℕ k l (pair X H) (pair Y K)) = coproduct-eq-is-finite : {l1 l2 : Level} {X : UU l1} {Y : UU l2} (P : is-finite X) (Q : is-finite Y) → - Id - ( (number-of-elements-is-finite P) +ℕ (number-of-elements-is-finite Q)) - ( number-of-elements-is-finite (is-finite-coproduct P Q)) + number-of-elements-is-finite P +ℕ number-of-elements-is-finite Q ＝ + number-of-elements-is-finite (is-finite-coproduct P Q) coproduct-eq-is-finite {X = X} {Y = Y} P Q = ap ( number-of-elements-has-finite-cardinality) diff --git a/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md b/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md index 879eb9fefd..42cde68ca2 100644 --- a/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md +++ b/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md @@ -222,9 +222,8 @@ abstract double-counting-Σ : {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (count-A : count A) (count-B : (x : A) → count (B x)) (count-C : count (Σ A B)) → - Id - ( number-of-elements-count count-C) - ( sum-count-ℕ count-A (λ x → number-of-elements-count (count-B x))) + number-of-elements-count count-C ＝ + sum-count-ℕ count-A (λ x → number-of-elements-count (count-B x)) double-counting-Σ count-A count-B count-C = ( double-counting count-C (count-Σ count-A count-B)) ∙ ( number-of-elements-count-Σ count-A count-B) @@ -233,11 +232,9 @@ abstract sum-number-of-elements-count-fiber-count-Σ : {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (e : count A) (f : count (Σ A B)) → - Id - ( sum-count-ℕ e - ( λ x → number-of-elements-count - (count-fiber-count-Σ-count-base e f x))) - ( number-of-elements-count f) + sum-count-ℕ e + ( λ x → number-of-elements-count (count-fiber-count-Σ-count-base e f x)) ＝ + number-of-elements-count f sum-number-of-elements-count-fiber-count-Σ e f = ( inv ( number-of-elements-count-Σ e (count-fiber-count-Σ-count-base e f))) ∙ @@ -247,10 +244,8 @@ abstract double-counting-fiber-count-Σ : {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (count-A : count A) (count-B : (x : A) → count (B x)) (count-C : count (Σ A B)) (x : A) → - Id - ( number-of-elements-count (count-B x)) - ( number-of-elements-count - ( count-fiber-count-Σ-count-base count-A count-C x)) + number-of-elements-count (count-B x) ＝ + number-of-elements-count (count-fiber-count-Σ-count-base count-A count-C x) double-counting-fiber-count-Σ count-A count-B count-C x = double-counting ( count-B x) @@ -260,11 +255,10 @@ abstract sum-number-of-elements-count-base-count-Σ : {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (b : (x : A) → B x) → (count-ΣAB : count (Σ A B)) (count-B : (x : A) → count (B x)) → - Id - ( sum-count-ℕ - ( count-base-count-Σ b count-ΣAB count-B) - ( λ x → number-of-elements-count (count-B x))) - ( number-of-elements-count count-ΣAB) + sum-count-ℕ + ( count-base-count-Σ b count-ΣAB count-B) + ( λ x → number-of-elements-count (count-B x)) ＝ + number-of-elements-count count-ΣAB sum-number-of-elements-count-base-count-Σ b count-ΣAB count-B = ( inv ( number-of-elements-count-Σ @@ -279,9 +273,8 @@ abstract {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (b : (x : A) → B x) → (count-A : count A) (count-B : (x : A) → count (B x)) (count-ΣAB : count (Σ A B)) → - Id - ( number-of-elements-count (count-base-count-Σ b count-ΣAB count-B)) - ( number-of-elements-count count-A) + number-of-elements-count (count-base-count-Σ b count-ΣAB count-B) ＝ + number-of-elements-count count-A double-counting-base-count-Σ b count-A count-B count-ΣAB = double-counting (count-base-count-Σ b count-ΣAB count-B) count-A @@ -291,11 +284,10 @@ abstract ( count-B : (x : A) → count (B x)) → ( count-nB : count (Σ A (λ x → is-zero-ℕ (number-of-elements-count (count-B x))))) → - Id - ( sum-count-ℕ - ( count-base-count-Σ' count-ΣAB count-B count-nB) - ( λ x → number-of-elements-count (count-B x))) - ( number-of-elements-count count-ΣAB) + sum-count-ℕ + ( count-base-count-Σ' count-ΣAB count-B count-nB) + ( λ x → number-of-elements-count (count-B x)) ＝ + number-of-elements-count count-ΣAB sum-number-of-elements-count-base-count-Σ' count-ΣAB count-B count-nB = ( inv ( number-of-elements-count-Σ @@ -313,10 +305,9 @@ abstract ( count-B : (x : A) → count (B x)) (count-ΣAB : count (Σ A B)) → ( count-nB : count (Σ A (λ x → is-zero-ℕ (number-of-elements-count (count-B x))))) → - Id - ( number-of-elements-count - ( count-base-count-Σ' count-ΣAB count-B count-nB)) - ( number-of-elements-count count-A) + number-of-elements-count + ( count-base-count-Σ' count-ΣAB count-B count-nB) ＝ + number-of-elements-count count-A double-counting-base-count-Σ' count-A count-B count-ΣAB count-nB = double-counting (count-base-count-Σ' count-ΣAB count-B count-nB) count-A ``` diff --git a/src/univalent-combinatorics/cyclic-finite-types.lagda.md b/src/univalent-combinatorics/cyclic-finite-types.lagda.md index bc93b6858d..28d8e49e6a 100644 --- a/src/univalent-combinatorics/cyclic-finite-types.lagda.md +++ b/src/univalent-combinatorics/cyclic-finite-types.lagda.md @@ -344,9 +344,8 @@ module _ ( refl-htpy) left-inverse-law-comp-equiv-Cyclic-Type : - Id - ( comp-equiv-Cyclic-Type k X Y X (inv-equiv-Cyclic-Type k X Y e) e) - ( id-equiv-Cyclic-Type k X) + comp-equiv-Cyclic-Type k X Y X (inv-equiv-Cyclic-Type k X Y e) e ＝ + id-equiv-Cyclic-Type k X left-inverse-law-comp-equiv-Cyclic-Type = eq-htpy-equiv-Cyclic-Type k X X ( comp-equiv-Cyclic-Type k X Y X (inv-equiv-Cyclic-Type k X Y e) e) @@ -354,9 +353,8 @@ module _ ( is-retraction-map-inv-equiv (equiv-equiv-Cyclic-Type k X Y e)) right-inverse-law-comp-equiv-Cyclic-Type : - Id - ( comp-equiv-Cyclic-Type k Y X Y e (inv-equiv-Cyclic-Type k X Y e)) - ( id-equiv-Cyclic-Type k Y) + comp-equiv-Cyclic-Type k Y X Y e (inv-equiv-Cyclic-Type k X Y e) ＝ + id-equiv-Cyclic-Type k Y right-inverse-law-comp-equiv-Cyclic-Type = eq-htpy-equiv-Cyclic-Type k Y Y ( comp-equiv-Cyclic-Type k Y X Y e (inv-equiv-Cyclic-Type k X Y e)) @@ -522,11 +520,10 @@ map-equiv-compute-Ω-Cyclic-Type k = map-equiv (equiv-compute-Ω-Cyclic-Type k) preserves-concat-equiv-eq-Cyclic-Type : (k : ℕ) (X Y Z : Cyclic-Type lzero k) (p : X ＝ Y) (q : Y ＝ Z) → - Id - ( equiv-eq-Cyclic-Type k X Z (p ∙ q)) - ( comp-equiv-Cyclic-Type k X Y Z - ( equiv-eq-Cyclic-Type k Y Z q) - ( equiv-eq-Cyclic-Type k X Y p)) + equiv-eq-Cyclic-Type k X Z (p ∙ q) ＝ + comp-equiv-Cyclic-Type k X Y Z + ( equiv-eq-Cyclic-Type k Y Z q) + ( equiv-eq-Cyclic-Type k X Y p) preserves-concat-equiv-eq-Cyclic-Type k X .X Z refl q = inv ( right-unit-law-comp-equiv-Cyclic-Type @@ -563,11 +560,10 @@ preserves-comp-Eq-equiv-Cyclic-Type k e f = preserves-concat-equiv-compute-Ω-Cyclic-Type : (k : ℕ) {p q : type-Ω (Cyclic-Type-Pointed-Type k)} → - Id - ( map-equiv (equiv-compute-Ω-Cyclic-Type k) (p ∙ q)) - ( add-ℤ-Mod k - ( map-equiv (equiv-compute-Ω-Cyclic-Type k) p) - ( map-equiv (equiv-compute-Ω-Cyclic-Type k) q)) + map-equiv (equiv-compute-Ω-Cyclic-Type k) (p ∙ q) ＝ + add-ℤ-Mod k + ( map-equiv (equiv-compute-Ω-Cyclic-Type k) p) + ( map-equiv (equiv-compute-Ω-Cyclic-Type k) q) preserves-concat-equiv-compute-Ω-Cyclic-Type k {p} {q} = ( ap ( Eq-equiv-Cyclic-Type k (ℤ-Mod-Cyclic-Type k)) diff --git a/src/univalent-combinatorics/decidable-propositions.lagda.md b/src/univalent-combinatorics/decidable-propositions.lagda.md index 320887a5c0..1ffb9e4602 100644 --- a/src/univalent-combinatorics/decidable-propositions.lagda.md +++ b/src/univalent-combinatorics/decidable-propositions.lagda.md @@ -80,9 +80,8 @@ number-of-elements-count-eq' d x y = cases-number-of-elements-count-eq : {l : Level} {X : UU l} (d : has-decidable-equality X) {x y : X} (e : is-decidable (x ＝ y)) → - Id - ( number-of-elements-count (cases-count-eq d e)) - ( cases-number-of-elements-count-eq' e) + number-of-elements-count (cases-count-eq d e) ＝ + cases-number-of-elements-count-eq' e cases-number-of-elements-count-eq d (inl p) = refl cases-number-of-elements-count-eq d (inr f) = refl diff --git a/src/univalent-combinatorics/embeddings-standard-finite-types.lagda.md b/src/univalent-combinatorics/embeddings-standard-finite-types.lagda.md index 3d9b26b588..f617d6a180 100644 --- a/src/univalent-combinatorics/embeddings-standard-finite-types.lagda.md +++ b/src/univalent-combinatorics/embeddings-standard-finite-types.lagda.md @@ -73,9 +73,8 @@ abstract (d : is-decidable (is-inl-Fin l (map-emb f (inr star)))) (x : Fin k) (e : is-decidable (is-inl-Fin l (map-emb f (inl x)))) (x' : Fin k) (e' : is-decidable (is-inl-Fin l (map-emb f (inl x')))) → - Id - ( cases-map-reduce-emb-Fin k l f d x e) - ( cases-map-reduce-emb-Fin k l f d x' e') → + ( cases-map-reduce-emb-Fin k l f d x e ＝ + cases-map-reduce-emb-Fin k l f d x' e') → x ＝ x' is-injective-cases-map-reduce-emb-Fin k l f (inl (pair t q)) x e x' e' p = is-injective-inl diff --git a/src/univalent-combinatorics/fibers-of-maps.lagda.md b/src/univalent-combinatorics/fibers-of-maps.lagda.md index 17f21e3818..619908516a 100644 --- a/src/univalent-combinatorics/fibers-of-maps.lagda.md +++ b/src/univalent-combinatorics/fibers-of-maps.lagda.md @@ -60,10 +60,9 @@ abstract sum-number-of-elements-count-fiber : {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) → (count-A : count A) (count-B : count B) → - Id - ( sum-count-ℕ count-B - ( λ x → number-of-elements-count (count-fiber f count-A count-B x))) - ( number-of-elements-count count-A) + sum-count-ℕ count-B + ( λ x → number-of-elements-count (count-fiber f count-A count-B x)) ＝ + number-of-elements-count count-A sum-number-of-elements-count-fiber f count-A count-B = sum-number-of-elements-count-fiber-count-Σ count-B ( count-equiv' (equiv-total-fiber f) count-A) @@ -72,9 +71,8 @@ abstract double-counting-fiber : {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) (count-A : count A) → (count-B : count B) (count-fiber-f : (y : B) → count (fiber f y)) (y : B) → - Id - ( number-of-elements-count (count-fiber-f y)) - ( number-of-elements-count (count-fiber f count-A count-B y)) + number-of-elements-count (count-fiber-f y) ＝ + number-of-elements-count (count-fiber f count-A count-B y) double-counting-fiber f count-A count-B count-fiber-f y = double-counting (count-fiber-f y) (count-fiber f count-A count-B y) ``` diff --git a/src/univalent-combinatorics/isotopies-latin-squares.lagda.md b/src/univalent-combinatorics/isotopies-latin-squares.lagda.md index 58d4aa23c2..dd9bd25b34 100644 --- a/src/univalent-combinatorics/isotopies-latin-squares.lagda.md +++ b/src/univalent-combinatorics/isotopies-latin-squares.lagda.md @@ -39,9 +39,8 @@ module _ Σ ( symbol-Latin-Square L ≃ symbol-Latin-Square K) ( λ e-symbol → ( x : row-Latin-Square L) (y : column-Latin-Square L) → - Id - ( map-equiv e-symbol (mul-Latin-Square L x y)) - ( mul-Latin-Square K - ( map-equiv e-row x) - ( map-equiv e-column y))))) + ( map-equiv e-symbol (mul-Latin-Square L x y)) ＝ + ( mul-Latin-Square K + ( map-equiv e-row x) + ( map-equiv e-column y))))) ``` diff --git a/src/univalent-combinatorics/sums-of-natural-numbers.lagda.md b/src/univalent-combinatorics/sums-of-natural-numbers.lagda.md index 6fff4532d6..9604974eda 100644 --- a/src/univalent-combinatorics/sums-of-natural-numbers.lagda.md +++ b/src/univalent-combinatorics/sums-of-natural-numbers.lagda.md @@ -35,9 +35,8 @@ abstract associative-sum-count-ℕ : {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (count-A : count A) (count-B : (x : A) → count (B x)) (f : (x : A) → B x → ℕ) → - Id - ( sum-count-ℕ count-A (λ x → sum-count-ℕ (count-B x) (f x))) - ( sum-count-ℕ (count-Σ count-A count-B) (ind-Σ f)) + sum-count-ℕ count-A (λ x → sum-count-ℕ (count-B x) (f x)) ＝ + sum-count-ℕ (count-Σ count-A count-B) (ind-Σ f) associative-sum-count-ℕ {l1} {l2} {A} {B} count-A count-B f = ( ( htpy-sum-count-ℕ count-A ( λ x → From d1e64616b28dcfe5833362ba43ed89a10acff616 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 1 Sep 2025 21:11:11 +0200 Subject: [PATCH 15/16] some more parentheses --- .../commutative-rings.lagda.md | 12 ++++-------- .../commutative-semirings.lagda.md | 12 ++++++------ .../euclidean-domains.lagda.md | 12 ++++-------- .../function-commutative-semirings.lagda.md | 3 ++- .../integral-domains.lagda.md | 12 ++++-------- .../products-commutative-rings.lagda.md | 12 ++++++------ .../addition-integer-fractions.lagda.md | 4 ++-- .../modular-arithmetic.lagda.md | 8 ++++---- .../multiplication-integer-fractions.lagda.md | 4 ++-- .../commutative-finite-rings.lagda.md | 16 ++++++++-------- .../dependent-products-finite-rings.lagda.md | 8 ++++---- src/finite-algebra/finite-fields.lagda.md | 16 ++++++---------- src/finite-algebra/finite-rings.lagda.md | 4 ++-- .../products-finite-rings.lagda.md | 12 ++++++------ src/group-theory/abelian-groups.lagda.md | 3 +-- .../quotients-abelian-groups.lagda.md | 4 ++-- ...ite-sequences-in-commutative-monoids.lagda.md | 10 ++++++---- ...inite-sequences-in-commutative-rings.lagda.md | 10 ++++++---- ...-sequences-in-commutative-semigroups.lagda.md | 10 ++++++---- ...e-sequences-in-commutative-semirings.lagda.md | 10 ++++++---- ...inite-sequences-in-euclidean-domains.lagda.md | 12 ++++-------- .../finite-sequences-in-monoids.lagda.md | 4 +--- .../finite-sequences-in-rings.lagda.md | 6 +++--- .../finite-sequences-in-semigroups.lagda.md | 8 ++++---- .../finite-sequences-in-semirings.lagda.md | 8 ++++---- 25 files changed, 103 insertions(+), 117 deletions(-) diff --git a/src/commutative-algebra/commutative-rings.lagda.md b/src/commutative-algebra/commutative-rings.lagda.md index f970d153a0..e18c8dd7f6 100644 --- a/src/commutative-algebra/commutative-rings.lagda.md +++ b/src/commutative-algebra/commutative-rings.lagda.md @@ -353,19 +353,15 @@ module _ left-distributive-mul-add-Commutative-Ring : (x y z : type-Commutative-Ring) → - ( mul-Commutative-Ring x (add-Commutative-Ring y z)) ＝ - ( add-Commutative-Ring - ( mul-Commutative-Ring x y) - ( mul-Commutative-Ring x z)) + mul-Commutative-Ring x (add-Commutative-Ring y z) ＝ + add-Commutative-Ring (mul-Commutative-Ring x y) (mul-Commutative-Ring x z) left-distributive-mul-add-Commutative-Ring = left-distributive-mul-add-Ring ring-Commutative-Ring right-distributive-mul-add-Commutative-Ring : (x y z : type-Commutative-Ring) → - ( mul-Commutative-Ring (add-Commutative-Ring x y) z) ＝ - ( add-Commutative-Ring - ( mul-Commutative-Ring x z) - ( mul-Commutative-Ring y z)) + mul-Commutative-Ring (add-Commutative-Ring x y) z ＝ + add-Commutative-Ring (mul-Commutative-Ring x z) (mul-Commutative-Ring y z) right-distributive-mul-add-Commutative-Ring = right-distributive-mul-add-Ring ring-Commutative-Ring diff --git a/src/commutative-algebra/commutative-semirings.lagda.md b/src/commutative-algebra/commutative-semirings.lagda.md index 78e21f5b30..e8d78cf89d 100644 --- a/src/commutative-algebra/commutative-semirings.lagda.md +++ b/src/commutative-algebra/commutative-semirings.lagda.md @@ -200,19 +200,19 @@ module _ left-distributive-mul-add-Commutative-Semiring : (x y z : type-Commutative-Semiring) → - ( mul-Commutative-Semiring x (add-Commutative-Semiring y z)) ＝ - ( add-Commutative-Semiring + mul-Commutative-Semiring x (add-Commutative-Semiring y z) ＝ + add-Commutative-Semiring ( mul-Commutative-Semiring x y) - ( mul-Commutative-Semiring x z)) + ( mul-Commutative-Semiring x z) left-distributive-mul-add-Commutative-Semiring = left-distributive-mul-add-Semiring semiring-Commutative-Semiring right-distributive-mul-add-Commutative-Semiring : (x y z : type-Commutative-Semiring) → - ( mul-Commutative-Semiring (add-Commutative-Semiring x y) z) ＝ - ( add-Commutative-Semiring + mul-Commutative-Semiring (add-Commutative-Semiring x y) z ＝ + add-Commutative-Semiring ( mul-Commutative-Semiring x z) - ( mul-Commutative-Semiring y z)) + ( mul-Commutative-Semiring y z) right-distributive-mul-add-Commutative-Semiring = right-distributive-mul-add-Semiring semiring-Commutative-Semiring diff --git a/src/commutative-algebra/euclidean-domains.lagda.md b/src/commutative-algebra/euclidean-domains.lagda.md index 3111ea0b91..510d02dab3 100644 --- a/src/commutative-algebra/euclidean-domains.lagda.md +++ b/src/commutative-algebra/euclidean-domains.lagda.md @@ -352,20 +352,16 @@ module _ left-distributive-mul-add-Euclidean-Domain : (x y z : type-Euclidean-Domain) → - ( mul-Euclidean-Domain x (add-Euclidean-Domain y z)) ＝ - ( add-Euclidean-Domain - ( mul-Euclidean-Domain x y) - ( mul-Euclidean-Domain x z)) + mul-Euclidean-Domain x (add-Euclidean-Domain y z) ＝ + add-Euclidean-Domain (mul-Euclidean-Domain x y) (mul-Euclidean-Domain x z) left-distributive-mul-add-Euclidean-Domain = left-distributive-mul-add-Integral-Domain integral-domain-Euclidean-Domain right-distributive-mul-add-Euclidean-Domain : (x y z : type-Euclidean-Domain) → - ( mul-Euclidean-Domain (add-Euclidean-Domain x y) z) ＝ - ( add-Euclidean-Domain - ( mul-Euclidean-Domain x z) - ( mul-Euclidean-Domain y z)) + mul-Euclidean-Domain (add-Euclidean-Domain x y) z ＝ + add-Euclidean-Domain (mul-Euclidean-Domain x z) (mul-Euclidean-Domain y z) right-distributive-mul-add-Euclidean-Domain = right-distributive-mul-add-Integral-Domain integral-domain-Euclidean-Domain diff --git a/src/commutative-algebra/function-commutative-semirings.lagda.md b/src/commutative-algebra/function-commutative-semirings.lagda.md index 186bc1dbc1..5180cdc24f 100644 --- a/src/commutative-algebra/function-commutative-semirings.lagda.md +++ b/src/commutative-algebra/function-commutative-semirings.lagda.md @@ -150,7 +150,8 @@ module _ right-distributive-mul-add-function-Commutative-Semiring : (f g h : type-function-Commutative-Semiring) → mul-function-Commutative-Semiring - ( add-function-Commutative-Semiring f g) h ＝ + ( add-function-Commutative-Semiring f g) + ( h) ＝ add-function-Commutative-Semiring ( mul-function-Commutative-Semiring f h) ( mul-function-Commutative-Semiring g h) diff --git a/src/commutative-algebra/integral-domains.lagda.md b/src/commutative-algebra/integral-domains.lagda.md index 5a5abcf20d..c4f6335dba 100644 --- a/src/commutative-algebra/integral-domains.lagda.md +++ b/src/commutative-algebra/integral-domains.lagda.md @@ -321,20 +321,16 @@ module _ left-distributive-mul-add-Integral-Domain : (x y z : type-Integral-Domain) → - ( mul-Integral-Domain x (add-Integral-Domain y z)) ＝ - ( add-Integral-Domain - ( mul-Integral-Domain x y) - ( mul-Integral-Domain x z)) + mul-Integral-Domain x (add-Integral-Domain y z) ＝ + add-Integral-Domain (mul-Integral-Domain x y) (mul-Integral-Domain x z) left-distributive-mul-add-Integral-Domain = left-distributive-mul-add-Commutative-Ring commutative-ring-Integral-Domain right-distributive-mul-add-Integral-Domain : (x y z : type-Integral-Domain) → - ( mul-Integral-Domain (add-Integral-Domain x y) z) ＝ - ( add-Integral-Domain - ( mul-Integral-Domain x z) - ( mul-Integral-Domain y z)) + mul-Integral-Domain (add-Integral-Domain x y) z ＝ + add-Integral-Domain (mul-Integral-Domain x z) (mul-Integral-Domain y z) right-distributive-mul-add-Integral-Domain = right-distributive-mul-add-Commutative-Ring commutative-ring-Integral-Domain diff --git a/src/commutative-algebra/products-commutative-rings.lagda.md b/src/commutative-algebra/products-commutative-rings.lagda.md index 916031cbac..57779dd7ae 100644 --- a/src/commutative-algebra/products-commutative-rings.lagda.md +++ b/src/commutative-algebra/products-commutative-rings.lagda.md @@ -162,10 +162,10 @@ module _ left-distributive-mul-add-product-Commutative-Ring : (x y z : type-product-Commutative-Ring) → - ( mul-product-Commutative-Ring x (add-product-Commutative-Ring y z)) ＝ - ( add-product-Commutative-Ring + mul-product-Commutative-Ring x (add-product-Commutative-Ring y z) ＝ + add-product-Commutative-Ring ( mul-product-Commutative-Ring x y) - ( mul-product-Commutative-Ring x z)) + ( mul-product-Commutative-Ring x z) left-distributive-mul-add-product-Commutative-Ring = left-distributive-mul-add-product-Ring ( ring-Commutative-Ring R1) @@ -173,10 +173,10 @@ module _ right-distributive-mul-add-product-Commutative-Ring : (x y z : type-product-Commutative-Ring) → - ( mul-product-Commutative-Ring (add-product-Commutative-Ring x y) z) ＝ - ( add-product-Commutative-Ring + mul-product-Commutative-Ring (add-product-Commutative-Ring x y) z ＝ + add-product-Commutative-Ring ( mul-product-Commutative-Ring x z) - ( mul-product-Commutative-Ring y z)) + ( mul-product-Commutative-Ring y z) right-distributive-mul-add-product-Commutative-Ring = right-distributive-mul-add-product-Ring ( ring-Commutative-Ring R1) diff --git a/src/elementary-number-theory/addition-integer-fractions.lagda.md b/src/elementary-number-theory/addition-integer-fractions.lagda.md index 2ae01f48aa..bfbaef8714 100644 --- a/src/elementary-number-theory/addition-integer-fractions.lagda.md +++ b/src/elementary-number-theory/addition-integer-fractions.lagda.md @@ -216,8 +216,8 @@ abstract distributive-neg-add-fraction-ℤ : (x y : fraction-ℤ) → sim-fraction-ℤ - (neg-fraction-ℤ (x +fraction-ℤ y)) - (neg-fraction-ℤ x +fraction-ℤ neg-fraction-ℤ y) + ( neg-fraction-ℤ (x +fraction-ℤ y)) + ( neg-fraction-ℤ x +fraction-ℤ neg-fraction-ℤ y) distributive-neg-add-fraction-ℤ (nx , dx , dxp) (ny , dy , dyp) = ap ( _*ℤ (dx *ℤ dy)) diff --git a/src/elementary-number-theory/modular-arithmetic.lagda.md b/src/elementary-number-theory/modular-arithmetic.lagda.md index b75b74c017..1c6bb54e19 100644 --- a/src/elementary-number-theory/modular-arithmetic.lagda.md +++ b/src/elementary-number-theory/modular-arithmetic.lagda.md @@ -390,16 +390,16 @@ right-unit-law-mul-ℤ-Mod (succ-ℕ k) = right-unit-law-mul-Fin k left-distributive-mul-add-ℤ-Mod : (k : ℕ) (x y z : ℤ-Mod k) → - ( mul-ℤ-Mod k x (add-ℤ-Mod k y z)) ＝ - ( add-ℤ-Mod k (mul-ℤ-Mod k x y) (mul-ℤ-Mod k x z)) + mul-ℤ-Mod k x (add-ℤ-Mod k y z) ＝ + add-ℤ-Mod k (mul-ℤ-Mod k x y) (mul-ℤ-Mod k x z) left-distributive-mul-add-ℤ-Mod zero-ℕ = left-distributive-mul-add-ℤ left-distributive-mul-add-ℤ-Mod (succ-ℕ k) = left-distributive-mul-add-Fin (succ-ℕ k) right-distributive-mul-add-ℤ-Mod : (k : ℕ) (x y z : ℤ-Mod k) → - ( mul-ℤ-Mod k (add-ℤ-Mod k x y) z) ＝ - ( add-ℤ-Mod k (mul-ℤ-Mod k x z) (mul-ℤ-Mod k y z)) + mul-ℤ-Mod k (add-ℤ-Mod k x y) z ＝ + add-ℤ-Mod k (mul-ℤ-Mod k x z) (mul-ℤ-Mod k y z) right-distributive-mul-add-ℤ-Mod zero-ℕ = right-distributive-mul-add-ℤ right-distributive-mul-add-ℤ-Mod (succ-ℕ k) = right-distributive-mul-add-Fin (succ-ℕ k) diff --git a/src/elementary-number-theory/multiplication-integer-fractions.lagda.md b/src/elementary-number-theory/multiplication-integer-fractions.lagda.md index 3e9c1700e9..f5ae8d1b34 100644 --- a/src/elementary-number-theory/multiplication-integer-fractions.lagda.md +++ b/src/elementary-number-theory/multiplication-integer-fractions.lagda.md @@ -122,8 +122,8 @@ commutative-mul-fraction-ℤ (nx , dx , dxp) (ny , dy , dyp) = left-distributive-mul-add-fraction-ℤ : (x y z : fraction-ℤ) → sim-fraction-ℤ - (mul-fraction-ℤ x (add-fraction-ℤ y z)) - (add-fraction-ℤ (mul-fraction-ℤ x y) (mul-fraction-ℤ x z)) + ( mul-fraction-ℤ x (add-fraction-ℤ y z)) + ( add-fraction-ℤ (mul-fraction-ℤ x y) (mul-fraction-ℤ x z)) left-distributive-mul-add-fraction-ℤ (nx , dx , dxp) (ny , dy , dyp) (nz , dz , dzp) = ( ap diff --git a/src/finite-algebra/commutative-finite-rings.lagda.md b/src/finite-algebra/commutative-finite-rings.lagda.md index 3a4e470b9c..3944575b56 100644 --- a/src/finite-algebra/commutative-finite-rings.lagda.md +++ b/src/finite-algebra/commutative-finite-rings.lagda.md @@ -152,8 +152,8 @@ module _ associative-add-Finite-Commutative-Ring : (x y z : type-Finite-Commutative-Ring) → - ( add-Finite-Commutative-Ring (add-Finite-Commutative-Ring x y) z) ＝ - ( add-Finite-Commutative-Ring x (add-Finite-Commutative-Ring y z)) + add-Finite-Commutative-Ring (add-Finite-Commutative-Ring x y) z ＝ + add-Finite-Commutative-Ring x (add-Finite-Commutative-Ring y z) associative-add-Finite-Commutative-Ring = associative-add-Finite-Ring finite-ring-Finite-Commutative-Ring @@ -366,19 +366,19 @@ module _ left-distributive-mul-add-Finite-Commutative-Ring : (x y z : type-Finite-Commutative-Ring) → - ( mul-Finite-Commutative-Ring x (add-Finite-Commutative-Ring y z)) ＝ - ( add-Finite-Commutative-Ring + mul-Finite-Commutative-Ring x (add-Finite-Commutative-Ring y z) ＝ + add-Finite-Commutative-Ring ( mul-Finite-Commutative-Ring x y) - ( mul-Finite-Commutative-Ring x z)) + ( mul-Finite-Commutative-Ring x z) left-distributive-mul-add-Finite-Commutative-Ring = left-distributive-mul-add-Finite-Ring finite-ring-Finite-Commutative-Ring right-distributive-mul-add-Finite-Commutative-Ring : (x y z : type-Finite-Commutative-Ring) → - ( mul-Finite-Commutative-Ring (add-Finite-Commutative-Ring x y) z) ＝ - ( add-Finite-Commutative-Ring + mul-Finite-Commutative-Ring (add-Finite-Commutative-Ring x y) z ＝ + add-Finite-Commutative-Ring ( mul-Finite-Commutative-Ring x z) - ( mul-Finite-Commutative-Ring y z)) + ( mul-Finite-Commutative-Ring y z) right-distributive-mul-add-Finite-Commutative-Ring = right-distributive-mul-add-Finite-Ring finite-ring-Finite-Commutative-Ring diff --git a/src/finite-algebra/dependent-products-finite-rings.lagda.md b/src/finite-algebra/dependent-products-finite-rings.lagda.md index 4bbf10587c..b6be07bb6a 100644 --- a/src/finite-algebra/dependent-products-finite-rings.lagda.md +++ b/src/finite-algebra/dependent-products-finite-rings.lagda.md @@ -92,8 +92,8 @@ module _ associative-add-Π-Finite-Ring : (x y z : type-Π-Finite-Ring) → - ( add-Π-Finite-Ring (add-Π-Finite-Ring x y) z) ＝ - ( add-Π-Finite-Ring x (add-Π-Finite-Ring y z)) + add-Π-Finite-Ring (add-Π-Finite-Ring x y) z ＝ + add-Π-Finite-Ring x (add-Π-Finite-Ring y z) associative-add-Π-Finite-Ring = associative-add-Π-Ring (type-Finite-Type I) (ring-Finite-Ring ∘ A) @@ -160,8 +160,8 @@ module _ right-distributive-mul-add-Π-Finite-Ring : (f g h : type-Π-Finite-Ring) → - ( mul-Π-Finite-Ring (add-Π-Finite-Ring f g) h) ＝ - ( add-Π-Finite-Ring (mul-Π-Finite-Ring f h) (mul-Π-Finite-Ring g h)) + mul-Π-Finite-Ring (add-Π-Finite-Ring f g) h ＝ + add-Π-Finite-Ring (mul-Π-Finite-Ring f h) (mul-Π-Finite-Ring g h) right-distributive-mul-add-Π-Finite-Ring = right-distributive-mul-add-Π-Ring ( type-Finite-Type I) diff --git a/src/finite-algebra/finite-fields.lagda.md b/src/finite-algebra/finite-fields.lagda.md index 72efaff9f4..384e6110d1 100644 --- a/src/finite-algebra/finite-fields.lagda.md +++ b/src/finite-algebra/finite-fields.lagda.md @@ -120,8 +120,8 @@ module _ associative-add-Finite-Field : (x y z : type-Finite-Field) → - ( add-Finite-Field (add-Finite-Field x y) z) ＝ - ( add-Finite-Field x (add-Finite-Field y z)) + add-Finite-Field (add-Finite-Field x y) z ＝ + add-Finite-Field x (add-Finite-Field y z) associative-add-Finite-Field = associative-add-Finite-Ring finite-ring-Finite-Field @@ -300,19 +300,15 @@ module _ left-distributive-mul-add-Finite-Field : (x y z : type-Finite-Field) → - ( mul-Finite-Field x (add-Finite-Field y z)) ＝ - ( add-Finite-Field - ( mul-Finite-Field x y) - ( mul-Finite-Field x z)) + mul-Finite-Field x (add-Finite-Field y z) ＝ + add-Finite-Field (mul-Finite-Field x y) (mul-Finite-Field x z) left-distributive-mul-add-Finite-Field = left-distributive-mul-add-Finite-Ring finite-ring-Finite-Field right-distributive-mul-add-Finite-Field : (x y z : type-Finite-Field) → - ( mul-Finite-Field (add-Finite-Field x y) z) ＝ - ( add-Finite-Field - ( mul-Finite-Field x z) - ( mul-Finite-Field y z)) + mul-Finite-Field (add-Finite-Field x y) z ＝ + add-Finite-Field (mul-Finite-Field x z) (mul-Finite-Field y z) right-distributive-mul-add-Finite-Field = right-distributive-mul-add-Finite-Ring finite-ring-Finite-Field diff --git a/src/finite-algebra/finite-rings.lagda.md b/src/finite-algebra/finite-rings.lagda.md index 77f1e9a2cb..673099a9af 100644 --- a/src/finite-algebra/finite-rings.lagda.md +++ b/src/finite-algebra/finite-rings.lagda.md @@ -139,8 +139,8 @@ module _ associative-add-Finite-Ring : (x y z : type-Finite-Ring R) → - ( add-Finite-Ring (add-Finite-Ring x y) z) ＝ - ( add-Finite-Ring x (add-Finite-Ring y z)) + add-Finite-Ring (add-Finite-Ring x y) z ＝ + add-Finite-Ring x (add-Finite-Ring y z) associative-add-Finite-Ring = associative-add-Ring (ring-Finite-Ring R) is-group-additive-semigroup-Finite-Ring : diff --git a/src/finite-algebra/products-finite-rings.lagda.md b/src/finite-algebra/products-finite-rings.lagda.md index 1f337cff1e..46486939cc 100644 --- a/src/finite-algebra/products-finite-rings.lagda.md +++ b/src/finite-algebra/products-finite-rings.lagda.md @@ -151,10 +151,10 @@ module _ left-distributive-mul-add-product-Finite-Ring : (x y z : type-product-Finite-Ring) → - ( mul-product-Finite-Ring x (add-product-Finite-Ring y z)) ＝ - ( add-product-Finite-Ring + mul-product-Finite-Ring x (add-product-Finite-Ring y z) ＝ + add-product-Finite-Ring ( mul-product-Finite-Ring x y) - ( mul-product-Finite-Ring x z)) + ( mul-product-Finite-Ring x z) left-distributive-mul-add-product-Finite-Ring = left-distributive-mul-add-product-Ring ( ring-Finite-Ring R1) @@ -162,10 +162,10 @@ module _ right-distributive-mul-add-product-Finite-Ring : (x y z : type-product-Finite-Ring) → - ( mul-product-Finite-Ring (add-product-Finite-Ring x y) z) ＝ - ( add-product-Finite-Ring + mul-product-Finite-Ring (add-product-Finite-Ring x y) z ＝ + add-product-Finite-Ring ( mul-product-Finite-Ring x z) - ( mul-product-Finite-Ring y z)) + ( mul-product-Finite-Ring y z) right-distributive-mul-add-product-Finite-Ring = right-distributive-mul-add-product-Ring ( ring-Finite-Ring R1) diff --git a/src/group-theory/abelian-groups.lagda.md b/src/group-theory/abelian-groups.lagda.md index 5b2831b180..84137f9179 100644 --- a/src/group-theory/abelian-groups.lagda.md +++ b/src/group-theory/abelian-groups.lagda.md @@ -195,8 +195,7 @@ module _ ( associative-add-Ab b a c) distributive-neg-add-Ab : - (x y : type-Ab) → - neg-Ab (add-Ab x y) ＝ add-Ab (neg-Ab x) (neg-Ab y) + (x y : type-Ab) → neg-Ab (add-Ab x y) ＝ add-Ab (neg-Ab x) (neg-Ab y) distributive-neg-add-Ab x y = ( distributive-inv-mul-Group group-Ab) ∙ ( commutative-add-Ab (neg-Ab y) (neg-Ab x)) diff --git a/src/group-theory/quotients-abelian-groups.lagda.md b/src/group-theory/quotients-abelian-groups.lagda.md index 51ce7f9b4a..ebd816089f 100644 --- a/src/group-theory/quotients-abelian-groups.lagda.md +++ b/src/group-theory/quotients-abelian-groups.lagda.md @@ -265,8 +265,8 @@ module _ associative-add-quotient-Ab : (x y z : type-quotient-Ab) → - ( add-quotient-Ab (add-quotient-Ab x y) z) ＝ - ( add-quotient-Ab x (add-quotient-Ab y z)) + add-quotient-Ab (add-quotient-Ab x y) z ＝ + add-quotient-Ab x (add-quotient-Ab y z) associative-add-quotient-Ab = associative-mul-quotient-Group ( group-Ab A) diff --git a/src/linear-algebra/finite-sequences-in-commutative-monoids.lagda.md b/src/linear-algebra/finite-sequences-in-commutative-monoids.lagda.md index d757475db9..7e32fa5a7f 100644 --- a/src/linear-algebra/finite-sequences-in-commutative-monoids.lagda.md +++ b/src/linear-algebra/finite-sequences-in-commutative-monoids.lagda.md @@ -97,10 +97,12 @@ module _ associative-add-fin-sequence-type-Commutative-Monoid : (n : ℕ) (v1 v2 v3 : fin-sequence-type-Commutative-Monoid M n) → - ( add-fin-sequence-type-Commutative-Monoid n - ( add-fin-sequence-type-Commutative-Monoid n v1 v2) v3) ＝ - ( add-fin-sequence-type-Commutative-Monoid n v1 - ( add-fin-sequence-type-Commutative-Monoid n v2 v3)) + add-fin-sequence-type-Commutative-Monoid n + ( add-fin-sequence-type-Commutative-Monoid n v1 v2) + ( v3) ＝ + add-fin-sequence-type-Commutative-Monoid n + ( v1) + ( add-fin-sequence-type-Commutative-Monoid n v2 v3) associative-add-fin-sequence-type-Commutative-Monoid = associative-add-fin-sequence-type-Monoid (monoid-Commutative-Monoid M) diff --git a/src/linear-algebra/finite-sequences-in-commutative-rings.lagda.md b/src/linear-algebra/finite-sequences-in-commutative-rings.lagda.md index eb6dd679df..bc58a7c70f 100644 --- a/src/linear-algebra/finite-sequences-in-commutative-rings.lagda.md +++ b/src/linear-algebra/finite-sequences-in-commutative-rings.lagda.md @@ -119,10 +119,12 @@ module _ associative-add-fin-sequence-type-Commutative-Ring : (n : ℕ) (v1 v2 v3 : fin-sequence-type-Commutative-Ring R n) → - ( add-fin-sequence-type-Commutative-Ring R n - ( add-fin-sequence-type-Commutative-Ring R n v1 v2) v3) ＝ - ( add-fin-sequence-type-Commutative-Ring R n v1 - ( add-fin-sequence-type-Commutative-Ring R n v2 v3)) + add-fin-sequence-type-Commutative-Ring R n + ( add-fin-sequence-type-Commutative-Ring R n v1 v2) + ( v3) ＝ + add-fin-sequence-type-Commutative-Ring R n + ( v1) + ( add-fin-sequence-type-Commutative-Ring R n v2 v3) associative-add-fin-sequence-type-Commutative-Ring = associative-add-fin-sequence-type-Ring (ring-Commutative-Ring R) ``` diff --git a/src/linear-algebra/finite-sequences-in-commutative-semigroups.lagda.md b/src/linear-algebra/finite-sequences-in-commutative-semigroups.lagda.md index 4339b8bf73..9ad49e6e66 100644 --- a/src/linear-algebra/finite-sequences-in-commutative-semigroups.lagda.md +++ b/src/linear-algebra/finite-sequences-in-commutative-semigroups.lagda.md @@ -86,10 +86,12 @@ module _ associative-add-fin-sequence-type-Commutative-Semigroup : (n : ℕ) (v1 v2 v3 : fin-sequence-type-Commutative-Semigroup M n) → - ( add-fin-sequence-type-Commutative-Semigroup n - ( add-fin-sequence-type-Commutative-Semigroup n v1 v2) v3) ＝ - ( add-fin-sequence-type-Commutative-Semigroup n v1 - ( add-fin-sequence-type-Commutative-Semigroup n v2 v3)) + add-fin-sequence-type-Commutative-Semigroup n + ( add-fin-sequence-type-Commutative-Semigroup n v1 v2) + ( v3) ＝ + add-fin-sequence-type-Commutative-Semigroup n + ( v1) + ( add-fin-sequence-type-Commutative-Semigroup n v2 v3) associative-add-fin-sequence-type-Commutative-Semigroup = associative-add-fin-sequence-type-Semigroup ( semigroup-Commutative-Semigroup M) diff --git a/src/linear-algebra/finite-sequences-in-commutative-semirings.lagda.md b/src/linear-algebra/finite-sequences-in-commutative-semirings.lagda.md index 5858b3dfa3..810075296f 100644 --- a/src/linear-algebra/finite-sequences-in-commutative-semirings.lagda.md +++ b/src/linear-algebra/finite-sequences-in-commutative-semirings.lagda.md @@ -104,10 +104,12 @@ module _ associative-add-fin-sequence-type-Commutative-Semiring : (n : ℕ) (v1 v2 v3 : fin-sequence-type-Commutative-Semiring R n) → - ( add-fin-sequence-type-Commutative-Semiring R n - ( add-fin-sequence-type-Commutative-Semiring R n v1 v2) v3) ＝ - ( add-fin-sequence-type-Commutative-Semiring R n v1 - ( add-fin-sequence-type-Commutative-Semiring R n v2 v3)) + add-fin-sequence-type-Commutative-Semiring R n + ( add-fin-sequence-type-Commutative-Semiring R n v1 v2) + ( v3) ＝ + add-fin-sequence-type-Commutative-Semiring R n + ( v1) + ( add-fin-sequence-type-Commutative-Semiring R n v2 v3) associative-add-fin-sequence-type-Commutative-Semiring = associative-add-fin-sequence-type-Semiring (semiring-Commutative-Semiring R) ``` diff --git a/src/linear-algebra/finite-sequences-in-euclidean-domains.lagda.md b/src/linear-algebra/finite-sequences-in-euclidean-domains.lagda.md index fa50780b53..2731e775d8 100644 --- a/src/linear-algebra/finite-sequences-in-euclidean-domains.lagda.md +++ b/src/linear-algebra/finite-sequences-in-euclidean-domains.lagda.md @@ -120,16 +120,12 @@ module _ associative-add-fin-sequence-type-Euclidean-Domain : (n : ℕ) (v1 v2 v3 : fin-sequence-type-Euclidean-Domain R n) → - ( add-fin-sequence-type-Euclidean-Domain - ( R) - ( n) + add-fin-sequence-type-Euclidean-Domain R n ( add-fin-sequence-type-Euclidean-Domain R n v1 v2) - ( v3)) ＝ - ( add-fin-sequence-type-Euclidean-Domain - ( R) - ( n) + ( v3) ＝ + add-fin-sequence-type-Euclidean-Domain R n ( v1) - ( add-fin-sequence-type-Euclidean-Domain R n v2 v3)) + ( add-fin-sequence-type-Euclidean-Domain R n v2 v3) associative-add-fin-sequence-type-Euclidean-Domain = associative-add-fin-sequence-type-Commutative-Ring ( commutative-ring-Euclidean-Domain R) diff --git a/src/linear-algebra/finite-sequences-in-monoids.lagda.md b/src/linear-algebra/finite-sequences-in-monoids.lagda.md index 401624ba8f..eb36d102ed 100644 --- a/src/linear-algebra/finite-sequences-in-monoids.lagda.md +++ b/src/linear-algebra/finite-sequences-in-monoids.lagda.md @@ -88,9 +88,7 @@ module _ associative-add-fin-sequence-type-Monoid : (n : ℕ) (v1 v2 v3 : fin-sequence-type-Monoid M n) → - ( add-fin-sequence-type-Monoid n - ( add-fin-sequence-type-Monoid n v1 v2) - ( v3)) ＝ + add-fin-sequence-type-Monoid n (add-fin-sequence-type-Monoid n v1 v2) v3 ＝ ( add-fin-sequence-type-Monoid n v1 (add-fin-sequence-type-Monoid n v2 v3)) associative-add-fin-sequence-type-Monoid = associative-add-fin-sequence-type-Semigroup (semigroup-Monoid M) diff --git a/src/linear-algebra/finite-sequences-in-rings.lagda.md b/src/linear-algebra/finite-sequences-in-rings.lagda.md index f0692887df..77f9dceacf 100644 --- a/src/linear-algebra/finite-sequences-in-rings.lagda.md +++ b/src/linear-algebra/finite-sequences-in-rings.lagda.md @@ -187,10 +187,10 @@ module _ associative-add-fin-sequence-type-Ring : (n : ℕ) (v1 v2 v3 : fin-sequence-type-Ring R n) → - ( add-fin-sequence-type-Ring R n + add-fin-sequence-type-Ring R n ( add-fin-sequence-type-Ring R n v1 v2) - ( v3)) ＝ - ( add-fin-sequence-type-Ring R n v1 (add-fin-sequence-type-Ring R n v2 v3)) + ( v3) ＝ + add-fin-sequence-type-Ring R n v1 (add-fin-sequence-type-Ring R n v2 v3) associative-add-fin-sequence-type-Ring = associative-add-Ring ∘ function-Ring R ∘ Fin ``` diff --git a/src/linear-algebra/finite-sequences-in-semigroups.lagda.md b/src/linear-algebra/finite-sequences-in-semigroups.lagda.md index 024679b333..383d1d9094 100644 --- a/src/linear-algebra/finite-sequences-in-semigroups.lagda.md +++ b/src/linear-algebra/finite-sequences-in-semigroups.lagda.md @@ -78,12 +78,12 @@ module _ associative-add-fin-sequence-type-Semigroup : (n : ℕ) (v1 v2 v3 : fin-sequence-type-Semigroup G n) → - ( add-fin-sequence-type-Semigroup n + add-fin-sequence-type-Semigroup n ( add-fin-sequence-type-Semigroup n v1 v2) - ( v3)) ＝ - ( add-fin-sequence-type-Semigroup n + ( v3) ＝ + add-fin-sequence-type-Semigroup n ( v1) - ( add-fin-sequence-type-Semigroup n v2 v3)) + ( add-fin-sequence-type-Semigroup n v2 v3) associative-add-fin-sequence-type-Semigroup n v1 v2 v3 = eq-htpy (λ i → associative-mul-Semigroup G (v1 i) (v2 i) (v3 i)) diff --git a/src/linear-algebra/finite-sequences-in-semirings.lagda.md b/src/linear-algebra/finite-sequences-in-semirings.lagda.md index 70c10129ed..ce8bbd6afb 100644 --- a/src/linear-algebra/finite-sequences-in-semirings.lagda.md +++ b/src/linear-algebra/finite-sequences-in-semirings.lagda.md @@ -102,14 +102,14 @@ module _ associative-add-fin-sequence-type-Semiring : (n : ℕ) (v1 v2 v3 : fin-sequence-type-Semiring R n) → - ( add-fin-sequence-type-Semiring R + add-fin-sequence-type-Semiring R ( n) ( add-fin-sequence-type-Semiring R n v1 v2) - ( v3)) ＝ - ( add-fin-sequence-type-Semiring R + ( v3) ＝ + add-fin-sequence-type-Semiring R ( n) ( v1) - ( add-fin-sequence-type-Semiring R n v2 v3)) + ( add-fin-sequence-type-Semiring R n v2 v3) associative-add-fin-sequence-type-Semiring = associative-add-fin-sequence-type-Commutative-Monoid ( additive-commutative-monoid-Semiring R) From ab73181c4eccbee281bd7081f874da77f2722216 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Fri, 5 Sep 2025 23:41:02 +0200 Subject: [PATCH 16/16] Update commutative-rings.lagda.md --- src/commutative-algebra/commutative-rings.lagda.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/commutative-algebra/commutative-rings.lagda.md b/src/commutative-algebra/commutative-rings.lagda.md index e18c8dd7f6..a12e07d1fe 100644 --- a/src/commutative-algebra/commutative-rings.lagda.md +++ b/src/commutative-algebra/commutative-rings.lagda.md @@ -654,7 +654,7 @@ module _ (l1 l2 : list type-Commutative-Ring) → add-list-Commutative-Ring (concat-list l1 l2) ＝ add-Commutative-Ring - (add-list-Commutative-Ring l1) + ( add-list-Commutative-Ring l1) ( add-list-Commutative-Ring l2) preserves-concat-add-list-Commutative-Ring = preserves-concat-add-list-Ring ring-Commutative-Ring