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Free-group lemmas (#106)
This commit is contained in:
Patrick Stevens
GitHub
@@ -4,6 +4,5 @@ open import LogicalFormulae
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module Decidable.Sets where
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record DecidableSet {a : _} (A : Set a) : Set a where
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field
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eq : (a b : A) → ((a ≡ b) || ((a ≡ b) → False))
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DecidableSet : {a : _} (A : Set a) → Set a
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DecidableSet A = (a b : A) → ((a ≡ b) || ((a ≡ b) → False))
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@@ -95,6 +95,8 @@ open import Setoids.Functions.Extension
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open import Setoids.Algebra.Lemmas
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open import Setoids.Intersection.Lemmas
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open import Setoids.Union.Lemmas
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open import Setoids.Cardinality.Infinite.Lemmas
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open import Setoids.Cardinality.Finite.Definition
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open import Sets.Cardinality.Infinite.Examples
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open import Sets.Cardinality.Infinite.Lemmas
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@@ -19,6 +19,7 @@ open import Sets.FinSetWithK
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open import Rings.Examples.Examples
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open import Rings.Examples.Proofs
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open import Groups.FreeGroup.Lemmas
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open import Groups.FreeGroup.UniversalProperty
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open import Groups.FreeGroup.Parity
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open import Groups.FreeProduct.UniversalProperty
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@@ -8,7 +8,7 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import Numbers.Naturals.Definition
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open import Numbers.Integers.Integers
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open import Numbers.Integers.Addition
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open import Sets.FinSet
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open import Sets.FinSet.Definition
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open import Groups.Homomorphisms.Definition
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open import Groups.Groups
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open import Groups.Subgroups.Definition
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@@ -17,6 +17,10 @@ data FreeCompletion {a : _} (A : Set a) : Set a where
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ofLetter : A → FreeCompletion A
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ofInv : A → FreeCompletion A
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freeCompletionMap : {a b : _} {A : Set a} {B : Set b} (f : A → B) (w : FreeCompletion A) → FreeCompletion B
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freeCompletionMap f (ofLetter x) = ofLetter (f x)
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freeCompletionMap f (ofInv x) = ofInv (f x)
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freeInverse : {a : _} {A : Set a} (l : FreeCompletion A) → FreeCompletion A
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freeInverse (ofLetter x) = ofInv x
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freeInverse (ofInv x) = ofLetter x
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@@ -27,8 +31,11 @@ ofLetterInjective refl = refl
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ofInvInjective : {a : _} {A : Set a} {x y : A} → (ofInv x ≡ ofInv y) → x ≡ y
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ofInvInjective refl = refl
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ofLetterOfInv : {a : _} {A : Set a} {x y : A} → ofLetter x ≡ ofInv y → False
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ofLetterOfInv ()
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decidableFreeCompletion : {a : _} {A : Set a} → DecidableSet A → DecidableSet (FreeCompletion A)
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decidableFreeCompletion {A = A} record { eq = dec } = record { eq = pr }
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decidableFreeCompletion {A = A} dec = pr
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where
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pr : (a b : FreeCompletion A) → (a ≡ b) || (a ≡ b → False)
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pr (ofLetter x) (ofLetter y) with dec x y
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@@ -41,17 +48,17 @@ decidableFreeCompletion {A = A} record { eq = dec } = record { eq = pr }
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... | inr x!=y = inr λ p → x!=y (ofInvInjective p)
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freeCompletionEqual : {a : _} {A : Set a} (dec : DecidableSet A) (x y : FreeCompletion A) → Bool
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freeCompletionEqual dec x y with DecidableSet.eq (decidableFreeCompletion dec) x y
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freeCompletionEqual dec x y with decidableFreeCompletion dec x y
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freeCompletionEqual dec x y | inl x₁ = BoolTrue
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freeCompletionEqual dec x y | inr x₁ = BoolFalse
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freeCompletionEqualFalse : {a : _} {A : Set a} (dec : DecidableSet A) {x y : FreeCompletion A} → ((x ≡ y) → False) → (freeCompletionEqual dec x y) ≡ BoolFalse
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freeCompletionEqualFalse dec {x = x} {y} x!=y with DecidableSet.eq (decidableFreeCompletion dec) x y
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freeCompletionEqualFalse dec {x = x} {y} x!=y with decidableFreeCompletion dec x y
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freeCompletionEqualFalse dec {x} {y} x!=y | inl x=y = exFalso (x!=y x=y)
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freeCompletionEqualFalse dec {x} {y} x!=y | inr _ = refl
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freeCompletionEqualFalse' : {a : _} {A : Set a} (dec : DecidableSet A) {x y : FreeCompletion A} → .((freeCompletionEqual dec x y) ≡ BoolFalse) → (x ≡ y) → False
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freeCompletionEqualFalse' dec {x} {y} pr with DecidableSet.eq (decidableFreeCompletion dec) x y
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freeCompletionEqualFalse' dec {x} {y} pr with decidableFreeCompletion dec x y
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freeCompletionEqualFalse' dec {x} {y} () | inl x₁
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freeCompletionEqualFalse' dec {x} {y} pr | inr ans = ans
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@@ -15,10 +15,10 @@ open import Groups.FreeGroup.Word decA
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prepend : ReducedWord → FreeCompletion A → ReducedWord
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prepend empty x = prependLetter x empty (wordEmpty refl)
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prepend (prependLetter (ofLetter y) w pr) (ofLetter x) = prependLetter (ofLetter x) (prependLetter (ofLetter y) w pr) (wordEnding (succIsPositive _) (freeCompletionEqualFalse decA {ofLetter x} {ofInv x} λ ()))
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prepend (prependLetter (ofInv y) w pr) (ofLetter x) with DecidableSet.eq decA x y
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prepend (prependLetter (ofInv y) w pr) (ofLetter x) with decA x y
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... | inl x=y = w
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... | inr x!=y = prependLetter (ofLetter x) (prependLetter (ofInv y) w pr) (wordEnding (succIsPositive _) (freeCompletionEqualFalse decA {ofLetter x} {ofLetter y} λ pr → x!=y (ofLetterInjective pr)))
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prepend (prependLetter (ofLetter y) w pr) (ofInv x) with DecidableSet.eq decA x y
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prepend (prependLetter (ofLetter y) w pr) (ofInv x) with decA x y
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... | inl x=y = w
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... | inr x!=y = prependLetter (ofInv x) (prependLetter (ofLetter y) w pr) (wordEnding (succIsPositive _) (freeCompletionEqualFalse decA λ pr → x!=y (ofInvInjective pr)))
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prepend (prependLetter (ofInv y) w pr) (ofInv x) = prependLetter (ofInv x) (prependLetter (ofInv y) w pr) (wordEnding (succIsPositive _) (freeCompletionEqualFalse decA {ofInv x} {ofLetter x} (λ ())))
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@@ -30,50 +30,50 @@ prependLetter letter a x +W b = prepend (a +W b) letter
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prependValid : (w : ReducedWord) (l : A) → (x : PrependIsValid w (ofLetter l)) → prepend w (ofLetter l) ≡ prependLetter (ofLetter l) w x
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prependValid empty l (wordEmpty refl) = refl
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prependValid (prependLetter (ofLetter l2) w x) l pr = prependLetterRefl
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prependValid (prependLetter (ofInv l2) w x) l pr with DecidableSet.eq decA l l2
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prependValid (prependLetter (ofInv l2) w x) l pr with decA l l2
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prependValid (prependLetter (ofInv l2) w x) .l2 (wordEnding _ x1) | inl refl = exFalso (freeCompletionEqualFalse' decA x1 refl)
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... | inr l!=l2 = prependLetterRefl
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prependValid' : (w : ReducedWord) (l : A) → (x : PrependIsValid w (ofInv l)) → prepend w (ofInv l) ≡ prependLetter (ofInv l) w x
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prependValid' empty l (wordEmpty refl) = refl
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prependValid' (prependLetter (ofLetter l2) w x) l pr with DecidableSet.eq decA l l2
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prependValid' (prependLetter (ofLetter l2) w x) l pr with decA l l2
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prependValid' (prependLetter (ofLetter l2) w x) .l2 (wordEnding _ x1) | inl refl = exFalso (freeCompletionEqualFalse' decA x1 refl)
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... | inr l!=l2 = prependLetterRefl
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prependValid' (prependLetter (ofInv l2) w x) l pr = prependLetterRefl
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prependInv : (w : ReducedWord) (l : A) → prepend (prepend w (ofLetter l)) (ofInv l) ≡ w
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prependInv empty l with DecidableSet.eq decA l l
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prependInv empty l with decA l l
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... | inl l=l = refl
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... | inr l!=l = exFalso (l!=l refl)
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prependInv (prependLetter (ofLetter l2) w x) l with DecidableSet.eq decA l l
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prependInv (prependLetter (ofLetter l2) w x) l with decA l l
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... | inl l=l = refl
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... | inr l!=l = exFalso (l!=l refl)
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prependInv (prependLetter (ofInv l2) w x) l with DecidableSet.eq decA l l2
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prependInv (prependLetter (ofInv l2) w x) l with decA l l2
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prependInv (prependLetter (ofInv l2) w x) .l2 | inl refl = prependValid' w l2 x
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... | inr l!=l2 with DecidableSet.eq decA l l
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... | inr l!=l2 with decA l l
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prependInv (prependLetter (ofInv l2) w x) l | inr l!=l2 | inl refl = refl
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prependInv (prependLetter (ofInv l2) w x) l | inr l!=l2 | inr bad = exFalso (bad refl)
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prependInv' : (w : ReducedWord) (l : A) → prepend (prepend w (ofInv l)) (ofLetter l) ≡ w
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prependInv' empty l with DecidableSet.eq decA l l
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prependInv' empty l with decA l l
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... | inl l=l = refl
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... | inr l!=l = exFalso (l!=l refl)
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prependInv' (prependLetter (ofLetter l2) w x) l with DecidableSet.eq decA l l2
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prependInv' (prependLetter (ofLetter l2) w x) l with decA l l2
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prependInv' (prependLetter (ofLetter l2) w x) .l2 | inl refl = prependValid w l2 x
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... | inr l!=l2 with DecidableSet.eq decA l l
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... | inr l!=l2 with decA l l
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... | inl refl = refl
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... | inr l!=l = exFalso (l!=l refl)
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prependInv' (prependLetter (ofInv l2) w x) l with DecidableSet.eq decA l l
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prependInv' (prependLetter (ofInv l2) w x) l with decA l l
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prependInv' (prependLetter (ofInv l2) w x) l | inl refl = refl
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prependInv' (prependLetter (ofInv l2) w x) l | inr l!=l = exFalso (l!=l refl)
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prependAndAdd : (a b : ReducedWord) (l : FreeCompletion A) → prepend (a +W b) l ≡ (prepend a l) +W b
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prependAndAdd empty b l = refl
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prependAndAdd (prependLetter (ofLetter x) w pr) b (ofLetter y) = refl
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prependAndAdd (prependLetter (ofLetter x) w pr) b (ofInv y) with DecidableSet.eq decA y x
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prependAndAdd (prependLetter (ofLetter x) w pr) b (ofInv y) with decA y x
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prependAndAdd (prependLetter (ofLetter x) w pr) b (ofInv .x) | inl refl = prependInv _ _
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... | inr y!=x = refl
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prependAndAdd (prependLetter (ofInv x) w pr) b (ofLetter y) with DecidableSet.eq decA y x
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prependAndAdd (prependLetter (ofInv x) w pr) b (ofLetter y) with decA y x
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prependAndAdd (prependLetter (ofInv x) w pr) b (ofLetter .x) | inl refl = prependInv' _ _
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... | inr y!=x = refl
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prependAndAdd (prependLetter (ofInv x) w pr) b (ofInv y) = refl
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@@ -96,13 +96,13 @@ invLeftW empty = refl
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invLeftW (prependLetter (ofLetter l) a x) rewrite equalityCommutative (+WAssoc (inverseW a) (prependLetter (ofInv l) empty (wordEmpty refl)) (prependLetter (ofLetter l) a x)) = t
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where
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t : (inverseW a +W (prepend (prependLetter (ofLetter l) a x) (ofInv l))) ≡ empty
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t with DecidableSet.eq decA l l
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t with decA l l
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... | inl refl = invLeftW a
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... | inr l!=l = exFalso (l!=l refl)
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invLeftW (prependLetter (ofInv l) a x) rewrite equalityCommutative (+WAssoc (inverseW a) (prependLetter (ofLetter l) empty (wordEmpty refl)) (prependLetter (ofInv l) a x)) = t
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where
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t : (inverseW a +W (prepend (prependLetter (ofInv l) a x) (ofLetter l))) ≡ empty
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t with DecidableSet.eq decA l l
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t with decA l l
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... | inl refl = invLeftW a
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... | inr l!=l = exFalso (l!=l refl)
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@@ -111,13 +111,13 @@ invRightW empty = refl
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invRightW (prependLetter (ofLetter l) a x) rewrite +WAssoc a (inverseW a) (prependLetter (ofInv l) empty (wordEmpty refl)) | invRightW a = t
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where
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t : prepend (prependLetter (ofInv l) empty (wordEmpty refl)) (ofLetter l) ≡ empty
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t with DecidableSet.eq decA l l
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t with decA l l
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... | inl refl = refl
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... | inr l!=l = exFalso (l!=l refl)
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invRightW (prependLetter (ofInv l) a x) rewrite +WAssoc a (inverseW a) (prependLetter (ofLetter l) empty (wordEmpty refl)) | invRightW a = t
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where
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t : prepend (prependLetter (ofLetter l) empty (wordEmpty refl)) (ofInv l) ≡ empty
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t with DecidableSet.eq decA l l
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t with decA l l
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... | inl refl = refl
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... | inr l!=l = exFalso (l!=l refl)
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@@ -1,11 +1,13 @@
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{-# OPTIONS --safe --warning=error #-}
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open import Sets.Cardinality.Infinite.Definition
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open import Sets.EquivalenceRelations
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open import Setoids.Setoids
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open import Groups.FreeGroup.Definition
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open import Groups.Homomorphisms.Definition
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open import Groups.Definition
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open import Decidable.Sets
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open import Numbers.Naturals.Semiring
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open import Numbers.Naturals.Order
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open import LogicalFormulae
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open import Semirings.Definition
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@@ -14,18 +16,116 @@ open import Groups.Isomorphisms.Definition
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open import Groups.FreeGroup.Word
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open import Groups.FreeGroup.Group
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open import Groups.FreeGroup.UniversalProperty
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open import Groups.Abelian.Definition
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open import Groups.QuotientGroup.Definition
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open import Groups.Lemmas
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open import Groups.Homomorphisms.Lemmas
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module Groups.FreeGroup.Lemmas {a : _} {A : Set a} (decA : DecidableSet A) where
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freeGroupFunctorWellDefined : {b : _} {B : Set b} (decB : DecidableSet B) → {f : A → B} → Bijection f → GroupsIsomorphic (freeGroup decA) (freeGroup decB)
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GroupsIsomorphic.isomorphism (freeGroupFunctorWellDefined decB {f} bij) = universalPropertyFunction decA (freeGroup decB) λ a → freeEmbedding decB (f a)
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GroupIso.groupHom (GroupsIsomorphic.proof (freeGroupFunctorWellDefined decB {f} bij)) = universalPropertyHom decA (freeGroup decB) λ a → freeEmbedding decB (f a)
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GroupIso.bij (GroupsIsomorphic.proof (freeGroupFunctorWellDefined decB {f} bij)) = {!!}
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freeGroupNonAbelian : AbelianGroup (freeGroup decA) → (a : A) → Sg (A → True) Bijection
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freeGroupNonAbelian record { commutative = commutative } a = (λ _ → record {}) , b
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where
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b : Bijection (λ _ → record {})
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Bijection.inj b {x} {y} _ with decA x y
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... | inl pr = pr
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... | inr neq = exFalso (neq (ofLetterInjective (prependLetterInjective' decA t)))
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where
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t : prependLetter {decA = decA} (ofLetter x) (prependLetter (ofLetter y) empty (wordEmpty refl)) (wordEnding (succIsPositive 0) refl) ≡ prependLetter (ofLetter y) (prependLetter (ofLetter x) empty (wordEmpty refl)) (wordEnding (succIsPositive 0) refl)
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t = commutative {prependLetter (ofLetter x) empty (wordEmpty refl)} {prependLetter (ofLetter y) empty (wordEmpty refl)}
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Bijection.surj b record {} = a , refl
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private
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subgroupGeneratedBySquares : {c d : _} {C : Set c} {S : Setoid {c} {d} C} {_+_ : C → C → C} (G : Group S _+_) → Set
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sub
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iso : {b : _} {B : Set b} (decB : DecidableSet B) → {f : A → B} → Bijection f → ReducedWord decA → ReducedWord decB
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iso decB {f} bij = universalPropertyFunction decA (freeGroup decB) λ a → freeEmbedding decB (f a)
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isoHom : {b : _} {B : Set b} (decB : DecidableSet B) → {f : A → B} → (bij : Bijection f) → GroupHom (freeGroup decA) (freeGroup decB) (iso decB bij)
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isoHom decB {f} bij = universalPropertyHom decA (freeGroup decB) λ a → iso decB bij (freeEmbedding decA a)
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iso2 : {b : _} {B : Set b} (decB : DecidableSet B) → {f : A → B} → Bijection f → ReducedWord decB → ReducedWord decA
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iso2 decB {f} bij = universalPropertyFunction decB (freeGroup decA) λ b → freeEmbedding decA (Invertible.inverse (bijectionImpliesInvertible bij) b)
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iso2Hom : {b : _} {B : Set b} (decB : DecidableSet B) → {f : A → B} → (bij : Bijection f) → GroupHom (freeGroup decB) (freeGroup decA) (iso2 decB bij)
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iso2Hom decB {f} bij = universalPropertyHom decB (freeGroup decA) λ b → iso2 decB bij (freeEmbedding decB b)
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fixesF : {b : _} {B : Set b} (decB : DecidableSet B) → {f : A → B} → (bij : Bijection f) → (x : A) → iso2 decB bij (iso decB bij (freeEmbedding decA x)) ≡ freeEmbedding decA x
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fixesF decB {f} bij x with Bijection.surj bij (f x)
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... | _ , pr rewrite Bijection.inj bij pr = refl
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fixesF' : {b : _} {B : Set b} (decB : DecidableSet B) → {f : A → B} → (bij : Bijection f) → (x : B) → iso decB bij (iso2 decB bij (freeEmbedding decB x)) ≡ freeEmbedding decB x
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fixesF' decB {f} bij x with Bijection.surj bij x
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... | _ , pr rewrite pr = refl
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uniq : {b : _} {B : Set b} (decB : DecidableSet B) → {f : A → B} → (bij : Bijection f) → (x : ReducedWord decA) → x ≡ universalPropertyFunction decA (freeGroup decA) (λ x → iso2 decB bij (iso decB bij (freeEmbedding decA x))) x
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uniq decB {f} bij x = universalPropertyUniqueness decA (freeGroup decA) (λ x → iso2 decB bij (iso decB bij (freeEmbedding decA x))) {id} (record { wellDefined = id ; groupHom = refl }) (fixesF decB bij) x
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uniqLemm : {b : _} {B : Set b} (decB : DecidableSet B) → {f : A → B} → (bij : Bijection f) → (x : ReducedWord decA) → iso2 decB bij (iso decB bij x) ≡ universalPropertyFunction decA (freeGroup decA) (λ x → iso2 decB bij (iso decB bij (freeEmbedding decA x))) x
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uniqLemm decB {f} bij x = universalPropertyUniqueness decA (freeGroup decA) (λ i → freeEmbedding decA (underlying (Bijection.surj bij (f i)))) {λ i → iso2 decB bij (iso decB bij i)} (groupHomsCompose (isoHom decB bij) (iso2Hom decB bij)) (λ _ → refl) x
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uniq! : {b : _} {B : Set b} (decB : DecidableSet B) → {f : A → B} → (bij : Bijection f) → (x : ReducedWord decA) → iso2 decB bij (iso decB bij x) ≡ x
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uniq! decB bij x = transitivity (uniqLemm decB bij x) (equalityCommutative (uniq decB bij x))
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uniq' : {b : _} {B : Set b} (decB : DecidableSet B) → {f : A → B} → (bij : Bijection f) → (x : ReducedWord decB) → x ≡ universalPropertyFunction decB (freeGroup decB) (λ x → iso decB bij (iso2 decB bij (freeEmbedding decB x))) x
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uniq' decB {f} bij x = universalPropertyUniqueness decB (freeGroup decB) (λ x → iso decB bij (iso2 decB bij (freeEmbedding decB x))) {id} (record { wellDefined = id ; groupHom = refl }) (fixesF' decB bij) x
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uniq'Lemm : {b : _} {B : Set b} (decB : DecidableSet B) → {f : A → B} → (bij : Bijection f) → (x : ReducedWord decB) → iso decB bij (iso2 decB bij x) ≡ universalPropertyFunction decB (freeGroup decB) (λ x → iso decB bij (iso2 decB bij (freeEmbedding decB x))) x
|
||||
uniq'Lemm decB {f} bij x = universalPropertyUniqueness decB (freeGroup decB) (λ i → freeEmbedding decB (f (Invertible.inverse (bijectionImpliesInvertible bij) i))) {λ i → iso decB bij (iso2 decB bij i)} (groupHomsCompose (iso2Hom decB bij) (isoHom decB bij)) (λ _ → refl) x
|
||||
uniq'! : {b : _} {B : Set b} (decB : DecidableSet B) → {f : A → B} → (bij : Bijection f) → (x : ReducedWord decB) → iso decB bij (iso2 decB bij x) ≡ x
|
||||
uniq'! decB bij x = transitivity (uniq'Lemm decB bij x) (equalityCommutative (uniq' decB bij x))
|
||||
|
||||
inBijection : {b : _} {B : Set b} (decB : DecidableSet B) {f : A → B} (bij : Bijection f) → Bijection (iso decB bij)
|
||||
inBijection decB bij = invertibleImpliesBijection (record { inverse = iso2 decB bij ; isLeft = uniq'! decB bij ; isRight = uniq! decB bij })
|
||||
|
||||
freeGroupFunctorWellDefined : {b : _} {B : Set b} (decB : DecidableSet B) → {f : A → B} → Bijection f → GroupsIsomorphic (freeGroup decA) (freeGroup decB)
|
||||
GroupsIsomorphic.isomorphism (freeGroupFunctorWellDefined decB {f} bij) = iso decB bij
|
||||
GroupIso.groupHom (GroupsIsomorphic.proof (freeGroupFunctorWellDefined decB {f} bij)) = universalPropertyHom decA (freeGroup decB) λ a → freeEmbedding decB (f a)
|
||||
SetoidInjection.wellDefined (SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof (freeGroupFunctorWellDefined decB {f} bij)))) refl = refl
|
||||
SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof (freeGroupFunctorWellDefined decB {f} bij)))) {x} {y} pr = Bijection.inj (inBijection decB bij) pr
|
||||
SetoidSurjection.wellDefined (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (freeGroupFunctorWellDefined decB {f} bij)))) refl = refl
|
||||
SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (freeGroupFunctorWellDefined decB {f} bij)))) {x} = Bijection.surj (inBijection decB bij) x
|
||||
|
||||
{-
|
||||
freeGroupFunctorInjective : {b : _} {B : Set b} (decB : DecidableSet B) → GroupsIsomorphic (freeGroup decA) (freeGroup decB) → Sg (A → B) (λ f → Bijection f)
|
||||
freeGroupFunctorInjective decB iso = {!!}
|
||||
|
||||
everyGroupQuotientOfFreeGroup : {b : _} → (S : Setoid {a} {b} A) → {_+_ : A → A → A} → (G : Group S _+_) → GroupsIsomorphic G (quotientGroupByHom (freeGroup decA) (universalPropertyHom decA {!!} {!!}))
|
||||
everyGroupQuotientOfFreeGroup = {!!}
|
||||
|
||||
everyFGGroupQuotientOfFGFreeGroup : {!!}
|
||||
everyFGGroupQuotientOfFGFreeGroup = {!!}
|
||||
|
||||
freeGroupTorsionFree : {!!}
|
||||
freeGroupTorsionFree = {!!}
|
||||
-}
|
||||
|
||||
private
|
||||
mapNToGrp : (a : A) → (n : ℕ) → ReducedWord decA
|
||||
mapNToGrpLen : (a : A) → (n : ℕ) → wordLength decA (mapNToGrp a n) ≡ n
|
||||
mapNToGrpFirstLetter : (a : A) → (n : ℕ) → .(pr : 0 <N wordLength decA (mapNToGrp a (succ n))) → firstLetter decA (mapNToGrp a (succ n)) pr ≡ (ofLetter a)
|
||||
lemma : (a : A) → (n : ℕ) → .(pr : 0 <N wordLength decA (mapNToGrp a (succ n))) → ofLetter a ≡ freeInverse (firstLetter decA (mapNToGrp a (succ n)) pr) → False
|
||||
|
||||
lemma a zero _ ()
|
||||
lemma a (succ n) _ ()
|
||||
|
||||
mapNToGrp a zero = empty
|
||||
mapNToGrp a 1 = prependLetter (ofLetter a) empty (wordEmpty refl)
|
||||
mapNToGrp a (succ (succ n)) = prependLetter (ofLetter a) (mapNToGrp a (succ n)) (wordEnding (identityOfIndiscernablesRight _<N_ (succIsPositive n) (equalityCommutative (mapNToGrpLen a (succ n)))) (freeCompletionEqualFalse decA λ p → lemma a n ((identityOfIndiscernablesRight _<N_ (succIsPositive n) (equalityCommutative (mapNToGrpLen a (succ n))))) p))
|
||||
|
||||
mapNToGrpFirstLetter a zero pr = refl
|
||||
mapNToGrpFirstLetter a (succ n) pr = refl
|
||||
|
||||
mapNToGrpLen a zero = refl
|
||||
mapNToGrpLen a (succ zero) = refl
|
||||
mapNToGrpLen a (succ (succ n)) = applyEquality succ (mapNToGrpLen a (succ n))
|
||||
|
||||
mapNToGrpInj : (a : A) → (x y : ℕ) → mapNToGrp a x ≡ mapNToGrp a y → x ≡ y
|
||||
mapNToGrpInj a zero zero pr = refl
|
||||
mapNToGrpInj a zero (succ zero) ()
|
||||
mapNToGrpInj a zero (succ (succ y)) ()
|
||||
mapNToGrpInj a (succ zero) zero ()
|
||||
mapNToGrpInj a (succ (succ x)) zero ()
|
||||
mapNToGrpInj a (succ zero) (succ zero) pr = refl
|
||||
mapNToGrpInj a (succ zero) (succ (succ y)) pr = exFalso (naughtE (transitivity (applyEquality (wordLength decA) (prependLetterInjective decA pr)) (mapNToGrpLen a (succ y))))
|
||||
mapNToGrpInj a (succ (succ x)) (succ 0) pr = exFalso (naughtE (transitivity (equalityCommutative (applyEquality (wordLength decA) (prependLetterInjective decA pr))) (mapNToGrpLen a (succ x))))
|
||||
mapNToGrpInj a (succ (succ x)) (succ (succ y)) pr = applyEquality succ (mapNToGrpInj a (succ x) (succ y) (prependLetterInjective decA pr))
|
||||
|
||||
freeGroupInfinite : (nonempty : A) → DedekindInfiniteSet (ReducedWord decA)
|
||||
DedekindInfiniteSet.inj (freeGroupInfinite nonempty) = mapNToGrp nonempty
|
||||
DedekindInfiniteSet.isInjection (freeGroupInfinite nonempty) {x} {y} = mapNToGrpInj nonempty x y
|
||||
|
||||
@@ -48,97 +48,97 @@ notWellDefined {BoolFalse} {BoolFalse} x=y = refl
|
||||
|
||||
parity : (x : A) → ReducedWord → Bool
|
||||
parity x empty = BoolFalse
|
||||
parity x (prependLetter (ofLetter y) w _) with DecidableSet.eq decA x y
|
||||
parity x (prependLetter (ofLetter y) w _) with decA x y
|
||||
parity x (prependLetter (ofLetter y) w _) | inl _ = not (parity x w)
|
||||
parity x (prependLetter (ofLetter y) w _) | inr _ = parity x w
|
||||
parity x (prependLetter (ofInv y) w _) with DecidableSet.eq decA x y
|
||||
parity x (prependLetter (ofInv y) w _) with decA x y
|
||||
parity x (prependLetter (ofInv y) w _) | inl _ = not (parity x w)
|
||||
parity x (prependLetter (ofInv y) w _) | inr _ = parity x w
|
||||
|
||||
parityPrepend : (a : A) (w : ReducedWord) (l : A) → ((a ≡ l) → False) → parity a (prepend w (ofLetter l)) ≡ parity a w
|
||||
parityPrepend a empty l notEq with DecidableSet.eq decA a l
|
||||
parityPrepend a empty l notEq with decA a l
|
||||
parityPrepend a empty l notEq | inl x = exFalso (notEq x)
|
||||
parityPrepend a empty l notEq | inr x = refl
|
||||
parityPrepend a (prependLetter (ofLetter r) w x) l notEq with DecidableSet.eq decA a l
|
||||
parityPrepend a (prependLetter (ofLetter r) w x) l notEq with decA a l
|
||||
parityPrepend a (prependLetter (ofLetter r) w x) l notEq | inl m = exFalso (notEq m)
|
||||
parityPrepend a (prependLetter (ofLetter r) w x) l notEq | inr _ = refl
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq with DecidableSet.eq decA a r
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r with DecidableSet.eq decA l r
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq with decA a r
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r with decA l r
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inl l=r = exFalso (notEq (transitivity a=r (equalityCommutative l=r)))
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r with DecidableSet.eq decA a l
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r with decA a l
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inl a=l = exFalso (notEq a=l)
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l with DecidableSet.eq decA a r
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l with decA a r
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l | inl x₁ = refl
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l | inr bad = exFalso (bad a=r)
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r with DecidableSet.eq decA l r
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r with decA l r
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inl x₁ = refl
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r with DecidableSet.eq decA a l
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r with decA a l
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inl a=l = exFalso (notEq a=l)
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l with DecidableSet.eq decA a r
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l with decA a r
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l | inl a=r = exFalso (a!=r a=r)
|
||||
parityPrepend a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l | inr x₁ = refl
|
||||
|
||||
parityPrepend' : (a : A) (w : ReducedWord) (l : A) → ((a ≡ l) → False) → parity a (prepend w (ofInv l)) ≡ parity a w
|
||||
parityPrepend' a empty l notEq with DecidableSet.eq decA a l
|
||||
parityPrepend' a empty l notEq with decA a l
|
||||
parityPrepend' a empty l notEq | inl x = exFalso (notEq x)
|
||||
parityPrepend' a empty l notEq | inr x = refl
|
||||
parityPrepend' a (prependLetter (ofLetter r) w x) l notEq with DecidableSet.eq decA l r
|
||||
parityPrepend' a (prependLetter (ofLetter r) w x) l notEq | inl m with DecidableSet.eq decA a r
|
||||
parityPrepend' a (prependLetter (ofLetter r) w x) l notEq with decA l r
|
||||
parityPrepend' a (prependLetter (ofLetter r) w x) l notEq | inl m with decA a r
|
||||
... | inl a=r = exFalso (notEq (transitivity a=r (equalityCommutative m)))
|
||||
... | inr a!=r = refl
|
||||
parityPrepend' a (prependLetter (ofLetter r) w x) l notEq | inr l!=r with DecidableSet.eq decA a l
|
||||
parityPrepend' a (prependLetter (ofLetter r) w x) l notEq | inr l!=r with decA a l
|
||||
... | inl a=l = exFalso (notEq a=l)
|
||||
... | inr a!=l = refl
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq with DecidableSet.eq decA a r
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r with DecidableSet.eq decA l r
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq with decA a r
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r with decA l r
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inl l=r = exFalso (notEq (transitivity a=r (equalityCommutative l=r)))
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r with DecidableSet.eq decA a l
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r with decA a l
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inl a=l = exFalso (notEq a=l)
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l with DecidableSet.eq decA a r
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l with decA a r
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l | inl x₁ = refl
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inl a=r | inr l!=r | inr a!=l | inr bad = exFalso (bad a=r)
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r with DecidableSet.eq decA l r
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inl l=r with DecidableSet.eq decA a l
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r with decA l r
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inl l=r with decA a l
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inl l=r | inl a=l = exFalso (notEq a=l)
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inl l=r | inr a!=l with DecidableSet.eq decA a r
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inl l=r | inr a!=l with decA a r
|
||||
... | inl a=r = exFalso (a!=r a=r)
|
||||
... | inr _ = refl
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r with DecidableSet.eq decA a l
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r with decA a l
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inl a=l = exFalso (notEq a=l)
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l with DecidableSet.eq decA a r
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l with decA a r
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l | inl a=r = exFalso (a!=r a=r)
|
||||
parityPrepend' a (prependLetter (ofInv r) w x) l notEq | inr a!=r | inr l!=r | inr a!=l | inr x₁ = refl
|
||||
|
||||
parityPrepend'' : (a : A) (w : ReducedWord) → parity a (prepend w (ofLetter a)) ≡ not (parity a w)
|
||||
parityPrepend'' a empty with DecidableSet.eq decA a a
|
||||
parityPrepend'' a empty with decA a a
|
||||
... | inl _ = refl
|
||||
... | inr bad = exFalso (bad refl)
|
||||
parityPrepend'' a (prependLetter (ofLetter l) w x) with DecidableSet.eq decA a a
|
||||
parityPrepend'' a (prependLetter (ofLetter l) w x) | inl _ with DecidableSet.eq decA a l
|
||||
parityPrepend'' a (prependLetter (ofLetter l) w x) with decA a a
|
||||
parityPrepend'' a (prependLetter (ofLetter l) w x) | inl _ with decA a l
|
||||
parityPrepend'' a (prependLetter (ofLetter l) w x) | inl _ | inl a=l = refl
|
||||
parityPrepend'' a (prependLetter (ofLetter l) w x) | inl _ | inr a!=l = refl
|
||||
parityPrepend'' a (prependLetter (ofLetter l) w x) | inr bad = exFalso (bad refl)
|
||||
parityPrepend'' a (prependLetter (ofInv l) w x) with DecidableSet.eq decA a l
|
||||
parityPrepend'' a (prependLetter (ofInv l) w x) with decA a l
|
||||
... | inl a=l = equalityCommutative (notNot (parity a w))
|
||||
parityPrepend'' a (prependLetter (ofInv l) w x) | inr a!=l with DecidableSet.eq decA a a
|
||||
parityPrepend'' a (prependLetter (ofInv l) w x) | inr a!=l | inl _ with DecidableSet.eq decA a l
|
||||
parityPrepend'' a (prependLetter (ofInv l) w x) | inr a!=l with decA a a
|
||||
parityPrepend'' a (prependLetter (ofInv l) w x) | inr a!=l | inl _ with decA a l
|
||||
... | inl a=l = exFalso (a!=l a=l)
|
||||
... | inr _ = refl
|
||||
parityPrepend'' a (prependLetter (ofInv l) w x) | inr a!=l | inr bad = exFalso (bad refl)
|
||||
|
||||
parityPrepend''' : (a : A) (w : ReducedWord) → parity a (prepend w (ofInv a)) ≡ not (parity a w)
|
||||
parityPrepend''' a empty with DecidableSet.eq decA a a
|
||||
parityPrepend''' a empty with decA a a
|
||||
... | inl _ = refl
|
||||
... | inr bad = exFalso (bad refl)
|
||||
parityPrepend''' a (prependLetter (ofLetter l) w x) with DecidableSet.eq decA a l
|
||||
parityPrepend''' a (prependLetter (ofLetter l) w x) with decA a l
|
||||
... | inl a=l = equalityCommutative (notNot _)
|
||||
parityPrepend''' a (prependLetter (ofLetter l) w x) | inr a!=l with DecidableSet.eq decA a a
|
||||
... | inl _ with DecidableSet.eq decA a l
|
||||
parityPrepend''' a (prependLetter (ofLetter l) w x) | inr a!=l with decA a a
|
||||
... | inl _ with decA a l
|
||||
parityPrepend''' a (prependLetter (ofLetter l) w x) | inr a!=l | inl _ | inl a=l = exFalso (a!=l a=l)
|
||||
parityPrepend''' a (prependLetter (ofLetter l) w x) | inr a!=l | inl _ | inr _ = refl
|
||||
parityPrepend''' a (prependLetter (ofLetter l) w x) | inr a!=l | inr bad = exFalso (bad refl)
|
||||
parityPrepend''' a (prependLetter (ofInv l) w x) with DecidableSet.eq decA a a
|
||||
parityPrepend''' a (prependLetter (ofInv l) w x) | inl _ with DecidableSet.eq decA a l
|
||||
parityPrepend''' a (prependLetter (ofInv l) w x) with decA a a
|
||||
parityPrepend''' a (prependLetter (ofInv l) w x) | inl _ with decA a l
|
||||
... | inl a=l = refl
|
||||
... | inr a!=l = refl
|
||||
parityPrepend''' a (prependLetter (ofInv l) w x) | inr bad = exFalso (bad refl)
|
||||
@@ -151,10 +151,10 @@ notXor BoolFalse BoolFalse = refl
|
||||
|
||||
parityHomIsHom : (a : A) → (x y : ReducedWord) → parity a (_+W_ x y) ≡ xor (parity a x) (parity a y)
|
||||
parityHomIsHom a empty y = refl
|
||||
parityHomIsHom a (prependLetter (ofLetter l) x _) y with DecidableSet.eq decA a l
|
||||
parityHomIsHom a (prependLetter (ofLetter l) x _) y with decA a l
|
||||
parityHomIsHom a (prependLetter (ofLetter .a) x _) y | inl refl = transitivity (parityPrepend'' a (x +W y)) (transitivity (notWellDefined (parityHomIsHom a x y)) (notXor (parity a x) (parity a y)))
|
||||
parityHomIsHom a (prependLetter (ofLetter l) x _) y | inr a!=l = transitivity (parityPrepend a (_+W_ x y) l a!=l) (parityHomIsHom a x y)
|
||||
parityHomIsHom a (prependLetter (ofInv l) x _) y with DecidableSet.eq decA a l
|
||||
parityHomIsHom a (prependLetter (ofInv l) x _) y with decA a l
|
||||
parityHomIsHom a (prependLetter (ofInv .a) x _) y | inl refl = transitivity (parityPrepend''' a (x +W y)) (transitivity (notWellDefined (parityHomIsHom a x y)) (notXor (parity a x) (parity a y)))
|
||||
parityHomIsHom a (prependLetter (ofInv l) x _) y | inr a!=l = transitivity (parityPrepend' a (x +W y) l a!=l) (parityHomIsHom a x y)
|
||||
|
||||
|
||||
@@ -32,7 +32,7 @@ private
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
prepLemma {S = S} G f (prependLetter (ofInv x) w pr) l with DecidableSet.eq decA l x
|
||||
prepLemma {S = S} G f (prependLetter (ofInv x) w pr) l with decA l x
|
||||
... | inl refl = transitive (symmetric identLeft) (transitive (+WellDefined (symmetric invRight) reflexive) (symmetric +Associative))
|
||||
where
|
||||
open Group G
|
||||
@@ -48,7 +48,7 @@ private
|
||||
where
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
prepLemma' {S = S} G f (prependLetter (ofLetter x) w pr) l with DecidableSet.eq decA l x
|
||||
prepLemma' {S = S} G f (prependLetter (ofLetter x) w pr) l with decA l x
|
||||
... | inl refl = symmetric (transitive +Associative (transitive (+WellDefined invLeft reflexive) identLeft))
|
||||
where
|
||||
open Group G
|
||||
|
||||
@@ -45,125 +45,12 @@ prependLetterInjective' refl = refl
|
||||
|
||||
badPrepend : {x : A} {w : ReducedWord} {pr : PrependIsValid w (ofInv x)} → (PrependIsValid (prependLetter (ofInv x) w pr) (ofLetter x)) → False
|
||||
badPrepend (wordEmpty ())
|
||||
badPrepend {x = x} (wordEnding pr bad) with DecidableSet.eq (decidableFreeCompletion decA) (ofLetter x) (ofLetter x)
|
||||
badPrepend {x = x} (wordEnding pr bad) with decidableFreeCompletion decA (ofLetter x) (ofLetter x)
|
||||
badPrepend {x} (wordEnding pr ()) | inl x₁
|
||||
badPrepend {x} (wordEnding pr bad) | inr pr2 = pr2 refl
|
||||
|
||||
badPrepend' : {x : A} {w : ReducedWord} {pr : PrependIsValid w (ofLetter x)} → (PrependIsValid (prependLetter (ofLetter x) w pr) (ofInv x)) → False
|
||||
badPrepend' (wordEmpty ())
|
||||
badPrepend' {x = x} (wordEnding pr x₁) with DecidableSet.eq (decidableFreeCompletion decA) (ofInv x) (ofInv x)
|
||||
badPrepend' {x = x} (wordEnding pr x₁) with decidableFreeCompletion decA (ofInv x) (ofInv x)
|
||||
badPrepend' {x} (wordEnding pr ()) | inl x₂
|
||||
badPrepend' {x} (wordEnding pr x₁) | inr pr2 = pr2 refl
|
||||
|
||||
data FreeGroupGenerators : Set a where
|
||||
χ : A → FreeGroupGenerators
|
||||
|
||||
freeGroupGenerators : (w : FreeGroupGenerators) → SymmetryGroupElements (reflSetoid (ReducedWord))
|
||||
freeGroupGenerators (χ x) = sym {f = f} bij
|
||||
where
|
||||
open DecidableSet decA
|
||||
f : ReducedWord → ReducedWord
|
||||
f empty = prependLetter (ofLetter x) empty (wordEmpty refl)
|
||||
f (prependLetter (ofLetter startLetter) w pr) = prependLetter (ofLetter x) (prependLetter (ofLetter startLetter) w pr) (wordEnding (succIsPositive _) ans)
|
||||
where
|
||||
ans : freeCompletionEqual decA (ofLetter x) (ofInv startLetter) ≡ BoolFalse
|
||||
ans with DecidableSet.eq (decidableFreeCompletion decA) (ofLetter x) (ofInv startLetter)
|
||||
... | bl = refl
|
||||
f (prependLetter (ofInv startLetter) w pr) with DecidableSet.eq decA startLetter x
|
||||
f (prependLetter (ofInv startLetter) w pr) | inl startLetter=x = w
|
||||
f (prependLetter (ofInv startLetter) w pr) | inr startLetter!=x = prependLetter (ofLetter x) (prependLetter (ofInv startLetter) w pr) (wordEnding (succIsPositive _) ans)
|
||||
where
|
||||
ans : freeCompletionEqual decA (ofLetter x) (ofLetter startLetter) ≡ BoolFalse
|
||||
ans with DecidableSet.eq (decidableFreeCompletion decA) (ofLetter x) (ofInv startLetter)
|
||||
ans | bl with DecidableSet.eq decA x startLetter
|
||||
ans | bl | inl x=sl = exFalso (startLetter!=x (equalityCommutative x=sl))
|
||||
ans | bl | inr x!=sl = refl
|
||||
bij : SetoidBijection (reflSetoid ReducedWord) (reflSetoid ReducedWord) f
|
||||
SetoidInjection.wellDefined (SetoidBijection.inj bij) x=y rewrite x=y = refl
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {empty} {empty} fx=fy = refl
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {empty} {prependLetter (ofLetter x₁) y pr1} ()
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {empty} {prependLetter (ofInv l) y pr1} fx=fy with DecidableSet.eq decA l x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {empty} {prependLetter (ofInv l) empty (wordEmpty x₁)} () | inl l=x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {empty} {prependLetter (ofInv l) empty (wordEnding () x₁)} fx=fy | inl l=x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {empty} {prependLetter (ofInv l) (prependLetter (ofLetter x₂) y x₁) (wordEmpty ())} fx=fy | inl l=x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {empty} {prependLetter (ofInv l) (prependLetter (ofLetter x₂) y x₁) (wordEnding pr bad)} fx=fy | inl refl with ofLetterInjective (prependLetterInjective' fx=fy)
|
||||
... | l=x2 = exFalso ((freeCompletionEqualFalse' decA bad) (applyEquality ofInv l=x2))
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {empty} {prependLetter (ofInv l) (prependLetter (ofInv x₂) y x₁) pr1} () | inl l=x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {empty} {prependLetter (ofInv l) y pr1} fx=fy | inr l!=x with prependLetterInjective fx=fy
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {empty} {prependLetter (ofInv l) y pr1} fx=fy | inr l!=x | ()
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofLetter x) w1 prX} {empty} fx=fy with prependLetterInjective fx=fy
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofLetter x) w1 prX} {empty} fx=fy | ()
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofLetter x) w1 prX} {prependLetter (ofLetter y) w2 prY} fx=fy = prependLetterInjective fx=fy
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofLetter a) w1 prX} {prependLetter (ofInv y) w2 prY} fx=fy with DecidableSet.eq decA y x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofLetter a) w1 prX} {prependLetter (ofInv y) empty prY} () | inl y=x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofLetter a) w1 prX} {prependLetter (ofInv y) (prependLetter (ofLetter b) w2 x₁) prY} fx=fy | inl y=x with ofLetterInjective (prependLetterInjective' fx=fy)
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofLetter a) w1 prX} {prependLetter (ofInv y) (prependLetter (ofLetter b) w2 x₁) (wordEmpty ())} fx=fy | inl y=x | x=b
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofLetter a) w1 prX} {prependLetter (ofInv y) (prependLetter (ofLetter b) w2 x₁) (wordEnding pr bad)} fx=fy | inl y=x | x=b rewrite x=b | y=x = exFalso (freeCompletionEqualFalse' decA bad refl)
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofLetter a) w1 prX} {prependLetter (ofInv y) (prependLetter (ofInv b) w2 x₁) prY} fx=fy | inl y=x with prependLetterInjective' fx=fy
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofLetter a) w1 prX} {prependLetter (ofInv y) (prependLetter (ofInv b) w2 x₁) prY} fx=fy | inl y=x | ()
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofLetter a) w1 prX} {prependLetter (ofInv y) w2 prY} fx=fy | inr y!=x with prependLetterInjective fx=fy
|
||||
... | bl = bl
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) w1 prX} {empty} fx=fy with DecidableSet.eq decA a x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) empty prX} {empty} () | inl a=x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) (prependLetter (ofLetter x₂) w1 x₁) prX} {empty} fx=fy | inl a=x with ofLetterInjective (prependLetterInjective' fx=fy)
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) (prependLetter (ofLetter x₂) w1 x₁) (wordEmpty ())} {empty} fx=fy | inl a=x | x2=x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) (prependLetter (ofLetter x₂) w1 x₁) (wordEnding pr x3)} {empty} fx=fy | inl a=x | x2=x with freeCompletionEqualFalse' decA x3
|
||||
... | bl = exFalso (bl (applyEquality ofInv (transitivity a=x (equalityCommutative x2=x))))
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) (prependLetter (ofInv x₂) w1 x₁) prX} {empty} () | inl a=x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) w1 prX} {empty} () | inr a!=x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) w1 prX} {prependLetter (ofLetter x₁) y x₂} fx=fy with DecidableSet.eq decA a x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) empty prX} {prependLetter (ofLetter b) y x₂} () | inl a=x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) (prependLetter (ofLetter x₃) w1 x₁) prX} {prependLetter (ofLetter b) y x₂} fx=fy | inl a=x with ofLetterInjective (prependLetterInjective' fx=fy)
|
||||
... | x3=x rewrite a=x | x3=x = exFalso (badPrepend' prX)
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) (prependLetter (ofInv x₃) w1 x₁) prX} {prependLetter (ofLetter b) y x₂} fx=fy | inl a=x with prependLetterInjective' fx=fy
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) (prependLetter (ofInv x₃) w1 x₁) prX} {prependLetter (ofLetter b) y x₂} fx=fy | inl a=x | ()
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) w1 prX} {prependLetter (ofLetter x₁) y x₂} () | inr a!=x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) w1 prX} {prependLetter (ofInv x₁) y x₂} fx=fy with DecidableSet.eq decA a x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) w1 prX} {prependLetter (ofInv b) y x₂} fx=fy | inl a=x with DecidableSet.eq decA b x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) w1 prX} {prependLetter (ofInv b) y x₂} fx=fy | inl a=x | inl b=x rewrite fx=fy | a=x | b=x = prependLetterRefl
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) empty prX} {prependLetter (ofInv b) y x₂} () | inl a=x | inr b!=x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) (prependLetter (ofLetter x₃) w1 x₁) prX} {prependLetter (ofInv b) y x₂} fx=fy | inl a=x | inr b!=x with ofLetterInjective (prependLetterInjective' fx=fy)
|
||||
... | x3=x rewrite a=x | x3=x = exFalso (badPrepend' prX)
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) (prependLetter (ofInv x₃) w1 x₁) prX} {prependLetter (ofInv b) y x₂} () | inl a=x | inr b!=x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) w1 prX} {prependLetter (ofInv b) y x₂} fx=fy | inr a!=x with DecidableSet.eq decA b x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) w1 prX} {prependLetter (ofInv b) empty x₂} () | inr a!=x | inl b=x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) w1 prX} {prependLetter (ofInv b) (prependLetter (ofLetter x₃) y x₁) x2} fx=fy | inr a!=x | inl b=x with ofLetterInjective (prependLetterInjective' fx=fy)
|
||||
... | x3=x rewrite (equalityCommutative x3=x) | b=x = exFalso (badPrepend' x2)
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) w1 prX} {prependLetter (ofInv b) (prependLetter (ofInv x₃) y x₁) x₂} () | inr a!=x | inl b=x
|
||||
SetoidInjection.injective (SetoidBijection.inj bij) {prependLetter (ofInv a) w1 prX} {prependLetter (ofInv b) y x₂} fx=fy | inr a!=x | inr b!=x = prependLetterInjective fx=fy
|
||||
SetoidSurjection.wellDefined (SetoidBijection.surj bij) x=y rewrite x=y = refl
|
||||
SetoidSurjection.surjective (SetoidBijection.surj bij) {empty} = prependLetter (ofInv x) empty (wordEmpty refl) , needed
|
||||
where
|
||||
needed : f (prependLetter (ofInv x) empty (wordEmpty refl)) ≡ empty
|
||||
needed with DecidableSet.eq decA x x
|
||||
needed | inl x₁ = refl
|
||||
needed | inr x!=x = exFalso (x!=x refl)
|
||||
SetoidSurjection.surjective (SetoidBijection.surj bij) {prependLetter (ofLetter l) empty pr} with DecidableSet.eq decA x l
|
||||
SetoidSurjection.surjective (SetoidBijection.surj bij) {prependLetter (ofLetter x) empty (wordEmpty refl)} | inl refl = empty , refl
|
||||
SetoidSurjection.surjective (SetoidBijection.surj bij) {prependLetter (ofLetter x) empty (wordEnding () x₁)} | inl refl
|
||||
SetoidSurjection.surjective (SetoidBijection.surj bij) {prependLetter (ofLetter l) empty pr} | inr x!=l = prependLetter (ofInv x) (prependLetter (ofLetter l) empty pr) (wordEnding (succIsPositive _) (freeCompletionEqualFalse decA (λ p → x!=l (ofInvInjective p)))) , needed
|
||||
where
|
||||
needed : f (prependLetter (ofInv x) (prependLetter (ofLetter l) empty pr) (wordEnding (succIsPositive 0) (freeCompletionEqualFalse decA {ofInv x} {ofInv l} λ p → x!=l (ofInvInjective p)))) ≡ prependLetter (ofLetter l) empty pr
|
||||
needed with DecidableSet.eq decA x x
|
||||
... | inl _ = refl
|
||||
... | inr bad = exFalso (bad refl)
|
||||
SetoidSurjection.surjective (SetoidBijection.surj bij) {prependLetter (ofLetter l) (prependLetter letter w pr2) pr} with DecidableSet.eq decA l x
|
||||
SetoidSurjection.surjective (SetoidBijection.surj bij) {prependLetter (ofLetter l) (prependLetter (ofLetter y) w pr2) pr} | inl l=x rewrite l=x = prependLetter (ofLetter y) w pr2 , prependLetterRefl
|
||||
SetoidSurjection.surjective (SetoidBijection.surj bij) {prependLetter (ofLetter l) (prependLetter (ofInv y) w pr2) pr} | inl l=x = prependLetter (ofInv y) w pr2 , needed
|
||||
where
|
||||
needed : f (prependLetter (ofInv y) w pr2) ≡ prependLetter (ofLetter l) (prependLetter (ofInv y) w pr2) pr
|
||||
needed with DecidableSet.eq decA y x
|
||||
needed | inl y=x rewrite l=x | y=x = exFalso (badPrepend pr)
|
||||
needed | inr y!=x rewrite l=x = prependLetterRefl
|
||||
SetoidSurjection.surjective (SetoidBijection.surj bij) {prependLetter (ofLetter l) (prependLetter letter w pr2) pr} | inr l!=x = prependLetter (ofInv x) (prependLetter (ofLetter l) (prependLetter letter w pr2) pr) (wordEnding (succIsPositive _) (freeCompletionEqualFalse decA λ p → l!=x (ofInvInjective (equalityCommutative p)))) , needed
|
||||
where
|
||||
needed : f (prependLetter (ofInv x) (prependLetter (ofLetter l) (prependLetter letter w pr2) pr) (wordEnding (succIsPositive (succ (wordLength w))) (freeCompletionEqualFalse decA (λ p → l!=x (ofInvInjective (equalityCommutative p)))))) ≡ prependLetter (ofLetter l) (prependLetter letter w pr2) pr
|
||||
needed with DecidableSet.eq decA x x
|
||||
needed | inl x₁ = refl
|
||||
needed | inr x!=x = exFalso (x!=x refl)
|
||||
SetoidSurjection.surjective (SetoidBijection.surj bij) {prependLetter (ofInv l) w pr} = prependLetter (ofInv x) (prependLetter (ofInv l) w pr) (wordEnding (succIsPositive _) (freeCompletionEqualFalse decA {ofInv x} {ofLetter l} λ ())) , needed
|
||||
where
|
||||
needed : f (prependLetter (ofInv x) (prependLetter (ofInv l) w pr) (wordEnding (succIsPositive (wordLength w)) (freeCompletionEqualFalse decA {ofInv x} {ofLetter l} λ ()))) ≡ (prependLetter (ofInv l) w pr)
|
||||
needed with DecidableSet.eq decA x x
|
||||
needed | inl x₁ = refl
|
||||
needed | inr x!=x = exFalso (x!=x refl)
|
||||
|
||||
47
Groups/FreeProduct/Lemmas.agda
Normal file
47
Groups/FreeProduct/Lemmas.agda
Normal file
@@ -0,0 +1,47 @@
|
||||
{-# OPTIONS --safe --warning=error #-}
|
||||
|
||||
open import Numbers.Naturals.Semiring
|
||||
open import Groups.FreeProduct.Definition
|
||||
open import Groups.FreeProduct.Setoid
|
||||
open import Groups.FreeProduct.Group
|
||||
open import Groups.Definition
|
||||
open import Groups.Homomorphisms.Definition
|
||||
open import Groups.Isomorphisms.Definition
|
||||
open import LogicalFormulae
|
||||
open import Numbers.Integers.Addition
|
||||
open import Numbers.Integers.Definition
|
||||
open import Groups.FreeGroup.Definition
|
||||
open import Groups.FreeGroup.Word
|
||||
open import Groups.FreeGroup.Group
|
||||
open import Groups.FreeGroup.UniversalProperty
|
||||
open import Setoids.Setoids
|
||||
|
||||
module Groups.FreeProduct.Lemmas {i : _} {I : Set i} (decidableIndex : (x y : I) → ((x ≡ y) || ((x ≡ y) → False))) where
|
||||
|
||||
private
|
||||
f : ReducedWord decidableIndex → ReducedSequence decidableIndex (λ _ → ℤDecideEquality) (λ _ → ℤGroup)
|
||||
f = universalPropertyFunction decidableIndex (FreeProductGroup decidableIndex (λ _ → ℤDecideEquality) (λ _ → ℤGroup)) λ i → nonempty i (ofEmpty i (nonneg 1) λ ())
|
||||
|
||||
freeProductIso : GroupHom (freeGroup decidableIndex) (FreeProductGroup decidableIndex (λ _ → ℤDecideEquality) (λ _ → ℤGroup)) f
|
||||
freeProductIso = universalPropertyHom decidableIndex (FreeProductGroup decidableIndex (λ _ → ℤDecideEquality) (λ _ → ℤGroup)) (λ i → nonempty i (ofEmpty i (nonneg 1) λ ()))
|
||||
|
||||
freeProductInj : (x y : ReducedWord decidableIndex) → (decidableIndex =RP λ _ → ℤDecideEquality) (λ _ → ℤGroup) (f x) (f y) → x ≡ y
|
||||
freeProductInj empty empty pr = refl
|
||||
freeProductInj empty (prependLetter (ofLetter x₁) y x) pr = exFalso {!!}
|
||||
freeProductInj empty (prependLetter (ofInv x₁) y x) pr = {!!}
|
||||
freeProductInj (prependLetter letter x x₁) y pr = {!!}
|
||||
|
||||
freeProductZ : GroupsIsomorphic (freeGroup decidableIndex) (FreeProductGroup decidableIndex (λ _ → ℤDecideEquality) (λ _ → ℤGroup))
|
||||
GroupsIsomorphic.isomorphism (freeProductZ) = universalPropertyFunction decidableIndex (FreeProductGroup decidableIndex (λ _ → ℤDecideEquality) (λ _ → ℤGroup)) λ i → nonempty i (ofEmpty i (nonneg 1) λ ())
|
||||
GroupIso.groupHom (GroupsIsomorphic.proof (freeProductZ)) = freeProductIso
|
||||
SetoidInjection.wellDefined (SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof (freeProductZ)))) = GroupHom.wellDefined (freeProductIso)
|
||||
SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof (freeProductZ)))) {x} {y} = freeProductInj x y
|
||||
SetoidSurjection.wellDefined (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (freeProductZ)))) = GroupHom.wellDefined freeProductIso
|
||||
SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (freeProductZ)))) {empty} = empty , record {}
|
||||
SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (freeProductZ)))) {nonempty i (ofEmpty .i (nonneg zero) nonZero)} = exFalso (nonZero refl)
|
||||
SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (freeProductZ)))) {nonempty i (ofEmpty .i (nonneg (succ x)) nonZero)} = prependLetter (ofLetter i) empty (wordEmpty refl) , {!!}
|
||||
SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (freeProductZ)))) {nonempty i (ofEmpty .i (negSucc x) nonZero)} = {!!}
|
||||
SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (freeProductZ)))) {nonempty i (prependLetter .i g nonZero x x₁)} = {!!}
|
||||
|
||||
freeProductNonAbelian : {!!}
|
||||
freeProductNonAbelian = {!!}
|
||||
16
Setoids/Cardinality/Finite/Definition.agda
Normal file
16
Setoids/Cardinality/Finite/Definition.agda
Normal file
@@ -0,0 +1,16 @@
|
||||
{-# OPTIONS --safe --warning=error --without-K #-}
|
||||
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
open import Functions
|
||||
open import Numbers.Naturals.Definition
|
||||
open import Sets.FinSet.Definition
|
||||
open import Setoids.Setoids
|
||||
|
||||
module Setoids.Cardinality.Finite.Definition where
|
||||
|
||||
record FiniteSetoid {a b : _} {A : Set a} (S : Setoid {a} {b} A) : Set (a ⊔ b) where
|
||||
field
|
||||
size : ℕ
|
||||
mapping : FinSet size → A
|
||||
bij : SetoidBijection (reflSetoid (FinSet size)) S mapping
|
||||
|
||||
18
Setoids/Cardinality/Infinite/Definition.agda
Normal file
18
Setoids/Cardinality/Infinite/Definition.agda
Normal file
@@ -0,0 +1,18 @@
|
||||
{-# OPTIONS --safe --warning=error --without-K #-}
|
||||
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
open import LogicalFormulae
|
||||
open import Functions
|
||||
open import Numbers.Naturals.Definition
|
||||
open import Sets.FinSet.Definition
|
||||
open import Setoids.Setoids
|
||||
|
||||
module Setoids.Cardinality.Infinite.Definition where
|
||||
|
||||
InfiniteSetoid : {a b : _} {A : Set a} (S : Setoid {a} {b} A) → Set (a ⊔ b)
|
||||
InfiniteSetoid {A = A} S = (n : ℕ) → (f : FinSet n → A) → (SetoidBijection (reflSetoid (FinSet n)) S f) → False
|
||||
|
||||
record DedekindInfiniteSetoid {a b : _} {A : Set a} (S : Setoid {a} {b} A) : Set (a ⊔ b) where
|
||||
field
|
||||
inj : ℕ → A
|
||||
isInjection : SetoidInjection (reflSetoid ℕ) S inj
|
||||
29
Setoids/Cardinality/Infinite/Lemmas.agda
Normal file
29
Setoids/Cardinality/Infinite/Lemmas.agda
Normal file
@@ -0,0 +1,29 @@
|
||||
{-# OPTIONS --safe --warning=error --without-K #-}
|
||||
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
open import LogicalFormulae
|
||||
open import Functions
|
||||
open import Numbers.Naturals.Definition
|
||||
open import Numbers.Naturals.Order
|
||||
open import Sets.FinSet.Definition
|
||||
open import Sets.FinSet.Lemmas
|
||||
open import Setoids.Setoids
|
||||
open import Setoids.Cardinality.Infinite.Definition
|
||||
open import Sets.Cardinality.Infinite.Lemmas
|
||||
open import Setoids.Subset
|
||||
open import Sets.EquivalenceRelations
|
||||
|
||||
module Setoids.Cardinality.Infinite.Lemmas where
|
||||
|
||||
finsetNotInfiniteSetoid : {n : ℕ} → InfiniteSetoid (reflSetoid (FinSet n)) → False
|
||||
finsetNotInfiniteSetoid {n} isInfinite = isInfinite n id (record { inj = record { wellDefined = id ; injective = id } ; surj = record { wellDefined = id ; surjective = λ {x} → x , refl } })
|
||||
|
||||
dedekindInfiniteImpliesInfiniteSetoid : {a b : _} {A : Set a} (S : Setoid {a} {b} A) → DedekindInfiniteSetoid S → InfiniteSetoid S
|
||||
dedekindInfiniteImpliesInfiniteSetoid S record { inj = inj ; isInjection = isInj } zero f isBij with SetoidInvertible.inverse (setoidBijectiveImpliesInvertible isBij) (inj 0)
|
||||
... | ()
|
||||
dedekindInfiniteImpliesInfiniteSetoid {A = A} S record { inj = inj ; isInjection = isInj } (succ n) f isBij = noInjectionNToFinite {f = t} tInjective
|
||||
where
|
||||
t : ℕ → FinSet (succ n)
|
||||
t n = SetoidInvertible.inverse (setoidBijectiveImpliesInvertible isBij) (inj n)
|
||||
tInjective : Injection t
|
||||
tInjective pr = SetoidInjection.injective isInj (SetoidInjection.injective (SetoidBijection.inj (setoidInvertibleImpliesBijective (inverseInvertible (setoidBijectiveImpliesInvertible isBij)))) pr)
|
||||
@@ -49,6 +49,9 @@ record SetoidInjection {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c}
|
||||
wellDefined : {x y : A} → x ∼A y → (f x) ∼B (f y)
|
||||
injective : {x y : A} → (f x) ∼B (f y) → x ∼A y
|
||||
|
||||
reflSetoidWellDefined : {a b c : _} {A : Set a} (S : Setoid {a} {b} A) (C : Set c) (f : C → A) → {x y : C} → x ≡ y → Setoid._∼_ S (f x) (f y)
|
||||
reflSetoidWellDefined S C f refl = Equivalence.reflexive (Setoid.eq S)
|
||||
|
||||
record SetoidSurjection {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f : A → B) : Set (a ⊔ b ⊔ c ⊔ d) where
|
||||
open Setoid S renaming (_∼_ to _∼A_)
|
||||
open Setoid T renaming (_∼_ to _∼B_)
|
||||
@@ -119,3 +122,10 @@ SetoidInjection.injective (SetoidBijection.inj (setoidInvertibleImpliesBijective
|
||||
open Setoid S
|
||||
SetoidSurjection.wellDefined (SetoidBijection.surj (setoidInvertibleImpliesBijective inv)) x~y = SetoidInvertible.fWellDefined inv x~y
|
||||
SetoidSurjection.surjective (SetoidBijection.surj (setoidInvertibleImpliesBijective inv)) {x} = SetoidInvertible.inverse inv x , SetoidInvertible.isLeft inv x
|
||||
|
||||
inverseInvertible : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {f : A → B} (inv1 : SetoidInvertible S T f) → SetoidInvertible T S (SetoidInvertible.inverse inv1)
|
||||
SetoidInvertible.fWellDefined (inverseInvertible record { fWellDefined = fWellDefined ; inverse = inverse ; inverseWellDefined = inverseWellDefined ; isLeft = isLeft ; isRight = isRight }) x=y = inverseWellDefined x=y
|
||||
SetoidInvertible.inverse (inverseInvertible {f = f} record { fWellDefined = fWellDefined ; inverse = inverse ; inverseWellDefined = inverseWellDefined ; isLeft = isLeft ; isRight = isRight }) = f
|
||||
SetoidInvertible.inverseWellDefined (inverseInvertible record { fWellDefined = fWellDefined ; inverse = inverse ; inverseWellDefined = inverseWellDefined ; isLeft = isLeft ; isRight = isRight }) = fWellDefined
|
||||
SetoidInvertible.isLeft (inverseInvertible record { fWellDefined = fWellDefined ; inverse = inverse ; inverseWellDefined = inverseWellDefined ; isLeft = isLeft ; isRight = isRight }) = isRight
|
||||
SetoidInvertible.isRight (inverseInvertible record { fWellDefined = fWellDefined ; inverse = inverse ; inverseWellDefined = inverseWellDefined ; isLeft = isLeft ; isRight = isRight }) = isLeft
|
||||
|
||||
@@ -1,6 +1,5 @@
|
||||
{-# OPTIONS --safe --warning=error --without-K #-}
|
||||
|
||||
|
||||
open import LogicalFormulae
|
||||
open import Functions
|
||||
open import Numbers.Naturals.Semiring
|
||||
|
||||
@@ -10,3 +10,7 @@ module Sets.Cardinality.Infinite.Definition where
|
||||
InfiniteSet : {a : _} (A : Set a) → Set a
|
||||
InfiniteSet A = (n : ℕ) → (f : FinSet n → A) → (Bijection f) → False
|
||||
|
||||
record DedekindInfiniteSet {a : _} (A : Set a) : Set a where
|
||||
field
|
||||
inj : ℕ → A
|
||||
isInjection : Injection inj
|
||||
|
||||
@@ -3,10 +3,30 @@
|
||||
open import Functions
|
||||
open import LogicalFormulae
|
||||
open import Numbers.Naturals.Definition
|
||||
open import Numbers.Naturals.Order
|
||||
open import Sets.Cardinality.Infinite.Definition
|
||||
open import Sets.FinSet.Definition
|
||||
open import Sets.FinSet.Lemmas
|
||||
|
||||
module Sets.Cardinality.Infinite.Lemmas where
|
||||
|
||||
finsetNotInfinite : {n : ℕ} → InfiniteSet (FinSet n) → False
|
||||
finsetNotInfinite {n} isInfinite = isInfinite n id idIsBijective
|
||||
|
||||
noInjectionNToFinite : {n : ℕ} → {f : ℕ → FinSet n} → Injection f → False
|
||||
noInjectionNToFinite {n} {f} inj = pigeonhole (le 0 refl) tInj
|
||||
where
|
||||
t : FinSet (succ n) → FinSet n
|
||||
t m = f (toNat m)
|
||||
tInj : Injection t
|
||||
tInj {x} {y} fx=fy = toNatInjective x y (inj fx=fy)
|
||||
|
||||
dedekindInfiniteImpliesInfinite : {a : _} (A : Set a) → DedekindInfiniteSet A → InfiniteSet A
|
||||
dedekindInfiniteImpliesInfinite A record { inj = inj ; isInjection = isInjection } zero f x with Invertible.inverse (bijectionImpliesInvertible x) (inj 0)
|
||||
... | ()
|
||||
dedekindInfiniteImpliesInfinite A record { inj = inj ; isInjection = isInj } (succ n) f isBij = noInjectionNToFinite {f = t} tInjective
|
||||
where
|
||||
t : ℕ → FinSet (succ n)
|
||||
t n = Invertible.inverse (bijectionImpliesInvertible isBij) (inj n)
|
||||
tInjective : Injection t
|
||||
tInjective pr = isInj ((Bijection.inj (invertibleImpliesBijection (inverseIsInvertible (bijectionImpliesInvertible isBij)))) pr)
|
||||
|
||||
Reference in New Issue
Block a user