module Data.Int.Properties where
Properties of integers🔗
This module establishes arithmetic and algebraic properties of the integers. Most of the proofs are straightforward case bashes; there are no further comments.
Differences🔗
nat-diff-zero : ∀ x → (x ℕ- x) ≡ 0 nat-diff-zero zero = refl nat-diff-zero (suc x) = nat-diff-zero x nat-diff-positive : ∀ a b → a ℕ- b ≡ 0 → a ≡ b nat-diff-positive a zero p = pos-injective p nat-diff-positive zero (suc b) p = absurd (negsuc≠pos p) nat-diff-positive (suc a) (suc b) p = ap suc (nat-diff-positive a b p) sucℤ-nat-diff : ∀ x y → sucℤ (x ℕ- y) ≡ suc x ℕ- y sucℤ-nat-diff zero zero = refl sucℤ-nat-diff zero (suc zero) = refl sucℤ-nat-diff zero (suc (suc y)) = refl sucℤ-nat-diff (suc x) zero = refl sucℤ-nat-diff (suc x) (suc y) = sucℤ-nat-diff x y predℤ-nat-diff : ∀ x y → predℤ (x ℕ- y) ≡ x ℕ- suc y predℤ-nat-diff zero zero = refl predℤ-nat-diff zero (suc y) = refl predℤ-nat-diff (suc x) zero = refl predℤ-nat-diff (suc x) (suc y) = predℤ-nat-diff x y nat-diff-monus : ∀ x y → y ≤ x → x ℕ- y ≡ pos (x - y) nat-diff-monus x y 0≤x = refl nat-diff-monus (suc x) (suc zero) (s≤s y≤x) = refl nat-diff-monus (suc x) (suc (suc y)) (s≤s y≤x) = nat-diff-monus x (suc y) y≤x nat-diff-bounded : ∀ a b c → a ≤ c → b ≤ c → abs (a ℕ- b) ≤ c nat-diff-bounded a zero c ac bc = ac nat-diff-bounded zero (suc b) c ac bc = bc nat-diff-bounded (suc a) (suc b) c ac bc = nat-diff-bounded a b c (<-weaken ac) (<-weaken bc)
Negations🔗
negℤ-negℤ : ∀ x → negℤ (negℤ x) ≡ x negℤ-negℤ (pos zero) = refl negℤ-negℤ (pos (suc x)) = refl negℤ-negℤ (negsuc x) = refl negℤ-injective : ∀ x y → negℤ x ≡ negℤ y → x ≡ y negℤ-injective x y p = sym (negℤ-negℤ x) ·· ap negℤ p ·· negℤ-negℤ y negℤ-predℤ : ∀ x → negℤ (predℤ x) ≡ sucℤ (negℤ x) negℤ-predℤ posz = refl negℤ-predℤ (possuc zero) = refl negℤ-predℤ (possuc (suc x)) = refl negℤ-predℤ (negsuc x) = refl negℤ-sucℤ : ∀ x → negℤ (sucℤ x) ≡ predℤ (negℤ x) negℤ-sucℤ posz = refl negℤ-sucℤ (possuc x) = refl negℤ-sucℤ (negsuc zero) = refl negℤ-sucℤ (negsuc (suc x)) = refl negℤ-distrib-max : ∀ x y → negℤ (maxℤ x y) ≡ minℤ (negℤ x) (negℤ y) negℤ-distrib-max posz posz = refl negℤ-distrib-max posz (possuc y) = refl negℤ-distrib-max posz (negsuc y) = refl negℤ-distrib-max (possuc x) posz = refl negℤ-distrib-max (possuc x) (possuc y) = refl negℤ-distrib-max (possuc x) (negsuc y) = refl negℤ-distrib-max (negsuc x) posz = refl negℤ-distrib-max (negsuc x) (possuc y) = refl negℤ-distrib-max (negsuc x) (negsuc y) = refl negℤ-distrib-min : ∀ x y → negℤ (minℤ x y) ≡ maxℤ (negℤ x) (negℤ y) negℤ-distrib-min posz posz = refl negℤ-distrib-min posz (possuc y) = refl negℤ-distrib-min posz (negsuc y) = refl negℤ-distrib-min (possuc x) posz = refl negℤ-distrib-min (possuc x) (possuc y) = refl negℤ-distrib-min (possuc x) (negsuc y) = refl negℤ-distrib-min (negsuc x) posz = refl negℤ-distrib-min (negsuc x) (possuc y) = refl negℤ-distrib-min (negsuc x) (negsuc y) = refl
Absolute value🔗
abs-positive : ∀ x → abs x ≡ 0 → x ≡ 0 abs-positive (pos x) p = ap pos p abs-positive (negsuc x) p = absurd (suc≠zero p) abs-negℤ : ∀ z → abs (negℤ z) ≡ abs z abs-negℤ (posz) = refl abs-negℤ (possuc _) = refl abs-negℤ (negsuc _) = refl
Rotations🔗
rot-pos : ∀ x y → rotℤ (pos x) (pos y) ≡ pos (x + y) rot-pos zero y = refl rot-pos (suc x) y = ap sucℤ (rot-pos x y) rot-negsuc : ∀ x y → rotℤ (negsuc x) (negsuc y) ≡ negsuc (suc (x + y)) rot-negsuc zero y = refl rot-negsuc (suc x) y = ap predℤ (rot-negsuc x y) rot-pos-neg : ∀ x y → sucℤ (rotℤ (pos x) (negsuc y)) ≡ x ℕ- y rot-pos-neg zero zero = refl rot-pos-neg zero (suc y) = refl rot-pos-neg (suc x) y = ap sucℤ (rot-pos-neg x y) ∙ sucℤ-nat-diff x y rot-neg-pos : ∀ x y → rotℤ (negsuc x) (pos y) ≡ y ℕ- suc x rot-neg-pos zero zero = refl rot-neg-pos zero (suc y) = refl rot-neg-pos (suc x) y = ap predℤ (rot-neg-pos x y) ∙ predℤ-nat-diff y (suc x) rot-sucl : ∀ x y → rotℤ (sucℤ x) y ≡ sucℤ (rotℤ x y) rot-sucl (pos x) y = refl rot-sucl (negsuc zero) y = sym (suc-predℤ y) rot-sucl (negsuc (suc x)) y = sym (suc-predℤ _) rot-predl : ∀ x y → rotℤ (predℤ x) y ≡ predℤ (rotℤ x y) rot-predl posz y = refl rot-predl (possuc x) y = sym (pred-sucℤ _) rot-predl (negsuc x) y = refl negℤ-distrib-rot : ∀ x y → negℤ (rotℤ x y) ≡ rotℤ (negℤ x) (negℤ y) negℤ-distrib-rot posz y = refl negℤ-distrib-rot (possuc zero) y = negℤ-sucℤ y negℤ-distrib-rot (possuc (suc x)) y = negℤ-sucℤ (sucℤ (rotℤ (pos x) y)) ∙ ap predℤ (negℤ-distrib-rot (pos (suc x)) y) negℤ-distrib-rot (negsuc zero) y = negℤ-predℤ y negℤ-distrib-rot (negsuc (suc x)) y = negℤ-predℤ (rotℤ (negsuc x) y) ∙ ap sucℤ (negℤ-distrib-rot (negsuc x) y) rotℤ-assoc : ∀ x y z → rotℤ x (rotℤ y z) ≡ rotℤ (rotℤ x y) z rotℤ-assoc (pos zero) y z = refl rotℤ-assoc (pos (suc x)) y z = sucℤ (rotℤ (pos x) (rotℤ y z)) ≡⟨ ap sucℤ (rotℤ-assoc (pos x) y z) ⟩≡ sucℤ (rotℤ (rotℤ (pos x) y) z) ≡˘⟨ rot-sucl (rotℤ (pos x) y) z ⟩≡˘ rotℤ (sucℤ (rotℤ (pos x) y)) z ∎ rotℤ-assoc (negsuc zero) y z = sym (rot-predl y z) rotℤ-assoc (negsuc (suc x)) y z = predℤ (rotℤ (negsuc x) (rotℤ y z)) ≡⟨ ap predℤ (rotℤ-assoc (negsuc x) y z) ⟩≡ predℤ (rotℤ (rotℤ (negsuc x) y) z) ≡˘⟨ rot-predl (rotℤ (negsuc x) y) z ⟩≡˘ rotℤ (predℤ (rotℤ (negsuc x) y)) z ∎
Addition🔗
+ℤ-zerol : ∀ x → 0 +ℤ x ≡ x +ℤ-zerol (pos x) = refl +ℤ-zerol (negsuc x) = refl +ℤ-zeror : ∀ x → x +ℤ 0 ≡ x +ℤ-zeror (pos x) = ap pos (+-zeror x) +ℤ-zeror (negsuc x) = refl rot-is-add : ∀ x y → x +ℤ y ≡ rotℤ x y rot-is-add (pos zero) x = +ℤ-zerol x rot-is-add (pos (suc x)) (pos y) = sym (ap sucℤ (rot-pos x y)) rot-is-add (pos (suc x)) (negsuc y) = sym (rot-pos-neg x y) rot-is-add (negsuc x) (pos y) = sym (rot-neg-pos x y) rot-is-add (negsuc x) (negsuc y) = sym (rot-negsuc x y) +ℤ-assoc : ∀ x y z → x +ℤ (y +ℤ z) ≡ (x +ℤ y) +ℤ z +ℤ-assoc x y z = x +ℤ (y +ℤ z) ≡⟨ ap (x +ℤ_) (rot-is-add y z) ⟩≡ x +ℤ rotℤ y z ≡⟨ rot-is-add x _ ⟩≡ rotℤ x (rotℤ y z) ≡⟨ rotℤ-assoc x y z ⟩≡ rotℤ (rotℤ x y) z ≡˘⟨ ap₂ rotℤ (rot-is-add x y) refl ⟩≡˘ rotℤ (x +ℤ y) z ≡˘⟨ rot-is-add (x +ℤ y) z ⟩≡˘ (x +ℤ y) +ℤ z ∎ +ℤ-invl : ∀ x → negℤ x +ℤ x ≡ 0 +ℤ-invl (pos zero) = refl +ℤ-invl (pos (suc x)) = nat-diff-zero x +ℤ-invl (negsuc x) = nat-diff-zero x +ℤ-invr : ∀ x → x +ℤ negℤ x ≡ 0 +ℤ-invr (pos zero) = refl +ℤ-invr (pos (suc x)) = nat-diff-zero x +ℤ-invr (negsuc x) = nat-diff-zero x +ℤ-commutative : ∀ x y → x +ℤ y ≡ y +ℤ x +ℤ-commutative (pos x) (pos y) = ap pos (+-commutative x y) +ℤ-commutative (pos x) (negsuc y) = refl +ℤ-commutative (negsuc x) (pos y) = refl +ℤ-commutative (negsuc x) (negsuc y) = ap negsuc (ap suc (+-commutative x y)) +ℤ-sucl : ∀ x y → sucℤ x +ℤ y ≡ sucℤ (x +ℤ y) +ℤ-sucl x y = rot-is-add (sucℤ x) y ·· rot-sucl x y ·· ap sucℤ (sym (rot-is-add x y)) +ℤ-sucr : ∀ x y → x +ℤ sucℤ y ≡ sucℤ (x +ℤ y) +ℤ-sucr x y = +ℤ-commutative x (sucℤ y) ·· +ℤ-sucl y x ·· ap sucℤ (+ℤ-commutative y x) +ℤ-predl : ∀ x y → predℤ x +ℤ y ≡ predℤ (x +ℤ y) +ℤ-predl x y = rot-is-add (predℤ x) y ·· rot-predl x y ·· ap predℤ (sym (rot-is-add x y)) +ℤ-predr : ∀ x y → x +ℤ predℤ y ≡ predℤ (x +ℤ y) +ℤ-predr x y = +ℤ-commutative x (predℤ y) ·· +ℤ-predl y x ·· ap predℤ (+ℤ-commutative y x) +ℤ-onel : ∀ x → 1 +ℤ x ≡ sucℤ x +ℤ-onel x = +ℤ-sucl 0 x ∙ ap sucℤ (+ℤ-zerol x) +ℤ-oner : ∀ x → x +ℤ 1 ≡ sucℤ x +ℤ-oner x = +ℤ-sucr x 0 ∙ ap sucℤ (+ℤ-zeror x) +ℤ-injectiver : ∀ k x y → k +ℤ x ≡ k +ℤ y → x ≡ y +ℤ-injectiver k x y p = sym (+ℤ-zerol x) ·· ap (_+ℤ x) (sym (+ℤ-invl k)) ·· sym (+ℤ-assoc (negℤ k) k x) ·· ap (negℤ k +ℤ_) p ·· +ℤ-assoc (negℤ k) k y ·· ap (_+ℤ y) (+ℤ-invl k) ·· +ℤ-zerol y +ℤ-injectivel : ∀ k x y → x +ℤ k ≡ y +ℤ k → x ≡ y +ℤ-injectivel k x y p = +ℤ-injectiver k x y $ +ℤ-commutative k x ·· p ·· +ℤ-commutative y k negℤ-distrib : ∀ x y → negℤ (x +ℤ y) ≡ (negℤ x) +ℤ (negℤ y) negℤ-distrib x y = ap negℤ (rot-is-add x y) ·· negℤ-distrib-rot x y ·· sym (rot-is-add (negℤ x) (negℤ y)) negℤ-+ℤ-negsuc : ∀ a b → negℤ (pos a) +ℤ negsuc b ≡ negsuc (a + b) negℤ-+ℤ-negsuc a b = +ℤ-commutative (negℤ (pos a)) (negsuc b) ·· sym (negℤ-distrib (possuc b) (pos a)) ·· ap negsuc (+-commutative b a) pos-pos : ∀ a b → pos a -ℤ pos b ≡ a ℕ- b pos-pos a zero = +ℤ-zeror _ pos-pos a (suc b) = refl -ℤ-swapl : ∀ a b c → a +ℤ b ≡ c → a ≡ c -ℤ b -ℤ-swapl a b c p = sym (+ℤ-zeror _) ∙ ap (a +ℤ_) (sym (+ℤ-invr b)) ·· +ℤ-assoc a _ _ ·· ap (_+ℤ negℤ b) p private distrib-lemma : ∀ x y z w → (x +ℤ y) +ℤ (z +ℤ w) ≡ (x +ℤ z) +ℤ (y +ℤ w) distrib-lemma x y z w = +ℤ-assoc (x +ℤ y) z w ·· ap (_+ℤ w) (sym (+ℤ-assoc x y z) ·· ap (x +ℤ_) (+ℤ-commutative y z) ·· +ℤ-assoc x z y) ·· sym (+ℤ-assoc (x +ℤ z) y w) -ℤ-cancelr : ∀ k x y → (x +ℤ k) -ℤ (y +ℤ k) ≡ x -ℤ y -ℤ-cancelr k x y = (x +ℤ k) -ℤ (y +ℤ k) ≡⟨ ap ((x +ℤ k) +ℤ_) (negℤ-distrib y k) ⟩≡ (x +ℤ k) +ℤ (negℤ y -ℤ k) ≡⟨ distrib-lemma x k (negℤ y) (negℤ k) ⟩≡ (x -ℤ y) +ℤ (k -ℤ k) ≡⟨ ap ((x -ℤ y) +ℤ_) (+ℤ-invr k) ⟩≡ (x -ℤ y) +ℤ 0 ≡⟨ +ℤ-zeror (x -ℤ y) ⟩≡ x -ℤ y ∎
Multiplication🔗
assign-pos : ∀ x → assign pos x ≡ pos x assign-pos zero = refl assign-pos (suc x) = refl assign-neg : ∀ x → assign neg x ≡ negℤ (pos x) assign-neg zero = refl assign-neg (suc x) = refl neg-sign : Sign → Sign neg-sign pos = neg neg-sign neg = pos neg-assign : ∀ {s} x → assign (neg-sign s) x ≡ negℤ (assign s x) neg-assign {pos} zero = refl neg-assign {pos} (suc x) = refl neg-assign {neg} zero = refl neg-assign {neg} (suc x) = refl assign-+ : ∀ {s} x y → assign s (x + y) ≡ assign s x +ℤ assign s y assign-+ {pos} zero y = sym (+ℤ-zerol _) assign-+ {pos} (suc x) zero = refl assign-+ {pos} (suc x) (suc y) = refl assign-+ {neg} zero y = sym (+ℤ-zerol _) assign-+ {neg} (suc x) zero = ap negsuc (+-zeror _) assign-+ {neg} (suc x) (suc y) = ap negsuc (+-sucr _ _) possuc≠assign-neg : ∀ {x y} → possuc x ≠assign neg y possuc≠assign-neg {x} {zero} p = suc≠zero (pos-injective p) possuc≠assign-neg {x} {suc y} p = pos≠negsuc p *ℤ-onel : ∀ x → 1 *ℤ x ≡ x *ℤ-onel (pos x) = ap (assign pos) (+-zeror x) ∙ assign-pos x *ℤ-onel (negsuc x) = ap negsuc (+-zeror x) *ℤ-oner : ∀ x → x *ℤ 1 ≡ x *ℤ-oner (pos x) = ap (assign pos) (*-oner x) ∙ assign-pos x *ℤ-oner (negsuc x) = ap negsuc (*-oner x) *ℤ-zerol : ∀ x → 0 *ℤ x ≡ 0 *ℤ-zerol (pos x) = refl *ℤ-zerol (negsuc x) = refl *ℤ-zeror : ∀ x → x *ℤ 0 ≡ 0 *ℤ-zeror (pos x) = ap (assign pos) (*-zeror x) *ℤ-zeror (negsuc x) = ap (assign neg) (*-zeror x) assign-*l : ∀ {s} x y → assign s (x * y) ≡ assign s x *ℤ pos y assign-*l {pos} zero y = sym (*ℤ-zerol (pos y)) assign-*l {pos} (suc x) y = refl assign-*l {neg} zero y = refl assign-*l {neg} (suc x) y = refl *ℤ-negl : ∀ x y → negℤ x *ℤ y ≡ negℤ (x *ℤ y) *ℤ-negl posz y = refl *ℤ-negl (possuc x) y with sign y ... | pos = neg-assign (suc x * abs y) ... | neg = neg-assign (suc x * abs y) *ℤ-negl (negsuc x) y with sign y ... | pos = neg-assign (suc x * abs y) ... | neg = neg-assign (suc x * abs y) private lemma : ∀ x y z → z + y * suc z + x * suc (z + y * suc z) ≡ z + (y + x * suc y) * suc z lemma x y z = nat! -- *ℤ-def : ∀ x y → x *ℤ y ≡ assign (sign× (sign x) (sign y)) (abs x * abs y) *ℤ-associative : ∀ x y z → x *ℤ (y *ℤ z) ≡ (x *ℤ y) *ℤ z *ℤ-associative posz y z = refl *ℤ-associative (pos x@(suc _)) posz z = ap (assign pos) (*-zeror x) ∙ sym (ap₂ _*ℤ_ (ap (assign pos) (*-zeror x)) refl) *ℤ-associative (negsuc x) posz z = ap (assign neg) (*-zeror x) ∙ sym (ap₂ _*ℤ_ (ap (assign neg) (*-zeror x)) refl) *ℤ-associative x (pos y) posz = ap₂ _*ℤ_ (λ i → x) (ap (assign pos) (*-zeror y)) ∙ *ℤ-zeror x ∙ sym (*ℤ-zeror (x *ℤ pos y)) *ℤ-associative x (negsuc y) posz = ap₂ _*ℤ_ (λ i → x) (ap (assign neg) (*-zeror y)) ∙ *ℤ-zeror x ∙ sym (*ℤ-zeror (x *ℤ negsuc y)) *ℤ-associative (possuc x) (possuc y) (possuc z) = ap possuc (lemma x y z) *ℤ-associative (possuc x) (possuc y) (negsuc z) = ap negsuc (lemma x y z) *ℤ-associative (possuc x) (negsuc y) (possuc z) = ap negsuc (lemma x y z) *ℤ-associative (possuc x) (negsuc y) (negsuc z) = ap possuc (lemma x y z) *ℤ-associative (negsuc x) (possuc y) (possuc z) = ap negsuc (lemma x y z) *ℤ-associative (negsuc x) (possuc y) (negsuc z) = ap possuc (lemma x y z) *ℤ-associative (negsuc x) (negsuc y) (possuc z) = ap possuc (lemma x y z) *ℤ-associative (negsuc x) (negsuc y) (negsuc z) = ap negsuc (lemma x y z) *ℤ-possucl : ∀ x y → possuc x *ℤ y ≡ y +ℤ (pos x *ℤ y) *ℤ-possucl x (pos y) = assign-pos (y + x * y) ∙ sym (ap (pos y +ℤ_) (assign-pos (x * y))) *ℤ-possucl zero (negsuc y) = ap negsuc (+-zeror y) *ℤ-possucl (suc x) (negsuc y) = ap negsuc p where p : y + suc (y + x * suc y) ≡ suc (y + (y + x * suc y)) p = nat! *ℤ-negsucl : ∀ x y → negsuc (suc x) *ℤ y ≡ negℤ y +ℤ (negsuc x *ℤ y) *ℤ-negsucl x posz = sym (+ℤ-zerol (assign neg (x * 0))) *ℤ-negsucl x (possuc y) = ap negsuc p where p : y + suc x * suc y ≡ suc (y + (y + x * suc y)) p = nat! *ℤ-negsucl x (negsuc y) = ap possuc p where p : y + suc x * suc y ≡ y + suc (y + x * suc y) p = nat! sign×-commutative : ∀ s s' → sign× s s' ≡ sign× s' s sign×-commutative pos pos = refl sign×-commutative pos neg = refl sign×-commutative neg pos = refl sign×-commutative neg neg = refl *ℤ-commutative : ∀ x y → x *ℤ y ≡ y *ℤ x *ℤ-commutative (pos x) (pos y) = ap (assign pos) (*-commutative x y) *ℤ-commutative (pos x) (negsuc y) = ap (assign neg) (*-commutative x (suc y)) *ℤ-commutative (negsuc x) (pos y) = ap (assign neg) (*-commutative (suc x) y) *ℤ-commutative (negsuc x) (negsuc y) = ap (assign pos) (*-commutative (suc x) (suc y)) dot-is-mul : ∀ x y → (dotℤ x y) ≡ (x *ℤ y) dot-is-mul posz y = refl dot-is-mul (possuc x) y = ap (y +ℤ_) (dot-is-mul (pos x) y) ∙ sym (*ℤ-possucl x y) dot-is-mul (negsuc zero) posz = refl dot-is-mul (negsuc zero) (possuc x) = ap negsuc (sym (+-zeror x)) dot-is-mul (negsuc zero) (negsuc x) = ap possuc (sym (+-zeror x)) dot-is-mul (negsuc (suc x)) y = sym (*ℤ-negsucl x y ∙ ap (negℤ y +ℤ_) (sym (dot-is-mul (negsuc x) y))) dot-distribr : ∀ x y z → dotℤ z (x +ℤ y) ≡ (dotℤ z x) +ℤ (dotℤ z y) dot-distribr x y posz = refl dot-distribr x y (possuc z) = ap ((x +ℤ y) +ℤ_) (dot-distribr x y (pos z)) ∙ distrib-lemma x y (dotℤ (pos z) x) (dotℤ (pos z) y) dot-distribr x y (negsuc zero) = negℤ-distrib x y dot-distribr x y (negsuc (suc z)) = ap₂ _+ℤ_ (negℤ-distrib x y) (dot-distribr x y (negsuc z)) ∙ distrib-lemma (negℤ x) (negℤ y) (dotℤ (negsuc z) x) (dotℤ (negsuc z) y) *ℤ-distribl : ∀ x y z → x *ℤ (y +ℤ z) ≡ (x *ℤ y) +ℤ (x *ℤ z) *ℤ-distribl x y z = sym (dot-is-mul x (y +ℤ z)) ·· dot-distribr y z x ·· ap₂ _+ℤ_ (dot-is-mul x y) (dot-is-mul x z) *ℤ-distribr : ∀ x y z → (y +ℤ z) *ℤ x ≡ (y *ℤ x) +ℤ (z *ℤ x) *ℤ-distribr x y z = *ℤ-commutative (y +ℤ z) x ·· *ℤ-distribl x y z ·· ap₂ _+ℤ_ (*ℤ-commutative x y) (*ℤ-commutative x z) *ℤ-distribr-minus : ∀ x y z → (y -ℤ z) *ℤ x ≡ (y *ℤ x) -ℤ (z *ℤ x) *ℤ-distribr-minus x y z = *ℤ-distribr x y (negℤ z) ∙ ap (y *ℤ x +ℤ_) (*ℤ-negl z x) *ℤ-sucr : ∀ x y → x *ℤ sucℤ y ≡ x *ℤ y +ℤ x *ℤ-sucr x y = x *ℤ sucℤ y ≡˘⟨ ap (x *ℤ_) (+ℤ-oner y) ⟩≡˘ x *ℤ (y +ℤ 1) ≡⟨ *ℤ-distribl x y 1 ⟩≡ x *ℤ y +ℤ x *ℤ 1 ≡⟨ ap (x *ℤ y +ℤ_) (*ℤ-oner x) ⟩≡ x *ℤ y +ℤ x ∎ *ℤ-injectiver-possuc : ∀ k x y → x *ℤ possuc k ≡ y *ℤ possuc k → x ≡ y *ℤ-injectiver-possuc k (pos x) (pos y) p = ap pos (*-suc-inj k x y (pos-injective (sym (assign-pos _) ·· p ·· assign-pos _))) *ℤ-injectiver-possuc k (pos x) (negsuc y) p = absurd (pos≠negsuc (sym (assign-pos (x * suc k)) ∙ p)) *ℤ-injectiver-possuc k (negsuc x) (pos y) p = absurd (negsuc≠pos (p ∙ assign-pos (y * suc k))) *ℤ-injectiver-possuc k (negsuc x) (negsuc y) p = ap (assign neg) (*-suc-inj k (suc x) (suc y) (ap suc (negsuc-injective p))) *ℤ-injectiver-negsuc : ∀ k x y → x *ℤ negsuc k ≡ y *ℤ negsuc k → x ≡ y *ℤ-injectiver-negsuc k (pos x) (pos y) p = ap pos (*-suc-inj k x y (pos-injective (negℤ-injective _ _ (sym (assign-neg _) ·· p ·· assign-neg _)))) *ℤ-injectiver-negsuc k posz (negsuc y) p = absurd (zero≠suc (pos-injective p)) *ℤ-injectiver-negsuc k (possuc x) (negsuc y) p = absurd (negsuc≠pos p) *ℤ-injectiver-negsuc k (negsuc x) posz p = absurd (suc≠zero (pos-injective p)) *ℤ-injectiver-negsuc k (negsuc x) (possuc y) p = absurd (pos≠negsuc p) *ℤ-injectiver-negsuc k (negsuc x) (negsuc y) p = ap (assign neg) (*-suc-inj k (suc x) (suc y) (ap suc (suc-inj (pos-injective p)))) *ℤ-injectiver : ∀ k x y → k ≠0 → x *ℤ k ≡ y *ℤ k → x ≡ y *ℤ-injectiver posz x y k≠0 p = absurd (k≠0 refl) *ℤ-injectiver (possuc k) x y k≠0 p = *ℤ-injectiver-possuc k x y p *ℤ-injectiver (negsuc k) x y k≠0 p = *ℤ-injectiver-negsuc k x y p *ℤ-is-zero : ∀ x y → (x *ℤ y) ≡ 0 → (x ≡ 0) ⊎ (y ≡ 0) *ℤ-is-zero posz (pos _) _ = inl refl *ℤ-is-zero (negsuc _) posz _ = inr refl *ℤ-is-zero posz (negsuc _) _ = inl refl *ℤ-is-zero (possuc _) posz _ = inr refl *ℤ-is-zero (possuc _) (possuc _) p = absurd (suc≠zero (pos-injective p)) *ℤ-is-zero (possuc _) (negsuc _) p = absurd (negsuc≠pos p) *ℤ-is-zero (negsuc _) (possuc _) p = absurd (negsuc≠pos p) *ℤ-is-zero (negsuc _) (negsuc _) p = absurd (suc≠zero (pos-injective p)) abs-assign : ∀ s n → abs (assign s n) ≡ n abs-assign pos zero = refl abs-assign pos (suc n) = refl abs-assign neg zero = refl abs-assign neg (suc n) = refl abs-*ℤ : ∀ x y → abs (x *ℤ y) ≡ abs x * abs y abs-*ℤ (pos x) (pos y) = abs-assign pos (x * y) abs-*ℤ (pos x) (negsuc y) = abs-assign neg (x * suc y) abs-*ℤ (negsuc x) (pos y) = abs-assign neg (y + x * y) abs-*ℤ (negsuc x) (negsuc y) = refl sign-*ℤ-positive : ∀ x t → Positive t → sign (x *ℤ t) ≡ sign x sign-*ℤ-positive (pos x) (possuc y) (pos _) = ap sign (assign-pos (x * suc y)) sign-*ℤ-positive (negsuc x) (possuc y) (pos _) = refl assign-sign : ∀ x → assign (sign x) (abs x) ≡ x assign-sign posz = refl assign-sign (possuc _) = refl assign-sign (negsuc _) = refl assign-pos-positive : ∀ x → Positive x → assign pos (abs x) ≡ x assign-pos-positive x@(possuc _) (pos _) = refl divides-*ℤ : ∀ {n k m} → k * n ≡ abs m → pos k *ℤ assign (sign m) n ≡ m divides-*ℤ {zero} {k} {pos x} p = ap (assign pos) (*-zeror k) ∙ ap Int.pos (sym (*-zeror k) ∙ p) divides-*ℤ {suc n} {k} {posz} p = ap (assign pos) p divides-*ℤ {suc n} {zero} {possuc x} p = ap pos p divides-*ℤ {suc n} {suc k} {possuc x} p = ap pos p divides-*ℤ {zero} {k} {negsuc x} p = absurd (zero≠suc (sym (*-zeror k) ∙ p)) divides-*ℤ {suc n} {zero} {negsuc x} p = absurd (zero≠suc p) divides-*ℤ {suc n} {suc k} {negsuc x} p = ap negsuc (suc-inj p)
Positivity🔗
*ℤ-positive : ∀ {x y} → Positive x → Positive y → Positive (x *ℤ y) *ℤ-positive (pos x) (pos y) = pos (y + x * suc y) *ℤ-nonzero : ∀ {x y} → x ≠0 → y ≠0 → (x *ℤ y) ≠0 *ℤ-nonzero {x} {y} x≠0 y≠0 p with *ℤ-is-zero x y p ... | inl x=0 = x≠0 x=0 ... | inr y=0 = y≠0 y=0 +ℤ-positive : ∀ {x y} → Positive x → Positive y → Positive (x +ℤ y) +ℤ-positive (pos x) (pos y) = pos (x + suc y) pos-positive : ∀ {x} → x ≠0 → Positive (pos x) pos-positive {zero} 0≠0 = absurd (0≠0 refl) pos-positive {suc n} _ = pos n positive→nonzero : ∀ {x} → Positive x → x ≠0 positive→nonzero .{possuc x} (pos x) p = suc≠zero (pos-injective p) *ℤ-positive-cancel : ∀ {x y} → Positive x → Positive (x *ℤ y) → Positive y *ℤ-positive-cancel {pos .(suc x)} {posz} (pos x) q = absurd (positive→nonzero q (ap (assign pos) (*-zeror x))) *ℤ-positive-cancel {pos .(suc x)} {possuc y} (pos x) q = pos y *ℤ-nonzero-cancel : ∀ {x y} → (x *ℤ y) ≠0 → x ≠0 *ℤ-nonzero-cancel {x} {y} xy≠0 x=0 = absurd (xy≠0 (ap (_*ℤ y) x=0))