@@ -5,7 +5,8 @@ open import WithK
open import Setoids.Setoids
open import Functions
open import Agda.Primitive using ( Level; lzero; lsuc; _⊔_)
open import Numbers.Naturals.Naturals
open import Numbers.Naturals.Semiring
open import Numbers.Naturals.Order
open import Numbers.Naturals.WithK
open import Sets.FinSet
open import Groups.Definition
@@ -13,201 +14,202 @@ open import Groups.SymmetricGroups.Definition
open import DecidableSet
module Groups.FreeGroups where
data FreeCompletion { a : _} ( A : Set a) : Set a where
ofLetter : A → FreeCompletion A
ofInv : A → FreeCompletion A
freeInverse : { a : _} { : Set a} ( l : FreeCompletion A) → FreeCompletion A
freeInverse ( ofLetter x) = ofInv x
freeInverse ( ofInv x) = ofLetter x
data FreeCompletion { a : _} ( : Set a) : Set a where
: A → FreeCompletion A
ofInv : A → FreeCompletion A
ofLetterInjectiv e : { a : _} { A : Set a} { x y : A} → ( ofLetter x ≡ ofLetter y ) → x ≡ y
Injective refl = refl
freeInvers e : { a : _} { A : Set a} ( l : FreeCompletion A ) → FreeCompletion A
freeInverse ( x) = ofInv x
( ofInv x) = ofLetter x
ofInv Injective : { a : _} { A : Set a} { x y : A} → ( ofInv x ≡ ofInv y) → x ≡ y
ofInv refl = refl
ofLetter Injective : { a : _} { A : Set a} { x y : A} → ( ofLetter x ≡ ofLetter y) → x ≡ y
ofLetter Injective 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)
ofInvInjective : { a : _} { A : Set a} { x y : A} → ( ofInv x ≡ ofInv y ) → x ≡ y
ofInvInjective refl = refl
f reeCompletionEqual : { a : _} { A : Set a} ( dec : DecidableSet A) ( x y : A) → Bool
fEqual dec x y with DecidableSet.eq ( decidableFreeCompletion dec) x y
freeCompletionEqual dec x y | inl x₁ = BoolTrue
freeCompletionEqual dec x y | inr x₁ = Bool False
decidableF reeCompletion : { a : _} { A : Set a} → DecidableSet A → DecidableSet ( A)
decidableF reeCompletion { 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)
freeCompletionEqualFalse : { a : _} { A : Set a} ( dec : DecidableSet A) { y : FreeCompletion A} → ( ( x ≡ y) → False) → ( freeCompletionEqual dec x y) ≡ BoolFalse
False dec { = x} { y} x!= with DecidableSet.eq ( decidableFreeCompletion dec) x y
False dec { x} { y} x!= | inl x=y = exFalso ( x!=y x=y)
False dec { x} { y} x!= | inr _ = refl
freeCompletionEqual : { a : _} { A : Set a} ( dec : DecidableSet A) ( y : FreeCompletion A) → Bool
dec x y with DecidableSet.eq ( decidableFreeCompletion dec) x y
dec x y | inl x₁ = BoolTrue
dec x y | inr x₁ = BoolFalse
freeCompletionEqualFalse' : { a : _} { A : Set a} ( dec : DecidableSet A) { x y : FreeCompletion A} → ( freeCompletionEqual dec x y) ≡ BoolFalse → ( x ≡ y) → False
' dec { x} { y} pr with DecidableSet.eq ( decidableFreeCompletion dec) x y
' dec { x} { y} ( ) | inl x₁
' dec { x} { y} pr | inr ans = ans
freeCompletionEqualFalse : { a : _} { A : Set a} ( dec : DecidableSet A) { x y : FreeCompletion A} → ( ( x ≡ y) → False) → ( freeCompletionEqual dec x y) ≡ BoolFalse
dec { x = x} { y} x!=y with DecidableSet.eq ( decidableFreeCompletion dec) x y
dec { x} { y} x!=y | inl x=y = exFalso ( x!=y x=y)
dec { x} { y} x!=y | inr _ = refl
data ReducedWord { a : _} { A : Set a} ( decA : DecidableSet A) : Set a
wordLength : { a : _} { A : Set a} { decA : DecidableSet A} → ReducedWord decA → ℕ
firstLetter : { a : _} { A : Set a} { A : DecidableSet A} → ( w : ReducedWord decA) → ( 0 <N wordLength w) → FreeCompletion A
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
dec { x} { y} pr | inr ans = ans
data PrependIsVali d { a : _} { A : Set a} ( decA : DecidableSet A) ( w : ReducedWord decA) ( l : FreeCompletion A) : Set a where
wordEmpty : wordLength w ≡ 0 → PrependIsVali d decA w l
wordEnding : ( pr : 0 <N wordLength w) → ( freeCompletionEqual decA l ( freeInverse ( firstLetter w pr) ) ≡ BoolFalse ) → P rependIsValid decA w l
data ReducedWor d { a : _} { A : Set a} ( decA : DecidableSet A) : Set a
wordLength : { a : _} { A : Set a} { decA : DecidableSet A} → ReducedWor decA → ℕ
firstLetter : { a : _} { A : Set a} { decA : DecidableSet A} → ( w : ReducedWord decA) → ( 0 <N wordLength w ) → F reeCompletion A
data ReducedWord { a} { A} decA where
e mpty : ReducedWor d decA
prependLetter : ( letter : F reeCompletion A) → ( w : ReducedWord decA) → PrependIsValid decA w letter → ReducedWord decA
data PrependIsValid { a : _} { A : Set a} ( decA : DecidableSet A) ( w : ReducedWord decA) ( l : FreeCompletion A) : Set a where
wordE mpty : wordLength w ≡ 0 → PrependIsVali d decA w l
wordEnding : ( pr : 0 <N wordLength w) → ( fEqual decA l ( freeInverse ( firstLetter w pr) ) ≡ BoolFalse ) → PrependIsValid decA w l
firstLetter { a} { A} empty ( le x ( ) )
firstLetter { a} { A} ( prependLetter letter w x) pr = letter
data ReducedWord { a} { A} decA where
empty : ReducedWord decA
prependLetter : ( letter : FreeCompletion A) → ( w : ReducedWord decA) → PrependIsValid decA w letter → ReducedWord decA
wordLength { a} { A} empty = 0
wordLength { a} { A} ( prependLetter letter w pr ) = succ ( wordLength w)
firstLetter { a} { A} empty ( le x ( ) )
firstLetter { a} { A} ( prependLetter letter w x ) pr = letter
data FreeGroupGenerators { a : _} ( : Set a) : Set a where
χ : A → FreeGroupGenerators A
wordLength { a } { } empty = 0
wordLength { a} { A} ( prependLetter letter w pr) = succ ( wordLength w)
prependLetterInjective : { a : _} { : Set a} { decA : DecidableSet A} { x : FreeCompletion A} { w1 w2 : ReducedWord decA} { pr1 : PrependIsValid decA w1 x} { pr2 : PrependIsValid decA w2 x} → prependLetter x w1 pr1 ≡ prependLetter x w2 pr2 → w1 ≡ w2
prependLetterInjective { x = x} { w1} { . w1} { pr1} { . pr1} refl = refl
data FreeGroupGenerators { a : _} ( : Set a) : Set a where
χ : A → FreeGroupGenerators A
prependLetterInjective' : { a : _} { A : Set a} { decA : DecidableSet A} { x y: FreeCompletion A} { w1 w2 : ReducedWord decA} { pr1 : PrependIsValid decA w1 x} { pr2 : PrependIsValid decA w2 y } → prependLetter x w1 pr1 ≡ prependLetter y w2 pr2 → x ≡ y
' refl = refl
prependLetterInjective : { a : _} { A : Set a} { decA : DecidableSet A} { x : FreeCompletion A} { w1 w2 : ReducedWord decA} { pr1 : PrependIsValid decA w1 x} { pr2 : PrependIsValid decA w2 x } → prependLetter x w1 pr1 ≡ prependLetter x w2 pr2 → w1 ≡ w2
{ x = x} { w1} { . w1} { pr1} { . pr1} refl = refl
prependLetterRefl : { a : _} { A : Set a} { decA : DecidableSet A} { x : FreeCompletion A} { w : ReducedWord decA} → { pr1 pr2 : PrependIsValid decA w x } → prependLetter x w pr1 ≡ prependLetter x w pr2
Refl { a} { A} { decA} { x} { empty} { wordEmpty refl} { wordEmpty refl} = refl
prependLetterRefl { a} { A} { decA} { x} { empty} { wordEmpty refl} { wordEnding ( le x₁ ( ) ) x₂}
prependLetterRefl { a} { A} { decA} { x} { empty} { wordEnding ( le x₂ ( ) ) x₁} { pr2}
prependLetterRefl { a} { A} { decA} { x} { prependLetter letter w x₁} { wordEmpty ( ) } { pr2}
prependLetterRefl { a} { A} { decA} { x} { prependLetter letter w x₁} { wordEnding pr x₂} { wordEmpty ( ) }
prependLetterRefl { a} { A} { decA} { x} { prependLetter letter w x₁} { wordEnding pr2 r2} { wordEnding pr1 r1} rewrite <NRefl pr1 ( succIsPositive ( wordLength w) ) | <NRefl pr2 ( succIsPositive ( wordLength w) ) | reflRefl r1 r2 = refl
prependLetterInjective' : { a : _} { A : Set a} { decA : DecidableSet A} { x y : FreeCompletion A} { w1 w2 : ReducedWord decA} { pr1 : PrependIsValid decA w1 x} { pr2 : PrependIsValid decA w2 y } → prependLetter x w1 pr1 ≡ prependLetter y w2 pr2 → x ≡ y
Injective' refl = refl
badPrepend : { a : _} { A : Set a} { decA : DecidableSet A} { x : A} { w : ReducedWord decA} { pr : PrependIsValid decA w ( ofInv x) } → ( PrependIsValid decA ( prependLetter ( ofInv x) w pr) ( ofLetter x) ) → False
badPrepend ( ( ) )
badPrepend { decA = } { x = x} ( nding pr bad) with DecidableSet.eq ( decidableFreeCompletion decA) ( ofLetter x) ( ofLetter x)
badPrepend { decA = } { x} ( pr ( ) ) | inl x₁
badPrepend { decA = decA} { x} ( wordEnding pr bad) | inr pr2 = pr2 refl
prependLetterRefl : { a : _} { A : Set a} { decA : DecidableSet A} { x : FreeCompletion A} { w : ReducedWord decA} → { pr1 pr2 : PrependIsValid decA w x} → prependLetter x w pr1 ≡ prependLetter x w pr2
prependLetterRefl { a} { A} { decA} { x} { empty} { refl} { wordEmpty refl} = refl
prependLetterRefl { a} { A} { decA} { x} { empty} { mpty refl} { wordEnding ( le x₁ ( ) ) x₂}
prependLetterRefl { a} { A} { decA} { x} { empty} { ( le x₂ ( ) ) x₁} { pr2}
prependLetterRefl { a} { A} { decA} { x} { prependLetter letter w x₁} { wordEmpty ( ) } { pr2}
{ a} { A} { decA} { x} { prependLetter letter w x₁} { wordEnding pr x₂} { wordEmpty ( ) }
{ a} { A} { decA} { x} { prependLetter letter w x₁} { wordEnding pr2 r2} { wordEnding pr1 r1} rewrite <NRefl pr1 ( succIsPositive ( wordLength w) ) | <NRefl pr2 ( succIsPositive ( wordLength w) ) | reflRefl r1 r2 = refl
badPrepend' : { a : _} { A : Set a} { decA : DecidableSet A} { x : A} { w : ReducedWord decA} { pr : PrependIsValid decA w ( ofLetter x) } → ( PrependIsValid decA ( prependLetter ( ofLetter x) w pr) ( ofInv x) ) → False
' ( wordEmpty ( ) )
' { decA = decA} { x = x} ( wordEnding pr x₁ ) with DecidableSet.eq ( decidableFreeCompletion decA) ( ofInv x) ( ofInv x)
' { decA = decA} { x} ( wordEnding pr ( ) ) | inl x₂
' { decA = decA} { x} ( wordEnding pr x₁ ) | inr pr2 = pr2 refl
badPrepend : { a : _} { A : Set a} { decA : DecidableSet A} { x : A} { w : ReducedWord decA} { pr : PrependIsValid decA w ( ofInv x) } → ( PrependIsValid decA ( prependLetter ( ofInv x) w pr) ( ofLetter x) ) → False
( wordEmpty ( ) )
{ decA = decA} { x = x} ( wordEnding pr bad ) with DecidableSet.eq ( decidableFreeCompletion decA) ( ofLetter x) ( ofLetter x)
{ decA = decA} { x} ( wordEnding pr ( ) ) | inl x₁
{ decA = decA} { x} ( wordEnding pr bad ) | inr pr2 = pr2 refl
freeGroupGenerators : { a : _} ( : Set a) ( : DecidableSet A) ( w : FreeGroupGenerators A ) → SymmetryGroupElements ( reflSetoid ( ReducedWord decA) )
freeGroupGenerators A decA ( χ x) = sym { f = f} bij
where
open DecidableSet.DecidableSet decA
f : ReducedWord decA → ReducedWord decA
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
( prependLetter ( ofInv startLetter) w pr) with DecidableSet.eq decA startLetter x
( 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
: fEqual 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 ( ! =x ( equalityCommutative x=sl) )
ans | bl | inr x!=sl = refl
bij : SetoidBijection ( reflSetoid ( ReducedWord decA) ) ( reflSetoid ( ReducedWord decA) ) 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
SetoidIn jection.injective ( SetoidBijection.inj bij) { empty} { prependLetter ( ofInv l) empty ( wordEnding ( le x ( ) ) x₁) } fx=fy | inl l=x
injective ( SetoidBijection.inj bij) { empty} { prependLetter ( ofInv l) ( prependLetter ( ofLetter x₂) y x₁) ( wordEmpty ( ) ) } fx=fy | inl l=x
( 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) )
( SetoidBijection.inj bij) { empty} { prependLetter ( ofInv l) ( prependLetter ( ofInv x₂) y x₁) pr1} ( ) | in l= x
( SetoidBijection.inj bij) { empty} { prependLetter ( ofInv l) y pr1} fx=fy | inr l! =x with prependLetterInjective fx=fy
( SetoidBijection.inj bij) { empty} { prependLetter ( ofInv l) y pr1 } fx=fy | inr l! =x | ( )
( SetoidBijection.inj bij) { prependLetter ( ofLetter x) w1 prX} { empty} fx=fy with prependLetterInjective fx=fy
( 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
( SetoidBijection.inj bij) { prependLetter ( ofLetter a ) w1 prX} { prependLetter ( ofInv y ) w2 prY } fx=fy with DecidableSet.eq decA y x
( SetoidBijection.inj bij) { prependLetter ( ofLetter a) w1 prX } { prependLetter ( ofInv y ) empt y prY } ( ) | inl y =x
( 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)
( SetoidBijection.inj bij) { prependLetter ( ofLetter a ) w1 prX} { prependLetter ( ofInv y) ( prependLetter ( ofLetter b) w2 x₁) ( wordEmpty ( ) ) } fx=fy | inl y=x | x=b
( 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 )
( 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
( SetoidBijection.inj bij) { prependLetter ( ofLetter a) w1 prX} { prependLetter ( ofInv y) ( prependLetter ( ofInv b) w2 x₁) prY} fx=fy | inl y=x | ( )
( SetoidBijection.inj bij) { prependLetter ( ofLetter a) w1 prX} { prependLetter ( ofInv y) w2 prY} fx=fy | inr y! =x with prependLetterInjective fx=fy
... | b l = bl
( SetoidBijection.inj bij) { prependLetter ( ofInv a) w1 prX} { empty} fx=fy with DecidableSet.eq decA a x
( SetoidBijection.inj bij) { prependLetter ( ofInv a) empty prX} { empt y} ( ) | inl a =x
( SetoidBijection.inj bij) { prependLetter ( ofInv a) ( ( ofLetter x₂ ) w1 x₁) prX} { empty } fx=fy | inl a =x with ofLetterInjective ( prependLetterInjective' fx=fy)
( SetoidBijection.inj bij) { prependLetter ( ofInv a) ( ( ofLetter x₂) w1 x₁) ( wordEmpty ( ) ) } { empty } fx=fy | inl a =x | x2=x
( SetoidBijection.inj bij) { prependLetter ( ofInv a) ( ( ofLetter x₂ ) w1 x₁) ( wordEnding pr x3) } { empty } fx=fy | inl a=x | x2 =x with f reeCompletionEqualFalse' decA x3
... | bl = exFalso ( bl ( applyEquality ofInv ( transitivity a=x ( equalityCommutative x2=x) ) ) )
( SetoidBijection.inj bij) { prependLetter ( ofInv a) ( prependLetter ( ofInv x₂) w1 x₁) prX} { empty} ( ) | inl a= x
( SetoidBijection.inj bij) { prependLetter ( ofInv a) w1 prX} { empty} ( ) | inr a! =x
( SetoidBijection.inj bij) { prependLetter ( ofInv a) w1 prX} { prependLetter ( ofLetter x₁ ) y x₂} fx=fy with DecidableSet.eq decA a x
( SetoidBijection.inj bij) { prependLetter ( ofInv a) empty prX} { prependLetter ( ofLetter b ) y x₂} ( ) | inl a=x
( SetoidBijection.inj bij) { prependLetter ( ofInv a) ( prependLetter ( ofLetter x₃ ) w1 x₁) prX} { prependLetter ( ofLette r b ) y x₂} fx=fy | inl a=x with ofLetterInjective ( prependLetterInjective' fx=fy)
... | x3=x rewrite a=x | x3=x = exFalso ( badPrepend' prX )
( 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
( SetoidBijection.inj bij) { prependLetter ( ofInv a) ( prependLetter ( ofInv x₃) w1 x₁) prX} { prependLetter ( ofLetter b) y x₂} fx=fy | inl a=x | ( )
( SetoidBijection.inj bij) { prependLetter ( ofInv a) w1 prX} { prependLetter ( ofLetter x₁) y x₂} ( ) | inr a!= x
( SetoidBijection.inj bij) { prependLetter ( ofInv a) w1 prX} { prependLetter ( ofInv x₁ ) y x₂} fx=fy with DecidableSet.eq decA a
( 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 rewrite fx=fy | a =x | b=x = prependLetterRefl
( SetoidBijection.inj bij) { prependLetter ( ofInv a) empty prX} { prependLetter ( ofInv b) y x₂} ( ) | inl a=x | inr b!=x
( 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 rewri te a=x | x3=x = exFalso ( badPrepend' prX)
( SetoidBijection.inj bij) { prependLetter ( ofInv a) ( prependLetter ( ofInv x₃) w1 x₁) prX} { prependLetter ( ofInv b ) y x₂} ( ) | inl a=x | inr b!=
( SetoidBijection.inj bij) { prependLetter ( ofInv a) w1 prX} { prependLetter ( ofInv b) y x₂} fx=fy | inr a! =x with DecidableSet.eq decA b x
( SetoidBijection.inj bij) { prependLetter ( ofInv a) w1 prX} { prependLetter ( ofInv b) empt y x₂} ( ) | inr a! =x | inl b=x
( 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 rewri te ( equalityCommutative x3=x) | b=x = exFalso ( badP repend' x2 )
SetoidInjection.injective ( SetoidBijection.inj bij) { prependLetter ( ofInv a) w1 prX} { prependLetter ( ofInv b) ( p rependLetter ( ofInv x₃) y x₁) x₂} ( ) | inr a!=x | inl b=x
( SetoidBijection.inj bij) { prependLetter ( ofInv a) w1 prX} { prependLetter ( ofInv b) y x₂} fx=fy | inr a! =x | inr b!=x = prependLetterInjective fx=fy
Sur jection.wellDefined ( SetoidBijection.sur j bij) x=y rewrite x=y = refl
Sur jection.sur jective ( SetoidBijection.sur j bij) { empty} = prependLetter ( ofInv x ) empty ( wordEmpty refl) , needed
where
needed : f ( prependLet ter ( ofInv x) empty ( wordEmpty refl) ) ≡ empty
needed with DecidableSet.eq decA x
needed | inl x₁ = refl
needed | inr x!=x = exFalso ( x!=x refl)
( 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) { ( ofLetter x) empty ( wordEnding ( le x₂ ( ) ) 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) ( wordEndg ( succIsPositive 0 ) ( freeCompletionEqualFalse decA { ofInv x} { ofInv l} λ p → l ( ofInvInjective p) ) ) ) ≡ prependLetter ( ofLetter l) empty pr
needed with DecidableSet.eq decA x x
... | inl _ = refl
... | inr bad = exFalso ( bad refl)
( SetoidBijection.surj bij) { prependLetter ( ofLetter l) ( prependLetter l etter 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) { ( ofLetter l) ( prependLetter ( y) w pr2) pr} | in l 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 rewri te l=x | y=x = exFalso ( badPrepend pr)
needed | inr y! =x rewrite l=x = prependLetterRefl
( 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) ( letter w pr2) pr) ( wordEnding ( succIsPositive ( succ ( wordLength w) ) ) ( freeCompletionEqualFalse decA ( λ p → l!=x ( ofInvInjective ( equalityCommutative p) ) ) ) ) ) ≡ prependLetter ( ofLetter l ) ( prependLetter letter pr2) pr
with DecidableSet.eq decA x x
| inl x₁ = refl
| inr x !=x = exFalso ( x!=x refl)
( SetoidBijection.surj bij) { prependLetter ( ofInv l) w pr} = prependLetter ( ofInv x) ( prependLetter ( ofInv l) w pr) ( wordEnding ( succIsPositive _) ( freeCompletionEqualFalse decA { ofInv x} { Letter 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 ( ) ) ) ) ≡ ( ( ofInv l) w pr)
with DecidableSet.eq decA x x
| inl x₁ = refl
| inr x!=x = exFalso ( x!=x refl)
badPrepend' : { a : _} { : Set a} { : DecidableSet A} { x : A} { w : ReducedWord decA} { pr : PrependIsValid decA w ( ofLetter x ) } → ( PrependIsValid decA ( p rependLetter ( ofLetter x) w pr) ( ofInv x) ) → False
badPrepend' ( wordEmpty ( ) )
badPrepend' { decA = decA} { x = x} ( wordEnding pr x₁) with DecidableSet.eq ( decidableFreeCompletion decA) ( ofInv x) ( ofInv x)
badPrepend' { decA = decA} { x} ( wordEnding pr ( ) ) | inl x₂
badPrepend' { decA = decA} { x} ( wordEnding pr x₁) | inr pr2 = pr2 refl
freeGroupGenerators : { a : _} ( A : Set a) ( decA : DecidableSet A) ( w : FreeGroupGenerators A) → SymmetryGroupElements ( reflSetoid ( ReducedWord decA) )
freeGroupGenerators A decA ( χ x) = sym { f = f} bij
where
open DecidableSet.DecidableSet decA
f : ReducedWord decA → ReducedWord decA
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 ( decidableF 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 = 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 : SetoidBi jection ( refl Setoid ( ReducedWord decA) ) ( reflSetoid ( ReducedWord decA) ) 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 ( le x ( ) ) 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 ) ( ( 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 | in l 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} { ( 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 p rependLetterInjective fx=fy
... | bl = bl
SetoidInjection.injective ( SetoidBijection.inj bij) { prependLetter ( ofInv a) w1 prX} { empty} fx=fy with DecidableSet.eq decA a
SetoidInjection.injective ( SetoidBijection.inj bij) { prependLetter ( ofInv a) empty prX} { empty} ( ) | inl a=x
SetoidInjection.injective ( SetoidBijection.inj bij) { prependLetter ( ofInv a) ( ( ofLetter x₂ ) w1 x₁) prX} { empty} fx=fy | inl a=x with ofLetterInjective ( prependLetterInjective' fx=fy)
SetoidInjection.injective ( SetoidBijection.inj bij) { prependLetter ( ofInv a) ( ( 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 p 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
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 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) { prependLet ter ( 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) ( prependLet ter ( ofLetter x₃) w1 x₁) prX} { prependLetter ( ofInv b) y x₂} fx=fy | inl a=x | inr b!=x with ofLetterInjective ( p rependLetterInjective' fx=fy )
... | x3=x rewrite a=x | x3=x = exFalso ( badP repend' 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
SetoidIn jection.injective ( SetoidBijection.in j bij) { prependLetter ( ofInv a) w1 prX} { prependLetter ( ofInv b) y x₂} ff y | inr a!=x with DecidableSet.eq decA b x
SetoidIn jection.in jective ( SetoidBijection.in j 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 rewri te ( 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 = p rependLetterInjective 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 ( ( ofInv x) empty ( wordEmpty refl) ) ≡ empty
needed with DecidableSet.eq decA x x
needed | inl x₁ = refl
needed | r 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 ,
SetoidSurjection.surjective ( SetoidBijection.surj bij) { prependLetter ( ofLetter x) empty ( wordEnding ( le x₂ ( ) ) x₁) } | inl refl
SetoidSurjection.surjective ( SetoidBijection.surj bij) { prependLetter ( ofLetter l) empty pr} | inr x!=l = prependLetter ( ofInv x) ( prependLetter ( ofL l) empty pr) ( wordEnding ( succIsPositive _) ( freeCompletionEqualFalse decA ( λ p → x!=l ( ofInvInjective p) ) ) ) , needed
where
needed : f ( prependLetter ( ofInv x) ( ( ofLetter l) empty pr) ( wordEnding ( succIsPositive 0 ) ( freeCompletionEqualFalse decA { 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 let ter 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 ,
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) ( ( 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 = p rependLetterRefl
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 ( InvInjective ( 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 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)
Patrick Stevens
GitHub