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Free group universal property (#102)

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Patrick Stevens
2020-03-22 13:08:10 +00:00
committed by GitHub
parent e0e627514e
commit d2d1c4d3f2
6 changed files with 350 additions and 94 deletions
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Groups/FreeGroup/Definition.agda Normal file
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{-# 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.Semiring
open import Numbers.Naturals.Order
open import Sets.FinSet.Definition
open import Groups.Definition
open import Groups.SymmetricGroups.Definition
open import Decidable.Sets
module Groups.FreeGroup.Definition where
data FreeCompletion {a : _} (A : Set a) : Set a where
ofLetter : A → FreeCompletion A
ofInv : A → FreeCompletion A
freeInverse : {a : _} {A : Set a} (l : FreeCompletion A) → FreeCompletion A
freeInverse (ofLetter x) = ofInv x
freeInverse (ofInv x) = ofLetter x
ofLetterInjective : {a : _} {A : Set a} {x y : A} → (ofLetter x ≡ ofLetter y) → x ≡ y
ofLetterInjective refl = refl
ofInvInjective : {a : _} {A : Set a} {x y : A} → (ofInv x ≡ ofInv y) → x ≡ y
ofInvInjective refl = refl
decidableFreeCompletion : {a : _} {A : Set a} → DecidableSet A → DecidableSet (FreeCompletion A)
decidableFreeCompletion {A = A} record { eq = dec } = record { eq = pr }
where
pr : (a b : FreeCompletion A) → (a ≡ b) || (a ≡ b → False)
pr (ofLetter x) (ofLetter y) with dec x y
... | inl refl = inl refl
... | inr x!=y = inr λ p → x!=y (ofLetterInjective p)
pr (ofLetter x) (ofInv y) = inr λ ()
pr (ofInv x) (ofLetter y) = inr λ ()
pr (ofInv x) (ofInv y) with dec x y
... | inl refl = inl refl
... | inr x!=y = inr λ p → x!=y (ofInvInjective p)
freeCompletionEqual : {a : _} {A : Set a} (dec : DecidableSet A) (x y : FreeCompletion A) → Bool
freeCompletionEqual dec x y with DecidableSet.eq (decidableFreeCompletion dec) x y
freeCompletionEqual dec x y | inl x₁ = BoolTrue
freeCompletionEqual dec x y | inr x₁ = BoolFalse
freeCompletionEqualFalse : {a : _} {A : Set a} (dec : DecidableSet A) {x y : FreeCompletion A} → ((x ≡ y) → False) → (freeCompletionEqual dec x y) ≡ BoolFalse
freeCompletionEqualFalse dec {x = x} {y} x!=y with DecidableSet.eq (decidableFreeCompletion dec) x y
freeCompletionEqualFalse dec {x} {y} x!=y | inl x=y = exFalso (x!=y x=y)
freeCompletionEqualFalse dec {x} {y} x!=y | inr _ = refl
freeCompletionEqualFalse' : {a : _} {A : Set a} (dec : DecidableSet A) {x y : FreeCompletion A} → .((freeCompletionEqual dec x y) ≡ BoolFalse) → (x ≡ y) → False
freeCompletionEqualFalse' dec {x} {y} pr with DecidableSet.eq (decidableFreeCompletion dec) x y
freeCompletionEqualFalse' dec {x} {y} () | inl x₁
freeCompletionEqualFalse' dec {x} {y} pr | inr ans = ans