From d13056f228cc10ab1125f01c5525b77ea88bb132 Mon Sep 17 00:00:00 2001 From: Louis Wasserman Date: Sat, 4 Oct 2025 22:28:32 -0700 Subject: [PATCH 01/12] Refactor the natural numbers and integers up to present code standards --- ...polynomials-commutative-semirings.lagda.md | 8 +- src/elementary-number-theory.lagda.md | 2 +- .../absolute-value-integers.lagda.md | 47 +- .../addition-integers.lagda.md | 18 +- .../archimedean-property-integers.lagda.md | 4 +- ...himedean-property-natural-numbers.lagda.md | 2 +- .../bezouts-lemma-natural-numbers.lagda.md | 10 +- .../binomial-coefficients.lagda.md | 2 +- .../binomial-theorem-natural-numbers.lagda.md | 2 +- .../congruence-integers.lagda.md | 78 +- .../congruence-natural-numbers.lagda.md | 2 +- .../distance-integers.lagda.md | 98 ++- .../distance-natural-numbers.lagda.md | 531 +++++------ .../divisibility-integers.lagda.md | 826 +++++++++--------- .../divisibility-natural-numbers.lagda.md | 596 +++++++------ .../equality-integers.lagda.md | 5 +- .../equality-natural-numbers.lagda.md | 17 +- ...uclidean-division-natural-numbers.lagda.md | 2 +- .../exponentiation-natural-numbers.lagda.md | 83 +- .../factorials.lagda.md | 71 +- ...st-common-divisor-natural-numbers.lagda.md | 304 ++++--- .../group-of-integers.lagda.md | 5 +- .../inequality-integers.lagda.md | 239 ++--- .../inequality-natural-numbers.lagda.md | 313 +++---- .../inequality-standard-finite-types.lagda.md | 117 ++- .../integers.lagda.md | 49 +- .../lower-bounds-natural-numbers.lagda.md | 10 +- .../maximum-natural-numbers.lagda.md | 178 ++-- .../minimum-natural-numbers.lagda.md | 141 +-- .../multiplication-integers.lagda.md | 729 ++++++++-------- .../multiplication-natural-numbers.lagda.md | 260 +++--- .../negative-integers.lagda.md | 38 +- .../nonnegative-integers.lagda.md | 38 +- .../nonpositive-integers.lagda.md | 40 +- .../parity-natural-numbers.lagda.md | 119 +-- ...l-numbers-ordered-by-divisibility.lagda.md | 3 +- .../positive-and-negative-integers.lagda.md | 54 +- .../positive-integers.lagda.md | 63 +- .../powers-integers.lagda.md | 6 +- .../prime-numbers.lagda.md | 231 ++--- .../reduced-integer-fractions.lagda.md | 4 +- .../relatively-prime-integers.lagda.md | 38 +- .../relatively-prime-natural-numbers.lagda.md | 155 ++-- .../ring-of-integers.lagda.md | 193 ++-- ...d => semiring-of-natural-numbers.lagda.md} | 4 +- .../squares-natural-numbers.lagda.md | 156 ++-- ...rict-inequality-integer-fractions.lagda.md | 2 +- .../strict-inequality-integers.lagda.md | 166 ++-- ...strict-inequality-natural-numbers.lagda.md | 303 ++++--- ...-inequality-standard-finite-types.lagda.md | 198 +++-- .../strong-induction-natural-numbers.lagda.md | 8 +- .../triangular-numbers.lagda.md | 2 +- .../unit-fractions-rational-numbers.lagda.md | 2 +- .../upper-bounds-natural-numbers.lagda.md | 10 +- ...rdering-principle-natural-numbers.lagda.md | 101 ++- ...g-principle-standard-finite-types.lagda.md | 7 +- ...roduction-to-homotopy-type-theory.lagda.md | 4 +- .../arithmetic-series-semirings.lagda.md | 2 +- .../classical-finite-types.lagda.md | 2 +- ...tations-complete-undirected-graph.lagda.md | 2 +- 60 files changed, 3589 insertions(+), 3111 deletions(-) rename src/elementary-number-theory/{commutative-semiring-of-natural-numbers.lagda.md => semiring-of-natural-numbers.lagda.md} (97%) diff --git a/src/commutative-algebra/polynomials-commutative-semirings.lagda.md b/src/commutative-algebra/polynomials-commutative-semirings.lagda.md index f656ac91ff..6e21dadfa9 100644 --- a/src/commutative-algebra/polynomials-commutative-semirings.lagda.md +++ b/src/commutative-algebra/polynomials-commutative-semirings.lagda.md @@ -704,7 +704,7 @@ module _ rec-coproduct ( λ j +## Idea + +Two [integers](elementary-number-theory.integers.md) are +{{#concept "congruent" WDID=Q3773677 WD="congruence of integers" Agda=cong-ℤ}} +mod an integer `k` if their +[difference](elementary-number-theory.difference-integers.md) is +[divisible](elementary-number-theory.divisibility-integers.md) by `k`. + ## Definitions ```agda @@ -40,6 +48,8 @@ is-cong-zero-ℤ k x = cong-ℤ k x zero-ℤ ## Properties +### `k | x` if and only if `x` is congruent to zero mod `k` + ```agda is-cong-zero-div-ℤ : (k x : ℤ) → div-ℤ k x → is-cong-zero-ℤ k x pr1 (is-cong-zero-div-ℤ k x (pair d p)) = d @@ -48,33 +58,55 @@ pr2 (is-cong-zero-div-ℤ k x (pair d p)) = p ∙ inv (right-unit-law-add-ℤ x) div-is-cong-zero-ℤ : (k x : ℤ) → is-cong-zero-ℤ k x → div-ℤ k x pr1 (div-is-cong-zero-ℤ k x (pair d p)) = d pr2 (div-is-cong-zero-ℤ k x (pair d p)) = p ∙ right-unit-law-add-ℤ x +``` +### If `k` is a unit, all integers are congruent mod `k` + +```agda is-indiscrete-cong-ℤ : (k : ℤ) → is-unit-ℤ k → (x y : ℤ) → cong-ℤ k x y is-indiscrete-cong-ℤ k H x y = div-is-unit-ℤ k (x -ℤ y) H +``` +### If `k` is zero and `x` and `y` are congruent mod `k`, `x = y` + +```agda is-discrete-cong-ℤ : (k : ℤ) → is-zero-ℤ k → (x y : ℤ) → cong-ℤ k x y → x ＝ y is-discrete-cong-ℤ .zero-ℤ refl x y K = eq-diff-ℤ (is-zero-div-zero-ℤ (x -ℤ y) K) +``` + +### If `x` and `succ-ℤ x` are congruent mod `k`, `k` is a unit + +```agda +abstract + is-unit-cong-succ-ℤ : (k x : ℤ) → cong-ℤ k x (succ-ℤ x) → is-unit-ℤ k + pr1 (is-unit-cong-succ-ℤ k x (pair y p)) = neg-ℤ y + pr2 (is-unit-cong-succ-ℤ k x (pair y p)) = + ( left-negative-law-mul-ℤ y k) ∙ + ( is-injective-neg-ℤ + ( ( neg-neg-ℤ (y *ℤ k)) ∙ + ( ( p) ∙ + ( ( ap (x +ℤ_) (neg-succ-ℤ x)) ∙ + ( ( right-predecessor-law-add-ℤ x (neg-ℤ x)) ∙ + ( ap pred-ℤ (right-inverse-law-add-ℤ x))))))) +``` + +### If `x` and `pred-ℤ x` are congruent mod `k`, `k` is a unit -is-unit-cong-succ-ℤ : (k x : ℤ) → cong-ℤ k x (succ-ℤ x) → is-unit-ℤ k -pr1 (is-unit-cong-succ-ℤ k x (pair y p)) = neg-ℤ y -pr2 (is-unit-cong-succ-ℤ k x (pair y p)) = - ( left-negative-law-mul-ℤ y k) ∙ - ( is-injective-neg-ℤ - ( ( neg-neg-ℤ (y *ℤ k)) ∙ - ( ( p) ∙ - ( ( ap (x +ℤ_) (neg-succ-ℤ x)) ∙ - ( ( right-predecessor-law-add-ℤ x (neg-ℤ x)) ∙ - ( ap pred-ℤ (right-inverse-law-add-ℤ x))))))) - -is-unit-cong-pred-ℤ : (k x : ℤ) → cong-ℤ k x (pred-ℤ x) → is-unit-ℤ k -pr1 (is-unit-cong-pred-ℤ k x (pair y p)) = y -pr2 (is-unit-cong-pred-ℤ k x (pair y p)) = - ( p) ∙ - ( ( ap (x +ℤ_) (neg-pred-ℤ x)) ∙ - ( ( right-successor-law-add-ℤ x (neg-ℤ x)) ∙ - ( ap succ-ℤ (right-inverse-law-add-ℤ x)))) +```agda +abstract + is-unit-cong-pred-ℤ : (k x : ℤ) → cong-ℤ k x (pred-ℤ x) → is-unit-ℤ k + pr1 (is-unit-cong-pred-ℤ k x (pair y p)) = y + pr2 (is-unit-cong-pred-ℤ k x (pair y p)) = + ( p) ∙ + ( ( ap (x +ℤ_) (neg-pred-ℤ x)) ∙ + ( ( right-successor-law-add-ℤ x (neg-ℤ x)) ∙ + ( ap succ-ℤ (right-inverse-law-add-ℤ x)))) +``` +### Congruence mod `k` is an equivalence relation + +```agda refl-cong-ℤ : (k : ℤ) → is-reflexive (cong-ℤ k) pr1 (refl-cong-ℤ k x) = zero-ℤ pr2 (refl-cong-ℤ k x) = left-zero-law-mul-ℤ k ∙ inv (is-zero-diff-ℤ' x) @@ -92,7 +124,11 @@ pr2 (transitive-cong-ℤ k x y z (pair e q) (pair d p)) = ( right-distributive-mul-add-ℤ d e k) ∙ ( ( ap-add-ℤ p q) ∙ ( triangle-diff-ℤ x y z)) +``` +### Concatenating congruence and equality + +```agda concatenate-eq-cong-ℤ : (k x x' y : ℤ) → x ＝ x' → cong-ℤ k x' y → cong-ℤ k x y concatenate-eq-cong-ℤ k x .x y refl = id @@ -114,7 +150,11 @@ concatenate-cong-cong-cong-ℤ : (k x y z w : ℤ) → cong-ℤ k x y → cong-ℤ k y z → cong-ℤ k z w → cong-ℤ k x w concatenate-cong-cong-cong-ℤ k x y z w H K L = transitive-cong-ℤ k x y w (transitive-cong-ℤ k y z w L K) H +``` +### If `-x` and `-y` are congruent mod `k`, so are `x` and `y` + +```agda cong-cong-neg-ℤ : (k x y : ℤ) → cong-ℤ k (neg-ℤ x) (neg-ℤ y) → cong-ℤ k x y pr1 (cong-cong-neg-ℤ k x y (pair d p)) = neg-ℤ d pr2 (cong-cong-neg-ℤ k x y (pair d p)) = @@ -131,6 +171,8 @@ pr2 (cong-neg-cong-ℤ k x y (pair d p)) = ( distributive-neg-add-ℤ x (neg-ℤ y))) ``` +### The canonical embedding of natural numbers preserves and reflects congruence + ```agda cong-int-cong-ℕ : (k x y : ℕ) → cong-ℕ k x y → cong-ℤ (int-ℕ k) (int-ℕ x) (int-ℕ y) diff --git a/src/elementary-number-theory/congruence-natural-numbers.lagda.md b/src/elementary-number-theory/congruence-natural-numbers.lagda.md index d4c0ff60a4..0abb907785 100644 --- a/src/elementary-number-theory/congruence-natural-numbers.lagda.md +++ b/src/elementary-number-theory/congruence-natural-numbers.lagda.md @@ -80,7 +80,7 @@ cong-identification-ℕ k {x} refl = refl-cong-ℕ k x symmetric-cong-ℕ : (k : ℕ) → is-symmetric (cong-ℕ k) pr1 (symmetric-cong-ℕ k x y (pair d p)) = d -pr2 (symmetric-cong-ℕ k x y (pair d p)) = p ∙ (symmetric-dist-ℕ x y) +pr2 (symmetric-cong-ℕ k x y (pair d p)) = p ∙ (commutative-dist-ℕ x y) cong-zero-ℕ' : (k : ℕ) → cong-ℕ k zero-ℕ k cong-zero-ℕ' k = diff --git a/src/elementary-number-theory/distance-integers.lagda.md b/src/elementary-number-theory/distance-integers.lagda.md index ee659aed25..922e42011b 100644 --- a/src/elementary-number-theory/distance-integers.lagda.md +++ b/src/elementary-number-theory/distance-integers.lagda.md @@ -26,7 +26,10 @@ open import foundation.identity-types ## Idea -The distance function between integers measures how far two integers are apart. +The +{{#concept "distance function" Disambiguation="between integers" Agda=dist-ℤ}} +between [integers](elementary-number-theory.integers.md) measures how far two +integers are apart. ## Definition @@ -37,43 +40,60 @@ dist-ℤ x y = abs-ℤ (x -ℤ y) ap-dist-ℤ : {x x' y y' : ℤ} → x ＝ x' → y ＝ y' → dist-ℤ x y ＝ dist-ℤ x' y' ap-dist-ℤ p q = ap-binary dist-ℤ p q +``` + +## Properties + +### Zero laws + +```agda +abstract + left-zero-law-dist-ℤ : (x : ℤ) → dist-ℤ zero-ℤ x ＝ abs-ℤ x + left-zero-law-dist-ℤ x = ap abs-ℤ (left-zero-law-diff-ℤ x) ∙ abs-neg-ℤ x + + right-zero-law-dist-ℤ : (x : ℤ) → dist-ℤ x zero-ℤ ＝ abs-ℤ x + right-zero-law-dist-ℤ x = ap abs-ℤ (right-zero-law-diff-ℤ x) +``` -left-zero-law-dist-ℤ : (x : ℤ) → dist-ℤ zero-ℤ x ＝ abs-ℤ x -left-zero-law-dist-ℤ x = ap abs-ℤ (left-zero-law-diff-ℤ x) ∙ abs-neg-ℤ x - -right-zero-law-dist-ℤ : (x : ℤ) → dist-ℤ x zero-ℤ ＝ abs-ℤ x -right-zero-law-dist-ℤ x = ap abs-ℤ (right-zero-law-diff-ℤ x) - -dist-int-ℕ : - (x y : ℕ) → dist-ℤ (int-ℕ x) (int-ℕ y) ＝ dist-ℕ x y -dist-int-ℕ zero-ℕ zero-ℕ = refl -dist-int-ℕ zero-ℕ (succ-ℕ y) = left-zero-law-dist-ℤ (int-ℕ (succ-ℕ y)) -dist-int-ℕ (succ-ℕ x) zero-ℕ = right-zero-law-dist-ℤ (int-ℕ (succ-ℕ x)) -dist-int-ℕ (succ-ℕ x) (succ-ℕ y) = - ( ( ap-dist-ℤ (inv (succ-int-ℕ x)) (inv (succ-int-ℕ y))) ∙ - ( ap abs-ℤ (diff-succ-ℤ (int-ℕ x) (int-ℕ y)))) ∙ - ( dist-int-ℕ x y) - -dist-abs-ℤ : - (x y : ℤ) → (H : is-nonnegative-ℤ x) → (K : is-nonnegative-ℤ y) → - dist-ℕ (abs-ℤ x) (abs-ℤ y) ＝ dist-ℤ x y -dist-abs-ℤ (inr (inl _)) y H K = - equational-reasoning - dist-ℕ 0 (abs-ℤ y) - ＝ abs-ℤ y - by left-unit-law-dist-ℕ (abs-ℤ y) - ＝ dist-ℤ (zero-ℤ) y - by inv (left-zero-law-dist-ℤ y) -dist-abs-ℤ (inr (inr x)) (inr (inl _)) H K = - equational-reasoning - dist-ℕ (abs-ℤ (inr (inr x))) 0 - ＝ succ-ℕ x - by right-unit-law-dist-ℕ (abs-ℤ (inr (inr x))) - ＝ dist-ℤ (inr (inr x)) zero-ℤ - by inv (right-zero-law-dist-ℤ (inr (inr x))) -dist-abs-ℤ (inr (inr x)) (inr (inr y)) H K = - equational-reasoning - dist-ℕ (succ-ℕ x) (succ-ℕ y) - ＝ dist-ℤ (int-ℕ (succ-ℕ x)) (int-ℕ (succ-ℕ y)) - by inv (dist-int-ℕ (succ-ℕ x) (succ-ℕ y)) +### The canonical embedding of natural numbers into the integers preserves distance + +```agda +abstract + dist-int-ℕ : + (x y : ℕ) → dist-ℤ (int-ℕ x) (int-ℕ y) ＝ dist-ℕ x y + dist-int-ℕ zero-ℕ zero-ℕ = refl + dist-int-ℕ zero-ℕ (succ-ℕ y) = left-zero-law-dist-ℤ (int-ℕ (succ-ℕ y)) + dist-int-ℕ (succ-ℕ x) zero-ℕ = right-zero-law-dist-ℤ (int-ℕ (succ-ℕ x)) + dist-int-ℕ (succ-ℕ x) (succ-ℕ y) = + ( ( ap-dist-ℤ (inv (succ-int-ℕ x)) (inv (succ-int-ℕ y))) ∙ + ( ap abs-ℤ (diff-succ-ℤ (int-ℕ x) (int-ℕ y)))) ∙ + ( dist-int-ℕ x y) +``` + +### If two integers are nonnegative, the distance between their absolute values is the distance between them + +```agda +abstract + dist-abs-ℤ : + (x y : ℤ) → (H : is-nonnegative-ℤ x) → (K : is-nonnegative-ℤ y) → + dist-ℕ (abs-ℤ x) (abs-ℤ y) ＝ dist-ℤ x y + dist-abs-ℤ (inr (inl _)) y H K = + equational-reasoning + dist-ℕ 0 (abs-ℤ y) + ＝ abs-ℤ y + by left-unit-law-dist-ℕ (abs-ℤ y) + ＝ dist-ℤ (zero-ℤ) y + by inv (left-zero-law-dist-ℤ y) + dist-abs-ℤ (inr (inr x)) (inr (inl _)) H K = + equational-reasoning + dist-ℕ (abs-ℤ (inr (inr x))) 0 + ＝ succ-ℕ x + by right-unit-law-dist-ℕ (abs-ℤ (inr (inr x))) + ＝ dist-ℤ (inr (inr x)) zero-ℤ + by inv (right-zero-law-dist-ℤ (inr (inr x))) + dist-abs-ℤ (inr (inr x)) (inr (inr y)) H K = + equational-reasoning + dist-ℕ (succ-ℕ x) (succ-ℕ y) + ＝ dist-ℤ (int-ℕ (succ-ℕ x)) (int-ℕ (succ-ℕ y)) + by inv (dist-int-ℕ (succ-ℕ x) (succ-ℕ y)) ``` diff --git a/src/elementary-number-theory/distance-natural-numbers.lagda.md b/src/elementary-number-theory/distance-natural-numbers.lagda.md index 9684f71377..b03c45d8d3 100644 --- a/src/elementary-number-theory/distance-natural-numbers.lagda.md +++ b/src/elementary-number-theory/distance-natural-numbers.lagda.md @@ -59,180 +59,192 @@ ap-dist-ℕ p q = ap-binary dist-ℕ p q ### Two natural numbers are equal if and only if their distance is zero ```agda -eq-dist-ℕ : (m n : ℕ) → is-zero-ℕ (dist-ℕ m n) → m ＝ n -eq-dist-ℕ zero-ℕ zero-ℕ p = refl -eq-dist-ℕ (succ-ℕ m) (succ-ℕ n) p = ap succ-ℕ (eq-dist-ℕ m n p) +abstract + eq-dist-ℕ : (m n : ℕ) → is-zero-ℕ (dist-ℕ m n) → m ＝ n + eq-dist-ℕ zero-ℕ zero-ℕ p = refl + eq-dist-ℕ (succ-ℕ m) (succ-ℕ n) p = ap succ-ℕ (eq-dist-ℕ m n p) -dist-eq-ℕ' : (n : ℕ) → is-zero-ℕ (dist-ℕ n n) -dist-eq-ℕ' zero-ℕ = refl -dist-eq-ℕ' (succ-ℕ n) = dist-eq-ℕ' n + dist-eq-ℕ' : (n : ℕ) → is-zero-ℕ (dist-ℕ n n) + dist-eq-ℕ' zero-ℕ = refl + dist-eq-ℕ' (succ-ℕ n) = dist-eq-ℕ' n -dist-eq-ℕ : (m n : ℕ) → m ＝ n → is-zero-ℕ (dist-ℕ m n) -dist-eq-ℕ m .m refl = dist-eq-ℕ' m + dist-eq-ℕ : (m n : ℕ) → m ＝ n → is-zero-ℕ (dist-ℕ m n) + dist-eq-ℕ m .m refl = dist-eq-ℕ' m -dist-neq-ℕ : (m n : ℕ) → m ≠ n → is-nonzero-ℕ (dist-ℕ m n) -dist-neq-ℕ m n = map-neg (eq-dist-ℕ m n) + dist-neq-ℕ : (m n : ℕ) → m ≠ n → is-nonzero-ℕ (dist-ℕ m n) + dist-neq-ℕ m n = map-neg (eq-dist-ℕ m n) -dist-neq-ℕ' : (m n : ℕ) → m ≠ n → is-successor-ℕ (dist-ℕ m n) -dist-neq-ℕ' m n np = is-successor-is-nonzero-ℕ (dist-neq-ℕ m n np) + dist-neq-ℕ' : (m n : ℕ) → m ≠ n → is-successor-ℕ (dist-ℕ m n) + dist-neq-ℕ' m n np = is-successor-is-nonzero-ℕ (dist-neq-ℕ m n np) +``` -is-one-dist-succ-ℕ : (x : ℕ) → is-one-ℕ (dist-ℕ x (succ-ℕ x)) -is-one-dist-succ-ℕ zero-ℕ = refl -is-one-dist-succ-ℕ (succ-ℕ x) = is-one-dist-succ-ℕ x +### The distance between a natural number and its successor is one -is-one-dist-succ-ℕ' : (x : ℕ) → is-one-ℕ (dist-ℕ (succ-ℕ x) x) -is-one-dist-succ-ℕ' zero-ℕ = refl -is-one-dist-succ-ℕ' (succ-ℕ x) = is-one-dist-succ-ℕ' x +```agda +abstract + is-one-dist-succ-ℕ : (x : ℕ) → is-one-ℕ (dist-ℕ x (succ-ℕ x)) + is-one-dist-succ-ℕ zero-ℕ = refl + is-one-dist-succ-ℕ (succ-ℕ x) = is-one-dist-succ-ℕ x + + is-one-dist-succ-ℕ' : (x : ℕ) → is-one-ℕ (dist-ℕ (succ-ℕ x) x) + is-one-dist-succ-ℕ' zero-ℕ = refl + is-one-dist-succ-ℕ' (succ-ℕ x) = is-one-dist-succ-ℕ' x ``` -### The distance function is symmetric +### The distance function is commutative ```agda -symmetric-dist-ℕ : - (m n : ℕ) → dist-ℕ m n ＝ dist-ℕ n m -symmetric-dist-ℕ zero-ℕ zero-ℕ = refl -symmetric-dist-ℕ zero-ℕ (succ-ℕ n) = refl -symmetric-dist-ℕ (succ-ℕ m) zero-ℕ = refl -symmetric-dist-ℕ (succ-ℕ m) (succ-ℕ n) = symmetric-dist-ℕ m n +abstract + commutative-dist-ℕ : + (m n : ℕ) → dist-ℕ m n ＝ dist-ℕ n m + commutative-dist-ℕ zero-ℕ zero-ℕ = refl + commutative-dist-ℕ zero-ℕ (succ-ℕ n) = refl + commutative-dist-ℕ (succ-ℕ m) zero-ℕ = refl + commutative-dist-ℕ (succ-ℕ m) (succ-ℕ n) = commutative-dist-ℕ m n ``` ### The distance from zero ```agda -left-unit-law-dist-ℕ : - (n : ℕ) → dist-ℕ zero-ℕ n ＝ n -left-unit-law-dist-ℕ zero-ℕ = refl -left-unit-law-dist-ℕ (succ-ℕ n) = refl - -right-unit-law-dist-ℕ : - (n : ℕ) → dist-ℕ n zero-ℕ ＝ n -right-unit-law-dist-ℕ zero-ℕ = refl -right-unit-law-dist-ℕ (succ-ℕ n) = refl +abstract + left-unit-law-dist-ℕ : + (n : ℕ) → dist-ℕ zero-ℕ n ＝ n + left-unit-law-dist-ℕ zero-ℕ = refl + left-unit-law-dist-ℕ (succ-ℕ n) = refl + + right-unit-law-dist-ℕ : + (n : ℕ) → dist-ℕ n zero-ℕ ＝ n + right-unit-law-dist-ℕ zero-ℕ = refl + right-unit-law-dist-ℕ (succ-ℕ n) = refl ``` ## The triangle inequality ```agda -triangle-inequality-dist-ℕ : - (m n k : ℕ) → (dist-ℕ m n) ≤-ℕ ((dist-ℕ m k) +ℕ (dist-ℕ k n)) -triangle-inequality-dist-ℕ zero-ℕ zero-ℕ zero-ℕ = star -triangle-inequality-dist-ℕ zero-ℕ zero-ℕ (succ-ℕ k) = star -triangle-inequality-dist-ℕ zero-ℕ (succ-ℕ n) zero-ℕ = - tr - ( leq-ℕ (succ-ℕ n)) - ( inv (left-unit-law-add-ℕ (succ-ℕ n))) - ( refl-leq-ℕ (succ-ℕ n)) -triangle-inequality-dist-ℕ zero-ℕ (succ-ℕ n) (succ-ℕ k) = - concatenate-eq-leq-eq-ℕ - ( inv (ap succ-ℕ (left-unit-law-dist-ℕ n))) - ( triangle-inequality-dist-ℕ zero-ℕ n k) - ( ( ap (succ-ℕ ∘ (_+ℕ (dist-ℕ k n))) (left-unit-law-dist-ℕ k)) ∙ - ( inv (left-successor-law-add-ℕ k (dist-ℕ k n)))) -triangle-inequality-dist-ℕ (succ-ℕ m) zero-ℕ zero-ℕ = refl-leq-ℕ (succ-ℕ m) -triangle-inequality-dist-ℕ (succ-ℕ m) zero-ℕ (succ-ℕ k) = - concatenate-eq-leq-eq-ℕ - ( inv (ap succ-ℕ (right-unit-law-dist-ℕ m))) - ( triangle-inequality-dist-ℕ m zero-ℕ k) - ( ap (succ-ℕ ∘ ((dist-ℕ m k) +ℕ_)) (right-unit-law-dist-ℕ k)) -triangle-inequality-dist-ℕ (succ-ℕ m) (succ-ℕ n) zero-ℕ = - concatenate-leq-eq-ℕ - ( dist-ℕ m n) - ( transitive-leq-ℕ +abstract + triangle-inequality-dist-ℕ : + (m n k : ℕ) → (dist-ℕ m n) ≤-ℕ ((dist-ℕ m k) +ℕ (dist-ℕ k n)) + triangle-inequality-dist-ℕ zero-ℕ zero-ℕ zero-ℕ = star + triangle-inequality-dist-ℕ zero-ℕ zero-ℕ (succ-ℕ k) = star + triangle-inequality-dist-ℕ zero-ℕ (succ-ℕ n) zero-ℕ = + tr + ( leq-ℕ (succ-ℕ n)) + ( inv (left-unit-law-add-ℕ (succ-ℕ n))) + ( refl-leq-ℕ (succ-ℕ n)) + triangle-inequality-dist-ℕ zero-ℕ (succ-ℕ n) (succ-ℕ k) = + concatenate-eq-leq-eq-ℕ + ( inv (ap succ-ℕ (left-unit-law-dist-ℕ n))) + ( triangle-inequality-dist-ℕ zero-ℕ n k) + ( ( ap (succ-ℕ ∘ (_+ℕ (dist-ℕ k n))) (left-unit-law-dist-ℕ k)) ∙ + ( inv (left-successor-law-add-ℕ k (dist-ℕ k n)))) + triangle-inequality-dist-ℕ (succ-ℕ m) zero-ℕ zero-ℕ = refl-leq-ℕ (succ-ℕ m) + triangle-inequality-dist-ℕ (succ-ℕ m) zero-ℕ (succ-ℕ k) = + concatenate-eq-leq-eq-ℕ + ( inv (ap succ-ℕ (right-unit-law-dist-ℕ m))) + ( triangle-inequality-dist-ℕ m zero-ℕ k) + ( ap (succ-ℕ ∘ ((dist-ℕ m k) +ℕ_)) (right-unit-law-dist-ℕ k)) + triangle-inequality-dist-ℕ (succ-ℕ m) (succ-ℕ n) zero-ℕ = + concatenate-leq-eq-ℕ ( dist-ℕ m n) - ( succ-ℕ ((dist-ℕ m zero-ℕ) +ℕ (dist-ℕ zero-ℕ n))) - ( succ-ℕ (succ-ℕ ((dist-ℕ m zero-ℕ) +ℕ (dist-ℕ zero-ℕ n)))) - ( succ-leq-ℕ (succ-ℕ ((dist-ℕ m zero-ℕ) +ℕ (dist-ℕ zero-ℕ n)))) ( transitive-leq-ℕ ( dist-ℕ m n) - ( (dist-ℕ m zero-ℕ) +ℕ (dist-ℕ zero-ℕ n)) ( succ-ℕ ((dist-ℕ m zero-ℕ) +ℕ (dist-ℕ zero-ℕ n))) - ( succ-leq-ℕ ((dist-ℕ m zero-ℕ) +ℕ (dist-ℕ zero-ℕ n))) - ( triangle-inequality-dist-ℕ m n zero-ℕ))) - ( ( ap - ( succ-ℕ ∘ succ-ℕ) - ( ap-add-ℕ (right-unit-law-dist-ℕ m) (left-unit-law-dist-ℕ n))) ∙ - ( inv (left-successor-law-add-ℕ m (succ-ℕ n)))) -triangle-inequality-dist-ℕ (succ-ℕ m) (succ-ℕ n) (succ-ℕ k) = - triangle-inequality-dist-ℕ m n k + ( succ-ℕ (succ-ℕ ((dist-ℕ m zero-ℕ) +ℕ (dist-ℕ zero-ℕ n)))) + ( succ-leq-ℕ (succ-ℕ ((dist-ℕ m zero-ℕ) +ℕ (dist-ℕ zero-ℕ n)))) + ( transitive-leq-ℕ + ( dist-ℕ m n) + ( (dist-ℕ m zero-ℕ) +ℕ (dist-ℕ zero-ℕ n)) + ( succ-ℕ ((dist-ℕ m zero-ℕ) +ℕ (dist-ℕ zero-ℕ n))) + ( succ-leq-ℕ ((dist-ℕ m zero-ℕ) +ℕ (dist-ℕ zero-ℕ n))) + ( triangle-inequality-dist-ℕ m n zero-ℕ))) + ( ( ap + ( succ-ℕ ∘ succ-ℕ) + ( ap-add-ℕ (right-unit-law-dist-ℕ m) (left-unit-law-dist-ℕ n))) ∙ + ( inv (left-successor-law-add-ℕ m (succ-ℕ n)))) + triangle-inequality-dist-ℕ (succ-ℕ m) (succ-ℕ n) (succ-ℕ k) = + triangle-inequality-dist-ℕ m n k ``` ### `dist-ℕ x y` is a solution in `z` to `x + z ＝ y` ```agda -is-additive-right-inverse-dist-ℕ : - (x y : ℕ) → x ≤-ℕ y → x +ℕ (dist-ℕ x y) ＝ y -is-additive-right-inverse-dist-ℕ zero-ℕ zero-ℕ H = refl -is-additive-right-inverse-dist-ℕ zero-ℕ (succ-ℕ y) star = - left-unit-law-add-ℕ (succ-ℕ y) -is-additive-right-inverse-dist-ℕ (succ-ℕ x) (succ-ℕ y) H = - ( left-successor-law-add-ℕ x (dist-ℕ x y)) ∙ - ( ap succ-ℕ (is-additive-right-inverse-dist-ℕ x y H)) - -rewrite-left-add-dist-ℕ : - (x y z : ℕ) → x +ℕ y ＝ z → x ＝ dist-ℕ y z -rewrite-left-add-dist-ℕ zero-ℕ zero-ℕ .zero-ℕ refl = refl -rewrite-left-add-dist-ℕ zero-ℕ (succ-ℕ y) .(succ-ℕ (zero-ℕ +ℕ y)) refl = - ( inv (dist-eq-ℕ' y)) ∙ - ( inv (ap (dist-ℕ (succ-ℕ y)) (left-unit-law-add-ℕ (succ-ℕ y)))) -rewrite-left-add-dist-ℕ (succ-ℕ x) zero-ℕ .(succ-ℕ x) refl = refl -rewrite-left-add-dist-ℕ (succ-ℕ x) (succ-ℕ y) ._ refl = - rewrite-left-add-dist-ℕ (succ-ℕ x) y ((succ-ℕ x) +ℕ y) refl - -rewrite-left-dist-add-ℕ : - (x y z : ℕ) → y ≤-ℕ z → x ＝ dist-ℕ y z → x +ℕ y ＝ z -rewrite-left-dist-add-ℕ .(dist-ℕ y z) y z H refl = - ( commutative-add-ℕ (dist-ℕ y z) y) ∙ - ( is-additive-right-inverse-dist-ℕ y z H) - -rewrite-right-add-dist-ℕ : - (x y z : ℕ) → x +ℕ y ＝ z → y ＝ dist-ℕ x z -rewrite-right-add-dist-ℕ x y z p = - rewrite-left-add-dist-ℕ y x z (commutative-add-ℕ y x ∙ p) - -rewrite-right-dist-add-ℕ : - (x y z : ℕ) → x ≤-ℕ z → y ＝ dist-ℕ x z → x +ℕ y ＝ z -rewrite-right-dist-add-ℕ x .(dist-ℕ x z) z H refl = - is-additive-right-inverse-dist-ℕ x z H - -is-difference-dist-ℕ : - (x y : ℕ) → x ≤-ℕ y → x +ℕ (dist-ℕ x y) ＝ y -is-difference-dist-ℕ zero-ℕ zero-ℕ H = refl -is-difference-dist-ℕ zero-ℕ (succ-ℕ y) H = left-unit-law-add-ℕ (succ-ℕ y) -is-difference-dist-ℕ (succ-ℕ x) (succ-ℕ y) H = - ( left-successor-law-add-ℕ x (dist-ℕ x y)) ∙ - ( ap succ-ℕ (is-difference-dist-ℕ x y H)) - -is-difference-dist-ℕ' : - (x y : ℕ) → x ≤-ℕ y → (dist-ℕ x y) +ℕ x ＝ y -is-difference-dist-ℕ' x y H = - ( commutative-add-ℕ (dist-ℕ x y) x) ∙ - ( is-difference-dist-ℕ x y H) +abstract + is-additive-right-inverse-dist-ℕ : + (x y : ℕ) → x ≤-ℕ y → x +ℕ (dist-ℕ x y) ＝ y + is-additive-right-inverse-dist-ℕ zero-ℕ zero-ℕ H = refl + is-additive-right-inverse-dist-ℕ zero-ℕ (succ-ℕ y) star = + left-unit-law-add-ℕ (succ-ℕ y) + is-additive-right-inverse-dist-ℕ (succ-ℕ x) (succ-ℕ y) H = + ( left-successor-law-add-ℕ x (dist-ℕ x y)) ∙ + ( ap succ-ℕ (is-additive-right-inverse-dist-ℕ x y H)) + + rewrite-left-add-dist-ℕ : + (x y z : ℕ) → x +ℕ y ＝ z → x ＝ dist-ℕ y z + rewrite-left-add-dist-ℕ zero-ℕ zero-ℕ .zero-ℕ refl = refl + rewrite-left-add-dist-ℕ zero-ℕ (succ-ℕ y) .(succ-ℕ (zero-ℕ +ℕ y)) refl = + ( inv (dist-eq-ℕ' y)) ∙ + ( inv (ap (dist-ℕ (succ-ℕ y)) (left-unit-law-add-ℕ (succ-ℕ y)))) + rewrite-left-add-dist-ℕ (succ-ℕ x) zero-ℕ .(succ-ℕ x) refl = refl + rewrite-left-add-dist-ℕ (succ-ℕ x) (succ-ℕ y) ._ refl = + rewrite-left-add-dist-ℕ (succ-ℕ x) y ((succ-ℕ x) +ℕ y) refl + + rewrite-left-dist-add-ℕ : + (x y z : ℕ) → y ≤-ℕ z → x ＝ dist-ℕ y z → x +ℕ y ＝ z + rewrite-left-dist-add-ℕ .(dist-ℕ y z) y z H refl = + ( commutative-add-ℕ (dist-ℕ y z) y) ∙ + ( is-additive-right-inverse-dist-ℕ y z H) + + rewrite-right-add-dist-ℕ : + (x y z : ℕ) → x +ℕ y ＝ z → y ＝ dist-ℕ x z + rewrite-right-add-dist-ℕ x y z p = + rewrite-left-add-dist-ℕ y x z (commutative-add-ℕ y x ∙ p) + + rewrite-right-dist-add-ℕ : + (x y z : ℕ) → x ≤-ℕ z → y ＝ dist-ℕ x z → x +ℕ y ＝ z + rewrite-right-dist-add-ℕ x .(dist-ℕ x z) z H refl = + is-additive-right-inverse-dist-ℕ x z H + + is-difference-dist-ℕ : + (x y : ℕ) → x ≤-ℕ y → x +ℕ (dist-ℕ x y) ＝ y + is-difference-dist-ℕ zero-ℕ zero-ℕ H = refl + is-difference-dist-ℕ zero-ℕ (succ-ℕ y) H = left-unit-law-add-ℕ (succ-ℕ y) + is-difference-dist-ℕ (succ-ℕ x) (succ-ℕ y) H = + ( left-successor-law-add-ℕ x (dist-ℕ x y)) ∙ + ( ap succ-ℕ (is-difference-dist-ℕ x y H)) + + is-difference-dist-ℕ' : + (x y : ℕ) → x ≤-ℕ y → (dist-ℕ x y) +ℕ x ＝ y + is-difference-dist-ℕ' x y H = + ( commutative-add-ℕ (dist-ℕ x y) x) ∙ + ( is-difference-dist-ℕ x y H) ``` ### The distance from `n` to `n + m` is `m` ```agda -dist-add-ℕ : (x y : ℕ) → dist-ℕ x (x +ℕ y) ＝ y -dist-add-ℕ x y = inv (rewrite-right-add-dist-ℕ x y (x +ℕ y) refl) +abstract + dist-add-ℕ : (x y : ℕ) → dist-ℕ x (x +ℕ y) ＝ y + dist-add-ℕ x y = inv (rewrite-right-add-dist-ℕ x y (x +ℕ y) refl) -dist-add-ℕ' : (x y : ℕ) → dist-ℕ (x +ℕ y) x ＝ y -dist-add-ℕ' x y = symmetric-dist-ℕ (x +ℕ y) x ∙ dist-add-ℕ x y + dist-add-ℕ' : (x y : ℕ) → dist-ℕ (x +ℕ y) x ＝ y + dist-add-ℕ' x y = commutative-dist-ℕ (x +ℕ y) x ∙ dist-add-ℕ x y ``` ### If three elements are ordered linearly, then their distances add up ```agda -triangle-equality-dist-ℕ : - (x y z : ℕ) → (x ≤-ℕ y) → (y ≤-ℕ z) → - (dist-ℕ x y) +ℕ (dist-ℕ y z) ＝ dist-ℕ x z -triangle-equality-dist-ℕ zero-ℕ zero-ℕ zero-ℕ H1 H2 = refl -triangle-equality-dist-ℕ zero-ℕ zero-ℕ (succ-ℕ z) star star = - ap succ-ℕ (left-unit-law-add-ℕ z) -triangle-equality-dist-ℕ zero-ℕ (succ-ℕ y) (succ-ℕ z) star H2 = - left-successor-law-add-ℕ y (dist-ℕ y z) ∙ - ap succ-ℕ (is-additive-right-inverse-dist-ℕ y z H2) -triangle-equality-dist-ℕ (succ-ℕ x) (succ-ℕ y) (succ-ℕ z) H1 H2 = - triangle-equality-dist-ℕ x y z H1 H2 +abstract + triangle-equality-dist-ℕ : + (x y z : ℕ) → (x ≤-ℕ y) → (y ≤-ℕ z) → + (dist-ℕ x y) +ℕ (dist-ℕ y z) ＝ dist-ℕ x z + triangle-equality-dist-ℕ zero-ℕ zero-ℕ zero-ℕ H1 H2 = refl + triangle-equality-dist-ℕ zero-ℕ zero-ℕ (succ-ℕ z) star star = + ap succ-ℕ (left-unit-law-add-ℕ z) + triangle-equality-dist-ℕ zero-ℕ (succ-ℕ y) (succ-ℕ z) star H2 = + left-successor-law-add-ℕ y (dist-ℕ y z) ∙ + ap succ-ℕ (is-additive-right-inverse-dist-ℕ y z H2) + triangle-equality-dist-ℕ (succ-ℕ x) (succ-ℕ y) (succ-ℕ z) H1 H2 = + triangle-equality-dist-ℕ x y z H1 H2 cases-dist-ℕ : (x y z : ℕ) → UU lzero @@ -241,155 +253,162 @@ cases-dist-ℕ x y z = ( ( (dist-ℕ y z) +ℕ (dist-ℕ x z) ＝ dist-ℕ x y) + ( (dist-ℕ x z) +ℕ (dist-ℕ x y) ＝ dist-ℕ y z)) -is-total-dist-ℕ : - (x y z : ℕ) → cases-dist-ℕ x y z -is-total-dist-ℕ x y z with order-three-elements-ℕ x y z -is-total-dist-ℕ x y z | inl (inl (pair H1 H2)) = - inl (triangle-equality-dist-ℕ x y z H1 H2) -is-total-dist-ℕ x y z | inl (inr (pair H1 H2)) = - inr - ( inl - ( ( commutative-add-ℕ (dist-ℕ y z) (dist-ℕ x z)) ∙ - ( ( ap ((dist-ℕ x z) +ℕ_) (symmetric-dist-ℕ y z)) ∙ - ( triangle-equality-dist-ℕ x z y H1 H2)))) -is-total-dist-ℕ x y z | inr (inl (inl (pair H1 H2))) = - inr - ( inl - ( ( ap ((dist-ℕ y z) +ℕ_) (symmetric-dist-ℕ x z)) ∙ - ( ( triangle-equality-dist-ℕ y z x H1 H2) ∙ - ( symmetric-dist-ℕ y x)))) -is-total-dist-ℕ x y z | inr (inl (inr (pair H1 H2))) = - inr - ( inr - ( ( ap ((dist-ℕ x z) +ℕ_) (symmetric-dist-ℕ x y)) ∙ - ( ( commutative-add-ℕ (dist-ℕ x z) (dist-ℕ y x)) ∙ - ( triangle-equality-dist-ℕ y x z H1 H2)))) -is-total-dist-ℕ x y z | inr (inr (inl (pair H1 H2))) = - inr - ( inr - ( ( ap (_+ℕ (dist-ℕ x y)) (symmetric-dist-ℕ x z)) ∙ - ( ( triangle-equality-dist-ℕ z x y H1 H2) ∙ - ( symmetric-dist-ℕ z y)))) -is-total-dist-ℕ x y z | inr (inr (inr (pair H1 H2))) = - inl - ( ( ap-add-ℕ (symmetric-dist-ℕ x y) (symmetric-dist-ℕ y z)) ∙ - ( ( commutative-add-ℕ (dist-ℕ y x) (dist-ℕ z y)) ∙ - ( ( triangle-equality-dist-ℕ z y x H1 H2) ∙ - ( symmetric-dist-ℕ z x)))) +abstract + is-total-dist-ℕ : + (x y z : ℕ) → cases-dist-ℕ x y z + is-total-dist-ℕ x y z with order-three-elements-ℕ x y z + is-total-dist-ℕ x y z | inl (inl (pair H1 H2)) = + inl (triangle-equality-dist-ℕ x y z H1 H2) + is-total-dist-ℕ x y z | inl (inr (pair H1 H2)) = + inr + ( inl + ( ( commutative-add-ℕ (dist-ℕ y z) (dist-ℕ x z)) ∙ + ( ( ap ((dist-ℕ x z) +ℕ_) (commutative-dist-ℕ y z)) ∙ + ( triangle-equality-dist-ℕ x z y H1 H2)))) + is-total-dist-ℕ x y z | inr (inl (inl (pair H1 H2))) = + inr + ( inl + ( ( ap ((dist-ℕ y z) +ℕ_) (commutative-dist-ℕ x z)) ∙ + ( ( triangle-equality-dist-ℕ y z x H1 H2) ∙ + ( commutative-dist-ℕ y x)))) + is-total-dist-ℕ x y z | inr (inl (inr (pair H1 H2))) = + inr + ( inr + ( ( ap ((dist-ℕ x z) +ℕ_) (commutative-dist-ℕ x y)) ∙ + ( ( commutative-add-ℕ (dist-ℕ x z) (dist-ℕ y x)) ∙ + ( triangle-equality-dist-ℕ y x z H1 H2)))) + is-total-dist-ℕ x y z | inr (inr (inl (pair H1 H2))) = + inr + ( inr + ( ( ap (_+ℕ (dist-ℕ x y)) (commutative-dist-ℕ x z)) ∙ + ( ( triangle-equality-dist-ℕ z x y H1 H2) ∙ + ( commutative-dist-ℕ z y)))) + is-total-dist-ℕ x y z | inr (inr (inr (pair H1 H2))) = + inl + ( ( ap-add-ℕ (commutative-dist-ℕ x y) (commutative-dist-ℕ y z)) ∙ + ( ( commutative-add-ℕ (dist-ℕ y x) (dist-ℕ z y)) ∙ + ( ( triangle-equality-dist-ℕ z y x H1 H2) ∙ + ( commutative-dist-ℕ z x)))) ``` -### If `x ≤ y` then the distance between `x` and `y` is less than `y` +### If `x ≤ y` then the distance between `x` and `y` is less than or equal to `y` ```agda -leq-dist-ℕ : - (x y : ℕ) → x ≤-ℕ y → dist-ℕ x y ≤-ℕ y -leq-dist-ℕ zero-ℕ zero-ℕ H = refl-leq-ℕ zero-ℕ -leq-dist-ℕ zero-ℕ (succ-ℕ y) H = refl-leq-ℕ y -leq-dist-ℕ (succ-ℕ x) (succ-ℕ y) H = - transitive-leq-ℕ (dist-ℕ x y) y (succ-ℕ y) (succ-leq-ℕ y) (leq-dist-ℕ x y H) +abstract + leq-dist-ℕ : + (x y : ℕ) → x ≤-ℕ y → dist-ℕ x y ≤-ℕ y + leq-dist-ℕ zero-ℕ zero-ℕ H = refl-leq-ℕ zero-ℕ + leq-dist-ℕ zero-ℕ (succ-ℕ y) H = refl-leq-ℕ y + leq-dist-ℕ (succ-ℕ x) (succ-ℕ y) H = + transitive-leq-ℕ (dist-ℕ x y) y (succ-ℕ y) (succ-leq-ℕ y) (leq-dist-ℕ x y H) ``` ### If `x < b` and `y < b`, then `dist-ℕ x y < b` ```agda -strict-upper-bound-dist-ℕ : - (b x y : ℕ) → x <-ℕ b → y <-ℕ b → dist-ℕ x y <-ℕ b -strict-upper-bound-dist-ℕ (succ-ℕ b) zero-ℕ y H K = K -strict-upper-bound-dist-ℕ (succ-ℕ b) (succ-ℕ x) zero-ℕ H K = H -strict-upper-bound-dist-ℕ (succ-ℕ b) (succ-ℕ x) (succ-ℕ y) H K = - preserves-le-succ-ℕ (dist-ℕ x y) b (strict-upper-bound-dist-ℕ b x y H K) +abstract + strict-upper-bound-dist-ℕ : + (b x y : ℕ) → x <-ℕ b → y <-ℕ b → dist-ℕ x y <-ℕ b + strict-upper-bound-dist-ℕ (succ-ℕ b) zero-ℕ y H K = K + strict-upper-bound-dist-ℕ (succ-ℕ b) (succ-ℕ x) zero-ℕ H K = H + strict-upper-bound-dist-ℕ (succ-ℕ b) (succ-ℕ x) (succ-ℕ y) H K = + preserves-le-succ-ℕ (dist-ℕ x y) b (strict-upper-bound-dist-ℕ b x y H K) ``` ### If `x < y` then `dist-ℕ x (succ-ℕ y) = succ-ℕ (dist-ℕ x y)` ```agda -right-successor-law-dist-ℕ : - (x y : ℕ) → x ≤-ℕ y → dist-ℕ x (succ-ℕ y) ＝ succ-ℕ (dist-ℕ x y) -right-successor-law-dist-ℕ zero-ℕ zero-ℕ star = refl -right-successor-law-dist-ℕ zero-ℕ (succ-ℕ y) star = refl -right-successor-law-dist-ℕ (succ-ℕ x) (succ-ℕ y) H = - right-successor-law-dist-ℕ x y H - -left-successor-law-dist-ℕ : - (x y : ℕ) → y ≤-ℕ x → dist-ℕ (succ-ℕ x) y ＝ succ-ℕ (dist-ℕ x y) -left-successor-law-dist-ℕ zero-ℕ zero-ℕ star = refl -left-successor-law-dist-ℕ (succ-ℕ x) zero-ℕ star = refl -left-successor-law-dist-ℕ (succ-ℕ x) (succ-ℕ y) H = - left-successor-law-dist-ℕ x y H +abstract + right-successor-law-dist-ℕ : + (x y : ℕ) → x ≤-ℕ y → dist-ℕ x (succ-ℕ y) ＝ succ-ℕ (dist-ℕ x y) + right-successor-law-dist-ℕ zero-ℕ zero-ℕ star = refl + right-successor-law-dist-ℕ zero-ℕ (succ-ℕ y) star = refl + right-successor-law-dist-ℕ (succ-ℕ x) (succ-ℕ y) H = + right-successor-law-dist-ℕ x y H + + left-successor-law-dist-ℕ : + (x y : ℕ) → y ≤-ℕ x → dist-ℕ (succ-ℕ x) y ＝ succ-ℕ (dist-ℕ x y) + left-successor-law-dist-ℕ zero-ℕ zero-ℕ star = refl + left-successor-law-dist-ℕ (succ-ℕ x) zero-ℕ star = refl + left-successor-law-dist-ℕ (succ-ℕ x) (succ-ℕ y) H = + left-successor-law-dist-ℕ x y H ``` ### `dist-ℕ` is translation invariant ```agda -translation-invariant-dist-ℕ : - (k m n : ℕ) → dist-ℕ (k +ℕ m) (k +ℕ n) ＝ dist-ℕ m n -translation-invariant-dist-ℕ zero-ℕ m n = - ap-dist-ℕ (left-unit-law-add-ℕ m) (left-unit-law-add-ℕ n) -translation-invariant-dist-ℕ (succ-ℕ k) m n = - ( ap-dist-ℕ (left-successor-law-add-ℕ k m) (left-successor-law-add-ℕ k n)) ∙ - ( translation-invariant-dist-ℕ k m n) - -translation-invariant-dist-ℕ' : - (k m n : ℕ) → dist-ℕ (m +ℕ k) (n +ℕ k) ＝ dist-ℕ m n -translation-invariant-dist-ℕ' k m n = - ( ap-dist-ℕ (commutative-add-ℕ m k) (commutative-add-ℕ n k)) ∙ - ( translation-invariant-dist-ℕ k m n) +abstract + translation-invariant-dist-ℕ : + (k m n : ℕ) → dist-ℕ (k +ℕ m) (k +ℕ n) ＝ dist-ℕ m n + translation-invariant-dist-ℕ zero-ℕ m n = + ap-dist-ℕ (left-unit-law-add-ℕ m) (left-unit-law-add-ℕ n) + translation-invariant-dist-ℕ (succ-ℕ k) m n = + ( ap-dist-ℕ (left-successor-law-add-ℕ k m) (left-successor-law-add-ℕ k n)) ∙ + ( translation-invariant-dist-ℕ k m n) + + translation-invariant-dist-ℕ' : + (k m n : ℕ) → dist-ℕ (m +ℕ k) (n +ℕ k) ＝ dist-ℕ m n + translation-invariant-dist-ℕ' k m n = + ( ap-dist-ℕ (commutative-add-ℕ m k) (commutative-add-ℕ n k)) ∙ + ( translation-invariant-dist-ℕ k m n) ``` ### `dist-ℕ` is linear with respect to scalar multiplication ```agda -left-distributive-mul-dist-ℕ : - (m n k : ℕ) → k *ℕ (dist-ℕ m n) ＝ dist-ℕ (k *ℕ m) (k *ℕ n) -left-distributive-mul-dist-ℕ zero-ℕ zero-ℕ zero-ℕ = refl -left-distributive-mul-dist-ℕ zero-ℕ zero-ℕ (succ-ℕ k) = - left-distributive-mul-dist-ℕ zero-ℕ zero-ℕ k -left-distributive-mul-dist-ℕ zero-ℕ (succ-ℕ n) zero-ℕ = refl -left-distributive-mul-dist-ℕ zero-ℕ (succ-ℕ n) (succ-ℕ k) = - ap - ( dist-ℕ' ((succ-ℕ k) *ℕ (succ-ℕ n))) - ( inv (right-zero-law-mul-ℕ (succ-ℕ k))) -left-distributive-mul-dist-ℕ (succ-ℕ m) zero-ℕ zero-ℕ = refl -left-distributive-mul-dist-ℕ (succ-ℕ m) zero-ℕ (succ-ℕ k) = - ap - ( dist-ℕ ((succ-ℕ k) *ℕ (succ-ℕ m))) - ( inv (right-zero-law-mul-ℕ (succ-ℕ k))) -left-distributive-mul-dist-ℕ (succ-ℕ m) (succ-ℕ n) zero-ℕ = refl -left-distributive-mul-dist-ℕ (succ-ℕ m) (succ-ℕ n) (succ-ℕ k) = - inv - ( ( ap-dist-ℕ - ( right-successor-law-mul-ℕ (succ-ℕ k) m) - ( right-successor-law-mul-ℕ (succ-ℕ k) n)) ∙ - ( ( translation-invariant-dist-ℕ - ( succ-ℕ k) - ( (succ-ℕ k) *ℕ m) - ( (succ-ℕ k) *ℕ n)) ∙ - ( inv (left-distributive-mul-dist-ℕ m n (succ-ℕ k))))) - -left-distributive-mul-dist-ℕ' : - (m n k : ℕ) → dist-ℕ (k *ℕ m) (k *ℕ n) ＝ k *ℕ (dist-ℕ m n) -left-distributive-mul-dist-ℕ' m n k = - inv (left-distributive-mul-dist-ℕ m n k) - -right-distributive-mul-dist-ℕ : - (x y k : ℕ) → (dist-ℕ x y) *ℕ k ＝ dist-ℕ (x *ℕ k) (y *ℕ k) -right-distributive-mul-dist-ℕ x y k = - ( commutative-mul-ℕ (dist-ℕ x y) k) ∙ - ( ( left-distributive-mul-dist-ℕ x y k) ∙ - ( ap-dist-ℕ (commutative-mul-ℕ k x) (commutative-mul-ℕ k y))) +abstract + left-distributive-mul-dist-ℕ : + (m n k : ℕ) → k *ℕ (dist-ℕ m n) ＝ dist-ℕ (k *ℕ m) (k *ℕ n) + left-distributive-mul-dist-ℕ zero-ℕ zero-ℕ zero-ℕ = refl + left-distributive-mul-dist-ℕ zero-ℕ zero-ℕ (succ-ℕ k) = + left-distributive-mul-dist-ℕ zero-ℕ zero-ℕ k + left-distributive-mul-dist-ℕ zero-ℕ (succ-ℕ n) zero-ℕ = refl + left-distributive-mul-dist-ℕ zero-ℕ (succ-ℕ n) (succ-ℕ k) = + ap + ( dist-ℕ' ((succ-ℕ k) *ℕ (succ-ℕ n))) + ( inv (right-zero-law-mul-ℕ (succ-ℕ k))) + left-distributive-mul-dist-ℕ (succ-ℕ m) zero-ℕ zero-ℕ = refl + left-distributive-mul-dist-ℕ (succ-ℕ m) zero-ℕ (succ-ℕ k) = + ap + ( dist-ℕ ((succ-ℕ k) *ℕ (succ-ℕ m))) + ( inv (right-zero-law-mul-ℕ (succ-ℕ k))) + left-distributive-mul-dist-ℕ (succ-ℕ m) (succ-ℕ n) zero-ℕ = refl + left-distributive-mul-dist-ℕ (succ-ℕ m) (succ-ℕ n) (succ-ℕ k) = + inv + ( ( ap-dist-ℕ + ( right-successor-law-mul-ℕ (succ-ℕ k) m) + ( right-successor-law-mul-ℕ (succ-ℕ k) n)) ∙ + ( ( translation-invariant-dist-ℕ + ( succ-ℕ k) + ( (succ-ℕ k) *ℕ m) + ( (succ-ℕ k) *ℕ n)) ∙ + ( inv (left-distributive-mul-dist-ℕ m n (succ-ℕ k))))) + + left-distributive-mul-dist-ℕ' : + (m n k : ℕ) → dist-ℕ (k *ℕ m) (k *ℕ n) ＝ k *ℕ (dist-ℕ m n) + left-distributive-mul-dist-ℕ' m n k = + inv (left-distributive-mul-dist-ℕ m n k) + + right-distributive-mul-dist-ℕ : + (x y k : ℕ) → (dist-ℕ x y) *ℕ k ＝ dist-ℕ (x *ℕ k) (y *ℕ k) + right-distributive-mul-dist-ℕ x y k = + ( commutative-mul-ℕ (dist-ℕ x y) k) ∙ + ( ( left-distributive-mul-dist-ℕ x y k) ∙ + ( ap-dist-ℕ (commutative-mul-ℕ k x) (commutative-mul-ℕ k y))) ``` ### The distance is the difference between the maximum and the minimum ```agda -eq-max-dist-min-ℕ : (x y : ℕ) → dist-ℕ x y +ℕ min-ℕ x y ＝ max-ℕ x y -eq-max-dist-min-ℕ zero-ℕ y = refl -eq-max-dist-min-ℕ (succ-ℕ x) zero-ℕ = refl -eq-max-dist-min-ℕ (succ-ℕ x) (succ-ℕ y) = ap succ-ℕ (eq-max-dist-min-ℕ x y) - -dist-max-min-ℕ : (x y : ℕ) → dist-ℕ x y ＝ dist-ℕ (max-ℕ x y) (min-ℕ x y) -dist-max-min-ℕ zero-ℕ zero-ℕ = refl -dist-max-min-ℕ zero-ℕ (succ-ℕ y) = refl -dist-max-min-ℕ (succ-ℕ x) zero-ℕ = refl -dist-max-min-ℕ (succ-ℕ x) (succ-ℕ y) = dist-max-min-ℕ x y +abstract + eq-max-dist-min-ℕ : (x y : ℕ) → dist-ℕ x y +ℕ min-ℕ x y ＝ max-ℕ x y + eq-max-dist-min-ℕ zero-ℕ y = refl + eq-max-dist-min-ℕ (succ-ℕ x) zero-ℕ = refl + eq-max-dist-min-ℕ (succ-ℕ x) (succ-ℕ y) = ap succ-ℕ (eq-max-dist-min-ℕ x y) + + dist-max-min-ℕ : (x y : ℕ) → dist-ℕ x y ＝ dist-ℕ (max-ℕ x y) (min-ℕ x y) + dist-max-min-ℕ zero-ℕ zero-ℕ = refl + dist-max-min-ℕ zero-ℕ (succ-ℕ y) = refl + dist-max-min-ℕ (succ-ℕ x) zero-ℕ = refl + dist-max-min-ℕ (succ-ℕ x) (succ-ℕ y) = dist-max-min-ℕ x y ``` diff --git a/src/elementary-number-theory/divisibility-integers.lagda.md b/src/elementary-number-theory/divisibility-integers.lagda.md index bed59408c3..2b38d5c502 100644 --- a/src/elementary-number-theory/divisibility-integers.lagda.md +++ b/src/elementary-number-theory/divisibility-integers.lagda.md @@ -67,32 +67,6 @@ div-ℤ d x = Σ ℤ (λ k → k *ℤ d ＝ x) quotient-div-ℤ : (x y : ℤ) → div-ℤ x y → ℤ quotient-div-ℤ x y H = pr1 H - -eq-quotient-div-ℤ : - (x y : ℤ) (H : div-ℤ x y) → (quotient-div-ℤ x y H) *ℤ x ＝ y -eq-quotient-div-ℤ x y H = pr2 H - -eq-quotient-div-ℤ' : - (x y : ℤ) (H : div-ℤ x y) → x *ℤ (quotient-div-ℤ x y H) ＝ y -eq-quotient-div-ℤ' x y H = - commutative-mul-ℤ x (quotient-div-ℤ x y H) ∙ eq-quotient-div-ℤ x y H - -div-quotient-div-ℤ : - (d x : ℤ) (H : div-ℤ d x) → div-ℤ (quotient-div-ℤ d x H) x -pr1 (div-quotient-div-ℤ d x (u , p)) = d -pr2 (div-quotient-div-ℤ d x (u , p)) = commutative-mul-ℤ d u ∙ p - -concatenate-eq-div-ℤ : - {x y z : ℤ} → x ＝ y → div-ℤ y z → div-ℤ x z -concatenate-eq-div-ℤ refl p = p - -concatenate-div-eq-ℤ : - {x y z : ℤ} → div-ℤ x y → y ＝ z → div-ℤ x z -concatenate-div-eq-ℤ p refl = p - -concatenate-eq-div-eq-ℤ : - {x y z w : ℤ} → x ＝ y → div-ℤ y z → z ＝ w → div-ℤ x w -concatenate-eq-div-eq-ℤ refl p refl = p ``` ### Unit integers @@ -132,11 +106,50 @@ sim-unit-ℤ x y = ¬ (is-zero-ℤ x × is-zero-ℤ y) → presim-unit-ℤ x y ## Properties +### If `d` divides `x`, the quotient times `d` equals `x` + +```agda +eq-quotient-div-ℤ : + (x y : ℤ) (H : div-ℤ x y) → (quotient-div-ℤ x y H) *ℤ x ＝ y +eq-quotient-div-ℤ x y H = pr2 H + +eq-quotient-div-ℤ' : + (x y : ℤ) (H : div-ℤ x y) → x *ℤ (quotient-div-ℤ x y H) ＝ y +eq-quotient-div-ℤ' x y H = + commutative-mul-ℤ x (quotient-div-ℤ x y H) ∙ eq-quotient-div-ℤ x y H +``` + +### If `d` divides `x`, the quotient also divides `x` + +```agda +div-quotient-div-ℤ : + (d x : ℤ) (H : div-ℤ d x) → div-ℤ (quotient-div-ℤ d x H) x +pr1 (div-quotient-div-ℤ d x (u , p)) = d +pr2 (div-quotient-div-ℤ d x (u , p)) = commutative-mul-ℤ d u ∙ p +``` + +### Concatenating equalities and divisibility + +```agda +concatenate-eq-div-ℤ : + {x y z : ℤ} → x ＝ y → div-ℤ y z → div-ℤ x z +concatenate-eq-div-ℤ refl p = p + +concatenate-div-eq-ℤ : + {x y z : ℤ} → div-ℤ x y → y ＝ z → div-ℤ x z +concatenate-div-eq-ℤ p refl = p + +concatenate-eq-div-eq-ℤ : + {x y z w : ℤ} → x ＝ y → div-ℤ y z → z ＝ w → div-ℤ x w +concatenate-eq-div-eq-ℤ refl p refl = p +``` + ### Divisibility by a nonzero integer is a property ```agda -is-prop-div-ℤ : (d x : ℤ) → is-nonzero-ℤ d → is-prop (div-ℤ d x) -is-prop-div-ℤ d x f = is-prop-map-is-emb (is-emb-right-mul-ℤ d f) x +abstract + is-prop-div-ℤ : (d x : ℤ) → is-nonzero-ℤ d → is-prop (div-ℤ d x) + is-prop-div-ℤ d x f = is-prop-map-is-emb (is-emb-right-mul-ℤ d f) x ``` ### The divisibility relation is a preorder @@ -144,16 +157,17 @@ is-prop-div-ℤ d x f = is-prop-map-is-emb (is-emb-right-mul-ℤ d f) x Note that the divisibility relation on the integers is not antisymmetric. ```agda -refl-div-ℤ : is-reflexive div-ℤ -pr1 (refl-div-ℤ x) = one-ℤ -pr2 (refl-div-ℤ x) = left-unit-law-mul-ℤ x - -transitive-div-ℤ : is-transitive div-ℤ -pr1 (transitive-div-ℤ x y z (pair e q) (pair d p)) = e *ℤ d -pr2 (transitive-div-ℤ x y z (pair e q) (pair d p)) = - ( associative-mul-ℤ e d x) ∙ - ( ( ap (e *ℤ_) p) ∙ - ( q)) +abstract + refl-div-ℤ : is-reflexive div-ℤ + pr1 (refl-div-ℤ x) = one-ℤ + pr2 (refl-div-ℤ x) = left-unit-law-mul-ℤ x + + transitive-div-ℤ : is-transitive div-ℤ + pr1 (transitive-div-ℤ x y z (pair e q) (pair d p)) = e *ℤ d + pr2 (transitive-div-ℤ x y z (pair e q) (pair d p)) = + ( associative-mul-ℤ e d x) ∙ + ( ( ap (e *ℤ_) p) ∙ + ( q)) ``` ### Every integer is divisible by `1` @@ -175,154 +189,165 @@ pr2 (div-zero-ℤ x) = left-zero-law-mul-ℤ x ### Every integer that is divisible by `0` is `0` ```agda -is-zero-div-zero-ℤ : - (x : ℤ) → div-ℤ zero-ℤ x → is-zero-ℤ x -is-zero-div-zero-ℤ x (pair d p) = inv p ∙ right-zero-law-mul-ℤ d +abstract + is-zero-div-zero-ℤ : + (x : ℤ) → div-ℤ zero-ℤ x → is-zero-ℤ x + is-zero-div-zero-ℤ x (pair d p) = inv p ∙ right-zero-law-mul-ℤ d ``` ### The quotient of `x` by one is `x` ```agda -eq-quotient-div-is-one-ℤ : - (k x : ℤ) → is-one-ℤ k → (H : div-ℤ k x) → quotient-div-ℤ k x H ＝ x -eq-quotient-div-is-one-ℤ .one-ℤ x refl H = - ap - ( quotient-div-ℤ one-ℤ x) - ( inv - ( eq-is-prop' - ( is-prop-div-ℤ one-ℤ x (λ ())) - ( div-one-ℤ x) - ( H))) +abstract + eq-quotient-div-is-one-ℤ : + (k x : ℤ) → is-one-ℤ k → (H : div-ℤ k x) → quotient-div-ℤ k x H ＝ x + eq-quotient-div-is-one-ℤ .one-ℤ x refl H = + ap + ( quotient-div-ℤ one-ℤ x) + ( inv + ( eq-is-prop' + ( is-prop-div-ℤ one-ℤ x (λ ())) + ( div-one-ℤ x) + ( H))) ``` ### If `k` divides `x` and `k` is `0` then `x` is `0` ```agda -is-zero-is-zero-div-ℤ : (x k : ℤ) → div-ℤ k x → is-zero-ℤ k → is-zero-ℤ x -is-zero-is-zero-div-ℤ x .zero-ℤ k-div-x refl = is-zero-div-zero-ℤ x k-div-x +abstract + is-zero-is-zero-div-ℤ : (x k : ℤ) → div-ℤ k x → is-zero-ℤ k → is-zero-ℤ x + is-zero-is-zero-div-ℤ x .zero-ℤ k-div-x refl = is-zero-div-zero-ℤ x k-div-x ``` ### If `x` divides both `y` and `z`, then it divides `y + z` ```agda -div-add-ℤ : (x y z : ℤ) → div-ℤ x y → div-ℤ x z → div-ℤ x (y +ℤ z) -pr1 (div-add-ℤ x y z (pair d p) (pair e q)) = d +ℤ e -pr2 (div-add-ℤ x y z (pair d p) (pair e q)) = - ( right-distributive-mul-add-ℤ d e x) ∙ - ( ap-add-ℤ p q) +abstract + div-add-ℤ : (x y z : ℤ) → div-ℤ x y → div-ℤ x z → div-ℤ x (y +ℤ z) + pr1 (div-add-ℤ x y z (pair d p) (pair e q)) = d +ℤ e + pr2 (div-add-ℤ x y z (pair d p) (pair e q)) = + ( right-distributive-mul-add-ℤ d e x) ∙ + ( ap-add-ℤ p q) ``` ### If `x` divides `y` then `x` divides any multiple of `y` ```agda -div-mul-ℤ : - (k x y : ℤ) → div-ℤ x y → div-ℤ x (k *ℤ y) -div-mul-ℤ k x y = transitive-div-ℤ x y (k *ℤ y) (k , refl) +abstract + div-mul-ℤ : + (k x y : ℤ) → div-ℤ x y → div-ℤ x (k *ℤ y) + div-mul-ℤ k x y = transitive-div-ℤ x y (k *ℤ y) (k , refl) ``` ### If `x` divides `y` then it divides `-y` ```agda -div-neg-ℤ : (x y : ℤ) → div-ℤ x y → div-ℤ x (neg-ℤ y) -pr1 (div-neg-ℤ x y (pair d p)) = neg-ℤ d -pr2 (div-neg-ℤ x y (pair d p)) = left-negative-law-mul-ℤ d x ∙ ap neg-ℤ p +abstract + div-neg-ℤ : (x y : ℤ) → div-ℤ x y → div-ℤ x (neg-ℤ y) + pr1 (div-neg-ℤ x y (pair d p)) = neg-ℤ d + pr2 (div-neg-ℤ x y (pair d p)) = left-negative-law-mul-ℤ d x ∙ ap neg-ℤ p ``` ### If `x` divides `y` then `-x` divides `y` ```agda -neg-div-ℤ : (x y : ℤ) → div-ℤ x y → div-ℤ (neg-ℤ x) y -pr1 (neg-div-ℤ x y (pair d p)) = neg-ℤ d -pr2 (neg-div-ℤ x y (pair d p)) = - equational-reasoning - (neg-ℤ d) *ℤ (neg-ℤ x) - ＝ neg-ℤ (d *ℤ (neg-ℤ x)) - by left-negative-law-mul-ℤ d (neg-ℤ x) - ＝ neg-ℤ (neg-ℤ (d *ℤ x)) - by ap neg-ℤ (right-negative-law-mul-ℤ d x) - ＝ (d *ℤ x) - by neg-neg-ℤ (d *ℤ x) - ＝ y - by p +abstract + neg-div-ℤ : (x y : ℤ) → div-ℤ x y → div-ℤ (neg-ℤ x) y + pr1 (neg-div-ℤ x y (pair d p)) = neg-ℤ d + pr2 (neg-div-ℤ x y (pair d p)) = + equational-reasoning + (neg-ℤ d) *ℤ (neg-ℤ x) + ＝ neg-ℤ (d *ℤ (neg-ℤ x)) + by left-negative-law-mul-ℤ d (neg-ℤ x) + ＝ neg-ℤ (neg-ℤ (d *ℤ x)) + by ap neg-ℤ (right-negative-law-mul-ℤ d x) + ＝ (d *ℤ x) + by neg-neg-ℤ (d *ℤ x) + ＝ y + by p ``` ### Multiplication preserves divisibility ```agda -preserves-div-mul-ℤ : - (k x y : ℤ) → div-ℤ x y → div-ℤ (k *ℤ x) (k *ℤ y) -pr1 (preserves-div-mul-ℤ k x y (pair q p)) = q -pr2 (preserves-div-mul-ℤ k x y (pair q p)) = - ( inv (associative-mul-ℤ q k x)) ∙ - ( ( ap (_*ℤ x) (commutative-mul-ℤ q k)) ∙ - ( ( associative-mul-ℤ k q x) ∙ - ( ap (k *ℤ_) p))) +abstract + preserves-div-mul-ℤ : + (k x y : ℤ) → div-ℤ x y → div-ℤ (k *ℤ x) (k *ℤ y) + pr1 (preserves-div-mul-ℤ k x y (pair q p)) = q + pr2 (preserves-div-mul-ℤ k x y (pair q p)) = + ( inv (associative-mul-ℤ q k x)) ∙ + ( ( ap (_*ℤ x) (commutative-mul-ℤ q k)) ∙ + ( ( associative-mul-ℤ k q x) ∙ + ( ap (k *ℤ_) p))) ``` ### Multiplication by a nonzero number reflects divisibility ```agda -reflects-div-mul-ℤ : - (k x y : ℤ) → is-nonzero-ℤ k → div-ℤ (k *ℤ x) (k *ℤ y) → div-ℤ x y -pr1 (reflects-div-mul-ℤ k x y H (pair q p)) = q -pr2 (reflects-div-mul-ℤ k x y H (pair q p)) = - is-injective-left-mul-ℤ k H - ( ( inv (associative-mul-ℤ k q x)) ∙ - ( ( ap (_*ℤ x) (commutative-mul-ℤ k q)) ∙ - ( ( associative-mul-ℤ q k x) ∙ - ( p)))) +abstract + reflects-div-mul-ℤ : + (k x y : ℤ) → is-nonzero-ℤ k → div-ℤ (k *ℤ x) (k *ℤ y) → div-ℤ x y + pr1 (reflects-div-mul-ℤ k x y H (pair q p)) = q + pr2 (reflects-div-mul-ℤ k x y H (pair q p)) = + is-injective-left-mul-ℤ k H + ( ( inv (associative-mul-ℤ k q x)) ∙ + ( ( ap (_*ℤ x) (commutative-mul-ℤ k q)) ∙ + ( ( associative-mul-ℤ q k x) ∙ + ( p)))) ``` ### If a nonzero number `d` divides `y`, then `dx` divides `y` if and only if `x` divides the quotient `y/d` ```agda -div-quotient-div-div-ℤ : - (x y d : ℤ) (H : div-ℤ d y) → is-nonzero-ℤ d → - div-ℤ (d *ℤ x) y → div-ℤ x (quotient-div-ℤ d y H) -div-quotient-div-div-ℤ x y d H f K = - reflects-div-mul-ℤ d x - ( quotient-div-ℤ d y H) - ( f) - ( tr (div-ℤ (d *ℤ x)) (inv (eq-quotient-div-ℤ' d y H)) K) - -div-div-quotient-div-ℤ : - (x y d : ℤ) (H : div-ℤ d y) → - div-ℤ x (quotient-div-ℤ d y H) → div-ℤ (d *ℤ x) y -div-div-quotient-div-ℤ x y d H K = - tr - ( div-ℤ (d *ℤ x)) - ( eq-quotient-div-ℤ' d y H) - ( preserves-div-mul-ℤ d x (quotient-div-ℤ d y H) K) +abstract + div-quotient-div-div-ℤ : + (x y d : ℤ) (H : div-ℤ d y) → is-nonzero-ℤ d → + div-ℤ (d *ℤ x) y → div-ℤ x (quotient-div-ℤ d y H) + div-quotient-div-div-ℤ x y d H f K = + reflects-div-mul-ℤ d x + ( quotient-div-ℤ d y H) + ( f) + ( tr (div-ℤ (d *ℤ x)) (inv (eq-quotient-div-ℤ' d y H)) K) + + div-div-quotient-div-ℤ : + (x y d : ℤ) (H : div-ℤ d y) → + div-ℤ x (quotient-div-ℤ d y H) → div-ℤ (d *ℤ x) y + div-div-quotient-div-ℤ x y d H K = + tr + ( div-ℤ (d *ℤ x)) + ( eq-quotient-div-ℤ' d y H) + ( preserves-div-mul-ℤ d x (quotient-div-ℤ d y H) K) ``` ### Comparison of divisibility on `ℕ` and on `ℤ` ```agda -div-int-div-ℕ : - {x y : ℕ} → div-ℕ x y → div-ℤ (int-ℕ x) (int-ℕ y) -pr1 (div-int-div-ℕ {x} {y} (pair d p)) = int-ℕ d -pr2 (div-int-div-ℕ {x} {y} (pair d p)) = mul-int-ℕ d x ∙ ap int-ℕ p - -div-div-int-ℕ : - {x y : ℕ} → div-ℤ (int-ℕ x) (int-ℕ y) → div-ℕ x y -div-div-int-ℕ {zero-ℕ} {y} (pair d p) = - div-eq-ℕ zero-ℕ y - ( inv (is-injective-int-ℕ (is-zero-div-zero-ℤ (int-ℕ y) (pair d p)))) -pr1 (div-div-int-ℕ {succ-ℕ x} {y} (pair d p)) = abs-ℤ d -pr2 (div-div-int-ℕ {succ-ℕ x} {y} (pair d p)) = - is-injective-int-ℕ - ( ( inv (mul-int-ℕ (abs-ℤ d) (succ-ℕ x))) ∙ - ( ( ap - ( _*ℤ (inr (inr x))) - { int-abs-ℤ d} - { d} - ( int-abs-is-nonnegative-ℤ d - ( is-nonnegative-left-factor-mul-ℤ - { d} - { inr (inr x)} - ( is-nonnegative-eq-ℤ (inv p) (is-nonnegative-int-ℕ y)) - ( star)))) ∙ - ( p))) +abstract + div-int-div-ℕ : + {x y : ℕ} → div-ℕ x y → div-ℤ (int-ℕ x) (int-ℕ y) + pr1 (div-int-div-ℕ {x} {y} (pair d p)) = int-ℕ d + pr2 (div-int-div-ℕ {x} {y} (pair d p)) = mul-int-ℕ d x ∙ ap int-ℕ p + + div-div-int-ℕ : + {x y : ℕ} → div-ℤ (int-ℕ x) (int-ℕ y) → div-ℕ x y + div-div-int-ℕ {zero-ℕ} {y} (pair d p) = + div-eq-ℕ zero-ℕ y + ( inv (is-injective-int-ℕ (is-zero-div-zero-ℤ (int-ℕ y) (pair d p)))) + pr1 (div-div-int-ℕ {succ-ℕ x} {y} (pair d p)) = abs-ℤ d + pr2 (div-div-int-ℕ {succ-ℕ x} {y} (pair d p)) = + is-injective-int-ℕ + ( ( inv (mul-int-ℕ (abs-ℤ d) (succ-ℕ x))) ∙ + ( ( ap + ( _*ℤ (inr (inr x))) + { int-abs-ℤ d} + { d} + ( int-abs-is-nonnegative-ℤ d + ( is-nonnegative-left-factor-mul-ℤ + { d} + { inr (inr x)} + ( is-nonnegative-eq-ℤ (inv p) (is-nonnegative-int-ℕ y)) + ( star)))) ∙ + ( p))) ``` ### An integer is a unit if and only if it is `1` or `-1` @@ -359,51 +384,53 @@ is-unit-is-one-or-neg-one-ℤ : is-unit-is-one-or-neg-one-ℤ x (inl p) = is-unit-is-one-ℤ x p is-unit-is-one-or-neg-one-ℤ x (inr p) = is-unit-is-neg-one-ℤ x p -is-one-or-neg-one-is-unit-ℤ : - (x : ℤ) → is-unit-ℤ x → is-one-or-neg-one-ℤ x -is-one-or-neg-one-is-unit-ℤ (inl zero-ℕ) (pair d p) = inr refl -is-one-or-neg-one-is-unit-ℤ (inl (succ-ℕ x)) (pair (inl zero-ℕ) p) = - ex-falso (Eq-eq-ℤ (inv p ∙ compute-mul-ℤ neg-one-ℤ (inl (succ-ℕ x)))) -is-one-or-neg-one-is-unit-ℤ (inl (succ-ℕ x)) (pair (inl (succ-ℕ d)) p) = - ex-falso (Eq-eq-ℤ (inv p ∙ compute-mul-ℤ (inl (succ-ℕ d)) (inl (succ-ℕ x)))) -is-one-or-neg-one-is-unit-ℤ (inl (succ-ℕ x)) (pair (inr (inl star)) p) = - ex-falso (Eq-eq-ℤ (inv p ∙ compute-mul-ℤ zero-ℤ (inl (succ-ℕ x)))) -is-one-or-neg-one-is-unit-ℤ (inl (succ-ℕ x)) (pair (inr (inr zero-ℕ)) p) = - ex-falso (Eq-eq-ℤ (inv p ∙ compute-mul-ℤ one-ℤ (inl (succ-ℕ x)))) -is-one-or-neg-one-is-unit-ℤ (inl (succ-ℕ x)) (pair (inr (inr (succ-ℕ d))) p) = - ex-falso - ( Eq-eq-ℤ (inv p ∙ compute-mul-ℤ (inr (inr (succ-ℕ d))) (inl (succ-ℕ x)))) -is-one-or-neg-one-is-unit-ℤ (inr (inl star)) (pair d p) = - ex-falso (Eq-eq-ℤ (inv (right-zero-law-mul-ℤ d) ∙ p)) -is-one-or-neg-one-is-unit-ℤ (inr (inr zero-ℕ)) (pair d p) = inl refl -is-one-or-neg-one-is-unit-ℤ (inr (inr (succ-ℕ x))) (pair (inl zero-ℕ) p) = - ex-falso (Eq-eq-ℤ (inv p ∙ compute-mul-ℤ neg-one-ℤ (inr (inr (succ-ℕ x))))) -is-one-or-neg-one-is-unit-ℤ (inr (inr (succ-ℕ x))) (pair (inl (succ-ℕ d)) p) = - ex-falso - ( Eq-eq-ℤ (inv p ∙ compute-mul-ℤ (inl (succ-ℕ d)) (inr (inr (succ-ℕ x))))) -is-one-or-neg-one-is-unit-ℤ (inr (inr (succ-ℕ x))) (pair (inr (inl star)) p) = - ex-falso (Eq-eq-ℤ (inv p ∙ compute-mul-ℤ zero-ℤ (inr (inr (succ-ℕ x))))) -is-one-or-neg-one-is-unit-ℤ (inr (inr (succ-ℕ x))) (pair (inr (inr zero-ℕ)) p) = - ex-falso (Eq-eq-ℤ (inv p ∙ compute-mul-ℤ one-ℤ (inr (inr (succ-ℕ x))))) -is-one-or-neg-one-is-unit-ℤ - (inr (inr (succ-ℕ x))) (pair (inr (inr (succ-ℕ d))) p) = - ex-falso - ( Eq-eq-ℤ - ( inv p ∙ compute-mul-ℤ (inr (inr (succ-ℕ d))) (inr (inr (succ-ℕ x))))) +abstract + is-one-or-neg-one-is-unit-ℤ : + (x : ℤ) → is-unit-ℤ x → is-one-or-neg-one-ℤ x + is-one-or-neg-one-is-unit-ℤ (inl zero-ℕ) (pair d p) = inr refl + is-one-or-neg-one-is-unit-ℤ (inl (succ-ℕ x)) (pair (inl zero-ℕ) p) = + ex-falso (Eq-eq-ℤ (inv p ∙ compute-mul-ℤ neg-one-ℤ (inl (succ-ℕ x)))) + is-one-or-neg-one-is-unit-ℤ (inl (succ-ℕ x)) (pair (inl (succ-ℕ d)) p) = + ex-falso (Eq-eq-ℤ (inv p ∙ compute-mul-ℤ (inl (succ-ℕ d)) (inl (succ-ℕ x)))) + is-one-or-neg-one-is-unit-ℤ (inl (succ-ℕ x)) (pair (inr (inl star)) p) = + ex-falso (Eq-eq-ℤ (inv p ∙ compute-mul-ℤ zero-ℤ (inl (succ-ℕ x)))) + is-one-or-neg-one-is-unit-ℤ (inl (succ-ℕ x)) (pair (inr (inr zero-ℕ)) p) = + ex-falso (Eq-eq-ℤ (inv p ∙ compute-mul-ℤ one-ℤ (inl (succ-ℕ x)))) + is-one-or-neg-one-is-unit-ℤ (inl (succ-ℕ x)) (pair (inr (inr (succ-ℕ d))) p) = + ex-falso + ( Eq-eq-ℤ (inv p ∙ compute-mul-ℤ (inr (inr (succ-ℕ d))) (inl (succ-ℕ x)))) + is-one-or-neg-one-is-unit-ℤ (inr (inl star)) (pair d p) = + ex-falso (Eq-eq-ℤ (inv (right-zero-law-mul-ℤ d) ∙ p)) + is-one-or-neg-one-is-unit-ℤ (inr (inr zero-ℕ)) (pair d p) = inl refl + is-one-or-neg-one-is-unit-ℤ (inr (inr (succ-ℕ x))) (pair (inl zero-ℕ) p) = + ex-falso (Eq-eq-ℤ (inv p ∙ compute-mul-ℤ neg-one-ℤ (inr (inr (succ-ℕ x))))) + is-one-or-neg-one-is-unit-ℤ (inr (inr (succ-ℕ x))) (pair (inl (succ-ℕ d)) p) = + ex-falso + ( Eq-eq-ℤ (inv p ∙ compute-mul-ℤ (inl (succ-ℕ d)) (inr (inr (succ-ℕ x))))) + is-one-or-neg-one-is-unit-ℤ (inr (inr (succ-ℕ x))) (pair (inr (inl star)) p) = + ex-falso (Eq-eq-ℤ (inv p ∙ compute-mul-ℤ zero-ℤ (inr (inr (succ-ℕ x))))) + is-one-or-neg-one-is-unit-ℤ + (inr (inr (succ-ℕ x))) (pair (inr (inr zero-ℕ)) p) = + ex-falso (Eq-eq-ℤ (inv p ∙ compute-mul-ℤ one-ℤ (inr (inr (succ-ℕ x))))) + is-one-or-neg-one-is-unit-ℤ + (inr (inr (succ-ℕ x))) (pair (inr (inr (succ-ℕ d))) p) = + ex-falso + ( Eq-eq-ℤ + ( inv p ∙ compute-mul-ℤ (inr (inr (succ-ℕ d))) (inr (inr (succ-ℕ x))))) ``` ### Units are idempotent ```agda -idempotent-is-unit-ℤ : {x : ℤ} → is-unit-ℤ x → x *ℤ x ＝ one-ℤ -idempotent-is-unit-ℤ {x} H = - f (is-one-or-neg-one-is-unit-ℤ x H) - where - f : is-one-or-neg-one-ℤ x → x *ℤ x ＝ one-ℤ - f (inl refl) = refl - f (inr refl) = refl - abstract + idempotent-is-unit-ℤ : {x : ℤ} → is-unit-ℤ x → x *ℤ x ＝ one-ℤ + idempotent-is-unit-ℤ {x} H = + f (is-one-or-neg-one-is-unit-ℤ x H) + where + f : is-one-or-neg-one-ℤ x → x *ℤ x ＝ one-ℤ + f (inl refl) = refl + f (inr refl) = refl + is-one-is-unit-int-ℕ : (x : ℕ) → is-unit-ℤ (int-ℕ x) → is-one-ℕ x is-one-is-unit-int-ℕ x H with is-one-or-neg-one-is-unit-ℤ (int-ℕ x) H ... | inl p = is-injective-int-ℕ p @@ -413,263 +440,274 @@ abstract ### The product `xy` is a unit if and only if both `x` and `y` are units ```agda -is-unit-mul-ℤ : - (x y : ℤ) → is-unit-ℤ x → is-unit-ℤ y → is-unit-ℤ (x *ℤ y) -pr1 (is-unit-mul-ℤ x y (pair d p) (pair e q)) = e *ℤ d -pr2 (is-unit-mul-ℤ x y (pair d p) (pair e q)) = - ( associative-mul-ℤ e d (x *ℤ y)) ∙ - ( ( ap - ( e *ℤ_) - ( ( inv (associative-mul-ℤ d x y)) ∙ - ( ap (_*ℤ y) p))) ∙ - ( q)) +abstract + is-unit-mul-ℤ : + (x y : ℤ) → is-unit-ℤ x → is-unit-ℤ y → is-unit-ℤ (x *ℤ y) + pr1 (is-unit-mul-ℤ x y (pair d p) (pair e q)) = e *ℤ d + pr2 (is-unit-mul-ℤ x y (pair d p) (pair e q)) = + ( associative-mul-ℤ e d (x *ℤ y)) ∙ + ( ( ap + ( e *ℤ_) + ( ( inv (associative-mul-ℤ d x y)) ∙ + ( ap (_*ℤ y) p))) ∙ + ( q)) mul-unit-ℤ : unit-ℤ → unit-ℤ → unit-ℤ pr1 (mul-unit-ℤ (pair x H) (pair y K)) = x *ℤ y pr2 (mul-unit-ℤ (pair x H) (pair y K)) = is-unit-mul-ℤ x y H K -is-unit-left-factor-mul-ℤ : - (x y : ℤ) → is-unit-ℤ (x *ℤ y) → is-unit-ℤ x -pr1 (is-unit-left-factor-mul-ℤ x y (pair d p)) = d *ℤ y -pr2 (is-unit-left-factor-mul-ℤ x y (pair d p)) = - associative-mul-ℤ d y x ∙ (ap (d *ℤ_) (commutative-mul-ℤ y x) ∙ p) - -is-unit-right-factor-ℤ : - (x y : ℤ) → is-unit-ℤ (x *ℤ y) → is-unit-ℤ y -is-unit-right-factor-ℤ x y (pair d p) = - is-unit-left-factor-mul-ℤ y x - ( pair d (ap (d *ℤ_) (commutative-mul-ℤ y x) ∙ p)) +abstract + is-unit-left-factor-mul-ℤ : + (x y : ℤ) → is-unit-ℤ (x *ℤ y) → is-unit-ℤ x + pr1 (is-unit-left-factor-mul-ℤ x y (pair d p)) = d *ℤ y + pr2 (is-unit-left-factor-mul-ℤ x y (pair d p)) = + associative-mul-ℤ d y x ∙ (ap (d *ℤ_) (commutative-mul-ℤ y x) ∙ p) + + is-unit-right-factor-ℤ : + (x y : ℤ) → is-unit-ℤ (x *ℤ y) → is-unit-ℤ y + is-unit-right-factor-ℤ x y (pair d p) = + is-unit-left-factor-mul-ℤ y x + ( pair d (ap (d *ℤ_) (commutative-mul-ℤ y x) ∙ p)) ``` ### The relations `presim-unit-ℤ` and `sim-unit-ℤ` are logically equivalent ```agda -sim-unit-presim-unit-ℤ : - {x y : ℤ} → presim-unit-ℤ x y → sim-unit-ℤ x y -sim-unit-presim-unit-ℤ {x} {y} H f = H - -presim-unit-sim-unit-ℤ : - {x y : ℤ} → sim-unit-ℤ x y → presim-unit-ℤ x y -presim-unit-sim-unit-ℤ {inl x} {inl y} H = H (λ t → Eq-eq-ℤ (pr1 t)) -presim-unit-sim-unit-ℤ {inl x} {inr y} H = H (λ t → Eq-eq-ℤ (pr1 t)) -presim-unit-sim-unit-ℤ {inr x} {inl y} H = H (λ t → Eq-eq-ℤ (pr2 t)) -pr1 (presim-unit-sim-unit-ℤ {inr (inl star)} {inr (inl star)} H) = one-unit-ℤ -pr2 (presim-unit-sim-unit-ℤ {inr (inl star)} {inr (inl star)} H) = refl -presim-unit-sim-unit-ℤ {inr (inl star)} {inr (inr y)} H = - H (λ t → Eq-eq-ℤ (pr2 t)) -presim-unit-sim-unit-ℤ {inr (inr x)} {inr (inl star)} H = - H (λ t → Eq-eq-ℤ (pr1 t)) -presim-unit-sim-unit-ℤ {inr (inr x)} {inr (inr y)} H = - H (λ t → Eq-eq-ℤ (pr1 t)) +abstract + sim-unit-presim-unit-ℤ : + {x y : ℤ} → presim-unit-ℤ x y → sim-unit-ℤ x y + sim-unit-presim-unit-ℤ {x} {y} H f = H + + presim-unit-sim-unit-ℤ : + {x y : ℤ} → sim-unit-ℤ x y → presim-unit-ℤ x y + presim-unit-sim-unit-ℤ {inl x} {inl y} H = H (λ t → Eq-eq-ℤ (pr1 t)) + presim-unit-sim-unit-ℤ {inl x} {inr y} H = H (λ t → Eq-eq-ℤ (pr1 t)) + presim-unit-sim-unit-ℤ {inr x} {inl y} H = H (λ t → Eq-eq-ℤ (pr2 t)) + pr1 (presim-unit-sim-unit-ℤ {inr (inl star)} {inr (inl star)} H) = one-unit-ℤ + pr2 (presim-unit-sim-unit-ℤ {inr (inl star)} {inr (inl star)} H) = refl + presim-unit-sim-unit-ℤ {inr (inl star)} {inr (inr y)} H = + H (λ t → Eq-eq-ℤ (pr2 t)) + presim-unit-sim-unit-ℤ {inr (inr x)} {inr (inl star)} H = + H (λ t → Eq-eq-ℤ (pr1 t)) + presim-unit-sim-unit-ℤ {inr (inr x)} {inr (inr y)} H = + H (λ t → Eq-eq-ℤ (pr1 t)) ``` ### The relations `presim-unit-ℤ` and `sim-unit-ℤ` relate `zero-ℤ` only to itself ```agda -is-nonzero-presim-unit-ℤ : - {x y : ℤ} → presim-unit-ℤ x y → is-nonzero-ℤ x → is-nonzero-ℤ y -is-nonzero-presim-unit-ℤ {x} {y} (pair (pair v (pair u α)) β) f p = - Eq-eq-ℤ (ap (_*ℤ u) (inv q) ∙ (commutative-mul-ℤ v u ∙ α)) - where - q : is-zero-ℤ v - q = is-injective-right-mul-ℤ x f {v} {zero-ℤ} (β ∙ p) - -is-nonzero-sim-unit-ℤ : - {x y : ℤ} → sim-unit-ℤ x y → is-nonzero-ℤ x → is-nonzero-ℤ y -is-nonzero-sim-unit-ℤ H f = - is-nonzero-presim-unit-ℤ (H (f ∘ pr1)) f - -is-zero-sim-unit-ℤ : - {x y : ℤ} → sim-unit-ℤ x y → is-zero-ℤ x → is-zero-ℤ y -is-zero-sim-unit-ℤ {x} {y} H p = - double-negation-elim-is-decidable - ( has-decidable-equality-ℤ y zero-ℤ) - ( λ g → g (inv (β g) ∙ (ap ((u g) *ℤ_) p ∙ right-zero-law-mul-ℤ (u g)))) - where - K : is-nonzero-ℤ y → presim-unit-ℤ x y - K g = H (λ (u , v) → g v) - u : is-nonzero-ℤ y → ℤ - u g = pr1 (pr1 (K g)) - v : is-nonzero-ℤ y → ℤ - v g = pr1 (pr2 (pr1 (K g))) - β : (g : is-nonzero-ℤ y) → (u g) *ℤ x ＝ y - β g = pr2 (K g) +abstract + is-nonzero-presim-unit-ℤ : + {x y : ℤ} → presim-unit-ℤ x y → is-nonzero-ℤ x → is-nonzero-ℤ y + is-nonzero-presim-unit-ℤ {x} {y} (pair (pair v (pair u α)) β) f p = + Eq-eq-ℤ (ap (_*ℤ u) (inv q) ∙ (commutative-mul-ℤ v u ∙ α)) + where + q : is-zero-ℤ v + q = is-injective-right-mul-ℤ x f {v} {zero-ℤ} (β ∙ p) + + is-nonzero-sim-unit-ℤ : + {x y : ℤ} → sim-unit-ℤ x y → is-nonzero-ℤ x → is-nonzero-ℤ y + is-nonzero-sim-unit-ℤ H f = + is-nonzero-presim-unit-ℤ (H (f ∘ pr1)) f + + is-zero-sim-unit-ℤ : + {x y : ℤ} → sim-unit-ℤ x y → is-zero-ℤ x → is-zero-ℤ y + is-zero-sim-unit-ℤ {x} {y} H p = + double-negation-elim-is-decidable + ( has-decidable-equality-ℤ y zero-ℤ) + ( λ g → g (inv (β g) ∙ (ap ((u g) *ℤ_) p ∙ right-zero-law-mul-ℤ (u g)))) + where + K : is-nonzero-ℤ y → presim-unit-ℤ x y + K g = H (λ (u , v) → g v) + u : is-nonzero-ℤ y → ℤ + u g = pr1 (pr1 (K g)) + v : is-nonzero-ℤ y → ℤ + v g = pr1 (pr2 (pr1 (K g))) + β : (g : is-nonzero-ℤ y) → (u g) *ℤ x ＝ y + β g = pr2 (K g) ``` ### The relations `presim-unit-ℤ` and `sim-unit-ℤ` are equivalence relations ```agda -refl-presim-unit-ℤ : is-reflexive presim-unit-ℤ -pr1 (refl-presim-unit-ℤ x) = one-unit-ℤ -pr2 (refl-presim-unit-ℤ x) = left-unit-law-mul-ℤ x - -refl-sim-unit-ℤ : is-reflexive sim-unit-ℤ -refl-sim-unit-ℤ x f = refl-presim-unit-ℤ x - -presim-unit-eq-ℤ : {x y : ℤ} → x ＝ y → presim-unit-ℤ x y -presim-unit-eq-ℤ {x} refl = refl-presim-unit-ℤ x - -sim-unit-eq-ℤ : {x y : ℤ} → x ＝ y → sim-unit-ℤ x y -sim-unit-eq-ℤ {x} refl = refl-sim-unit-ℤ x - -symmetric-presim-unit-ℤ : is-symmetric presim-unit-ℤ -symmetric-presim-unit-ℤ x y (pair (pair u H) p) = - f (is-one-or-neg-one-is-unit-ℤ u H) - where - f : is-one-or-neg-one-ℤ u → presim-unit-ℤ y x - pr1 (f (inl refl)) = one-unit-ℤ - pr2 (f (inl refl)) = inv p - pr1 (f (inr refl)) = neg-one-unit-ℤ - pr2 (f (inr refl)) = inv (inv (neg-neg-ℤ x) ∙ ap (neg-one-ℤ *ℤ_) p) - -symmetric-sim-unit-ℤ : is-symmetric sim-unit-ℤ -symmetric-sim-unit-ℤ x y H f = - symmetric-presim-unit-ℤ x y (H (λ p → f (pair (pr2 p) (pr1 p)))) - -is-nonzero-sim-unit-ℤ' : - {x y : ℤ} → sim-unit-ℤ x y → is-nonzero-ℤ y → is-nonzero-ℤ x -is-nonzero-sim-unit-ℤ' {x} {y} H = - is-nonzero-sim-unit-ℤ (symmetric-sim-unit-ℤ x y H) - -is-zero-sim-unit-ℤ' : - {x y : ℤ} → sim-unit-ℤ x y → is-zero-ℤ y → is-zero-ℤ x -is-zero-sim-unit-ℤ' {x} {y} H = is-zero-sim-unit-ℤ (symmetric-sim-unit-ℤ x y H) - -transitive-presim-unit-ℤ : is-transitive presim-unit-ℤ -transitive-presim-unit-ℤ x y z (pair (pair v K) q) (pair (pair u H) p) = - f (is-one-or-neg-one-is-unit-ℤ u H) (is-one-or-neg-one-is-unit-ℤ v K) - where - f : is-one-or-neg-one-ℤ u → is-one-or-neg-one-ℤ v → presim-unit-ℤ x z - pr1 (f (inl refl) (inl refl)) = one-unit-ℤ - pr2 (f (inl refl) (inl refl)) = p ∙ q - pr1 (f (inl refl) (inr refl)) = neg-one-unit-ℤ - pr2 (f (inl refl) (inr refl)) = ap neg-ℤ p ∙ q - pr1 (f (inr refl) (inl refl)) = neg-one-unit-ℤ - pr2 (f (inr refl) (inl refl)) = p ∙ q - pr1 (f (inr refl) (inr refl)) = one-unit-ℤ - pr2 (f (inr refl) (inr refl)) = inv (neg-neg-ℤ x) ∙ (ap neg-ℤ p ∙ q) - -transitive-sim-unit-ℤ : is-transitive sim-unit-ℤ -transitive-sim-unit-ℤ x y z K H f = - transitive-presim-unit-ℤ x y z - ( K (λ (p , q) → f (is-zero-sim-unit-ℤ' H p , q))) - ( H (λ (p , q) → f (p , is-zero-sim-unit-ℤ K q))) +abstract + refl-presim-unit-ℤ : is-reflexive presim-unit-ℤ + pr1 (refl-presim-unit-ℤ x) = one-unit-ℤ + pr2 (refl-presim-unit-ℤ x) = left-unit-law-mul-ℤ x + + refl-sim-unit-ℤ : is-reflexive sim-unit-ℤ + refl-sim-unit-ℤ x f = refl-presim-unit-ℤ x + + presim-unit-eq-ℤ : {x y : ℤ} → x ＝ y → presim-unit-ℤ x y + presim-unit-eq-ℤ {x} refl = refl-presim-unit-ℤ x + + sim-unit-eq-ℤ : {x y : ℤ} → x ＝ y → sim-unit-ℤ x y + sim-unit-eq-ℤ {x} refl = refl-sim-unit-ℤ x + + symmetric-presim-unit-ℤ : is-symmetric presim-unit-ℤ + symmetric-presim-unit-ℤ x y (pair (pair u H) p) = + f (is-one-or-neg-one-is-unit-ℤ u H) + where + f : is-one-or-neg-one-ℤ u → presim-unit-ℤ y x + pr1 (f (inl refl)) = one-unit-ℤ + pr2 (f (inl refl)) = inv p + pr1 (f (inr refl)) = neg-one-unit-ℤ + pr2 (f (inr refl)) = inv (inv (neg-neg-ℤ x) ∙ ap (neg-one-ℤ *ℤ_) p) + + symmetric-sim-unit-ℤ : is-symmetric sim-unit-ℤ + symmetric-sim-unit-ℤ x y H f = + symmetric-presim-unit-ℤ x y (H (λ p → f (pair (pr2 p) (pr1 p)))) + + is-nonzero-sim-unit-ℤ' : + {x y : ℤ} → sim-unit-ℤ x y → is-nonzero-ℤ y → is-nonzero-ℤ x + is-nonzero-sim-unit-ℤ' {x} {y} H = + is-nonzero-sim-unit-ℤ (symmetric-sim-unit-ℤ x y H) + + is-zero-sim-unit-ℤ' : + {x y : ℤ} → sim-unit-ℤ x y → is-zero-ℤ y → is-zero-ℤ x + is-zero-sim-unit-ℤ' {x} {y} H = + is-zero-sim-unit-ℤ (symmetric-sim-unit-ℤ x y H) + + transitive-presim-unit-ℤ : is-transitive presim-unit-ℤ + transitive-presim-unit-ℤ x y z (pair (pair v K) q) (pair (pair u H) p) = + f (is-one-or-neg-one-is-unit-ℤ u H) (is-one-or-neg-one-is-unit-ℤ v K) + where + f : is-one-or-neg-one-ℤ u → is-one-or-neg-one-ℤ v → presim-unit-ℤ x z + pr1 (f (inl refl) (inl refl)) = one-unit-ℤ + pr2 (f (inl refl) (inl refl)) = p ∙ q + pr1 (f (inl refl) (inr refl)) = neg-one-unit-ℤ + pr2 (f (inl refl) (inr refl)) = ap neg-ℤ p ∙ q + pr1 (f (inr refl) (inl refl)) = neg-one-unit-ℤ + pr2 (f (inr refl) (inl refl)) = p ∙ q + pr1 (f (inr refl) (inr refl)) = one-unit-ℤ + pr2 (f (inr refl) (inr refl)) = inv (neg-neg-ℤ x) ∙ (ap neg-ℤ p ∙ q) + + transitive-sim-unit-ℤ : is-transitive sim-unit-ℤ + transitive-sim-unit-ℤ x y z K H f = + transitive-presim-unit-ℤ x y z + ( K (λ (p , q) → f (is-zero-sim-unit-ℤ' H p , q))) + ( H (λ (p , q) → f (p , is-zero-sim-unit-ℤ K q))) ``` ### `sim-unit-ℤ x y` holds if and only if `x|y` and `y|x` ```agda -antisymmetric-div-ℤ : - (x y : ℤ) → div-ℤ x y → div-ℤ y x → sim-unit-ℤ x y -antisymmetric-div-ℤ x y (pair d p) (pair e q) H = - f (is-decidable-is-zero-ℤ x) - where - f : is-decidable (is-zero-ℤ x) → presim-unit-ℤ x y - f (inl refl) = presim-unit-eq-ℤ (inv (right-zero-law-mul-ℤ d) ∙ p) - pr1 (pr1 (f (inr g))) = d - pr1 (pr2 (pr1 (f (inr g)))) = e - pr2 (pr2 (pr1 (f (inr g)))) = - is-injective-left-mul-ℤ x g - ( ( commutative-mul-ℤ x (e *ℤ d)) ∙ - ( ( associative-mul-ℤ e d x) ∙ - ( ( ap (e *ℤ_) p) ∙ - ( q ∙ inv (right-unit-law-mul-ℤ x))))) - pr2 (f (inr g)) = p +abstract + antisymmetric-div-ℤ : + (x y : ℤ) → div-ℤ x y → div-ℤ y x → sim-unit-ℤ x y + antisymmetric-div-ℤ x y (pair d p) (pair e q) H = + f (is-decidable-is-zero-ℤ x) + where + f : is-decidable (is-zero-ℤ x) → presim-unit-ℤ x y + f (inl refl) = presim-unit-eq-ℤ (inv (right-zero-law-mul-ℤ d) ∙ p) + pr1 (pr1 (f (inr g))) = d + pr1 (pr2 (pr1 (f (inr g)))) = e + pr2 (pr2 (pr1 (f (inr g)))) = + is-injective-left-mul-ℤ x g + ( ( commutative-mul-ℤ x (e *ℤ d)) ∙ + ( ( associative-mul-ℤ e d x) ∙ + ( ( ap (e *ℤ_) p) ∙ + ( q ∙ inv (right-unit-law-mul-ℤ x))))) + pr2 (f (inr g)) = p ``` ### `sim-unit-ℤ |x| x` holds ```agda -sim-unit-abs-ℤ : (x : ℤ) → sim-unit-ℤ (int-abs-ℤ x) x -pr1 (sim-unit-abs-ℤ (inl x) f) = neg-one-unit-ℤ -pr2 (sim-unit-abs-ℤ (inl x) f) = refl -sim-unit-abs-ℤ (inr (inl star)) = refl-sim-unit-ℤ zero-ℤ -sim-unit-abs-ℤ (inr (inr x)) = refl-sim-unit-ℤ (inr (inr x)) - -div-presim-unit-ℤ : - {x y x' y' : ℤ} → presim-unit-ℤ x x' → presim-unit-ℤ y y' → - div-ℤ x y → div-ℤ x' y' -pr1 (div-presim-unit-ℤ {x} {y} {x'} {y'} (pair u q) (pair v r) (pair d p)) = - ((int-unit-ℤ v) *ℤ d) *ℤ (int-unit-ℤ u) -pr2 (div-presim-unit-ℤ {x} {y} {x'} {y'} (pair u q) (pair v r) (pair d p)) = - ( ap ((((int-unit-ℤ v) *ℤ d) *ℤ (int-unit-ℤ u)) *ℤ_) (inv q)) ∙ - ( ( associative-mul-ℤ - ( (int-unit-ℤ v) *ℤ d) - ( int-unit-ℤ u) - ( (int-unit-ℤ u) *ℤ x)) ∙ - ( ( ap - ( ((int-unit-ℤ v) *ℤ d) *ℤ_) - ( ( inv (associative-mul-ℤ (int-unit-ℤ u) (int-unit-ℤ u) x)) ∙ - ( ap (_*ℤ x) (idempotent-is-unit-ℤ (is-unit-int-unit-ℤ u))))) ∙ - ( ( associative-mul-ℤ (int-unit-ℤ v) d x) ∙ - ( ( ap ((int-unit-ℤ v) *ℤ_) p) ∙ - ( r))))) - -div-sim-unit-ℤ : - {x y x' y' : ℤ} → sim-unit-ℤ x x' → sim-unit-ℤ y y' → - div-ℤ x y → div-ℤ x' y' -div-sim-unit-ℤ {x} {y} {x'} {y'} H K = - div-presim-unit-ℤ (presim-unit-sim-unit-ℤ H) (presim-unit-sim-unit-ℤ K) - -div-int-abs-div-ℤ : - {x y : ℤ} → div-ℤ x y → div-ℤ (int-abs-ℤ x) y -div-int-abs-div-ℤ {x} {y} = - div-sim-unit-ℤ - ( symmetric-sim-unit-ℤ (int-abs-ℤ x) x (sim-unit-abs-ℤ x)) - ( refl-sim-unit-ℤ y) - -div-div-int-abs-ℤ : - {x y : ℤ} → div-ℤ (int-abs-ℤ x) y → div-ℤ x y -div-div-int-abs-ℤ {x} {y} = - div-sim-unit-ℤ (sim-unit-abs-ℤ x) (refl-sim-unit-ℤ y) +abstract + sim-unit-abs-ℤ : (x : ℤ) → sim-unit-ℤ (int-abs-ℤ x) x + pr1 (sim-unit-abs-ℤ (inl x) f) = neg-one-unit-ℤ + pr2 (sim-unit-abs-ℤ (inl x) f) = refl + sim-unit-abs-ℤ (inr (inl star)) = refl-sim-unit-ℤ zero-ℤ + sim-unit-abs-ℤ (inr (inr x)) = refl-sim-unit-ℤ (inr (inr x)) + + div-presim-unit-ℤ : + {x y x' y' : ℤ} → presim-unit-ℤ x x' → presim-unit-ℤ y y' → + div-ℤ x y → div-ℤ x' y' + pr1 (div-presim-unit-ℤ {x} {y} {x'} {y'} (pair u q) (pair v r) (pair d p)) = + ((int-unit-ℤ v) *ℤ d) *ℤ (int-unit-ℤ u) + pr2 (div-presim-unit-ℤ {x} {y} {x'} {y'} (pair u q) (pair v r) (pair d p)) = + ( ap ((((int-unit-ℤ v) *ℤ d) *ℤ (int-unit-ℤ u)) *ℤ_) (inv q)) ∙ + ( ( associative-mul-ℤ + ( (int-unit-ℤ v) *ℤ d) + ( int-unit-ℤ u) + ( (int-unit-ℤ u) *ℤ x)) ∙ + ( ( ap + ( ((int-unit-ℤ v) *ℤ d) *ℤ_) + ( ( inv (associative-mul-ℤ (int-unit-ℤ u) (int-unit-ℤ u) x)) ∙ + ( ap (_*ℤ x) (idempotent-is-unit-ℤ (is-unit-int-unit-ℤ u))))) ∙ + ( ( associative-mul-ℤ (int-unit-ℤ v) d x) ∙ + ( ( ap ((int-unit-ℤ v) *ℤ_) p) ∙ + ( r))))) + + div-sim-unit-ℤ : + {x y x' y' : ℤ} → sim-unit-ℤ x x' → sim-unit-ℤ y y' → + div-ℤ x y → div-ℤ x' y' + div-sim-unit-ℤ {x} {y} {x'} {y'} H K = + div-presim-unit-ℤ (presim-unit-sim-unit-ℤ H) (presim-unit-sim-unit-ℤ K) + + div-int-abs-div-ℤ : + {x y : ℤ} → div-ℤ x y → div-ℤ (int-abs-ℤ x) y + div-int-abs-div-ℤ {x} {y} = + div-sim-unit-ℤ + ( symmetric-sim-unit-ℤ (int-abs-ℤ x) x (sim-unit-abs-ℤ x)) + ( refl-sim-unit-ℤ y) + + div-div-int-abs-ℤ : + {x y : ℤ} → div-ℤ (int-abs-ℤ x) y → div-ℤ x y + div-div-int-abs-ℤ {x} {y} = + div-sim-unit-ℤ (sim-unit-abs-ℤ x) (refl-sim-unit-ℤ y) ``` ### If we have that `sim-unit-ℤ x y`, then they must differ only by sign ```agda -is-plus-or-minus-sim-unit-ℤ : - {x y : ℤ} → sim-unit-ℤ x y → is-plus-or-minus-ℤ x y -is-plus-or-minus-sim-unit-ℤ {x} {y} H with ( is-decidable-is-zero-ℤ x) -is-plus-or-minus-sim-unit-ℤ {x} {y} H | inl z = - inl (z ∙ inv (is-zero-sim-unit-ℤ H z)) -is-plus-or-minus-sim-unit-ℤ {x} {y} H | inr nz - with - ( is-one-or-neg-one-is-unit-ℤ - ( int-unit-ℤ (pr1 (H (λ u → nz (pr1 u))))) - ( is-unit-int-unit-ℤ (pr1 (H (λ u → nz (pr1 u)))))) -is-plus-or-minus-sim-unit-ℤ {x} {y} H | inr nz | inl pos = - inl - ( equational-reasoning - x - ＝ one-ℤ *ℤ x - by (inv (left-unit-law-mul-ℤ x)) - ＝ (int-unit-ℤ (pr1 (H (λ u → nz (pr1 u))))) *ℤ x - by inv (ap (_*ℤ x) pos) - ＝ y - by pr2 (H (λ u → nz (pr1 u)))) -is-plus-or-minus-sim-unit-ℤ {x} {y} H | inr nz | inr p = - inr - ( equational-reasoning - neg-ℤ x - ＝ (int-unit-ℤ (pr1 (H (λ u → nz (pr1 u))))) *ℤ x - by ap (_*ℤ x) (inv p) - ＝ y - by pr2 (H (λ u → nz (pr1 u)))) +abstract + is-plus-or-minus-sim-unit-ℤ : + {x y : ℤ} → sim-unit-ℤ x y → is-plus-or-minus-ℤ x y + is-plus-or-minus-sim-unit-ℤ {x} {y} H with ( is-decidable-is-zero-ℤ x) + is-plus-or-minus-sim-unit-ℤ {x} {y} H | inl z = + inl (z ∙ inv (is-zero-sim-unit-ℤ H z)) + is-plus-or-minus-sim-unit-ℤ {x} {y} H | inr nz + with + ( is-one-or-neg-one-is-unit-ℤ + ( int-unit-ℤ (pr1 (H (λ u → nz (pr1 u))))) + ( is-unit-int-unit-ℤ (pr1 (H (λ u → nz (pr1 u)))))) + is-plus-or-minus-sim-unit-ℤ {x} {y} H | inr nz | inl pos = + inl + ( equational-reasoning + x + ＝ one-ℤ *ℤ x + by (inv (left-unit-law-mul-ℤ x)) + ＝ (int-unit-ℤ (pr1 (H (λ u → nz (pr1 u))))) *ℤ x + by inv (ap (_*ℤ x) pos) + ＝ y + by pr2 (H (λ u → nz (pr1 u)))) + is-plus-or-minus-sim-unit-ℤ {x} {y} H | inr nz | inr p = + inr + ( equational-reasoning + neg-ℤ x + ＝ (int-unit-ℤ (pr1 (H (λ u → nz (pr1 u))))) *ℤ x + by ap (_*ℤ x) (inv p) + ＝ y + by pr2 (H (λ u → nz (pr1 u)))) ``` ### If `sim-unit-ℤ x y` holds and both `x` and `y` have the same sign, then `x ＝ y` ```agda -eq-sim-unit-is-nonnegative-ℤ : - {a b : ℤ} → is-nonnegative-ℤ a → is-nonnegative-ℤ b → sim-unit-ℤ a b → a ＝ b -eq-sim-unit-is-nonnegative-ℤ {a} {b} H H' K = - rec-coproduct - ( id) - ( λ K' → - eq-is-zero-ℤ - ( is-zero-is-nonnegative-neg-is-nonnegative-ℤ H - ( is-nonnegative-eq-ℤ (inv K') H')) - ( is-zero-is-nonnegative-neg-is-nonnegative-ℤ H' - ( is-nonnegative-eq-ℤ (inv (neg-neg-ℤ a) ∙ ap neg-ℤ K') H))) - ( is-plus-or-minus-sim-unit-ℤ K) +abstract + eq-sim-unit-is-nonnegative-ℤ : + {a b : ℤ} → is-nonnegative-ℤ a → is-nonnegative-ℤ b → sim-unit-ℤ a b → + a ＝ b + eq-sim-unit-is-nonnegative-ℤ {a} {b} H H' K = + rec-coproduct + ( id) + ( λ K' → + eq-is-zero-ℤ + ( is-zero-is-nonnegative-neg-is-nonnegative-ℤ H + ( is-nonnegative-eq-ℤ (inv K') H')) + ( is-zero-is-nonnegative-neg-is-nonnegative-ℤ H' + ( is-nonnegative-eq-ℤ (inv (neg-neg-ℤ a) ∙ ap neg-ℤ K') H))) + ( is-plus-or-minus-sim-unit-ℤ K) ``` diff --git a/src/elementary-number-theory/divisibility-natural-numbers.lagda.md b/src/elementary-number-theory/divisibility-natural-numbers.lagda.md index 9bf81abc22..8beca6d17a 100644 --- a/src/elementary-number-theory/divisibility-natural-numbers.lagda.md +++ b/src/elementary-number-theory/divisibility-natural-numbers.lagda.md @@ -54,20 +54,6 @@ div-ℕ m n = Σ ℕ (λ k → k *ℕ m ＝ n) quotient-div-ℕ : (x y : ℕ) → div-ℕ x y → ℕ quotient-div-ℕ x y H = pr1 H - -eq-quotient-div-ℕ : - (x y : ℕ) (H : div-ℕ x y) → (quotient-div-ℕ x y H) *ℕ x ＝ y -eq-quotient-div-ℕ x y H = pr2 H - -eq-quotient-div-ℕ' : - (x y : ℕ) (H : div-ℕ x y) → x *ℕ (quotient-div-ℕ x y H) ＝ y -eq-quotient-div-ℕ' x y H = - commutative-mul-ℕ x (quotient-div-ℕ x y H) ∙ eq-quotient-div-ℕ x y H - -div-quotient-div-ℕ : - (d x : ℕ) (H : div-ℕ d x) → div-ℕ (quotient-div-ℕ d x H) x -pr1 (div-quotient-div-ℕ d x (u , p)) = d -pr2 (div-quotient-div-ℕ d x (u , p)) = commutative-mul-ℕ d u ∙ p ``` ### Concatenating equality and divisibility @@ -88,33 +74,57 @@ concatenate-eq-div-eq-ℕ refl p refl = p ## Properties +### If `x` divides `y`, then the quotient times `x` is `y` + +```agda +eq-quotient-div-ℕ : + (x y : ℕ) (H : div-ℕ x y) → (quotient-div-ℕ x y H) *ℕ x ＝ y +eq-quotient-div-ℕ x y H = pr2 H + +eq-quotient-div-ℕ' : + (x y : ℕ) (H : div-ℕ x y) → x *ℕ (quotient-div-ℕ x y H) ＝ y +eq-quotient-div-ℕ' x y H = + commutative-mul-ℕ x (quotient-div-ℕ x y H) ∙ eq-quotient-div-ℕ x y H +``` + +### If `x` divides `y`, the quotient also divides `y` + +```agda +div-quotient-div-ℕ : + (d x : ℕ) (H : div-ℕ d x) → div-ℕ (quotient-div-ℕ d x H) x +pr1 (div-quotient-div-ℕ d x (u , p)) = d +pr2 (div-quotient-div-ℕ d x (u , p)) = commutative-mul-ℕ d u ∙ p +``` + ### The quotients of a natural number `n` by two natural numbers `p` and `q` are equal if `p` and `q` are equal ```agda -eq-quotient-div-eq-div-ℕ : - (x y z : ℕ) → is-nonzero-ℕ x → x ＝ y → - (H : div-ℕ x z) → (I : div-ℕ y z) → - quotient-div-ℕ x z H ＝ quotient-div-ℕ y z I -eq-quotient-div-eq-div-ℕ x y z n e H I = - is-injective-left-mul-ℕ - ( x) - ( n) - ( tr - ( λ p → - x *ℕ (quotient-div-ℕ x z H) ＝ - p *ℕ (quotient-div-ℕ y z I)) - ( inv e) - ( commutative-mul-ℕ x (quotient-div-ℕ x z H) ∙ - ( eq-quotient-div-ℕ x z H ∙ - ( inv (eq-quotient-div-ℕ y z I) ∙ - commutative-mul-ℕ (quotient-div-ℕ y z I) y)))) +abstract + eq-quotient-div-eq-div-ℕ : + (x y z : ℕ) → is-nonzero-ℕ x → x ＝ y → + (H : div-ℕ x z) → (I : div-ℕ y z) → + quotient-div-ℕ x z H ＝ quotient-div-ℕ y z I + eq-quotient-div-eq-div-ℕ x y z n e H I = + is-injective-left-mul-ℕ + ( x) + ( n) + ( tr + ( λ p → + x *ℕ (quotient-div-ℕ x z H) ＝ + p *ℕ (quotient-div-ℕ y z I)) + ( inv e) + ( commutative-mul-ℕ x (quotient-div-ℕ x z H) ∙ + ( eq-quotient-div-ℕ x z H ∙ + ( inv (eq-quotient-div-ℕ y z I) ∙ + commutative-mul-ℕ (quotient-div-ℕ y z I) y)))) ``` ### Divisibility by a nonzero natural number is a property ```agda -is-prop-div-ℕ : (k x : ℕ) → is-nonzero-ℕ k → is-prop (div-ℕ k x) -is-prop-div-ℕ k x f = is-prop-map-is-emb (is-emb-right-mul-ℕ k f) x +abstract + is-prop-div-ℕ : (k x : ℕ) → is-nonzero-ℕ k → is-prop (div-ℕ k x) + is-prop-div-ℕ k x f = is-prop-map-is-emb (is-emb-right-mul-ℕ k f) x ``` ### The divisibility relation is a partial order on the natural numbers @@ -127,63 +137,68 @@ pr2 (refl-div-ℕ x) = left-unit-law-mul-ℕ x div-eq-ℕ : (x y : ℕ) → x ＝ y → div-ℕ x y div-eq-ℕ x .x refl = refl-div-ℕ x -antisymmetric-div-ℕ : is-antisymmetric div-ℕ -antisymmetric-div-ℕ zero-ℕ zero-ℕ H K = refl -antisymmetric-div-ℕ zero-ℕ (succ-ℕ y) (pair k p) K = - inv (right-zero-law-mul-ℕ k) ∙ p -antisymmetric-div-ℕ (succ-ℕ x) zero-ℕ H (pair l q) = - inv q ∙ right-zero-law-mul-ℕ l -antisymmetric-div-ℕ (succ-ℕ x) (succ-ℕ y) (pair k p) (pair l q) = - ( inv (left-unit-law-mul-ℕ (succ-ℕ x))) ∙ - ( ( ap - ( _*ℕ (succ-ℕ x)) - ( inv - ( is-one-right-is-one-mul-ℕ l k - ( is-one-is-left-unit-mul-ℕ (l *ℕ k) x - ( ( associative-mul-ℕ l k (succ-ℕ x)) ∙ - ( ap (l *ℕ_) p ∙ q)))))) ∙ - ( p)) - -transitive-div-ℕ : is-transitive div-ℕ -pr1 (transitive-div-ℕ x y z (pair l q) (pair k p)) = l *ℕ k -pr2 (transitive-div-ℕ x y z (pair l q) (pair k p)) = - associative-mul-ℕ l k x ∙ (ap (l *ℕ_) p ∙ q) +abstract + antisymmetric-div-ℕ : is-antisymmetric div-ℕ + antisymmetric-div-ℕ zero-ℕ zero-ℕ H K = refl + antisymmetric-div-ℕ zero-ℕ (succ-ℕ y) (pair k p) K = + inv (right-zero-law-mul-ℕ k) ∙ p + antisymmetric-div-ℕ (succ-ℕ x) zero-ℕ H (pair l q) = + inv q ∙ right-zero-law-mul-ℕ l + antisymmetric-div-ℕ (succ-ℕ x) (succ-ℕ y) (pair k p) (pair l q) = + ( inv (left-unit-law-mul-ℕ (succ-ℕ x))) ∙ + ( ( ap + ( _*ℕ (succ-ℕ x)) + ( inv + ( is-one-right-is-one-mul-ℕ l k + ( is-one-is-left-unit-mul-ℕ (l *ℕ k) x + ( ( associative-mul-ℕ l k (succ-ℕ x)) ∙ + ( ap (l *ℕ_) p ∙ q)))))) ∙ + ( p)) + + transitive-div-ℕ : is-transitive div-ℕ + pr1 (transitive-div-ℕ x y z (pair l q) (pair k p)) = l *ℕ k + pr2 (transitive-div-ℕ x y z (pair l q) (pair k p)) = + associative-mul-ℕ l k x ∙ (ap (l *ℕ_) p ∙ q) ``` ### If `x` is nonzero and `d | x`, then `d ≤ x` ```agda -leq-div-succ-ℕ : (d x : ℕ) → div-ℕ d (succ-ℕ x) → leq-ℕ d (succ-ℕ x) -leq-div-succ-ℕ d x (pair (succ-ℕ k) p) = - concatenate-leq-eq-ℕ d (leq-mul-ℕ' k d) p - -leq-div-ℕ : (d x : ℕ) → is-nonzero-ℕ x → div-ℕ d x → leq-ℕ d x -leq-div-ℕ d x f H with is-successor-is-nonzero-ℕ f -... | (pair y refl) = leq-div-succ-ℕ d y H - -leq-quotient-div-ℕ : - (d x : ℕ) → is-nonzero-ℕ x → (H : div-ℕ d x) → leq-ℕ (quotient-div-ℕ d x H) x -leq-quotient-div-ℕ d x f H = - leq-div-ℕ (quotient-div-ℕ d x H) x f (div-quotient-div-ℕ d x H) - -leq-quotient-div-ℕ' : - (d x : ℕ) → is-nonzero-ℕ d → (H : div-ℕ d x) → leq-ℕ (quotient-div-ℕ d x H) x -leq-quotient-div-ℕ' d zero-ℕ f (zero-ℕ , p) = star -leq-quotient-div-ℕ' d zero-ℕ f (succ-ℕ n , p) = - f (is-zero-right-is-zero-add-ℕ _ d p) -leq-quotient-div-ℕ' d (succ-ℕ x) f H = - leq-quotient-div-ℕ d (succ-ℕ x) (is-nonzero-succ-ℕ x) H +abstract + leq-div-succ-ℕ : (d x : ℕ) → div-ℕ d (succ-ℕ x) → leq-ℕ d (succ-ℕ x) + leq-div-succ-ℕ d x (pair (succ-ℕ k) p) = + concatenate-leq-eq-ℕ d (leq-mul-ℕ' k d) p + + leq-div-ℕ : (d x : ℕ) → is-nonzero-ℕ x → div-ℕ d x → leq-ℕ d x + leq-div-ℕ d x f H with is-successor-is-nonzero-ℕ f + ... | (pair y refl) = leq-div-succ-ℕ d y H + + leq-quotient-div-ℕ : + (d x : ℕ) → is-nonzero-ℕ x → (H : div-ℕ d x) → + leq-ℕ (quotient-div-ℕ d x H) x + leq-quotient-div-ℕ d x f H = + leq-div-ℕ (quotient-div-ℕ d x H) x f (div-quotient-div-ℕ d x H) + + leq-quotient-div-ℕ' : + (d x : ℕ) → is-nonzero-ℕ d → (H : div-ℕ d x) → + leq-ℕ (quotient-div-ℕ d x H) x + leq-quotient-div-ℕ' d zero-ℕ f (zero-ℕ , p) = star + leq-quotient-div-ℕ' d zero-ℕ f (succ-ℕ n , p) = + f (is-zero-right-is-zero-add-ℕ _ d p) + leq-quotient-div-ℕ' d (succ-ℕ x) f H = + leq-quotient-div-ℕ d (succ-ℕ x) (is-nonzero-succ-ℕ x) H ``` ### If `x` is nonzero, if `d | x` and `d ≠ x`, then `d < x` ```agda -le-div-succ-ℕ : - (d x : ℕ) → div-ℕ d (succ-ℕ x) → d ≠ succ-ℕ x → le-ℕ d (succ-ℕ x) -le-div-succ-ℕ d x H f = le-leq-neq-ℕ (leq-div-succ-ℕ d x H) f +abstract + le-div-succ-ℕ : + (d x : ℕ) → div-ℕ d (succ-ℕ x) → d ≠ succ-ℕ x → le-ℕ d (succ-ℕ x) + le-div-succ-ℕ d x H f = le-leq-neq-ℕ (leq-div-succ-ℕ d x H) f -le-div-ℕ : (d x : ℕ) → is-nonzero-ℕ x → div-ℕ d x → d ≠ x → le-ℕ d x -le-div-ℕ d x H K f = le-leq-neq-ℕ (leq-div-ℕ d x H K) f + le-div-ℕ : (d x : ℕ) → is-nonzero-ℕ x → div-ℕ d x → d ≠ x → le-ℕ d x + le-div-ℕ d x H K f = le-leq-neq-ℕ (leq-div-ℕ d x H K) f ``` ### `1` divides any number @@ -232,294 +247,313 @@ is-zero-is-zero-div-ℕ .zero-ℕ y d refl = is-zero-div-zero-ℕ y d ### Any divisor of a nonzero number is nonzero ```agda -is-nonzero-div-ℕ : - (d x : ℕ) → div-ℕ d x → is-nonzero-ℕ x → is-nonzero-ℕ d -is-nonzero-div-ℕ .zero-ℕ x H K refl = K (is-zero-div-zero-ℕ x H) +abstract + is-nonzero-div-ℕ : + (d x : ℕ) → div-ℕ d x → is-nonzero-ℕ x → is-nonzero-ℕ d + is-nonzero-div-ℕ .zero-ℕ x H K refl = K (is-zero-div-zero-ℕ x H) ``` ### Any divisor of a number at least `1` is at least `1` ```agda -leq-one-div-ℕ : - (d x : ℕ) → div-ℕ d x → leq-ℕ 1 x → leq-ℕ 1 d -leq-one-div-ℕ d x H L = - leq-one-is-nonzero-ℕ d (is-nonzero-div-ℕ d x H (is-nonzero-leq-one-ℕ x L)) +abstract + leq-one-div-ℕ : + (d x : ℕ) → div-ℕ d x → leq-ℕ 1 x → leq-ℕ 1 d + leq-one-div-ℕ d x H L = + leq-one-is-nonzero-ℕ d (is-nonzero-div-ℕ d x H (is-nonzero-leq-one-ℕ x L)) ``` ### If `x < d` and `d | x`, then `x` must be `0` ```agda -is-zero-div-ℕ : - (d x : ℕ) → le-ℕ x d → div-ℕ d x → is-zero-ℕ x -is-zero-div-ℕ d zero-ℕ H D = refl -is-zero-div-ℕ d (succ-ℕ x) H (pair (succ-ℕ k) p) = - ex-falso - ( contradiction-le-ℕ - ( succ-ℕ x) d H - ( concatenate-leq-eq-ℕ d - ( leq-add-ℕ' d (k *ℕ d)) p)) +abstract + is-zero-div-ℕ : + (d x : ℕ) → le-ℕ x d → div-ℕ d x → is-zero-ℕ x + is-zero-div-ℕ d zero-ℕ H D = refl + is-zero-div-ℕ d (succ-ℕ x) H (pair (succ-ℕ k) p) = + ex-falso + ( contradiction-le-ℕ + ( succ-ℕ x) d H + ( concatenate-leq-eq-ℕ d + ( leq-add-ℕ' d (k *ℕ d)) p)) ``` ### If `x` divides `y` then `x` divides any multiple of `y` ```agda -div-mul-ℕ : - (k x y : ℕ) → div-ℕ x y → div-ℕ x (k *ℕ y) -div-mul-ℕ k x y H = - transitive-div-ℕ x y (k *ℕ y) (pair k refl) H +abstract + div-mul-ℕ : + (k x y : ℕ) → div-ℕ x y → div-ℕ x (k *ℕ y) + div-mul-ℕ k x y H = + transitive-div-ℕ x y (k *ℕ y) (pair k refl) H -div-mul-ℕ' : - (k x y : ℕ) → div-ℕ x y → div-ℕ x (y *ℕ k) -div-mul-ℕ' k x y H = - tr (div-ℕ x) (commutative-mul-ℕ k y) (div-mul-ℕ k x y H) + div-mul-ℕ' : + (k x y : ℕ) → div-ℕ x y → div-ℕ x (y *ℕ k) + div-mul-ℕ' k x y H = + tr (div-ℕ x) (commutative-mul-ℕ k y) (div-mul-ℕ k x y H) ``` ### A 3-for-2 property of division with respect to addition ```agda -div-add-ℕ : - (d x y : ℕ) → div-ℕ d x → div-ℕ d y → div-ℕ d (x +ℕ y) -pr1 (div-add-ℕ d x y (pair n p) (pair m q)) = n +ℕ m -pr2 (div-add-ℕ d x y (pair n p) (pair m q)) = - ( right-distributive-mul-add-ℕ n m d) ∙ - ( ap-add-ℕ p q) - -div-left-summand-ℕ : - (d x y : ℕ) → div-ℕ d y → div-ℕ d (x +ℕ y) → div-ℕ d x -div-left-summand-ℕ zero-ℕ x y (pair m q) (pair n p) = - pair zero-ℕ - ( ( inv (right-zero-law-mul-ℕ n)) ∙ - ( p ∙ (ap (x +ℕ_) ((inv q) ∙ (right-zero-law-mul-ℕ m))))) -pr1 (div-left-summand-ℕ (succ-ℕ d) x y (pair m q) (pair n p)) = dist-ℕ m n -pr2 (div-left-summand-ℕ (succ-ℕ d) x y (pair m q) (pair n p)) = - is-injective-right-add-ℕ (m *ℕ (succ-ℕ d)) - ( ( inv - ( ( right-distributive-mul-add-ℕ m (dist-ℕ m n) (succ-ℕ d)) ∙ - ( commutative-add-ℕ - ( m *ℕ (succ-ℕ d)) - ( (dist-ℕ m n) *ℕ (succ-ℕ d))))) ∙ - ( ( ap - ( _*ℕ (succ-ℕ d)) - ( is-additive-right-inverse-dist-ℕ m n - ( reflects-order-mul-ℕ d m n - ( concatenate-eq-leq-eq-ℕ q - ( leq-add-ℕ' y x) - ( inv p))))) ∙ - ( p ∙ (ap (x +ℕ_) (inv q))))) - -div-right-summand-ℕ : - (d x y : ℕ) → div-ℕ d x → div-ℕ d (x +ℕ y) → div-ℕ d y -div-right-summand-ℕ d x y H1 H2 = - div-left-summand-ℕ d y x H1 (concatenate-div-eq-ℕ H2 (commutative-add-ℕ x y)) +abstract + div-add-ℕ : + (d x y : ℕ) → div-ℕ d x → div-ℕ d y → div-ℕ d (x +ℕ y) + pr1 (div-add-ℕ d x y (pair n p) (pair m q)) = n +ℕ m + pr2 (div-add-ℕ d x y (pair n p) (pair m q)) = + ( right-distributive-mul-add-ℕ n m d) ∙ + ( ap-add-ℕ p q) + + div-left-summand-ℕ : + (d x y : ℕ) → div-ℕ d y → div-ℕ d (x +ℕ y) → div-ℕ d x + div-left-summand-ℕ zero-ℕ x y (pair m q) (pair n p) = + pair zero-ℕ + ( ( inv (right-zero-law-mul-ℕ n)) ∙ + ( p ∙ (ap (x +ℕ_) ((inv q) ∙ (right-zero-law-mul-ℕ m))))) + pr1 (div-left-summand-ℕ (succ-ℕ d) x y (pair m q) (pair n p)) = dist-ℕ m n + pr2 (div-left-summand-ℕ (succ-ℕ d) x y (pair m q) (pair n p)) = + is-injective-right-add-ℕ (m *ℕ (succ-ℕ d)) + ( ( inv + ( ( right-distributive-mul-add-ℕ m (dist-ℕ m n) (succ-ℕ d)) ∙ + ( commutative-add-ℕ + ( m *ℕ (succ-ℕ d)) + ( (dist-ℕ m n) *ℕ (succ-ℕ d))))) ∙ + ( ( ap + ( _*ℕ (succ-ℕ d)) + ( is-additive-right-inverse-dist-ℕ m n + ( reflects-order-mul-ℕ d m n + ( concatenate-eq-leq-eq-ℕ q + ( leq-add-ℕ' y x) + ( inv p))))) ∙ + ( p ∙ (ap (x +ℕ_) (inv q))))) + + div-right-summand-ℕ : + (d x y : ℕ) → div-ℕ d x → div-ℕ d (x +ℕ y) → div-ℕ d y + div-right-summand-ℕ d x y H1 H2 = + div-left-summand-ℕ d y x H1 + ( concatenate-div-eq-ℕ H2 (commutative-add-ℕ x y)) ``` ### If `d` divides both `x` and `x + 1`, then `d ＝ 1` ```agda -is-one-div-ℕ : (x y : ℕ) → div-ℕ x y → div-ℕ x (succ-ℕ y) → is-one-ℕ x -is-one-div-ℕ x y H K = is-one-div-one-ℕ x (div-right-summand-ℕ x y 1 H K) +abstract + is-one-div-ℕ : (x y : ℕ) → div-ℕ x y → div-ℕ x (succ-ℕ y) → is-one-ℕ x + is-one-div-ℕ x y H K = is-one-div-one-ℕ x (div-right-summand-ℕ x y 1 H K) ``` ### Multiplication preserves divisibility ```agda -preserves-div-mul-ℕ : - (k x y : ℕ) → div-ℕ x y → div-ℕ (k *ℕ x) (k *ℕ y) -pr1 (preserves-div-mul-ℕ k x y (pair q p)) = q -pr2 (preserves-div-mul-ℕ k x y (pair q p)) = - ( inv (associative-mul-ℕ q k x)) ∙ - ( ( ap (_*ℕ x) (commutative-mul-ℕ q k)) ∙ - ( ( associative-mul-ℕ k q x) ∙ - ( ap (k *ℕ_) p))) +abstract + preserves-div-mul-ℕ : + (k x y : ℕ) → div-ℕ x y → div-ℕ (k *ℕ x) (k *ℕ y) + pr1 (preserves-div-mul-ℕ k x y (pair q p)) = q + pr2 (preserves-div-mul-ℕ k x y (pair q p)) = + ( inv (associative-mul-ℕ q k x)) ∙ + ( ( ap (_*ℕ x) (commutative-mul-ℕ q k)) ∙ + ( ( associative-mul-ℕ k q x) ∙ + ( ap (k *ℕ_) p))) ``` ### Multiplication by a nonzero number reflects divisibility ```agda -reflects-div-mul-ℕ : - (k x y : ℕ) → is-nonzero-ℕ k → div-ℕ (k *ℕ x) (k *ℕ y) → div-ℕ x y -pr1 (reflects-div-mul-ℕ k x y H (pair q p)) = q -pr2 (reflects-div-mul-ℕ k x y H (pair q p)) = - is-injective-left-mul-ℕ k H - ( ( inv (associative-mul-ℕ k q x)) ∙ - ( ( ap (_*ℕ x) (commutative-mul-ℕ k q)) ∙ - ( ( associative-mul-ℕ q k x) ∙ - ( p)))) +abstract + reflects-div-mul-ℕ : + (k x y : ℕ) → is-nonzero-ℕ k → div-ℕ (k *ℕ x) (k *ℕ y) → div-ℕ x y + pr1 (reflects-div-mul-ℕ k x y H (pair q p)) = q + pr2 (reflects-div-mul-ℕ k x y H (pair q p)) = + is-injective-left-mul-ℕ k H + ( ( inv (associative-mul-ℕ k q x)) ∙ + ( ( ap (_*ℕ x) (commutative-mul-ℕ k q)) ∙ + ( ( associative-mul-ℕ q k x) ∙ + ( p)))) ``` ### If a nonzero number `d` divides `y`, then `dx` divides `y` if and only if `x` divides the quotient `y/d` ```agda -div-quotient-div-div-ℕ : - (x y d : ℕ) (H : div-ℕ d y) → is-nonzero-ℕ d → - div-ℕ (d *ℕ x) y → div-ℕ x (quotient-div-ℕ d y H) -div-quotient-div-div-ℕ x y d H f K = - reflects-div-mul-ℕ d x - ( quotient-div-ℕ d y H) - ( f) - ( tr (div-ℕ (d *ℕ x)) (inv (eq-quotient-div-ℕ' d y H)) K) - -div-div-quotient-div-ℕ : - (x y d : ℕ) (H : div-ℕ d y) → - div-ℕ x (quotient-div-ℕ d y H) → div-ℕ (d *ℕ x) y -div-div-quotient-div-ℕ x y d H K = - tr - ( div-ℕ (d *ℕ x)) - ( eq-quotient-div-ℕ' d y H) - ( preserves-div-mul-ℕ d x (quotient-div-ℕ d y H) K) +abstract + div-quotient-div-div-ℕ : + (x y d : ℕ) (H : div-ℕ d y) → is-nonzero-ℕ d → + div-ℕ (d *ℕ x) y → div-ℕ x (quotient-div-ℕ d y H) + div-quotient-div-div-ℕ x y d H f K = + reflects-div-mul-ℕ d x + ( quotient-div-ℕ d y H) + ( f) + ( tr (div-ℕ (d *ℕ x)) (inv (eq-quotient-div-ℕ' d y H)) K) + + div-div-quotient-div-ℕ : + (x y d : ℕ) (H : div-ℕ d y) → + div-ℕ x (quotient-div-ℕ d y H) → div-ℕ (d *ℕ x) y + div-div-quotient-div-ℕ x y d H K = + tr + ( div-ℕ (d *ℕ x)) + ( eq-quotient-div-ℕ' d y H) + ( preserves-div-mul-ℕ d x (quotient-div-ℕ d y H) K) ``` ### If `d` divides a nonzero number `x`, then the quotient `x/d` is also nonzero ```agda -is-nonzero-quotient-div-ℕ : - {d x : ℕ} (H : div-ℕ d x) → - is-nonzero-ℕ x → is-nonzero-ℕ (quotient-div-ℕ d x H) -is-nonzero-quotient-div-ℕ {d} {.(k *ℕ d)} (pair k refl) = - is-nonzero-left-factor-mul-ℕ k d +abstract + is-nonzero-quotient-div-ℕ : + {d x : ℕ} (H : div-ℕ d x) → + is-nonzero-ℕ x → is-nonzero-ℕ (quotient-div-ℕ d x H) + is-nonzero-quotient-div-ℕ {d} {.(k *ℕ d)} (pair k refl) = + is-nonzero-left-factor-mul-ℕ k d ``` ### If `d` divides a number `1 ≤ x`, then `1 ≤ x/d` ```agda -leq-one-quotient-div-ℕ : - (d x : ℕ) (H : div-ℕ d x) → leq-ℕ 1 x → leq-ℕ 1 (quotient-div-ℕ d x H) -leq-one-quotient-div-ℕ d x H K = - leq-one-div-ℕ - ( quotient-div-ℕ d x H) - ( x) - ( div-quotient-div-ℕ d x H) - ( K) +abstract + leq-one-quotient-div-ℕ : + (d x : ℕ) (H : div-ℕ d x) → leq-ℕ 1 x → leq-ℕ 1 (quotient-div-ℕ d x H) + leq-one-quotient-div-ℕ d x H K = + leq-one-div-ℕ + ( quotient-div-ℕ d x H) + ( x) + ( div-quotient-div-ℕ d x H) + ( K) ``` ### `a/a ＝ 1` ```agda -is-idempotent-quotient-div-ℕ : - (a : ℕ) → is-nonzero-ℕ a → (H : div-ℕ a a) → is-one-ℕ (quotient-div-ℕ a a H) -is-idempotent-quotient-div-ℕ zero-ℕ nz (u , p) = ex-falso (nz refl) -is-idempotent-quotient-div-ℕ (succ-ℕ a) nz (u , p) = - is-one-is-left-unit-mul-ℕ u a p +abstract + is-idempotent-quotient-div-ℕ : + (a : ℕ) → is-nonzero-ℕ a → (H : div-ℕ a a) → is-one-ℕ (quotient-div-ℕ a a H) + is-idempotent-quotient-div-ℕ zero-ℕ nz (u , p) = ex-falso (nz refl) + is-idempotent-quotient-div-ℕ (succ-ℕ a) nz (u , p) = + is-one-is-left-unit-mul-ℕ u a p ``` ### If `b` divides `a` and `c` divides `b` and `c` is nonzero, then `a/b · b/c ＝ a/c` ```agda -simplify-mul-quotient-div-ℕ : - {a b c : ℕ} → is-nonzero-ℕ c → - (H : div-ℕ b a) (K : div-ℕ c b) (L : div-ℕ c a) → - ( (quotient-div-ℕ b a H) *ℕ (quotient-div-ℕ c b K)) ＝ - ( quotient-div-ℕ c a L) -simplify-mul-quotient-div-ℕ {a} {b} {c} nz H K L = - is-injective-right-mul-ℕ c nz - ( equational-reasoning - (a/b *ℕ b/c) *ℕ c - ＝ a/b *ℕ (b/c *ℕ c) - by associative-mul-ℕ a/b b/c c - ＝ a/b *ℕ b - by ap (a/b *ℕ_) (eq-quotient-div-ℕ c b K) - ＝ a - by eq-quotient-div-ℕ b a H - ＝ a/c *ℕ c - by inv (eq-quotient-div-ℕ c a L)) - where - a/b : ℕ - a/b = quotient-div-ℕ b a H - b/c : ℕ - b/c = quotient-div-ℕ c b K - a/c : ℕ - a/c = quotient-div-ℕ c a L +abstract + simplify-mul-quotient-div-ℕ : + {a b c : ℕ} → is-nonzero-ℕ c → + (H : div-ℕ b a) (K : div-ℕ c b) (L : div-ℕ c a) → + ( (quotient-div-ℕ b a H) *ℕ (quotient-div-ℕ c b K)) ＝ + ( quotient-div-ℕ c a L) + simplify-mul-quotient-div-ℕ {a} {b} {c} nz H K L = + is-injective-right-mul-ℕ c nz + ( equational-reasoning + (a/b *ℕ b/c) *ℕ c + ＝ a/b *ℕ (b/c *ℕ c) + by associative-mul-ℕ a/b b/c c + ＝ a/b *ℕ b + by ap (a/b *ℕ_) (eq-quotient-div-ℕ c b K) + ＝ a + by eq-quotient-div-ℕ b a H + ＝ a/c *ℕ c + by inv (eq-quotient-div-ℕ c a L)) + where + a/b : ℕ + a/b = quotient-div-ℕ b a H + b/c : ℕ + b/c = quotient-div-ℕ c b K + a/c : ℕ + a/c = quotient-div-ℕ c a L ``` ### If `d | a` and `d` is nonzero, then `x | a/d` if and only if `xd | a` ```agda -simplify-div-quotient-div-ℕ : - {a d x : ℕ} → is-nonzero-ℕ d → (H : div-ℕ d a) → - (div-ℕ x (quotient-div-ℕ d a H)) ↔ (div-ℕ (x *ℕ d) a) -pr1 (pr1 (simplify-div-quotient-div-ℕ nz H) (u , p)) = u -pr2 (pr1 (simplify-div-quotient-div-ℕ {a} {d} {x} nz H) (u , p)) = - equational-reasoning - u *ℕ (x *ℕ d) - ＝ (u *ℕ x) *ℕ d - by inv (associative-mul-ℕ u x d) - ＝ (quotient-div-ℕ d a H) *ℕ d - by ap (_*ℕ d) p - ＝ a - by eq-quotient-div-ℕ d a H -pr1 (pr2 (simplify-div-quotient-div-ℕ nz H) (u , p)) = u -pr2 (pr2 (simplify-div-quotient-div-ℕ {a} {d} {x} nz H) (u , p)) = - is-injective-right-mul-ℕ d nz - ( equational-reasoning - (u *ℕ x) *ℕ d - ＝ u *ℕ (x *ℕ d) - by associative-mul-ℕ u x d - ＝ a - by p - ＝ (quotient-div-ℕ d a H) *ℕ d - by inv (eq-quotient-div-ℕ d a H)) +abstract + simplify-div-quotient-div-ℕ : + {a d x : ℕ} → is-nonzero-ℕ d → (H : div-ℕ d a) → + (div-ℕ x (quotient-div-ℕ d a H)) ↔ (div-ℕ (x *ℕ d) a) + pr1 (pr1 (simplify-div-quotient-div-ℕ nz H) (u , p)) = u + pr2 (pr1 (simplify-div-quotient-div-ℕ {a} {d} {x} nz H) (u , p)) = + equational-reasoning + u *ℕ (x *ℕ d) + ＝ (u *ℕ x) *ℕ d + by inv (associative-mul-ℕ u x d) + ＝ (quotient-div-ℕ d a H) *ℕ d + by ap (_*ℕ d) p + ＝ a + by eq-quotient-div-ℕ d a H + pr1 (pr2 (simplify-div-quotient-div-ℕ nz H) (u , p)) = u + pr2 (pr2 (simplify-div-quotient-div-ℕ {a} {d} {x} nz H) (u , p)) = + is-injective-right-mul-ℕ d nz + ( equational-reasoning + (u *ℕ x) *ℕ d + ＝ u *ℕ (x *ℕ d) + by associative-mul-ℕ u x d + ＝ a + by p + ＝ (quotient-div-ℕ d a H) *ℕ d + by inv (eq-quotient-div-ℕ d a H)) ``` ### Suppose `H : b | a` and `K : c | b`, where `c` is nonzero. If `d` divides `b/c` then `d` divides `a/c` ```agda -div-quotient-div-div-quotient-div-ℕ : - {a b c d : ℕ} → is-nonzero-ℕ c → (H : div-ℕ b a) - (K : div-ℕ c b) (L : div-ℕ c a) → - div-ℕ d (quotient-div-ℕ c b K) → - div-ℕ d (quotient-div-ℕ c a L) -div-quotient-div-div-quotient-div-ℕ {a} {b} {c} {d} nz H K L M = - tr - ( div-ℕ d) - ( simplify-mul-quotient-div-ℕ nz H K L) - ( div-mul-ℕ - ( quotient-div-ℕ b a H) - ( d) - ( quotient-div-ℕ c b K) - ( M)) +abstract + div-quotient-div-div-quotient-div-ℕ : + {a b c d : ℕ} → is-nonzero-ℕ c → (H : div-ℕ b a) + (K : div-ℕ c b) (L : div-ℕ c a) → + div-ℕ d (quotient-div-ℕ c b K) → + div-ℕ d (quotient-div-ℕ c a L) + div-quotient-div-div-quotient-div-ℕ {a} {b} {c} {d} nz H K L M = + tr + ( div-ℕ d) + ( simplify-mul-quotient-div-ℕ nz H K L) + ( div-mul-ℕ + ( quotient-div-ℕ b a H) + ( d) + ( quotient-div-ℕ c b K) + ( M)) ``` ### If `x` is nonzero and `d | x`, then `x/d ＝ 1` if and only if `d ＝ x` ```agda -is-one-quotient-div-ℕ : - (d x : ℕ) → is-nonzero-ℕ x → (H : div-ℕ d x) → (d ＝ x) → - is-one-ℕ (quotient-div-ℕ d x H) -is-one-quotient-div-ℕ d .d f H refl = is-idempotent-quotient-div-ℕ d f H +abstract + is-one-quotient-div-ℕ : + (d x : ℕ) → is-nonzero-ℕ x → (H : div-ℕ d x) → (d ＝ x) → + is-one-ℕ (quotient-div-ℕ d x H) + is-one-quotient-div-ℕ d .d f H refl = is-idempotent-quotient-div-ℕ d f H -eq-is-one-quotient-div-ℕ : - (d x : ℕ) → (H : div-ℕ d x) → is-one-ℕ (quotient-div-ℕ d x H) → d ＝ x -eq-is-one-quotient-div-ℕ d x (.1 , q) refl = inv (left-unit-law-mul-ℕ d) ∙ q + eq-is-one-quotient-div-ℕ : + (d x : ℕ) → (H : div-ℕ d x) → is-one-ℕ (quotient-div-ℕ d x H) → d ＝ x + eq-is-one-quotient-div-ℕ d x (.1 , q) refl = inv (left-unit-law-mul-ℕ d) ∙ q ``` ### If `x` is nonzero and `d | x`, then `x/d ＝ x` if and only if `d ＝ 1` ```agda -compute-quotient-div-is-one-ℕ : - (d x : ℕ) → (H : div-ℕ d x) → is-one-ℕ d → quotient-div-ℕ d x H ＝ x -compute-quotient-div-is-one-ℕ .1 x (u , q) refl = - inv (right-unit-law-mul-ℕ u) ∙ q +abstract + compute-quotient-div-is-one-ℕ : + (d x : ℕ) → (H : div-ℕ d x) → is-one-ℕ d → quotient-div-ℕ d x H ＝ x + compute-quotient-div-is-one-ℕ .1 x (u , q) refl = + inv (right-unit-law-mul-ℕ u) ∙ q -is-one-divisor-ℕ : - (d x : ℕ) → is-nonzero-ℕ x → (H : div-ℕ d x) → - quotient-div-ℕ d x H ＝ x → is-one-ℕ d -is-one-divisor-ℕ d x f (.x , q) refl = - is-injective-left-mul-ℕ x f (q ∙ inv (right-unit-law-mul-ℕ x)) + is-one-divisor-ℕ : + (d x : ℕ) → is-nonzero-ℕ x → (H : div-ℕ d x) → + quotient-div-ℕ d x H ＝ x → is-one-ℕ d + is-one-divisor-ℕ d x f (.x , q) refl = + is-injective-left-mul-ℕ x f (q ∙ inv (right-unit-law-mul-ℕ x)) ``` ### If `x` is nonzero and `d | x` is a nontrivial divisor, then `x/d < x` ```agda -le-quotient-div-ℕ : - (d x : ℕ) → is-nonzero-ℕ x → (H : div-ℕ d x) → ¬ (is-one-ℕ d) → - le-ℕ (quotient-div-ℕ d x H) x -le-quotient-div-ℕ d x f H g = - map-left-unit-law-coproduct-is-empty - ( quotient-div-ℕ d x H ＝ x) - ( le-ℕ (quotient-div-ℕ d x H) x) - ( map-neg (is-one-divisor-ℕ d x f H) g) - ( eq-or-le-leq-ℕ - ( quotient-div-ℕ d x H) - ( x) - ( leq-quotient-div-ℕ d x f H)) +abstract + le-quotient-div-ℕ : + (d x : ℕ) → is-nonzero-ℕ x → (H : div-ℕ d x) → ¬ (is-one-ℕ d) → + le-ℕ (quotient-div-ℕ d x H) x + le-quotient-div-ℕ d x f H g = + map-left-unit-law-coproduct-is-empty + ( quotient-div-ℕ d x H ＝ x) + ( le-ℕ (quotient-div-ℕ d x H) x) + ( map-neg (is-one-divisor-ℕ d x f H) g) + ( eq-or-le-leq-ℕ + ( quotient-div-ℕ d x H) + ( x) + ( leq-quotient-div-ℕ d x f H)) ``` diff --git a/src/elementary-number-theory/equality-integers.lagda.md b/src/elementary-number-theory/equality-integers.lagda.md index 0638b0bb24..0b2ef3070d 100644 --- a/src/elementary-number-theory/equality-integers.lagda.md +++ b/src/elementary-number-theory/equality-integers.lagda.md @@ -33,8 +33,9 @@ open import foundation.universe-levels ## Idea -An equality predicate is defined by pattern matching on the integers. Then we -show that this predicate characterizes the identit type of the integers +We define equality on the [integers](elementary-number-theory.integers.md) by +pattern matching, and show that this characterizes the +[identity type](foundation.identity-types.md) on the integers. ## Definition diff --git a/src/elementary-number-theory/equality-natural-numbers.lagda.md b/src/elementary-number-theory/equality-natural-numbers.lagda.md index 78bff66b5f..6b37e60205 100644 --- a/src/elementary-number-theory/equality-natural-numbers.lagda.md +++ b/src/elementary-number-theory/equality-natural-numbers.lagda.md @@ -33,6 +33,13 @@ open import foundation-core.torsorial-type-families +## Idea + +We define equality on the +[natural numbers](elementary-number-theory.natural-numbers.md) by pattern +matching, and show that this characterizes the +[identity type](foundation.identity-types.md) on the natural numbers. + ## Definitions ### Observational equality on the natural numbers @@ -86,8 +93,9 @@ pr2 ℕ-Set = is-set-ℕ ### The property of being zero ```agda -is-prop-is-zero-ℕ : (n : ℕ) → is-prop (is-zero-ℕ n) -is-prop-is-zero-ℕ n = is-set-ℕ n zero-ℕ +abstract + is-prop-is-zero-ℕ : (n : ℕ) → is-prop (is-zero-ℕ n) + is-prop-is-zero-ℕ n = is-set-ℕ n zero-ℕ is-zero-ℕ-Prop : ℕ → Prop lzero pr1 (is-zero-ℕ-Prop n) = is-zero-ℕ n @@ -97,8 +105,9 @@ pr2 (is-zero-ℕ-Prop n) = is-prop-is-zero-ℕ n ### The property of being one ```agda -is-prop-is-one-ℕ : (n : ℕ) → is-prop (is-one-ℕ n) -is-prop-is-one-ℕ n = is-set-ℕ n 1 +abstract + is-prop-is-one-ℕ : (n : ℕ) → is-prop (is-one-ℕ n) + is-prop-is-one-ℕ n = is-set-ℕ n 1 is-one-ℕ-Prop : ℕ → Prop lzero pr1 (is-one-ℕ-Prop n) = is-one-ℕ n diff --git a/src/elementary-number-theory/euclidean-division-natural-numbers.lagda.md b/src/elementary-number-theory/euclidean-division-natural-numbers.lagda.md index 1ef17f8247..f07171b144 100644 --- a/src/elementary-number-theory/euclidean-division-natural-numbers.lagda.md +++ b/src/elementary-number-theory/euclidean-division-natural-numbers.lagda.md @@ -131,7 +131,7 @@ eq-euclidean-division-ℕ (succ-ℕ k) x = ( ap ( _+ℕ (remainder-euclidean-division-ℕ (succ-ℕ k) x)) ( ( pr2 (cong-euclidean-division-ℕ (succ-ℕ k) x)) ∙ - ( symmetric-dist-ℕ x + ( commutative-dist-ℕ x ( remainder-euclidean-division-ℕ (succ-ℕ k) x)))) ∙ ( is-difference-dist-ℕ' (remainder-euclidean-division-ℕ (succ-ℕ k) x) x ( leq-nat-mod-succ-ℕ k x)) diff --git a/src/elementary-number-theory/exponentiation-natural-numbers.lagda.md b/src/elementary-number-theory/exponentiation-natural-numbers.lagda.md index abbf1734ae..0e1b525344 100644 --- a/src/elementary-number-theory/exponentiation-natural-numbers.lagda.md +++ b/src/elementary-number-theory/exponentiation-natural-numbers.lagda.md @@ -10,10 +10,10 @@ module elementary-number-theory.exponentiation-natural-numbers where open import commutative-algebra.powers-of-elements-commutative-semirings open import elementary-number-theory.addition-natural-numbers -open import elementary-number-theory.commutative-semiring-of-natural-numbers open import elementary-number-theory.multiplication-natural-numbers open import elementary-number-theory.natural-numbers open import elementary-number-theory.products-of-natural-numbers +open import elementary-number-theory.semiring-of-natural-numbers open import foundation.action-on-identifications-functions open import foundation.identity-types @@ -45,44 +45,61 @@ power-ℕ = power-Commutative-Semiring ℕ-Commutative-Semiring ## Properties -### Tarski's high school arithmetic laws for exponentiation +### 1^n = 1 for all n ```agda -annihilation-law-exp-ℕ : (n : ℕ) → 1 ^ℕ n ＝ 1 -annihilation-law-exp-ℕ zero-ℕ = refl -annihilation-law-exp-ℕ (succ-ℕ n) = - right-unit-law-mul-ℕ (1 ^ℕ n) ∙ annihilation-law-exp-ℕ n - -left-distributive-exp-add-ℕ : - (x y z : ℕ) → x ^ℕ (y +ℕ z) ＝ (x ^ℕ y) *ℕ (x ^ℕ z) -left-distributive-exp-add-ℕ x y zero-ℕ = inv (right-unit-law-mul-ℕ (x ^ℕ y)) -left-distributive-exp-add-ℕ x y (succ-ℕ z) = - ( ap (_*ℕ x) (left-distributive-exp-add-ℕ x y z)) ∙ - ( associative-mul-ℕ (x ^ℕ y) (x ^ℕ z) x) - -right-distributive-exp-mul-ℕ : - (x y z : ℕ) → (x *ℕ y) ^ℕ z ＝ (x ^ℕ z) *ℕ (y ^ℕ z) -right-distributive-exp-mul-ℕ x y zero-ℕ = refl -right-distributive-exp-mul-ℕ x y (succ-ℕ z) = - ( ap (_*ℕ (x *ℕ y)) (right-distributive-exp-mul-ℕ x y z)) ∙ - ( interchange-law-mul-mul-ℕ (x ^ℕ z) (y ^ℕ z) x y) - -exp-mul-ℕ : (x y z : ℕ) → x ^ℕ (y *ℕ z) ＝ (x ^ℕ y) ^ℕ z -exp-mul-ℕ x zero-ℕ z = inv (annihilation-law-exp-ℕ z) -exp-mul-ℕ x (succ-ℕ y) z = - ( left-distributive-exp-add-ℕ x (y *ℕ z) z) ∙ - ( ( ap (_*ℕ (x ^ℕ z)) (exp-mul-ℕ x y z)) ∙ - ( inv (right-distributive-exp-mul-ℕ (x ^ℕ y) x z))) +abstract + annihilation-law-exp-ℕ : (n : ℕ) → 1 ^ℕ n ＝ 1 + annihilation-law-exp-ℕ zero-ℕ = refl + annihilation-law-exp-ℕ (succ-ℕ n) = + right-unit-law-mul-ℕ (1 ^ℕ n) ∙ annihilation-law-exp-ℕ n ``` -### The exponent $m^n$ is always nonzero +### `xʸ⁺ᶻ = xʸ xᶻ` ```agda -is-nonzero-exp-ℕ : - (m n : ℕ) → is-nonzero-ℕ m → is-nonzero-ℕ (m ^ℕ n) -is-nonzero-exp-ℕ m zero-ℕ p = is-nonzero-one-ℕ -is-nonzero-exp-ℕ m (succ-ℕ n) p = - is-nonzero-mul-ℕ (m ^ℕ n) m (is-nonzero-exp-ℕ m n p) p +abstract + left-distributive-exp-add-ℕ : + (x y z : ℕ) → x ^ℕ (y +ℕ z) ＝ (x ^ℕ y) *ℕ (x ^ℕ z) + left-distributive-exp-add-ℕ x y zero-ℕ = inv (right-unit-law-mul-ℕ (x ^ℕ y)) + left-distributive-exp-add-ℕ x y (succ-ℕ z) = + ( ap (_*ℕ x) (left-distributive-exp-add-ℕ x y z)) ∙ + ( associative-mul-ℕ (x ^ℕ y) (x ^ℕ z) x) +``` + +### `(xy)ᶻ = xᶻ yᶻ` + +```agda +abstract + right-distributive-exp-mul-ℕ : + (x y z : ℕ) → (x *ℕ y) ^ℕ z ＝ (x ^ℕ z) *ℕ (y ^ℕ z) + right-distributive-exp-mul-ℕ x y zero-ℕ = refl + right-distributive-exp-mul-ℕ x y (succ-ℕ z) = + ( ap (_*ℕ (x *ℕ y)) (right-distributive-exp-mul-ℕ x y z)) ∙ + ( interchange-law-mul-mul-ℕ (x ^ℕ z) (y ^ℕ z) x y) +``` + +### `xʸᶻ = (xʸ)ᶻ` + +```agda +abstract + exp-mul-ℕ : (x y z : ℕ) → x ^ℕ (y *ℕ z) ＝ (x ^ℕ y) ^ℕ z + exp-mul-ℕ x zero-ℕ z = inv (annihilation-law-exp-ℕ z) + exp-mul-ℕ x (succ-ℕ y) z = + ( left-distributive-exp-add-ℕ x (y *ℕ z) z) ∙ + ( ( ap (_*ℕ (x ^ℕ z)) (exp-mul-ℕ x y z)) ∙ + ( inv (right-distributive-exp-mul-ℕ (x ^ℕ y) x z))) +``` + +### If `m` is nonzero, `mⁿ` is nonzero + +```agda +abstract + is-nonzero-exp-ℕ : + (m n : ℕ) → is-nonzero-ℕ m → is-nonzero-ℕ (m ^ℕ n) + is-nonzero-exp-ℕ m zero-ℕ p = is-nonzero-one-ℕ + is-nonzero-exp-ℕ m (succ-ℕ n) p = + is-nonzero-mul-ℕ (m ^ℕ n) m (is-nonzero-exp-ℕ m n p) p ``` ### The exponent $m^n$ is equal to the $n$-fold product of $m$ diff --git a/src/elementary-number-theory/factorials.lagda.md b/src/elementary-number-theory/factorials.lagda.md index 05851e426e..2d1a9326cc 100644 --- a/src/elementary-number-theory/factorials.lagda.md +++ b/src/elementary-number-theory/factorials.lagda.md @@ -37,40 +37,51 @@ factorial-ℕ (succ-ℕ m) = (factorial-ℕ m) *ℕ (succ-ℕ m) ## Properties +### If `x ≤ n` and `x` is nonzero, `x | n!` + +```agda +abstract + div-factorial-ℕ : + (n x : ℕ) → leq-ℕ x n → is-nonzero-ℕ x → div-ℕ x (factorial-ℕ n) + div-factorial-ℕ zero-ℕ zero-ℕ l H = ex-falso (H refl) + div-factorial-ℕ (succ-ℕ n) x l H with + decide-leq-succ-ℕ x n l + ... | inl l' = + transitive-div-ℕ x + ( factorial-ℕ n) + ( factorial-ℕ (succ-ℕ n)) + ( pair (succ-ℕ n) (commutative-mul-ℕ (succ-ℕ n) (factorial-ℕ n))) + ( div-factorial-ℕ n x l' H) + ... | inr refl = pair (factorial-ℕ n) refl +``` + +### All factorials are nonzero + ```agda -div-factorial-ℕ : - (n x : ℕ) → leq-ℕ x n → is-nonzero-ℕ x → div-ℕ x (factorial-ℕ n) -div-factorial-ℕ zero-ℕ zero-ℕ l H = ex-falso (H refl) -div-factorial-ℕ (succ-ℕ n) x l H with - decide-leq-succ-ℕ x n l -... | inl l' = - transitive-div-ℕ x - ( factorial-ℕ n) - ( factorial-ℕ (succ-ℕ n)) - ( pair (succ-ℕ n) (commutative-mul-ℕ (succ-ℕ n) (factorial-ℕ n))) - ( div-factorial-ℕ n x l' H) -... | inr refl = pair (factorial-ℕ n) refl +abstract + is-nonzero-factorial-ℕ : + (x : ℕ) → is-nonzero-ℕ (factorial-ℕ x) + is-nonzero-factorial-ℕ zero-ℕ = Eq-eq-ℕ + is-nonzero-factorial-ℕ (succ-ℕ x) = + is-nonzero-mul-ℕ + ( factorial-ℕ x) + ( succ-ℕ x) + ( is-nonzero-factorial-ℕ x) + ( is-nonzero-succ-ℕ x) ``` +### `n ≤ n!` + ```agda -is-nonzero-factorial-ℕ : - (x : ℕ) → is-nonzero-ℕ (factorial-ℕ x) -is-nonzero-factorial-ℕ zero-ℕ = Eq-eq-ℕ -is-nonzero-factorial-ℕ (succ-ℕ x) = - is-nonzero-mul-ℕ - ( factorial-ℕ x) - ( succ-ℕ x) - ( is-nonzero-factorial-ℕ x) - ( is-nonzero-succ-ℕ x) - -leq-factorial-ℕ : - (n : ℕ) → leq-ℕ n (factorial-ℕ n) -leq-factorial-ℕ zero-ℕ = leq-zero-ℕ 1 -leq-factorial-ℕ (succ-ℕ n) = - leq-mul-is-nonzero-ℕ' - ( factorial-ℕ n) - ( succ-ℕ n) - ( is-nonzero-factorial-ℕ n) +abstract + leq-factorial-ℕ : + (n : ℕ) → leq-ℕ n (factorial-ℕ n) + leq-factorial-ℕ zero-ℕ = leq-zero-ℕ 1 + leq-factorial-ℕ (succ-ℕ n) = + leq-mul-is-nonzero-ℕ' + ( factorial-ℕ n) + ( succ-ℕ n) + ( is-nonzero-factorial-ℕ n) ``` ## External links diff --git a/src/elementary-number-theory/greatest-common-divisor-natural-numbers.lagda.md b/src/elementary-number-theory/greatest-common-divisor-natural-numbers.lagda.md index bc14f911c8..419d6fba1a 100644 --- a/src/elementary-number-theory/greatest-common-divisor-natural-numbers.lagda.md +++ b/src/elementary-number-theory/greatest-common-divisor-natural-numbers.lagda.md @@ -78,88 +78,95 @@ is-multiple-of-gcd-ℕ a b n = ### Reflexivity for common divisors ```agda -refl-is-common-divisor-ℕ : - (x : ℕ) → is-common-divisor-ℕ x x x -pr1 (refl-is-common-divisor-ℕ x) = refl-div-ℕ x -pr2 (refl-is-common-divisor-ℕ x) = refl-div-ℕ x +abstract + refl-is-common-divisor-ℕ : + (x : ℕ) → is-common-divisor-ℕ x x x + pr1 (refl-is-common-divisor-ℕ x) = refl-div-ℕ x + pr2 (refl-is-common-divisor-ℕ x) = refl-div-ℕ x ``` ### Being a common divisor is decidable ```agda -is-decidable-is-common-divisor-ℕ : - (a b : ℕ) → is-decidable-family (is-common-divisor-ℕ a b) -is-decidable-is-common-divisor-ℕ a b x = - is-decidable-product - ( is-decidable-div-ℕ x a) - ( is-decidable-div-ℕ x b) +abstract + is-decidable-is-common-divisor-ℕ : + (a b : ℕ) → is-decidable-family (is-common-divisor-ℕ a b) + is-decidable-is-common-divisor-ℕ a b x = + is-decidable-product + ( is-decidable-div-ℕ x a) + ( is-decidable-div-ℕ x b) ``` ### Any greatest common divisor is a common divisor ```agda -is-common-divisor-is-gcd-ℕ : - (a b d : ℕ) → is-gcd-ℕ a b d → is-common-divisor-ℕ a b d -is-common-divisor-is-gcd-ℕ a b d H = pr2 (H d) (refl-div-ℕ d) +abstract + is-common-divisor-is-gcd-ℕ : + (a b d : ℕ) → is-gcd-ℕ a b d → is-common-divisor-ℕ a b d + is-common-divisor-is-gcd-ℕ a b d H = pr2 (H d) (refl-div-ℕ d) ``` ### Uniqueness of greatest common divisors ```agda -uniqueness-is-gcd-ℕ : - (a b d d' : ℕ) → is-gcd-ℕ a b d → is-gcd-ℕ a b d' → d ＝ d' -uniqueness-is-gcd-ℕ a b d d' H H' = - antisymmetric-div-ℕ d d' - ( pr1 (H' d) (is-common-divisor-is-gcd-ℕ a b d H)) - ( pr1 (H d') (is-common-divisor-is-gcd-ℕ a b d' H')) +abstract + uniqueness-is-gcd-ℕ : + (a b d d' : ℕ) → is-gcd-ℕ a b d → is-gcd-ℕ a b d' → d ＝ d' + uniqueness-is-gcd-ℕ a b d d' H H' = + antisymmetric-div-ℕ d d' + ( pr1 (H' d) (is-common-divisor-is-gcd-ℕ a b d H)) + ( pr1 (H d') (is-common-divisor-is-gcd-ℕ a b d' H')) ``` ### If `d` is a common divisor of `a` and `b`, and `a + b ≠ 0`, then `d ≤ a + b` ```agda -leq-sum-is-common-divisor-ℕ' : - (a b d : ℕ) → - is-successor-ℕ (a +ℕ b) → is-common-divisor-ℕ a b d → leq-ℕ d (a +ℕ b) -leq-sum-is-common-divisor-ℕ' a zero-ℕ d (pair k p) H = - concatenate-leq-eq-ℕ d - ( leq-div-succ-ℕ d k (concatenate-div-eq-ℕ (pr1 H) p)) - ( inv p) -leq-sum-is-common-divisor-ℕ' a (succ-ℕ b) d (pair k p) H = - leq-div-succ-ℕ d (a +ℕ b) (div-add-ℕ d a (succ-ℕ b) (pr1 H) (pr2 H)) - -leq-sum-is-common-divisor-ℕ : - (a b d : ℕ) → - is-nonzero-ℕ (a +ℕ b) → is-common-divisor-ℕ a b d → leq-ℕ d (a +ℕ b) -leq-sum-is-common-divisor-ℕ a b d H = - leq-sum-is-common-divisor-ℕ' a b d (is-successor-is-nonzero-ℕ H) +abstract + leq-sum-is-common-divisor-ℕ' : + (a b d : ℕ) → + is-successor-ℕ (a +ℕ b) → is-common-divisor-ℕ a b d → leq-ℕ d (a +ℕ b) + leq-sum-is-common-divisor-ℕ' a zero-ℕ d (pair k p) H = + concatenate-leq-eq-ℕ d + ( leq-div-succ-ℕ d k (concatenate-div-eq-ℕ (pr1 H) p)) + ( inv p) + leq-sum-is-common-divisor-ℕ' a (succ-ℕ b) d (pair k p) H = + leq-div-succ-ℕ d (a +ℕ b) (div-add-ℕ d a (succ-ℕ b) (pr1 H) (pr2 H)) + + leq-sum-is-common-divisor-ℕ : + (a b d : ℕ) → + is-nonzero-ℕ (a +ℕ b) → is-common-divisor-ℕ a b d → leq-ℕ d (a +ℕ b) + leq-sum-is-common-divisor-ℕ a b d H = + leq-sum-is-common-divisor-ℕ' a b d (is-successor-is-nonzero-ℕ H) ``` ### Being a multiple of the greatest common divisor is decidable ```agda -is-decidable-is-multiple-of-gcd-ℕ : - (a b : ℕ) → is-decidable-family (is-multiple-of-gcd-ℕ a b) -is-decidable-is-multiple-of-gcd-ℕ a b n = - is-decidable-function-type' - ( is-decidable-neg (is-decidable-is-zero-ℕ (a +ℕ b))) - ( λ np → - is-decidable-product - ( is-decidable-neg (is-decidable-is-zero-ℕ n)) - ( is-decidable-bounded-Π-ℕ - ( is-common-divisor-ℕ a b) - ( λ x → div-ℕ x n) - ( is-decidable-is-common-divisor-ℕ a b) - ( λ x → is-decidable-div-ℕ x n) - ( a +ℕ b) - ( λ x → leq-sum-is-common-divisor-ℕ a b x np))) +abstract + is-decidable-is-multiple-of-gcd-ℕ : + (a b : ℕ) → is-decidable-family (is-multiple-of-gcd-ℕ a b) + is-decidable-is-multiple-of-gcd-ℕ a b n = + is-decidable-function-type' + ( is-decidable-neg (is-decidable-is-zero-ℕ (a +ℕ b))) + ( λ np → + is-decidable-product + ( is-decidable-neg (is-decidable-is-zero-ℕ n)) + ( is-decidable-bounded-Π-ℕ + ( is-common-divisor-ℕ a b) + ( λ x → div-ℕ x n) + ( is-decidable-is-common-divisor-ℕ a b) + ( λ x → is-decidable-div-ℕ x n) + ( a +ℕ b) + ( λ x → leq-sum-is-common-divisor-ℕ a b x np))) ``` ### The sum of `a` and `b` is a multiple of the greatest common divisor of `a` and `b` ```agda -sum-is-multiple-of-gcd-ℕ : (a b : ℕ) → is-multiple-of-gcd-ℕ a b (a +ℕ b) -pr1 (sum-is-multiple-of-gcd-ℕ a b np) = np -pr2 (sum-is-multiple-of-gcd-ℕ a b np) x H = div-add-ℕ x a b (pr1 H) (pr2 H) +abstract + sum-is-multiple-of-gcd-ℕ : (a b : ℕ) → is-multiple-of-gcd-ℕ a b (a +ℕ b) + pr1 (sum-is-multiple-of-gcd-ℕ a b np) = np + pr2 (sum-is-multiple-of-gcd-ℕ a b np) x H = div-add-ℕ x a b (pr1 H) (pr2 H) ``` ### The greatest common divisor exists @@ -187,40 +194,42 @@ is-lower-bound-gcd-ℕ a b = pr2 (pr2 (GCD-ℕ a b)) ### `a + b = 0` if and only if `gcd a b = 0` ```agda -is-zero-gcd-ℕ : - (a b : ℕ) → is-zero-ℕ (a +ℕ b) → is-zero-ℕ (gcd-ℕ a b) -is-zero-gcd-ℕ a b p = - is-zero-leq-zero-ℕ - ( gcd-ℕ a b) - ( concatenate-leq-eq-ℕ +abstract + is-zero-gcd-ℕ : + (a b : ℕ) → is-zero-ℕ (a +ℕ b) → is-zero-ℕ (gcd-ℕ a b) + is-zero-gcd-ℕ a b p = + is-zero-leq-zero-ℕ ( gcd-ℕ a b) - ( is-lower-bound-gcd-ℕ a b - ( a +ℕ b) - ( sum-is-multiple-of-gcd-ℕ a b)) - ( p)) - -is-zero-gcd-zero-zero-ℕ : is-zero-ℕ (gcd-ℕ zero-ℕ zero-ℕ) -is-zero-gcd-zero-zero-ℕ = is-zero-gcd-ℕ zero-ℕ zero-ℕ refl - -is-zero-add-is-zero-gcd-ℕ : - (a b : ℕ) → is-zero-ℕ (gcd-ℕ a b) → is-zero-ℕ (a +ℕ b) -is-zero-add-is-zero-gcd-ℕ a b H = - double-negation-elim-is-decidable - ( is-decidable-is-zero-ℕ (a +ℕ b)) - ( λ f → pr1 (is-multiple-of-gcd-gcd-ℕ a b f) H) + ( concatenate-leq-eq-ℕ + ( gcd-ℕ a b) + ( is-lower-bound-gcd-ℕ a b + ( a +ℕ b) + ( sum-is-multiple-of-gcd-ℕ a b)) + ( p)) + + is-zero-gcd-zero-zero-ℕ : is-zero-ℕ (gcd-ℕ zero-ℕ zero-ℕ) + is-zero-gcd-zero-zero-ℕ = is-zero-gcd-ℕ zero-ℕ zero-ℕ refl + + is-zero-add-is-zero-gcd-ℕ : + (a b : ℕ) → is-zero-ℕ (gcd-ℕ a b) → is-zero-ℕ (a +ℕ b) + is-zero-add-is-zero-gcd-ℕ a b H = + double-negation-elim-is-decidable + ( is-decidable-is-zero-ℕ (a +ℕ b)) + ( λ f → pr1 (is-multiple-of-gcd-gcd-ℕ a b f) H) ``` ### If at least one of `a` and `b` is nonzero, then their gcd is nonzero ```agda -is-nonzero-gcd-ℕ : - (a b : ℕ) → is-nonzero-ℕ (a +ℕ b) → is-nonzero-ℕ (gcd-ℕ a b) -is-nonzero-gcd-ℕ a b ne = pr1 (is-multiple-of-gcd-gcd-ℕ a b ne) - -is-successor-gcd-ℕ : - (a b : ℕ) → is-nonzero-ℕ (a +ℕ b) → is-successor-ℕ (gcd-ℕ a b) -is-successor-gcd-ℕ a b ne = - is-successor-is-nonzero-ℕ (is-nonzero-gcd-ℕ a b ne) +abstract + is-nonzero-gcd-ℕ : + (a b : ℕ) → is-nonzero-ℕ (a +ℕ b) → is-nonzero-ℕ (gcd-ℕ a b) + is-nonzero-gcd-ℕ a b ne = pr1 (is-multiple-of-gcd-gcd-ℕ a b ne) + + is-successor-gcd-ℕ : + (a b : ℕ) → is-nonzero-ℕ (a +ℕ b) → is-successor-ℕ (gcd-ℕ a b) + is-successor-gcd-ℕ a b ne = + is-successor-is-nonzero-ℕ (is-nonzero-gcd-ℕ a b ne) ``` ### Any common divisor is also a divisor of the greatest common divisor @@ -237,15 +246,16 @@ div-gcd-is-common-divisor-ℕ a b x H with ### If every common divisor divides a number `r < gcd a b`, then `r = 0` ```agda -is-zero-is-common-divisor-le-gcd-ℕ : - (a b r : ℕ) → le-ℕ r (gcd-ℕ a b) → - ((x : ℕ) → is-common-divisor-ℕ a b x → div-ℕ x r) → is-zero-ℕ r -is-zero-is-common-divisor-le-gcd-ℕ a b r l d with is-decidable-is-zero-ℕ r -... | inl H = H -... | inr x = - ex-falso - ( contradiction-le-ℕ r (gcd-ℕ a b) l - ( is-lower-bound-gcd-ℕ a b r (λ np → pair x d))) +abstract + is-zero-is-common-divisor-le-gcd-ℕ : + (a b r : ℕ) → le-ℕ r (gcd-ℕ a b) → + ((x : ℕ) → is-common-divisor-ℕ a b x → div-ℕ x r) → is-zero-ℕ r + is-zero-is-common-divisor-le-gcd-ℕ a b r l d with is-decidable-is-zero-ℕ r + ... | inl H = H + ... | inr x = + ex-falso + ( contradiction-le-ℕ r (gcd-ℕ a b) l + ( is-lower-bound-gcd-ℕ a b r (λ np → pair x d))) ``` ### Any divisor of `gcd a b` is a common divisor of `a` and `b` @@ -344,17 +354,18 @@ pr2 (is-gcd-gcd-ℕ a b x) = is-common-divisor-div-gcd-ℕ a b x ### The gcd is commutative ```agda -is-commutative-gcd-ℕ : (a b : ℕ) → gcd-ℕ a b ＝ gcd-ℕ b a -is-commutative-gcd-ℕ a b = - antisymmetric-div-ℕ - ( gcd-ℕ a b) - ( gcd-ℕ b a) - ( pr1 (is-gcd-gcd-ℕ b a (gcd-ℕ a b)) (σ (is-common-divisor-gcd-ℕ a b))) - ( pr1 (is-gcd-gcd-ℕ a b (gcd-ℕ b a)) (σ (is-common-divisor-gcd-ℕ b a))) - where - σ : {A B : UU lzero} → A × B → B × A - pr1 (σ (pair x y)) = y - pr2 (σ (pair x y)) = x +abstract + is-commutative-gcd-ℕ : (a b : ℕ) → gcd-ℕ a b ＝ gcd-ℕ b a + is-commutative-gcd-ℕ a b = + antisymmetric-div-ℕ + ( gcd-ℕ a b) + ( gcd-ℕ b a) + ( pr1 (is-gcd-gcd-ℕ b a (gcd-ℕ a b)) (σ (is-common-divisor-gcd-ℕ a b))) + ( pr1 (is-gcd-gcd-ℕ a b (gcd-ℕ b a)) (σ (is-common-divisor-gcd-ℕ b a))) + where + σ : {A B : UU lzero} → A × B → B × A + pr1 (σ (pair x y)) = y + pr2 (σ (pair x y)) = x ``` ### If `d` is a common divisor of `a` and `b`, then `kd` is a common divisor of `ka` and `kb` @@ -381,43 +392,47 @@ reflects-is-common-divisor-mul-ℕ k a b d H = ### `gcd-ℕ 1 b ＝ 1` ```agda -is-one-is-gcd-one-ℕ : {b x : ℕ} → is-gcd-ℕ 1 b x → is-one-ℕ x -is-one-is-gcd-one-ℕ {b} {x} H = - is-one-div-one-ℕ x (pr1 (is-common-divisor-is-gcd-ℕ 1 b x H)) +abstract + is-one-is-gcd-one-ℕ : {b x : ℕ} → is-gcd-ℕ 1 b x → is-one-ℕ x + is-one-is-gcd-one-ℕ {b} {x} H = + is-one-div-one-ℕ x (pr1 (is-common-divisor-is-gcd-ℕ 1 b x H)) -is-one-gcd-one-ℕ : (b : ℕ) → is-one-ℕ (gcd-ℕ 1 b) -is-one-gcd-one-ℕ b = is-one-is-gcd-one-ℕ (is-gcd-gcd-ℕ 1 b) + is-one-gcd-one-ℕ : (b : ℕ) → is-one-ℕ (gcd-ℕ 1 b) + is-one-gcd-one-ℕ b = is-one-is-gcd-one-ℕ (is-gcd-gcd-ℕ 1 b) ``` ### `gcd-ℕ a 1 ＝ 1` ```agda -is-one-is-gcd-one-ℕ' : {a x : ℕ} → is-gcd-ℕ a 1 x → is-one-ℕ x -is-one-is-gcd-one-ℕ' {a} {x} H = - is-one-div-one-ℕ x (pr2 (is-common-divisor-is-gcd-ℕ a 1 x H)) +abstract + is-one-is-gcd-one-ℕ' : {a x : ℕ} → is-gcd-ℕ a 1 x → is-one-ℕ x + is-one-is-gcd-one-ℕ' {a} {x} H = + is-one-div-one-ℕ x (pr2 (is-common-divisor-is-gcd-ℕ a 1 x H)) -is-one-gcd-one-ℕ' : (a : ℕ) → is-one-ℕ (gcd-ℕ a 1) -is-one-gcd-one-ℕ' a = is-one-is-gcd-one-ℕ' (is-gcd-gcd-ℕ a 1) + is-one-gcd-one-ℕ' : (a : ℕ) → is-one-ℕ (gcd-ℕ a 1) + is-one-gcd-one-ℕ' a = is-one-is-gcd-one-ℕ' (is-gcd-gcd-ℕ a 1) ``` ### `gcd-ℕ 0 b ＝ b` ```agda -is-id-is-gcd-zero-ℕ : {b x : ℕ} → gcd-ℕ 0 b ＝ x → x ＝ b -is-id-is-gcd-zero-ℕ {b} {x} H = antisymmetric-div-ℕ x b - (pr2 (is-common-divisor-is-gcd-ℕ 0 b x - (tr (λ t → is-gcd-ℕ 0 b t) H (is-gcd-gcd-ℕ 0 b)))) - (tr (λ t → div-ℕ b t) H - (div-gcd-is-common-divisor-ℕ 0 b b - (pair' (div-zero-ℕ b) (refl-div-ℕ b)))) +abstract + is-id-is-gcd-zero-ℕ : {b x : ℕ} → gcd-ℕ 0 b ＝ x → x ＝ b + is-id-is-gcd-zero-ℕ {b} {x} H = antisymmetric-div-ℕ x b + (pr2 (is-common-divisor-is-gcd-ℕ 0 b x + (tr (λ t → is-gcd-ℕ 0 b t) H (is-gcd-gcd-ℕ 0 b)))) + (tr (λ t → div-ℕ b t) H + (div-gcd-is-common-divisor-ℕ 0 b b + (pair' (div-zero-ℕ b) (refl-div-ℕ b)))) ``` ### `gcd-ℕ a 0 ＝ a` ```agda -is-id-is-gcd-zero-ℕ' : {a x : ℕ} → gcd-ℕ a 0 ＝ x → x ＝ a -is-id-is-gcd-zero-ℕ' {a} {x} H = is-id-is-gcd-zero-ℕ {a} {x} - ((is-commutative-gcd-ℕ 0 a) ∙ H) +abstract + is-id-is-gcd-zero-ℕ' : {a x : ℕ} → gcd-ℕ a 0 ＝ x → x ＝ a + is-id-is-gcd-zero-ℕ' {a} {x} H = is-id-is-gcd-zero-ℕ {a} {x} + ((is-commutative-gcd-ℕ 0 a) ∙ H) ``` ### Consider a common divisor `d` of `a` and `b` and let `e` be a divisor of `d`. Then any divisor of `d/e` is a common divisor of `a/e` and `b/e` @@ -456,32 +471,33 @@ pr2 (pr2 (simplify-is-common-divisor-quotient-div-ℕ nz H) K) = ### The greatest common divisor of `a/d` and `b/d` is `gcd(a,b)/d` ```agda -is-gcd-quotient-div-gcd-ℕ : - {a b d : ℕ} → is-nonzero-ℕ d → (H : is-common-divisor-ℕ a b d) → - is-gcd-ℕ - ( quotient-div-ℕ d a (pr1 H)) - ( quotient-div-ℕ d b (pr2 H)) - ( quotient-div-ℕ d - ( gcd-ℕ a b) - ( div-gcd-is-common-divisor-ℕ a b d H)) -is-gcd-quotient-div-gcd-ℕ {a} {b} {d} nz H x = - logical-equivalence-reasoning - is-common-divisor-ℕ +abstract + is-gcd-quotient-div-gcd-ℕ : + {a b d : ℕ} → is-nonzero-ℕ d → (H : is-common-divisor-ℕ a b d) → + is-gcd-ℕ ( quotient-div-ℕ d a (pr1 H)) ( quotient-div-ℕ d b (pr2 H)) - ( x) - ↔ is-common-divisor-ℕ a b (x *ℕ d) - by simplify-is-common-divisor-quotient-div-ℕ nz H - ↔ div-ℕ (x *ℕ d) (gcd-ℕ a b) - by is-gcd-gcd-ℕ a b (x *ℕ d) - ↔ div-ℕ x - ( quotient-div-ℕ d - ( gcd-ℕ a b) - ( div-gcd-is-common-divisor-ℕ a b d H)) - by - inv-iff - ( simplify-div-quotient-div-ℕ nz - ( div-gcd-is-common-divisor-ℕ a b d H)) + ( quotient-div-ℕ d + ( gcd-ℕ a b) + ( div-gcd-is-common-divisor-ℕ a b d H)) + is-gcd-quotient-div-gcd-ℕ {a} {b} {d} nz H x = + logical-equivalence-reasoning + is-common-divisor-ℕ + ( quotient-div-ℕ d a (pr1 H)) + ( quotient-div-ℕ d b (pr2 H)) + ( x) + ↔ is-common-divisor-ℕ a b (x *ℕ d) + by simplify-is-common-divisor-quotient-div-ℕ nz H + ↔ div-ℕ (x *ℕ d) (gcd-ℕ a b) + by is-gcd-gcd-ℕ a b (x *ℕ d) + ↔ div-ℕ x + ( quotient-div-ℕ d + ( gcd-ℕ a b) + ( div-gcd-is-common-divisor-ℕ a b d H)) + by + inv-iff + ( simplify-div-quotient-div-ℕ nz + ( div-gcd-is-common-divisor-ℕ a b d H)) ``` ## References diff --git a/src/elementary-number-theory/group-of-integers.lagda.md b/src/elementary-number-theory/group-of-integers.lagda.md index 358f281d2d..1cd08f5e7f 100644 --- a/src/elementary-number-theory/group-of-integers.lagda.md +++ b/src/elementary-number-theory/group-of-integers.lagda.md @@ -22,8 +22,9 @@ open import group-theory.semigroups ## Idea -The type of integers, equipped with a zero-element, addition, and negatives, -forms a group. +The type of [integers](elementary-number-theory.integers.md), equipped with +zero, [addition](elementary-number-theory.addition-integers.md), and negation, +forms a [group](group-theory.groups.md). ## Definition diff --git a/src/elementary-number-theory/inequality-integers.lagda.md b/src/elementary-number-theory/inequality-integers.lagda.md index ebb650550e..09988385be 100644 --- a/src/elementary-number-theory/inequality-integers.lagda.md +++ b/src/elementary-number-theory/inequality-integers.lagda.md @@ -83,20 +83,21 @@ leq-zero-one-ℤ = star ### Inequality on the integers is reflexive, antisymmetric and transitive ```agda -refl-leq-ℤ : (k : ℤ) → leq-ℤ k k -refl-leq-ℤ k = tr is-nonnegative-ℤ (inv (right-inverse-law-add-ℤ k)) star - -antisymmetric-leq-ℤ : {x y : ℤ} → leq-ℤ x y → leq-ℤ y x → x ＝ y -antisymmetric-leq-ℤ {x} {y} H K = - eq-diff-ℤ - ( is-zero-is-nonnegative-neg-is-nonnegative-ℤ K - ( is-nonnegative-eq-ℤ (inv (distributive-neg-diff-ℤ x y)) H)) - -transitive-leq-ℤ : (k l m : ℤ) → leq-ℤ l m → leq-ℤ k l → leq-ℤ k m -transitive-leq-ℤ k l m H K = - is-nonnegative-eq-ℤ - ( triangle-diff-ℤ m l k) - ( is-nonnegative-add-ℤ H K) +abstract + refl-leq-ℤ : (k : ℤ) → leq-ℤ k k + refl-leq-ℤ k = tr is-nonnegative-ℤ (inv (right-inverse-law-add-ℤ k)) star + + antisymmetric-leq-ℤ : {x y : ℤ} → leq-ℤ x y → leq-ℤ y x → x ＝ y + antisymmetric-leq-ℤ {x} {y} H K = + eq-diff-ℤ + ( is-zero-is-nonnegative-neg-is-nonnegative-ℤ K + ( is-nonnegative-eq-ℤ (inv (distributive-neg-diff-ℤ x y)) H)) + + transitive-leq-ℤ : (k l m : ℤ) → leq-ℤ l m → leq-ℤ k l → leq-ℤ k m + transitive-leq-ℤ k l m H K = + is-nonnegative-eq-ℤ + ( triangle-diff-ℤ m l k) + ( is-nonnegative-add-ℤ H K) ``` ### Inequality on the integers is decidable @@ -141,16 +142,17 @@ linear-leq-ℤ x y = ### An integer is lesser than its successor ```agda -succ-leq-ℤ : (k : ℤ) → leq-ℤ k (succ-ℤ k) -succ-leq-ℤ k = - is-nonnegative-eq-ℤ - ( inv - ( ( left-successor-law-add-ℤ k (neg-ℤ k)) ∙ - ( ap succ-ℤ (right-inverse-law-add-ℤ k)))) - ( star) - -leq-ℤ-succ-leq-ℤ : (k l : ℤ) → leq-ℤ k l → leq-ℤ k (succ-ℤ l) -leq-ℤ-succ-leq-ℤ k l = transitive-leq-ℤ k l (succ-ℤ l) (succ-leq-ℤ l) +abstract + succ-leq-ℤ : (k : ℤ) → leq-ℤ k (succ-ℤ k) + succ-leq-ℤ k = + is-nonnegative-eq-ℤ + ( inv + ( ( left-successor-law-add-ℤ k (neg-ℤ k)) ∙ + ( ap succ-ℤ (right-inverse-law-add-ℤ k)))) + ( star) + + leq-ℤ-succ-leq-ℤ : (k l : ℤ) → leq-ℤ k l → leq-ℤ k (succ-ℤ l) + leq-ℤ-succ-leq-ℤ k l = transitive-leq-ℤ k l (succ-ℤ l) (succ-leq-ℤ l) ``` ### Chaining rules for equality and inequality @@ -172,25 +174,26 @@ concatenate-eq-leq-ℤ y refl H = H ### Addition on the integers preserves inequality ```agda -preserves-leq-left-add-ℤ : - (z x y : ℤ) → leq-ℤ x y → leq-ℤ (x +ℤ z) (y +ℤ z) -preserves-leq-left-add-ℤ z x y = - is-nonnegative-eq-ℤ (inv (right-translation-diff-ℤ y x z)) - -preserves-leq-right-add-ℤ : - (z x y : ℤ) → leq-ℤ x y → leq-ℤ (z +ℤ x) (z +ℤ y) -preserves-leq-right-add-ℤ z x y = - is-nonnegative-eq-ℤ (inv (left-translation-diff-ℤ y x z)) - -preserves-leq-add-ℤ : - {a b c d : ℤ} → leq-ℤ a b → leq-ℤ c d → leq-ℤ (a +ℤ c) (b +ℤ d) -preserves-leq-add-ℤ {a} {b} {c} {d} H K = - transitive-leq-ℤ - ( a +ℤ c) - ( b +ℤ c) - ( b +ℤ d) - ( preserves-leq-right-add-ℤ b c d K) - ( preserves-leq-left-add-ℤ c a b H) +abstract + preserves-leq-left-add-ℤ : + (z x y : ℤ) → leq-ℤ x y → leq-ℤ (x +ℤ z) (y +ℤ z) + preserves-leq-left-add-ℤ z x y = + is-nonnegative-eq-ℤ (inv (right-translation-diff-ℤ y x z)) + + preserves-leq-right-add-ℤ : + (z x y : ℤ) → leq-ℤ x y → leq-ℤ (z +ℤ x) (z +ℤ y) + preserves-leq-right-add-ℤ z x y = + is-nonnegative-eq-ℤ (inv (left-translation-diff-ℤ y x z)) + + preserves-leq-add-ℤ : + {a b c d : ℤ} → leq-ℤ a b → leq-ℤ c d → leq-ℤ (a +ℤ c) (b +ℤ d) + preserves-leq-add-ℤ {a} {b} {c} {d} H K = + transitive-leq-ℤ + ( a +ℤ c) + ( b +ℤ c) + ( b +ℤ d) + ( preserves-leq-right-add-ℤ b c d K) + ( preserves-leq-left-add-ℤ c a b H) right-add-hom-leq-ℤ : (z : ℤ) → hom-Poset ℤ-Poset ℤ-Poset pr1 (right-add-hom-leq-ℤ z) x = x +ℤ z @@ -204,31 +207,33 @@ pr2 (left-add-hom-leq-ℤ z) = preserves-leq-right-add-ℤ z ### Addition on the integers reflects inequality ```agda -reflects-leq-left-add-ℤ : - (z x y : ℤ) → leq-ℤ (x +ℤ z) (y +ℤ z) → leq-ℤ x y -reflects-leq-left-add-ℤ z x y = - is-nonnegative-eq-ℤ (right-translation-diff-ℤ y x z) - -reflects-leq-right-add-ℤ : - (z x y : ℤ) → leq-ℤ (z +ℤ x) (z +ℤ y) → leq-ℤ x y -reflects-leq-right-add-ℤ z x y = - is-nonnegative-eq-ℤ (left-translation-diff-ℤ y x z) +abstract + reflects-leq-left-add-ℤ : + (z x y : ℤ) → leq-ℤ (x +ℤ z) (y +ℤ z) → leq-ℤ x y + reflects-leq-left-add-ℤ z x y = + is-nonnegative-eq-ℤ (right-translation-diff-ℤ y x z) + + reflects-leq-right-add-ℤ : + (z x y : ℤ) → leq-ℤ (z +ℤ x) (z +ℤ y) → leq-ℤ x y + reflects-leq-right-add-ℤ z x y = + is-nonnegative-eq-ℤ (left-translation-diff-ℤ y x z) ``` ### The inclusion of ℕ into ℤ preserves inequality ```agda -leq-int-ℕ : (x y : ℕ) → leq-ℕ x y → leq-ℤ (int-ℕ x) (int-ℕ y) -leq-int-ℕ zero-ℕ y H = - tr - ( is-nonnegative-ℤ) - ( inv (right-unit-law-add-ℤ (int-ℕ y))) - ( is-nonnegative-int-ℕ y) -leq-int-ℕ (succ-ℕ x) (succ-ℕ y) H = tr (is-nonnegative-ℤ) - ( inv (diff-succ-ℤ (int-ℕ y) (int-ℕ x)) ∙ - ( ap (_-ℤ (succ-ℤ (int-ℕ x))) (succ-int-ℕ y) ∙ - ap ((int-ℕ (succ-ℕ y)) -ℤ_) (succ-int-ℕ x))) - ( leq-int-ℕ x y H) +abstract + leq-int-ℕ : (x y : ℕ) → leq-ℕ x y → leq-ℤ (int-ℕ x) (int-ℕ y) + leq-int-ℕ zero-ℕ y H = + tr + ( is-nonnegative-ℤ) + ( inv (right-unit-law-add-ℤ (int-ℕ y))) + ( is-nonnegative-int-ℕ y) + leq-int-ℕ (succ-ℕ x) (succ-ℕ y) H = tr (is-nonnegative-ℤ) + ( inv (diff-succ-ℤ (int-ℕ y) (int-ℕ x)) ∙ + ( ap (_-ℤ (succ-ℤ (int-ℕ x))) (succ-int-ℕ y) ∙ + ap ((int-ℕ (succ-ℕ y)) -ℤ_) (succ-int-ℕ x))) + ( leq-int-ℕ x y H) ``` ### An integer `x` is nonnegative if and only if `0 ≤ x` @@ -307,64 +312,66 @@ module _ ### Negation of integers reverses inequality ```agda -neg-leq-ℤ : (x y : ℤ) → leq-ℤ x y → leq-ℤ (neg-ℤ y) (neg-ℤ x) -neg-leq-ℤ x y = - tr - ( is-nonnegative-ℤ) - ( ap (_+ℤ neg-ℤ x) (inv (neg-neg-ℤ y)) ∙ - commutative-add-ℤ (neg-ℤ (neg-ℤ y)) (neg-ℤ x)) +abstract + neg-leq-ℤ : (x y : ℤ) → leq-ℤ x y → leq-ℤ (neg-ℤ y) (neg-ℤ x) + neg-leq-ℤ x y = + tr + ( is-nonnegative-ℤ) + ( ap (_+ℤ neg-ℤ x) (inv (neg-neg-ℤ y)) ∙ + commutative-add-ℤ (neg-ℤ (neg-ℤ y)) (neg-ℤ x)) ``` ### Transposing additions over inequalities of integers ```agda -leq-transpose-right-diff-ℤ : (x y z : ℤ) → x ≤-ℤ (y -ℤ z) → x +ℤ z ≤-ℤ y -leq-transpose-right-diff-ℤ x y z x≤y-z = - leq-transpose-is-section-hom-Poset - ( ℤ-Poset) - ( ℤ-Poset) - ( right-add-hom-leq-ℤ z) - ( _-ℤ z) - ( is-section-right-add-neg-ℤ z) - ( x) - ( y) - ( x≤y-z) - -leq-transpose-right-add-ℤ : (x y z : ℤ) → x ≤-ℤ y +ℤ z → x -ℤ z ≤-ℤ y -leq-transpose-right-add-ℤ x y z x≤y+z = - leq-transpose-is-section-hom-Poset - ( ℤ-Poset) - ( ℤ-Poset) - ( right-add-hom-leq-ℤ (neg-ℤ z)) - ( _+ℤ z) - ( is-retraction-right-add-neg-ℤ z) - ( x) - ( y) - ( x≤y+z) - -leq-transpose-left-add-ℤ : (x y z : ℤ) → x +ℤ y ≤-ℤ z → x ≤-ℤ z -ℤ y -leq-transpose-left-add-ℤ x y z x+y≤z = - leq-transpose-is-retraction-hom-Poset - ( ℤ-Poset) - ( ℤ-Poset) - ( _+ℤ y) - ( right-add-hom-leq-ℤ (neg-ℤ y)) - ( is-retraction-right-add-neg-ℤ y) - ( x) - ( z) - ( x+y≤z) - -leq-transpose-left-diff-ℤ : (x y z : ℤ) → x -ℤ y ≤-ℤ z → x ≤-ℤ z +ℤ y -leq-transpose-left-diff-ℤ x y z x-y≤z = - leq-transpose-is-retraction-hom-Poset - ( ℤ-Poset) - ( ℤ-Poset) - ( _-ℤ y) - ( right-add-hom-leq-ℤ y) - ( is-section-right-add-neg-ℤ y) - ( x) - ( z) - ( x-y≤z) +abstract + leq-transpose-right-diff-ℤ : (x y z : ℤ) → x ≤-ℤ (y -ℤ z) → x +ℤ z ≤-ℤ y + leq-transpose-right-diff-ℤ x y z x≤y-z = + leq-transpose-is-section-hom-Poset + ( ℤ-Poset) + ( ℤ-Poset) + ( right-add-hom-leq-ℤ z) + ( _-ℤ z) + ( is-section-right-add-neg-ℤ z) + ( x) + ( y) + ( x≤y-z) + + leq-transpose-right-add-ℤ : (x y z : ℤ) → x ≤-ℤ y +ℤ z → x -ℤ z ≤-ℤ y + leq-transpose-right-add-ℤ x y z x≤y+z = + leq-transpose-is-section-hom-Poset + ( ℤ-Poset) + ( ℤ-Poset) + ( right-add-hom-leq-ℤ (neg-ℤ z)) + ( _+ℤ z) + ( is-retraction-right-add-neg-ℤ z) + ( x) + ( y) + ( x≤y+z) + + leq-transpose-left-add-ℤ : (x y z : ℤ) → x +ℤ y ≤-ℤ z → x ≤-ℤ z -ℤ y + leq-transpose-left-add-ℤ x y z x+y≤z = + leq-transpose-is-retraction-hom-Poset + ( ℤ-Poset) + ( ℤ-Poset) + ( _+ℤ y) + ( right-add-hom-leq-ℤ (neg-ℤ y)) + ( is-retraction-right-add-neg-ℤ y) + ( x) + ( z) + ( x+y≤z) + + leq-transpose-left-diff-ℤ : (x y z : ℤ) → x -ℤ y ≤-ℤ z → x ≤-ℤ z +ℤ y + leq-transpose-left-diff-ℤ x y z x-y≤z = + leq-transpose-is-retraction-hom-Poset + ( ℤ-Poset) + ( ℤ-Poset) + ( _-ℤ y) + ( right-add-hom-leq-ℤ y) + ( is-section-right-add-neg-ℤ y) + ( x) + ( z) + ( x-y≤z) leq-iff-transpose-left-add-ℤ : (x y z : ℤ) → (x +ℤ y ≤-ℤ z) ↔ (x ≤-ℤ z -ℤ y) pr1 (leq-iff-transpose-left-add-ℤ x y z) = leq-transpose-left-add-ℤ x y z diff --git a/src/elementary-number-theory/inequality-natural-numbers.lagda.md b/src/elementary-number-theory/inequality-natural-numbers.lagda.md index f536153a61..4a7adfcad5 100644 --- a/src/elementary-number-theory/inequality-natural-numbers.lagda.md +++ b/src/elementary-number-theory/inequality-natural-numbers.lagda.md @@ -67,12 +67,13 @@ data leq-ℕ' : ℕ → ℕ → UU lzero where ### Inequality on the natural numbers is a proposition ```agda -is-prop-leq-ℕ : - (m n : ℕ) → is-prop (leq-ℕ m n) -is-prop-leq-ℕ zero-ℕ zero-ℕ = is-prop-unit -is-prop-leq-ℕ zero-ℕ (succ-ℕ n) = is-prop-unit -is-prop-leq-ℕ (succ-ℕ m) zero-ℕ = is-prop-empty -is-prop-leq-ℕ (succ-ℕ m) (succ-ℕ n) = is-prop-leq-ℕ m n +abstract + is-prop-leq-ℕ : + (m n : ℕ) → is-prop (leq-ℕ m n) + is-prop-leq-ℕ zero-ℕ zero-ℕ = is-prop-unit + is-prop-leq-ℕ zero-ℕ (succ-ℕ n) = is-prop-unit + is-prop-leq-ℕ (succ-ℕ m) zero-ℕ = is-prop-empty + is-prop-leq-ℕ (succ-ℕ m) (succ-ℕ n) = is-prop-leq-ℕ m n leq-ℕ-Prop : ℕ → ℕ → Prop lzero pr1 (leq-ℕ-Prop m n) = leq-ℕ m n @@ -120,10 +121,11 @@ leq-eq-ℕ m .m refl = refl-leq-ℕ m ### Inequality on the natural numbers is transitive ```agda -transitive-leq-ℕ : is-transitive leq-ℕ -transitive-leq-ℕ zero-ℕ m l p q = star -transitive-leq-ℕ (succ-ℕ n) (succ-ℕ m) (succ-ℕ l) p q = - transitive-leq-ℕ n m l p q +abstract + transitive-leq-ℕ : is-transitive leq-ℕ + transitive-leq-ℕ zero-ℕ m l p q = star + transitive-leq-ℕ (succ-ℕ n) (succ-ℕ m) (succ-ℕ l) p q = + transitive-leq-ℕ n m l p q ``` ### Inequality on the natural numbers is antisymmetric @@ -208,7 +210,7 @@ leq-zero-ℕ : leq-zero-ℕ n = star ``` -### Any natural number less than zero is zero +### Any natural number less than or equal to zero is zero ```agda is-zero-leq-zero-ℕ : @@ -220,25 +222,27 @@ is-zero-leq-zero-ℕ' : is-zero-leq-zero-ℕ' zero-ℕ star = refl ``` -### Any number is nonzero natural number if it is at least `1` +### A natural number is nonzero if and only if it is at least `1` ```agda -leq-one-is-nonzero-ℕ : - (x : ℕ) → is-nonzero-ℕ x → leq-ℕ 1 x -leq-one-is-nonzero-ℕ zero-ℕ H = ex-falso (H refl) -leq-one-is-nonzero-ℕ (succ-ℕ x) H = star - -is-nonzero-leq-one-ℕ : - (x : ℕ) → leq-ℕ 1 x → is-nonzero-ℕ x -is-nonzero-leq-one-ℕ .zero-ℕ () refl +abstract + leq-one-is-nonzero-ℕ : + (x : ℕ) → is-nonzero-ℕ x → leq-ℕ 1 x + leq-one-is-nonzero-ℕ zero-ℕ H = ex-falso (H refl) + leq-one-is-nonzero-ℕ (succ-ℕ x) H = star + + is-nonzero-leq-one-ℕ : + (x : ℕ) → leq-ℕ 1 x → is-nonzero-ℕ x + is-nonzero-leq-one-ℕ .zero-ℕ () refl ``` ### Any natural number is less than or equal to its own successor ```agda -succ-leq-ℕ : (n : ℕ) → n ≤-ℕ (succ-ℕ n) -succ-leq-ℕ zero-ℕ = star -succ-leq-ℕ (succ-ℕ n) = succ-leq-ℕ n +abstract + succ-leq-ℕ : (n : ℕ) → n ≤-ℕ (succ-ℕ n) + succ-leq-ℕ zero-ℕ = star + succ-leq-ℕ (succ-ℕ n) = succ-leq-ℕ n ``` ### Any natural number less than or equal to `n+1` is either less than or equal to `n` or it is `n+1` @@ -257,18 +261,20 @@ decide-leq-succ-ℕ (succ-ℕ m) (succ-ℕ n) l = ### If `m` is less than `n`, then it is less than `n+1` ```agda -preserves-leq-succ-ℕ : - (m n : ℕ) → m ≤-ℕ n → m ≤-ℕ (succ-ℕ n) -preserves-leq-succ-ℕ m n p = transitive-leq-ℕ m n (succ-ℕ n) (succ-leq-ℕ n) p +abstract + preserves-leq-succ-ℕ : + (m n : ℕ) → m ≤-ℕ n → m ≤-ℕ (succ-ℕ n) + preserves-leq-succ-ℕ m n p = transitive-leq-ℕ m n (succ-ℕ n) (succ-leq-ℕ n) p ``` ### The successor of `n` is not less than or equal to `n` ```agda -neg-succ-leq-ℕ : - (n : ℕ) → ¬ (leq-ℕ (succ-ℕ n) n) -neg-succ-leq-ℕ zero-ℕ = id -neg-succ-leq-ℕ (succ-ℕ n) = neg-succ-leq-ℕ n +abstract + neg-succ-leq-ℕ : + (n : ℕ) → ¬ (leq-ℕ (succ-ℕ n) n) + neg-succ-leq-ℕ zero-ℕ = id + neg-succ-leq-ℕ (succ-ℕ n) = neg-succ-leq-ℕ n ``` ### If `m ≤ n + 1` then either `m ≤ n` or `m = n + 1` @@ -292,73 +298,77 @@ cases-leq-succ-reflexive-leq-ℕ {succ-ℕ n} = ### `m ≤ n` if and only if `n + 1 ≰ m` ```agda -contradiction-leq-ℕ : (m n : ℕ) → m ≤-ℕ n → ¬ ((succ-ℕ n) ≤-ℕ m) -contradiction-leq-ℕ (succ-ℕ m) (succ-ℕ n) H K = contradiction-leq-ℕ m n H K +abstract + contradiction-leq-ℕ : (m n : ℕ) → m ≤-ℕ n → ¬ ((succ-ℕ n) ≤-ℕ m) + contradiction-leq-ℕ (succ-ℕ m) (succ-ℕ n) H K = contradiction-leq-ℕ m n H K -contradiction-leq-ℕ' : (m n : ℕ) → (succ-ℕ n) ≤-ℕ m → ¬ (m ≤-ℕ n) -contradiction-leq-ℕ' m n K H = contradiction-leq-ℕ m n H K + contradiction-leq-ℕ' : (m n : ℕ) → (succ-ℕ n) ≤-ℕ m → ¬ (m ≤-ℕ n) + contradiction-leq-ℕ' m n K H = contradiction-leq-ℕ m n H K ``` ### Addition preserves inequality of natural numbers ```agda -preserves-leq-left-add-ℕ : - (k m n : ℕ) → m ≤-ℕ n → (m +ℕ k) ≤-ℕ (n +ℕ k) -preserves-leq-left-add-ℕ zero-ℕ m n = id -preserves-leq-left-add-ℕ (succ-ℕ k) m n H = preserves-leq-left-add-ℕ k m n H - -preserves-leq-right-add-ℕ : (k m n : ℕ) → m ≤-ℕ n → (k +ℕ m) ≤-ℕ (k +ℕ n) -preserves-leq-right-add-ℕ k m n H = - concatenate-eq-leq-eq-ℕ - ( commutative-add-ℕ k m) - ( preserves-leq-left-add-ℕ k m n H) - ( commutative-add-ℕ n k) - -preserves-leq-add-ℕ : - {m m' n n' : ℕ} → m ≤-ℕ m' → n ≤-ℕ n' → (m +ℕ n) ≤-ℕ (m' +ℕ n') -preserves-leq-add-ℕ {m} {m'} {n} {n'} H K = - transitive-leq-ℕ - ( m +ℕ n) - ( m' +ℕ n) - ( m' +ℕ n') - ( preserves-leq-right-add-ℕ m' n n' K) - ( preserves-leq-left-add-ℕ n m m' H) +abstract + preserves-leq-left-add-ℕ : + (k m n : ℕ) → m ≤-ℕ n → (m +ℕ k) ≤-ℕ (n +ℕ k) + preserves-leq-left-add-ℕ zero-ℕ m n = id + preserves-leq-left-add-ℕ (succ-ℕ k) m n H = preserves-leq-left-add-ℕ k m n H + + preserves-leq-right-add-ℕ : (k m n : ℕ) → m ≤-ℕ n → (k +ℕ m) ≤-ℕ (k +ℕ n) + preserves-leq-right-add-ℕ k m n H = + concatenate-eq-leq-eq-ℕ + ( commutative-add-ℕ k m) + ( preserves-leq-left-add-ℕ k m n H) + ( commutative-add-ℕ n k) + + preserves-leq-add-ℕ : + {m m' n n' : ℕ} → m ≤-ℕ m' → n ≤-ℕ n' → (m +ℕ n) ≤-ℕ (m' +ℕ n') + preserves-leq-add-ℕ {m} {m'} {n} {n'} H K = + transitive-leq-ℕ + ( m +ℕ n) + ( m' +ℕ n) + ( m' +ℕ n') + ( preserves-leq-right-add-ℕ m' n n' K) + ( preserves-leq-left-add-ℕ n m m' H) ``` ### Addition reflects inequality of natural numbers ```agda -reflects-leq-left-add-ℕ : - (k m n : ℕ) → (m +ℕ k) ≤-ℕ (n +ℕ k) → m ≤-ℕ n -reflects-leq-left-add-ℕ zero-ℕ m n = id -reflects-leq-left-add-ℕ (succ-ℕ k) m n = reflects-leq-left-add-ℕ k m n - -reflects-leq-right-add-ℕ : - (k m n : ℕ) → (k +ℕ m) ≤-ℕ (k +ℕ n) → m ≤-ℕ n -reflects-leq-right-add-ℕ k m n H = - reflects-leq-left-add-ℕ k m n - ( concatenate-eq-leq-eq-ℕ - ( commutative-add-ℕ m k) - ( H) - ( commutative-add-ℕ k n)) +abstract + reflects-leq-left-add-ℕ : + (k m n : ℕ) → (m +ℕ k) ≤-ℕ (n +ℕ k) → m ≤-ℕ n + reflects-leq-left-add-ℕ zero-ℕ m n = id + reflects-leq-left-add-ℕ (succ-ℕ k) m n = reflects-leq-left-add-ℕ k m n + + reflects-leq-right-add-ℕ : + (k m n : ℕ) → (k +ℕ m) ≤-ℕ (k +ℕ n) → m ≤-ℕ n + reflects-leq-right-add-ℕ k m n H = + reflects-leq-left-add-ℕ k m n + ( concatenate-eq-leq-eq-ℕ + ( commutative-add-ℕ m k) + ( H) + ( commutative-add-ℕ k n)) ``` ### `m ≤ m + n` for any two natural numbers `m` and `n` ```agda -leq-add-ℕ : (m n : ℕ) → m ≤-ℕ (m +ℕ n) -leq-add-ℕ m zero-ℕ = refl-leq-ℕ m -leq-add-ℕ m (succ-ℕ n) = - transitive-leq-ℕ - ( m) - ( m +ℕ n) - ( succ-ℕ (m +ℕ n)) - ( succ-leq-ℕ (m +ℕ n)) - ( leq-add-ℕ m n) - -leq-add-ℕ' : (m n : ℕ) → m ≤-ℕ (n +ℕ m) -leq-add-ℕ' m n = - concatenate-leq-eq-ℕ m (leq-add-ℕ m n) (commutative-add-ℕ m n) +abstract + leq-add-ℕ : (m n : ℕ) → m ≤-ℕ (m +ℕ n) + leq-add-ℕ m zero-ℕ = refl-leq-ℕ m + leq-add-ℕ m (succ-ℕ n) = + transitive-leq-ℕ + ( m) + ( m +ℕ n) + ( succ-ℕ (m +ℕ n)) + ( succ-leq-ℕ (m +ℕ n)) + ( leq-add-ℕ m n) + + leq-add-ℕ' : (m n : ℕ) → m ≤-ℕ (n +ℕ m) + leq-add-ℕ' m n = + concatenate-leq-eq-ℕ m (leq-add-ℕ m n) (commutative-add-ℕ m n) ``` ### We have `n ≤ m` if and only if there is a number `l` such that `l+n=m` @@ -380,85 +390,88 @@ leq-subtraction-ℕ (succ-ℕ n) (succ-ℕ m) l p = ### Multiplication preserves inequality of natural numbers ```agda -preserves-leq-left-mul-ℕ : - (k m n : ℕ) → m ≤-ℕ n → (m *ℕ k) ≤-ℕ (n *ℕ k) -preserves-leq-left-mul-ℕ k zero-ℕ n p = star -preserves-leq-left-mul-ℕ k (succ-ℕ m) (succ-ℕ n) p = - preserves-leq-left-add-ℕ k - ( m *ℕ k) - ( n *ℕ k) - ( preserves-leq-left-mul-ℕ k m n p) - -preserves-leq-right-mul-ℕ : - (k m n : ℕ) → m ≤-ℕ n → (k *ℕ m) ≤-ℕ (k *ℕ n) -preserves-leq-right-mul-ℕ k m n H = - concatenate-eq-leq-eq-ℕ - ( commutative-mul-ℕ k m) - ( preserves-leq-left-mul-ℕ k m n H) - ( commutative-mul-ℕ n k) - -preserves-leq-mul-ℕ : - (m m' n n' : ℕ) → m ≤-ℕ m' → n ≤-ℕ n' → (m *ℕ n) ≤-ℕ (m' *ℕ n') -preserves-leq-mul-ℕ m m' n n' H K = - transitive-leq-ℕ - ( m *ℕ n) - ( m' *ℕ n) - ( m' *ℕ n') - ( preserves-leq-right-mul-ℕ m' n n' K) - ( preserves-leq-left-mul-ℕ n m m' H) +abstract + preserves-leq-left-mul-ℕ : + (k m n : ℕ) → m ≤-ℕ n → (m *ℕ k) ≤-ℕ (n *ℕ k) + preserves-leq-left-mul-ℕ k zero-ℕ n p = star + preserves-leq-left-mul-ℕ k (succ-ℕ m) (succ-ℕ n) p = + preserves-leq-left-add-ℕ k + ( m *ℕ k) + ( n *ℕ k) + ( preserves-leq-left-mul-ℕ k m n p) + + preserves-leq-right-mul-ℕ : + (k m n : ℕ) → m ≤-ℕ n → (k *ℕ m) ≤-ℕ (k *ℕ n) + preserves-leq-right-mul-ℕ k m n H = + concatenate-eq-leq-eq-ℕ + ( commutative-mul-ℕ k m) + ( preserves-leq-left-mul-ℕ k m n H) + ( commutative-mul-ℕ n k) + + preserves-leq-mul-ℕ : + (m m' n n' : ℕ) → m ≤-ℕ m' → n ≤-ℕ n' → (m *ℕ n) ≤-ℕ (m' *ℕ n') + preserves-leq-mul-ℕ m m' n n' H K = + transitive-leq-ℕ + ( m *ℕ n) + ( m' *ℕ n) + ( m' *ℕ n') + ( preserves-leq-right-mul-ℕ m' n n' K) + ( preserves-leq-left-mul-ℕ n m m' H) ``` ### Multiplication by a nonzero natural number reflects inequality of natural numbers ```agda -reflects-order-mul-ℕ : - (k m n : ℕ) → (m *ℕ (succ-ℕ k)) ≤-ℕ (n *ℕ (succ-ℕ k)) → m ≤-ℕ n -reflects-order-mul-ℕ k zero-ℕ n p = star -reflects-order-mul-ℕ k (succ-ℕ m) (succ-ℕ n) p = - reflects-order-mul-ℕ k m n - ( reflects-leq-left-add-ℕ - ( succ-ℕ k) - ( m *ℕ (succ-ℕ k)) - ( n *ℕ (succ-ℕ k)) - ( p)) - -reflects-order-mul-ℕ' : - (k m n : ℕ) → ((succ-ℕ k) *ℕ m) ≤-ℕ ((succ-ℕ k) *ℕ n) → m ≤-ℕ n -reflects-order-mul-ℕ' k m n H = - reflects-order-mul-ℕ k m n - ( concatenate-eq-leq-eq-ℕ - ( commutative-mul-ℕ m (succ-ℕ k)) - ( H) - ( commutative-mul-ℕ (succ-ℕ k) n)) +abstract + reflects-order-mul-ℕ : + (k m n : ℕ) → (m *ℕ (succ-ℕ k)) ≤-ℕ (n *ℕ (succ-ℕ k)) → m ≤-ℕ n + reflects-order-mul-ℕ k zero-ℕ n p = star + reflects-order-mul-ℕ k (succ-ℕ m) (succ-ℕ n) p = + reflects-order-mul-ℕ k m n + ( reflects-leq-left-add-ℕ + ( succ-ℕ k) + ( m *ℕ (succ-ℕ k)) + ( n *ℕ (succ-ℕ k)) + ( p)) + + reflects-order-mul-ℕ' : + (k m n : ℕ) → ((succ-ℕ k) *ℕ m) ≤-ℕ ((succ-ℕ k) *ℕ n) → m ≤-ℕ n + reflects-order-mul-ℕ' k m n H = + reflects-order-mul-ℕ k m n + ( concatenate-eq-leq-eq-ℕ + ( commutative-mul-ℕ m (succ-ℕ k)) + ( H) + ( commutative-mul-ℕ (succ-ℕ k) n)) ``` ### Any number `x` is less than or equal to a nonzero multiple of itself ```agda -leq-mul-ℕ : - (k x : ℕ) → x ≤-ℕ (x *ℕ (succ-ℕ k)) -leq-mul-ℕ k x = - concatenate-eq-leq-ℕ - ( x *ℕ (succ-ℕ k)) - ( inv (right-unit-law-mul-ℕ x)) - ( preserves-leq-right-mul-ℕ x 1 (succ-ℕ k) (leq-zero-ℕ k)) - -leq-mul-ℕ' : - (k x : ℕ) → x ≤-ℕ ((succ-ℕ k) *ℕ x) -leq-mul-ℕ' k x = - concatenate-leq-eq-ℕ x - ( leq-mul-ℕ k x) - ( commutative-mul-ℕ x (succ-ℕ k)) - -leq-mul-is-nonzero-ℕ : - (k x : ℕ) → is-nonzero-ℕ k → x ≤-ℕ (x *ℕ k) -leq-mul-is-nonzero-ℕ k x H with is-successor-is-nonzero-ℕ H -... | (l , refl) = leq-mul-ℕ l x - -leq-mul-is-nonzero-ℕ' : - (k x : ℕ) → is-nonzero-ℕ k → x ≤-ℕ (k *ℕ x) -leq-mul-is-nonzero-ℕ' k x H with is-successor-is-nonzero-ℕ H -... | (l , refl) = leq-mul-ℕ' l x +abstract + leq-mul-ℕ : + (k x : ℕ) → x ≤-ℕ (x *ℕ (succ-ℕ k)) + leq-mul-ℕ k x = + concatenate-eq-leq-ℕ + ( x *ℕ (succ-ℕ k)) + ( inv (right-unit-law-mul-ℕ x)) + ( preserves-leq-right-mul-ℕ x 1 (succ-ℕ k) (leq-zero-ℕ k)) + + leq-mul-ℕ' : + (k x : ℕ) → x ≤-ℕ ((succ-ℕ k) *ℕ x) + leq-mul-ℕ' k x = + concatenate-leq-eq-ℕ x + ( leq-mul-ℕ k x) + ( commutative-mul-ℕ x (succ-ℕ k)) + + leq-mul-is-nonzero-ℕ : + (k x : ℕ) → is-nonzero-ℕ k → x ≤-ℕ (x *ℕ k) + leq-mul-is-nonzero-ℕ k x H with is-successor-is-nonzero-ℕ H + ... | (l , refl) = leq-mul-ℕ l x + + leq-mul-is-nonzero-ℕ' : + (k x : ℕ) → is-nonzero-ℕ k → x ≤-ℕ (k *ℕ x) + leq-mul-is-nonzero-ℕ' k x H with is-successor-is-nonzero-ℕ H + ... | (l , refl) = leq-mul-ℕ' l x ``` ## See also diff --git a/src/elementary-number-theory/inequality-standard-finite-types.lagda.md b/src/elementary-number-theory/inequality-standard-finite-types.lagda.md index d21985ad6f..da2ee242df 100644 --- a/src/elementary-number-theory/inequality-standard-finite-types.lagda.md +++ b/src/elementary-number-theory/inequality-standard-finite-types.lagda.md @@ -31,6 +31,13 @@ open import univalent-combinatorics.standard-finite-types +## Idea + +Inequality on the +[standard finite types](univalent-combinatorics.standard-finite-types.md) is +defined inductively with the rules that all elements of `Fin (succ-ℕ n)` are +`≤ neg-one-Fin n`, and otherwise `inl-Fin a ≤ inl-Fin b` if `a ≤ b`. + ## Definitions ```agda @@ -50,76 +57,94 @@ abstract leq-Fin-Prop : (k : ℕ) → Fin k → Fin k → Prop lzero pr1 (leq-Fin-Prop k x y) = leq-Fin k x y pr2 (leq-Fin-Prop k x y) = is-prop-leq-Fin k x y +``` + +## Properties +### For all `k : Fin (succ-ℕ n)`, `k ≤ neg-one-Fin n` + +```agda leq-neg-one-Fin : (k : ℕ) (x : Fin (succ-ℕ k)) → leq-Fin (succ-ℕ k) x (neg-one-Fin k) leq-neg-one-Fin k x = star +``` -refl-leq-Fin : (k : ℕ) → is-reflexive (leq-Fin k) -refl-leq-Fin (succ-ℕ k) (inl x) = refl-leq-Fin k x -refl-leq-Fin (succ-ℕ k) (inr star) = star +### Inequality on the standard finite types is a partial order -antisymmetric-leq-Fin : - (k : ℕ) (x y : Fin k) → leq-Fin k x y → leq-Fin k y x → x ＝ y -antisymmetric-leq-Fin (succ-ℕ k) (inl x) (inl y) H K = - ap inl (antisymmetric-leq-Fin k x y H K) -antisymmetric-leq-Fin (succ-ℕ k) (inr star) (inr star) H K = refl +```agda +abstract + refl-leq-Fin : (k : ℕ) → is-reflexive (leq-Fin k) + refl-leq-Fin (succ-ℕ k) (inl x) = refl-leq-Fin k x + refl-leq-Fin (succ-ℕ k) (inr star) = star + + antisymmetric-leq-Fin : + (k : ℕ) (x y : Fin k) → leq-Fin k x y → leq-Fin k y x → x ＝ y + antisymmetric-leq-Fin (succ-ℕ k) (inl x) (inl y) H K = + ap inl (antisymmetric-leq-Fin k x y H K) + antisymmetric-leq-Fin (succ-ℕ k) (inr star) (inr star) H K = refl + + transitive-leq-Fin : + (k : ℕ) → is-transitive (leq-Fin k) + transitive-leq-Fin (succ-ℕ k) (inl x) (inl y) (inl z) H K = + transitive-leq-Fin k x y z H K + transitive-leq-Fin (succ-ℕ k) (inl x) (inl y) (inr star) H K = star + transitive-leq-Fin (succ-ℕ k) (inl x) (inr star) (inr star) H K = star + transitive-leq-Fin (succ-ℕ k) (inr star) (inr star) (inr star) H K = star + +Fin-Preorder : ℕ → Preorder lzero lzero +pr1 (Fin-Preorder k) = Fin k +pr1 (pr2 (Fin-Preorder k)) = leq-Fin-Prop k +pr1 (pr2 (pr2 (Fin-Preorder k))) = refl-leq-Fin k +pr2 (pr2 (pr2 (Fin-Preorder k))) = transitive-leq-Fin k + +Fin-Poset : ℕ → Poset lzero lzero +pr1 (Fin-Poset k) = Fin-Preorder k +pr2 (Fin-Poset k) = antisymmetric-leq-Fin k +``` -transitive-leq-Fin : - (k : ℕ) → is-transitive (leq-Fin k) -transitive-leq-Fin (succ-ℕ k) (inl x) (inl y) (inl z) H K = - transitive-leq-Fin k x y z H K -transitive-leq-Fin (succ-ℕ k) (inl x) (inl y) (inr star) H K = star -transitive-leq-Fin (succ-ℕ k) (inl x) (inr star) (inr star) H K = star -transitive-leq-Fin (succ-ℕ k) (inr star) (inr star) (inr star) H K = star +### Concatenation laws with equality +```agda concatenate-eq-leq-eq-Fin : (k : ℕ) {x1 x2 x3 x4 : Fin k} → x1 ＝ x2 → leq-Fin k x2 x3 → x3 ＝ x4 → leq-Fin k x1 x4 concatenate-eq-leq-eq-Fin k refl H refl = H +``` + +### `inl-Fin k ≤ succ-Fin (inl-Fin k)` +```agda leq-succ-Fin : (k : ℕ) (x : Fin k) → leq-Fin (succ-ℕ k) (inl-Fin k x) (succ-Fin (succ-ℕ k) (inl-Fin k x)) leq-succ-Fin (succ-ℕ k) (inl x) = leq-succ-Fin k x leq-succ-Fin (succ-ℕ k) (inr star) = star - -preserves-leq-nat-Fin : - (k : ℕ) {x y : Fin k} → leq-Fin k x y → leq-ℕ (nat-Fin k x) (nat-Fin k y) -preserves-leq-nat-Fin (succ-ℕ k) {inl x} {inl y} H = - preserves-leq-nat-Fin k H -preserves-leq-nat-Fin (succ-ℕ k) {inl x} {inr star} H = - leq-le-ℕ (nat-Fin k x) k (strict-upper-bound-nat-Fin k x) -preserves-leq-nat-Fin (succ-ℕ k) {inr star} {inr star} H = - refl-leq-ℕ k - -reflects-leq-nat-Fin : - (k : ℕ) {x y : Fin k} → leq-ℕ (nat-Fin k x) (nat-Fin k y) → leq-Fin k x y -reflects-leq-nat-Fin (succ-ℕ k) {inl x} {inl y} H = - reflects-leq-nat-Fin k H -reflects-leq-nat-Fin (succ-ℕ k) {inr star} {inl y} H = - ex-falso - ( contradiction-le-ℕ (nat-Fin k y) k (strict-upper-bound-nat-Fin k y) H) -reflects-leq-nat-Fin (succ-ℕ k) {inl x} {inr star} H = star -reflects-leq-nat-Fin (succ-ℕ k) {inr star} {inr star} H = star ``` -### The partial order on the standard finite types +### The canonical embedding of the standard finite types in the natural numbers preserves and reflects inequality ```agda -Fin-Preorder : ℕ → Preorder lzero lzero -pr1 (Fin-Preorder k) = Fin k -pr1 (pr2 (Fin-Preorder k)) = leq-Fin-Prop k -pr1 (pr2 (pr2 (Fin-Preorder k))) = refl-leq-Fin k -pr2 (pr2 (pr2 (Fin-Preorder k))) = transitive-leq-Fin k - -Fin-Poset : ℕ → Poset lzero lzero -pr1 (Fin-Poset k) = Fin-Preorder k -pr2 (Fin-Poset k) = antisymmetric-leq-Fin k +abstract + preserves-leq-nat-Fin : + (k : ℕ) {x y : Fin k} → leq-Fin k x y → leq-ℕ (nat-Fin k x) (nat-Fin k y) + preserves-leq-nat-Fin (succ-ℕ k) {inl x} {inl y} H = + preserves-leq-nat-Fin k H + preserves-leq-nat-Fin (succ-ℕ k) {inl x} {inr star} H = + leq-le-ℕ (nat-Fin k x) k (strict-upper-bound-nat-Fin k x) + preserves-leq-nat-Fin (succ-ℕ k) {inr star} {inr star} H = + refl-leq-ℕ k + + reflects-leq-nat-Fin : + (k : ℕ) {x y : Fin k} → leq-ℕ (nat-Fin k x) (nat-Fin k y) → leq-Fin k x y + reflects-leq-nat-Fin (succ-ℕ k) {inl x} {inl y} H = + reflects-leq-nat-Fin k H + reflects-leq-nat-Fin (succ-ℕ k) {inr star} {inl y} H = + ex-falso + ( contradiction-le-ℕ (nat-Fin k y) k (strict-upper-bound-nat-Fin k y) H) + reflects-leq-nat-Fin (succ-ℕ k) {inl x} {inr star} H = star + reflects-leq-nat-Fin (succ-ℕ k) {inr star} {inr star} H = star ``` -## Properties - ### Ordering on the standard finite types is decidable ```agda diff --git a/src/elementary-number-theory/integers.lagda.md b/src/elementary-number-theory/integers.lagda.md index 2c1ea175d6..f3a2464be5 100644 --- a/src/elementary-number-theory/integers.lagda.md +++ b/src/elementary-number-theory/integers.lagda.md @@ -21,6 +21,7 @@ open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.injective-maps +open import foundation.involutions open import foundation.negated-equality open import foundation.negation open import foundation.propositions @@ -185,10 +186,8 @@ abstract abstract is-equiv-succ-ℤ : is-equiv succ-ℤ - pr1 (pr1 is-equiv-succ-ℤ) = pred-ℤ - pr2 (pr1 is-equiv-succ-ℤ) = is-section-pred-ℤ - pr1 (pr2 is-equiv-succ-ℤ) = pred-ℤ - pr2 (pr2 is-equiv-succ-ℤ) = is-retraction-pred-ℤ + is-equiv-succ-ℤ = + is-equiv-is-invertible pred-ℤ is-section-pred-ℤ is-retraction-pred-ℤ equiv-succ-ℤ : ℤ ≃ ℤ pr1 equiv-succ-ℤ = succ-ℤ @@ -196,10 +195,8 @@ pr2 equiv-succ-ℤ = is-equiv-succ-ℤ abstract is-equiv-pred-ℤ : is-equiv pred-ℤ - pr1 (pr1 is-equiv-pred-ℤ) = succ-ℤ - pr2 (pr1 is-equiv-pred-ℤ) = is-retraction-pred-ℤ - pr1 (pr2 is-equiv-pred-ℤ) = succ-ℤ - pr2 (pr2 is-equiv-pred-ℤ) = is-section-pred-ℤ + is-equiv-pred-ℤ = + is-equiv-is-invertible succ-ℤ is-retraction-pred-ℤ is-section-pred-ℤ equiv-pred-ℤ : ℤ ≃ ℤ pr1 equiv-pred-ℤ = pred-ℤ @@ -211,8 +208,7 @@ pr2 equiv-pred-ℤ = is-equiv-pred-ℤ ```agda abstract is-injective-succ-ℤ : is-injective succ-ℤ - is-injective-succ-ℤ {x} {y} p = - inv (is-retraction-pred-ℤ x) ∙ ap pred-ℤ p ∙ is-retraction-pred-ℤ y + is-injective-succ-ℤ = is-injective-is-equiv is-equiv-succ-ℤ has-no-fixed-points-succ-ℤ : (x : ℤ) → succ-ℤ x ≠ x has-no-fixed-points-succ-ℤ (inl zero-ℕ) () @@ -220,7 +216,7 @@ abstract has-no-fixed-points-succ-ℤ (inr (inl star)) () ``` -### The negative function is an involution +### The negation function is an involution ```agda abstract @@ -228,18 +224,23 @@ abstract neg-neg-ℤ (inl n) = refl neg-neg-ℤ (inr (inl star)) = refl neg-neg-ℤ (inr (inr n)) = refl +``` + +### The negation function is an equivalence +```agda abstract is-equiv-neg-ℤ : is-equiv neg-ℤ - pr1 (pr1 is-equiv-neg-ℤ) = neg-ℤ - pr2 (pr1 is-equiv-neg-ℤ) = neg-neg-ℤ - pr1 (pr2 is-equiv-neg-ℤ) = neg-ℤ - pr2 (pr2 is-equiv-neg-ℤ) = neg-neg-ℤ + is-equiv-neg-ℤ = is-equiv-is-involution neg-neg-ℤ equiv-neg-ℤ : ℤ ≃ ℤ pr1 equiv-neg-ℤ = neg-ℤ pr2 equiv-neg-ℤ = is-equiv-neg-ℤ +``` + +### The negation function is an embedding +```agda abstract is-emb-neg-ℤ : is-emb neg-ℤ is-emb-neg-ℤ = is-emb-is-equiv is-equiv-neg-ℤ @@ -247,20 +248,34 @@ abstract emb-neg-ℤ : ℤ ↪ ℤ pr1 emb-neg-ℤ = neg-ℤ pr2 emb-neg-ℤ = is-emb-neg-ℤ +``` +### The negation of the predecessor of `x` is the successor of the negation of `x` + +```agda abstract neg-pred-ℤ : (k : ℤ) → neg-ℤ (pred-ℤ k) ＝ succ-ℤ (neg-ℤ k) neg-pred-ℤ (inl x) = refl neg-pred-ℤ (inr (inl star)) = refl neg-pred-ℤ (inr (inr zero-ℕ)) = refl neg-pred-ℤ (inr (inr (succ-ℕ x))) = refl +``` + +### The negation of the successor of `x` is the predecessor of the negation of `x` +```agda +abstract neg-succ-ℤ : (x : ℤ) → neg-ℤ (succ-ℤ x) ＝ pred-ℤ (neg-ℤ x) neg-succ-ℤ (inl zero-ℕ) = refl neg-succ-ℤ (inl (succ-ℕ x)) = refl neg-succ-ℤ (inr (inl star)) = refl neg-succ-ℤ (inr (inr x)) = refl +``` + +### The predecessor of the negation of `x` is the negation of the successor of `x` +```agda +abstract pred-neg-ℤ : (k : ℤ) → pred-ℤ (neg-ℤ k) ＝ neg-ℤ (succ-ℤ k) pred-neg-ℤ (inl zero-ℕ) = refl @@ -269,12 +284,12 @@ abstract pred-neg-ℤ (inr (inr x)) = refl ``` -### The negative function is injective +### The negation function is injective ```agda abstract is-injective-neg-ℤ : is-injective neg-ℤ - is-injective-neg-ℤ {x} {y} p = inv (neg-neg-ℤ x) ∙ ap neg-ℤ p ∙ neg-neg-ℤ y + is-injective-neg-ℤ = is-injective-is-equiv is-equiv-neg-ℤ ``` ### The integer successor of a natural number is the successor of the natural number diff --git a/src/elementary-number-theory/lower-bounds-natural-numbers.lagda.md b/src/elementary-number-theory/lower-bounds-natural-numbers.lagda.md index f0b34d2825..2af4f9c643 100644 --- a/src/elementary-number-theory/lower-bounds-natural-numbers.lagda.md +++ b/src/elementary-number-theory/lower-bounds-natural-numbers.lagda.md @@ -26,16 +26,18 @@ open import foundation.universe-levels ## Idea -A lower bound for a type family `P` over the natural numbers is a natural number -`n` such that `P x → n ≤ x` for all `x : ℕ`. +A +{{#concept "lower bound" Disambiguation="for a type family over ℕ" Agda=is-lower-bound-ℕ}} +for a type family `P` over the +[natural numbers](elementary-number-theory.natural-numbers.md) is a natural +number `n` such that `P x → n ≤ x` for all `x : ℕ`. ## Definition ```agda is-lower-bound-ℕ : {l : Level} (P : ℕ → UU l) (n : ℕ) → UU l -is-lower-bound-ℕ P n = - (m : ℕ) → P m → leq-ℕ n m +is-lower-bound-ℕ P n = (m : ℕ) → P m → leq-ℕ n m ``` ## Properties diff --git a/src/elementary-number-theory/maximum-natural-numbers.lagda.md b/src/elementary-number-theory/maximum-natural-numbers.lagda.md index 95dee8dced..e6befe6a6f 100644 --- a/src/elementary-number-theory/maximum-natural-numbers.lagda.md +++ b/src/elementary-number-theory/maximum-natural-numbers.lagda.md @@ -57,51 +57,52 @@ max-Fin-ℕ (succ-ℕ n) f = max-ℕ (f (inr star)) (max-Fin-ℕ n (λ k → f ( ## Properties -### Maximum is a least upper bound +### The maximum is a least upper bound ```agda -leq-max-ℕ : - (k m n : ℕ) → m ≤-ℕ k → n ≤-ℕ k → (max-ℕ m n) ≤-ℕ k -leq-max-ℕ zero-ℕ zero-ℕ zero-ℕ H K = star -leq-max-ℕ (succ-ℕ k) zero-ℕ zero-ℕ H K = star -leq-max-ℕ (succ-ℕ k) zero-ℕ (succ-ℕ n) H K = K -leq-max-ℕ (succ-ℕ k) (succ-ℕ m) zero-ℕ H K = H -leq-max-ℕ (succ-ℕ k) (succ-ℕ m) (succ-ℕ n) H K = leq-max-ℕ k m n H K - -leq-left-leq-max-ℕ : - (k m n : ℕ) → (max-ℕ m n) ≤-ℕ k → m ≤-ℕ k -leq-left-leq-max-ℕ k zero-ℕ zero-ℕ H = star -leq-left-leq-max-ℕ k zero-ℕ (succ-ℕ n) H = star -leq-left-leq-max-ℕ k (succ-ℕ m) zero-ℕ H = H -leq-left-leq-max-ℕ (succ-ℕ k) (succ-ℕ m) (succ-ℕ n) H = - leq-left-leq-max-ℕ k m n H - -leq-right-leq-max-ℕ : - (k m n : ℕ) → (max-ℕ m n) ≤-ℕ k → n ≤-ℕ k -leq-right-leq-max-ℕ k zero-ℕ zero-ℕ H = star -leq-right-leq-max-ℕ k zero-ℕ (succ-ℕ n) H = H -leq-right-leq-max-ℕ k (succ-ℕ m) zero-ℕ H = star -leq-right-leq-max-ℕ (succ-ℕ k) (succ-ℕ m) (succ-ℕ n) H = - leq-right-leq-max-ℕ k m n H - -left-leq-max-ℕ : (m n : ℕ) → leq-ℕ m (max-ℕ m n) -left-leq-max-ℕ m n = - leq-left-leq-max-ℕ (max-ℕ m n) m n (refl-leq-ℕ (max-ℕ m n)) - -right-leq-max-ℕ : (m n : ℕ) → leq-ℕ n (max-ℕ m n) -right-leq-max-ℕ m n = - leq-right-leq-max-ℕ (max-ℕ m n) m n (refl-leq-ℕ (max-ℕ m n)) - -is-least-upper-bound-max-ℕ : - (m n : ℕ) → is-least-binary-upper-bound-Poset ℕ-Poset m n (max-ℕ m n) -is-least-upper-bound-max-ℕ m n = - prove-is-least-binary-upper-bound-Poset - ( ℕ-Poset) - { m} - { n} - { max-ℕ m n} - ( left-leq-max-ℕ m n , right-leq-max-ℕ m n) - ( λ x (H , K) → leq-max-ℕ x m n H K) +abstract + leq-max-ℕ : + (k m n : ℕ) → m ≤-ℕ k → n ≤-ℕ k → (max-ℕ m n) ≤-ℕ k + leq-max-ℕ zero-ℕ zero-ℕ zero-ℕ H K = star + leq-max-ℕ (succ-ℕ k) zero-ℕ zero-ℕ H K = star + leq-max-ℕ (succ-ℕ k) zero-ℕ (succ-ℕ n) H K = K + leq-max-ℕ (succ-ℕ k) (succ-ℕ m) zero-ℕ H K = H + leq-max-ℕ (succ-ℕ k) (succ-ℕ m) (succ-ℕ n) H K = leq-max-ℕ k m n H K + + leq-left-leq-max-ℕ : + (k m n : ℕ) → (max-ℕ m n) ≤-ℕ k → m ≤-ℕ k + leq-left-leq-max-ℕ k zero-ℕ zero-ℕ H = star + leq-left-leq-max-ℕ k zero-ℕ (succ-ℕ n) H = star + leq-left-leq-max-ℕ k (succ-ℕ m) zero-ℕ H = H + leq-left-leq-max-ℕ (succ-ℕ k) (succ-ℕ m) (succ-ℕ n) H = + leq-left-leq-max-ℕ k m n H + + leq-right-leq-max-ℕ : + (k m n : ℕ) → (max-ℕ m n) ≤-ℕ k → n ≤-ℕ k + leq-right-leq-max-ℕ k zero-ℕ zero-ℕ H = star + leq-right-leq-max-ℕ k zero-ℕ (succ-ℕ n) H = H + leq-right-leq-max-ℕ k (succ-ℕ m) zero-ℕ H = star + leq-right-leq-max-ℕ (succ-ℕ k) (succ-ℕ m) (succ-ℕ n) H = + leq-right-leq-max-ℕ k m n H + + left-leq-max-ℕ : (m n : ℕ) → leq-ℕ m (max-ℕ m n) + left-leq-max-ℕ m n = + leq-left-leq-max-ℕ (max-ℕ m n) m n (refl-leq-ℕ (max-ℕ m n)) + + right-leq-max-ℕ : (m n : ℕ) → leq-ℕ n (max-ℕ m n) + right-leq-max-ℕ m n = + leq-right-leq-max-ℕ (max-ℕ m n) m n (refl-leq-ℕ (max-ℕ m n)) + + is-least-upper-bound-max-ℕ : + (m n : ℕ) → is-least-binary-upper-bound-Poset ℕ-Poset m n (max-ℕ m n) + is-least-upper-bound-max-ℕ m n = + prove-is-least-binary-upper-bound-Poset + ( ℕ-Poset) + { m} + { n} + { max-ℕ m n} + ( left-leq-max-ℕ m n , right-leq-max-ℕ m n) + ( λ x (H , K) → leq-max-ℕ x m n H K) ``` ### The maximum computes to the greater of the two numbers @@ -127,14 +128,15 @@ abstract ### Associativity of `max-ℕ` ```agda -associative-max-ℕ : - (x y z : ℕ) → max-ℕ (max-ℕ x y) z ＝ max-ℕ x (max-ℕ y z) -associative-max-ℕ zero-ℕ y z = refl -associative-max-ℕ (succ-ℕ x) zero-ℕ zero-ℕ = refl -associative-max-ℕ (succ-ℕ x) zero-ℕ (succ-ℕ z) = refl -associative-max-ℕ (succ-ℕ x) (succ-ℕ y) zero-ℕ = refl -associative-max-ℕ (succ-ℕ x) (succ-ℕ y) (succ-ℕ z) = - ap succ-ℕ (associative-max-ℕ x y z) +abstract + associative-max-ℕ : + (x y z : ℕ) → max-ℕ (max-ℕ x y) z ＝ max-ℕ x (max-ℕ y z) + associative-max-ℕ zero-ℕ y z = refl + associative-max-ℕ (succ-ℕ x) zero-ℕ zero-ℕ = refl + associative-max-ℕ (succ-ℕ x) zero-ℕ (succ-ℕ z) = refl + associative-max-ℕ (succ-ℕ x) (succ-ℕ y) zero-ℕ = refl + associative-max-ℕ (succ-ℕ x) (succ-ℕ y) (succ-ℕ z) = + ap succ-ℕ (associative-max-ℕ x y z) ``` ### Unit laws for `max-ℕ` @@ -151,54 +153,58 @@ right-unit-law-max-ℕ (succ-ℕ x) = refl ### Commutativity of `max-ℕ` ```agda -commutative-max-ℕ : (x y : ℕ) → max-ℕ x y ＝ max-ℕ y x -commutative-max-ℕ zero-ℕ zero-ℕ = refl -commutative-max-ℕ zero-ℕ (succ-ℕ y) = refl -commutative-max-ℕ (succ-ℕ x) zero-ℕ = refl -commutative-max-ℕ (succ-ℕ x) (succ-ℕ y) = ap succ-ℕ (commutative-max-ℕ x y) +abstract + commutative-max-ℕ : (x y : ℕ) → max-ℕ x y ＝ max-ℕ y x + commutative-max-ℕ zero-ℕ zero-ℕ = refl + commutative-max-ℕ zero-ℕ (succ-ℕ y) = refl + commutative-max-ℕ (succ-ℕ x) zero-ℕ = refl + commutative-max-ℕ (succ-ℕ x) (succ-ℕ y) = ap succ-ℕ (commutative-max-ℕ x y) ``` ### `max-ℕ` is idempotent ```agda -idempotent-max-ℕ : (x : ℕ) → max-ℕ x x ＝ x -idempotent-max-ℕ zero-ℕ = refl -idempotent-max-ℕ (succ-ℕ x) = ap succ-ℕ (idempotent-max-ℕ x) +abstract + idempotent-max-ℕ : (x : ℕ) → max-ℕ x x ＝ x + idempotent-max-ℕ zero-ℕ = refl + idempotent-max-ℕ (succ-ℕ x) = ap succ-ℕ (idempotent-max-ℕ x) ``` ### Successor diagonal laws for `max-ℕ` ```agda -left-successor-diagonal-law-max-ℕ : (x : ℕ) → max-ℕ (succ-ℕ x) x ＝ succ-ℕ x -left-successor-diagonal-law-max-ℕ zero-ℕ = refl -left-successor-diagonal-law-max-ℕ (succ-ℕ x) = - ap succ-ℕ (left-successor-diagonal-law-max-ℕ x) - -right-successor-diagonal-law-max-ℕ : (x : ℕ) → max-ℕ x (succ-ℕ x) ＝ succ-ℕ x -right-successor-diagonal-law-max-ℕ zero-ℕ = refl -right-successor-diagonal-law-max-ℕ (succ-ℕ x) = - ap succ-ℕ (right-successor-diagonal-law-max-ℕ x) +abstract + left-successor-diagonal-law-max-ℕ : (x : ℕ) → max-ℕ (succ-ℕ x) x ＝ succ-ℕ x + left-successor-diagonal-law-max-ℕ zero-ℕ = refl + left-successor-diagonal-law-max-ℕ (succ-ℕ x) = + ap succ-ℕ (left-successor-diagonal-law-max-ℕ x) + + right-successor-diagonal-law-max-ℕ : (x : ℕ) → max-ℕ x (succ-ℕ x) ＝ succ-ℕ x + right-successor-diagonal-law-max-ℕ zero-ℕ = refl + right-successor-diagonal-law-max-ℕ (succ-ℕ x) = + ap succ-ℕ (right-successor-diagonal-law-max-ℕ x) ``` ### Addition distributes over `max-ℕ` ```agda -left-distributive-add-max-ℕ : - (x y z : ℕ) → x +ℕ (max-ℕ y z) ＝ max-ℕ (x +ℕ y) (x +ℕ z) -left-distributive-add-max-ℕ zero-ℕ y z = - ( left-unit-law-add-ℕ (max-ℕ y z)) ∙ - ( ap-max-ℕ (inv (left-unit-law-add-ℕ y)) (inv (left-unit-law-add-ℕ z))) -left-distributive-add-max-ℕ (succ-ℕ x) y z = - ( left-successor-law-add-ℕ x (max-ℕ y z)) ∙ - ( ( ap succ-ℕ (left-distributive-add-max-ℕ x y z)) ∙ - ( ap-max-ℕ - ( inv (left-successor-law-add-ℕ x y)) - ( inv (left-successor-law-add-ℕ x z)))) - -right-distributive-add-max-ℕ : - (x y z : ℕ) → (max-ℕ x y) +ℕ z ＝ max-ℕ (x +ℕ z) (y +ℕ z) -right-distributive-add-max-ℕ x y z = - ( commutative-add-ℕ (max-ℕ x y) z) ∙ - ( ( left-distributive-add-max-ℕ z x y) ∙ - ( ap-max-ℕ (commutative-add-ℕ z x) (commutative-add-ℕ z y))) +abstract + left-distributive-add-max-ℕ : + (x y z : ℕ) → x +ℕ (max-ℕ y z) ＝ max-ℕ (x +ℕ y) (x +ℕ z) + left-distributive-add-max-ℕ zero-ℕ y z = + ( left-unit-law-add-ℕ (max-ℕ y z)) ∙ + ( ap-max-ℕ (inv (left-unit-law-add-ℕ y)) (inv (left-unit-law-add-ℕ z))) + left-distributive-add-max-ℕ (succ-ℕ x) y z = + ( left-successor-law-add-ℕ x (max-ℕ y z)) ∙ + ( ( ap succ-ℕ (left-distributive-add-max-ℕ x y z)) ∙ + ( ap-max-ℕ + ( inv (left-successor-law-add-ℕ x y)) + ( inv (left-successor-law-add-ℕ x z)))) + + right-distributive-add-max-ℕ : + (x y z : ℕ) → (max-ℕ x y) +ℕ z ＝ max-ℕ (x +ℕ z) (y +ℕ z) + right-distributive-add-max-ℕ x y z = + ( commutative-add-ℕ (max-ℕ x y) z) ∙ + ( ( left-distributive-add-max-ℕ z x y) ∙ + ( ap-max-ℕ (commutative-add-ℕ z x) (commutative-add-ℕ z y))) ``` diff --git a/src/elementary-number-theory/minimum-natural-numbers.lagda.md b/src/elementary-number-theory/minimum-natural-numbers.lagda.md index 1f1d263d93..7e72adee71 100644 --- a/src/elementary-number-theory/minimum-natural-numbers.lagda.md +++ b/src/elementary-number-theory/minimum-natural-numbers.lagda.md @@ -52,40 +52,41 @@ min-Fin-ℕ (succ-ℕ n) f = min-ℕ (f (inr star)) (min-Fin-ℕ n (λ k → f ( ### The minimum of two natural numbers is a greatest lower bound ```agda -leq-min-ℕ : - (k m n : ℕ) → k ≤-ℕ m → k ≤-ℕ n → k ≤-ℕ (min-ℕ m n) -leq-min-ℕ zero-ℕ zero-ℕ zero-ℕ H K = star -leq-min-ℕ zero-ℕ zero-ℕ (succ-ℕ n) H K = star -leq-min-ℕ zero-ℕ (succ-ℕ m) zero-ℕ H K = star -leq-min-ℕ zero-ℕ (succ-ℕ m) (succ-ℕ n) H K = star -leq-min-ℕ (succ-ℕ k) (succ-ℕ m) (succ-ℕ n) H K = leq-min-ℕ k m n H K - -leq-left-leq-min-ℕ : - (k m n : ℕ) → k ≤-ℕ (min-ℕ m n) → k ≤-ℕ m -leq-left-leq-min-ℕ zero-ℕ zero-ℕ zero-ℕ H = star -leq-left-leq-min-ℕ zero-ℕ zero-ℕ (succ-ℕ n) H = star -leq-left-leq-min-ℕ zero-ℕ (succ-ℕ m) zero-ℕ H = star -leq-left-leq-min-ℕ zero-ℕ (succ-ℕ m) (succ-ℕ n) H = star -leq-left-leq-min-ℕ (succ-ℕ k) (succ-ℕ m) (succ-ℕ n) H = - leq-left-leq-min-ℕ k m n H - -leq-right-leq-min-ℕ : - (k m n : ℕ) → k ≤-ℕ (min-ℕ m n) → k ≤-ℕ n -leq-right-leq-min-ℕ zero-ℕ zero-ℕ zero-ℕ H = star -leq-right-leq-min-ℕ zero-ℕ zero-ℕ (succ-ℕ n) H = star -leq-right-leq-min-ℕ zero-ℕ (succ-ℕ m) zero-ℕ H = star -leq-right-leq-min-ℕ zero-ℕ (succ-ℕ m) (succ-ℕ n) H = star -leq-right-leq-min-ℕ (succ-ℕ k) (succ-ℕ m) (succ-ℕ n) H = - leq-right-leq-min-ℕ k m n H - -is-greatest-lower-bound-min-ℕ : - (l m : ℕ) → is-greatest-binary-lower-bound-Poset ℕ-Poset l m (min-ℕ l m) -is-greatest-lower-bound-min-ℕ l m = - prove-is-greatest-binary-lower-bound-Poset - ( ℕ-Poset) - ( leq-left-leq-min-ℕ (min-ℕ l m) l m (refl-leq-ℕ (min-ℕ l m)) , - leq-right-leq-min-ℕ (min-ℕ l m) l m (refl-leq-ℕ (min-ℕ l m))) - ( λ x (H , K) → leq-min-ℕ x l m H K) +abstract + leq-min-ℕ : + (k m n : ℕ) → k ≤-ℕ m → k ≤-ℕ n → k ≤-ℕ (min-ℕ m n) + leq-min-ℕ zero-ℕ zero-ℕ zero-ℕ H K = star + leq-min-ℕ zero-ℕ zero-ℕ (succ-ℕ n) H K = star + leq-min-ℕ zero-ℕ (succ-ℕ m) zero-ℕ H K = star + leq-min-ℕ zero-ℕ (succ-ℕ m) (succ-ℕ n) H K = star + leq-min-ℕ (succ-ℕ k) (succ-ℕ m) (succ-ℕ n) H K = leq-min-ℕ k m n H K + + leq-left-leq-min-ℕ : + (k m n : ℕ) → k ≤-ℕ (min-ℕ m n) → k ≤-ℕ m + leq-left-leq-min-ℕ zero-ℕ zero-ℕ zero-ℕ H = star + leq-left-leq-min-ℕ zero-ℕ zero-ℕ (succ-ℕ n) H = star + leq-left-leq-min-ℕ zero-ℕ (succ-ℕ m) zero-ℕ H = star + leq-left-leq-min-ℕ zero-ℕ (succ-ℕ m) (succ-ℕ n) H = star + leq-left-leq-min-ℕ (succ-ℕ k) (succ-ℕ m) (succ-ℕ n) H = + leq-left-leq-min-ℕ k m n H + + leq-right-leq-min-ℕ : + (k m n : ℕ) → k ≤-ℕ (min-ℕ m n) → k ≤-ℕ n + leq-right-leq-min-ℕ zero-ℕ zero-ℕ zero-ℕ H = star + leq-right-leq-min-ℕ zero-ℕ zero-ℕ (succ-ℕ n) H = star + leq-right-leq-min-ℕ zero-ℕ (succ-ℕ m) zero-ℕ H = star + leq-right-leq-min-ℕ zero-ℕ (succ-ℕ m) (succ-ℕ n) H = star + leq-right-leq-min-ℕ (succ-ℕ k) (succ-ℕ m) (succ-ℕ n) H = + leq-right-leq-min-ℕ k m n H + + is-greatest-lower-bound-min-ℕ : + (l m : ℕ) → is-greatest-binary-lower-bound-Poset ℕ-Poset l m (min-ℕ l m) + is-greatest-lower-bound-min-ℕ l m = + prove-is-greatest-binary-lower-bound-Poset + ( ℕ-Poset) + ( leq-left-leq-min-ℕ (min-ℕ l m) l m (refl-leq-ℕ (min-ℕ l m)) , + leq-right-leq-min-ℕ (min-ℕ l m) l m (refl-leq-ℕ (min-ℕ l m))) + ( λ x (H , K) → leq-min-ℕ x l m H K) ``` ### The minimum computes to the lesser of the two numbers @@ -111,14 +112,15 @@ abstract ### Associativity of `min-ℕ` ```agda -associative-min-ℕ : - (x y z : ℕ) → min-ℕ (min-ℕ x y) z ＝ min-ℕ x (min-ℕ y z) -associative-min-ℕ zero-ℕ y z = refl -associative-min-ℕ (succ-ℕ x) zero-ℕ zero-ℕ = refl -associative-min-ℕ (succ-ℕ x) zero-ℕ (succ-ℕ z) = refl -associative-min-ℕ (succ-ℕ x) (succ-ℕ y) zero-ℕ = refl -associative-min-ℕ (succ-ℕ x) (succ-ℕ y) (succ-ℕ z) = - ap succ-ℕ (associative-min-ℕ x y z) +abstract + associative-min-ℕ : + (x y z : ℕ) → min-ℕ (min-ℕ x y) z ＝ min-ℕ x (min-ℕ y z) + associative-min-ℕ zero-ℕ y z = refl + associative-min-ℕ (succ-ℕ x) zero-ℕ zero-ℕ = refl + associative-min-ℕ (succ-ℕ x) zero-ℕ (succ-ℕ z) = refl + associative-min-ℕ (succ-ℕ x) (succ-ℕ y) zero-ℕ = refl + associative-min-ℕ (succ-ℕ x) (succ-ℕ y) (succ-ℕ z) = + ap succ-ℕ (associative-min-ℕ x y z) ``` ### Zero laws for `min-ℕ` @@ -135,40 +137,43 @@ right-zero-law-min-ℕ (succ-ℕ x) = refl ### Commutativity of `min-ℕ` ```agda -commutative-min-ℕ : (x y : ℕ) → min-ℕ x y ＝ min-ℕ y x -commutative-min-ℕ zero-ℕ zero-ℕ = refl -commutative-min-ℕ zero-ℕ (succ-ℕ y) = refl -commutative-min-ℕ (succ-ℕ x) zero-ℕ = refl -commutative-min-ℕ (succ-ℕ x) (succ-ℕ y) = ap succ-ℕ (commutative-min-ℕ x y) +abstract + commutative-min-ℕ : (x y : ℕ) → min-ℕ x y ＝ min-ℕ y x + commutative-min-ℕ zero-ℕ zero-ℕ = refl + commutative-min-ℕ zero-ℕ (succ-ℕ y) = refl + commutative-min-ℕ (succ-ℕ x) zero-ℕ = refl + commutative-min-ℕ (succ-ℕ x) (succ-ℕ y) = ap succ-ℕ (commutative-min-ℕ x y) ``` ### `min-ℕ` is idempotent ```agda -idempotent-min-ℕ : (x : ℕ) → min-ℕ x x ＝ x -idempotent-min-ℕ zero-ℕ = refl -idempotent-min-ℕ (succ-ℕ x) = ap succ-ℕ (idempotent-min-ℕ x) +abstract + idempotent-min-ℕ : (x : ℕ) → min-ℕ x x ＝ x + idempotent-min-ℕ zero-ℕ = refl + idempotent-min-ℕ (succ-ℕ x) = ap succ-ℕ (idempotent-min-ℕ x) ``` ### Addition distributes over `min-ℕ` ```agda -left-distributive-add-min-ℕ : - (x y z : ℕ) → x +ℕ (min-ℕ y z) ＝ min-ℕ (x +ℕ y) (x +ℕ z) -left-distributive-add-min-ℕ zero-ℕ y z = - ( left-unit-law-add-ℕ (min-ℕ y z)) ∙ - ( ap-min-ℕ (inv (left-unit-law-add-ℕ y)) (inv (left-unit-law-add-ℕ z))) -left-distributive-add-min-ℕ (succ-ℕ x) y z = - ( left-successor-law-add-ℕ x (min-ℕ y z)) ∙ - ( ( ap succ-ℕ (left-distributive-add-min-ℕ x y z)) ∙ - ( ap-min-ℕ - ( inv (left-successor-law-add-ℕ x y)) - ( inv (left-successor-law-add-ℕ x z)))) - -right-distributive-add-min-ℕ : - (x y z : ℕ) → (min-ℕ x y) +ℕ z ＝ min-ℕ (x +ℕ z) (y +ℕ z) -right-distributive-add-min-ℕ x y z = - ( commutative-add-ℕ (min-ℕ x y) z) ∙ - ( ( left-distributive-add-min-ℕ z x y) ∙ - ( ap-min-ℕ (commutative-add-ℕ z x) (commutative-add-ℕ z y))) +abstract + left-distributive-add-min-ℕ : + (x y z : ℕ) → x +ℕ (min-ℕ y z) ＝ min-ℕ (x +ℕ y) (x +ℕ z) + left-distributive-add-min-ℕ zero-ℕ y z = + ( left-unit-law-add-ℕ (min-ℕ y z)) ∙ + ( ap-min-ℕ (inv (left-unit-law-add-ℕ y)) (inv (left-unit-law-add-ℕ z))) + left-distributive-add-min-ℕ (succ-ℕ x) y z = + ( left-successor-law-add-ℕ x (min-ℕ y z)) ∙ + ( ( ap succ-ℕ (left-distributive-add-min-ℕ x y z)) ∙ + ( ap-min-ℕ + ( inv (left-successor-law-add-ℕ x y)) + ( inv (left-successor-law-add-ℕ x z)))) + + right-distributive-add-min-ℕ : + (x y z : ℕ) → (min-ℕ x y) +ℕ z ＝ min-ℕ (x +ℕ z) (y +ℕ z) + right-distributive-add-min-ℕ x y z = + ( commutative-add-ℕ (min-ℕ x y) z) ∙ + ( ( left-distributive-add-min-ℕ z x y) ∙ + ( ap-min-ℕ (commutative-add-ℕ z x) (commutative-add-ℕ z y))) ``` diff --git a/src/elementary-number-theory/multiplication-integers.lagda.md b/src/elementary-number-theory/multiplication-integers.lagda.md index d8c89edf3b..7d77ed97dc 100644 --- a/src/elementary-number-theory/multiplication-integers.lagda.md +++ b/src/elementary-number-theory/multiplication-integers.lagda.md @@ -98,446 +98,465 @@ is-plus-or-minus-ℤ x y = (x ＝ y) + (neg-one-ℤ *ℤ x ＝ y) ### Multiplication by zero is zero ```agda -left-zero-law-mul-ℤ : (k : ℤ) → zero-ℤ *ℤ k ＝ zero-ℤ -left-zero-law-mul-ℤ k = refl - -right-zero-law-mul-ℤ : (k : ℤ) → k *ℤ zero-ℤ ＝ zero-ℤ -right-zero-law-mul-ℤ (inl zero-ℕ) = refl -right-zero-law-mul-ℤ (inl (succ-ℕ n)) = - right-zero-law-mul-ℤ (inl n) -right-zero-law-mul-ℤ (inr (inl star)) = refl -right-zero-law-mul-ℤ (inr (inr zero-ℕ)) = refl -right-zero-law-mul-ℤ (inr (inr (succ-ℕ n))) = - right-zero-law-mul-ℤ (inr (inr n)) +abstract + left-zero-law-mul-ℤ : (k : ℤ) → zero-ℤ *ℤ k ＝ zero-ℤ + left-zero-law-mul-ℤ k = refl + + right-zero-law-mul-ℤ : (k : ℤ) → k *ℤ zero-ℤ ＝ zero-ℤ + right-zero-law-mul-ℤ (inl zero-ℕ) = refl + right-zero-law-mul-ℤ (inl (succ-ℕ n)) = + right-zero-law-mul-ℤ (inl n) + right-zero-law-mul-ℤ (inr (inl star)) = refl + right-zero-law-mul-ℤ (inr (inr zero-ℕ)) = refl + right-zero-law-mul-ℤ (inr (inr (succ-ℕ n))) = + right-zero-law-mul-ℤ (inr (inr n)) ``` ### Unit laws ```agda -left-unit-law-mul-ℤ : (k : ℤ) → one-ℤ *ℤ k ＝ k -left-unit-law-mul-ℤ k = refl - -right-unit-law-mul-ℤ : (k : ℤ) → k *ℤ one-ℤ ＝ k -right-unit-law-mul-ℤ (inl zero-ℕ) = refl -right-unit-law-mul-ℤ (inl (succ-ℕ n)) = - ap ((neg-one-ℤ) +ℤ_) (right-unit-law-mul-ℤ (inl n)) -right-unit-law-mul-ℤ (inr (inl star)) = refl -right-unit-law-mul-ℤ (inr (inr zero-ℕ)) = refl -right-unit-law-mul-ℤ (inr (inr (succ-ℕ n))) = - ap (one-ℤ +ℤ_) (right-unit-law-mul-ℤ (inr (inr n))) +abstract + left-unit-law-mul-ℤ : (k : ℤ) → one-ℤ *ℤ k ＝ k + left-unit-law-mul-ℤ k = refl + + right-unit-law-mul-ℤ : (k : ℤ) → k *ℤ one-ℤ ＝ k + right-unit-law-mul-ℤ (inl zero-ℕ) = refl + right-unit-law-mul-ℤ (inl (succ-ℕ n)) = + ap ((neg-one-ℤ) +ℤ_) (right-unit-law-mul-ℤ (inl n)) + right-unit-law-mul-ℤ (inr (inl star)) = refl + right-unit-law-mul-ℤ (inr (inr zero-ℕ)) = refl + right-unit-law-mul-ℤ (inr (inr (succ-ℕ n))) = + ap (one-ℤ +ℤ_) (right-unit-law-mul-ℤ (inr (inr n))) ``` ### Multiplication of an integer by `-1` is equal to the negative ```agda -left-neg-unit-law-mul-ℤ : (k : ℤ) → neg-one-ℤ *ℤ k ＝ neg-ℤ k -left-neg-unit-law-mul-ℤ k = refl - -right-neg-unit-law-mul-ℤ : (k : ℤ) → k *ℤ neg-one-ℤ ＝ neg-ℤ k -right-neg-unit-law-mul-ℤ (inl zero-ℕ) = refl -right-neg-unit-law-mul-ℤ (inl (succ-ℕ n)) = - ap (one-ℤ +ℤ_) (right-neg-unit-law-mul-ℤ (inl n)) -right-neg-unit-law-mul-ℤ (inr (inl star)) = refl -right-neg-unit-law-mul-ℤ (inr (inr zero-ℕ)) = refl -right-neg-unit-law-mul-ℤ (inr (inr (succ-ℕ n))) = - ap (neg-one-ℤ +ℤ_) (right-neg-unit-law-mul-ℤ (inr (inr n))) +abstract + left-neg-unit-law-mul-ℤ : (k : ℤ) → neg-one-ℤ *ℤ k ＝ neg-ℤ k + left-neg-unit-law-mul-ℤ k = refl + + right-neg-unit-law-mul-ℤ : (k : ℤ) → k *ℤ neg-one-ℤ ＝ neg-ℤ k + right-neg-unit-law-mul-ℤ (inl zero-ℕ) = refl + right-neg-unit-law-mul-ℤ (inl (succ-ℕ n)) = + ap (one-ℤ +ℤ_) (right-neg-unit-law-mul-ℤ (inl n)) + right-neg-unit-law-mul-ℤ (inr (inl star)) = refl + right-neg-unit-law-mul-ℤ (inr (inr zero-ℕ)) = refl + right-neg-unit-law-mul-ℤ (inr (inr (succ-ℕ n))) = + ap (neg-one-ℤ +ℤ_) (right-neg-unit-law-mul-ℤ (inr (inr n))) ``` ### Multiplication by the successor or the predecessor of an integer ```agda -left-successor-law-mul-ℤ : - (k l : ℤ) → (succ-ℤ k) *ℤ l ＝ l +ℤ (k *ℤ l) -left-successor-law-mul-ℤ (inl zero-ℕ) l = - inv (right-inverse-law-add-ℤ l) -left-successor-law-mul-ℤ (inl (succ-ℕ n)) l = - ( ( inv (left-unit-law-add-ℤ ((inl n) *ℤ l))) ∙ +abstract + left-successor-law-mul-ℤ : + (k l : ℤ) → (succ-ℤ k) *ℤ l ＝ l +ℤ (k *ℤ l) + left-successor-law-mul-ℤ (inl zero-ℕ) l = + inv (right-inverse-law-add-ℤ l) + left-successor-law-mul-ℤ (inl (succ-ℕ n)) l = + ( ( inv (left-unit-law-add-ℤ ((inl n) *ℤ l))) ∙ + ( ap + ( _+ℤ ((inl n) *ℤ l)) + ( inv (right-inverse-law-add-ℤ l)))) ∙ + ( associative-add-ℤ l (neg-ℤ l) ((inl n) *ℤ l)) + left-successor-law-mul-ℤ (inr (inl star)) l = + inv (right-unit-law-add-ℤ l) + left-successor-law-mul-ℤ (inr (inr n)) l = refl + + left-successor-law-mul-ℤ' : + (k l : ℤ) → (succ-ℤ k) *ℤ l ＝ (k *ℤ l) +ℤ l + left-successor-law-mul-ℤ' k l = + left-successor-law-mul-ℤ k l ∙ + commutative-add-ℤ l (k *ℤ l) + + left-predecessor-law-mul-ℤ : + (k l : ℤ) → (pred-ℤ k) *ℤ l ＝ (neg-ℤ l) +ℤ (k *ℤ l) + left-predecessor-law-mul-ℤ (inl n) l = refl + left-predecessor-law-mul-ℤ (inr (inl star)) l = + ( left-neg-unit-law-mul-ℤ l) ∙ + ( inv (right-unit-law-add-ℤ (neg-ℤ l))) + left-predecessor-law-mul-ℤ (inr (inr zero-ℕ)) l = + inv (left-inverse-law-add-ℤ l) + left-predecessor-law-mul-ℤ (inr (inr (succ-ℕ x))) l = ( ap - ( _+ℤ ((inl n) *ℤ l)) - ( inv (right-inverse-law-add-ℤ l)))) ∙ - ( associative-add-ℤ l (neg-ℤ l) ((inl n) *ℤ l)) -left-successor-law-mul-ℤ (inr (inl star)) l = - inv (right-unit-law-add-ℤ l) -left-successor-law-mul-ℤ (inr (inr n)) l = refl - -left-successor-law-mul-ℤ' : - (k l : ℤ) → (succ-ℤ k) *ℤ l ＝ (k *ℤ l) +ℤ l -left-successor-law-mul-ℤ' k l = - left-successor-law-mul-ℤ k l ∙ - commutative-add-ℤ l (k *ℤ l) - -left-predecessor-law-mul-ℤ : - (k l : ℤ) → (pred-ℤ k) *ℤ l ＝ (neg-ℤ l) +ℤ (k *ℤ l) -left-predecessor-law-mul-ℤ (inl n) l = refl -left-predecessor-law-mul-ℤ (inr (inl star)) l = - ( left-neg-unit-law-mul-ℤ l) ∙ - ( inv (right-unit-law-add-ℤ (neg-ℤ l))) -left-predecessor-law-mul-ℤ (inr (inr zero-ℕ)) l = - inv (left-inverse-law-add-ℤ l) -left-predecessor-law-mul-ℤ (inr (inr (succ-ℕ x))) l = - ( ap - ( _+ℤ ((in-pos-ℤ x) *ℤ l)) - ( inv (left-inverse-law-add-ℤ l))) ∙ - ( associative-add-ℤ (neg-ℤ l) l ((in-pos-ℤ x) *ℤ l)) - -left-predecessor-law-mul-ℤ' : - (k l : ℤ) → (pred-ℤ k) *ℤ l ＝ (k *ℤ l) +ℤ (neg-ℤ l) -left-predecessor-law-mul-ℤ' k l = - left-predecessor-law-mul-ℤ k l ∙ - commutative-add-ℤ (neg-ℤ l) (k *ℤ l) - -right-successor-law-mul-ℤ : - (k l : ℤ) → k *ℤ (succ-ℤ l) ＝ k +ℤ (k *ℤ l) -right-successor-law-mul-ℤ (inl zero-ℕ) l = inv (pred-neg-ℤ l) -right-successor-law-mul-ℤ (inl (succ-ℕ n)) l = - ( left-predecessor-law-mul-ℤ (inl n) (succ-ℤ l)) ∙ - ( ( ap ((neg-ℤ (succ-ℤ l)) +ℤ_) (right-successor-law-mul-ℤ (inl n) l)) ∙ - ( ( inv (associative-add-ℤ (neg-ℤ (succ-ℤ l)) (inl n) ((inl n) *ℤ l))) ∙ - ( ( ap - ( _+ℤ ((inl n) *ℤ l)) - { x = (neg-ℤ (succ-ℤ l)) +ℤ (inl n)} - { y = (inl (succ-ℕ n)) +ℤ (neg-ℤ l)} - ( ( right-successor-law-add-ℤ (neg-ℤ (succ-ℤ l)) (inl (succ-ℕ n))) ∙ - ( ( ap succ-ℤ - ( commutative-add-ℤ (neg-ℤ (succ-ℤ l)) (inl (succ-ℕ n)))) ∙ - ( ( inv - ( right-successor-law-add-ℤ - ( inl (succ-ℕ n)) - ( neg-ℤ (succ-ℤ l)))) ∙ - ( ap - ( (inl (succ-ℕ n)) +ℤ_) - ( ( ap succ-ℤ (inv (pred-neg-ℤ l))) ∙ - ( is-section-pred-ℤ (neg-ℤ l)))))))) ∙ - ( associative-add-ℤ (inl (succ-ℕ n)) (neg-ℤ l) ((inl n) *ℤ l))))) -right-successor-law-mul-ℤ (inr (inl star)) l = refl -right-successor-law-mul-ℤ (inr (inr zero-ℕ)) l = refl -right-successor-law-mul-ℤ (inr (inr (succ-ℕ n))) l = - ( left-successor-law-mul-ℤ (in-pos-ℤ n) (succ-ℤ l)) ∙ - ( ( ap ((succ-ℤ l) +ℤ_) (right-successor-law-mul-ℤ (inr (inr n)) l)) ∙ - ( ( inv (associative-add-ℤ (succ-ℤ l) (in-pos-ℤ n) ((in-pos-ℤ n) *ℤ l))) ∙ - ( ( ap - ( _+ℤ ((in-pos-ℤ n) *ℤ l)) - { x = (succ-ℤ l) +ℤ (in-pos-ℤ n)} - { y = (in-pos-ℤ (succ-ℕ n)) +ℤ l} - ( ( left-successor-law-add-ℤ l (in-pos-ℤ n)) ∙ - ( ( ap succ-ℤ (commutative-add-ℤ l (in-pos-ℤ n))) ∙ - ( inv (left-successor-law-add-ℤ (in-pos-ℤ n) l))))) ∙ - ( associative-add-ℤ (inr (inr (succ-ℕ n))) l ((inr (inr n)) *ℤ l))))) - -right-successor-law-mul-ℤ' : - (k l : ℤ) → k *ℤ (succ-ℤ l) ＝ (k *ℤ l) +ℤ k -right-successor-law-mul-ℤ' k l = - right-successor-law-mul-ℤ k l ∙ - commutative-add-ℤ k (k *ℤ l) - -right-predecessor-law-mul-ℤ : - (k l : ℤ) → k *ℤ (pred-ℤ l) ＝ (neg-ℤ k) +ℤ (k *ℤ l) -right-predecessor-law-mul-ℤ (inl zero-ℕ) l = - ( left-neg-unit-law-mul-ℤ (pred-ℤ l)) ∙ - ( neg-pred-ℤ l) -right-predecessor-law-mul-ℤ (inl (succ-ℕ n)) l = - ( left-predecessor-law-mul-ℤ (inl n) (pred-ℤ l)) ∙ - ( ( ap ((neg-ℤ (pred-ℤ l)) +ℤ_) (right-predecessor-law-mul-ℤ (inl n) l)) ∙ - ( ( inv - ( associative-add-ℤ (neg-ℤ (pred-ℤ l)) (in-pos-ℤ n) ((inl n) *ℤ l))) ∙ - ( ( ap - ( _+ℤ ((inl n) *ℤ l)) - { x = (neg-ℤ (pred-ℤ l)) +ℤ (inr (inr n))} - { y = (neg-ℤ (inl (succ-ℕ n))) +ℤ (neg-ℤ l)} - ( ( ap (_+ℤ (in-pos-ℤ n)) (neg-pred-ℤ l)) ∙ - ( ( left-successor-law-add-ℤ (neg-ℤ l) (in-pos-ℤ n)) ∙ - ( ( ap succ-ℤ (commutative-add-ℤ (neg-ℤ l) (in-pos-ℤ n))) ∙ - ( inv (left-successor-law-add-ℤ (in-pos-ℤ n) (neg-ℤ l))))))) ∙ - ( associative-add-ℤ (in-pos-ℤ (succ-ℕ n)) (neg-ℤ l) ((inl n) *ℤ l))))) -right-predecessor-law-mul-ℤ (inr (inl star)) l = refl -right-predecessor-law-mul-ℤ (inr (inr zero-ℕ)) l = refl -right-predecessor-law-mul-ℤ (inr (inr (succ-ℕ n))) l = - ( left-successor-law-mul-ℤ (in-pos-ℤ n) (pred-ℤ l)) ∙ - ( ( ap ((pred-ℤ l) +ℤ_) (right-predecessor-law-mul-ℤ (inr (inr n)) l)) ∙ - ( ( inv (associative-add-ℤ (pred-ℤ l) (inl n) ((inr (inr n)) *ℤ l))) ∙ - ( ( ap - ( _+ℤ ((in-pos-ℤ n) *ℤ l)) - { x = (pred-ℤ l) +ℤ (inl n)} - { y = (neg-ℤ (in-pos-ℤ (succ-ℕ n))) +ℤ l} - ( ( left-predecessor-law-add-ℤ l (inl n)) ∙ - ( ( ap pred-ℤ (commutative-add-ℤ l (inl n))) ∙ - ( inv (left-predecessor-law-add-ℤ (inl n) l))))) ∙ - ( associative-add-ℤ (inl (succ-ℕ n)) l ((inr (inr n)) *ℤ l))))) - -right-predecessor-law-mul-ℤ' : - (k l : ℤ) → k *ℤ (pred-ℤ l) ＝ (k *ℤ l) +ℤ (neg-ℤ k) -right-predecessor-law-mul-ℤ' k l = - right-predecessor-law-mul-ℤ k l ∙ - commutative-add-ℤ (neg-ℤ k) (k *ℤ l) + ( _+ℤ ((in-pos-ℤ x) *ℤ l)) + ( inv (left-inverse-law-add-ℤ l))) ∙ + ( associative-add-ℤ (neg-ℤ l) l ((in-pos-ℤ x) *ℤ l)) + + left-predecessor-law-mul-ℤ' : + (k l : ℤ) → (pred-ℤ k) *ℤ l ＝ (k *ℤ l) +ℤ (neg-ℤ l) + left-predecessor-law-mul-ℤ' k l = + left-predecessor-law-mul-ℤ k l ∙ + commutative-add-ℤ (neg-ℤ l) (k *ℤ l) + + right-successor-law-mul-ℤ : + (k l : ℤ) → k *ℤ (succ-ℤ l) ＝ k +ℤ (k *ℤ l) + right-successor-law-mul-ℤ (inl zero-ℕ) l = inv (pred-neg-ℤ l) + right-successor-law-mul-ℤ (inl (succ-ℕ n)) l = + ( left-predecessor-law-mul-ℤ (inl n) (succ-ℤ l)) ∙ + ( ( ap ((neg-ℤ (succ-ℤ l)) +ℤ_) (right-successor-law-mul-ℤ (inl n) l)) ∙ + ( ( inv (associative-add-ℤ (neg-ℤ (succ-ℤ l)) (inl n) ((inl n) *ℤ l))) ∙ + ( ( ap + ( _+ℤ ((inl n) *ℤ l)) + { x = (neg-ℤ (succ-ℤ l)) +ℤ (inl n)} + { y = (inl (succ-ℕ n)) +ℤ (neg-ℤ l)} + ( ( right-successor-law-add-ℤ (neg-ℤ (succ-ℤ l)) (inl (succ-ℕ n))) ∙ + ( ( ap succ-ℤ + ( commutative-add-ℤ (neg-ℤ (succ-ℤ l)) (inl (succ-ℕ n)))) ∙ + ( ( inv + ( right-successor-law-add-ℤ + ( inl (succ-ℕ n)) + ( neg-ℤ (succ-ℤ l)))) ∙ + ( ap + ( (inl (succ-ℕ n)) +ℤ_) + ( ( ap succ-ℤ (inv (pred-neg-ℤ l))) ∙ + ( is-section-pred-ℤ (neg-ℤ l)))))))) ∙ + ( associative-add-ℤ (inl (succ-ℕ n)) (neg-ℤ l) ((inl n) *ℤ l))))) + right-successor-law-mul-ℤ (inr (inl star)) l = refl + right-successor-law-mul-ℤ (inr (inr zero-ℕ)) l = refl + right-successor-law-mul-ℤ (inr (inr (succ-ℕ n))) l = + ( left-successor-law-mul-ℤ (in-pos-ℤ n) (succ-ℤ l)) ∙ + ( ( ap ((succ-ℤ l) +ℤ_) (right-successor-law-mul-ℤ (inr (inr n)) l)) ∙ + ( ( inv (associative-add-ℤ (succ-ℤ l) (in-pos-ℤ n) ((in-pos-ℤ n) *ℤ l))) ∙ + ( ( ap + ( _+ℤ ((in-pos-ℤ n) *ℤ l)) + { x = (succ-ℤ l) +ℤ (in-pos-ℤ n)} + { y = (in-pos-ℤ (succ-ℕ n)) +ℤ l} + ( ( left-successor-law-add-ℤ l (in-pos-ℤ n)) ∙ + ( ( ap succ-ℤ (commutative-add-ℤ l (in-pos-ℤ n))) ∙ + ( inv (left-successor-law-add-ℤ (in-pos-ℤ n) l))))) ∙ + ( associative-add-ℤ (inr (inr (succ-ℕ n))) l ((inr (inr n)) *ℤ l))))) + + right-successor-law-mul-ℤ' : + (k l : ℤ) → k *ℤ (succ-ℤ l) ＝ (k *ℤ l) +ℤ k + right-successor-law-mul-ℤ' k l = + right-successor-law-mul-ℤ k l ∙ + commutative-add-ℤ k (k *ℤ l) + + right-predecessor-law-mul-ℤ : + (k l : ℤ) → k *ℤ (pred-ℤ l) ＝ (neg-ℤ k) +ℤ (k *ℤ l) + right-predecessor-law-mul-ℤ (inl zero-ℕ) l = + ( left-neg-unit-law-mul-ℤ (pred-ℤ l)) ∙ + ( neg-pred-ℤ l) + right-predecessor-law-mul-ℤ (inl (succ-ℕ n)) l = + ( left-predecessor-law-mul-ℤ (inl n) (pred-ℤ l)) ∙ + ( ( ap ((neg-ℤ (pred-ℤ l)) +ℤ_) (right-predecessor-law-mul-ℤ (inl n) l)) ∙ + ( ( inv + ( associative-add-ℤ (neg-ℤ (pred-ℤ l)) (in-pos-ℤ n) ((inl n) *ℤ l))) ∙ + ( ( ap + ( _+ℤ ((inl n) *ℤ l)) + { x = (neg-ℤ (pred-ℤ l)) +ℤ (inr (inr n))} + { y = (neg-ℤ (inl (succ-ℕ n))) +ℤ (neg-ℤ l)} + ( ( ap (_+ℤ (in-pos-ℤ n)) (neg-pred-ℤ l)) ∙ + ( ( left-successor-law-add-ℤ (neg-ℤ l) (in-pos-ℤ n)) ∙ + ( ( ap succ-ℤ (commutative-add-ℤ (neg-ℤ l) (in-pos-ℤ n))) ∙ + ( inv (left-successor-law-add-ℤ (in-pos-ℤ n) (neg-ℤ l))))))) ∙ + ( associative-add-ℤ (in-pos-ℤ (succ-ℕ n)) (neg-ℤ l) ((inl n) *ℤ l))))) + right-predecessor-law-mul-ℤ (inr (inl star)) l = refl + right-predecessor-law-mul-ℤ (inr (inr zero-ℕ)) l = refl + right-predecessor-law-mul-ℤ (inr (inr (succ-ℕ n))) l = + ( left-successor-law-mul-ℤ (in-pos-ℤ n) (pred-ℤ l)) ∙ + ( ( ap ((pred-ℤ l) +ℤ_) (right-predecessor-law-mul-ℤ (inr (inr n)) l)) ∙ + ( ( inv (associative-add-ℤ (pred-ℤ l) (inl n) ((inr (inr n)) *ℤ l))) ∙ + ( ( ap + ( _+ℤ ((in-pos-ℤ n) *ℤ l)) + { x = (pred-ℤ l) +ℤ (inl n)} + { y = (neg-ℤ (in-pos-ℤ (succ-ℕ n))) +ℤ l} + ( ( left-predecessor-law-add-ℤ l (inl n)) ∙ + ( ( ap pred-ℤ (commutative-add-ℤ l (inl n))) ∙ + ( inv (left-predecessor-law-add-ℤ (inl n) l))))) ∙ + ( associative-add-ℤ (inl (succ-ℕ n)) l ((inr (inr n)) *ℤ l))))) + + right-predecessor-law-mul-ℤ' : + (k l : ℤ) → k *ℤ (pred-ℤ l) ＝ (k *ℤ l) +ℤ (neg-ℤ k) + right-predecessor-law-mul-ℤ' k l = + right-predecessor-law-mul-ℤ k l ∙ + commutative-add-ℤ (neg-ℤ k) (k *ℤ l) ``` ### Multiplication on the integers distributes on the right over addition ```agda -right-distributive-mul-add-ℤ : - (k l m : ℤ) → (k +ℤ l) *ℤ m ＝ (k *ℤ m) +ℤ (l *ℤ m) -right-distributive-mul-add-ℤ (inl zero-ℕ) l m = - ( left-predecessor-law-mul-ℤ l m) ∙ - ( ap - ( _+ℤ (l *ℤ m)) - ( inv - ( ( left-predecessor-law-mul-ℤ zero-ℤ m) ∙ - ( right-unit-law-add-ℤ (neg-ℤ m))))) -right-distributive-mul-add-ℤ (inl (succ-ℕ x)) l m = - ( left-predecessor-law-mul-ℤ ((inl x) +ℤ l) m) ∙ - ( ( ap ((neg-ℤ m) +ℤ_) (right-distributive-mul-add-ℤ (inl x) l m)) ∙ - ( inv (associative-add-ℤ (neg-ℤ m) ((inl x) *ℤ m) (l *ℤ m)))) -right-distributive-mul-add-ℤ (inr (inl star)) l m = refl -right-distributive-mul-add-ℤ (inr (inr zero-ℕ)) l m = - left-successor-law-mul-ℤ l m -right-distributive-mul-add-ℤ (inr (inr (succ-ℕ n))) l m = - ( left-successor-law-mul-ℤ ((in-pos-ℤ n) +ℤ l) m) ∙ - ( ( ap (m +ℤ_) (right-distributive-mul-add-ℤ (inr (inr n)) l m)) ∙ - ( inv (associative-add-ℤ m ((in-pos-ℤ n) *ℤ m) (l *ℤ m)))) +abstract + right-distributive-mul-add-ℤ : + (k l m : ℤ) → (k +ℤ l) *ℤ m ＝ (k *ℤ m) +ℤ (l *ℤ m) + right-distributive-mul-add-ℤ (inl zero-ℕ) l m = + ( left-predecessor-law-mul-ℤ l m) ∙ + ( ap + ( _+ℤ (l *ℤ m)) + ( inv + ( ( left-predecessor-law-mul-ℤ zero-ℤ m) ∙ + ( right-unit-law-add-ℤ (neg-ℤ m))))) + right-distributive-mul-add-ℤ (inl (succ-ℕ x)) l m = + ( left-predecessor-law-mul-ℤ ((inl x) +ℤ l) m) ∙ + ( ( ap ((neg-ℤ m) +ℤ_) (right-distributive-mul-add-ℤ (inl x) l m)) ∙ + ( inv (associative-add-ℤ (neg-ℤ m) ((inl x) *ℤ m) (l *ℤ m)))) + right-distributive-mul-add-ℤ (inr (inl star)) l m = refl + right-distributive-mul-add-ℤ (inr (inr zero-ℕ)) l m = + left-successor-law-mul-ℤ l m + right-distributive-mul-add-ℤ (inr (inr (succ-ℕ n))) l m = + ( left-successor-law-mul-ℤ ((in-pos-ℤ n) +ℤ l) m) ∙ + ( ( ap (m +ℤ_) (right-distributive-mul-add-ℤ (inr (inr n)) l m)) ∙ + ( inv (associative-add-ℤ m ((in-pos-ℤ n) *ℤ m) (l *ℤ m)))) ``` ### Left multiplication by the negative of an integer is the negative of the multiplication ```agda -left-negative-law-mul-ℤ : - (k l : ℤ) → (neg-ℤ k) *ℤ l ＝ neg-ℤ (k *ℤ l) -left-negative-law-mul-ℤ (inl zero-ℕ) l = - ( left-unit-law-mul-ℤ l) ∙ - ( inv (neg-neg-ℤ l)) -left-negative-law-mul-ℤ (inl (succ-ℕ n)) l = - ( ap (_*ℤ l) (neg-pred-ℤ (inl n))) ∙ - ( ( left-successor-law-mul-ℤ (neg-ℤ (inl n)) l) ∙ - ( ( ap (l +ℤ_) (left-negative-law-mul-ℤ (inl n) l)) ∙ - ( right-negative-law-add-ℤ l ((inl n) *ℤ l)))) -left-negative-law-mul-ℤ (inr (inl star)) l = refl -left-negative-law-mul-ℤ (inr (inr zero-ℕ)) l = refl -left-negative-law-mul-ℤ (inr (inr (succ-ℕ n))) l = - ( left-predecessor-law-mul-ℤ (inl n) l) ∙ - ( ( ap ((neg-ℤ l) +ℤ_) (left-negative-law-mul-ℤ (inr (inr n)) l)) ∙ - ( inv (distributive-neg-add-ℤ l ((in-pos-ℤ n) *ℤ l)))) +abstract + left-negative-law-mul-ℤ : + (k l : ℤ) → (neg-ℤ k) *ℤ l ＝ neg-ℤ (k *ℤ l) + left-negative-law-mul-ℤ (inl zero-ℕ) l = + ( left-unit-law-mul-ℤ l) ∙ + ( inv (neg-neg-ℤ l)) + left-negative-law-mul-ℤ (inl (succ-ℕ n)) l = + ( ap (_*ℤ l) (neg-pred-ℤ (inl n))) ∙ + ( ( left-successor-law-mul-ℤ (neg-ℤ (inl n)) l) ∙ + ( ( ap (l +ℤ_) (left-negative-law-mul-ℤ (inl n) l)) ∙ + ( right-negative-law-add-ℤ l ((inl n) *ℤ l)))) + left-negative-law-mul-ℤ (inr (inl star)) l = refl + left-negative-law-mul-ℤ (inr (inr zero-ℕ)) l = refl + left-negative-law-mul-ℤ (inr (inr (succ-ℕ n))) l = + ( left-predecessor-law-mul-ℤ (inl n) l) ∙ + ( ( ap ((neg-ℤ l) +ℤ_) (left-negative-law-mul-ℤ (inr (inr n)) l)) ∙ + ( inv (distributive-neg-add-ℤ l ((in-pos-ℤ n) *ℤ l)))) ``` ### Multiplication on the integers is associative ```agda -associative-mul-ℤ : - (k l m : ℤ) → (k *ℤ l) *ℤ m ＝ k *ℤ (l *ℤ m) -associative-mul-ℤ (inl zero-ℕ) l m = - left-negative-law-mul-ℤ l m -associative-mul-ℤ (inl (succ-ℕ n)) l m = - ( right-distributive-mul-add-ℤ (neg-ℤ l) ((inl n) *ℤ l) m) ∙ - ( ( ap (((neg-ℤ l) *ℤ m) +ℤ_) (associative-mul-ℤ (inl n) l m)) ∙ - ( ap - ( _+ℤ ((inl n) *ℤ (l *ℤ m))) - ( left-negative-law-mul-ℤ l m))) -associative-mul-ℤ (inr (inl star)) l m = refl -associative-mul-ℤ (inr (inr zero-ℕ)) l m = refl -associative-mul-ℤ (inr (inr (succ-ℕ n))) l m = - ( right-distributive-mul-add-ℤ l ((in-pos-ℤ n) *ℤ l) m) ∙ - ( ap ((l *ℤ m) +ℤ_) (associative-mul-ℤ (inr (inr n)) l m)) +abstract + associative-mul-ℤ : + (k l m : ℤ) → (k *ℤ l) *ℤ m ＝ k *ℤ (l *ℤ m) + associative-mul-ℤ (inl zero-ℕ) l m = + left-negative-law-mul-ℤ l m + associative-mul-ℤ (inl (succ-ℕ n)) l m = + ( right-distributive-mul-add-ℤ (neg-ℤ l) ((inl n) *ℤ l) m) ∙ + ( ( ap (((neg-ℤ l) *ℤ m) +ℤ_) (associative-mul-ℤ (inl n) l m)) ∙ + ( ap + ( _+ℤ ((inl n) *ℤ (l *ℤ m))) + ( left-negative-law-mul-ℤ l m))) + associative-mul-ℤ (inr (inl star)) l m = refl + associative-mul-ℤ (inr (inr zero-ℕ)) l m = refl + associative-mul-ℤ (inr (inr (succ-ℕ n))) l m = + ( right-distributive-mul-add-ℤ l ((in-pos-ℤ n) *ℤ l) m) ∙ + ( ap ((l *ℤ m) +ℤ_) (associative-mul-ℤ (inr (inr n)) l m)) ``` ### Multiplication on the integers is commutative ```agda -commutative-mul-ℤ : - (k l : ℤ) → k *ℤ l ＝ l *ℤ k -commutative-mul-ℤ (inl zero-ℕ) l = inv (right-neg-unit-law-mul-ℤ l) -commutative-mul-ℤ (inl (succ-ℕ n)) l = - ( ap ((neg-ℤ l) +ℤ_) (commutative-mul-ℤ (inl n) l)) ∙ - ( inv (right-predecessor-law-mul-ℤ l (inl n))) -commutative-mul-ℤ (inr (inl star)) l = inv (right-zero-law-mul-ℤ l) -commutative-mul-ℤ (inr (inr zero-ℕ)) l = inv (right-unit-law-mul-ℤ l) -commutative-mul-ℤ (inr (inr (succ-ℕ n))) l = - ( ap (l +ℤ_) (commutative-mul-ℤ (inr (inr n)) l)) ∙ - ( inv (right-successor-law-mul-ℤ l (in-pos-ℤ n))) +abstract + commutative-mul-ℤ : + (k l : ℤ) → k *ℤ l ＝ l *ℤ k + commutative-mul-ℤ (inl zero-ℕ) l = inv (right-neg-unit-law-mul-ℤ l) + commutative-mul-ℤ (inl (succ-ℕ n)) l = + ( ap ((neg-ℤ l) +ℤ_) (commutative-mul-ℤ (inl n) l)) ∙ + ( inv (right-predecessor-law-mul-ℤ l (inl n))) + commutative-mul-ℤ (inr (inl star)) l = inv (right-zero-law-mul-ℤ l) + commutative-mul-ℤ (inr (inr zero-ℕ)) l = inv (right-unit-law-mul-ℤ l) + commutative-mul-ℤ (inr (inr (succ-ℕ n))) l = + ( ap (l +ℤ_) (commutative-mul-ℤ (inr (inr n)) l)) ∙ + ( inv (right-successor-law-mul-ℤ l (in-pos-ℤ n))) ``` ### Multiplication on the integers distributes on the left over addition ```agda -left-distributive-mul-add-ℤ : - (m k l : ℤ) → m *ℤ (k +ℤ l) ＝ (m *ℤ k) +ℤ (m *ℤ l) -left-distributive-mul-add-ℤ m k l = - commutative-mul-ℤ m (k +ℤ l) ∙ - ( ( right-distributive-mul-add-ℤ k l m) ∙ - ( ap-add-ℤ (commutative-mul-ℤ k m) (commutative-mul-ℤ l m))) +abstract + left-distributive-mul-add-ℤ : + (m k l : ℤ) → m *ℤ (k +ℤ l) ＝ (m *ℤ k) +ℤ (m *ℤ l) + left-distributive-mul-add-ℤ m k l = + ( commutative-mul-ℤ m (k +ℤ l)) ∙ + ( right-distributive-mul-add-ℤ k l m) ∙ + ( ap-add-ℤ (commutative-mul-ℤ k m) (commutative-mul-ℤ l m)) ``` ### Right multiplication by the negative of an integer is the negative of the multiplication ```agda -right-negative-law-mul-ℤ : - (k l : ℤ) → k *ℤ (neg-ℤ l) ＝ neg-ℤ (k *ℤ l) -right-negative-law-mul-ℤ k l = - ( ( commutative-mul-ℤ k (neg-ℤ l)) ∙ - ( left-negative-law-mul-ℤ l k)) ∙ - ( ap neg-ℤ (commutative-mul-ℤ l k)) +abstract + right-negative-law-mul-ℤ : + (k l : ℤ) → k *ℤ (neg-ℤ l) ＝ neg-ℤ (k *ℤ l) + right-negative-law-mul-ℤ k l = + ( commutative-mul-ℤ k (neg-ℤ l)) ∙ + ( left-negative-law-mul-ℤ l k) ∙ + ( ap neg-ℤ (commutative-mul-ℤ l k)) ``` ### The multiplication of the negatives of two integers is equal to their multiplication ```agda -double-negative-law-mul-ℤ : (k l : ℤ) → (neg-ℤ k) *ℤ (neg-ℤ l) ＝ k *ℤ l -double-negative-law-mul-ℤ k l = - equational-reasoning - (neg-ℤ k) *ℤ (neg-ℤ l) - ＝ neg-ℤ (k *ℤ (neg-ℤ l)) - by left-negative-law-mul-ℤ k (neg-ℤ l) - ＝ neg-ℤ (neg-ℤ (k *ℤ l)) - by ap neg-ℤ (right-negative-law-mul-ℤ k l) - ＝ k *ℤ l - by neg-neg-ℤ (k *ℤ l) +abstract + double-negative-law-mul-ℤ : (k l : ℤ) → (neg-ℤ k) *ℤ (neg-ℤ l) ＝ k *ℤ l + double-negative-law-mul-ℤ k l = + equational-reasoning + (neg-ℤ k) *ℤ (neg-ℤ l) + ＝ neg-ℤ (k *ℤ (neg-ℤ l)) + by left-negative-law-mul-ℤ k (neg-ℤ l) + ＝ neg-ℤ (neg-ℤ (k *ℤ l)) + by ap neg-ℤ (right-negative-law-mul-ℤ k l) + ＝ k *ℤ l + by neg-neg-ℤ (k *ℤ l) ``` ### Interchange law ```agda -interchange-law-mul-mul-ℤ : - (x y u v : ℤ) → (x *ℤ y) *ℤ (u *ℤ v) ＝ (x *ℤ u) *ℤ (y *ℤ v) -interchange-law-mul-mul-ℤ = - interchange-law-commutative-and-associative - mul-ℤ - commutative-mul-ℤ - associative-mul-ℤ +abstract + interchange-law-mul-mul-ℤ : + (x y u v : ℤ) → (x *ℤ y) *ℤ (u *ℤ v) ＝ (x *ℤ u) *ℤ (y *ℤ v) + interchange-law-mul-mul-ℤ = + interchange-law-commutative-and-associative + mul-ℤ + commutative-mul-ℤ + associative-mul-ℤ ``` -### Computing multiplication of integers that come from natural numbers +### The canonical embedding of natural numbers in the integers preserves multiplication ```agda -mul-int-ℕ : (x y : ℕ) → (int-ℕ x) *ℤ (int-ℕ y) ＝ int-ℕ (x *ℕ y) -mul-int-ℕ zero-ℕ y = refl -mul-int-ℕ (succ-ℕ x) y = - ( ap (_*ℤ (int-ℕ y)) (inv (succ-int-ℕ x))) ∙ - ( ( left-successor-law-mul-ℤ (int-ℕ x) (int-ℕ y)) ∙ - ( ( ( ap ((int-ℕ y) +ℤ_) (mul-int-ℕ x y)) ∙ - ( add-int-ℕ y (x *ℕ y))) ∙ - ( ap int-ℕ (commutative-add-ℕ y (x *ℕ y))))) - -compute-mul-ℤ : (x y : ℤ) → x *ℤ y ＝ explicit-mul-ℤ x y -compute-mul-ℤ (inl zero-ℕ) (inl y) = - inv (ap int-ℕ (left-unit-law-mul-ℕ (succ-ℕ y))) -compute-mul-ℤ (inl (succ-ℕ x)) (inl y) = - ( ( ap ((int-ℕ (succ-ℕ y)) +ℤ_) (compute-mul-ℤ (inl x) (inl y))) ∙ - ( commutative-add-ℤ - ( int-ℕ (succ-ℕ y)) - ( int-ℕ ((succ-ℕ x) *ℕ (succ-ℕ y))))) ∙ - ( add-int-ℕ ((succ-ℕ x) *ℕ (succ-ℕ y)) (succ-ℕ y)) -compute-mul-ℤ (inl zero-ℕ) (inr (inl star)) = refl -compute-mul-ℤ (inl zero-ℕ) (inr (inr x)) = ap inl (inv (left-unit-law-add-ℕ x)) -compute-mul-ℤ (inl (succ-ℕ x)) (inr (inl star)) = right-zero-law-mul-ℤ (inl x) -compute-mul-ℤ (inl (succ-ℕ x)) (inr (inr y)) = - ( ( ( ap ((inl y) +ℤ_) (compute-mul-ℤ (inl x) (inr (inr y)))) ∙ - ( inv - ( distributive-neg-add-ℤ - ( inr (inr y)) - ( int-ℕ ((succ-ℕ x) *ℕ (succ-ℕ y)))))) ∙ - ( ap - ( neg-ℤ) +abstract + mul-int-ℕ : (x y : ℕ) → (int-ℕ x) *ℤ (int-ℕ y) ＝ int-ℕ (x *ℕ y) + mul-int-ℕ zero-ℕ y = refl + mul-int-ℕ (succ-ℕ x) y = + ( ap (_*ℤ (int-ℕ y)) (inv (succ-int-ℕ x))) ∙ + ( ( left-successor-law-mul-ℤ (int-ℕ x) (int-ℕ y)) ∙ + ( ( ( ap ((int-ℕ y) +ℤ_) (mul-int-ℕ x y)) ∙ + ( add-int-ℕ y (x *ℕ y))) ∙ + ( ap int-ℕ (commutative-add-ℕ y (x *ℕ y))))) + + compute-mul-ℤ : (x y : ℤ) → x *ℤ y ＝ explicit-mul-ℤ x y + compute-mul-ℤ (inl zero-ℕ) (inl y) = + inv (ap int-ℕ (left-unit-law-mul-ℕ (succ-ℕ y))) + compute-mul-ℤ (inl (succ-ℕ x)) (inl y) = + ( ( ap ((int-ℕ (succ-ℕ y)) +ℤ_) (compute-mul-ℤ (inl x) (inl y))) ∙ ( commutative-add-ℤ ( int-ℕ (succ-ℕ y)) - ( int-ℕ ((succ-ℕ x) *ℕ (succ-ℕ y)))))) ∙ - ( ap neg-ℤ (add-int-ℕ ((succ-ℕ x) *ℕ (succ-ℕ y)) (succ-ℕ y))) -compute-mul-ℤ (inr (inl star)) (inl y) = refl -compute-mul-ℤ (inr (inr zero-ℕ)) (inl y) = ap inl (inv (left-unit-law-add-ℕ y)) -compute-mul-ℤ (inr (inr (succ-ℕ x))) (inl y) = - ( ap ((inl y) +ℤ_) (compute-mul-ℤ (inr (inr x)) (inl y))) ∙ - ( ( ( inv - ( distributive-neg-add-ℤ - ( inr (inr y)) - ( inr (inr ((x *ℕ (succ-ℕ y)) +ℕ y))))) ∙ + ( int-ℕ ((succ-ℕ x) *ℕ (succ-ℕ y))))) ∙ + ( add-int-ℕ ((succ-ℕ x) *ℕ (succ-ℕ y)) (succ-ℕ y)) + compute-mul-ℤ (inl zero-ℕ) (inr (inl star)) = refl + compute-mul-ℤ (inl zero-ℕ) (inr (inr x)) = + ap inl (inv (left-unit-law-add-ℕ x)) + compute-mul-ℤ (inl (succ-ℕ x)) (inr (inl star)) = right-zero-law-mul-ℤ (inl x) + compute-mul-ℤ (inl (succ-ℕ x)) (inr (inr y)) = + ( ( ( ap ((inl y) +ℤ_) (compute-mul-ℤ (inl x) (inr (inr y)))) ∙ + ( inv + ( distributive-neg-add-ℤ + ( inr (inr y)) + ( int-ℕ ((succ-ℕ x) *ℕ (succ-ℕ y)))))) ∙ ( ap ( neg-ℤ) - ( ( add-int-ℕ (succ-ℕ y) ((succ-ℕ x) *ℕ (succ-ℕ y))) ∙ - ( ap - ( inr ∘ inr) - ( left-successor-law-add-ℕ y ((x *ℕ (succ-ℕ y)) +ℕ y)))))) ∙ - ( ap inl (commutative-add-ℕ y ((succ-ℕ x) *ℕ (succ-ℕ y))))) -compute-mul-ℤ (inr (inl star)) (inr (inl star)) = refl -compute-mul-ℤ (inr (inl star)) (inr (inr y)) = refl -compute-mul-ℤ (inr (inr zero-ℕ)) (inr (inl star)) = refl -compute-mul-ℤ (inr (inr (succ-ℕ x))) (inr (inl star)) = - right-zero-law-mul-ℤ (inr (inr (succ-ℕ x))) -compute-mul-ℤ (inr (inr zero-ℕ)) (inr (inr y)) = - ap - ( inr ∘ inr) - ( inv - ( ( ap (_+ℕ y) (left-zero-law-mul-ℕ (succ-ℕ y))) ∙ - ( left-unit-law-add-ℕ y))) -compute-mul-ℤ (inr (inr (succ-ℕ x))) (inr (inr y)) = - ( ap ((inr (inr y)) +ℤ_) (compute-mul-ℤ (inr (inr x)) (inr (inr y)))) ∙ - ( ( add-int-ℕ (succ-ℕ y) ((succ-ℕ x) *ℕ (succ-ℕ y))) ∙ - ( ap int-ℕ (commutative-add-ℕ (succ-ℕ y) ((succ-ℕ x) *ℕ (succ-ℕ y))))) + ( commutative-add-ℤ + ( int-ℕ (succ-ℕ y)) + ( int-ℕ ((succ-ℕ x) *ℕ (succ-ℕ y)))))) ∙ + ( ap neg-ℤ (add-int-ℕ ((succ-ℕ x) *ℕ (succ-ℕ y)) (succ-ℕ y))) + compute-mul-ℤ (inr (inl star)) (inl y) = refl + compute-mul-ℤ (inr (inr zero-ℕ)) (inl y) = + ap inl (inv (left-unit-law-add-ℕ y)) + compute-mul-ℤ (inr (inr (succ-ℕ x))) (inl y) = + ( ap ((inl y) +ℤ_) (compute-mul-ℤ (inr (inr x)) (inl y))) ∙ + ( ( ( inv + ( distributive-neg-add-ℤ + ( inr (inr y)) + ( inr (inr ((x *ℕ (succ-ℕ y)) +ℕ y))))) ∙ + ( ap + ( neg-ℤ) + ( ( add-int-ℕ (succ-ℕ y) ((succ-ℕ x) *ℕ (succ-ℕ y))) ∙ + ( ap + ( inr ∘ inr) + ( left-successor-law-add-ℕ y ((x *ℕ (succ-ℕ y)) +ℕ y)))))) ∙ + ( ap inl (commutative-add-ℕ y ((succ-ℕ x) *ℕ (succ-ℕ y))))) + compute-mul-ℤ (inr (inl star)) (inr (inl star)) = refl + compute-mul-ℤ (inr (inl star)) (inr (inr y)) = refl + compute-mul-ℤ (inr (inr zero-ℕ)) (inr (inl star)) = refl + compute-mul-ℤ (inr (inr (succ-ℕ x))) (inr (inl star)) = + right-zero-law-mul-ℤ (inr (inr (succ-ℕ x))) + compute-mul-ℤ (inr (inr zero-ℕ)) (inr (inr y)) = + ap + ( inr ∘ inr) + ( inv + ( ( ap (_+ℕ y) (left-zero-law-mul-ℕ (succ-ℕ y))) ∙ + ( left-unit-law-add-ℕ y))) + compute-mul-ℤ (inr (inr (succ-ℕ x))) (inr (inr y)) = + ( ap ((inr (inr y)) +ℤ_) (compute-mul-ℤ (inr (inr x)) (inr (inr y)))) ∙ + ( ( add-int-ℕ (succ-ℕ y) ((succ-ℕ x) *ℕ (succ-ℕ y))) ∙ + ( ap int-ℕ (commutative-add-ℕ (succ-ℕ y) ((succ-ℕ x) *ℕ (succ-ℕ y))))) ``` ### Multiplication on integers distributes over the difference ```agda -left-distributive-mul-diff-ℤ : - (z x y : ℤ) → z *ℤ (x -ℤ y) ＝ (z *ℤ x) -ℤ (z *ℤ y) -left-distributive-mul-diff-ℤ z x y = - ( left-distributive-mul-add-ℤ z x (neg-ℤ y)) ∙ - ( ap ((z *ℤ x) +ℤ_) (right-negative-law-mul-ℤ z y)) - -right-distributive-mul-diff-ℤ : - (x y z : ℤ) → (x -ℤ y) *ℤ z ＝ (x *ℤ z) -ℤ (y *ℤ z) -right-distributive-mul-diff-ℤ x y z = - ( right-distributive-mul-add-ℤ x (neg-ℤ y) z) ∙ - ( ap ((x *ℤ z) +ℤ_) (left-negative-law-mul-ℤ y z)) +abstract + left-distributive-mul-diff-ℤ : + (z x y : ℤ) → z *ℤ (x -ℤ y) ＝ (z *ℤ x) -ℤ (z *ℤ y) + left-distributive-mul-diff-ℤ z x y = + ( left-distributive-mul-add-ℤ z x (neg-ℤ y)) ∙ + ( ap ((z *ℤ x) +ℤ_) (right-negative-law-mul-ℤ z y)) + + right-distributive-mul-diff-ℤ : + (x y z : ℤ) → (x -ℤ y) *ℤ z ＝ (x *ℤ z) -ℤ (y *ℤ z) + right-distributive-mul-diff-ℤ x y z = + ( right-distributive-mul-add-ℤ x (neg-ℤ y) z) ∙ + ( ap ((x *ℤ z) +ℤ_) (left-negative-law-mul-ℤ y z)) ``` ### If the product of two integers is zero, one of the factors is zero ```agda -is-zero-is-zero-mul-ℤ : - (x y : ℤ) → is-zero-ℤ (x *ℤ y) → (is-zero-ℤ x) + (is-zero-ℤ y) -is-zero-is-zero-mul-ℤ (inl x) (inl y) H = - ex-falso (Eq-eq-ℤ (inv (compute-mul-ℤ (inl x) (inl y)) ∙ H)) -is-zero-is-zero-mul-ℤ (inl x) (inr (inl star)) H = inr refl -is-zero-is-zero-mul-ℤ (inl x) (inr (inr y)) H = - ex-falso (Eq-eq-ℤ (inv (compute-mul-ℤ (inl x) (inr (inr y))) ∙ H)) -is-zero-is-zero-mul-ℤ (inr (inl star)) y H = inl refl -is-zero-is-zero-mul-ℤ (inr (inr x)) (inl y) H = - ex-falso (Eq-eq-ℤ (inv (compute-mul-ℤ (inr (inr x)) (inl y)) ∙ H)) -is-zero-is-zero-mul-ℤ (inr (inr x)) (inr (inl star)) H = inr refl -is-zero-is-zero-mul-ℤ (inr (inr x)) (inr (inr y)) H = - ex-falso (Eq-eq-ℤ (inv (compute-mul-ℤ (inr (inr x)) (inr (inr y))) ∙ H)) +abstract + is-zero-is-zero-mul-ℤ : + (x y : ℤ) → is-zero-ℤ (x *ℤ y) → (is-zero-ℤ x) + (is-zero-ℤ y) + is-zero-is-zero-mul-ℤ (inl x) (inl y) H = + ex-falso (Eq-eq-ℤ (inv (compute-mul-ℤ (inl x) (inl y)) ∙ H)) + is-zero-is-zero-mul-ℤ (inl x) (inr (inl star)) H = inr refl + is-zero-is-zero-mul-ℤ (inl x) (inr (inr y)) H = + ex-falso (Eq-eq-ℤ (inv (compute-mul-ℤ (inl x) (inr (inr y))) ∙ H)) + is-zero-is-zero-mul-ℤ (inr (inl star)) y H = inl refl + is-zero-is-zero-mul-ℤ (inr (inr x)) (inl y) H = + ex-falso (Eq-eq-ℤ (inv (compute-mul-ℤ (inr (inr x)) (inl y)) ∙ H)) + is-zero-is-zero-mul-ℤ (inr (inr x)) (inr (inl star)) H = inr refl + is-zero-is-zero-mul-ℤ (inr (inr x)) (inr (inr y)) H = + ex-falso (Eq-eq-ℤ (inv (compute-mul-ℤ (inr (inr x)) (inr (inr y))) ∙ H)) ``` ### Injectivity of multiplication by a nonzero integer ```agda -is-injective-left-mul-ℤ : - (x : ℤ) → is-nonzero-ℤ x → is-injective (x *ℤ_) -is-injective-left-mul-ℤ x f {y} {z} p = - eq-diff-ℤ - ( map-left-unit-law-coproduct-is-empty - ( is-zero-ℤ x) - ( is-zero-ℤ (y -ℤ z)) +abstract + is-injective-left-mul-ℤ : + (x : ℤ) → is-nonzero-ℤ x → is-injective (x *ℤ_) + is-injective-left-mul-ℤ x f {y} {z} p = + eq-diff-ℤ + ( map-left-unit-law-coproduct-is-empty + ( is-zero-ℤ x) + ( is-zero-ℤ (y -ℤ z)) + ( f) + ( is-zero-is-zero-mul-ℤ x + ( y -ℤ z) + ( left-distributive-mul-diff-ℤ x y z ∙ is-zero-diff-ℤ p))) + + is-injective-right-mul-ℤ : + (x : ℤ) → is-nonzero-ℤ x → is-injective (_*ℤ x) + is-injective-right-mul-ℤ x f {y} {z} p = + is-injective-left-mul-ℤ + ( x) ( f) - ( is-zero-is-zero-mul-ℤ x - ( y -ℤ z) - ( left-distributive-mul-diff-ℤ x y z ∙ is-zero-diff-ℤ p))) - -is-injective-right-mul-ℤ : - (x : ℤ) → is-nonzero-ℤ x → is-injective (_*ℤ x) -is-injective-right-mul-ℤ x f {y} {z} p = - is-injective-left-mul-ℤ - ( x) - ( f) - ( commutative-mul-ℤ x y ∙ (p ∙ commutative-mul-ℤ z x)) + ( commutative-mul-ℤ x y ∙ (p ∙ commutative-mul-ℤ z x)) ``` ### Multiplication by a nonzero integer is an embedding ```agda -is-emb-left-mul-ℤ : (x : ℤ) → is-nonzero-ℤ x → is-emb (x *ℤ_) -is-emb-left-mul-ℤ x f = - is-emb-is-injective is-set-ℤ (is-injective-left-mul-ℤ x f) - -is-emb-right-mul-ℤ : (x : ℤ) → is-nonzero-ℤ x → is-emb (_*ℤ x) -is-emb-right-mul-ℤ x f = - is-emb-is-injective is-set-ℤ (is-injective-right-mul-ℤ x f) +abstract + is-emb-left-mul-ℤ : (x : ℤ) → is-nonzero-ℤ x → is-emb (x *ℤ_) + is-emb-left-mul-ℤ x f = + is-emb-is-injective is-set-ℤ (is-injective-left-mul-ℤ x f) + + is-emb-right-mul-ℤ : (x : ℤ) → is-nonzero-ℤ x → is-emb (_*ℤ x) + is-emb-right-mul-ℤ x f = + is-emb-is-injective is-set-ℤ (is-injective-right-mul-ℤ x f) ``` ## See also diff --git a/src/elementary-number-theory/multiplication-natural-numbers.lagda.md b/src/elementary-number-theory/multiplication-natural-numbers.lagda.md index c4d6dd3e5f..335c42ad42 100644 --- a/src/elementary-number-theory/multiplication-natural-numbers.lagda.md +++ b/src/elementary-number-theory/multiplication-natural-numbers.lagda.md @@ -68,6 +68,8 @@ triple-ℕ x = 3 *ℕ x ## Properties +### Zero laws of multiplication + ```agda abstract left-zero-law-mul-ℕ : @@ -79,7 +81,11 @@ abstract right-zero-law-mul-ℕ zero-ℕ = refl right-zero-law-mul-ℕ (succ-ℕ x) = ( right-unit-law-add-ℕ (x *ℕ zero-ℕ)) ∙ (right-zero-law-mul-ℕ x) +``` +### Unit laws of multiplication + +```agda abstract right-unit-law-mul-ℕ : (x : ℕ) → x *ℕ 1 ＝ x @@ -90,7 +96,11 @@ abstract (x : ℕ) → 1 *ℕ x ＝ x left-unit-law-mul-ℕ zero-ℕ = refl left-unit-law-mul-ℕ (succ-ℕ x) = ap succ-ℕ (left-unit-law-mul-ℕ x) +``` + +### Successor laws of multiplication +```agda abstract left-successor-law-mul-ℕ : (x y : ℕ) → (succ-ℕ x) *ℕ y ＝ (x *ℕ y) +ℕ y @@ -103,7 +113,11 @@ abstract ( ( ap (λ t → succ-ℕ (t +ℕ y)) (right-successor-law-mul-ℕ x y)) ∙ ( ap succ-ℕ (associative-add-ℕ x (x *ℕ y) y))) ∙ ( inv (left-successor-law-add-ℕ x ((x *ℕ y) +ℕ y))) +``` + +### Commutativity of multiplication +```agda abstract commutative-mul-ℕ : (x y : ℕ) → x *ℕ y ＝ y *ℕ x @@ -112,7 +126,11 @@ abstract ( commutative-add-ℕ (x *ℕ y) y) ∙ ( ( ap (y +ℕ_) (commutative-mul-ℕ x y)) ∙ ( inv (right-successor-law-mul-ℕ y x))) +``` + +### Distributivity of multiplication over addition +```agda abstract left-distributive-mul-add-ℕ : (x y z : ℕ) → x *ℕ (y +ℕ z) ＝ (x *ℕ y) +ℕ (x *ℕ z) @@ -128,7 +146,6 @@ abstract ( associative-add-ℕ y (x *ℕ z) z)))) ∙ ( inv (associative-add-ℕ (x *ℕ y) y ((x *ℕ z) +ℕ z)))))) -abstract right-distributive-mul-add-ℕ : (x y z : ℕ) → (x +ℕ y) *ℕ z ＝ (x *ℕ z) +ℕ (y *ℕ z) right-distributive-mul-add-ℕ x y z = @@ -136,7 +153,11 @@ abstract ( ( left-distributive-mul-add-ℕ z x y) ∙ ( ( ap (_+ℕ (z *ℕ y)) (commutative-mul-ℕ z x)) ∙ ( ap ((x *ℕ z) +ℕ_) (commutative-mul-ℕ z y)))) +``` + +### Associativity of multiplication +```agda abstract associative-mul-ℕ : (x y z : ℕ) → (x *ℕ y) *ℕ z ＝ x *ℕ (y *ℕ z) @@ -144,7 +165,11 @@ abstract associative-mul-ℕ (succ-ℕ x) y z = ( right-distributive-mul-add-ℕ (x *ℕ y) y z) ∙ ( ap (_+ℕ (y *ℕ z)) (associative-mul-ℕ x y z)) +``` + +### Multiplying a natural number by two is adding it to itself +```agda left-two-law-mul-ℕ : (x : ℕ) → 2 *ℕ x ＝ x +ℕ x left-two-law-mul-ℕ x = @@ -156,44 +181,62 @@ right-two-law-mul-ℕ : right-two-law-mul-ℕ x = ( right-successor-law-mul-ℕ x 1) ∙ ( ap (x +ℕ_) (right-unit-law-mul-ℕ x)) +``` + +### Interchange laws of multiplication +```agda interchange-law-mul-mul-ℕ : interchange-law mul-ℕ mul-ℕ interchange-law-mul-mul-ℕ = interchange-law-commutative-and-associative mul-ℕ commutative-mul-ℕ associative-mul-ℕ +``` -is-injective-right-mul-succ-ℕ : - (k : ℕ) → is-injective (_*ℕ (succ-ℕ k)) -is-injective-right-mul-succ-ℕ k {zero-ℕ} {zero-ℕ} p = refl -is-injective-right-mul-succ-ℕ k {succ-ℕ m} {succ-ℕ n} p = - ap succ-ℕ - ( is-injective-right-mul-succ-ℕ k {m} {n} - ( is-injective-right-add-ℕ - ( succ-ℕ k) - ( ( inv (left-successor-law-mul-ℕ m (succ-ℕ k))) ∙ - ( ( p) ∙ - ( left-successor-law-mul-ℕ n (succ-ℕ k)))))) - -is-injective-right-mul-ℕ : (k : ℕ) → is-nonzero-ℕ k → is-injective (_*ℕ k) -is-injective-right-mul-ℕ k H p with - is-successor-is-nonzero-ℕ H -... | pair l refl = is-injective-right-mul-succ-ℕ l p - -is-injective-left-mul-succ-ℕ : - (k : ℕ) → is-injective ((succ-ℕ k) *ℕ_) -is-injective-left-mul-succ-ℕ k {m} {n} p = - is-injective-right-mul-succ-ℕ k - ( ( commutative-mul-ℕ m (succ-ℕ k)) ∙ - ( p ∙ commutative-mul-ℕ (succ-ℕ k) n)) - -is-injective-left-mul-ℕ : - (k : ℕ) → is-nonzero-ℕ k → is-injective (k *ℕ_) -is-injective-left-mul-ℕ k H p with - is-successor-is-nonzero-ℕ H -... | pair l refl = is-injective-left-mul-succ-ℕ l p +### Right multiplication by a nonzero natural number is injective +```agda +abstract + is-injective-right-mul-succ-ℕ : + (k : ℕ) → is-injective (_*ℕ (succ-ℕ k)) + is-injective-right-mul-succ-ℕ k {zero-ℕ} {zero-ℕ} p = refl + is-injective-right-mul-succ-ℕ k {succ-ℕ m} {succ-ℕ n} p = + ap succ-ℕ + ( is-injective-right-mul-succ-ℕ k {m} {n} + ( is-injective-right-add-ℕ + ( succ-ℕ k) + ( ( inv (left-successor-law-mul-ℕ m (succ-ℕ k))) ∙ + ( ( p) ∙ + ( left-successor-law-mul-ℕ n (succ-ℕ k)))))) + + is-injective-right-mul-ℕ : (k : ℕ) → is-nonzero-ℕ k → is-injective (_*ℕ k) + is-injective-right-mul-ℕ k H p with + is-successor-is-nonzero-ℕ H + ... | pair l refl = is-injective-right-mul-succ-ℕ l p +``` + +### Left multiplication by a nonzero natural number is injective + +```agda +abstract + is-injective-left-mul-succ-ℕ : + (k : ℕ) → is-injective ((succ-ℕ k) *ℕ_) + is-injective-left-mul-succ-ℕ k {m} {n} p = + is-injective-right-mul-succ-ℕ k + ( ( commutative-mul-ℕ m (succ-ℕ k)) ∙ + ( p ∙ commutative-mul-ℕ (succ-ℕ k) n)) + + is-injective-left-mul-ℕ : + (k : ℕ) → is-nonzero-ℕ k → is-injective (k *ℕ_) + is-injective-left-mul-ℕ k H p with + is-successor-is-nonzero-ℕ H + ... | pair l refl = is-injective-left-mul-succ-ℕ l p +``` + +### Multiplication by a nonzero natural number is an embedding + +```agda is-emb-left-mul-ℕ : (x : ℕ) → is-nonzero-ℕ x → is-emb (x *ℕ_) is-emb-left-mul-ℕ x H = is-emb-is-injective is-set-ℕ (is-injective-left-mul-ℕ x H) @@ -201,88 +244,105 @@ is-emb-left-mul-ℕ x H = is-emb-right-mul-ℕ : (x : ℕ) → is-nonzero-ℕ x → is-emb (_*ℕ x) is-emb-right-mul-ℕ x H = is-emb-is-injective is-set-ℕ (is-injective-right-mul-ℕ x H) +``` + +### The product of two natural numbers is nonzero if and only if both are nonzero -is-nonzero-mul-ℕ : - (x y : ℕ) → is-nonzero-ℕ x → is-nonzero-ℕ y → is-nonzero-ℕ (x *ℕ y) -is-nonzero-mul-ℕ x y H K p = - K (is-injective-left-mul-ℕ x H (p ∙ (inv (right-zero-law-mul-ℕ x)))) +```agda +abstract + is-nonzero-mul-ℕ : + (x y : ℕ) → is-nonzero-ℕ x → is-nonzero-ℕ y → is-nonzero-ℕ (x *ℕ y) + is-nonzero-mul-ℕ x y H K p = + K (is-injective-left-mul-ℕ x H (p ∙ (inv (right-zero-law-mul-ℕ x)))) + + is-nonzero-left-factor-mul-ℕ : + (x y : ℕ) → is-nonzero-ℕ (x *ℕ y) → is-nonzero-ℕ x + is-nonzero-left-factor-mul-ℕ .zero-ℕ y H refl = H (left-zero-law-mul-ℕ y) + + is-nonzero-right-factor-mul-ℕ : + (x y : ℕ) → is-nonzero-ℕ (x *ℕ y) → is-nonzero-ℕ y + is-nonzero-right-factor-mul-ℕ x .zero-ℕ H refl = H (right-zero-law-mul-ℕ x) +``` + +### If `(x+1)y = x+1`, then `y = 1` + +```agda +abstract + is-one-is-right-unit-mul-ℕ : + (x y : ℕ) → (succ-ℕ x) *ℕ y ＝ succ-ℕ x → is-one-ℕ y + is-one-is-right-unit-mul-ℕ x y p = + is-injective-left-mul-succ-ℕ x (p ∙ inv (right-unit-law-mul-ℕ (succ-ℕ x))) + + is-one-is-left-unit-mul-ℕ : + (x y : ℕ) → x *ℕ (succ-ℕ y) ＝ succ-ℕ y → is-one-ℕ x + is-one-is-left-unit-mul-ℕ x y p = + is-injective-right-mul-succ-ℕ y (p ∙ inv (left-unit-law-mul-ℕ (succ-ℕ y))) + + is-one-right-is-one-mul-ℕ : + (x y : ℕ) → is-one-ℕ (x *ℕ y) → is-one-ℕ y + is-one-right-is-one-mul-ℕ zero-ℕ zero-ℕ p = p + is-one-right-is-one-mul-ℕ zero-ℕ (succ-ℕ y) () + is-one-right-is-one-mul-ℕ (succ-ℕ x) zero-ℕ p = + is-one-right-is-one-mul-ℕ x zero-ℕ p + is-one-right-is-one-mul-ℕ (succ-ℕ x) (succ-ℕ y) p = + ap + ( succ-ℕ) + ( is-zero-right-is-zero-add-ℕ (x *ℕ (succ-ℕ y)) y + ( is-injective-succ-ℕ p)) +``` -is-nonzero-left-factor-mul-ℕ : - (x y : ℕ) → is-nonzero-ℕ (x *ℕ y) → is-nonzero-ℕ x -is-nonzero-left-factor-mul-ℕ .zero-ℕ y H refl = H (left-zero-law-mul-ℕ y) +### The product of two natural numbers is one if and only if both are one -is-nonzero-right-factor-mul-ℕ : - (x y : ℕ) → is-nonzero-ℕ (x *ℕ y) → is-nonzero-ℕ y -is-nonzero-right-factor-mul-ℕ x .zero-ℕ H refl = H (right-zero-law-mul-ℕ x) +```agda +abstract + is-one-left-is-one-mul-ℕ : + (x y : ℕ) → is-one-ℕ (x *ℕ y) → is-one-ℕ x + is-one-left-is-one-mul-ℕ x y p = + is-one-right-is-one-mul-ℕ y x (commutative-mul-ℕ y x ∙ p) + + is-one-mul-ℕ : + (x y : ℕ) → is-one-ℕ x → is-one-ℕ y → is-one-ℕ (x *ℕ y) + is-one-mul-ℕ .1 .1 refl refl = refl ``` -We conclude that $y = 1$ if $(x+1)y = x+1$. +### For all `m` and `n`, `m + 1 ≠ (m+1)(n+2)` ```agda -is-one-is-right-unit-mul-ℕ : - (x y : ℕ) → (succ-ℕ x) *ℕ y ＝ succ-ℕ x → is-one-ℕ y -is-one-is-right-unit-mul-ℕ x y p = - is-injective-left-mul-succ-ℕ x (p ∙ inv (right-unit-law-mul-ℕ (succ-ℕ x))) - -is-one-is-left-unit-mul-ℕ : - (x y : ℕ) → x *ℕ (succ-ℕ y) ＝ succ-ℕ y → is-one-ℕ x -is-one-is-left-unit-mul-ℕ x y p = - is-injective-right-mul-succ-ℕ y (p ∙ inv (left-unit-law-mul-ℕ (succ-ℕ y))) - -is-one-right-is-one-mul-ℕ : - (x y : ℕ) → is-one-ℕ (x *ℕ y) → is-one-ℕ y -is-one-right-is-one-mul-ℕ zero-ℕ zero-ℕ p = p -is-one-right-is-one-mul-ℕ zero-ℕ (succ-ℕ y) () -is-one-right-is-one-mul-ℕ (succ-ℕ x) zero-ℕ p = - is-one-right-is-one-mul-ℕ x zero-ℕ p -is-one-right-is-one-mul-ℕ (succ-ℕ x) (succ-ℕ y) p = - ap - ( succ-ℕ) - ( is-zero-right-is-zero-add-ℕ (x *ℕ (succ-ℕ y)) y - ( is-injective-succ-ℕ p)) - -is-one-left-is-one-mul-ℕ : - (x y : ℕ) → is-one-ℕ (x *ℕ y) → is-one-ℕ x -is-one-left-is-one-mul-ℕ x y p = - is-one-right-is-one-mul-ℕ y x (commutative-mul-ℕ y x ∙ p) - -is-one-mul-ℕ : - (x y : ℕ) → is-one-ℕ x → is-one-ℕ y → is-one-ℕ (x *ℕ y) -is-one-mul-ℕ .1 .1 refl refl = refl - -neq-mul-ℕ : - (m n : ℕ) → succ-ℕ m ≠ (succ-ℕ m *ℕ (succ-ℕ (succ-ℕ n))) -neq-mul-ℕ m n p = - neq-add-ℕ - ( succ-ℕ m) - ( ( m *ℕ (succ-ℕ n)) +ℕ n) - ( ( p) ∙ - ( ( right-successor-law-mul-ℕ (succ-ℕ m) (succ-ℕ n)) ∙ - ( ap ((succ-ℕ m) +ℕ_) (left-successor-law-mul-ℕ m (succ-ℕ n))))) +abstract + neq-mul-ℕ : + (m n : ℕ) → succ-ℕ m ≠ (succ-ℕ m *ℕ (succ-ℕ (succ-ℕ n))) + neq-mul-ℕ m n p = + neq-add-ℕ + ( succ-ℕ m) + ( ( m *ℕ (succ-ℕ n)) +ℕ n) + ( ( p) ∙ + ( ( right-successor-law-mul-ℕ (succ-ℕ m) (succ-ℕ n)) ∙ + ( ap ((succ-ℕ m) +ℕ_) (left-successor-law-mul-ℕ m (succ-ℕ n))))) ``` ### Either of the factors is zero if and only if the product is zero ```agda -is-zero-summand-is-zero-mul-ℕ : - (x y : ℕ) → is-zero-ℕ (x *ℕ y) → is-zero-ℕ x + is-zero-ℕ y -is-zero-summand-is-zero-mul-ℕ 0 y H = inl refl -is-zero-summand-is-zero-mul-ℕ (succ-ℕ x) 0 H = inr refl - -is-zero-mul-ℕ-is-zero-left-summand : - (x y : ℕ) → is-zero-ℕ x → is-zero-ℕ (x *ℕ y) -is-zero-mul-ℕ-is-zero-left-summand .0 y refl = left-zero-law-mul-ℕ y - -is-zero-mul-ℕ-is-zero-right-summand : - (x y : ℕ) → is-zero-ℕ y → is-zero-ℕ (x *ℕ y) -is-zero-mul-ℕ-is-zero-right-summand x .0 refl = right-zero-law-mul-ℕ x - -is-zero-mul-ℕ-is-zero-summand : - (x y : ℕ) → is-zero-ℕ x + is-zero-ℕ y → is-zero-ℕ (x *ℕ y) -is-zero-mul-ℕ-is-zero-summand x y (inl H) = - is-zero-mul-ℕ-is-zero-left-summand x y H -is-zero-mul-ℕ-is-zero-summand x y (inr H) = - is-zero-mul-ℕ-is-zero-right-summand x y H +abstract + is-zero-summand-is-zero-mul-ℕ : + (x y : ℕ) → is-zero-ℕ (x *ℕ y) → is-zero-ℕ x + is-zero-ℕ y + is-zero-summand-is-zero-mul-ℕ 0 y H = inl refl + is-zero-summand-is-zero-mul-ℕ (succ-ℕ x) 0 H = inr refl + + is-zero-mul-ℕ-is-zero-left-summand : + (x y : ℕ) → is-zero-ℕ x → is-zero-ℕ (x *ℕ y) + is-zero-mul-ℕ-is-zero-left-summand .0 y refl = left-zero-law-mul-ℕ y + + is-zero-mul-ℕ-is-zero-right-summand : + (x y : ℕ) → is-zero-ℕ y → is-zero-ℕ (x *ℕ y) + is-zero-mul-ℕ-is-zero-right-summand x .0 refl = right-zero-law-mul-ℕ x + + is-zero-mul-ℕ-is-zero-summand : + (x y : ℕ) → is-zero-ℕ x + is-zero-ℕ y → is-zero-ℕ (x *ℕ y) + is-zero-mul-ℕ-is-zero-summand x y (inl H) = + is-zero-mul-ℕ-is-zero-left-summand x y H + is-zero-mul-ℕ-is-zero-summand x y (inr H) = + is-zero-mul-ℕ-is-zero-right-summand x y H ``` ## See also diff --git a/src/elementary-number-theory/negative-integers.lagda.md b/src/elementary-number-theory/negative-integers.lagda.md index 7bc2494ac5..c773139bdb 100644 --- a/src/elementary-number-theory/negative-integers.lagda.md +++ b/src/elementary-number-theory/negative-integers.lagda.md @@ -148,24 +148,26 @@ eq-nat-negative-pred-negative-ℤ : nat-negative-ℤ (pred-negative-ℤ x) ＝ succ-ℕ (nat-negative-ℤ x) eq-nat-negative-pred-negative-ℤ (inl x , H) = refl -is-section-nat-negative-ℤ : - (x : negative-ℤ) → negative-int-ℕ (nat-negative-ℤ x) ＝ x -is-section-nat-negative-ℤ (inl zero-ℕ , H) = refl -is-section-nat-negative-ℤ (inl (succ-ℕ x) , H) = - ap pred-negative-ℤ (is-section-nat-negative-ℤ (inl x , H)) - -is-retraction-nat-negative-ℤ : - (n : ℕ) → nat-negative-ℤ (negative-int-ℕ n) ＝ n -is-retraction-nat-negative-ℤ zero-ℕ = refl -is-retraction-nat-negative-ℤ (succ-ℕ n) = - eq-nat-negative-pred-negative-ℤ (negative-int-ℕ n) ∙ - ap succ-ℕ (is-retraction-nat-negative-ℤ n) - -is-equiv-negative-int-ℕ : is-equiv negative-int-ℕ -pr1 (pr1 is-equiv-negative-int-ℕ) = nat-negative-ℤ -pr2 (pr1 is-equiv-negative-int-ℕ) = is-section-nat-negative-ℤ -pr1 (pr2 is-equiv-negative-int-ℕ) = nat-negative-ℤ -pr2 (pr2 is-equiv-negative-int-ℕ) = is-retraction-nat-negative-ℤ +abstract + is-section-nat-negative-ℤ : + (x : negative-ℤ) → negative-int-ℕ (nat-negative-ℤ x) ＝ x + is-section-nat-negative-ℤ (inl zero-ℕ , H) = refl + is-section-nat-negative-ℤ (inl (succ-ℕ x) , H) = + ap pred-negative-ℤ (is-section-nat-negative-ℤ (inl x , H)) + + is-retraction-nat-negative-ℤ : + (n : ℕ) → nat-negative-ℤ (negative-int-ℕ n) ＝ n + is-retraction-nat-negative-ℤ zero-ℕ = refl + is-retraction-nat-negative-ℤ (succ-ℕ n) = + eq-nat-negative-pred-negative-ℤ (negative-int-ℕ n) ∙ + ap succ-ℕ (is-retraction-nat-negative-ℤ n) + + is-equiv-negative-int-ℕ : is-equiv negative-int-ℕ + is-equiv-negative-int-ℕ = + is-equiv-is-invertible + ( nat-negative-ℤ) + ( is-section-nat-negative-ℤ) + ( is-retraction-nat-negative-ℤ) equiv-negative-int-ℕ : ℕ ≃ negative-ℤ pr1 equiv-negative-int-ℕ = negative-int-ℕ diff --git a/src/elementary-number-theory/nonnegative-integers.lagda.md b/src/elementary-number-theory/nonnegative-integers.lagda.md index 9c06ff7fb4..5022258b3c 100644 --- a/src/elementary-number-theory/nonnegative-integers.lagda.md +++ b/src/elementary-number-theory/nonnegative-integers.lagda.md @@ -165,27 +165,23 @@ eq-nat-nonnegative-succ-nonnnegative-ℤ : eq-nat-nonnegative-succ-nonnnegative-ℤ (inr (inl x) , H) = refl eq-nat-nonnegative-succ-nonnnegative-ℤ (inr (inr x) , H) = refl -is-section-nat-nonnegative-ℤ : - (x : nonnegative-ℤ) → nonnegative-int-ℕ (nat-nonnegative-ℤ x) ＝ x -is-section-nat-nonnegative-ℤ ((inr (inl star)) , H) = refl -is-section-nat-nonnegative-ℤ ((inr (inr x)) , H) = refl - -is-retraction-nat-nonnegative-ℤ : - (n : ℕ) → nat-nonnegative-ℤ (nonnegative-int-ℕ n) ＝ n -is-retraction-nat-nonnegative-ℤ zero-ℕ = refl -is-retraction-nat-nonnegative-ℤ (succ-ℕ n) = refl - -is-equiv-nat-nonnegative-ℤ : is-equiv nat-nonnegative-ℤ -pr1 (pr1 is-equiv-nat-nonnegative-ℤ) = nonnegative-int-ℕ -pr2 (pr1 is-equiv-nat-nonnegative-ℤ) = is-retraction-nat-nonnegative-ℤ -pr1 (pr2 is-equiv-nat-nonnegative-ℤ) = nonnegative-int-ℕ -pr2 (pr2 is-equiv-nat-nonnegative-ℤ) = is-section-nat-nonnegative-ℤ - -is-equiv-nonnegative-int-ℕ : is-equiv nonnegative-int-ℕ -pr1 (pr1 is-equiv-nonnegative-int-ℕ) = nat-nonnegative-ℤ -pr2 (pr1 is-equiv-nonnegative-int-ℕ) = is-section-nat-nonnegative-ℤ -pr1 (pr2 is-equiv-nonnegative-int-ℕ) = nat-nonnegative-ℤ -pr2 (pr2 is-equiv-nonnegative-int-ℕ) = is-retraction-nat-nonnegative-ℤ +abstract + is-section-nat-nonnegative-ℤ : + (x : nonnegative-ℤ) → nonnegative-int-ℕ (nat-nonnegative-ℤ x) ＝ x + is-section-nat-nonnegative-ℤ ((inr (inl star)) , H) = refl + is-section-nat-nonnegative-ℤ ((inr (inr x)) , H) = refl + + is-retraction-nat-nonnegative-ℤ : + (n : ℕ) → nat-nonnegative-ℤ (nonnegative-int-ℕ n) ＝ n + is-retraction-nat-nonnegative-ℤ zero-ℕ = refl + is-retraction-nat-nonnegative-ℤ (succ-ℕ n) = refl + + is-equiv-nonnegative-int-ℕ : is-equiv nonnegative-int-ℕ + is-equiv-nonnegative-int-ℕ = + is-equiv-is-invertible + ( nat-nonnegative-ℤ) + ( is-section-nat-nonnegative-ℤ) + ( is-retraction-nat-nonnegative-ℤ) equiv-nonnegative-int-ℕ : ℕ ≃ nonnegative-ℤ pr1 equiv-nonnegative-int-ℕ = nonnegative-int-ℕ diff --git a/src/elementary-number-theory/nonpositive-integers.lagda.md b/src/elementary-number-theory/nonpositive-integers.lagda.md index 2fb4ab4cb4..e366a33d77 100644 --- a/src/elementary-number-theory/nonpositive-integers.lagda.md +++ b/src/elementary-number-theory/nonpositive-integers.lagda.md @@ -160,25 +160,27 @@ eq-nat-nonpositive-pred-nonpositive-ℤ : eq-nat-nonpositive-pred-nonpositive-ℤ (inl x , H) = refl eq-nat-nonpositive-pred-nonpositive-ℤ (inr (inl x) , H) = refl -is-section-nat-nonpositive-ℤ : - (x : nonpositive-ℤ) → nonpositive-int-ℕ (nat-nonpositive-ℤ x) ＝ x -is-section-nat-nonpositive-ℤ (inl zero-ℕ , H) = refl -is-section-nat-nonpositive-ℤ (inl (succ-ℕ x) , H) = - ap pred-nonpositive-ℤ (is-section-nat-nonpositive-ℤ (inl x , H)) -is-section-nat-nonpositive-ℤ (inr (inl x) , H) = refl - -is-retraction-nat-nonpositive-ℤ : - (n : ℕ) → nat-nonpositive-ℤ (nonpositive-int-ℕ n) ＝ n -is-retraction-nat-nonpositive-ℤ zero-ℕ = refl -is-retraction-nat-nonpositive-ℤ (succ-ℕ n) = - eq-nat-nonpositive-pred-nonpositive-ℤ (nonpositive-int-ℕ n) ∙ - ap succ-ℕ (is-retraction-nat-nonpositive-ℤ n) - -is-equiv-nonpositive-int-ℕ : is-equiv nonpositive-int-ℕ -pr1 (pr1 is-equiv-nonpositive-int-ℕ) = nat-nonpositive-ℤ -pr2 (pr1 is-equiv-nonpositive-int-ℕ) = is-section-nat-nonpositive-ℤ -pr1 (pr2 is-equiv-nonpositive-int-ℕ) = nat-nonpositive-ℤ -pr2 (pr2 is-equiv-nonpositive-int-ℕ) = is-retraction-nat-nonpositive-ℤ +abstract + is-section-nat-nonpositive-ℤ : + (x : nonpositive-ℤ) → nonpositive-int-ℕ (nat-nonpositive-ℤ x) ＝ x + is-section-nat-nonpositive-ℤ (inl zero-ℕ , H) = refl + is-section-nat-nonpositive-ℤ (inl (succ-ℕ x) , H) = + ap pred-nonpositive-ℤ (is-section-nat-nonpositive-ℤ (inl x , H)) + is-section-nat-nonpositive-ℤ (inr (inl x) , H) = refl + + is-retraction-nat-nonpositive-ℤ : + (n : ℕ) → nat-nonpositive-ℤ (nonpositive-int-ℕ n) ＝ n + is-retraction-nat-nonpositive-ℤ zero-ℕ = refl + is-retraction-nat-nonpositive-ℤ (succ-ℕ n) = + eq-nat-nonpositive-pred-nonpositive-ℤ (nonpositive-int-ℕ n) ∙ + ap succ-ℕ (is-retraction-nat-nonpositive-ℤ n) + + is-equiv-nonpositive-int-ℕ : is-equiv nonpositive-int-ℕ + is-equiv-nonpositive-int-ℕ = + is-equiv-is-invertible + ( nat-nonpositive-ℤ) + ( is-section-nat-nonpositive-ℤ) + ( is-retraction-nat-nonpositive-ℤ) equiv-nonpositive-int-ℕ : ℕ ≃ nonpositive-ℤ pr1 equiv-nonpositive-int-ℕ = nonpositive-int-ℕ diff --git a/src/elementary-number-theory/parity-natural-numbers.lagda.md b/src/elementary-number-theory/parity-natural-numbers.lagda.md index 00663e7fa5..503b524542 100644 --- a/src/elementary-number-theory/parity-natural-numbers.lagda.md +++ b/src/elementary-number-theory/parity-natural-numbers.lagda.md @@ -68,76 +68,82 @@ is-odd-one-ℕ H = Eq-eq-ℕ (is-one-div-one-ℕ 2 H) ### A natural number `x` is even if and only if `x + 2` is even ```agda -is-even-is-even-succ-succ-ℕ : - (n : ℕ) → is-even-ℕ (succ-ℕ (succ-ℕ n)) → is-even-ℕ n -pr1 (is-even-is-even-succ-succ-ℕ n (succ-ℕ d , p)) = d -pr2 (is-even-is-even-succ-succ-ℕ n (succ-ℕ d , p)) = - is-injective-succ-ℕ (is-injective-succ-ℕ p) - -is-even-succ-succ-is-even-ℕ : - (n : ℕ) → is-even-ℕ n → is-even-ℕ (succ-ℕ (succ-ℕ n)) -pr1 (is-even-succ-succ-is-even-ℕ n (d , p)) = succ-ℕ d -pr2 (is-even-succ-succ-is-even-ℕ n (d , p)) = ap (succ-ℕ ∘ succ-ℕ) p +abstract + is-even-is-even-succ-succ-ℕ : + (n : ℕ) → is-even-ℕ (succ-ℕ (succ-ℕ n)) → is-even-ℕ n + pr1 (is-even-is-even-succ-succ-ℕ n (succ-ℕ d , p)) = d + pr2 (is-even-is-even-succ-succ-ℕ n (succ-ℕ d , p)) = + is-injective-succ-ℕ (is-injective-succ-ℕ p) + + is-even-succ-succ-is-even-ℕ : + (n : ℕ) → is-even-ℕ n → is-even-ℕ (succ-ℕ (succ-ℕ n)) + pr1 (is-even-succ-succ-is-even-ℕ n (d , p)) = succ-ℕ d + pr2 (is-even-succ-succ-is-even-ℕ n (d , p)) = ap (succ-ℕ ∘ succ-ℕ) p ``` ### A natural number `x` is odd if and only if `x + 2` is odd ```agda -is-odd-is-odd-succ-succ-ℕ : - (n : ℕ) → is-odd-ℕ (succ-ℕ (succ-ℕ n)) → is-odd-ℕ n -is-odd-is-odd-succ-succ-ℕ n = map-neg (is-even-succ-succ-is-even-ℕ n) +abstract + is-odd-is-odd-succ-succ-ℕ : + (n : ℕ) → is-odd-ℕ (succ-ℕ (succ-ℕ n)) → is-odd-ℕ n + is-odd-is-odd-succ-succ-ℕ n = map-neg (is-even-succ-succ-is-even-ℕ n) -is-odd-succ-succ-is-odd-ℕ : - (n : ℕ) → is-odd-ℕ n → is-odd-ℕ (succ-ℕ (succ-ℕ n)) -is-odd-succ-succ-is-odd-ℕ n = map-neg (is-even-is-even-succ-succ-ℕ n) + is-odd-succ-succ-is-odd-ℕ : + (n : ℕ) → is-odd-ℕ n → is-odd-ℕ (succ-ℕ (succ-ℕ n)) + is-odd-succ-succ-is-odd-ℕ n = map-neg (is-even-is-even-succ-succ-ℕ n) ``` ### If a natural number `x` is odd, then `x + 1` is even ```agda -is-even-succ-is-odd-ℕ : - (n : ℕ) → is-odd-ℕ n → is-even-ℕ (succ-ℕ n) -is-even-succ-is-odd-ℕ zero-ℕ p = ex-falso (p is-even-zero-ℕ) -is-even-succ-is-odd-ℕ (succ-ℕ zero-ℕ) p = (1 , refl) -is-even-succ-is-odd-ℕ (succ-ℕ (succ-ℕ n)) p = - is-even-succ-succ-is-even-ℕ - ( succ-ℕ n) - ( is-even-succ-is-odd-ℕ n (is-odd-is-odd-succ-succ-ℕ n p)) +abstract + is-even-succ-is-odd-ℕ : + (n : ℕ) → is-odd-ℕ n → is-even-ℕ (succ-ℕ n) + is-even-succ-is-odd-ℕ zero-ℕ p = ex-falso (p is-even-zero-ℕ) + is-even-succ-is-odd-ℕ (succ-ℕ zero-ℕ) p = (1 , refl) + is-even-succ-is-odd-ℕ (succ-ℕ (succ-ℕ n)) p = + is-even-succ-succ-is-even-ℕ + ( succ-ℕ n) + ( is-even-succ-is-odd-ℕ n (is-odd-is-odd-succ-succ-ℕ n p)) ``` ### If a natural number `x` is even, then `x + 1` is odd ```agda -is-odd-succ-is-even-ℕ : - (n : ℕ) → is-even-ℕ n → is-odd-ℕ (succ-ℕ n) -is-odd-succ-is-even-ℕ zero-ℕ p = is-odd-one-ℕ -is-odd-succ-is-even-ℕ (succ-ℕ zero-ℕ) p = ex-falso (is-odd-one-ℕ p) -is-odd-succ-is-even-ℕ (succ-ℕ (succ-ℕ n)) p = - is-odd-succ-succ-is-odd-ℕ - ( succ-ℕ n) - ( is-odd-succ-is-even-ℕ n (is-even-is-even-succ-succ-ℕ n p)) +abstract + is-odd-succ-is-even-ℕ : + (n : ℕ) → is-even-ℕ n → is-odd-ℕ (succ-ℕ n) + is-odd-succ-is-even-ℕ zero-ℕ p = is-odd-one-ℕ + is-odd-succ-is-even-ℕ (succ-ℕ zero-ℕ) p = ex-falso (is-odd-one-ℕ p) + is-odd-succ-is-even-ℕ (succ-ℕ (succ-ℕ n)) p = + is-odd-succ-succ-is-odd-ℕ + ( succ-ℕ n) + ( is-odd-succ-is-even-ℕ n (is-even-is-even-succ-succ-ℕ n p)) ``` ### If a natural number `x + 1` is odd, then `x` is even ```agda -is-even-is-odd-succ-ℕ : - (n : ℕ) → is-odd-ℕ (succ-ℕ n) → is-even-ℕ n -is-even-is-odd-succ-ℕ n p = - is-even-is-even-succ-succ-ℕ - ( n) - ( is-even-succ-is-odd-ℕ (succ-ℕ n) p) +abstract + is-even-is-odd-succ-ℕ : + (n : ℕ) → is-odd-ℕ (succ-ℕ n) → is-even-ℕ n + is-even-is-odd-succ-ℕ n p = + is-even-is-even-succ-succ-ℕ + ( n) + ( is-even-succ-is-odd-ℕ (succ-ℕ n) p) ``` ### If a natural number `x + 1` is even, then `x` is odd ```agda -is-odd-is-even-succ-ℕ : - (n : ℕ) → is-even-ℕ (succ-ℕ n) → is-odd-ℕ n -is-odd-is-even-succ-ℕ n p = - is-odd-is-odd-succ-succ-ℕ - ( n) - ( is-odd-succ-is-even-ℕ (succ-ℕ n) p) +abstract + is-odd-is-even-succ-ℕ : + (n : ℕ) → is-even-ℕ (succ-ℕ n) → is-odd-ℕ n + is-odd-is-even-succ-ℕ n p = + is-odd-is-odd-succ-succ-ℕ + ( n) + ( is-odd-succ-is-even-ℕ (succ-ℕ n) p) ``` ### A natural number `x` is odd if and only if there is a natural number `y` such that `succ-ℕ (y *ℕ 2) ＝ x` @@ -146,18 +152,19 @@ is-odd-is-even-succ-ℕ n p = has-odd-expansion : ℕ → UU lzero has-odd-expansion x = Σ ℕ (λ y → (succ-ℕ (y *ℕ 2)) ＝ x) -is-odd-has-odd-expansion : (n : ℕ) → has-odd-expansion n → is-odd-ℕ n -is-odd-has-odd-expansion .(succ-ℕ (m *ℕ 2)) (m , refl) = - is-odd-succ-is-even-ℕ (m *ℕ 2) (m , refl) - -has-odd-expansion-is-odd : (n : ℕ) → is-odd-ℕ n → has-odd-expansion n -has-odd-expansion-is-odd zero-ℕ p = ex-falso (p is-even-zero-ℕ) -has-odd-expansion-is-odd (succ-ℕ zero-ℕ) p = 0 , refl -has-odd-expansion-is-odd (succ-ℕ (succ-ℕ n)) p = - ( succ-ℕ (pr1 s)) , ap (succ-ℕ ∘ succ-ℕ) (pr2 s) - where - s : has-odd-expansion n - s = has-odd-expansion-is-odd n (is-odd-is-odd-succ-succ-ℕ n p) +abstract + is-odd-has-odd-expansion : (n : ℕ) → has-odd-expansion n → is-odd-ℕ n + is-odd-has-odd-expansion .(succ-ℕ (m *ℕ 2)) (m , refl) = + is-odd-succ-is-even-ℕ (m *ℕ 2) (m , refl) + + has-odd-expansion-is-odd : (n : ℕ) → is-odd-ℕ n → has-odd-expansion n + has-odd-expansion-is-odd zero-ℕ p = ex-falso (p is-even-zero-ℕ) + has-odd-expansion-is-odd (succ-ℕ zero-ℕ) p = 0 , refl + has-odd-expansion-is-odd (succ-ℕ (succ-ℕ n)) p = + ( succ-ℕ (pr1 s)) , ap (succ-ℕ ∘ succ-ℕ) (pr2 s) + where + s : has-odd-expansion n + s = has-odd-expansion-is-odd n (is-odd-is-odd-succ-succ-ℕ n p) ``` ### If `x` is odd and `y` is even, `x ≠ y` diff --git a/src/elementary-number-theory/poset-of-natural-numbers-ordered-by-divisibility.lagda.md b/src/elementary-number-theory/poset-of-natural-numbers-ordered-by-divisibility.lagda.md index 4977a5e18b..0373090536 100644 --- a/src/elementary-number-theory/poset-of-natural-numbers-ordered-by-divisibility.lagda.md +++ b/src/elementary-number-theory/poset-of-natural-numbers-ordered-by-divisibility.lagda.md @@ -30,7 +30,8 @@ open import order-theory.preorders ## Idea -The **poset of natural numbers ordered by divisibility** consists of the +The **[poset](order-theory.posets.md) of natural numbers ordered by +divisibility** consists of the [natural numbers](elementary-number-theory.natural-numbers.md) and its ordering is defined by `m ≤ n := m | n`, i.e., by [divisibility](elementary-number-theory.divisibility-natural-numbers.md). diff --git a/src/elementary-number-theory/positive-and-negative-integers.lagda.md b/src/elementary-number-theory/positive-and-negative-integers.lagda.md index 9f35831163..c403156aad 100644 --- a/src/elementary-number-theory/positive-and-negative-integers.lagda.md +++ b/src/elementary-number-theory/positive-and-negative-integers.lagda.md @@ -223,35 +223,37 @@ neg-negative-ℤ (x , H) = (neg-ℤ x , is-positive-neg-is-negative-ℤ H) #### Nonpositivity is negated positivity ```agda -is-not-positive-is-nonpositive-ℤ : - {x : ℤ} → is-nonpositive-ℤ x → ¬ (is-positive-ℤ x) -is-not-positive-is-nonpositive-ℤ {inr (inl x)} _ q = q -is-not-positive-is-nonpositive-ℤ {inr (inr x)} p _ = p - -is-nonpositive-is-not-positive-ℤ : - {x : ℤ} → ¬ (is-positive-ℤ x) → is-nonpositive-ℤ x -is-nonpositive-is-not-positive-ℤ {x} H = - rec-coproduct - ( λ K → ex-falso (H K)) - ( id) - ( decide-is-positive-is-nonpositive-ℤ) +abstract + is-not-positive-is-nonpositive-ℤ : + {x : ℤ} → is-nonpositive-ℤ x → ¬ (is-positive-ℤ x) + is-not-positive-is-nonpositive-ℤ {inr (inl x)} _ q = q + is-not-positive-is-nonpositive-ℤ {inr (inr x)} p _ = p + + is-nonpositive-is-not-positive-ℤ : + {x : ℤ} → ¬ (is-positive-ℤ x) → is-nonpositive-ℤ x + is-nonpositive-is-not-positive-ℤ {x} H = + rec-coproduct + ( λ K → ex-falso (H K)) + ( id) + ( decide-is-positive-is-nonpositive-ℤ) ``` #### Nonnegativity is negated negativity ```agda -is-not-negative-is-nonnegative-ℤ : - {x : ℤ} → is-nonnegative-ℤ x → ¬ (is-negative-ℤ x) -is-not-negative-is-nonnegative-ℤ {x} H K = - is-not-positive-is-nonpositive-ℤ - ( is-nonpositive-neg-is-nonnegative-ℤ H) - ( is-positive-neg-is-negative-ℤ K) - -is-nonnegative-is-not-negative-ℤ : - {x : ℤ} → ¬ (is-negative-ℤ x) → is-nonnegative-ℤ x -is-nonnegative-is-not-negative-ℤ {x} H = - rec-coproduct - ( λ K → ex-falso (H K)) - ( id) - ( decide-is-negative-is-nonnegative-ℤ) +abstract + is-not-negative-is-nonnegative-ℤ : + {x : ℤ} → is-nonnegative-ℤ x → ¬ (is-negative-ℤ x) + is-not-negative-is-nonnegative-ℤ {x} H K = + is-not-positive-is-nonpositive-ℤ + ( is-nonpositive-neg-is-nonnegative-ℤ H) + ( is-positive-neg-is-negative-ℤ K) + + is-nonnegative-is-not-negative-ℤ : + {x : ℤ} → ¬ (is-negative-ℤ x) → is-nonnegative-ℤ x + is-nonnegative-is-not-negative-ℤ {x} H = + rec-coproduct + ( λ K → ex-falso (H K)) + ( id) + ( decide-is-negative-is-nonnegative-ℤ) ``` diff --git a/src/elementary-number-theory/positive-integers.lagda.md b/src/elementary-number-theory/positive-integers.lagda.md index 11ba8a6bff..4d20d11351 100644 --- a/src/elementary-number-theory/positive-integers.lagda.md +++ b/src/elementary-number-theory/positive-integers.lagda.md @@ -66,10 +66,11 @@ is-positive-ℤ (inl x) = empty is-positive-ℤ (inr (inl x)) = empty is-positive-ℤ (inr (inr x)) = unit -is-prop-is-positive-ℤ : (x : ℤ) → is-prop (is-positive-ℤ x) -is-prop-is-positive-ℤ (inl x) = is-prop-empty -is-prop-is-positive-ℤ (inr (inl x)) = is-prop-empty -is-prop-is-positive-ℤ (inr (inr x)) = is-prop-unit +abstract + is-prop-is-positive-ℤ : (x : ℤ) → is-prop (is-positive-ℤ x) + is-prop-is-positive-ℤ (inl x) = is-prop-empty + is-prop-is-positive-ℤ (inr (inl x)) = is-prop-empty + is-prop-is-positive-ℤ (inr (inr x)) = is-prop-unit subtype-positive-ℤ : subtype lzero ℤ subtype-positive-ℤ x = (is-positive-ℤ x , is-prop-is-positive-ℤ x) @@ -142,9 +143,10 @@ positive-ℤ-Set = positive-ℤ , is-set-positive-ℤ ### The successor of a positive integer is positive ```agda -is-positive-succ-is-positive-ℤ : - {x : ℤ} → is-positive-ℤ x → is-positive-ℤ (succ-ℤ x) -is-positive-succ-is-positive-ℤ {inr (inr x)} H = H +abstract + is-positive-succ-is-positive-ℤ : + {x : ℤ} → is-positive-ℤ x → is-positive-ℤ (succ-ℤ x) + is-positive-succ-is-positive-ℤ {inr (inr x)} H = H succ-positive-ℤ : positive-ℤ → positive-ℤ succ-positive-ℤ (x , H) = (succ-ℤ x , is-positive-succ-is-positive-ℤ H) @@ -153,10 +155,11 @@ succ-positive-ℤ (x , H) = (succ-ℤ x , is-positive-succ-is-positive-ℤ H) ### The integer image of a nonzero natural number is positive ```agda -is-positive-int-is-nonzero-ℕ : - (x : ℕ) → is-nonzero-ℕ x → is-positive-ℤ (int-ℕ x) -is-positive-int-is-nonzero-ℕ zero-ℕ H = ex-falso (H refl) -is-positive-int-is-nonzero-ℕ (succ-ℕ x) H = star +abstract + is-positive-int-is-nonzero-ℕ : + (x : ℕ) → is-nonzero-ℕ x → is-positive-ℤ (int-ℕ x) + is-positive-int-is-nonzero-ℕ zero-ℕ H = ex-falso (H refl) + is-positive-int-is-nonzero-ℕ (succ-ℕ x) H = star positive-int-ℕ⁺ : ℕ⁺ → positive-ℤ positive-int-ℕ⁺ (n , n≠0) = int-ℕ n , is-positive-int-is-nonzero-ℕ n n≠0 @@ -197,24 +200,26 @@ eq-nat-positive-succ-positive-ℤ : nat-positive-ℤ (succ-positive-ℤ x) ＝ succ-ℕ (nat-positive-ℤ x) eq-nat-positive-succ-positive-ℤ (inr (inr x) , H) = refl -is-section-nat-positive-ℤ : - (x : positive-ℤ) → positive-int-ℕ (nat-positive-ℤ x) ＝ x -is-section-nat-positive-ℤ (inr (inr zero-ℕ) , H) = refl -is-section-nat-positive-ℤ (inr (inr (succ-ℕ x)) , H) = - ap succ-positive-ℤ (is-section-nat-positive-ℤ ( inr (inr x) , H)) - -is-retraction-nat-positive-ℤ : - (n : ℕ) → nat-positive-ℤ (positive-int-ℕ n) ＝ n -is-retraction-nat-positive-ℤ zero-ℕ = refl -is-retraction-nat-positive-ℤ (succ-ℕ n) = - eq-nat-positive-succ-positive-ℤ (positive-int-ℕ n) ∙ - ap succ-ℕ (is-retraction-nat-positive-ℤ n) - -is-equiv-positive-int-ℕ : is-equiv positive-int-ℕ -pr1 (pr1 is-equiv-positive-int-ℕ) = nat-positive-ℤ -pr2 (pr1 is-equiv-positive-int-ℕ) = is-section-nat-positive-ℤ -pr1 (pr2 is-equiv-positive-int-ℕ) = nat-positive-ℤ -pr2 (pr2 is-equiv-positive-int-ℕ) = is-retraction-nat-positive-ℤ +abstract + is-section-nat-positive-ℤ : + (x : positive-ℤ) → positive-int-ℕ (nat-positive-ℤ x) ＝ x + is-section-nat-positive-ℤ (inr (inr zero-ℕ) , H) = refl + is-section-nat-positive-ℤ (inr (inr (succ-ℕ x)) , H) = + ap succ-positive-ℤ (is-section-nat-positive-ℤ ( inr (inr x) , H)) + + is-retraction-nat-positive-ℤ : + (n : ℕ) → nat-positive-ℤ (positive-int-ℕ n) ＝ n + is-retraction-nat-positive-ℤ zero-ℕ = refl + is-retraction-nat-positive-ℤ (succ-ℕ n) = + eq-nat-positive-succ-positive-ℤ (positive-int-ℕ n) ∙ + ap succ-ℕ (is-retraction-nat-positive-ℤ n) + + is-equiv-positive-int-ℕ : is-equiv positive-int-ℕ + is-equiv-positive-int-ℕ = + is-equiv-is-invertible + ( nat-positive-ℤ) + ( is-section-nat-positive-ℤ) + ( is-retraction-nat-positive-ℤ) equiv-positive-int-ℕ : ℕ ≃ positive-ℤ pr1 equiv-positive-int-ℕ = positive-int-ℕ diff --git a/src/elementary-number-theory/powers-integers.lagda.md b/src/elementary-number-theory/powers-integers.lagda.md index 05ec3f9e69..19e2ab1984 100644 --- a/src/elementary-number-theory/powers-integers.lagda.md +++ b/src/elementary-number-theory/powers-integers.lagda.md @@ -21,8 +21,10 @@ open import foundation.identity-types ## Idea -The power operation on the integers is the map `n x ↦ xⁿ`, which is defined by -iteratively multiplying `x` with itself `n` times. +The +{{#concept "power operation" WDID=Q33456 WD="exponentiation" Disambiguation="on the integers" Agda=power-ℤ}} +on the [integers](elementary-number-theory.integers.md) is the map `n x ↦ xⁿ`, +which is defined by iteratively multiplying `x` with itself `n` times. ## Definition diff --git a/src/elementary-number-theory/prime-numbers.lagda.md b/src/elementary-number-theory/prime-numbers.lagda.md index 930d240933..b2906e4cf5 100644 --- a/src/elementary-number-theory/prime-numbers.lagda.md +++ b/src/elementary-number-theory/prime-numbers.lagda.md @@ -90,163 +90,174 @@ has-unique-proper-divisor-ℕ n = is-torsorial (is-proper-divisor-ℕ n) ### The number `0` is not a prime ```agda -is-nonzero-is-prime-ℕ : - (n : ℕ) → is-prime-ℕ n → is-nonzero-ℕ n -is-nonzero-is-prime-ℕ n H p = - is-not-one-two-ℕ - ( pr1 - ( H 2) - ( tr (λ k → 2 ≠ k) (inv p) ( is-nonzero-two-ℕ) , - tr (div-ℕ 2) (inv p) (0 , refl))) +abstract + is-nonzero-is-prime-ℕ : + (n : ℕ) → is-prime-ℕ n → is-nonzero-ℕ n + is-nonzero-is-prime-ℕ n H p = + is-not-one-two-ℕ + ( pr1 + ( H 2) + ( tr (λ k → 2 ≠ k) (inv p) ( is-nonzero-two-ℕ) , + tr (div-ℕ 2) (inv p) (0 , refl))) ``` ### The number `1` is not a prime ```agda -is-not-one-is-prime-ℕ : (n : ℕ) → is-prime-ℕ n → is-not-one-ℕ n -is-not-one-is-prime-ℕ n H p = pr1 (pr2 (H 1) refl) (inv p) +abstract + is-not-one-is-prime-ℕ : (n : ℕ) → is-prime-ℕ n → is-not-one-ℕ n + is-not-one-is-prime-ℕ n H p = pr1 (pr2 (H 1) refl) (inv p) ``` ### A prime is strictly greater than `1` ```agda -le-one-is-prime-ℕ : - (n : ℕ) → is-prime-ℕ n → le-ℕ 1 n -le-one-is-prime-ℕ 0 x = ex-falso (is-nonzero-is-prime-ℕ 0 x refl) -le-one-is-prime-ℕ 1 x = ex-falso (is-not-one-is-prime-ℕ 1 x refl) -le-one-is-prime-ℕ (succ-ℕ (succ-ℕ n)) x = star +abstract + le-one-is-prime-ℕ : + (n : ℕ) → is-prime-ℕ n → le-ℕ 1 n + le-one-is-prime-ℕ 0 x = ex-falso (is-nonzero-is-prime-ℕ 0 x refl) + le-one-is-prime-ℕ 1 x = ex-falso (is-not-one-is-prime-ℕ 1 x refl) + le-one-is-prime-ℕ (succ-ℕ (succ-ℕ n)) x = star ``` ### Being a prime is a proposition ```agda -is-prop-is-prime-ℕ : - (n : ℕ) → is-prop (is-prime-ℕ n) -is-prop-is-prime-ℕ n = - is-prop-Π - ( λ x → - is-prop-product - ( is-prop-Π (λ p → is-set-ℕ x 1)) - ( is-prop-Π (λ p → is-prop-is-proper-divisor-ℕ n x))) +abstract + is-prop-is-prime-ℕ : + (n : ℕ) → is-prop (is-prime-ℕ n) + is-prop-is-prime-ℕ n = + is-prop-Π + ( λ x → + is-prop-product + ( is-prop-Π (λ p → is-set-ℕ x 1)) + ( is-prop-Π (λ p → is-prop-is-proper-divisor-ℕ n x))) is-prime-ℕ-Prop : (n : ℕ) → Prop lzero pr1 (is-prime-ℕ-Prop n) = is-prime-ℕ n pr2 (is-prime-ℕ-Prop n) = is-prop-is-prime-ℕ n -is-prop-has-unique-proper-divisor-ℕ : - (n : ℕ) → is-prop (has-unique-proper-divisor-ℕ n) -is-prop-has-unique-proper-divisor-ℕ n = is-property-is-contr +abstract + is-prop-has-unique-proper-divisor-ℕ : + (n : ℕ) → is-prop (has-unique-proper-divisor-ℕ n) + is-prop-has-unique-proper-divisor-ℕ n = is-property-is-contr ``` ### The three definitions of primes are equivalent ```agda -is-prime-easy-is-prime-ℕ : (n : ℕ) → is-prime-ℕ n → is-prime-easy-ℕ n -pr1 (is-prime-easy-is-prime-ℕ n H) = is-not-one-is-prime-ℕ n H -pr2 (is-prime-easy-is-prime-ℕ n H) x = pr1 (H x) - -is-prime-is-prime-easy-ℕ : (n : ℕ) → is-prime-easy-ℕ n → is-prime-ℕ n -pr1 (is-prime-is-prime-easy-ℕ n H x) = pr2 H x -pr1 (pr2 (is-prime-is-prime-easy-ℕ n H .(succ-ℕ zero-ℕ)) refl) q = pr1 H (inv q) -pr2 (pr2 (is-prime-is-prime-easy-ℕ n H .(succ-ℕ zero-ℕ)) refl) = div-one-ℕ n - -has-unique-proper-divisor-is-prime-ℕ : - (n : ℕ) → is-prime-ℕ n → has-unique-proper-divisor-ℕ n -has-unique-proper-divisor-is-prime-ℕ n H = - fundamental-theorem-id' - ( λ x p → pr2 (H x) (inv p)) - ( λ x → - is-equiv-has-converse-is-prop - ( is-set-ℕ 1 x) - ( is-prop-is-proper-divisor-ℕ n x) - ( λ p → inv (pr1 (H x) p))) - -is-prime-has-unique-proper-divisor-ℕ : - (n : ℕ) → has-unique-proper-divisor-ℕ n → is-prime-ℕ n -pr1 (is-prime-has-unique-proper-divisor-ℕ n H x) K = - ap pr1 - ( eq-is-contr H - { pair x K} - { pair 1 (is-proper-divisor-one-is-proper-divisor-ℕ K)}) -pr2 (is-prime-has-unique-proper-divisor-ℕ n H .1) refl = - is-proper-divisor-one-is-proper-divisor-ℕ (pr2 (center H)) +abstract + is-prime-easy-is-prime-ℕ : (n : ℕ) → is-prime-ℕ n → is-prime-easy-ℕ n + pr1 (is-prime-easy-is-prime-ℕ n H) = is-not-one-is-prime-ℕ n H + pr2 (is-prime-easy-is-prime-ℕ n H) x = forward-implication (H x) + + is-prime-is-prime-easy-ℕ : (n : ℕ) → is-prime-easy-ℕ n → is-prime-ℕ n + pr1 (is-prime-is-prime-easy-ℕ n H x) = pr2 H x + pr1 (pr2 (is-prime-is-prime-easy-ℕ n H .(succ-ℕ zero-ℕ)) refl) q = + pr1 H (inv q) + pr2 (pr2 (is-prime-is-prime-easy-ℕ n H .(succ-ℕ zero-ℕ)) refl) = div-one-ℕ n + + has-unique-proper-divisor-is-prime-ℕ : + (n : ℕ) → is-prime-ℕ n → has-unique-proper-divisor-ℕ n + has-unique-proper-divisor-is-prime-ℕ n H = + fundamental-theorem-id' + ( λ x p → backward-implication (H x) (inv p)) + ( λ x → + is-equiv-has-converse-is-prop + ( is-set-ℕ 1 x) + ( is-prop-is-proper-divisor-ℕ n x) + ( λ p → inv (forward-implication (H x) p))) + + is-prime-has-unique-proper-divisor-ℕ : + (n : ℕ) → has-unique-proper-divisor-ℕ n → is-prime-ℕ n + pr1 (is-prime-has-unique-proper-divisor-ℕ n H x) K = + ap pr1 + ( eq-is-contr H + { pair x K} + { pair 1 (is-proper-divisor-one-is-proper-divisor-ℕ K)}) + pr2 (is-prime-has-unique-proper-divisor-ℕ n H .1) refl = + is-proper-divisor-one-is-proper-divisor-ℕ (pr2 (center H)) ``` ### Being a prime is decidable ```agda -is-decidable-is-prime-easy-ℕ : (n : ℕ) → is-decidable (is-prime-easy-ℕ n) -is-decidable-is-prime-easy-ℕ zero-ℕ = - inr - ( λ H → - is-not-one-two-ℕ (pr2 H 2 (is-proper-divisor-zero-succ-ℕ 1))) -is-decidable-is-prime-easy-ℕ (succ-ℕ n) = - is-decidable-product - ( is-decidable-neg (is-decidable-is-one-ℕ (succ-ℕ n))) - ( is-decidable-bounded-Π-ℕ - ( is-proper-divisor-ℕ (succ-ℕ n)) - ( is-one-ℕ) - ( is-decidable-is-proper-divisor-ℕ (succ-ℕ n)) - ( is-decidable-is-one-ℕ) - ( succ-ℕ n) - ( λ x H → leq-div-succ-ℕ x n (pr2 H))) - -is-decidable-is-prime-ℕ : (n : ℕ) → is-decidable (is-prime-ℕ n) -is-decidable-is-prime-ℕ n = - is-decidable-iff - ( is-prime-is-prime-easy-ℕ n) - ( is-prime-easy-is-prime-ℕ n) - ( is-decidable-is-prime-easy-ℕ n) +abstract + is-decidable-is-prime-easy-ℕ : (n : ℕ) → is-decidable (is-prime-easy-ℕ n) + is-decidable-is-prime-easy-ℕ zero-ℕ = + inr + ( λ H → + is-not-one-two-ℕ (pr2 H 2 (is-proper-divisor-zero-succ-ℕ 1))) + is-decidable-is-prime-easy-ℕ (succ-ℕ n) = + is-decidable-product + ( is-decidable-neg (is-decidable-is-one-ℕ (succ-ℕ n))) + ( is-decidable-bounded-Π-ℕ + ( is-proper-divisor-ℕ (succ-ℕ n)) + ( is-one-ℕ) + ( is-decidable-is-proper-divisor-ℕ (succ-ℕ n)) + ( is-decidable-is-one-ℕ) + ( succ-ℕ n) + ( λ x H → leq-div-succ-ℕ x n (pr2 H))) + + is-decidable-is-prime-ℕ : (n : ℕ) → is-decidable (is-prime-ℕ n) + is-decidable-is-prime-ℕ n = + is-decidable-iff + ( is-prime-is-prime-easy-ℕ n) + ( is-prime-easy-is-prime-ℕ n) + ( is-decidable-is-prime-easy-ℕ n) ``` ### The number `2` is a prime ```agda -is-one-is-proper-divisor-two-ℕ : is-one-is-proper-divisor-ℕ 2 -is-one-is-proper-divisor-two-ℕ zero-ℕ (pair f (pair k p)) = - ex-falso (f (inv (right-zero-law-mul-ℕ k) ∙ p)) -is-one-is-proper-divisor-two-ℕ (succ-ℕ zero-ℕ) (pair f H) = refl -is-one-is-proper-divisor-two-ℕ (succ-ℕ (succ-ℕ zero-ℕ)) (pair f H) = - ex-falso (f refl) -is-one-is-proper-divisor-two-ℕ (succ-ℕ (succ-ℕ (succ-ℕ x))) (pair f H) = - ex-falso (leq-div-succ-ℕ (succ-ℕ (succ-ℕ (succ-ℕ x))) 1 H) - -is-prime-easy-two-ℕ : is-prime-easy-ℕ 2 -pr1 is-prime-easy-two-ℕ = Eq-eq-ℕ -pr2 is-prime-easy-two-ℕ = is-one-is-proper-divisor-two-ℕ - -is-prime-two-ℕ : is-prime-ℕ 2 -is-prime-two-ℕ = - is-prime-is-prime-easy-ℕ 2 is-prime-easy-two-ℕ +abstract + is-one-is-proper-divisor-two-ℕ : is-one-is-proper-divisor-ℕ 2 + is-one-is-proper-divisor-two-ℕ zero-ℕ (pair f (pair k p)) = + ex-falso (f (inv (right-zero-law-mul-ℕ k) ∙ p)) + is-one-is-proper-divisor-two-ℕ (succ-ℕ zero-ℕ) (pair f H) = refl + is-one-is-proper-divisor-two-ℕ (succ-ℕ (succ-ℕ zero-ℕ)) (pair f H) = + ex-falso (f refl) + is-one-is-proper-divisor-two-ℕ (succ-ℕ (succ-ℕ (succ-ℕ x))) (pair f H) = + ex-falso (leq-div-succ-ℕ (succ-ℕ (succ-ℕ (succ-ℕ x))) 1 H) + + is-prime-easy-two-ℕ : is-prime-easy-ℕ 2 + pr1 is-prime-easy-two-ℕ = Eq-eq-ℕ + pr2 is-prime-easy-two-ℕ = is-one-is-proper-divisor-two-ℕ + + is-prime-two-ℕ : is-prime-ℕ 2 + is-prime-two-ℕ = + is-prime-is-prime-easy-ℕ 2 is-prime-easy-two-ℕ ``` ### If a prime number `p` divides a nonzero number `x`, then `x/p < x` ```agda -le-quotient-div-is-prime-ℕ : - (p x : ℕ) → is-nonzero-ℕ x → is-prime-ℕ p → - (H : div-ℕ p x) → le-ℕ (quotient-div-ℕ p x H) x -le-quotient-div-is-prime-ℕ p x N P H = - le-quotient-div-ℕ p x N H (is-not-one-is-prime-ℕ p P) +abstract + le-quotient-div-is-prime-ℕ : + (p x : ℕ) → is-nonzero-ℕ x → is-prime-ℕ p → + (H : div-ℕ p x) → le-ℕ (quotient-div-ℕ p x H) x + le-quotient-div-is-prime-ℕ p x N P H = + le-quotient-div-ℕ p x N H (is-not-one-is-prime-ℕ p P) ``` ### If a prime number `p` divides a number `x + 1`, then `(x + 1)/p ≤ x` ```agda -leq-quotient-div-is-prime-ℕ : - (p x : ℕ) → is-prime-ℕ p → - (H : div-ℕ p (succ-ℕ x)) → leq-ℕ (quotient-div-ℕ p (succ-ℕ x) H) x -leq-quotient-div-is-prime-ℕ p x P H = - leq-le-succ-ℕ - ( quotient-div-ℕ p (succ-ℕ x) H) - ( x) - ( le-quotient-div-is-prime-ℕ p (succ-ℕ x) (is-nonzero-succ-ℕ x) P H) +abstract + leq-quotient-div-is-prime-ℕ : + (p x : ℕ) → is-prime-ℕ p → + (H : div-ℕ p (succ-ℕ x)) → leq-ℕ (quotient-div-ℕ p (succ-ℕ x) H) x + leq-quotient-div-is-prime-ℕ p x P H = + leq-le-succ-ℕ + ( quotient-div-ℕ p (succ-ℕ x) H) + ( x) + ( le-quotient-div-is-prime-ℕ p (succ-ℕ x) (is-nonzero-succ-ℕ x) P H) ``` ## See also -- The fundamental theorem of arithmetic asserts that every positive natural - number can be written uniquely as a product of primes. This theorem is proven - in - [`fundamental-theorem-of-arithmetic`](elementary-number-theory.fundamental-theorem-of-arithmetic.md). +- The + [fundamental theorem of arithmetic](elementary-number-theory.fundamental-theorem-of-arithmetic.md) + asserts that every positive natural number can be written uniquely as a + product of primes. diff --git a/src/elementary-number-theory/reduced-integer-fractions.lagda.md b/src/elementary-number-theory/reduced-integer-fractions.lagda.md index a532182a02..2694b62a82 100644 --- a/src/elementary-number-theory/reduced-integer-fractions.lagda.md +++ b/src/elementary-number-theory/reduced-integer-fractions.lagda.md @@ -48,7 +48,7 @@ coprime. ```agda is-reduced-fraction-ℤ : fraction-ℤ → UU lzero is-reduced-fraction-ℤ x = - is-relative-prime-ℤ (numerator-fraction-ℤ x) (denominator-fraction-ℤ x) + is-relatively-prime-ℤ (numerator-fraction-ℤ x) (denominator-fraction-ℤ x) ``` ## Properties and constructions @@ -59,7 +59,7 @@ is-reduced-fraction-ℤ x = is-prop-is-reduced-fraction-ℤ : (x : fraction-ℤ) → is-prop (is-reduced-fraction-ℤ x) is-prop-is-reduced-fraction-ℤ x = - is-prop-is-relative-prime-ℤ + is-prop-is-relatively-prime-ℤ ( numerator-fraction-ℤ x) ( denominator-fraction-ℤ x) ``` diff --git a/src/elementary-number-theory/relatively-prime-integers.lagda.md b/src/elementary-number-theory/relatively-prime-integers.lagda.md index 56ab84992a..626759c15c 100644 --- a/src/elementary-number-theory/relatively-prime-integers.lagda.md +++ b/src/elementary-number-theory/relatively-prime-integers.lagda.md @@ -21,14 +21,17 @@ open import foundation.universe-levels ## Idea -Two integers are said to be relatively prime if their greatest common divisor -is 1. +Two [integers](elementary-number-theory.integers.md) `x` and `y` are said to be +{{#concept "relatively prime" Disambiguation="integers" Agda=is-relatively-prime-ℤ}} +if their +[greatest common divisor](elementary-number-theory.greatest-common-divisor-integers.md) +is `1`. ## Definition ```agda -is-relative-prime-ℤ : ℤ → ℤ → UU lzero -is-relative-prime-ℤ x y = is-one-ℤ (gcd-ℤ x y) +is-relatively-prime-ℤ : ℤ → ℤ → UU lzero +is-relatively-prime-ℤ x y = is-one-ℤ (gcd-ℤ x y) ``` ## Properties @@ -36,22 +39,25 @@ is-relative-prime-ℤ x y = is-one-ℤ (gcd-ℤ x y) ### Being relatively prime is a proposition ```agda -is-prop-is-relative-prime-ℤ : (x y : ℤ) → is-prop (is-relative-prime-ℤ x y) -is-prop-is-relative-prime-ℤ x y = is-set-ℤ (gcd-ℤ x y) one-ℤ +abstract + is-prop-is-relatively-prime-ℤ : + (x y : ℤ) → is-prop (is-relatively-prime-ℤ x y) + is-prop-is-relatively-prime-ℤ x y = is-set-ℤ (gcd-ℤ x y) one-ℤ ``` ### Two integers are relatively prime if and only if their absolute values are relatively prime natural numbers ```agda -is-relatively-prime-abs-is-relatively-prime-ℤ : - {a b : ℤ} → is-relative-prime-ℤ a b → - is-relatively-prime-ℕ (abs-ℤ a) (abs-ℤ b) -is-relatively-prime-abs-is-relatively-prime-ℤ {a} {b} H = is-injective-int-ℕ H - -is-relatively-prime-is-relatively-prime-abs-ℤ : - {a b : ℤ} → is-relatively-prime-ℕ (abs-ℤ a) (abs-ℤ b) → - is-relative-prime-ℤ a b -is-relatively-prime-is-relatively-prime-abs-ℤ {a} {b} H = ap int-ℕ H +abstract + is-relatively-prime-abs-is-relatively-prime-ℤ : + {a b : ℤ} → is-relatively-prime-ℤ a b → + is-relatively-prime-ℕ (abs-ℤ a) (abs-ℤ b) + is-relatively-prime-abs-is-relatively-prime-ℤ {a} {b} H = is-injective-int-ℕ H + + is-relatively-prime-is-relatively-prime-abs-ℤ : + {a b : ℤ} → is-relatively-prime-ℕ (abs-ℤ a) (abs-ℤ b) → + is-relatively-prime-ℤ a b + is-relatively-prime-is-relatively-prime-abs-ℤ {a} {b} H = ap int-ℕ H ``` ### For any two integers `a` and `b` that are not both `0`, the integers `a/gcd(a,b)` and `b/gcd(a,b)` are relatively prime @@ -60,7 +66,7 @@ is-relatively-prime-is-relatively-prime-abs-ℤ {a} {b} H = ap int-ℕ H {- relatively-prime-quotient-div-ℤ : {a b : ℤ} → (is-nonzero-ℤ a + is-nonzero-ℤ b) → - is-relative-prime-ℤ + is-relatively-prime-ℤ ( quotient-div-ℤ (gcd-ℤ a b) a (div-left-gcd-ℤ a b)) ( quotient-div-ℤ (gcd-ℤ a b) b (div-right-gcd-ℤ a b)) relatively-prime-quotient-div-ℤ H = diff --git a/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md b/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md index 01f59c2f4b..1f4b26c526 100644 --- a/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md +++ b/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md @@ -28,8 +28,12 @@ open import foundation.universe-levels ## Idea -Two natural numbers `x` and `y` are said to be relatively prime if their -greatest common divisor is `1`. +Two [natural numbers](elementary-number-theory.natural-numbers.md) `x` and `y` +are said to be +{{#concept "relatively prime" WDID=Q104752 WD="coprime" Disambiguation="natural numbers" Agda=is-relatively-prime-ℕ}} +if their +[greatest common divisor](elementary-number-theory.greatest-common-divisor-natural-numbers.md) +is `1`. ## Definition @@ -43,8 +47,10 @@ is-relatively-prime-ℕ x y = is-one-ℕ (gcd-ℕ x y) ### Being relatively prime is a proposition ```agda -is-prop-is-relatively-prime-ℕ : (x y : ℕ) → is-prop (is-relatively-prime-ℕ x y) -is-prop-is-relatively-prime-ℕ x y = is-set-ℕ (gcd-ℕ x y) 1 +abstract + is-prop-is-relatively-prime-ℕ : + (x y : ℕ) → is-prop (is-relatively-prime-ℕ x y) + is-prop-is-relatively-prime-ℕ x y = is-set-ℕ (gcd-ℕ x y) 1 is-relatively-prime-ℕ-Prop : ℕ → ℕ → Prop lzero pr1 (is-relatively-prime-ℕ-Prop x y) = is-relatively-prime-ℕ x y @@ -76,70 +82,74 @@ pr2 (is-relatively-prime-ℕ-Decidable-Prop x y) = ### `a` and `b` are relatively prime if and only if any common divisor is equal to `1` ```agda -is-one-is-common-divisor-is-relatively-prime-ℕ : - (x y d : ℕ) → - is-relatively-prime-ℕ x y → is-common-divisor-ℕ x y d → is-one-ℕ d -is-one-is-common-divisor-is-relatively-prime-ℕ x y d H K = - is-one-div-one-ℕ d - ( tr - ( div-ℕ d) - ( H) - ( div-gcd-is-common-divisor-ℕ x y d K)) - -is-relatively-prime-is-one-is-common-divisor-ℕ : - (x y : ℕ) → - ((d : ℕ) → is-common-divisor-ℕ x y d → is-one-ℕ d) → is-relatively-prime-ℕ x y -is-relatively-prime-is-one-is-common-divisor-ℕ x y H = - H (gcd-ℕ x y) (is-common-divisor-gcd-ℕ x y) +abstract + is-one-is-common-divisor-is-relatively-prime-ℕ : + (x y d : ℕ) → + is-relatively-prime-ℕ x y → is-common-divisor-ℕ x y d → is-one-ℕ d + is-one-is-common-divisor-is-relatively-prime-ℕ x y d H K = + is-one-div-one-ℕ d + ( tr + ( div-ℕ d) + ( H) + ( div-gcd-is-common-divisor-ℕ x y d K)) + + is-relatively-prime-is-one-is-common-divisor-ℕ : + (x y : ℕ) → + ((d : ℕ) → is-common-divisor-ℕ x y d → is-one-ℕ d) → + is-relatively-prime-ℕ x y + is-relatively-prime-is-one-is-common-divisor-ℕ x y H = + H (gcd-ℕ x y) (is-common-divisor-gcd-ℕ x y) ``` ### If `a` and `b` are relatively prime, then so are any divisors of `a` and `b` ```agda -is-relatively-prime-div-ℕ : - (a b c d : ℕ) → div-ℕ c a → div-ℕ d b → - is-relatively-prime-ℕ a b → is-relatively-prime-ℕ c d -is-relatively-prime-div-ℕ a b c d H K L = - is-one-is-common-divisor-is-relatively-prime-ℕ a b - ( gcd-ℕ c d) - ( L) - ( transitive-div-ℕ (gcd-ℕ c d) c a H (div-left-factor-gcd-ℕ c d) , - transitive-div-ℕ (gcd-ℕ c d) d b K (div-right-factor-gcd-ℕ c d)) +abstract + is-relatively-prime-div-ℕ : + (a b c d : ℕ) → div-ℕ c a → div-ℕ d b → + is-relatively-prime-ℕ a b → is-relatively-prime-ℕ c d + is-relatively-prime-div-ℕ a b c d H K L = + is-one-is-common-divisor-is-relatively-prime-ℕ a b + ( gcd-ℕ c d) + ( L) + ( transitive-div-ℕ (gcd-ℕ c d) c a H (div-left-factor-gcd-ℕ c d) , + transitive-div-ℕ (gcd-ℕ c d) d b K (div-right-factor-gcd-ℕ c d)) ``` ### For any two natural numbers `a` and `b` such that `a + b ≠ 0`, the numbers `a/gcd(a,b)` and `b/gcd(a,b)` are relatively prime ```agda -is-relatively-prime-quotient-div-gcd-ℕ : - (a b : ℕ) → is-nonzero-ℕ (a +ℕ b) → - is-relatively-prime-ℕ - ( quotient-div-ℕ (gcd-ℕ a b) a (div-left-factor-gcd-ℕ a b)) - ( quotient-div-ℕ (gcd-ℕ a b) b (div-right-factor-gcd-ℕ a b)) -is-relatively-prime-quotient-div-gcd-ℕ a b nz = - ( uniqueness-is-gcd-ℕ - ( quotient-div-ℕ (gcd-ℕ a b) a (div-left-factor-gcd-ℕ a b)) - ( quotient-div-ℕ (gcd-ℕ a b) b (div-right-factor-gcd-ℕ a b)) - ( gcd-ℕ +abstract + is-relatively-prime-quotient-div-gcd-ℕ : + (a b : ℕ) → is-nonzero-ℕ (a +ℕ b) → + is-relatively-prime-ℕ ( quotient-div-ℕ (gcd-ℕ a b) a (div-left-factor-gcd-ℕ a b)) - ( quotient-div-ℕ (gcd-ℕ a b) b (div-right-factor-gcd-ℕ a b))) - ( quotient-div-ℕ - ( gcd-ℕ a b) + ( quotient-div-ℕ (gcd-ℕ a b) b (div-right-factor-gcd-ℕ a b)) + is-relatively-prime-quotient-div-gcd-ℕ a b nz = + ( uniqueness-is-gcd-ℕ + ( quotient-div-ℕ (gcd-ℕ a b) a (div-left-factor-gcd-ℕ a b)) + ( quotient-div-ℕ (gcd-ℕ a b) b (div-right-factor-gcd-ℕ a b)) + ( gcd-ℕ + ( quotient-div-ℕ (gcd-ℕ a b) a (div-left-factor-gcd-ℕ a b)) + ( quotient-div-ℕ (gcd-ℕ a b) b (div-right-factor-gcd-ℕ a b))) + ( quotient-div-ℕ + ( gcd-ℕ a b) + ( gcd-ℕ a b) + ( div-gcd-is-common-divisor-ℕ a b + ( gcd-ℕ a b) + ( is-common-divisor-gcd-ℕ a b))) + ( is-gcd-gcd-ℕ + ( quotient-div-ℕ (gcd-ℕ a b) a (div-left-factor-gcd-ℕ a b)) + ( quotient-div-ℕ (gcd-ℕ a b) b (div-right-factor-gcd-ℕ a b))) + ( is-gcd-quotient-div-gcd-ℕ + ( is-nonzero-gcd-ℕ a b nz) + ( is-common-divisor-gcd-ℕ a b))) ∙ + ( is-idempotent-quotient-div-ℕ ( gcd-ℕ a b) + ( is-nonzero-gcd-ℕ a b nz) ( div-gcd-is-common-divisor-ℕ a b ( gcd-ℕ a b) ( is-common-divisor-gcd-ℕ a b))) - ( is-gcd-gcd-ℕ - ( quotient-div-ℕ (gcd-ℕ a b) a (div-left-factor-gcd-ℕ a b)) - ( quotient-div-ℕ (gcd-ℕ a b) b (div-right-factor-gcd-ℕ a b))) - ( is-gcd-quotient-div-gcd-ℕ - ( is-nonzero-gcd-ℕ a b nz) - ( is-common-divisor-gcd-ℕ a b))) ∙ - ( is-idempotent-quotient-div-ℕ - ( gcd-ℕ a b) - ( is-nonzero-gcd-ℕ a b nz) - ( div-gcd-is-common-divisor-ℕ a b - ( gcd-ℕ a b) - ( is-common-divisor-gcd-ℕ a b))) ``` ### If `a` and `b` are prime and distinct, then they are relatively prime @@ -152,23 +162,24 @@ module _ (n : a ≠ b) where - is-one-is-common-divisor-is-prime-ℕ : - (d : ℕ) → is-common-divisor-ℕ a b d → is-one-ℕ d - is-one-is-common-divisor-is-prime-ℕ d c = - pr1 - ( pa d) - ( ( λ e → - is-not-one-is-prime-ℕ - ( a) - ( pa) - ( pr1 (pb a) (n , (tr (λ x → div-ℕ x b) e (pr2 c))))) , - ( pr1 c)) - - is-relatively-prime-is-prime-ℕ : - is-relatively-prime-ℕ a b - is-relatively-prime-is-prime-ℕ = - is-relatively-prime-is-one-is-common-divisor-ℕ - ( a) - ( b) - ( is-one-is-common-divisor-is-prime-ℕ) + abstract + is-one-is-common-divisor-is-prime-ℕ : + (d : ℕ) → is-common-divisor-ℕ a b d → is-one-ℕ d + is-one-is-common-divisor-is-prime-ℕ d c = + pr1 + ( pa d) + ( ( λ e → + is-not-one-is-prime-ℕ + ( a) + ( pa) + ( pr1 (pb a) (n , (tr (λ x → div-ℕ x b) e (pr2 c))))) , + ( pr1 c)) + + is-relatively-prime-is-prime-ℕ : + is-relatively-prime-ℕ a b + is-relatively-prime-is-prime-ℕ = + is-relatively-prime-is-one-is-common-divisor-ℕ + ( a) + ( b) + ( is-one-is-common-divisor-is-prime-ℕ) ``` diff --git a/src/elementary-number-theory/ring-of-integers.lagda.md b/src/elementary-number-theory/ring-of-integers.lagda.md index 80b29723c7..9ce97dd158 100644 --- a/src/elementary-number-theory/ring-of-integers.lagda.md +++ b/src/elementary-number-theory/ring-of-integers.lagda.md @@ -34,6 +34,19 @@ open import ring-theory.trivial-rings +## Idea + +The type of [integers](elementary-number-theory.integers.md) equipped with +[addition](elementary-number-theory.addition-integers.md) and +[multiplication](elementary-number-theory.multiplication-integers.md) is a +[commutative ring](commutative-algebra.commutative-ring.md). + +The {{#concept "ring of integers" Agda=ℤ-Ring}} is the **initial** +[ring](ring-theory.rings.md): for any ring `R`, there's a unique +[ring homomorphism](ring-theory.homomorphisms-rings.md) from `ℤ` to `R`, i.e., +the type of ring homomorphisms `hom-Ring ℤ-Ring R` is +[contractible](foundation.contractible-types.md). + ## Definition ```agda @@ -64,35 +77,46 @@ is-nontrivial-ℤ-Ring H = is-nonzero-one-ℤ (inv H) ### The integer multiples in `ℤ-Ring` coincide with multiplication in `ℤ` ```agda -is-mul-integer-multiple-ℤ-Ring : - (k l : ℤ) → integer-multiple-Ring ℤ-Ring k l ＝ mul-ℤ k l -is-mul-integer-multiple-ℤ-Ring (inl zero-ℕ) l = - ( integer-multiple-neg-one-Ring ℤ-Ring l) ∙ - ( inv (left-neg-unit-law-mul-ℤ l)) -is-mul-integer-multiple-ℤ-Ring (inl (succ-ℕ k)) l = - ( integer-multiple-pred-Ring ℤ-Ring (inl k) l) ∙ - ( ap - ( λ t → right-subtraction-Ring ℤ-Ring t l) - ( is-mul-integer-multiple-ℤ-Ring (inl k) l)) ∙ - ( commutative-add-ℤ (mul-ℤ (inl k) l) (neg-ℤ l)) ∙ - ( inv (left-predecessor-law-mul-ℤ (inl k) l)) -is-mul-integer-multiple-ℤ-Ring (inr (inl _)) l = - ( integer-multiple-zero-Ring ℤ-Ring l) ∙ - ( inv (left-zero-law-mul-ℤ l)) -is-mul-integer-multiple-ℤ-Ring (inr (inr zero-ℕ)) l = - ( integer-multiple-one-Ring ℤ-Ring l) ∙ - ( inv (left-unit-law-mul-ℤ l)) -is-mul-integer-multiple-ℤ-Ring (inr (inr (succ-ℕ k))) l = - ( integer-multiple-succ-Ring ℤ-Ring (inr (inr k)) l) ∙ - ( ap (add-ℤ' l) (is-mul-integer-multiple-ℤ-Ring (inr (inr k)) l)) ∙ - ( commutative-add-ℤ _ l) ∙ - ( inv (left-successor-law-mul-ℤ (inr (inr k)) l)) - -is-integer-multiple-ℤ : - (k : ℤ) → integer-multiple-Ring ℤ-Ring k one-ℤ ＝ k -is-integer-multiple-ℤ k = - ( is-mul-integer-multiple-ℤ-Ring k one-ℤ) ∙ - ( right-unit-law-mul-ℤ k) +abstract + integer-multiple-one-ℤ-Ring : + (k : ℤ) → integer-multiple-Ring ℤ-Ring k one-ℤ ＝ k + integer-multiple-one-ℤ-Ring (inl zero-ℕ) = refl + integer-multiple-one-ℤ-Ring (inl (succ-ℕ n)) = + ap pred-ℤ (integer-multiple-one-ℤ-Ring (inl n)) + integer-multiple-one-ℤ-Ring (inr (inl _)) = refl + integer-multiple-one-ℤ-Ring (inr (inr zero-ℕ)) = refl + integer-multiple-one-ℤ-Ring (inr (inr (succ-ℕ n))) = + ap succ-ℤ (integer-multiple-one-ℤ-Ring (inr (inr n))) + + compute-integer-multiple-ℤ-Ring : + (k l : ℤ) → integer-multiple-Ring ℤ-Ring k l ＝ mul-ℤ k l + compute-integer-multiple-ℤ-Ring (inl zero-ℕ) l = + ( integer-multiple-neg-one-Ring ℤ-Ring l) ∙ + ( inv (left-neg-unit-law-mul-ℤ l)) + compute-integer-multiple-ℤ-Ring (inl (succ-ℕ k)) l = + ( integer-multiple-pred-Ring ℤ-Ring (inl k) l) ∙ + ( ap + ( λ t → right-subtraction-Ring ℤ-Ring t l) + ( compute-integer-multiple-ℤ-Ring (inl k) l)) ∙ + ( commutative-add-ℤ (mul-ℤ (inl k) l) (neg-ℤ l)) ∙ + ( inv (left-predecessor-law-mul-ℤ (inl k) l)) + compute-integer-multiple-ℤ-Ring (inr (inl _)) l = + ( integer-multiple-zero-Ring ℤ-Ring l) ∙ + ( inv (left-zero-law-mul-ℤ l)) + compute-integer-multiple-ℤ-Ring (inr (inr zero-ℕ)) l = + ( integer-multiple-one-Ring ℤ-Ring l) ∙ + ( inv (left-unit-law-mul-ℤ l)) + compute-integer-multiple-ℤ-Ring (inr (inr (succ-ℕ k))) l = + ( integer-multiple-succ-Ring ℤ-Ring (inr (inr k)) l) ∙ + ( ap (add-ℤ' l) (compute-integer-multiple-ℤ-Ring (inr (inr k)) l)) ∙ + ( commutative-add-ℤ _ l) ∙ + ( inv (left-successor-law-mul-ℤ (inr (inr k)) l)) + + is-integer-multiple-ℤ : + (k : ℤ) → integer-multiple-Ring ℤ-Ring k one-ℤ ＝ k + is-integer-multiple-ℤ k = + ( compute-integer-multiple-ℤ-Ring k one-ℤ) ∙ + ( right-unit-law-mul-ℤ k) ``` ### The ring of integers is the initial ring @@ -111,30 +135,31 @@ module _ map-initial-hom-Ring = map-hom-Group ℤ-Group (group-Ring R) hom-group-initial-hom-Ring - preserves-add-initial-hom-Ring : - (k l : ℤ) → - map-initial-hom-Ring (add-ℤ k l) ＝ - add-Ring R (map-initial-hom-Ring k) (map-initial-hom-Ring l) - preserves-add-initial-hom-Ring k l = - preserves-mul-hom-Group - ( ℤ-Group) - ( group-Ring R) - ( hom-group-initial-hom-Ring) - { k} - { l} - - preserves-one-initial-hom-Ring : map-initial-hom-Ring one-ℤ ＝ one-Ring R - preserves-one-initial-hom-Ring = integer-multiple-one-Ring R (one-Ring R) - - preserves-mul-initial-hom-Ring : - (k l : ℤ) → - map-initial-hom-Ring (mul-ℤ k l) ＝ - mul-Ring R (map-initial-hom-Ring k) (map-initial-hom-Ring l) - preserves-mul-initial-hom-Ring k l = - ( ap map-initial-hom-Ring (commutative-mul-ℤ k l)) ∙ - ( integer-multiple-mul-Ring R l k (one-Ring R)) ∙ - ( ap (integer-multiple-Ring R l) (inv (right-unit-law-mul-Ring R _))) ∙ - ( inv (right-integer-multiple-law-mul-Ring R l _ _)) + abstract + preserves-add-initial-hom-Ring : + (k l : ℤ) → + map-initial-hom-Ring (add-ℤ k l) ＝ + add-Ring R (map-initial-hom-Ring k) (map-initial-hom-Ring l) + preserves-add-initial-hom-Ring k l = + preserves-mul-hom-Group + ( ℤ-Group) + ( group-Ring R) + ( hom-group-initial-hom-Ring) + { k} + { l} + + preserves-one-initial-hom-Ring : map-initial-hom-Ring one-ℤ ＝ one-Ring R + preserves-one-initial-hom-Ring = integer-multiple-one-Ring R (one-Ring R) + + preserves-mul-initial-hom-Ring : + (k l : ℤ) → + map-initial-hom-Ring (mul-ℤ k l) ＝ + mul-Ring R (map-initial-hom-Ring k) (map-initial-hom-Ring l) + preserves-mul-initial-hom-Ring k l = + ( ap map-initial-hom-Ring (commutative-mul-ℤ k l)) ∙ + ( integer-multiple-mul-Ring R l k (one-Ring R)) ∙ + ( ap (integer-multiple-Ring R l) (inv (right-unit-law-mul-Ring R _))) ∙ + ( inv (right-integer-multiple-law-mul-Ring R l _ _)) initial-hom-Ring : hom-Ring ℤ-Ring R pr1 initial-hom-Ring = hom-group-initial-hom-Ring @@ -142,27 +167,28 @@ module _ pr2 (pr2 initial-hom-Ring) = preserves-one-initial-hom-Ring ``` -#### Any ring homomorphisms from `ℤ` to `R` is equal to the homomorphism `initial-hom-Ring` +#### Any ring homomorphism from `ℤ` to `R` is equal to the homomorphism `initial-hom-Ring` ```agda module _ {l : Level} (R : Ring l) where - htpy-initial-hom-Ring : - (f : hom-Ring ℤ-Ring R) → htpy-hom-Ring ℤ-Ring R (initial-hom-Ring R) f - htpy-initial-hom-Ring f k = - ( inv - ( ( preserves-integer-multiples-hom-Ring ℤ-Ring R f k one-ℤ) ∙ - ( ap - ( integer-multiple-Ring R k) - ( preserves-one-hom-Ring ℤ-Ring R f)))) ∙ - ( ap (map-hom-Ring ℤ-Ring R f) (is-integer-multiple-ℤ k)) - - contraction-initial-hom-Ring : - (f : hom-Ring ℤ-Ring R) → initial-hom-Ring R ＝ f - contraction-initial-hom-Ring f = - eq-htpy-hom-Ring ℤ-Ring R (initial-hom-Ring R) f (htpy-initial-hom-Ring f) + abstract + htpy-initial-hom-Ring : + (f : hom-Ring ℤ-Ring R) → htpy-hom-Ring ℤ-Ring R (initial-hom-Ring R) f + htpy-initial-hom-Ring f k = + ( inv + ( ( preserves-integer-multiples-hom-Ring ℤ-Ring R f k one-ℤ) ∙ + ( ap + ( integer-multiple-Ring R k) + ( preserves-one-hom-Ring ℤ-Ring R f)))) ∙ + ( ap (map-hom-Ring ℤ-Ring R f) (is-integer-multiple-ℤ k)) + + contraction-initial-hom-Ring : + (f : hom-Ring ℤ-Ring R) → initial-hom-Ring R ＝ f + contraction-initial-hom-Ring f = + eq-htpy-hom-Ring ℤ-Ring R (initial-hom-Ring R) f (htpy-initial-hom-Ring f) ``` #### The ring of integers is the initial ring @@ -173,37 +199,6 @@ pr1 (is-initial-ℤ-Ring S) = initial-hom-Ring S pr2 (is-initial-ℤ-Ring S) f = contraction-initial-hom-Ring S f ``` -### Integer multiplication in the ring of integers coincides with multiplication of integers - -```agda -integer-multiple-one-ℤ-Ring : - (k : ℤ) → integer-multiple-Ring ℤ-Ring k one-ℤ ＝ k -integer-multiple-one-ℤ-Ring (inl zero-ℕ) = refl -integer-multiple-one-ℤ-Ring (inl (succ-ℕ n)) = - ap pred-ℤ (integer-multiple-one-ℤ-Ring (inl n)) -integer-multiple-one-ℤ-Ring (inr (inl _)) = refl -integer-multiple-one-ℤ-Ring (inr (inr zero-ℕ)) = refl -integer-multiple-one-ℤ-Ring (inr (inr (succ-ℕ n))) = - ap succ-ℤ (integer-multiple-one-ℤ-Ring (inr (inr n))) - -compute-integer-multiple-ℤ-Ring : - (k l : ℤ) → integer-multiple-Ring ℤ-Ring k l ＝ mul-ℤ k l -compute-integer-multiple-ℤ-Ring k l = - equational-reasoning - integer-multiple-Ring ℤ-Ring k l - ＝ integer-multiple-Ring ℤ-Ring k (integer-multiple-Ring ℤ-Ring l one-ℤ) - by - ap - ( integer-multiple-Ring ℤ-Ring k) - ( inv (integer-multiple-one-ℤ-Ring l)) - ＝ integer-multiple-Ring ℤ-Ring (mul-ℤ k l) one-ℤ - by - inv (integer-multiple-mul-Ring ℤ-Ring k l one-ℤ) - ＝ mul-ℤ k l - by - integer-multiple-one-ℤ-Ring _ -``` - ### The image of `ℤ` in a ring is commutative ```agda diff --git a/src/elementary-number-theory/commutative-semiring-of-natural-numbers.lagda.md b/src/elementary-number-theory/semiring-of-natural-numbers.lagda.md similarity index 97% rename from src/elementary-number-theory/commutative-semiring-of-natural-numbers.lagda.md rename to src/elementary-number-theory/semiring-of-natural-numbers.lagda.md index 2346b3328f..f24cb45c4e 100644 --- a/src/elementary-number-theory/commutative-semiring-of-natural-numbers.lagda.md +++ b/src/elementary-number-theory/semiring-of-natural-numbers.lagda.md @@ -1,7 +1,7 @@ -# The commutative semiring of natural numbers +# The semiring of natural numbers ```agda -module elementary-number-theory.commutative-semiring-of-natural-numbers where +module elementary-number-theory.semiring-of-natural-numbers where ```
Imports diff --git a/src/elementary-number-theory/squares-natural-numbers.lagda.md b/src/elementary-number-theory/squares-natural-numbers.lagda.md index f8afd7063e..75f66ed2fd 100644 --- a/src/elementary-number-theory/squares-natural-numbers.lagda.md +++ b/src/elementary-number-theory/squares-natural-numbers.lagda.md @@ -68,31 +68,32 @@ square-root-ℕ _ (root , _) = root ### Squares of successors ```agda -square-succ-ℕ : - (n : ℕ) → - square-ℕ (succ-ℕ n) ＝ succ-ℕ ((succ-ℕ (succ-ℕ n)) *ℕ n) -square-succ-ℕ n = - ( right-successor-law-mul-ℕ (succ-ℕ n) n) ∙ - ( commutative-add-ℕ (succ-ℕ n) ((succ-ℕ n) *ℕ n)) - -square-succ-succ-ℕ : - (n : ℕ) → - square-ℕ (succ-ℕ (succ-ℕ n)) ＝ square-ℕ n +ℕ 2 *ℕ n +ℕ 2 *ℕ n +ℕ 4 -square-succ-succ-ℕ n = - equational-reasoning - square-ℕ (n +ℕ 2) - ＝ (n +ℕ 2) *ℕ n +ℕ (n +ℕ 2) *ℕ 2 - by (left-distributive-mul-add-ℕ (n +ℕ 2) n 2) - ＝ square-ℕ n +ℕ 2 *ℕ n +ℕ 2 *ℕ (n +ℕ 2) - by - ( ap-add-ℕ {(n +ℕ 2) *ℕ n} {(n +ℕ 2) *ℕ 2} - ( right-distributive-mul-add-ℕ n 2 n) - ( commutative-mul-ℕ (n +ℕ 2) 2)) - ＝ square-ℕ n +ℕ 2 *ℕ n +ℕ 2 *ℕ n +ℕ 4 - by - ( ap-add-ℕ {square-ℕ n +ℕ 2 *ℕ n} {2 *ℕ (n +ℕ 2)} - ( refl) - ( left-distributive-mul-add-ℕ 2 n 2)) +abstract + square-succ-ℕ : + (n : ℕ) → + square-ℕ (succ-ℕ n) ＝ succ-ℕ ((succ-ℕ (succ-ℕ n)) *ℕ n) + square-succ-ℕ n = + ( right-successor-law-mul-ℕ (succ-ℕ n) n) ∙ + ( commutative-add-ℕ (succ-ℕ n) ((succ-ℕ n) *ℕ n)) + + square-succ-succ-ℕ : + (n : ℕ) → + square-ℕ (succ-ℕ (succ-ℕ n)) ＝ square-ℕ n +ℕ 2 *ℕ n +ℕ 2 *ℕ n +ℕ 4 + square-succ-succ-ℕ n = + equational-reasoning + square-ℕ (n +ℕ 2) + ＝ (n +ℕ 2) *ℕ n +ℕ (n +ℕ 2) *ℕ 2 + by (left-distributive-mul-add-ℕ (n +ℕ 2) n 2) + ＝ square-ℕ n +ℕ 2 *ℕ n +ℕ 2 *ℕ (n +ℕ 2) + by + ( ap-add-ℕ {(n +ℕ 2) *ℕ n} {(n +ℕ 2) *ℕ 2} + ( right-distributive-mul-add-ℕ n 2 n) + ( commutative-mul-ℕ (n +ℕ 2) 2)) + ＝ square-ℕ n +ℕ 2 *ℕ n +ℕ 2 *ℕ n +ℕ 4 + by + ( ap-add-ℕ {square-ℕ n +ℕ 2 *ℕ n} {2 *ℕ (n +ℕ 2)} + ( refl) + ( left-distributive-mul-add-ℕ 2 n 2)) ``` ### `n > √n` for `n > 1` @@ -101,58 +102,59 @@ The idea is to assume `n = m + 2 ≤ sqrt(m + 2)` for some `m : ℕ` and derive contradiction by squaring both sides of the inequality ```agda -greater-than-square-root-ℕ : - (n root : ℕ) → ¬ ((n +ℕ 2 ≤-ℕ root) × (n +ℕ 2 ＝ square-ℕ root)) -greater-than-square-root-ℕ n root (pf-leq , pf-eq) = - reflects-leq-left-add-ℕ - ( square-ℕ root) - ( square-ℕ n +ℕ 2 *ℕ n +ℕ n +ℕ 2) - ( 0) - ( tr - ( leq-ℕ (square-ℕ n +ℕ 2 *ℕ n +ℕ n +ℕ 2 +ℕ square-ℕ root)) - ( inv (left-unit-law-add-ℕ (square-ℕ root))) - ( concatenate-eq-leq-ℕ - ( square-ℕ root) - ( inv - ( lemma ∙ - ( ap-add-ℕ {square-ℕ n +ℕ 2 *ℕ n +ℕ n +ℕ 2} {n +ℕ 2} - ( refl) - ( pf-eq)))) - ( preserves-leq-mul-ℕ (n +ℕ 2) root (n +ℕ 2) root pf-leq pf-leq))) - where - lemma : square-ℕ (n +ℕ 2) ＝ square-ℕ n +ℕ 2 *ℕ n +ℕ n +ℕ 2 +ℕ n +ℕ 2 - lemma = - equational-reasoning - square-ℕ (n +ℕ 2) - ＝ square-ℕ n +ℕ 2 *ℕ n +ℕ 2 *ℕ n +ℕ 4 - by (square-succ-succ-ℕ n) - ＝ square-ℕ n +ℕ 2 *ℕ n +ℕ (n +ℕ 2 +ℕ n +ℕ 2) - by - ( ap-add-ℕ {square-ℕ n +ℕ 2 *ℕ n} {2 *ℕ n +ℕ 4} - ( refl) - ( equational-reasoning - 2 *ℕ n +ℕ 4 - ＝ n +ℕ n +ℕ 2 +ℕ 2 - by - ( ap-add-ℕ {2 *ℕ n} {4} - ( left-two-law-mul-ℕ n) - ( refl)) - ＝ n +ℕ 2 +ℕ 2 +ℕ n - by (commutative-add-ℕ n (n +ℕ 2 +ℕ 2)) - ＝ n +ℕ 2 +ℕ (2 +ℕ n) - by (associative-add-ℕ (n +ℕ 2) 2 n) - ＝ n +ℕ 2 +ℕ (n +ℕ 2) - by - ( ap-add-ℕ {n +ℕ 2} {2 +ℕ n} - ( refl) - ( commutative-add-ℕ 2 n)))) - ＝ square-ℕ n +ℕ 2 *ℕ n +ℕ n +ℕ 2 +ℕ n +ℕ 2 - by - ( inv - ( associative-add-ℕ - ( square-ℕ n +ℕ 2 *ℕ n) - ( n +ℕ 2) - ( n +ℕ 2))) +abstract + greater-than-square-root-ℕ : + (n root : ℕ) → ¬ ((n +ℕ 2 ≤-ℕ root) × (n +ℕ 2 ＝ square-ℕ root)) + greater-than-square-root-ℕ n root (pf-leq , pf-eq) = + reflects-leq-left-add-ℕ + ( square-ℕ root) + ( square-ℕ n +ℕ 2 *ℕ n +ℕ n +ℕ 2) + ( 0) + ( tr + ( leq-ℕ (square-ℕ n +ℕ 2 *ℕ n +ℕ n +ℕ 2 +ℕ square-ℕ root)) + ( inv (left-unit-law-add-ℕ (square-ℕ root))) + ( concatenate-eq-leq-ℕ + ( square-ℕ root) + ( inv + ( lemma ∙ + ( ap-add-ℕ {square-ℕ n +ℕ 2 *ℕ n +ℕ n +ℕ 2} {n +ℕ 2} + ( refl) + ( pf-eq)))) + ( preserves-leq-mul-ℕ (n +ℕ 2) root (n +ℕ 2) root pf-leq pf-leq))) + where + lemma : square-ℕ (n +ℕ 2) ＝ square-ℕ n +ℕ 2 *ℕ n +ℕ n +ℕ 2 +ℕ n +ℕ 2 + lemma = + equational-reasoning + square-ℕ (n +ℕ 2) + ＝ square-ℕ n +ℕ 2 *ℕ n +ℕ 2 *ℕ n +ℕ 4 + by (square-succ-succ-ℕ n) + ＝ square-ℕ n +ℕ 2 *ℕ n +ℕ (n +ℕ 2 +ℕ n +ℕ 2) + by + ( ap-add-ℕ {square-ℕ n +ℕ 2 *ℕ n} {2 *ℕ n +ℕ 4} + ( refl) + ( equational-reasoning + 2 *ℕ n +ℕ 4 + ＝ n +ℕ n +ℕ 2 +ℕ 2 + by + ( ap-add-ℕ {2 *ℕ n} {4} + ( left-two-law-mul-ℕ n) + ( refl)) + ＝ n +ℕ 2 +ℕ 2 +ℕ n + by (commutative-add-ℕ n (n +ℕ 2 +ℕ 2)) + ＝ n +ℕ 2 +ℕ (2 +ℕ n) + by (associative-add-ℕ (n +ℕ 2) 2 n) + ＝ n +ℕ 2 +ℕ (n +ℕ 2) + by + ( ap-add-ℕ {n +ℕ 2} {2 +ℕ n} + ( refl) + ( commutative-add-ℕ 2 n)))) + ＝ square-ℕ n +ℕ 2 *ℕ n +ℕ n +ℕ 2 +ℕ n +ℕ 2 + by + ( inv + ( associative-add-ℕ + ( square-ℕ n +ℕ 2 *ℕ n) + ( n +ℕ 2) + ( n +ℕ 2))) ``` ### Squareness in ℕ is decidable diff --git a/src/elementary-number-theory/strict-inequality-integer-fractions.lagda.md b/src/elementary-number-theory/strict-inequality-integer-fractions.lagda.md index d58fbf1f4c..ecfd106600 100644 --- a/src/elementary-number-theory/strict-inequality-integer-fractions.lagda.md +++ b/src/elementary-number-theory/strict-inequality-integer-fractions.lagda.md @@ -61,7 +61,7 @@ is [strictly less](elementary-number-theory.strict-inequality-integers.md) than ```agda le-fraction-ℤ-Prop : fraction-ℤ → fraction-ℤ → Prop lzero le-fraction-ℤ-Prop (m , n , p) (m' , n' , p') = - le-ℤ-Prop (m *ℤ n') (m' *ℤ n) + le-prop-ℤ (m *ℤ n') (m' *ℤ n) le-fraction-ℤ : fraction-ℤ → fraction-ℤ → UU lzero le-fraction-ℤ x y = type-Prop (le-fraction-ℤ-Prop x y) diff --git a/src/elementary-number-theory/strict-inequality-integers.lagda.md b/src/elementary-number-theory/strict-inequality-integers.lagda.md index 4156161355..eac42a980e 100644 --- a/src/elementary-number-theory/strict-inequality-integers.lagda.md +++ b/src/elementary-number-theory/strict-inequality-integers.lagda.md @@ -57,14 +57,15 @@ on the integers. ### Strict inequality on the integers ```agda -le-ℤ-Prop : ℤ → ℤ → Prop lzero -le-ℤ-Prop x y = subtype-positive-ℤ (y -ℤ x) +le-prop-ℤ : ℤ → ℤ → Prop lzero +le-prop-ℤ x y = subtype-positive-ℤ (y -ℤ x) le-ℤ : ℤ → ℤ → UU lzero -le-ℤ x y = type-Prop (le-ℤ-Prop x y) +le-ℤ x y = type-Prop (le-prop-ℤ x y) -is-prop-le-ℤ : (x y : ℤ) → is-prop (le-ℤ x y) -is-prop-le-ℤ x y = is-prop-type-Prop (le-ℤ-Prop x y) +abstract + is-prop-le-ℤ : (x y : ℤ) → is-prop (le-ℤ x y) + is-prop-le-ℤ x y = is-prop-type-Prop (le-prop-ℤ x y) ``` ## Properties @@ -72,8 +73,9 @@ is-prop-le-ℤ x y = is-prop-type-Prop (le-ℤ-Prop x y) ### Strict inequality on the integers implies inequality ```agda -leq-le-ℤ : {x y : ℤ} → le-ℤ x y → leq-ℤ x y -leq-le-ℤ {x} {y} = is-nonnegative-is-positive-ℤ +abstract + leq-le-ℤ : {x y : ℤ} → le-ℤ x y → leq-ℤ x y + leq-le-ℤ {x} {y} = is-nonnegative-is-positive-ℤ ``` ### Strict inequality on the integers is decidable @@ -92,23 +94,25 @@ le-ℤ-Decidable-Prop x y = ### Strict inequality on the integers is transitive ```agda -transitive-le-ℤ : (k l m : ℤ) → le-ℤ l m → le-ℤ k l → le-ℤ k m -transitive-le-ℤ k l m H K = - is-positive-eq-ℤ - ( triangle-diff-ℤ m l k) - ( is-positive-add-ℤ H K) +abstract + transitive-le-ℤ : (k l m : ℤ) → le-ℤ l m → le-ℤ k l → le-ℤ k m + transitive-le-ℤ k l m H K = + is-positive-eq-ℤ + ( triangle-diff-ℤ m l k) + ( is-positive-add-ℤ H K) ``` ### Strict inequality on the integers is asymmetric ```agda -asymmetric-le-ℤ : (x y : ℤ) → le-ℤ x y → ¬ (le-ℤ y x) -asymmetric-le-ℤ x y p = - is-not-positive-is-nonpositive-ℤ - ( is-nonpositive-eq-ℤ - ( distributive-neg-diff-ℤ y x) - ( is-nonpositive-neg-is-nonnegative-ℤ - ( is-nonnegative-is-positive-ℤ p))) +abstract + asymmetric-le-ℤ : (x y : ℤ) → le-ℤ x y → ¬ (le-ℤ y x) + asymmetric-le-ℤ x y p = + is-not-positive-is-nonpositive-ℤ + ( is-nonpositive-eq-ℤ + ( distributive-neg-diff-ℤ y x) + ( is-nonpositive-neg-is-nonnegative-ℤ + ( is-nonnegative-is-positive-ℤ p))) ``` ### Strict inequality on the integers is connected @@ -128,21 +132,23 @@ connected-le-ℤ x y H = ### Any integer is strictly greater than its predecessor ```agda -le-pred-ℤ : (x : ℤ) → le-ℤ (pred-ℤ x) x -le-pred-ℤ x = - is-positive-eq-ℤ - ( inv (right-predecessor-law-diff-ℤ x x ∙ ap succ-ℤ (is-zero-diff-ℤ' x))) - ( is-positive-int-positive-ℤ one-positive-ℤ) +abstract + le-pred-ℤ : (x : ℤ) → le-ℤ (pred-ℤ x) x + le-pred-ℤ x = + is-positive-eq-ℤ + ( inv (right-predecessor-law-diff-ℤ x x ∙ ap succ-ℤ (is-zero-diff-ℤ' x))) + ( is-positive-int-positive-ℤ one-positive-ℤ) ``` ### Any integer is strictly lesser than its successor ```agda -le-succ-ℤ : (x : ℤ) → le-ℤ x (succ-ℤ x) -le-succ-ℤ x = - is-positive-eq-ℤ - ( inv (left-successor-law-diff-ℤ x x ∙ ap succ-ℤ (is-zero-diff-ℤ' x))) - ( is-positive-int-positive-ℤ one-positive-ℤ) +abstract + le-succ-ℤ : (x : ℤ) → le-ℤ x (succ-ℤ x) + le-succ-ℤ x = + is-positive-eq-ℤ + ( inv (left-successor-law-diff-ℤ x x ∙ ap succ-ℤ (is-zero-diff-ℤ' x))) + ( is-positive-int-positive-ℤ one-positive-ℤ) ``` ### Strict inequality on the integers is invariant by translation @@ -162,39 +168,41 @@ module _ ### Addition on the integers preserves strict inequality ```agda -preserves-le-left-add-ℤ : - (z x y : ℤ) → le-ℤ x y → le-ℤ (x +ℤ z) (y +ℤ z) -preserves-le-left-add-ℤ z x y = - is-positive-eq-ℤ (inv (right-translation-diff-ℤ y x z)) - -preserves-le-right-add-ℤ : - (z x y : ℤ) → le-ℤ x y → le-ℤ (z +ℤ x) (z +ℤ y) -preserves-le-right-add-ℤ z x y = - is-positive-eq-ℤ (inv (left-translation-diff-ℤ y x z)) - -preserves-le-add-ℤ : - {a b c d : ℤ} → le-ℤ a b → le-ℤ c d → le-ℤ (a +ℤ c) (b +ℤ d) -preserves-le-add-ℤ {a} {b} {c} {d} H K = - transitive-le-ℤ - ( a +ℤ c) - ( b +ℤ c) - ( b +ℤ d) - ( preserves-le-right-add-ℤ b c d K) - ( preserves-le-left-add-ℤ c a b H) +abstract + preserves-le-left-add-ℤ : + (z x y : ℤ) → le-ℤ x y → le-ℤ (x +ℤ z) (y +ℤ z) + preserves-le-left-add-ℤ z x y = + is-positive-eq-ℤ (inv (right-translation-diff-ℤ y x z)) + + preserves-le-right-add-ℤ : + (z x y : ℤ) → le-ℤ x y → le-ℤ (z +ℤ x) (z +ℤ y) + preserves-le-right-add-ℤ z x y = + is-positive-eq-ℤ (inv (left-translation-diff-ℤ y x z)) + + preserves-le-add-ℤ : + {a b c d : ℤ} → le-ℤ a b → le-ℤ c d → le-ℤ (a +ℤ c) (b +ℤ d) + preserves-le-add-ℤ {a} {b} {c} {d} H K = + transitive-le-ℤ + ( a +ℤ c) + ( b +ℤ c) + ( b +ℤ d) + ( preserves-le-right-add-ℤ b c d K) + ( preserves-le-left-add-ℤ c a b H) ``` ### Addition on the integers reflects strict inequality ```agda -reflects-le-left-add-ℤ : - (z x y : ℤ) → le-ℤ (x +ℤ z) (y +ℤ z) → le-ℤ x y -reflects-le-left-add-ℤ z x y = - is-positive-eq-ℤ (right-translation-diff-ℤ y x z) - -reflects-le-right-add-ℤ : - (z x y : ℤ) → le-ℤ (z +ℤ x) (z +ℤ y) → le-ℤ x y -reflects-le-right-add-ℤ z x y = - is-positive-eq-ℤ (left-translation-diff-ℤ y x z) +abstract + reflects-le-left-add-ℤ : + (z x y : ℤ) → le-ℤ (x +ℤ z) (y +ℤ z) → le-ℤ x y + reflects-le-left-add-ℤ z x y = + is-positive-eq-ℤ (right-translation-diff-ℤ y x z) + + reflects-le-right-add-ℤ : + (z x y : ℤ) → le-ℤ (z +ℤ x) (z +ℤ y) → le-ℤ x y + reflects-le-right-add-ℤ z x y = + is-positive-eq-ℤ (left-translation-diff-ℤ y x z) ``` ### An integer `x` is positive if and only if `0 < x` @@ -273,30 +281,32 @@ module _ ### The inclusion of natural numbers preserves strict inequality ```agda -le-natural-le-ℤ : (m n : ℕ) → le-ℕ m n → le-ℤ (int-ℕ m) (int-ℕ n) -le-natural-le-ℤ zero-ℕ (succ-ℕ n) star = - le-zero-is-positive-ℤ - (int-ℕ (succ-ℕ n)) - (is-positive-int-is-nonzero-ℕ (succ-ℕ n) λ ()) -le-natural-le-ℤ (succ-ℕ m) (succ-ℕ n) m +## Idea + +We define the +{{#concept "standard strict ordering" Disambiguation="natural numbers" Agda=le-ℕ}} +on the [natural numbers](elementary-number-theory.natural-numbers.md) and prove +its basic properties. + ## Definition ### The standard strict inequality on the natural numbers ```agda -le-ℕ-Prop : ℕ → ℕ → Prop lzero -le-ℕ-Prop m zero-ℕ = empty-Prop -le-ℕ-Prop zero-ℕ (succ-ℕ m) = unit-Prop -le-ℕ-Prop (succ-ℕ n) (succ-ℕ m) = le-ℕ-Prop n m +le-prop-ℕ : ℕ → ℕ → Prop lzero +le-prop-ℕ m zero-ℕ = empty-Prop +le-prop-ℕ zero-ℕ (succ-ℕ m) = unit-Prop +le-prop-ℕ (succ-ℕ n) (succ-ℕ m) = le-prop-ℕ n m le-ℕ : ℕ → ℕ → UU lzero -le-ℕ n m = type-Prop (le-ℕ-Prop n m) +le-ℕ n m = type-Prop (le-prop-ℕ n m) is-prop-le-ℕ : (n : ℕ) → (m : ℕ) → is-prop (le-ℕ n m) -is-prop-le-ℕ n m = is-prop-type-Prop (le-ℕ-Prop n m) +is-prop-le-ℕ n m = is-prop-type-Prop (le-prop-ℕ n m) infix 30 _<-ℕ_ _<-ℕ_ = le-ℕ @@ -111,60 +119,66 @@ is-nonzero-le-ℕ m .zero-ℕ () refl ### Any nonzero natural number is strictly greater than `0` ```agda -le-is-nonzero-ℕ : (n : ℕ) → is-nonzero-ℕ n → le-ℕ zero-ℕ n -le-is-nonzero-ℕ zero-ℕ H = H refl -le-is-nonzero-ℕ (succ-ℕ n) H = star +abstract + le-is-nonzero-ℕ : (n : ℕ) → is-nonzero-ℕ n → le-ℕ zero-ℕ n + le-is-nonzero-ℕ zero-ℕ H = H refl + le-is-nonzero-ℕ (succ-ℕ n) H = star ``` ### No natural number is strictly less than zero ```agda -contradiction-le-zero-ℕ : - (m : ℕ) → (le-ℕ m zero-ℕ) → empty -contradiction-le-zero-ℕ zero-ℕ () -contradiction-le-zero-ℕ (succ-ℕ m) () +abstract + contradiction-le-zero-ℕ : + (m : ℕ) → (le-ℕ m zero-ℕ) → empty + contradiction-le-zero-ℕ zero-ℕ () + contradiction-le-zero-ℕ (succ-ℕ m) () ``` ### No successor is strictly less than one ```agda -contradiction-le-one-ℕ : - (n : ℕ) → le-ℕ (succ-ℕ n) 1 → empty -contradiction-le-one-ℕ zero-ℕ () -contradiction-le-one-ℕ (succ-ℕ n) () +abstract + contradiction-le-one-ℕ : + (n : ℕ) → le-ℕ (succ-ℕ n) 1 → empty + contradiction-le-one-ℕ zero-ℕ () + contradiction-le-one-ℕ (succ-ℕ n) () ``` -### The strict inequality on the natural numbers is anti-reflexive +### The strict inequality on the natural numbers is irreflexive ```agda -anti-reflexive-le-ℕ : (n : ℕ) → ¬ (n <-ℕ n) -anti-reflexive-le-ℕ zero-ℕ () -anti-reflexive-le-ℕ (succ-ℕ n) = anti-reflexive-le-ℕ n +abstract + irreflexive-le-ℕ : (n : ℕ) → ¬ (n <-ℕ n) + irreflexive-le-ℕ zero-ℕ () + irreflexive-le-ℕ (succ-ℕ n) = irreflexive-le-ℕ n ``` ### If `x < y` then `x ≠ y` ```agda -neq-le-ℕ : {x y : ℕ} → le-ℕ x y → x ≠ y -neq-le-ℕ {zero-ℕ} {succ-ℕ y} H = is-nonzero-succ-ℕ y ∘ inv -neq-le-ℕ {succ-ℕ x} {succ-ℕ y} H p = neq-le-ℕ H (is-injective-succ-ℕ p) +abstract + neq-le-ℕ : {x y : ℕ} → le-ℕ x y → x ≠ y + neq-le-ℕ {x} H refl = irreflexive-le-ℕ x H ``` ### The strict inequality on the natural numbers is antisymmetric ```agda -antisymmetric-le-ℕ : (m n : ℕ) → le-ℕ m n → le-ℕ n m → m ＝ n -antisymmetric-le-ℕ (succ-ℕ m) (succ-ℕ n) p q = - ap succ-ℕ (antisymmetric-le-ℕ m n p q) +abstract + antisymmetric-le-ℕ : (m n : ℕ) → le-ℕ m n → le-ℕ n m → m ＝ n + antisymmetric-le-ℕ (succ-ℕ m) (succ-ℕ n) p q = + ap succ-ℕ (antisymmetric-le-ℕ m n p q) ``` ### The strict inequality on the natural numbers is transitive ```agda -transitive-le-ℕ : (n m l : ℕ) → (le-ℕ n m) → (le-ℕ m l) → (le-ℕ n l) -transitive-le-ℕ zero-ℕ (succ-ℕ m) (succ-ℕ l) p q = star -transitive-le-ℕ (succ-ℕ n) (succ-ℕ m) (succ-ℕ l) p q = - transitive-le-ℕ n m l p q +abstract + transitive-le-ℕ : (n m l : ℕ) → (le-ℕ n m) → (le-ℕ m l) → (le-ℕ n l) + transitive-le-ℕ zero-ℕ (succ-ℕ m) (succ-ℕ l) p q = star + transitive-le-ℕ (succ-ℕ n) (succ-ℕ m) (succ-ℕ l) p q = + transitive-le-ℕ n m l p q ``` ### The strictly preordered set of natural numbers ordered by strict inequality @@ -173,23 +187,24 @@ transitive-le-ℕ (succ-ℕ n) (succ-ℕ m) (succ-ℕ l) p q = strictly-preordered-set-ℕ : Strictly-Preordered-Set lzero lzero pr1 strictly-preordered-set-ℕ = ℕ-Set pr2 strictly-preordered-set-ℕ = - le-ℕ-Prop , anti-reflexive-le-ℕ , λ n m l I J → transitive-le-ℕ n m l J I + le-prop-ℕ , irreflexive-le-ℕ , λ n m l I J → transitive-le-ℕ n m l J I ``` ### A sharper variant of transitivity ```agda -transitive-le-ℕ' : - (k l m : ℕ) → (le-ℕ k l) → (le-ℕ l (succ-ℕ m)) → le-ℕ k m -transitive-le-ℕ' zero-ℕ zero-ℕ m () s -transitive-le-ℕ' (succ-ℕ k) zero-ℕ m () s -transitive-le-ℕ' zero-ℕ (succ-ℕ l) zero-ℕ star s = - ex-falso (contradiction-le-one-ℕ l s) -transitive-le-ℕ' (succ-ℕ k) (succ-ℕ l) zero-ℕ t s = - ex-falso (contradiction-le-one-ℕ l s) -transitive-le-ℕ' zero-ℕ (succ-ℕ l) (succ-ℕ m) star s = star -transitive-le-ℕ' (succ-ℕ k) (succ-ℕ l) (succ-ℕ m) t s = - transitive-le-ℕ' k l m t s +abstract + transitive-le-ℕ' : + (k l m : ℕ) → (le-ℕ k l) → (le-ℕ l (succ-ℕ m)) → le-ℕ k m + transitive-le-ℕ' zero-ℕ zero-ℕ m () s + transitive-le-ℕ' (succ-ℕ k) zero-ℕ m () s + transitive-le-ℕ' zero-ℕ (succ-ℕ l) zero-ℕ star s = + ex-falso (contradiction-le-one-ℕ l s) + transitive-le-ℕ' (succ-ℕ k) (succ-ℕ l) zero-ℕ t s = + ex-falso (contradiction-le-one-ℕ l s) + transitive-le-ℕ' zero-ℕ (succ-ℕ l) (succ-ℕ m) star s = star + transitive-le-ℕ' (succ-ℕ k) (succ-ℕ l) (succ-ℕ m) t s = + transitive-le-ℕ' k l m t s ``` ### The strict inequality on the natural numbers is linear @@ -215,125 +230,138 @@ subtraction-le-ℕ (succ-ℕ n) (succ-ℕ m) p = P : Σ ℕ (λ l' → (is-nonzero-ℕ l') × (l' +ℕ n ＝ m)) P = subtraction-le-ℕ n m p -le-subtraction-ℕ : (n m l : ℕ) → is-nonzero-ℕ l → l +ℕ n ＝ m → le-ℕ n m -le-subtraction-ℕ zero-ℕ m l q p = - tr (λ x → le-ℕ zero-ℕ x) p (le-is-nonzero-ℕ l q) -le-subtraction-ℕ (succ-ℕ n) (succ-ℕ m) l q p = - le-subtraction-ℕ n m l q (is-injective-succ-ℕ p) +abstract + le-subtraction-ℕ : (n m l : ℕ) → is-nonzero-ℕ l → l +ℕ n ＝ m → le-ℕ n m + le-subtraction-ℕ zero-ℕ m l q p = + tr (λ x → le-ℕ zero-ℕ x) p (le-is-nonzero-ℕ l q) + le-subtraction-ℕ (succ-ℕ n) (succ-ℕ m) l q p = + le-subtraction-ℕ n m l q (is-injective-succ-ℕ p) ``` ### Any natural number is strictly less than its successor ```agda -succ-le-ℕ : (n : ℕ) → le-ℕ n (succ-ℕ n) -succ-le-ℕ zero-ℕ = star -succ-le-ℕ (succ-ℕ n) = succ-le-ℕ n +abstract + succ-le-ℕ : (n : ℕ) → le-ℕ n (succ-ℕ n) + succ-le-ℕ zero-ℕ = star + succ-le-ℕ (succ-ℕ n) = succ-le-ℕ n ``` ### The successor function preserves strict inequality on the right ```agda -preserves-le-succ-ℕ : - (m n : ℕ) → le-ℕ m n → le-ℕ m (succ-ℕ n) -preserves-le-succ-ℕ m n H = - transitive-le-ℕ m n (succ-ℕ n) H (succ-le-ℕ n) +abstract + preserves-le-succ-ℕ : + (m n : ℕ) → le-ℕ m n → le-ℕ m (succ-ℕ n) + preserves-le-succ-ℕ m n H = + transitive-le-ℕ m n (succ-ℕ n) H (succ-le-ℕ n) ``` ### Concatenating strict and nonstrict inequalities ```agda -concatenate-leq-le-ℕ : - {x y z : ℕ} → x ≤-ℕ y → le-ℕ y z → le-ℕ x z -concatenate-leq-le-ℕ {zero-ℕ} {zero-ℕ} {succ-ℕ z} H K = star -concatenate-leq-le-ℕ {zero-ℕ} {succ-ℕ y} {succ-ℕ z} H K = star -concatenate-leq-le-ℕ {succ-ℕ x} {succ-ℕ y} {succ-ℕ z} H K = - concatenate-leq-le-ℕ {x} {y} {z} H K +abstract + concatenate-leq-le-ℕ : + {x y z : ℕ} → x ≤-ℕ y → le-ℕ y z → le-ℕ x z + concatenate-leq-le-ℕ {zero-ℕ} {zero-ℕ} {succ-ℕ z} H K = star + concatenate-leq-le-ℕ {zero-ℕ} {succ-ℕ y} {succ-ℕ z} H K = star + concatenate-leq-le-ℕ {succ-ℕ x} {succ-ℕ y} {succ-ℕ z} H K = + concatenate-leq-le-ℕ {x} {y} {z} H K -concatenate-le-leq-ℕ : - {x y z : ℕ} → le-ℕ x y → y ≤-ℕ z → le-ℕ x z -concatenate-le-leq-ℕ {zero-ℕ} {succ-ℕ y} {succ-ℕ z} H K = star -concatenate-le-leq-ℕ {succ-ℕ x} {succ-ℕ y} {succ-ℕ z} H K = - concatenate-le-leq-ℕ {x} {y} {z} H K + concatenate-le-leq-ℕ : + {x y z : ℕ} → le-ℕ x y → y ≤-ℕ z → le-ℕ x z + concatenate-le-leq-ℕ {zero-ℕ} {succ-ℕ y} {succ-ℕ z} H K = star + concatenate-le-leq-ℕ {succ-ℕ x} {succ-ℕ y} {succ-ℕ z} H K = + concatenate-le-leq-ℕ {x} {y} {z} H K ``` ### If `m < n` then `n ≰ m` ```agda -contradiction-le-ℕ : (m n : ℕ) → le-ℕ m n → ¬ (n ≤-ℕ m) -contradiction-le-ℕ zero-ℕ (succ-ℕ n) H K = K -contradiction-le-ℕ (succ-ℕ m) (succ-ℕ n) H = contradiction-le-ℕ m n H +abstract + contradiction-le-ℕ : (m n : ℕ) → le-ℕ m n → ¬ (n ≤-ℕ m) + contradiction-le-ℕ zero-ℕ (succ-ℕ n) H K = K + contradiction-le-ℕ (succ-ℕ m) (succ-ℕ n) H = contradiction-le-ℕ m n H ``` ### If `n ≤ m` then `m ≮ n` ```agda -contradiction-le-ℕ' : (m n : ℕ) → n ≤-ℕ m → ¬ (le-ℕ m n) -contradiction-le-ℕ' m n K H = contradiction-le-ℕ m n H K +abstract + contradiction-le-ℕ' : (m n : ℕ) → n ≤-ℕ m → ¬ (le-ℕ m n) + contradiction-le-ℕ' m n K H = contradiction-le-ℕ m n H K ``` ### If `m ≮ n` then `n ≤ m` ```agda -leq-not-le-ℕ : (m n : ℕ) → ¬ (le-ℕ m n) → n ≤-ℕ m -leq-not-le-ℕ zero-ℕ zero-ℕ H = star -leq-not-le-ℕ zero-ℕ (succ-ℕ n) H = ex-falso (H star) -leq-not-le-ℕ (succ-ℕ m) zero-ℕ H = star -leq-not-le-ℕ (succ-ℕ m) (succ-ℕ n) H = leq-not-le-ℕ m n H +abstract + leq-not-le-ℕ : (m n : ℕ) → ¬ (le-ℕ m n) → n ≤-ℕ m + leq-not-le-ℕ zero-ℕ zero-ℕ H = star + leq-not-le-ℕ zero-ℕ (succ-ℕ n) H = ex-falso (H star) + leq-not-le-ℕ (succ-ℕ m) zero-ℕ H = star + leq-not-le-ℕ (succ-ℕ m) (succ-ℕ n) H = leq-not-le-ℕ m n H ``` ### If `n ≰ m` then `m < n` ```agda -le-not-leq-ℕ : (m n : ℕ) → ¬ (n ≤-ℕ m) → le-ℕ m n -le-not-leq-ℕ zero-ℕ zero-ℕ H = ex-falso (H star) -le-not-leq-ℕ zero-ℕ (succ-ℕ n) H = star -le-not-leq-ℕ (succ-ℕ m) zero-ℕ H = ex-falso (H star) -le-not-leq-ℕ (succ-ℕ m) (succ-ℕ n) H = le-not-leq-ℕ m n H +abstract + le-not-leq-ℕ : (m n : ℕ) → ¬ (n ≤-ℕ m) → le-ℕ m n + le-not-leq-ℕ zero-ℕ zero-ℕ H = ex-falso (H star) + le-not-leq-ℕ zero-ℕ (succ-ℕ n) H = star + le-not-leq-ℕ (succ-ℕ m) zero-ℕ H = ex-falso (H star) + le-not-leq-ℕ (succ-ℕ m) (succ-ℕ n) H = le-not-leq-ℕ m n H ``` ### If `x < y` then `x ≤ y` ```agda -leq-le-ℕ : - (x y : ℕ) → le-ℕ x y → x ≤-ℕ y -leq-le-ℕ zero-ℕ (succ-ℕ y) H = star -leq-le-ℕ (succ-ℕ x) (succ-ℕ y) H = leq-le-ℕ x y H +abstract + leq-le-ℕ : + (x y : ℕ) → le-ℕ x y → x ≤-ℕ y + leq-le-ℕ zero-ℕ (succ-ℕ y) H = star + leq-le-ℕ (succ-ℕ x) (succ-ℕ y) H = leq-le-ℕ x y H ``` ### If `x < y + 1` then `x ≤ y` ```agda -leq-le-succ-ℕ : - (x y : ℕ) → le-ℕ x (succ-ℕ y) → x ≤-ℕ y -leq-le-succ-ℕ zero-ℕ y H = star -leq-le-succ-ℕ (succ-ℕ x) (succ-ℕ y) H = leq-le-succ-ℕ x y H +abstract + leq-le-succ-ℕ : + (x y : ℕ) → le-ℕ x (succ-ℕ y) → x ≤-ℕ y + leq-le-succ-ℕ zero-ℕ y H = star + leq-le-succ-ℕ (succ-ℕ x) (succ-ℕ y) H = leq-le-succ-ℕ x y H ``` ### If `x < y` then `x + 1 ≤ y` ```agda -leq-succ-le-ℕ : - (x y : ℕ) → le-ℕ x y → leq-ℕ (succ-ℕ x) y -leq-succ-le-ℕ zero-ℕ (succ-ℕ y) H = star -leq-succ-le-ℕ (succ-ℕ x) (succ-ℕ y) H = leq-succ-le-ℕ x y H +abstract + leq-succ-le-ℕ : + (x y : ℕ) → le-ℕ x y → leq-ℕ (succ-ℕ x) y + leq-succ-le-ℕ zero-ℕ (succ-ℕ y) H = star + leq-succ-le-ℕ (succ-ℕ x) (succ-ℕ y) H = leq-succ-le-ℕ x y H ``` ### If `x + 1 ≤ y` then `x < y` ```agda -le-leq-succ-ℕ : - (x y : ℕ) → leq-ℕ (succ-ℕ x) y → le-ℕ x y -le-leq-succ-ℕ zero-ℕ (succ-ℕ y) H = star -le-leq-succ-ℕ (succ-ℕ x) (succ-ℕ y) H = le-leq-succ-ℕ x y H +abstract + le-leq-succ-ℕ : + (x y : ℕ) → leq-ℕ (succ-ℕ x) y → le-ℕ x y + le-leq-succ-ℕ zero-ℕ (succ-ℕ y) H = star + le-leq-succ-ℕ (succ-ℕ x) (succ-ℕ y) H = le-leq-succ-ℕ x y H ``` ### If `x ≤ y` then `x < y + 1` ```agda -le-succ-leq-ℕ : - (x y : ℕ) → leq-ℕ x y → le-ℕ x (succ-ℕ y) -le-succ-leq-ℕ zero-ℕ y H = star -le-succ-leq-ℕ (succ-ℕ x) (succ-ℕ y) H = le-succ-leq-ℕ x y H +abstract + le-succ-leq-ℕ : + (x y : ℕ) → leq-ℕ x y → le-ℕ x (succ-ℕ y) + le-succ-leq-ℕ zero-ℕ y H = star + le-succ-leq-ℕ (succ-ℕ x) (succ-ℕ y) H = le-succ-leq-ℕ x y H ``` ### `x ≤ y` if and only if `(x ＝ y) + (x < y)` @@ -350,20 +378,20 @@ eq-or-le-leq-ℕ' : (x y : ℕ) → leq-ℕ x y → ((y ＝ x) + (le-ℕ x y)) eq-or-le-leq-ℕ' x y H = map-coproduct inv id (eq-or-le-leq-ℕ x y H) -leq-eq-or-le-ℕ : - (x y : ℕ) → ((x ＝ y) + (le-ℕ x y)) → leq-ℕ x y -leq-eq-or-le-ℕ x .x (inl refl) = refl-leq-ℕ x -leq-eq-or-le-ℕ x y (inr l) = leq-le-ℕ x y l +abstract + leq-eq-or-le-ℕ : + (x y : ℕ) → ((x ＝ y) + (le-ℕ x y)) → leq-ℕ x y + leq-eq-or-le-ℕ x .x (inl refl) = refl-leq-ℕ x + leq-eq-or-le-ℕ x y (inr l) = leq-le-ℕ x y l ``` ### If `x ≤ y` and `x ≠ y` then `x < y` ```agda -le-leq-neq-ℕ : {x y : ℕ} → x ≤-ℕ y → x ≠ y → le-ℕ x y -le-leq-neq-ℕ {zero-ℕ} {zero-ℕ} l f = ex-falso (f refl) -le-leq-neq-ℕ {zero-ℕ} {succ-ℕ y} l f = star -le-leq-neq-ℕ {succ-ℕ x} {succ-ℕ y} l f = - le-leq-neq-ℕ {x} {y} l (λ p → f (ap succ-ℕ p)) +abstract + le-leq-neq-ℕ : {x y : ℕ} → x ≤-ℕ y → x ≠ y → le-ℕ x y + le-leq-neq-ℕ {x} {y} x≤y x≠y = + rec-coproduct (ex-falso ∘ x≠y) id (eq-or-le-leq-ℕ x y x≤y) ``` ### For any two natural numbers `x` and `y`, either `x < y` or `y ≤ x` @@ -378,8 +406,9 @@ decide-le-leq-ℕ (succ-ℕ x) (succ-ℕ y) = decide-le-leq-ℕ x y ### If `1 < x` and `1 < y`, then `1 < xy` ```agda -le-one-mul-ℕ : (x y : ℕ) → 1 <-ℕ x → 1 <-ℕ y → 1 <-ℕ (x *ℕ y) -le-one-mul-ℕ (succ-ℕ (succ-ℕ x)) (succ-ℕ (succ-ℕ y)) star star = star +abstract + le-one-mul-ℕ : (x y : ℕ) → 1 <-ℕ x → 1 <-ℕ y → 1 <-ℕ (x *ℕ y) + le-one-mul-ℕ (succ-ℕ (succ-ℕ x)) (succ-ℕ (succ-ℕ y)) star star = star ``` ### Strict inequality on the natural numbers is invariant by translation @@ -398,27 +427,28 @@ eq-translate-left-le-ℕ z x y = ### Addition on the natural numbers preserves strict inequality ```agda -preserves-le-right-add-ℕ : (z x y : ℕ) → le-ℕ x y → le-ℕ (x +ℕ z) (y +ℕ z) -preserves-le-right-add-ℕ zero-ℕ x y I = I -preserves-le-right-add-ℕ (succ-ℕ z) x y I = preserves-le-right-add-ℕ z x y I +abstract + preserves-le-right-add-ℕ : (z x y : ℕ) → le-ℕ x y → le-ℕ (x +ℕ z) (y +ℕ z) + preserves-le-right-add-ℕ zero-ℕ x y I = I + preserves-le-right-add-ℕ (succ-ℕ z) x y I = preserves-le-right-add-ℕ z x y I -preserves-le-left-add-ℕ : (z x y : ℕ) → le-ℕ x y → le-ℕ (z +ℕ x) (z +ℕ y) -preserves-le-left-add-ℕ z x y I = - binary-tr - le-ℕ - (commutative-add-ℕ x z) - (commutative-add-ℕ y z) - (preserves-le-right-add-ℕ z x y I) + preserves-le-left-add-ℕ : (z x y : ℕ) → le-ℕ x y → le-ℕ (z +ℕ x) (z +ℕ y) + preserves-le-left-add-ℕ z x y I = + binary-tr + ( le-ℕ) + ( commutative-add-ℕ x z) + ( commutative-add-ℕ y z) + ( preserves-le-right-add-ℕ z x y I) -preserves-le-add-ℕ : - {a b c d : ℕ} → le-ℕ a b → le-ℕ c d → le-ℕ (a +ℕ c) (b +ℕ d) -preserves-le-add-ℕ {a} {b} {c} {d} H K = - transitive-le-ℕ - (a +ℕ c) - (b +ℕ c) - (b +ℕ d) - (preserves-le-right-add-ℕ c a b H) - (preserves-le-left-add-ℕ b c d K) + preserves-le-add-ℕ : + {a b c d : ℕ} → le-ℕ a b → le-ℕ c d → le-ℕ (a +ℕ c) (b +ℕ d) + preserves-le-add-ℕ {a} {b} {c} {d} H K = + transitive-le-ℕ + ( a +ℕ c) + ( b +ℕ c) + ( b +ℕ d) + ( preserves-le-right-add-ℕ c a b H) + ( preserves-le-left-add-ℕ b c d K) ``` ### There is an equivalence between natural numbers less than `succ-ℕ n` and natural numbers less than or equal to `n` @@ -426,10 +456,9 @@ preserves-le-add-ℕ {a} {b} {c} {d} H K = ```agda equiv-le-succ-ℕ-leq-ℕ : (n : ℕ) → Σ ℕ (λ k → le-ℕ k (succ-ℕ n)) ≃ Σ ℕ (λ k → leq-ℕ k n) -pr1 (equiv-le-succ-ℕ-leq-ℕ n) (k , k +## Idea + +Strict inequality on the +[standard finite types](univalent-combinatorics.standard-finite-types.md) is +defined inductively with the rule that `inl-Fin k < neg-one-Fin n`, and +`inl-Fin a < inl-Fin b` if `a < b`. + ## Definitions ### The strict inequality relation on the standard finite types ```agda -le-Fin-Prop : (k : ℕ) → Fin k → Fin k → Prop lzero -le-Fin-Prop (succ-ℕ k) (inl x) (inl y) = le-Fin-Prop k x y -le-Fin-Prop (succ-ℕ k) (inl x) (inr star) = unit-Prop -le-Fin-Prop (succ-ℕ k) (inr star) y = empty-Prop +le-prop-Fin : (k : ℕ) → Fin k → Fin k → Prop lzero +le-prop-Fin (succ-ℕ k) (inl x) (inl y) = le-prop-Fin k x y +le-prop-Fin (succ-ℕ k) (inl x) (inr star) = unit-Prop +le-prop-Fin (succ-ℕ k) (inr star) y = empty-Prop le-Fin : (k : ℕ) → Fin k → Fin k → UU lzero -le-Fin k x y = type-Prop (le-Fin-Prop k x y) +le-Fin k x y = type-Prop (le-prop-Fin k x y) -is-prop-le-Fin : - (k : ℕ) (x y : Fin k) → is-prop (le-Fin k x y) -is-prop-le-Fin k x y = is-prop-type-Prop (le-Fin-Prop k x y) +abstract + is-prop-le-Fin : + (k : ℕ) (x y : Fin k) → is-prop (le-Fin k x y) + is-prop-le-Fin k x y = is-prop-type-Prop (le-prop-Fin k x y) ``` ### The predicate on maps between standard finite types of preserving strict inequality @@ -52,14 +60,15 @@ preserves-le-Fin n m f = le-Fin n a b → le-Fin m (f a) (f b) -is-prop-preserves-le-Fin : - (n m : ℕ) (f : Fin n → Fin m) → - is-prop (preserves-le-Fin n m f) -is-prop-preserves-le-Fin n m f = - is-prop-Π λ a → - is-prop-Π λ b → - is-prop-Π λ p → - is-prop-le-Fin m (f a) (f b) +abstract + is-prop-preserves-le-Fin : + (n m : ℕ) (f : Fin n → Fin m) → + is-prop (preserves-le-Fin n m f) + is-prop-preserves-le-Fin n m f = + is-prop-Π λ a → + is-prop-Π λ b → + is-prop-Π λ p → + is-prop-le-Fin m (f a) (f b) ``` ### A map `Fin (succ-ℕ m) → Fin (succ-ℕ n)` preserving strict inequality restricts to a map `Fin m → Fin n` @@ -89,95 +98,100 @@ restriction-preserves-le-Fin m n f pf x = #### The induced map is indeed a restriction ```agda -inl-restriction-preserves-le-Fin' : - (m n : ℕ) (f : Fin (succ-ℕ m) → Fin (succ-ℕ n)) → - (pf : preserves-le-Fin (succ-ℕ m) (succ-ℕ n) f) → - (x : Fin m) → - (rx : Fin (succ-ℕ n)) → - (px : f (inl x) ＝ rx) → - inl-Fin n (restriction-preserves-le-Fin' m n f pf x rx px) ＝ f (inl-Fin m x) -inl-restriction-preserves-le-Fin' (succ-ℕ m) n f pf x (inl a) px = inv px -inl-restriction-preserves-le-Fin' (succ-ℕ m) n f pf x (inr a) px = - ex-falso - ( tr (λ - → le-Fin (succ-ℕ n) - (f (inr star))) px - ( pf (inl x) (inr star) star)) - -inl-restriction-preserves-le-Fin : - (m n : ℕ) (f : Fin (succ-ℕ m) → Fin (succ-ℕ n)) → - (pf : preserves-le-Fin (succ-ℕ m) (succ-ℕ n) f) → - (x : Fin m) → - inl-Fin n (restriction-preserves-le-Fin m n f pf x) ＝ f (inl-Fin m x) -inl-restriction-preserves-le-Fin m n f pf x = - inl-restriction-preserves-le-Fin' m n f pf x (f (inl x)) refl +abstract + inl-restriction-preserves-le-Fin' : + (m n : ℕ) (f : Fin (succ-ℕ m) → Fin (succ-ℕ n)) → + (pf : preserves-le-Fin (succ-ℕ m) (succ-ℕ n) f) → + (x : Fin m) → + (rx : Fin (succ-ℕ n)) → + (px : f (inl x) ＝ rx) → + inl-Fin n (restriction-preserves-le-Fin' m n f pf x rx px) ＝ + f (inl-Fin m x) + inl-restriction-preserves-le-Fin' (succ-ℕ m) n f pf x (inl a) px = inv px + inl-restriction-preserves-le-Fin' (succ-ℕ m) n f pf x (inr a) px = + ex-falso + ( tr (λ - → le-Fin (succ-ℕ n) - (f (inr star))) px + ( pf (inl x) (inr star) star)) + + inl-restriction-preserves-le-Fin : + (m n : ℕ) (f : Fin (succ-ℕ m) → Fin (succ-ℕ n)) → + (pf : preserves-le-Fin (succ-ℕ m) (succ-ℕ n) f) → + (x : Fin m) → + inl-Fin n (restriction-preserves-le-Fin m n f pf x) ＝ f (inl-Fin m x) + inl-restriction-preserves-le-Fin m n f pf x = + inl-restriction-preserves-le-Fin' m n f pf x (f (inl x)) refl ``` #### The induced map preserves strict inequality ```agda -preserves-le-restriction-preserves-le-Fin' : - (m n : ℕ) (f : Fin (succ-ℕ m) → Fin (succ-ℕ n)) → - (pf : preserves-le-Fin (succ-ℕ m) (succ-ℕ n) f) → - ( a : Fin m) - ( b : Fin m) → - ( ra : Fin (succ-ℕ n)) → - ( pa : f (inl a) ＝ ra) → - ( rb : Fin (succ-ℕ n)) → - ( pb : f (inl b) ＝ rb) → - le-Fin m a b → - le-Fin n - (restriction-preserves-le-Fin' m n f pf a ra pa) - (restriction-preserves-le-Fin' m n f pf b rb pb) -preserves-le-restriction-preserves-le-Fin' - (succ-ℕ m) n f pf a b (inl x) pa (inl y) pb H = - tr (le-Fin (succ-ℕ n) (inl x)) pb - ( tr (λ - → le-Fin (succ-ℕ n) - (f (inl b))) pa - ( pf (inl a) (inl b) H)) -preserves-le-restriction-preserves-le-Fin' - (succ-ℕ m) n f pf a b (inl x) pa (inr y) pb H = - ex-falso - ( tr (λ - → le-Fin (succ-ℕ n) - (f (inr star))) pb - ( pf (inl b) (inr star) star)) -preserves-le-restriction-preserves-le-Fin' - (succ-ℕ m) n f pf a b (inr x) pa y pb H = - ex-falso - ( tr (λ - → le-Fin (succ-ℕ n) - (f (inr star))) pa - ( pf (inl a) (inr star) star)) - -preserves-le-restriction-preserves-le-Fin : - (m n : ℕ) (f : Fin (succ-ℕ m) → Fin (succ-ℕ n)) → - (pf : preserves-le-Fin (succ-ℕ m) (succ-ℕ n) f) → - preserves-le-Fin m n (restriction-preserves-le-Fin m n f pf) -preserves-le-restriction-preserves-le-Fin m n f pf a b H = - preserves-le-restriction-preserves-le-Fin' m n f pf a b - ( f (inl a)) refl (f (inl b)) refl H +abstract + preserves-le-restriction-preserves-le-Fin' : + (m n : ℕ) (f : Fin (succ-ℕ m) → Fin (succ-ℕ n)) → + (pf : preserves-le-Fin (succ-ℕ m) (succ-ℕ n) f) → + ( a : Fin m) + ( b : Fin m) → + ( ra : Fin (succ-ℕ n)) → + ( pa : f (inl a) ＝ ra) → + ( rb : Fin (succ-ℕ n)) → + ( pb : f (inl b) ＝ rb) → + le-Fin m a b → + le-Fin n + (restriction-preserves-le-Fin' m n f pf a ra pa) + (restriction-preserves-le-Fin' m n f pf b rb pb) + preserves-le-restriction-preserves-le-Fin' + (succ-ℕ m) n f pf a b (inl x) pa (inl y) pb H = + tr (le-Fin (succ-ℕ n) (inl x)) pb + ( tr (λ - → le-Fin (succ-ℕ n) - (f (inl b))) pa + ( pf (inl a) (inl b) H)) + preserves-le-restriction-preserves-le-Fin' + (succ-ℕ m) n f pf a b (inl x) pa (inr y) pb H = + ex-falso + ( tr (λ - → le-Fin (succ-ℕ n) - (f (inr star))) pb + ( pf (inl b) (inr star) star)) + preserves-le-restriction-preserves-le-Fin' + (succ-ℕ m) n f pf a b (inr x) pa y pb H = + ex-falso + ( tr (λ - → le-Fin (succ-ℕ n) - (f (inr star))) pa + ( pf (inl a) (inr star) star)) + + preserves-le-restriction-preserves-le-Fin : + (m n : ℕ) (f : Fin (succ-ℕ m) → Fin (succ-ℕ n)) → + (pf : preserves-le-Fin (succ-ℕ m) (succ-ℕ n) f) → + preserves-le-Fin m n (restriction-preserves-le-Fin m n f pf) + preserves-le-restriction-preserves-le-Fin m n f pf a b H = + preserves-le-restriction-preserves-le-Fin' m n f pf a b + ( f (inl a)) refl (f (inl b)) refl H ``` ### A strict inequality preserving map implies an inequality of cardinalities ```agda -leq-preserves-le-Fin : - (m n : ℕ) → (f : Fin m → Fin n) → - preserves-le-Fin m n f → leq-ℕ m n -leq-preserves-le-Fin 0 0 f pf = star -leq-preserves-le-Fin 0 (succ-ℕ n) f pf = star -leq-preserves-le-Fin (succ-ℕ m) 0 f pf = f (inr star) -leq-preserves-le-Fin (succ-ℕ 0) (succ-ℕ n) f pf = star -leq-preserves-le-Fin (succ-ℕ (succ-ℕ m)) (succ-ℕ n) f pf = - leq-preserves-le-Fin (succ-ℕ m) n - ( restriction-preserves-le-Fin (succ-ℕ m) n f pf) - ( preserves-le-restriction-preserves-le-Fin (succ-ℕ m) n f pf) +abstract + leq-preserves-le-Fin : + (m n : ℕ) → (f : Fin m → Fin n) → + preserves-le-Fin m n f → leq-ℕ m n + leq-preserves-le-Fin 0 0 f pf = star + leq-preserves-le-Fin 0 (succ-ℕ n) f pf = star + leq-preserves-le-Fin (succ-ℕ m) 0 f pf = f (inr star) + leq-preserves-le-Fin (succ-ℕ 0) (succ-ℕ n) f pf = star + leq-preserves-le-Fin (succ-ℕ (succ-ℕ m)) (succ-ℕ n) f pf = + leq-preserves-le-Fin (succ-ℕ m) n + ( restriction-preserves-le-Fin (succ-ℕ m) n f pf) + ( preserves-le-restriction-preserves-le-Fin (succ-ℕ m) n f pf) ``` ### Composition of strict inequality preserving maps ```agda -comp-preserves-le-Fin : - (m n o : ℕ) - (g : Fin n → Fin o) - (f : Fin m → Fin n) → - preserves-le-Fin m n f → - preserves-le-Fin n o g → - preserves-le-Fin m o (g ∘ f) -comp-preserves-le-Fin m n o g f pf pg a b H = - pg (f a) (f b) (pf a b H) +abstract + comp-preserves-le-Fin : + (m n o : ℕ) + (g : Fin n → Fin o) + (f : Fin m → Fin n) → + preserves-le-Fin m n f → + preserves-le-Fin n o g → + preserves-le-Fin m o (g ∘ f) + comp-preserves-le-Fin m n o g f pf pg a b H = + pg (f a) (f b) (pf a b H) ``` diff --git a/src/elementary-number-theory/strong-induction-natural-numbers.lagda.md b/src/elementary-number-theory/strong-induction-natural-numbers.lagda.md index 6a0bf777c5..3ceb4f55f3 100644 --- a/src/elementary-number-theory/strong-induction-natural-numbers.lagda.md +++ b/src/elementary-number-theory/strong-induction-natural-numbers.lagda.md @@ -27,8 +27,12 @@ open import foundation.universe-levels ## Idea -The strong induction principle allows one to assume in the inductive step that -the inductive hypothesis is satisfied at all smaller values. +The +{{#concept "strong induction principle" WDID=Q54506667 WD="principle of complete induction" Agda=strong-ind-ℕ}} +on the [natural numbers](elementary-number-theory.natural-numbers.md) allows one +to assume in the inductive step that the inductive hypothesis is satisfied at +all +[smaller values](elementary-number-theory.strict-inequality-natural-numbers.md). ## Definition diff --git a/src/elementary-number-theory/triangular-numbers.lagda.md b/src/elementary-number-theory/triangular-numbers.lagda.md index 474f4dc44d..f98fcd8a70 100644 --- a/src/elementary-number-theory/triangular-numbers.lagda.md +++ b/src/elementary-number-theory/triangular-numbers.lagda.md @@ -8,10 +8,10 @@ module elementary-number-theory.triangular-numbers where ```agda open import elementary-number-theory.addition-natural-numbers -open import elementary-number-theory.commutative-semiring-of-natural-numbers open import elementary-number-theory.divisibility-natural-numbers open import elementary-number-theory.multiplication-natural-numbers open import elementary-number-theory.natural-numbers +open import elementary-number-theory.semiring-of-natural-numbers open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types diff --git a/src/elementary-number-theory/unit-fractions-rational-numbers.lagda.md b/src/elementary-number-theory/unit-fractions-rational-numbers.lagda.md index a95f01a51c..984edbabd5 100644 --- a/src/elementary-number-theory/unit-fractions-rational-numbers.lagda.md +++ b/src/elementary-number-theory/unit-fractions-rational-numbers.lagda.md @@ -151,7 +151,7 @@ opaque ( le-ℤ) ( left-unit-law-mul-ℤ (int-ℕ m)) ( left-unit-law-mul-ℤ (int-ℕ n)) - ( le-natural-le-ℤ m n mImports ```agda -open import elementary-number-theory.commutative-semiring-of-natural-numbers open import elementary-number-theory.natural-numbers +open import elementary-number-theory.semiring-of-natural-numbers open import elementary-number-theory.triangular-numbers open import foundation.action-on-identifications-binary-functions diff --git a/src/univalent-combinatorics/classical-finite-types.lagda.md b/src/univalent-combinatorics/classical-finite-types.lagda.md index 8732b061aa..acf531c692 100644 --- a/src/univalent-combinatorics/classical-finite-types.lagda.md +++ b/src/univalent-combinatorics/classical-finite-types.lagda.md @@ -165,7 +165,7 @@ is-section-classical-standard-Fin-reverse (succ-ℕ n) (zero-ℕ , k Date: Sat, 4 Oct 2025 22:37:38 -0700 Subject: [PATCH 02/12] Unabstract divisibility --- .../divisibility-integers.lagda.md | 192 +++++++++--------- .../divisibility-natural-numbers.lagda.md | 88 ++++---- 2 files changed, 135 insertions(+), 145 deletions(-) diff --git a/src/elementary-number-theory/divisibility-integers.lagda.md b/src/elementary-number-theory/divisibility-integers.lagda.md index 2b38d5c502..a3bbce1005 100644 --- a/src/elementary-number-theory/divisibility-integers.lagda.md +++ b/src/elementary-number-theory/divisibility-integers.lagda.md @@ -157,17 +157,16 @@ abstract Note that the divisibility relation on the integers is not antisymmetric. ```agda -abstract - refl-div-ℤ : is-reflexive div-ℤ - pr1 (refl-div-ℤ x) = one-ℤ - pr2 (refl-div-ℤ x) = left-unit-law-mul-ℤ x - - transitive-div-ℤ : is-transitive div-ℤ - pr1 (transitive-div-ℤ x y z (pair e q) (pair d p)) = e *ℤ d - pr2 (transitive-div-ℤ x y z (pair e q) (pair d p)) = - ( associative-mul-ℤ e d x) ∙ - ( ( ap (e *ℤ_) p) ∙ - ( q)) +refl-div-ℤ : is-reflexive div-ℤ +pr1 (refl-div-ℤ x) = one-ℤ +pr2 (refl-div-ℤ x) = left-unit-law-mul-ℤ x + +transitive-div-ℤ : is-transitive div-ℤ +pr1 (transitive-div-ℤ x y z (pair e q) (pair d p)) = e *ℤ d +pr2 (transitive-div-ℤ x y z (pair e q) (pair d p)) = + ( associative-mul-ℤ e d x) ∙ + ( ( ap (e *ℤ_) p) ∙ + ( q)) ``` ### Every integer is divisible by `1` @@ -222,63 +221,58 @@ abstract ### If `x` divides both `y` and `z`, then it divides `y + z` ```agda -abstract - div-add-ℤ : (x y z : ℤ) → div-ℤ x y → div-ℤ x z → div-ℤ x (y +ℤ z) - pr1 (div-add-ℤ x y z (pair d p) (pair e q)) = d +ℤ e - pr2 (div-add-ℤ x y z (pair d p) (pair e q)) = - ( right-distributive-mul-add-ℤ d e x) ∙ - ( ap-add-ℤ p q) +div-add-ℤ : (x y z : ℤ) → div-ℤ x y → div-ℤ x z → div-ℤ x (y +ℤ z) +pr1 (div-add-ℤ x y z (pair d p) (pair e q)) = d +ℤ e +pr2 (div-add-ℤ x y z (pair d p) (pair e q)) = + ( right-distributive-mul-add-ℤ d e x) ∙ + ( ap-add-ℤ p q) ``` ### If `x` divides `y` then `x` divides any multiple of `y` ```agda -abstract - div-mul-ℤ : - (k x y : ℤ) → div-ℤ x y → div-ℤ x (k *ℤ y) - div-mul-ℤ k x y = transitive-div-ℤ x y (k *ℤ y) (k , refl) +div-mul-ℤ : + (k x y : ℤ) → div-ℤ x y → div-ℤ x (k *ℤ y) +div-mul-ℤ k x y = transitive-div-ℤ x y (k *ℤ y) (k , refl) ``` ### If `x` divides `y` then it divides `-y` ```agda -abstract - div-neg-ℤ : (x y : ℤ) → div-ℤ x y → div-ℤ x (neg-ℤ y) - pr1 (div-neg-ℤ x y (pair d p)) = neg-ℤ d - pr2 (div-neg-ℤ x y (pair d p)) = left-negative-law-mul-ℤ d x ∙ ap neg-ℤ p +div-neg-ℤ : (x y : ℤ) → div-ℤ x y → div-ℤ x (neg-ℤ y) +pr1 (div-neg-ℤ x y (pair d p)) = neg-ℤ d +pr2 (div-neg-ℤ x y (pair d p)) = left-negative-law-mul-ℤ d x ∙ ap neg-ℤ p ``` ### If `x` divides `y` then `-x` divides `y` ```agda -abstract - neg-div-ℤ : (x y : ℤ) → div-ℤ x y → div-ℤ (neg-ℤ x) y - pr1 (neg-div-ℤ x y (pair d p)) = neg-ℤ d - pr2 (neg-div-ℤ x y (pair d p)) = - equational-reasoning - (neg-ℤ d) *ℤ (neg-ℤ x) - ＝ neg-ℤ (d *ℤ (neg-ℤ x)) - by left-negative-law-mul-ℤ d (neg-ℤ x) - ＝ neg-ℤ (neg-ℤ (d *ℤ x)) - by ap neg-ℤ (right-negative-law-mul-ℤ d x) - ＝ (d *ℤ x) - by neg-neg-ℤ (d *ℤ x) - ＝ y - by p +neg-div-ℤ : (x y : ℤ) → div-ℤ x y → div-ℤ (neg-ℤ x) y +pr1 (neg-div-ℤ x y (pair d p)) = neg-ℤ d +pr2 (neg-div-ℤ x y (pair d p)) = + equational-reasoning + (neg-ℤ d) *ℤ (neg-ℤ x) + ＝ neg-ℤ (d *ℤ (neg-ℤ x)) + by left-negative-law-mul-ℤ d (neg-ℤ x) + ＝ neg-ℤ (neg-ℤ (d *ℤ x)) + by ap neg-ℤ (right-negative-law-mul-ℤ d x) + ＝ (d *ℤ x) + by neg-neg-ℤ (d *ℤ x) + ＝ y + by p ``` ### Multiplication preserves divisibility ```agda -abstract - preserves-div-mul-ℤ : - (k x y : ℤ) → div-ℤ x y → div-ℤ (k *ℤ x) (k *ℤ y) - pr1 (preserves-div-mul-ℤ k x y (pair q p)) = q - pr2 (preserves-div-mul-ℤ k x y (pair q p)) = - ( inv (associative-mul-ℤ q k x)) ∙ - ( ( ap (_*ℤ x) (commutative-mul-ℤ q k)) ∙ - ( ( associative-mul-ℤ k q x) ∙ - ( ap (k *ℤ_) p))) +preserves-div-mul-ℤ : + (k x y : ℤ) → div-ℤ x y → div-ℤ (k *ℤ x) (k *ℤ y) +pr1 (preserves-div-mul-ℤ k x y (pair q p)) = q +pr2 (preserves-div-mul-ℤ k x y (pair q p)) = + ( inv (associative-mul-ℤ q k x)) ∙ + ( ap (_*ℤ x) (commutative-mul-ℤ q k)) ∙ + ( associative-mul-ℤ k q x) ∙ + ( ap (k *ℤ_) p) ``` ### Multiplication by a nonzero number reflects divisibility @@ -291,63 +285,61 @@ abstract pr2 (reflects-div-mul-ℤ k x y H (pair q p)) = is-injective-left-mul-ℤ k H ( ( inv (associative-mul-ℤ k q x)) ∙ - ( ( ap (_*ℤ x) (commutative-mul-ℤ k q)) ∙ - ( ( associative-mul-ℤ q k x) ∙ - ( p)))) + ( ap (_*ℤ x) (commutative-mul-ℤ k q)) ∙ + ( associative-mul-ℤ q k x) ∙ + ( p)) ``` ### If a nonzero number `d` divides `y`, then `dx` divides `y` if and only if `x` divides the quotient `y/d` ```agda -abstract - div-quotient-div-div-ℤ : - (x y d : ℤ) (H : div-ℤ d y) → is-nonzero-ℤ d → - div-ℤ (d *ℤ x) y → div-ℤ x (quotient-div-ℤ d y H) - div-quotient-div-div-ℤ x y d H f K = - reflects-div-mul-ℤ d x - ( quotient-div-ℤ d y H) - ( f) - ( tr (div-ℤ (d *ℤ x)) (inv (eq-quotient-div-ℤ' d y H)) K) - - div-div-quotient-div-ℤ : - (x y d : ℤ) (H : div-ℤ d y) → - div-ℤ x (quotient-div-ℤ d y H) → div-ℤ (d *ℤ x) y - div-div-quotient-div-ℤ x y d H K = - tr - ( div-ℤ (d *ℤ x)) - ( eq-quotient-div-ℤ' d y H) - ( preserves-div-mul-ℤ d x (quotient-div-ℤ d y H) K) -``` - -### Comparison of divisibility on `ℕ` and on `ℤ` - -```agda -abstract - div-int-div-ℕ : - {x y : ℕ} → div-ℕ x y → div-ℤ (int-ℕ x) (int-ℕ y) - pr1 (div-int-div-ℕ {x} {y} (pair d p)) = int-ℕ d - pr2 (div-int-div-ℕ {x} {y} (pair d p)) = mul-int-ℕ d x ∙ ap int-ℕ p - - div-div-int-ℕ : - {x y : ℕ} → div-ℤ (int-ℕ x) (int-ℕ y) → div-ℕ x y - div-div-int-ℕ {zero-ℕ} {y} (pair d p) = - div-eq-ℕ zero-ℕ y - ( inv (is-injective-int-ℕ (is-zero-div-zero-ℤ (int-ℕ y) (pair d p)))) - pr1 (div-div-int-ℕ {succ-ℕ x} {y} (pair d p)) = abs-ℤ d - pr2 (div-div-int-ℕ {succ-ℕ x} {y} (pair d p)) = - is-injective-int-ℕ - ( ( inv (mul-int-ℕ (abs-ℤ d) (succ-ℕ x))) ∙ - ( ( ap - ( _*ℤ (inr (inr x))) - { int-abs-ℤ d} - { d} - ( int-abs-is-nonnegative-ℤ d - ( is-nonnegative-left-factor-mul-ℤ - { d} - { inr (inr x)} - ( is-nonnegative-eq-ℤ (inv p) (is-nonnegative-int-ℕ y)) - ( star)))) ∙ - ( p))) +div-quotient-div-div-ℤ : + (x y d : ℤ) (H : div-ℤ d y) → is-nonzero-ℤ d → + div-ℤ (d *ℤ x) y → div-ℤ x (quotient-div-ℤ d y H) +div-quotient-div-div-ℤ x y d H f K = + reflects-div-mul-ℤ d x + ( quotient-div-ℤ d y H) + ( f) + ( tr (div-ℤ (d *ℤ x)) (inv (eq-quotient-div-ℤ' d y H)) K) + +div-div-quotient-div-ℤ : + (x y d : ℤ) (H : div-ℤ d y) → + div-ℤ x (quotient-div-ℤ d y H) → div-ℤ (d *ℤ x) y +div-div-quotient-div-ℤ x y d H K = + tr + ( div-ℤ (d *ℤ x)) + ( eq-quotient-div-ℤ' d y H) + ( preserves-div-mul-ℤ d x (quotient-div-ℤ d y H) K) +``` + +### The canonical embedding of natural numbers preserves and reflects divisibility + +```agda +div-int-div-ℕ : + {x y : ℕ} → div-ℕ x y → div-ℤ (int-ℕ x) (int-ℕ y) +pr1 (div-int-div-ℕ {x} {y} (pair d p)) = int-ℕ d +pr2 (div-int-div-ℕ {x} {y} (pair d p)) = mul-int-ℕ d x ∙ ap int-ℕ p + +div-div-int-ℕ : + {x y : ℕ} → div-ℤ (int-ℕ x) (int-ℕ y) → div-ℕ x y +div-div-int-ℕ {zero-ℕ} {y} (pair d p) = + div-eq-ℕ zero-ℕ y + ( inv (is-injective-int-ℕ (is-zero-div-zero-ℤ (int-ℕ y) (pair d p)))) +pr1 (div-div-int-ℕ {succ-ℕ x} {y} (pair d p)) = abs-ℤ d +pr2 (div-div-int-ℕ {succ-ℕ x} {y} (pair d p)) = + is-injective-int-ℕ + ( ( inv (mul-int-ℕ (abs-ℤ d) (succ-ℕ x))) ∙ + ( ( ap + ( _*ℤ (inr (inr x))) + { int-abs-ℤ d} + { d} + ( int-abs-is-nonnegative-ℤ d + ( is-nonnegative-left-factor-mul-ℤ + { d} + { inr (inr x)} + ( is-nonnegative-eq-ℤ (inv p) (is-nonnegative-int-ℕ y)) + ( star)))) ∙ + ( p))) ``` ### An integer is a unit if and only if it is `1` or `-1` diff --git a/src/elementary-number-theory/divisibility-natural-numbers.lagda.md b/src/elementary-number-theory/divisibility-natural-numbers.lagda.md index 8beca6d17a..5aa6db8692 100644 --- a/src/elementary-number-theory/divisibility-natural-numbers.lagda.md +++ b/src/elementary-number-theory/divisibility-natural-numbers.lagda.md @@ -281,57 +281,55 @@ abstract ### If `x` divides `y` then `x` divides any multiple of `y` ```agda -abstract - div-mul-ℕ : - (k x y : ℕ) → div-ℕ x y → div-ℕ x (k *ℕ y) - div-mul-ℕ k x y H = - transitive-div-ℕ x y (k *ℕ y) (pair k refl) H +div-mul-ℕ : + (k x y : ℕ) → div-ℕ x y → div-ℕ x (k *ℕ y) +div-mul-ℕ k x y H = + transitive-div-ℕ x y (k *ℕ y) (pair k refl) H - div-mul-ℕ' : - (k x y : ℕ) → div-ℕ x y → div-ℕ x (y *ℕ k) - div-mul-ℕ' k x y H = - tr (div-ℕ x) (commutative-mul-ℕ k y) (div-mul-ℕ k x y H) +div-mul-ℕ' : + (k x y : ℕ) → div-ℕ x y → div-ℕ x (y *ℕ k) +div-mul-ℕ' k x y H = + tr (div-ℕ x) (commutative-mul-ℕ k y) (div-mul-ℕ k x y H) ``` ### A 3-for-2 property of division with respect to addition ```agda -abstract - div-add-ℕ : - (d x y : ℕ) → div-ℕ d x → div-ℕ d y → div-ℕ d (x +ℕ y) - pr1 (div-add-ℕ d x y (pair n p) (pair m q)) = n +ℕ m - pr2 (div-add-ℕ d x y (pair n p) (pair m q)) = - ( right-distributive-mul-add-ℕ n m d) ∙ - ( ap-add-ℕ p q) - - div-left-summand-ℕ : - (d x y : ℕ) → div-ℕ d y → div-ℕ d (x +ℕ y) → div-ℕ d x - div-left-summand-ℕ zero-ℕ x y (pair m q) (pair n p) = - pair zero-ℕ - ( ( inv (right-zero-law-mul-ℕ n)) ∙ - ( p ∙ (ap (x +ℕ_) ((inv q) ∙ (right-zero-law-mul-ℕ m))))) - pr1 (div-left-summand-ℕ (succ-ℕ d) x y (pair m q) (pair n p)) = dist-ℕ m n - pr2 (div-left-summand-ℕ (succ-ℕ d) x y (pair m q) (pair n p)) = - is-injective-right-add-ℕ (m *ℕ (succ-ℕ d)) - ( ( inv - ( ( right-distributive-mul-add-ℕ m (dist-ℕ m n) (succ-ℕ d)) ∙ - ( commutative-add-ℕ - ( m *ℕ (succ-ℕ d)) - ( (dist-ℕ m n) *ℕ (succ-ℕ d))))) ∙ - ( ( ap - ( _*ℕ (succ-ℕ d)) - ( is-additive-right-inverse-dist-ℕ m n - ( reflects-order-mul-ℕ d m n - ( concatenate-eq-leq-eq-ℕ q - ( leq-add-ℕ' y x) - ( inv p))))) ∙ - ( p ∙ (ap (x +ℕ_) (inv q))))) - - div-right-summand-ℕ : - (d x y : ℕ) → div-ℕ d x → div-ℕ d (x +ℕ y) → div-ℕ d y - div-right-summand-ℕ d x y H1 H2 = - div-left-summand-ℕ d y x H1 - ( concatenate-div-eq-ℕ H2 (commutative-add-ℕ x y)) +div-add-ℕ : + (d x y : ℕ) → div-ℕ d x → div-ℕ d y → div-ℕ d (x +ℕ y) +pr1 (div-add-ℕ d x y (pair n p) (pair m q)) = n +ℕ m +pr2 (div-add-ℕ d x y (pair n p) (pair m q)) = + ( right-distributive-mul-add-ℕ n m d) ∙ + ( ap-add-ℕ p q) + +div-left-summand-ℕ : + (d x y : ℕ) → div-ℕ d y → div-ℕ d (x +ℕ y) → div-ℕ d x +div-left-summand-ℕ zero-ℕ x y (pair m q) (pair n p) = + pair zero-ℕ + ( ( inv (right-zero-law-mul-ℕ n)) ∙ + ( p ∙ (ap (x +ℕ_) ((inv q) ∙ (right-zero-law-mul-ℕ m))))) +pr1 (div-left-summand-ℕ (succ-ℕ d) x y (pair m q) (pair n p)) = dist-ℕ m n +pr2 (div-left-summand-ℕ (succ-ℕ d) x y (pair m q) (pair n p)) = + is-injective-right-add-ℕ (m *ℕ (succ-ℕ d)) + ( ( inv + ( ( right-distributive-mul-add-ℕ m (dist-ℕ m n) (succ-ℕ d)) ∙ + ( commutative-add-ℕ + ( m *ℕ (succ-ℕ d)) + ( (dist-ℕ m n) *ℕ (succ-ℕ d))))) ∙ + ( ( ap + ( _*ℕ (succ-ℕ d)) + ( is-additive-right-inverse-dist-ℕ m n + ( reflects-order-mul-ℕ d m n + ( concatenate-eq-leq-eq-ℕ q + ( leq-add-ℕ' y x) + ( inv p))))) ∙ + ( p ∙ (ap (x +ℕ_) (inv q))))) + +div-right-summand-ℕ : + (d x y : ℕ) → div-ℕ d x → div-ℕ d (x +ℕ y) → div-ℕ d y +div-right-summand-ℕ d x y H1 H2 = + div-left-summand-ℕ d y x H1 + ( concatenate-div-eq-ℕ H2 (commutative-add-ℕ x y)) ``` ### If `d` divides both `x` and `x + 1`, then `d ＝ 1` From 7cd6e075a6c074c617c213bcfc881264097e5700 Mon Sep 17 00:00:00 2001 From: Louis Wasserman Date: Sat, 4 Oct 2025 22:38:02 -0700 Subject: [PATCH 03/12] Unabstract divisibility --- src/elementary-number-theory/prime-numbers.lagda.md | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/src/elementary-number-theory/prime-numbers.lagda.md b/src/elementary-number-theory/prime-numbers.lagda.md index b2906e4cf5..5b753260c0 100644 --- a/src/elementary-number-theory/prime-numbers.lagda.md +++ b/src/elementary-number-theory/prime-numbers.lagda.md @@ -221,13 +221,13 @@ abstract is-one-is-proper-divisor-two-ℕ (succ-ℕ (succ-ℕ (succ-ℕ x))) (pair f H) = ex-falso (leq-div-succ-ℕ (succ-ℕ (succ-ℕ (succ-ℕ x))) 1 H) - is-prime-easy-two-ℕ : is-prime-easy-ℕ 2 - pr1 is-prime-easy-two-ℕ = Eq-eq-ℕ - pr2 is-prime-easy-two-ℕ = is-one-is-proper-divisor-two-ℕ +is-prime-easy-two-ℕ : is-prime-easy-ℕ 2 +pr1 is-prime-easy-two-ℕ = Eq-eq-ℕ +pr2 is-prime-easy-two-ℕ = is-one-is-proper-divisor-two-ℕ - is-prime-two-ℕ : is-prime-ℕ 2 - is-prime-two-ℕ = - is-prime-is-prime-easy-ℕ 2 is-prime-easy-two-ℕ +is-prime-two-ℕ : is-prime-ℕ 2 +is-prime-two-ℕ = + is-prime-is-prime-easy-ℕ 2 is-prime-easy-two-ℕ ``` ### If a prime number `p` divides a nonzero number `x`, then `x/p < x` From 9ac853658d9cbbf26b1b0e7cfadfc49284524532 Mon Sep 17 00:00:00 2001 From: Louis Wasserman Date: Sat, 4 Oct 2025 22:41:23 -0700 Subject: [PATCH 04/12] Unabstract deciding if a natural number is prime --- .../divisibility-natural-numbers.lagda.md | 2 +- .../inequality-natural-numbers.lagda.md | 16 +++---- .../prime-numbers.lagda.md | 45 +++++++++---------- ...roduction-to-homotopy-type-theory.lagda.md | 4 +- 4 files changed, 33 insertions(+), 34 deletions(-) diff --git a/src/elementary-number-theory/divisibility-natural-numbers.lagda.md b/src/elementary-number-theory/divisibility-natural-numbers.lagda.md index 5aa6db8692..65535f82f1 100644 --- a/src/elementary-number-theory/divisibility-natural-numbers.lagda.md +++ b/src/elementary-number-theory/divisibility-natural-numbers.lagda.md @@ -319,7 +319,7 @@ pr2 (div-left-summand-ℕ (succ-ℕ d) x y (pair m q) (pair n p)) = ( ( ap ( _*ℕ (succ-ℕ d)) ( is-additive-right-inverse-dist-ℕ m n - ( reflects-order-mul-ℕ d m n + ( reflects-leq-mul-ℕ d m n ( concatenate-eq-leq-eq-ℕ q ( leq-add-ℕ' y x) ( inv p))))) ∙ diff --git a/src/elementary-number-theory/inequality-natural-numbers.lagda.md b/src/elementary-number-theory/inequality-natural-numbers.lagda.md index 4a7adfcad5..c767ca2479 100644 --- a/src/elementary-number-theory/inequality-natural-numbers.lagda.md +++ b/src/elementary-number-theory/inequality-natural-numbers.lagda.md @@ -423,28 +423,28 @@ abstract ```agda abstract - reflects-order-mul-ℕ : + reflects-leq-mul-ℕ : (k m n : ℕ) → (m *ℕ (succ-ℕ k)) ≤-ℕ (n *ℕ (succ-ℕ k)) → m ≤-ℕ n - reflects-order-mul-ℕ k zero-ℕ n p = star - reflects-order-mul-ℕ k (succ-ℕ m) (succ-ℕ n) p = - reflects-order-mul-ℕ k m n + reflects-leq-mul-ℕ k zero-ℕ n p = star + reflects-leq-mul-ℕ k (succ-ℕ m) (succ-ℕ n) p = + reflects-leq-mul-ℕ k m n ( reflects-leq-left-add-ℕ ( succ-ℕ k) ( m *ℕ (succ-ℕ k)) ( n *ℕ (succ-ℕ k)) ( p)) - reflects-order-mul-ℕ' : + reflects-leq-mul-ℕ' : (k m n : ℕ) → ((succ-ℕ k) *ℕ m) ≤-ℕ ((succ-ℕ k) *ℕ n) → m ≤-ℕ n - reflects-order-mul-ℕ' k m n H = - reflects-order-mul-ℕ k m n + reflects-leq-mul-ℕ' k m n H = + reflects-leq-mul-ℕ k m n ( concatenate-eq-leq-eq-ℕ ( commutative-mul-ℕ m (succ-ℕ k)) ( H) ( commutative-mul-ℕ (succ-ℕ k) n)) ``` -### Any number `x` is less than or equal to a nonzero multiple of itself +### Any number `x` is less than or equal to any nonzero multiple of itself ```agda abstract diff --git a/src/elementary-number-theory/prime-numbers.lagda.md b/src/elementary-number-theory/prime-numbers.lagda.md index 5b753260c0..6f6b84ed1f 100644 --- a/src/elementary-number-theory/prime-numbers.lagda.md +++ b/src/elementary-number-theory/prime-numbers.lagda.md @@ -183,29 +183,28 @@ abstract ### Being a prime is decidable ```agda -abstract - is-decidable-is-prime-easy-ℕ : (n : ℕ) → is-decidable (is-prime-easy-ℕ n) - is-decidable-is-prime-easy-ℕ zero-ℕ = - inr - ( λ H → - is-not-one-two-ℕ (pr2 H 2 (is-proper-divisor-zero-succ-ℕ 1))) - is-decidable-is-prime-easy-ℕ (succ-ℕ n) = - is-decidable-product - ( is-decidable-neg (is-decidable-is-one-ℕ (succ-ℕ n))) - ( is-decidable-bounded-Π-ℕ - ( is-proper-divisor-ℕ (succ-ℕ n)) - ( is-one-ℕ) - ( is-decidable-is-proper-divisor-ℕ (succ-ℕ n)) - ( is-decidable-is-one-ℕ) - ( succ-ℕ n) - ( λ x H → leq-div-succ-ℕ x n (pr2 H))) - - is-decidable-is-prime-ℕ : (n : ℕ) → is-decidable (is-prime-ℕ n) - is-decidable-is-prime-ℕ n = - is-decidable-iff - ( is-prime-is-prime-easy-ℕ n) - ( is-prime-easy-is-prime-ℕ n) - ( is-decidable-is-prime-easy-ℕ n) +is-decidable-is-prime-easy-ℕ : (n : ℕ) → is-decidable (is-prime-easy-ℕ n) +is-decidable-is-prime-easy-ℕ zero-ℕ = + inr + ( λ H → + is-not-one-two-ℕ (pr2 H 2 (is-proper-divisor-zero-succ-ℕ 1))) +is-decidable-is-prime-easy-ℕ (succ-ℕ n) = + is-decidable-product + ( is-decidable-neg (is-decidable-is-one-ℕ (succ-ℕ n))) + ( is-decidable-bounded-Π-ℕ + ( is-proper-divisor-ℕ (succ-ℕ n)) + ( is-one-ℕ) + ( is-decidable-is-proper-divisor-ℕ (succ-ℕ n)) + ( is-decidable-is-one-ℕ) + ( succ-ℕ n) + ( λ x H → leq-div-succ-ℕ x n (pr2 H))) + +is-decidable-is-prime-ℕ : (n : ℕ) → is-decidable (is-prime-ℕ n) +is-decidable-is-prime-ℕ n = + is-decidable-iff + ( is-prime-is-prime-easy-ℕ n) + ( is-prime-easy-is-prime-ℕ n) + ( is-decidable-is-prime-easy-ℕ n) ``` ### The number `2` is a prime diff --git a/src/literature/introduction-to-homotopy-type-theory.lagda.md b/src/literature/introduction-to-homotopy-type-theory.lagda.md index 449a1c996f..71ba5d050c 100644 --- a/src/literature/introduction-to-homotopy-type-theory.lagda.md +++ b/src/literature/introduction-to-homotopy-type-theory.lagda.md @@ -971,12 +971,12 @@ _ = -- (d) open import elementary-number-theory.inequality-natural-numbers using ( preserves-leq-left-mul-ℕ - ; reflects-order-mul-ℕ) + ; reflects-leq-mul-ℕ) _ : (m n k : ℕ) → (m ≤-ℕ n) ↔ (m *ℕ (succ-ℕ k) ≤-ℕ n *ℕ (succ-ℕ k)) _ = λ m n k → - (preserves-leq-left-mul-ℕ (succ-ℕ k) m n , reflects-order-mul-ℕ k m n) + (preserves-leq-left-mul-ℕ (succ-ℕ k) m n , reflects-leq-mul-ℕ k m n) -- (e) open import elementary-number-theory.minimum-natural-numbers using From 53bd6e8927b003b08b917fa1ff77f2d3a23c46e1 Mon Sep 17 00:00:00 2001 From: Louis Wasserman Date: Sat, 4 Oct 2025 22:45:04 -0700 Subject: [PATCH 05/12] Fix link --- src/elementary-number-theory/ring-of-integers.lagda.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/elementary-number-theory/ring-of-integers.lagda.md b/src/elementary-number-theory/ring-of-integers.lagda.md index 9ce97dd158..7077abe51f 100644 --- a/src/elementary-number-theory/ring-of-integers.lagda.md +++ b/src/elementary-number-theory/ring-of-integers.lagda.md @@ -39,7 +39,7 @@ open import ring-theory.trivial-rings The type of [integers](elementary-number-theory.integers.md) equipped with [addition](elementary-number-theory.addition-integers.md) and [multiplication](elementary-number-theory.multiplication-integers.md) is a -[commutative ring](commutative-algebra.commutative-ring.md). +[commutative ring](commutative-algebra.commutative-rings.md). The {{#concept "ring of integers" Agda=ℤ-Ring}} is the **initial** [ring](ring-theory.rings.md): for any ring `R`, there's a unique From bb9677753883e847dca412f09393496b63c544a3 Mon Sep 17 00:00:00 2001 From: Louis Wasserman Date: Tue, 7 Oct 2025 07:59:15 -0700 Subject: [PATCH 06/12] Apply suggestions from code review Co-authored-by: Fredrik Bakke --- src/elementary-number-theory/equality-integers.lagda.md | 2 +- .../equality-natural-numbers.lagda.md | 2 +- .../exponentiation-natural-numbers.lagda.md | 4 ++-- 3 files changed, 4 insertions(+), 4 deletions(-) diff --git a/src/elementary-number-theory/equality-integers.lagda.md b/src/elementary-number-theory/equality-integers.lagda.md index 0b2ef3070d..a679a3e7dc 100644 --- a/src/elementary-number-theory/equality-integers.lagda.md +++ b/src/elementary-number-theory/equality-integers.lagda.md @@ -33,7 +33,7 @@ open import foundation.universe-levels ## Idea -We define equality on the [integers](elementary-number-theory.integers.md) by +We define {{#concept "equality" Disambiguation="of integers" Agda=Eq-ℤ}} on the [integers](elementary-number-theory.integers.md) by pattern matching, and show that this characterizes the [identity type](foundation.identity-types.md) on the integers. diff --git a/src/elementary-number-theory/equality-natural-numbers.lagda.md b/src/elementary-number-theory/equality-natural-numbers.lagda.md index 6b37e60205..16a6323959 100644 --- a/src/elementary-number-theory/equality-natural-numbers.lagda.md +++ b/src/elementary-number-theory/equality-natural-numbers.lagda.md @@ -35,7 +35,7 @@ open import foundation-core.torsorial-type-families ## Idea -We define equality on the +We define {{#concept "equality" Disambiguation="of natural numbers" Agda=Eq-ℕ}} on the [natural numbers](elementary-number-theory.natural-numbers.md) by pattern matching, and show that this characterizes the [identity type](foundation.identity-types.md) on the natural numbers. diff --git a/src/elementary-number-theory/exponentiation-natural-numbers.lagda.md b/src/elementary-number-theory/exponentiation-natural-numbers.lagda.md index 0e1b525344..c5ad0692fc 100644 --- a/src/elementary-number-theory/exponentiation-natural-numbers.lagda.md +++ b/src/elementary-number-theory/exponentiation-natural-numbers.lagda.md @@ -87,8 +87,8 @@ abstract exp-mul-ℕ x zero-ℕ z = inv (annihilation-law-exp-ℕ z) exp-mul-ℕ x (succ-ℕ y) z = ( left-distributive-exp-add-ℕ x (y *ℕ z) z) ∙ - ( ( ap (_*ℕ (x ^ℕ z)) (exp-mul-ℕ x y z)) ∙ - ( inv (right-distributive-exp-mul-ℕ (x ^ℕ y) x z))) + ( ap (_*ℕ (x ^ℕ z)) (exp-mul-ℕ x y z)) ∙ + ( inv (right-distributive-exp-mul-ℕ (x ^ℕ y) x z)) ``` ### If `m` is nonzero, `mⁿ` is nonzero From cd12463e222f932b9e7a7f230cbc574db10a2909 Mon Sep 17 00:00:00 2001 From: Louis Wasserman Date: Tue, 7 Oct 2025 08:02:42 -0700 Subject: [PATCH 07/12] Respond to review comment --- .../exponentiation-natural-numbers.lagda.md | 9 ++++++--- 1 file changed, 6 insertions(+), 3 deletions(-) diff --git a/src/elementary-number-theory/exponentiation-natural-numbers.lagda.md b/src/elementary-number-theory/exponentiation-natural-numbers.lagda.md index 0e1b525344..fad036a8a8 100644 --- a/src/elementary-number-theory/exponentiation-natural-numbers.lagda.md +++ b/src/elementary-number-theory/exponentiation-natural-numbers.lagda.md @@ -23,9 +23,9 @@ open import foundation.identity-types ## Idea -The {{#concept "exponential" Agda=exp-ℕ WD="exponentiation" WDID=Q33456}} $m^n$ -of two [natural numbers](elementary-number-theory.natural-numbers.md) is the -number obtained by multiplying $m$ with itself $n$ times. +The {{#concept "exponent" Agda=exp-ℕ WD="exponentiation" WDID=Q33456}} $m^n$ of +two [natural numbers](elementary-number-theory.natural-numbers.md) is the number +obtained by multiplying $m$ with itself $n$ times. ## Definition @@ -45,6 +45,9 @@ power-ℕ = power-Commutative-Semiring ℕ-Commutative-Semiring ## Properties +These properties are the axioms associated with +[Tarski's high school algebra problem](https://en.wikipedia.org/wiki/Tarski%27s_high_school_algebra_problem). + ### 1^n = 1 for all n ```agda From 168b1c7e5ae159e850b8cf33699beb65ec2cf946 Mon Sep 17 00:00:00 2001 From: Louis Wasserman Date: Tue, 7 Oct 2025 13:02:28 -0700 Subject: [PATCH 08/12] make pre-commit --- src/elementary-number-theory/equality-integers.lagda.md | 7 ++++--- .../equality-natural-numbers.lagda.md | 4 ++-- 2 files changed, 6 insertions(+), 5 deletions(-) diff --git a/src/elementary-number-theory/equality-integers.lagda.md b/src/elementary-number-theory/equality-integers.lagda.md index a679a3e7dc..ce58d1823c 100644 --- a/src/elementary-number-theory/equality-integers.lagda.md +++ b/src/elementary-number-theory/equality-integers.lagda.md @@ -33,9 +33,10 @@ open import foundation.universe-levels ## Idea -We define {{#concept "equality" Disambiguation="of integers" Agda=Eq-ℤ}} on the [integers](elementary-number-theory.integers.md) by -pattern matching, and show that this characterizes the -[identity type](foundation.identity-types.md) on the integers. +We define {{#concept "equality" Disambiguation="of integers" Agda=Eq-ℤ}} on the +[integers](elementary-number-theory.integers.md) by pattern matching, and show +that this characterizes the [identity type](foundation.identity-types.md) on the +integers. ## Definition diff --git a/src/elementary-number-theory/equality-natural-numbers.lagda.md b/src/elementary-number-theory/equality-natural-numbers.lagda.md index 16a6323959..04c1cd7a07 100644 --- a/src/elementary-number-theory/equality-natural-numbers.lagda.md +++ b/src/elementary-number-theory/equality-natural-numbers.lagda.md @@ -35,8 +35,8 @@ open import foundation-core.torsorial-type-families ## Idea -We define {{#concept "equality" Disambiguation="of natural numbers" Agda=Eq-ℕ}} on the -[natural numbers](elementary-number-theory.natural-numbers.md) by pattern +We define {{#concept "equality" Disambiguation="of natural numbers" Agda=Eq-ℕ}} +on the [natural numbers](elementary-number-theory.natural-numbers.md) by pattern matching, and show that this characterizes the [identity type](foundation.identity-types.md) on the natural numbers. From 31a6b3b90b27e8b309fc7f21d8e1ecad8f7ff63d Mon Sep 17 00:00:00 2001 From: Louis Wasserman Date: Fri, 10 Oct 2025 16:52:42 -0700 Subject: [PATCH 09/12] Apply suggestions from code review Co-authored-by: Fredrik Bakke --- .../divisibility-natural-numbers.lagda.md | 2 +- .../poset-of-natural-numbers-ordered-by-divisibility.lagda.md | 3 +-- .../relatively-prime-integers.lagda.md | 4 ++-- .../relatively-prime-natural-numbers.lagda.md | 4 ++-- 4 files changed, 6 insertions(+), 7 deletions(-) diff --git a/src/elementary-number-theory/divisibility-natural-numbers.lagda.md b/src/elementary-number-theory/divisibility-natural-numbers.lagda.md index 65535f82f1..c78f716cc8 100644 --- a/src/elementary-number-theory/divisibility-natural-numbers.lagda.md +++ b/src/elementary-number-theory/divisibility-natural-numbers.lagda.md @@ -189,7 +189,7 @@ abstract leq-quotient-div-ℕ d (succ-ℕ x) (is-nonzero-succ-ℕ x) H ``` -### If `x` is nonzero, if `d | x` and `d ≠ x`, then `d < x` +### If `x` is nonzero, `d | x`, and `d ≠ x`, then `d < x` ```agda abstract diff --git a/src/elementary-number-theory/poset-of-natural-numbers-ordered-by-divisibility.lagda.md b/src/elementary-number-theory/poset-of-natural-numbers-ordered-by-divisibility.lagda.md index 0373090536..4cdbbe5c39 100644 --- a/src/elementary-number-theory/poset-of-natural-numbers-ordered-by-divisibility.lagda.md +++ b/src/elementary-number-theory/poset-of-natural-numbers-ordered-by-divisibility.lagda.md @@ -30,8 +30,7 @@ open import order-theory.preorders ## Idea -The **[poset](order-theory.posets.md) of natural numbers ordered by -divisibility** consists of the +The {{#concept "poset of natural numbers ordered by divisibility" Agda=ℕ-Div-Poset}} consists of the [natural numbers](elementary-number-theory.natural-numbers.md) and its ordering is defined by `m ≤ n := m | n`, i.e., by [divisibility](elementary-number-theory.divisibility-natural-numbers.md). diff --git a/src/elementary-number-theory/relatively-prime-integers.lagda.md b/src/elementary-number-theory/relatively-prime-integers.lagda.md index 626759c15c..6090b9bf26 100644 --- a/src/elementary-number-theory/relatively-prime-integers.lagda.md +++ b/src/elementary-number-theory/relatively-prime-integers.lagda.md @@ -22,8 +22,8 @@ open import foundation.universe-levels ## Idea Two [integers](elementary-number-theory.integers.md) `x` and `y` are said to be -{{#concept "relatively prime" Disambiguation="integers" Agda=is-relatively-prime-ℤ}} -if their +{{#concept "relatively prime" Disambiguation="integers" WDID=Q104752 WD="coprime" Agda=is-relatively-prime-ℤ}}, +or **coprime**, if their [greatest common divisor](elementary-number-theory.greatest-common-divisor-integers.md) is `1`. diff --git a/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md b/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md index 1f4b26c526..8bb8114192 100644 --- a/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md +++ b/src/elementary-number-theory/relatively-prime-natural-numbers.lagda.md @@ -30,8 +30,8 @@ open import foundation.universe-levels Two [natural numbers](elementary-number-theory.natural-numbers.md) `x` and `y` are said to be -{{#concept "relatively prime" WDID=Q104752 WD="coprime" Disambiguation="natural numbers" Agda=is-relatively-prime-ℕ}} -if their +{{#concept "relatively prime" WDID=Q104752 WD="coprime" Disambiguation="natural numbers" Agda=is-relatively-prime-ℕ}}, +or **coprime**, if their [greatest common divisor](elementary-number-theory.greatest-common-divisor-natural-numbers.md) is `1`. From 2f239fcccbd50571f2dbebe916dcdc4867ed336a Mon Sep 17 00:00:00 2001 From: Louis Wasserman Date: Fri, 10 Oct 2025 16:53:23 -0700 Subject: [PATCH 10/12] Apply suggestions from code review Co-authored-by: Fredrik Bakke --- src/elementary-number-theory/distance-natural-numbers.lagda.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/elementary-number-theory/distance-natural-numbers.lagda.md b/src/elementary-number-theory/distance-natural-numbers.lagda.md index b03c45d8d3..b1831be249 100644 --- a/src/elementary-number-theory/distance-natural-numbers.lagda.md +++ b/src/elementary-number-theory/distance-natural-numbers.lagda.md @@ -165,7 +165,7 @@ abstract triangle-inequality-dist-ℕ m n k ``` -### `dist-ℕ x y` is a solution in `z` to `x + z ＝ y` +### `dist-ℕ x y` is a solution in `z` to `x + z ＝ y` when `x ≤ y` ```agda abstract From cd1404ba409deb814e3c0565d3af51ed5678d6b4 Mon Sep 17 00:00:00 2001 From: Louis Wasserman Date: Fri, 10 Oct 2025 16:53:56 -0700 Subject: [PATCH 11/12] Respond to review comments --- src/order-theory/well-founded-relations.lagda.md | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/src/order-theory/well-founded-relations.lagda.md b/src/order-theory/well-founded-relations.lagda.md index 7ae600f45c..366a51ad21 100644 --- a/src/order-theory/well-founded-relations.lagda.md +++ b/src/order-theory/well-founded-relations.lagda.md @@ -32,8 +32,9 @@ if all elements of `X` are Well-founded relations satisfy an induction principle: In order to construct an element of `P x` for all `x : X` it suffices to construct an element of `P y` -for all elements `y ∈ x`. More precisely, the **well-founded induction -principle** is a function +for all elements `y ∈ x`. More precisely, the +{{#concept "well-founded induction principle" WDID=Q14402036 WD="well-founded induction" Agda=ind-Well-Founded-Relation}} +is a function ```text (x : X) → ((y : Y) → y ∈ x → P y) → P x. From 4ad347255e8cb3fbef0e58b4a6a33a6adeed4734 Mon Sep 17 00:00:00 2001 From: Louis Wasserman Date: Sat, 11 Oct 2025 21:41:00 -0700 Subject: [PATCH 12/12] Respond to review comments: --- ...set-of-natural-numbers-ordered-by-divisibility.lagda.md | 7 ++++--- .../semiring-of-natural-numbers.lagda.md | 6 +++++- 2 files changed, 9 insertions(+), 4 deletions(-) diff --git a/src/elementary-number-theory/poset-of-natural-numbers-ordered-by-divisibility.lagda.md b/src/elementary-number-theory/poset-of-natural-numbers-ordered-by-divisibility.lagda.md index 4cdbbe5c39..1e5513ff91 100644 --- a/src/elementary-number-theory/poset-of-natural-numbers-ordered-by-divisibility.lagda.md +++ b/src/elementary-number-theory/poset-of-natural-numbers-ordered-by-divisibility.lagda.md @@ -30,9 +30,10 @@ open import order-theory.preorders ## Idea -The {{#concept "poset of natural numbers ordered by divisibility" Agda=ℕ-Div-Poset}} consists of the -[natural numbers](elementary-number-theory.natural-numbers.md) and its ordering -is defined by `m ≤ n := m | n`, i.e., by +The +{{#concept "poset of natural numbers ordered by divisibility" Agda=ℕ-Div-Poset}} +consists of the [natural numbers](elementary-number-theory.natural-numbers.md) +and its ordering is defined by `m ≤ n := m | n`, i.e., by [divisibility](elementary-number-theory.divisibility-natural-numbers.md). The divisibility relation `m | n` on the natural numbers, however, is only diff --git a/src/elementary-number-theory/semiring-of-natural-numbers.lagda.md b/src/elementary-number-theory/semiring-of-natural-numbers.lagda.md index f24cb45c4e..8c163917bb 100644 --- a/src/elementary-number-theory/semiring-of-natural-numbers.lagda.md +++ b/src/elementary-number-theory/semiring-of-natural-numbers.lagda.md @@ -45,7 +45,7 @@ i.e., the type of semiring homomorphisms `hom-Semiring ℕ-Semiring R` is ## Definition -### The commutative semiring of natural numbers +### The semiring of natural numbers ```agda has-mul-ℕ-Commutative-Monoid : @@ -63,7 +63,11 @@ pr1 ℕ-Semiring = ℕ-Commutative-Monoid pr1 (pr2 ℕ-Semiring) = has-mul-ℕ-Commutative-Monoid pr1 (pr2 (pr2 ℕ-Semiring)) = left-zero-law-mul-ℕ pr2 (pr2 (pr2 ℕ-Semiring)) = right-zero-law-mul-ℕ +``` +### The commutative semiring of natural numbers + +```agda ℕ-Commutative-Semiring : Commutative-Semiring lzero pr1 ℕ-Commutative-Semiring = ℕ-Semiring pr2 ℕ-Commutative-Semiring = commutative-mul-ℕ