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Cleanup finset and modulo (#92)

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Patrick Stevens
2020-01-01 10:14:55 +00:00
committed by GitHub
parent b6ef9b46f2
commit 019a9d9a07
66 changed files with 1154 additions and 1431 deletions
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46
Functions.agda
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@@ -11,13 +11,11 @@ module Functions where
_∘_ : {a b c : _} {A : Set a} {B : Set b} {C : Set c} → (f : B → C) → (g : A → B) → (A → C)
g ∘ f = λ a → g (f a)
record Injection {a b : _} {A : Set a} {B : Set b} (f : A → B) : Set (a ⊔ b) where
field
property : {x y : A} → (f x ≡ f y) → x ≡ y
Injection : {a b : _} {A : Set a} {B : Set b} (f : A → B) → Set (a ⊔ b)
Injection {A = A} f = {x y : A} → (f x ≡ f y) → x ≡ y
record Surjection {a b : _} {A : Set a} {B : Set b} (f : A → B) : Set (a ⊔ b) where
field
property : (b : B) → Sg A (λ a → f a ≡ b)
Surjection : {a b : _} {A : Set a} {B : Set b} (f : A → B) → Set (a ⊔ b)
Surjection {A = A} {B = B} f = (b : B) → Sg A (λ a → f a ≡ b)
record Bijection {a b : _} {A : Set a} {B : Set b} (f : A → B) : Set (a ⊔ b) where
field
@@ -31,28 +29,28 @@ module Functions where
isRight : (a : A) → inverse (f a) ≡ a
invertibleImpliesBijection : {a b : _} {A : Set a} {B : Set b} {f : A → B} → Invertible f → Bijection f
Injection.property (Bijection.inj (invertibleImpliesBijection {a} {b} {A} {B} {f} record { inverse = inverse ; isLeft = isLeft ; isRight = isRight })) {x} {y} fx=fy = ans
Bijection.inj (invertibleImpliesBijection {a} {b} {A} {B} {f} record { inverse = inverse ; isLeft = isLeft ; isRight = isRight }) {x} {y} fx=fy = ans
where
bl : inverse (f x) ≡ inverse (f y)
bl = applyEquality inverse fx=fy
ans : x ≡ y
ans rewrite equalityCommutative (isRight x) | equalityCommutative (isRight y) = bl
Surjection.property (Bijection.surj (invertibleImpliesBijection {a} {b} {A} {B} {f} record { inverse = inverse ; isLeft = isLeft ; isRight = isRight })) y = (inverse y , isLeft y)
Bijection.surj (invertibleImpliesBijection {a} {b} {A} {B} {f} record { inverse = inverse ; isLeft = isLeft ; isRight = isRight }) y = (inverse y , isLeft y)
bijectionImpliesInvertible : {a b : _} {A : Set a} {B : Set b} {f : A → B} → Bijection f → Invertible f
Invertible.inverse (bijectionImpliesInvertible record { inj = inj ; surj = record { property = property } }) b = underlying (property b)
Invertible.isLeft (bijectionImpliesInvertible {f = f} record { inj = inj ; surj = surj }) b with Surjection.property surj b
Invertible.inverse (bijectionImpliesInvertible record { inj = inj ; surj = surj }) b = underlying (surj b)
Invertible.isLeft (bijectionImpliesInvertible {f = f} record { inj = inj ; surj = surj }) b with surj b
Invertible.isLeft (bijectionImpliesInvertible {f = f} record { inj = inj ; surj = surj }) b | a , prop = prop
Invertible.isRight (bijectionImpliesInvertible {f = f} record { inj = inj ; surj = surj }) a with Surjection.property surj (f a)
Invertible.isRight (bijectionImpliesInvertible {f = f} record { inj = record { property = property } ; surj = surj }) a | a₁ , b = property b
Invertible.isRight (bijectionImpliesInvertible {f = f} record { inj = inj ; surj = surj }) a with surj (f a)
Invertible.isRight (bijectionImpliesInvertible {f = f} record { inj = property ; surj = surj }) a | a₁ , b = property b
injComp : {a b c : _} {A : Set a} {B : Set b} {C : Set c} {f : A → B} {g : B → C} → Injection f → Injection g → Injection (g ∘ f)
Injection.property (injComp {f = f} {g} record { property = propF } record { property = propG }) pr = propF (propG pr)
injComp {f = f} {g} propF propG pr = propF (propG pr)
surjComp : {a b c : _} {A : Set a} {B : Set b} {C : Set c} {f : A → B} {g : B → C} → Surjection f → Surjection g → Surjection (g ∘ f)
Surjection.property (surjComp {f = f} {g} record { property = propF } record { property = propG }) c with propG c
Surjection.property (surjComp {f = f} {g} record { property = propF } record { property = propG }) c | b , pr with propF b
Surjection.property (surjComp {f = f} {g} record { property = propF } record { property = propG }) c | b , pr | a , pr2 = a , pr'
surjComp {f = f} {g} propF propG c with propG c
surjComp {f = f} {g} propF propG c | b , pr with propF b
surjComp {f = f} {g} propF propG c | b , pr | a , pr2 = a , pr'
where
pr' : g (f a) ≡ c
pr' rewrite pr2 = pr
@@ -62,18 +60,18 @@ module Functions where
Bijection.surj (bijectionComp bijF bijG) = surjComp (Bijection.surj bijF) (Bijection.surj bijG)
compInjRightInj : {a b c : _} {A : Set a} {B : Set b} {C : Set c} {f : A → B} {g : B → C} → Injection (g ∘ f) → Injection f
Injection.property (compInjRightInj {f = f} {g} record { property = property }) {x} {y} fx=fy = property (applyEquality g fx=fy)
compInjRightInj {f = f} {g} property {x} {y} fx=fy = property (applyEquality g fx=fy)
compSurjLeftSurj : {a b c : _} {A : Set a} {B : Set b} {C : Set c} {f : A → B} {g : B → C} → Surjection (g ∘ f) → Surjection g
Surjection.property (compSurjLeftSurj {f = f} {g} record { property = property }) b with property b
Surjection.property (compSurjLeftSurj {f = f} {g} record { property = property }) b | a , b1 = f a , b1
compSurjLeftSurj {f = f} {g} property b with property b
compSurjLeftSurj {f = f} {g} property b | a , b1 = f a , b1
injectionPreservedUnderExtensionalEq : {a b : _} {A : Set a} {B : Set b} {f g : A → B} → Injection f → ({x : A} → f x ≡ g x) → Injection g
Injection.property (injectionPreservedUnderExtensionalEq {A = A} {B} {f} {g} record { property = property } prop) {x} {y} gx=gy = property (transitivity (prop {x}) (transitivity gx=gy (equalityCommutative (prop {y}))))
injectionPreservedUnderExtensionalEq {A = A} {B} {f} {g} property prop {x} {y} gx=gy = property (transitivity (prop {x}) (transitivity gx=gy (equalityCommutative (prop {y}))))
surjectionPreservedUnderExtensionalEq : {a b : _} {A : Set a} {B : Set b} {f g : A → B} → Surjection f → ({x : A} → f x ≡ g x) → Surjection g
Surjection.property (surjectionPreservedUnderExtensionalEq {f = f} {g} surj ext) b with (Surjection.property surj b)
Surjection.property (surjectionPreservedUnderExtensionalEq {f = f} {g} surj ext) b | a , pr = a , transitivity (equalityCommutative ext) pr
surjectionPreservedUnderExtensionalEq {f = f} {g} surj ext b with surj b
surjectionPreservedUnderExtensionalEq {f = f} {g} surj ext b | a , pr = a , transitivity (equalityCommutative ext) pr
bijectionPreservedUnderExtensionalEq : {a b : _} {A : Set a} {B : Set b} {f g : A → B} → Bijection f → ({x : A} → f x ≡ g x) → Bijection g
Bijection.inj (bijectionPreservedUnderExtensionalEq record { inj = inj ; surj = surj } ext) = injectionPreservedUnderExtensionalEq inj ext
@@ -88,8 +86,8 @@ module Functions where
id a = a
idIsBijective : {a : _} {A : Set a} → Bijection (id {a} {A})
Injection.property (Bijection.inj idIsBijective) pr = pr
Surjection.property (Bijection.surj idIsBijective) b = b , refl
Bijection.inj idIsBijective pr = pr
Bijection.surj idIsBijective b = b , refl
functionCompositionExtensionallyAssociative : {a b c d : _} {A : Set a} {B : Set b} {C : Set c} {D : Set d} → (f : A → B) → (g : B → C) → (h : C → D) → (x : A) → (h ∘ (g ∘ f)) x ≡ ((h ∘ g) ∘ f) x
functionCompositionExtensionallyAssociative f g h x = refl