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Tidy up groups (#64)
This commit is contained in:
Patrick Stevens
GitHub
@@ -2,26 +2,19 @@
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open import LogicalFormulae
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import Numbers.Naturals
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open import Numbers.Naturals.Naturals
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open import Numbers.Naturals.Order
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open import Vectors
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open import Semirings.Definition
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open import Categories.Definition
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module Categories.Category where
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postulate
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extensionality : {a b : _} {S : Set a}{T : S → Set b} {f g : (x : S) → T x} → ((x : S) → f x ≡ g x) → f ≡ g
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record Category {a b : _} : Set (lsuc (a ⊔ b)) where
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field
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objects : Set a
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arrows : objects → objects → Set b
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id : (x : objects) → arrows x x
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_∘_ : {x y z : objects} → arrows y z → arrows x y → arrows x z
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rightId : {x y : objects} → (f : arrows x y) → (id y) ∘ f ≡ f
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leftId : {x y : objects} → (f : arrows x y) → f ∘ (id x) ≡ f
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associative : {x y z w : objects} → (f : arrows z w) → (g : arrows y z) → (h : arrows x y) → (f ∘ (g ∘ h)) ≡ (f ∘ g) ∘ h
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dual : {a b : _} → Category {a} {b} → Category {a} {b}
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dual record { objects = objects ; arrows = arrows ; id = id ; _∘_ = _∘_ ; rightId = rightId ; leftId = leftId ; associative = associative } = record { objects = objects ; arrows = λ i j → arrows j i ; id = id ; _∘_ = λ {x y z} g f → f ∘ g ; rightId = λ {x y} f → leftId f ; leftId = λ {x y} f → rightId f ; associative = λ {x y z w} f g h → equalityCommutative (associative h g f) }
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dual record { objects = objects ; arrows = arrows ; id = id ; _∘_ = _∘_ ; rightId = rightId ; leftId = leftId ; compositionAssociative = associative } = record { objects = objects ; arrows = λ i j → arrows j i ; id = id ; _∘_ = λ {x y z} g f → f ∘ g ; rightId = λ {x y} f → leftId f ; leftId = λ {x y} f → rightId f ; associative = λ {x y z w} f g h → equalityCommutative (associative h g f) }
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SET : {a : _} → Category {lsuc a} {a}
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SET {a} = record { objects = Set a ; arrows = λ i j → (i → j) ; id = λ X x → x ; _∘_ = λ g f x → g (f x) ; rightId = λ f → refl ; leftId = λ f → refl ; associative = λ f g h → refl }
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@@ -55,7 +48,7 @@ NatPreorder = record { objects = ℕ ; arrows = λ m n → m ≤N n ; id = λ x
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leqUnique (inr a=b1) (inr a=b2) rewrite a=b1 | a=b2 = refl
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NatMonoid : Category {lzero} {lzero}
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NatMonoid = record { objects = True ; arrows = λ _ _ → ℕ ; id = λ x → 0 ; _∘_ = λ f g → f +N g ; rightId = λ f → refl ; leftId = λ f → addZeroRight f ; associative = λ a b c → equalityCommutative (additionNIsAssociative a b c) }
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NatMonoid = record { objects = True ; arrows = λ _ _ → ℕ ; id = λ x → 0 ; _∘_ = λ f g → f +N g ; rightId = λ f → refl ; leftId = λ f → Semiring.sumZeroRight ℕSemiring f ; associative = λ a b c → Semiring.+Associative ℕSemiring a b c }
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DISCRETE : {a : _} (X : Set a) → Category {a} {a}
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DISCRETE X = record { objects = X ; arrows = _≡_ ; id = λ x → refl ; _∘_ = λ f g → transitivity g f ; rightId = λ f → ≡Unique (transitivity f refl) f ; leftId = λ f → ≡Unique (transitivity refl f) f ; associative = λ f g h → ≡Unique (transitivity (transitivity h g) f) (transitivity h (transitivity g f)) }
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20
Categories/Definition.agda
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20
Categories/Definition.agda
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@@ -0,0 +1,20 @@
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{-# OPTIONS --warning=error --safe --without-K #-}
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open import LogicalFormulae
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import Numbers.Naturals.Naturals
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open import Numbers.Naturals.Order
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open import Vectors
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open import Semirings.Definition
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module Categories.Definition where
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record Category {a b : _} : Set (lsuc (a ⊔ b)) where
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field
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objects : Set a
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arrows : objects → objects → Set b
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id : (x : objects) → arrows x x
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_∘_ : {x y z : objects} → arrows y z → arrows x y → arrows x z
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rightId : {x y : objects} → (f : arrows x y) → (id y) ∘ f ≡ f
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leftId : {x y : objects} → (f : arrows x y) → f ∘ (id x) ≡ f
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compositionAssociative : {x y z w : objects} → (f : arrows z w) → (g : arrows y z) → (h : arrows x y) → (f ∘ (g ∘ h)) ≡ (f ∘ g) ∘ h
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30
Categories/Examples.agda
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30
Categories/Examples.agda
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@@ -0,0 +1,30 @@
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{-# OPTIONS --warning=error --safe --without-K #-}
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open import LogicalFormulae
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import Numbers.Naturals.Naturals
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open import Numbers.Naturals.Order
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open import Vectors
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open import Semirings.Definition
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open import Categories.Definition
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open import Groups.Definition
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module Categories.Examples where
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SET : {a : _} → Category {lsuc a} {a}
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Category.objects (SET {a}) = Set a
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Category.arrows (SET {a}) = λ a b → (a → b)
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Category.id (SET {a}) = λ x → λ y → y
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Category._∘_ (SET {a}) = λ f g → λ x → f (g x)
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Category.rightId (SET {a}) = λ f → refl
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Category.leftId (SET {a}) = λ f → refl
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Category.compositionAssociative (SET {a}) = λ f g h → refl
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GROUP : {a : _} → Category {lsuc a} {a}
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Category.objects (GROUP {a}) = Group {! !} {! !}
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Category.arrows (GROUP {a}) = {! !}
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Category.id (GROUP {a}) = {! !}
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Category._∘_ (GROUP {a}) = {! !}
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Category.rightId (GROUP {a}) = {! !}
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Category.leftId (GROUP {a}) = {! !}
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Category.compositionAssociative (GROUP {a}) = {! !}
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@@ -13,6 +13,11 @@ open import Numbers.Integers.Integers
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open import Lists.Lists
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open import Groups.Groups
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open import Groups.Abelian.Lemmas
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open import Groups.DirectSum.Definition
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open import Groups.FiniteGroups.Definition
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open import Groups.Homomorphisms.Lemmas
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open import Groups.Isomorphisms.Lemmas
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open import Groups.FinitePermutations
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open import Groups.Lemmas
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open import Groups.Groups2
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@@ -62,4 +67,3 @@ open import Fields.CauchyCompletion.Group
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open import Fields.CauchyCompletion.Ring
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module Everything.Safe where
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@@ -4,14 +4,14 @@
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open import Numbers.Primes.PrimeNumbers
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open import Numbers.Primes.IntegerFactorisation
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open import Numbers.Rationals
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open import Numbers.RationalsLemmas
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open import Numbers.Rationals.Definition
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open import Numbers.Rationals.Lemmas
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open import Logic.PropositionalLogic
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open import Logic.PropositionalLogicExamples
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open import Logic.PropositionalAxiomsTautology
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open import Numbers.Modulo.IntegersModN
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open import Numbers.Modulo.Group
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open import Sets.FinSetWithK
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@@ -23,5 +23,6 @@ open import Groups.Examples.ExampleSheet1
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open import Groups.LectureNotes.Lecture1
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open import LectureNotes.NumbersAndSets.Lecture1
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open import LectureNotes.Groups.Lecture1
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module Everything.WithK where
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@@ -7,6 +7,7 @@ open import Rings.Lemmas
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open import Rings.Orders.Partial.Definition
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open import Rings.Orders.Total.Definition
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open import Groups.Definition
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open import Groups.Lemmas
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open import Groups.Groups
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open import Fields.Fields
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open import Sets.EquivalenceRelations
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@@ -9,6 +9,7 @@ open import Rings.Orders.Partial.Definition
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open import Rings.Orders.Total.Definition
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open import Groups.Definition
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open import Groups.Groups
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open import Groups.Lemmas
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open import Fields.Fields
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open import Sets.EquivalenceRelations
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open import Sequences
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@@ -123,7 +124,7 @@ abstract
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... | f with totality 0G (am + inverse aN)
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r | am-aN<e/4 | inl (inl 0<am-aN) = SetoidPartialOrder.<Transitive pOrder (<WellDefined (Equivalence.transitive eq (invContravariant additiveGroup) (Equivalence.transitive eq groupIsAbelian (+WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup _)))) (Equivalence.reflexive eq) (lemm2' _ 0<am-aN)) 0<e/4
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r | f | inl (inr x) = <WellDefined (Equivalence.transitive eq (invContravariant additiveGroup) (Equivalence.transitive eq groupIsAbelian (+WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup _)))) (Equivalence.reflexive eq) f
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r | am-aN<e/4 | inr 0=am-aN = <WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq 0=am-aN) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq (invIdentity additiveGroup)) (inverseWellDefined additiveGroup 0=am-aN)) (inverseWellDefined additiveGroup groupIsAbelian)) (invContravariant additiveGroup)) (+WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup _)))) (Equivalence.reflexive eq) am-aN<e/4
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r | am-aN<e/4 | inr 0=am-aN = <WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq 0=am-aN) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq (invIdent additiveGroup)) (inverseWellDefined additiveGroup 0=am-aN)) (inverseWellDefined additiveGroup groupIsAbelian)) (invContravariant additiveGroup)) (+WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup _)))) (Equivalence.reflexive eq) am-aN<e/4
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q : ((inverse (index (CauchyCompletion.elts a) m)) + (index (CauchyCompletion.elts a) (succ N) + e/2)) < (e/4 + e/2)
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q = <WellDefined (Equivalence.symmetric eq +Associative) (Equivalence.reflexive eq) (orderRespectsAddition r e/2)
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@@ -8,6 +8,7 @@ open import Rings.Orders.Partial.Definition
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open import Rings.Orders.Total.Definition
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open import Groups.Definition
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open import Groups.Groups
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open import Groups.Lemmas
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open import Fields.Fields
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open import Fields.Orders.Total.Definition
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open import Sets.EquivalenceRelations
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@@ -7,6 +7,8 @@ open import Rings.Lemmas
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open import Rings.Orders.Partial.Definition
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open import Rings.Orders.Total.Definition
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open import Groups.Definition
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open import Groups.Lemmas
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open import Groups.Homomorphisms.Definition
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open import Groups.Groups
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open import Fields.Fields
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open import Sets.EquivalenceRelations
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@@ -50,7 +52,7 @@ CinvRight : {a : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a +C
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CinvRight {a} ε 0<e = 0 , ans
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where
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ans : {m : ℕ} → (0 <N m) → abs (index (apply _+_ (CauchyCompletion.elts (a +C (-C a))) (map inverse (CauchyCompletion.elts (injection 0G)))) m) < ε
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ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a +C (-C a))) (map inverse (CauchyCompletion.elts (injection 0G))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (map inverse (CauchyCompletion.elts a)) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts a) inverse m) | equalityCommutative (mapAndIndex (constSequence 0G) inverse m) | indexAndConst 0G m = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.transitive eq (+WellDefined invRight (invIdentity (Ring.additiveGroup R))) identRight)) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
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ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a +C (-C a))) (map inverse (CauchyCompletion.elts (injection 0G))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (map inverse (CauchyCompletion.elts a)) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts a) inverse m) | equalityCommutative (mapAndIndex (constSequence 0G) inverse m) | indexAndConst 0G m = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.transitive eq (+WellDefined invRight (invIdent (Ring.additiveGroup R))) identRight)) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
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CGroup : Group cauchyCompletionSetoid _+C_
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Group.+WellDefined CGroup {a} {b} {c} {d} x y = additionWellDefined {a} {c} {b} {d} x y
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@@ -8,6 +8,7 @@ open import Rings.Orders.Partial.Definition
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open import Rings.Orders.Total.Definition
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open import Groups.Definition
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open import Groups.Groups
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open import Groups.Lemmas
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open import Fields.Fields
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open import Sets.EquivalenceRelations
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open import Sequences
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@@ -8,6 +8,7 @@ open import Rings.Orders.Partial.Definition
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open import Rings.Orders.Total.Definition
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open import Groups.Definition
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open import Groups.Groups
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open import Groups.Lemmas
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open import Fields.Fields
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open import Sets.EquivalenceRelations
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open import Sequences
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@@ -8,6 +8,7 @@ open import Rings.Orders.Partial.Definition
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open import Rings.Orders.Total.Definition
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open import Groups.Definition
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open import Groups.Groups
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open import Groups.Lemmas
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open import Fields.Fields
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open import Sets.EquivalenceRelations
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open import Sequences
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@@ -2,6 +2,7 @@
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open import LogicalFormulae
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open import Groups.Groups
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open import Groups.Homomorphisms.Definition
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open import Groups.Definition
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open import Groups.Lemmas
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open import Rings.Definition
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@@ -2,6 +2,7 @@
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open import LogicalFormulae
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open import Groups.Groups
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open import Groups.Lemmas
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open import Groups.Definition
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open import Rings.Definition
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open import Rings.Orders.Partial.Definition
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17
Groups/Abelian/Definition.agda
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17
Groups/Abelian/Definition.agda
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@@ -0,0 +1,17 @@
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{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Setoids.Setoids
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open import Functions
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import Numbers.Naturals.Naturals
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open import Sets.FinSet
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open import Groups.Definition
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open import Sets.EquivalenceRelations
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module Groups.Abelian.Definition where
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record AbelianGroup {a} {b} {A : Set a} {S : Setoid {a} {b} A} {_·_ : A → A → A} (G : Group S _·_) : Set (lsuc a ⊔ b) where
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open Setoid S
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field
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commutative : {a b : A} → (a · b) ∼ (b · a)
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29
Groups/Abelian/Lemmas.agda
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29
Groups/Abelian/Lemmas.agda
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@@ -0,0 +1,29 @@
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{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Setoids.Setoids
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open import Functions
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import Numbers.Naturals.Naturals
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open import Sets.FinSet
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open import Groups.Definition
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open import Sets.EquivalenceRelations
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open import Groups.Abelian.Definition
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open import Groups.Homomorphisms.Definition
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open import Groups.DirectSum.Definition
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open import Groups.Subgroups.Definition
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open import Groups.Isomorphisms.Definition
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module Groups.Abelian.Lemmas where
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directSumAbelianGroup : {m n o p : _} → {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} {underG : Group S _·A_} {underH : Group T _·B_} (G : AbelianGroup underG) (h : AbelianGroup underH) → (AbelianGroup (directSum underG underH))
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AbelianGroup.commutative (directSumAbelianGroup {A = A} {B} G H) = AbelianGroup.commutative G ,, AbelianGroup.commutative H
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subgroupOfAbelianIsAbelian : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : B → A} {fHom : GroupHom H G f} → Subgroup G H fHom → AbelianGroup G → AbelianGroup H
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AbelianGroup.commutative (subgroupOfAbelianIsAbelian {S = S} {_+B_ = _+B_} {fHom = fHom} record { fInj = fInj } record { commutative = commutative }) {x} {y} = SetoidInjection.injective fInj (transitive (GroupHom.groupHom fHom) (transitive commutative (symmetric (GroupHom.groupHom fHom))))
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where
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open Setoid S
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open Equivalence eq
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abelianIsGroupProperty : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} → GroupsIsomorphic G H → AbelianGroup H → AbelianGroup G
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abelianIsGroupProperty iso abH = subgroupOfAbelianIsAbelian {fHom = GroupIso.groupHom (GroupsIsomorphic.proof iso)} (record { fInj = SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof iso)) }) abH
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22
Groups/DirectSum/Definition.agda
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22
Groups/DirectSum/Definition.agda
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@@ -0,0 +1,22 @@
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{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Setoids.Setoids
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open import Functions
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import Numbers.Naturals.Naturals
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open import Sets.FinSet
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open import Groups.Definition
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open import Sets.EquivalenceRelations
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module Groups.DirectSum.Definition where
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directSum : {m n o p : _} → {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (h : Group T _·B_) → Group (directSumSetoid S T) (λ x1 y1 → (((_&&_.fst x1) ·A (_&&_.fst y1)) ,, ((_&&_.snd x1) ·B (_&&_.snd y1))))
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Group.+WellDefined (directSum {A = A} {B} g h) (s ,, t) (u ,, v) = Group.+WellDefined g s u ,, Group.+WellDefined h t v
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Group.0G (directSum {A = A} {B} g h) = (Group.0G g ,, Group.0G h)
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Group.inverse (directSum {A = A} {B} g h) (g1 ,, h1) = (Group.inverse g g1) ,, (Group.inverse h h1)
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Group.+Associative (directSum {A = A} {B} g h) = Group.+Associative g ,, Group.+Associative h
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Group.identRight (directSum {A = A} {B} g h) = Group.identRight g ,, Group.identRight h
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Group.identLeft (directSum {A = A} {B} g h) = Group.identLeft g ,, Group.identLeft h
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Group.invLeft (directSum {A = A} {B} g h) = Group.invLeft g ,, Group.invLeft h
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Group.invRight (directSum {A = A} {B} g h) = Group.invRight g ,, Group.invRight h
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@@ -8,6 +8,12 @@ open import Numbers.Naturals.Naturals
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open import Numbers.Integers.Integers
|
||||
open import Sets.FinSet
|
||||
open import Groups.Definition
|
||||
open import Groups.Homomorphisms.Definition
|
||||
open import Groups.Homomorphisms.Lemmas
|
||||
open import Groups.Isomorphisms.Definition
|
||||
open import Groups.Abelian.Definition
|
||||
open import Groups.Subgroups.Definition
|
||||
open import Groups.Lemmas
|
||||
open import Groups.Groups
|
||||
open import Rings.Definition
|
||||
open import Rings.Lemmas
|
||||
|
||||
20
Groups/FiniteGroups/Definition.agda
Normal file
20
Groups/FiniteGroups/Definition.agda
Normal file
@@ -0,0 +1,20 @@
|
||||
{-# OPTIONS --safe --warning=error --without-K #-}
|
||||
|
||||
open import LogicalFormulae
|
||||
open import Setoids.Setoids
|
||||
open import Functions
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Sets.FinSet
|
||||
open import Groups.Definition
|
||||
open import Sets.EquivalenceRelations
|
||||
|
||||
module Groups.FiniteGroups.Definition where
|
||||
|
||||
record FiniteGroup {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_·_ : A → A → A} (G : Group S _·_) {c : _} (quotientSet : Set c) : Set (a ⊔ b ⊔ c) where
|
||||
field
|
||||
toSet : SetoidToSet S quotientSet
|
||||
finite : FiniteSet quotientSet
|
||||
|
||||
groupOrder : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_·_ : A → A → A} {underG : Group S _·_} {c : _} {quotientSet : Set c} (G : FiniteGroup underG quotientSet) → ℕ
|
||||
groupOrder record { toSet = toSet ; finite = record { size = size ; mapping = mapping ; bij = bij } } = size
|
||||
@@ -8,123 +8,14 @@ open import Numbers.Naturals.Naturals
|
||||
open import Sets.FinSet
|
||||
open import Groups.Definition
|
||||
open import Sets.EquivalenceRelations
|
||||
open import Groups.Homomorphisms.Definition
|
||||
open import Groups.Lemmas
|
||||
open import Groups.Homomorphisms.Lemmas
|
||||
|
||||
module Groups.Groups where
|
||||
|
||||
reflGroupWellDefined : {lvl : _} {A : Set lvl} {m n x y : A} {op : A → A → A} → m ≡ x → n ≡ y → (op m n) ≡ (op x y)
|
||||
reflGroupWellDefined {lvl} {A} {m} {n} {.m} {.n} {op} refl refl = refl
|
||||
|
||||
record AbelianGroup {a} {b} {A : Set a} {S : Setoid {a} {b} A} {_·_ : A → A → A} (G : Group S _·_) : Set (lsuc a ⊔ b) where
|
||||
open Setoid S
|
||||
field
|
||||
commutative : {a b : A} → (a · b) ∼ (b · a)
|
||||
|
||||
groupsHaveLeftCancellation : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → (x y z : A) → (Setoid._∼_ S (x · y) (x · z)) → (Setoid._∼_ S y z)
|
||||
groupsHaveLeftCancellation {_·_ = _·_} {S} g x y z pr = o
|
||||
where
|
||||
open Group g renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
transitive = Equivalence.transitive (Setoid.eq S)
|
||||
symmetric = Equivalence.symmetric (Setoid.eq S)
|
||||
j : ((x ^-1) · x) · y ∼ (x ^-1) · (x · z)
|
||||
j = Equivalence.transitive eq (symmetric (+Associative {x ^-1} {x} {y})) (+WellDefined ~refl pr)
|
||||
k : ((x ^-1) · x) · y ∼ ((x ^-1) · x) · z
|
||||
k = transitive j +Associative
|
||||
l : 0G · y ∼ ((x ^-1) · x) · z
|
||||
l = transitive (+WellDefined (symmetric invLeft) ~refl) k
|
||||
m : 0G · y ∼ 0G · z
|
||||
m = transitive l (+WellDefined invLeft ~refl)
|
||||
n : y ∼ 0G · z
|
||||
n = transitive (symmetric identLeft) m
|
||||
o : y ∼ z
|
||||
o = transitive n identLeft
|
||||
|
||||
groupsHaveRightCancellation : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → (x y z : A) → (Setoid._∼_ S (y · x) (z · x)) → (Setoid._∼_ S y z)
|
||||
groupsHaveRightCancellation {_·_ = _·_} {S} g x y z pr = transitive m identRight
|
||||
where
|
||||
open Group g renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
transitive = Equivalence.transitive (Setoid.eq S)
|
||||
symmetric = Equivalence.symmetric (Setoid.eq S)
|
||||
i : (y · x) · (x ^-1) ∼ (z · x) · (x ^-1)
|
||||
i = +WellDefined pr ~refl
|
||||
j : y · (x · (x ^-1)) ∼ (z · x) · (x ^-1)
|
||||
j = transitive +Associative i
|
||||
j' : y · 0G ∼ (z · x) · (x ^-1)
|
||||
j' = transitive (+WellDefined ~refl (symmetric invRight)) j
|
||||
k : y ∼ (z · x) · (x ^-1)
|
||||
k = transitive (symmetric identRight) j'
|
||||
l : y ∼ z · (x · (x ^-1))
|
||||
l = transitive k (symmetric +Associative)
|
||||
m : y ∼ z · 0G
|
||||
m = transitive l (+WellDefined ~refl invRight)
|
||||
|
||||
replaceGroupOp : {l m : _} {A : Set l} {S : Setoid {l} {m} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b c d w x y z : A} → (Setoid._∼_ S a c) → (Setoid._∼_ S b d) → (Setoid._∼_ S w y) → (Setoid._∼_ S x z) → Setoid._∼_ S (a · b) (w · x) → Setoid._∼_ S (c · d) (y · z)
|
||||
replaceGroupOp {S = S} {_·_} G a~c b~d w~y x~z pr = transitive (symmetric (+WellDefined a~c b~d)) (transitive pr (+WellDefined w~y x~z))
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
replaceGroupOpRight : {l m : _} {A : Set l} {S : Setoid {l} {m} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b c x : A} → (Setoid._∼_ S a (b · c)) → (Setoid._∼_ S c x) → (Setoid._∼_ S a (b · x))
|
||||
replaceGroupOpRight {S = S} {_·_} G a~bc c~x = transitive a~bc (+WellDefined reflexive c~x)
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
inverseWellDefined : {l m : _} {A : Set l} {S : Setoid {l} {m} A} {_·_ : A → A → A} (G : Group S _·_) → {x y : A} → (Setoid._∼_ S x y) → Setoid._∼_ S (Group.inverse G x) (Group.inverse G y)
|
||||
inverseWellDefined {S = S} {_·_} G {x} {y} x~y = groupsHaveRightCancellation G x (inverse x) (inverse y) q
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
p : inverse x · x ∼ inverse y · y
|
||||
p = transitive invLeft (symmetric invLeft)
|
||||
q : inverse x · x ∼ inverse y · x
|
||||
q = replaceGroupOpRight G {_·_ (inverse x) x} {inverse y} {y} {x} p (symmetric x~y)
|
||||
|
||||
rightInversesAreUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (x : A) → (y : A) → Setoid._∼_ S (y · x) (Group.0G G) → Setoid._∼_ S y (Group.inverse G x)
|
||||
rightInversesAreUnique {S = S} {_·_} G x y f = transitive i (transitive j (transitive k (transitive l m)))
|
||||
where
|
||||
open Group G renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
i : y ∼ y · 0G
|
||||
j : y · 0G ∼ y · (x · (x ^-1))
|
||||
k : y · (x · (x ^-1)) ∼ (y · x) · (x ^-1)
|
||||
l : (y · x) · (x ^-1) ∼ 0G · (x ^-1)
|
||||
m : 0G · (x ^-1) ∼ x ^-1
|
||||
i = symmetric identRight
|
||||
j = +WellDefined ~refl (symmetric invRight)
|
||||
k = +Associative
|
||||
l = +WellDefined f ~refl
|
||||
m = identLeft
|
||||
|
||||
leftInversesAreUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {x : A} → {y : A} → Setoid._∼_ S (x · y) (Group.0G G) → Setoid._∼_ S y (Group.inverse G x)
|
||||
leftInversesAreUnique {S = S} {_·_} G {x} {y} f = rightInversesAreUnique G x y l
|
||||
where
|
||||
open Group G renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
i : y · (x · y) ∼ y · 0G
|
||||
i' : y · (x · y) ∼ y
|
||||
j : (y · x) · y ∼ y
|
||||
k : (y · x) · y ∼ 0G · y
|
||||
l : y · x ∼ 0G
|
||||
i = +WellDefined ~refl f
|
||||
i' = transitive i identRight
|
||||
j = transitive (symmetric +Associative) i'
|
||||
k = transitive j (symmetric identLeft)
|
||||
l = groupsHaveRightCancellation G y (y · x) 0G k
|
||||
|
||||
invTwice : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (x : A) → Setoid._∼_ S (Group.inverse G (Group.inverse G x)) x
|
||||
invTwice {S = S} {_·_} G x = symmetric (rightInversesAreUnique G (x ^-1) x invRight)
|
||||
where
|
||||
open Group G renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
fourWay+Associative : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {r s t u : A} → (Setoid._∼_ S) (r · ((s · t) · u)) ((r · s) · (t · u))
|
||||
fourWay+Associative {S = S} {_·_} G {r} {s} {t} {u} = transitive p1 (transitive p2 p3)
|
||||
where
|
||||
@@ -152,153 +43,6 @@ fourWay+Associative'' {S = S} {_·_ = _·_} G = transitive +Associative (symmetr
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
invContravariant : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {x y : A} → (Setoid._∼_ S (Group.inverse G (x · y)) ((Group.inverse G y) · (Group.inverse G x)))
|
||||
invContravariant {S = S} {_·_} G {x} {y} = ans
|
||||
where
|
||||
open Group G renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
otherInv = (y ^-1) · (x ^-1)
|
||||
many+Associatives : x · ((y · (y ^-1)) · (x ^-1)) ∼ (x · y) · ((y ^-1) · (x ^-1))
|
||||
oneMult : (x · y) · otherInv ∼ x · (x ^-1)
|
||||
many+Associatives = fourWay+Associative G
|
||||
oneMult = symmetric (transitive (Group.+WellDefined G reflexive (transitive (symmetric (Group.identLeft G)) (Group.+WellDefined G (symmetric (Group.invRight G)) reflexive))) many+Associatives)
|
||||
otherInvIsInverse : (x · y) · otherInv ∼ 0G
|
||||
otherInvIsInverse = transitive oneMult (Group.invRight G)
|
||||
ans : (x · y) ^-1 ∼ (y ^-1) · (x ^-1)
|
||||
ans = symmetric (leftInversesAreUnique G otherInvIsInverse)
|
||||
|
||||
invIdentity : {l m : _} → {A : Set l} → {S : Setoid {l} {m} A} → {_·_ : A → A → A} → (G : Group S _·_) → Setoid._∼_ S ((Group.inverse G) (Group.0G G)) (Group.0G G)
|
||||
invIdentity {S = S} G = transitive (symmetric identLeft) invRight
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
transferToRight : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S (a · (Group.inverse G b)) (Group.0G G) → Setoid._∼_ S a b
|
||||
transferToRight {S = S} {_·_ = _·_} G {a} {b} ab-1 = transitive (symmetric (invTwice G a)) (transitive u (invTwice G b))
|
||||
where
|
||||
open Setoid S
|
||||
open Group G
|
||||
open Equivalence eq
|
||||
t : inverse a ∼ inverse b
|
||||
t = symmetric (leftInversesAreUnique G ab-1)
|
||||
u : inverse (inverse a) ∼ inverse (inverse b)
|
||||
u = inverseWellDefined G t
|
||||
|
||||
transferToRight' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S (a · b) (Group.0G G) → Setoid._∼_ S a (Group.inverse G b)
|
||||
transferToRight' {S = S} {_·_ = _·_} G {a} {b} ab-1 = transferToRight G lemma
|
||||
where
|
||||
open Setoid S
|
||||
open Group G
|
||||
open Equivalence eq
|
||||
lemma : a · (inverse (inverse b)) ∼ 0G
|
||||
lemma = transitive (+WellDefined reflexive (invTwice G b)) ab-1
|
||||
|
||||
transferToRight'' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S a b → Setoid._∼_ S (a · (Group.inverse G b)) (Group.0G G)
|
||||
transferToRight'' {S = S} {_·_ = _·_} G {a} {b} a~b = transitive (Group.+WellDefined G a~b reflexive) invRight
|
||||
where
|
||||
open Group G
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
directSumGroup : {m n o p : _} → {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (h : Group T _·B_) → Group (directSumSetoid S T) (λ x1 y1 → (((_&&_.fst x1) ·A (_&&_.fst y1)) ,, ((_&&_.snd x1) ·B (_&&_.snd y1))))
|
||||
Group.+WellDefined (directSumGroup {A = A} {B} g h) (s ,, t) (u ,, v) = Group.+WellDefined g s u ,, Group.+WellDefined h t v
|
||||
Group.0G (directSumGroup {A = A} {B} g h) = (Group.0G g ,, Group.0G h)
|
||||
Group.inverse (directSumGroup {A = A} {B} g h) (g1 ,, h1) = (Group.inverse g g1) ,, (Group.inverse h h1)
|
||||
Group.+Associative (directSumGroup {A = A} {B} g h) = Group.+Associative g ,, Group.+Associative h
|
||||
Group.identRight (directSumGroup {A = A} {B} g h) = Group.identRight g ,, Group.identRight h
|
||||
Group.identLeft (directSumGroup {A = A} {B} g h) = Group.identLeft g ,, Group.identLeft h
|
||||
Group.invLeft (directSumGroup {A = A} {B} g h) = Group.invLeft g ,, Group.invLeft h
|
||||
Group.invRight (directSumGroup {A = A} {B} g h) = Group.invRight g ,, Group.invRight h
|
||||
|
||||
directSumAbelianGroup : {m n o p : _} → {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} {underG : Group S _·A_} {underH : Group T _·B_} (G : AbelianGroup underG) (h : AbelianGroup underH) → (AbelianGroup (directSumGroup underG underH))
|
||||
AbelianGroup.commutative (directSumAbelianGroup {A = A} {B} G H) = AbelianGroup.commutative G ,, AbelianGroup.commutative H
|
||||
|
||||
record GroupHom {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
|
||||
open Group H
|
||||
open Setoid T
|
||||
field
|
||||
groupHom : {x y : A} → f (x ·A y) ∼ (f x) ·B (f y)
|
||||
wellDefined : {x y : A} → Setoid._∼_ S x y → f x ∼ f y
|
||||
|
||||
record InjectiveGroupHom {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {underf : A → B} (f : GroupHom G H underf) : Set (m ⊔ n ⊔ o ⊔ p) where
|
||||
open Setoid S renaming (_∼_ to _∼A_)
|
||||
open Setoid T renaming (_∼_ to _∼B_)
|
||||
field
|
||||
injective : SetoidInjection S T underf
|
||||
|
||||
imageOfIdentityIsIdentity : {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {B : Set n} {T : Setoid {n} {p} B} {_·A_ : A → A → A} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {f : A → B} → (hom : GroupHom G H f) → Setoid._∼_ T (f (Group.0G G)) (Group.0G H)
|
||||
imageOfIdentityIsIdentity {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} {G = G} {H = H} {f = f} hom = Equivalence.symmetric (Setoid.eq T) t
|
||||
where
|
||||
open Group H
|
||||
open Setoid T
|
||||
id2 : Setoid._∼_ S (Group.0G G) ((Group.0G G) ·A (Group.0G G))
|
||||
id2 = Equivalence.symmetric (Setoid.eq S) (Group.identRight G)
|
||||
r : f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
|
||||
s : 0G ·B f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
|
||||
t : 0G ∼ f (Group.0G G)
|
||||
t = groupsHaveRightCancellation H (f (Group.0G G)) 0G (f (Group.0G G)) s
|
||||
s = Equivalence.transitive (Setoid.eq T) identLeft r
|
||||
r = Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined hom id2) (GroupHom.groupHom hom)
|
||||
|
||||
groupHomsCompose : {m n o r s t : _} {A : Set m} {S : Setoid {m} {r} A} {_+A_ : A → A → A} {B : Set n} {T : Setoid {n} {s} B} {_+B_ : B → B → B} {C : Set o} {U : Setoid {o} {t} C} {_+C_ : C → C → C} {G : Group S _+A_} {H : Group T _+B_} {I : Group U _+C_} {f : A → B} {g : B → C} (fHom : GroupHom G H f) (gHom : GroupHom H I g) → GroupHom G I (g ∘ f)
|
||||
GroupHom.wellDefined (groupHomsCompose {G = G} {H} {I} {f} {g} fHom gHom) {x} {y} pr = GroupHom.wellDefined gHom (GroupHom.wellDefined fHom pr)
|
||||
GroupHom.groupHom (groupHomsCompose {S = S} {_+A_ = _·A_} {T = T} {U = U} {_+C_ = _·C_} {G = G} {H} {I} {f} {g} fHom gHom) {x} {y} = answer
|
||||
where
|
||||
open Group I
|
||||
answer : (Setoid._∼_ U) ((g ∘ f) (x ·A y)) ((g ∘ f) x ·C (g ∘ f) y)
|
||||
answer = (Equivalence.transitive (Setoid.eq U)) (GroupHom.wellDefined gHom (GroupHom.groupHom fHom {x} {y}) ) (GroupHom.groupHom gHom {f x} {f y})
|
||||
|
||||
record GroupIso {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
|
||||
open Setoid S renaming (_∼_ to _∼G_)
|
||||
open Setoid T renaming (_∼_ to _∼H_)
|
||||
field
|
||||
groupHom : GroupHom G H f
|
||||
bij : SetoidBijection S T f
|
||||
|
||||
record GroupsIsomorphic {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) : Set (m ⊔ n ⊔ o ⊔ p) where
|
||||
field
|
||||
isomorphism : A → B
|
||||
proof : GroupIso G H isomorphism
|
||||
|
||||
groupIsosCompose : {m n o r s t : _} {A : Set m} {S : Setoid {m} {r} A} {_+A_ : A → A → A} {B : Set n} {T : Setoid {n} {s} B} {_+B_ : B → B → B} {C : Set o} {U : Setoid {o} {t} C} {_+C_ : C → C → C} {G : Group S _+A_} {H : Group T _+B_} {I : Group U _+C_} {f : A → B} {g : B → C} (fHom : GroupIso G H f) (gHom : GroupIso H I g) → GroupIso G I (g ∘ f)
|
||||
GroupIso.groupHom (groupIsosCompose fHom gHom) = groupHomsCompose (GroupIso.groupHom fHom) (GroupIso.groupHom gHom)
|
||||
GroupIso.bij (groupIsosCompose {A = A} {S = S} {T = T} {C = C} {U = U} {f = f} {g = g} fHom gHom) = record { inj = record { injective = λ pr → (SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij fHom))) (SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij gHom)) pr) ; wellDefined = +WellDefined } ; surj = record { surjective = surj ; wellDefined = +WellDefined } }
|
||||
where
|
||||
open Setoid S renaming (_∼_ to _∼A_)
|
||||
open Setoid U renaming (_∼_ to _∼C_)
|
||||
+WellDefined : {x y : A} → (x ∼A y) → (g (f x) ∼C g (f y))
|
||||
+WellDefined = GroupHom.wellDefined (groupHomsCompose (GroupIso.groupHom fHom) (GroupIso.groupHom gHom))
|
||||
surj : {x : C} → Sg A (λ a → (g (f a) ∼C x))
|
||||
surj {x} with SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij gHom)) {x}
|
||||
surj {x} | a , prA with SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij fHom)) {a}
|
||||
... | b , prB = b , transitive (GroupHom.wellDefined (GroupIso.groupHom gHom) prB) prA
|
||||
where
|
||||
open Equivalence (Setoid.eq U)
|
||||
|
||||
record FiniteGroup {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_·_ : A → A → A} (G : Group S _·_) {c : _} (quotientSet : Set c) : Set (a ⊔ b ⊔ c) where
|
||||
field
|
||||
toSet : SetoidToSet S quotientSet
|
||||
finite : FiniteSet quotientSet
|
||||
|
||||
groupOrder : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_·_ : A → A → A} {underG : Group S _·_} {c : _} {quotientSet : Set c} (G : FiniteGroup underG quotientSet) → ℕ
|
||||
groupOrder record { toSet = toSet ; finite = record { size = size ; mapping = mapping ; bij = bij } } = size
|
||||
|
||||
--groupIsoInvertible : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d}} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : A → B} → (iso : GroupIso G H f) → GroupIso H G (Invertible.inverse (bijectionImpliesInvertible (GroupIso.bijective iso)))
|
||||
--GroupHom.groupHom (GroupIso.groupHom (groupIsoInvertible {G = G} {H} {f} iso)) {x} {y} = {!!}
|
||||
-- where
|
||||
-- open Group G renaming (_·_ to _+G_)
|
||||
-- open Group H renaming (_·_ to _+H_)
|
||||
--GroupHom.wellDefined (GroupIso.groupHom (groupIsoInvertible {G = G} {H} {f} iso)) {x} {y} x∼y = {!GroupHom.wellDefined x∼y!}
|
||||
--GroupIso.bijective (groupIsoInvertible {G = G} {H} {f} iso) = invertibleImpliesBijection (inverseIsInvertible (bijectionImpliesInvertible (GroupIso.bijective iso)))
|
||||
|
||||
homRespectsInverse : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {underF : A → B} → (f : GroupHom G H underF) → {x : A} → Setoid._∼_ T (underF (Group.inverse G x)) (Group.inverse H (underF x))
|
||||
homRespectsInverse {T = T} {_·A_ = _·A_} {_·B_ = _·B_} {G = G} {H = H} {underF = f} fHom {x} = rightInversesAreUnique H (f x) (f (Group.inverse G x)) (transitive (symmetric (GroupHom.groupHom fHom)) (transitive (GroupHom.wellDefined fHom (Group.invLeft G)) (imageOfIdentityIsIdentity fHom)))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
|
||||
quotientGroupSetoid : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → (Setoid {a} {d} A)
|
||||
quotientGroupSetoid {A = A} {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H} {f} fHom = ansSetoid
|
||||
where
|
||||
@@ -311,7 +55,7 @@ quotientGroupSetoid {A = A} {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H
|
||||
Equivalence.symmetric (Setoid.eq ansSetoid) {m} {n} pr = i
|
||||
where
|
||||
g : f (Group.inverse G (m ·A Group.inverse G n)) ∼ 0G
|
||||
g = transitive (homRespectsInverse fHom {m ·A Group.inverse G n}) (transitive (inverseWellDefined H pr) (invIdentity H))
|
||||
g = transitive (homRespectsInverse fHom {m ·A Group.inverse G n}) (transitive (inverseWellDefined H pr) (invIdent H))
|
||||
h : f (Group.inverse G (Group.inverse G n) ·A Group.inverse G m) ∼ 0G
|
||||
h = transitive (GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (invContravariant G))) g
|
||||
i : f (n ·A Group.inverse G m) ∼ 0G
|
||||
@@ -382,20 +126,15 @@ Group.invLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf =
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
ans : f ((inverse x ·A x) ·A (inverse 0G)) ∼ (Group.0G H)
|
||||
ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.symmetric (Setoid.eq S) (invIdentity G)) (Equivalence.reflexive (Setoid.eq S)) ((Equivalence.reflexive (Setoid.eq S))) ((Equivalence.reflexive (Setoid.eq S)))) (identRight {0G}))) (imageOfIdentityIsIdentity fHom)
|
||||
ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.symmetric (Setoid.eq S) (invIdent G)) (Equivalence.reflexive (Setoid.eq S)) ((Equivalence.reflexive (Setoid.eq S))) ((Equivalence.reflexive (Setoid.eq S)))) (identRight {0G}))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.invRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
|
||||
where
|
||||
open Group G
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
ans : f ((x ·A inverse x) ·A (inverse 0G)) ∼ (Group.0G H)
|
||||
ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invRight G)) (Equivalence.symmetric (Setoid.eq S) (invIdentity G)) (Equivalence.reflexive (Setoid.eq S)) (Equivalence.reflexive (Setoid.eq S)) (Equivalence.reflexive (Setoid.eq S))) (identRight {0G}))) (imageOfIdentityIsIdentity fHom)
|
||||
ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invRight G)) (Equivalence.symmetric (Setoid.eq S) (invIdent G)) (Equivalence.reflexive (Setoid.eq S)) (Equivalence.reflexive (Setoid.eq S)) (Equivalence.reflexive (Setoid.eq S))) (identRight {0G}))) (imageOfIdentityIsIdentity fHom)
|
||||
|
||||
record Subgroup {a} {b} {c} {d} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) {f : B → A} (hom : GroupHom H G f) : Set (a ⊔ b ⊔ c ⊔ d) where
|
||||
open Setoid T renaming (_∼_ to _∼G_)
|
||||
open Setoid S renaming (_∼_ to _∼H_)
|
||||
field
|
||||
fInj : SetoidInjection T S f
|
||||
{-
|
||||
quotientHom : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {f : A → B} → (fHom : GroupHom G H f) → A → A
|
||||
quotientHom {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {f = f} fHom a = {!!}
|
||||
@@ -422,11 +161,3 @@ quotientIsSubgroup : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A
|
||||
quotientIsSubgroup = {!!}
|
||||
|
||||
-}
|
||||
subgroupOfAbelianIsAbelian : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : B → A} {fHom : GroupHom H G f} → Subgroup G H fHom → AbelianGroup G → AbelianGroup H
|
||||
AbelianGroup.commutative (subgroupOfAbelianIsAbelian {S = S} {_+B_ = _+B_} {fHom = fHom} record { fInj = fInj } record { commutative = commutative }) {x} {y} = SetoidInjection.injective fInj (transitive (GroupHom.groupHom fHom) (transitive commutative (symmetric (GroupHom.groupHom fHom))))
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
abelianIsGroupProperty : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} → GroupsIsomorphic G H → AbelianGroup H → AbelianGroup G
|
||||
abelianIsGroupProperty iso abH = subgroupOfAbelianIsAbelian {fHom = GroupIso.groupHom (GroupsIsomorphic.proof iso)} (record { fInj = SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof iso)) }) abH
|
||||
|
||||
@@ -10,6 +10,12 @@ open import Sets.FinSet
|
||||
open import Functions
|
||||
open import Sets.EquivalenceRelations
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Groups.Homomorphisms.Definition
|
||||
open import Groups.Homomorphisms.Lemmas
|
||||
open import Groups.Isomorphisms.Definition
|
||||
open import Groups.Subgroups.Definition
|
||||
open import Groups.Lemmas
|
||||
open import Groups.Abelian.Definition
|
||||
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
|
||||
@@ -106,7 +112,7 @@ module Groups.Groups2 where
|
||||
groupKernelGroup : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → Group (groupKernel G hom) (groupKernelGroupOp G hom)
|
||||
Group.+WellDefined (groupKernelGroup G fHom) {kerOfElt x prX} {kerOfElt y prY} {kerOfElt a prA} {kerOfElt b prB} = Group.+WellDefined G
|
||||
Group.0G (groupKernelGroup G fHom) = kerOfElt (Group.0G G) (imageOfIdentityIsIdentity fHom)
|
||||
Group.inverse (groupKernelGroup {T = T} G {H = H} fHom) (kerOfElt x prX) = kerOfElt (Group.inverse G x) (transitive (homRespectsInverse fHom) (transitive (inverseWellDefined H prX) (invIdentity H)))
|
||||
Group.inverse (groupKernelGroup {T = T} G {H = H} fHom) (kerOfElt x prX) = kerOfElt (Group.inverse G x) (transitive (homRespectsInverse fHom) (transitive (inverseWellDefined H prX) (invIdent H)))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
@@ -161,4 +167,3 @@ module Groups.Groups2 where
|
||||
identityHom : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+A_ : A → A → A} (G : Group S _+A_) → GroupHom G G id
|
||||
GroupHom.groupHom (identityHom {S = S} G) = Equivalence.reflexive (Setoid.eq S)
|
||||
GroupHom.wellDefined (identityHom G) = id
|
||||
|
||||
|
||||
25
Groups/Homomorphisms/Definition.agda
Normal file
25
Groups/Homomorphisms/Definition.agda
Normal file
@@ -0,0 +1,25 @@
|
||||
{-# OPTIONS --safe --warning=error --without-K #-}
|
||||
|
||||
open import LogicalFormulae
|
||||
open import Setoids.Setoids
|
||||
open import Functions
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Sets.FinSet
|
||||
open import Groups.Definition
|
||||
open import Sets.EquivalenceRelations
|
||||
|
||||
module Groups.Homomorphisms.Definition where
|
||||
|
||||
record GroupHom {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
|
||||
open Group H
|
||||
open Setoid T
|
||||
field
|
||||
groupHom : {x y : A} → f (x ·A y) ∼ (f x) ·B (f y)
|
||||
wellDefined : {x y : A} → Setoid._∼_ S x y → f x ∼ f y
|
||||
|
||||
record InjectiveGroupHom {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {underf : A → B} (f : GroupHom G H underf) : Set (m ⊔ n ⊔ o ⊔ p) where
|
||||
open Setoid S renaming (_∼_ to _∼A_)
|
||||
open Setoid T renaming (_∼_ to _∼B_)
|
||||
field
|
||||
injective : SetoidInjection S T underf
|
||||
42
Groups/Homomorphisms/Lemmas.agda
Normal file
42
Groups/Homomorphisms/Lemmas.agda
Normal file
@@ -0,0 +1,42 @@
|
||||
{-# OPTIONS --safe --warning=error --without-K #-}
|
||||
|
||||
open import LogicalFormulae
|
||||
open import Setoids.Setoids
|
||||
open import Functions
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Sets.FinSet
|
||||
open import Groups.Definition
|
||||
open import Sets.EquivalenceRelations
|
||||
open import Groups.Homomorphisms.Definition
|
||||
open import Groups.Lemmas
|
||||
|
||||
module Groups.Homomorphisms.Lemmas where
|
||||
|
||||
imageOfIdentityIsIdentity : {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {B : Set n} {T : Setoid {n} {p} B} {_·A_ : A → A → A} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {f : A → B} → (hom : GroupHom G H f) → Setoid._∼_ T (f (Group.0G G)) (Group.0G H)
|
||||
imageOfIdentityIsIdentity {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} {G = G} {H = H} {f = f} hom = Equivalence.symmetric (Setoid.eq T) t
|
||||
where
|
||||
open Group H
|
||||
open Setoid T
|
||||
id2 : Setoid._∼_ S (Group.0G G) ((Group.0G G) ·A (Group.0G G))
|
||||
id2 = Equivalence.symmetric (Setoid.eq S) (Group.identRight G)
|
||||
r : f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
|
||||
s : 0G ·B f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
|
||||
t : 0G ∼ f (Group.0G G)
|
||||
t = groupsHaveRightCancellation H (f (Group.0G G)) 0G (f (Group.0G G)) s
|
||||
s = Equivalence.transitive (Setoid.eq T) identLeft r
|
||||
r = Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined hom id2) (GroupHom.groupHom hom)
|
||||
|
||||
groupHomsCompose : {m n o r s t : _} {A : Set m} {S : Setoid {m} {r} A} {_+A_ : A → A → A} {B : Set n} {T : Setoid {n} {s} B} {_+B_ : B → B → B} {C : Set o} {U : Setoid {o} {t} C} {_+C_ : C → C → C} {G : Group S _+A_} {H : Group T _+B_} {I : Group U _+C_} {f : A → B} {g : B → C} (fHom : GroupHom G H f) (gHom : GroupHom H I g) → GroupHom G I (g ∘ f)
|
||||
GroupHom.wellDefined (groupHomsCompose {G = G} {H} {I} {f} {g} fHom gHom) {x} {y} pr = GroupHom.wellDefined gHom (GroupHom.wellDefined fHom pr)
|
||||
GroupHom.groupHom (groupHomsCompose {S = S} {_+A_ = _·A_} {T = T} {U = U} {_+C_ = _·C_} {G = G} {H} {I} {f} {g} fHom gHom) {x} {y} = answer
|
||||
where
|
||||
open Group I
|
||||
answer : (Setoid._∼_ U) ((g ∘ f) (x ·A y)) ((g ∘ f) x ·C (g ∘ f) y)
|
||||
answer = (Equivalence.transitive (Setoid.eq U)) (GroupHom.wellDefined gHom (GroupHom.groupHom fHom {x} {y}) ) (GroupHom.groupHom gHom {f x} {f y})
|
||||
|
||||
homRespectsInverse : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {underF : A → B} → (f : GroupHom G H underF) → {x : A} → Setoid._∼_ T (underF (Group.inverse G x)) (Group.inverse H (underF x))
|
||||
homRespectsInverse {T = T} {_·A_ = _·A_} {_·B_ = _·B_} {G = G} {H = H} {underF = f} fHom {x} = rightInversesAreUnique H (f x) (f (Group.inverse G x)) (transitive (symmetric (GroupHom.groupHom fHom)) (transitive (GroupHom.wellDefined fHom (Group.invLeft G)) (imageOfIdentityIsIdentity fHom)))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
25
Groups/Isomorphisms/Definition.agda
Normal file
25
Groups/Isomorphisms/Definition.agda
Normal file
@@ -0,0 +1,25 @@
|
||||
{-# OPTIONS --safe --warning=error --without-K #-}
|
||||
|
||||
open import LogicalFormulae
|
||||
open import Setoids.Setoids
|
||||
open import Functions
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Sets.FinSet
|
||||
open import Groups.Definition
|
||||
open import Groups.Homomorphisms.Definition
|
||||
open import Sets.EquivalenceRelations
|
||||
|
||||
module Groups.Isomorphisms.Definition where
|
||||
|
||||
record GroupIso {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
|
||||
open Setoid S renaming (_∼_ to _∼G_)
|
||||
open Setoid T renaming (_∼_ to _∼H_)
|
||||
field
|
||||
groupHom : GroupHom G H f
|
||||
bij : SetoidBijection S T f
|
||||
|
||||
record GroupsIsomorphic {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) : Set (m ⊔ n ⊔ o ⊔ p) where
|
||||
field
|
||||
isomorphism : A → B
|
||||
proof : GroupIso G H isomorphism
|
||||
38
Groups/Isomorphisms/Lemmas.agda
Normal file
38
Groups/Isomorphisms/Lemmas.agda
Normal file
@@ -0,0 +1,38 @@
|
||||
{-# OPTIONS --safe --warning=error --without-K #-}
|
||||
|
||||
open import LogicalFormulae
|
||||
open import Setoids.Setoids
|
||||
open import Functions
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Sets.FinSet
|
||||
open import Groups.Definition
|
||||
open import Sets.EquivalenceRelations
|
||||
open import Groups.Isomorphisms.Definition
|
||||
open import Groups.Homomorphisms.Definition
|
||||
open import Groups.Homomorphisms.Lemmas
|
||||
|
||||
module Groups.Isomorphisms.Lemmas where
|
||||
|
||||
groupIsosCompose : {m n o r s t : _} {A : Set m} {S : Setoid {m} {r} A} {_+A_ : A → A → A} {B : Set n} {T : Setoid {n} {s} B} {_+B_ : B → B → B} {C : Set o} {U : Setoid {o} {t} C} {_+C_ : C → C → C} {G : Group S _+A_} {H : Group T _+B_} {I : Group U _+C_} {f : A → B} {g : B → C} (fHom : GroupIso G H f) (gHom : GroupIso H I g) → GroupIso G I (g ∘ f)
|
||||
GroupIso.groupHom (groupIsosCompose fHom gHom) = groupHomsCompose (GroupIso.groupHom fHom) (GroupIso.groupHom gHom)
|
||||
GroupIso.bij (groupIsosCompose {A = A} {S = S} {T = T} {C = C} {U = U} {f = f} {g = g} fHom gHom) = record { inj = record { injective = λ pr → (SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij fHom))) (SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij gHom)) pr) ; wellDefined = +WellDefined } ; surj = record { surjective = surj ; wellDefined = +WellDefined } }
|
||||
where
|
||||
open Setoid S renaming (_∼_ to _∼A_)
|
||||
open Setoid U renaming (_∼_ to _∼C_)
|
||||
+WellDefined : {x y : A} → (x ∼A y) → (g (f x) ∼C g (f y))
|
||||
+WellDefined = GroupHom.wellDefined (groupHomsCompose (GroupIso.groupHom fHom) (GroupIso.groupHom gHom))
|
||||
surj : {x : C} → Sg A (λ a → (g (f a) ∼C x))
|
||||
surj {x} with SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij gHom)) {x}
|
||||
surj {x} | a , prA with SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij fHom)) {a}
|
||||
... | b , prB = b , transitive (GroupHom.wellDefined (GroupIso.groupHom gHom) prB) prA
|
||||
where
|
||||
open Equivalence (Setoid.eq U)
|
||||
|
||||
--groupIsoInvertible : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d}} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : A → B} → (iso : GroupIso G H f) → GroupIso H G (Invertible.inverse (bijectionImpliesInvertible (GroupIso.bijective iso)))
|
||||
--GroupHom.groupHom (GroupIso.groupHom (groupIsoInvertible {G = G} {H} {f} iso)) {x} {y} = {!!}
|
||||
-- where
|
||||
-- open Group G renaming (_·_ to _+G_)
|
||||
-- open Group H renaming (_·_ to _+H_)
|
||||
--GroupHom.wellDefined (GroupIso.groupHom (groupIsoInvertible {G = G} {H} {f} iso)) {x} {y} x∼y = {!GroupHom.wellDefined x∼y!}
|
||||
--GroupIso.bijective (groupIsoInvertible {G = G} {H} {f} iso) = invertibleImpliesBijection (inverseIsInvertible (bijectionImpliesInvertible (GroupIso.bijective iso)))
|
||||
@@ -6,12 +6,15 @@ open import Functions
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Numbers.Integers.Integers
|
||||
open import Numbers.Rationals
|
||||
open import Numbers.Rationals.Definition
|
||||
open import Sets.FinSet
|
||||
open import Groups.Definition
|
||||
open import Groups.Groups
|
||||
open import Groups.Abelian.Definition
|
||||
open import Groups.FiniteGroups.Definition
|
||||
open import Rings.Definition
|
||||
open import Numbers.Modulo.IntegersModN
|
||||
open import Numbers.Modulo.Group
|
||||
open import Numbers.Modulo.Definition
|
||||
open import Semirings.Definition
|
||||
|
||||
module Groups.LectureNotes.Lecture1 where
|
||||
|
||||
@@ -5,35 +5,138 @@ open import Setoids.Setoids
|
||||
open import Functions
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Groups.Groups
|
||||
open import Groups.Definition
|
||||
open import Sets.EquivalenceRelations
|
||||
|
||||
module Groups.Lemmas where
|
||||
invInv : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → {x : A} → Setoid._∼_ S (Group.inverse G (Group.inverse G x)) x
|
||||
invInv {S = S} G {x} = symmetric (transferToRight' G invRight)
|
||||
where
|
||||
open Setoid S
|
||||
open Group G
|
||||
open Equivalence eq
|
||||
module Groups.Lemmas {a b : _} {A : Set a} {_·_ : A → A → A} {S : Setoid {a} {b} A} (G : Group S _·_) where
|
||||
|
||||
invIdent : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → Setoid._∼_ S (Group.inverse G (Group.0G G)) (Group.0G G)
|
||||
invIdent {S = S} G = symmetric (transferToRight' G (Group.identLeft G))
|
||||
where
|
||||
open Setoid S
|
||||
open Group G
|
||||
open Equivalence eq
|
||||
open Setoid S
|
||||
open Group G
|
||||
open Equivalence eq
|
||||
|
||||
swapInv : {a b : _} {A : Set a} {_+_ : A → A → A} {S : Setoid {a} {b} A} (G : Group S _+_) → {x y : A} → Setoid._∼_ S (Group.inverse G x) y → Setoid._∼_ S x (Group.inverse G y)
|
||||
swapInv {S = S} G {x} {y} -x=y = transitive (symmetric (invInv G)) (inverseWellDefined G -x=y)
|
||||
where
|
||||
open Setoid S
|
||||
open Group G
|
||||
open Equivalence eq
|
||||
groupsHaveLeftCancellation : (x y z : A) → (x · y) ∼ (x · z) → y ∼ z
|
||||
groupsHaveLeftCancellation x y z pr = o
|
||||
where
|
||||
_^-1 = inverse
|
||||
j : ((x ^-1) · x) · y ∼ (x ^-1) · (x · z)
|
||||
j = transitive (symmetric (+Associative {x ^-1} {x} {y})) (+WellDefined ~refl pr)
|
||||
k : ((x ^-1) · x) · y ∼ ((x ^-1) · x) · z
|
||||
k = transitive j +Associative
|
||||
l : 0G · y ∼ ((x ^-1) · x) · z
|
||||
l = transitive (+WellDefined (symmetric invLeft) ~refl) k
|
||||
m : 0G · y ∼ 0G · z
|
||||
m = transitive l (+WellDefined invLeft ~refl)
|
||||
n : y ∼ 0G · z
|
||||
n = transitive (symmetric identLeft) m
|
||||
o : y ∼ z
|
||||
o = transitive n identLeft
|
||||
|
||||
identityIsUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (e : A) → ((b : A) → (Setoid._∼_ S (b · e) b)) → (Setoid._∼_ S e (Group.0G G))
|
||||
identityIsUnique {S = S} {_·_} g thing fb = transitive (symmetric identLeft) (fb 0G)
|
||||
where
|
||||
open Group g renaming (inverse to _^-1)
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
groupsHaveRightCancellation : (x y z : A) → (y · x) ∼ (z · x) → y ∼ z
|
||||
groupsHaveRightCancellation x y z pr = transitive m identRight
|
||||
where
|
||||
_^-1 = inverse
|
||||
i : (y · x) · (x ^-1) ∼ (z · x) · (x ^-1)
|
||||
i = +WellDefined pr ~refl
|
||||
j : y · (x · (x ^-1)) ∼ (z · x) · (x ^-1)
|
||||
j = transitive +Associative i
|
||||
j' : y · 0G ∼ (z · x) · (x ^-1)
|
||||
j' = transitive (+WellDefined ~refl (symmetric invRight)) j
|
||||
k : y ∼ (z · x) · (x ^-1)
|
||||
k = transitive (symmetric identRight) j'
|
||||
l : y ∼ z · (x · (x ^-1))
|
||||
l = transitive k (symmetric +Associative)
|
||||
m : y ∼ z · 0G
|
||||
m = transitive l (+WellDefined ~refl invRight)
|
||||
|
||||
rightInversesAreUnique : (x : A) → (y : A) → (y · x) ∼ 0G → y ∼ (inverse x)
|
||||
rightInversesAreUnique x y f = transitive i (transitive j (transitive k (transitive l m)))
|
||||
where
|
||||
_^-1 = inverse
|
||||
i : y ∼ y · 0G
|
||||
j : y · 0G ∼ y · (x · (x ^-1))
|
||||
k : y · (x · (x ^-1)) ∼ (y · x) · (x ^-1)
|
||||
l : (y · x) · (x ^-1) ∼ 0G · (x ^-1)
|
||||
m : 0G · (x ^-1) ∼ x ^-1
|
||||
i = symmetric identRight
|
||||
j = +WellDefined ~refl (symmetric invRight)
|
||||
k = +Associative
|
||||
l = +WellDefined f ~refl
|
||||
m = identLeft
|
||||
|
||||
leftInversesAreUnique : {x : A} → {y : A} → (x · y) ∼ 0G → y ∼ (inverse x)
|
||||
leftInversesAreUnique {x} {y} f = rightInversesAreUnique x y l
|
||||
where
|
||||
_^-1 = inverse
|
||||
i : y · (x · y) ∼ y · 0G
|
||||
i' : y · (x · y) ∼ y
|
||||
j : (y · x) · y ∼ y
|
||||
k : (y · x) · y ∼ 0G · y
|
||||
l : y · x ∼ 0G
|
||||
i = +WellDefined ~refl f
|
||||
i' = transitive i identRight
|
||||
j = transitive (symmetric +Associative) i'
|
||||
k = transitive j (symmetric identLeft)
|
||||
l = groupsHaveRightCancellation y (y · x) 0G k
|
||||
|
||||
|
||||
invTwice : (x : A) → (inverse (inverse x)) ∼ x
|
||||
invTwice x = symmetric (rightInversesAreUnique (x ^-1) x invRight)
|
||||
where
|
||||
_^-1 = inverse
|
||||
|
||||
replaceGroupOp : {a b c d w x y z : A} → (Setoid._∼_ S a c) → (Setoid._∼_ S b d) → (Setoid._∼_ S w y) → (Setoid._∼_ S x z) → Setoid._∼_ S (a · b) (w · x) → Setoid._∼_ S (c · d) (y · z)
|
||||
replaceGroupOp a~c b~d w~y x~z pr = transitive (symmetric (+WellDefined a~c b~d)) (transitive pr (+WellDefined w~y x~z))
|
||||
|
||||
replaceGroupOpRight : {a b c x : A} → (Setoid._∼_ S a (b · c)) → (Setoid._∼_ S c x) → (Setoid._∼_ S a (b · x))
|
||||
replaceGroupOpRight a~bc c~x = transitive a~bc (+WellDefined reflexive c~x)
|
||||
|
||||
inverseWellDefined : {x y : A} → (x ∼ y) → (inverse x) ∼ (inverse y)
|
||||
inverseWellDefined {x} {y} x~y = groupsHaveRightCancellation x (inverse x) (inverse y) q
|
||||
where
|
||||
p : inverse x · x ∼ inverse y · y
|
||||
p = transitive invLeft (symmetric invLeft)
|
||||
q : inverse x · x ∼ inverse y · x
|
||||
q = replaceGroupOpRight {_·_ (inverse x) x} {inverse y} {y} {x} p (symmetric x~y)
|
||||
|
||||
transferToRight : {a b : A} → (a · (inverse b)) ∼ 0G → a ∼ b
|
||||
transferToRight {a} {b} ab-1 = transitive (symmetric (invTwice a)) (transitive u (invTwice b))
|
||||
where
|
||||
t : inverse a ∼ inverse b
|
||||
t = symmetric (leftInversesAreUnique ab-1)
|
||||
u : inverse (inverse a) ∼ inverse (inverse b)
|
||||
u = inverseWellDefined t
|
||||
|
||||
transferToRight' : {a b : A} → (a · b) ∼ 0G → a ∼ (inverse b)
|
||||
transferToRight' {a} {b} ab-1 = transferToRight lemma
|
||||
where
|
||||
lemma : a · (inverse (inverse b)) ∼ 0G
|
||||
lemma = transitive (+WellDefined reflexive (invTwice b)) ab-1
|
||||
|
||||
transferToRight'' : {a b : A} → Setoid._∼_ S a b → (a · (inverse b)) ∼ 0G
|
||||
transferToRight'' {a} {b} a~b = transitive (+WellDefined a~b reflexive) invRight
|
||||
|
||||
invInv : {x : A} → (inverse (inverse x)) ∼ x
|
||||
invInv {x} = symmetric (transferToRight' invRight)
|
||||
|
||||
invIdent : (inverse 0G) ∼ 0G
|
||||
invIdent = symmetric (transferToRight' identLeft)
|
||||
|
||||
swapInv : {x y : A} → (inverse x) ∼ y → x ∼ (inverse y)
|
||||
swapInv {x} {y} -x=y = transitive (symmetric invInv) (inverseWellDefined -x=y)
|
||||
|
||||
identityIsUnique : (e : A) → ((b : A) → ((b · e) ∼ b)) → (e ∼ 0G)
|
||||
identityIsUnique thing fb = transitive (symmetric identLeft) (fb 0G)
|
||||
|
||||
invContravariant : {x y : A} → (Setoid._∼_ S (Group.inverse G (x · y)) ((Group.inverse G y) · (Group.inverse G x)))
|
||||
invContravariant {x} {y} = ans
|
||||
where
|
||||
_^-1 = inverse
|
||||
otherInv = (y ^-1) · (x ^-1)
|
||||
many+Associatives : x · ((y · (y ^-1)) · (x ^-1)) ∼ (x · y) · ((y ^-1) · (x ^-1))
|
||||
oneMult : (x · y) · otherInv ∼ x · (x ^-1)
|
||||
many+Associatives = transitive +Associative (transitive (+WellDefined +Associative reflexive) (symmetric +Associative))
|
||||
oneMult = symmetric (transitive (+WellDefined reflexive (transitive (symmetric identLeft) (+WellDefined (symmetric invRight) reflexive))) many+Associatives)
|
||||
otherInvIsInverse : (x · y) · otherInv ∼ 0G
|
||||
otherInvIsInverse = transitive oneMult invRight
|
||||
ans : (x · y) ^-1 ∼ (y ^-1) · (x ^-1)
|
||||
ans = symmetric (leftInversesAreUnique otherInvIsInverse)
|
||||
|
||||
18
Groups/Subgroups/Definition.agda
Normal file
18
Groups/Subgroups/Definition.agda
Normal file
@@ -0,0 +1,18 @@
|
||||
{-# OPTIONS --safe --warning=error --without-K #-}
|
||||
|
||||
open import LogicalFormulae
|
||||
open import Setoids.Setoids
|
||||
open import Functions
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Sets.FinSet
|
||||
open import Groups.Definition
|
||||
open import Groups.Homomorphisms.Definition
|
||||
|
||||
module Groups.Subgroups.Definition where
|
||||
|
||||
record Subgroup {a} {b} {c} {d} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) {f : B → A} (hom : GroupHom H G f) : Set (a ⊔ b ⊔ c ⊔ d) where
|
||||
open Setoid T renaming (_∼_ to _∼G_)
|
||||
open Setoid S renaming (_∼_ to _∼H_)
|
||||
field
|
||||
fInj : SetoidInjection T S f
|
||||
205
LectureNotes/Groups/Lecture1.agda
Normal file
205
LectureNotes/Groups/Lecture1.agda
Normal file
@@ -0,0 +1,205 @@
|
||||
{-# OPTIONS --warning=error --safe #-}
|
||||
|
||||
open import Functions
|
||||
open import Sets.FinSet
|
||||
open import LogicalFormulae
|
||||
open import Groups.Definition
|
||||
open import Groups.Groups
|
||||
open import Groups.FiniteGroups.Definition
|
||||
open import Groups.Abelian.Definition
|
||||
open import Setoids.Setoids
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Numbers.Integers.Integers
|
||||
open import Numbers.Rationals.Definition
|
||||
open import Rings.Definition
|
||||
open import Fields.FieldOfFractions.Setoid
|
||||
open import Semirings.Definition
|
||||
open import Numbers.Modulo.Definition
|
||||
open import Numbers.Modulo.Group
|
||||
|
||||
module LectureNotes.Groups.Lecture1 where
|
||||
|
||||
-- Examples
|
||||
|
||||
groupExample1 : Group (reflSetoid ℤ) (_+Z_)
|
||||
groupExample1 = ℤGroup
|
||||
|
||||
groupExample2 : Group (fieldOfFractionsSetoid ℤIntDom) (_+Q_)
|
||||
groupExample2 = Ring.additiveGroup ℚRing
|
||||
|
||||
groupExample3 : Group (reflSetoid ℤ) (_-Z_) → False
|
||||
groupExample3 record { +Associative = multAssoc } with multAssoc {nonneg 3} {nonneg 2} {nonneg 1}
|
||||
groupExample3 record { +WellDefined = wellDefined } | ()
|
||||
|
||||
negSuccInjective : {a b : ℕ} → (negSucc a ≡ negSucc b) → a ≡ b
|
||||
negSuccInjective {a} {.a} refl = refl
|
||||
|
||||
nonnegInjective : {a b : ℕ} → (nonneg a ≡ nonneg b) → a ≡ b
|
||||
nonnegInjective {a} {.a} refl = refl
|
||||
|
||||
integersTimesNotGroup : Group (reflSetoid ℤ) (_*Z_) → False
|
||||
integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg zero) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with multIdentLeft {negSucc 1}
|
||||
... | ()
|
||||
integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ zero)) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with invLeft {nonneg zero}
|
||||
... | bl with inverse (nonneg zero)
|
||||
integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ zero)) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | () | nonneg zero
|
||||
integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ zero)) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | p | nonneg (succ x) = naughtE (nonnegInjective (transitivity (applyEquality nonneg (equalityCommutative (Semiring.productZeroRight ℕSemiring x))) p))
|
||||
integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ zero)) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | () | negSucc x
|
||||
integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ (succ x))) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with succInjective (negSuccInjective (multIdentLeft {negSucc 1}))
|
||||
... | ()
|
||||
integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (negSucc x) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with multIdentLeft {nonneg 2}
|
||||
integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (negSucc x) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | ()
|
||||
|
||||
-- TODO: Q is not a group with *Q
|
||||
-- TODO: Q without 0 is a group with *Q
|
||||
-- TODO: {1, -1} is a group with *
|
||||
|
||||
ℤnIsGroup : (n : ℕ) → (0<n : 0 <N n) → _
|
||||
ℤnIsGroup n pr = ℤnGroup n pr
|
||||
|
||||
-- Groups example 8.9 from lecture 1
|
||||
data Weird : Set where
|
||||
e : Weird
|
||||
a : Weird
|
||||
b : Weird
|
||||
c : Weird
|
||||
|
||||
_+W_ : Weird → Weird → Weird
|
||||
e +W t = t
|
||||
a +W e = a
|
||||
a +W a = e
|
||||
a +W b = c
|
||||
a +W c = b
|
||||
b +W e = b
|
||||
b +W a = c
|
||||
b +W b = e
|
||||
b +W c = a
|
||||
c +W e = c
|
||||
c +W a = b
|
||||
c +W b = a
|
||||
c +W c = e
|
||||
|
||||
+WAssoc : {x y z : Weird} → (x +W (y +W z)) ≡ (x +W y) +W z
|
||||
+WAssoc {e} {y} {z} = refl
|
||||
+WAssoc {a} {e} {z} = refl
|
||||
+WAssoc {a} {a} {e} = refl
|
||||
+WAssoc {a} {a} {a} = refl
|
||||
+WAssoc {a} {a} {b} = refl
|
||||
+WAssoc {a} {a} {c} = refl
|
||||
+WAssoc {a} {b} {e} = refl
|
||||
+WAssoc {a} {b} {a} = refl
|
||||
+WAssoc {a} {b} {b} = refl
|
||||
+WAssoc {a} {b} {c} = refl
|
||||
+WAssoc {a} {c} {e} = refl
|
||||
+WAssoc {a} {c} {a} = refl
|
||||
+WAssoc {a} {c} {b} = refl
|
||||
+WAssoc {a} {c} {c} = refl
|
||||
+WAssoc {b} {e} {z} = refl
|
||||
+WAssoc {b} {a} {e} = refl
|
||||
+WAssoc {b} {a} {a} = refl
|
||||
+WAssoc {b} {a} {b} = refl
|
||||
+WAssoc {b} {a} {c} = refl
|
||||
+WAssoc {b} {b} {e} = refl
|
||||
+WAssoc {b} {b} {a} = refl
|
||||
+WAssoc {b} {b} {b} = refl
|
||||
+WAssoc {b} {b} {c} = refl
|
||||
+WAssoc {b} {c} {e} = refl
|
||||
+WAssoc {b} {c} {a} = refl
|
||||
+WAssoc {b} {c} {b} = refl
|
||||
+WAssoc {b} {c} {c} = refl
|
||||
+WAssoc {c} {e} {z} = refl
|
||||
+WAssoc {c} {a} {e} = refl
|
||||
+WAssoc {c} {a} {a} = refl
|
||||
+WAssoc {c} {a} {b} = refl
|
||||
+WAssoc {c} {a} {c} = refl
|
||||
+WAssoc {c} {b} {e} = refl
|
||||
+WAssoc {c} {b} {a} = refl
|
||||
+WAssoc {c} {b} {b} = refl
|
||||
+WAssoc {c} {b} {c} = refl
|
||||
+WAssoc {c} {c} {e} = refl
|
||||
+WAssoc {c} {c} {a} = refl
|
||||
+WAssoc {c} {c} {b} = refl
|
||||
+WAssoc {c} {c} {c} = refl
|
||||
|
||||
weirdGroup : Group (reflSetoid Weird) _+W_
|
||||
Group.+WellDefined weirdGroup = reflGroupWellDefined
|
||||
Group.0G weirdGroup = e
|
||||
Group.inverse weirdGroup t = t
|
||||
Group.+Associative weirdGroup {r} {s} {t} = +WAssoc {r} {s} {t}
|
||||
Group.identRight weirdGroup {e} = refl
|
||||
Group.identRight weirdGroup {a} = refl
|
||||
Group.identRight weirdGroup {b} = refl
|
||||
Group.identRight weirdGroup {c} = refl
|
||||
Group.identLeft weirdGroup {e} = refl
|
||||
Group.identLeft weirdGroup {a} = refl
|
||||
Group.identLeft weirdGroup {b} = refl
|
||||
Group.identLeft weirdGroup {c} = refl
|
||||
Group.invLeft weirdGroup {e} = refl
|
||||
Group.invLeft weirdGroup {a} = refl
|
||||
Group.invLeft weirdGroup {b} = refl
|
||||
Group.invLeft weirdGroup {c} = refl
|
||||
Group.invRight weirdGroup {e} = refl
|
||||
Group.invRight weirdGroup {a} = refl
|
||||
Group.invRight weirdGroup {b} = refl
|
||||
Group.invRight weirdGroup {c} = refl
|
||||
|
||||
weirdAb : AbelianGroup weirdGroup
|
||||
AbelianGroup.commutative weirdAb {e} {e} = refl
|
||||
AbelianGroup.commutative weirdAb {e} {a} = refl
|
||||
AbelianGroup.commutative weirdAb {e} {b} = refl
|
||||
AbelianGroup.commutative weirdAb {e} {c} = refl
|
||||
AbelianGroup.commutative weirdAb {a} {e} = refl
|
||||
AbelianGroup.commutative weirdAb {a} {a} = refl
|
||||
AbelianGroup.commutative weirdAb {a} {b} = refl
|
||||
AbelianGroup.commutative weirdAb {a} {c} = refl
|
||||
AbelianGroup.commutative weirdAb {b} {e} = refl
|
||||
AbelianGroup.commutative weirdAb {b} {a} = refl
|
||||
AbelianGroup.commutative weirdAb {b} {b} = refl
|
||||
AbelianGroup.commutative weirdAb {b} {c} = refl
|
||||
AbelianGroup.commutative weirdAb {c} {e} = refl
|
||||
AbelianGroup.commutative weirdAb {c} {a} = refl
|
||||
AbelianGroup.commutative weirdAb {c} {b} = refl
|
||||
AbelianGroup.commutative weirdAb {c} {c} = refl
|
||||
|
||||
weirdProjection : Weird → FinSet 4
|
||||
weirdProjection a = ofNat 0 (le 3 refl)
|
||||
weirdProjection b = ofNat 1 (le 2 refl)
|
||||
weirdProjection c = ofNat 2 (le 1 refl)
|
||||
weirdProjection e = ofNat 3 (le zero refl)
|
||||
|
||||
weirdProjectionSurj : Surjection weirdProjection
|
||||
Surjection.property weirdProjectionSurj fzero = a , refl
|
||||
Surjection.property weirdProjectionSurj (fsucc fzero) = b , refl
|
||||
Surjection.property weirdProjectionSurj (fsucc (fsucc fzero)) = c , refl
|
||||
Surjection.property weirdProjectionSurj (fsucc (fsucc (fsucc fzero))) = e , refl
|
||||
Surjection.property weirdProjectionSurj (fsucc (fsucc (fsucc (fsucc ()))))
|
||||
|
||||
weirdProjectionInj : (x y : Weird) → weirdProjection x ≡ weirdProjection y → Setoid._∼_ (reflSetoid Weird) x y
|
||||
weirdProjectionInj e e fx=fy = refl
|
||||
weirdProjectionInj e a ()
|
||||
weirdProjectionInj e b ()
|
||||
weirdProjectionInj e c ()
|
||||
weirdProjectionInj a e ()
|
||||
weirdProjectionInj a a fx=fy = refl
|
||||
weirdProjectionInj a b ()
|
||||
weirdProjectionInj a c ()
|
||||
weirdProjectionInj b e ()
|
||||
weirdProjectionInj b a ()
|
||||
weirdProjectionInj b b fx=fy = refl
|
||||
weirdProjectionInj b c ()
|
||||
weirdProjectionInj c e ()
|
||||
weirdProjectionInj c a ()
|
||||
weirdProjectionInj c b ()
|
||||
weirdProjectionInj c c fx=fy = refl
|
||||
|
||||
weirdFinite : FiniteGroup weirdGroup (FinSet 4)
|
||||
SetoidToSet.project (FiniteGroup.toSet weirdFinite) = weirdProjection
|
||||
SetoidToSet.wellDefined (FiniteGroup.toSet weirdFinite) x y = applyEquality weirdProjection
|
||||
SetoidToSet.surj (FiniteGroup.toSet weirdFinite) = weirdProjectionSurj
|
||||
SetoidToSet.inj (FiniteGroup.toSet weirdFinite) = weirdProjectionInj
|
||||
FiniteSet.size (FiniteGroup.finite weirdFinite) = 4
|
||||
FiniteSet.mapping (FiniteGroup.finite weirdFinite) = id
|
||||
FiniteSet.bij (FiniteGroup.finite weirdFinite) = idIsBijective
|
||||
|
||||
weirdOrder : groupOrder weirdFinite ≡ 4
|
||||
weirdOrder = refl
|
||||
@@ -5,6 +5,7 @@ open import Numbers.Naturals.Naturals
|
||||
open import Numbers.Integers.Definition
|
||||
open import Semirings.Definition
|
||||
open import Groups.Groups
|
||||
open import Groups.Abelian.Definition
|
||||
open import Groups.Definition
|
||||
open import Setoids.Setoids
|
||||
|
||||
|
||||
102
Numbers/Modulo/Addition.agda
Normal file
102
Numbers/Modulo/Addition.agda
Normal file
@@ -0,0 +1,102 @@
|
||||
{-# OPTIONS --safe --warning=error #-}
|
||||
-- These are explicitly with-K, because we currently encode an element of Zn as
|
||||
-- a natural together with a proof that it is small.
|
||||
|
||||
open import LogicalFormulae
|
||||
open import Groups.Definition
|
||||
open import Groups.Groups
|
||||
open import Groups.Abelian.Definition
|
||||
open import Groups.FiniteGroups.Definition
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Numbers.Naturals.Addition -- TODO remove this dependency
|
||||
open import Numbers.Primes.PrimeNumbers
|
||||
open import Setoids.Setoids
|
||||
open import Sets.FinSet
|
||||
open import Sets.FinSetWithK
|
||||
open import Functions
|
||||
open import Numbers.Naturals.WithK
|
||||
open import Semirings.Definition
|
||||
open import Orders
|
||||
open import Numbers.Modulo.Definition
|
||||
|
||||
module Numbers.Modulo.Addition where
|
||||
|
||||
cancelSumFromInequality : {a b c : ℕ} → a +N b <N a +N c → b <N c
|
||||
cancelSumFromInequality {a} {b} {c} (le x proof) = le x help
|
||||
where
|
||||
help : succ x +N b ≡ c
|
||||
help rewrite Semiring.+Associative ℕSemiring (succ x) a b | Semiring.commutative ℕSemiring x a | equalityCommutative (Semiring.+Associative ℕSemiring (succ a) x b) | Semiring.commutative ℕSemiring a (x +N b) | Semiring.commutative ℕSemiring (succ (x +N b)) a = canSubtractFromEqualityLeft {a} proof
|
||||
|
||||
_+n_ : {n : ℕ} {pr : 0 <N n} → ℤn n pr → ℤn n pr → ℤn n pr
|
||||
_+n_ {0} {le x ()} a b
|
||||
_+n_ {succ n} {pr} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } with orderIsTotal (a +N b) (succ n)
|
||||
_+n_ {succ n} {pr} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inl (a+b<n)) = record { x = a +N b ; xLess = a+b<n }
|
||||
_+n_ {succ n} {pr} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inr (n<a+b)) = record { x = subtractionNResult.result sub ; xLess = pr2 }
|
||||
where
|
||||
sub : subtractionNResult (succ n) (a +N b) (inl n<a+b)
|
||||
sub = -N (inl n<a+b)
|
||||
pr1 : a +N b <N succ n +N succ n
|
||||
pr1 = addStrongInequalities aLess bLess
|
||||
pr1' : succ n +N (subtractionNResult.result sub) <N succ n +N succ n
|
||||
pr1' rewrite subtractionNResult.pr sub = pr1
|
||||
pr2 : subtractionNResult.result sub <N succ n
|
||||
pr2 = cancelSumFromInequality pr1'
|
||||
_+n_ {succ n} {pr} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inr (a+b=n) = record { x = 0 ; xLess = succIsPositive n }
|
||||
|
||||
plusZnIdentityRight : {n : ℕ} → {pr : 0 <N n} → (a : ℤn n pr) → (a +n record { x = 0 ; xLess = pr }) ≡ a
|
||||
plusZnIdentityRight {zero} {()} a
|
||||
plusZnIdentityRight {succ x} {_} record { x = a ; xLess = aLess } with orderIsTotal (a +N 0) (succ x)
|
||||
plusZnIdentityRight {succ x} {_} record { x = a ; xLess = aLess } | inl (inl a<sx) = equalityZn _ _ (Semiring.commutative ℕSemiring a 0)
|
||||
plusZnIdentityRight {succ x} {_} record { x = a ; xLess = aLess } | inl (inr sx<a) = exFalso (f aLess sx<a)
|
||||
where
|
||||
f : (aL : a <N succ x) → (sx<a : succ x <N a +N 0) → False
|
||||
f aL sx<a rewrite Semiring.commutative ℕSemiring a 0 = orderIsIrreflexive aL sx<a
|
||||
plusZnIdentityRight {succ x} {_} record { x = a ; xLess = aLess } | inr a=sx = exFalso (f aLess a=sx)
|
||||
where
|
||||
f : (aL : a <N succ x) → (a=sx : a +N 0 ≡ succ x) → False
|
||||
f aL a=sx rewrite Semiring.commutative ℕSemiring a 0 | a=sx = TotalOrder.irreflexive ℕTotalOrder aL
|
||||
|
||||
|
||||
plusZnIdentityLeft : {n : ℕ} → {pr : 0 <N n} → (a : ℤn n pr) → (record { x = 0 ; xLess = pr }) +n a ≡ a
|
||||
plusZnIdentityLeft {zero} {()}
|
||||
plusZnIdentityLeft {succ n} {pr} record { x = x ; xLess = xLess } with orderIsTotal x (succ n)
|
||||
plusZnIdentityLeft {succ n} {pr} record { x = x ; xLess = xLess } | inl (inl x<succn) rewrite <NRefl x<succn xLess = refl
|
||||
plusZnIdentityLeft {succ n} {pr} record { x = x ; xLess = xLess } | inl (inr succn<x) = exFalso (TotalOrder.irreflexive ℕTotalOrder (TotalOrder.<Transitive ℕTotalOrder succn<x xLess))
|
||||
plusZnIdentityLeft {succ n} {pr} record { x = x ; xLess = xLess } | inr x=succn rewrite x=succn = exFalso (TotalOrder.irreflexive ℕTotalOrder xLess)
|
||||
|
||||
subLemma : {a b c : ℕ} → (pr1 : a <N b) → (pr2 : a <N c) → (pr : b ≡ c) → (subtractionNResult.result (-N (inl pr1))) ≡ (subtractionNResult.result (-N (inl pr2)))
|
||||
subLemma {a} {b} {c} a<b a<c b=c rewrite b=c | <NRefl a<b a<c = refl
|
||||
|
||||
plusZnCommutative : {n : ℕ} → {pr : 0 <N n} → (a b : ℤn n pr) → (a +n b) ≡ b +n a
|
||||
plusZnCommutative {zero} {()} a b
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } with orderIsTotal (a +N b) (succ n)
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inl a+b<sn) with orderIsTotal (b +N a) (succ n)
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inl a+b<sn) | inl (inl b+a<sn) = equalityZn _ _ (Semiring.commutative ℕSemiring a b)
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inl a+b<sn) | inl (inr sn<b+a) = exFalso (f a+b<sn sn<b+a)
|
||||
where
|
||||
f : (a +N b) <N succ n → succ n <N b +N a → False
|
||||
f pr1 pr2 rewrite Semiring.commutative ℕSemiring b a = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr2 pr1)
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inl a+b<sn) | inr b+a=sn = exFalso (f a+b<sn b+a=sn)
|
||||
where
|
||||
f : (a +N b) <N succ n → b +N a ≡ succ n → False
|
||||
f pr1 eq rewrite Semiring.commutative ℕSemiring b a | eq = TotalOrder.irreflexive ℕTotalOrder pr1
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inr sn<a+b) with orderIsTotal (b +N a) (succ n)
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inr sn<a+b) | inl (inl b+a<sn) = exFalso (f sn<a+b b+a<sn)
|
||||
where
|
||||
f : succ n <N a +N b → b +N a <N succ n → False
|
||||
f pr1 pr2 rewrite Semiring.commutative ℕSemiring b a = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive sn<a+b b+a<sn)
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inr sn<a+b) | inl (inr sn<b+a) = equalityZn _ _ (subLemma sn<a+b sn<b+a (Semiring.commutative ℕSemiring a b))
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inr sn<a+b) | inr sn=b+a = exFalso (f sn<a+b sn=b+a)
|
||||
where
|
||||
f : succ n <N a +N b → b +N a ≡ succ n → False
|
||||
f pr eq rewrite Semiring.commutative ℕSemiring b a | eq = TotalOrder.irreflexive ℕTotalOrder pr
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inr sn=a+b with orderIsTotal (b +N a) (succ n)
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inr sn=a+b | inl (inl b+a<sn) = exFalso f
|
||||
where
|
||||
f : False
|
||||
f rewrite Semiring.commutative ℕSemiring b a | sn=a+b = TotalOrder.irreflexive ℕTotalOrder b+a<sn
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inr sn=a+b | inl (inr sn<b+a) = exFalso f
|
||||
where
|
||||
f : False
|
||||
f rewrite Semiring.commutative ℕSemiring b a | sn=a+b = TotalOrder.irreflexive ℕTotalOrder sn<b+a
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inr sn=a+b | inr sn=b+a = equalityZn _ _ refl
|
||||
28
Numbers/Modulo/Definition.agda
Normal file
28
Numbers/Modulo/Definition.agda
Normal file
@@ -0,0 +1,28 @@
|
||||
{-# OPTIONS --safe --warning=error #-}
|
||||
-- These are explicitly with-K, because we currently encode an element of Zn as
|
||||
-- a natural together with a proof that it is small.
|
||||
|
||||
open import LogicalFormulae
|
||||
open import Groups.Definition
|
||||
open import Groups.Groups
|
||||
open import Groups.Abelian.Definition
|
||||
open import Groups.FiniteGroups.Definition
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Numbers.Naturals.Addition -- TODO remove this dependency
|
||||
open import Numbers.Primes.PrimeNumbers
|
||||
open import Setoids.Setoids
|
||||
open import Sets.FinSet
|
||||
open import Sets.FinSetWithK
|
||||
open import Functions
|
||||
open import Numbers.Naturals.WithK
|
||||
open import Semirings.Definition
|
||||
open import Orders
|
||||
|
||||
module Numbers.Modulo.Definition where
|
||||
record ℤn (n : ℕ) (pr : 0 <N n) : Set where
|
||||
field
|
||||
x : ℕ
|
||||
xLess : x <N n
|
||||
|
||||
equalityZn : {n : ℕ} {pr : 0 <N n} → (a b : ℤn n pr) → (ℤn.x a ≡ ℤn.x b) → a ≡ b
|
||||
equalityZn record { x = a ; xLess = aLess } record { x = .a ; xLess = bLess } refl rewrite <NRefl aLess bLess = refl
|
||||
622
Numbers/Modulo/Group.agda
Normal file
622
Numbers/Modulo/Group.agda
Normal file
@@ -0,0 +1,622 @@
|
||||
{-# OPTIONS --safe --warning=error #-}
|
||||
-- These are explicitly with-K, because we currently encode an element of Zn as
|
||||
-- a natural together with a proof that it is small.
|
||||
|
||||
open import LogicalFormulae
|
||||
open import Groups.Definition
|
||||
open import Groups.Groups
|
||||
open import Groups.Abelian.Definition
|
||||
open import Groups.FiniteGroups.Definition
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Numbers.Naturals.Addition -- TODO remove this dependency
|
||||
open import Numbers.Primes.PrimeNumbers
|
||||
open import Setoids.Setoids
|
||||
open import Sets.FinSet
|
||||
open import Sets.FinSetWithK
|
||||
open import Functions
|
||||
open import Numbers.Naturals.WithK
|
||||
open import Semirings.Definition
|
||||
open import Orders
|
||||
open import Numbers.Modulo.Definition
|
||||
open import Numbers.Modulo.Addition
|
||||
|
||||
module Numbers.Modulo.Group where
|
||||
|
||||
help30 : {a b c n : ℕ} → (c <N n) → (a +N b ≡ n) → (n<b+c : n <N b +N c) → (n <N a +N subtractionNResult.result (-N (inl n<b+c))) → False
|
||||
help30 {a} {b} {c} {n} c<n a+b=n n<b+c x = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr5 c<n)
|
||||
where
|
||||
pr : n +N n <N a +N (subtractionNResult.result (-N (inl n<b+c)) +N n)
|
||||
pr = identityOfIndiscernablesRight _<N_ (additionPreservesInequality n x) (equalityCommutative (Semiring.+Associative ℕSemiring a _ n))
|
||||
pr2 : n +N n <N a +N (b +N c)
|
||||
pr2 = identityOfIndiscernablesRight _<N_ pr (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
pr3 : n +N n <N (a +N b) +N c
|
||||
pr3 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = pr2
|
||||
pr4 : n +N n <N c +N n
|
||||
pr4 rewrite Semiring.commutative ℕSemiring c n = identityOfIndiscernablesRight _<N_ pr3 (applyEquality (_+N c) a+b=n)
|
||||
pr5 : n <N c
|
||||
pr5 = subtractionPreservesInequality n pr4
|
||||
|
||||
help31 : {a b c n : ℕ} → (a +N b ≡ n) → (n<b+c : n <N b +N c) → (a +N subtractionNResult.result (-N (inl n<b+c))) ≡ c
|
||||
help31 {a} {b} {c} {n} a+b=n n<b+c = canSubtractFromEqualityLeft pr2
|
||||
where
|
||||
pr1 : a +N (n +N subtractionNResult.result (-N (inl n<b+c))) ≡ n +N c
|
||||
pr1 rewrite addMinus' (inl n<b+c) | Semiring.+Associative ℕSemiring a b c | a+b=n = refl
|
||||
pr2 : n +N (a +N subtractionNResult.result (-N (inl n<b+c))) ≡ n +N c
|
||||
pr2 = identityOfIndiscernablesLeft _≡_ pr1 (lemm a n (subtractionNResult.result (-N (inl n<b+c))))
|
||||
where
|
||||
lemm : (a b c : ℕ) → a +N (b +N c) ≡ b +N (a +N c)
|
||||
lemm a b c rewrite Semiring.+Associative ℕSemiring a b c | Semiring.commutative ℕSemiring a b | equalityCommutative (Semiring.+Associative ℕSemiring b a c) = refl
|
||||
|
||||
help7 : {a b c n : ℕ} → b +N c ≡ n → a <N n → (n<a+b : n <N a +N b) → (subtractionNResult.result (-N (inl n<a+b)) +N c ≡ n) → False
|
||||
help7 {a} {b} {c} {n} b+c=n a<n n<a+b x = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesLeft _<N_ a<n pr5)
|
||||
where
|
||||
pr : (n +N subtractionNResult.result (-N (inl n<a+b))) +N c ≡ n +N n
|
||||
pr = identityOfIndiscernablesLeft _≡_ (applyEquality (n +N_) x) (Semiring.+Associative ℕSemiring n _ c)
|
||||
pr2 : (a +N b) +N c ≡ n +N n
|
||||
pr2 = identityOfIndiscernablesLeft _≡_ pr (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
pr3 : a +N (b +N c) ≡ n +N n
|
||||
pr3 rewrite Semiring.+Associative ℕSemiring a b c = pr2
|
||||
pr4 : a +N n ≡ n +N n
|
||||
pr4 = identityOfIndiscernablesLeft _≡_ pr3 (applyEquality (a +N_) b+c=n)
|
||||
pr5 : a ≡ n
|
||||
pr5 = canSubtractFromEqualityRight pr4
|
||||
|
||||
help29 : {a b c n : ℕ} → (c <N n) → (n<b+c : n <N b +N c) → (a +N subtractionNResult.result (-N (inl n<b+c))) ≡ n → a +N b ≡ n → False
|
||||
help29 {a} {b} {c} {n} c<n n<b+c pr a+b=n = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesLeft _<N_ c<n p4)
|
||||
where
|
||||
p1 : a +N (subtractionNResult.result (-N (inl n<b+c)) +N n) ≡ n +N n
|
||||
p1 = identityOfIndiscernablesLeft _≡_ (applyEquality (_+N n) pr) (equalityCommutative (Semiring.+Associative ℕSemiring a _ n))
|
||||
p2 : (a +N b) +N c ≡ n +N n
|
||||
p2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = identityOfIndiscernablesLeft _≡_ p1 (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
p3 : n +N c ≡ n +N n
|
||||
p3 = identityOfIndiscernablesLeft _≡_ p2 (applyEquality (_+N c) a+b=n)
|
||||
p4 : c ≡ n
|
||||
p4 = canSubtractFromEqualityLeft p3
|
||||
|
||||
help28 : {a b c n : ℕ} → (n<a+'b+c : n <N a +N (b +N c)) → (a +N b ≡ n) → subtractionNResult.result (-N (inl n<a+'b+c)) ≡ c
|
||||
help28 {a} {b} {c} {n} pr a+b=n = canSubtractFromEqualityLeft p2
|
||||
where
|
||||
p1 : a +N (b +N c) ≡ n +N c
|
||||
p1 rewrite Semiring.+Associative ℕSemiring a b c | a+b=n = refl
|
||||
p2 : n +N subtractionNResult.result (-N (inl pr)) ≡ n +N c
|
||||
p2 = identityOfIndiscernablesLeft _≡_ p1 (equalityCommutative (addMinus' (inl pr)))
|
||||
|
||||
help27 : {a b c n : ℕ} → (a +N b ≡ succ n) → (a +N (b +N c) <N succ n) → a +N (b +N c) ≡ c
|
||||
help27 {a} {b} {zero} {n} a+b=sn a+b+c<sn rewrite Semiring.commutative ℕSemiring b 0 | a+b=sn = exFalso (TotalOrder.irreflexive ℕTotalOrder a+b+c<sn)
|
||||
help27 {a} {b} {succ c} {n} a+b=sn a+b+c<sn rewrite Semiring.+Associative ℕSemiring a b (succ c) | a+b=sn = exFalso (cannotAddAndEnlarge' a+b+c<sn)
|
||||
|
||||
help26 : {a b c n : ℕ} → (a +N b ≡ n) → (a +N (b +N c) ≡ n) → 0 ≡ c
|
||||
help26 {a} {b} {c} {n} a+b=n a+b+c=n rewrite Semiring.+Associative ℕSemiring a b c | a+b=n | Semiring.commutative ℕSemiring n c = equalityCommutative (canSubtractFromEqualityRight a+b+c=n)
|
||||
|
||||
help25 : {a b c n : ℕ} → (a +N b ≡ n) → (b +N c ≡ n) → (a +N 0 ≡ c)
|
||||
help25 {a} {b} {c} {n} a+b=n b+c=n rewrite Semiring.commutative ℕSemiring a 0 | Semiring.commutative ℕSemiring a b | equalityCommutative a+b=n = equalityCommutative (canSubtractFromEqualityLeft b+c=n)
|
||||
|
||||
help24 : {a n : ℕ} → (a <N n) → (n <N a +N 0) → False
|
||||
help24 {a} {n} a<n n<a+0 rewrite Semiring.commutative ℕSemiring a 0 = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive a<n n<a+0)
|
||||
|
||||
help23 : {a n : ℕ} → (a <N n) → (a +N 0 ≡ n) → False
|
||||
help23 {a} {n} a<n a+0=n rewrite Semiring.commutative ℕSemiring a 0 | a+0=n = TotalOrder.irreflexive ℕTotalOrder a<n
|
||||
|
||||
help22 : {a b c n : ℕ} → (a +N b ≡ n) → (c ≡ n) → (n<b+c : n <N b +N c) → (n <N a +N subtractionNResult.result (-N (inl n<b+c))) → False
|
||||
help22 {a} {b} {c} {.c} a+b=n refl n<b+c pr = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesRight _<N_ pr4 a+b=n)
|
||||
where
|
||||
pr1 : c +N c <N a +N (subtractionNResult.result (-N (inl n<b+c)) +N c)
|
||||
pr1 = identityOfIndiscernablesRight _<N_ (additionPreservesInequality c pr) (equalityCommutative (Semiring.+Associative ℕSemiring a _ c))
|
||||
pr2 : c +N c <N a +N (b +N c)
|
||||
pr2 = identityOfIndiscernablesRight _<N_ pr1 (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
pr3 : c +N c <N (a +N b) +N c
|
||||
pr3 = identityOfIndiscernablesRight _<N_ pr2 (Semiring.+Associative ℕSemiring a b c)
|
||||
pr4 : c <N a +N b
|
||||
pr4 = subtractionPreservesInequality c pr3
|
||||
|
||||
help21 : {a b c n : ℕ} → (a +N b ≡ n) → (b +N c ≡ n) → (c ≡ n) → (a <N n) → False
|
||||
help21 {a} {b} {c} {.c} a+b=n b+c=n refl a<n = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesLeft _<N_ a<n a=c)
|
||||
where
|
||||
b=0 : b ≡ 0
|
||||
a=c : a ≡ c
|
||||
a=c = identityOfIndiscernablesLeft _≡_ a+b=n lemm
|
||||
where
|
||||
lemm : a +N b ≡ a
|
||||
lemm rewrite b=0 | Semiring.commutative ℕSemiring a 0 = refl
|
||||
b=0' : (b c : ℕ) → (b +N c ≡ c) → b ≡ 0
|
||||
b=0' zero c b+c=c = refl
|
||||
b=0' (succ b) zero b+c=c = exFalso (naughtE (equalityCommutative b+c=c))
|
||||
b=0' (succ b) (succ c) b+c=c = b=0' (succ b) c bl2
|
||||
where
|
||||
bl2 : succ b +N c ≡ c
|
||||
bl2 rewrite Semiring.commutative ℕSemiring b c | Semiring.commutative ℕSemiring (succ c) b = succInjective b+c=c
|
||||
b=0 = b=0' b c b+c=n
|
||||
|
||||
help20 : {a b c n : ℕ} → (c ≡ n) → (a +N b ≡ n) → (n<b+c : n <N b +N c) → (a +N subtractionNResult.result (-N (inl n<b+c)) <N n) → False
|
||||
help20 {a} {b} {c} {n} c=n a+b=n n<b+c pr = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesLeft _<N_ pr5 c=n)
|
||||
where
|
||||
pr1 : a +N (subtractionNResult.result (-N (inl n<b+c)) +N n) <N n +N n
|
||||
pr1 = identityOfIndiscernablesLeft _<N_ (additionPreservesInequality n pr) (equalityCommutative (Semiring.+Associative ℕSemiring a _ n))
|
||||
pr2 : a +N (b +N c) <N n +N n
|
||||
pr2 = identityOfIndiscernablesLeft _<N_ pr1 (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
pr3 : (a +N b) +N c <N n +N n
|
||||
pr3 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = pr2
|
||||
pr4 : c +N n <N n +N n
|
||||
pr4 = identityOfIndiscernablesLeft _<N_ pr3 (transitivity (applyEquality (_+N c) a+b=n) (Semiring.commutative ℕSemiring n c))
|
||||
pr5 : c <N n
|
||||
pr5 = subtractionPreservesInequality n pr4
|
||||
|
||||
help19 : {a b c n : ℕ} → (b+c<n : b +N c <N n) → (n<a+b : n <N a +N b) → (a <N n) → (subtractionNResult.result (-N (inl n<a+b)) +N c ≡ n) → False
|
||||
help19 {a} {b} {c} {n} b+c<n n<a+b a<n pr = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesLeft _<N_ r q')
|
||||
where
|
||||
p : (n +N subtractionNResult.result (-N (inl n<a+b))) +N c ≡ n +N n
|
||||
p = identityOfIndiscernablesLeft _≡_ (applyEquality (n +N_ ) pr) (Semiring.+Associative ℕSemiring n _ c)
|
||||
q : (a +N b) +N c ≡ n +N n
|
||||
q = identityOfIndiscernablesLeft _≡_ p (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
q' : a +N (b +N c) ≡ n +N n
|
||||
q' = identityOfIndiscernablesLeft _≡_ q (equalityCommutative (Semiring.+Associative ℕSemiring a b c))
|
||||
r : a +N (b +N c) <N n +N n
|
||||
r = addStrongInequalities a<n b+c<n
|
||||
|
||||
help18 : {a b c n : ℕ} → (b+c<n : b +N c <N n) → (n<a+b : n <N a +N b) → (a <N n) → (n <N subtractionNResult.result (-N (inl n<a+b)) +N c) → False
|
||||
help18 {a} {b} {c} {n} b+c<n n<a+b a<n pr = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive p4 a<n)
|
||||
where
|
||||
p : n +N n <N (n +N subtractionNResult.result (-N (inl n<a+b))) +N c
|
||||
p = identityOfIndiscernablesRight _<N_ (additionPreservesInequalityOnLeft n pr) (Semiring.+Associative ℕSemiring n _ c)
|
||||
p' : n +N n <N (a +N b) +N c
|
||||
p' = identityOfIndiscernablesRight _<N_ p (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
p2 : n +N n <N a +N (b +N c)
|
||||
p2 = identityOfIndiscernablesRight _<N_ p' (equalityCommutative (Semiring.+Associative ℕSemiring a b c))
|
||||
p3 : n +N n <N a +N n
|
||||
p3 = orderIsTransitive p2 (additionPreservesInequalityOnLeft a b+c<n)
|
||||
p4 : n <N a
|
||||
p4 = subtractionPreservesInequality n p3
|
||||
|
||||
help17 : {a b c n : ℕ} → (n<b+c : n <N b +N c) → (n<a+b : n <N a +N b) → (a +N subtractionNResult.result (-N (inl n<b+c)) <N n) → (subtractionNResult.result (-N (inl n<a+b)) +N c) ≡ n → False
|
||||
help17 {a} {b} {c} {n} n<b+c n<a+b p1 p2 = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesLeft _<N_ pr1'' pr3)
|
||||
where
|
||||
pr1' : a +N (subtractionNResult.result (-N (inl n<b+c)) +N n) <N n +N n
|
||||
pr1' = identityOfIndiscernablesLeft _<N_ (additionPreservesInequality n p1) (equalityCommutative (Semiring.+Associative ℕSemiring a _ n))
|
||||
pr1'' : a +N (b +N c) <N n +N n
|
||||
pr1'' = identityOfIndiscernablesLeft _<N_ pr1' (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
pr2' : (n +N subtractionNResult.result (-N (inl n<a+b))) +N c ≡ n +N n
|
||||
pr2' = identityOfIndiscernablesLeft _≡_ (applyEquality (n +N_) p2) (Semiring.+Associative ℕSemiring n _ c)
|
||||
pr2'' : (a +N b) +N c ≡ n +N n
|
||||
pr2'' = identityOfIndiscernablesLeft _≡_ pr2' (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
pr3 : a +N (b +N c) ≡ n +N n
|
||||
pr3 = identityOfIndiscernablesLeft _≡_ pr2'' (equalityCommutative (Semiring.+Associative ℕSemiring a b c))
|
||||
|
||||
help16 : {a b c n : ℕ} → (n<b+c : n <N b +N c) → (n<a+b : n <N a +N b) → (a +N subtractionNResult.result (-N (inl n<b+c))) <N n → (pr : n <N subtractionNResult.result (-N (inl n<a+b)) +N c) → a +N subtractionNResult.result (-N (inl n<b+c)) ≡ subtractionNResult.result (-N (inl pr))
|
||||
help16 {a} {b} {c} {n} n<b+c n<a+b pr1 pr2 = exFalso (TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr3 pr1''))
|
||||
where
|
||||
pr1' : a +N (subtractionNResult.result (-N (inl n<b+c)) +N n) <N n +N n
|
||||
pr1' = identityOfIndiscernablesLeft _<N_ (additionPreservesInequality n pr1) (equalityCommutative (Semiring.+Associative ℕSemiring a _ n))
|
||||
pr1'' : a +N (b +N c) <N n +N n
|
||||
pr1'' = identityOfIndiscernablesLeft _<N_ pr1' (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
pr2' : n +N n <N (n +N subtractionNResult.result (-N (inl n<a+b))) +N c
|
||||
pr2' = identityOfIndiscernablesRight _<N_ (additionPreservesInequalityOnLeft n pr2) (Semiring.+Associative ℕSemiring n _ c)
|
||||
pr2'' : n +N n <N (a +N b) +N c
|
||||
pr2'' = identityOfIndiscernablesRight _<N_ pr2' (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
pr3 : n +N n <N a +N (b +N c)
|
||||
pr3 = identityOfIndiscernablesRight _<N_ pr2'' (equalityCommutative (Semiring.+Associative ℕSemiring a b c))
|
||||
|
||||
help15 : {a b c n : ℕ} → (n<b+c : n <N b +N c) → (n<a+b : n <N a +N b) → (n <N a +N subtractionNResult.result (-N (inl n<b+c))) → (subtractionNResult.result (-N (inl n<a+b)) +N c) <N n → False
|
||||
help15 {a} {b} {c} {n} n<b+c n<a+b pr1 pr2 = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive p2'' p1')
|
||||
where
|
||||
p1 : (n +N subtractionNResult.result (-N (inl n<a+b))) +N c <N n +N n
|
||||
p1 = identityOfIndiscernablesLeft _<N_ (additionPreservesInequalityOnLeft n pr2) (Semiring.+Associative ℕSemiring n _ c)
|
||||
p1' : (a +N b) +N c <N n +N n
|
||||
p1' = identityOfIndiscernablesLeft _<N_ p1 (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
p2 : n +N n <N a +N (subtractionNResult.result (-N (inl n<b+c)) +N n)
|
||||
p2 = identityOfIndiscernablesRight _<N_ (additionPreservesInequality n pr1) (equalityCommutative (Semiring.+Associative ℕSemiring a _ n))
|
||||
p2' : n +N n <N a +N (b +N c)
|
||||
p2' = identityOfIndiscernablesRight _<N_ p2 (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
p2'' : n +N n <N (a +N b) +N c
|
||||
p2'' = identityOfIndiscernablesRight _<N_ p2' (Semiring.+Associative ℕSemiring a b c)
|
||||
|
||||
help14 : {a b c n : ℕ} → (n<b+c : n <N b +N c) → (n<a+b : n <N a +N b) → (pr1 : n <N a +N subtractionNResult.result (-N (inl n<b+c))) → (pr2 : n <N subtractionNResult.result (-N (inl n<a+b)) +N c) → subtractionNResult.result (-N (inl pr1)) ≡ subtractionNResult.result (-N (inl pr2))
|
||||
help14 {a} {b} {c} {n} n<b+c n<a+b pr1 pr2 = equivalentSubtraction _ _ _ _ pr1 pr2 (transitivity (Semiring.+Associative ℕSemiring n _ c) (transitivity (applyEquality (_+N c) (addMinus' (inl n<a+b))) (transitivity (equalityCommutative (Semiring.+Associative ℕSemiring a b c)) (equalityCommutative p2))))
|
||||
where
|
||||
p1 : (a +N subtractionNResult.result (-N (inl n<b+c))) +N n ≡ a +N (subtractionNResult.result (-N (inl n<b+c)) +N n)
|
||||
p1 = equalityCommutative (Semiring.+Associative ℕSemiring a _ n)
|
||||
p2 : (a +N subtractionNResult.result (-N (inl n<b+c))) +N n ≡ a +N (b +N c)
|
||||
p2 = identityOfIndiscernablesRight _≡_ p1 (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
|
||||
help13 : {a b c n : ℕ} → (n<b+c : n <N b +N c) → (n<a+b : n <N a +N b) → (n <N a +N subtractionNResult.result (-N (inl n<b+c))) → (subtractionNResult.result (-N (inl n<a+b)) +N c ≡ n) → False
|
||||
help13 {a} {b} {c} {n} n<b+c n<a+b pr1 pr2 = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesRight _<N_ lemm1' lemm3)
|
||||
where
|
||||
lemm1 : n +N n <N a +N (subtractionNResult.result (-N (inl n<b+c)) +N n)
|
||||
lemm1 = identityOfIndiscernablesRight _<N_ (additionPreservesInequality n pr1) (equalityCommutative (Semiring.+Associative ℕSemiring a _ n))
|
||||
lemm1' : n +N n <N a +N (b +N c)
|
||||
lemm1' = identityOfIndiscernablesRight _<N_ lemm1 (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
lemm2 : (n +N subtractionNResult.result (-N (inl n<a+b))) +N c ≡ n +N n
|
||||
lemm2 = identityOfIndiscernablesLeft _≡_ (applyEquality (n +N_) pr2) (Semiring.+Associative ℕSemiring n _ c)
|
||||
lemm2' : (a +N b) +N c ≡ n +N n
|
||||
lemm2' = identityOfIndiscernablesLeft _≡_ lemm2 (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
lemm3 : a +N (b +N c) ≡ n +N n
|
||||
lemm3 rewrite Semiring.+Associative ℕSemiring a b c = lemm2'
|
||||
|
||||
help12 : {a b c n : ℕ} → (n<b+c : n <N b +N c) → (n<a+b : n <N a +N b) → (a +N subtractionNResult.result (-N (inl n<b+c))) ≡ n → subtractionNResult.result (-N (inl n<a+b)) +N c <N n → False
|
||||
help12 {a} {b} {c} {n} n<b+c n<a+b pr1 pr2 = TotalOrder.irreflexive ℕTotalOrder lemm4
|
||||
where
|
||||
pr : {a b c : ℕ} → a +N (b +N c) ≡ b +N (a +N c)
|
||||
pr {a} {b} {c} rewrite Semiring.+Associative ℕSemiring a b c | Semiring.commutative ℕSemiring a b | equalityCommutative (Semiring.+Associative ℕSemiring b a c) = refl
|
||||
lemm1 : (n +N subtractionNResult.result (-N (inl n<a+b))) +N c <N n +N n
|
||||
lemm1 = identityOfIndiscernablesLeft _<N_ (additionPreservesInequalityOnLeft n pr2) (Semiring.+Associative ℕSemiring n _ c)
|
||||
lemm2 : (a +N b) +N c <N n +N n
|
||||
lemm2 = identityOfIndiscernablesLeft _<N_ lemm1 (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
lemm1' : a +N (subtractionNResult.result (-N (inl n<b+c)) +N n) ≡ n +N n
|
||||
lemm1' = identityOfIndiscernablesLeft _≡_ (applyEquality (_+N n) pr1) (equalityCommutative (Semiring.+Associative ℕSemiring a _ n))
|
||||
lemm2' : a +N (b +N c) ≡ n +N n
|
||||
lemm2' = identityOfIndiscernablesLeft _≡_ lemm1' (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
lemm3 : (a +N b) +N c ≡ n +N n
|
||||
lemm3 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = lemm2'
|
||||
lemm4 : (a +N b) +N c <N (a +N b) +N c
|
||||
lemm4 = identityOfIndiscernablesRight _<N_ lemm2 (equalityCommutative lemm3)
|
||||
|
||||
help11 : {a b c n : ℕ} → (a <N n) → (b +N c ≡ n) → (n<a+b : n <N a +N b) → (n <N subtractionNResult.result (-N (inl n<a+b)) +N c) → False
|
||||
help11 {a} {b} {c} {n} a<n b+c=n n<a+b pr1 = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive a<n lemm5)
|
||||
where
|
||||
pr : {a b c : ℕ} → a +N (b +N c) ≡ b +N (a +N c)
|
||||
pr {a} {b} {c} rewrite Semiring.+Associative ℕSemiring a b c | Semiring.commutative ℕSemiring a b | equalityCommutative (Semiring.+Associative ℕSemiring b a c) = refl
|
||||
lemm : n +N n <N (n +N subtractionNResult.result (-N (inl n<a+b))) +N c
|
||||
lemm = identityOfIndiscernablesRight _<N_ (additionPreservesInequalityOnLeft n pr1) (Semiring.+Associative ℕSemiring n _ c)
|
||||
lemm2 : n +N n <N (a +N b) +N c
|
||||
lemm2 = identityOfIndiscernablesRight _<N_ lemm (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
lemm3 : n +N n <N a +N (b +N c)
|
||||
lemm3 = identityOfIndiscernablesRight _<N_ lemm2 (equalityCommutative (Semiring.+Associative ℕSemiring a b c))
|
||||
lemm4 : n +N n <N a +N n
|
||||
lemm4 = identityOfIndiscernablesRight _<N_ lemm3 (applyEquality (a +N_) b+c=n)
|
||||
lemm5 : n <N a
|
||||
lemm5 = subtractionPreservesInequality n lemm4
|
||||
|
||||
help10 : {a b c n : ℕ} → (n<b+c : n <N b +N c) → (n<a+b : n <N a +N b) → (a +N subtractionNResult.result (-N (inl n<b+c)) ≡ n) → (n <N subtractionNResult.result (-N (inl n<a+b)) +N c) → False
|
||||
help10 {a} {b} {c} {n} n<b+c n<a+b pr1 pr2 = TotalOrder.irreflexive ℕTotalOrder lemm6
|
||||
where
|
||||
pr : {a b c : ℕ} → a +N (b +N c) ≡ b +N (a +N c)
|
||||
pr {a} {b} {c} rewrite Semiring.+Associative ℕSemiring a b c | Semiring.commutative ℕSemiring a b | equalityCommutative (Semiring.+Associative ℕSemiring b a c) = refl
|
||||
lemm : a +N (n +N subtractionNResult.result (-N (inl n<b+c))) ≡ n +N n
|
||||
lemm = identityOfIndiscernablesLeft _≡_ (applyEquality (n +N_) pr1) (pr {n} {a})
|
||||
lemm2 : a +N (b +N c) ≡ n +N n
|
||||
lemm2 = identityOfIndiscernablesLeft _≡_ lemm (applyEquality (a +N_) (addMinus' (inl n<b+c)))
|
||||
lemm3 : n +N n <N (n +N subtractionNResult.result (-N (inl n<a+b))) +N c
|
||||
lemm3 = identityOfIndiscernablesRight _<N_ (additionPreservesInequalityOnLeft n pr2) (Semiring.+Associative ℕSemiring n _ c)
|
||||
lemm4 : n +N n <N (a +N b) +N c
|
||||
lemm4 = identityOfIndiscernablesRight _<N_ lemm3 (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
lemm5 : n +N n <N a +N (b +N c)
|
||||
lemm5 = identityOfIndiscernablesRight _<N_ lemm4 (equalityCommutative (Semiring.+Associative ℕSemiring a b c))
|
||||
lemm6 : a +N (b +N c) <N a +N (b +N c)
|
||||
lemm6 = identityOfIndiscernablesLeft _<N_ lemm5 (equalityCommutative lemm2)
|
||||
|
||||
help9 : {a n : ℕ} → (a +N 0 ≡ n) → (a <N n) → False
|
||||
help9 {a} {n} n=a+0 a<n rewrite Semiring.commutative ℕSemiring a 0 | n=a+0 = TotalOrder.irreflexive ℕTotalOrder a<n
|
||||
|
||||
help8 : {a n : ℕ} → (n <N a +N 0) → (a <N n) → False
|
||||
help8 {a} {n} n<a+0 a<n rewrite Semiring.commutative ℕSemiring a 0 = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive a<n n<a+0)
|
||||
|
||||
help6 : {a b c n : ℕ} → (b +N c ≡ n) → (n<a+b : n <N a +N b) → (a +N 0 ≡ subtractionNResult.result (-N (inl n<a+b)) +N c)
|
||||
help6 {a} {b} {c} {n} b+c=n n<a+b rewrite Semiring.commutative ℕSemiring a 0 = canSubtractFromEqualityLeft {n} lem'
|
||||
where
|
||||
lem : n +N a ≡ (n +N subtractionNResult.result (-N (inl n<a+b))) +N c
|
||||
lem rewrite addMinus' (inl n<a+b) | equalityCommutative (Semiring.+Associative ℕSemiring a b c) | b+c=n = Semiring.commutative ℕSemiring n a
|
||||
lem' : n +N a ≡ n +N (subtractionNResult.result (-N (inl n<a+b)) +N c)
|
||||
lem' = identityOfIndiscernablesRight _≡_ lem (equalityCommutative (Semiring.+Associative ℕSemiring n _ c))
|
||||
|
||||
help5 : {a b c n : ℕ} → (n<b+c : n <N b +N c) → (n<a+b : n <N a +N b) → a +N subtractionNResult.result (-N (inl n<b+c)) ≡ subtractionNResult.result (-N (inl n<a+b)) +N c
|
||||
help5 {a} {b} {c} {n} n<b+c n<a+b = canSubtractFromEqualityLeft {n} lemma''
|
||||
where
|
||||
lemma : a +N (n +N subtractionNResult.result (-N (inl n<b+c))) ≡ (n +N subtractionNResult.result (-N (inl n<a+b))) +N c
|
||||
lemma rewrite addMinus' (inl n<b+c) | addMinus' (inl n<a+b) = Semiring.+Associative ℕSemiring a b c
|
||||
lemma' : a +N (n +N subtractionNResult.result (-N (inl n<b+c))) ≡ n +N (subtractionNResult.result (-N (inl n<a+b)) +N c)
|
||||
lemma' = identityOfIndiscernablesRight _≡_ lemma (equalityCommutative (Semiring.+Associative ℕSemiring n _ c))
|
||||
lemma'' : n +N (a +N subtractionNResult.result (-N (inl n<b+c))) ≡ n +N (subtractionNResult.result (-N (inl n<a+b)) +N c)
|
||||
lemma'' = identityOfIndiscernablesLeft _≡_ lemma' (pr {a} {n} {subtractionNResult.result (-N (inl n<b+c))})
|
||||
where
|
||||
pr : {a b c : ℕ} → a +N (b +N c) ≡ b +N (a +N c)
|
||||
pr {a} {b} {c} rewrite Semiring.+Associative ℕSemiring a b c | Semiring.commutative ℕSemiring a b | equalityCommutative (Semiring.+Associative ℕSemiring b a c) = refl
|
||||
|
||||
help4 : {a b c n : ℕ} → (n<a+'b+c : n <N a +N (b +N c)) → (n<a+b : n <N a +N b) → (subtractionNResult.result (-N (inl n<a+'b+c)) ≡ subtractionNResult.result (-N (inl n<a+b)) +N c)
|
||||
help4 {a} {b} {c} {n} n<a+'b+c n<a+b = canSubtractFromEqualityLeft lemma'
|
||||
where
|
||||
lemma : (n +N subtractionNResult.result (-N (inl n<a+'b+c))) ≡ (n +N subtractionNResult.result (-N (inl n<a+b))) +N c
|
||||
lemma rewrite addMinus' (inl n<a+'b+c) | addMinus' (inl n<a+b) = Semiring.+Associative ℕSemiring a b c
|
||||
lemma' : n +N subtractionNResult.result (-N (inl n<a+'b+c)) ≡ n +N (subtractionNResult.result (-N (inl n<a+b)) +N c)
|
||||
lemma' = identityOfIndiscernablesRight _≡_ lemma (equalityCommutative (Semiring.+Associative ℕSemiring n (subtractionNResult.result (-N (inl n<a+b))) c))
|
||||
|
||||
help3 : {a b c n : ℕ} → (a <N n) → (b <N n) → (c <N n) → (a +N b <N n) → (pr : n <N b +N c) → a +N subtractionNResult.result (-N (inl pr)) ≡ n → False
|
||||
help3 {a} {b} {c} {n} a<n b<n c<n a+b<n n<b+c pr = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive (inter4 inter3) c<n)
|
||||
where
|
||||
inter : a +N (n +N subtractionNResult.result (-N (inl n<b+c))) ≡ n +N n
|
||||
inter = identityOfIndiscernablesLeft _≡_ (applyEquality (n +N_) pr) (lemma n a (subtractionNResult.result (-N (inl n<b+c))))
|
||||
where
|
||||
lemma : (a b c : ℕ) → a +N (b +N c) ≡ b +N (a +N c)
|
||||
lemma a b c rewrite Semiring.+Associative ℕSemiring a b c | Semiring.+Associative ℕSemiring b a c = applyEquality (_+N c) (Semiring.commutative ℕSemiring a b)
|
||||
inter2 : n +N n ≡ a +N (b +N c)
|
||||
inter2 = equalityCommutative (identityOfIndiscernablesLeft _≡_ inter (applyEquality (a +N_) (addMinus' (inl n<b+c))))
|
||||
inter3 : n +N n <N n +N c
|
||||
inter3 rewrite inter2 | Semiring.+Associative ℕSemiring a b c = additionPreservesInequality c a+b<n
|
||||
inter4 : (n +N n <N n +N c) → n <N c
|
||||
inter4 pr rewrite Semiring.commutative ℕSemiring n c = subtractionPreservesInequality n pr
|
||||
|
||||
help2 : {a b c n : ℕ} → (sn<b+c : succ n <N b +N c) → (sn<a+b+c : succ n <N (a +N b) +N c) → a +N subtractionNResult.result (-N (inl sn<b+c)) ≡ subtractionNResult.result (-N (inl sn<a+b+c))
|
||||
help2 {a} {b} {c} {n} sn<b+c sn<a+b+c = res inter
|
||||
where
|
||||
inter : a +N (subtractionNResult.result (-N (inl sn<b+c)) +N succ n) ≡ subtractionNResult.result (-N (inl sn<a+b+c)) +N succ n
|
||||
inter rewrite addMinus (inl sn<b+c) | addMinus (inl sn<a+b+c) = Semiring.+Associative ℕSemiring a b c
|
||||
res : (a +N (subtractionNResult.result (-N (inl sn<b+c)) +N succ n) ≡ subtractionNResult.result (-N (inl sn<a+b+c)) +N succ n) → a +N subtractionNResult.result (-N (inl sn<b+c)) ≡ subtractionNResult.result (-N (inl sn<a+b+c))
|
||||
res pr = canSubtractFromEqualityRight {_} {succ n} (identityOfIndiscernablesLeft _≡_ pr (Semiring.+Associative ℕSemiring a (subtractionNResult.result (-N (inl sn<b+c))) (succ n)))
|
||||
|
||||
help1 : {a b c n : ℕ} → (sn<b+c : succ n <N b +N c) → (pr1 : succ n <N a +N subtractionNResult.result (-N (inl sn<b+c))) → (a +N b <N succ n) → (a <N succ n) → (b <N succ n) → (c <N succ n) → False
|
||||
help1 {a} {b} {c} {n} sn<b+c pr1 a+b<sn a<sn b<sn c<sn with -N (inl sn<b+c)
|
||||
help1 {a} {b} {c} {n} sn<b+c pr1 a+b<sn a<sn b<sn c<sn | record { result = b+c-sn ; pr = Prb+c-sn } = ans
|
||||
where
|
||||
b+c-nNonzero : b+c-sn ≡ 0 → False
|
||||
b+c-nNonzero pr rewrite (equalityCommutative Prb+c-sn) | pr | Semiring.commutative ℕSemiring n 0 = TotalOrder.irreflexive ℕTotalOrder sn<b+c
|
||||
2sn<a+b+c' : succ n +N succ n <N succ n +N (a +N b+c-sn)
|
||||
2sn<a+b+c' = additionPreservesInequalityOnLeft (succ n) pr1
|
||||
2sn<a+b+c'' : succ n +N succ n <N a +N (succ n +N b+c-sn)
|
||||
2sn<a+b+c'' rewrite Semiring.+Associative ℕSemiring a (succ n) b+c-sn | Semiring.commutative ℕSemiring a (succ n) | equalityCommutative (Semiring.+Associative ℕSemiring (succ n) a b+c-sn) = 2sn<a+b+c'
|
||||
eep : succ n +N succ n <N a +N (b +N c)
|
||||
eep rewrite equalityCommutative Prb+c-sn = 2sn<a+b+c''
|
||||
eep2 : a +N (b +N c) <N succ n +N c
|
||||
eep2 rewrite Semiring.+Associative ℕSemiring a b c = additionPreservesInequality c a+b<sn
|
||||
eep2' : a +N (b +N c) <N succ n +N succ n
|
||||
eep2' = orderIsTransitive eep2 (additionPreservesInequalityOnLeft (succ n) c<sn)
|
||||
ans : False
|
||||
ans = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive eep eep2')
|
||||
|
||||
plusZnAssociative : {n : ℕ} → {pr : 0 <N n} → (a b c : ℤn n pr) → a +n (b +n c) ≡ ((a +n b) +n c)
|
||||
plusZnAssociative {zero} {()}
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess} record { x = c ; xLess = cLess } with orderIsTotal (a +N b) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) with orderIsTotal ((a +N b) +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) with orderIsTotal (b +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inl b+c<sn) with orderIsTotal (a +N (b +N c)) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inl b+c<sn) | inl (inl a+'b+c<sn) = equalityZn _ _ (Semiring.+Associative ℕSemiring a b c)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inl b+c<sn) | inl (inr sn<a+'b+c) = exFalso (false {succ n} a+b+c<sn sn<a+'b+c)
|
||||
where
|
||||
false : {x : ℕ} → (a +N b) +N c <N succ n → succ n <N a +N (b +N c) → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr1 pr2)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inl b+c<sn) | inr sn=a+b+c = exFalso (false a+b+c<sn sn=a+b+c)
|
||||
where
|
||||
false : {x : ℕ} → (a +N b) +N c <N x → (a +N (b +N c)) ≡ x → False
|
||||
false p1 p2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | p2 = TotalOrder.irreflexive ℕTotalOrder p1
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) with orderIsTotal (a +N (b +N c)) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) | inl (inl _) with orderIsTotal (a +N subtractionNResult.result (-N (inl sn<b+c))) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) | inl (inl _) | inl (inl x) with -N (inl sn<b+c)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) | inl (inl a+'b+c<sn) | inl (inl x) | record { result = result ; pr = pr } = exFalso (false a+'b+c<sn pr)
|
||||
where
|
||||
false : a +N (b +N c) <N succ n → succ n +N result ≡ b +N c → False
|
||||
false pr1 pr2 rewrite equalityCommutative pr2 | Semiring.+Associative ℕSemiring a (succ n) result | Semiring.commutative ℕSemiring a (succ n) | equalityCommutative (Semiring.+Associative ℕSemiring (succ n) a result) = cannotAddAndEnlarge' pr1
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) | inl (inl _) | inl (inr x) with -N (inl sn<b+c)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) | inl (inl a+'b+c<sn) | inl (inr x) | record { result = result ; pr = pr } = exFalso false
|
||||
where
|
||||
lemma : a +N (succ n +N result) ≡ a +N (b +N c)
|
||||
lemma' : a +N (succ n +N result) <N succ n
|
||||
lemma'' : succ n +N (a +N result) <N succ n
|
||||
lemma'' = identityOfIndiscernablesLeft _<N_ lemma' (transitivity (Semiring.+Associative ℕSemiring a (succ n) result) (transitivity (applyEquality (λ t → t +N result) (Semiring.commutative ℕSemiring a (succ n))) (equalityCommutative (Semiring.+Associative ℕSemiring (succ n) a result))))
|
||||
lemma = applyEquality (λ i → a +N i) pr
|
||||
lemma' rewrite lemma = a+'b+c<sn
|
||||
false : False
|
||||
false = cannotAddAndEnlarge' lemma''
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) | inl (inl _) | inr x with -N (inl sn<b+c)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) | inl (inl a+'b+c<sn) | inr x | record { result = result ; pr = pr } = exFalso false
|
||||
where
|
||||
lemma : a +N (succ n +N result) <N succ n
|
||||
lemma rewrite pr = a+'b+c<sn
|
||||
lemma' : succ n +N (a +N result) <N succ n
|
||||
lemma' = identityOfIndiscernablesLeft _<N_ lemma (transitivity (Semiring.+Associative ℕSemiring a (succ n) result) (transitivity (applyEquality (λ t → t +N result) (Semiring.commutative ℕSemiring a (succ n))) (equalityCommutative (Semiring.+Associative ℕSemiring (succ n) a result))))
|
||||
false : False
|
||||
false = cannotAddAndEnlarge' lemma'
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) | inl (inr x) = equalityZn _ _ (exFalso (false {succ n} a+b+c<sn x))
|
||||
where
|
||||
false : {x : ℕ} → (a +N b) +N c <N succ n → succ n <N a +N (b +N c) → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr1 pr2)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) | inr x = exFalso (false a+b+c<sn x)
|
||||
where
|
||||
false : (a +N b) +N c <N succ n → a +N (b +N c) ≡ succ n → False
|
||||
false pr1 pr2 rewrite equalityCommutative pr2 | equalityCommutative (Semiring.+Associative ℕSemiring a b c) = TotalOrder.irreflexive ℕTotalOrder pr1
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inr sn=b+c = exFalso (false a+b+c<sn sn=b+c)
|
||||
where
|
||||
false : (a +N b) +N c <N succ n → b +N c ≡ succ n → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr2 | Semiring.commutative ℕSemiring a (succ n) = cannotAddAndEnlarge' a+b+c<sn
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) with orderIsTotal (b +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inl b+c<sn) with orderIsTotal (a +N (b +N c)) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inl b+c<sn) | inl (inl a+'b+c<sn) = exFalso (false sn<a+b+c a+'b+c<sn)
|
||||
where
|
||||
false : (succ n <N (a +N b) +N c) → a +N (b +N c) <N succ n → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr1 pr2)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inl b+c<sn) | inl (inr sn<a+'b+c) = equalityZn _ _ ans
|
||||
where
|
||||
lemma : succ n +N ((a +N b) +N c) ≡ (a +N (b +N c)) +N succ n
|
||||
lemma rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = Semiring.commutative ℕSemiring (succ n) _
|
||||
ans : subtractionNResult.result (-N (inl sn<a+'b+c)) ≡ subtractionNResult.result (-N (inl sn<a+b+c))
|
||||
ans = equivalentSubtraction _ _ _ _ sn<a+'b+c sn<a+b+c lemma
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inl b+c<sn) | inr x = exFalso (false sn<a+b+c x)
|
||||
where
|
||||
false : succ n <N (a +N b) +N c → (a +N (b +N c)) ≡ succ n → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr2 = TotalOrder.irreflexive ℕTotalOrder sn<a+b+c
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inr sn<b+c) with orderIsTotal (a +N (b +N c)) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inr sn<b+c) | inl (inl a+b+c<sn) = exFalso (false sn<a+b+c a+b+c<sn)
|
||||
where
|
||||
false : succ n <N (a +N b) +N c → (a +N (b +N c)) <N succ n → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr1 pr2)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inr sn<b+c) | inl (inr _) with orderIsTotal (a +N subtractionNResult.result (-N (inl sn<b+c))) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inr sn<b+c) | inl (inr _) | inl (inl x) = equalityZn _ _ (help2 {a} {b} {c} {n} sn<b+c sn<a+b+c)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inr sn<b+c) | inl (inr _) | inl (inr x) = equalityZn _ _ (exFalso (help1 sn<b+c x a+b<sn aLess bLess cLess))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inr sn<b+c) | inl (inr _) | inr x = exFalso (help3 aLess bLess cLess a+b<sn sn<b+c x)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inr sn<b+c) | inr a+b+c=sn = exFalso (false sn<a+b+c a+b+c=sn)
|
||||
where
|
||||
false : (succ n <N (a +N b) +N c) → (a +N (b +N c) ≡ succ n) → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr2 = TotalOrder.irreflexive ℕTotalOrder pr1
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inr sn=b+c with orderIsTotal (a +N (b +N c)) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inr sn=b+c | inl (inl x) = exFalso (false sn<a+b+c x)
|
||||
where
|
||||
false : succ n <N (a +N b) +N c → a +N (b +N c) <N succ n → False
|
||||
false p1 p2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive p1 p2)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inr sn=b+c | inl (inr _) with orderIsTotal (a +N 0) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inr sn=b+c | inl (inr _) | inl (inl x) = equalityZn _ _ (ans sn=b+c)
|
||||
where
|
||||
ans : b +N c ≡ succ n → a +N 0 ≡ subtractionNResult.result (-N (inl sn<a+b+c))
|
||||
ans pr with -N (inl sn<a+b+c)
|
||||
ans b+c=sn | record { result = result ; pr = pr1 } rewrite Semiring.commutative ℕSemiring a 0 | equalityCommutative (Semiring.+Associative ℕSemiring a b c) | b+c=sn | Semiring.commutative ℕSemiring (succ n) result = equalityCommutative (canSubtractFromEqualityRight pr1)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inr sn=b+c | inl (inr _) | inl (inr x) = exFalso (false b a+b<sn x)
|
||||
where
|
||||
false : (b : ℕ) → a +N b <N succ n → succ n <N a +N 0 → False
|
||||
false zero pr1 pr2 rewrite Semiring.commutative ℕSemiring a 0 = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr1 pr2)
|
||||
false (succ b) pr1 pr2 rewrite Semiring.commutative ℕSemiring a 0 = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr2 (orderIsTransitive (le b (Semiring.commutative ℕSemiring (succ b) a)) pr1))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inr sn=b+c | inl (inr _) | inr x = exFalso (false b a+b<sn x)
|
||||
where
|
||||
false : (b : ℕ) → a +N b <N succ n → a +N 0 ≡ succ n → False
|
||||
false zero pr1 pr2 rewrite pr2 = TotalOrder.irreflexive ℕTotalOrder pr1
|
||||
false (succ b) pr1 pr2 rewrite Semiring.commutative ℕSemiring a 0 | pr2 = cannotAddAndEnlarge' pr1
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inr sn=b+c | inr x = exFalso (false sn<a+b+c x)
|
||||
where
|
||||
false : succ n <N (a +N b) +N c → a +N (b +N c) ≡ succ n → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr2 = TotalOrder.irreflexive ℕTotalOrder pr1
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c with orderIsTotal (b +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c | inl (inl b+c<sn) with orderIsTotal (a +N (b +N c)) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c | inl (inl b+c<sn) | inl (inl a+'b+c<sn) = exFalso (false sn=a+b+c a+'b+c<sn)
|
||||
where
|
||||
false : (a +N b) +N c ≡ succ n → a +N (b +N c) <N succ n → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr1 = TotalOrder.irreflexive ℕTotalOrder pr2
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c | inl (inl b+c<sn) | inl (inr sn<a+'b+c) = exFalso (false sn=a+b+c sn<a+'b+c)
|
||||
where
|
||||
false : (a +N b) +N c ≡ succ n → succ n <N a +N (b +N c) → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr1 = TotalOrder.irreflexive ℕTotalOrder pr2
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c | inl (inl b+c<sn) | inr _ = equalityZn _ _ refl
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c | inl (inr sn<b+c) = exFalso (false a sn=a+b+c sn<b+c)
|
||||
where
|
||||
false : (a : ℕ) → (a +N b) +N c ≡ succ n → succ n <N b +N c → False
|
||||
false zero pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | equalityCommutative pr1 = TotalOrder.irreflexive ℕTotalOrder pr2
|
||||
false (succ a) pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | equalityCommutative pr1 = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr2 (le a refl))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c | inr b+c=sn with orderIsTotal (a +N 0) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c | inr b+c=sn | inl (inl a+0<sn) = equalityZn _ _ ans
|
||||
where
|
||||
a=0 : (a : ℕ) → (a +N b) +N c ≡ succ n → b +N c ≡ succ n → a ≡ 0
|
||||
a=0 zero pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr2 = refl
|
||||
a=0 (succ a) pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr2 = canSubtractFromEqualityRight pr1
|
||||
ans : a +N 0 ≡ 0
|
||||
ans rewrite Semiring.commutative ℕSemiring a 0 = a=0 a sn=a+b+c b+c=sn
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c | inr b+c=sn | inl (inr sn<a+0) = exFalso (false sn<a+0 sn=a+b+c b+c=sn)
|
||||
where
|
||||
false : succ n <N a +N 0 → (a +N b) +N c ≡ succ n → b +N c ≡ succ n → False
|
||||
false pr1 pr2 pr3 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr3 | Semiring.commutative ℕSemiring a 0 = zeroNeverGreater {succ n} (identityOfIndiscernablesRight _<N_ pr1 (a=0 a pr2))
|
||||
where
|
||||
a=0 : (a : ℕ) → (a +N succ n ≡ succ n) → a ≡ 0
|
||||
a=0 zero pr = refl
|
||||
a=0 (succ a) pr = canSubtractFromEqualityRight pr
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c | inr b+c=sn | inr sn=a+0 = exFalso (false sn=a+b+c b+c=sn sn=a+0)
|
||||
where
|
||||
false : (a +N b) +N c ≡ succ n → b +N c ≡ succ n → a +N 0 ≡ succ n → False
|
||||
false pr1 pr2 pr3 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr2 | equalityCommutative pr3 | Semiring.commutative ℕSemiring a 0 = naughtE (identityOfIndiscernablesLeft _≡_ pr3 (a=0 a pr1))
|
||||
where
|
||||
a=0 : (a : ℕ) → (a +N a ≡ a) → a ≡ 0
|
||||
a=0 zero pr = refl
|
||||
a=0 (succ a) pr = exFalso (naughtE {a} (equalityCommutative (canSubtractFromEqualityRight pr)))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) with orderIsTotal (b +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inl b+c<sn) with orderIsTotal (a +N (b +N c)) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inl b+c<sn) | inl (inl a+'b+c<sn) = exFalso (false sn<a+b a+'b+c<sn)
|
||||
where
|
||||
false : succ n <N a +N b → a +N (b +N c) <N succ n → False
|
||||
false pr1 pr2 rewrite Semiring.+Associative ℕSemiring a b c = cannotAddAndEnlarge' (orderIsTransitive pr2 pr1)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inl b+c<sn) | inl (inr sn<a+'b+c) with orderIsTotal (subtractionNResult.result (-N (inl sn<a+b)) +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inl b+c<sn) | inl (inr sn<a+'b+c) | inl (inl x) = equalityZn _ _ (help4 {a} {b} {c} {succ n} sn<a+'b+c sn<a+b)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inl b+c<sn) | inl (inr sn<a+'b+c) | inl (inr x) = exFalso (help18 {a} {b} {c} {succ n} b+c<sn sn<a+b aLess x)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inl b+c<sn) | inl (inr sn<a+'b+c) | inr x = exFalso (help19 {a} {b} {c} {succ n} b+c<sn sn<a+b aLess x)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inl b+c<sn) | inr a+'b+c=sn = exFalso (false sn<a+b a+'b+c=sn)
|
||||
where
|
||||
false : (succ n <N a +N b) → a +N (b +N c) ≡ succ n → False
|
||||
false pr1 pr2 rewrite Semiring.+Associative ℕSemiring a b c | equalityCommutative pr2 = cannotAddAndEnlarge' pr1
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) with orderIsTotal (a +N subtractionNResult.result (-N (inl sn<b+c))) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inl (inl x) with orderIsTotal (subtractionNResult.result (-N (inl sn<a+b)) +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inl (inl x) | inl (inl x₁) = equalityZn _ _ (help5 {a} {b} {c} {succ n} sn<b+c sn<a+b)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inl (inl x) | inl (inr x1) = equalityZn _ _ (help16 {a} {b} {c} {succ n} sn<b+c sn<a+b x x1)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inl (inl x) | inr x1 = exFalso (help17 {a} {b} {c} {succ n} sn<b+c sn<a+b x x1)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inl (inr x) with orderIsTotal (subtractionNResult.result (-N (inl sn<a+b)) +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inl (inr x) | inl (inl x1) = equalityZn _ _ (exFalso (help15 {a} {b} {c} {succ n} sn<b+c sn<a+b x x1))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inl (inr x) | inl (inr x1) = equalityZn _ _ (help14 {a} {b} {c} {succ n} sn<b+c sn<a+b x x1)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inl (inr x) | inr x1 = equalityZn _ _ (exFalso (help13 {a} {b} {c} {succ n} sn<b+c sn<a+b x x1))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inr x with orderIsTotal (subtractionNResult.result (-N (inl sn<a+b)) +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inr x | inl (inl x1) = equalityZn _ _ (exFalso (help12 {a} {b} {c} {succ n} sn<b+c sn<a+b x x1))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inr x | inl (inr x1) = equalityZn _ _ (exFalso (help10 {a} {b} {c} {succ n} sn<b+c sn<a+b x x1))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inr x | inr x₁ = equalityZn _ _ refl
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inr b+c=sn with orderIsTotal (a +N 0) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inr b+c=sn | inl (inl a+0<sn) with orderIsTotal (subtractionNResult.result (-N (inl sn<a+b)) +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inr b+c=sn | inl (inl a+0<sn) | inl (inl x) = equalityZn _ _ (help6 {a} {b} {c} {succ n} b+c=sn sn<a+b)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inr b+c=sn | inl (inl _) | inl (inr x) = equalityZn _ _ (exFalso (help11 {a} {b} {c} {succ n} aLess b+c=sn sn<a+b x))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inr b+c=sn | inl (inl a+0<sn) | inr x = equalityZn _ _ (exFalso (help7 {a} {b} {c} {succ n} b+c=sn aLess sn<a+b x))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inr b+c=sn | inl (inr sn<a+0) = exFalso (help8 {a} {succ n} sn<a+0 aLess)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inr b+c=sn | inr a+0=sn = exFalso (help9 {a} {succ n} a+0=sn aLess)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b with orderIsTotal c (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) with orderIsTotal (b +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inl (inl b+c<sn) with orderIsTotal (a +N (b +N c)) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inl (inl b+c<sn) | inl (inl a+'b+c<sn) = equalityZn _ _ (help27 {a} {b} {c} {n} sn=a+b a+'b+c<sn)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inl (inl b+c<sn) | inl (inr sn<a+'b+c) = equalityZn _ _ (help28 {a} {b} {c} {succ n} sn<a+'b+c sn=a+b)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inl (inl b+c<sn) | inr a+'b+c=sn = equalityZn _ _ (help26 {a} {b} {c} {succ n} sn=a+b a+'b+c=sn)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inl (inr sn<b+c) with orderIsTotal (a +N subtractionNResult.result (-N (inl sn<b+c))) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inl (inr sn<b+c) | inl (inl x) = equalityZn _ _ (help31 {a} {b} {c} {succ n} sn=a+b sn<b+c)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inl (inr sn<b+c) | inl (inr x) = exFalso (help30 {a} {b} {c} {succ n} cLess sn=a+b sn<b+c x)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inl (inr sn<b+c) | inr x = exFalso (help29 {a} {b} {c} {succ n} cLess sn<b+c x sn=a+b)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inr b+c=sn with orderIsTotal (a +N 0) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inr b+c=sn | inl (inl a+0<sn) = equalityZn _ _ (help25 {a} {b} {c} {succ n} sn=a+b b+c=sn)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inr b+c=sn | inl (inr sn<a+0) = exFalso (help24 {a} {succ n} aLess sn<a+0)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inr b+c=sn | inr a+0=sn = exFalso (help23 {a} {succ n} aLess a+0=sn)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inr sn<c) = exFalso (TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive sn<c cLess))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inr c=sn with orderIsTotal (b +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inr c=sn | inl (inl b+c<sn) = exFalso (help b+c<sn)
|
||||
where
|
||||
help : (b +N c <N succ n) → False
|
||||
help b+c<sn rewrite equalityCommutative c=sn | Semiring.commutative ℕSemiring b c = cannotAddAndEnlarge' b+c<sn
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inr c=sn | inl (inr sn<b+c) with orderIsTotal (a +N subtractionNResult.result (-N (inl sn<b+c))) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inr c=sn | inl (inr sn<b+c) | inl (inl x) = exFalso (help20 {a} {b} {c} {succ n} c=sn sn=a+b sn<b+c x)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inr c=sn | inl (inr sn<b+c) | inl (inr x) = exFalso (help22 {a} {b} {c} {succ n} sn=a+b c=sn sn<b+c x)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inr c=sn | inl (inr sn<b+c) | inr x = equalityZn _ _ refl
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inr c=sn | inr b+c=sn = exFalso (help21 {a} {b} {c} {succ n} sn=a+b b+c=sn c=sn aLess)
|
||||
|
||||
subLess : {a b : ℕ} → (0<a : 0 <N a) → (a<b : a <N b) → subtractionNResult.result (-N (inl a<b)) <N b
|
||||
subLess {zero} {b} 0<a a<b = exFalso (TotalOrder.irreflexive ℕTotalOrder 0<a)
|
||||
subLess {succ a} {b} _ a<b with -N (inl a<b)
|
||||
... | record { result = b-a ; pr = pr } = record { x = a ; proof = pr }
|
||||
|
||||
inverseZn : {n : ℕ} → {pr : 0 <N n} → (a : ℤn n pr) → Sg (ℤn n pr) (λ i → i +n a ≡ record { x = 0 ; xLess = pr } )
|
||||
inverseZn {zero} {()}
|
||||
inverseZn {succ n} {0<n} record { x = zero ; xLess = zeroLess } = record { x = zero ; xLess = zeroLess } , plusZnIdentityLeft _
|
||||
inverseZn {succ n} {0<n} record { x = (succ a) ; xLess = aLess } = ans , pr
|
||||
where
|
||||
ans = record { x = subtractionNResult.result (-N (inl (canRemoveSuccFrom<N aLess))) ; xLess = subLess (succIsPositive a) aLess }
|
||||
pr : ans +n record { x = (succ a) ; xLess = aLess } ≡ record { x = 0 ; xLess = 0<n }
|
||||
pr with orderIsTotal (subtractionNResult.result (-N (inl (canRemoveSuccFrom<N aLess))) +N (succ a)) (succ n)
|
||||
... | inl (inl x) = exFalso f
|
||||
where
|
||||
h : subtractionNResult.result (-N (inl (canRemoveSuccFrom<N aLess))) +N succ a ≡ succ n
|
||||
h with -N (inl (canRemoveSuccFrom<N aLess))
|
||||
h | record { result = result ; pr = pr } rewrite equalityCommutative pr = Semiring.commutative ℕSemiring result (succ a)
|
||||
f : False
|
||||
f = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesLeft _<N_ x h)
|
||||
... | inl (inr x) = exFalso f
|
||||
where
|
||||
h : subtractionNResult.result (-N (inl (canRemoveSuccFrom<N aLess))) +N succ a ≡ succ n
|
||||
h = addMinus {succ a} {succ n} (inl aLess)
|
||||
f : False
|
||||
f rewrite h = TotalOrder.irreflexive ℕTotalOrder x
|
||||
... | (inr n-a+sa=sn) = equalityZn _ _ refl
|
||||
|
||||
ℤnGroup : (n : ℕ) → (pr : 0 <N n) → Group (reflSetoid (ℤn n pr)) _+n_
|
||||
ℤnGroup n pr = record { 0G = record { x = 0 ; xLess = pr } ; inverse = λ a → underlying (inverseZn a) ; +Associative = λ {a} {b} {c} → plusZnAssociative a b c ; identRight = λ {a} → plusZnIdentityRight a ; identLeft = λ {a} → plusZnIdentityLeft a ; invLeft = λ {a} → helpInvLeft a ; invRight = λ {a} → helpInvRight a ; +WellDefined = reflGroupWellDefined }
|
||||
where
|
||||
helpInvLeft : (a : ℤn n pr) → underlying (inverseZn a) +n a ≡ record { x = 0 ; xLess = pr }
|
||||
helpInvLeft a with inverseZn a
|
||||
... | -a , pr-a = pr-a
|
||||
helpInvRight : (a : ℤn n pr) → a +n underlying (inverseZn a) ≡ record { x = 0 ; xLess = pr }
|
||||
helpInvRight a rewrite plusZnCommutative a (underlying (inverseZn a)) = helpInvLeft a
|
||||
|
||||
ℤnAbGroup : (n : ℕ) → (pr : 0 <N n) → AbelianGroup (ℤnGroup n pr)
|
||||
AbelianGroup.commutative (ℤnAbGroup n pr) {a} {b} = plusZnCommutative a b
|
||||
|
||||
ℤnFinite : (n : ℕ) → (pr : 0 <N n) → FiniteGroup (ℤnGroup n pr) (FinSet n)
|
||||
SetoidToSet.project (FiniteGroup.toSet (ℤnFinite n pr)) record { x = x ; xLess = xLess } = ofNat x xLess
|
||||
SetoidToSet.wellDefined (FiniteGroup.toSet (ℤnFinite n pr)) x y x=y rewrite x=y = refl
|
||||
Surjection.property (SetoidToSet.surj (FiniteGroup.toSet (ℤnFinite n pr))) b = record { x = toNat b ; xLess = toNatSmaller b } , ofNatToNat b
|
||||
SetoidToSet.inj (FiniteGroup.toSet (ℤnFinite zero ())) x y x=y
|
||||
SetoidToSet.inj (FiniteGroup.toSet (ℤnFinite (succ n) pr)) record { x = x ; xLess = xLess } record { x = y ; xLess = yLess } x=y = equalityZn _ _ b
|
||||
where
|
||||
b : x ≡ y
|
||||
b = ofNatInjective x y xLess yLess x=y
|
||||
FiniteSet.size (FiniteGroup.finite (ℤnFinite n pr)) = n
|
||||
FiniteSet.mapping (FiniteGroup.finite (ℤnFinite n pr)) = id
|
||||
FiniteSet.bij (FiniteGroup.finite (ℤnFinite n pr)) = idIsBijective
|
||||
@@ -1,705 +0,0 @@
|
||||
{-# OPTIONS --safe --warning=error #-}
|
||||
-- These are explicitly with-K, because we currently encode an element of Zn as
|
||||
-- a natural together with a proof that it is small.
|
||||
|
||||
open import LogicalFormulae
|
||||
open import Groups.Definition
|
||||
open import Groups.Groups
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Numbers.Naturals.Addition -- TODO remove this dependency
|
||||
open import Numbers.Primes.PrimeNumbers
|
||||
open import Setoids.Setoids
|
||||
open import Sets.FinSet
|
||||
open import Sets.FinSetWithK
|
||||
open import Functions
|
||||
open import Numbers.Naturals.WithK
|
||||
open import Semirings.Definition
|
||||
open import Orders
|
||||
|
||||
module Numbers.Modulo.IntegersModN where
|
||||
record ℤn (n : ℕ) (pr : 0 <N n) : Set where
|
||||
field
|
||||
x : ℕ
|
||||
xLess : x <N n
|
||||
|
||||
equalityZn : {n : ℕ} {pr : 0 <N n} → (a b : ℤn n pr) → (ℤn.x a ≡ ℤn.x b) → a ≡ b
|
||||
equalityZn record { x = a ; xLess = aLess } record { x = .a ; xLess = bLess } refl rewrite <NRefl aLess bLess = refl
|
||||
|
||||
cancelSumFromInequality : {a b c : ℕ} → a +N b <N a +N c → b <N c
|
||||
cancelSumFromInequality {a} {b} {c} (le x proof) = le x help
|
||||
where
|
||||
help : succ x +N b ≡ c
|
||||
help rewrite Semiring.+Associative ℕSemiring (succ x) a b | Semiring.commutative ℕSemiring x a | equalityCommutative (Semiring.+Associative ℕSemiring (succ a) x b) | Semiring.commutative ℕSemiring a (x +N b) | Semiring.commutative ℕSemiring (succ (x +N b)) a = canSubtractFromEqualityLeft {a} proof
|
||||
|
||||
_+n_ : {n : ℕ} {pr : 0 <N n} → ℤn n pr → ℤn n pr → ℤn n pr
|
||||
_+n_ {0} {le x ()} a b
|
||||
_+n_ {succ n} {pr} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } with orderIsTotal (a +N b) (succ n)
|
||||
_+n_ {succ n} {pr} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inl (a+b<n)) = record { x = a +N b ; xLess = a+b<n }
|
||||
_+n_ {succ n} {pr} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inr (n<a+b)) = record { x = subtractionNResult.result sub ; xLess = pr2 }
|
||||
where
|
||||
sub : subtractionNResult (succ n) (a +N b) (inl n<a+b)
|
||||
sub = -N (inl n<a+b)
|
||||
pr1 : a +N b <N succ n +N succ n
|
||||
pr1 = addStrongInequalities aLess bLess
|
||||
pr1' : succ n +N (subtractionNResult.result sub) <N succ n +N succ n
|
||||
pr1' rewrite subtractionNResult.pr sub = pr1
|
||||
pr2 : subtractionNResult.result sub <N succ n
|
||||
pr2 = cancelSumFromInequality pr1'
|
||||
_+n_ {succ n} {pr} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inr (a+b=n) = record { x = 0 ; xLess = succIsPositive n }
|
||||
|
||||
plusZnIdentityRight : {n : ℕ} → {pr : 0 <N n} → (a : ℤn n pr) → (a +n record { x = 0 ; xLess = pr }) ≡ a
|
||||
plusZnIdentityRight {zero} {()} a
|
||||
plusZnIdentityRight {succ x} {_} record { x = a ; xLess = aLess } with orderIsTotal (a +N 0) (succ x)
|
||||
plusZnIdentityRight {succ x} {_} record { x = a ; xLess = aLess } | inl (inl a<sx) = equalityZn _ _ (Semiring.commutative ℕSemiring a 0)
|
||||
plusZnIdentityRight {succ x} {_} record { x = a ; xLess = aLess } | inl (inr sx<a) = exFalso (f aLess sx<a)
|
||||
where
|
||||
f : (aL : a <N succ x) → (sx<a : succ x <N a +N 0) → False
|
||||
f aL sx<a rewrite Semiring.commutative ℕSemiring a 0 = orderIsIrreflexive aL sx<a
|
||||
plusZnIdentityRight {succ x} {_} record { x = a ; xLess = aLess } | inr a=sx = exFalso (f aLess a=sx)
|
||||
where
|
||||
f : (aL : a <N succ x) → (a=sx : a +N 0 ≡ succ x) → False
|
||||
f aL a=sx rewrite Semiring.commutative ℕSemiring a 0 | a=sx = TotalOrder.irreflexive ℕTotalOrder aL
|
||||
|
||||
|
||||
plusZnIdentityLeft : {n : ℕ} → {pr : 0 <N n} → (a : ℤn n pr) → (record { x = 0 ; xLess = pr }) +n a ≡ a
|
||||
plusZnIdentityLeft {zero} {()}
|
||||
plusZnIdentityLeft {succ n} {pr} record { x = x ; xLess = xLess } with orderIsTotal x (succ n)
|
||||
plusZnIdentityLeft {succ n} {pr} record { x = x ; xLess = xLess } | inl (inl x<succn) rewrite <NRefl x<succn xLess = refl
|
||||
plusZnIdentityLeft {succ n} {pr} record { x = x ; xLess = xLess } | inl (inr succn<x) = exFalso (TotalOrder.irreflexive ℕTotalOrder (TotalOrder.<Transitive ℕTotalOrder succn<x xLess))
|
||||
plusZnIdentityLeft {succ n} {pr} record { x = x ; xLess = xLess } | inr x=succn rewrite x=succn = exFalso (TotalOrder.irreflexive ℕTotalOrder xLess)
|
||||
|
||||
subLemma : {a b c : ℕ} → (pr1 : a <N b) → (pr2 : a <N c) → (pr : b ≡ c) → (subtractionNResult.result (-N (inl pr1))) ≡ (subtractionNResult.result (-N (inl pr2)))
|
||||
subLemma {a} {b} {c} a<b a<c b=c rewrite b=c | <NRefl a<b a<c = refl
|
||||
|
||||
plusZnCommutative : {n : ℕ} → {pr : 0 <N n} → (a b : ℤn n pr) → (a +n b) ≡ b +n a
|
||||
plusZnCommutative {zero} {()} a b
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } with orderIsTotal (a +N b) (succ n)
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inl a+b<sn) with orderIsTotal (b +N a) (succ n)
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inl a+b<sn) | inl (inl b+a<sn) = equalityZn _ _ (Semiring.commutative ℕSemiring a b)
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inl a+b<sn) | inl (inr sn<b+a) = exFalso (f a+b<sn sn<b+a)
|
||||
where
|
||||
f : (a +N b) <N succ n → succ n <N b +N a → False
|
||||
f pr1 pr2 rewrite Semiring.commutative ℕSemiring b a = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr2 pr1)
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inl a+b<sn) | inr b+a=sn = exFalso (f a+b<sn b+a=sn)
|
||||
where
|
||||
f : (a +N b) <N succ n → b +N a ≡ succ n → False
|
||||
f pr1 eq rewrite Semiring.commutative ℕSemiring b a | eq = TotalOrder.irreflexive ℕTotalOrder pr1
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inr sn<a+b) with orderIsTotal (b +N a) (succ n)
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inr sn<a+b) | inl (inl b+a<sn) = exFalso (f sn<a+b b+a<sn)
|
||||
where
|
||||
f : succ n <N a +N b → b +N a <N succ n → False
|
||||
f pr1 pr2 rewrite Semiring.commutative ℕSemiring b a = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive sn<a+b b+a<sn)
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inr sn<a+b) | inl (inr sn<b+a) = equalityZn _ _ (subLemma sn<a+b sn<b+a (Semiring.commutative ℕSemiring a b))
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inl (inr sn<a+b) | inr sn=b+a = exFalso (f sn<a+b sn=b+a)
|
||||
where
|
||||
f : succ n <N a +N b → b +N a ≡ succ n → False
|
||||
f pr eq rewrite Semiring.commutative ℕSemiring b a | eq = TotalOrder.irreflexive ℕTotalOrder pr
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inr sn=a+b with orderIsTotal (b +N a) (succ n)
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inr sn=a+b | inl (inl b+a<sn) = exFalso f
|
||||
where
|
||||
f : False
|
||||
f rewrite Semiring.commutative ℕSemiring b a | sn=a+b = TotalOrder.irreflexive ℕTotalOrder b+a<sn
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inr sn=a+b | inl (inr sn<b+a) = exFalso f
|
||||
where
|
||||
f : False
|
||||
f rewrite Semiring.commutative ℕSemiring b a | sn=a+b = TotalOrder.irreflexive ℕTotalOrder sn<b+a
|
||||
plusZnCommutative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } | inr sn=a+b | inr sn=b+a = equalityZn _ _ refl
|
||||
|
||||
help30 : {a b c n : ℕ} → (c <N n) → (a +N b ≡ n) → (n<b+c : n <N b +N c) → (n <N a +N subtractionNResult.result (-N (inl n<b+c))) → False
|
||||
help30 {a} {b} {c} {n} c<n a+b=n n<b+c x = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr5 c<n)
|
||||
where
|
||||
pr : n +N n <N a +N (subtractionNResult.result (-N (inl n<b+c)) +N n)
|
||||
pr = identityOfIndiscernablesRight _<N_ (additionPreservesInequality n x) (equalityCommutative (Semiring.+Associative ℕSemiring a _ n))
|
||||
pr2 : n +N n <N a +N (b +N c)
|
||||
pr2 = identityOfIndiscernablesRight _<N_ pr (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
pr3 : n +N n <N (a +N b) +N c
|
||||
pr3 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = pr2
|
||||
pr4 : n +N n <N c +N n
|
||||
pr4 rewrite Semiring.commutative ℕSemiring c n = identityOfIndiscernablesRight _<N_ pr3 (applyEquality (_+N c) a+b=n)
|
||||
pr5 : n <N c
|
||||
pr5 = subtractionPreservesInequality n pr4
|
||||
|
||||
help31 : {a b c n : ℕ} → (a +N b ≡ n) → (n<b+c : n <N b +N c) → (a +N subtractionNResult.result (-N (inl n<b+c))) ≡ c
|
||||
help31 {a} {b} {c} {n} a+b=n n<b+c = canSubtractFromEqualityLeft pr2
|
||||
where
|
||||
pr1 : a +N (n +N subtractionNResult.result (-N (inl n<b+c))) ≡ n +N c
|
||||
pr1 rewrite addMinus' (inl n<b+c) | Semiring.+Associative ℕSemiring a b c | a+b=n = refl
|
||||
pr2 : n +N (a +N subtractionNResult.result (-N (inl n<b+c))) ≡ n +N c
|
||||
pr2 = identityOfIndiscernablesLeft _≡_ pr1 (lemm a n (subtractionNResult.result (-N (inl n<b+c))))
|
||||
where
|
||||
lemm : (a b c : ℕ) → a +N (b +N c) ≡ b +N (a +N c)
|
||||
lemm a b c rewrite Semiring.+Associative ℕSemiring a b c | Semiring.commutative ℕSemiring a b | equalityCommutative (Semiring.+Associative ℕSemiring b a c) = refl
|
||||
|
||||
help7 : {a b c n : ℕ} → b +N c ≡ n → a <N n → (n<a+b : n <N a +N b) → (subtractionNResult.result (-N (inl n<a+b)) +N c ≡ n) → False
|
||||
help7 {a} {b} {c} {n} b+c=n a<n n<a+b x = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesLeft _<N_ a<n pr5)
|
||||
where
|
||||
pr : (n +N subtractionNResult.result (-N (inl n<a+b))) +N c ≡ n +N n
|
||||
pr = identityOfIndiscernablesLeft _≡_ (applyEquality (n +N_) x) (Semiring.+Associative ℕSemiring n _ c)
|
||||
pr2 : (a +N b) +N c ≡ n +N n
|
||||
pr2 = identityOfIndiscernablesLeft _≡_ pr (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
pr3 : a +N (b +N c) ≡ n +N n
|
||||
pr3 rewrite Semiring.+Associative ℕSemiring a b c = pr2
|
||||
pr4 : a +N n ≡ n +N n
|
||||
pr4 = identityOfIndiscernablesLeft _≡_ pr3 (applyEquality (a +N_) b+c=n)
|
||||
pr5 : a ≡ n
|
||||
pr5 = canSubtractFromEqualityRight pr4
|
||||
|
||||
help29 : {a b c n : ℕ} → (c <N n) → (n<b+c : n <N b +N c) → (a +N subtractionNResult.result (-N (inl n<b+c))) ≡ n → a +N b ≡ n → False
|
||||
help29 {a} {b} {c} {n} c<n n<b+c pr a+b=n = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesLeft _<N_ c<n p4)
|
||||
where
|
||||
p1 : a +N (subtractionNResult.result (-N (inl n<b+c)) +N n) ≡ n +N n
|
||||
p1 = identityOfIndiscernablesLeft _≡_ (applyEquality (_+N n) pr) (equalityCommutative (Semiring.+Associative ℕSemiring a _ n))
|
||||
p2 : (a +N b) +N c ≡ n +N n
|
||||
p2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = identityOfIndiscernablesLeft _≡_ p1 (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
p3 : n +N c ≡ n +N n
|
||||
p3 = identityOfIndiscernablesLeft _≡_ p2 (applyEquality (_+N c) a+b=n)
|
||||
p4 : c ≡ n
|
||||
p4 = canSubtractFromEqualityLeft p3
|
||||
|
||||
help28 : {a b c n : ℕ} → (n<a+'b+c : n <N a +N (b +N c)) → (a +N b ≡ n) → subtractionNResult.result (-N (inl n<a+'b+c)) ≡ c
|
||||
help28 {a} {b} {c} {n} pr a+b=n = canSubtractFromEqualityLeft p2
|
||||
where
|
||||
p1 : a +N (b +N c) ≡ n +N c
|
||||
p1 rewrite Semiring.+Associative ℕSemiring a b c | a+b=n = refl
|
||||
p2 : n +N subtractionNResult.result (-N (inl pr)) ≡ n +N c
|
||||
p2 = identityOfIndiscernablesLeft _≡_ p1 (equalityCommutative (addMinus' (inl pr)))
|
||||
|
||||
help27 : {a b c n : ℕ} → (a +N b ≡ succ n) → (a +N (b +N c) <N succ n) → a +N (b +N c) ≡ c
|
||||
help27 {a} {b} {zero} {n} a+b=sn a+b+c<sn rewrite Semiring.commutative ℕSemiring b 0 | a+b=sn = exFalso (TotalOrder.irreflexive ℕTotalOrder a+b+c<sn)
|
||||
help27 {a} {b} {succ c} {n} a+b=sn a+b+c<sn rewrite Semiring.+Associative ℕSemiring a b (succ c) | a+b=sn = exFalso (cannotAddAndEnlarge' a+b+c<sn)
|
||||
|
||||
help26 : {a b c n : ℕ} → (a +N b ≡ n) → (a +N (b +N c) ≡ n) → 0 ≡ c
|
||||
help26 {a} {b} {c} {n} a+b=n a+b+c=n rewrite Semiring.+Associative ℕSemiring a b c | a+b=n | Semiring.commutative ℕSemiring n c = equalityCommutative (canSubtractFromEqualityRight a+b+c=n)
|
||||
|
||||
help25 : {a b c n : ℕ} → (a +N b ≡ n) → (b +N c ≡ n) → (a +N 0 ≡ c)
|
||||
help25 {a} {b} {c} {n} a+b=n b+c=n rewrite Semiring.commutative ℕSemiring a 0 | Semiring.commutative ℕSemiring a b | equalityCommutative a+b=n = equalityCommutative (canSubtractFromEqualityLeft b+c=n)
|
||||
|
||||
help24 : {a n : ℕ} → (a <N n) → (n <N a +N 0) → False
|
||||
help24 {a} {n} a<n n<a+0 rewrite Semiring.commutative ℕSemiring a 0 = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive a<n n<a+0)
|
||||
|
||||
help23 : {a n : ℕ} → (a <N n) → (a +N 0 ≡ n) → False
|
||||
help23 {a} {n} a<n a+0=n rewrite Semiring.commutative ℕSemiring a 0 | a+0=n = TotalOrder.irreflexive ℕTotalOrder a<n
|
||||
|
||||
help22 : {a b c n : ℕ} → (a +N b ≡ n) → (c ≡ n) → (n<b+c : n <N b +N c) → (n <N a +N subtractionNResult.result (-N (inl n<b+c))) → False
|
||||
help22 {a} {b} {c} {.c} a+b=n refl n<b+c pr = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesRight _<N_ pr4 a+b=n)
|
||||
where
|
||||
pr1 : c +N c <N a +N (subtractionNResult.result (-N (inl n<b+c)) +N c)
|
||||
pr1 = identityOfIndiscernablesRight _<N_ (additionPreservesInequality c pr) (equalityCommutative (Semiring.+Associative ℕSemiring a _ c))
|
||||
pr2 : c +N c <N a +N (b +N c)
|
||||
pr2 = identityOfIndiscernablesRight _<N_ pr1 (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
pr3 : c +N c <N (a +N b) +N c
|
||||
pr3 = identityOfIndiscernablesRight _<N_ pr2 (Semiring.+Associative ℕSemiring a b c)
|
||||
pr4 : c <N a +N b
|
||||
pr4 = subtractionPreservesInequality c pr3
|
||||
|
||||
help21 : {a b c n : ℕ} → (a +N b ≡ n) → (b +N c ≡ n) → (c ≡ n) → (a <N n) → False
|
||||
help21 {a} {b} {c} {.c} a+b=n b+c=n refl a<n = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesLeft _<N_ a<n a=c)
|
||||
where
|
||||
b=0 : b ≡ 0
|
||||
a=c : a ≡ c
|
||||
a=c = identityOfIndiscernablesLeft _≡_ a+b=n lemm
|
||||
where
|
||||
lemm : a +N b ≡ a
|
||||
lemm rewrite b=0 | Semiring.commutative ℕSemiring a 0 = refl
|
||||
b=0' : (b c : ℕ) → (b +N c ≡ c) → b ≡ 0
|
||||
b=0' zero c b+c=c = refl
|
||||
b=0' (succ b) zero b+c=c = exFalso (naughtE (equalityCommutative b+c=c))
|
||||
b=0' (succ b) (succ c) b+c=c = b=0' (succ b) c bl2
|
||||
where
|
||||
bl2 : succ b +N c ≡ c
|
||||
bl2 rewrite Semiring.commutative ℕSemiring b c | Semiring.commutative ℕSemiring (succ c) b = succInjective b+c=c
|
||||
b=0 = b=0' b c b+c=n
|
||||
|
||||
help20 : {a b c n : ℕ} → (c ≡ n) → (a +N b ≡ n) → (n<b+c : n <N b +N c) → (a +N subtractionNResult.result (-N (inl n<b+c)) <N n) → False
|
||||
help20 {a} {b} {c} {n} c=n a+b=n n<b+c pr = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesLeft _<N_ pr5 c=n)
|
||||
where
|
||||
pr1 : a +N (subtractionNResult.result (-N (inl n<b+c)) +N n) <N n +N n
|
||||
pr1 = identityOfIndiscernablesLeft _<N_ (additionPreservesInequality n pr) (equalityCommutative (Semiring.+Associative ℕSemiring a _ n))
|
||||
pr2 : a +N (b +N c) <N n +N n
|
||||
pr2 = identityOfIndiscernablesLeft _<N_ pr1 (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
pr3 : (a +N b) +N c <N n +N n
|
||||
pr3 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = pr2
|
||||
pr4 : c +N n <N n +N n
|
||||
pr4 = identityOfIndiscernablesLeft _<N_ pr3 (transitivity (applyEquality (_+N c) a+b=n) (Semiring.commutative ℕSemiring n c))
|
||||
pr5 : c <N n
|
||||
pr5 = subtractionPreservesInequality n pr4
|
||||
|
||||
help19 : {a b c n : ℕ} → (b+c<n : b +N c <N n) → (n<a+b : n <N a +N b) → (a <N n) → (subtractionNResult.result (-N (inl n<a+b)) +N c ≡ n) → False
|
||||
help19 {a} {b} {c} {n} b+c<n n<a+b a<n pr = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesLeft _<N_ r q')
|
||||
where
|
||||
p : (n +N subtractionNResult.result (-N (inl n<a+b))) +N c ≡ n +N n
|
||||
p = identityOfIndiscernablesLeft _≡_ (applyEquality (n +N_ ) pr) (Semiring.+Associative ℕSemiring n _ c)
|
||||
q : (a +N b) +N c ≡ n +N n
|
||||
q = identityOfIndiscernablesLeft _≡_ p (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
q' : a +N (b +N c) ≡ n +N n
|
||||
q' = identityOfIndiscernablesLeft _≡_ q (equalityCommutative (Semiring.+Associative ℕSemiring a b c))
|
||||
r : a +N (b +N c) <N n +N n
|
||||
r = addStrongInequalities a<n b+c<n
|
||||
|
||||
help18 : {a b c n : ℕ} → (b+c<n : b +N c <N n) → (n<a+b : n <N a +N b) → (a <N n) → (n <N subtractionNResult.result (-N (inl n<a+b)) +N c) → False
|
||||
help18 {a} {b} {c} {n} b+c<n n<a+b a<n pr = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive p4 a<n)
|
||||
where
|
||||
p : n +N n <N (n +N subtractionNResult.result (-N (inl n<a+b))) +N c
|
||||
p = identityOfIndiscernablesRight _<N_ (additionPreservesInequalityOnLeft n pr) (Semiring.+Associative ℕSemiring n _ c)
|
||||
p' : n +N n <N (a +N b) +N c
|
||||
p' = identityOfIndiscernablesRight _<N_ p (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
p2 : n +N n <N a +N (b +N c)
|
||||
p2 = identityOfIndiscernablesRight _<N_ p' (equalityCommutative (Semiring.+Associative ℕSemiring a b c))
|
||||
p3 : n +N n <N a +N n
|
||||
p3 = orderIsTransitive p2 (additionPreservesInequalityOnLeft a b+c<n)
|
||||
p4 : n <N a
|
||||
p4 = subtractionPreservesInequality n p3
|
||||
|
||||
help17 : {a b c n : ℕ} → (n<b+c : n <N b +N c) → (n<a+b : n <N a +N b) → (a +N subtractionNResult.result (-N (inl n<b+c)) <N n) → (subtractionNResult.result (-N (inl n<a+b)) +N c) ≡ n → False
|
||||
help17 {a} {b} {c} {n} n<b+c n<a+b p1 p2 = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesLeft _<N_ pr1'' pr3)
|
||||
where
|
||||
pr1' : a +N (subtractionNResult.result (-N (inl n<b+c)) +N n) <N n +N n
|
||||
pr1' = identityOfIndiscernablesLeft _<N_ (additionPreservesInequality n p1) (equalityCommutative (Semiring.+Associative ℕSemiring a _ n))
|
||||
pr1'' : a +N (b +N c) <N n +N n
|
||||
pr1'' = identityOfIndiscernablesLeft _<N_ pr1' (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
pr2' : (n +N subtractionNResult.result (-N (inl n<a+b))) +N c ≡ n +N n
|
||||
pr2' = identityOfIndiscernablesLeft _≡_ (applyEquality (n +N_) p2) (Semiring.+Associative ℕSemiring n _ c)
|
||||
pr2'' : (a +N b) +N c ≡ n +N n
|
||||
pr2'' = identityOfIndiscernablesLeft _≡_ pr2' (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
pr3 : a +N (b +N c) ≡ n +N n
|
||||
pr3 = identityOfIndiscernablesLeft _≡_ pr2'' (equalityCommutative (Semiring.+Associative ℕSemiring a b c))
|
||||
|
||||
help16 : {a b c n : ℕ} → (n<b+c : n <N b +N c) → (n<a+b : n <N a +N b) → (a +N subtractionNResult.result (-N (inl n<b+c))) <N n → (pr : n <N subtractionNResult.result (-N (inl n<a+b)) +N c) → a +N subtractionNResult.result (-N (inl n<b+c)) ≡ subtractionNResult.result (-N (inl pr))
|
||||
help16 {a} {b} {c} {n} n<b+c n<a+b pr1 pr2 = exFalso (TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr3 pr1''))
|
||||
where
|
||||
pr1' : a +N (subtractionNResult.result (-N (inl n<b+c)) +N n) <N n +N n
|
||||
pr1' = identityOfIndiscernablesLeft _<N_ (additionPreservesInequality n pr1) (equalityCommutative (Semiring.+Associative ℕSemiring a _ n))
|
||||
pr1'' : a +N (b +N c) <N n +N n
|
||||
pr1'' = identityOfIndiscernablesLeft _<N_ pr1' (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
pr2' : n +N n <N (n +N subtractionNResult.result (-N (inl n<a+b))) +N c
|
||||
pr2' = identityOfIndiscernablesRight _<N_ (additionPreservesInequalityOnLeft n pr2) (Semiring.+Associative ℕSemiring n _ c)
|
||||
pr2'' : n +N n <N (a +N b) +N c
|
||||
pr2'' = identityOfIndiscernablesRight _<N_ pr2' (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
pr3 : n +N n <N a +N (b +N c)
|
||||
pr3 = identityOfIndiscernablesRight _<N_ pr2'' (equalityCommutative (Semiring.+Associative ℕSemiring a b c))
|
||||
|
||||
help15 : {a b c n : ℕ} → (n<b+c : n <N b +N c) → (n<a+b : n <N a +N b) → (n <N a +N subtractionNResult.result (-N (inl n<b+c))) → (subtractionNResult.result (-N (inl n<a+b)) +N c) <N n → False
|
||||
help15 {a} {b} {c} {n} n<b+c n<a+b pr1 pr2 = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive p2'' p1')
|
||||
where
|
||||
p1 : (n +N subtractionNResult.result (-N (inl n<a+b))) +N c <N n +N n
|
||||
p1 = identityOfIndiscernablesLeft _<N_ (additionPreservesInequalityOnLeft n pr2) (Semiring.+Associative ℕSemiring n _ c)
|
||||
p1' : (a +N b) +N c <N n +N n
|
||||
p1' = identityOfIndiscernablesLeft _<N_ p1 (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
p2 : n +N n <N a +N (subtractionNResult.result (-N (inl n<b+c)) +N n)
|
||||
p2 = identityOfIndiscernablesRight _<N_ (additionPreservesInequality n pr1) (equalityCommutative (Semiring.+Associative ℕSemiring a _ n))
|
||||
p2' : n +N n <N a +N (b +N c)
|
||||
p2' = identityOfIndiscernablesRight _<N_ p2 (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
p2'' : n +N n <N (a +N b) +N c
|
||||
p2'' = identityOfIndiscernablesRight _<N_ p2' (Semiring.+Associative ℕSemiring a b c)
|
||||
|
||||
help14 : {a b c n : ℕ} → (n<b+c : n <N b +N c) → (n<a+b : n <N a +N b) → (pr1 : n <N a +N subtractionNResult.result (-N (inl n<b+c))) → (pr2 : n <N subtractionNResult.result (-N (inl n<a+b)) +N c) → subtractionNResult.result (-N (inl pr1)) ≡ subtractionNResult.result (-N (inl pr2))
|
||||
help14 {a} {b} {c} {n} n<b+c n<a+b pr1 pr2 = equivalentSubtraction _ _ _ _ pr1 pr2 (transitivity (Semiring.+Associative ℕSemiring n _ c) (transitivity (applyEquality (_+N c) (addMinus' (inl n<a+b))) (transitivity (equalityCommutative (Semiring.+Associative ℕSemiring a b c)) (equalityCommutative p2))))
|
||||
where
|
||||
p1 : (a +N subtractionNResult.result (-N (inl n<b+c))) +N n ≡ a +N (subtractionNResult.result (-N (inl n<b+c)) +N n)
|
||||
p1 = equalityCommutative (Semiring.+Associative ℕSemiring a _ n)
|
||||
p2 : (a +N subtractionNResult.result (-N (inl n<b+c))) +N n ≡ a +N (b +N c)
|
||||
p2 = identityOfIndiscernablesRight _≡_ p1 (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
|
||||
help13 : {a b c n : ℕ} → (n<b+c : n <N b +N c) → (n<a+b : n <N a +N b) → (n <N a +N subtractionNResult.result (-N (inl n<b+c))) → (subtractionNResult.result (-N (inl n<a+b)) +N c ≡ n) → False
|
||||
help13 {a} {b} {c} {n} n<b+c n<a+b pr1 pr2 = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesRight _<N_ lemm1' lemm3)
|
||||
where
|
||||
lemm1 : n +N n <N a +N (subtractionNResult.result (-N (inl n<b+c)) +N n)
|
||||
lemm1 = identityOfIndiscernablesRight _<N_ (additionPreservesInequality n pr1) (equalityCommutative (Semiring.+Associative ℕSemiring a _ n))
|
||||
lemm1' : n +N n <N a +N (b +N c)
|
||||
lemm1' = identityOfIndiscernablesRight _<N_ lemm1 (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
lemm2 : (n +N subtractionNResult.result (-N (inl n<a+b))) +N c ≡ n +N n
|
||||
lemm2 = identityOfIndiscernablesLeft _≡_ (applyEquality (n +N_) pr2) (Semiring.+Associative ℕSemiring n _ c)
|
||||
lemm2' : (a +N b) +N c ≡ n +N n
|
||||
lemm2' = identityOfIndiscernablesLeft _≡_ lemm2 (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
lemm3 : a +N (b +N c) ≡ n +N n
|
||||
lemm3 rewrite Semiring.+Associative ℕSemiring a b c = lemm2'
|
||||
|
||||
help12 : {a b c n : ℕ} → (n<b+c : n <N b +N c) → (n<a+b : n <N a +N b) → (a +N subtractionNResult.result (-N (inl n<b+c))) ≡ n → subtractionNResult.result (-N (inl n<a+b)) +N c <N n → False
|
||||
help12 {a} {b} {c} {n} n<b+c n<a+b pr1 pr2 = TotalOrder.irreflexive ℕTotalOrder lemm4
|
||||
where
|
||||
pr : {a b c : ℕ} → a +N (b +N c) ≡ b +N (a +N c)
|
||||
pr {a} {b} {c} rewrite Semiring.+Associative ℕSemiring a b c | Semiring.commutative ℕSemiring a b | equalityCommutative (Semiring.+Associative ℕSemiring b a c) = refl
|
||||
lemm1 : (n +N subtractionNResult.result (-N (inl n<a+b))) +N c <N n +N n
|
||||
lemm1 = identityOfIndiscernablesLeft _<N_ (additionPreservesInequalityOnLeft n pr2) (Semiring.+Associative ℕSemiring n _ c)
|
||||
lemm2 : (a +N b) +N c <N n +N n
|
||||
lemm2 = identityOfIndiscernablesLeft _<N_ lemm1 (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
lemm1' : a +N (subtractionNResult.result (-N (inl n<b+c)) +N n) ≡ n +N n
|
||||
lemm1' = identityOfIndiscernablesLeft _≡_ (applyEquality (_+N n) pr1) (equalityCommutative (Semiring.+Associative ℕSemiring a _ n))
|
||||
lemm2' : a +N (b +N c) ≡ n +N n
|
||||
lemm2' = identityOfIndiscernablesLeft _≡_ lemm1' (applyEquality (a +N_) (addMinus (inl n<b+c)))
|
||||
lemm3 : (a +N b) +N c ≡ n +N n
|
||||
lemm3 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = lemm2'
|
||||
lemm4 : (a +N b) +N c <N (a +N b) +N c
|
||||
lemm4 = identityOfIndiscernablesRight _<N_ lemm2 (equalityCommutative lemm3)
|
||||
|
||||
help11 : {a b c n : ℕ} → (a <N n) → (b +N c ≡ n) → (n<a+b : n <N a +N b) → (n <N subtractionNResult.result (-N (inl n<a+b)) +N c) → False
|
||||
help11 {a} {b} {c} {n} a<n b+c=n n<a+b pr1 = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive a<n lemm5)
|
||||
where
|
||||
pr : {a b c : ℕ} → a +N (b +N c) ≡ b +N (a +N c)
|
||||
pr {a} {b} {c} rewrite Semiring.+Associative ℕSemiring a b c | Semiring.commutative ℕSemiring a b | equalityCommutative (Semiring.+Associative ℕSemiring b a c) = refl
|
||||
lemm : n +N n <N (n +N subtractionNResult.result (-N (inl n<a+b))) +N c
|
||||
lemm = identityOfIndiscernablesRight _<N_ (additionPreservesInequalityOnLeft n pr1) (Semiring.+Associative ℕSemiring n _ c)
|
||||
lemm2 : n +N n <N (a +N b) +N c
|
||||
lemm2 = identityOfIndiscernablesRight _<N_ lemm (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
lemm3 : n +N n <N a +N (b +N c)
|
||||
lemm3 = identityOfIndiscernablesRight _<N_ lemm2 (equalityCommutative (Semiring.+Associative ℕSemiring a b c))
|
||||
lemm4 : n +N n <N a +N n
|
||||
lemm4 = identityOfIndiscernablesRight _<N_ lemm3 (applyEquality (a +N_) b+c=n)
|
||||
lemm5 : n <N a
|
||||
lemm5 = subtractionPreservesInequality n lemm4
|
||||
|
||||
help10 : {a b c n : ℕ} → (n<b+c : n <N b +N c) → (n<a+b : n <N a +N b) → (a +N subtractionNResult.result (-N (inl n<b+c)) ≡ n) → (n <N subtractionNResult.result (-N (inl n<a+b)) +N c) → False
|
||||
help10 {a} {b} {c} {n} n<b+c n<a+b pr1 pr2 = TotalOrder.irreflexive ℕTotalOrder lemm6
|
||||
where
|
||||
pr : {a b c : ℕ} → a +N (b +N c) ≡ b +N (a +N c)
|
||||
pr {a} {b} {c} rewrite Semiring.+Associative ℕSemiring a b c | Semiring.commutative ℕSemiring a b | equalityCommutative (Semiring.+Associative ℕSemiring b a c) = refl
|
||||
lemm : a +N (n +N subtractionNResult.result (-N (inl n<b+c))) ≡ n +N n
|
||||
lemm = identityOfIndiscernablesLeft _≡_ (applyEquality (n +N_) pr1) (pr {n} {a})
|
||||
lemm2 : a +N (b +N c) ≡ n +N n
|
||||
lemm2 = identityOfIndiscernablesLeft _≡_ lemm (applyEquality (a +N_) (addMinus' (inl n<b+c)))
|
||||
lemm3 : n +N n <N (n +N subtractionNResult.result (-N (inl n<a+b))) +N c
|
||||
lemm3 = identityOfIndiscernablesRight _<N_ (additionPreservesInequalityOnLeft n pr2) (Semiring.+Associative ℕSemiring n _ c)
|
||||
lemm4 : n +N n <N (a +N b) +N c
|
||||
lemm4 = identityOfIndiscernablesRight _<N_ lemm3 (applyEquality (_+N c) (addMinus' (inl n<a+b)))
|
||||
lemm5 : n +N n <N a +N (b +N c)
|
||||
lemm5 = identityOfIndiscernablesRight _<N_ lemm4 (equalityCommutative (Semiring.+Associative ℕSemiring a b c))
|
||||
lemm6 : a +N (b +N c) <N a +N (b +N c)
|
||||
lemm6 = identityOfIndiscernablesLeft _<N_ lemm5 (equalityCommutative lemm2)
|
||||
|
||||
help9 : {a n : ℕ} → (a +N 0 ≡ n) → (a <N n) → False
|
||||
help9 {a} {n} n=a+0 a<n rewrite Semiring.commutative ℕSemiring a 0 | n=a+0 = TotalOrder.irreflexive ℕTotalOrder a<n
|
||||
|
||||
help8 : {a n : ℕ} → (n <N a +N 0) → (a <N n) → False
|
||||
help8 {a} {n} n<a+0 a<n rewrite Semiring.commutative ℕSemiring a 0 = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive a<n n<a+0)
|
||||
|
||||
help6 : {a b c n : ℕ} → (b +N c ≡ n) → (n<a+b : n <N a +N b) → (a +N 0 ≡ subtractionNResult.result (-N (inl n<a+b)) +N c)
|
||||
help6 {a} {b} {c} {n} b+c=n n<a+b rewrite Semiring.commutative ℕSemiring a 0 = canSubtractFromEqualityLeft {n} lem'
|
||||
where
|
||||
lem : n +N a ≡ (n +N subtractionNResult.result (-N (inl n<a+b))) +N c
|
||||
lem rewrite addMinus' (inl n<a+b) | equalityCommutative (Semiring.+Associative ℕSemiring a b c) | b+c=n = Semiring.commutative ℕSemiring n a
|
||||
lem' : n +N a ≡ n +N (subtractionNResult.result (-N (inl n<a+b)) +N c)
|
||||
lem' = identityOfIndiscernablesRight _≡_ lem (equalityCommutative (Semiring.+Associative ℕSemiring n _ c))
|
||||
|
||||
help5 : {a b c n : ℕ} → (n<b+c : n <N b +N c) → (n<a+b : n <N a +N b) → a +N subtractionNResult.result (-N (inl n<b+c)) ≡ subtractionNResult.result (-N (inl n<a+b)) +N c
|
||||
help5 {a} {b} {c} {n} n<b+c n<a+b = canSubtractFromEqualityLeft {n} lemma''
|
||||
where
|
||||
lemma : a +N (n +N subtractionNResult.result (-N (inl n<b+c))) ≡ (n +N subtractionNResult.result (-N (inl n<a+b))) +N c
|
||||
lemma rewrite addMinus' (inl n<b+c) | addMinus' (inl n<a+b) = Semiring.+Associative ℕSemiring a b c
|
||||
lemma' : a +N (n +N subtractionNResult.result (-N (inl n<b+c))) ≡ n +N (subtractionNResult.result (-N (inl n<a+b)) +N c)
|
||||
lemma' = identityOfIndiscernablesRight _≡_ lemma (equalityCommutative (Semiring.+Associative ℕSemiring n _ c))
|
||||
lemma'' : n +N (a +N subtractionNResult.result (-N (inl n<b+c))) ≡ n +N (subtractionNResult.result (-N (inl n<a+b)) +N c)
|
||||
lemma'' = identityOfIndiscernablesLeft _≡_ lemma' (pr {a} {n} {subtractionNResult.result (-N (inl n<b+c))})
|
||||
where
|
||||
pr : {a b c : ℕ} → a +N (b +N c) ≡ b +N (a +N c)
|
||||
pr {a} {b} {c} rewrite Semiring.+Associative ℕSemiring a b c | Semiring.commutative ℕSemiring a b | equalityCommutative (Semiring.+Associative ℕSemiring b a c) = refl
|
||||
|
||||
help4 : {a b c n : ℕ} → (n<a+'b+c : n <N a +N (b +N c)) → (n<a+b : n <N a +N b) → (subtractionNResult.result (-N (inl n<a+'b+c)) ≡ subtractionNResult.result (-N (inl n<a+b)) +N c)
|
||||
help4 {a} {b} {c} {n} n<a+'b+c n<a+b = canSubtractFromEqualityLeft lemma'
|
||||
where
|
||||
lemma : (n +N subtractionNResult.result (-N (inl n<a+'b+c))) ≡ (n +N subtractionNResult.result (-N (inl n<a+b))) +N c
|
||||
lemma rewrite addMinus' (inl n<a+'b+c) | addMinus' (inl n<a+b) = Semiring.+Associative ℕSemiring a b c
|
||||
lemma' : n +N subtractionNResult.result (-N (inl n<a+'b+c)) ≡ n +N (subtractionNResult.result (-N (inl n<a+b)) +N c)
|
||||
lemma' = identityOfIndiscernablesRight _≡_ lemma (equalityCommutative (Semiring.+Associative ℕSemiring n (subtractionNResult.result (-N (inl n<a+b))) c))
|
||||
|
||||
help3 : {a b c n : ℕ} → (a <N n) → (b <N n) → (c <N n) → (a +N b <N n) → (pr : n <N b +N c) → a +N subtractionNResult.result (-N (inl pr)) ≡ n → False
|
||||
help3 {a} {b} {c} {n} a<n b<n c<n a+b<n n<b+c pr = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive (inter4 inter3) c<n)
|
||||
where
|
||||
inter : a +N (n +N subtractionNResult.result (-N (inl n<b+c))) ≡ n +N n
|
||||
inter = identityOfIndiscernablesLeft _≡_ (applyEquality (n +N_) pr) (lemma n a (subtractionNResult.result (-N (inl n<b+c))))
|
||||
where
|
||||
lemma : (a b c : ℕ) → a +N (b +N c) ≡ b +N (a +N c)
|
||||
lemma a b c rewrite Semiring.+Associative ℕSemiring a b c | Semiring.+Associative ℕSemiring b a c = applyEquality (_+N c) (Semiring.commutative ℕSemiring a b)
|
||||
inter2 : n +N n ≡ a +N (b +N c)
|
||||
inter2 = equalityCommutative (identityOfIndiscernablesLeft _≡_ inter (applyEquality (a +N_) (addMinus' (inl n<b+c))))
|
||||
inter3 : n +N n <N n +N c
|
||||
inter3 rewrite inter2 | Semiring.+Associative ℕSemiring a b c = additionPreservesInequality c a+b<n
|
||||
inter4 : (n +N n <N n +N c) → n <N c
|
||||
inter4 pr rewrite Semiring.commutative ℕSemiring n c = subtractionPreservesInequality n pr
|
||||
|
||||
help2 : {a b c n : ℕ} → (sn<b+c : succ n <N b +N c) → (sn<a+b+c : succ n <N (a +N b) +N c) → a +N subtractionNResult.result (-N (inl sn<b+c)) ≡ subtractionNResult.result (-N (inl sn<a+b+c))
|
||||
help2 {a} {b} {c} {n} sn<b+c sn<a+b+c = res inter
|
||||
where
|
||||
inter : a +N (subtractionNResult.result (-N (inl sn<b+c)) +N succ n) ≡ subtractionNResult.result (-N (inl sn<a+b+c)) +N succ n
|
||||
inter rewrite addMinus (inl sn<b+c) | addMinus (inl sn<a+b+c) = Semiring.+Associative ℕSemiring a b c
|
||||
res : (a +N (subtractionNResult.result (-N (inl sn<b+c)) +N succ n) ≡ subtractionNResult.result (-N (inl sn<a+b+c)) +N succ n) → a +N subtractionNResult.result (-N (inl sn<b+c)) ≡ subtractionNResult.result (-N (inl sn<a+b+c))
|
||||
res pr = canSubtractFromEqualityRight {_} {succ n} (identityOfIndiscernablesLeft _≡_ pr (Semiring.+Associative ℕSemiring a (subtractionNResult.result (-N (inl sn<b+c))) (succ n)))
|
||||
|
||||
help1 : {a b c n : ℕ} → (sn<b+c : succ n <N b +N c) → (pr1 : succ n <N a +N subtractionNResult.result (-N (inl sn<b+c))) → (a +N b <N succ n) → (a <N succ n) → (b <N succ n) → (c <N succ n) → False
|
||||
help1 {a} {b} {c} {n} sn<b+c pr1 a+b<sn a<sn b<sn c<sn with -N (inl sn<b+c)
|
||||
help1 {a} {b} {c} {n} sn<b+c pr1 a+b<sn a<sn b<sn c<sn | record { result = b+c-sn ; pr = Prb+c-sn } = ans
|
||||
where
|
||||
b+c-nNonzero : b+c-sn ≡ 0 → False
|
||||
b+c-nNonzero pr rewrite (equalityCommutative Prb+c-sn) | pr | Semiring.commutative ℕSemiring n 0 = TotalOrder.irreflexive ℕTotalOrder sn<b+c
|
||||
2sn<a+b+c' : succ n +N succ n <N succ n +N (a +N b+c-sn)
|
||||
2sn<a+b+c' = additionPreservesInequalityOnLeft (succ n) pr1
|
||||
2sn<a+b+c'' : succ n +N succ n <N a +N (succ n +N b+c-sn)
|
||||
2sn<a+b+c'' rewrite Semiring.+Associative ℕSemiring a (succ n) b+c-sn | Semiring.commutative ℕSemiring a (succ n) | equalityCommutative (Semiring.+Associative ℕSemiring (succ n) a b+c-sn) = 2sn<a+b+c'
|
||||
eep : succ n +N succ n <N a +N (b +N c)
|
||||
eep rewrite equalityCommutative Prb+c-sn = 2sn<a+b+c''
|
||||
eep2 : a +N (b +N c) <N succ n +N c
|
||||
eep2 rewrite Semiring.+Associative ℕSemiring a b c = additionPreservesInequality c a+b<sn
|
||||
eep2' : a +N (b +N c) <N succ n +N succ n
|
||||
eep2' = orderIsTransitive eep2 (additionPreservesInequalityOnLeft (succ n) c<sn)
|
||||
ans : False
|
||||
ans = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive eep eep2')
|
||||
|
||||
plusZnAssociative : {n : ℕ} → {pr : 0 <N n} → (a b c : ℤn n pr) → a +n (b +n c) ≡ ((a +n b) +n c)
|
||||
plusZnAssociative {zero} {()}
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess} record { x = c ; xLess = cLess } with orderIsTotal (a +N b) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) with orderIsTotal ((a +N b) +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) with orderIsTotal (b +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inl b+c<sn) with orderIsTotal (a +N (b +N c)) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inl b+c<sn) | inl (inl a+'b+c<sn) = equalityZn _ _ (Semiring.+Associative ℕSemiring a b c)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inl b+c<sn) | inl (inr sn<a+'b+c) = exFalso (false {succ n} a+b+c<sn sn<a+'b+c)
|
||||
where
|
||||
false : {x : ℕ} → (a +N b) +N c <N succ n → succ n <N a +N (b +N c) → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr1 pr2)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inl b+c<sn) | inr sn=a+b+c = exFalso (false a+b+c<sn sn=a+b+c)
|
||||
where
|
||||
false : {x : ℕ} → (a +N b) +N c <N x → (a +N (b +N c)) ≡ x → False
|
||||
false p1 p2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | p2 = TotalOrder.irreflexive ℕTotalOrder p1
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) with orderIsTotal (a +N (b +N c)) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) | inl (inl _) with orderIsTotal (a +N subtractionNResult.result (-N (inl sn<b+c))) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) | inl (inl _) | inl (inl x) with -N (inl sn<b+c)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) | inl (inl a+'b+c<sn) | inl (inl x) | record { result = result ; pr = pr } = exFalso (false a+'b+c<sn pr)
|
||||
where
|
||||
false : a +N (b +N c) <N succ n → succ n +N result ≡ b +N c → False
|
||||
false pr1 pr2 rewrite equalityCommutative pr2 | Semiring.+Associative ℕSemiring a (succ n) result | Semiring.commutative ℕSemiring a (succ n) | equalityCommutative (Semiring.+Associative ℕSemiring (succ n) a result) = cannotAddAndEnlarge' pr1
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) | inl (inl _) | inl (inr x) with -N (inl sn<b+c)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) | inl (inl a+'b+c<sn) | inl (inr x) | record { result = result ; pr = pr } = exFalso false
|
||||
where
|
||||
lemma : a +N (succ n +N result) ≡ a +N (b +N c)
|
||||
lemma' : a +N (succ n +N result) <N succ n
|
||||
lemma'' : succ n +N (a +N result) <N succ n
|
||||
lemma'' = identityOfIndiscernablesLeft _<N_ lemma' (transitivity (Semiring.+Associative ℕSemiring a (succ n) result) (transitivity (applyEquality (λ t → t +N result) (Semiring.commutative ℕSemiring a (succ n))) (equalityCommutative (Semiring.+Associative ℕSemiring (succ n) a result))))
|
||||
lemma = applyEquality (λ i → a +N i) pr
|
||||
lemma' rewrite lemma = a+'b+c<sn
|
||||
false : False
|
||||
false = cannotAddAndEnlarge' lemma''
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) | inl (inl _) | inr x with -N (inl sn<b+c)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) | inl (inl a+'b+c<sn) | inr x | record { result = result ; pr = pr } = exFalso false
|
||||
where
|
||||
lemma : a +N (succ n +N result) <N succ n
|
||||
lemma rewrite pr = a+'b+c<sn
|
||||
lemma' : succ n +N (a +N result) <N succ n
|
||||
lemma' = identityOfIndiscernablesLeft _<N_ lemma (transitivity (Semiring.+Associative ℕSemiring a (succ n) result) (transitivity (applyEquality (λ t → t +N result) (Semiring.commutative ℕSemiring a (succ n))) (equalityCommutative (Semiring.+Associative ℕSemiring (succ n) a result))))
|
||||
false : False
|
||||
false = cannotAddAndEnlarge' lemma'
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) | inl (inr x) = equalityZn _ _ (exFalso (false {succ n} a+b+c<sn x))
|
||||
where
|
||||
false : {x : ℕ} → (a +N b) +N c <N succ n → succ n <N a +N (b +N c) → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr1 pr2)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inl (inr sn<b+c) | inr x = exFalso (false a+b+c<sn x)
|
||||
where
|
||||
false : (a +N b) +N c <N succ n → a +N (b +N c) ≡ succ n → False
|
||||
false pr1 pr2 rewrite equalityCommutative pr2 | equalityCommutative (Semiring.+Associative ℕSemiring a b c) = TotalOrder.irreflexive ℕTotalOrder pr1
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inl a+b+c<sn) | inr sn=b+c = exFalso (false a+b+c<sn sn=b+c)
|
||||
where
|
||||
false : (a +N b) +N c <N succ n → b +N c ≡ succ n → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr2 | Semiring.commutative ℕSemiring a (succ n) = cannotAddAndEnlarge' a+b+c<sn
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) with orderIsTotal (b +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inl b+c<sn) with orderIsTotal (a +N (b +N c)) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inl b+c<sn) | inl (inl a+'b+c<sn) = exFalso (false sn<a+b+c a+'b+c<sn)
|
||||
where
|
||||
false : (succ n <N (a +N b) +N c) → a +N (b +N c) <N succ n → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr1 pr2)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inl b+c<sn) | inl (inr sn<a+'b+c) = equalityZn _ _ ans
|
||||
where
|
||||
lemma : succ n +N ((a +N b) +N c) ≡ (a +N (b +N c)) +N succ n
|
||||
lemma rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = Semiring.commutative ℕSemiring (succ n) _
|
||||
ans : subtractionNResult.result (-N (inl sn<a+'b+c)) ≡ subtractionNResult.result (-N (inl sn<a+b+c))
|
||||
ans = equivalentSubtraction _ _ _ _ sn<a+'b+c sn<a+b+c lemma
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inl b+c<sn) | inr x = exFalso (false sn<a+b+c x)
|
||||
where
|
||||
false : succ n <N (a +N b) +N c → (a +N (b +N c)) ≡ succ n → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr2 = TotalOrder.irreflexive ℕTotalOrder sn<a+b+c
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inr sn<b+c) with orderIsTotal (a +N (b +N c)) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inr sn<b+c) | inl (inl a+b+c<sn) = exFalso (false sn<a+b+c a+b+c<sn)
|
||||
where
|
||||
false : succ n <N (a +N b) +N c → (a +N (b +N c)) <N succ n → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr1 pr2)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inr sn<b+c) | inl (inr _) with orderIsTotal (a +N subtractionNResult.result (-N (inl sn<b+c))) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inr sn<b+c) | inl (inr _) | inl (inl x) = equalityZn _ _ (help2 {a} {b} {c} {n} sn<b+c sn<a+b+c)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inr sn<b+c) | inl (inr _) | inl (inr x) = equalityZn _ _ (exFalso (help1 sn<b+c x a+b<sn aLess bLess cLess))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inr sn<b+c) | inl (inr _) | inr x = exFalso (help3 aLess bLess cLess a+b<sn sn<b+c x)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inl (inr sn<b+c) | inr a+b+c=sn = exFalso (false sn<a+b+c a+b+c=sn)
|
||||
where
|
||||
false : (succ n <N (a +N b) +N c) → (a +N (b +N c) ≡ succ n) → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr2 = TotalOrder.irreflexive ℕTotalOrder pr1
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inr sn=b+c with orderIsTotal (a +N (b +N c)) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inr sn=b+c | inl (inl x) = exFalso (false sn<a+b+c x)
|
||||
where
|
||||
false : succ n <N (a +N b) +N c → a +N (b +N c) <N succ n → False
|
||||
false p1 p2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive p1 p2)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inr sn=b+c | inl (inr _) with orderIsTotal (a +N 0) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inr sn=b+c | inl (inr _) | inl (inl x) = equalityZn _ _ (ans sn=b+c)
|
||||
where
|
||||
ans : b +N c ≡ succ n → a +N 0 ≡ subtractionNResult.result (-N (inl sn<a+b+c))
|
||||
ans pr with -N (inl sn<a+b+c)
|
||||
ans b+c=sn | record { result = result ; pr = pr1 } rewrite Semiring.commutative ℕSemiring a 0 | equalityCommutative (Semiring.+Associative ℕSemiring a b c) | b+c=sn | Semiring.commutative ℕSemiring (succ n) result = equalityCommutative (canSubtractFromEqualityRight pr1)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inr sn=b+c | inl (inr _) | inl (inr x) = exFalso (false b a+b<sn x)
|
||||
where
|
||||
false : (b : ℕ) → a +N b <N succ n → succ n <N a +N 0 → False
|
||||
false zero pr1 pr2 rewrite Semiring.commutative ℕSemiring a 0 = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr1 pr2)
|
||||
false (succ b) pr1 pr2 rewrite Semiring.commutative ℕSemiring a 0 = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr2 (orderIsTransitive (le b (Semiring.commutative ℕSemiring (succ b) a)) pr1))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inr sn=b+c | inl (inr _) | inr x = exFalso (false b a+b<sn x)
|
||||
where
|
||||
false : (b : ℕ) → a +N b <N succ n → a +N 0 ≡ succ n → False
|
||||
false zero pr1 pr2 rewrite pr2 = TotalOrder.irreflexive ℕTotalOrder pr1
|
||||
false (succ b) pr1 pr2 rewrite Semiring.commutative ℕSemiring a 0 | pr2 = cannotAddAndEnlarge' pr1
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inl (inr sn<a+b+c) | inr sn=b+c | inr x = exFalso (false sn<a+b+c x)
|
||||
where
|
||||
false : succ n <N (a +N b) +N c → a +N (b +N c) ≡ succ n → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr2 = TotalOrder.irreflexive ℕTotalOrder pr1
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c with orderIsTotal (b +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c | inl (inl b+c<sn) with orderIsTotal (a +N (b +N c)) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c | inl (inl b+c<sn) | inl (inl a+'b+c<sn) = exFalso (false sn=a+b+c a+'b+c<sn)
|
||||
where
|
||||
false : (a +N b) +N c ≡ succ n → a +N (b +N c) <N succ n → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr1 = TotalOrder.irreflexive ℕTotalOrder pr2
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c | inl (inl b+c<sn) | inl (inr sn<a+'b+c) = exFalso (false sn=a+b+c sn<a+'b+c)
|
||||
where
|
||||
false : (a +N b) +N c ≡ succ n → succ n <N a +N (b +N c) → False
|
||||
false pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr1 = TotalOrder.irreflexive ℕTotalOrder pr2
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c | inl (inl b+c<sn) | inr _ = equalityZn _ _ refl
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c | inl (inr sn<b+c) = exFalso (false a sn=a+b+c sn<b+c)
|
||||
where
|
||||
false : (a : ℕ) → (a +N b) +N c ≡ succ n → succ n <N b +N c → False
|
||||
false zero pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | equalityCommutative pr1 = TotalOrder.irreflexive ℕTotalOrder pr2
|
||||
false (succ a) pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | equalityCommutative pr1 = TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive pr2 (le a refl))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c | inr b+c=sn with orderIsTotal (a +N 0) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c | inr b+c=sn | inl (inl a+0<sn) = equalityZn _ _ ans
|
||||
where
|
||||
a=0 : (a : ℕ) → (a +N b) +N c ≡ succ n → b +N c ≡ succ n → a ≡ 0
|
||||
a=0 zero pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr2 = refl
|
||||
a=0 (succ a) pr1 pr2 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr2 = canSubtractFromEqualityRight pr1
|
||||
ans : a +N 0 ≡ 0
|
||||
ans rewrite Semiring.commutative ℕSemiring a 0 = a=0 a sn=a+b+c b+c=sn
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c | inr b+c=sn | inl (inr sn<a+0) = exFalso (false sn<a+0 sn=a+b+c b+c=sn)
|
||||
where
|
||||
false : succ n <N a +N 0 → (a +N b) +N c ≡ succ n → b +N c ≡ succ n → False
|
||||
false pr1 pr2 pr3 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr3 | Semiring.commutative ℕSemiring a 0 = zeroNeverGreater {succ n} (identityOfIndiscernablesRight _<N_ pr1 (a=0 a pr2))
|
||||
where
|
||||
a=0 : (a : ℕ) → (a +N succ n ≡ succ n) → a ≡ 0
|
||||
a=0 zero pr = refl
|
||||
a=0 (succ a) pr = canSubtractFromEqualityRight pr
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inl a+b<sn) | inr sn=a+b+c | inr b+c=sn | inr sn=a+0 = exFalso (false sn=a+b+c b+c=sn sn=a+0)
|
||||
where
|
||||
false : (a +N b) +N c ≡ succ n → b +N c ≡ succ n → a +N 0 ≡ succ n → False
|
||||
false pr1 pr2 pr3 rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | pr2 | equalityCommutative pr3 | Semiring.commutative ℕSemiring a 0 = naughtE (identityOfIndiscernablesLeft _≡_ pr3 (a=0 a pr1))
|
||||
where
|
||||
a=0 : (a : ℕ) → (a +N a ≡ a) → a ≡ 0
|
||||
a=0 zero pr = refl
|
||||
a=0 (succ a) pr = exFalso (naughtE {a} (equalityCommutative (canSubtractFromEqualityRight pr)))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) with orderIsTotal (b +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inl b+c<sn) with orderIsTotal (a +N (b +N c)) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inl b+c<sn) | inl (inl a+'b+c<sn) = exFalso (false sn<a+b a+'b+c<sn)
|
||||
where
|
||||
false : succ n <N a +N b → a +N (b +N c) <N succ n → False
|
||||
false pr1 pr2 rewrite Semiring.+Associative ℕSemiring a b c = cannotAddAndEnlarge' (orderIsTransitive pr2 pr1)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inl b+c<sn) | inl (inr sn<a+'b+c) with orderIsTotal (subtractionNResult.result (-N (inl sn<a+b)) +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inl b+c<sn) | inl (inr sn<a+'b+c) | inl (inl x) = equalityZn _ _ (help4 {a} {b} {c} {succ n} sn<a+'b+c sn<a+b)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inl b+c<sn) | inl (inr sn<a+'b+c) | inl (inr x) = exFalso (help18 {a} {b} {c} {succ n} b+c<sn sn<a+b aLess x)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inl b+c<sn) | inl (inr sn<a+'b+c) | inr x = exFalso (help19 {a} {b} {c} {succ n} b+c<sn sn<a+b aLess x)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inl b+c<sn) | inr a+'b+c=sn = exFalso (false sn<a+b a+'b+c=sn)
|
||||
where
|
||||
false : (succ n <N a +N b) → a +N (b +N c) ≡ succ n → False
|
||||
false pr1 pr2 rewrite Semiring.+Associative ℕSemiring a b c | equalityCommutative pr2 = cannotAddAndEnlarge' pr1
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) with orderIsTotal (a +N subtractionNResult.result (-N (inl sn<b+c))) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inl (inl x) with orderIsTotal (subtractionNResult.result (-N (inl sn<a+b)) +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inl (inl x) | inl (inl x₁) = equalityZn _ _ (help5 {a} {b} {c} {succ n} sn<b+c sn<a+b)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inl (inl x) | inl (inr x1) = equalityZn _ _ (help16 {a} {b} {c} {succ n} sn<b+c sn<a+b x x1)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inl (inl x) | inr x1 = exFalso (help17 {a} {b} {c} {succ n} sn<b+c sn<a+b x x1)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inl (inr x) with orderIsTotal (subtractionNResult.result (-N (inl sn<a+b)) +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inl (inr x) | inl (inl x1) = equalityZn _ _ (exFalso (help15 {a} {b} {c} {succ n} sn<b+c sn<a+b x x1))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inl (inr x) | inl (inr x1) = equalityZn _ _ (help14 {a} {b} {c} {succ n} sn<b+c sn<a+b x x1)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inl (inr x) | inr x1 = equalityZn _ _ (exFalso (help13 {a} {b} {c} {succ n} sn<b+c sn<a+b x x1))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inr x with orderIsTotal (subtractionNResult.result (-N (inl sn<a+b)) +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inr x | inl (inl x1) = equalityZn _ _ (exFalso (help12 {a} {b} {c} {succ n} sn<b+c sn<a+b x x1))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inr x | inl (inr x1) = equalityZn _ _ (exFalso (help10 {a} {b} {c} {succ n} sn<b+c sn<a+b x x1))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inl (inr sn<b+c) | inr x | inr x₁ = equalityZn _ _ refl
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inr b+c=sn with orderIsTotal (a +N 0) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inr b+c=sn | inl (inl a+0<sn) with orderIsTotal (subtractionNResult.result (-N (inl sn<a+b)) +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inr b+c=sn | inl (inl a+0<sn) | inl (inl x) = equalityZn _ _ (help6 {a} {b} {c} {succ n} b+c=sn sn<a+b)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inr b+c=sn | inl (inl _) | inl (inr x) = equalityZn _ _ (exFalso (help11 {a} {b} {c} {succ n} aLess b+c=sn sn<a+b x))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inr b+c=sn | inl (inl a+0<sn) | inr x = equalityZn _ _ (exFalso (help7 {a} {b} {c} {succ n} b+c=sn aLess sn<a+b x))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inr b+c=sn | inl (inr sn<a+0) = exFalso (help8 {a} {succ n} sn<a+0 aLess)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inl (inr sn<a+b) | inr b+c=sn | inr a+0=sn = exFalso (help9 {a} {succ n} a+0=sn aLess)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b with orderIsTotal c (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) with orderIsTotal (b +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inl (inl b+c<sn) with orderIsTotal (a +N (b +N c)) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inl (inl b+c<sn) | inl (inl a+'b+c<sn) = equalityZn _ _ (help27 {a} {b} {c} {n} sn=a+b a+'b+c<sn)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inl (inl b+c<sn) | inl (inr sn<a+'b+c) = equalityZn _ _ (help28 {a} {b} {c} {succ n} sn<a+'b+c sn=a+b)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inl (inl b+c<sn) | inr a+'b+c=sn = equalityZn _ _ (help26 {a} {b} {c} {succ n} sn=a+b a+'b+c=sn)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inl (inr sn<b+c) with orderIsTotal (a +N subtractionNResult.result (-N (inl sn<b+c))) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inl (inr sn<b+c) | inl (inl x) = equalityZn _ _ (help31 {a} {b} {c} {succ n} sn=a+b sn<b+c)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inl (inr sn<b+c) | inl (inr x) = exFalso (help30 {a} {b} {c} {succ n} cLess sn=a+b sn<b+c x)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inl (inr sn<b+c) | inr x = exFalso (help29 {a} {b} {c} {succ n} cLess sn<b+c x sn=a+b)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inr b+c=sn with orderIsTotal (a +N 0) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inr b+c=sn | inl (inl a+0<sn) = equalityZn _ _ (help25 {a} {b} {c} {succ n} sn=a+b b+c=sn)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inr b+c=sn | inl (inr sn<a+0) = exFalso (help24 {a} {succ n} aLess sn<a+0)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inl c<sn) | inr b+c=sn | inr a+0=sn = exFalso (help23 {a} {succ n} aLess a+0=sn)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inl (inr sn<c) = exFalso (TotalOrder.irreflexive ℕTotalOrder (orderIsTransitive sn<c cLess))
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inr c=sn with orderIsTotal (b +N c) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inr c=sn | inl (inl b+c<sn) = exFalso (help b+c<sn)
|
||||
where
|
||||
help : (b +N c <N succ n) → False
|
||||
help b+c<sn rewrite equalityCommutative c=sn | Semiring.commutative ℕSemiring b c = cannotAddAndEnlarge' b+c<sn
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inr c=sn | inl (inr sn<b+c) with orderIsTotal (a +N subtractionNResult.result (-N (inl sn<b+c))) (succ n)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inr c=sn | inl (inr sn<b+c) | inl (inl x) = exFalso (help20 {a} {b} {c} {succ n} c=sn sn=a+b sn<b+c x)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inr c=sn | inl (inr sn<b+c) | inl (inr x) = exFalso (help22 {a} {b} {c} {succ n} sn=a+b c=sn sn<b+c x)
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inr c=sn | inl (inr sn<b+c) | inr x = equalityZn _ _ refl
|
||||
plusZnAssociative {succ n} {_} record { x = a ; xLess = aLess } record { x = b ; xLess = bLess } record { x = c ; xLess = cLess } | inr sn=a+b | inr c=sn | inr b+c=sn = exFalso (help21 {a} {b} {c} {succ n} sn=a+b b+c=sn c=sn aLess)
|
||||
|
||||
subLess : {a b : ℕ} → (0<a : 0 <N a) → (a<b : a <N b) → subtractionNResult.result (-N (inl a<b)) <N b
|
||||
subLess {zero} {b} 0<a a<b = exFalso (TotalOrder.irreflexive ℕTotalOrder 0<a)
|
||||
subLess {succ a} {b} _ a<b with -N (inl a<b)
|
||||
... | record { result = b-a ; pr = pr } = record { x = a ; proof = pr }
|
||||
|
||||
inverseZn : {n : ℕ} → {pr : 0 <N n} → (a : ℤn n pr) → Sg (ℤn n pr) (λ i → i +n a ≡ record { x = 0 ; xLess = pr } )
|
||||
inverseZn {zero} {()}
|
||||
inverseZn {succ n} {0<n} record { x = zero ; xLess = zeroLess } = record { x = zero ; xLess = zeroLess } , plusZnIdentityLeft _
|
||||
inverseZn {succ n} {0<n} record { x = (succ a) ; xLess = aLess } = ans , pr
|
||||
where
|
||||
ans = record { x = subtractionNResult.result (-N (inl (canRemoveSuccFrom<N aLess))) ; xLess = subLess (succIsPositive a) aLess }
|
||||
pr : ans +n record { x = (succ a) ; xLess = aLess } ≡ record { x = 0 ; xLess = 0<n }
|
||||
pr with orderIsTotal (subtractionNResult.result (-N (inl (canRemoveSuccFrom<N aLess))) +N (succ a)) (succ n)
|
||||
... | inl (inl x) = exFalso f
|
||||
where
|
||||
h : subtractionNResult.result (-N (inl (canRemoveSuccFrom<N aLess))) +N succ a ≡ succ n
|
||||
h with -N (inl (canRemoveSuccFrom<N aLess))
|
||||
h | record { result = result ; pr = pr } rewrite equalityCommutative pr = Semiring.commutative ℕSemiring result (succ a)
|
||||
f : False
|
||||
f = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesLeft _<N_ x h)
|
||||
... | inl (inr x) = exFalso f
|
||||
where
|
||||
h : subtractionNResult.result (-N (inl (canRemoveSuccFrom<N aLess))) +N succ a ≡ succ n
|
||||
h = addMinus {succ a} {succ n} (inl aLess)
|
||||
f : False
|
||||
f rewrite h = TotalOrder.irreflexive ℕTotalOrder x
|
||||
... | (inr n-a+sa=sn) = equalityZn _ _ refl
|
||||
|
||||
ℤnGroup : (n : ℕ) → (pr : 0 <N n) → Group (reflSetoid (ℤn n pr)) _+n_
|
||||
ℤnGroup n pr = record { 0G = record { x = 0 ; xLess = pr } ; inverse = λ a → underlying (inverseZn a) ; +Associative = λ {a} {b} {c} → plusZnAssociative a b c ; identRight = λ {a} → plusZnIdentityRight a ; identLeft = λ {a} → plusZnIdentityLeft a ; invLeft = λ {a} → helpInvLeft a ; invRight = λ {a} → helpInvRight a ; +WellDefined = reflGroupWellDefined }
|
||||
where
|
||||
helpInvLeft : (a : ℤn n pr) → underlying (inverseZn a) +n a ≡ record { x = 0 ; xLess = pr }
|
||||
helpInvLeft a with inverseZn a
|
||||
... | -a , pr-a = pr-a
|
||||
helpInvRight : (a : ℤn n pr) → a +n underlying (inverseZn a) ≡ record { x = 0 ; xLess = pr }
|
||||
helpInvRight a rewrite plusZnCommutative a (underlying (inverseZn a)) = helpInvLeft a
|
||||
|
||||
ℤnAbGroup : (n : ℕ) → (pr : 0 <N n) → AbelianGroup (ℤnGroup n pr)
|
||||
AbelianGroup.commutative (ℤnAbGroup n pr) {a} {b} = plusZnCommutative a b
|
||||
|
||||
ℤnFinite : (n : ℕ) → (pr : 0 <N n) → FiniteGroup (ℤnGroup n pr) (FinSet n)
|
||||
SetoidToSet.project (FiniteGroup.toSet (ℤnFinite n pr)) record { x = x ; xLess = xLess } = ofNat x xLess
|
||||
SetoidToSet.wellDefined (FiniteGroup.toSet (ℤnFinite n pr)) x y x=y rewrite x=y = refl
|
||||
Surjection.property (SetoidToSet.surj (FiniteGroup.toSet (ℤnFinite n pr))) b = record { x = toNat b ; xLess = toNatSmaller b } , ofNatToNat b
|
||||
SetoidToSet.inj (FiniteGroup.toSet (ℤnFinite zero ())) x y x=y
|
||||
SetoidToSet.inj (FiniteGroup.toSet (ℤnFinite (succ n) pr)) record { x = x ; xLess = xLess } record { x = y ; xLess = yLess } x=y = equalityZn _ _ b
|
||||
where
|
||||
b : x ≡ y
|
||||
b = ofNatInjective x y xLess yLess x=y
|
||||
FiniteSet.size (FiniteGroup.finite (ℤnFinite n pr)) = n
|
||||
FiniteSet.mapping (FiniteGroup.finite (ℤnFinite n pr)) = id
|
||||
FiniteSet.bij (FiniteGroup.finite (ℤnFinite n pr)) = idIsBijective
|
||||
@@ -5,6 +5,7 @@ open import Numbers.Naturals.Naturals
|
||||
open import Numbers.Integers.Integers
|
||||
open import Groups.Groups
|
||||
open import Groups.Definition
|
||||
open import Groups.Lemmas
|
||||
open import Rings.Definition
|
||||
open import Rings.Orders.Total.Definition
|
||||
open import Rings.Orders.Partial.Definition
|
||||
@@ -15,7 +16,7 @@ open import Setoids.Orders
|
||||
open import Functions
|
||||
open import Sets.EquivalenceRelations
|
||||
|
||||
module Numbers.Rationals where
|
||||
module Numbers.Rationals.Definition where
|
||||
|
||||
open import Fields.FieldOfFractions.Setoid ℤIntDom
|
||||
open import Fields.FieldOfFractions.Addition ℤIntDom
|
||||
@@ -16,11 +16,11 @@ open import Setoids.Setoids
|
||||
open import Setoids.Orders
|
||||
open import Functions
|
||||
open import Sets.EquivalenceRelations
|
||||
open import Numbers.Rationals
|
||||
open import Numbers.Rationals.Definition
|
||||
open import Semirings.Definition
|
||||
open import Orders
|
||||
|
||||
module Numbers.RationalsLemmas where
|
||||
module Numbers.Rationals.Lemmas where
|
||||
|
||||
open PartiallyOrderedRing ℤPOrderedRing
|
||||
open import Rings.Orders.Total.Lemmas ℤOrderedRing
|
||||
@@ -2,6 +2,7 @@
|
||||
|
||||
open import LogicalFormulae
|
||||
open import Groups.Groups
|
||||
open import Groups.Homomorphisms.Definition
|
||||
open import Groups.Definition
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Setoids.Orders
|
||||
|
||||
@@ -5,7 +5,8 @@ open import Groups.Groups
|
||||
open import Functions
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Numbers.Integers.Integers
|
||||
open import Numbers.Modulo.IntegersModN
|
||||
open import Numbers.Modulo.Group
|
||||
open import Numbers.Modulo.Definition
|
||||
open import Rings.Examples.Proofs
|
||||
open import Numbers.Primes.PrimeNumbers
|
||||
|
||||
|
||||
@@ -9,7 +9,8 @@ open import Rings.Definition
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Numbers.Integers.Integers
|
||||
open import Numbers.Primes.PrimeNumbers
|
||||
open import Numbers.Modulo.IntegersModN
|
||||
open import Numbers.Modulo.Definition
|
||||
open import Numbers.Modulo.Group
|
||||
|
||||
module Rings.Examples.Proofs where
|
||||
nToZn' : (n : ℕ) (pr : 0 <N n) (x : ℕ) → ℤn n pr
|
||||
|
||||
@@ -2,6 +2,7 @@
|
||||
|
||||
open import LogicalFormulae
|
||||
open import Groups.Groups
|
||||
open import Groups.Lemmas
|
||||
open import Groups.Definition
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Orders
|
||||
|
||||
@@ -50,7 +50,7 @@ abstract
|
||||
lemm3 a b pr1 pr2 with transferToRight' additiveGroup (Equivalence.symmetric eq pr1)
|
||||
... | a=-b with Equivalence.transitive eq pr2 a=-b
|
||||
... | 0=-b with inverseWellDefined additiveGroup 0=-b
|
||||
... | -0=--b = Equivalence.transitive eq (Equivalence.symmetric eq (invIdentity additiveGroup)) (Equivalence.transitive eq -0=--b (invTwice additiveGroup b))
|
||||
... | -0=--b = Equivalence.transitive eq (Equivalence.symmetric eq (invIdent additiveGroup)) (Equivalence.transitive eq -0=--b (invTwice additiveGroup b))
|
||||
|
||||
charNot2ImpliesNontrivial : ((1R + 1R) ∼ 0R → False) → (0R ∼ 1R) → False
|
||||
charNot2ImpliesNontrivial charNot2 0=1 = charNot2 (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq 0=1) (Equivalence.symmetric eq 0=1)) identRight)
|
||||
|
||||
@@ -90,10 +90,10 @@ abstract
|
||||
triangleInequality a b | inl (inr a+b<0) | inl (inr a<0) with totality 0G b
|
||||
triangleInequality a b | inl (inr a+b<0) | inl (inr a<0) | inl (inl 0<b) = inl (<WellDefined (Equivalence.symmetric eq (invContravariant additiveGroup)) groupIsAbelian (orderRespectsAddition (SetoidPartialOrder.<Transitive pOrder (lemm2' _ 0<b) 0<b) (inverse a)))
|
||||
triangleInequality a b | inl (inr a+b<0) | inl (inr a<0) | inl (inr b<0) = inr (Equivalence.transitive eq (invContravariant additiveGroup) groupIsAbelian)
|
||||
triangleInequality a b | inl (inr a+b<0) | inl (inr a<0) | inr 0=b = inr (Equivalence.transitive eq (invContravariant additiveGroup) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (+WellDefined (Equivalence.transitive eq (inverseWellDefined additiveGroup (Equivalence.symmetric eq 0=b)) (invIdentity additiveGroup)) (Equivalence.reflexive eq)) identLeft) (Equivalence.symmetric eq identRight)) (+WellDefined (Equivalence.reflexive eq) 0=b)))
|
||||
triangleInequality a b | inl (inr a+b<0) | inl (inr a<0) | inr 0=b = inr (Equivalence.transitive eq (invContravariant additiveGroup) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (+WellDefined (Equivalence.transitive eq (inverseWellDefined additiveGroup (Equivalence.symmetric eq 0=b)) (invIdent additiveGroup)) (Equivalence.reflexive eq)) identLeft) (Equivalence.symmetric eq identRight)) (+WellDefined (Equivalence.reflexive eq) 0=b)))
|
||||
triangleInequality a b | inl (inr a+b<0) | inr 0=a with totality 0G b
|
||||
triangleInequality a b | inl (inr a+b<0) | inr 0=a | inl (inl 0<b) = exFalso (irreflexive {0G} (SetoidPartialOrder.<Transitive pOrder 0<b (<WellDefined (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq 0=a) (Equivalence.reflexive eq)) identLeft) (Equivalence.reflexive eq) a+b<0)))
|
||||
triangleInequality a b | inl (inr a+b<0) | inr 0=a | inl (inr b<0) = inr (Equivalence.transitive eq (invContravariant additiveGroup) (Equivalence.transitive eq groupIsAbelian (+WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq (inverseWellDefined additiveGroup 0=a)) (invIdentity additiveGroup)) 0=a) (Equivalence.reflexive eq))))
|
||||
triangleInequality a b | inl (inr a+b<0) | inr 0=a | inl (inr b<0) = inr (Equivalence.transitive eq (invContravariant additiveGroup) (Equivalence.transitive eq groupIsAbelian (+WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq (inverseWellDefined additiveGroup 0=a)) (invIdent additiveGroup)) 0=a) (Equivalence.reflexive eq))))
|
||||
triangleInequality a b | inl (inr a+b<0) | inr 0=a | inr 0=b = exFalso (irreflexive {0G} (<WellDefined (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq 0=a) (Equivalence.symmetric eq 0=b)) identLeft) (Equivalence.reflexive eq) a+b<0))
|
||||
triangleInequality a b | inr 0=a+b with totality 0G a
|
||||
triangleInequality a b | inr 0=a+b | inl (inl 0<a) with totality 0G b
|
||||
@@ -169,7 +169,7 @@ abstract
|
||||
f : inverse 0G ∼ inverse (y + inverse x)
|
||||
f = inverseWellDefined additiveGroup 0=y-x
|
||||
g : 0G ∼ (inverse y) + x
|
||||
g = Equivalence.transitive eq (symmetric (invIdentity additiveGroup)) (Equivalence.transitive eq f (Equivalence.transitive eq (Equivalence.transitive eq (invContravariant additiveGroup) groupIsAbelian) (+WellDefined reflexive (invInv additiveGroup))))
|
||||
g = Equivalence.transitive eq (symmetric (invIdent additiveGroup)) (Equivalence.transitive eq f (Equivalence.transitive eq (Equivalence.transitive eq (invContravariant additiveGroup) groupIsAbelian) (+WellDefined reflexive (invInv additiveGroup))))
|
||||
x=y : x ∼ y
|
||||
x=y = transferToRight additiveGroup (symmetric (Equivalence.transitive eq g groupIsAbelian))
|
||||
q'' : (0R + x) < ((y + Group.inverse additiveGroup x) + x)
|
||||
|
||||
Reference in New Issue
Block a user