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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.ZCohomology.Groups.Sn where
open import Cubical.ZCohomology.Base
open import Cubical.ZCohomology.Properties
open import Cubical.ZCohomology.MayerVietorisUnreduced
open import Cubical.ZCohomology.Groups.Unit
open import Cubical.ZCohomology.Groups.Connected
open import Cubical.ZCohomology.KcompPrelims
open import Cubical.ZCohomology.Groups.Prelims
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Function
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Pointed
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.GroupoidLaws
open import Cubical.HITs.Pushout
open import Cubical.HITs.Sn
open import Cubical.HITs.S1
open import Cubical.HITs.Susp
open import Cubical.HITs.SetTruncation renaming (rec to sRec ; elim to sElim ; elim2 to sElim2)
open import Cubical.HITs.PropositionalTruncation renaming (rec to pRec ; elim to pElim ; elim2 to pElim2 ; ∥_∥ to ∥_∥₁ ; ∣_∣ to ∣_∣₁)
open import Cubical.HITs.Nullification
open import Cubical.Data.Sigma
open import Cubical.Data.Int renaming (_+_ to _+ℤ_; +-comm to +ℤ-comm ; +-assoc to +ℤ-assoc)
open import Cubical.Data.Nat
open import Cubical.HITs.Truncation renaming (elim to trElim ; map to trMap ; rec to trRec)
open import Cubical.Data.Unit
open import Cubical.Algebra.Group
infixr 31 _□_
_□_ : _
_□_ = compGroupEquiv
open GroupEquiv
open vSES
Sn-connected : (n : ℕ) (x : S₊ (suc n)) → ∥ north ≡ x ∥₁
Sn-connected n = suspToPropRec north (λ _ → propTruncIsProp) ∣ refl ∣₁
H⁰-Sⁿ≅ℤ : (n : ℕ) → GroupEquiv (coHomGr 0 (S₊ (suc n))) intGroup
H⁰-Sⁿ≅ℤ n = H⁰-connected north (Sn-connected n)
-- ----------------------------------------------------------------------
--- We will need to switch between Sⁿ defined using suspensions and using pushouts
--- in order to apply Mayer Vietoris.
coHomPushout≅coHomSn : (n m : ℕ) → GroupEquiv (coHomGr m (S₊ (suc n))) (coHomGr m (Pushout {A = S₊ n} (λ _ → tt) λ _ → tt))
coHomPushout≅coHomSn n m = transport (λ i → GroupEquiv (coHomGr m (PushoutSusp≡Susp {A = S₊ n} i))
(coHomGr m (Pushout {A = S₊ n} (λ _ → tt) λ _ → tt))) (idGroupEquiv _)
-------------------------- H⁰(S⁰) -----------------------------
S0→Int : (a : Int × Int) → S₊ 0 → Int
S0→Int a north = fst a
S0→Int a south = snd a
H⁰-S⁰≅ℤ×ℤ : GroupEquiv (coHomGr 0 (S₊ 0)) (dirProd intGroup intGroup)
eq H⁰-S⁰≅ℤ×ℤ =
isoToEquiv (iso (sRec (isOfHLevelΣ 2 isSetInt λ _ → isSetInt) λ f → (f north) , (f south))
(λ a → ∣ S0→Int a ∣₂)
(λ _ → refl)
(sElim (λ _ → isOfHLevelPath 2 setTruncIsSet _ _)
λ f → cong ∣_∣₂ (funExt (λ {north → refl ; south → refl}))))
isHom H⁰-S⁰≅ℤ×ℤ =
sElim2 (λ _ _ → isOfHLevelPath 2 (isOfHLevelΣ 2 isSetInt (λ _ → isSetInt)) _ _)
λ a b i → addLemma (a north) (b north) i , addLemma (a south) (b south) i
-- --------------------------H¹(S¹) -----------------------------------
{-
In order to apply Mayer-Vietoris, we need the following lemma.
Given the following diagram
a ↦ (a , 0) ψ ϕ
A --> A × A -------> B ---> C
If ψ is an isomorphism and ϕ is surjective with ker ϕ ≡ {ψ (a , a) ∣ a ∈ A}, then C ≅ B
-}
diagonalIso : ∀ {ℓ ℓ' ℓ''} {A : Group {ℓ}} (B : Group {ℓ'}) {C : Group {ℓ''}}
(ψ : GroupEquiv (dirProd A A) B) (ϕ : GroupHom B C)
→ isSurjective _ _ ϕ
→ ((x : ⟨ B ⟩) → isInKer B C ϕ x
→ ∃[ y ∈ ⟨ A ⟩ ] x ≡ fst (eq ψ) (y , y))
→ ((x : ⟨ B ⟩) → (∃[ y ∈ ⟨ A ⟩ ] x ≡ fst (eq ψ) (y , y))
→ isInKer B C ϕ x)
→ GroupEquiv A C
diagonalIso {A = A} B {C = C} ψ ϕ issurj ker→diag diag→ker =
BijectionIsoToGroupEquiv
(bij-iso (compGroupHom fstProj (compGroupHom (grouphom (fst (eq ψ)) (isHom ψ)) ϕ))
(λ a inker
→ pRec (isSetCarrier A _ _)
(λ {(a' , id) → cong fst (sym (secEq (eq ψ) (a , 0g A)) ∙∙ cong (invEq (eq ψ)) id ∙∙ secEq (eq ψ) (a' , a'))
∙ cong snd (sym (secEq (eq ψ) (a' , a')) ∙∙ cong (invEq (eq ψ)) (sym id) ∙∙ secEq (eq ψ) (a , 0g A))})
(ker→diag _ inker))
λ c → pRec propTruncIsProp
(λ { (b , id) → ∣ (fst (ψ⁻ b) A.+ (A.- snd (ψ⁻ b))) -- (fst (ψ⁻ b) A.+ (A.- snd (ψ⁻ b)))
, (sym (Group.rid C _)
∙∙ cong ((fun ϕ) (equivFun (eq ψ) (fst (ψ⁻ b) A.+ (A.- snd (ψ⁻ b)) , 0g A)) C.+_)
(sym (diag→ker (equivFun (eq ψ) ((snd (ψ⁻ b)) , (snd (ψ⁻ b))))
∣ (snd (ψ⁻ b)) , refl ∣₁))
∙∙ sym ((isHom ϕ) _ _))
∙∙ cong (fun ϕ) (sym ((isHom ψ) _ _)
∙∙ cong (equivFun (eq ψ)) (ΣPathP (sym (Group.assoc A _ _ _)
∙∙ cong (fst (ψ⁻ b) A.+_) (Group.invl A _)
∙∙ Group.rid A _
, (Group.lid A _)))
∙∙ retEq (eq ψ) b)
∙∙ id ∣₁ })
(issurj c))
where
open Group
open IsGroup
open GroupHom
module A = Group A
module B = Group B
module C = Group C
module A×A = Group (dirProd A A)
module ψ = GroupEquiv ψ
module ϕ = GroupHom ϕ
ψ⁻ = fst (invEquiv (eq ψ))
fstProj : GroupHom A (dirProd A A)
fun fstProj a = a , 0g A
isHom fstProj g0 g1 i = (g0 A.+ g1) , Group.lid A (0g A) (~ i)
H¹-S¹≅ℤ : GroupEquiv intGroup (coHomGr 1 (S₊ 1))
H¹-S¹≅ℤ =
diagonalIso (coHomGr 0 (S₊ 0))
(invGroupEquiv H⁰-S⁰≅ℤ×ℤ)
(I.d 0)
(λ x → I.Ker-i⊂Im-d 0 x
(ΣPathP (isOfHLevelSuc 0 (isContrHⁿ-Unit 0) _ _
, isOfHLevelSuc 0 (isContrHⁿ-Unit 0) _ _)))
((sElim (λ _ → isOfHLevelΠ 2 λ _ → isOfHLevelSuc 1 propTruncIsProp)
(λ x inker
→ pRec propTruncIsProp
(λ {((f , g) , id') → helper x f g id' inker})
((I.Ker-d⊂Im-Δ 0 ∣ x ∣₂ inker)))))
((sElim (λ _ → isOfHLevelΠ 2 λ _ → isOfHLevelPath 2 setTruncIsSet _ _)
λ F surj
→ pRec (setTruncIsSet _ _)
(λ { (x , id) → I.Im-Δ⊂Ker-d 0 ∣ F ∣₂
∣ (∣ (λ _ → x) ∣₂ , ∣ (λ _ → 0) ∣₂) ,
(cong ∣_∣₂ (funExt (surjHelper x))) ∙ sym id ∣₁ })
surj) )
□ invGroupEquiv (coHomPushout≅coHomSn 0 1)
where
module I = MV Unit Unit (S₊ 0) (λ _ → tt) (λ _ → tt)
surjHelper : (x : Int) (x₁ : S₊ 0) → x +ₖ (-ₖ pos 0) ≡ S0→Int (x , x) x₁
surjHelper x north = cong (x +ₖ_) (-0ₖ) ∙ rUnitₖ x
surjHelper x south = cong (x +ₖ_) (-0ₖ) ∙ rUnitₖ x
helper : (F : S₊ 0 → Int) (f g : ∥ (Unit → Int) ∥₂)
(id : GroupHom.fun (I.Δ 0) (f , g) ≡ ∣ F ∣₂)
→ isInKer (coHomGr 0 (S₊ 0))
(coHomGr 1 (Pushout (λ _ → tt) (λ _ → tt)))
(I.d 0)
∣ F ∣₂
→ ∃[ x ∈ Int ] ∣ F ∣₂ ≡ equivFun (invEquiv (eq H⁰-S⁰≅ℤ×ℤ)) (x , x)
helper F =
sElim2 (λ _ _ → isOfHLevelΠ 2 λ _ → isOfHLevelΠ 2 λ _ → isOfHLevelSuc 1 propTruncIsProp)
λ f g id inker
→ pRec propTruncIsProp
(λ ((a , b) , id2)
→ sElim2 {B = λ f g → GroupHom.fun (I.Δ 0) (f , g) ≡ ∣ F ∣₂ → _ }
(λ _ _ → isOfHLevelΠ 2 λ _ → isOfHLevelSuc 1 propTruncIsProp)
(λ f g id → ∣ (helper2 f g .fst) , (sym id ∙ sym (helper2 f g .snd)) ∣₁)
a b id2)
(MV.Ker-d⊂Im-Δ _ _ (S₊ 0) (λ _ → tt) (λ _ → tt) 0 ∣ F ∣₂ inker)
where
helper2 : (f g : Unit → Int)
→ Σ[ x ∈ Int ] (equivFun (invEquiv (eq H⁰-S⁰≅ℤ×ℤ))) (x , x)
≡ GroupHom.fun (I.Δ 0) (∣ f ∣₂ , ∣ g ∣₂)
helper2 f g = (f _ +ₖ (-ₖ g _) ) , cong ∣_∣₂ (funExt (λ {north → refl ; south → refl}))
------------------------- H¹(S⁰) ≅ 0 -------------------------------
private
Hⁿ-S0≃Kₙ×Kₙ : (n : ℕ) → Iso (S₊ 0 → coHomK (suc n)) (coHomK (suc n) × coHomK (suc n))
Iso.fun (Hⁿ-S0≃Kₙ×Kₙ n) f = (f north) , (f south)
Iso.inv (Hⁿ-S0≃Kₙ×Kₙ n) (a , b) north = a
Iso.inv (Hⁿ-S0≃Kₙ×Kₙ n) (a , b) south = b
Iso.rightInv (Hⁿ-S0≃Kₙ×Kₙ n) a = refl
Iso.leftInv (Hⁿ-S0≃Kₙ×Kₙ n) b = funExt λ {north → refl ; south → refl}
isContrHⁿ-S0 : (n : ℕ) → isContr (coHom (suc n) (S₊ 0))
isContrHⁿ-S0 n =
transport (λ i → isContr ∥ isoToPath (Hⁿ-S0≃Kₙ×Kₙ n) (~ i) ∥₂)
(transport (λ i → isContr (isoToPath (setTruncOfProdIso {A = coHomK (suc n)} {B = coHomK (suc n)} ) (~ i)))
((∣ 0ₖ ∣₂ , ∣ 0ₖ ∣₂)
, prodElim (λ _ → isOfHLevelSuc 1 (isOfHLevelΣ 2 setTruncIsSet (λ _ → setTruncIsSet) _ _))
(elim2 (λ _ _ → isProp→isOfHLevelSuc (2 + n) (isOfHLevelΣ 2 setTruncIsSet (λ _ → setTruncIsSet) _ _))
(suspToPropRec2 north (λ _ _ → isOfHLevelΣ 2 setTruncIsSet (λ _ → setTruncIsSet) _ _) refl))))
H¹-S⁰≅0 : (n : ℕ) → GroupEquiv (coHomGr (suc n) (S₊ 0)) trivialGroup
H¹-S⁰≅0 n = contrGroup≅trivialGroup (isContrHⁿ-S0 n)
------------------------- H²(S¹) ≅ 0 -------------------------------
H²-S¹≅0 : GroupEquiv (coHomGr 2 (S₊ 1)) trivialGroup
H²-S¹≅0 =
coHomPushout≅coHomSn 0 2
□ (invGroupEquiv (vSES→GroupEquiv _ _ vSES-helper))
□ (H¹-S⁰≅0 0)
where
module I = MV Unit Unit (S₊ 0) (λ _ → tt) (λ _ → tt)
vSES-helper : vSES (coHomGr 1 (S₊ 0)) (coHomGr 2 (Pushout (λ _ → tt) (λ _ → tt)))
_ _
isTrivialLeft vSES-helper = isOfHLevelSuc 0 (isOfHLevelΣ 0 (isContrHⁿ-Unit 0) (λ _ → isContrHⁿ-Unit 0))
isTrivialRight vSES-helper = isOfHLevelSuc 0 (isOfHLevelΣ 0 (isContrHⁿ-Unit 1) (λ _ → isContrHⁿ-Unit 1))
left vSES-helper = I.Δ 1
right vSES-helper = I.i 2
vSES.ϕ vSES-helper = I.d 1
Ker-ϕ⊂Im-left vSES-helper = I.Ker-d⊂Im-Δ 1
Ker-right⊂Im-ϕ vSES-helper = sElim (λ _ → isOfHLevelΠ 2 λ _ → isOfHLevelSuc 1 propTruncIsProp) -- doesn't terminate without elimination
λ a → I.Ker-i⊂Im-d 1 ∣ a ∣₂
--------------- H¹(Sⁿ), n ≥ 1 --------------------------------------------
H¹-Sⁿ≅0 : (n : ℕ) → GroupEquiv (coHomGr 1 (S₊ (2 + n))) trivialGroup
H¹-Sⁿ≅0 n = coHomPushout≅coHomSn (1 + n) 1
□ BijectionIsoToGroupEquiv
(bij-iso (I.i 1)
helper
λ x → ∣ 0ₕ , isOfHLevelSuc 0 (isOfHLevelΣ 0 (isContrHⁿ-Unit zero) (λ _ → isContrHⁿ-Unit zero)) _ x ∣₁)
□ dirProdEquiv (Hⁿ-Unit≅0 zero) (Hⁿ-Unit≅0 zero)
□ lUnitGroupIso
where
module I = MV Unit Unit (S₊ (1 + n)) (λ _ → tt) (λ _ → tt)
surj-helper : (x : ⟨ coHomGr 0 (S₊ _) ⟩) → isInIm _ _ (I.Δ 0) x
surj-helper =
sElim (λ _ → isOfHLevelSuc 1 propTruncIsProp)
λ f → ∣ (∣ (λ _ → f north) ∣₂ , 0ₕ)
, (cong ∣_∣₂ (funExt (suspToPropRec north
(λ _ → isSetInt _ _)
(cong (f north +ₖ_) -0ₖ ∙ rUnitₖ (f north))))) ∣₁
helper : isInjective _ _ (I.i 1)
helper =
sElim (λ _ → isOfHLevelΠ 2 λ _ → isOfHLevelSuc 1 (setTruncIsSet _ _)) -- useless elimination speeds things up significantly
λ x inker → pRec (setTruncIsSet _ _)
(sigmaElim (λ _ → isOfHLevelPath 2 setTruncIsSet _ _)
λ a id → sym id ∙ I.Im-Δ⊂Ker-d 0 ∣ a ∣₂ (surj-helper _))
(I.Ker-i⊂Im-d 0 ∣ x ∣₂ inker)
--------- Direct proof of H¹(S¹) ≅ ℤ without Mayer-Vietoris -------
-- The strategy is to use the proof that ΩS¹ ≃ ℤ. Since we only have this for S¹ with the base/loop definition
-- we begin with some functions translating between H¹(S₊ 1) and ∥ S¹ → S¹ ∥₀. The latter type is easy to characterise,
-- by (S¹ → S¹) ≃ S¹ × ℤ (see Cubical.ZCohomology.Groups.Prelims). Truncating this leaves only ℤ, since S¹ is connected.
-- The translation mentioned above uses the basechange function. We use basechange-lemma (Cubical.ZCohomology.Groups.Prelims) to prove the basechange2⁻ preserves
-- path composition (in a more general sense than what is proved in basechange2⁻-morph)
-- We can now give the group equivalence. The first bit is just a big composition of our previously defined translations and is pretty uninteresting.
-- The harder step is proving that the equivalence is a morphism. This relies heavily on the fact that addition the cohomology groups essentially is defined using an
-- application of cong₂, which allows us to use basechange-lemma.
coHom1S1≃ℤ : GroupEquiv (coHomGr 1 (S₊ 1)) intGroup
eq coHom1S1≃ℤ = isoToEquiv theIso
where
F = Iso.fun S¹→S¹≡S¹×Int
F⁻ = Iso.inv S¹→S¹≡S¹×Int
G = Iso.fun S1→S1≡S¹→S¹
G⁻ = Iso.inv S1→S1≡S¹→S¹
theIso : Iso ⟨ coHomGr 1 (S₊ 1) ⟩ ⟨ intGroup ⟩
Iso.fun theIso = sRec isSetInt (λ f → snd (F (G f)))
Iso.inv theIso a = ∣ G⁻ (F⁻ (base , a)) ∣₂
Iso.rightInv theIso a =
(cong (snd ∘ F) (Iso.rightInv S1→S1≡S¹→S¹ (F⁻ (base , a)))
∙ cong snd (Iso.rightInv S¹→S¹≡S¹×Int (base , a)))
Iso.leftInv theIso =
sElim (λ _ → isOfHLevelPath 2 setTruncIsSet _ _)
λ f → cong ((sRec setTruncIsSet (λ x → ∣ G⁻ x ∣₂))
∘ sRec setTruncIsSet λ x → ∣ F⁻ (x , (snd (F (G f)))) ∣₂)
(Iso.inv PathIdTrunc₀Iso (isConnectedS¹ (fst (F (G f)))))
∙∙ cong (∣_∣₂ ∘ G⁻) (Iso.leftInv S¹→S¹≡S¹×Int (G f))
∙∙ cong ∣_∣₂ (Iso.leftInv S1→S1≡S¹→S¹ f)
isHom coHom1S1≃ℤ =
sElim2 (λ _ _ → isOfHLevelPath 2 isSetInt _ _)
λ f g → (λ i → winding (guy (ΩKn+1→Kn (Kn→ΩKn+1 1 (f (S¹→S1 base)) ∙ Kn→ΩKn+1 1 (g (S¹→S1 base))))
(λ i → pre-guy (ΩKn+1→Kn (Kn→ΩKn+1 1 (f (S¹→S1 (loop i))) ∙ Kn→ΩKn+1 1 (g (S¹→S1 (loop i))))))))
∙∙ cong winding (helper (f (S¹→S1 base)) (g (S¹→S1 base)) f g refl refl)
∙∙ winding-hom (guy (f north)
(λ i → pre-guy (f (S¹→S1 (loop i)))))
(guy (g north)
(λ i → pre-guy (g (S¹→S1 (loop i)))))
where
pre-guy = S¹map ∘ trMap S1→S¹
guy = basechange2⁻ ∘ pre-guy
helper : (x y : coHomK 1) (f g : S₊ 1 → coHomK 1)
→ (f (S¹→S1 base)) ≡ x
→ (g (S¹→S1 base)) ≡ y
→ (guy (ΩKn+1→Kn (Kn→ΩKn+1 1 (f (S¹→S1 base)) ∙ Kn→ΩKn+1 1 (g (S¹→S1 base))))
(λ i → S¹map (trMap S1→S¹ (ΩKn+1→Kn (Kn→ΩKn+1 1 (f (S¹→S1 (loop i))) ∙ Kn→ΩKn+1 1 (g (S¹→S1 (loop i))))))))
≡ (guy (f (S¹→S1 base))
(λ i → pre-guy (f (S¹→S1 (loop i)))))
∙ (guy (g (S¹→S1 base))
(λ i → pre-guy ((g (S¹→S1 (loop i))))))
helper = elim2 (λ _ _ → isGroupoidΠ4 λ _ _ _ _ → isOfHLevelPath 3 (isOfHLevelSuc 3 (isGroupoidS¹) base base) _ _)
(suspToPropRec2 {A = S₊ 0} north
(λ _ _ → isPropΠ4 λ _ _ _ _ → isGroupoidS¹ _ _ _ _)
λ f g reflf reflg →
(basechange-lemma
base
base
(λ x → S¹map (trMap S1→S¹ (ΩKn+1→Kn x)))
(λ x → Kn→ΩKn+1 1 (f (S¹→S1 x)))
((λ x → Kn→ΩKn+1 1 (g (S¹→S1 x))))
(cong (Kn→ΩKn+1 1) reflf ∙ Kn→ΩKn+10ₖ 1)
(cong (Kn→ΩKn+1 1) reflg ∙ Kn→ΩKn+10ₖ 1))
∙ λ j → guy (Iso.leftInv (Iso3-Kn-ΩKn+1 1) (f (S¹→S1 base)) j)
(λ i → pre-guy (Iso.leftInv (Iso3-Kn-ΩKn+1 1) (f (S¹→S1 (loop i))) j))
∙ guy (Iso.leftInv (Iso3-Kn-ΩKn+1 1) (g (S¹→S1 base)) j)
(λ i → pre-guy (Iso.leftInv (Iso3-Kn-ΩKn+1 1) (g (S¹→S1 (loop i))) j)))
---------------------------- Hⁿ(Sⁿ) ≅ ℤ , n ≥ 1 -------------------
Hⁿ-Sⁿ≅ℤ : (n : ℕ) → GroupEquiv intGroup (coHomGr (suc n) (S₊ (suc n)))
Hⁿ-Sⁿ≅ℤ zero = invGroupEquiv coHom1S1≃ℤ
Hⁿ-Sⁿ≅ℤ (suc n) =
Hⁿ-Sⁿ≅ℤ n
□ vSES→GroupEquiv _ _ theIso
□ invGroupEquiv (coHomPushout≅coHomSn (suc n) (suc (suc n)))
where
module I = MV Unit Unit (S₊ (suc n)) (λ _ → tt) (λ _ → tt)
theIso : vSES (coHomGr (suc n) (S₊ (suc n))) (coHomGr (suc (suc n))
(Pushout {A = S₊ (suc n)} (λ _ → tt) (λ _ → tt)))
_
_
isTrivialLeft theIso p q = ΣPathP (isOfHLevelSuc 0 (isContrHⁿ-Unit n) (fst p) (fst q)
, isOfHLevelSuc 0 (isContrHⁿ-Unit n) (snd p) (snd q))
isTrivialRight theIso p q = ΣPathP (isOfHLevelSuc 0 (isContrHⁿ-Unit (suc n)) (fst p) (fst q)
, isOfHLevelSuc 0 (isContrHⁿ-Unit (suc n)) (snd p) (snd q))
left theIso = I.Δ (suc n)
right theIso = I.i (2 + n)
vSES.ϕ theIso = I.d (suc n)
Ker-ϕ⊂Im-left theIso = I.Ker-d⊂Im-Δ (suc n)
Ker-right⊂Im-ϕ theIso = I.Ker-i⊂Im-d (suc n)
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open import Agda.Builtin.Nat
foo : Nat → Nat → Nat
foo 0 m = {!m!}
foo (suc n) m = {!!}
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{-# OPTIONS --without-K #-}
open import lib.Base
open import lib.NType
open import lib.Relation
open import lib.types.Bool
open import lib.types.Int
module lib.types.List where
infixr 80 _::_
data List {i} (A : Type i) : Type i where
nil : List A
_::_ : A → List A → List A
data HList {i} : List (Type i) → Type (lsucc i) where
nil : HList nil
_::_ : {A : Type i} {L : List (Type i)} → A → HList L → HList (A :: L)
hlist-curry-type : ∀ {i j} (L : List (Type i))
(B : HList L → Type (lmax i j)) → Type (lmax i j)
hlist-curry-type nil B = B nil
hlist-curry-type {j = j} (A :: L) B =
(x : A) → hlist-curry-type {j = j} L (λ xs → B (x :: xs))
hlist-curry : ∀ {i j} {L : List (Type i)} {B : HList L → Type (lmax i j)}
(f : Π (HList L) B) → hlist-curry-type {j = j} L B
hlist-curry {L = nil} f = f nil
hlist-curry {L = A :: _} f = λ x → hlist-curry (λ xs → f (x :: xs))
infixr 80 _++_
_++_ : ∀ {i} {A : Type i} → List A → List A → List A
nil ++ l = l
(x :: l₁) ++ l₂ = x :: (l₁ ++ l₂)
++-nil-r : ∀ {i} {A : Type i} (l : List A) → l ++ nil == l
++-nil-r nil = idp
++-nil-r (a :: l) = ap (a ::_) $ ++-nil-r l
-- [any] in Haskell
data Any {i j} {A : Type i} (P : A → Type j) : List A → Type (lmax i j) where
here : ∀ {a} {l} → P a → Any P (a :: l)
there : ∀ {a} {l} → Any P l → Any P (a :: l)
infix 80 _∈_
_∈_ : ∀ {i} {A : Type i} → A → List A → Type i
a ∈ l = Any (_== a) l
Any-dec : ∀ {i j} {A : Type i} (P : A → Type j) → (∀ a → Dec (P a)) → (∀ l → Dec (Any P l))
Any-dec P _ nil = inr λ{()}
Any-dec P dec (a :: l) with dec a
... | inl p = inl $ here p
... | inr p⊥ with Any-dec P dec l
... | inl ∃p = inl $ there ∃p
... | inr ∃p⊥ = inr λ{(here p) → p⊥ p; (there ∃p) → ∃p⊥ ∃p}
∈-dec : ∀ {i} {A : Type i} → has-dec-eq A → ∀ a l → Dec (a ∈ l)
∈-dec dec a l = Any-dec (_== a) (λ a' → dec a' a) l
Any-++-l : ∀ {i j} {A : Type i} (P : A → Type j)
→ ∀ l₁ l₂ → Any P l₁ → Any P (l₁ ++ l₂)
Any-++-l P _ _ (here p) = here p
Any-++-l P _ _ (there ∃p) = there (Any-++-l P _ _ ∃p)
∈-++-l : ∀ {i} {A : Type i} (a : A) → ∀ l₁ l₂ → a ∈ l₁ → a ∈ (l₁ ++ l₂)
∈-++-l a = Any-++-l (_== a)
Any-++-r : ∀ {i j} {A : Type i} (P : A → Type j)
→ ∀ l₁ l₂ → Any P l₂ → Any P (l₁ ++ l₂)
Any-++-r P nil _ ∃p = ∃p
Any-++-r P (a :: l) _ ∃p = there (Any-++-r P l _ ∃p)
∈-++-r : ∀ {i} {A : Type i} (a : A) → ∀ l₁ l₂ → a ∈ l₂ → a ∈ (l₁ ++ l₂)
∈-++-r a = Any-++-r (_== a)
Any-++ : ∀ {i j} {A : Type i} (P : A → Type j)
→ ∀ l₁ l₂ → Any P (l₁ ++ l₂) → (Any P l₁) ⊔ (Any P l₂)
Any-++ P nil l₂ ∃p = inr ∃p
Any-++ P (a :: l₁) l₂ (here p) = inl (here p)
Any-++ P (a :: l₁) l₂ (there ∃p) with Any-++ P l₁ l₂ ∃p
... | inl ∃p₁ = inl (there ∃p₁)
... | inr ∃p₂ = inr ∃p₂
∈-++ : ∀ {i} {A : Type i} (a : A) → ∀ l₁ l₂ → a ∈ (l₁ ++ l₂) → (a ∈ l₁) ⊔ (a ∈ l₂)
∈-++ a = Any-++ (_== a)
-- [map] in Haskell
map : ∀ {i j} {A : Type i} {B : Type j} → (A → B) → (List A → List B)
map f nil = nil
map f (a :: l) = f a :: map f l
-- [foldr] in Haskell
foldr : ∀ {i j} {A : Type i} {B : Type j} → (A → B → B) → B → List A → B
foldr f b nil = b
foldr f b (a :: l) = f a (foldr f b l)
-- [concat] in Haskell
concat : ∀ {i} {A : Type i} → List (List A) → List A
concat l = foldr _++_ nil l
ℤsum = foldr _ℤ+_ 0
-- [filter] in Haskell
-- Note that [Bool] is currently defined as a coproduct.
filter : ∀ {i j k} {A : Type i} {Keep : A → Type j} {Drop : A → Type k}
→ ((a : A) → Keep a ⊔ Drop a) → List A → List A
filter p nil = nil
filter p (a :: l) with p a
... | inl _ = a :: filter p l
... | inr _ = filter p l
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{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Setoids.Setoids
open import Rings.Definition
open import Rings.Orders.Partial.Definition
open import Sets.EquivalenceRelations
open import Sequences
open import Setoids.Orders.Partial.Definition
open import Functions.Definition
open import LogicalFormulae
open import Numbers.Naturals.Semiring
open import Groups.Definition
module Rings.Orders.Partial.Bounded {m n o : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {m} {o} A} {pOrder : SetoidPartialOrder S _<_} {R : Ring S _+_ _*_} (pRing : PartiallyOrderedRing R pOrder) where
open Group (Ring.additiveGroup R)
open import Groups.Lemmas (Ring.additiveGroup R)
open Setoid S
open Equivalence eq
open SetoidPartialOrder pOrder
BoundedAbove : Sequence A → Set (m ⊔ o)
BoundedAbove x = Sg A (λ K → (n : ℕ) → index x n < K)
BoundedBelow : Sequence A → Set (m ⊔ o)
BoundedBelow x = Sg A (λ K → (n : ℕ) → K < index x n)
Bounded : Sequence A → Set (m ⊔ o)
Bounded x = Sg A (λ K → (n : ℕ) → ((Group.inverse (Ring.additiveGroup R) K) < index x n) && (index x n < K))
boundNonzero : {s : Sequence A} → (b : Bounded s) → underlying b ∼ 0G → False
boundNonzero {s} (a , b) isEq with b 0
... | bad1 ,, bad2 = irreflexive (<Transitive bad1 (<WellDefined reflexive (transitive isEq (symmetric (transitive (inverseWellDefined isEq) invIdent))) bad2))
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Transitive closures
--
-- This module is DEPRECATED. Please use the
-- Relation.Binary.Construct.Closure.Transitive module directly.
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.Plus where
open import Relation.Binary.Construct.Closure.Transitive public
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module LiftGet where
open import Data.Unit using (⊤ ; tt)
open import Data.Nat using (ℕ ; suc)
open import Data.Vec using (Vec ; toList ; fromList) renaming ([] to []V ; _∷_ to _∷V_ ; map to mapV)
open import Data.List using (List ; [] ; _∷_ ; length ; replicate ; map)
open import Data.List.Properties using (length-map)
open import Data.Product using (∃ ; _,_ ; proj₁ ; proj₂)
open import Function using (_∘_ ; flip ; const)
open import Relation.Binary.Core using (_≡_)
open import Relation.Binary.PropositionalEquality using (_≗_ ; sym ; cong ; refl ; subst ; trans ; proof-irrelevance ; module ≡-Reasoning)
open import Relation.Binary.HeterogeneousEquality as H using (module ≅-Reasoning ; _≅_ ; ≅-to-≡ ; ≡-to-≅ ; ≡-subst-removable) renaming (refl to het-refl ; sym to het-sym ; cong to het-cong ; reflexive to het-reflexive)
import FreeTheorems
open import Generic using (length-replicate ; toList-fromList ; toList-subst)
open FreeTheorems.ListList using (get-type) renaming (free-theorem to free-theoremL ; Get to GetL ; module Get to GetL)
open FreeTheorems.VecVec using () renaming (get-type to getV-type ; Get to GetV ; module Get to GetV)
getVec-to-getList : {getlen : ℕ → ℕ} → (getV-type getlen) → get-type
getVec-to-getList get = toList ∘ get ∘ fromList
fromList∘map : {α β : Set} → (f : α → β) → (l : List α) → fromList (map f l) ≅ mapV f (fromList l)
fromList∘map f [] = het-refl
fromList∘map f (x ∷ xs) = H.cong₂ (λ n → _∷V_ {n = n} (f x)) (H.reflexive (length-map f xs)) (fromList∘map f xs)
toList∘map : {α β : Set} {n : ℕ} → (f : α → β) → (v : Vec α n) → toList (mapV f v) ≡ map f (toList v)
toList∘map f []V = refl
toList∘map f (x ∷V xs) = cong (_∷_ (f x)) (toList∘map f xs)
GetV-to-GetL : GetV → GetL
GetV-to-GetL getrecord = record { get = toList ∘ get ∘ fromList; free-theorem = ft }
where open GetV getrecord
open ≡-Reasoning
ft : {α β : Set} → (f : α → β) → (xs : List α) → toList (get (fromList (map f xs))) ≡ map f (toList (get (fromList xs)))
ft f xs = begin
toList (get (fromList (map f xs)))
≅⟨ H.cong₂ {B = Vec _} (λ n → toList ∘ get) (het-reflexive (length-map f xs)) (fromList∘map f xs) ⟩
toList (get (mapV f (fromList xs)))
≡⟨ cong toList (free-theorem f (fromList xs)) ⟩
toList (mapV f (get (fromList xs)))
≡⟨ toList∘map f (get (fromList xs)) ⟩
map f (toList (get (fromList xs))) ∎
getList-to-getlen : get-type → ℕ → ℕ
getList-to-getlen get = length ∘ get ∘ flip replicate tt
replicate-length : {A : Set} → (l : List A) → map (const tt) l ≡ replicate (length l) tt
replicate-length [] = refl
replicate-length (_ ∷ l) = cong (_∷_ tt) (replicate-length l)
getList-length : (get : get-type) → {B : Set} → (l : List B) → length (get l) ≡ getList-to-getlen get (length l)
getList-length get l = begin
length (get l)
≡⟨ sym (length-map (const tt) (get l)) ⟩
length (map (const tt) (get l))
≡⟨ cong length (sym (free-theoremL get (const tt) l)) ⟩
length (get (map (const tt) l))
≡⟨ cong (length ∘ get) (replicate-length l) ⟩
length (get (replicate (length l) tt)) ∎
where open ≡-Reasoning
length-toList : {A : Set} {n : ℕ} → (v : Vec A n) → length (toList v) ≡ n
length-toList []V = refl
length-toList (x ∷V xs) = cong suc (length-toList xs)
getList-to-getVec-length-property : (get : get-type) → {C : Set} → {m : ℕ} → (v : Vec C m) → length (get (toList v)) ≡ length (get (replicate m tt))
getList-to-getVec-length-property get {_} {m} v = begin
length (get (toList v))
≡⟨ getList-length get (toList v) ⟩
length (get (replicate (length (toList v)) tt))
≡⟨ cong (length ∘ get ∘ flip replicate tt) (length-toList v) ⟩
length (get (replicate m tt)) ∎
where open ≡-Reasoning
getList-to-getVec : get-type → ∃ λ (getlen : ℕ → ℕ) → (getV-type getlen)
getList-to-getVec get = getlen , get'
where getlen : ℕ → ℕ
getlen = getList-to-getlen get
get' : {C : Set} {m : ℕ} → Vec C m → Vec C (getlen m)
get' {C} v = subst (Vec C) (getList-to-getVec-length-property get v) (fromList (get (toList v)))
private
module GetV-Implementation (getrecord : GetL) where
open GetL getrecord
getlen = length ∘ get ∘ flip replicate tt
length-property : {C : Set} {m : ℕ} → (s : Vec C m) → length (get (toList s)) ≡ getlen m
length-property = getList-to-getVec-length-property get
getV : {C : Set} {m : ℕ} → Vec C m → Vec C (getlen m)
getV s = subst (Vec _) (length-property s) (fromList (get (toList s)))
ft : {α β : Set} (f : α → β) {n : ℕ} (v : Vec α n) → getV (mapV f v) ≡ mapV f (getV v)
ft f v = ≅-to-≡ (begin
subst (Vec _) (length-property (mapV f v)) (fromList (get (toList (mapV f v))))
≅⟨ ≡-subst-removable (Vec _) (length-property (mapV f v)) (fromList (get (toList (mapV f v)))) ⟩
fromList (get (toList (mapV f v)))
≅⟨ het-cong (fromList ∘ get) (het-reflexive (toList∘map f v)) ⟩
fromList (get (map f (toList v)))
≅⟨ het-cong fromList (het-reflexive (free-theorem f (toList v))) ⟩
fromList (map f (get (toList v)))
≅⟨ fromList∘map f (get (toList v)) ⟩
mapV f (fromList (get (toList v)))
≅⟨ H.cong₂ (λ n → mapV {n = n} f) (H.reflexive (length-property v)) (H.sym (≡-subst-removable (Vec _) (length-property v) (fromList (get (toList v))))) ⟩
mapV f (subst (Vec _) (length-property v) (fromList (get (toList v)))) ∎)
where open ≅-Reasoning
GetL-to-GetV : GetL → GetV
GetL-to-GetV getrecord = record { getlen = getlen; get = getV; free-theorem = ft }
where open GetV-Implementation getrecord
get-commut-1-≅ : (get : get-type) {A : Set} → (l : List A) → fromList (get l) ≅ proj₂ (getList-to-getVec get) (fromList l)
get-commut-1-≅ get l = begin
fromList (get l)
≅⟨ het-cong (fromList ∘ get) (≡-to-≅ (sym (toList-fromList l))) ⟩
fromList (get (toList (fromList l)))
≅⟨ het-sym (≡-subst-removable (Vec _) (getList-to-getVec-length-property get (fromList l)) (fromList (get (toList (fromList l))))) ⟩
subst (Vec _) (getList-to-getVec-length-property get (fromList l)) (fromList (get (toList (fromList l)))) ∎
where open ≅-Reasoning
get-commut-1 : (get : get-type) {A : Set} → (l : List A) → fromList (get l) ≡ subst (Vec A) (sym (getList-length get l)) (proj₂ (getList-to-getVec get) (fromList l))
get-commut-1 get {A} l = ≅-to-≡ (begin
fromList (get l)
≅⟨ get-commut-1-≅ get l ⟩
proj₂ (getList-to-getVec get) (fromList l)
≅⟨ het-sym (≡-subst-removable (Vec _) (sym (getList-length get l)) (proj₂ (getList-to-getVec get) (fromList l))) ⟩
subst (Vec _) (sym (getList-length get l)) (proj₂ (getList-to-getVec get) (fromList l)) ∎)
where open ≅-Reasoning
get-trafo-1 : (get : get-type) → {B : Set} → getVec-to-getList (proj₂ (getList-to-getVec get)) {B} ≗ get {B}
get-trafo-1 get {B} l = begin
getVec-to-getList (proj₂ (getList-to-getVec get)) l
≡⟨ refl ⟩
toList (proj₂ (getList-to-getVec get) (fromList l))
≡⟨ refl ⟩
toList (subst (Vec B) (getList-to-getVec-length-property get (fromList l)) (fromList (get (toList (fromList l)))))
≡⟨ toList-subst (fromList (get (toList (fromList l)))) (getList-to-getVec-length-property get (fromList l)) ⟩
toList (fromList (get (toList (fromList l))))
≡⟨ toList-fromList (get (toList (fromList l))) ⟩
get (toList (fromList l))
≡⟨ cong get (toList-fromList l) ⟩
get l ∎
where open ≡-Reasoning
GetLVL-identity : (G : GetL) → {A : Set} → GetL.get (GetV-to-GetL (GetL-to-GetV G)) ≗ GetL.get G {A}
GetLVL-identity G = get-trafo-1 (GetL.get G)
vec-len : {A : Set} {n : ℕ} → Vec A n → ℕ
vec-len {_} {n} _ = n
fromList-toList : {A : Set} {n : ℕ} → (v : Vec A n) → fromList (toList v) ≅ v
fromList-toList []V = het-refl
fromList-toList (x ∷V xs) = H.cong₂ (λ n → _∷V_ {n = n} x) (het-reflexive (length-toList xs)) (fromList-toList xs)
get-commut-2 : {getlen : ℕ → ℕ} → (get : getV-type getlen) → {B : Set} {n : ℕ} → (toList ∘ get {B} {n}) ≗ (getVec-to-getList get) ∘ toList
get-commut-2 get {B} v = sym (≅-to-≡ (H.cong₂ (λ n → toList ∘ get {n = n}) (H.reflexive (length-toList v)) (fromList-toList v)))
get-trafo-2-getlen : {getlen : ℕ → ℕ} → (get : getV-type getlen) → proj₁ (getList-to-getVec (getVec-to-getList get)) ≗ getlen
get-trafo-2-getlen {getlen} get n = begin
proj₁ (getList-to-getVec (getVec-to-getList get)) n
≡⟨ refl ⟩
length (toList (get (fromList (replicate n tt))))
≡⟨ length-toList (get (fromList (replicate n tt))) ⟩
vec-len (get (fromList (replicate n tt)))
≡⟨ cong getlen (length-replicate n) ⟩
getlen n ∎
where open ≡-Reasoning
get-trafo-2-get-≅ : {getlen : ℕ → ℕ} → (get : getV-type getlen) → {B : Set} {n : ℕ} → (v : Vec B n) → proj₂ (getList-to-getVec (getVec-to-getList get)) v ≅ get v
get-trafo-2-get-≅ {getlen} get v = begin
subst (Vec _) (getList-to-getVec-length-property (getVec-to-getList get) v) (fromList (toList (get (fromList (toList v)))))
≅⟨ ≡-subst-removable (Vec _) (getList-to-getVec-length-property (getVec-to-getList get) v) (fromList (toList (get (fromList (toList v))))) ⟩
fromList (toList (get (fromList (toList v))))
≅⟨ fromList-toList (get (fromList (toList v))) ⟩
get (fromList (toList v))
≅⟨ H.cong₂ (λ n → get {n = n}) (H.reflexive (length-toList v)) (fromList-toList v) ⟩
get v ∎
where open ≅-Reasoning
get-trafo-2-get : {getlen : ℕ → ℕ} → (get : getV-type getlen) → {B : Set} {n : ℕ} → proj₂ (getList-to-getVec (getVec-to-getList get)) ≗ subst (Vec B) (sym (get-trafo-2-getlen get n)) ∘ get
get-trafo-2-get get v = ≅-to-≡ (begin
proj₂ (getList-to-getVec (getVec-to-getList get)) v
≅⟨ get-trafo-2-get-≅ get v ⟩
get v
≅⟨ het-sym (≡-subst-removable (Vec _) (sym (get-trafo-2-getlen get (vec-len v))) (get v)) ⟩
subst (Vec _) (sym (get-trafo-2-getlen get (vec-len v))) (get v) ∎)
where open ≅-Reasoning
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module TestQuote where
{- test of reflection, implementing a trivial prover. -}
open import Common.Reflection
open import Common.Prelude
open import Common.Level
_==_ : Term → Term → Bool
def x [] == def y [] = primQNameEquality x y
_ == _ = false
data Thm : Set where
triv : Thm
`Thm = def (quote Thm) []
⟦_⟧ : Term → Set
⟦ goal ⟧ with goal == `Thm
... | true = Thm
... | false = ⊥
Hyp : Term → Set → Set
Hyp goal A with goal == `Thm
... | true = ⊤
... | false = A
solve : (goal : Term) → Hyp goal ⟦ goal ⟧ → ⟦ goal ⟧
solve goal h with goal == `Thm
... | true = triv
... | false = h
test₁ : Thm
test₁ = quoteGoal t in solve t _
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{-# OPTIONS --without-K --safe #-}
module Cham.Context where
open import Cham.Name
open import Cham.Label
open import Data.Product
infix 5 _⊢_
data Context : Set where
∅ : Context
_⊢_ : Context → Label → Context
_,_ : Context → Context → Context
rename : (Name → Label) → Context → Context
rename ϕ ∅ = ∅
rename ϕ (Γ ⊢ (N ⁺)) = rename ϕ Γ ⊢ ϕ N
rename ϕ (Γ ⊢ (N ⁻)) = rename ϕ Γ ⊢ ϕ N
rename ϕ (Γ₁ , Γ₂) = rename ϕ Γ₁ , rename ϕ Γ₂
permeate : Context → Context → Context
permeate ∅ Γ₂ = Γ₂
permeate (Γ₁ ⊢ N) Γ₂ = permeate Γ₁ (Γ₂ ⊢ N)
permeate (Γ₁ , Γ₂) Γ₃ = Γ₁ , Γ₂ , Γ₃
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{-# OPTIONS --safe #-}
module Cubical.Algebra.Group.Exact where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Function
open import Cubical.Data.Unit
open import Cubical.Algebra.Group.Base
open import Cubical.Algebra.Group.Morphisms
open import Cubical.Algebra.Group.MorphismProperties
open import Cubical.Algebra.Group.GroupPath
open import Cubical.Algebra.Group.Instances.Unit
open import Cubical.HITs.PropositionalTruncation as PT
-- TODO : Define exact sequences
-- (perhaps short, finite, ℕ-indexed and ℤ-indexed)
SES→isEquiv : ∀ {ℓ ℓ'} {L R : Group ℓ-zero}
→ {G : Group ℓ} {H : Group ℓ'}
→ UnitGroup₀ ≡ L
→ UnitGroup₀ ≡ R
→ (lhom : GroupHom L G) (midhom : GroupHom G H) (rhom : GroupHom H R)
→ ((x : _) → isInKer midhom x → isInIm lhom x)
→ ((x : _) → isInKer rhom x → isInIm midhom x)
→ isEquiv (fst midhom)
SES→isEquiv {R = R} {G = G} {H = H} =
J (λ L _ → UnitGroup₀ ≡ R →
(lhom : GroupHom L G) (midhom : GroupHom G H)
(rhom : GroupHom H R) →
((x : fst G) → isInKer midhom x → isInIm lhom x) →
((x : fst H) → isInKer rhom x → isInIm midhom x) →
isEquiv (fst midhom))
((J (λ R _ → (lhom : GroupHom UnitGroup₀ G) (midhom : GroupHom G H)
(rhom : GroupHom H R) →
((x : fst G) → isInKer midhom x → isInIm lhom x) →
((x : _) → isInKer rhom x → isInIm midhom x) →
isEquiv (fst midhom))
main))
where
main : (lhom : GroupHom UnitGroup₀ G) (midhom : GroupHom G H)
(rhom : GroupHom H UnitGroup₀) →
((x : fst G) → isInKer midhom x → isInIm lhom x) →
((x : fst H) → isInKer rhom x → isInIm midhom x) →
isEquiv (fst midhom)
main lhom midhom rhom lexact rexact =
BijectionIsoToGroupEquiv {G = G} {H = H}
bijIso' .fst .snd
where
bijIso' : BijectionIso G H
BijectionIso.fun bijIso' = midhom
BijectionIso.inj bijIso' x inker =
PT.rec (GroupStr.is-set (snd G) _ _)
(λ s → sym (snd s) ∙ IsGroupHom.pres1 (snd lhom))
(lexact _ inker)
BijectionIso.surj bijIso' x = rexact x refl
-- exact sequence of 4 groups. Useful for the proof of π₄S³
record Exact4 {ℓ ℓ' ℓ'' ℓ''' : Level} (G : Group ℓ)
(H : Group ℓ') (L : Group ℓ'') (R : Group ℓ''')
(G→H : GroupHom G H) (H→L : GroupHom H L) (L→R : GroupHom L R)
: Type (ℓ-max ℓ (ℓ-max ℓ' (ℓ-max ℓ'' ℓ'''))) where
field
ImG→H⊂KerH→L : (x : fst H) → isInIm G→H x → isInKer H→L x
KerH→L⊂ImG→H : (x : fst H) → isInKer H→L x → isInIm G→H x
ImH→L⊂KerL→R : (x : fst L) → isInIm H→L x → isInKer L→R x
KerL→R⊂ImH→L : (x : fst L) → isInKer L→R x → isInIm H→L x
open Exact4
extendExact4Surjective : {ℓ ℓ' ℓ'' ℓ''' ℓ'''' : Level}
(G : Group ℓ) (H : Group ℓ') (L : Group ℓ'') (R : Group ℓ''') (S : Group ℓ'''')
(G→H : GroupHom G H) (H→L : GroupHom H L) (L→R : GroupHom L R) (R→S : GroupHom R S)
→ isSurjective G→H
→ Exact4 H L R S H→L L→R R→S
→ Exact4 G L R S (compGroupHom G→H H→L) L→R R→S
ImG→H⊂KerH→L (extendExact4Surjective G H L R S G→H H→L L→R R→S surj ex) x =
PT.rec (GroupStr.is-set (snd R) _ _)
(uncurry λ g → J (λ x _ → isInKer L→R x)
(ImG→H⊂KerH→L ex (fst H→L (fst G→H g))
∣ (fst G→H g) , refl ∣₁))
KerH→L⊂ImG→H (extendExact4Surjective G H L R S G→H H→L L→R R→S surj ex) x ker =
PT.rec squash₁
(uncurry λ y → J (λ x _ → isInIm (compGroupHom G→H H→L) x)
(PT.map (uncurry
(λ y → J (λ y _ → Σ[ g ∈ fst G ] fst H→L (fst G→H g) ≡ H→L .fst y)
(y , refl))) (surj y)))
(KerH→L⊂ImG→H ex x ker)
ImH→L⊂KerL→R (extendExact4Surjective G H L R S G→H H→L L→R R→S surj ex) =
ImH→L⊂KerL→R ex
KerL→R⊂ImH→L (extendExact4Surjective G H L R S G→H H→L L→R R→S surj ex) =
KerL→R⊂ImH→L ex
-- Useful lemma in the proof of π₄S³≅ℤ
transportExact4 : {ℓ ℓ' ℓ'' : Level}
{G G₂ : Group ℓ} {H H₂ : Group ℓ'} {L L₂ : Group ℓ''} {R : Group₀}
(G≡G₂ : G ≡ G₂) (H≡H₂ : H ≡ H₂) (L≡L₂ : L ≡ L₂)
→ UnitGroup₀ ≡ R
→ (G→H : GroupHom G H) (G₂→H₂ : GroupHom G₂ H₂)
(H→L : GroupHom H L) (H₂→L₂ : GroupHom H₂ L₂)
(L→R : GroupHom L R)
→ Exact4 G H L R G→H H→L L→R
→ PathP (λ i → GroupHom (G≡G₂ i) (H≡H₂ i)) G→H G₂→H₂
→ PathP (λ i → GroupHom (H≡H₂ i) (L≡L₂ i)) H→L H₂→L₂
→ Exact4 G₂ H₂ L₂ UnitGroup₀ G₂→H₂ H₂→L₂ (→UnitHom L₂)
transportExact4 {G = G} {G₂ = G₂} {H = H} {H₂ = H₂} {L = L} {L₂ = L₂} {R = R} =
J4 (λ G₂ H₂ L₂ R G≡G₂ H≡H₂ L≡L₂ Unit≡R
→ (G→H : GroupHom G H) (G₂→H₂ : GroupHom G₂ H₂)
(H→L : GroupHom H L) (H₂→L₂ : GroupHom H₂ L₂)
(L→R : GroupHom L R)
→ Exact4 G H L R G→H H→L L→R
→ PathP (λ i → GroupHom (G≡G₂ i) (H≡H₂ i)) G→H G₂→H₂
→ PathP (λ i → GroupHom (H≡H₂ i) (L≡L₂ i)) H→L H₂→L₂
→ Exact4 G₂ H₂ L₂ UnitGroup₀ G₂→H₂ H₂→L₂ (→UnitHom L₂))
(λ G→H G₂→H₂ H→L H₂→L₂ L→R ex pp1 pp2
→ J4 (λ G₂→H₂ H₂→L₂ (x : Unit) (y : Unit)
pp1 pp2 (_ : tt ≡ x) (_ : tt ≡ x)
→ Exact4 G H L UnitGroup₀ G₂→H₂ H₂→L₂ (→UnitHom L))
ex G₂→H₂ H₂→L₂ tt tt pp1 pp2 refl refl )
G₂ H₂ L₂ R
where
J4 : ∀ {ℓ ℓ₂ ℓ₃ ℓ₄ ℓ'} {A : Type ℓ}
{A₂ : Type ℓ₂} {A₃ : Type ℓ₃} {A₄ : Type ℓ₄}
{x : A} {x₂ : A₂} {x₃ : A₃} {x₄ : A₄}
(B : (y : A) (z : A₂) (w : A₃) (u : A₄)
→ x ≡ y → x₂ ≡ z → x₃ ≡ w → x₄ ≡ u → Type ℓ')
→ B x x₂ x₃ x₄ refl refl refl refl
→ (y : A) (z : A₂) (w : A₃) (u : A₄)
(p : x ≡ y) (q : x₂ ≡ z) (r : x₃ ≡ w) (s : x₄ ≡ u)
→ B y z w u p q r s
J4 {x = x} {x₂ = x₂} {x₃ = x₃} {x₄ = x₄} B b y z w u =
J (λ y p → (q : x₂ ≡ z) (r : x₃ ≡ w) (s : x₄ ≡ u) →
B y z w u p q r s)
(J (λ z q → (r : x₃ ≡ w) (s : x₄ ≡ u) → B x z w u refl q r s)
(J (λ w r → (s : x₄ ≡ u) → B x x₂ w u refl refl r s)
(J (λ u s → B x x₂ x₃ u refl refl refl s) b)))
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module _ where
open import Agda.Primitive
open import Agda.Builtin.Equality
open import Agda.Builtin.Nat
variable
ℓ : Level
A B C : Set ℓ
infixr 1 _×_ _,_
record _×_ {ℓ₁ ℓ₂} (A : Set ℓ₁) (B : Set ℓ₂) : Set (ℓ₁ ⊔ ℓ₂) where
constructor _,_
field
fst : A
snd : B
module _ (x : A) (y : B) where
f : C → A × B × C
f z = x , y , z
check : (x : B) (y : C) (z : A) → f x y z ≡ (x , y , z)
check x y z = refl
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{-# OPTIONS --warning=error --safe --without-K #-}
-- This file contains everything that can be compiled in --safe mode.
open import Numbers.Naturals.Naturals
open import Numbers.Naturals.Division
open import Numbers.BinaryNaturals.Definition
open import Numbers.BinaryNaturals.Multiplication
open import Numbers.BinaryNaturals.Order
open import Numbers.BinaryNaturals.Subtraction
open import Numbers.Primes.PrimeNumbers
open import Numbers.Primes.IntegerFactorisation
open import Numbers.Rationals.Lemmas
open import Numbers.Modulo.Group
open import Numbers.Integers.Integers
open import Numbers.Integers.RingStructure.EuclideanDomain
open import Numbers.Integers.RingStructure.Archimedean
open import Numbers.ClassicalReals.Examples
open import Numbers.ClassicalReals.RealField.Lemmas
open import Lists.Lists
open import Lists.Filter.AllTrue
open import Groups.Groups
open import Groups.Abelian.Lemmas
open import Groups.DirectSum.Definition
open import Groups.QuotientGroup.Definition
open import Groups.QuotientGroup.Lemmas
open import Groups.FiniteGroups.Definition
open import Groups.Homomorphisms.Lemmas
open import Groups.Homomorphisms.Lemmas2
open import Groups.Homomorphisms.Examples
open import Groups.Isomorphisms.Lemmas
open import Groups.FinitePermutations
open import Groups.Lemmas
open import Groups.FirstIsomorphismTheorem
open import Groups.SymmetricGroups.Definition
open import Groups.Actions.Stabiliser
open import Groups.Actions.Orbit
open import Groups.SymmetricGroups.Lemmas
open import Groups.ActionIsSymmetry
open import Groups.Cyclic.Definition
open import Groups.Cyclic.DefinitionLemmas
open import Groups.Polynomials.Examples
open import Groups.Cosets
open import Groups.Subgroups.Normal.Examples
open import Groups.Examples.ExampleSheet1
open import Groups.LectureNotes.Lecture1
open import Fields.Fields
open import Fields.Orders.Partial.Definition
open import Fields.Orders.Total.Definition
open import Fields.Orders.Partial.Archimedean
open import Fields.Orders.LeastUpperBounds.Examples
open import Fields.Orders.Lemmas
open import Fields.FieldOfFractions.Field
open import Fields.FieldOfFractions.Lemmas
open import Fields.FieldOfFractions.Order
--open import Fields.FieldOfFractions.Archimedean
open import Rings.Definition
open import Rings.Lemmas
open import Rings.Orders.Partial.Definition
open import Rings.Orders.Total.Lemmas
open import Rings.Orders.Partial.Lemmas
open import Rings.IntegralDomains.Definition
open import Rings.DirectSum
open import Rings.Polynomial.Ring
open import Rings.Polynomial.Evaluation
open import Rings.Ideals.Definition
open import Rings.Isomorphisms.Definition
open import Rings.Quotients.Definition
open import Rings.Cosets
open import Rings.EuclideanDomains.Definition
open import Rings.EuclideanDomains.Examples
open import Rings.Homomorphisms.Image
open import Rings.Homomorphisms.Kernel
open import Rings.Ideals.FirstIsomorphismTheorem
open import Rings.Ideals.Lemmas
open import Rings.Ideals.Prime.Definition
open import Rings.Ideals.Prime.Lemmas
open import Rings.Ideals.Principal.Definition
open import Rings.IntegralDomains.Examples
open import Rings.PrincipalIdealDomains.Lemmas
open import Rings.Subrings.Definition
open import Rings.Ideals.Maximal.Lemmas
open import Rings.Primes.Lemmas
open import Rings.Irreducibles.Definition
open import Rings.Divisible.Lemmas
open import Rings.Associates.Lemmas
open import Rings.InitialRing
open import Rings.Homomorphisms.Lemmas
open import Rings.UniqueFactorisationDomains.Definition
open import Rings.Examples.Examples
open import Setoids.Setoids
open import Setoids.DirectSum
open import Setoids.Lists
open import Setoids.Orders.Total.Lemmas
open import Setoids.Functions.Definition
open import Setoids.Functions.Extension
open import Setoids.Algebra.Lemmas
open import Setoids.Intersection.Lemmas
open import Setoids.Union.Lemmas
open import Setoids.Cardinality.Infinite.Lemmas
open import Setoids.Cardinality.Finite.Definition
open import Sets.Cardinality.Infinite.Lemmas
open import Sets.Cardinality.Countable.Lemmas
open import Sets.Cardinality.Finite.Lemmas
open import Sets.Cardinality
open import Sets.FinSet.Lemmas
open import Decidable.Sets
open import Decidable.Reduction
open import Decidable.Relations
open import Vectors
open import Vectors.VectorSpace
open import KeyValue.KeyValue
open import KeyValue.LinearStore.Definition
open import Maybe
open import Orders.Total.Lemmas
open import Orders.WellFounded.Induction
open import ClassicalLogic.ClassicalFive
open import Monoids.Definition
open import Semirings.Definition
open import Semirings.Solver
open import Categories.Definition
open import Categories.Functor.Definition
open import Categories.Functor.Lemmas
open import Categories.Dual.Definition
open import Graphs.PathGraph
open import Graphs.CycleGraph
open import Graphs.UnionGraph
open import Graphs.CompleteGraph
open import Graphs.Colouring
open import Graphs.Bipartite
open import Graphs.Complement
open import Graphs.InducedSubgraph
open import LectureNotes.MetAndTop.Chapter1
open import LectureNotes.NumbersAndSets.Lecture1
open import Modules.Examples
open import Modules.PolynomialModule
open import Modules.Lemmas
open import Modules.DirectSum
open import Computability.LambdaCalculus.ChurchNumeral
open import ProjectEuler.Problem1
module Everything.Safe where
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module Issue2756 where
module M where
postulate A : Set
module M′ = M
B : Set
B = M′.A
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------------------------------------------------------------------------
-- A variant of Nat.Wrapper.Cubical, defined using --erased-cubical
------------------------------------------------------------------------
{-# OPTIONS --erased-cubical --safe #-}
open import Equality.Path as P
open import Prelude hiding (zero; suc; _+_)
open import Bijection equality-with-J using (_↔_)
module Nat.Wrapper.Cubical.Erased
-- The underlying representation of natural numbers.
(Nat′ : Type)
-- A bijection between this representation and the unary natural
-- numbers.
(Nat′↔ℕ : Nat′ ↔ ℕ)
where
open import Equality.Path.Univalence
open import Logical-equivalence using (_⇔_)
import Equivalence equality-with-J as Eq
open import Equivalence.Erased.Cubical equality-with-paths as EEq
using (_≃ᴱ_)
import Equivalence.Erased.Contractible-preimages.Cubical
equality-with-paths
as ECP
open import Erased.Cubical equality-with-paths
open import Function-universe equality-with-J as F hiding (_∘_)
open import H-level equality-with-J
open import H-level.Closure equality-with-J
open import H-level.Truncation.Propositional.Erased equality-with-paths
as Trunc
import Nat equality-with-J as Nat
import Univalence-axiom equality-with-J as U
open import Nat.Wrapper equality-with-J Nat′ Nat′↔ℕ as NW
open NW.[]-cong instance-of-[]-cong-axiomatisation
private
variable
A : Type
m n : A
------------------------------------------------------------------------
-- Could Nat have been defined using the propositional truncation
-- operator (with an erased higher constructor) instead of Erased?
-- Could Nat have been defined using ∥_∥ᴱ instead of Erased? Let us
-- try.
-- Given a truncated natural number we can kind of apply Nat-[_] to
-- it, because Nat-[_] is a family of contractible types. (The code
-- uses erased univalence.)
Nat-[]′ : ∥ ℕ ∥ᴱ → ∃ λ (A : Type) → Contractible A
Nat-[]′ = Trunc.rec λ where
.truncation-is-propositionʳ →
U.∃-H-level-H-level-1+ ext univ 0
.∣∣ʳ n →
Nat-[ n ]
, propositional⇒inhabited⇒contractible
Nat-[]-propositional
( _↔_.from Nat′↔ℕ n
, [ _↔_.right-inverse-of Nat′↔ℕ n ]
)
Nat-[_]′ : ∥ ℕ ∥ᴱ → Type
Nat-[ n ]′ = proj₁ (Nat-[]′ n)
-- Thus we can form a variant of Nat.
Nat-with-∥∥ᴱ : Type
Nat-with-∥∥ᴱ = Σ ∥ ℕ ∥ᴱ Nat-[_]′
-- However, this variant is equivalent (with erased proofs) to the
-- unit type.
Nat-with-∥∥ᴱ≃ᴱ⊤ : Nat-with-∥∥ᴱ ≃ᴱ ⊤
Nat-with-∥∥ᴱ≃ᴱ⊤ =
_⇔_.to EEq.Contractibleᴱ⇔≃ᴱ⊤ $
ECP.Contractibleᴱ-Σ
(ECP.inhabited→Is-proposition→Contractibleᴱ
∣ 0 ∣ truncation-is-proposition)
(ECP.Contractible→Contractibleᴱ ∘ proj₂ ∘ Nat-[]′)
-- And thus it is not isomorphic to the natural numbers.
¬-Nat-with-∥∥ᴱ↔ℕ : ¬ (Nat-with-∥∥ᴱ ↔ ℕ)
¬-Nat-with-∥∥ᴱ↔ℕ =
Stable-¬
[ Nat-with-∥∥ᴱ ↔ ℕ ↝⟨ F._∘ inverse (from-equivalence (EEq.≃ᴱ→≃ Nat-with-∥∥ᴱ≃ᴱ⊤)) ⟩
⊤ ↔ ℕ ↝⟨ (λ hyp → _↔_.injective (inverse hyp) refl) ⟩
0 ≡ 1 ↝⟨ Nat.0≢+ ⟩□
⊥ □
]
------------------------------------------------------------------------
-- Addition of "wrapped" numbers is commutative and associative
module _ (o : Operations) where
open Operations-for-Nat o
open Operations-for-Nat-correct o
private
-- A lemma used several times below.
from[to+to]≡+ :
∀ m →
_↔_.from Nat↔ℕ (_↔_.to Nat↔ℕ m Prelude.+ _↔_.to Nat↔ℕ n) ≡ m + n
from[to+to]≡+ {n = n} m =
_↔_.from Nat↔ℕ (_↔_.to Nat↔ℕ m Prelude.+ _↔_.to Nat↔ℕ n) ≡⟨ cong (_↔_.from Nat↔ℕ) $ sym $ to-ℕ-+ m n ⟩
_↔_.from Nat↔ℕ (_↔_.to Nat↔ℕ (m + n)) ≡⟨ _↔_.left-inverse-of Nat↔ℕ _ ⟩∎
m + n ∎
-- First two "traditional" proofs.
-- Addition is commutative.
+-comm-traditional : ∀ m {n} → m + n ≡ n + m
+-comm-traditional m {n = n} =
m + n ≡⟨ sym $ from[to+to]≡+ m ⟩
_↔_.from Nat↔ℕ (_↔_.to Nat↔ℕ m Prelude.+ _↔_.to Nat↔ℕ n) ≡⟨ cong (_↔_.from Nat↔ℕ) $ Nat.+-comm (_↔_.to Nat↔ℕ m) ⟩
_↔_.from Nat↔ℕ (_↔_.to Nat↔ℕ n Prelude.+ _↔_.to Nat↔ℕ m) ≡⟨ from[to+to]≡+ n ⟩∎
n + m ∎
-- Addition is associative.
+-assoc-traditional : ∀ m {n o} → m + (n + o) ≡ (m + n) + o
+-assoc-traditional m {n = n} {o = o} =
m + (n + o) ≡⟨ cong (m +_) $ sym $ from[to+to]≡+ n ⟩
m + (_↔_.from Nat↔ℕ (_↔_.to Nat↔ℕ n Prelude.+ _↔_.to Nat↔ℕ o)) ≡⟨ sym $ from[to+to]≡+ m ⟩
_↔_.from Nat↔ℕ (_↔_.to Nat↔ℕ m
Prelude.+
_↔_.to Nat↔ℕ (_↔_.from Nat↔ℕ
(_↔_.to Nat↔ℕ n Prelude.+ _↔_.to Nat↔ℕ o))) ≡⟨ cong (λ n → _↔_.from Nat↔ℕ (_↔_.to Nat↔ℕ m Prelude.+ n)) $
_↔_.right-inverse-of Nat↔ℕ _ ⟩
_↔_.from Nat↔ℕ
(_↔_.to Nat↔ℕ m
Prelude.+
(_↔_.to Nat↔ℕ n Prelude.+ _↔_.to Nat↔ℕ o)) ≡⟨ cong (_↔_.from Nat↔ℕ) $ Nat.+-assoc (_↔_.to Nat↔ℕ m) ⟩
_↔_.from Nat↔ℕ
((_↔_.to Nat↔ℕ m Prelude.+ _↔_.to Nat↔ℕ n)
Prelude.+
_↔_.to Nat↔ℕ o) ≡⟨ cong (λ n → _↔_.from Nat↔ℕ (n Prelude.+ _↔_.to Nat↔ℕ o)) $ sym $
_↔_.right-inverse-of Nat↔ℕ _ ⟩
_↔_.from Nat↔ℕ
(_↔_.to Nat↔ℕ (_↔_.from Nat↔ℕ
(_↔_.to Nat↔ℕ m Prelude.+ _↔_.to Nat↔ℕ n))
Prelude.+
_↔_.to Nat↔ℕ o) ≡⟨ from[to+to]≡+ (_↔_.from Nat↔ℕ (_↔_.to Nat↔ℕ m Prelude.+ _↔_.to Nat↔ℕ n)) ⟩
(_↔_.from Nat↔ℕ (_↔_.to Nat↔ℕ m Prelude.+ _↔_.to Nat↔ℕ n)) + o ≡⟨ cong (_+ o) $ from[to+to]≡+ {n = n} m ⟩∎
(m + n) + o ∎
-- The following proofs are instead based on a technique used by
-- Vezzosi, Mörtberg and Abel in "Cubical Agda: A Dependently Typed
-- Programming Language with Univalence and Higher Inductive Types".
-- The type of unary natural numbers is equal to the type of wrapped
-- natural numbers (in erased contexts).
@0 ℕ≡Nat : ℕ ≡ Nat
ℕ≡Nat = sym (≃⇒≡ (Eq.↔⇒≃ Nat↔ℕ))
-- Addition of unary natural numbers is, in a certain sense, equal
-- to addition of wrapped natural numbers (in erased contexts).
@0 +≡+ : P.[ (λ i → ℕ≡Nat i → ℕ≡Nat i → ℕ≡Nat i) ] Prelude._+_ ≡ _+_
+≡+ =
Prelude._+_ ≡⟨ (λ i → transport
(λ j → ℕ≡Nat (min i j) → ℕ≡Nat (min i j) → ℕ≡Nat (min i j))
(- i) Prelude._+_) ⟩h
transport (λ i → ℕ≡Nat i → ℕ≡Nat i → ℕ≡Nat i) 0̲ Prelude._+_ ≡⟨⟩
(λ m n → _↔_.from Nat↔ℕ (_↔_.to Nat↔ℕ m Prelude.+ _↔_.to Nat↔ℕ n)) ≡⟨ (⟨ext⟩ λ m → ⟨ext⟩ λ _ → from[to+to]≡+ m) ⟩∎
_+_ ∎
-- Addition is commutative (in erased contexts).
@0 +-comm-cubical : ∀ m {n} → m + n ≡ n + m
+-comm-cubical =
transport
(λ i → (m {n} : ℕ≡Nat i) → +≡+ i m n ≡ +≡+ i n m)
0̲
Nat.+-comm
-- Addition is associative (in erased contexts).
@0 +-assoc-cubical : ∀ m {n o} → m + (n + o) ≡ (m + n) + o
+-assoc-cubical =
transport
(λ i → (m {n o} : ℕ≡Nat i) →
+≡+ i m (+≡+ i n o) ≡ +≡+ i (+≡+ i m n) o)
0̲
Nat.+-assoc
-- This proof technique seems to scale better than the one used
-- above, at least for examples of the kind used here. However, when
-- --erased-cubical is used it only works in erased contexts.
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module reverse-++-distrib where
import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl; sym; cong)
open Eq.≡-Reasoning
open import lists using (List; []; _∷_; [_]; _++_; ++-assoc; ++-identityʳ; reverse)
-- 結合したリストの逆順は、逆順にしたリストの逆順の結合と等しいことの証明
reverse-++-distrib : ∀ {A : Set} → (xs ys : List A)
→ reverse (xs ++ ys) ≡ reverse ys ++ reverse xs
reverse-++-distrib [] ys =
begin
reverse ([] ++ ys)
≡⟨⟩
reverse ys
≡⟨ sym (++-identityʳ (reverse ys)) ⟩
reverse ys ++ []
≡⟨⟩
reverse ys ++ reverse []
∎
reverse-++-distrib (x ∷ xs) ys =
begin
reverse (x ∷ xs ++ ys)
≡⟨⟩
reverse (xs ++ ys) ++ [ x ]
≡⟨ cong (_++ [ x ]) (reverse-++-distrib xs ys) ⟩
(reverse ys ++ reverse xs) ++ [ x ]
≡⟨ ++-assoc (reverse ys) (reverse xs) [ x ] ⟩
reverse ys ++ (reverse xs ++ [ x ])
≡⟨⟩
reverse ys ++ reverse (x ∷ xs)
∎
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{-# OPTIONS --safe #-}
open import Definition.Typed.EqualityRelation
module Definition.LogicalRelation.Substitution.ProofIrrelevance {{eqrel : EqRelSet}} where
open EqRelSet {{...}}
open import Definition.Untyped as U hiding (wk)
open import Definition.Untyped.Properties using (wkSingleSubstId)
open import Definition.Typed
open import Definition.Typed.Weakening
open import Definition.Typed.Properties
open import Definition.LogicalRelation
open import Definition.LogicalRelation.Properties
open import Definition.LogicalRelation.Substitution
open import Tools.Product
open import Tools.Unit
open import Tools.Empty
open import Tools.Nat
import Tools.PropositionalEquality as PE
import Data.Nat as Nat
~-quasirefl : ∀ {Γ n n′ A r} → Γ ⊢ n ~ n′ ∷ A ^ r → Γ ⊢ n ~ n ∷ A ^ r
~-quasirefl p = ~-trans p (~-sym p)
≅-quasirefl : ∀ {Γ n n′ A r} → Γ ⊢ n ≅ n′ ∷ A ^ r → Γ ⊢ n ≅ n ∷ A ^ r
≅-quasirefl p = ≅ₜ-trans p (≅ₜ-sym p)
proof-irrelevanceRel : ∀ {Γ A t u l l′} ([A] : Γ ⊩⟨ l ⟩ A ^ [ % , l′ ])
→ Γ ⊩⟨ l ⟩ t ∷ A ^ [ % , l′ ] / [A]
→ Γ ⊩⟨ l ⟩ u ∷ A ^ [ % , l′ ] / [A]
→ Γ ⊩⟨ l ⟩ t ≡ u ∷ A ^ [ % , l′ ] / [A]
proof-irrelevanceRel (Emptyᵣ x)
(Emptyₜ (ne ⊢t))
(Emptyₜ (ne ⊢t₁)) = Emptyₜ₌ (ne ⊢t ⊢t₁)
proof-irrelevanceRel (ne x)
(neₜ ⊢t)
(neₜ ⊢t₁) =
neₜ₌ ⊢t ⊢t₁
proof-irrelevanceRel {Γ} {l = l} (Πᵣ′ rF lF lG _ _ F G D ⊢F ⊢G A≡A [F] [G] G-ext) [f] [f₁] =
[f] , [f₁]
proof-irrelevanceRel {Γ} {l = l} (∃ᵣ′ F G D ⊢F ⊢G A≡A [F] [G] G-ext) [f] [f₁] =
[f] , [f₁]
proof-irrelevanceRel (emb emb< [A]) [t] [u] = proof-irrelevanceRel [A] [t] [u]
proof-irrelevanceRel (emb ∞< [A]) [t] [u] = proof-irrelevanceRel [A] [t] [u]
proof-irrelevanceᵛ : ∀ {Γ A t u l l′} ([Γ] : ⊩ᵛ Γ) ([A] : Γ ⊩ᵛ⟨ l ⟩ A ^ [ % , l′ ] / [Γ])
→ Γ ⊩ᵛ⟨ l ⟩ t ∷ A ^ [ % , l′ ] / [Γ] / [A]
→ Γ ⊩ᵛ⟨ l ⟩ u ∷ A ^ [ % , l′ ] / [Γ] / [A]
→ Γ ⊩ᵛ⟨ l ⟩ t ≡ u ∷ A ^ [ % , l′ ] / [Γ] / [A]
proof-irrelevanceᵛ [Γ] [A] [t] [u] {σ = σ} ⊢Δ [σ] =
proof-irrelevanceRel (proj₁ ([A] ⊢Δ [σ])) (proj₁ ([t] ⊢Δ [σ])) (proj₁ ([u] ⊢Δ [σ]))
validityIrr : ∀ {l A t Γ l'} ([Γ] : ⊩ᵛ Γ) ([A] : Γ ⊩ᵛ⟨ l ⟩ A ^ [ % , l' ] / [Γ])
(⊢t : ∀ {Δ σ} (⊢Δ : ⊢ Δ) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ) → Δ ⊢ subst σ t ∷ subst σ A ^ [ % , l' ]) → Γ ⊩ᵛ⟨ l ⟩ t ∷ A ^ [ % , l' ] / [Γ] / [A]
validityIrr [Γ] [A] ⊢t {Δ} {σ} ⊢Δ [σ] =
let [Aσ] = proj₁ ([A] ⊢Δ [σ])
[tσ] = logRelIrr [Aσ] (⊢t ⊢Δ [σ])
-- [tσ] = proj₁ ([t] ⊢Δ [σ])
in logRelIrr [Aσ] (⊢t ⊢Δ [σ]) ,
λ [σ′] [σ≡σ′] → let [Aσ′] = proj₁ ([A] ⊢Δ [σ′])
[tσ′] = logRelIrr [Aσ′] (⊢t ⊢Δ [σ′])
[Aσ≡σ′] = proj₂ ([A] ⊢Δ [σ]) [σ′] [σ≡σ′]
in logRelIrrEq [Aσ] (⊢t ⊢Δ [σ]) (escapeTerm [Aσ] (convTerm₂ [Aσ] [Aσ′] [Aσ≡σ′] [tσ′]))
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{-# OPTIONS --without-K --safe #-}
module Tools.List where
infixr 30 _∷_
data List (A : Set) : Set where
[] : List A
_∷_ : A → List A → List A
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-- Andreas, 2014-07-02 wondering about the ETA pragma (legacy?)
open import Common.Equality
data Prod (A B : Set) : Set where
pair : A → B → Prod A B
{-# ETA Prod #-}
-- The ETA pragma does not exist anymore.
fst : {A B : Set} → Prod A B → A
fst (pair a b) = a
snd : {A B : Set} → Prod A B → B
snd (pair a b) = b
-- Just an illusion...
test : {A B : Set} (x : Prod A B) → x ≡ pair (fst x) (snd x)
test x = refl
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{-# POLARITY F #-}
{-# POLARITY G #-}
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----------------------------------------------------------------------
-- Copyright: 2013, Jan Stolarek, Lodz University of Technology --
-- --
-- License: See LICENSE file in root of the repo --
-- Repo address: https://github.com/jstolarek/dep-typed-wbl-heaps --
-- --
-- Definition of datatypes that represent ordering and functions --
-- that operate on them. These datatypes are based on ideas --
-- introduced in "Why Dependent Types Matter". --
----------------------------------------------------------------------
module Basics.Ordering where
open import Basics.Nat hiding (_≥_)
-- The ≥ type is a proof of greater-equal relation between two natural
-- numbers. It proves that: a) any number natural is greater or equal
-- to zero and b) any two natural numbers are in ≥ relation if their
-- predecessors are also in that relation.
data _≥_ : Nat → Nat → Set where
ge0 : { y : Nat} → y ≥ zero
geS : {x y : Nat} → x ≥ y → suc x ≥ suc y
infixl 4 _≥_
-- Order datatype tells whether two numbers are in ≥ relation or
-- not. In that sense it is an equivalent of Bool datatype. Unlike
-- Bool however, Order supplies a proof of WHY the numbers are (or are
-- not) in the ≥ relation.
data Order : Nat → Nat → Set where
ge : {x : Nat} {y : Nat} → x ≥ y → Order x y
le : {x : Nat} {y : Nat} → y ≥ x → Order x y
-- order function takes two natural numbers and compares them,
-- returning the result of comparison together with a proof of the
-- result (result and its proof are encoded by Order datatype).
order : (a : Nat) → (b : Nat) → Order a b
order a zero = ge ge0
order zero (suc b) = le ge0
order (suc a) (suc b) with order a b
order (suc a) (suc b) | ge a≥b = ge (geS a≥b)
order (suc a) (suc b) | le b≥a = le (geS b≥a)
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module Numeral.Natural.Oper.Comparisons where
import Lvl
open import Data.Boolean
import Data.Boolean.Operators
open Data.Boolean.Operators.Programming
open import Numeral.Natural
open import Numeral.Sign
ℕbool : Bool → ℕ
ℕbool = if_then 1 else 0
-- Compare
_⋚?_ : ℕ → ℕ → (−|0|+)
𝟎 ⋚? 𝟎 = 𝟎
𝟎 ⋚? 𝐒(b) = ➖
𝐒(a) ⋚? 𝟎 = ➕
𝐒(a) ⋚? 𝐒(b) = a ⋚? b
-- Equality check
_≡?_ : ℕ → ℕ → Bool
a ≡? b = elim₃ 𝐹 𝑇 𝐹 (a ⋚? b)
{-# BUILTIN NATEQUALS _≡?_ #-}
-- Non-equality check
_≢?_ : ℕ → ℕ → Bool
x ≢? y = !(x ≡? y)
-- Positivity check
positive? : ℕ → Bool
positive? (𝟎) = 𝐹
positive? (𝐒(_)) = 𝑇
-- Zero check
zero? : ℕ → Bool
zero? n = !(positive? n)
-- Lesser-than check
_<?_ : ℕ → ℕ → Bool
_ <? 𝟎 = 𝐹
𝟎 <? 𝐒(_) = 𝑇
𝐒(x) <? 𝐒(y) = (x <? y)
{-# BUILTIN NATLESS _<?_ #-}
-- Lesser-than or equals check
_≤?_ : ℕ → ℕ → Bool
x ≤? y = x <? 𝐒(y)
-- Greater-than check
_>?_ : ℕ → ℕ → Bool
x >? y = y <? x
-- Greater-than or equals check
_≥?_ : ℕ → ℕ → Bool
x ≥? y = y ≤? x
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties satisfied by preorders
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Relation.Binary
module Relation.Binary.Properties.Preorder
{p₁ p₂ p₃} (P : Preorder p₁ p₂ p₃) where
open import Function
open import Data.Product as Prod
open Relation.Binary.Preorder P
-- The inverse relation is also a preorder.
invIsPreorder : IsPreorder _≈_ (flip _∼_)
invIsPreorder = record
{ isEquivalence = isEquivalence
; reflexive = reflexive ∘ Eq.sym
; trans = flip trans
}
invPreorder : Preorder p₁ p₂ p₃
invPreorder = record { isPreorder = invIsPreorder }
------------------------------------------------------------------------
-- For every preorder there is an induced equivalence
InducedEquivalence : Setoid _ _
InducedEquivalence = record
{ _≈_ = λ x y → x ∼ y × y ∼ x
; isEquivalence = record
{ refl = (refl , refl)
; sym = swap
; trans = Prod.zip trans (flip trans)
}
}
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module Data.List.Functions.Multi where
import Lvl
open import Data
open import Data.List
open import Data.List.Functions hiding (separate)
open import Data.Option
open import Data.Tuple
open import Data.Tuple.Raise
import Data.Tuple.Raiseᵣ.Functions as Raise
open import Numeral.Finite
open import Numeral.Natural
open import Numeral.Natural.Oper.Modulo
open import Type
private variable ℓ : Lvl.Level
private variable T : Type{ℓ}
-- TODO: Also called zip in other languages
-- module _ where
-- open import Data.Tuple.Raise as Tuple using (_^_)
-- open import Function.Multi as Multi using (_⇉_)
--map₊ : ∀{n}{As : Type{ℓ} ^ n}{B} → (As ⇉ B) → (Tuple.map List(As) ⇉ List(B))
-- map₊ {n = 𝟎} = const ∅
-- map₊ {n = 𝐒(𝟎)} = map
-- map₊ {n = 𝐒(𝐒(n))} _ ∅ = Multi.const ∅
-- map₊ {n = 𝐒(𝐒(n))} f (x ⊰ l) = {!!}
{-
separate : (n : ℕ) → List(T) → (List(T) ^ n)
separate(0) _ = <>
separate(1) l = l
separate(𝐒(𝐒(n))) l = {!!}
-- Raise.repeat (𝐒(𝐒(n))) ∅
-- (x ⊰ l) = Raise.map₂ {!skip!} {!!} (∅ , separate(𝐒(n)) l)
-}
-- Separates a list by a function assigning which list index it should lie in.
-- Example:
-- separateBy(_mod 3) [0,1,2,3,4,5,6,7,8,9] = [[0,3,6,9] , [1,4,7] , [2,5,8]]
separateBy : ∀{n} → (T → 𝕟(n)) → List(T) → (List(T) ^ n)
separateBy f ∅ = Raise.repeat _ ∅
separateBy f (x ⊰ l) = Raise.updateAt (f(x)) (x ⊰_) (separateBy f l)
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{-# OPTIONS --without-K --rewriting #-}
open import HoTT
open import homotopy.SuspSmash
module homotopy.SuspSmashComm where
module Σ∧Σ-PathElim
{i} {j} {k} {A : Type i} {B : Type j} {C : Type k}
(f g : ⊙Susp A ∧ ⊙Susp B → C)
(n-n : f (smin north north) == g (smin north north))
(n-s : f (smin north south) == g (smin north south))
(s-n : f (smin south north) == g (smin south north))
(s-s : f (smin south south) == g (smin south south))
(n-m : ∀ b → Square n-n (ap (λ sb → f (smin north sb)) (merid b))
(ap (λ sb → g (smin north sb)) (merid b)) n-s)
(s-m : ∀ b → Square s-n (ap (λ sb → f (smin south sb)) (merid b))
(ap (λ sb → g (smin south sb)) (merid b)) s-s)
(m-n : ∀ a → Square n-n (ap (λ sa → f (smin sa north)) (merid a))
(ap (λ sa → g (smin sa north)) (merid a)) s-n)
(m-s : ∀ a → Square n-s (ap (λ sa → f (smin sa south)) (merid a))
(ap (λ sa → g (smin sa south)) (merid a)) s-s)
(m-m : ∀ a b →
Cube (m-n a)
(m-s a)
(n-m b)
(natural-square (λ sb → ap (λ sa → f (smin sa sb)) (merid a)) (merid b))
(natural-square (λ sb → ap (λ sa → g (smin sa sb)) (merid a)) (merid b))
(s-m b))
(basel : f smbasel == g smbasel)
(baser : f smbaser == g smbaser)
(gluel-north : Square n-n (ap f (smgluel north)) (ap g (smgluel north)) basel)
(gluel-south : Square s-n (ap f (smgluel south)) (ap g (smgluel south)) basel)
(gluel-merid : ∀ a →
Cube gluel-north
gluel-south
(m-n a)
(natural-square (λ sa → ap f (smgluel sa)) (merid a))
(natural-square (λ sa → ap g (smgluel sa)) (merid a))
(natural-square (λ sa → basel) (merid a)))
(gluer-north : Square n-n (ap f (smgluer north)) (ap g (smgluer north)) baser)
(gluer-south : Square n-s (ap f (smgluer south)) (ap g (smgluer south)) baser)
(gluer-merid : ∀ b →
Cube gluer-north
gluer-south
(n-m b)
(natural-square (λ a → ap f (smgluer a)) (merid b))
(natural-square (λ a → ap g (smgluer a)) (merid b))
(natural-square (λ a → baser) (merid b))
)
where
private
module M =
SuspDoublePathElim
(λ sa sb → f (smin sa sb))
(λ sa sb → g (smin sa sb))
n-n n-s s-n s-s n-m s-m m-n m-s m-m
module P =
SmashElim
{P = λ sa∧sb → f sa∧sb == g sa∧sb}
M.p
basel
baser
(λ sa → ↓-='-from-square $
Susp-elim
{P = λ sa → Square (M.p sa north) (ap f (smgluel sa)) (ap g (smgluel sa)) basel}
gluel-north
gluel-south
(λ a → cube-to-↓-square $
cube-shift-back (! (M.merid-north-square-β a)) $
gluel-merid a)
sa)
(λ sb → ↓-='-from-square $
Susp-elim
{P = λ sb → Square (M.p north sb) (ap f (smgluer sb)) (ap g (smgluer sb)) baser}
gluer-north
gluer-south
(λ b → cube-to-↓-square $
cube-shift-back (! (M.north-merid-square-β b)) $
gluer-merid b)
sb)
open P using (f) public
module _ {i j} (X : Ptd i) (Y : Ptd j) where
private
f : ⊙Susp (de⊙ X) ∧ ⊙Susp (de⊙ Y) → Susp (Susp (X ∧ Y))
f = Susp-fmap (∧Σ-out X Y) ∘ Σ∧-out X (⊙Susp (de⊙ Y))
g : ⊙Susp (de⊙ X) ∧ ⊙Susp (de⊙ Y) → Susp (Susp (X ∧ Y))
g = Susp-flip ∘ Susp-fmap (Σ∧-out X Y) ∘ ∧Σ-out (⊙Susp (de⊙ X)) Y
f-merid-x : ∀ x sy →
ap (λ sx → f (smin sx sy)) (merid x)
=-=
merid (Susp-fmap (smin x) sy)
f-merid-x x sy =
ap (Susp-fmap (∧Σ-out X Y) ∘ Susp-fmap (λ x' → smin x' sy)) (merid x)
=⟪ ap-∘ (Susp-fmap (∧Σ-out X Y)) (Susp-fmap (λ x' → smin x' sy)) (merid x) ⟫
ap (Susp-fmap (∧Σ-out X Y)) (ap (Susp-fmap (λ x' → smin x' sy)) (merid x))
=⟪ ap (ap (Susp-fmap (∧Σ-out X Y)))
(SuspFmap.merid-β (λ x' → smin x' sy) x) ⟫
ap (Susp-fmap (∧Σ-out X Y)) (merid (smin x sy))
=⟪ SuspFmap.merid-β (∧Σ-out X Y) (smin x sy) ⟫
merid (Susp-fmap (smin x) sy) ∎∎
g-merid-y : ∀ sx y →
ap (g ∘ smin sx) (merid y)
=-=
! (merid (Susp-fmap (λ x' → smin x' y) sx))
g-merid-y sx y =
ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y) ∘ Susp-fmap (smin sx)) (merid y)
=⟪ ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin sx)) (merid y) ⟫
ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (ap (Susp-fmap (smin sx)) (merid y))
=⟪ ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(SuspFmap.merid-β (smin sx) y) ⟫
ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (merid (smin sx y))
=⟪ ap-∘ Susp-flip (Susp-fmap (Σ∧-out X Y)) (merid (smin sx y)) ⟫
ap Susp-flip (ap (Susp-fmap (Σ∧-out X Y)) (merid (smin sx y)))
=⟪ ap (ap Susp-flip) (SuspFmap.merid-β (Σ∧-out X Y) (smin sx y)) ⟫
ap Susp-flip (merid (Susp-fmap (λ x' → smin x' y) sx))
=⟪ SuspFlip.merid-β (Susp-fmap (λ x' → smin x' y) sx) ⟫
! (merid (Susp-fmap (λ x' → smin x' y) sx)) ∎∎
n-n : f (smin north north) == g (smin north north)
n-n = merid north
n-s : f (smin north south) == g (smin north south)
n-s = idp
s-n : f (smin south north) == g (smin south north)
s-n = idp
s-s : f (smin south south) == g (smin south south)
s-s = ! (merid south)
n-m : ∀ y → Square n-n (ap (f ∘ smin north) (merid y))
(ap (g ∘ smin north) (merid y)) n-s
n-m y = ap-cst north (merid y) ∙v⊡ (lb-square (merid north) ⊡v∙ ! (↯ (g-merid-y north y)))
s-m : ∀ y → Square s-n (ap (f ∘ smin south) (merid y))
(ap (g ∘ smin south) (merid y)) s-s
s-m y = ap-cst south (merid y) ∙v⊡ br-square (! (merid south)) ⊡v∙ ! (↯ (g-merid-y south y))
m-n : ∀ x → Square n-n (ap (λ sx → f (smin sx north)) (merid x))
(ap (λ sx → g (smin sx north)) (merid x)) s-n
m-n x = ↯ (f-merid-x x north) ∙v⊡ (lt-square (merid north) ⊡v∙ ! (ap-cst south (merid x)))
m-s : ∀ x → Square n-s (ap (λ sx → f (smin sx south)) (merid x))
(ap (λ sx → g (smin sx south)) (merid x)) s-s
m-s x = ↯ (f-merid-x x south) ∙v⊡ (tr-square (merid south) ⊡v∙ ! (ap-cst north (merid x)))
m-m : ∀ x y →
Cube (m-n x)
(m-s x)
(n-m y)
(natural-square (λ sy → ap (λ sx → f (smin sx sy)) (merid x)) (merid y))
(natural-square (λ sy → ap (λ sx → g (smin sx sy)) (merid x)) (merid y))
(s-m y)
m-m x y =
custom-cube-∙v⊡
(↯ (f-merid-x x north))
(↯ (f-merid-x x south))
(ap-cst north (merid y))
(ap-cst south (merid y)) $
cube-shift-top (! top-path) $
custom-cube-⊡v∙
(! (ap-cst south (merid x)))
(ap-cst north (merid x))
(! (↯ (g-merid-y north y)))
(↯ (g-merid-y south y)) $
cube-shift-bot (! bot-path) $
custom-cube (merid north) (merid south) (ap merid (merid (smin x y)))
where
custom-cube-∙v⊡ : ∀ {i} {A : Type i} {a₀₀₀ a₀₁₀ a₁₀₀ a₁₁₀ a₀₀₁ a₀₁₁ a₁₀₁ a₁₁₁ : A}
{p₀₋₀ : a₀₀₀ == a₀₁₀}
{p₋₀₀ p₋₀₀' : a₀₀₀ == a₁₀₀} (q₋₀₀ : p₋₀₀ == p₋₀₀')
{p₋₁₀ : a₀₁₀ == a₁₁₀}
{p₁₋₀ : a₁₀₀ == a₁₁₀}
{sq₋₋₀ : Square p₀₋₀ p₋₀₀' p₋₁₀ p₁₋₀} -- left
{p₀₋₁ : a₀₀₁ == a₀₁₁}
{p₋₀₁ p₋₀₁' : a₀₀₁ == a₁₀₁} (q₋₀₁ : p₋₀₁ == p₋₀₁')
{p₋₁₁ : a₀₁₁ == a₁₁₁}
{p₁₋₁ : a₁₀₁ == a₁₁₁}
{sq₋₋₁ : Square p₀₋₁ p₋₀₁' p₋₁₁ p₁₋₁} -- right
{p₀₀₋ p₀₀₋' : a₀₀₀ == a₀₀₁} (q₀₀₋ : p₀₀₋ == p₀₀₋')
{p₀₁₋ : a₀₁₀ == a₀₁₁}
{p₁₀₋ p₁₀₋' : a₁₀₀ == a₁₀₁} (q₁₀₋ : p₁₀₋ == p₁₀₋')
{p₁₁₋ : a₁₁₀ == a₁₁₁}
{sq₀₋₋ : Square p₀₋₀ p₀₀₋' p₀₁₋ p₀₋₁} -- back
{sq₋₀₋ : Square p₋₀₀ p₀₀₋ p₁₀₋ p₋₀₁} -- top
{sq₋₁₋ : Square p₋₁₀ p₀₁₋ p₁₁₋ p₋₁₁} -- bottom
{sq₁₋₋ : Square p₁₋₀ p₁₀₋' p₁₁₋ p₁₋₁} -- front
→ Cube sq₋₋₀ sq₋₋₁ sq₀₋₋ (! q₀₀₋ ∙v⊡ (! q₋₀₀ ∙h⊡ sq₋₀₋ ⊡h∙ q₋₀₁) ⊡v∙ q₁₀₋) sq₋₁₋ sq₁₋₋
→ Cube (q₋₀₀ ∙v⊡ sq₋₋₀) (q₋₀₁ ∙v⊡ sq₋₋₁) (q₀₀₋ ∙v⊡ sq₀₋₋) sq₋₀₋ sq₋₁₋ (q₁₀₋ ∙v⊡ sq₁₋₋)
custom-cube-∙v⊡ idp idp idp idp c = c
custom-cube-⊡v∙ : ∀ {i} {A : Type i} {a₀₀₀ a₀₁₀ a₁₀₀ a₁₁₀ a₀₀₁ a₀₁₁ a₁₀₁ a₁₁₁ : A}
{p₀₋₀ : a₀₀₀ == a₀₁₀}
{p₋₀₀ : a₀₀₀ == a₁₀₀}
{p₋₁₀ p₋₁₀' : a₀₁₀ == a₁₁₀} (q₋₁₀ : p₋₁₀ == p₋₁₀')
{p₁₋₀ : a₁₀₀ == a₁₁₀}
{sq₋₋₀ : Square p₀₋₀ p₋₀₀ p₋₁₀ p₁₋₀} -- left
{p₀₋₁ : a₀₀₁ == a₀₁₁}
{p₋₀₁ : a₀₀₁ == a₁₀₁}
{p₋₁₁ p₋₁₁' : a₀₁₁ == a₁₁₁} (q₋₁₁ : p₋₁₁' == p₋₁₁)
{p₁₋₁ : a₁₀₁ == a₁₁₁}
{sq₋₋₁ : Square p₀₋₁ p₋₀₁ p₋₁₁ p₁₋₁} -- right
{p₀₀₋ : a₀₀₀ == a₀₀₁}
{p₀₁₋ p₀₁₋' : a₀₁₀ == a₀₁₁} (q₀₁₋ : p₀₁₋ == p₀₁₋')
{p₁₀₋ : a₁₀₀ == a₁₀₁}
{p₁₁₋ p₁₁₋' : a₁₁₀ == a₁₁₁} (q₁₁₋ : p₁₁₋' == p₁₁₋)
{sq₀₋₋ : Square p₀₋₀ p₀₀₋ p₀₁₋ p₀₋₁} -- back
{sq₋₀₋ : Square p₋₀₀ p₀₀₋ p₁₀₋ p₋₀₁} -- top
{sq₋₁₋ : Square p₋₁₀' p₀₁₋' p₁₁₋' p₋₁₁'} -- bottom
{sq₁₋₋ : Square p₁₋₀ p₁₀₋ p₁₁₋ p₁₋₁} -- front
→ Cube sq₋₋₀ sq₋₋₁ sq₀₋₋ sq₋₀₋ (square-symmetry (q₋₁₀ ∙v⊡ (q₀₁₋ ∙h⊡ square-symmetry sq₋₁₋ ⊡h∙ q₁₁₋) ⊡v∙ q₋₁₁)) sq₁₋₋
→ Cube (sq₋₋₀ ⊡v∙ q₋₁₀) (sq₋₋₁ ⊡v∙ ! q₋₁₁) (sq₀₋₋ ⊡v∙ q₀₁₋) sq₋₀₋ sq₋₁₋ (sq₁₋₋ ⊡v∙ ! q₁₁₋)
custom-cube-⊡v∙ idp idp idp idp {sq₋₁₋ = sq₋₁₋} c =
cube-shift-bot (square-sym-inv sq₋₁₋) c
custom-cube : ∀ {i} {S : Type i} {n s : S}
(p q : n == s) (r : p == q)
→ Cube (lt-square p)
(tr-square q)
(lb-square p)
(horiz-degen-square r)
(vert-degen-square (ap ! r))
(br-square (! q))
custom-cube p@idp [email protected] r@idp = idc
top-path :
! (ap-cst north (merid y)) ∙v⊡
(! (↯ (f-merid-x x north)) ∙h⊡
natural-square (λ sy → ap (λ sx → f (smin sx sy)) (merid x)) (merid y) ⊡h∙
↯ (f-merid-x x south)) ⊡v∙
ap-cst south (merid y)
==
horiz-degen-square (ap merid (merid (smin x y)))
top-path =
! (ap-cst north (merid y)) ∙v⊡
(! (↯ (f-merid-x x north)) ∙h⊡
natural-square (λ sy → ap (λ sx → f (smin sx sy)) (merid x)) (merid y) ⊡h∙
↯ (f-merid-x x south)) ⊡v∙
ap-cst south (merid y)
=⟨ ap (λ u → ! (ap-cst north (merid y)) ∙v⊡ u ⊡v∙ ap-cst south (merid y)) $
natural-square-path
(λ sy → merid (Susp-fmap (smin x) sy))
(λ sy → ap (λ sx → f (smin sx sy)) (merid x))
(λ sy → ↯ (f-merid-x x sy))
(merid y) ⟩
! (ap-cst north (merid y)) ∙v⊡
natural-square (λ sy → merid (Susp-fmap (smin x) sy)) (merid y) ⊡v∙
ap-cst south (merid y)
=⟨ natural-square-cst
north
south
(λ sy → merid (Susp-fmap (smin x) sy))
(merid y) ⟩
horiz-degen-square (ap (merid ∘ Susp-fmap (smin x)) (merid y))
=⟨ ap horiz-degen-square (ap-∘ merid (Susp-fmap (smin x)) (merid y)) ⟩
horiz-degen-square (ap merid (ap (Susp-fmap (smin x)) (merid y)))
=⟨ ap (horiz-degen-square ∘ ap merid) (SuspFmap.merid-β (smin x) y) ⟩
horiz-degen-square (ap merid (merid (smin x y))) =∎
bot-path :
square-symmetry
(! (ap-cst south (merid x)) ∙v⊡
(! (↯ (g-merid-y north y)) ∙h⊡
square-symmetry (natural-square
(λ sy → ap (λ sx → g (smin sx sy)) (merid x)) (merid y)) ⊡h∙
↯ (g-merid-y south y)) ⊡v∙
ap-cst north (merid x))
==
vert-degen-square (ap ! (ap merid (merid (smin x y))))
bot-path =
square-symmetry
(! (ap-cst south (merid x)) ∙v⊡
(! (↯ (g-merid-y north y)) ∙h⊡
square-symmetry (natural-square
(λ sy → ap (λ sx → g (smin sx sy)) (merid x)) (merid y)) ⊡h∙
↯ (g-merid-y south y)) ⊡v∙
ap-cst north (merid x))
=⟨ ap (λ u → square-symmetry
(! (ap-cst south (merid x)) ∙v⊡
(! (↯ (g-merid-y north y)) ∙h⊡
u ⊡h∙
↯ (g-merid-y south y)) ⊡v∙
ap-cst north (merid x))) $
natural-square-symmetry
(λ sx sy → g (smin sx sy))
(merid x)
(merid y) ⟩
square-symmetry
(! (ap-cst south (merid x)) ∙v⊡
(! (↯ (g-merid-y north y)) ∙h⊡
natural-square (λ sx → ap (g ∘ smin sx) (merid y)) (merid x) ⊡h∙
↯ (g-merid-y south y)) ⊡v∙
ap-cst north (merid x))
=⟨ ap (λ u → square-symmetry
(! (ap-cst south (merid x)) ∙v⊡ u ⊡v∙ ap-cst north (merid x))) $
natural-square-path
(λ sx → ! (merid (Susp-fmap (λ x' → smin x' y) sx)))
(λ sx → ap (g ∘ smin sx) (merid y))
(λ sx → ↯ (g-merid-y sx y))
(merid x) ⟩
square-symmetry
(! (ap-cst south (merid x)) ∙v⊡
natural-square (λ sx → ! (merid (Susp-fmap (λ x' → smin x' y) sx))) (merid x) ⊡v∙
ap-cst north (merid x))
=⟨ ap square-symmetry $
natural-square-cst
south
north
(λ sx → ! (merid (Susp-fmap (λ x' → smin x' y) sx)))
(merid x) ⟩
square-symmetry
(horiz-degen-square
(ap (λ sx → ! (merid (Susp-fmap (λ x' → smin x' y) sx))) (merid x)))
=⟨ horiz-degen-square-symmetry
(ap (λ sx → ! (merid (Susp-fmap (λ x' → smin x' y) sx))) (merid x)) ⟩
vert-degen-square (ap (! ∘ merid ∘ Susp-fmap (λ x' → smin x' y)) (merid x))
=⟨ ap vert-degen-square (ap-∘ (! ∘ merid) (Susp-fmap (λ x' → smin x' y)) (merid x)) ⟩
vert-degen-square (ap (! ∘ merid) (ap (Susp-fmap (λ x' → smin x' y)) (merid x)))
=⟨ ap (vert-degen-square ∘ ap (! ∘ merid)) (SuspFmap.merid-β (λ x' → smin x' y) x) ⟩
vert-degen-square (ap (! ∘ merid) (merid (smin x y)))
=⟨ ap vert-degen-square (ap-∘ ! merid (merid (smin x y))) ⟩
vert-degen-square (ap ! (ap merid (merid (smin x y)))) =∎
basel : f smbasel == g smbasel
basel = merid north
baser : f smbaser == g smbaser
baser = merid north
f-smgluel : ∀ sx → ap f (smgluel sx) == ap (Susp-fmap (∧Σ-out X Y)) (Σ∧OutSmgluel.f X (⊙Susp (de⊙ Y)) sx)
f-smgluel sx =
ap f (smgluel sx)
=⟨ ap-∘ (Susp-fmap (∧Σ-out X Y)) (Σ∧-out X (⊙Susp (de⊙ Y))) (smgluel sx) ⟩
ap (Susp-fmap (∧Σ-out X Y)) (ap (Σ∧-out X (⊙Susp (de⊙ Y))) (smgluel sx))
=⟨ ap (ap (Susp-fmap (∧Σ-out X Y)))
(Σ∧Out.smgluel-β X (⊙Susp (de⊙ Y)) sx) ⟩
ap (Susp-fmap (∧Σ-out X Y)) (Σ∧OutSmgluel.f X (⊙Susp (de⊙ Y)) sx) =∎
f-smgluel-north : ap f (smgluel north) == idp
f-smgluel-north = f-smgluel north
f-smgluel-south : ap f (smgluel south) =-= ! (merid north)
f-smgluel-south =
ap f (smgluel south)
=⟪ f-smgluel south ⟫
ap (Susp-fmap (∧Σ-out X Y)) (! (merid (smin (pt X) north)))
=⟪ ap-! (Susp-fmap (∧Σ-out X Y)) (merid (smin (pt X) north)) ⟫
! (ap (Susp-fmap (∧Σ-out X Y)) (merid (smin (pt X) north)))
=⟪ ap ! (SuspFmap.merid-β (∧Σ-out X Y) (smin (pt X) north)) ⟫
! (merid north) ∎∎
g-smgluel : ∀ sx → ap g (smgluel sx) == idp
g-smgluel sx =
ap g (smgluel sx)
=⟨ ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (∧Σ-out (⊙Susp (de⊙ X)) Y) (smgluel sx) ⟩
ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (ap (∧Σ-out (⊙Susp (de⊙ X)) Y) (smgluel sx))
=⟨ ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(∧ΣOut.smgluel-β (⊙Susp (de⊙ X)) Y sx) ⟩
idp =∎
smgluel-north : Square n-n (ap f (smgluel north)) (ap g (smgluel north)) basel
smgluel-north = f-smgluel-north ∙v⊡ hid-square ⊡v∙ ! (g-smgluel north)
smgluel-south : Square s-n (ap f (smgluel south)) (ap g (smgluel south)) basel
smgluel-south = ↯ f-smgluel-south ∙v⊡ rt-square (merid north) ⊡v∙ ! (g-smgluel south)
custom-cube-⊡v∙ : ∀ {i} {A : Type i} {a₀₀₀ a₀₁₀ a₁₀₀ a₁₁₀ a₀₀₁ a₀₁₁ a₁₀₁ a₁₁₁ : A}
{p₀₋₀ : a₀₀₀ == a₀₁₀}
{p₋₀₀ : a₀₀₀ == a₁₀₀}
{p₋₁₀ p₋₁₀' : a₀₁₀ == a₁₁₀} (q₋₁₀ : p₋₁₀ == p₋₁₀')
{p₁₋₀ : a₁₀₀ == a₁₁₀}
{sq₋₋₀ : Square p₀₋₀ p₋₀₀ p₋₁₀ p₁₋₀} -- left
{p₀₋₁ : a₀₀₁ == a₀₁₁}
{p₋₀₁ : a₀₀₁ == a₁₀₁}
{p₋₁₁ p₋₁₁' : a₀₁₁ == a₁₁₁} (q₋₁₁ : p₋₁₁' == p₋₁₁)
{p₁₋₁ : a₁₀₁ == a₁₁₁}
{sq₋₋₁ : Square p₀₋₁ p₋₀₁ p₋₁₁ p₁₋₁} -- right
{p₀₀₋ : a₀₀₀ == a₀₀₁}
{p₀₁₋ p₀₁₋' : a₀₁₀ == a₀₁₁} (q₀₁₋ : p₀₁₋ == p₀₁₋')
{p₁₀₋ : a₁₀₀ == a₁₀₁}
{p₁₁₋ p₁₁₋' : a₁₁₀ == a₁₁₁} (q₁₁₋ : p₁₁₋ == p₁₁₋')
{sq₀₋₋ : Square p₀₋₀ p₀₀₋ p₀₁₋ p₀₋₁} -- back
{sq₋₀₋ : Square p₋₀₀ p₀₀₋ p₁₀₋ p₋₀₁} -- top
{sq₋₁₋ : Square p₋₁₀' p₀₁₋' p₁₁₋ p₋₁₁'} -- bottom
{sq₁₋₋ : Square p₁₋₀ p₁₀₋ p₁₁₋ p₁₋₁} -- front
→ Cube sq₋₋₀ sq₋₋₁ sq₀₋₋ sq₋₀₋ (q₀₁₋ ∙v⊡ (q₋₁₀ ∙h⊡ sq₋₁₋ ⊡h∙ q₋₁₁) ⊡v∙ q₁₁₋) (sq₁₋₋ ⊡v∙ q₁₁₋)
→ Cube (sq₋₋₀ ⊡v∙ q₋₁₀) (sq₋₋₁ ⊡v∙ ! q₋₁₁) (sq₀₋₋ ⊡v∙ q₀₁₋) sq₋₀₋ sq₋₁₋ sq₁₋₋
custom-cube-⊡v∙ idp idp idp idp c = c
smgluel-merid : ∀ x →
Cube smgluel-north
smgluel-south
(m-n x)
(natural-square (λ sx → ap f (smgluel sx)) (merid x))
(natural-square (λ sx → ap g (smgluel sx)) (merid x))
(natural-square (λ sx → basel) (merid x))
smgluel-merid x =
custom-cube-∙v⊡
f-smgluel-north
(f-smgluel south) (↯ (tail f-smgluel-south))
(ap-∘ (Susp-fmap (∧Σ-out X Y)) (Susp-fmap (λ x' → smin x' north)) (merid x))
(↯ (tail (f-merid-x x north)))
(ap-cst north (merid x)) $
cube-shift-top (! (top-path x)) $
custom-cube-⊡v∙
(! (g-smgluel north))
(g-smgluel south)
(! (ap-cst south (merid x)))
(ap-cst south (merid x)) $
cube-shift-front (! (front-path x)) $
cube-shift-bot (! (bot-path x)) $
custom-cube (merid north)
where
custom-cube-∙v⊡ : ∀ {i} {A : Type i} {a₀₀₀ a₀₁₀ a₁₀₀ a₁₁₀ a₀₀₁ a₀₁₁ a₁₀₁ a₁₁₁ : A}
{p₀₋₀ : a₀₀₀ == a₀₁₀}
{p₋₀₀ p₋₀₀' : a₀₀₀ == a₁₀₀} (q₋₀₀ : p₋₀₀ == p₋₀₀')
{p₋₁₀ : a₀₁₀ == a₁₁₀}
{p₁₋₀ : a₁₀₀ == a₁₁₀}
{sq₋₋₀ : Square p₀₋₀ p₋₀₀' p₋₁₀ p₁₋₀} -- left
{p₀₋₁ : a₀₀₁ == a₀₁₁}
{p₋₀₁ p₋₀₁' p₋₀₁'' : a₀₀₁ == a₁₀₁} (q₋₀₁ : p₋₀₁ == p₋₀₁') (q₋₀₁' : p₋₀₁' == p₋₀₁'')
{p₋₁₁ : a₀₁₁ == a₁₁₁}
{p₁₋₁ : a₁₀₁ == a₁₁₁}
{sq₋₋₁ : Square p₀₋₁ p₋₀₁'' p₋₁₁ p₁₋₁} -- right
{p₀₀₋ p₀₀₋' p₀₀₋'' : a₀₀₀ == a₀₀₁} (q₀₀₋ : p₀₀₋ == p₀₀₋') (q₀₀₋' : p₀₀₋' == p₀₀₋'')
{p₀₁₋ : a₀₁₀ == a₀₁₁}
{p₁₀₋ p₁₀₋' : a₁₀₀ == a₁₀₁} (q₁₀₋ : p₁₀₋ == p₁₀₋')
{p₁₁₋ : a₁₁₀ == a₁₁₁}
{sq₀₋₋ : Square p₀₋₀ p₀₀₋'' p₀₁₋ p₀₋₁} -- back
{sq₋₀₋ : Square p₋₀₀ p₀₀₋ p₁₀₋ p₋₀₁} -- top
{sq₋₁₋ : Square p₋₁₀ p₀₁₋ p₁₁₋ p₋₁₁} -- bottom
{sq₁₋₋ : Square p₁₋₀ p₁₀₋ p₁₁₋ p₁₋₁} -- front
→ Cube sq₋₋₀ sq₋₋₁ sq₀₋₋
((! q₀₀₋' ∙v⊡ (! q₀₀₋ ∙v⊡ (! q₋₀₀ ∙h⊡ sq₋₀₋ ⊡h∙ q₋₀₁)) ⊡v∙ q₁₀₋) ⊡h∙ q₋₀₁')
sq₋₁₋ (! q₁₀₋ ∙v⊡ sq₁₋₋)
→ Cube (q₋₀₀ ∙v⊡ sq₋₋₀) ((q₋₀₁ ∙ q₋₀₁') ∙v⊡ sq₋₋₁) ((q₀₀₋ ∙ q₀₀₋') ∙v⊡ sq₀₋₋) sq₋₀₋ sq₋₁₋ sq₁₋₋
custom-cube-∙v⊡ idp idp idp idp idp idp c = c
custom-cube : ∀ {i} {S : Type i} {n s : S} (p : n == s)
→ Cube hid-square (rt-square p) (lt-square p) (tr-square p) ids hid-square
custom-cube p@idp = idc
top-path : ∀ x →
(! (↯ (tail (f-merid-x x north))) ∙v⊡
(! (ap-∘ (Susp-fmap (∧Σ-out X Y)) (Susp-fmap (λ x' → smin x' north)) (merid x)) ∙v⊡
(! f-smgluel-north ∙h⊡
natural-square (λ sx → ap f (smgluel sx)) (merid x) ⊡h∙
f-smgluel south)) ⊡v∙
ap-cst north (merid x)) ⊡h∙
↯ (tail f-smgluel-south)
==
tr-square (merid north)
top-path x =
(! (↯ (tail (f-merid-x x north))) ∙v⊡
(! (ap-∘ (Susp-fmap (∧Σ-out X Y)) (Susp-fmap (λ x' → smin x' north)) (merid x)) ∙v⊡
(! f-smgluel-north ∙h⊡
natural-square (λ sx → ap f (smgluel sx)) (merid x) ⊡h∙
f-smgluel south)) ⊡v∙
ap-cst north (merid x)) ⊡h∙
↯ (tail f-smgluel-south)
=⟨ ap (λ u → (! (↯ (tail (f-merid-x x north))) ∙v⊡
(! (ap-∘ (Susp-fmap (∧Σ-out X Y)) (Susp-fmap (λ x' → smin x' north)) (merid x)) ∙v⊡
u) ⊡v∙
ap-cst north (merid x)) ⊡h∙
↯ (tail f-smgluel-south)) $
natural-square-path
(ap (Susp-fmap (∧Σ-out X Y)) ∘ Σ∧OutSmgluel.f X (⊙Susp (de⊙ Y)))
(λ sx → ap f (smgluel sx))
f-smgluel
(merid x) ⟩
(! (↯ (tail (f-merid-x x north))) ∙v⊡
(! (ap-∘ (Susp-fmap (∧Σ-out X Y)) (Susp-fmap (λ x' → smin x' north)) (merid x)) ∙v⊡
natural-square (ap (Susp-fmap (∧Σ-out X Y)) ∘ Σ∧OutSmgluel.f X (⊙Susp (de⊙ Y))) (merid x)) ⊡v∙
ap-cst north (merid x)) ⊡h∙
↯ (tail f-smgluel-south)
=⟨ ap (λ v → (! (↯ (tail (f-merid-x x north))) ∙v⊡
v ⊡v∙
ap-cst north (merid x)) ⊡h∙
↯ (tail f-smgluel-south)) $
! (ap-∘ (Susp-fmap (∧Σ-out X Y)) (Susp-fmap (λ x' → smin x' north)) (merid x)) ∙v⊡
natural-square (ap (Susp-fmap (∧Σ-out X Y)) ∘ Σ∧OutSmgluel.f X (⊙Susp (de⊙ Y))) (merid x)
=⟨ ap (! (ap-∘ (Susp-fmap (∧Σ-out X Y)) (Susp-fmap (λ x' → smin x' north)) (merid x)) ∙v⊡_) $
natural-square-ap (Susp-fmap (∧Σ-out X Y)) (Σ∧OutSmgluel.f X (⊙Susp (de⊙ Y))) (merid x) ⟩
! (ap-∘ (Susp-fmap (∧Σ-out X Y)) (Susp-fmap (λ x' → smin x' north)) (merid x)) ∙v⊡
ap-∘ (Susp-fmap (∧Σ-out X Y)) (Susp-fmap (λ x' → smin x' north)) (merid x) ∙v⊡
ap-square (Susp-fmap (∧Σ-out X Y)) (natural-square (Σ∧OutSmgluel.f X (⊙Susp (de⊙ Y))) (merid x)) ⊡v∙
∘-ap (Susp-fmap (∧Σ-out X Y)) (λ _ → north) (merid x)
=⟨ ! $ ∙v⊡-assoc
(! (ap-∘ (Susp-fmap (∧Σ-out X Y)) (Susp-fmap (λ x' → smin x' north)) (merid x)))
(ap-∘ (Susp-fmap (∧Σ-out X Y)) (Susp-fmap (λ x' → smin x' north)) (merid x))
(ap-square (Susp-fmap (∧Σ-out X Y)) (natural-square (Σ∧OutSmgluel.f X (⊙Susp (de⊙ Y))) (merid x)) ⊡v∙
∘-ap (Susp-fmap (∧Σ-out X Y)) (λ _ → north) (merid x)) ⟩
(! (ap-∘ (Susp-fmap (∧Σ-out X Y)) (Susp-fmap (λ x' → smin x' north)) (merid x)) ∙
ap-∘ (Susp-fmap (∧Σ-out X Y)) (Susp-fmap (λ x' → smin x' north)) (merid x)) ∙v⊡
ap-square (Susp-fmap (∧Σ-out X Y)) (natural-square (Σ∧OutSmgluel.f X (⊙Susp (de⊙ Y))) (merid x)) ⊡v∙
∘-ap (Susp-fmap (∧Σ-out X Y)) (λ _ → north) (merid x)
=⟨ ap (_∙v⊡ ap-square (Susp-fmap (∧Σ-out X Y)) (natural-square (Σ∧OutSmgluel.f X (⊙Susp (de⊙ Y))) (merid x)) ⊡v∙
∘-ap (Susp-fmap (∧Σ-out X Y)) (λ _ → north) (merid x)) $
!-inv-l (ap-∘ (Susp-fmap (∧Σ-out X Y)) (Susp-fmap (λ x' → smin x' north)) (merid x)) ⟩
ap-square (Susp-fmap (∧Σ-out X Y)) (natural-square (Σ∧OutSmgluel.f X (⊙Susp (de⊙ Y))) (merid x)) ⊡v∙
∘-ap (Susp-fmap (∧Σ-out X Y)) (λ _ → north) (merid x) =∎ ⟩
(! (↯ (tail (f-merid-x x north))) ∙v⊡
(ap-square (Susp-fmap (∧Σ-out X Y)) (natural-square (Σ∧OutSmgluel.f X (⊙Susp (de⊙ Y))) (merid x)) ⊡v∙
∘-ap (Susp-fmap (∧Σ-out X Y)) (λ _ → north) (merid x)) ⊡v∙
ap-cst north (merid x)) ⊡h∙
↯ (tail f-smgluel-south)
=⟨ ap (λ u → (! (↯ (tail (f-merid-x x north))) ∙v⊡
u) ⊡h∙
↯ (tail f-smgluel-south)) $
(ap-square (Susp-fmap (∧Σ-out X Y)) (natural-square (Σ∧OutSmgluel.f X (⊙Susp (de⊙ Y))) (merid x)) ⊡v∙
∘-ap (Susp-fmap (∧Σ-out X Y)) (λ _ → north) (merid x)) ⊡v∙
ap-cst north (merid x)
=⟨ ap (λ u → (ap-square (Susp-fmap (∧Σ-out X Y)) u ⊡v∙
∘-ap (Susp-fmap (∧Σ-out X Y)) (λ _ → north) (merid x)) ⊡v∙
ap-cst north (merid x)) $
Σ∧OutSmgluel.merid-square-β X (⊙Susp (de⊙ Y)) x ⟩
(ap-square (Susp-fmap (∧Σ-out X Y)) (Σ∧-out-smgluel-merid X (⊙Susp (de⊙ Y)) x) ⊡v∙
∘-ap (Susp-fmap (∧Σ-out X Y)) (λ _ → north) (merid x)) ⊡v∙
ap-cst north (merid x)
=⟨ ⊡v∙-assoc
(ap-square (Susp-fmap (∧Σ-out X Y)) (Σ∧-out-smgluel-merid X (⊙Susp (de⊙ Y)) x))
(∘-ap (Susp-fmap (∧Σ-out X Y)) (λ _ → north) (merid x))
(ap-cst north (merid x)) ⟩
ap-square (Susp-fmap (∧Σ-out X Y)) (Σ∧-out-smgluel-merid X (⊙Susp (de⊙ Y)) x) ⊡v∙
(∘-ap (Susp-fmap (∧Σ-out X Y)) (λ _ → north) (merid x) ∙
ap-cst north (merid x))
=⟨ ap (ap-square (Susp-fmap (∧Σ-out X Y)) (Σ∧-out-smgluel-merid X (⊙Susp (de⊙ Y)) x) ⊡v∙_) $
=ₛ-out $ ap-∘-cst-coh (Susp-fmap (∧Σ-out X Y)) north (merid x) ⟩
ap-square (Susp-fmap (∧Σ-out X Y)) (Σ∧-out-smgluel-merid X (⊙Susp (de⊙ Y)) x) ⊡v∙
ap (ap (Susp-fmap (∧Σ-out X Y))) (ap-cst north (merid x))
=⟨ ap-square-⊡v∙
(Susp-fmap (∧Σ-out X Y))
(Σ∧-out-smgluel-merid X (⊙Susp (de⊙ Y)) x)
(ap-cst north (merid x)) ⟩
ap-square
(Susp-fmap (∧Σ-out X Y))
(Σ∧-out-smgluel-merid X (⊙Susp (de⊙ Y)) x ⊡v∙ ap-cst north (merid x))
=⟨ ap (ap-square (Susp-fmap (∧Σ-out X Y))) $
⊡v∙-assoc
((SuspFmap.merid-β (λ x' → smin x' north) x ∙ ap merid (∧-norm-l x)) ∙v⊡
tr-square (merid (smin (pt X) north)))
(! (ap-cst north (merid x)))
(ap-cst north (merid x)) ⟩
ap-square
(Susp-fmap (∧Σ-out X Y))
(((SuspFmap.merid-β (λ x' → smin x' north) x ∙ ap merid (∧-norm-l x)) ∙v⊡
tr-square (merid (smin (pt X) north))) ⊡v∙
(! (ap-cst north (merid x)) ∙ ap-cst north (merid x)))
=⟨ ap (λ u → ap-square (Susp-fmap (∧Σ-out X Y)) $
((SuspFmap.merid-β (λ x' → smin x' north) x ∙
ap merid (∧-norm-l x)) ∙v⊡
tr-square (merid (smin (pt X) north))) ⊡v∙
u) $
!-inv-l (ap-cst north (merid x)) ⟩
ap-square
(Susp-fmap (∧Σ-out X Y))
((SuspFmap.merid-β (λ x' → smin x' north) x ∙ ap merid (∧-norm-l x)) ∙v⊡
tr-square (merid (smin (pt X) north)))
=⟨ ! $ ap-square-∙v⊡
(Susp-fmap (∧Σ-out X Y))
(SuspFmap.merid-β (λ x' → smin x' north) x ∙ ap merid (∧-norm-l x))
(tr-square (merid (smin (pt X) north))) ⟩
ap (ap (Susp-fmap (∧Σ-out X Y)))
(SuspFmap.merid-β (λ x' → smin x' north) x ∙ ap merid (∧-norm-l x)) ∙v⊡
ap-square (Susp-fmap (∧Σ-out X Y)) (tr-square (merid (smin (pt X) north))) =∎
⟩
(! (↯ (tail (f-merid-x x north))) ∙v⊡
ap (ap (Susp-fmap (∧Σ-out X Y)))
(SuspFmap.merid-β (λ x' → smin x' north) x ∙ ap merid (∧-norm-l x)) ∙v⊡
ap-square (Susp-fmap (∧Σ-out X Y)) (tr-square (merid (smin (pt X) north)))) ⊡h∙
↯ (tail f-smgluel-south)
=⟨ ! $ ap (_⊡h∙ ↯ (tail f-smgluel-south)) $
∙v⊡-assoc (! (↯ (tail (f-merid-x x north))))
(ap (ap (Susp-fmap (∧Σ-out X Y)))
(SuspFmap.merid-β (λ x' → smin x' north) x ∙ ap merid (∧-norm-l x)))
(ap-square (Susp-fmap (∧Σ-out X Y)) (tr-square (merid (smin (pt X) north)))) ⟩
((! (↯ (tail (f-merid-x x north))) ∙
ap (ap (Susp-fmap (∧Σ-out X Y)))
(SuspFmap.merid-β (λ x' → smin x' north) x ∙ ap merid (∧-norm-l x))) ∙v⊡
ap-square (Susp-fmap (∧Σ-out X Y)) (tr-square (merid (smin (pt X) north)))) ⊡h∙
↯ (tail f-smgluel-south)
=⟨ ap (λ u → (u ∙v⊡ ap-square (Susp-fmap (∧Σ-out X Y)) (tr-square (merid (smin (pt X) north)))) ⊡h∙
↯ (tail f-smgluel-south)) $ =ₛ-out $
! (↯ (tail (f-merid-x x north))) ◃∙
ap (ap (Susp-fmap (∧Σ-out X Y)))
(SuspFmap.merid-β (λ x' → smin x' north) x ∙ ap merid (∧-norm-l x)) ◃∎
=ₛ⟨ 0 & 1 & !-∙-seq (tail (f-merid-x x north)) ⟩
! (SuspFmap.merid-β (∧Σ-out X Y) (smin x north)) ◃∙
! (ap (ap (Susp-fmap (∧Σ-out X Y))) (SuspFmap.merid-β (λ x' → smin x' north) x)) ◃∙
ap (ap (Susp-fmap (∧Σ-out X Y)))
(SuspFmap.merid-β (λ x' → smin x' north) x ∙ ap merid (∧-norm-l x)) ◃∎
=ₛ⟨ 2 & 1 &
ap-seq-∙
(ap (Susp-fmap (∧Σ-out X Y)))
(SuspFmap.merid-β (λ x' → smin x' north) x ◃∙
ap merid (∧-norm-l x) ◃∎) ⟩
! (SuspFmap.merid-β (∧Σ-out X Y) (smin x north)) ◃∙
! (ap (ap (Susp-fmap (∧Σ-out X Y))) (SuspFmap.merid-β (λ x' → smin x' north) x)) ◃∙
ap (ap (Susp-fmap (∧Σ-out X Y))) (SuspFmap.merid-β (λ x' → smin x' north) x) ◃∙
ap (ap (Susp-fmap (∧Σ-out X Y))) (ap merid (∧-norm-l x)) ◃∎
=ₛ⟨ 1 & 2 & seq-!-inv-l $
ap (ap (Susp-fmap (∧Σ-out X Y)))
(SuspFmap.merid-β (λ x' → smin x' north) x) ◃∎ ⟩
! (SuspFmap.merid-β (∧Σ-out X Y) (smin x north)) ◃∙
ap (ap (Susp-fmap (∧Σ-out X Y))) (ap merid (∧-norm-l x)) ◃∎
=ₛ₁⟨ 1 & 1 & ∘-ap (ap (Susp-fmap (∧Σ-out X Y))) merid (∧-norm-l x) ⟩
! (SuspFmap.merid-β (∧Σ-out X Y) (smin x north)) ◃∙
ap (ap (Susp-fmap (∧Σ-out X Y)) ∘ merid) (∧-norm-l x) ◃∎
=ₛ⟨ !ₛ $
homotopy-naturality
(merid ∘ ∧Σ-out X Y)
(ap (Susp-fmap (∧Σ-out X Y)) ∘ merid)
(! ∘ SuspFmap.merid-β (∧Σ-out X Y))
(∧-norm-l x) ⟩
ap (merid ∘ ∧Σ-out X Y) (∧-norm-l x) ◃∙
! (SuspFmap.merid-β (∧Σ-out X Y) (smin (pt X) north)) ◃∎
=ₛ⟨ 0 & 1 & =ₛ-in {t = []} $
ap-∘ merid (∧Σ-out X Y) (∧-norm-l x) ∙
ap (ap merid) (∧Σ-out-∧-norm-l X Y x) ⟩
! (SuspFmap.merid-β (∧Σ-out X Y) (smin (pt X) north)) ◃∎ ∎ₛ ⟩
(! (SuspFmap.merid-β (∧Σ-out X Y) (smin (pt X) north)) ∙v⊡
ap-square (Susp-fmap (∧Σ-out X Y)) (tr-square (merid (smin (pt X) north)))) ⊡h∙
↯ (tail f-smgluel-south)
=⟨ ∙v⊡-⊡h∙-comm
(! (SuspFmap.merid-β (∧Σ-out X Y) (smin (pt X) north)))
(ap-square (Susp-fmap (∧Σ-out X Y)) (tr-square (merid (smin (pt X) north))))
(↯ (tail f-smgluel-south)) ⟩
! (SuspFmap.merid-β (∧Σ-out X Y) (smin (pt X) north)) ∙v⊡
ap-square (Susp-fmap (∧Σ-out X Y)) (tr-square (merid (smin (pt X) north))) ⊡h∙
↯ (tail f-smgluel-south)
=⟨ ! $ ap (! (SuspFmap.merid-β (∧Σ-out X Y) (smin (pt X) north)) ∙v⊡_) $
⊡h∙-assoc
(ap-square (Susp-fmap (∧Σ-out X Y)) (tr-square (merid (smin (pt X) north))))
(ap-! (Susp-fmap (∧Σ-out X Y)) (merid (smin (pt X) north)))
(ap ! (SuspFmap.merid-β (∧Σ-out X Y) (smin (pt X) north))) ⟩
! (SuspFmap.merid-β (∧Σ-out X Y) (smin (pt X) north)) ∙v⊡
ap-square (Susp-fmap (∧Σ-out X Y)) (tr-square (merid (smin (pt X) north))) ⊡h∙
ap-! (Susp-fmap (∧Σ-out X Y)) (merid (smin (pt X) north)) ⊡h∙
ap ! (SuspFmap.merid-β (∧Σ-out X Y) (smin (pt X) north))
=⟨ ap (λ u → ! (SuspFmap.merid-β (∧Σ-out X Y) (smin (pt X) north)) ∙v⊡
u ⊡h∙
ap ! (SuspFmap.merid-β (∧Σ-out X Y) (smin (pt X) north))) $
ap-tr-square (Susp-fmap (∧Σ-out X Y)) (merid (smin (pt X) north)) ⟩
! (SuspFmap.merid-β (∧Σ-out X Y) (smin (pt X) north)) ∙v⊡
tr-square (ap (Susp-fmap (∧Σ-out X Y)) (merid (smin (pt X) north))) ⊡h∙
ap ! (SuspFmap.merid-β (∧Σ-out X Y) (smin (pt X) north))
=⟨ tr-square-∙v⊡-⊡h∙ (SuspFmap.merid-β (∧Σ-out X Y) (smin (pt X) north)) ⟩
tr-square (merid north) =∎
front-path : ∀ (x : de⊙ X) →
(! (ap-cst north (merid x)) ∙v⊡
natural-square (λ sx → basel) (merid x)) ⊡v∙
ap-cst south (merid x)
==
hid-square
front-path x =
(! (ap-cst north (merid x)) ∙v⊡
natural-square (λ sx → basel) (merid x)) ⊡v∙
ap-cst south (merid x)
=⟨ ∙v⊡-⊡v∙-comm
(! (ap-cst north (merid x)))
(natural-square (λ sx → basel) (merid x))
(ap-cst south (merid x)) ⟩
! (ap-cst north (merid x)) ∙v⊡
natural-square (λ sx → basel) (merid x) ⊡v∙
ap-cst south (merid x)
=⟨ natural-square-cst north south (λ sx → basel) (merid x) ⟩
horiz-degen-square (ap (λ sx → basel) (merid x))
=⟨ ap horiz-degen-square (ap-cst basel (merid x)) ⟩
horiz-degen-square idp
=⟨ horiz-degen-square-idp ⟩
hid-square =∎
bot-path : ∀ (x : de⊙ X) →
! (ap-cst south (merid x)) ∙v⊡
(! (g-smgluel north) ∙h⊡
natural-square (λ sx → ap g (smgluel sx)) (merid x) ⊡h∙
g-smgluel south) ⊡v∙
ap-cst south (merid x)
==
ids
bot-path x =
! (ap-cst south (merid x)) ∙v⊡
(! (g-smgluel north) ∙h⊡
natural-square (λ sx → ap g (smgluel sx)) (merid x) ⊡h∙
g-smgluel south) ⊡v∙
ap-cst south (merid x)
=⟨ ap (λ u → ! (ap-cst south (merid x)) ∙v⊡ u ⊡v∙ ap-cst south (merid x)) $
natural-square-path
(λ sx → idp)
(λ sx → ap g (smgluel sx))
g-smgluel
(merid x) ⟩
! (ap-cst south (merid x)) ∙v⊡
natural-square (λ sx → idp) (merid x) ⊡v∙
ap-cst south (merid x)
=⟨ natural-square-cst
south
south
(λ sx → idp)
(merid x) ⟩
disc-to-square (ap (λ sx → idp) (merid x))
=⟨ ap disc-to-square (ap-cst idp (merid x)) ⟩
ids =∎
f-smgluer : ∀ sy → ap f (smgluer sy) == idp
f-smgluer sy =
ap f (smgluer sy)
=⟨ ap-∘ (Susp-fmap (∧Σ-out X Y)) (Σ∧-out X (⊙Susp (de⊙ Y))) (smgluer sy) ⟩
ap (Susp-fmap (∧Σ-out X Y)) (ap (Σ∧-out X (⊙Susp (de⊙ Y))) (smgluer sy))
=⟨ ap (ap (Susp-fmap (∧Σ-out X Y)))
(Σ∧Out.smgluer-β X (⊙Susp (de⊙ Y)) sy) ⟩
idp =∎
g-smgluer : ∀ sy →
ap g (smgluer sy)
==
ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (∧ΣOutSmgluer.f (⊙Susp (de⊙ X)) Y sy)
g-smgluer sy =
ap g (smgluer sy)
=⟨ ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (∧Σ-out (⊙Susp (de⊙ X)) Y) (smgluer sy) ⟩
ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (ap (∧Σ-out (⊙Susp (de⊙ X)) Y) (smgluer sy))
=⟨ ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(∧ΣOut.smgluer-β (⊙Susp (de⊙ X)) Y sy) ⟩
ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (∧ΣOutSmgluer.f (⊙Susp (de⊙ X)) Y sy) =∎
g-smgluer-north : ap g (smgluer north) == idp
g-smgluer-north = g-smgluer north
g-smgluer-south-step :
ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y)))
==
! (merid north)
g-smgluer-south-step = ↯ $
ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y)))
=⟪ ap-∘ Susp-flip (Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y))) ⟫
ap Susp-flip (ap (Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y))))
=⟪ ap (ap Susp-flip) (SuspFmap.merid-β (Σ∧-out X Y) (smin north (pt Y))) ⟫
ap Susp-flip (merid north)
=⟪ SuspFlip.merid-β north ⟫
! (merid north) ∎∎
g-smgluer-south : ap g (smgluer south) =-= merid north
g-smgluer-south =
ap g (smgluer south)
=⟪ g-smgluer south ⟫
ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (! (merid (smin north (pt Y))))
=⟪ ap-! (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y))) ⟫
! (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y))))
=⟪ ap ! g-smgluer-south-step ⟫
! (! (merid north))
=⟪ !-! (merid north) ⟫
merid north ∎∎
smgluer-north : Square n-n (ap f (smgluer north)) (ap g (smgluer north)) baser
smgluer-north = f-smgluer north ∙v⊡ hid-square ⊡v∙ ! g-smgluer-north
smgluer-south : Square n-s (ap f (smgluer south)) (ap g (smgluer south)) baser
smgluer-south = f-smgluer south ∙v⊡ br-square (merid north) ⊡v∙ ! (↯ (g-smgluer-south))
smgluer-merid : ∀ (y : de⊙ Y) →
Cube smgluer-north
smgluer-south
(n-m y)
(natural-square (λ a → ap f (smgluer a)) (merid y))
(natural-square (λ a → ap g (smgluer a)) (merid y))
(natural-square (λ a → baser) (merid y))
smgluer-merid y =
smgluer-merid-custom-cube-∙v⊡
(f-smgluer north)
(f-smgluer south)
(ap-cst north (merid y))
(ap-cst north (merid y)) $
cube-shift-top (! top-path) $
custom-cube-⊡v∙
(! g-smgluer-north)
(↯ g-smgluer-south)
(! (↯ (g-merid-y north y)))
(ap-cst south (merid y)) $
cube-shift-front (! front-path) $
cube-shift-bot (! bot-path) $
custom-cube (merid north)
where
smgluer-merid-custom-cube-∙v⊡ :
∀ {i} {A : Type i} {a₀₀₀ a₀₁₀ a₁₀₀ a₁₁₀ a₀₀₁ a₀₁₁ a₁₀₁ a₁₁₁ : A}
{p₀₋₀ : a₀₀₀ == a₀₁₀}
{p₋₀₀ p₋₀₀' : a₀₀₀ == a₁₀₀} (q₋₀₀ : p₋₀₀' == p₋₀₀)
{p₋₁₀ : a₀₁₀ == a₁₁₀}
{p₁₋₀ : a₁₀₀ == a₁₁₀}
{sq₋₋₀ : Square p₀₋₀ p₋₀₀ p₋₁₀ p₁₋₀} -- left
{p₀₋₁ : a₀₀₁ == a₀₁₁}
{p₋₀₁ p₋₀₁' : a₀₀₁ == a₁₀₁} (q₋₀₁ : p₋₀₁' == p₋₀₁)
{p₋₁₁ : a₀₁₁ == a₁₁₁}
{p₁₋₁ : a₁₀₁ == a₁₁₁}
{sq₋₋₁ : Square p₀₋₁ p₋₀₁ p₋₁₁ p₁₋₁} -- right
{p₀₀₋ p₀₀₋' : a₀₀₀ == a₀₀₁} (q₀₀₋ : p₀₀₋' == p₀₀₋)
{p₀₁₋ : a₀₁₀ == a₀₁₁}
{p₁₀₋ p₁₀₋' : a₁₀₀ == a₁₀₁} (q₁₀₋ : p₁₀₋ == p₁₀₋')
{p₁₁₋ : a₁₁₀ == a₁₁₁}
{sq₀₋₋ : Square p₀₋₀ p₀₀₋ p₀₁₋ p₀₋₁} -- back
{sq₋₀₋ : Square p₋₀₀' p₀₀₋' p₁₀₋ p₋₀₁'} -- top
{sq₋₁₋ : Square p₋₁₀ p₀₁₋ p₁₁₋ p₋₁₁} -- bottom
{sq₁₋₋ : Square p₁₋₀ p₁₀₋ p₁₁₋ p₁₋₁} -- front
→ Cube sq₋₋₀ sq₋₋₁ sq₀₋₋ (! q₀₀₋ ∙v⊡ (! q₋₀₀ ∙h⊡ sq₋₀₋ ⊡h∙ q₋₀₁) ⊡v∙ q₁₀₋) sq₋₁₋ (! q₁₀₋ ∙v⊡ sq₁₋₋)
→ Cube (q₋₀₀ ∙v⊡ sq₋₋₀) (q₋₀₁ ∙v⊡ sq₋₋₁) (q₀₀₋ ∙v⊡ sq₀₋₋) sq₋₀₋ sq₋₁₋ sq₁₋₋
smgluer-merid-custom-cube-∙v⊡ idp idp idp idp c = c
top-path :
! (ap-cst north (merid y)) ∙v⊡
(! (f-smgluer north) ∙h⊡
natural-square (λ sy → ap f (smgluer sy)) (merid y) ⊡h∙
f-smgluer south) ⊡v∙
ap-cst north (merid y)
==
ids
top-path =
! (ap-cst north (merid y)) ∙v⊡
(! (f-smgluer north) ∙h⊡
natural-square (λ sy → ap f (smgluer sy)) (merid y) ⊡h∙
f-smgluer south) ⊡v∙
ap-cst north (merid y)
=⟨ ap (λ u → ! (ap-cst north (merid y)) ∙v⊡ u ⊡v∙ ap-cst north (merid y)) $
natural-square-path
(λ sy → idp)
(λ sy → ap f (smgluer sy))
f-smgluer
(merid y) ⟩
! (ap-cst north (merid y)) ∙v⊡
natural-square (λ sy → idp) (merid y) ⊡v∙
ap-cst north (merid y)
=⟨ natural-square-cst
north
north
(λ sy → idp)
(merid y) ⟩
disc-to-square (ap (λ sy → idp) (merid y))
=⟨ ap disc-to-square (ap-cst idp (merid y)) ⟩
ids =∎
front-path :
(! (ap-cst north (merid y)) ∙v⊡
natural-square (λ sy → baser) (merid y)) ⊡v∙
ap-cst south (merid y)
==
hid-square
front-path =
(! (ap-cst north (merid y)) ∙v⊡
natural-square (λ sy → baser) (merid y)) ⊡v∙
ap-cst south (merid y)
=⟨ ∙v⊡-⊡v∙-comm (! (ap-cst north (merid y)))
(natural-square (λ sy → baser) (merid y))
(ap-cst south (merid y)) ⟩
! (ap-cst north (merid y)) ∙v⊡
natural-square (λ sy → baser) (merid y) ⊡v∙
ap-cst south (merid y)
=⟨ natural-square-cst
north
south
(λ sy → baser)
(merid y) ⟩
horiz-degen-square (ap (λ sy → baser) (merid y))
=⟨ ap horiz-degen-square (ap-cst baser (merid y)) ⟩
horiz-degen-square idp
=⟨ horiz-degen-square-idp ⟩
hid-square =∎
bot-path :
! (↯ (g-merid-y north y)) ∙v⊡
(! g-smgluer-north ∙h⊡
natural-square (λ sy → ap g (smgluer sy)) (merid y) ⊡h∙
↯ g-smgluer-south) ⊡v∙
ap-cst south (merid y)
==
rt-square (merid north)
bot-path =
! (↯ (g-merid-y north y)) ∙v⊡
(! g-smgluer-north ∙h⊡
natural-square (λ sy → ap g (smgluer sy)) (merid y) ⊡h∙
↯ g-smgluer-south) ⊡v∙
ap-cst south (merid y)
=⟨ ap (λ u → ! (↯ (g-merid-y north y)) ∙v⊡ u ⊡v∙ ap-cst south (merid y)) $
! g-smgluer-north ∙h⊡
natural-square (λ sy → ap g (smgluer sy)) (merid y) ⊡h∙
↯ g-smgluer-south
=⟨ ! $ ap (! g-smgluer-north ∙h⊡_) $
⊡h∙-assoc
(natural-square (λ sy → ap g (smgluer sy)) (merid y))
(g-smgluer south)
(↯ (tail g-smgluer-south)) ⟩
! g-smgluer-north ∙h⊡
natural-square (λ sy → ap g (smgluer sy)) (merid y) ⊡h∙
g-smgluer south ⊡h∙
↯ (tail g-smgluer-south)
=⟨ ! $ ∙h⊡-⊡h∙-comm
(! g-smgluer-north)
(natural-square (λ sy → ap g (smgluer sy)) (merid y) ⊡h∙ g-smgluer south)
(↯ (tail g-smgluer-south)) ⟩
(! g-smgluer-north ∙h⊡
natural-square (λ sy → ap g (smgluer sy)) (merid y) ⊡h∙
g-smgluer south) ⊡h∙
↯ (tail g-smgluer-south)
=⟨ ap (_⊡h∙ ↯ (tail g-smgluer-south)) $
natural-square-path
(ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) ∘ ∧ΣOutSmgluer.f (⊙Susp (de⊙ X)) Y)
(λ sy → ap g (smgluer sy))
g-smgluer
(merid y) ⟩
natural-square (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) ∘ ∧ΣOutSmgluer.f (⊙Susp (de⊙ X)) Y) (merid y) ⊡h∙
↯ (tail g-smgluer-south)
=⟨ ap (_⊡h∙ ↯ (tail g-smgluer-south)) $
natural-square-ap
(Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(∧ΣOutSmgluer.f (⊙Susp (de⊙ X)) Y)
(merid y) ⟩
(ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(natural-square (∧ΣOutSmgluer.f (⊙Susp (de⊙ X)) Y) (merid y)) ⊡v∙
∘-ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (λ _ → north) (merid y)) ⊡h∙
↯ (tail g-smgluer-south)
=⟨ ap (λ u → (ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) u ⊡v∙
∘-ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (λ _ → north) (merid y)) ⊡h∙
↯ (tail g-smgluer-south)) $
∧ΣOutSmgluer.merid-square-β (⊙Susp (de⊙ X)) Y y ⟩
(ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(∧Σ-out-smgluer-merid (⊙Susp (de⊙ X)) Y y) ⊡v∙
∘-ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (λ _ → north) (merid y)) ⊡h∙
↯ (tail g-smgluer-south)
=⟨ ap (λ u → (ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙v⊡
u ⊡v∙
∘-ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (λ _ → north) (merid y)) ⊡h∙
↯ (tail g-smgluer-south)) $ ! $
ap-square-⊡v∙
(Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
((SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y)) ∙v⊡
tr-square (merid (smin north (pt Y))))
(! (ap-cst north (merid y))) ⟩
(ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
((SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y)) ∙v⊡
tr-square (merid (smin north (pt Y)))) ⊡v∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))) (! (ap-cst north (merid y))) ⊡v∙
∘-ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (λ _ → north) (merid y)) ⊡h∙
↯ (tail g-smgluer-south)
=⟨ ap (λ u → (ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙v⊡ u) ⊡h∙
↯ (tail g-smgluer-south)) $
⊡v∙-assoc
(ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
((SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y)) ∙v⊡
tr-square (merid (smin north (pt Y)))))
(ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))) (! (ap-cst north (merid y))))
(∘-ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (λ _ → north) (merid y)) ⟩
(ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
((SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y)) ∙v⊡
tr-square (merid (smin north (pt Y)))) ⊡v∙
(ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))) (! (ap-cst north (merid y))) ∙
∘-ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (λ _ → north) (merid y))) ⊡h∙
↯ (tail g-smgluer-south)
=⟨ ap (λ u → (ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
((SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y)) ∙v⊡
tr-square (merid (smin north (pt Y)))) ⊡v∙
u) ⊡h∙
↯ (tail g-smgluer-south)) $ =ₛ-out $
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))) (! (ap-cst north (merid y))) ◃∙
∘-ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (λ _ → north) (merid y) ◃∎
=ₛ⟨ 1 & 1 &
post-rotate-in {p = _ ◃∎} $
ap-∘-cst-coh (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) north (merid y) ⟩
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))) (! (ap-cst north (merid y))) ◃∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))) (ap-cst north (merid y)) ◃∙
! (ap-cst south (merid y)) ◃∎
=ₛ⟨ 0 & 2 & ap-seq-=ₛ (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))) $
seq-!-inv-l (ap-cst north (merid y) ◃∎) ⟩
! (ap-cst south (merid y)) ◃∎ ∎ₛ ⟩
(ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
((SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y)) ∙v⊡
tr-square (merid (smin north (pt Y)))) ⊡v∙
! (ap-cst south (merid y))) ⊡h∙
↯ (tail g-smgluer-south)
=⟨ ap (λ u → (ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙v⊡ u ⊡v∙
(! (ap-cst south (merid y)))) ⊡h∙
↯ (tail g-smgluer-south)) $ ! $
ap-square-∙v⊡
(Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y))
(tr-square (merid (smin north (pt Y)))) ⟩
(ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙v⊡
(ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y)) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y))))) ⊡v∙
! (ap-cst south (merid y))) ⊡h∙
↯ (tail g-smgluer-south)
=⟨ ! $ ap (_⊡h∙ ↯ (tail g-smgluer-south)) $
∙v⊡-⊡v∙-comm
(ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y))
(ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y)) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y)))))
(! (ap-cst south (merid y))) ⟩
(ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙v⊡
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y)) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y))))) ⊡v∙
! (ap-cst south (merid y)) ⊡h∙
↯ (tail g-smgluer-south)
=⟨ ! $ ap (λ u → u ⊡v∙ ! (ap-cst south (merid y)) ⊡h∙ ↯ (tail g-smgluer-south)) $
∙v⊡-assoc
(ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y))
(ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y)))
(ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y))))) ⟩
((ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y))) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y))))) ⊡v∙
! (ap-cst south (merid y)) ⊡h∙
↯ (tail g-smgluer-south) =∎
⟩
! (↯ (g-merid-y north y)) ∙v⊡
(((ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y))) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y))))) ⊡v∙
! (ap-cst south (merid y)) ⊡h∙
↯ (tail g-smgluer-south)) ⊡v∙
ap-cst south (merid y)
=⟨ ap (! (↯ (g-merid-y north y)) ∙v⊡_) $ ! $
⊡v∙-⊡h∙-comm
(((ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y))) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y))))) ⊡v∙
! (ap-cst south (merid y)))
(ap-cst south (merid y))
(↯ (tail g-smgluer-south)) ⟩
! (↯ (g-merid-y north y)) ∙v⊡
(((ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y))) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y))))) ⊡v∙
! (ap-cst south (merid y)) ⊡v∙
ap-cst south (merid y)) ⊡h∙
↯ (tail g-smgluer-south)
=⟨ ap (λ u → ! (↯ (g-merid-y north y)) ∙v⊡ u ⊡h∙ ↯ (tail g-smgluer-south)) $
⊡v∙-assoc
((ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y))) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y)))))
(! (ap-cst south (merid y)))
(ap-cst south (merid y)) ⟩
! (↯ (g-merid-y north y)) ∙v⊡
(((ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y))) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y))))) ⊡v∙
(! (ap-cst south (merid y)) ∙ ap-cst south (merid y))) ⊡h∙
↯ (tail g-smgluer-south)
=⟨ ap (λ u → ! (↯ (g-merid-y north y)) ∙v⊡
(((ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y))) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y))))) ⊡v∙
u) ⊡h∙
↯ (tail g-smgluer-south)) $
!-inv-l (ap-cst south (merid y)) ⟩
! (↯ (g-merid-y north y)) ∙v⊡
((ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y))) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y))))) ⊡h∙
↯ (tail g-smgluer-south)
=⟨ ap (! (↯ (g-merid-y north y)) ∙v⊡_) $
∙v⊡-⊡h∙-comm
(ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y)))
(ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y)))))
(↯ (tail g-smgluer-south)) ⟩
! (↯ (g-merid-y north y)) ∙v⊡
(ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y))) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y)))) ⊡h∙
↯ (tail g-smgluer-south)
=⟨ ! $
∙v⊡-assoc
(! (↯ (g-merid-y north y)))
(ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y)))
(ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y)))) ⊡h∙
↯ (tail g-smgluer-south)) ⟩
(! (↯ (g-merid-y north y)) ∙
ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y))) ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y)))) ⊡h∙
↯ (tail g-smgluer-south)
=⟨ ap (_∙v⊡ ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y)))) ⊡h∙
↯ (tail g-smgluer-south)) $ =ₛ-out $
! (↯ (g-merid-y north y)) ◃∙
ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ◃∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(SuspFmap.merid-β (smin north) y ∙ ap merid (∧-norm-r y)) ◃∎
=ₛ⟨ 2 & 1 & ap-seq-∙ (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))) $
(SuspFmap.merid-β (smin north) y ◃∙
ap merid (∧-norm-r y) ◃∎) ⟩
! (↯ (g-merid-y north y)) ◃∙
ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ◃∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))) (SuspFmap.merid-β (smin north) y) ◃∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))) (ap merid (∧-norm-r y)) ◃∎
=ₛ⟨ 0 & 1 & !-∙-seq (g-merid-y north y) ⟩
! (SuspFlip.merid-β north) ◃∙
! (ap (ap Susp-flip) (SuspFmap.merid-β (Σ∧-out X Y) (smin north y))) ◃∙
! (ap-∘ Susp-flip (Susp-fmap (Σ∧-out X Y)) (merid (smin north y))) ◃∙
! (ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))) (SuspFmap.merid-β (smin north) y)) ◃∙
! (ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y)) ◃∙
ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ◃∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))) (SuspFmap.merid-β (smin north) y) ◃∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))) (ap merid (∧-norm-r y)) ◃∎
=ₛ⟨ 3 & 4 & seq-!-inv-l $
ap-∘ (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (Susp-fmap (smin north)) (merid y) ◃∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))) (SuspFmap.merid-β (smin north) y) ◃∎ ⟩
! (SuspFlip.merid-β north) ◃∙
! (ap (ap Susp-flip) (SuspFmap.merid-β (Σ∧-out X Y) (smin north y))) ◃∙
! (ap-∘ Susp-flip (Susp-fmap (Σ∧-out X Y)) (merid (smin north y))) ◃∙
ap (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))) (ap merid (∧-norm-r y)) ◃∎
=ₛ⟨ 2 & 2 & !ₛ $
homotopy-naturality
(ap Susp-flip ∘ ap (Susp-fmap (Σ∧-out X Y)))
(ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)))
(λ p → ! (ap-∘ Susp-flip (Susp-fmap (Σ∧-out X Y)) p))
(ap merid (∧-norm-r y)) ⟩
! (SuspFlip.merid-β north) ◃∙
! (ap (ap Susp-flip) (SuspFmap.merid-β (Σ∧-out X Y) (smin north y))) ◃∙
ap (ap Susp-flip ∘ ap (Susp-fmap (Σ∧-out X Y))) (ap merid (∧-norm-r y)) ◃∙
! (ap-∘ Susp-flip (Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y)))) ◃∎
=ₛ₁⟨ 2 & 1 &
∘-ap (ap Susp-flip ∘ ap (Susp-fmap (Σ∧-out X Y))) merid (∧-norm-r y) ⟩
! (SuspFlip.merid-β north) ◃∙
! (ap (ap Susp-flip) (SuspFmap.merid-β (Σ∧-out X Y) (smin north y))) ◃∙
ap (ap Susp-flip ∘ ap (Susp-fmap (Σ∧-out X Y)) ∘ merid) (∧-norm-r y) ◃∙
! (ap-∘ Susp-flip (Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y)))) ◃∎
=ₛ⟨ 1 & 2 & !ₛ $
homotopy-naturality
(ap Susp-flip ∘ merid ∘ Σ∧Out.f X Y)
(ap Susp-flip ∘ ap (Susp-fmap (Σ∧-out X Y)) ∘ merid)
(λ w → ! (ap (ap Susp-flip) (SuspFmap.merid-β (Σ∧-out X Y) w)))
(∧-norm-r y) ⟩
! (SuspFlip.merid-β north) ◃∙
ap (ap Susp-flip ∘ merid ∘ Σ∧-out X Y) (∧-norm-r y) ◃∙
! (ap (ap Susp-flip) (SuspFmap.merid-β (Σ∧-out X Y) (smin north (pt Y)))) ◃∙
! (ap-∘ Susp-flip (Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y)))) ◃∎
=ₛ⟨ 1 & 1 & =ₛ-in {t = []} $
ap-∘ (ap Susp-flip ∘ merid) (Σ∧-out X Y) (∧-norm-r y) ∙
ap (ap (ap Susp-flip ∘ merid)) (Σ∧-out-∧-norm-r X Y y) ⟩
! (SuspFlip.merid-β north) ◃∙
! (ap (ap Susp-flip) (SuspFmap.merid-β (Σ∧-out X Y) (smin north (pt Y)))) ◃∙
! (ap-∘ Susp-flip (Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y)))) ◃∎
=ₛ⟨ ∙-!-seq $
ap-∘ Susp-flip (Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y))) ◃∙
ap (ap Susp-flip) (SuspFmap.merid-β (Σ∧-out X Y) (smin north (pt Y))) ◃∙
SuspFlip.merid-β north ◃∎ ⟩
! g-smgluer-south-step ◃∎ ∎ₛ
⟩
! g-smgluer-south-step ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y)))) ⊡h∙
↯ (tail g-smgluer-south)
=⟨ ! $ ap (! g-smgluer-south-step ∙v⊡_) $
⊡h∙-assoc
(ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y)))))
(ap-! (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y))))
(ap ! g-smgluer-south-step ∙ !-! (merid north)) ⟩
! g-smgluer-south-step ∙v⊡
ap-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y))
(tr-square (merid (smin north (pt Y)))) ⊡h∙
ap-! (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y))) ⊡h∙
(ap ! g-smgluer-south-step ∙ !-! (merid north))
=⟨ ap (λ u → ! g-smgluer-south-step ∙v⊡ u ⊡h∙
(ap ! g-smgluer-south-step ∙ !-! (merid north))) $
ap-tr-square (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y))) ⟩
! g-smgluer-south-step ∙v⊡
tr-square (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y)))) ⊡h∙
(ap ! g-smgluer-south-step ∙ !-! (merid north))
=⟨ ! $ ap (! g-smgluer-south-step ∙v⊡_) $
⊡h∙-assoc
(tr-square (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y)))))
(ap ! g-smgluer-south-step)
(!-! (merid north)) ⟩
! g-smgluer-south-step ∙v⊡
tr-square (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y)))) ⊡h∙
ap ! g-smgluer-south-step ⊡h∙
!-! (merid north)
=⟨ ! $ ∙v⊡-⊡h∙-comm
(! g-smgluer-south-step)
(tr-square (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y)))) ⊡h∙
ap ! g-smgluer-south-step)
(!-! (merid north)) ⟩
(! g-smgluer-south-step ∙v⊡
tr-square (ap (Susp-flip ∘ Susp-fmap (Σ∧-out X Y)) (merid (smin north (pt Y)))) ⊡h∙
ap ! g-smgluer-south-step) ⊡h∙
!-! (merid north)
=⟨ ap (_⊡h∙ !-! (merid north)) $
tr-square-∙v⊡-⊡h∙ g-smgluer-south-step ⟩
tr-square (! (merid north)) ⊡h∙ !-! (merid north)
=⟨ tr-square-! (merid north) ⟩
rt-square (merid north) =∎
custom-cube : ∀ {i} {S : Type i} {n s : S} (p : n == s)
→ Cube hid-square (br-square p) (lb-square p) ids (rt-square p) hid-square
custom-cube p@idp = idc
module ∧Σ-Σ∧-Out =
Σ∧Σ-PathElim
f g
n-n n-s s-n s-s
n-m s-m m-n m-s
m-m
basel
baser
smgluel-north
smgluel-south
smgluel-merid
smgluer-north
smgluer-south
smgluer-merid
∧Σ-Σ∧-out : Susp-fmap (∧Σ-out X Y) ∘ Σ∧-out X (⊙Susp (de⊙ Y))
∼ Susp-flip ∘ Susp-fmap (Σ∧-out X Y) ∘ ∧Σ-out (⊙Susp (de⊙ X)) Y
∧Σ-Σ∧-out = ∧Σ-Σ∧-Out.f
⊙∧Σ-Σ∧-out : ⊙Susp-fmap (∧Σ-out X Y) ◃⊙∘
⊙Σ∧-out X (⊙Susp (de⊙ Y)) ◃⊙idf
=⊙∘
⊙Susp-flip (⊙Susp (X ∧ Y)) ◃⊙∘
⊙Susp-fmap (Σ∧-out X Y) ◃⊙∘
⊙∧Σ-out (⊙Susp (de⊙ X)) Y ◃⊙idf
⊙∧Σ-Σ∧-out =
⊙seq-λ= ∧Σ-Σ∧-out $ !ₛ $
merid north ◃∙
idp ◃∙
idp ◃∙
! (merid north) ◃∎
=ₛ⟨ 1 & 1 & expand [] ⟩
merid north ◃∙
idp ◃∙
! (merid north) ◃∎
=ₛ⟨ 1 & 1 & expand [] ⟩
merid north ◃∙
! (merid north) ◃∎
=ₛ₁⟨ !-inv-r (merid north) ⟩
idp ◃∎
=ₛ⟨ 0 & 0 & contract ⟩
idp ◃∙
idp ◃∎ ∎ₛ
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{-# OPTIONS --type-in-type #-}
open import Data.Unit
open import Data.Product hiding ( curry ; uncurry )
open import Data.List hiding ( concat )
open import Data.String
open import Relation.Binary.PropositionalEquality
module Spire.Examples.PropositionalDesc where
----------------------------------------------------------------------
elimEq : (A : Set) (x : A) (P : (y : A) → x ≡ y → Set)
→ P x refl
→ (y : A) (p : x ≡ y) → P y p
elimEq A .x P prefl x refl = prefl
----------------------------------------------------------------------
Label : Set
Label = String
Enum : Set
Enum = List Label
data Tag : Enum → Set where
here : ∀{l E} → Tag (l ∷ E)
there : ∀{l E} → Tag E → Tag (l ∷ E)
Cases : (E : Enum) (P : Tag E → Set) → Set
Cases [] P = ⊤
Cases (l ∷ E) P = P here × Cases E λ t → P (there t)
case : (E : Enum) (P : Tag E → Set) (cs : Cases E P) (t : Tag E) → P t
case (l ∷ E) P (c , cs) here = c
case (l ∷ E) P (c , cs) (there t) = case E (λ t → P (there t)) cs t
UncurriedCases : (E : Enum) (P : Tag E → Set) (X : Set)
→ Set
UncurriedCases E P X = Cases E P → X
CurriedCases : (E : Enum) (P : Tag E → Set) (X : Set)
→ Set
CurriedCases [] P X = X
CurriedCases (l ∷ E) P X = P here → CurriedCases E (λ t → P (there t)) X
curryCases : (E : Enum) (P : Tag E → Set) (X : Set)
(f : UncurriedCases E P X) → CurriedCases E P X
curryCases [] P X f = f tt
curryCases (l ∷ E) P X f = λ c → curryCases E (λ t → P (there t)) X (λ cs → f (c , cs))
uncurryCases : (E : Enum) (P : Tag E → Set) (X : Set)
(f : CurriedCases E P X) → UncurriedCases E P X
uncurryCases [] P X x tt = x
uncurryCases (l ∷ E) P X f (c , cs) = uncurryCases E (λ t → P (there t)) X (f c) cs
----------------------------------------------------------------------
data Desc (I : Set) : Set₁ where
`End : (i : I) → Desc I
`Rec : (i : I) (D : Desc I) → Desc I
`Arg : (A : Set) (B : A → Desc I) → Desc I
`RecFun : (A : Set) (B : A → I) (D : Desc I) → Desc I
ISet : Set → Set₁
ISet I = I → Set
El : (I : Set) (D : Desc I) (X : ISet I) → ISet I
El I (`End j) X i = j ≡ i
El I (`Rec j D) X i = X j × El I D X i
El I (`Arg A B) X i = Σ A (λ a → El I (B a) X i)
El I (`RecFun A B D) X i = ((a : A) → X (B a)) × El I D X i
Hyps : (I : Set) (D : Desc I) (X : ISet I) (P : (i : I) → X i → Set) (i : I) (xs : El I D X i) → Set
Hyps I (`End j) X P i q = ⊤
Hyps I (`Rec j D) X P i (x , xs) = P j x × Hyps I D X P i xs
Hyps I (`Arg A B) X P i (a , b) = Hyps I (B a) X P i b
Hyps I (`RecFun A B D) X P i (f , xs) = ((a : A) → P (B a) (f a)) × Hyps I D X P i xs
caseD : (E : Enum) (I : Set) (cs : Cases E (λ _ → Desc I)) (t : Tag E) → Desc I
caseD E I cs t = case E (λ _ → Desc I) cs t
----------------------------------------------------------------------
TagDesc : (I : Set) → Set
TagDesc I = Σ Enum (λ E → Cases E (λ _ → Desc I))
toCase : (I : Set) (E,cs : TagDesc I) → Tag (proj₁ E,cs) → Desc I
toCase I (E , cs) = case E (λ _ → Desc I) cs
toDesc : (I : Set) → TagDesc I → Desc I
toDesc I (E , cs) = `Arg (Tag E) (toCase I (E , cs))
----------------------------------------------------------------------
UncurriedEl : (I : Set) (D : Desc I) (X : ISet I) → Set
UncurriedEl I D X = {i : I} → El I D X i → X i
CurriedEl : (I : Set) (D : Desc I) (X : ISet I) → Set
CurriedEl I (`End i) X = X i
CurriedEl I (`Rec j D) X = (x : X j) → CurriedEl I D X
CurriedEl I (`Arg A B) X = (a : A) → CurriedEl I (B a) X
CurriedEl I (`RecFun A B D) X = ((a : A) → X (B a)) → CurriedEl I D X
curryEl : (I : Set) (D : Desc I) (X : ISet I)
(cn : UncurriedEl I D X) → CurriedEl I D X
curryEl I (`End i) X cn = cn refl
curryEl I (`Rec i D) X cn = λ x → curryEl I D X (λ xs → cn (x , xs))
curryEl I (`Arg A B) X cn = λ a → curryEl I (B a) X (λ xs → cn (a , xs))
curryEl I (`RecFun A B D) X cn = λ f → curryEl I D X (λ xs → cn (f , xs))
uncurryEl : (I : Set) (D : Desc I) (X : ISet I)
(cn : CurriedEl I D X) → UncurriedEl I D X
uncurryEl I (`End i) X cn refl = cn
uncurryEl I (`Rec i D) X cn (x , xs) = uncurryEl I D X (cn x) xs
uncurryEl I (`Arg A B) X cn (a , xs) = uncurryEl I (B a) X (cn a) xs
uncurryEl I (`RecFun A B D) X cn (f , xs) = uncurryEl I D X (cn f) xs
data μ (I : Set) (D : Desc I) : I → Set where
con : UncurriedEl I D (μ I D)
con2 : (I : Set) (D : Desc I) → CurriedEl I D (μ I D)
con2 I D = curryEl I D (μ I D) con
----------------------------------------------------------------------
UncurriedHyps : (I : Set) (D : Desc I) (X : ISet I)
(P : (i : I) → X i → Set)
(cn : UncurriedEl I D X)
→ Set
UncurriedHyps I D X P cn =
(i : I) (xs : El I D X i) → Hyps I D X P i xs → P i (cn xs)
CurriedHyps : (I : Set) (D : Desc I) (X : ISet I)
(P : (i : I) → X i → Set)
(cn : UncurriedEl I D X)
→ Set
CurriedHyps I (`End i) X P cn =
P i (cn refl)
CurriedHyps I (`Rec i D) X P cn =
(x : X i) → P i x → CurriedHyps I D X P (λ xs → cn (x , xs))
CurriedHyps I (`Arg A B) X P cn =
(a : A) → CurriedHyps I (B a) X P (λ xs → cn (a , xs))
CurriedHyps I (`RecFun A B D) X P cn =
(f : (a : A) → X (B a)) (ihf : (a : A) → P (B a) (f a)) → CurriedHyps I D X P (λ xs → cn (f , xs))
curryHyps : (I : Set) (D : Desc I) (X : ISet I)
(P : (i : I) → X i → Set)
(cn : UncurriedEl I D X)
(pf : UncurriedHyps I D X P cn)
→ CurriedHyps I D X P cn
curryHyps I (`End i) X P cn pf =
pf i refl tt
curryHyps I (`Rec i D) X P cn pf =
λ x ih → curryHyps I D X P (λ xs → cn (x , xs)) (λ i xs ihs → pf i (x , xs) (ih , ihs))
curryHyps I (`Arg A B) X P cn pf =
λ a → curryHyps I (B a) X P (λ xs → cn (a , xs)) (λ i xs ihs → pf i (a , xs) ihs)
curryHyps I (`RecFun A B D) X P cn pf =
λ f ihf → curryHyps I D X P (λ xs → cn (f , xs)) (λ i xs ihs → pf i (f , xs) (ihf , ihs))
uncurryHyps : (I : Set) (D : Desc I) (X : ISet I)
(P : (i : I) → X i → Set)
(cn : UncurriedEl I D X)
(pf : CurriedHyps I D X P cn)
→ UncurriedHyps I D X P cn
uncurryHyps I (`End .i) X P cn pf i refl tt =
pf
uncurryHyps I (`Rec j D) X P cn pf i (x , xs) (ih , ihs) =
uncurryHyps I D X P (λ ys → cn (x , ys)) (pf x ih) i xs ihs
uncurryHyps I (`Arg A B) X P cn pf i (a , xs) ihs =
uncurryHyps I (B a) X P (λ ys → cn (a , ys)) (pf a) i xs ihs
uncurryHyps I (`RecFun A B D) X P cn pf i (f , xs) (ihf , ihs) =
uncurryHyps I D X P (λ ys → cn (f , ys)) (pf f ihf) i xs ihs
----------------------------------------------------------------------
ind :
(I : Set)
(D : Desc I)
(P : (i : I) → μ I D i → Set)
(pcon : UncurriedHyps I D (μ I D) P con)
(i : I)
(x : μ I D i)
→ P i x
hyps :
(I : Set)
(D₁ : Desc I)
(P : (i : I) → μ I D₁ i → Set)
(pcon : UncurriedHyps I D₁ (μ I D₁) P con)
(D₂ : Desc I)
(i : I)
(xs : El I D₂ (μ I D₁) i)
→ Hyps I D₂ (μ I D₁) P i xs
ind I D P pcon i (con xs) = pcon i xs (hyps I D P pcon D i xs)
hyps I D P pcon (`End j) i q = tt
hyps I D P pcon (`Rec j A) i (x , xs) = ind I D P pcon j x , hyps I D P pcon A i xs
hyps I D P pcon (`Arg A B) i (a , b) = hyps I D P pcon (B a) i b
hyps I D P pcon (`RecFun A B E) i (f , xs) = (λ a → ind I D P pcon (B a) (f a)) , hyps I D P pcon E i xs
----------------------------------------------------------------------
ind2 :
(I : Set)
(D : Desc I)
(P : (i : I) → μ I D i → Set)
(pcon : CurriedHyps I D (μ I D) P con)
(i : I)
(x : μ I D i)
→ P i x
ind2 I D P pcon i x = ind I D P (uncurryHyps I D (μ I D) P con pcon) i x
elim :
(I : Set)
(TD : TagDesc I)
→ let
D = toDesc I TD
E = proj₁ TD
Cs = toCase I TD
in (P : (i : I) → μ I D i → Set)
→ let
Q = λ t → CurriedHyps I (Cs t) (μ I D) P (λ xs → con (t , xs))
X = (i : I) (x : μ I D i) → P i x
in UncurriedCases E Q X
elim I TD P cs i x =
let
D = toDesc I TD
E = proj₁ TD
Cs = toCase I TD
Q = λ t → CurriedHyps I (Cs t) (μ I D) P (λ xs → con (t , xs))
p = case E Q cs
in ind2 I D P p i x
elim2 :
(I : Set)
(TD : TagDesc I)
→ let
D = toDesc I TD
E = proj₁ TD
Cs = toCase I TD
in (P : (i : I) → μ I D i → Set)
→ let
Q = λ t → CurriedHyps I (Cs t) (μ I D) P (λ xs → con (t , xs))
X = (i : I) (x : μ I D i) → P i x
in CurriedCases E Q X
elim2 I TD P =
let
D = toDesc I TD
E = proj₁ TD
Cs = toCase I TD
Q = λ t → CurriedHyps I (Cs t) (μ I D) P (λ xs → con (t , xs))
X = (i : I) (x : μ I D i) → P i x
in curryCases E Q X (elim I TD P)
----------------------------------------------------------------------
module Sugared where
data ℕT : Set where `zero `suc : ℕT
data VecT : Set where `nil `cons : VecT
ℕD : Desc ⊤
ℕD = `Arg ℕT λ
{ `zero → `End tt
; `suc → `Rec tt (`End tt)
}
ℕ : ⊤ → Set
ℕ = μ ⊤ ℕD
zero : ℕ tt
zero = con (`zero , refl)
suc : ℕ tt → ℕ tt
suc n = con (`suc , n , refl)
VecD : (A : Set) → Desc (ℕ tt)
VecD A = `Arg VecT λ
{ `nil → `End zero
; `cons → `Arg (ℕ tt) λ n → `Arg A λ _ → `Rec n (`End (suc n))
}
Vec : (A : Set) (n : ℕ tt) → Set
Vec A n = μ (ℕ tt) (VecD A) n
nil : (A : Set) → Vec A zero
nil A = con (`nil , refl)
cons : (A : Set) (n : ℕ tt) (x : A) (xs : Vec A n) → Vec A (suc n)
cons A n x xs = con (`cons , n , x , xs , refl)
----------------------------------------------------------------------
add : ℕ tt → ℕ tt → ℕ tt
add = ind ⊤ ℕD (λ _ _ → ℕ tt → ℕ tt)
(λ
{ tt (`zero , q) tt n → n
; tt (`suc , m , q) (ih , tt) n → suc (ih n)
}
)
tt
mult : ℕ tt → ℕ tt → ℕ tt
mult = ind ⊤ ℕD (λ _ _ → ℕ tt → ℕ tt)
(λ
{ tt (`zero , q) tt n → zero
; tt (`suc , m , q) (ih , tt) n → add n (ih n)
}
)
tt
append : (A : Set) (m : ℕ tt) (xs : Vec A m) (n : ℕ tt) (ys : Vec A n) → Vec A (add m n)
append A = ind (ℕ tt) (VecD A) (λ m xs → (n : ℕ tt) (ys : Vec A n) → Vec A (add m n))
(λ
{ .(con (`zero , refl)) (`nil , refl) ih n ys → ys
; .(con (`suc , m , refl)) (`cons , m , x , xs , refl) (ih , tt) n ys → cons A (add m n) x (ih n ys)
}
)
concat : (A : Set) (m n : ℕ tt) (xss : Vec (Vec A m) n) → Vec A (mult n m)
concat A m = ind (ℕ tt) (VecD (Vec A m)) (λ n xss → Vec A (mult n m))
(λ
{ .(con (`zero , refl)) (`nil , refl) tt → nil A
; .(con (`suc , n , refl)) (`cons , n , xs , xss , refl) (ih , tt) → append A m xs (mult n m) ih
}
)
----------------------------------------------------------------------
module Desugared where
ℕT : Enum
ℕT = "zero" ∷ "suc" ∷ []
VecT : Enum
VecT = "nil" ∷ "cons" ∷ []
ℕTD : TagDesc ⊤
ℕTD = ℕT
, `End tt
, `Rec tt (`End tt)
, tt
ℕCs : Tag ℕT → Desc ⊤
ℕCs = toCase ⊤ ℕTD
ℕD : Desc ⊤
ℕD = toDesc ⊤ ℕTD
ℕ : ⊤ → Set
ℕ = μ ⊤ ℕD
zero : ℕ tt
zero = con (here , refl)
suc : ℕ tt → ℕ tt
suc n = con (there here , n , refl)
zero2 : ℕ tt
zero2 = con2 ⊤ ℕD here
suc2 : ℕ tt → ℕ tt
suc2 = con2 ⊤ ℕD (there here)
VecTD : (A : Set) → TagDesc (ℕ tt)
VecTD A = VecT
, `End zero
, `Arg (ℕ tt) (λ n → `Arg A λ _ → `Rec n (`End (suc n)))
, tt
VecCs : (A : Set) → Tag VecT → Desc (ℕ tt)
VecCs A = toCase (ℕ tt) (VecTD A)
VecD : (A : Set) → Desc (ℕ tt)
VecD A = toDesc (ℕ tt) (VecTD A)
Vec : (A : Set) (n : ℕ tt) → Set
Vec A n = μ (ℕ tt) (VecD A) n
nil : (A : Set) → Vec A zero
nil A = con (here , refl)
cons : (A : Set) (n : ℕ tt) (x : A) (xs : Vec A n) → Vec A (suc n)
cons A n x xs = con (there here , n , x , xs , refl)
nil2 : (A : Set) → Vec A zero
nil2 A = con2 (ℕ tt) (VecD A) here
cons2 : (A : Set) (n : ℕ tt) (x : A) (xs : Vec A n) → Vec A (suc n)
cons2 A = con2 (ℕ tt) (VecD A) (there here)
----------------------------------------------------------------------
module Induction where
add : ℕ tt → ℕ tt → ℕ tt
add = ind ⊤ ℕD (λ _ _ → ℕ tt → ℕ tt)
(λ u t,c → case ℕT
(λ t → (c : El ⊤ (ℕCs t) ℕ u)
(ih : Hyps ⊤ ℕD ℕ (λ u n → ℕ u → ℕ u) u (t , c))
→ ℕ u → ℕ u
)
( (λ q ih n → n)
, (λ m,q ih,tt n → suc (proj₁ ih,tt n))
, tt
)
(proj₁ t,c)
(proj₂ t,c)
)
tt
mult : ℕ tt → ℕ tt → ℕ tt
mult = ind ⊤ ℕD (λ _ _ → ℕ tt → ℕ tt)
(λ u t,c → case ℕT
(λ t → (c : El ⊤ (ℕCs t) ℕ u)
(ih : Hyps ⊤ ℕD ℕ (λ u n → ℕ u → ℕ u) u (t , c))
→ ℕ u → ℕ u
)
( (λ q ih n → zero)
, (λ m,q ih,tt n → add n (proj₁ ih,tt n))
, tt
)
(proj₁ t,c)
(proj₂ t,c)
)
tt
append : (A : Set) (m : ℕ tt) (xs : Vec A m) (n : ℕ tt) (ys : Vec A n) → Vec A (add m n)
append A = ind (ℕ tt) (VecD A) (λ m xs → (n : ℕ tt) (ys : Vec A n) → Vec A (add m n))
(λ m t,c → case VecT
(λ t → (c : El (ℕ tt) (VecCs A t) (Vec A) m)
(ih : Hyps (ℕ tt) (VecD A) (Vec A) (λ m xs → (n : ℕ tt) (ys : Vec A n) → Vec A (add m n)) m (t , c))
(n : ℕ tt) (ys : Vec A n) → Vec A (add m n)
)
( (λ q ih n ys → subst (λ m → Vec A (add m n)) q ys)
, (λ m',x,xs,q ih,tt n ys →
let m' = proj₁ m',x,xs,q
x = proj₁ (proj₂ m',x,xs,q)
q = proj₂ (proj₂ (proj₂ m',x,xs,q))
ih = proj₁ ih,tt
in
subst (λ m → Vec A (add m n)) q (cons A (add m' n) x (ih n ys))
)
, tt
)
(proj₁ t,c)
(proj₂ t,c)
)
concat : (A : Set) (m n : ℕ tt) (xss : Vec (Vec A m) n) → Vec A (mult n m)
concat A m = ind (ℕ tt) (VecD (Vec A m)) (λ n xss → Vec A (mult n m))
(λ n t,c → case VecT
(λ t → (c : El (ℕ tt) (VecCs (Vec A m) t) (Vec (Vec A m)) n)
(ih : Hyps (ℕ tt) (VecD (Vec A m)) (Vec (Vec A m)) (λ n xss → Vec A (mult n m)) n (t , c))
→ Vec A (mult n m)
)
( (λ q ih → subst (λ n → Vec A (mult n m)) q (nil A))
, (λ n',xs,xss,q ih,tt →
let n' = proj₁ n',xs,xss,q
xs = proj₁ (proj₂ n',xs,xss,q)
q = proj₂ (proj₂ (proj₂ n',xs,xss,q))
ih = proj₁ ih,tt
in
subst (λ n → Vec A (mult n m)) q (append A m xs (mult n' m) ih)
)
, tt
)
(proj₁ t,c)
(proj₂ t,c)
)
----------------------------------------------------------------------
module Eliminator where
elimℕ : (P : (ℕ tt) → Set)
(pzero : P zero)
(psuc : (m : ℕ tt) → P m → P (suc m))
(n : ℕ tt)
→ P n
elimℕ P pzero psuc = ind ⊤ ℕD (λ u n → P n)
(λ u t,c → case ℕT
(λ t → (c : El ⊤ (ℕCs t) ℕ u)
(ih : Hyps ⊤ ℕD ℕ (λ u n → P n) u (t , c))
→ P (con (t , c))
)
( (λ q ih →
elimEq ⊤ tt (λ u q → P (con (here , q)))
pzero
u q
)
, (λ n,q ih,tt →
elimEq ⊤ tt (λ u q → P (con (there here , proj₁ n,q , q)))
(psuc (proj₁ n,q) (proj₁ ih,tt))
u (proj₂ n,q)
)
, tt
)
(proj₁ t,c)
(proj₂ t,c)
)
tt
elimVec : (A : Set) (P : (n : ℕ tt) → Vec A n → Set)
(pnil : P zero (nil A))
(pcons : (n : ℕ tt) (a : A) (xs : Vec A n) → P n xs → P (suc n) (cons A n a xs))
(n : ℕ tt)
(xs : Vec A n)
→ P n xs
elimVec A P pnil pcons = ind (ℕ tt) (VecD A) (λ n xs → P n xs)
(λ n t,c → case VecT
(λ t → (c : El (ℕ tt) (VecCs A t) (Vec A) n)
(ih : Hyps (ℕ tt) (VecD A) (Vec A) (λ n xs → P n xs) n (t , c))
→ P n (con (t , c))
)
( (λ q ih →
elimEq (ℕ tt) zero (λ n q → P n (con (here , q)))
pnil
n q
)
, (λ n',x,xs,q ih,tt →
let n' = proj₁ n',x,xs,q
x = proj₁ (proj₂ n',x,xs,q)
xs = proj₁ (proj₂ (proj₂ n',x,xs,q))
q = proj₂ (proj₂ (proj₂ n',x,xs,q))
ih = proj₁ ih,tt
in
elimEq (ℕ tt) (suc n') (λ n q → P n (con (there here , n' , x , xs , q)))
(pcons n' x xs ih )
n q
)
, tt
)
(proj₁ t,c)
(proj₂ t,c)
)
----------------------------------------------------------------------
add : ℕ tt → ℕ tt → ℕ tt
add = elimℕ (λ _ → ℕ tt → ℕ tt)
(λ n → n)
(λ m ih n → suc (ih n))
mult : ℕ tt → ℕ tt → ℕ tt
mult = elimℕ (λ _ → ℕ tt → ℕ tt)
(λ n → zero)
(λ m ih n → add n (ih n))
append : (A : Set) (m : ℕ tt) (xs : Vec A m) (n : ℕ tt) (ys : Vec A n) → Vec A (add m n)
append A = elimVec A (λ m xs → (n : ℕ tt) (ys : Vec A n) → Vec A (add m n))
(λ n ys → ys)
(λ m x xs ih n ys → cons A (add m n) x (ih n ys))
concat : (A : Set) (m n : ℕ tt) (xss : Vec (Vec A m) n) → Vec A (mult n m)
concat A m = elimVec (Vec A m) (λ n xss → Vec A (mult n m))
(nil A)
(λ n xs xss ih → append A m xs (mult n m) ih)
----------------------------------------------------------------------
module GenericEliminator where
add : ℕ tt → ℕ tt → ℕ tt
add = elim2 ⊤ ℕTD _
(λ n → n)
(λ m ih n → suc (ih n))
tt
mult : ℕ tt → ℕ tt → ℕ tt
mult = elim2 ⊤ ℕTD _
(λ n → zero)
(λ m ih n → add n (ih n))
tt
append : (A : Set) (m : ℕ tt) (xs : Vec A m) (n : ℕ tt) (ys : Vec A n) → Vec A (add m n)
append A = elim2 (ℕ tt) (VecTD A) _
(λ n ys → ys)
(λ m x xs ih n ys → cons A (add m n) x (ih n ys))
concat : (A : Set) (m n : ℕ tt) (xss : Vec (Vec A m) n) → Vec A (mult n m)
concat A m = elim2 (ℕ tt) (VecTD (Vec A m)) _
(nil A)
(λ n xs xss ih → append A m xs (mult n m) ih)
----------------------------------------------------------------------
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{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
open import Numbers.Naturals.Semiring
open import Numbers.Naturals.Order
open import LogicalFormulae
open import Orders.Total.Definition
open import Orders.Partial.Definition
open import Setoids.Setoids
open import Functions.Definition
open import Sequences
open import Setoids.Orders.Partial.Definition
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Setoids.Orders.Partial.Sequences {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_<_ : Rel {a} {c} A} (p : SetoidPartialOrder S _<_) where
open SetoidPartialOrder p
WeaklyIncreasing' : (Sequence A) → Set (b ⊔ c)
WeaklyIncreasing' s = (m n : ℕ) → (m <N n) → (index s m) <= (index s n)
WeaklyIncreasing : (Sequence A) → Set (b ⊔ c)
WeaklyIncreasing s = (m : ℕ) → (index s m) <= index s (succ m)
tailRespectsWeaklyIncreasing : (a : Sequence A) → WeaklyIncreasing a → WeaklyIncreasing (Sequence.tail a)
tailRespectsWeaklyIncreasing a incr m = incr (succ m)
weaklyIncreasingImplies' : (a : Sequence A) → WeaklyIncreasing a → WeaklyIncreasing' a
weaklyIncreasingImplies' a x zero (succ zero) m<n = x 0
weaklyIncreasingImplies' a x zero (succ (succ n)) m<n = <=Transitive (weaklyIncreasingImplies' a x zero (succ n) (succIsPositive n)) (x (succ n))
weaklyIncreasingImplies' a x (succ m) (succ n) m<n = weaklyIncreasingImplies' (Sequence.tail a) (tailRespectsWeaklyIncreasing a x) m n (canRemoveSuccFrom<N m<n)
weaklyIncreasing'Implies : (a : Sequence A) → WeaklyIncreasing' a → WeaklyIncreasing a
weaklyIncreasing'Implies a incr m = incr m (succ m) (le 0 refl)
StrictlyIncreasing' : (Sequence A) → Set (c)
StrictlyIncreasing' s = (m n : ℕ) → (m <N n) → (index s m) < (index s n)
StrictlyIncreasing : (Sequence A) → Set (c)
StrictlyIncreasing s = (m : ℕ) → (index s m) < index s (succ m)
tailRespectsStrictlyIncreasing : (a : Sequence A) → StrictlyIncreasing a → StrictlyIncreasing (Sequence.tail a)
tailRespectsStrictlyIncreasing a incr m = incr (succ m)
strictlyIncreasingImplies' : (a : Sequence A) → StrictlyIncreasing a → StrictlyIncreasing' a
strictlyIncreasingImplies' a x zero (succ zero) m<n = x 0
strictlyIncreasingImplies' a x zero (succ (succ n)) m<n = <Transitive (strictlyIncreasingImplies' a x zero (succ n) (succIsPositive n)) (x (succ n))
strictlyIncreasingImplies' a x (succ m) (succ n) m<n = strictlyIncreasingImplies' (Sequence.tail a) (tailRespectsStrictlyIncreasing a x) m n (canRemoveSuccFrom<N m<n)
strictlyIncreasing'Implies : (a : Sequence A) → StrictlyIncreasing' a → StrictlyIncreasing a
strictlyIncreasing'Implies a incr m = incr m (succ m) (le 0 refl)
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open import Level using (Level; suc; zero; _⊔_)
open import Function using (const)
open import Algebra
open import Algebra.Structures
open import Algebra.OrderedMonoid
open import Algebra.Pregroup
open import Algebra.FunctionProperties as FunctionProperties using (Op₁; Op₂)
open import Data.Product using (_×_; _,_; proj₁; proj₂)
open import Relation.Binary
open import Relation.Binary.PartialOrderReasoning as ≤-Reasoning using ()
module Algebra.PregroupPlus where
record IsPregroupPlus
{a ℓ₁ ℓ₂} {A : Set a} (≈ : Rel A ℓ₁) (≤ : Rel A ℓ₂)
(∙ : Op₂ A) (ε₁ : A) (ˡ : Op₁ A) (ʳ : Op₁ A)
(+ : Op₂ A) (ε₀ : A) (ᵘ : Op₁ A) (ᵈ : Op₁ A)
: Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
field
∙-isPregroup : IsPregroup ≈ ≤ ∙ ε₁ ˡ ʳ
+-isPregroup : IsPregroup ≈ ≤ + ε₀ ᵘ ᵈ
open IsPregroup ∙-isPregroup public
using (ˡ-cong; ʳ-cong; ˡ-contract; ʳ-contract
;ˡ-expand; ʳ-expand; ∙-cong)
renaming (assoc to ∙-assoc; identity to ∙-identity
;compatibility to ∙-compatibility)
open IsPregroup +-isPregroup public
using (≤-refl; ≤-reflexive; ≤-trans; ≤-resp-≈; refl; sym; trans; antisym)
renaming (ˡ-cong to ᵈ-cong; ʳ-cong to ᵘ-cong
;ˡ-contract to ᵈ-contract; ʳ-contract to ᵘ-contract
;ˡ-expand to ᵈ-expand; ʳ-expand to ᵘ-expand
;∙-cong to +-cong; assoc to +-assoc; identity to +-identity
;compatibility to +-compatibility)
record PregroupPlus c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
infixl 7 _∙_
infixl 6 _+_
infix 4 _≈_
infix 4 _≤_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ₁
_≤_ : Rel Carrier ℓ₂
_∙_ : Op₂ Carrier
ε₁ : Carrier
_ˡ : Op₁ Carrier
_ʳ : Op₁ Carrier
_+_ : Op₂ Carrier
ε₀ : Carrier
_ᵘ : Op₁ Carrier
_ᵈ : Op₁ Carrier
isPregroupPlus : IsPregroupPlus _≈_ _≤_ _∙_ ε₁ _ˡ _ʳ _+_ ε₀ _ᵘ _ᵈ
open IsPregroupPlus isPregroupPlus public
∙-pregroup : Pregroup _ _ _
∙-pregroup = record { isPregroup = ∙-isPregroup }
open Pregroup ∙-pregroup public using (
ˡ-unique; ˡ-identity; ˡ-distrib; ˡ-contra; ˡʳ-cancel;
ʳ-unique; ʳ-identity; ʳ-distrib; ʳ-contra; ʳˡ-cancel)
+-pregroup : Pregroup _ _ _
+-pregroup = record { isPregroup = +-isPregroup }
open Pregroup +-pregroup public using () renaming (
ˡ-unique to ᵈ-unique; ˡ-identity to ᵈ-identity; ˡ-distrib to ᵈ-distrib;
ˡ-contra to ᵈ-contra; ˡʳ-cancel to ᵈᵘ-cancel;
ʳ-unique to ᵘ-unique; ʳ-identity to ᵘ-identity; ʳ-distrib to ᵘ-distrib;
ʳ-contra to ᵘ-contra; ʳˡ-cancel to ᵘᵈ-calcel)
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-- Everything is strictly positive, but Agda doesn't see this.
{-# OPTIONS --no-positivity-check #-}
module Generic.Core where
open import Generic.Lib.Prelude public
infix 4 _≤ℓ_
infixr 5 _⇒_ _⊛_
_≤ℓ_ : Level -> Level -> Set
α ≤ℓ β = α ⊔ β ≡ β
mutual
Binder : ∀ {ι} α β γ -> ArgInfo -> ι ⊔ lsuc (α ⊔ β) ≡ γ -> Set ι -> Set γ
Binder α β γ i q I = Coerce q (∃ λ (A : Set α) -> < relevance i > A -> Desc I β)
data Desc {ι} (I : Set ι) β : Set (ι ⊔ lsuc β) where
var : I -> Desc I β
π : ∀ {α} i
-> (q : α ≤ℓ β)
-> Binder α β _ i (cong (λ αβ -> ι ⊔ lsuc αβ) q) I
-> Desc I β
_⊛_ : Desc I β -> Desc I β -> Desc I β
pattern DPi i A D = π i refl (coerce (A , D))
{-# DISPLAY π i refl (coerce (A , D)) = DPi i A D #-}
pattern explRelDPi A D = DPi explRelInfo A D
pattern explIrrDPi A D = DPi explIrrInfo A D
pattern implRelDPi A D = DPi implRelInfo A D
pattern implIrrDPi A D = DPi implIrrInfo A D
pattern instRelDPi A D = DPi instRelInfo A D
pattern instIrrDPi A D = DPi instIrrInfo A D
{-# DISPLAY DPi explRelInfo A D = explRelDPi A D #-}
{-# DISPLAY DPi explIrrInfo A D = explIrrDPi A D #-}
{-# DISPLAY DPi implRelInfo A D = implRelDPi A D #-}
{-# DISPLAY DPi implIrrInfo A D = implIrrDPi A D #-}
{-# DISPLAY DPi instRelInfo A D = instRelDPi A D #-}
{-# DISPLAY DPi instIrrInfo A D = instIrrDPi A D #-}
_⇒_ : ∀ {ι α β} {I : Set ι} {{q : α ≤ℓ β}} -> Set α -> Desc I β -> Desc I β
_⇒_ {{q}} A D = π (explRelInfo) q (qcoerce (A , λ _ -> D))
mutual
⟦_⟧ : ∀ {ι β} {I : Set ι} -> Desc I β -> (I -> Set β) -> Set β
⟦ var i ⟧ B = B i
⟦ π i q C ⟧ B = ⟦ i / C ⟧ᵇ q B
⟦ D ⊛ E ⟧ B = ⟦ D ⟧ B × ⟦ E ⟧ B
⟦_/_⟧ᵇ : ∀ {α ι β γ q} {I : Set ι} i
-> Binder α β γ i q I -> α ≤ℓ β -> (I -> Set β) -> Set β
⟦ i / coerce (A , D) ⟧ᵇ q B = Coerce′ q $ Pi i A λ x -> ⟦ D x ⟧ B
mutual
Extend : ∀ {ι β} {I : Set ι} -> Desc I β -> (I -> Set β) -> I -> Set β
Extend (var i) B j = Lift _ (i ≡ j)
Extend (π i q C) B j = Extendᵇ i C q B j
Extend (D ⊛ E) B j = ⟦ D ⟧ B × Extend E B j
Extendᵇ : ∀ {ι α β γ q} {I : Set ι} i
-> Binder α β γ i q I -> α ≤ℓ β -> (I -> Set β) -> I -> Set β
Extendᵇ i (coerce (A , D)) q B j = Coerce′ q $ ∃ λ x -> Extend (D x) B j
module _ {ι β} {I : Set ι} (D : Data (Desc I β)) where
mutual
data μ j : Set β where
node : Node D j -> μ j
Node : Data (Desc I β) -> I -> Set β
Node D j = Any (λ C -> Extend C μ j) (consTypes D)
mutual
Cons : ∀ {ι β} {I : Set ι} -> (I -> Set β) -> Desc I β -> Set β
Cons B (var i) = B i
Cons B (π i q C) = Consᵇ B i C q
Cons B (D ⊛ E) = ⟦ D ⟧ B -> Cons B E
Consᵇ : ∀ {ι α β γ q} {I : Set ι}
-> (I -> Set β) -> ∀ i -> Binder α β γ i q I -> α ≤ℓ β -> Set β
Consᵇ B i (coerce (A , D)) q = Coerce′ q $ Pi i A λ x -> Cons B (D x)
cons : ∀ {ι β} {I : Set ι} {D} -> (D₀ : Data (Desc I β)) -> D ∈ consTypes D₀ -> Cons (μ D₀) D
cons {D = D} D₀ p = go D λ e ->
node (mapAny (consTypes D₀) (λ q -> subst (λ E -> Extend E _ _) q e) p) where
mutual
go : ∀ {ι β} {I : Set ι} {B : I -> Set β}
-> (D : Desc I β) -> (∀ {j} -> Extend D B j -> B j) -> Cons B D
go (var i) k = k lrefl
go (π a q C) k = goᵇ a C k
go (D ⊛ E) k = λ x -> go E (k ∘ _,_ x)
goᵇ : ∀ {ι α β γ q q′} {I : Set ι} {B : I -> Set β} i
-> (C : Binder α β γ i q′ I) -> (∀ {j} -> Extendᵇ i C q B j -> B j) -> Consᵇ B i C q
goᵇ {q = q} i (coerce (A , D)) k =
coerce′ q $ lamPi i λ x -> go (D x) (k ∘ coerce′ q ∘ _,_ x)
allCons : ∀ {ι β} {I : Set ι} -> (D : Data (Desc I β)) -> All (Cons (μ D)) (consTypes D)
allCons D = allIn _ (cons D)
node-inj : ∀ {i β} {I : Set i} {D : Data (Desc I β)} {j} {e₁ e₂ : Node D D j}
-> node {D = D} e₁ ≡ node e₂ -> e₁ ≡ e₂
node-inj refl = refl
μ′ : ∀ {β} -> Data (Desc ⊤₀ β) -> Set β
μ′ D = μ D tt
pos : ∀ {β} -> Desc ⊤₀ β
pos = var tt
pattern #₀ p = node (inj₁ p)
pattern #₁ p = node (inj₂ (inj₁ p))
pattern #₂ p = node (inj₂ (inj₂ (inj₁ p)))
pattern #₃ p = node (inj₂ (inj₂ (inj₂ (inj₁ p))))
pattern #₄ p = node (inj₂ (inj₂ (inj₂ (inj₂ (inj₁ p)))))
pattern #₅ p = node (inj₂ (inj₂ (inj₂ (inj₂ (inj₂ (inj₁ p))))))
pattern !#₀ p = node p
pattern !#₁ p = node (inj₂ p)
pattern !#₂ p = node (inj₂ (inj₂ p))
pattern !#₃ p = node (inj₂ (inj₂ (inj₂ p)))
pattern !#₄ p = node (inj₂ (inj₂ (inj₂ (inj₂ p))))
pattern !#₅ p = node (inj₂ (inj₂ (inj₂ (inj₂ (inj₂ p)))))
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{-# OPTIONS --cubical --safe --postfix-projections #-}
module Data.Rational.Unnormalised where
open import Prelude
open import Data.Integer using (ℤ; ⁺)
import Data.Integer as ℤ
import Data.Nat as ℕ
open import Data.Nat.DivMod using (nonZero)
infixl 7 _/_ _/suc_
record ℚ : Type where
constructor _/suc_
field
num : ℤ
den-pred : ℕ
den : ℕ
den = suc den-pred
open ℚ public
_/_ : (n : ℤ) → (d : ℕ) → ⦃ d≢0 : T (nonZero d) ⦄ → ℚ
n / suc d = n /suc d
{-# DISPLAY _/suc_ n d = n / suc d #-}
infixl 6 _+_
_+_ : ℚ → ℚ → ℚ
(x + y) .num = num x ℤ.* ⁺ (den y) ℤ.+ num y ℤ.* ⁺ (den x)
(x + y) .den-pred = x .den-pred ℕ.+ y .den-pred ℕ.+ x .den-pred ℕ.* y .den-pred
infixl 7 _*_
_*_ : ℚ → ℚ → ℚ
(x * y) .num = x .num ℤ.* y .num
(x * y) .den-pred = x .den-pred ℕ.+ y .den-pred ℕ.+ x .den-pred ℕ.* y .den-pred
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------------------------------------------------------------------------
-- The Agda standard library
--
-- This module is DEPRECATED. Please use
-- Data.List.Relation.Binary.Permutation.Propositional.Properties
-- directly.
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.List.Relation.Binary.Permutation.Inductive.Properties where
{-# WARNING_ON_IMPORT
"Data.List.Relation.Binary.Permutation.Inductive.Properties was deprecated in v1.1.
Use Data.List.Relation.Binary.Permutation.Propositional.Properties instead."
#-}
open import Data.List.Relation.Binary.Permutation.Propositional.Properties public
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-- This example comes from the discussion on Issue423.
module SolveNeutralApplication where
postulate
A : Set
a b : A
T : A → Set
mkT : ∀ a → T a
data Bool : Set where
true false : Bool
f : Bool → A → A
f true x = a
f false x = b
-- We can solve the constraint
-- f x _4 == f x y
-- with
-- _4 := y
-- since the application of f is neutral.
g : (x : Bool)(y : A) → T (f x y)
g x y = mkT (f x _)
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-- ASR (31 December 2015). The error message for this test was changed
-- by fixing Issue 1763.
module Issue586 where
{-# NO_TERMINATION_CHECK #-}
Foo : Set
Foo = Foo
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-- Andreas, 2017-07-29, issue and test case by Nisse
{-# OPTIONS --profile=interactive #-}
A : Set
A = {!Set!} -- Give to provoke error
-- This issue concerns the AgdaInfo buffer,
-- the behavior on the command line might be different.
-- ERROR WAS (note the "Total 0ms"):
-- Set₁ != Set
-- when checking that the expression Set has type SetTotal 0ms
-- Expected:
-- Set₁ != Set
-- when checking that the expression Set has type Set
-- Issue2602.out:
-- ...
-- ((last . 1) . (agda2-goals-action '(0)))
-- (agda2-verbose "Total 0ms ")
-- (agda2-info-action "*Error*" "1,1-4 Set₁ != Set when checking that the expression Set has type Set" nil)
-- ...
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import cedille-options
open import general-util
module untyped-spans (options : cedille-options.options) {F : Set → Set} {{monadF : monad F}} where
open import lib
open import ctxt
open import cedille-types
open import conversion
open import spans options {F}
open import syntax-util
open import to-string options
open import is-free
{-# TERMINATING #-}
untyped-term-spans : term → spanM ⊤
untyped-type-spans : type → spanM ⊤
untyped-kind-spans : kind → spanM ⊤
untyped-tk-spans : tk → spanM ⊤
untyped-liftingType-spans : liftingType → spanM ⊤
untyped-optTerm-spans : optTerm → spanM (posinfo → posinfo)
untyped-optType-spans : optType → spanM ⊤
untyped-optGuide-spans : optGuide → spanM (𝕃 tagged-val)
untyped-lterms-spans : lterms → spanM ⊤
untyped-optClass-spans : optClass → spanM ⊤
untyped-defTermOrType-spans : posinfo → (ctxt → posinfo → var → (atk : tk) → (if tk-is-type atk then term else type) → span) → defTermOrType → spanM ⊤ → spanM ⊤
untyped-var-spans : posinfo → var → (ctxt → posinfo → var → checking-mode → 𝕃 tagged-val → err-m → span) → spanM ⊤ → spanM ⊤
untyped-caseArgs-spans : caseArgs → (body : term) → spanM (𝕃 tagged-val)
untyped-case-spans : case → (ℕ → ℕ → err-m) → spanM ((ℕ → ℕ → err-m) × 𝕃 tagged-val)
untyped-cases-spans : cases → spanM (err-m × 𝕃 tagged-val)
untyped-var-spans pi x f m = get-ctxt λ Γ → with-ctxt (ctxt-var-decl-loc pi x Γ) (get-ctxt λ Γ → spanM-add (f Γ pi x untyped [] nothing) ≫span m)
untyped-term-spans (App t me t') = untyped-term-spans t ≫span untyped-term-spans t' ≫span spanM-add (App-span ff t t' untyped [] nothing)
untyped-term-spans (AppTp t T) = untyped-term-spans t ≫span untyped-type-spans T ≫span spanM-add (AppTp-span t T untyped [] nothing)
untyped-term-spans (Beta pi ot ot') = untyped-optTerm-spans ot ≫=span λ f → untyped-optTerm-spans ot' ≫=span λ f' → spanM-add (Beta-span pi (f' (f (posinfo-plus pi 1))) untyped [] nothing)
untyped-term-spans (Chi pi mT t) = untyped-optType-spans mT ≫span untyped-term-spans t ≫span get-ctxt λ Γ → spanM-add (Chi-span Γ pi mT t untyped [] nothing)
untyped-term-spans (Delta pi mT t) = untyped-optType-spans mT ≫span untyped-term-spans t ≫span get-ctxt λ Γ → spanM-add (Delta-span Γ pi mT t untyped [] nothing)
untyped-term-spans (Epsilon pi lr mm t) = untyped-term-spans t ≫span spanM-add (Epsilon-span pi lr mm t untyped [] nothing)
untyped-term-spans (Hole pi) = get-ctxt λ Γ → spanM-add (hole-span Γ pi nothing [])
untyped-term-spans (IotaPair pi t t' og pi') = untyped-term-spans t ≫span untyped-term-spans t' ≫span untyped-optGuide-spans og ≫=span λ tvs → spanM-add (IotaPair-span pi pi' untyped tvs nothing)
untyped-term-spans (IotaProj t n pi) = untyped-term-spans t ≫span spanM-add (IotaProj-span t pi untyped [] nothing)
untyped-term-spans (Lam pi me pi' x oc t) =
untyped-optClass-spans oc
≫span get-ctxt λ Γ → spanM-add (Lam-span Γ untyped pi pi' me x (Tkt $ TpHole pi) t [] occursCheck)
≫span untyped-var-spans pi' x Var-span (untyped-term-spans t)
where
occursCheck = maybe-if (me && is-free-in skip-erased x t) ≫maybe just "The bound variable occurs free in the erasure of the body (not allowed)"
untyped-term-spans (Let pi fe d t) =
untyped-defTermOrType-spans pi (λ Γ pi' x atk val → Let-span Γ untyped pi pi' fe x atk val t [] nothing) d (untyped-term-spans t)
-- ≫span get-ctxt λ Γ → spanM-add (Let-span Γ untyped pi d t [] nothing)
untyped-term-spans (Open pi o pi' x t) = get-ctxt λ Γ → spanM-add (Open-span Γ o pi' x t untyped [] nothing) ≫span spanM-add (Var-span Γ pi' x untyped [] (maybe-not (ctxt-lookup-term-loc Γ x) ≫maybe just "This term variable is not currently in scope")) ≫span untyped-term-spans t
untyped-term-spans (Parens pi t pi') = untyped-term-spans t
untyped-term-spans (Phi pi t t' t'' pi') = untyped-term-spans t ≫span untyped-term-spans t' ≫span untyped-term-spans t'' ≫span spanM-add (Phi-span pi pi' untyped [] nothing)
untyped-term-spans (Rho pi op on t og t') = untyped-term-spans t ≫span untyped-term-spans t' ≫span untyped-optGuide-spans og ≫=span λ tvs → spanM-add (mk-span "Rho" pi (term-end-pos t') (ll-data-term :: checking-data untyped :: tvs) nothing)
untyped-term-spans (Sigma pi t) = untyped-term-spans t ≫span get-ctxt λ Γ → spanM-add (mk-span "Sigma" pi (term-end-pos t) (ll-data-term :: [ checking-data untyped ]) nothing)
untyped-term-spans (Theta pi θ t ls) = untyped-term-spans t ≫span untyped-lterms-spans ls ≫span get-ctxt λ Γ → spanM-add (Theta-span Γ pi θ t ls untyped [] nothing)
untyped-term-spans (Var pi x) = get-ctxt λ Γ →
spanM-add (Var-span Γ pi x untyped [] (if ctxt-binds-var Γ x then nothing else just "This variable is not currently in scope."))
untyped-term-spans (Mu pi pi' x t ot pi'' cs pi''') = get-ctxt λ Γ → untyped-term-spans t ≫span with-ctxt (ctxt-var-decl x Γ) (get-ctxt λ Γ → spanM-add (Var-span Γ pi' x untyped [ binder-data (ctxt-var-decl-loc pi' x Γ) pi' x (Tkt (TpHole pi')) NotErased nothing pi'' pi''' ] nothing) ≫span untyped-cases-spans cs) ≫=span uncurry λ e ts → spanM-add (Mu-span Γ pi (just x) pi''' (optType-elim ot nothing just) untyped ts e)
untyped-term-spans (Mu' pi ot t oT pi' cs pi'') = get-ctxt λ Γ → untyped-optTerm-spans ot ≫span untyped-term-spans t ≫span untyped-optType-spans oT ≫span untyped-cases-spans cs ≫=span uncurry λ e ts → spanM-add (Mu-span Γ pi nothing pi'' (optType-elim oT nothing just) untyped ts e)
untyped-caseArgs-spans [] t = untyped-term-spans t ≫span spanMr []
untyped-caseArgs-spans (c :: cs) t with caseArg-to-var c
...| pi , x , me , ll =
let e? = maybe-if (me && is-free-in skip-erased x (caseArgs-to-lams cs t)) ≫maybe
just "The bound variable occurs free in the erasure of the body (not allowed)"
f = if ll then Var-span else TpVar-span in
get-ctxt λ Γ →
spanM-add (f (ctxt-var-decl-loc pi x Γ) pi x untyped [] e?) ≫span
with-ctxt (ctxt-var-decl x Γ) (untyped-caseArgs-spans cs t) ≫=span λ ts →
spanMr (binder-data (ctxt-var-decl x Γ) pi x (if ll then Tkt (TpHole pi) else Tkk star) me nothing (term-start-pos t) (term-end-pos t) :: ts)
untyped-case-spans (Case pi x cas t) fₑ =
get-ctxt λ Γ →
let m = untyped-caseArgs-spans cas t
x' = unqual-all (ctxt-get-qualif Γ) $ unqual-local x
eᵤ = just $ "This is not a valid constructor name"
eₗ = just $ "Constructor's datatype has a different number of constructors than " ^ x'
eᵢ = just $ "This constructor overlaps with " ^ x' in
case qual-lookup Γ x of λ where
(just (as , ctr-def ps? T Cₗ cᵢ cₐ , _ , _)) →
spanM-add (Var-span Γ pi x untyped [] $ fₑ Cₗ cᵢ) ≫span m ≫=span λ s →
spanMr ((λ Cₗ' cᵢ' → if Cₗ =ℕ Cₗ' then if cᵢ =ℕ cᵢ' then eᵢ else nothing else eₗ) , s)
_ →
spanM-add (Var-span Γ pi x untyped [] eᵤ) ≫span m ≫=span λ s →
spanMr ((λ _ _ → nothing) , s)
untyped-cases-spans ms =
let eₗ = just $ "Constructor's datatype should have " ^ ℕ-to-string (length ms) ^
" constructor" ^ (if 1 =ℕ length ms then "" else "s") in
(λ c → foldr c (λ _ → spanMr (nothing , [])) ms λ Cₗ cᵢ → if Cₗ =ℕ length ms then nothing else eₗ)
λ c m fₑ → untyped-case-spans c fₑ ≫=span uncurry λ e s →
m e ≫=span (spanMr ∘ map-snd (s ++_))
untyped-type-spans (Abs pi me pi' x atk T) = untyped-tk-spans atk ≫span untyped-var-spans pi' x (if tk-is-type atk then Var-span else TpVar-span) (get-ctxt λ Γ → spanM-add (TpQuant-span Γ (~ me) pi pi' x atk T untyped [] nothing) ≫span untyped-type-spans T)
untyped-type-spans (Iota pi pi' x T T') = untyped-type-spans T ≫span untyped-var-spans pi' x TpVar-span (get-ctxt λ Γ → spanM-add (Iota-span Γ pi pi' x T' untyped [] nothing) ≫span untyped-type-spans T')
untyped-type-spans (Lft pi pi' x t lT) = untyped-liftingType-spans lT ≫span untyped-var-spans pi' x Var-span (get-ctxt λ Γ → spanM-add (Lft-span Γ pi pi' x t untyped [] nothing) ≫span untyped-term-spans t)
untyped-type-spans (NoSpans T pi) = spanMok
untyped-type-spans (TpApp T T') = untyped-type-spans T ≫span untyped-type-spans T' ≫span spanM-add (TpApp-span T T' untyped [] nothing)
untyped-type-spans (TpAppt T t) = untyped-type-spans T ≫span untyped-term-spans t ≫span spanM-add (TpAppt-span T t untyped [] nothing)
untyped-type-spans (TpArrow T a T') = untyped-type-spans T ≫span untyped-type-spans T' ≫span spanM-add (TpArrow-span T T' untyped [] nothing)
untyped-type-spans (TpEq pi t t' pi') = untyped-term-spans t ≫span untyped-term-spans t' ≫span spanM-add (TpEq-span pi t t' pi' untyped [] nothing)
untyped-type-spans (TpHole pi) = get-ctxt λ Γ → spanM-add (tp-hole-span Γ pi nothing [])
untyped-type-spans (TpLambda pi pi' x atk T) = untyped-tk-spans atk ≫span untyped-var-spans pi' x TpVar-span (get-ctxt λ Γ → spanM-add (TpLambda-span Γ pi pi' x atk T untyped [] nothing) ≫span untyped-type-spans T)
untyped-type-spans (TpParens pi T pi') = untyped-type-spans T
untyped-type-spans (TpVar pi x) = get-ctxt λ Γ →
spanM-add (TpVar-span Γ pi x untyped [] (if ctxt-binds-var Γ x then nothing else just "This variable is not currently in scope."))
untyped-type-spans (TpLet pi d T) =
untyped-defTermOrType-spans pi (λ Γ pi' x atk val → TpLet-span Γ untyped pi pi' x atk val T [] nothing) d (untyped-type-spans T)
--≫span get-ctxt λ Γ → spanM-add (TpLet-span Γ untyped pi d T [] nothing)
untyped-kind-spans (KndArrow k k') = untyped-kind-spans k ≫span untyped-kind-spans k' ≫span spanM-add (KndArrow-span k k' untyped nothing)
untyped-kind-spans (KndParens pi k pi') = untyped-kind-spans k
untyped-kind-spans (KndPi pi pi' x atk k) = untyped-tk-spans atk ≫span untyped-var-spans pi' x (if tk-is-type atk then Var-span else TpVar-span) (get-ctxt λ Γ → spanM-add (KndPi-span Γ pi pi' x atk k untyped nothing) ≫span untyped-kind-spans k)
untyped-kind-spans (KndTpArrow T k) = untyped-type-spans T ≫span untyped-kind-spans k ≫span spanM-add (KndTpArrow-span T k untyped nothing)
untyped-kind-spans (KndVar pi x as) = get-ctxt λ Γ →
spanM-add (KndVar-span Γ (pi , x) (kvar-end-pos pi x as) [] untyped [] (if ctxt-binds-var Γ x then nothing else just "This variable is not currently in scope."))
untyped-kind-spans (Star pi) = spanM-add (Star-span pi untyped nothing)
untyped-liftingType-spans lT = spanMok -- Unimplemented
untyped-tk-spans (Tkt T) = untyped-type-spans T
untyped-tk-spans (Tkk k) = untyped-kind-spans k
untyped-optTerm-spans NoTerm = spanMr λ pi → pi
untyped-optTerm-spans (SomeTerm t pi) = untyped-term-spans t ≫span spanMr λ _ → pi
untyped-optType-spans NoType = spanMok
untyped-optType-spans (SomeType T) = untyped-type-spans T
untyped-optGuide-spans NoGuide = spanMr []
untyped-optGuide-spans (Guide pi x T) = untyped-var-spans pi x Var-span (untyped-type-spans T) ≫span get-ctxt λ Γ → spanMr [ binder-data Γ pi x (Tkt $ TpHole pi) NotErased nothing (type-start-pos T) (type-end-pos T) ]
untyped-lterms-spans [] = spanMok
untyped-lterms-spans ((Lterm me t) :: ls) = untyped-term-spans t ≫span untyped-lterms-spans ls
untyped-optClass-spans NoClass = spanMok
untyped-optClass-spans (SomeClass atk) = untyped-tk-spans atk
untyped-defTermOrType-spans pi s (DefTerm pi' x NoType t) m =
untyped-term-spans t ≫span
get-ctxt λ Γ → with-ctxt (ctxt-var-decl-loc pi' x Γ) $
get-ctxt λ Γ → spanM-add (s Γ pi' x (Tkt $ TpHole pi') t) ≫span
spanM-add (Var-span Γ pi' x untyped [] nothing) ≫span m
untyped-defTermOrType-spans pi s (DefTerm pi' x (SomeType tp) t) m =
untyped-type-spans tp ≫span
untyped-term-spans t ≫span
get-ctxt λ Γ → with-ctxt (ctxt-var-decl-loc pi' x Γ) $
get-ctxt λ Γ → spanM-add (s Γ pi' x (Tkt $ TpHole pi') t) ≫span
spanM-add (Var-span Γ pi' x untyped [] nothing) ≫span m
untyped-defTermOrType-spans pi s (DefType pi' x k tp) m =
untyped-kind-spans k ≫span
untyped-type-spans tp ≫span
get-ctxt λ Γ → with-ctxt (ctxt-var-decl-loc pi' x Γ) $
get-ctxt λ Γ → spanM-add (s Γ pi' x (Tkk k) tp) ≫span
spanM-add (TpVar-span Γ pi' x untyped [] nothing) ≫span m
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module Builtin where
open import Agda.Primitive
data Bool : Set where
false : Bool
true : Bool
not : Bool -> Bool
not true = false
not false = true
_||_ : Bool -> Bool -> Bool
true || _ = true
false || x = x
_&&_ : Bool -> Bool -> Bool
true && x = x
false && _ = false
{-# BUILTIN BOOL Bool #-}
{-# BUILTIN TRUE true #-}
{-# BUILTIN FALSE false #-}
data Nat : Set where
zero : Nat
suc : Nat -> Nat
{-# BUILTIN NATURAL Nat #-}
zero' : Nat
zero' = 0
one : Nat
one = 1
data Int : Set where
pos : Nat → Int
negsuc : Nat → Int
postulate
String : Set
Float : Set
Char : Set
{-# BUILTIN INTEGER Int #-}
{-# BUILTIN INTEGERPOS pos #-}
{-# BUILTIN INTEGERNEGSUC negsuc #-}
{-# BUILTIN STRING String #-}
{-# BUILTIN FLOAT Float #-}
{-# BUILTIN CHAR Char #-}
data Maybe (A : Set) : Set where
just : A → Maybe A
nothing : Maybe A
{-# BUILTIN MAYBE Maybe #-}
record Sigma {a b} (A : Set a) (B : A → Set b) : Set (a ⊔ b) where
constructor _,_
field
fst : A
snd : B fst
open Sigma public
infixr 4 _,_
{-# BUILTIN SIGMA Sigma #-}
infixr 10 _::_
data List (A : Set) : Set where
nil : List A
_::_ : A -> List A -> List A
{-# BUILTIN LIST List #-}
postulate Word64 : Set
{-# BUILTIN WORD64 Word64 #-}
primitive
-- Integer functions
primShowInteger : Int -> String
-- Floating point function
primFloatInequality : Float → Float → Bool
primFloatEquality : Float → Float → Bool
primFloatLess : Float → Float → Bool
primFloatIsInfinite : Float → Bool
primFloatIsNaN : Float → Bool
primFloatIsNegativeZero : Float → Bool
primFloatToWord64 : Float → Word64
primNatToFloat : Nat → Float
primIntToFloat : Int → Float
primFloatRound : Float → Maybe Int
primFloatFloor : Float → Maybe Int
primFloatCeiling : Float → Maybe Int
primFloatToRatio : Float → (Sigma Int λ _ → Int)
primRatioToFloat : Int → Int → Float
primFloatDecode : Float → Maybe (Sigma Int λ _ → Int)
primFloatEncode : Int → Int → Maybe Float
primShowFloat : Float → String
primFloatPlus : Float → Float → Float
primFloatMinus : Float → Float → Float
primFloatTimes : Float → Float → Float
primFloatDiv : Float → Float → Float
primFloatNegate : Float → Float
primFloatSqrt : Float → Float
primFloatExp : Float → Float
primFloatLog : Float → Float
primFloatSin : Float → Float
primFloatCos : Float → Float
primFloatTan : Float → Float
primFloatASin : Float → Float
primFloatACos : Float → Float
primFloatATan : Float → Float
primFloatATan2 : Float → Float → Float
primFloatSinh : Float → Float
primFloatCosh : Float → Float
primFloatTanh : Float → Float
primFloatASinh : Float → Float
primFloatACosh : Float → Float
primFloatATanh : Float → Float
primFloatPow : Float → Float → Float
-- Character functions
primCharEquality : Char -> Char -> Bool
primIsLower : Char -> Bool
primIsDigit : Char -> Bool
primIsAlpha : Char -> Bool
primIsSpace : Char -> Bool
primIsAscii : Char -> Bool
primIsLatin1 : Char -> Bool
primIsPrint : Char -> Bool
primIsHexDigit : Char -> Bool
primToUpper : Char -> Char
primToLower : Char -> Char
primCharToNat : Char -> Nat
primNatToChar : Nat -> Char -- partial
primShowChar : Char -> String
-- String functions
primStringToList : String -> List Char
primStringFromList : List Char -> String
primStringAppend : String -> String -> String
primStringEquality : String -> String -> Bool
primShowString : String -> String
isLower : Char -> Bool
isLower = primIsLower
isAlpha : Char -> Bool
isAlpha = primIsAlpha
isUpper : Char -> Bool
isUpper c = isAlpha c && not (isLower c)
infixl 14 _/_
nat0 = primCharToNat '\0'
int0 = pos nat0
_/_ = primFloatDiv
pi : Float
pi = 3.141592653589793
sin : Float -> Float
sin = primFloatSin
cos : Float -> Float
cos = primFloatCos
tan : Float -> Float
tan = primFloatTan
reverse : {A : Set} -> List A -> List A
reverse xs = rev xs nil
where
rev : {A : Set} -> List A -> List A -> List A
rev nil ys = ys
rev (x :: xs) ys = rev xs (x :: ys)
infixr 20 _∘_
_∘_ : {A B C : Set} -> (B -> C) -> (A -> B) -> A -> C
f ∘ g = \x -> f (g x)
map : {A B : Set} -> (A -> B) -> List A -> List B
map f nil = nil
map f (x :: xs) = f x :: map f xs
stringAsList : (List Char -> List Char) -> String -> String
stringAsList f = primStringFromList ∘ f ∘ primStringToList
revStr : String -> String
revStr = stringAsList reverse
mapStr : (Char -> Char) -> String -> String
mapStr f = stringAsList (map f)
-- Testing unicode literals
uString = "åäö⊢ξ∀"
uChar = '∀'
data _≡_ {a} {A : Set a} (x : A) : A → Set a where
refl : x ≡ x
thm-show-pos : primShowInteger (pos 42) ≡ "42"
thm-show-pos = refl
thm-show-neg : primShowInteger (negsuc 41) ≡ "-42"
thm-show-neg = refl
thm-floor : primFloatFloor 4.2 ≡ just (pos 4)
thm-floor = refl
thm-ceiling : primFloatCeiling -5.1 ≡ just (negsuc 4)
thm-ceiling = refl
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module PiFrac.Interp where
open import Data.Empty
open import Data.Unit hiding (_≟_)
open import Data.Sum
open import Data.Product
open import Data.Maybe
open import Relation.Binary.Core
open import Relation.Binary
open import Relation.Nullary
open import Relation.Binary.PropositionalEquality
open import Function using (_∘_)
open import PiFrac.Syntax
open import PiFrac.Opsem
interp : {t₁ t₂ : 𝕌} → (t₁ ↔ t₂) → ⟦ t₁ ⟧ → Maybe ⟦ t₂ ⟧
interp unite₊l (inj₂ v) = just v
interp uniti₊l v = just (inj₂ v)
interp swap⋆ (v₁ , v₂) = just (v₂ , v₁)
interp swap₊ (inj₁ v) = just (inj₂ v)
interp swap₊ (inj₂ v) = just (inj₁ v)
interp assocl₊ (inj₁ v) = just (inj₁ (inj₁ v))
interp assocl₊ (inj₂ (inj₁ v)) = just (inj₁ (inj₂ v))
interp assocl₊ (inj₂ (inj₂ v)) = just (inj₂ v)
interp assocr₊ (inj₁ (inj₁ v)) = just (inj₁ v)
interp assocr₊ (inj₁ (inj₂ v)) = just (inj₂ (inj₁ v))
interp assocr₊ (inj₂ v) = just (inj₂ (inj₂ v))
interp unite⋆l v = just (proj₂ v)
interp uniti⋆l v = just (tt , v)
interp assocl⋆ (v₁ , v₂ , v₃) = just ((v₁ , v₂) , v₃)
interp assocr⋆ ((v₁ , v₂) , v₃) = just (v₁ , v₂ , v₃)
interp dist (inj₁ v₁ , v₃) = just (inj₁ (v₁ , v₃))
interp dist (inj₂ v₂ , v₃) = just (inj₂ (v₂ , v₃))
interp factor (inj₁ (v₁ , v₃)) = just (inj₁ v₁ , v₃)
interp factor (inj₂ (v₂ , v₃)) = just (inj₂ v₂ , v₃)
interp id↔ v = just v
interp (c₁ ⊕ c₂) (inj₁ v) = interp c₁ v >>= just ∘ inj₁
interp (c₁ ⊕ c₂) (inj₂ v) = interp c₂ v >>= just ∘ inj₂
interp (c₁ ⊗ c₂) (v₁ , v₂) = interp c₁ v₁ >>=
(λ v₁' → interp c₂ v₂ >>=
λ v₂' → just (v₁' , v₂'))
interp (c₁ ⨾ c₂) v = interp c₁ v >>= interp c₂
interp (ηₓ v) tt = just (v , ↻)
interp (εₓ v) (v' , ○) with v ≟ v'
... | yes _ = just tt
... | no _ = nothing
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------------------------------------------------------------------------------
-- Elimination properties for the inequalities
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
module FOTC.Data.Nat.Inequalities.EliminationPropertiesATP where
open import FOTC.Base
open import FOTC.Data.Nat
open import FOTC.Data.Nat.Inequalities
------------------------------------------------------------------------------
0<0→⊥ : ¬ (zero < zero)
0<0→⊥ h = prf
where postulate prf : ⊥
{-# ATP prove prf #-}
x<0→⊥ : ∀ {n} → N n → ¬ (n < zero)
x<0→⊥ nzero h = prf
where postulate prf : ⊥
{-# ATP prove prf #-}
x<0→⊥ (nsucc Nn) h = prf
where postulate prf : ⊥
{-# ATP prove prf #-}
x<x→⊥ : ∀ {n} → N n → ¬ (n < n)
x<x→⊥ nzero h = prf
where postulate prf : ⊥
{-# ATP prove prf #-}
x<x→⊥ (nsucc {n} Nn) h = prf (x<x→⊥ Nn)
where postulate prf : ¬ (n < n) → ⊥
{-# ATP prove prf #-}
0>x→⊥ : ∀ {n} → N n → ¬ (zero > n)
0>x→⊥ nzero h = prf
where postulate prf : ⊥
{-# ATP prove prf #-}
0>x→⊥ (nsucc Nn) h = prf
where postulate prf : ⊥
{-# ATP prove prf #-}
S≤0→⊥ : ∀ {n} → N n → ¬ (succ₁ n ≤ zero)
S≤0→⊥ nzero h = prf
where postulate prf : ⊥
{-# ATP prove prf #-}
S≤0→⊥ (nsucc {n} Nn) h = prf
where postulate prf : ⊥
{-# ATP prove prf #-}
0≥S→⊥ : ∀ {n} → N n → ¬ (zero ≥ succ₁ n)
0≥S→⊥ Nn h = prf
where postulate prf : ⊥
{-# ATP prove prf S≤0→⊥ #-}
postulate S≯0→⊥ : ∀ {n} → ¬ (succ₁ n ≯ zero)
{-# ATP prove S≯0→⊥ #-}
x<y→y<x→⊥ : ∀ {m n} → N m → N n → m < n → ¬ (n < m)
x<y→y<x→⊥ nzero Nn h₁ h₂ = prf
where postulate prf : ⊥
{-# ATP prove prf 0>x→⊥ #-}
x<y→y<x→⊥ (nsucc Nm) nzero h₁ h₂ = prf
where postulate prf : ⊥
{-# ATP prove prf 0>x→⊥ #-}
x<y→y<x→⊥ (nsucc {m} Nm) (nsucc {n} Nn) h₁ h₂ = prf (x<y→y<x→⊥ Nm Nn m<n)
where
postulate m<n : m < n
{-# ATP prove m<n #-}
postulate prf : ¬ (n < m) → ⊥
{-# ATP prove prf #-}
x>y→x≤y→⊥ : ∀ {m n} → N m → N n → m > n → ¬ (m ≤ n)
x>y→x≤y→⊥ nzero Nn h₁ h₂ = prf
where postulate prf : ⊥
{-# ATP prove prf x<0→⊥ #-}
x>y→x≤y→⊥ (nsucc Nm) nzero h₁ h₂ = prf
where postulate prf : ⊥
{-# ATP prove prf x<0→⊥ #-}
x>y→x≤y→⊥ (nsucc {m} Nm) (nsucc {n} Nn) h₁ h₂ = prf (x>y→x≤y→⊥ Nm Nn m>n)
where
postulate m>n : m > n
{-# ATP prove m>n #-}
postulate prf : ¬ (m ≤ n) → ⊥
{-# ATP prove prf #-}
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{-# OPTIONS --without-K --rewriting #-}
open import HoTT
open import cohomology.Theory
module cohomology.Sphere {i} (OT : OrdinaryTheory i) where
open OrdinaryTheory OT
abstract
C-Sphere-≠-is-trivial : (n : ℤ) (m : ℕ) → (n ≠ ℕ-to-ℤ m)
→ is-trivialᴳ (C n (⊙Lift (⊙Sphere m)))
C-Sphere-≠-is-trivial n O n≠0 = C-dimension n≠0
C-Sphere-≠-is-trivial n (S m) n≠Sm = iso-preserves'-trivial
(C n (⊙Lift (⊙Sphere (S m)))
≃ᴳ⟨ C-emap n $ ⊙Susp-Lift-econv (⊙Sphere m) ⟩
C n (⊙Susp (⊙Lift (⊙Sphere m)))
≃ᴳ⟨ transportᴳ-iso (λ n → C n (⊙Susp (⊙Lift (⊙Sphere m)))) (succ-pred n) ⁻¹ᴳ ⟩
C (succ (pred n)) (⊙Susp (⊙Lift (⊙Sphere m)))
≃ᴳ⟨ C-Susp (pred n) (⊙Lift (⊙Sphere m)) ⟩
C (pred n) (⊙Lift (⊙Sphere m))
≃ᴳ∎)
(C-Sphere-≠-is-trivial (pred n) m (pred-≠ n≠Sm))
C-Sphere-diag : (m : ℕ) → C (ℕ-to-ℤ m) (⊙Lift (⊙Sphere m)) ≃ᴳ C2 0
C-Sphere-diag O = idiso _
C-Sphere-diag (S m) =
C (ℕ-to-ℤ (S m)) (⊙Lift (⊙Sphere (S m)))
≃ᴳ⟨ C-emap (ℕ-to-ℤ (S m)) (⊙Susp-Lift-econv (⊙Sphere m)) ⟩
C (ℕ-to-ℤ (S m)) (⊙Susp (⊙Lift (⊙Sphere m)))
≃ᴳ⟨ C-Susp (ℕ-to-ℤ m) (⊙Lift (⊙Sphere m)) ⟩
C (ℕ-to-ℤ m) (⊙Lift (⊙Sphere m))
≃ᴳ⟨ C-Sphere-diag m ⟩
C 0 (⊙Lift ⊙S⁰)
≃ᴳ∎
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{-# OPTIONS --without-K #-}
open import HoTT
open import cohomology.Exactness
open import cohomology.Theory
{- Cohomology functor sends constant functions to constant functions -}
module cohomology.ConstantFunction {i} (CT : CohomologyTheory i) where
open import cohomology.Unit CT
open CohomologyTheory CT
module _ (n : ℤ) {X Y : Ptd i} where
CF-cst : CF-hom n (⊙cst {X = X} {Y = Y}) == cst-hom
CF-cst =
CF-hom n (⊙cst {X = PLU} ⊙∘ ⊙cst {X = X})
=⟨ CF-comp n ⊙cst ⊙cst ⟩
(CF-hom n (⊙cst {X = X})) ∘ᴳ (CF-hom n (⊙cst {X = PLU}))
=⟨ hom= (CF-hom n (⊙cst {X = PLU})) cst-hom
(λ= (λ _ → prop-has-all-paths (C-Unit-is-prop n) _ _))
|in-ctx (λ w → CF-hom n (⊙cst {X = X} {Y = PLU}) ∘ᴳ w) ⟩
(CF-hom n (⊙cst {X = X} {Y = PLU})) ∘ᴳ cst-hom
=⟨ pre∘-cst-hom (CF-hom n (⊙cst {X = X} {Y = PLU})) ⟩
cst-hom ∎
where
PLU = ⊙Lift {j = i} ⊙Unit
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-- Andreas, 2018-10-29, issue #3331
-- Document that using and renaming lists cannot overlap.
open import Agda.Builtin.Bool using (true) renaming (true to tt)
-- Expected error:
-- Repeated name in import directive: true
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Many properties which hold for _∼_ also hold for _∼_ on f
------------------------------------------------------------------------
open import Relation.Binary
module Relation.Binary.On where
open import Function
open import Data.Product
module _ {a b} {A : Set a} {B : Set b} (f : B → A) where
implies : ∀ {ℓ₁ ℓ₂} (≈ : Rel A ℓ₁) (∼ : Rel A ℓ₂) →
≈ ⇒ ∼ → (≈ on f) ⇒ (∼ on f)
implies _ _ impl = impl
reflexive : ∀ {ℓ} (∼ : Rel A ℓ) → Reflexive ∼ → Reflexive (∼ on f)
reflexive _ refl = refl
irreflexive : ∀ {ℓ₁ ℓ₂} (≈ : Rel A ℓ₁) (∼ : Rel A ℓ₂) →
Irreflexive ≈ ∼ → Irreflexive (≈ on f) (∼ on f)
irreflexive _ _ irrefl = irrefl
symmetric : ∀ {ℓ} (∼ : Rel A ℓ) → Symmetric ∼ → Symmetric (∼ on f)
symmetric _ sym = sym
transitive : ∀ {ℓ} (∼ : Rel A ℓ) → Transitive ∼ → Transitive (∼ on f)
transitive _ trans = trans
antisymmetric : ∀ {ℓ₁ ℓ₂} (≈ : Rel A ℓ₁) (≤ : Rel A ℓ₂) →
Antisymmetric ≈ ≤ → Antisymmetric (≈ on f) (≤ on f)
antisymmetric _ _ antisym = antisym
asymmetric : ∀ {ℓ} (< : Rel A ℓ) → Asymmetric < → Asymmetric (< on f)
asymmetric _ asym = asym
respects : ∀ {ℓ p} (∼ : Rel A ℓ) (P : A → Set p) →
P Respects ∼ → (P ∘ f) Respects (∼ on f)
respects _ _ resp = resp
respects₂ : ∀ {ℓ₁ ℓ₂} (∼₁ : Rel A ℓ₁) (∼₂ : Rel A ℓ₂) →
∼₁ Respects₂ ∼₂ → (∼₁ on f) Respects₂ (∼₂ on f)
respects₂ _ _ (resp₁ , resp₂) =
((λ {_} {_} {_} → resp₁) , λ {_} {_} {_} → resp₂)
decidable : ∀ {ℓ} (∼ : Rel A ℓ) → Decidable ∼ → Decidable (∼ on f)
decidable _ dec = λ x y → dec (f x) (f y)
total : ∀ {ℓ} (∼ : Rel A ℓ) → Total ∼ → Total (∼ on f)
total _ tot = λ x y → tot (f x) (f y)
trichotomous : ∀ {ℓ₁ ℓ₂} (≈ : Rel A ℓ₁) (< : Rel A ℓ₂) →
Trichotomous ≈ < → Trichotomous (≈ on f) (< on f)
trichotomous _ _ compare = λ x y → compare (f x) (f y)
isEquivalence : ∀ {ℓ} {≈ : Rel A ℓ} →
IsEquivalence ≈ → IsEquivalence (≈ on f)
isEquivalence {≈ = ≈} eq = record
{ refl = reflexive ≈ Eq.refl
; sym = symmetric ≈ Eq.sym
; trans = transitive ≈ Eq.trans
}
where module Eq = IsEquivalence eq
isPreorder : ∀ {ℓ₁ ℓ₂} {≈ : Rel A ℓ₁} {∼ : Rel A ℓ₂} →
IsPreorder ≈ ∼ → IsPreorder (≈ on f) (∼ on f)
isPreorder {≈ = ≈} {∼} pre = record
{ isEquivalence = isEquivalence Pre.isEquivalence
; reflexive = implies ≈ ∼ Pre.reflexive
; trans = transitive ∼ Pre.trans
}
where module Pre = IsPreorder pre
isDecEquivalence : ∀ {ℓ} {≈ : Rel A ℓ} →
IsDecEquivalence ≈ → IsDecEquivalence (≈ on f)
isDecEquivalence {≈ = ≈} dec = record
{ isEquivalence = isEquivalence Dec.isEquivalence
; _≟_ = decidable ≈ Dec._≟_
}
where module Dec = IsDecEquivalence dec
isPartialOrder : ∀ {ℓ₁ ℓ₂} {≈ : Rel A ℓ₁} {≤ : Rel A ℓ₂} →
IsPartialOrder ≈ ≤ →
IsPartialOrder (≈ on f) (≤ on f)
isPartialOrder {≈ = ≈} {≤} po = record
{ isPreorder = isPreorder Po.isPreorder
; antisym = antisymmetric ≈ ≤ Po.antisym
}
where module Po = IsPartialOrder po
isDecPartialOrder : ∀ {ℓ₁ ℓ₂} {≈ : Rel A ℓ₁} {≤ : Rel A ℓ₂} →
IsDecPartialOrder ≈ ≤ →
IsDecPartialOrder (≈ on f) (≤ on f)
isDecPartialOrder dpo = record
{ isPartialOrder = isPartialOrder DPO.isPartialOrder
; _≟_ = decidable _ DPO._≟_
; _≤?_ = decidable _ DPO._≤?_
}
where module DPO = IsDecPartialOrder dpo
isStrictPartialOrder : ∀ {ℓ₁ ℓ₂} {≈ : Rel A ℓ₁} {< : Rel A ℓ₂} →
IsStrictPartialOrder ≈ < →
IsStrictPartialOrder (≈ on f) (< on f)
isStrictPartialOrder {≈ = ≈} {<} spo = record
{ isEquivalence = isEquivalence Spo.isEquivalence
; irrefl = irreflexive ≈ < Spo.irrefl
; trans = transitive < Spo.trans
; <-resp-≈ = respects₂ < ≈ Spo.<-resp-≈
}
where module Spo = IsStrictPartialOrder spo
isTotalOrder : ∀ {ℓ₁ ℓ₂} {≈ : Rel A ℓ₁} {≤ : Rel A ℓ₂} →
IsTotalOrder ≈ ≤ →
IsTotalOrder (≈ on f) (≤ on f)
isTotalOrder {≈ = ≈} {≤} to = record
{ isPartialOrder = isPartialOrder To.isPartialOrder
; total = total ≤ To.total
}
where module To = IsTotalOrder to
isDecTotalOrder : ∀ {ℓ₁ ℓ₂} {≈ : Rel A ℓ₁} {≤ : Rel A ℓ₂} →
IsDecTotalOrder ≈ ≤ →
IsDecTotalOrder (≈ on f) (≤ on f)
isDecTotalOrder {≈ = ≈} {≤} dec = record
{ isTotalOrder = isTotalOrder Dec.isTotalOrder
; _≟_ = decidable ≈ Dec._≟_
; _≤?_ = decidable ≤ Dec._≤?_
}
where module Dec = IsDecTotalOrder dec
isStrictTotalOrder : ∀ {ℓ₁ ℓ₂} {≈ : Rel A ℓ₁} {< : Rel A ℓ₂} →
IsStrictTotalOrder ≈ < →
IsStrictTotalOrder (≈ on f) (< on f)
isStrictTotalOrder {≈ = ≈} {<} sto = record
{ isEquivalence = isEquivalence Sto.isEquivalence
; trans = transitive < Sto.trans
; compare = trichotomous ≈ < Sto.compare
; <-resp-≈ = respects₂ < ≈ Sto.<-resp-≈
}
where module Sto = IsStrictTotalOrder sto
preorder : ∀ {p₁ p₂ p₃ b} {B : Set b} (P : Preorder p₁ p₂ p₃) →
(B → Preorder.Carrier P) → Preorder _ _ _
preorder P f = record
{ isPreorder = isPreorder f (Preorder.isPreorder P)
}
setoid : ∀ {s₁ s₂ b} {B : Set b} (S : Setoid s₁ s₂) →
(B → Setoid.Carrier S) → Setoid _ _
setoid S f = record
{ isEquivalence = isEquivalence f (Setoid.isEquivalence S)
}
decSetoid : ∀ {d₁ d₂ b} {B : Set b} (D : DecSetoid d₁ d₂) →
(B → DecSetoid.Carrier D) → DecSetoid _ _
decSetoid D f = record
{ isDecEquivalence = isDecEquivalence f (DecSetoid.isDecEquivalence D)
}
poset : ∀ {p₁ p₂ p₃ b} {B : Set b} (P : Poset p₁ p₂ p₃) →
(B → Poset.Carrier P) → Poset _ _ _
poset P f = record
{ isPartialOrder = isPartialOrder f (Poset.isPartialOrder P)
}
decPoset : ∀ {d₁ d₂ d₃ b} {B : Set b} (D : DecPoset d₁ d₂ d₃) →
(B → DecPoset.Carrier D) → DecPoset _ _ _
decPoset D f = record
{ isDecPartialOrder =
isDecPartialOrder f (DecPoset.isDecPartialOrder D)
}
strictPartialOrder :
∀ {s₁ s₂ s₃ b} {B : Set b} (S : StrictPartialOrder s₁ s₂ s₃) →
(B → StrictPartialOrder.Carrier S) → StrictPartialOrder _ _ _
strictPartialOrder S f = record
{ isStrictPartialOrder =
isStrictPartialOrder f (StrictPartialOrder.isStrictPartialOrder S)
}
totalOrder : ∀ {t₁ t₂ t₃ b} {B : Set b} (T : TotalOrder t₁ t₂ t₃) →
(B → TotalOrder.Carrier T) → TotalOrder _ _ _
totalOrder T f = record
{ isTotalOrder = isTotalOrder f (TotalOrder.isTotalOrder T)
}
decTotalOrder :
∀ {d₁ d₂ d₃ b} {B : Set b} (D : DecTotalOrder d₁ d₂ d₃) →
(B → DecTotalOrder.Carrier D) → DecTotalOrder _ _ _
decTotalOrder D f = record
{ isDecTotalOrder = isDecTotalOrder f (DecTotalOrder.isDecTotalOrder D)
}
strictTotalOrder :
∀ {s₁ s₂ s₃ b} {B : Set b} (S : StrictTotalOrder s₁ s₂ s₃) →
(B → StrictTotalOrder.Carrier S) → StrictTotalOrder _ _ _
strictTotalOrder S f = record
{ isStrictTotalOrder =
isStrictTotalOrder f (StrictTotalOrder.isStrictTotalOrder S)
}
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module options where
open import lib
open import options-types public
----------------------------------------------------------------------------------
-- Run-rewriting rules
----------------------------------------------------------------------------------
data gratr2-nt : Set where
_ws-plus-34 : gratr2-nt
_ws : gratr2-nt
_str-bool : gratr2-nt
_start : gratr2-nt
_squote : gratr2-nt
_posinfo : gratr2-nt
_paths : gratr2-nt
_path-star-1 : gratr2-nt
_path : gratr2-nt
_ows-star-35 : gratr2-nt
_ows : gratr2-nt
_opts : gratr2-nt
_opt : gratr2-nt
_numpunct-bar-8 : gratr2-nt
_numpunct-bar-7 : gratr2-nt
_numpunct : gratr2-nt
_numone-range-5 : gratr2-nt
_numone : gratr2-nt
_num-plus-6 : gratr2-nt
_num : gratr2-nt
_comment-star-30 : gratr2-nt
_comment : gratr2-nt
_aws-bar-33 : gratr2-nt
_aws-bar-32 : gratr2-nt
_aws-bar-31 : gratr2-nt
_aws : gratr2-nt
_anychar-bar-9 : gratr2-nt
_anychar-bar-29 : gratr2-nt
_anychar-bar-28 : gratr2-nt
_anychar-bar-27 : gratr2-nt
_anychar-bar-26 : gratr2-nt
_anychar-bar-25 : gratr2-nt
_anychar-bar-24 : gratr2-nt
_anychar-bar-23 : gratr2-nt
_anychar-bar-22 : gratr2-nt
_anychar-bar-21 : gratr2-nt
_anychar-bar-20 : gratr2-nt
_anychar-bar-19 : gratr2-nt
_anychar-bar-18 : gratr2-nt
_anychar-bar-17 : gratr2-nt
_anychar-bar-16 : gratr2-nt
_anychar-bar-15 : gratr2-nt
_anychar-bar-14 : gratr2-nt
_anychar-bar-13 : gratr2-nt
_anychar-bar-12 : gratr2-nt
_anychar-bar-11 : gratr2-nt
_anychar-bar-10 : gratr2-nt
_anychar : gratr2-nt
_alpha-range-3 : gratr2-nt
_alpha-range-2 : gratr2-nt
_alpha-bar-4 : gratr2-nt
_alpha : gratr2-nt
gratr2-nt-eq : gratr2-nt → gratr2-nt → 𝔹
gratr2-nt-eq _ws-plus-34 _ws-plus-34 = tt
gratr2-nt-eq _ws _ws = tt
gratr2-nt-eq _str-bool _str-bool = tt
gratr2-nt-eq _start _start = tt
gratr2-nt-eq _squote _squote = tt
gratr2-nt-eq _posinfo _posinfo = tt
gratr2-nt-eq _paths _paths = tt
gratr2-nt-eq _path-star-1 _path-star-1 = tt
gratr2-nt-eq _path _path = tt
gratr2-nt-eq _ows-star-35 _ows-star-35 = tt
gratr2-nt-eq _ows _ows = tt
gratr2-nt-eq _opts _opts = tt
gratr2-nt-eq _opt _opt = tt
gratr2-nt-eq _numpunct-bar-8 _numpunct-bar-8 = tt
gratr2-nt-eq _numpunct-bar-7 _numpunct-bar-7 = tt
gratr2-nt-eq _numpunct _numpunct = tt
gratr2-nt-eq _numone-range-5 _numone-range-5 = tt
gratr2-nt-eq _numone _numone = tt
gratr2-nt-eq _num-plus-6 _num-plus-6 = tt
gratr2-nt-eq _num _num = tt
gratr2-nt-eq _comment-star-30 _comment-star-30 = tt
gratr2-nt-eq _comment _comment = tt
gratr2-nt-eq _aws-bar-33 _aws-bar-33 = tt
gratr2-nt-eq _aws-bar-32 _aws-bar-32 = tt
gratr2-nt-eq _aws-bar-31 _aws-bar-31 = tt
gratr2-nt-eq _aws _aws = tt
gratr2-nt-eq _anychar-bar-9 _anychar-bar-9 = tt
gratr2-nt-eq _anychar-bar-29 _anychar-bar-29 = tt
gratr2-nt-eq _anychar-bar-28 _anychar-bar-28 = tt
gratr2-nt-eq _anychar-bar-27 _anychar-bar-27 = tt
gratr2-nt-eq _anychar-bar-26 _anychar-bar-26 = tt
gratr2-nt-eq _anychar-bar-25 _anychar-bar-25 = tt
gratr2-nt-eq _anychar-bar-24 _anychar-bar-24 = tt
gratr2-nt-eq _anychar-bar-23 _anychar-bar-23 = tt
gratr2-nt-eq _anychar-bar-22 _anychar-bar-22 = tt
gratr2-nt-eq _anychar-bar-21 _anychar-bar-21 = tt
gratr2-nt-eq _anychar-bar-20 _anychar-bar-20 = tt
gratr2-nt-eq _anychar-bar-19 _anychar-bar-19 = tt
gratr2-nt-eq _anychar-bar-18 _anychar-bar-18 = tt
gratr2-nt-eq _anychar-bar-17 _anychar-bar-17 = tt
gratr2-nt-eq _anychar-bar-16 _anychar-bar-16 = tt
gratr2-nt-eq _anychar-bar-15 _anychar-bar-15 = tt
gratr2-nt-eq _anychar-bar-14 _anychar-bar-14 = tt
gratr2-nt-eq _anychar-bar-13 _anychar-bar-13 = tt
gratr2-nt-eq _anychar-bar-12 _anychar-bar-12 = tt
gratr2-nt-eq _anychar-bar-11 _anychar-bar-11 = tt
gratr2-nt-eq _anychar-bar-10 _anychar-bar-10 = tt
gratr2-nt-eq _anychar _anychar = tt
gratr2-nt-eq _alpha-range-3 _alpha-range-3 = tt
gratr2-nt-eq _alpha-range-2 _alpha-range-2 = tt
gratr2-nt-eq _alpha-bar-4 _alpha-bar-4 = tt
gratr2-nt-eq _alpha _alpha = tt
gratr2-nt-eq _ _ = ff
open import rtn gratr2-nt
options-start : gratr2-nt → 𝕃 gratr2-rule
options-start _ws-plus-34 = (just "P132" , nothing , just _ws-plus-34 , inj₁ _aws :: inj₁ _ws-plus-34 :: []) :: (just "P131" , nothing , just _ws-plus-34 , inj₁ _aws :: []) :: []
options-start _ws = (just "P133" , nothing , just _ws , inj₁ _ws-plus-34 :: []) :: []
options-start _str-bool = (just "StrBoolTrue" , nothing , just _str-bool , inj₂ 't' :: inj₂ 'r' :: inj₂ 'u' :: inj₂ 'e' :: []) :: (just "StrBoolFalse" , nothing , just _str-bool , inj₂ 'f' :: inj₂ 'a' :: inj₂ 'l' :: inj₂ 's' :: inj₂ 'e' :: []) :: []
options-start _start = (just "File" , nothing , just _start , inj₁ _opts :: inj₁ _ows :: []) :: []
options-start _squote = (just "P0" , nothing , just _squote , inj₂ '\"' :: []) :: []
options-start _posinfo = (just "Posinfo" , nothing , just _posinfo , []) :: []
options-start _paths = (just "PathsNil" , nothing , just _paths , []) :: (just "PathsCons" , nothing , just _paths , inj₁ _ws :: inj₁ _path :: inj₁ _paths :: []) :: []
options-start _path-star-1 = (just "P2" , nothing , just _path-star-1 , inj₁ _anychar :: inj₁ _path-star-1 :: []) :: (just "P1" , nothing , just _path-star-1 , []) :: []
options-start _path = (just "P3" , nothing , just _path , inj₁ _squote :: inj₁ _path-star-1 :: inj₁ _squote :: []) :: []
options-start _ows-star-35 = (just "P135" , nothing , just _ows-star-35 , inj₁ _aws :: inj₁ _ows-star-35 :: []) :: (just "P134" , nothing , just _ows-star-35 , []) :: []
options-start _ows = (just "P136" , nothing , just _ows , inj₁ _ows-star-35 :: []) :: []
options-start _opts = (just "OptsNil" , nothing , just _opts , []) :: (just "OptsCons" , nothing , just _opts , inj₁ _ows :: inj₁ _opt :: inj₁ _opts :: []) :: []
options-start _opt = (just "UseCedeFiles" , nothing , just _opt , inj₂ 'u' :: inj₂ 's' :: inj₂ 'e' :: inj₂ '-' :: inj₂ 'c' :: inj₂ 'e' :: inj₂ 'd' :: inj₂ 'e' :: inj₂ '-' :: inj₂ 'f' :: inj₂ 'i' :: inj₂ 'l' :: inj₂ 'e' :: inj₂ 's' :: inj₁ _ows :: inj₂ '=' :: inj₁ _ows :: inj₁ _str-bool :: inj₁ _ows :: inj₂ '.' :: inj₁ _ows :: []) :: (just "ShowQualifiedVars" , nothing , just _opt , inj₂ 's' :: inj₂ 'h' :: inj₂ 'o' :: inj₂ 'w' :: inj₂ '-' :: inj₂ 'q' :: inj₂ 'u' :: inj₂ 'a' :: inj₂ 'l' :: inj₂ 'i' :: inj₂ 'f' :: inj₂ 'i' :: inj₂ 'e' :: inj₂ 'd' :: inj₂ '-' :: inj₂ 'v' :: inj₂ 'a' :: inj₂ 'r' :: inj₂ 's' :: inj₁ _ows :: inj₂ '=' :: inj₁ _ows :: inj₁ _str-bool :: inj₁ _ows :: inj₂ '.' :: inj₁ _ows :: []) :: (just "MakeRktFiles" , nothing , just _opt , inj₂ 'm' :: inj₂ 'a' :: inj₂ 'k' :: inj₂ 'e' :: inj₂ '-' :: inj₂ 'r' :: inj₂ 'k' :: inj₂ 't' :: inj₂ '-' :: inj₂ 'f' :: inj₂ 'i' :: inj₂ 'l' :: inj₂ 'e' :: inj₂ 's' :: inj₁ _ows :: inj₂ '=' :: inj₁ _ows :: inj₁ _str-bool :: inj₁ _ows :: inj₂ '.' :: inj₁ _ows :: []) :: (just "Lib" , nothing , just _opt , inj₂ 'i' :: inj₂ 'm' :: inj₂ 'p' :: inj₂ 'o' :: inj₂ 'r' :: inj₂ 't' :: inj₂ '-' :: inj₂ 'd' :: inj₂ 'i' :: inj₂ 'r' :: inj₂ 'e' :: inj₂ 'c' :: inj₂ 't' :: inj₂ 'o' :: inj₂ 'r' :: inj₂ 'i' :: inj₂ 'e' :: inj₂ 's' :: inj₁ _ows :: inj₂ '=' :: inj₁ _ows :: inj₁ _paths :: inj₁ _ows :: inj₂ '.' :: []) :: (just "GenerateLogs" , nothing , just _opt , inj₂ 'g' :: inj₂ 'e' :: inj₂ 'n' :: inj₂ 'e' :: inj₂ 'r' :: inj₂ 'a' :: inj₂ 't' :: inj₂ 'e' :: inj₂ '-' :: inj₂ 'l' :: inj₂ 'o' :: inj₂ 'g' :: inj₂ 's' :: inj₁ _ows :: inj₂ '=' :: inj₁ _ows :: inj₁ _str-bool :: inj₁ _ows :: inj₂ '.' :: inj₁ _ows :: []) :: []
options-start _numpunct-bar-8 = (just "P76" , nothing , just _numpunct-bar-8 , inj₁ _numpunct-bar-7 :: []) :: (just "P75" , nothing , just _numpunct-bar-8 , inj₁ _numone :: []) :: []
options-start _numpunct-bar-7 = (just "P74" , nothing , just _numpunct-bar-7 , inj₂ '-' :: []) :: (just "P73" , nothing , just _numpunct-bar-7 , inj₂ '\'' :: []) :: []
options-start _numpunct = (just "P77" , nothing , just _numpunct , inj₁ _numpunct-bar-8 :: []) :: []
options-start _numone-range-5 = (just "P68" , nothing , just _numone-range-5 , inj₂ '9' :: []) :: (just "P67" , nothing , just _numone-range-5 , inj₂ '8' :: []) :: (just "P66" , nothing , just _numone-range-5 , inj₂ '7' :: []) :: (just "P65" , nothing , just _numone-range-5 , inj₂ '6' :: []) :: (just "P64" , nothing , just _numone-range-5 , inj₂ '5' :: []) :: (just "P63" , nothing , just _numone-range-5 , inj₂ '4' :: []) :: (just "P62" , nothing , just _numone-range-5 , inj₂ '3' :: []) :: (just "P61" , nothing , just _numone-range-5 , inj₂ '2' :: []) :: (just "P60" , nothing , just _numone-range-5 , inj₂ '1' :: []) :: (just "P59" , nothing , just _numone-range-5 , inj₂ '0' :: []) :: []
options-start _numone = (just "P69" , nothing , just _numone , inj₁ _numone-range-5 :: []) :: []
options-start _num-plus-6 = (just "P71" , nothing , just _num-plus-6 , inj₁ _numone :: inj₁ _num-plus-6 :: []) :: (just "P70" , nothing , just _num-plus-6 , inj₁ _numone :: []) :: []
options-start _num = (just "P72" , nothing , just _num , inj₁ _num-plus-6 :: []) :: []
options-start _comment-star-30 = (just "P122" , nothing , just _comment-star-30 , inj₁ _anychar :: inj₁ _comment-star-30 :: []) :: (just "P121" , nothing , just _comment-star-30 , []) :: []
options-start _comment = (just "P123" , nothing , just _comment , inj₂ '%' :: inj₁ _comment-star-30 :: inj₂ '\n' :: []) :: []
options-start _aws-bar-33 = (just "P129" , nothing , just _aws-bar-33 , inj₁ _aws-bar-32 :: []) :: (just "P128" , nothing , just _aws-bar-33 , inj₂ '\n' :: []) :: []
options-start _aws-bar-32 = (just "P127" , nothing , just _aws-bar-32 , inj₁ _aws-bar-31 :: []) :: (just "P126" , nothing , just _aws-bar-32 , inj₂ '\t' :: []) :: []
options-start _aws-bar-31 = (just "P125" , nothing , just _aws-bar-31 , inj₁ _comment :: []) :: (just "P124" , nothing , just _aws-bar-31 , inj₂ ' ' :: []) :: []
options-start _aws = (just "P130" , nothing , just _aws , inj₁ _aws-bar-33 :: []) :: []
options-start _anychar-bar-9 = (just "P79" , nothing , just _anychar-bar-9 , inj₂ '_' :: []) :: (just "P78" , nothing , just _anychar-bar-9 , inj₂ '/' :: []) :: []
options-start _anychar-bar-29 = (just "P119" , nothing , just _anychar-bar-29 , inj₁ _anychar-bar-28 :: []) :: (just "P118" , nothing , just _anychar-bar-29 , inj₁ _alpha :: []) :: []
options-start _anychar-bar-28 = (just "P117" , nothing , just _anychar-bar-28 , inj₁ _anychar-bar-27 :: []) :: (just "P116" , nothing , just _anychar-bar-28 , inj₁ _numpunct :: []) :: []
options-start _anychar-bar-27 = (just "P115" , nothing , just _anychar-bar-27 , inj₁ _anychar-bar-26 :: []) :: (just "P114" , nothing , just _anychar-bar-27 , inj₂ '\t' :: []) :: []
options-start _anychar-bar-26 = (just "P113" , nothing , just _anychar-bar-26 , inj₁ _anychar-bar-25 :: []) :: (just "P112" , nothing , just _anychar-bar-26 , inj₂ ' ' :: []) :: []
options-start _anychar-bar-25 = (just "P111" , nothing , just _anychar-bar-25 , inj₁ _anychar-bar-24 :: []) :: (just "P110" , nothing , just _anychar-bar-25 , inj₂ '%' :: []) :: []
options-start _anychar-bar-24 = (just "P109" , nothing , just _anychar-bar-24 , inj₁ _anychar-bar-23 :: []) :: (just "P108" , nothing , just _anychar-bar-24 , inj₂ '(' :: []) :: []
options-start _anychar-bar-23 = (just "P107" , nothing , just _anychar-bar-23 , inj₁ _anychar-bar-22 :: []) :: (just "P106" , nothing , just _anychar-bar-23 , inj₂ ')' :: []) :: []
options-start _anychar-bar-22 = (just "P105" , nothing , just _anychar-bar-22 , inj₁ _anychar-bar-21 :: []) :: (just "P104" , nothing , just _anychar-bar-22 , inj₂ ':' :: []) :: []
options-start _anychar-bar-21 = (just "P103" , nothing , just _anychar-bar-21 , inj₁ _anychar-bar-20 :: []) :: (just "P102" , nothing , just _anychar-bar-21 , inj₂ '.' :: []) :: []
options-start _anychar-bar-20 = (just "P101" , nothing , just _anychar-bar-20 , inj₁ _anychar-bar-19 :: []) :: (just "P100" , nothing , just _anychar-bar-20 , inj₂ '[' :: []) :: []
options-start _anychar-bar-19 = (just "P99" , nothing , just _anychar-bar-19 , inj₁ _anychar-bar-18 :: []) :: (just "P98" , nothing , just _anychar-bar-19 , inj₂ ']' :: []) :: []
options-start _anychar-bar-18 = (just "P97" , nothing , just _anychar-bar-18 , inj₁ _anychar-bar-17 :: []) :: (just "P96" , nothing , just _anychar-bar-18 , inj₂ ',' :: []) :: []
options-start _anychar-bar-17 = (just "P95" , nothing , just _anychar-bar-17 , inj₁ _anychar-bar-16 :: []) :: (just "P94" , nothing , just _anychar-bar-17 , inj₂ '!' :: []) :: []
options-start _anychar-bar-16 = (just "P93" , nothing , just _anychar-bar-16 , inj₁ _anychar-bar-15 :: []) :: (just "P92" , nothing , just _anychar-bar-16 , inj₂ '{' :: []) :: []
options-start _anychar-bar-15 = (just "P91" , nothing , just _anychar-bar-15 , inj₁ _anychar-bar-14 :: []) :: (just "P90" , nothing , just _anychar-bar-15 , inj₂ '}' :: []) :: []
options-start _anychar-bar-14 = (just "P89" , nothing , just _anychar-bar-14 , inj₁ _anychar-bar-13 :: []) :: (just "P88" , nothing , just _anychar-bar-14 , inj₂ '-' :: []) :: []
options-start _anychar-bar-13 = (just "P87" , nothing , just _anychar-bar-13 , inj₁ _anychar-bar-12 :: []) :: (just "P86" , nothing , just _anychar-bar-13 , inj₂ '=' :: []) :: []
options-start _anychar-bar-12 = (just "P85" , nothing , just _anychar-bar-12 , inj₁ _anychar-bar-11 :: []) :: (just "P84" , nothing , just _anychar-bar-12 , inj₂ '+' :: []) :: []
options-start _anychar-bar-11 = (just "P83" , nothing , just _anychar-bar-11 , inj₁ _anychar-bar-10 :: []) :: (just "P82" , nothing , just _anychar-bar-11 , inj₂ '<' :: []) :: []
options-start _anychar-bar-10 = (just "P81" , nothing , just _anychar-bar-10 , inj₁ _anychar-bar-9 :: []) :: (just "P80" , nothing , just _anychar-bar-10 , inj₂ '>' :: []) :: []
options-start _anychar = (just "P120" , nothing , just _anychar , inj₁ _anychar-bar-29 :: []) :: []
options-start _alpha-range-3 = (just "P55" , nothing , just _alpha-range-3 , inj₂ 'Z' :: []) :: (just "P54" , nothing , just _alpha-range-3 , inj₂ 'Y' :: []) :: (just "P53" , nothing , just _alpha-range-3 , inj₂ 'X' :: []) :: (just "P52" , nothing , just _alpha-range-3 , inj₂ 'W' :: []) :: (just "P51" , nothing , just _alpha-range-3 , inj₂ 'V' :: []) :: (just "P50" , nothing , just _alpha-range-3 , inj₂ 'U' :: []) :: (just "P49" , nothing , just _alpha-range-3 , inj₂ 'T' :: []) :: (just "P48" , nothing , just _alpha-range-3 , inj₂ 'S' :: []) :: (just "P47" , nothing , just _alpha-range-3 , inj₂ 'R' :: []) :: (just "P46" , nothing , just _alpha-range-3 , inj₂ 'Q' :: []) :: (just "P45" , nothing , just _alpha-range-3 , inj₂ 'P' :: []) :: (just "P44" , nothing , just _alpha-range-3 , inj₂ 'O' :: []) :: (just "P43" , nothing , just _alpha-range-3 , inj₂ 'N' :: []) :: (just "P42" , nothing , just _alpha-range-3 , inj₂ 'M' :: []) :: (just "P41" , nothing , just _alpha-range-3 , inj₂ 'L' :: []) :: (just "P40" , nothing , just _alpha-range-3 , inj₂ 'K' :: []) :: (just "P39" , nothing , just _alpha-range-3 , inj₂ 'J' :: []) :: (just "P38" , nothing , just _alpha-range-3 , inj₂ 'I' :: []) :: (just "P37" , nothing , just _alpha-range-3 , inj₂ 'H' :: []) :: (just "P36" , nothing , just _alpha-range-3 , inj₂ 'G' :: []) :: (just "P35" , nothing , just _alpha-range-3 , inj₂ 'F' :: []) :: (just "P34" , nothing , just _alpha-range-3 , inj₂ 'E' :: []) :: (just "P33" , nothing , just _alpha-range-3 , inj₂ 'D' :: []) :: (just "P32" , nothing , just _alpha-range-3 , inj₂ 'C' :: []) :: (just "P31" , nothing , just _alpha-range-3 , inj₂ 'B' :: []) :: (just "P30" , nothing , just _alpha-range-3 , inj₂ 'A' :: []) :: []
options-start _alpha-range-2 = (just "P9" , nothing , just _alpha-range-2 , inj₂ 'f' :: []) :: (just "P8" , nothing , just _alpha-range-2 , inj₂ 'e' :: []) :: (just "P7" , nothing , just _alpha-range-2 , inj₂ 'd' :: []) :: (just "P6" , nothing , just _alpha-range-2 , inj₂ 'c' :: []) :: (just "P5" , nothing , just _alpha-range-2 , inj₂ 'b' :: []) :: (just "P4" , nothing , just _alpha-range-2 , inj₂ 'a' :: []) :: (just "P29" , nothing , just _alpha-range-2 , inj₂ 'z' :: []) :: (just "P28" , nothing , just _alpha-range-2 , inj₂ 'y' :: []) :: (just "P27" , nothing , just _alpha-range-2 , inj₂ 'x' :: []) :: (just "P26" , nothing , just _alpha-range-2 , inj₂ 'w' :: []) :: (just "P25" , nothing , just _alpha-range-2 , inj₂ 'v' :: []) :: (just "P24" , nothing , just _alpha-range-2 , inj₂ 'u' :: []) :: (just "P23" , nothing , just _alpha-range-2 , inj₂ 't' :: []) :: (just "P22" , nothing , just _alpha-range-2 , inj₂ 's' :: []) :: (just "P21" , nothing , just _alpha-range-2 , inj₂ 'r' :: []) :: (just "P20" , nothing , just _alpha-range-2 , inj₂ 'q' :: []) :: (just "P19" , nothing , just _alpha-range-2 , inj₂ 'p' :: []) :: (just "P18" , nothing , just _alpha-range-2 , inj₂ 'o' :: []) :: (just "P17" , nothing , just _alpha-range-2 , inj₂ 'n' :: []) :: (just "P16" , nothing , just _alpha-range-2 , inj₂ 'm' :: []) :: (just "P15" , nothing , just _alpha-range-2 , inj₂ 'l' :: []) :: (just "P14" , nothing , just _alpha-range-2 , inj₂ 'k' :: []) :: (just "P13" , nothing , just _alpha-range-2 , inj₂ 'j' :: []) :: (just "P12" , nothing , just _alpha-range-2 , inj₂ 'i' :: []) :: (just "P11" , nothing , just _alpha-range-2 , inj₂ 'h' :: []) :: (just "P10" , nothing , just _alpha-range-2 , inj₂ 'g' :: []) :: []
options-start _alpha-bar-4 = (just "P57" , nothing , just _alpha-bar-4 , inj₁ _alpha-range-3 :: []) :: (just "P56" , nothing , just _alpha-bar-4 , inj₁ _alpha-range-2 :: []) :: []
options-start _alpha = (just "P58" , nothing , just _alpha , inj₁ _alpha-bar-4 :: []) :: []
options-return : maybe gratr2-nt → 𝕃 gratr2-rule
options-return _ = []
options-rtn : gratr2-rtn
options-rtn = record { start = _start ; _eq_ = gratr2-nt-eq ; gratr2-start = options-start ; gratr2-return = options-return }
open import run ptr
open noderiv
------------------------------------------
-- Length-decreasing rules
------------------------------------------
len-dec-rewrite : Run → maybe (Run × ℕ)
len-dec-rewrite {- File-} ((Id "File") :: (ParseTree (parsed-opts x0)) :: _::_(ParseTree parsed-ows) rest) = just (ParseTree (parsed-start (norm-start (File x0))) ::' rest , 3)
len-dec-rewrite {- GenerateLogs-} ((Id "GenerateLogs") :: (InputChar 'g') :: (InputChar 'e') :: (InputChar 'n') :: (InputChar 'e') :: (InputChar 'r') :: (InputChar 'a') :: (InputChar 't') :: (InputChar 'e') :: (InputChar '-') :: (InputChar 'l') :: (InputChar 'o') :: (InputChar 'g') :: (InputChar 's') :: (ParseTree parsed-ows) :: (InputChar '=') :: (ParseTree parsed-ows) :: (ParseTree (parsed-str-bool x0)) :: (ParseTree parsed-ows) :: (InputChar '.') :: _::_(ParseTree parsed-ows) rest) = just (ParseTree (parsed-opt (norm-opt (GenerateLogs x0))) ::' rest , 21)
len-dec-rewrite {- Lib-} ((Id "Lib") :: (InputChar 'i') :: (InputChar 'm') :: (InputChar 'p') :: (InputChar 'o') :: (InputChar 'r') :: (InputChar 't') :: (InputChar '-') :: (InputChar 'd') :: (InputChar 'i') :: (InputChar 'r') :: (InputChar 'e') :: (InputChar 'c') :: (InputChar 't') :: (InputChar 'o') :: (InputChar 'r') :: (InputChar 'i') :: (InputChar 'e') :: (InputChar 's') :: (ParseTree parsed-ows) :: (InputChar '=') :: (ParseTree parsed-ows) :: (ParseTree (parsed-paths x0)) :: (ParseTree parsed-ows) :: _::_(InputChar '.') rest) = just (ParseTree (parsed-opt (norm-opt (Lib x0))) ::' rest , 25)
len-dec-rewrite {- MakeRktFiles-} ((Id "MakeRktFiles") :: (InputChar 'm') :: (InputChar 'a') :: (InputChar 'k') :: (InputChar 'e') :: (InputChar '-') :: (InputChar 'r') :: (InputChar 'k') :: (InputChar 't') :: (InputChar '-') :: (InputChar 'f') :: (InputChar 'i') :: (InputChar 'l') :: (InputChar 'e') :: (InputChar 's') :: (ParseTree parsed-ows) :: (InputChar '=') :: (ParseTree parsed-ows) :: (ParseTree (parsed-str-bool x0)) :: (ParseTree parsed-ows) :: (InputChar '.') :: _::_(ParseTree parsed-ows) rest) = just (ParseTree (parsed-opt (norm-opt (MakeRktFiles x0))) ::' rest , 22)
len-dec-rewrite {- OptsCons-} ((Id "OptsCons") :: (ParseTree parsed-ows) :: (ParseTree (parsed-opt x0)) :: _::_(ParseTree (parsed-opts x1)) rest) = just (ParseTree (parsed-opts (norm-opts (OptsCons x0 x1))) ::' rest , 4)
len-dec-rewrite {- P0-} ((Id "P0") :: _::_(InputChar '\"') rest) = just (ParseTree parsed-squote ::' rest , 2)
len-dec-rewrite {- P10-} ((Id "P10") :: _::_(InputChar 'g') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'g'))) ::' rest , 2)
len-dec-rewrite {- P100-} ((Id "P100") :: _::_(InputChar '[') rest) = just (ParseTree (parsed-anychar-bar-20 (string-append 0 (char-to-string '['))) ::' rest , 2)
len-dec-rewrite {- P101-} ((Id "P101") :: _::_(ParseTree (parsed-anychar-bar-19 x0)) rest) = just (ParseTree (parsed-anychar-bar-20 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P102-} ((Id "P102") :: _::_(InputChar '.') rest) = just (ParseTree (parsed-anychar-bar-21 (string-append 0 (char-to-string '.'))) ::' rest , 2)
len-dec-rewrite {- P103-} ((Id "P103") :: _::_(ParseTree (parsed-anychar-bar-20 x0)) rest) = just (ParseTree (parsed-anychar-bar-21 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P104-} ((Id "P104") :: _::_(InputChar ':') rest) = just (ParseTree (parsed-anychar-bar-22 (string-append 0 (char-to-string ':'))) ::' rest , 2)
len-dec-rewrite {- P105-} ((Id "P105") :: _::_(ParseTree (parsed-anychar-bar-21 x0)) rest) = just (ParseTree (parsed-anychar-bar-22 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P106-} ((Id "P106") :: _::_(InputChar ')') rest) = just (ParseTree (parsed-anychar-bar-23 (string-append 0 (char-to-string ')'))) ::' rest , 2)
len-dec-rewrite {- P107-} ((Id "P107") :: _::_(ParseTree (parsed-anychar-bar-22 x0)) rest) = just (ParseTree (parsed-anychar-bar-23 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P108-} ((Id "P108") :: _::_(InputChar '(') rest) = just (ParseTree (parsed-anychar-bar-24 (string-append 0 (char-to-string '('))) ::' rest , 2)
len-dec-rewrite {- P109-} ((Id "P109") :: _::_(ParseTree (parsed-anychar-bar-23 x0)) rest) = just (ParseTree (parsed-anychar-bar-24 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P11-} ((Id "P11") :: _::_(InputChar 'h') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'h'))) ::' rest , 2)
len-dec-rewrite {- P110-} ((Id "P110") :: _::_(InputChar '%') rest) = just (ParseTree (parsed-anychar-bar-25 (string-append 0 (char-to-string '%'))) ::' rest , 2)
len-dec-rewrite {- P111-} ((Id "P111") :: _::_(ParseTree (parsed-anychar-bar-24 x0)) rest) = just (ParseTree (parsed-anychar-bar-25 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P112-} ((Id "P112") :: _::_(InputChar ' ') rest) = just (ParseTree (parsed-anychar-bar-26 (string-append 0 (char-to-string ' '))) ::' rest , 2)
len-dec-rewrite {- P113-} ((Id "P113") :: _::_(ParseTree (parsed-anychar-bar-25 x0)) rest) = just (ParseTree (parsed-anychar-bar-26 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P114-} ((Id "P114") :: _::_(InputChar '\t') rest) = just (ParseTree (parsed-anychar-bar-27 (string-append 0 (char-to-string '\t'))) ::' rest , 2)
len-dec-rewrite {- P115-} ((Id "P115") :: _::_(ParseTree (parsed-anychar-bar-26 x0)) rest) = just (ParseTree (parsed-anychar-bar-27 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P116-} ((Id "P116") :: _::_(ParseTree (parsed-numpunct x0)) rest) = just (ParseTree (parsed-anychar-bar-28 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P117-} ((Id "P117") :: _::_(ParseTree (parsed-anychar-bar-27 x0)) rest) = just (ParseTree (parsed-anychar-bar-28 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P118-} ((Id "P118") :: _::_(ParseTree (parsed-alpha x0)) rest) = just (ParseTree (parsed-anychar-bar-29 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P119-} ((Id "P119") :: _::_(ParseTree (parsed-anychar-bar-28 x0)) rest) = just (ParseTree (parsed-anychar-bar-29 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P12-} ((Id "P12") :: _::_(InputChar 'i') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'i'))) ::' rest , 2)
len-dec-rewrite {- P120-} ((Id "P120") :: _::_(ParseTree (parsed-anychar-bar-29 x0)) rest) = just (ParseTree (parsed-anychar (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P122-} ((Id "P122") :: (ParseTree (parsed-anychar x0)) :: _::_(ParseTree parsed-comment-star-30) rest) = just (ParseTree parsed-comment-star-30 ::' rest , 3)
len-dec-rewrite {- P123-} ((Id "P123") :: (InputChar '%') :: (ParseTree parsed-comment-star-30) :: _::_(InputChar '\n') rest) = just (ParseTree parsed-comment ::' rest , 4)
len-dec-rewrite {- P124-} ((Id "P124") :: _::_(InputChar ' ') rest) = just (ParseTree parsed-aws-bar-31 ::' rest , 2)
len-dec-rewrite {- P125-} ((Id "P125") :: _::_(ParseTree parsed-comment) rest) = just (ParseTree parsed-aws-bar-31 ::' rest , 2)
len-dec-rewrite {- P126-} ((Id "P126") :: _::_(InputChar '\t') rest) = just (ParseTree parsed-aws-bar-32 ::' rest , 2)
len-dec-rewrite {- P127-} ((Id "P127") :: _::_(ParseTree parsed-aws-bar-31) rest) = just (ParseTree parsed-aws-bar-32 ::' rest , 2)
len-dec-rewrite {- P128-} ((Id "P128") :: _::_(InputChar '\n') rest) = just (ParseTree parsed-aws-bar-33 ::' rest , 2)
len-dec-rewrite {- P129-} ((Id "P129") :: _::_(ParseTree parsed-aws-bar-32) rest) = just (ParseTree parsed-aws-bar-33 ::' rest , 2)
len-dec-rewrite {- P13-} ((Id "P13") :: _::_(InputChar 'j') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'j'))) ::' rest , 2)
len-dec-rewrite {- P130-} ((Id "P130") :: _::_(ParseTree parsed-aws-bar-33) rest) = just (ParseTree parsed-aws ::' rest , 2)
len-dec-rewrite {- P131-} ((Id "P131") :: _::_(ParseTree parsed-aws) rest) = just (ParseTree parsed-ws-plus-34 ::' rest , 2)
len-dec-rewrite {- P132-} ((Id "P132") :: (ParseTree parsed-aws) :: _::_(ParseTree parsed-ws-plus-34) rest) = just (ParseTree parsed-ws-plus-34 ::' rest , 3)
len-dec-rewrite {- P133-} ((Id "P133") :: _::_(ParseTree parsed-ws-plus-34) rest) = just (ParseTree parsed-ws ::' rest , 2)
len-dec-rewrite {- P135-} ((Id "P135") :: (ParseTree parsed-aws) :: _::_(ParseTree parsed-ows-star-35) rest) = just (ParseTree parsed-ows-star-35 ::' rest , 3)
len-dec-rewrite {- P136-} ((Id "P136") :: _::_(ParseTree parsed-ows-star-35) rest) = just (ParseTree parsed-ows ::' rest , 2)
len-dec-rewrite {- P14-} ((Id "P14") :: _::_(InputChar 'k') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'k'))) ::' rest , 2)
len-dec-rewrite {- P15-} ((Id "P15") :: _::_(InputChar 'l') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'l'))) ::' rest , 2)
len-dec-rewrite {- P16-} ((Id "P16") :: _::_(InputChar 'm') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'm'))) ::' rest , 2)
len-dec-rewrite {- P17-} ((Id "P17") :: _::_(InputChar 'n') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'n'))) ::' rest , 2)
len-dec-rewrite {- P18-} ((Id "P18") :: _::_(InputChar 'o') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'o'))) ::' rest , 2)
len-dec-rewrite {- P19-} ((Id "P19") :: _::_(InputChar 'p') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'p'))) ::' rest , 2)
len-dec-rewrite {- P2-} ((Id "P2") :: (ParseTree (parsed-anychar x0)) :: _::_(ParseTree (parsed-path-star-1 x1)) rest) = just (ParseTree (parsed-path-star-1 (string-append 1 x0 x1)) ::' rest , 3)
len-dec-rewrite {- P20-} ((Id "P20") :: _::_(InputChar 'q') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'q'))) ::' rest , 2)
len-dec-rewrite {- P21-} ((Id "P21") :: _::_(InputChar 'r') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'r'))) ::' rest , 2)
len-dec-rewrite {- P22-} ((Id "P22") :: _::_(InputChar 's') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 's'))) ::' rest , 2)
len-dec-rewrite {- P23-} ((Id "P23") :: _::_(InputChar 't') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 't'))) ::' rest , 2)
len-dec-rewrite {- P24-} ((Id "P24") :: _::_(InputChar 'u') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'u'))) ::' rest , 2)
len-dec-rewrite {- P25-} ((Id "P25") :: _::_(InputChar 'v') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'v'))) ::' rest , 2)
len-dec-rewrite {- P26-} ((Id "P26") :: _::_(InputChar 'w') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'w'))) ::' rest , 2)
len-dec-rewrite {- P27-} ((Id "P27") :: _::_(InputChar 'x') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'x'))) ::' rest , 2)
len-dec-rewrite {- P28-} ((Id "P28") :: _::_(InputChar 'y') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'y'))) ::' rest , 2)
len-dec-rewrite {- P29-} ((Id "P29") :: _::_(InputChar 'z') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'z'))) ::' rest , 2)
len-dec-rewrite {- P3-} ((Id "P3") :: (ParseTree parsed-squote) :: (ParseTree (parsed-path-star-1 x0)) :: _::_(ParseTree parsed-squote) rest) = just (ParseTree (parsed-path (string-append 0 x0)) ::' rest , 4)
len-dec-rewrite {- P30-} ((Id "P30") :: _::_(InputChar 'A') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'A'))) ::' rest , 2)
len-dec-rewrite {- P31-} ((Id "P31") :: _::_(InputChar 'B') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'B'))) ::' rest , 2)
len-dec-rewrite {- P32-} ((Id "P32") :: _::_(InputChar 'C') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'C'))) ::' rest , 2)
len-dec-rewrite {- P33-} ((Id "P33") :: _::_(InputChar 'D') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'D'))) ::' rest , 2)
len-dec-rewrite {- P34-} ((Id "P34") :: _::_(InputChar 'E') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'E'))) ::' rest , 2)
len-dec-rewrite {- P35-} ((Id "P35") :: _::_(InputChar 'F') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'F'))) ::' rest , 2)
len-dec-rewrite {- P36-} ((Id "P36") :: _::_(InputChar 'G') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'G'))) ::' rest , 2)
len-dec-rewrite {- P37-} ((Id "P37") :: _::_(InputChar 'H') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'H'))) ::' rest , 2)
len-dec-rewrite {- P38-} ((Id "P38") :: _::_(InputChar 'I') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'I'))) ::' rest , 2)
len-dec-rewrite {- P39-} ((Id "P39") :: _::_(InputChar 'J') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'J'))) ::' rest , 2)
len-dec-rewrite {- P4-} ((Id "P4") :: _::_(InputChar 'a') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'a'))) ::' rest , 2)
len-dec-rewrite {- P40-} ((Id "P40") :: _::_(InputChar 'K') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'K'))) ::' rest , 2)
len-dec-rewrite {- P41-} ((Id "P41") :: _::_(InputChar 'L') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'L'))) ::' rest , 2)
len-dec-rewrite {- P42-} ((Id "P42") :: _::_(InputChar 'M') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'M'))) ::' rest , 2)
len-dec-rewrite {- P43-} ((Id "P43") :: _::_(InputChar 'N') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'N'))) ::' rest , 2)
len-dec-rewrite {- P44-} ((Id "P44") :: _::_(InputChar 'O') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'O'))) ::' rest , 2)
len-dec-rewrite {- P45-} ((Id "P45") :: _::_(InputChar 'P') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'P'))) ::' rest , 2)
len-dec-rewrite {- P46-} ((Id "P46") :: _::_(InputChar 'Q') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'Q'))) ::' rest , 2)
len-dec-rewrite {- P47-} ((Id "P47") :: _::_(InputChar 'R') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'R'))) ::' rest , 2)
len-dec-rewrite {- P48-} ((Id "P48") :: _::_(InputChar 'S') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'S'))) ::' rest , 2)
len-dec-rewrite {- P49-} ((Id "P49") :: _::_(InputChar 'T') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'T'))) ::' rest , 2)
len-dec-rewrite {- P5-} ((Id "P5") :: _::_(InputChar 'b') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'b'))) ::' rest , 2)
len-dec-rewrite {- P50-} ((Id "P50") :: _::_(InputChar 'U') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'U'))) ::' rest , 2)
len-dec-rewrite {- P51-} ((Id "P51") :: _::_(InputChar 'V') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'V'))) ::' rest , 2)
len-dec-rewrite {- P52-} ((Id "P52") :: _::_(InputChar 'W') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'W'))) ::' rest , 2)
len-dec-rewrite {- P53-} ((Id "P53") :: _::_(InputChar 'X') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'X'))) ::' rest , 2)
len-dec-rewrite {- P54-} ((Id "P54") :: _::_(InputChar 'Y') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'Y'))) ::' rest , 2)
len-dec-rewrite {- P55-} ((Id "P55") :: _::_(InputChar 'Z') rest) = just (ParseTree (parsed-alpha-range-3 (string-append 0 (char-to-string 'Z'))) ::' rest , 2)
len-dec-rewrite {- P56-} ((Id "P56") :: _::_(ParseTree (parsed-alpha-range-2 x0)) rest) = just (ParseTree (parsed-alpha-bar-4 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P57-} ((Id "P57") :: _::_(ParseTree (parsed-alpha-range-3 x0)) rest) = just (ParseTree (parsed-alpha-bar-4 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P58-} ((Id "P58") :: _::_(ParseTree (parsed-alpha-bar-4 x0)) rest) = just (ParseTree (parsed-alpha (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P59-} ((Id "P59") :: _::_(InputChar '0') rest) = just (ParseTree (parsed-numone-range-5 (string-append 0 (char-to-string '0'))) ::' rest , 2)
len-dec-rewrite {- P6-} ((Id "P6") :: _::_(InputChar 'c') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'c'))) ::' rest , 2)
len-dec-rewrite {- P60-} ((Id "P60") :: _::_(InputChar '1') rest) = just (ParseTree (parsed-numone-range-5 (string-append 0 (char-to-string '1'))) ::' rest , 2)
len-dec-rewrite {- P61-} ((Id "P61") :: _::_(InputChar '2') rest) = just (ParseTree (parsed-numone-range-5 (string-append 0 (char-to-string '2'))) ::' rest , 2)
len-dec-rewrite {- P62-} ((Id "P62") :: _::_(InputChar '3') rest) = just (ParseTree (parsed-numone-range-5 (string-append 0 (char-to-string '3'))) ::' rest , 2)
len-dec-rewrite {- P63-} ((Id "P63") :: _::_(InputChar '4') rest) = just (ParseTree (parsed-numone-range-5 (string-append 0 (char-to-string '4'))) ::' rest , 2)
len-dec-rewrite {- P64-} ((Id "P64") :: _::_(InputChar '5') rest) = just (ParseTree (parsed-numone-range-5 (string-append 0 (char-to-string '5'))) ::' rest , 2)
len-dec-rewrite {- P65-} ((Id "P65") :: _::_(InputChar '6') rest) = just (ParseTree (parsed-numone-range-5 (string-append 0 (char-to-string '6'))) ::' rest , 2)
len-dec-rewrite {- P66-} ((Id "P66") :: _::_(InputChar '7') rest) = just (ParseTree (parsed-numone-range-5 (string-append 0 (char-to-string '7'))) ::' rest , 2)
len-dec-rewrite {- P67-} ((Id "P67") :: _::_(InputChar '8') rest) = just (ParseTree (parsed-numone-range-5 (string-append 0 (char-to-string '8'))) ::' rest , 2)
len-dec-rewrite {- P68-} ((Id "P68") :: _::_(InputChar '9') rest) = just (ParseTree (parsed-numone-range-5 (string-append 0 (char-to-string '9'))) ::' rest , 2)
len-dec-rewrite {- P69-} ((Id "P69") :: _::_(ParseTree (parsed-numone-range-5 x0)) rest) = just (ParseTree (parsed-numone (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P7-} ((Id "P7") :: _::_(InputChar 'd') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'd'))) ::' rest , 2)
len-dec-rewrite {- P70-} ((Id "P70") :: _::_(ParseTree (parsed-numone x0)) rest) = just (ParseTree (parsed-num-plus-6 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P71-} ((Id "P71") :: (ParseTree (parsed-numone x0)) :: _::_(ParseTree (parsed-num-plus-6 x1)) rest) = just (ParseTree (parsed-num-plus-6 (string-append 1 x0 x1)) ::' rest , 3)
len-dec-rewrite {- P72-} ((Id "P72") :: _::_(ParseTree (parsed-num-plus-6 x0)) rest) = just (ParseTree (parsed-num (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P73-} ((Id "P73") :: _::_(InputChar '\'') rest) = just (ParseTree (parsed-numpunct-bar-7 (string-append 0 (char-to-string '\''))) ::' rest , 2)
len-dec-rewrite {- P74-} ((Id "P74") :: _::_(InputChar '-') rest) = just (ParseTree (parsed-numpunct-bar-7 (string-append 0 (char-to-string '-'))) ::' rest , 2)
len-dec-rewrite {- P75-} ((Id "P75") :: _::_(ParseTree (parsed-numone x0)) rest) = just (ParseTree (parsed-numpunct-bar-8 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P76-} ((Id "P76") :: _::_(ParseTree (parsed-numpunct-bar-7 x0)) rest) = just (ParseTree (parsed-numpunct-bar-8 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P77-} ((Id "P77") :: _::_(ParseTree (parsed-numpunct-bar-8 x0)) rest) = just (ParseTree (parsed-numpunct (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P78-} ((Id "P78") :: _::_(InputChar '/') rest) = just (ParseTree (parsed-anychar-bar-9 (string-append 0 (char-to-string '/'))) ::' rest , 2)
len-dec-rewrite {- P79-} ((Id "P79") :: _::_(InputChar '_') rest) = just (ParseTree (parsed-anychar-bar-9 (string-append 0 (char-to-string '_'))) ::' rest , 2)
len-dec-rewrite {- P8-} ((Id "P8") :: _::_(InputChar 'e') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'e'))) ::' rest , 2)
len-dec-rewrite {- P80-} ((Id "P80") :: _::_(InputChar '>') rest) = just (ParseTree (parsed-anychar-bar-10 (string-append 0 (char-to-string '>'))) ::' rest , 2)
len-dec-rewrite {- P81-} ((Id "P81") :: _::_(ParseTree (parsed-anychar-bar-9 x0)) rest) = just (ParseTree (parsed-anychar-bar-10 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P82-} ((Id "P82") :: _::_(InputChar '<') rest) = just (ParseTree (parsed-anychar-bar-11 (string-append 0 (char-to-string '<'))) ::' rest , 2)
len-dec-rewrite {- P83-} ((Id "P83") :: _::_(ParseTree (parsed-anychar-bar-10 x0)) rest) = just (ParseTree (parsed-anychar-bar-11 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P84-} ((Id "P84") :: _::_(InputChar '+') rest) = just (ParseTree (parsed-anychar-bar-12 (string-append 0 (char-to-string '+'))) ::' rest , 2)
len-dec-rewrite {- P85-} ((Id "P85") :: _::_(ParseTree (parsed-anychar-bar-11 x0)) rest) = just (ParseTree (parsed-anychar-bar-12 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P86-} ((Id "P86") :: _::_(InputChar '=') rest) = just (ParseTree (parsed-anychar-bar-13 (string-append 0 (char-to-string '='))) ::' rest , 2)
len-dec-rewrite {- P87-} ((Id "P87") :: _::_(ParseTree (parsed-anychar-bar-12 x0)) rest) = just (ParseTree (parsed-anychar-bar-13 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P88-} ((Id "P88") :: _::_(InputChar '-') rest) = just (ParseTree (parsed-anychar-bar-14 (string-append 0 (char-to-string '-'))) ::' rest , 2)
len-dec-rewrite {- P89-} ((Id "P89") :: _::_(ParseTree (parsed-anychar-bar-13 x0)) rest) = just (ParseTree (parsed-anychar-bar-14 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P9-} ((Id "P9") :: _::_(InputChar 'f') rest) = just (ParseTree (parsed-alpha-range-2 (string-append 0 (char-to-string 'f'))) ::' rest , 2)
len-dec-rewrite {- P90-} ((Id "P90") :: _::_(InputChar '}') rest) = just (ParseTree (parsed-anychar-bar-15 (string-append 0 (char-to-string '}'))) ::' rest , 2)
len-dec-rewrite {- P91-} ((Id "P91") :: _::_(ParseTree (parsed-anychar-bar-14 x0)) rest) = just (ParseTree (parsed-anychar-bar-15 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P92-} ((Id "P92") :: _::_(InputChar '{') rest) = just (ParseTree (parsed-anychar-bar-16 (string-append 0 (char-to-string '{'))) ::' rest , 2)
len-dec-rewrite {- P93-} ((Id "P93") :: _::_(ParseTree (parsed-anychar-bar-15 x0)) rest) = just (ParseTree (parsed-anychar-bar-16 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P94-} ((Id "P94") :: _::_(InputChar '!') rest) = just (ParseTree (parsed-anychar-bar-17 (string-append 0 (char-to-string '!'))) ::' rest , 2)
len-dec-rewrite {- P95-} ((Id "P95") :: _::_(ParseTree (parsed-anychar-bar-16 x0)) rest) = just (ParseTree (parsed-anychar-bar-17 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P96-} ((Id "P96") :: _::_(InputChar ',') rest) = just (ParseTree (parsed-anychar-bar-18 (string-append 0 (char-to-string ','))) ::' rest , 2)
len-dec-rewrite {- P97-} ((Id "P97") :: _::_(ParseTree (parsed-anychar-bar-17 x0)) rest) = just (ParseTree (parsed-anychar-bar-18 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- P98-} ((Id "P98") :: _::_(InputChar ']') rest) = just (ParseTree (parsed-anychar-bar-19 (string-append 0 (char-to-string ']'))) ::' rest , 2)
len-dec-rewrite {- P99-} ((Id "P99") :: _::_(ParseTree (parsed-anychar-bar-18 x0)) rest) = just (ParseTree (parsed-anychar-bar-19 (string-append 0 x0)) ::' rest , 2)
len-dec-rewrite {- PathsCons-} ((Id "PathsCons") :: (ParseTree parsed-ws) :: (ParseTree (parsed-path x0)) :: _::_(ParseTree (parsed-paths x1)) rest) = just (ParseTree (parsed-paths (norm-paths (PathsCons x0 x1))) ::' rest , 4)
len-dec-rewrite {- ShowQualifiedVars-} ((Id "ShowQualifiedVars") :: (InputChar 's') :: (InputChar 'h') :: (InputChar 'o') :: (InputChar 'w') :: (InputChar '-') :: (InputChar 'q') :: (InputChar 'u') :: (InputChar 'a') :: (InputChar 'l') :: (InputChar 'i') :: (InputChar 'f') :: (InputChar 'i') :: (InputChar 'e') :: (InputChar 'd') :: (InputChar '-') :: (InputChar 'v') :: (InputChar 'a') :: (InputChar 'r') :: (InputChar 's') :: (ParseTree parsed-ows) :: (InputChar '=') :: (ParseTree parsed-ows) :: (ParseTree (parsed-str-bool x0)) :: (ParseTree parsed-ows) :: (InputChar '.') :: _::_(ParseTree parsed-ows) rest) = just (ParseTree (parsed-opt (norm-opt (ShowQualifiedVars x0))) ::' rest , 27)
len-dec-rewrite {- StrBoolFalse-} ((Id "StrBoolFalse") :: (InputChar 'f') :: (InputChar 'a') :: (InputChar 'l') :: (InputChar 's') :: _::_(InputChar 'e') rest) = just (ParseTree (parsed-str-bool (norm-str-bool StrBoolFalse)) ::' rest , 6)
len-dec-rewrite {- StrBoolTrue-} ((Id "StrBoolTrue") :: (InputChar 't') :: (InputChar 'r') :: (InputChar 'u') :: _::_(InputChar 'e') rest) = just (ParseTree (parsed-str-bool (norm-str-bool StrBoolTrue)) ::' rest , 5)
len-dec-rewrite {- UseCedeFiles-} ((Id "UseCedeFiles") :: (InputChar 'u') :: (InputChar 's') :: (InputChar 'e') :: (InputChar '-') :: (InputChar 'c') :: (InputChar 'e') :: (InputChar 'd') :: (InputChar 'e') :: (InputChar '-') :: (InputChar 'f') :: (InputChar 'i') :: (InputChar 'l') :: (InputChar 'e') :: (InputChar 's') :: (ParseTree parsed-ows) :: (InputChar '=') :: (ParseTree parsed-ows) :: (ParseTree (parsed-str-bool x0)) :: (ParseTree parsed-ows) :: (InputChar '.') :: _::_(ParseTree parsed-ows) rest) = just (ParseTree (parsed-opt (norm-opt (UseCedeFiles x0))) ::' rest , 22)
len-dec-rewrite {- OptsNil-} (_::_(Id "OptsNil") rest) = just (ParseTree (parsed-opts (norm-opts OptsNil)) ::' rest , 1)
len-dec-rewrite {- P1-} (_::_(Id "P1") rest) = just (ParseTree (parsed-path-star-1 empty-string) ::' rest , 1)
len-dec-rewrite {- P121-} (_::_(Id "P121") rest) = just (ParseTree parsed-comment-star-30 ::' rest , 1)
len-dec-rewrite {- P134-} (_::_(Id "P134") rest) = just (ParseTree parsed-ows-star-35 ::' rest , 1)
len-dec-rewrite {- PathsNil-} (_::_(Id "PathsNil") rest) = just (ParseTree (parsed-paths (norm-paths PathsNil)) ::' rest , 1)
len-dec-rewrite {- Posinfo-} (_::_(Posinfo n) rest) = just (ParseTree (parsed-posinfo (ℕ-to-string n)) ::' rest , 1)
len-dec-rewrite x = nothing
rrs : rewriteRules
rrs = record { len-dec-rewrite = len-dec-rewrite }
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{- Byzantine Fault Tolerant Consensus Verification in Agda, version 0.9.
Copyright (c) 2020, 2021, Oracle and/or its affiliates.
Licensed under the Universal Permissive License v 1.0 as shown at https://opensource.oracle.com/licenses/upl
-}
open import LibraBFT.Prelude
open import LibraBFT.Lemmas
open import LibraBFT.Base.Types
open import LibraBFT.Abstract.Types.EpochConfig
open WithAbsVote
-- This module defines RecordChains and related types and utility definitions
module LibraBFT.Abstract.RecordChain
(UID : Set)
(_≟UID_ : (u₀ u₁ : UID) → Dec (u₀ ≡ u₁))
(NodeId : Set)
(𝓔 : EpochConfig UID NodeId)
(𝓥 : VoteEvidence UID NodeId 𝓔)
where
open import LibraBFT.Abstract.Records UID _≟UID_ NodeId 𝓔 𝓥
open import LibraBFT.Abstract.Records.Extends UID _≟UID_ NodeId 𝓔 𝓥
open import LibraBFT.Abstract.Types UID NodeId
open EpochConfig 𝓔
-- One way of looking at a 'RecordChain r' is as a path from the epoch's
-- initial record (I) to r. For generality, we express this in two steps.
data RecordChainFrom (o : Record) : Record → Set where
empty : RecordChainFrom o o
step : ∀ {r r'}
→ (rc : RecordChainFrom o r)
→ r ← r'
→ RecordChainFrom o r'
RecordChain : Record → Set
RecordChain = RecordChainFrom I
prevBlock : ∀{q} → RecordChain (Q q) → Block
prevBlock (step {r = B b} _ (B←Q _ _)) = b
prevQCorI : ∀{b} → RecordChain (B b) → Record
prevQCorI (step _ (I←B _ _)) = I
prevQCorI (step {r = Q q} _ (Q←B _ _)) = Q q
-- Returns the unique identifier of the parent tagged to be either
-- the initial block or a B or Q.
parentUID : ∀{b} → RecordChain (B b) → TypedUID
parentUID (step {r = I} _ _) = id-I
parentUID (step {r = Q q} _ _) = id-B∨Q (qCertBlockId q)
-- Defition of 'previous_round' as in Section 5.5 (see Abstract.Properties).
currRound : ∀{r} → RecordChain r → Round
currRound empty = 0
currRound (step {r = r} _ _) = round r
-- TODO-1: it would be cleaner to define prevRound only for RecordChains ending in a Block.
prevRound : ∀{r} → RecordChain r → Round
prevRound empty = 0
prevRound (step rc (I←B x vr)) = 0
prevRound (step rc (Q←B x vr)) = currRound rc
prevRound (step rc (B←Q x vr)) = prevRound rc
-- Extensional definition of prevRound; useful to 'rewrite' on.
prevRound-ss : ∀{b₁ q b}(rc : RecordChain (B b₁))
→ (ext₁ : B b₁ ← Q q)
→ (ext₀ : Q q ← B b)
→ prevRound (step (step rc ext₁) ext₀) ≡ bRound b₁
prevRound-ss rc (B←Q _ _) (Q←B _ _) = refl
----------------------
-- RecordChain Equality and Irrelevance
--
-- Distributing a record relation pointwise
-- over record chains. Let rc₀ and rc₁ be as illustrated
-- below; a value of type ≈RC-pw, named prf is shown
-- in between them.
--
-- rc₀ : B₀ ← C₀ ← B₁ ← C₁ ← ⋯ ← Bₖ ← Cₖ
--
-- prf ≈ ≈ ≈ ≈ ≈ ≈
--
-- rc₁ : 𝓑₀ ← 𝓒₀ ← 𝓑₁ ← 𝓒₁ ← ⋯ ← 𝓑ₖ ← 𝓒ₖ
--
--
data ≈RC-pw {ℓ}(_≈_ : Rel Record ℓ)
: ∀{o₀ o₁ r₀ r₁} → RecordChainFrom o₀ r₀ → RecordChainFrom o₁ r₁ → Set ℓ where
eq-empty : ∀{o₀ o₁} → o₀ ≈ o₁ → ≈RC-pw _≈_ (empty {o = o₀}) (empty {o = o₁})
eq-step : ∀{o₀ o₁ r₀ r₁ s₀ s₁}
→ (rc₀ : RecordChainFrom o₀ s₀)
→ (rc₁ : RecordChainFrom o₁ s₁)
→ r₀ ≈ r₁
→ (ext₀ : s₀ ← r₀)
→ (ext₁ : s₁ ← r₁)
→ ≈RC-pw _≈_ rc₀ rc₁
→ ≈RC-pw _≈_ (step rc₀ ext₀) (step rc₁ ext₁)
-- RecordChain equivalence is then defined in terms of
-- record equivalence (i.e., we don't care about the set of
-- votes for the QCs in the chain); borrowing the illustration
-- above, we now have:
--
-- rc₀ : B₀ ← C₀ ← B₁ ← C₁ ← ⋯ ← Bₖ ← Cₖ
--
-- prf ≡ ≈QC ≡ ≈QC ≡ ≈QC
--
-- rc₁ : 𝓑₀ ← 𝓒₀ ← 𝓑₁ ← 𝓒₁ ← ⋯ ← 𝓑ₖ ← 𝓒ₖ
--
-- It is easy to see that if rc₀ ≈RC rc₁, then they contain
-- the same blocks (propositionally!) but potentially
-- different /sets of votes/ certifying said blocks.
_≈RC_ : ∀{o₀ o₁ r₀ r₁} → RecordChainFrom o₀ r₀ → RecordChainFrom o₁ r₁ → Set
_≈RC_ = ≈RC-pw _≈Rec_
≈RC-head : ∀{o₀ o₁ r₀ r₁}{rc₀ : RecordChainFrom o₀ r₀}{rc₁ : RecordChainFrom o₁ r₁}
→ rc₀ ≈RC rc₁ → o₀ ≈Rec o₁
≈RC-head (eq-empty x) = x
≈RC-head (eq-step _ _ _ _ _ x) = ≈RC-head x
-- Heterogeneous irrelevance of _≈RC_ happens only modulo
-- propositional non-injectivity of block ids.
≈RC-refl : ∀{r₀ r₁}(rc₀ : RecordChain r₀)(rc₁ : RecordChain r₁)
→ r₀ ≈Rec r₁
→ NonInjective _≡_ bId ⊎ (rc₀ ≈RC rc₁)
≈RC-refl empty empty hyp
= inj₂ (eq-empty hyp)
≈RC-refl (step r0 x) (step r1 x₁) hyp
= (←-≈Rec x x₁ hyp ⊎⟫= ≈RC-refl r0 r1)
⊎⟫= (inj₂ ∘ eq-step r0 r1 hyp x x₁)
≈RC-refl empty (step r1 (I←B x x₁)) ()
≈RC-refl empty (step r1 (Q←B x x₁)) ()
≈RC-refl empty (step r1 (B←Q x x₁)) ()
≈RC-refl (step r0 (I←B x x₁)) empty ()
≈RC-refl (step r0 (Q←B x x₁)) empty ()
≈RC-refl (step r0 (B←Q x x₁)) empty ()
≈RC-sym : ∀{o₀ o₁ r₀ r₁}{rc₀ : RecordChainFrom o₀ r₀}{rc₁ : RecordChainFrom o₁ r₁}
→ rc₀ ≈RC rc₁ → rc₁ ≈RC rc₀
≈RC-sym (eq-empty x) = eq-empty (≈Rec-sym x)
≈RC-sym (eq-step rc₀ rc₁ x ext₀ ext₁ hyp) = eq-step rc₁ rc₀ (≈Rec-sym x) ext₁ ext₀ (≈RC-sym hyp)
≈RC-trans : ∀ {r₀ r₁ r₂}
→ {rc₀ : RecordChain r₀}{rc₁ : RecordChain r₁}{rc₂ : RecordChain r₂}
→ rc₀ ≈RC rc₁ → rc₁ ≈RC rc₂ → rc₀ ≈RC rc₂
≈RC-trans (eq-empty x) q = q
≈RC-trans (eq-step rc₀ rc₁ x ext₀ ext₁ p) (eq-step .rc₁ rc₂ x₁ .ext₁ ext₂ q)
= eq-step rc₀ rc₂ (≈Rec-trans x x₁) ext₀ ext₂ (≈RC-trans p q)
-- Heterogeneous irrelevance proves that two record chains that end at the same record
-- have the same blocks and equivalent QCs.
RecordChain-irrelevant : ∀{r}(rc₀ : RecordChain r)(rc₁ : RecordChain r)
→ NonInjective _≡_ bId ⊎ rc₀ ≈RC rc₁
RecordChain-irrelevant rc0 rc1 = ≈RC-refl rc0 rc1 ≈Rec-refl
-------------------------------------------------
-- Sub RecordChains
-- A value of type '⊆RC-pw _≈_ rc1 rc2' establishes that rc1 is
-- a suffix of rc2 modulo _≈_.
data ⊆RC-pw {ℓ}(_≈_ : Rel Record ℓ)
: ∀{o₀ o₁ r₀ r₁} → RecordChainFrom o₀ r₀ → RecordChainFrom o₁ r₁ → Set ℓ where
sub-empty : ∀{o₀ o₁ s₁}{r₁ : RecordChainFrom o₁ s₁} → o₀ ≈ s₁ → ⊆RC-pw _≈_ (empty {o = o₀}) r₁
sub-step : ∀{o₀ o₁ r₀ r₁ s₀ s₁}
→ (rc₀ : RecordChainFrom o₀ s₀)
→ (rc₁ : RecordChainFrom o₁ s₁)
→ r₀ ≈ r₁
→ (ext₀ : s₀ ← r₀)
→ (ext₁ : s₁ ← r₁)
→ ⊆RC-pw _≈_ rc₀ rc₁
→ ⊆RC-pw _≈_ (step rc₀ ext₀) (step rc₁ ext₁)
_⊆RC_ : ∀{o₀ o₁ r₀ r₁} → RecordChainFrom o₀ r₀ → RecordChainFrom o₁ r₁ → Set
_⊆RC_ = ⊆RC-pw _≈Rec_
-- The ⊆RC relation is used to establish irrelevance of suffixes
RecordChainFrom-irrelevant : ∀{o₀ o₁ r₀ r₁}(rc₀ : RecordChainFrom o₀ r₀)(rc₁ : RecordChainFrom o₁ r₁)
→ r₀ ≈Rec r₁
→ NonInjective _≡_ bId ⊎ (rc₀ ⊆RC rc₁ ⊎ rc₁ ⊆RC rc₀)
RecordChainFrom-irrelevant empty empty hyp = inj₂ (inj₁ (sub-empty hyp))
RecordChainFrom-irrelevant empty (step rc1 x) hyp = inj₂ (inj₁ (sub-empty hyp))
RecordChainFrom-irrelevant (step rc0 x) empty hyp = inj₂ (inj₂ (sub-empty (≈Rec-sym hyp)))
RecordChainFrom-irrelevant (step rc0 x) (step rc1 x₁) hyp
= (←-≈Rec x x₁ hyp ⊎⟫= RecordChainFrom-irrelevant rc0 rc1)
⊎⟫= (inj₂ ∘ either (inj₁ ∘ sub-step rc0 rc1 hyp x x₁) (inj₂ ∘ sub-step rc1 rc0 (≈Rec-sym hyp) x₁ x))
-- If a chain from the initial record is a suffix from a second chain,
-- then the second chain is also from the initial record.
RecordChain-glb : ∀{o' r r'}{rc : RecordChain r}{rcf : RecordChainFrom o' r'}
→ rc ⊆RC rcf
→ rc ≈RC rcf
RecordChain-glb {rcf = step _ ()} (sub-empty eq-I)
RecordChain-glb {rcf = empty} (sub-empty eq-I) = eq-empty eq-I
RecordChain-glb (sub-step rc₀ rc₁ x ext₀ ext₁ sub) = eq-step rc₀ rc₁ x ext₀ ext₁ (RecordChain-glb sub)
-------------------------------------------------
-- Id congruences over RecordChain equivalences
parentUID-≈ : ∀{b₀ b₁}(rc₀ : RecordChain (B b₀))(rc₁ : RecordChain (B b₁))
→ rc₀ ≈RC rc₁
→ parentUID rc₀ ≡ parentUID rc₁
parentUID-≈ _ _ (eq-step _ _ (eq-B refl) _ _ (eq-empty x)) = refl
parentUID-≈ _ _ (eq-step _ _ (eq-B refl) _ _ (eq-step _ _ (eq-Q refl) _ _ _)) = refl
--------------------------
-- RecordChain elements
data _∈RC-simple_ {o : Record}(r₀ : Record) : ∀{r₁} → RecordChainFrom o r₁ → Set where
here : ∀{rc : RecordChainFrom o r₀} → r₀ ∈RC-simple rc
there : ∀{r₁ r₂}{rc : RecordChainFrom o r₁}(p : r₁ ← r₂)
→ r₀ ∈RC-simple rc
→ r₀ ∈RC-simple (step rc p)
-- States that a given record belongs in a record chain.
data _∈RC_ {o : Record}(r₀ : Record) : ∀{r₁} → RecordChainFrom o r₁ → Set where
here : ∀{rc : RecordChainFrom o r₀} → r₀ ∈RC rc
there : ∀{r₁ r₂}{rc : RecordChainFrom o r₁}(p : r₁ ← r₂)
→ r₀ ∈RC rc
→ r₀ ∈RC (step rc p)
-- This is an important rule. It is the equivalent of a
-- /congruence/ on record chains and enables us to prove
-- the 𝕂-chain-∈RC property.
transp : ∀{r}{rc₀ : RecordChainFrom o r}{rc₁ : RecordChainFrom o r}
→ r₀ ∈RC rc₀ → rc₀ ≈RC rc₁ → r₀ ∈RC rc₁
∈RC-empty-I : ∀{r} → r ∈RC (empty {o = I}) → r ≡ I
∈RC-empty-I here = refl
∈RC-empty-I (transp old (eq-empty x)) = ∈RC-empty-I old
b∉RCempty : ∀ {b} → B b ∈RC empty → ⊥
b∉RCempty xx with ∈RC-empty-I xx
...| ()
transp-B∈RC : ∀{r r' b}{rc : RecordChain r}{rc' : RecordChain r'}
→ rc ≈RC rc' → B b ∈RC rc → B b ∈RC rc'
transp-B∈RC rc≈rc' (transp b∈rc x) = transp-B∈RC (≈RC-trans x rc≈rc') b∈rc
transp-B∈RC (eq-step rc₀ rc₁ (eq-B refl) ext₀ ext₁ rc≈rc') here = here
transp-B∈RC (eq-step rc₀ rc₁ x .p ext₁ rc≈rc') (there p b∈rc) = there ext₁ (transp-B∈RC rc≈rc' b∈rc)
-- A k-chain (paper Section 5.2; see Abstract.Properties) is a sequence of
-- blocks and quorum certificates for said blocks:
--
-- B₀ ← C₀ ← B₁ ← C₁ ← ⋯ ← Bₖ ← Cₖ
--
-- such that for each Bᵢ some predicate R is satisfies for Bᵢ and Bᵢ₊₁.
-- The first parameter R enables predicate definitions to avoid the need
-- to find a predecessor for B₀ (see Contig definition below).
--
-- The 𝕂-chain datatype captures exactly that structure.
--
data 𝕂-chain (R : ℕ → Record → Record → Set)
: (k : ℕ){o r : Record} → RecordChainFrom o r → Set where
0-chain : ∀{o r}{rc : RecordChainFrom o r} → 𝕂-chain R 0 rc
s-chain : ∀{k o r}{rc : RecordChainFrom o r}{b : Block}{q : QC}
→ (r←b : r ← B b)
→ (prf : R k r (B b))
→ (b←q : B b ← Q q)
→ 𝕂-chain R k rc
→ 𝕂-chain R (suc k) (step (step rc r←b) b←q)
-- Simple 𝕂-chains do not impose any restricton on its records.
Simple : ℕ → Record → Record → Set
Simple _ _ _ = Unit
-- Contiguous 𝕂-chains are those in which all adjacent pairs of Records have contiguous rounds.
Contig : ℕ → Record → Record → Set
Contig 0 _ _ = Unit
Contig (suc _) r r' = round r' ≡ suc (round r)
-- The 'Contig' relation is substitutive in a rather restrictive setting,
-- but this is enough for our purposes.
Contig-subst : ∀{k r r' b}{rc : RecordChain r}{rc' : RecordChain r'}
→ rc ≈RC rc'
→ r ← (B b)
→ Contig k r (B b) → Contig k r' (B b)
Contig-subst {zero} _ _ _ = unit
Contig-subst {suc k} (eq-empty x) _ cr = cr
Contig-subst {suc k} (eq-step .(step rc₀ ext₂) .(step rc₁ ext₃) (eq-Q x) (B←Q refl refl) (B←Q refl refl)
(eq-step rc₀ rc₁ (eq-B refl) ext₂ ext₃ rc≈rc')) (Q←B h0 h1) cr = cr
-- Consequently, contiguous 𝕂-chains are substitutive w.r.t. _≈RC_
transp-𝕂-chain : ∀{k r r'}{rc : RecordChain r}{rc' : RecordChain r'}
→ rc ≈RC rc'
→ 𝕂-chain Contig k rc
→ 𝕂-chain Contig k rc'
transp-𝕂-chain rc≈rc' 0-chain = 0-chain
transp-𝕂-chain (eq-step .(step rc₀ r←b) .(step rc₁ r←b') (eq-Q refl) _ ext₁
(eq-step rc₀ rc₁ (eq-B refl) .r←b r←b' rc≈rc'))
(s-chain r←b prf (B←Q _ _) kc) =
s-chain r←b' (Contig-subst rc≈rc' r←b prf) ext₁ (transp-𝕂-chain rc≈rc' kc)
-- TODO-2: Consider duplicating the above for the ⊆RC relation.
-- I believe we can generalize all of this via the same function, but
-- it's nontrivial to change the whole module.
--
-- Arguably, the right way to deal with all of this would be to
-- design the bare record chain type as the most expressive possible:
--
-- data RC : ℕ → Record → Record
--
-- Then, a value r of type 'RC 12 I (B b)' represents a record chain with
-- 12 steps from I to (B b). K-chains become trivial to express:
--
-- K-chain k = RC (2*k) ...
Contig-subst-⊆ : ∀{k o o' r r' b}{rc : RecordChainFrom o r}{rc' : RecordChainFrom o' r'}
→ (kc : 𝕂-chain Contig k rc)
→ rc ⊆RC rc'
→ r ← (B b)
→ Contig k r (B b) → Contig k r' (B b)
Contig-subst-⊆ 0-chain _ ext hyp = unit
Contig-subst-⊆ (s-chain r←b prf _ kc) (sub-step _ _ (eq-Q x) (B←Q refl refl) (B←Q refl refl)
(sub-step rc₀ rc₁ (eq-B refl) .r←b ext₂ sub)) ext hyp
= hyp
transp-𝕂-chain-⊆ : ∀{k o o' r r'}{rc : RecordChainFrom o r}{rc' : RecordChainFrom o' r'}
→ rc ⊆RC rc'
→ 𝕂-chain Contig k rc
→ 𝕂-chain Contig k rc'
transp-𝕂-chain-⊆ rc⊆rc' 0-chain = 0-chain
transp-𝕂-chain-⊆ (sub-step .(step rc₀ r←b) .(step rc₁ r←b') (eq-Q refl) _ ext₁
(sub-step rc₀ rc₁ (eq-B refl) .r←b r←b' rc⊆rc'))
(s-chain r←b prf (B←Q _ _) kc) =
s-chain r←b' (Contig-subst-⊆ kc rc⊆rc' r←b prf) ext₁ (transp-𝕂-chain-⊆ rc⊆rc' kc)
𝕂-chain-contig : (k : ℕ){o r : Record} → RecordChainFrom o r → Set
𝕂-chain-contig = 𝕂-chain Contig
𝕂-chain-to-Simple : ∀{R k r}{rc : RecordChain r} → 𝕂-chain R k rc → 𝕂-chain Simple k rc
𝕂-chain-to-Simple 0-chain = 0-chain
𝕂-chain-to-Simple (s-chain r←b prf b←q kc) = s-chain r←b unit b←q (𝕂-chain-to-Simple kc)
-- Every record chain that ends in a QC is a 𝕂-chain
to-kchain : ∀{q}(rc : RecordChain (Q q)) → ∃[ k ] (𝕂-chain Simple k rc)
to-kchain (step (step {B b} rc ()) x@(B←Q _ _))
to-kchain (step (step {I} rc x₁) x@(B←Q _ _))
= 1 , s-chain x₁ unit x 0-chain
to-kchain (step (step {Q q} rc x₁) x@(B←Q _ _))
with to-kchain rc
...| k , kc = suc k , s-chain x₁ unit x kc
kchainForget : ∀{P k r}{rc : RecordChain r}(c : 𝕂-chain P k rc) → RecordChain r
kchainForget {rc = rc} _ = rc
-- Returns the round of the block heading the k-chain.
kchainHeadRound : ∀{k r P}{rc : RecordChain r} → 𝕂-chain P k rc → Round
kchainHeadRound (0-chain {r = r}) = round r
kchainHeadRound (s-chain r←b _ b←q kk) = kchainHeadRound kk
kchainBlock : ∀{k o r P}{rc : RecordChainFrom o r} → Fin k → 𝕂-chain P k rc → Block
kchainBlock zero (s-chain {b = b} _ _ _ _) = b
kchainBlock (suc x) (s-chain r←b _ b←q kk) = kchainBlock x kk
kchainBlock-toSimple-≡
: ∀{k r P}{rc : RecordChain r}(x : Fin k)(c : 𝕂-chain P k rc)
→ kchainBlock x (𝕂-chain-to-Simple c) ≡ kchainBlock x c
kchainBlock-toSimple-≡ zero (s-chain _ _ _ _) = refl
kchainBlock-toSimple-≡ (suc x) (s-chain _ _ _ kk) = kchainBlock-toSimple-≡ x kk
_b⟦_⟧ : ∀{k o r P}{rc : RecordChainFrom o r} → 𝕂-chain P k rc → Fin k → Block
chain b⟦ ix ⟧ = kchainBlock ix chain
kchainBlock-≈RC : ∀{k r r'}{rc : RecordChain r}{rc' : RecordChain r'}
→ (c3 : 𝕂-chain Contig k rc)(ix : Fin k)
→ (rc≈rc' : rc ≈RC rc')
→ c3 b⟦ ix ⟧ ≡ transp-𝕂-chain rc≈rc' c3 b⟦ ix ⟧
kchainBlock-≈RC 0-chain () _
kchainBlock-≈RC (s-chain r←b prf (B←Q _ _) kc) ix
(eq-step .(step rc₀ r←b) .(step rc₁ r←b') (eq-Q refl) _ ext₁
(eq-step rc₀ rc₁ (eq-B refl) .r←b r←b' rc≈rc'))
with ix
...| zero = refl
...| suc ix' = kchainBlock-≈RC kc ix' rc≈rc'
kchainBlock-⊆RC : ∀{k o o' r r'}{rc : RecordChainFrom o r}{rc' : RecordChainFrom o' r'}
→ (c3 : 𝕂-chain Contig k rc)(ix : Fin k)
→ (rc⊆rc' : rc ⊆RC rc')
→ c3 b⟦ ix ⟧ ≡ transp-𝕂-chain-⊆ rc⊆rc' c3 b⟦ ix ⟧
kchainBlock-⊆RC 0-chain () _
kchainBlock-⊆RC (s-chain r←b prf (B←Q _ _) kc) ix
(sub-step .(step rc₀ r←b) .(step rc₁ r←b') (eq-Q refl) _ ext₁
(sub-step rc₀ rc₁ (eq-B refl) .r←b r←b' rc≈rc'))
with ix
...| zero = refl
...| suc ix' = kchainBlock-⊆RC kc ix' rc≈rc'
kchainQC : ∀{k r P}{rc : RecordChain r} → Fin k → 𝕂-chain P k rc → QC
kchainQC zero (s-chain {q = q} _ _ _ _) = q
kchainQC (suc x) (s-chain r←b _ b←q kk) = kchainQC x kk
kchain-to-RecordChain-at-b⟦⟧
: ∀{P k r}{rc : RecordChain r}(c : 𝕂-chain P k rc)(ix : Fin k)
→ RecordChain (B (c b⟦ ix ⟧))
kchain-to-RecordChain-at-b⟦⟧ 0-chain ()
kchain-to-RecordChain-at-b⟦⟧ (s-chain {rc = rc} r←b x b←q c) zero
= (step rc r←b)
kchain-to-RecordChain-at-b⟦⟧ (s-chain r←b x b←q c) (suc zz)
= kchain-to-RecordChain-at-b⟦⟧ c zz
kchainBlockRoundZero-lemma
: ∀{k q P}{rc : RecordChain (Q q)}(c : 𝕂-chain P (suc k) rc)
→ getRound (kchainBlock zero c) ≡ getRound q
kchainBlockRoundZero-lemma (s-chain r←b prf (B←Q r h) c) = sym r
kchainBlockRound≤ : ∀{k r P}{rc : RecordChain r}(x y : Fin k)(kc : 𝕂-chain P k rc)
→ x ≤Fin y → getRound (kchainBlock y kc) ≤ getRound (kchainBlock x kc)
kchainBlockRound≤ zero zero (s-chain r←b prf b←q kc) hyp = ≤-refl
kchainBlockRound≤ zero (suc y) (s-chain (Q←B r r←b) prf b←q (s-chain r←b₁ prf₁ (B←Q refl b←q₁) kc)) hyp
= ≤-trans (kchainBlockRound≤ zero y (s-chain r←b₁ prf₁ (B←Q refl b←q₁) kc) z≤n) (<⇒≤ r)
kchainBlockRound≤ (suc x) (suc y) (s-chain r←b prf b←q kc) (s≤s hyp)
= kchainBlockRound≤ x y kc hyp
kchain-round-≤-lemma'
: ∀{k q}{rc : RecordChain (Q q)}(c3 : 𝕂-chain Contig k rc)(ix : Fin k)
→ getRound (c3 b⟦ ix ⟧) ≤ getRound q
kchain-round-≤-lemma' (s-chain r←b x (B←Q refl b←q) c3) zero = ≤-refl
kchain-round-≤-lemma' (s-chain (I←B prf imp) unit (B←Q refl _) 0-chain) (suc ())
kchain-round-≤-lemma' (s-chain (Q←B prf imp) x (B←Q refl _) c2) (suc ix)
= ≤-trans (kchain-round-≤-lemma' c2 ix) (<⇒≤ prf)
rc-←-maxRound : ∀{r r'}(rc : RecordChain r') → r ∈RC rc → round r ≤ round r'
rc-←-maxRound rc here = ≤-refl
rc-←-maxRound rc (transp n x) = rc-←-maxRound _ n
rc-←-maxRound .(step _ p) (there p n) = ≤-trans (rc-←-maxRound _ n) (←-round-≤ p)
kchainBlock-correct
: ∀{P k q b}{rc : RecordChain (B b)}{b←q : B b ← Q q}
→ (kc : 𝕂-chain P k (step rc b←q))
→ (x : Fin k) → (B (kc b⟦ x ⟧)) ∈RC rc
kchainBlock-correct (s-chain r←b prf b←q kc) zero = here
kchainBlock-correct (s-chain r←b prf b←q (s-chain r←b₁ prf₁ b←q₁ kc)) (suc x)
= there r←b (there b←q₁ (kchainBlock-correct (s-chain r←b₁ prf₁ b←q₁ kc) x))
-- This is an extended form of RecordChain-irrelevance.
-- Let rc be:
--
-- B₀ ← C₀ ← B₁ ← C₁ ← ⋯ ← Bₙ ← Cₙ
--
-- The (c : 𝕂-chain P k rc) is a predicate on the shape
-- of rc, establishing that it must be of the following shape:
-- (where consecutive blocks satisfy P)
--
-- B₀ ← C₀ ← B₁ ← C₁ ← ⋯ ← Bₙ₋ₖ ← Cₙ₋ₖ ⋯ ← Bₙ₋₁ ← Cₙ₋₁ ← Bₙ ← Cₙ
-- /\ /\ /
-- ⋯ P ₋⌟ ⌞₋₋₋₋ P ₋₋₋₋⌟ ⌞₋₋₋₋ P ₋₋₋⌟
--
-- This property states that any other RecordChain that contains one
-- block b of the kchain above, it also contains the prefix of the
-- kchain leading to b.
--
𝕂-chain-∈RC : ∀{r k P}{rc : RecordChain r}
→ (c : 𝕂-chain P k rc)
→ (x y : Fin k)
→ x ≤Fin y
→ {b : Block}(prf : kchainBlock x c ≡ b)
→ (rc₁ : RecordChain (B b))
→ NonInjective _≡_ bId ⊎ (B (kchainBlock y c) ∈RC rc₁)
𝕂-chain-∈RC (s-chain r←b prf b←q c) zero y z≤n refl rc1
with RecordChain-irrelevant (step (kchainForget c) r←b) rc1
...| inj₁ hb = inj₁ hb
...| inj₂ res = inj₂ (transp (kchainBlock-correct (s-chain r←b prf b←q c) y) res)
𝕂-chain-∈RC (s-chain r←b prf b←q c) (suc x) (suc y) (s≤s x≤y) hyp rc1
= 𝕂-chain-∈RC c x y x≤y hyp rc1
-----------------
-- Commit Rule --
-- A block (and everything preceeding it) is said to match the commit rule
-- when the block is the head of a contiguious 3-chain. Here we define an auxiliary
-- datatype to make definitions more readable.
data CommitRuleFrom {o r : Record}(rc : RecordChainFrom o r)(b : Block) : Set where
commit-rule : (c3 : 𝕂-chain Contig 3 rc)
→ b ≡ c3 b⟦ suc (suc zero) ⟧
→ CommitRuleFrom rc b
CommitRule : ∀{r} → RecordChain r → Block → Set
CommitRule = CommitRuleFrom
transp-CR : ∀{b q}{rc rc' : RecordChain (Q q)}
→ rc ≈RC rc'
→ CommitRule rc b
→ CommitRule rc' b
transp-CR rc≈rc' (commit-rule c3 x) =
commit-rule (transp-𝕂-chain rc≈rc' c3)
(trans x (kchainBlock-≈RC c3 (suc (suc zero)) rc≈rc'))
crf⇒cr : ∀ {o q b}
→ (rcf : RecordChainFrom o (Q q))
→ (rc : RecordChain (Q q))
→ CommitRuleFrom rcf b
→ NonInjective-≡ bId ⊎ CommitRule {Q q} rc b
crf⇒cr rcf rc (commit-rule c3 prf)
with RecordChainFrom-irrelevant rcf rc ≈Rec-refl
...| inj₁ hb = inj₁ hb
...| inj₂ (inj₁ rcf⊆rc) = inj₂ (commit-rule (transp-𝕂-chain-⊆ rcf⊆rc c3)
(trans prf (kchainBlock-⊆RC c3 (suc (suc zero)) rcf⊆rc)))
...| inj₂ (inj₂ rc⊆rcf)
with ≈RC-sym (RecordChain-glb rc⊆rcf)
...| rcf≈rc
with ≈RC-head rcf≈rc
...| eq-I = inj₂ (transp-CR rcf≈rc (commit-rule c3 prf))
vote≡⇒QPrevId≡ : {q q' : QC} {v v' : Vote}
→ v ∈ qcVotes q
→ v' ∈ qcVotes q'
→ v ≡ v'
→ qCertBlockId q ≡ qCertBlockId q'
vote≡⇒QPrevId≡ {q} {q'} v∈q v'∈q' refl
with witness v∈q (qVotes-C2 q) | witness v'∈q' (qVotes-C2 q')
... | refl | refl = refl
vote≡⇒QRound≡ : {q q' : QC} {v v' : Vote}
→ v ∈ qcVotes q
→ v' ∈ qcVotes q'
→ v ≡ v'
→ getRound q ≡ getRound q'
vote≡⇒QRound≡ {q} {q'} v∈q v'∈q' refl
with witness v∈q (qVotes-C3 q) | witness v'∈q' (qVotes-C3 q')
... | refl | refl = refl
¬bRound≡0 : ∀{b} → RecordChain (B b) → ¬ (getRound b ≡ 0)
¬bRound≡0 (step s (I←B () h)) refl
¬bRound≡0 (step s (Q←B () h)) refl
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{-# OPTIONS --without-K --rewriting #-}
open import HoTT
-- an attempt to speed up [QuotGroup (im-nprop ...)]
-- which removes most intermediate constructions
module groups.Cokernel {i j}
{G : Group i} {H : Group j}
(φ : G →ᴳ H) (H-ab : is-abelian H) where
-- G ---φ--→ᴳ H
private
module G = Group G
module H = AbGroup (H , H-ab)
module φ = GroupHom φ
coker-rel : Rel H.El (lmax i j)
coker-rel h₁ h₂ = Trunc -1 (hfiber φ.f (H.diff h₁ h₂))
private
coker-El : Type (lmax i j)
coker-El = SetQuot coker-rel
coker-struct : GroupStructure coker-El
coker-struct = record {M} where
module M where
ident : coker-El
ident = q[ H.ident ]
inv : coker-El → coker-El
inv = SetQuot-rec SetQuot-level inv' inv-rel
where
inv' : H.El → coker-El
inv' h = q[ H.inv h ]
abstract
inv-rel : ∀ {h₁ h₂} → coker-rel h₁ h₂ → inv' h₁ == inv' h₂
inv-rel {h₁} {h₂} = Trunc-rec (SetQuot-level _ _)
λ{(g , φg=h₁h₂⁻¹) → quot-rel
[ G.inv g , φ.pres-inv g ∙ ap H.inv φg=h₁h₂⁻¹
∙ H.inv-diff h₁ h₂ ∙ H.comm h₂ (H.inv h₁)
∙ ap (H.comp (H.inv h₁)) (! (H.inv-inv h₂)) ]}
comp : coker-El → coker-El → coker-El
comp = SetQuot-rec level comp' comp-rel where
abstract
level : is-set (coker-El → coker-El)
level = →-is-set SetQuot-level
comp' : H.El → coker-El → coker-El
comp' h₁ = SetQuot-rec SetQuot-level comp'' comp'-rel where
comp'' : H.El → coker-El
comp'' h₂ = q[ H.comp h₁ h₂ ]
abstract
comp'-rel : ∀ {h₂ h₂'} → coker-rel h₂ h₂' → comp'' h₂ == comp'' h₂'
comp'-rel {h₂} {h₂'} = Trunc-rec (SetQuot-level _ _)
λ{(g , φg=h₂h₂'⁻¹) → quot-rel
[ g , ! ( ap (H.comp (H.comp h₁ h₂)) (H.inv-comp h₁ h₂')
∙ H.assoc h₁ h₂ (H.comp (H.inv h₂') (H.inv h₁))
∙ ap (H.comp h₁) (! $ H.assoc h₂ (H.inv h₂') (H.inv h₁))
∙ H.comm h₁ (H.comp (H.comp h₂ (H.inv h₂')) (H.inv h₁))
∙ H.assoc (H.comp h₂ (H.inv h₂')) (H.inv h₁) h₁
∙ ap2 H.comp (! φg=h₂h₂'⁻¹) (H.inv-l h₁)
∙ H.unit-r (φ.f g) )]}
abstract
comp-rel : ∀ {h₁ h₁'} → coker-rel h₁ h₁' → comp' h₁ == comp' h₁'
comp-rel {h₁} {h₁'} = Trunc-rec (level _ _)
λ{(g , φg=h₁h₁'⁻¹) → λ= $ SetQuot-elim (λ _ → =-preserves-set SetQuot-level)
(λ h₂ → quot-rel
[ g , ! ( ap (H.comp (H.comp h₁ h₂)) (H.inv-comp h₁' h₂)
∙ H.assoc h₁ h₂ (H.comp (H.inv h₂) (H.inv h₁'))
∙ ap (H.comp h₁) ( ! (H.assoc h₂ (H.inv h₂) (H.inv h₁'))
∙ ap (λ h → H.comp h (H.inv h₁')) (H.inv-r h₂)
∙ H.unit-l (H.inv h₁'))
∙ ! φg=h₁h₁'⁻¹ )])
(λ _ → prop-has-all-paths-↓ (SetQuot-level _ _))}
abstract
unit-l : ∀ cok → comp ident cok == cok
unit-l = SetQuot-elim
(λ _ → =-preserves-set SetQuot-level)
(λ h → ap q[_] $ H.unit-l h)
(λ _ → prop-has-all-paths-↓ (SetQuot-level _ _))
assoc : ∀ cok₁ cok₂ cok₃ → comp (comp cok₁ cok₂) cok₃ == comp cok₁ (comp cok₂ cok₃)
assoc = SetQuot-elim
(λ _ → Π-is-set λ _ → Π-is-set λ _ → =-preserves-set SetQuot-level)
(λ h₁ → SetQuot-elim
(λ _ → Π-is-set λ _ → =-preserves-set SetQuot-level)
(λ h₂ → SetQuot-elim
(λ _ → =-preserves-set SetQuot-level)
(λ h₃ → ap q[_] $ H.assoc h₁ h₂ h₃)
(λ _ → prop-has-all-paths-↓ (SetQuot-level _ _)))
(λ _ → prop-has-all-paths-↓ (Π-is-prop λ _ → SetQuot-level _ _)))
(λ _ → prop-has-all-paths-↓ (Π-is-prop λ _ → Π-is-prop λ _ → SetQuot-level _ _))
inv-l : ∀ cok → comp (inv cok) cok == ident
inv-l = SetQuot-elim
(λ _ → =-preserves-set SetQuot-level)
(λ h → ap q[_] $ H.inv-l h)
(λ _ → prop-has-all-paths-↓ (SetQuot-level _ _))
Coker : Group (lmax i j)
Coker = group _ SetQuot-level coker-struct
module Coker = Group Coker
{- correctness -}
Coker-β : Coker ≃ᴳ QuotGroup (im-npropᴳ φ H-ab)
Coker-β = ≃-to-≃ᴳ (ide _) to-pres-comp where
abstract
to-pres-comp : preserves-comp Coker.comp
(QuotGroup.comp (im-npropᴳ φ H-ab)) (idf _)
to-pres-comp = SetQuot-elim
(λ _ → Π-is-set λ _ → =-preserves-set SetQuot-level)
(λ _ → SetQuot-elim
(λ _ → =-preserves-set SetQuot-level)
(λ _ → idp)
(λ _ → prop-has-all-paths-↓ (SetQuot-level _ _)))
(λ _ → prop-has-all-paths-↓ (Π-is-prop λ _ → SetQuot-level _ _))
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{-# OPTIONS --safe --warning=error --without-K #-}
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import LogicalFormulae
open import Sets.EquivalenceRelations
open import Setoids.Setoids
module Setoids.Intersection.Definition {a b : _} {A : Set a} (S : Setoid {a} {b} A) where
open Setoid S
open Equivalence eq
open import Setoids.Subset S
intersectionPredicate : {c d : _} {pred1 : A → Set c} {pred2 : A → Set d} (s1 : subset pred1) (s2 : subset pred2) → A → Set (c ⊔ d)
intersectionPredicate {pred1 = pred1} {pred2} s1 s2 a = pred1 a && pred2 a
intersection : {c d : _} {pred1 : A → Set c} {pred2 : A → Set d} (s1 : subset pred1) (s2 : subset pred2) → subset (intersectionPredicate s1 s2)
intersection s1 s2 {x1} {x2} x1=x2 (inS1 ,, inS2) = s1 x1=x2 inS1 ,, s2 x1=x2 inS2
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{-# OPTIONS --cubical --safe #-}
module Control.Monad.State where
open import Prelude
record StatePair {s a} (S : Type s) (A : Type a) : Type (a ℓ⊔ s) where
constructor state-pair
field
value : A
state : S
open StatePair public
State : ∀ {s a} → Type s → Type a → Type (s ℓ⊔ a)
State S A = S → StatePair S A
private
variable
s : Level
S : Type s
pure : A → State S A
pure = state-pair
_<*>_ : State S (A → B) → State S A → State S B
(fs <*> xs) s =
let state-pair f s′ = fs s
state-pair x s″ = xs s′
in state-pair (f x) s″
_>>=_ : State S A → (A → State S B) → State S B
(xs >>= f) s = let state-pair x s′ = xs s in f x s′
get : State S S
value (get s) = s
state (get s) = s
modify : (S → S) → State S ⊤
value (modify f s) = tt
state (modify f s) = f s
put : S → State S ⊤
value (put s _) = tt
state (put s _) = s
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open import Oscar.Prelude
open import Oscar.Class
open import Oscar.Class.Unit
module Oscar.Class.SimilarityM where
module SimilarityM
{𝔞} {𝔄 : Ø 𝔞}
{𝔟} {𝔅 : Ø 𝔟}
{𝔣} {𝔉 : Ø 𝔣}
{𝔞̇ 𝔟̇}
(_∼₁_ : 𝔄 → 𝔄 → Ø 𝔞̇)
(_∼₂_ : 𝔅 → 𝔅 → Ø 𝔟̇) (let _∼₂_ = _∼₂_; infix 4 _∼₂_)
(_◃_ : 𝔉 → 𝔄 → 𝔅) (let _◃_ = _◃_; infix 16 _◃_)
x y
= ℭLASS (_◃_ , _∼₁_ , _∼₂_ , x , y) (∀ f → x ∼₁ y → f ◃ x ∼₂ f ◃ y)
module _
{𝔞} {𝔄 : Ø 𝔞}
{𝔟} {𝔅 : Ø 𝔟}
{𝔣} {𝔉 : Ø 𝔣}
{𝔞̇ 𝔟̇}
{∼₁ : 𝔄 → 𝔄 → Ø 𝔞̇}
{∼₂ : 𝔅 → 𝔅 → Ø 𝔟̇}
{◃ : 𝔉 → 𝔄 → 𝔅}
{x y}
where
similarityM = SimilarityM.method ∼₁ ∼₂ ◃ x y
instance SimilarityM--Unit : ⦃ _ : SimilarityM.class ∼₁ ∼₂ ◃ x y ⦄ → Unit.class (SimilarityM.type ∼₁ ∼₂ ◃ x y)
SimilarityM--Unit .⋆ = similarityM
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{-
Definition of a homogeneous pointed type, and proofs that pi, product, path, and discrete types are homogeneous
Portions of this file adapted from Nicolai Kraus' code here:
https://bitbucket.org/nicolaikraus/agda/src/e30d70c72c6af8e62b72eefabcc57623dd921f04/trunc-inverse.lagda
-}
{-# OPTIONS --safe #-}
module Cubical.Foundations.Pointed.Homogeneous where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Univalence
open import Cubical.Foundations.Path
open import Cubical.Data.Sigma
open import Cubical.Data.Empty as ⊥
open import Cubical.Relation.Nullary
open import Cubical.Foundations.GroupoidLaws
open import Cubical.Foundations.Pointed.Base
open import Cubical.Foundations.Pointed.Properties
open import Cubical.Structures.Pointed
isHomogeneous : ∀ {ℓ} → Pointed ℓ → Type (ℓ-suc ℓ)
isHomogeneous {ℓ} (A , x) = ∀ y → Path (Pointed ℓ) (A , x) (A , y)
-- Pointed functions into a homogeneous type are equal as soon as they are equal
-- as unpointed functions
→∙Homogeneous≡ : ∀ {ℓ ℓ'} {A∙ : Pointed ℓ} {B∙ : Pointed ℓ'} {f∙ g∙ : A∙ →∙ B∙}
(h : isHomogeneous B∙) → f∙ .fst ≡ g∙ .fst → f∙ ≡ g∙
→∙Homogeneous≡ {A∙ = A∙@(_ , a₀)} {B∙@(B , _)} {f∙@(_ , f₀)} {g∙@(_ , g₀)} h p =
subst (λ Q∙ → PathP (λ i → A∙ →∙ Q∙ i) f∙ g∙) (sym (flipSquare fix)) badPath
where
badPath : PathP (λ i → A∙ →∙ (B , (sym f₀ ∙∙ funExt⁻ p a₀ ∙∙ g₀) i)) f∙ g∙
badPath i .fst = p i
badPath i .snd j = doubleCompPath-filler (sym f₀) (funExt⁻ p a₀) g₀ j i
fix : PathP (λ i → B∙ ≡ (B , (sym f₀ ∙∙ funExt⁻ p a₀ ∙∙ g₀) i)) refl refl
fix i =
hcomp
(λ j → λ
{ (i = i0) → lCancel (h (pt B∙)) j
; (i = i1) → lCancel (h (pt B∙)) j
})
(sym (h (pt B∙)) ∙ h ((sym f₀ ∙∙ funExt⁻ p a₀ ∙∙ g₀) i))
→∙Homogeneous≡Path : ∀ {ℓ ℓ'} {A∙ : Pointed ℓ} {B∙ : Pointed ℓ'} {f∙ g∙ : A∙ →∙ B∙}
(h : isHomogeneous B∙) → (p q : f∙ ≡ g∙) → cong fst p ≡ cong fst q → p ≡ q
→∙Homogeneous≡Path {A∙ = A∙@(A , a₀)} {B∙@(B , b)} {f∙@(f , f₀)} {g∙@(g , g₀)} h p q r =
transport (λ k
→ PathP (λ i
→ PathP (λ j → (A , a₀) →∙ newPath-refl p q r i j (~ k))
(f , f₀) (g , g₀)) p q)
(badPath p q r)
where
newPath : (p q : f∙ ≡ g∙) (r : cong fst p ≡ cong fst q)
→ Square (refl {x = b}) refl refl refl
newPath p q r i j =
hcomp (λ k → λ {(i = i0) → cong snd p j k
; (i = i1) → cong snd q j k
; (j = i0) → f₀ k
; (j = i1) → g₀ k})
(r i j a₀)
newPath-refl : (p q : f∙ ≡ g∙) (r : cong fst p ≡ cong fst q)
→ PathP (λ i → (PathP (λ j → B∙ ≡ (B , newPath p q r i j))) refl refl) refl refl
newPath-refl p q r i j k =
hcomp (λ w → λ { (i = i0) → lCancel (h b) w k
; (i = i1) → lCancel (h b) w k
; (j = i0) → lCancel (h b) w k
; (j = i1) → lCancel (h b) w k
; (k = i0) → lCancel (h b) w k
; (k = i1) → B , newPath p q r i j})
((sym (h b) ∙ h (newPath p q r i j)) k)
badPath : (p q : f∙ ≡ g∙) (r : cong fst p ≡ cong fst q)
→ PathP (λ i →
PathP (λ j → A∙ →∙ (B , newPath p q r i j))
(f , f₀) (g , g₀))
p q
fst (badPath p q r i j) = r i j
snd (badPath p q s i j) k =
hcomp (λ r → λ { (i = i0) → snd (p j) (r ∧ k)
; (i = i1) → snd (q j) (r ∧ k)
; (j = i0) → f₀ (k ∧ r)
; (j = i1) → g₀ (k ∧ r)
; (k = i0) → s i j a₀})
(s i j a₀)
isHomogeneousPi : ∀ {ℓ ℓ'} {A : Type ℓ} {B∙ : A → Pointed ℓ'}
→ (∀ a → isHomogeneous (B∙ a)) → isHomogeneous (Πᵘ∙ A B∙)
isHomogeneousPi h f i .fst = ∀ a → typ (h a (f a) i)
isHomogeneousPi h f i .snd a = pt (h a (f a) i)
isHomogeneousΠ∙ : ∀ {ℓ ℓ'} (A : Pointed ℓ) (B : typ A → Type ℓ')
→ (b₀ : B (pt A))
→ ((a : typ A) (x : B a) → isHomogeneous (B a , x))
→ (f : Π∙ A B b₀)
→ isHomogeneous (Π∙ A B b₀ , f)
fst (isHomogeneousΠ∙ A B b₀ h f g i) =
Σ[ r ∈ ((a : typ A) → fst ((h a (fst f a) (fst g a)) i)) ]
r (pt A) ≡ hcomp (λ k → λ {(i = i0) → snd f k
; (i = i1) → snd g k})
(snd (h (pt A) (fst f (pt A)) (fst g (pt A)) i))
snd (isHomogeneousΠ∙ A B b₀ h f g i) =
(λ a → snd (h a (fst f a) (fst g a) i))
, λ j → hcomp (λ k → λ { (i = i0) → snd f (k ∧ j)
; (i = i1) → snd g (k ∧ j)
; (j = i0) → snd (h (pt A) (fst f (pt A))
(fst g (pt A)) i)})
(snd (h (pt A) (fst f (pt A)) (fst g (pt A)) i))
isHomogeneous→∙ : ∀ {ℓ ℓ'} {A∙ : Pointed ℓ} {B∙ : Pointed ℓ'}
→ isHomogeneous B∙ → isHomogeneous (A∙ →∙ B∙ ∙)
isHomogeneous→∙ {A∙ = A∙} {B∙} h f∙ =
ΣPathP
( (λ i → Π∙ A∙ (λ a → T a i) (t₀ i))
, PathPIsoPath _ _ _ .Iso.inv
(→∙Homogeneous≡ h
(PathPIsoPath (λ i → (a : typ A∙) → T a i) (λ _ → pt B∙) _ .Iso.fun
(λ i a → pt (h (f∙ .fst a) i))))
)
where
T : ∀ a → typ B∙ ≡ typ B∙
T a i = typ (h (f∙ .fst a) i)
t₀ : PathP (λ i → T (pt A∙) i) (pt B∙) (pt B∙)
t₀ = cong pt (h (f∙ .fst (pt A∙))) ▷ f∙ .snd
isHomogeneousProd : ∀ {ℓ ℓ'} {A∙ : Pointed ℓ} {B∙ : Pointed ℓ'}
→ isHomogeneous A∙ → isHomogeneous B∙ → isHomogeneous (A∙ ×∙ B∙)
isHomogeneousProd hA hB (a , b) i .fst = typ (hA a i) × typ (hB b i)
isHomogeneousProd hA hB (a , b) i .snd .fst = pt (hA a i)
isHomogeneousProd hA hB (a , b) i .snd .snd = pt (hB b i)
isHomogeneousPath : ∀ {ℓ} (A : Type ℓ) {x y : A} (p : x ≡ y) → isHomogeneous ((x ≡ y) , p)
isHomogeneousPath A {x} {y} p q
= pointed-sip ((x ≡ y) , p) ((x ≡ y) , q) (eqv , compPathr-cancel p q)
where eqv : (x ≡ y) ≃ (x ≡ y)
eqv = compPathlEquiv (q ∙ sym p)
module HomogeneousDiscrete {ℓ} {A∙ : Pointed ℓ} (dA : Discrete (typ A∙)) (y : typ A∙) where
-- switches pt A∙ with y
switch : typ A∙ → typ A∙
switch x with dA x (pt A∙)
... | yes _ = y
... | no _ with dA x y
... | yes _ = pt A∙
... | no _ = x
switch-ptA∙ : switch (pt A∙) ≡ y
switch-ptA∙ with dA (pt A∙) (pt A∙)
... | yes _ = refl
... | no ¬p = ⊥.rec (¬p refl)
switch-idp : ∀ x → switch (switch x) ≡ x
switch-idp x with dA x (pt A∙)
switch-idp x | yes p with dA y (pt A∙)
switch-idp x | yes p | yes q = q ∙ sym p
switch-idp x | yes p | no _ with dA y y
switch-idp x | yes p | no _ | yes _ = sym p
switch-idp x | yes p | no _ | no ¬p = ⊥.rec (¬p refl)
switch-idp x | no ¬p with dA x y
switch-idp x | no ¬p | yes p with dA y (pt A∙)
switch-idp x | no ¬p | yes p | yes q = ⊥.rec (¬p (p ∙ q))
switch-idp x | no ¬p | yes p | no _ with dA (pt A∙) (pt A∙)
switch-idp x | no ¬p | yes p | no _ | yes _ = sym p
switch-idp x | no ¬p | yes p | no _ | no ¬q = ⊥.rec (¬q refl)
switch-idp x | no ¬p | no ¬q with dA x (pt A∙)
switch-idp x | no ¬p | no ¬q | yes p = ⊥.rec (¬p p)
switch-idp x | no ¬p | no ¬q | no _ with dA x y
switch-idp x | no ¬p | no ¬q | no _ | yes q = ⊥.rec (¬q q)
switch-idp x | no ¬p | no ¬q | no _ | no _ = refl
switch-eqv : typ A∙ ≃ typ A∙
switch-eqv = isoToEquiv (iso switch switch switch-idp switch-idp)
isHomogeneousDiscrete : ∀ {ℓ} {A∙ : Pointed ℓ} (dA : Discrete (typ A∙)) → isHomogeneous A∙
isHomogeneousDiscrete {ℓ} {A∙} dA y
= pointed-sip (typ A∙ , pt A∙) (typ A∙ , y) (switch-eqv , switch-ptA∙)
where open HomogeneousDiscrete {ℓ} {A∙} dA y
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module tests.PrintBool where
open import Prelude.IO
open import Prelude.Bool
open import Prelude.Char
open import Prelude.List
open import Prelude.Unit
open import Prelude.String
isNewline : Char -> Bool
isNewline '\n' = true
isNewline _ = false
sequence : {A : Set} -> List (IO A) -> IO (List A)
sequence [] = return []
sequence (x :: xs) =
r <- x ,
rs <- sequence xs ,
return (r :: rs)
mapM : {A B : Set} -> (A -> IO B) -> List A -> IO (List B)
mapM f xs = sequence (map f xs)
printList : List Char -> IO Unit
printList xs =
mapM printChar xs ,,
printChar '\n'
main : IO Unit
main =
printChar 'a' ,,
printList ('a' :: 'b' :: 'c' :: []) ,,
putStrLn "printBool" ,,
printBool (isNewline '\n') ,,
printBool (isNewline 'a') ,,
return unit
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{-# OPTIONS --prop --type-in-type #-}
open import Agda.Primitive
data Squash (A : Setω) : Prop where
squash : A → Squash A
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------------------------------------------------------------------------------
-- Testing the translation of higher-order functions
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --universal-quantified-functions #-}
{-# OPTIONS --without-K #-}
-- We can use the Agda pragma @--universal-quantified-functions@ to
-- translate higher-order functions. The canonical examples are the
-- conversion rules for the λ-abstraction and the fixed-point
-- operator.
module NonFOLHigherOrderFunctions where
infixl 6 _∙_
infix 4 _≡_
postulate
D : Set
_≡_ : D → D → Set
lam : (D → D) → D
_∙_ : D → D → D
fix : (D → D) → D
postulate beta : (f : D → D) → (a : D) → (lam f) ∙ a ≡ f a
{-# ATP axiom beta #-}
postulate fix-f : (f : D → D) → fix f ≡ f (fix f)
{-# ATP axiom fix-f #-}
-- We need to have at least one conjecture to generate a TPTP file.
postulate refl : ∀ d → d ≡ d
{-# ATP prove refl #-}
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module IO where
open import Base
postulate
IO : Set -> Set
getLine : IO String
putStrLn : String -> IO Unit
mapM₋ : {A : Set} -> (A -> IO Unit) -> List A -> IO Unit
bindIO : {A B : Set} -> IO A -> (A -> IO B) -> IO B
returnIO : {A : Set} -> A -> IO A
{-# COMPILED putStrLn putStrLn #-}
{-# COMPILED mapM₋ (\_ -> mapM_ :: (a -> IO ()) -> [a] -> IO ()) #-}
-- we need to throw away the type argument to mapM_
-- and resolve the overloading explicitly (since Alonzo
-- output is sprinkled with unsafeCoerce#).
{-# COMPILED bindIO (\_ _ -> (>>=) :: IO a -> (a -> IO b) -> IO b) #-}
{-# COMPILED returnIO (\_ -> return :: a -> IO a) #-}
{-# COMPILED getLine getLine #-}
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module Oscar.Data.Fin.ThickAndThin where
open import Oscar.Data.Fin
open import Oscar.Class.ThickAndThin
import Data.Fin as F
import Data.Fin.Properties as F
instance ThickAndThinFin : ThickAndThin Fin
ThickAndThin.thin ThickAndThinFin = F.thin
ThickAndThin.thick ThickAndThinFin = F.thick
ThickAndThin.thin-injective ThickAndThinFin z = F.thin-injective {z = z}
ThickAndThin.thick∘thin=id ThickAndThinFin = F.thick-thin
ThickAndThin.check ThickAndThinFin = F.check
ThickAndThin.thin-check-id ThickAndThinFin = F.thin-check-id
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{- Byzantine Fault Tolerant Consensus Verification in Agda, version 0.9.
Copyright (c) 2021, Oracle and/or its affiliates.
Licensed under the Universal Permissive License v 1.0 as shown at https://opensource.oracle.com/licenses/upl
-}
open import LibraBFT.Base.Types
open import LibraBFT.ImplShared.Base.Types
open import LibraBFT.ImplShared.Consensus.Types
open import LibraBFT.Impl.OBM.Logging.Logging
open import Optics.All
open import Util.Prelude
module LibraBFT.Impl.Consensus.ConsensusTypes.VoteData where
verify : VoteData → Either ErrLog Unit
verify self = do
lcheck (self ^∙ vdParent ∙ biEpoch == self ^∙ vdProposed ∙ biEpoch)
("parent and proposed epochs do not match" ∷ [])
lcheck (⌊ self ^∙ vdParent ∙ biRound <? self ^∙ vdProposed ∙ biRound ⌋)
("proposed round is less than parent round" ∷ [])
-- lcheck (self^.vdParent.biTimestamp <= self^.vdProposed.biTimestamp)
-- ["proposed happened before parent"]
lcheck (⌊ (self ^∙ vdParent ∙ biVersion) ≤?-Version (self ^∙ vdProposed ∙ biVersion) ⌋)
("proposed version is less than parent version" ∷ [])
-- , lsVersion (self^.vdProposed.biVersion), lsVersion (self^.vdParent.biVersion)]
new : BlockInfo → BlockInfo → VoteData
new = VoteData∙new
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module Dummy where
-- Run this in Acme: go build && ./acme-agda -v
data ℕ : Set where
zero : ℕ
succ : ℕ -> ℕ
_+_ : ℕ → ℕ → ℕ
m + n = {!!}
_*_ : ℕ → ℕ → ℕ
m * n = {!m !} | {
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{-# OPTIONS --cubical --safe #-}
module Label where
open import Cubical.Core.Everything
open import Cubical.Foundations.Prelude using (isProp; transport)
open import Cubical.Data.Nat using (ℕ; zero; suc; isSetℕ)
open import Cubical.Data.Nat.Order using (_<_; _≤_; ≤-refl; <-weaken; ≤<-trans; m≤n-isProp; <-asym)
open import Cubical.Data.Maybe using (Maybe; nothing; just)
open import Cubical.Data.Empty using () renaming (rec to ⊥-elim)
Label : Set
Label = ℕ
data Record (A : Set) : Label -> Set where
nil : forall {l} -> Record A l
cons : forall {l} -> Record A l -> (l' : Label) -> A -> .(l < l') -> Record A l'
data _∈_ {A : Set} (l₁ : Label) {l : Label} : Record A l -> Set where
here : forall {l'} {r : Record A l'} {x lt} -> l₁ ≡ l -> l₁ ∈ cons r l x lt
there : forall {l'} {r : Record A l'} {x lt} -> l₁ ∈ r -> l₁ ∈ cons r l x lt
find : forall {A} {l} -> (l₁ : Label) -> (r : Record A l) -> l₁ ∈ r -> A
find l₁ (cons _ _ x _) (here e) = x
find l₁ (cons r _ _ _) (there l₁∈r) = find l₁ r l₁∈r
∈-implies-≤ : forall {A} {l l'} {r : Record A l'} -> l ∈ r -> l ≤ l'
∈-implies-≤ {l = l} (here e) = transport (λ i -> l ≤ e i) ≤-refl
∈-implies-≤ (there {lt = lt} l∈r) = <-weaken (≤<-trans (∈-implies-≤ l∈r) lt)
l∈r-isProp : forall {A} l {l'} (r : Record A l') -> isProp (l ∈ r)
l∈r-isProp l {l'} (cons _ _ _ _) (here {lt = a} e1) (here {lt = b} e2) = λ i -> here {lt = m≤n-isProp a b i} (isSetℕ l l' e1 e2 i)
l∈r-isProp l (cons {l = l₁} r _ _ _) (here {lt = k} e) (there y) = ⊥-elim (<-asym k (transport (λ i -> e i ≤ l₁) (∈-implies-≤ y)))
l∈r-isProp l (cons {l = l₁} r _ _ _) (there {lt = k} x) (here e) = ⊥-elim (<-asym k (transport (λ i -> e i ≤ l₁) (∈-implies-≤ x)))
l∈r-isProp l (cons r _ _ _) (there {lt = k1} x) (there {lt = k2} y) = let a = l∈r-isProp l r x y in λ i → there {lt = m≤n-isProp k1 k2 i} (a i)
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
{-
This file defines
sucPred : ∀ (i : Int) → sucInt (predInt i) ≡ i
predSuc : ∀ (i : Int) → predInt (sucInt i) ≡ i
discreteInt : discrete Int
isSetInt : isSet Int
addition of Int is defined _+_ : Int → Int → Int
as well as its commutativity and associativity
+-comm : ∀ (m n : Int) → m + n ≡ n + m
+-assoc : ∀ (m n o : Int) → m + (n + o) ≡ (m + n) + o
An alternate definition of _+_ is defined via ua,
namely _+'_, which helps us to easily prove
isEquivAddInt : (m : Int) → isEquiv (λ n → n + m)
-}
module Cubical.Data.Int.Properties where
open import Cubical.Core.Everything
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Transport
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Univalence
open import Cubical.Data.Empty
open import Cubical.Data.Nat hiding (+-assoc ; +-comm) renaming (_·_ to _·ℕ_; _+_ to _+ℕ_)
open import Cubical.Data.Bool
open import Cubical.Data.Sum
open import Cubical.Data.Int.Base
open import Cubical.Relation.Nullary
open import Cubical.Relation.Nullary.DecidableEq
abs : Int → ℕ
abs (pos n) = n
abs (negsuc n) = suc n
sgn : Int → Bool
sgn (pos n) = true
sgn (negsuc n) = false
sucPred : ∀ i → sucInt (predInt i) ≡ i
sucPred (pos zero) = refl
sucPred (pos (suc n)) = refl
sucPred (negsuc n) = refl
predSuc : ∀ i → predInt (sucInt i) ≡ i
predSuc (pos n) = refl
predSuc (negsuc zero) = refl
predSuc (negsuc (suc n)) = refl
-- TODO: define multiplication
injPos : ∀ {a b : ℕ} → pos a ≡ pos b → a ≡ b
injPos {a} h = subst T h refl
where
T : Int → Type₀
T (pos x) = a ≡ x
T (negsuc _) = ⊥
injNegsuc : ∀ {a b : ℕ} → negsuc a ≡ negsuc b → a ≡ b
injNegsuc {a} h = subst T h refl
where
T : Int → Type₀
T (pos _) = ⊥
T (negsuc x) = a ≡ x
posNotnegsuc : ∀ (a b : ℕ) → ¬ (pos a ≡ negsuc b)
posNotnegsuc a b h = subst T h 0
where
T : Int → Type₀
T (pos _) = ℕ
T (negsuc _) = ⊥
negsucNotpos : ∀ (a b : ℕ) → ¬ (negsuc a ≡ pos b)
negsucNotpos a b h = subst T h 0
where
T : Int → Type₀
T (pos _) = ⊥
T (negsuc _) = ℕ
discreteInt : Discrete Int
discreteInt (pos n) (pos m) with discreteℕ n m
... | yes p = yes (cong pos p)
... | no p = no (λ x → p (injPos x))
discreteInt (pos n) (negsuc m) = no (posNotnegsuc n m)
discreteInt (negsuc n) (pos m) = no (negsucNotpos n m)
discreteInt (negsuc n) (negsuc m) with discreteℕ n m
... | yes p = yes (cong negsuc p)
... | no p = no (λ x → p (injNegsuc x))
isSetInt : isSet Int
isSetInt = Discrete→isSet discreteInt
isEven : Int → Bool
isEven (pos zero) = true
isEven (pos (suc zero)) = false
isEven (pos (suc (suc n))) = isEven (pos n)
isEven (negsuc zero) = false
isEven (negsuc (suc n)) = isEven (pos n)
_ℕ-_ : ℕ → ℕ → Int
a ℕ- 0 = pos a
0 ℕ- suc b = negsuc b
suc a ℕ- suc b = a ℕ- b
_+pos_ : Int → ℕ → Int
z +pos 0 = z
z +pos (suc n) = sucInt (z +pos n)
_+negsuc_ : Int → ℕ → Int
z +negsuc 0 = predInt z
z +negsuc (suc n) = predInt (z +negsuc n)
_+_ : Int → Int → Int
m + pos n = m +pos n
m + negsuc n = m +negsuc n
-_ : Int → Int
- pos zero = pos zero
- pos (suc n) = negsuc n
- negsuc n = pos (suc n)
_-_ : Int → Int → Int
m - n = m + (- n)
-pos : ∀ n → - (pos n) ≡ neg n
-pos zero = refl
-pos (suc n) = refl
-neg : ∀ n → - (neg n) ≡ pos n
-neg zero = refl
-neg (suc n) = refl
-Involutive : ∀ z → - (- z) ≡ z
-Involutive (pos n) = (- (- pos n)) ≡⟨ cong -_ (-pos n) ⟩
- (neg n) ≡⟨ -neg n ⟩
pos n ∎
-Involutive (negsuc n) = refl
sucInt+pos : ∀ n m → sucInt (m +pos n) ≡ (sucInt m) +pos n
sucInt+pos zero m = refl
sucInt+pos (suc n) m = cong sucInt (sucInt+pos n m)
predInt+negsuc : ∀ n m → predInt (m +negsuc n) ≡ (predInt m) +negsuc n
predInt+negsuc zero m = refl
predInt+negsuc (suc n) m = cong predInt (predInt+negsuc n m)
sucInt+negsuc : ∀ n m → sucInt (m +negsuc n) ≡ (sucInt m) +negsuc n
sucInt+negsuc zero m = (sucPred _) ∙ (sym (predSuc _))
sucInt+negsuc (suc n) m = _ ≡⟨ sucPred _ ⟩
m +negsuc n ≡[ i ]⟨ predSuc m (~ i) +negsuc n ⟩
(predInt (sucInt m)) +negsuc n ≡⟨ sym (predInt+negsuc n (sucInt m)) ⟩
predInt (sucInt m +negsuc n) ∎
predInt+pos : ∀ n m → predInt (m +pos n) ≡ (predInt m) +pos n
predInt+pos zero m = refl
predInt+pos (suc n) m = _ ≡⟨ predSuc _ ⟩
m +pos n ≡[ i ]⟨ sucPred m (~ i) + pos n ⟩
(sucInt (predInt m)) +pos n ≡⟨ sym (sucInt+pos n (predInt m))⟩
(predInt m) +pos (suc n) ∎
predInt-pos : ∀ n → predInt(- (pos n)) ≡ negsuc n
predInt-pos zero = refl
predInt-pos (suc n) = refl
predInt+ : ∀ m n → predInt (m + n) ≡ (predInt m) + n
predInt+ m (pos n) = predInt+pos n m
predInt+ m (negsuc n) = predInt+negsuc n m
+predInt : ∀ m n → predInt (m + n) ≡ m + (predInt n)
+predInt m (pos zero) = refl
+predInt m (pos (suc n)) = (predSuc (m + pos n)) ∙ (cong (_+_ m) (sym (predSuc (pos n))))
+predInt m (negsuc n) = refl
sucInt+ : ∀ m n → sucInt (m + n) ≡ (sucInt m) + n
sucInt+ m (pos n) = sucInt+pos n m
sucInt+ m (negsuc n) = sucInt+negsuc n m
+sucInt : ∀ m n → sucInt (m + n) ≡ m + (sucInt n)
+sucInt m (pos n) = refl
+sucInt m (negsuc zero) = sucPred _
+sucInt m (negsuc (suc n)) = (sucPred (m +negsuc n)) ∙ (cong (_+_ m) (sym (sucPred (negsuc n))))
pos0+ : ∀ z → z ≡ pos 0 + z
pos0+ (pos zero) = refl
pos0+ (pos (suc n)) = cong sucInt (pos0+ (pos n))
pos0+ (negsuc zero) = refl
pos0+ (negsuc (suc n)) = cong predInt (pos0+ (negsuc n))
negsuc0+ : ∀ z → predInt z ≡ negsuc 0 + z
negsuc0+ (pos zero) = refl
negsuc0+ (pos (suc n)) = (sym (sucPred (pos n))) ∙ (cong sucInt (negsuc0+ _))
negsuc0+ (negsuc zero) = refl
negsuc0+ (negsuc (suc n)) = cong predInt (negsuc0+ (negsuc n))
ind-comm : {A : Type₀} (_∙_ : A → A → A) (f : ℕ → A) (g : A → A)
(p : ∀ {n} → f (suc n) ≡ g (f n))
(g∙ : ∀ a b → g (a ∙ b) ≡ g a ∙ b)
(∙g : ∀ a b → g (a ∙ b) ≡ a ∙ g b)
(base : ∀ z → z ∙ f 0 ≡ f 0 ∙ z)
→ ∀ z n → z ∙ f n ≡ f n ∙ z
ind-comm _∙_ f g p g∙ ∙g base z 0 = base z
ind-comm _∙_ f g p g∙ ∙g base z (suc n) =
z ∙ f (suc n) ≡[ i ]⟨ z ∙ p {n} i ⟩
z ∙ g (f n) ≡⟨ sym ( ∙g z (f n)) ⟩
g (z ∙ f n) ≡⟨ cong g IH ⟩
g (f n ∙ z) ≡⟨ g∙ (f n) z ⟩
g (f n) ∙ z ≡[ i ]⟨ p {n} (~ i) ∙ z ⟩
f (suc n) ∙ z ∎
where
IH = ind-comm _∙_ f g p g∙ ∙g base z n
ind-assoc : {A : Type₀} (_·_ : A → A → A) (f : ℕ → A)
(g : A → A) (p : ∀ a b → g (a · b) ≡ a · (g b))
(q : ∀ {c} → f (suc c) ≡ g (f c))
(base : ∀ m n → (m · n) · (f 0) ≡ m · (n · (f 0)))
(m n : A) (o : ℕ)
→ m · (n · (f o)) ≡ (m · n) · (f o)
ind-assoc _·_ f g p q base m n 0 = sym (base m n)
ind-assoc _·_ f g p q base m n (suc o) =
m · (n · (f (suc o))) ≡[ i ]⟨ m · (n · q {o} i) ⟩
m · (n · (g (f o))) ≡[ i ]⟨ m · (p n (f o) (~ i)) ⟩
m · (g (n · (f o))) ≡⟨ sym (p m (n · (f o)))⟩
g (m · (n · (f o))) ≡⟨ cong g IH ⟩
g ((m · n) · (f o)) ≡⟨ p (m · n) (f o) ⟩
(m · n) · (g (f o)) ≡[ i ]⟨ (m · n) · q {o} (~ i) ⟩
(m · n) · (f (suc o)) ∎
where
IH = ind-assoc _·_ f g p q base m n o
+-comm : ∀ m n → m + n ≡ n + m
+-comm m (pos n) = ind-comm _+_ pos sucInt refl sucInt+ +sucInt pos0+ m n
+-comm m (negsuc n) = ind-comm _+_ negsuc predInt refl predInt+ +predInt negsuc0+ m n
+-assoc : ∀ m n o → m + (n + o) ≡ (m + n) + o
+-assoc m n (pos o) = ind-assoc _+_ pos sucInt +sucInt refl (λ _ _ → refl) m n o
+-assoc m n (negsuc o) = ind-assoc _+_ negsuc predInt +predInt refl +predInt m n o
-- Compose sucPathInt with itself n times. Transporting along this
-- will be addition, transporting with it backwards will be subtraction.
-- Use this to define _+'_ for which is easier to prove
-- isEquiv (λ n → n +' m) since _+'_ is defined by transport
sucPathInt : Int ≡ Int
sucPathInt = ua (sucInt , isoToIsEquiv (iso sucInt predInt sucPred predSuc))
addEq : ℕ → Int ≡ Int
addEq zero = refl
addEq (suc n) = (addEq n) ∙ sucPathInt
predPathInt : Int ≡ Int
predPathInt = ua (predInt , isoToIsEquiv (iso predInt sucInt predSuc sucPred))
subEq : ℕ → Int ≡ Int
subEq zero = refl
subEq (suc n) = (subEq n) ∙ predPathInt
_+'_ : Int → Int → Int
m +' pos n = transport (addEq n) m
m +' negsuc n = transport (subEq (suc n)) m
+'≡+ : _+'_ ≡ _+_
+'≡+ i m (pos zero) = m
+'≡+ i m (pos (suc n)) = sucInt (+'≡+ i m (pos n))
+'≡+ i m (negsuc zero) = predInt m
+'≡+ i m (negsuc (suc n)) = predInt (+'≡+ i m (negsuc n)) --
-- compPath (λ i → (+'≡+ i (predInt m) (negsuc n))) (sym (predInt+negsuc n m)) i
isEquivAddInt' : (m : Int) → isEquiv (λ n → n +' m)
isEquivAddInt' (pos n) = isEquivTransport (addEq n)
isEquivAddInt' (negsuc n) = isEquivTransport (subEq (suc n))
isEquivAddInt : (m : Int) → isEquiv (λ n → n + m)
isEquivAddInt = subst (λ add → (m : Int) → isEquiv (λ n → add n m)) +'≡+ isEquivAddInt'
-- below is an alternate proof of isEquivAddInt for comparison
-- We also have two useful lemma here.
minusPlus : ∀ m n → (n - m) + m ≡ n
minusPlus (pos zero) n = refl
minusPlus (pos 1) = sucPred
minusPlus (pos (suc (suc m))) n =
sucInt ((n +negsuc (suc m)) +pos (suc m)) ≡⟨ sucInt+pos (suc m) _ ⟩
sucInt (n +negsuc (suc m)) +pos (suc m) ≡[ i ]⟨ sucPred (n +negsuc m) i +pos (suc m) ⟩
(n - pos (suc m)) +pos (suc m) ≡⟨ minusPlus (pos (suc m)) n ⟩
n ∎
minusPlus (negsuc zero) = predSuc
minusPlus (negsuc (suc m)) n =
predInt (sucInt (sucInt (n +pos m)) +negsuc m) ≡⟨ predInt+negsuc m _ ⟩
predInt (sucInt (sucInt (n +pos m))) +negsuc m ≡[ i ]⟨ predSuc (sucInt (n +pos m)) i +negsuc m ⟩
sucInt (n +pos m) +negsuc m ≡⟨ minusPlus (negsuc m) n ⟩
n ∎
plusMinus : ∀ m n → (n + m) - m ≡ n
plusMinus (pos zero) n = refl
plusMinus (pos (suc m)) = minusPlus (negsuc m)
plusMinus (negsuc m) = minusPlus (pos (suc m))
private
alternateProof : (m : Int) → isEquiv (λ n → n + m)
alternateProof m = isoToIsEquiv (iso (λ n → n + m)
(λ n → n - m)
(minusPlus m)
(plusMinus m))
-Cancel : ∀ z → z - z ≡ pos zero
-Cancel z = z - z ≡⟨ cong (_- z) (pos0+ z) ⟩
(pos zero + z) - z ≡⟨ plusMinus z (pos zero) ⟩
pos zero ∎
pos+ : ∀ m n → pos (m +ℕ n) ≡ pos m + pos n
pos+ zero zero = refl
pos+ zero (suc n) =
pos (zero +ℕ suc n) ≡⟨ +-comm (pos (suc n)) (pos zero) ⟩
pos zero + pos (suc n) ∎
pos+ (suc m) zero =
pos (suc (m +ℕ zero)) ≡⟨ cong pos (cong suc (+-zero m)) ⟩
pos (suc m) + pos zero ∎
pos+ (suc m) (suc n) =
pos (suc m +ℕ suc n) ≡⟨ cong pos (cong suc (+-suc m n)) ⟩
sucInt (pos (suc (m +ℕ n))) ≡⟨ cong sucInt (cong sucInt (pos+ m n)) ⟩
sucInt (sucInt (pos m + pos n)) ≡⟨ sucInt+ (pos m) (sucInt (pos n)) ⟩
pos (suc m) + pos (suc n) ∎
negsuc+ : ∀ m n → negsuc (m +ℕ n) ≡ negsuc m - pos n
negsuc+ zero zero = refl
negsuc+ zero (suc n) =
negsuc (zero +ℕ suc n) ≡⟨ negsuc0+ (negsuc n) ⟩
negsuc zero + negsuc n ≡⟨ cong (negsuc zero +_) (-pos (suc n)) ⟩
negsuc zero - pos (suc n) ∎
negsuc+ (suc m) zero =
negsuc (suc m +ℕ zero) ≡⟨ cong negsuc (cong suc (+-zero m)) ⟩
negsuc (suc m) - pos zero ∎
negsuc+ (suc m) (suc n) =
negsuc (suc m +ℕ suc n) ≡⟨ cong negsuc (sym (+-suc m (suc n))) ⟩
negsuc (m +ℕ suc (suc n)) ≡⟨ negsuc+ m (suc (suc n)) ⟩
negsuc m - pos (suc (suc n)) ≡⟨ sym (+predInt (negsuc m) (negsuc n)) ⟩
predInt (negsuc m + negsuc n ) ≡⟨ predInt+ (negsuc m) (negsuc n) ⟩
negsuc (suc m) - pos (suc n) ∎
neg+ : ∀ m n → neg (m +ℕ n) ≡ neg m + neg n
neg+ zero zero = refl
neg+ zero (suc n) = neg (zero +ℕ suc n) ≡⟨ +-comm (neg (suc n)) (pos zero) ⟩
neg zero + neg (suc n) ∎
neg+ (suc m) zero = neg (suc (m +ℕ zero)) ≡⟨ cong neg (cong suc (+-zero m)) ⟩
neg (suc m) + neg zero ∎
neg+ (suc m) (suc n) = neg (suc m +ℕ suc n) ≡⟨ negsuc+ m (suc n) ⟩
neg (suc m) + neg (suc n) ∎
ℕ-AntiComm : ∀ m n → m ℕ- n ≡ - (n ℕ- m)
ℕ-AntiComm zero zero = refl
ℕ-AntiComm zero (suc n) = refl
ℕ-AntiComm (suc m) zero = refl
ℕ-AntiComm (suc m) (suc n) = suc m ℕ- suc n ≡⟨ ℕ-AntiComm m n ⟩
- (suc n ℕ- suc m) ∎
pos- : ∀ m n → m ℕ- n ≡ pos m - pos n
pos- zero zero = refl
pos- zero (suc n) = zero ℕ- suc n ≡⟨ +-comm (negsuc n) (pos zero) ⟩
pos zero - pos (suc n) ∎
pos- (suc m) zero = refl
pos- (suc m) (suc n) =
suc m ℕ- suc n ≡⟨ pos- m n ⟩
pos m - pos n ≡⟨ sym (sucPred (pos m - pos n)) ⟩
sucInt (predInt (pos m - pos n)) ≡⟨ cong sucInt (+predInt (pos m) (- pos n)) ⟩
sucInt (pos m + predInt (- (pos n))) ≡⟨ cong sucInt (cong (pos m +_) (predInt-pos n)) ⟩
sucInt (pos m + negsuc n) ≡⟨ sucInt+negsuc n (pos m) ⟩
pos (suc m) - pos (suc n) ∎
-AntiComm : ∀ m n → m - n ≡ - (n - m)
-AntiComm (pos n) (pos m) = pos n - pos m ≡⟨ sym (pos- n m) ⟩
n ℕ- m ≡⟨ ℕ-AntiComm n m ⟩
- (m ℕ- n) ≡⟨ cong -_ (pos- m n) ⟩
- (pos m - pos n) ∎
-AntiComm (pos n) (negsuc m) =
pos n - negsuc m ≡⟨ +-comm (pos n) (pos (suc m)) ⟩
pos (suc m) + pos n ≡⟨ sym (pos+ (suc m) n) ⟩
pos (suc m +ℕ n) ≡⟨ sym (-neg (suc m +ℕ n)) ⟩
- neg (suc m +ℕ n) ≡⟨ cong -_ (neg+ (suc m) n) ⟩
- (neg (suc m) + neg n) ≡⟨ cong -_ (cong (negsuc m +_) (sym (-pos n))) ⟩
- (negsuc m - pos n) ∎
-AntiComm (negsuc n) (pos m) =
negsuc n - pos m ≡⟨ sym (negsuc+ n m) ⟩
negsuc (n +ℕ m) ≡⟨ cong -_ (pos+ (suc n) m) ⟩
- (pos (suc n) + pos m) ≡⟨ cong -_ (+-comm (pos (suc n)) (pos m)) ⟩
- (pos m - negsuc n) ∎
-AntiComm (negsuc n) (negsuc m) =
negsuc n - negsuc m ≡⟨ +-comm (negsuc n) (pos (suc m)) ⟩
pos (suc m) + negsuc n ≡⟨ sym (pos- (suc m) (suc n)) ⟩
suc m ℕ- suc n ≡⟨ ℕ-AntiComm (suc m) (suc n) ⟩
- (suc n ℕ- suc m) ≡⟨ cong -_ (pos- (suc n) (suc m)) ⟩
- (pos (suc n) - pos (suc m)) ≡⟨ cong -_ (+-comm (pos (suc n)) (negsuc m)) ⟩
- (negsuc m - negsuc n) ∎
-Dist+ : ∀ m n → - (m + n) ≡ (- m) + (- n)
-Dist+ (pos n) (pos m) =
- (pos n + pos m) ≡⟨ cong -_ (sym (pos+ n m)) ⟩
- pos (n +ℕ m) ≡⟨ -pos (n +ℕ m) ⟩
neg (n +ℕ m) ≡⟨ neg+ n m ⟩
neg n + neg m ≡⟨ cong (neg n +_) (sym (-pos m)) ⟩
neg n + (- pos m) ≡⟨ cong (_+ (- pos m)) (sym (-pos n)) ⟩
(- pos n) + (- pos m) ∎
-Dist+ (pos n) (negsuc m) =
- (pos n + negsuc m) ≡⟨ sym (-AntiComm (pos (suc m)) (pos n)) ⟩
pos (suc m) - pos n ≡⟨ +-comm (pos (suc m)) (- pos n) ⟩
(- pos n) + (- negsuc m) ∎
-Dist+ (negsuc n) (pos m) =
- (negsuc n + pos m) ≡⟨ cong -_ (+-comm (negsuc n) (pos m)) ⟩
- (pos m + negsuc n) ≡⟨ sym (-AntiComm (- negsuc n) (pos m)) ⟩
(- negsuc n) + (- pos m) ∎
-Dist+ (negsuc n) (negsuc m) =
- (negsuc n + negsuc m) ≡⟨ cong -_ (sym (neg+ (suc n) (suc m))) ⟩
- neg (suc n +ℕ suc m) ≡⟨ pos+ (suc n) (suc m) ⟩
(- negsuc n) + (- negsuc m) ∎
_·_ : Int → Int → Int
pos zero · m = pos zero
pos (suc n) · m = m + (pos n · m)
negsuc zero · m = (- m)
negsuc (suc n) · m = (- m) + ((negsuc n) · m)
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module *-distrib-+ where
import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; cong; sym)
open Eq.≡-Reasoning using (begin_; _≡⟨⟩_; step-≡; _∎)
open import Data.Nat using (ℕ; zero; suc; _+_; _*_)
open import Induction′ using (+-assoc; +-comm; +-suc)
-- 積が和に対して分配的であることの証明
*-distrib-+ : ∀ (m n p : ℕ) → (m + n) * p ≡ m * p + n * p
*-distrib-+ zero n p =
begin
(zero + n) * p
≡⟨⟩
n * p
≡⟨⟩
zero * p + n * p
∎
*-distrib-+ (suc m) n p =
begin
((suc m) + n) * p
≡⟨ cong (_* p) (+-comm (suc m) n) ⟩
(n + (suc m)) * p
≡⟨ cong (_* p) (+-suc n m) ⟩
(suc (n + m)) * p
≡⟨⟩
p + ((n + m) * p)
≡⟨ cong (p +_) (*-distrib-+ n m p) ⟩
p + (n * p + m * p)
≡⟨ cong (p +_) (+-comm (n * p) (m * p)) ⟩
p + (m * p + n * p)
≡⟨ sym (+-assoc p (m * p) (n * p)) ⟩
(p + (m * p)) + n * p
≡⟨⟩
(suc m) * p + n * p
∎
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{-# OPTIONS --cubical --safe #-}
module Data.List.Kleene.Relation.Unary where
open import Data.List.Kleene
open import Prelude
open import Data.Fin
open import Relation.Nullary
private
variable
p : Level
◇⁺ : ∀ {A : Type a} (P : A → Type p) → A ⁺ → Type _
◇⁺ P xs = ∃[ i ] P (xs !⁺ i)
◇⋆ : ∀ {A : Type a} (P : A → Type p) → A ⋆ → Type _
◇⋆ P xs = ∃[ i ] P (xs !⋆ i)
module Exists {a} {A : Type a} {p} (P : A → Type p) where
push : ∀ {x xs} → ◇⋆ P xs → ◇⁺ P (x & xs)
push (n , x∈xs) = (fs n , x∈xs)
pull : ∀ {x xs} → ¬ P x → ◇⁺ P (x & xs) → ◇⋆ P xs
pull ¬Px (f0 , p∈xs) = ⊥-elim (¬Px p∈xs)
pull ¬Px (fs n , p∈xs) = (n , p∈xs)
◻⁺ : {A : Type a} (P : A → Type p) → A ⁺ → Type _
◻⁺ P xs = ∀ i → P (xs !⁺ i)
◻⋆ : {A : Type a} (P : A → Type p) → A ⋆ → Type _
◻⋆ P xs = ∀ i → P (xs !⋆ i)
module Forall {a} {A : Type a} {p} (P : A → Type p) where
push⁺ : ∀ {x xs} → P x → ◻⋆ P xs → ◻⁺ P (x & xs)
push⁺ px pxs f0 = px
push⁺ px pxs (fs n) = pxs n
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------------------------------------------------------------------------------
-- Totality properties respect to OrdList (flatten-OrdList-helper)
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
module FOTC.Program.SortList.Properties.Totality.OrdList.FlattenI where
open import FOTC.Base
open import FOTC.Data.Nat.Type
open import FOTC.Program.SortList.SortList
------------------------------------------------------------------------------
-- See the combined proof.
postulate
flatten-OrdList-helper : ∀ {t₁ i t₂} → Tree t₁ → N i → Tree t₂ →
OrdTree (node t₁ i t₂) →
≤-Lists (flatten t₁) (flatten t₂)
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{-# OPTIONS --universe-polymorphism #-}
module Categories.Functor.Product where
open import Categories.Category
open import Categories.Functor using (Functor)
import Categories.Object.Product as Product
import Categories.Object.BinaryProducts as BinaryProducts
-- Ugh, we should start bundling things (categories with binary products, in this case) up consistently
_[_][_×-] : ∀ {o ℓ e} → (C : Category o ℓ e) → BinaryProducts.BinaryProducts C → Category.Obj C → Functor C C
C [ P ][ O ×-] = record
{ F₀ = λ x → Product.A×B (product {O} {x})
; F₁ = λ f → ⟨ π₁ , f ∘ π₂ ⟩
; identity = λ {x} → identity′ {x}
; homomorphism = λ {x} {y} {z} {f} {g} → homomorphism′ {x} {y} {z} {f} {g}
; F-resp-≡ = λ f≡g → ⟨⟩-cong₂ refl (∘-resp-≡ˡ f≡g)
}
where
open Category C
open Equiv
open Product C
open BinaryProducts.BinaryProducts P
.identity′ : {A : Obj} → ⟨ π₁ , id ∘ π₂ ⟩ ≡ id
identity′ =
begin
⟨ π₁ , id ∘ π₂ ⟩
≈⟨ ⟨⟩-cong₂ refl identityˡ ⟩
⟨ π₁ , π₂ ⟩
≈⟨ η ⟩
id
∎
where open HomReasoning
.homomorphism′ : {X Y Z : Obj} {f : X ⇒ Y} {g : Y ⇒ Z} → ⟨ π₁ , (g ∘ f) ∘ π₂ ⟩ ≡ ⟨ π₁ , g ∘ π₂ ⟩ ∘ ⟨ π₁ , f ∘ π₂ ⟩
homomorphism′ {f = f} {g} =
begin
⟨ π₁ , (g ∘ f) ∘ π₂ ⟩
↓⟨ ⟨⟩-cong₂ refl assoc ⟩
⟨ π₁ , g ∘ (f ∘ π₂) ⟩
↑⟨ ⟨⟩-cong₂ refl (∘-resp-≡ʳ commute₂) ⟩
⟨ π₁ , g ∘ (π₂ ∘ ⟨ π₁ , f ∘ π₂ ⟩) ⟩
↑⟨ ⟨⟩-cong₂ commute₁ assoc ⟩
⟨ π₁ ∘ ⟨ π₁ , f ∘ π₂ ⟩ , (g ∘ π₂) ∘ ⟨ π₁ , f ∘ π₂ ⟩ ⟩
↑⟨ ⟨⟩∘ ⟩
⟨ π₁ , g ∘ π₂ ⟩ ∘ ⟨ π₁ , f ∘ π₂ ⟩
∎
where open HomReasoning
_[_][-×_] : ∀ {o ℓ e} → (C : Category o ℓ e) → BinaryProducts.BinaryProducts C → Category.Obj C → Functor C C
C [ P ][-× O ] = record
{ F₀ = λ x → Product.A×B (product {x} {O})
; F₁ = λ f → ⟨ f ∘ π₁ , π₂ ⟩
; identity = λ {x} → identity′ {x}
; homomorphism = λ {x} {y} {z} {f} {g} → homomorphism′ {x} {y} {z} {f} {g}
; F-resp-≡ = λ f≡g → ⟨⟩-cong₂ (∘-resp-≡ˡ f≡g) refl
}
where
open Category C
open Equiv
open Product C
open BinaryProducts.BinaryProducts P
.identity′ : {A : Obj} → ⟨ id ∘ π₁ , π₂ ⟩ ≡ id
identity′ =
begin
⟨ id ∘ π₁ , π₂ ⟩
≈⟨ ⟨⟩-cong₂ identityˡ refl ⟩
⟨ π₁ , π₂ ⟩
≈⟨ η ⟩
id
∎
where open HomReasoning
.homomorphism′ : {X Y Z : Obj} {f : X ⇒ Y} {g : Y ⇒ Z} → ⟨ (g ∘ f) ∘ π₁ , π₂ ⟩ ≡ ⟨ g ∘ π₁ , π₂ ⟩ ∘ ⟨ f ∘ π₁ , π₂ ⟩
homomorphism′ {f = f} {g} =
begin
⟨ (g ∘ f) ∘ π₁ , π₂ ⟩
↓⟨ ⟨⟩-cong₂ assoc refl ⟩
⟨ g ∘ (f ∘ π₁) , π₂ ⟩
↑⟨ ⟨⟩-cong₂ (∘-resp-≡ʳ commute₁) refl ⟩
⟨ g ∘ (π₁ ∘ ⟨ f ∘ π₁ , π₂ ⟩) , π₂ ⟩
↑⟨ ⟨⟩-cong₂ assoc commute₂ ⟩
⟨ (g ∘ π₁) ∘ ⟨ f ∘ π₁ , π₂ ⟩ , π₂ ∘ ⟨ f ∘ π₁ , π₂ ⟩ ⟩
↑⟨ ⟨⟩∘ ⟩
⟨ g ∘ π₁ , π₂ ⟩ ∘ ⟨ f ∘ π₁ , π₂ ⟩
∎
where open HomReasoning
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module Sodium where
open import Category.Functor
open import Category.Applicative
open import Category.Monad
open import Data.Unit
open import Data.Bool
open import Data.Maybe
open import Data.Product
open import IO.Primitive
postulate
-- Core.
Reactive : Set → Set
sync : ∀ {A} → Reactive A → IO A
Event : Set → Set
Behaviour : Set → Set
listen : ∀ {A} → Event A → (A → IO ⊤) → Reactive (IO ⊤)
filterJust : ∀ {A} → Event (Maybe A) → Event A
hold : ∀ {A} → A → Event A → Reactive (Behaviour A)
updates value : ∀ {A} → Behaviour A → Event A
snapshot : ∀ {A B C} → (A → B → C) → Event A → Behaviour B → Event C
-- Derived.
gate : ∀ {A} → Event A → Behaviour Bool → Event A
filterE : ∀ {A} → (A → Bool) → Event A → Event A
collectE : ∀ {A B} {S : Set} → (A → S → (B × S)) → S →
Event A → Reactive (Event B)
collect : ∀ {A B} {S : Set} → (A → S → (B × S)) → S →
Behaviour A → Reactive (Behaviour B)
accum : ∀ {A} → A → Event (A → A) → Reactive (Behaviour A)
-- IO.
executeSyncIO executeAsyncIO : ∀ {A} → Event (IO A) → Event A
-- Instances.
reactiveMonad : RawMonad Reactive
behaviourApplicative : RawApplicative Behaviour
eventFunctor : RawFunctor Event
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postulate
F : (Set → Set) → Set
syntax F (λ x → y) = [ x ] y
X : Set
X = [ ? ] x | {
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{-# OPTIONS --safe #-}
module SafeFlagPostulate where
data Empty : Set where
postulate inhabitant : Empty
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{-# OPTIONS --cubical --safe --guardedness #-}
module Cubical.Codata.Stream.Properties where
open import Cubical.Core.Everything
open import Cubical.Data.Nat
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Univalence
open import Cubical.Codata.Stream.Base
open Stream
mapS : ∀ {A B} → (A → B) → Stream A → Stream B
head (mapS f xs) = f (head xs)
tail (mapS f xs) = mapS f (tail xs)
even : ∀ {A} → Stream A → Stream A
head (even a) = head a
tail (even a) = even (tail (tail a))
odd : ∀ {A} → Stream A → Stream A
head (odd a) = head (tail a)
tail (odd a) = odd (tail (tail a))
merge : ∀ {A} → Stream A → Stream A → Stream A
head (merge a _) = head a
head (tail (merge _ b)) = head b
tail (tail (merge a b)) = merge (tail a) (tail b)
mapS-id : ∀ {A} {xs : Stream A} → mapS (λ x → x) xs ≡ xs
head (mapS-id {xs = xs} i) = head xs
tail (mapS-id {xs = xs} i) = mapS-id {xs = tail xs} i
Stream-η : ∀ {A} {xs : Stream A} → xs ≡ (head xs , tail xs)
head (Stream-η {A} {xs} i) = head xs
tail (Stream-η {A} {xs} i) = tail xs
elimS : ∀ {A} (P : Stream A → Type₀) (c : ∀ x xs → P (x , xs)) (xs : Stream A) → P xs
elimS P c xs = transp (λ i → P (Stream-η {xs = xs} (~ i))) i0
(c (head xs) (tail xs))
odd≡even∘tail : ∀ {A} → (a : Stream A) → odd a ≡ even (tail a)
head (odd≡even∘tail a i) = head (tail a)
tail (odd≡even∘tail a i) = odd≡even∘tail (tail (tail a)) i
mergeEvenOdd≡id : ∀ {A} → (a : Stream A) → merge (even a) (odd a) ≡ a
head (mergeEvenOdd≡id a i) = head a
head (tail (mergeEvenOdd≡id a i)) = head (tail a)
tail (tail (mergeEvenOdd≡id a i)) = mergeEvenOdd≡id (tail (tail a)) i
module Equality≅Bisimulation where
-- Bisimulation
record _≈_ {A : Type₀} (x y : Stream A) : Type₀ where
coinductive
field
≈head : head x ≡ head y
≈tail : tail x ≈ tail y
open _≈_
bisim : {A : Type₀} → {x y : Stream A} → x ≈ y → x ≡ y
head (bisim x≈y i) = ≈head x≈y i
tail (bisim x≈y i) = bisim (≈tail x≈y) i
misib : {A : Type₀} → {x y : Stream A} → x ≡ y → x ≈ y
≈head (misib p) = λ i → head (p i)
≈tail (misib p) = misib (λ i → tail (p i))
iso1 : {A : Type₀} → {x y : Stream A} → (p : x ≡ y) → bisim (misib p) ≡ p
head (iso1 p i j) = head (p j)
tail (iso1 p i j) = iso1 (λ i → tail (p i)) i j
iso2 : {A : Type₀} → {x y : Stream A} → (p : x ≈ y) → misib (bisim p) ≡ p
≈head (iso2 p i) = ≈head p
≈tail (iso2 p i) = iso2 (≈tail p) i
path≃bisim : {A : Type₀} → {x y : Stream A} → (x ≡ y) ≃ (x ≈ y)
path≃bisim = isoToEquiv (iso misib bisim iso2 iso1)
path≡bisim : {A : Type₀} → {x y : Stream A} → (x ≡ y) ≡ (x ≈ y)
path≡bisim = ua path≃bisim
-- misib can be implemented by transport as well.
refl≈ : {A : Type₀} {x : Stream A} → x ≈ x
≈head refl≈ = refl
≈tail refl≈ = refl≈
cast : ∀ {A : Type₀} {x y : Stream A} (p : x ≡ y) → x ≈ y
cast {x = x} p = transport (λ i → x ≈ p i) refl≈
misib-refl : ∀ {A : Type₀} {x : Stream A} → misib {x = x} refl ≡ refl≈
≈head (misib-refl i) = refl
≈tail (misib-refl i) = misib-refl i
misibTransp : ∀ {A : Type₀} {x y : Stream A} (p : x ≡ y) → cast p ≡ misib p
misibTransp p = J (λ _ p → cast p ≡ misib p) ((transportRefl refl≈) ∙ (sym misib-refl)) p
module Stream≅Nat→ {A : Type₀} where
lookup : {A : Type₀} → Stream A → ℕ → A
lookup xs zero = head xs
lookup xs (suc n) = lookup (tail xs) n
tabulate : {A : Type₀} → (ℕ → A) → Stream A
head (tabulate f) = f zero
tail (tabulate f) = tabulate (λ n → f (suc n))
lookup∘tabulate : (λ (x : _ → A) → lookup (tabulate x)) ≡ (λ x → x)
lookup∘tabulate i f zero = f zero
lookup∘tabulate i f (suc n) = lookup∘tabulate i (λ n → f (suc n)) n
tabulate∘lookup : (λ (x : Stream A) → tabulate (lookup x)) ≡ (λ x → x)
head (tabulate∘lookup i xs) = head xs
tail (tabulate∘lookup i xs) = tabulate∘lookup i (tail xs)
Stream≡Nat→ : Stream A ≡ (ℕ → A)
Stream≡Nat→ = isoToPath (iso lookup tabulate (λ f i → lookup∘tabulate i f) (λ xs i → tabulate∘lookup i xs))
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#-}
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module Monoids where
open import Library
record Monoid {a} : Set (lsuc a) where
field S : Set a
ε : S
_•_ : S → S → S
lid : ∀{m} → ε • m ≅ m
rid : ∀{m} → m • ε ≅ m
ass : ∀{m n o} → (m • n) • o ≅ m • (n • o)
infix 10 _•_
Nat+Mon : Monoid
Nat+Mon = record {
S = ℕ;
ε = zero;
_•_ = _+_;
lid = refl;
rid = ≡-to-≅ $ +-right-identity _;
ass = λ{m} → ≡-to-≅ $ +-assoc m _ _}
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{-# OPTIONS --without-K --safe #-}
module Categories.Category.Species where
-- The Category of Species, as the Functor category from Core (FinSetoids) to Setoids.
-- Setoids used here because that's what fits best in this setting.
-- The constructions of the theory of Species are in Species.Construction
open import Level
open import Categories.Category.Core using (Category)
open import Categories.Category.Construction.Functors
open import Categories.Category.Construction.Core using (Core)
open import Categories.Category.Instance.FinSetoids using (FinSetoids)
open import Categories.Category.Instance.Setoids using (Setoids)
private
variable
o ℓ o′ ℓ′ : Level
-- note how Species, as a category, raises levels.
Species : (o ℓ o′ ℓ′ : Level) → Category (suc (o ⊔ ℓ ⊔ o′ ⊔ ℓ′)) (suc (o ⊔ ℓ) ⊔ (o′ ⊔ ℓ′)) (suc (o ⊔ ℓ) ⊔ o′ ⊔ ℓ′)
Species o ℓ o′ ℓ′ = Functors (Core (FinSetoids o ℓ)) (Setoids o′ ℓ′)
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module Issue451 where
infix 10 _==_
data _==_ {A : Set} (x : A) : (y : A) -> Set where
refl : x == x
postulate
Nat : Set
data G : Nat -> Nat -> Set where
I : (place : Nat) -> G place place
s : (n m : Nat) -> G n m
mul : (l m n : Nat) -> G m n -> G l m -> G l n
mul a b .b (I .b) x = x
mul a .a b x (I .a) = x
mul a b c (s .b .c) (s .a .b) = s a c
postulate
a b c : Nat
f : G a b
bad : mul a a b (s a b) (I a) == s a b
bad = refl
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module guarded-recursion where
import guarded-recursion.prelude
import guarded-recursion.model
import guarded-recursion.model.Agda
import guarded-recursion.embedding
import guarded-recursion.compute
import guarded-recursion.clocks
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open import Oscar.Prelude
open import Oscar.Class.Successor₀
open import Oscar.Class.Successor₁
open import Oscar.Class.Injectivity
open import Oscar.Class.Thickandthin
open import Oscar.Class.Congruity
open import Oscar.Class.Fmap
open import Oscar.Data.¶
open import Oscar.Data.Fin
open import Oscar.Data.Proposequality
open import Oscar.Data.Maybe
import Oscar.Property.Monad.Maybe
import Oscar.Class.Congruity.Proposequality
module Oscar.Property.Thickandthin.FinFinProposequalityMaybeProposequality where
instance
𝓢uccessor₀¶ : 𝓢uccessor₀ ¶
𝓢uccessor₀¶ .𝓢uccessor₀.successor₀ = ↑_
[𝓢uccessor₁]Fin : [𝓢uccessor₁] Fin
[𝓢uccessor₁]Fin = ∁
𝓢uccessor₁Fin : 𝓢uccessor₁ Fin
𝓢uccessor₁Fin .𝓢uccessor₁.successor₁ = ↑_
[𝓘njectivity₁]Fin : ∀ {m} → [𝓘njectivity₁] (λ (_ : Fin m) → Fin (⇑₀ m)) Proposequality Proposequality
[𝓘njectivity₁]Fin = ∁
𝓘njectivity₁Fin : ∀ {m} → 𝓘njectivity₁ (λ (_ : Fin m) → Fin (⇑₀ m)) Proposequality Proposequality
𝓘njectivity₁Fin .𝓘njectivity₁.injectivity₁ ∅ = ∅
[𝓣hick]Fin,Fin : [𝓣hick] Fin Fin
[𝓣hick]Fin,Fin = ∁
𝓣hickFin,Fin : 𝓣hick Fin Fin
𝓣hickFin,Fin .𝓣hick.thick {∅} () ∅
𝓣hickFin,Fin .𝓣hick.thick {↑ _} _ ∅ = ∅
𝓣hickFin,Fin .𝓣hick.thick ∅ (↑ y) = y
𝓣hickFin,Fin .𝓣hick.thick (↑ x) (↑ y) = ↑ thick x y
[𝓣hin]Fin,Fin : [𝓣hin] Fin Fin
[𝓣hin]Fin,Fin = ∁
𝓣hinFin,Fin : 𝓣hin Fin Fin
𝓣hinFin,Fin .𝓣hin.thin ∅ = ↑_
𝓣hinFin,Fin .𝓣hin.thin (↑ x) ∅ = ∅
𝓣hinFin,Fin .𝓣hin.thin (↑ x) (↑ y) = ↑ (thin x y)
[𝓘njectivity₂,₁]ThinFinFin : ∀ {m} → [𝓘njectivity₂,₁] (𝔱hin Fin Fin m) Proposequality Proposequality
[𝓘njectivity₂,₁]ThinFinFin = ∁
𝓘njectivity₂,₁ThinFinFin : ∀ {m} → 𝓘njectivity₂,₁ (𝔱hin Fin Fin m) Proposequality Proposequality
𝓘njectivity₂,₁ThinFinFin .𝓘njectivity₂,₁.injectivity₂,₁ ∅ ∅ = ∅
𝓘njectivity₂,₁ThinFinFin .𝓘njectivity₂,₁.injectivity₂,₁ (↑ _) {∅} {∅} _ = ∅
𝓘njectivity₂,₁ThinFinFin .𝓘njectivity₂,₁.injectivity₂,₁ (↑ _) {∅} {↑ _} ()
𝓘njectivity₂,₁ThinFinFin .𝓘njectivity₂,₁.injectivity₂,₁ (↑ _) {↑ _} {∅} ()
𝓘njectivity₂,₁ThinFinFin .𝓘njectivity₂,₁.injectivity₂,₁ (↑ x) {↑ _} {↑ _} = congruity ↑_ ∘ injectivity₂,₁ x ∘ injectivity₁[ Proposequality ]
[𝓒heck]FinFinMaybe : [𝓒heck] Fin Fin Maybe
[𝓒heck]FinFinMaybe = ∁
𝓒heckFinFinMaybe : 𝓒heck Fin Fin Maybe
𝓒heckFinFinMaybe .𝓒heck.check ∅ ∅ = ∅
𝓒heckFinFinMaybe .𝓒heck.check ∅ (↑ y) = ↑ y
𝓒heckFinFinMaybe .𝓒heck.check {∅} (↑ ()) _
𝓒heckFinFinMaybe .𝓒heck.check {↑ _} (↑ x) ∅ = ↑ ∅
𝓒heckFinFinMaybe .𝓒heck.check {↑ _} (↑ x) (↑ y) = fmap′ ¶⟨<_⟩.↑_ $ check x y
[𝓣hick/thin=1]FinFin : [𝓣hick/thin=1] Fin Fin Proposequality
[𝓣hick/thin=1]FinFin = ∁
𝓣hick/thin=1FinFin : 𝓣hick/thin=1 Fin Fin Proposequality
𝓣hick/thin=1FinFin .𝓣hick/thin=1.thick/thin=1 x ∅ = ∅
𝓣hick/thin=1FinFin .𝓣hick/thin=1.thick/thin=1 ∅ (↑ y) = ∅
𝓣hick/thin=1FinFin .𝓣hick/thin=1.thick/thin=1 (↑ x) (↑ y) = congruity ↑_ (thick/thin=1 x y)
[𝓒heck/thin=1FinFin] : [𝓒heck/thin=1] Fin Fin Maybe Proposequality
[𝓒heck/thin=1FinFin] = ∁
𝓒heck/thin=1FinFin : 𝓒heck/thin=1 Fin Fin Maybe Proposequality
𝓒heck/thin=1FinFin .𝓒heck/thin=1.check/thin=1 ∅ y = ∅
𝓒heck/thin=1FinFin .𝓒heck/thin=1.check/thin=1 (↑ x) ∅ = ∅
𝓒heck/thin=1FinFin .𝓒heck/thin=1.check/thin=1 (↑ x) (↑ y) rewrite check/thin=1 {_≈_ = Proposequality⟦ Maybe _ ⟧} x y = ∅
IsThickandthinFinFin : IsThickandthin Fin Fin Proposequality Maybe Proposequality
IsThickandthinFinFin = ∁
ThickandthinFinFin : Thickandthin _ _ _ _ _ _
ThickandthinFinFin = ∁ Fin Fin Proposequality Maybe Proposequality
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module UniDB.Subst.Subs where
open import UniDB.Subst.Core
open import UniDB.Morph.Subs
--------------------------------------------------------------------------------
module _ {T : STX} {{vrT : Vr T}} {{wkT : Wk T}} {{apTT : Ap T T}} where
instance
iCompSubs : Comp (Subs T)
_⊙_ {{iCompSubs}} (refl) ξ₂ = ξ₂
_⊙_ {{iCompSubs}} (step ξ₁ t) ξ₂ = step (ξ₁ ⊙ ξ₂) (ap {T} ξ₂ t )
_⊙_ {{iCompSubs}} (skip ξ₁) refl = skip ξ₁
_⊙_ {{iCompSubs}} (skip ξ₁) (step ξ₂ t) = step (ξ₁ ⊙ ξ₂) t
_⊙_ {{iCompSubs}} (skip ξ₁) (skip ξ₂) = skip (ξ₁ ⊙ ξ₂)
--------------------------------------------------------------------------------
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{-# OPTIONS --safe --warning=error --without-K #-}
open import Numbers.ClassicalReals.RealField
open import LogicalFormulae
open import Setoids.Subset
open import Setoids.Setoids
open import Sets.EquivalenceRelations
open import Rings.Orders.Total.Definition
open import Rings.Orders.Partial.Definition
open import Rings.Definition
open import Fields.Fields
open import Groups.Definition
open import Numbers.Naturals.Semiring
open import Numbers.Naturals.Order
open import Numbers.Naturals.Order.Lemmas
open import Setoids.Orders.Partial.Definition
open import Setoids.Orders.Total.Definition
module Numbers.ClassicalReals.Examples (ℝ : RealField) where
open RealField ℝ
open Setoid S
open Equivalence eq
open Ring R
open Field F
open SetoidPartialOrder pOrder
open import Fields.Orders.LeastUpperBounds.Definition pOrder
open import Rings.Orders.Total.Lemmas orderedRing
open import Rings.Orders.Partial.Lemmas pOrderedRing
open Group additiveGroup
open PartiallyOrderedRing pOrderedRing
open SetoidTotalOrder (TotallyOrderedRing.total orderedRing)
open import Rings.InitialRing R
open import Fields.Orders.Lemmas oField
open import Rings.Lemmas R
open import Groups.Lemmas additiveGroup
open import Numbers.Intervals.Definition pOrderedRing
open import Numbers.Intervals.Arithmetic pOrderedRing
open import Fields.Lemmas F
squarePositive : (a : A) → (((a ∼ 0R) → False) && (0R < (a * a))) || (a ∼ 0R)
squarePositive a with totality 0R a
squarePositive a | inl (inl x) = inl ((λ a=0 → irreflexive (<WellDefined reflexive a=0 x)) ,, orderRespectsMultiplication x x)
squarePositive a | inl (inr x) = inl ((λ a=0 → irreflexive (<WellDefined a=0 reflexive x)) ,, <WellDefined reflexive twoNegativesTimes (orderRespectsMultiplication (<WellDefined invIdent reflexive (ringSwapNegatives' x)) (<WellDefined invIdent reflexive (ringSwapNegatives' x))))
squarePositive a | inr 0=a = inr (symmetric 0=a)
fraction<1 : (n m : ℕ) → n <N m → (0<m : fromN m ∼ 0G → False) → ((fromN n) * underlying (allInvertible (fromN m) 0<m)) < 1R
fraction<1 n m n<m 0<m with allInvertible (fromN m) 0<m
... | 1/m , prM = <WellDefined reflexive (transitive *Commutative prM) (ringCanMultiplyByPositive (inversePositiveIsPositive prM (fromNPreservesOrder (0<1 nontrivial) {0} {m} (zeroLeast n<m))) (fromNPreservesOrder (0<1 nontrivial) n<m))
1/2 : A
1/2 = underlying (allInvertible (1R + 1R) (orderedImpliesCharNot2 nontrivial))
pr1/2 : (1/2 * (1R + 1R)) ∼ 1R
pr1/2 with allInvertible (1R + 1R) (orderedImpliesCharNot2 nontrivial)
... | x , pr = pr
pr1/2' : (1/2 + 1/2) ∼ 1R
pr1/2' = transitive (symmetric (transitive *DistributesOver+ (+WellDefined (transitive *Commutative identIsIdent) (transitive *Commutative identIsIdent)))) pr1/2
1/2<1 : 1/2 < 1R
1/2<1 = <WellDefined (transitive (*WellDefined identRight (allInvertibleWellDefined (+WellDefined reflexive identRight))) identIsIdent) reflexive (fraction<1 1 2 (le 0 refl) λ p → charZero 1 (symmetric p))
3/2<2 : (fromN 3 * 1/2) < (1R + 1R)
3/2<2 = <WellDefined (transitive (+WellDefined reflexive (transitive (symmetric pr1/2) (transitive *Commutative (*WellDefined (+WellDefined reflexive (symmetric identRight)) reflexive)))) (symmetric *DistributesOver+')) reflexive (orderRespectsAddition (<WellDefined (symmetric identIsIdent) reflexive 1/2<1) 1R)
2<9/4 : (1R + 1R) < ((fromN 3 * 1/2) * (fromN 3 * 1/2))
2<9/4 = <WellDefined reflexive (symmetric *Associative) (halveInequality _ _ 1/2 pr1/2' (<WellDefined reflexive (transitive (symmetric *Associative) *Commutative) (halveInequality _ _ 1/2 pr1/2' (<WellDefined (transitive (fromNPreserves* 4 2) (transitive (*WellDefined (+WellDefined reflexive (+WellDefined reflexive (+WellDefined reflexive identRight))) (+WellDefined reflexive identRight)) (transitive *DistributesOver+ (+WellDefined (transitive *Commutative (transitive identIsIdent +Associative)) (transitive *Commutative (transitive identIsIdent +Associative)))))) (fromNPreserves* 3 3) (fromNPreservesOrder (0<1 nontrivial) (le {8} 0 refl))))))
0<2 : 0R < (1R + 1R)
0<2 = <WellDefined identLeft reflexive (ringAddInequalities (0<1 nontrivial) (0<1 nontrivial))
0<3/2 : 0R < (fromN 3 * 1/2)
0<3/2 = orderRespectsMultiplication (fromNPreservesOrder (0<1 nontrivial) (le {0} {3} 2 refl)) (inversePositiveIsPositive pr1/2 0<2)
3/2ub : {r : A} → (r * r) < (1R + 1R) → (r < (fromN 3 * 1/2))
3/2ub {r} r^2<2 with totality r (fromN 3 * 1/2)
3/2ub {r} r^2<2 | inl (inl r<3/2) = r<3/2
3/2ub {r} r^2<2 | inl (inr 3/2<r) = exFalso (irreflexive (<Transitive r^2<2 (<Transitive 2<9/4 (ringMultiplyPositives 0<3/2 0<3/2 3/2<r 3/2<r))))
3/2ub {r} r^2<2 | inr r=3/2 = exFalso (irreflexive (<Transitive r^2<2 (<WellDefined reflexive (symmetric (*WellDefined r=3/2 r=3/2)) 2<9/4)))
2/6<1 : ((1R + 1R) * underlying (allInvertible (fromN 6) (charZero' 5))) < 1R
2/6<1 with allInvertible (fromN 6) (charZero' 5)
2/6<1 | 1/6 , pr1/6 = <WellDefined reflexive (transitive *Commutative pr1/6) (ringCanMultiplyByPositive (inversePositiveIsPositive pr1/6 (fromNPreservesOrder (0<1 nontrivial) {0} {6} (le 5 refl))) (<WellDefined (+WellDefined reflexive identRight) reflexive (fromNPreservesOrder (0<1 nontrivial) {2} {6} (le 3 refl))))
2<2*2 : (1R + 1R) < ((1R + 1R) * (1R + 1R))
2<2*2 = (<WellDefined (+WellDefined reflexive identRight) (transitive (fromNPreserves* 2 2) (*WellDefined (+WellDefined reflexive identRight) (+WellDefined reflexive identRight))) (fromNPreservesOrder (0<1 nontrivial) {2} {4} (le 1 refl)))
square<2Means<2 : (u : A) → (u * u) < (1R + 1R) → u < (1R + 1R)
square<2Means<2 u u^2<2 with totality u (1R + 1R)
square<2Means<2 u u^2<2 | inl (inl x) = x
square<2Means<2 u u^2<2 | inl (inr x) = exFalso (irreflexive (<Transitive u^2<2 (<Transitive 2<2*2 (ringMultiplyPositives 0<2 0<2 x x))))
square<2Means<2 u u^2<2 | inr u=2 = exFalso (irreflexive (<Transitive (<WellDefined (*WellDefined u=2 u=2) reflexive u^2<2) 2<2*2))
2=2*2-2 : (fromN 2) ∼ (((1R + 1R) * (1R + 1R)) + inverse (fromN 2))
2=2*2-2 = transitive (+WellDefined reflexive identRight) (transitive (transitive (transitive (transitive (symmetric identRight) (+WellDefined reflexive (symmetric invRight))) +Associative) (+WellDefined (symmetric (+WellDefined (+WellDefined identIsIdent identIsIdent) (+WellDefined identIsIdent identIsIdent))) reflexive)) (+WellDefined (transitive (+WellDefined (symmetric *DistributesOver+') (symmetric *DistributesOver+')) (symmetric *DistributesOver+)) (inverseWellDefined (symmetric (+WellDefined reflexive identRight)))))
sqrtRespectsInequality : {x y : A} → (x * x) < (y * y) → 0R < y → x < y
sqrtRespectsInequality {x} {y} x^2<y^2 _ with totality x y
sqrtRespectsInequality {x} {y} x^2<y^2 _ | inl (inl x<y) = x<y
sqrtRespectsInequality {x} {y} x^2<y^2 0<y | inl (inr y<x) = exFalso (irreflexive (<Transitive x^2<y^2 (ringMultiplyPositives 0<y 0<y y<x y<x)))
sqrtRespectsInequality {x} {y} x^2<y^2 _ | inr x=y = exFalso (irreflexive (<WellDefined (*WellDefined x=y x=y) reflexive x^2<y^2))
sqrt2 : Sg A (λ i → (i * i) ∼ (1R + 1R))
sqrt2 = sqrt2' , sqrt2IsSqrt2
where
pred : A → Set c
pred a = (a * a) < (1R + 1R)
sub : subset S pred
sub {y} x=y x^2<2 = <WellDefined (*WellDefined x=y x=y) reflexive x^2<2
abstract
2ub : UpperBound sub (1R + 1R)
2ub y y^2<2 with totality y (1R + 1R)
2ub y y^2<2 | inl (inl y<2) = inl y<2
2ub y y^2<2 | inl (inr 2<y) = exFalso (irreflexive (<Transitive y^2<2 (<Transitive s r)))
where
r : ((1R + 1R) * (1R + 1R)) < (y * y)
r = ringMultiplyPositives 0<2 0<2 2<y 2<y
s : (1R + 1R) < ((1R + 1R) * (1R + 1R))
s = <WellDefined reflexive (symmetric *DistributesOver+) (<WellDefined reflexive (+WellDefined (transitive (symmetric identIsIdent) *Commutative) (transitive (symmetric identIsIdent) *Commutative)) (<WellDefined identLeft reflexive (orderRespectsAddition 0<2 (1R + 1R))))
2ub y y^2<2 | inr y=2 = inr y=2
abstract
sup : Sg A (LeastUpperBound sub)
sup = lub sub (0R , <Transitive (<WellDefined (symmetric timesZero) (symmetric identLeft) (0<1 nontrivial)) (orderRespectsAddition (0<1 nontrivial) 1R)) ((1R + 1R) , 2ub)
sqrt2' : A
sqrt2' = underlying sup
sqrt2IsSqrt2 : (sqrt2' * sqrt2') ∼ (1R + 1R)
sqrt2IsSqrt2 with totality (sqrt2' * sqrt2') (1R + 1R)
sqrt2IsSqrt2 | inl (inl sup^2<2) with sup
sqrt2IsSqrt2 | inl (inl sup^2<2) | sqrt2' , record { upperBound = upperBound ; leastUpperBound = leastUpperBound } = exFalso bad
where
abstract
t : A
t = ((fromN 2) + inverse (sqrt2' * sqrt2')) * (underlying (allInvertible (fromN 6) (charZero' 5)))
pr' : (underlying (allInvertible (fromN 6) (charZero' 5)) * fromN 6) ∼ 1R
pr' with allInvertible (fromN 6) (charZero' 5)
... | x , p = p
crudeBound : isInInterval (sqrt2' * sqrt2') record { minBound = 0R ; maxBound = 1R + 1R }
crudeBound with squarePositive sqrt2'
crudeBound | inl (_ ,, snd) = snd ,, sup^2<2
crudeBound | inr sqrt2=0 with upperBound 1R (<WellDefined (transitive identRight (symmetric identIsIdent)) (+WellDefined reflexive identRight) (fromNPreservesOrder (0<1 nontrivial) {1} {2} (le 0 refl)))
crudeBound | inr sqrt2=0 | inl 1<sqrt2 = exFalso (irreflexive (<Transitive 1<sqrt2 (<WellDefined (symmetric sqrt2=0) reflexive (0<1 nontrivial))))
crudeBound | inr sqrt2=0 | inr 1=sqrt2 = exFalso (nontrivial (transitive (symmetric sqrt2=0) (symmetric 1=sqrt2)))
crudeBound' : isInInterval (inverse (sqrt2' * sqrt2')) record { minBound = inverse (1R + 1R) ; maxBound = inverse 0R }
crudeBound' = intervalInverseContains crudeBound
numeratorBound : isInInterval (inverse (sqrt2' * sqrt2') + fromN 2) record { minBound = 0R ; maxBound = 1R + 1R }
numeratorBound = intervalWellDefined (transitive groupIsAbelian (transferToRight'' (transitive +Associative identRight)) ,, transitive (+WellDefined invIdent reflexive) (transitive identLeft (+WellDefined reflexive identRight))) (intervalConstantSumContains (fromN 2) crudeBound')
numeratorBound' : isInInterval (fromN 2 + inverse (sqrt2' * sqrt2')) record { minBound = 0R ; maxBound = 1R + 1R }
numeratorBound' = intervalWellDefined' groupIsAbelian numeratorBound
tBound : isInInterval t record { minBound = 0R ; maxBound = (1R + 1R) * (underlying (allInvertible (fromN 6) (charZero' 5))) }
tBound = intervalWellDefined (transitive *Commutative timesZero ,, reflexive) (intervalConstantProductContains (inversePositiveIsPositive pr' (fromNPreservesOrder (0<1 nontrivial) {0} {6} (le 5 refl))) numeratorBound')
u : ((((fromN 5) * ((fromN 2) + inverse (sqrt2' * sqrt2'))) * underlying (allInvertible (fromN 6) (charZero' 5))) + (sqrt2' * sqrt2')) < ((fromN 2 + inverse (sqrt2' * sqrt2')) + (sqrt2' * sqrt2'))
u = orderRespectsAddition (<WellDefined (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) identIsIdent (ringCanMultiplyByPositive {(fromN 5) * underlying (allInvertible (fromN 6) (charZero' 5))} {1R} {fromN 2 + inverse (sqrt2' * sqrt2')} (moveInequality (<WellDefined reflexive (transitive (symmetric identRight) (symmetric +Associative)) sup^2<2)) (fraction<1 5 6 (le zero refl) λ pr → charZero' 5 pr))) (sqrt2' * sqrt2')
tBound' : (((fromN 5) * t) + (sqrt2' * sqrt2')) < (1R + 1R)
tBound' = <WellDefined (+WellDefined (symmetric *Associative) reflexive) (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invLeft) (transitive identRight (+WellDefined reflexive identRight)))) u
sqrt2<2 : sqrt2' < (1R + 1R)
sqrt2<2 with leastUpperBound (fromN 3 * 1/2) (λ r s → inl (3/2ub s))
sqrt2<2 | inl x = <Transitive x 3/2<2
sqrt2<2 | inr x = <WellDefined (symmetric x) reflexive 3/2<2
2sqrt2<4 : (sqrt2' + sqrt2') < fromN 4
2sqrt2<4 = <WellDefined (transitive *DistributesOver+ (+WellDefined (transitive *Commutative identIsIdent) (transitive *Commutative identIsIdent))) (transitive (transitive (*WellDefined (symmetric identRight) (symmetric identRight)) (*WellDefined (symmetric +Associative) (symmetric +Associative))) (symmetric (fromNPreserves* 2 2))) (ringCanMultiplyByPositive {sqrt2'} {1R + 1R} {1R + 1R} (<WellDefined identLeft reflexive (ringAddInequalities (0<1 nontrivial) (0<1 nontrivial))) sqrt2<2)
t<1 : t < 1R
t<1 = <Transitive (_&&_.snd tBound) 2/6<1
newElementIsElement : ((sqrt2' + t) * (sqrt2' + t)) < (1R + 1R)
newElementIsElement = <WellDefined (symmetric *DistributesOver+') reflexive (<WellDefined (+WellDefined (symmetric *DistributesOver+) reflexive) reflexive (<WellDefined (transitive groupIsAbelian +Associative) reflexive (<Transitive (orderRespectsAddition (<WellDefined (+WellDefined *Commutative reflexive) reflexive (<WellDefined (transitive *Commutative *DistributesOver+) reflexive (ringCanMultiplyByPositive (_&&_.fst tBound) (<WellDefined (symmetric +Associative) reflexive (<Transitive (orderRespectsAddition 2sqrt2<4 t) (<WellDefined groupIsAbelian reflexive (<WellDefined reflexive (reflexive {fromN 5}) (orderRespectsAddition t<1 (fromN 4))))))))) (sqrt2' * sqrt2')) tBound')))
bad : False
bad with upperBound (sqrt2' + t) newElementIsElement
bad | inl x = irreflexive (<Transitive x (<WellDefined identLeft groupIsAbelian (orderRespectsAddition (_&&_.fst tBound) sqrt2')))
bad | inr x = irreflexive (<WellDefined identLeft (transitive groupIsAbelian x) (orderRespectsAddition (_&&_.fst tBound) sqrt2'))
sqrt2IsSqrt2 | inl (inr 2<sup^2) with sup
sqrt2IsSqrt2 | inl (inr 2<sup^2) | sqrt2' , record { upperBound = upperBound ; leastUpperBound = leastUpperBound } = exFalso bad
where
abstract
1<sqrt2 : 1R < sqrt2'
1<sqrt2 with upperBound 1R (<WellDefined (transitive identRight (symmetric identIsIdent)) (+WellDefined reflexive identRight) (fromNPreservesOrder (0<1 nontrivial) {1} {2} (le zero refl)))
1<sqrt2 | inl x = x
1<sqrt2 | inr x = exFalso (irreflexive (<Transitive 2<sup^2 (<WellDefined (transitive identRight (transitive (symmetric identIsIdent) (*WellDefined x x))) (+WellDefined reflexive identRight) (fromNPreservesOrder (0<1 nontrivial) {1} {2} (le 0 refl)))))
0<sqrt2 : 0R < sqrt2'
0<sqrt2 = <Transitive (0<1 nontrivial) 1<sqrt2
sqrt2<2 : sqrt2' < (1R + 1R)
sqrt2<2 with leastUpperBound (fromN 3 * 1/2) (λ r s → inl (3/2ub s))
sqrt2<2 | inl x = <Transitive x 3/2<2
sqrt2<2 | inr x = <WellDefined (symmetric x) reflexive 3/2<2
2sqrt2<4 : (sqrt2' + sqrt2') < fromN 4
2sqrt2<4 = <WellDefined (transitive *DistributesOver+ (+WellDefined (transitive *Commutative identIsIdent) (transitive *Commutative identIsIdent))) (transitive (transitive (*WellDefined (symmetric identRight) (symmetric identRight)) (*WellDefined (symmetric +Associative) (symmetric +Associative))) (symmetric (fromNPreserves* 2 2))) (ringCanMultiplyByPositive {sqrt2'} {1R + 1R} {1R + 1R} (<WellDefined identLeft reflexive (ringAddInequalities (0<1 nontrivial) (0<1 nontrivial))) sqrt2<2)
t : A
t = ((sqrt2' * sqrt2') + inverse (fromN 2)) * underlying (allInvertible (fromN 4) (charZero' 3))
pr1 : inverse (fromN 4 * t) ∼ (inverse (sqrt2' * sqrt2') + (fromN 2))
pr1 with allInvertible (fromN 4) (charZero' 3)
... | 1/4 , pr1/4 = transitive (transitive (transitive (transitive (symmetric ringMinusExtracts) (transitive (transitive (*WellDefined reflexive (transitive (symmetric ringMinusExtracts') (transitive *Commutative (*WellDefined reflexive (transitive invContravariant (transitive groupIsAbelian (+WellDefined reflexive invInv))))))) *Associative) *Commutative)) (*WellDefined reflexive (transitive *Commutative pr1/4))) *Commutative) identIsIdent
t<sqrt2 : t < sqrt2'
t<sqrt2 with allInvertible (fromN 4) (charZero' 3)
... | 1/4 , pr4 = <WellDefined reflexive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive (transitive *Commutative pr4)) (transitive *Commutative identIsIdent))) (ringCanMultiplyByPositive (inversePositiveIsPositive pr4 (fromNPreservesOrder (0<1 nontrivial) {0} {4} (le 3 refl))) (<Transitive (orderRespectsAddition (ringMultiplyPositives 0<sqrt2 0<sqrt2 sqrt2<2 sqrt2<2) (inverse (1R + (1R + 0G)))) (<WellDefined {fromN 2} 2=2*2-2 reflexive (<Transitive {fromN 2} {sqrt2' * fromN 2} (<WellDefined identIsIdent reflexive (ringCanMultiplyByPositive (fromNPreservesOrder (0<1 nontrivial) {0} (le 1 refl)) 1<sqrt2)) (<WellDefined *Commutative *Commutative (ringCanMultiplyByPositive 0<sqrt2 (fromNPreservesOrder (0<1 nontrivial) {2} (le 1 refl))))))))
0<t : 0R < t
0<t with allInvertible (fromN 4) (charZero' 3)
... | 1/4 , pr4 = orderRespectsMultiplication (<WellDefined reflexive (+WellDefined reflexive (inverseWellDefined (+WellDefined reflexive (symmetric identRight)))) (moveInequality 2<sup^2)) (inversePositiveIsPositive pr4 (fromNPreservesOrder (0<1 nontrivial) {0} {4} (le 3 refl)))
anotherUpperBound : A
anotherUpperBound = (sqrt2' + inverse t) * (sqrt2' + inverse t)
abstract
u : anotherUpperBound ∼ ((sqrt2' * sqrt2') + (inverse ((t * sqrt2') + (t * sqrt2')) + (t * t)))
u = transitive *DistributesOver+' (transitive (+WellDefined *DistributesOver+ reflexive) (transitive (symmetric +Associative) (+WellDefined reflexive (transitive (+WellDefined ringMinusExtracts *DistributesOver+) (transitive +Associative (+WellDefined (transitive (+WellDefined (inverseWellDefined *Commutative) ringMinusExtracts') (symmetric invContravariant)) twoNegativesTimes))))))
w : ((sqrt2' * sqrt2') + inverse ((fromN 4) * t)) < anotherUpperBound
w = <WellDefined reflexive (symmetric u) (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (<WellDefined (ringMinusExtracts') (+WellDefined (inverseWellDefined (*DistributesOver+)) reflexive) (<WellDefined reflexive (+WellDefined ringMinusExtracts reflexive) (<WellDefined reflexive *DistributesOver+ (<WellDefined reflexive *Commutative (ringCanMultiplyByPositive 0<t (<WellDefined identRight reflexive (ringAddInequalities (ringSwapNegatives' (<WellDefined reflexive (transitive (+WellDefined (+WellDefined reflexive (symmetric identRight)) (+WellDefined reflexive (symmetric identRight))) (symmetric (fromNPreserves+ 2 2))) (ringAddInequalities sqrt2<2 sqrt2<2))) 0<t))))))) (sqrt2' * sqrt2')))
w' : ((sqrt2' * sqrt2') + inverse ((fromN 4) * t)) ∼ (1R + 1R)
w' = transitive (+WellDefined reflexive pr1) (transitive +Associative (transitive (+WellDefined invRight reflexive) (transitive identLeft (+WellDefined reflexive identRight))))
anotherUpperBoundBounds : (1R + 1R) < anotherUpperBound
anotherUpperBoundBounds = <WellDefined w' reflexive w
aubIsBound : UpperBound sub (sqrt2' + inverse t)
aubIsBound y y^2<2 with totality y (sqrt2' + inverse t)
aubIsBound y y^2<2 | inl (inl x) = inl x
aubIsBound y y^2<2 | inl (inr x) = exFalso (irreflexive (<Transitive (<Transitive anotherUpperBoundBounds (ringMultiplyPositives (moveInequality t<sqrt2) (moveInequality t<sqrt2) x x)) y^2<2))
aubIsBound y y^2<2 | inr x = inr x
bad : False
bad with leastUpperBound (sqrt2' + inverse t) aubIsBound
bad | inl x = irreflexive (<Transitive x (<WellDefined groupIsAbelian identLeft (orderRespectsAddition (ringMinusFlipsOrder 0<t) sqrt2')))
bad | inr x = irreflexive (<WellDefined (symmetric (transitive x groupIsAbelian)) identLeft (orderRespectsAddition (ringMinusFlipsOrder 0<t) sqrt2'))
sqrt2IsSqrt2 | inr x = x
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{-# OPTIONS --safe #-}
module Cubical.Algebra.Field where
open import Cubical.Algebra.Field.Base public
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module _ where
import Agda.Primitive.Cubical as C
-- Cannot alias module exporting primitives with typechecking constraints.
module M = C
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{-# OPTIONS --without-K --exact-split --safe #-}
module Fragment.Setoid.Morphism where
open import Fragment.Setoid.Morphism.Base public
open import Fragment.Setoid.Morphism.Setoid public
open import Fragment.Setoid.Morphism.Properties public
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------------------------------------------------------------------------------
-- Totality properties respect to Tree
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
module FOTC.Program.SortList.Properties.Totality.TreeATP where
open import FOTC.Base
open import FOTC.Base.List
open import FOTC.Data.Nat.Inequalities.PropertiesATP
open import FOTC.Data.Nat.List.Type
open import FOTC.Data.Nat.Type
open import FOTC.Program.SortList.SortList
open import FOTC.Data.Nat.Inequalities
------------------------------------------------------------------------------
toTree-Tree : ∀ {item t} → N item → Tree t → Tree (toTree · item · t)
toTree-Tree {item} Nitem tnil = prf
where postulate prf : Tree (toTree · item · nil)
{-# ATP prove prf #-}
toTree-Tree {item} Nitem (ttip {i} Ni) = prf (x>y∨x≤y Ni Nitem)
where postulate prf : i > item ∨ i ≤ item → Tree (toTree · item · tip i)
{-# ATP prove prf x>y→x≰y #-}
toTree-Tree {item} Nitem (tnode {t₁} {i} {t₂} Tt₁ Ni Tt₂) =
prf (x>y∨x≤y Ni Nitem) (toTree-Tree Nitem Tt₁) (toTree-Tree Nitem Tt₂)
where
postulate prf : i > item ∨ i ≤ item →
Tree (toTree · item · t₁) →
Tree (toTree · item · t₂) →
Tree (toTree · item · node t₁ i t₂)
{-# ATP prove prf x>y→x≰y #-}
makeTree-Tree : ∀ {is} → ListN is → Tree (makeTree is)
makeTree-Tree lnnil = prf
where postulate prf : Tree (makeTree [])
{-# ATP prove prf #-}
makeTree-Tree (lncons {i} {is} Nn Lis) = prf (makeTree-Tree Lis)
where postulate prf : Tree (makeTree is) → Tree (makeTree (i ∷ is))
{-# ATP prove prf toTree-Tree #-}
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{-# OPTIONS --allow-unsolved-metas #-}
{-# OPTIONS --show-implicit #-}
open import Oscar.Prelude
open import Oscar.Class
-- classes
open import Oscar.Class.Transitivity
-- data
open import Oscar.Data.Substitunction
open import Oscar.Data.Term
module Test.EquivalentCandidates-2 where
module _
{a}
where
instance
𝓣ransitivityFunction₁ : Transitivity.class Function⟦ a ⟧
𝓣ransitivityFunction₁ .⋆ f g = g ∘ f
𝓣ransitivityFunction₂ : Transitivity.class Function⟦ a ⟧
𝓣ransitivityFunction₂ .⋆ f g = g ∘ f
module _ (𝔓 : Ø₀) where
open Substitunction 𝔓
open Term 𝔓
test-1 test-2 test-3 test-4 test-5 test-6 :
∀ {m n} (f : Substitunction m n) → Substitunction m n
test-1 f = transitivity {!!} {!!}
test-2 f = transitivity {𝔒 = Ø₀} {_∼_ = Function⟦ ∅̂ ⟧} {!!} {!!}
test-3 f = transitivity f {!!}
test-4 {m} {n} f = transitivity {𝔒 = Ø₀} f (¡ {𝔒 = Term n})
test-5 f = transitivity {_∼_ = Function⟦ ∅̂ ⟧} f {!!}
test-6 {m} {n} f = transitivity {𝔒 = Ø₀} ⦃ {!!} ⦄ f (¡ {𝔒 = Term n})
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{-# OPTIONS --without-K #-}
open import lib.Base
module test.fail.Test1 where
module _ where
private
data #I-aux : Type₀ where
#zero : #I-aux
#one : #I-aux
data #I : Type₀ where
#i : #I-aux → (Unit → Unit) → #I
I : Type₀
I = #I
zero : I
zero = #i #zero _
one : I
one = #i #one _
postulate
seg : zero == one
absurd : zero ≠ one
absurd () -- fails
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data ℕ : Set where
zero : ℕ
suc : ℕ → ℕ
{-# BUILTIN NATURAL ℕ #-}
_+_ : ℕ → ℕ → ℕ
zero + b = b
suc a + b = suc (a + b)
infix 100 _+_
data _≡_ : ℕ → ℕ → Set where
refl : {n : ℕ} → n ≡ n
sym : {n m : ℕ} → n ≡ m → m ≡ n
sym refl = refl
trans : {m n o : ℕ} → m ≡ n → n ≡ o → m ≡ o
trans refl p₂ = p₂
suc-inj : {m n : ℕ} → suc m ≡ suc n → m ≡ n
suc-inj refl = refl
data ⊥ : Set where
¬ : Set → Set
¬ A = A → ⊥
_≢_ : ℕ → ℕ → Set
m ≢ n = ¬(m ≡ n)
zero-img : ∀ {m} → suc m ≢ 0
zero-img ()
induction : (P : ℕ → Set) → P 0 → (∀ {n} → P n → P (suc n)) → (∀ m → P m)
induction pred base hypo zero = base
induction pred base hypo (suc m) = hypo (induction pred base hypo m)
cong : ∀ {m n} → (f : ℕ → ℕ) → m ≡ n → f m ≡ f n
cong f refl = refl
assoc : ∀ m n o → m + (n + o) ≡ (m + n) + o
assoc zero n o = refl
assoc (suc m) n o = cong suc (assoc m n o)
n+zero : ∀ n → n ≡ n + 0
n+zero zero = refl
n+zero (suc n) = cong suc (n+zero n)
suc+ : ∀ m n → suc (n + m) ≡ (n + suc m)
suc+ m zero = refl
suc+ m (suc n) = cong suc (suc+ m n)
comm : ∀ m n → m + n ≡ n + m
comm zero n = n+zero n
comm (suc m) n = trans (cong suc (comm m n)) (suc+ m n)
-- Where to go from here?
-- Agda:
-- • Aaron Stump - Verified Functional Programming in Agda, https://svn.divms.uiowa.edu/repos/clc/projects/agda/book/book.pdf
-- • Conor McBride - Dependently Typed Metaprogramming (in Agda), http://cs.ioc.ee/ewscs/2014/mcbride/mcbride-deptypedmetaprog.pdf
-- Type theory:
-- • So you want to learn type theory, http://purelytheoretical.com/sywtltt.html
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module UnifyTermF (FunctionName : Set) where
open import Data.Fin using (Fin; suc; zero)
open import Data.Nat using (ℕ; suc; zero)
open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong₂; cong; sym; trans)
open import Function using (_∘_; flip)
open import Relation.Nullary using (¬_; Dec; yes; no)
open import Data.Product using (∃; _,_; _×_)
open import Data.Empty using (⊥-elim)
open import Data.Vec using (Vec; []; _∷_) renaming (map to mapV)
data Term (n : ℕ) : Set where
i : (x : Fin n) -> Term n
leaf : Term n
_fork_ : (s t : Term n) -> Term n
function : FunctionName → ∀ {f} → Vec (Term n) f → Term n
Term-function-inj-FunctionName : ∀ {fn₁ fn₂} {n N₁ N₂} {ts₁ : Vec (Term n) N₁} {ts₂ : Vec (Term n) N₂} → Term.function fn₁ ts₁ ≡ Term.function fn₂ ts₂ → fn₁ ≡ fn₂
Term-function-inj-FunctionName refl = refl
Term-function-inj-VecSize : ∀ {fn₁ fn₂} {n N₁ N₂} {ts₁ : Vec (Term n) N₁} {ts₂ : Vec (Term n) N₂} → Term.function fn₁ ts₁ ≡ Term.function fn₂ ts₂ → N₁ ≡ N₂
Term-function-inj-VecSize refl = refl
Term-function-inj-Vector : ∀ {fn₁ fn₂} {n N} {ts₁ : Vec (Term n) N} {ts₂ : Vec (Term n) N} → Term.function fn₁ ts₁ ≡ Term.function fn₂ ts₂ → ts₁ ≡ ts₂
Term-function-inj-Vector refl = refl
Term-fork-inj-left : ∀ {n} {l₁ r₁ l₂ r₂ : Term n} → l₁ fork r₁ ≡ l₂ fork r₂ → l₁ ≡ l₂
Term-fork-inj-left refl = refl
Term-fork-inj-right : ∀ {n} {l₁ r₁ l₂ r₂ : Term n} → l₁ fork r₁ ≡ l₂ fork r₂ → r₁ ≡ r₂
Term-fork-inj-right refl = refl
open import Relation.Binary.HeterogeneousEquality using (_≅_; refl)
Term-function-inj-HetVector : ∀ {fn₁ fn₂} {n N₁ N₂} {ts₁ : Vec (Term n) N₁} {ts₂ : Vec (Term n) N₂} → Term.function fn₁ ts₁ ≡ Term.function fn₂ ts₂ → ts₁ ≅ ts₂
Term-function-inj-HetVector refl = refl
_~>_ : (m n : ℕ) -> Set
m ~> n = Fin m -> Term n
▹ : ∀ {m n} -> (r : Fin m -> Fin n) -> Fin m -> Term n
▹ r = i ∘ r
Property⋆ : (m : ℕ) -> Set1
Property⋆ m = ∀ {n} -> (Fin m -> Term n) -> Set
record Substitution (T : ℕ → Set) : Set where
field
_◃_ : ∀ {m n} -> (f : m ~> n) -> T m -> T n
Unifies⋆ : ∀ {m} (s t : T m) -> Property⋆ m
Unifies⋆ s t f = f ◃ s ≡ f ◃ t
open Substitution ⦃ … ⦄ public
{-# DISPLAY Substitution._◃_ _ = _◃_ #-}
mutual
instance SubstitutionTerm : Substitution Term
Substitution._◃_ SubstitutionTerm = _◃′_ where
_◃′_ : ∀ {m n} -> (f : m ~> n) -> Term m -> Term n
f ◃′ i x = f x
f ◃′ leaf = leaf
f ◃′ (s fork t) = (f ◃ s) fork (f ◃ t)
f ◃′ (function fn ts) = function fn (f ◃ ts)
instance SubstitutionVecTerm : ∀ {N} → Substitution (flip Vec N ∘ Term )
Substitution._◃_ (SubstitutionVecTerm {N}) = _◃′_ where
_◃′_ : ∀ {m n} -> (f : m ~> n) -> Vec (Term m) N -> Vec (Term n) N
f ◃′ [] = []
f ◃′ (t ∷ ts) = f ◃ t ∷ f ◃ ts
_≐_ : {m n : ℕ} -> (Fin m -> Term n) -> (Fin m -> Term n) -> Set
f ≐ g = ∀ x -> f x ≡ g x
record SubstitutionExtensionality (T : ℕ → Set) ⦃ _ : Substitution T ⦄ : Set₁ where
field
◃ext : ∀ {m n} {f g : Fin m -> Term n} -> f ≐ g -> (t : T m) -> f ◃ t ≡ g ◃ t
open SubstitutionExtensionality ⦃ … ⦄ public
mutual
instance SubstitutionExtensionalityTerm : SubstitutionExtensionality Term
SubstitutionExtensionality.◃ext SubstitutionExtensionalityTerm = ◃ext′ where
◃ext′ : ∀ {m n} {f g : Fin m -> Term n} -> f ≐ g -> ∀ t -> f ◃ t ≡ g ◃ t
◃ext′ p (i x) = p x
◃ext′ p leaf = refl
◃ext′ p (s fork t) = cong₂ _fork_ (◃ext p s) (◃ext p t)
◃ext′ p (function fn ts) = cong (function fn) (◃ext p ts)
instance SubstitutionExtensionalityVecTerm : ∀ {N} → SubstitutionExtensionality (flip Vec N ∘ Term)
SubstitutionExtensionality.◃ext (SubstitutionExtensionalityVecTerm {N}) = λ x → ◃ext′ x where
◃ext′ : ∀ {m n} {f g : Fin m -> Term n} -> f ≐ g -> ∀ {N} (t : Vec (Term m) N) -> f ◃ t ≡ g ◃ t
◃ext′ p [] = refl
◃ext′ p (t ∷ ts) = cong₂ _∷_ (◃ext p t) (◃ext p ts)
_◇_ : ∀ {l m n : ℕ } -> (f : Fin m -> Term n) (g : Fin l -> Term m) -> Fin l -> Term n
f ◇ g = (f ◃_) ∘ g
≐-cong : ∀ {m n o} {f : m ~> n} {g} (h : _ ~> o) -> f ≐ g -> (h ◇ f) ≐ (h ◇ g)
≐-cong h f≐g t = cong (h ◃_) (f≐g t)
≐-sym : ∀ {m n} {f : m ~> n} {g} -> f ≐ g -> g ≐ f
≐-sym f≐g = sym ∘ f≐g
open import Prelude using (it)
module Sub where
record Fact1 (T : ℕ → Set) ⦃ _ : Substitution T ⦄ : Set where
field
fact1 : ∀ {n} -> (t : T n) -> i ◃ t ≡ t
open Fact1 ⦃ … ⦄ public
mutual
instance Fact1Term : Fact1 Term
Fact1.fact1 Fact1Term (i x) = refl
Fact1.fact1 Fact1Term leaf = refl
Fact1.fact1 Fact1Term (s fork t) = cong₂ _fork_ (fact1 s) (fact1 t)
Fact1.fact1 Fact1Term (function fn ts) = cong (function fn) (fact1 ts)
instance Fact1TermVec : ∀ {N} → Fact1 (flip Vec N ∘ Term)
Fact1.fact1 Fact1TermVec [] = refl
Fact1.fact1 Fact1TermVec (t ∷ ts) = cong₂ _∷_ (fact1 t) (fact1 ts)
record Fact2 (T : ℕ → Set) ⦃ _ : Substitution T ⦄ : Set where
field
-- ⦃ s ⦄ : Substitution T
fact2 : ∀ {l m n} -> {f : Fin m -> Term n} {g : _} (t : T l) → (f ◇ g) ◃ t ≡ f ◃ (g ◃ t)
open Fact2 ⦃ … ⦄ public
mutual
instance Fact2Term : Fact2 Term
-- Fact2.s Fact2Term = SubstitutionTerm
Fact2.fact2 Fact2Term (i x) = refl
Fact2.fact2 Fact2Term leaf = refl
Fact2.fact2 Fact2Term (s fork t) = cong₂ _fork_ (fact2 s) (fact2 t)
Fact2.fact2 Fact2Term {f = f} {g = g} (function fn ts) = cong (function fn) (fact2 {f = f} {g = g} ts) -- fact2 ts
instance Fact2TermVec : ∀ {N} → Fact2 (flip Vec N ∘ Term)
-- Fact2.s Fact2TermVec = SubstitutionVecTerm
Fact2.fact2 Fact2TermVec [] = refl
Fact2.fact2 Fact2TermVec (t ∷ ts) = cong₂ _∷_ (fact2 t) (fact2 ts)
fact3 : ∀ {l m n} (f : Fin m -> Term n) (r : Fin l -> Fin m) -> (f ◇ (▹ r)) ≡ (f ∘ r)
fact3 f r = refl
◃ext' : ∀ {m n o} {f : Fin m -> Term n}{g : Fin m -> Term o}{h} -> f ≐ (h ◇ g) -> ∀ (t : Term _) -> f ◃ t ≡ h ◃ (g ◃ t)
◃ext' p t = trans (◃ext p t) (Sub.fact2 t)
open import Agda.Primitive
Injectivity : ∀ {a} {A : Set a} {b} {B : Set b} (f : A → B) → Set (a ⊔ b)
Injectivity f = ∀ {x y} → f x ≡ f y → x ≡ y
Injectivity₂ : ∀ {a} {A : Set a} {b} {B : Set b} {c} {C : Set c} (f : A → B → C) → Set (a ⊔ b ⊔ c)
Injectivity₂ f = ∀ {w x y z} → f w x ≡ f y z → x ≡ z
record Injective {a} {A : Set a} {b} {B : Set b} (f : A → B) : Set (a ⊔ b) where
field injectivity : ∀ x y → f x ≡ f y → x ≡ y
open Injective public -- ⦃ … ⦄ public
record Thin (T : ℕ → Set) : Set where
field
thin : ∀ {n} -> Fin (suc n) → T n → T (suc n)
thinfact1 : ∀ {n} (f : Fin (suc n)) → Injectivity (thin f)
term-i-inj : ∀ {n} → Injectivity (Term.i {n})
term-i-inj refl = refl
term-fork-l-inj : ∀ {n} → Injectivity₂ (flip (_fork_ {n}))
term-fork-l-inj refl = refl
term-fork-r-inj : ∀ {n} → Injectivity₂ (_fork_ {n})
term-fork-r-inj refl = refl
open Thin ⦃ … ⦄ public
p : ∀ {n} -> Fin (suc (suc n)) -> Fin (suc n)
p (suc x) = x
p zero = zero
instance ThinFin : Thin Fin
Thin.thin ThinFin zero y = suc y
Thin.thin ThinFin (suc x) zero = zero
Thin.thin ThinFin (suc x) (suc y) = suc (thin x y)
Thin.thinfact1 ThinFin zero refl = refl
Thin.thinfact1 ThinFin (suc x) {zero} {zero} r = refl
Thin.thinfact1 ThinFin (suc x) {zero} {(suc z)} ()
Thin.thinfact1 ThinFin (suc x) {(suc y)} {zero} ()
Thin.thinfact1 ThinFin (suc x) {(suc y)} {(suc z)} r =
cong suc (thinfact1 x (cong p r))
{- TODO defining using the below leads to termination checker problem -}
tfact1 : ∀ {n} (x : Fin (suc n)) (y : Fin _) (z : Fin n) -> thin x y ≡ thin x z -> y ≡ z
tfact1 zero y .y refl = refl
tfact1 (suc x) zero zero r = refl
tfact1 (suc x) zero (suc z) ()
tfact1 (suc x) (suc y) zero ()
tfact1 (suc x) (suc y) (suc z) r = cong suc (tfact1 x y z (cong p r))
mutual
mapTerm : ∀ {n m} → (Fin n → Fin m) → Term n → Term m
mapTerm x (i x₁) = i (x x₁)
mapTerm x leaf = leaf
mapTerm x (x₁ fork x₂) = mapTerm x x₁ fork mapTerm x x₂
mapTerm x (function x₁ x₂) = function x₁ (mapTerms x x₂)
mapTerms : ∀ {n m} → (Fin n → Fin m) → ∀ {N} → Vec (Term n) N → Vec (Term m) N
mapTerms x [] = []
mapTerms x (x₁ ∷ x₂) = mapTerm x x₁ ∷ mapTerms x x₂
mutual
thinfact1Term : ∀ {n} (f : Fin (suc n)) → Injectivity (mapTerm (thin f))
thinfact1Term x₁ {i x} {i x₃} x₂ = cong i (thinfact1 x₁ (term-i-inj x₂))
thinfact1Term x₁ {i x} {leaf} ()
thinfact1Term x₁ {i x} {y fork y₁} ()
thinfact1Term x₁ {i x} {function x₂ x₃} ()
thinfact1Term x₁ {leaf} {i x} ()
thinfact1Term x₁ {leaf} {leaf} x₂ = refl
thinfact1Term x₁ {leaf} {y fork y₁} ()
thinfact1Term x₁ {leaf} {function x x₂} ()
thinfact1Term x₁ {x fork x₂} {i x₃} ()
thinfact1Term x₁ {x fork x₂} {leaf} ()
thinfact1Term x₁ {x fork x₂} {y fork y₁} x₃ = cong₂ _fork_ (thinfact1Term x₁ (term-fork-l-inj x₃)) ((thinfact1Term x₁ (term-fork-r-inj x₃)))
thinfact1Term x₁ {x fork x₂} {function x₃ x₄} ()
thinfact1Term x₁ {function x x₂} {i x₃} ()
thinfact1Term x₁ {function x x₂} {leaf} ()
thinfact1Term x₁ {function x x₂} {y fork y₁} ()
thinfact1Term x₁ {function f1 {n} ts1} {function f2 ts2} r rewrite Term-function-inj-FunctionName r with Term-function-inj-VecSize r
thinfact1Term x₁ {function f1 {n} ts1} {function f2 ts2} r | refl with Term-function-inj-Vector r
thinfact1Term {m} x₁ {function f1 {n} ts1} {function f2 {.n} ts2} r | refl | w = cong (function f2) (((thinfact1Terms x₁ w)))
thinfact1Terms : ∀ {N} {n} (f : Fin (suc n)) → Injectivity (mapTerms (thin f) {N})
thinfact1Terms {.0} f {[]} {[]} x₁ = refl
thinfact1Terms {.(suc _)} f {x ∷ x₁} {x₂ ∷ y} x₃ = cong₂ _∷_ (thinfact1Term f (cong Data.Vec.head x₃)) (thinfact1Terms f (cong Data.Vec.tail x₃))
mutual
instance ThinTerm : Thin Term
Thin.thin ThinTerm = mapTerm ∘ thin
Thin.thinfact1 ThinTerm = thinfact1Term
instance ThinTermVec : ∀ {N} → Thin (flip Vec N ∘ Term)
Thin.thin ThinTermVec x x₁ = mapTerms (thin x) x₁
Thin.thinfact1 ThinTermVec = thinfact1Terms
module ThinFact where
fact2 : ∀ {n} x (y : Fin n) -> ¬ thin x y ≡ x
fact2 zero y ()
fact2 (suc x) zero ()
fact2 (suc x) (suc y) r = fact2 x y (cong p r)
fact3 : ∀{n} x (y : Fin (suc n)) -> ¬ x ≡ y -> ∃ λ y' -> thin x y' ≡ y
fact3 zero zero ne = ⊥-elim (ne refl)
fact3 zero (suc y) _ = y , refl
fact3 {zero} (suc ()) _ _
fact3 {suc n} (suc x) zero ne = zero , refl
fact3 {suc n} (suc x) (suc y) ne with y | fact3 x y (ne ∘ cong suc)
... | .(thin x y') | y' , refl = suc y' , refl
open import Data.Maybe
open import Category.Functor
open import Category.Monad
import Level
open RawMonad (Data.Maybe.monad {Level.zero})
thick : ∀ {n} -> (x y : Fin (suc n)) -> Maybe (Fin n)
thick zero zero = nothing
thick zero (suc y) = just y
thick {zero} (suc ()) _
thick {suc _} (suc x) zero = just zero
thick {suc _} (suc x) (suc y) = suc <$> (thick x y)
open import Data.Sum
_≡Fin_ : ∀ {n} -> (x y : Fin n) -> Dec (x ≡ y)
zero ≡Fin zero = yes refl
zero ≡Fin suc y = no λ ()
suc x ≡Fin zero = no λ ()
suc {suc _} x ≡Fin suc y with x ≡Fin y
... | yes r = yes (cong suc r)
... | no r = no λ e -> r (cong p e)
suc {zero} () ≡Fin _
module Thick where
half1 : ∀ {n} (x : Fin (suc n)) -> thick x x ≡ nothing
half1 zero = refl
half1 {suc _} (suc x) = cong (_<$>_ suc) (half1 x)
half1 {zero} (suc ())
half2 : ∀ {n} (x : Fin (suc n)) y -> ∀ y' -> thin x y' ≡ y -> thick x y ≡ just y'
half2 zero zero y' ()
half2 zero (suc y) .y refl = refl
half2 {suc n} (suc x) zero zero refl = refl
half2 {suc _} (suc _) zero (suc _) ()
half2 {suc n} (suc x) (suc y) zero ()
half2 {suc n} (suc x) (suc .(thin x y')) (suc y') refl with thick x (thin x y') | half2 x (thin x y') y' refl
... | .(just y') | refl = refl
half2 {zero} (suc ()) _ _ _
fact1 : ∀ {n} (x : Fin (suc n)) y r
-> thick x y ≡ r
-> x ≡ y × r ≡ nothing ⊎ ∃ λ y' -> thin x y' ≡ y × r ≡ just y'
fact1 x y .(thick x y) refl with x ≡Fin y
fact1 x .x ._ refl | yes refl = inj₁ (refl , half1 x)
... | no el with ThinFact.fact3 x y el
... | y' , thinxy'=y = inj₂ (y' , ( thinxy'=y , half2 x y y' thinxy'=y ))
record Check (T : ℕ → Set) : Set where
field
check : ∀{n} (x : Fin (suc n)) (t : T (suc n)) -> Maybe (T n)
open Check ⦃ … ⦄ public
_<*>_ = _⊛_
mutual
instance CheckTerm : Check Term
Check.check CheckTerm x (i y) = i <$> thick x y
Check.check CheckTerm x leaf = just leaf
Check.check CheckTerm x (s fork t) = _fork_ <$> check x s ⊛ check x t
Check.check CheckTerm x (function fn ts) = ⦇ (function fn) (check x ts) ⦈
instance CheckTermVec : ∀ {N} → Check (flip Vec N ∘ Term)
Check.check CheckTermVec x [] = just []
Check.check CheckTermVec x (t ∷ ts) = ⦇ check x t ∷ check x ts ⦈
_for_ : ∀ {n} (t' : Term n) (x : Fin (suc n)) -> Fin (suc n) -> Term n
(t' for x) y = maybe′ i t' (thick x y)
data AList : ℕ -> ℕ -> Set where
anil : ∀ {n} -> AList n n
_asnoc_/_ : ∀ {m n} (σ : AList m n) (t' : Term m) (x : Fin (suc m))
-> AList (suc m) n
sub : ∀ {m n} (σ : AList m n) -> Fin m -> Term n
sub anil = i
sub (σ asnoc t' / x) = sub σ ◇ (t' for x)
_++_ : ∀ {l m n} (ρ : AList m n) (σ : AList l m) -> AList l n
ρ ++ anil = ρ
ρ ++ (σ asnoc t' / x) = (ρ ++ σ) asnoc t' / x
++-assoc : ∀ {l m n o} (ρ : AList l m) (σ : AList n _) (τ : AList o _) -> ρ ++ (σ ++ τ) ≡ (ρ ++ σ) ++ τ
++-assoc ρ σ anil = refl
++-assoc ρ σ (τ asnoc t / x) = cong (λ s -> s asnoc t / x) (++-assoc ρ σ τ)
module SubList where
anil-id-l : ∀ {m n} (σ : AList m n) -> anil ++ σ ≡ σ
anil-id-l anil = refl
anil-id-l (σ asnoc t' / x) = cong (λ σ -> σ asnoc t' / x) (anil-id-l σ)
fact1 : ∀ {l m n} (ρ : AList m n) (σ : AList l m) -> sub (ρ ++ σ) ≐ (sub ρ ◇ sub σ)
fact1 ρ anil v = refl
fact1 {suc l} {m} {n} r (s asnoc t' / x) v = trans hyp-on-terms ◃-assoc
where
t = (t' for x) v
hyp-on-terms = ◃ext (fact1 r s) t
◃-assoc = Sub.fact2 t
_∃asnoc_/_ : ∀ {m} (a : ∃ (AList m)) (t' : Term m) (x : Fin (suc m))
-> ∃ (AList (suc m))
(n , σ) ∃asnoc t' / x = n , σ asnoc t' / x
flexFlex : ∀ {m} (x y : Fin m) -> ∃ (AList m)
flexFlex {suc m} x y with thick x y
... | just y' = m , anil asnoc i y' / x
... | nothing = suc m , anil
flexFlex {zero} () _
flexRigid : ∀ {m} (x : Fin m) (t : Term m) -> Maybe (∃(AList m))
flexRigid {suc m} x t with check x t
... | just t' = just (m , anil asnoc t' / x)
... | nothing = nothing
flexRigid {zero} () _
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module L.Base.Sigma.Core where
-- Import the Σ record with constructors fst, snd
open import Agda.Builtin.Sigma public
split : ∀{a b c} {A : Set a} {B : A → Set b} (C : Σ A B → Set c)
→ ((x : A)(y : B x) → C (x , y)) → (p : Σ A B) → C p
split C g (a , b) = g a b
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module Prelude.List where
open import Prelude.Bool
open import Prelude.Nat
open import Prelude.Fin
infixr 40 _::_
data List (A : Set) : Set where
[] : List A
_::_ : A -> List A -> List A
{-# BUILTIN LIST List #-}
{-# BUILTIN NIL [] #-}
{-# BUILTIN CONS _::_ #-}
infixr 30 _++_
_++_ : {A : Set} -> List A -> List A -> List A
[] ++ ys = ys
(x :: xs) ++ ys = x :: (xs ++ ys)
snoc : {A : Set} -> List A -> A -> List A
snoc [] e = e :: []
snoc (x :: xs) e = x :: snoc xs e
length : {A : Set} -> List A -> Nat
length [] = 0
length (x :: xs) = 1 + length xs
zipWith : ∀ {A B C} -> (A -> B -> C) -> List A -> List B -> List C
zipWith f (x :: xs) (y :: ys) = f x y :: zipWith f xs ys
zipWith _ _ _ = []
map : ∀ {A B} -> (A -> B) -> List A -> List B
map _ [] = []
map f (x :: xs) = f x :: map f xs
mapIgen : ∀ {A B} -> (A -> B) -> List A -> List B
mapIgen = map
_!_ : ∀ {A} -> (xs : List A) -> Fin (length {A} xs) -> A
_!_ {A} (x :: xs) (fz .{length xs}) = x
_!_ {A} (x :: xs) (fs .{length xs} n) = _!_ {A} xs n
_!_ {A} [] ()
_[_]=_ : {A : Set} -> (xs : List A) -> Fin (length xs) -> A -> List A
(a :: as) [ fz ]= e = e :: as
(a :: as) [ fs n ]= e = a :: (as [ n ]= e)
[] [ () ]= e
listEq : {A : Set} -> (A -> A -> Bool) -> List A -> List A -> Bool
listEq _ [] [] = true
listEq _==_ (a :: as) (b :: bs) with a == b
... | true = listEq _==_ as bs
... | false = false
listEq _ _ _ = false
tail : {A : Set} -> List A -> List A
tail [] = []
tail (x :: xs) = xs
reverse : {A : Set} -> List A -> List A
reverse [] = []
reverse (x :: xs) = reverse xs ++ (x :: [])
init : {A : Set} -> List A -> List A
init xs = reverse (tail (reverse xs))
filter : {A : Set} -> (A -> Bool) -> List A -> List A
filter p [] = []
filter p (a :: as) with p a
... | true = a :: filter p as
... | false = filter p as
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{-# OPTIONS --without-K --safe #-}
-- Monoidal natural isomorphisms between lax and strong braided
-- monoidal functors.
--
-- NOTE. Braided monoidal natural isomorphisms are really just
-- monoidal natural isomorphisms that happen to go between braided
-- monoidal functors. No additional conditions are necessary.
-- Nevertheless, the definitions in this module are useful when one is
-- working in a braided monoidal setting. They also help Agda's type
-- checker by bundling the (braided monoidal) categories and functors
-- involved.
module Categories.NaturalTransformation.NaturalIsomorphism.Monoidal.Braided
where
open import Level
open import Relation.Binary using (IsEquivalence)
open import Categories.Category.Monoidal using (BraidedMonoidalCategory)
import Categories.Functor.Monoidal.Braided as BMF
open import Categories.Functor.Monoidal.Properties using () renaming
( idF-BraidedMonoidal to idFˡ ; idF-StrongBraidedMonoidal to idFˢ
; ∘-BraidedMonoidal to _∘Fˡ_ ; ∘-StrongBraidedMonoidal to _∘Fˢ_
)
open import Categories.NaturalTransformation.NaturalIsomorphism as NI
using (NaturalIsomorphism)
import Categories.NaturalTransformation.NaturalIsomorphism.Monoidal as MNI
module Lax where
open BMF.Lax using (BraidedMonoidalFunctor)
open MNI.Lax using (IsMonoidalNaturalIsomorphism)
open BraidedMonoidalFunctor using () renaming (F to UF; monoidalFunctor to MF)
private module U = MNI.Lax
module _ {o ℓ e o′ ℓ′ e′}
{C : BraidedMonoidalCategory o ℓ e}
{D : BraidedMonoidalCategory o′ ℓ′ e′} where
-- Monoidal natural isomorphisms between lax braided monoidal functors.
record BraidedMonoidalNaturalIsomorphism
(F G : BraidedMonoidalFunctor C D) : Set (o ⊔ ℓ ⊔ ℓ′ ⊔ e′) where
field
U : NaturalIsomorphism (UF F) (UF G)
F⇒G-isMonoidal : IsMonoidalNaturalIsomorphism (MF F) (MF G) U
⌊_⌋ : U.MonoidalNaturalIsomorphism (MF F) (MF G)
⌊_⌋ = record { U = U ; F⇒G-isMonoidal = F⇒G-isMonoidal }
open U.MonoidalNaturalIsomorphism ⌊_⌋ public hiding (U; F⇒G-isMonoidal)
infix 4 _≃_
_≃_ = BraidedMonoidalNaturalIsomorphism
-- "Strengthening"
⌈_⌉ : {F G : BraidedMonoidalFunctor C D} →
U.MonoidalNaturalIsomorphism (MF F) (MF G) → F ≃ G
⌈ α ⌉ = record { U = U ; F⇒G-isMonoidal = F⇒G-isMonoidal }
where open U.MonoidalNaturalIsomorphism α
open BraidedMonoidalNaturalIsomorphism
-- Identity and compositions
infixr 9 _ⓘᵥ_
id : {F : BraidedMonoidalFunctor C D} → F ≃ F
id = ⌈ U.id ⌉
_ⓘᵥ_ : {F G H : BraidedMonoidalFunctor C D} → G ≃ H → F ≃ G → F ≃ H
α ⓘᵥ β = ⌈ ⌊ α ⌋ U.ⓘᵥ ⌊ β ⌋ ⌉
isEquivalence : IsEquivalence _≃_
isEquivalence = record
{ refl = id
; sym = λ α → record
{ U = NI.sym (U α)
; F⇒G-isMonoidal = F⇐G-isMonoidal α
}
; trans = λ α β → β ⓘᵥ α
}
where
open BraidedMonoidalNaturalIsomorphism
module _ {o ℓ e o′ ℓ′ e′ o″ ℓ″ e″}
{C : BraidedMonoidalCategory o ℓ e}
{D : BraidedMonoidalCategory o′ ℓ′ e′}
{E : BraidedMonoidalCategory o″ ℓ″ e″} where
infixr 9 _ⓘₕ_ _ⓘˡ_ _ⓘʳ_
_ⓘₕ_ : {F G : BraidedMonoidalFunctor C D}
{H I : BraidedMonoidalFunctor D E} →
H ≃ I → F ≃ G → (H ∘Fˡ F) ≃ (I ∘Fˡ G)
-- NOTE: this definition is clearly equivalent to
--
-- α ⓘₕ β = ⌈ ⌊ α ⌋ U.ⓘₕ ⌊ β ⌋ ⌉
--
-- but the latter takes an unreasonably long time to typecheck,
-- while the unfolded version typechecks almost immediately.
α ⓘₕ β = record
{ U = C.U
; F⇒G-isMonoidal = record
{ ε-compat = C.ε-compat
; ⊗-homo-compat = C.⊗-homo-compat }
}
where module C = U.MonoidalNaturalIsomorphism (⌊ α ⌋ U.ⓘₕ ⌊ β ⌋)
_ⓘˡ_ : {F G : BraidedMonoidalFunctor C D}
(H : BraidedMonoidalFunctor D E) → F ≃ G → (H ∘Fˡ F) ≃ (H ∘Fˡ G)
H ⓘˡ α = id {F = H} ⓘₕ α
_ⓘʳ_ : {G H : BraidedMonoidalFunctor D E} →
G ≃ H → (F : BraidedMonoidalFunctor C D) → (G ∘Fˡ F) ≃ (H ∘Fˡ F)
α ⓘʳ F = α ⓘₕ id {F = F}
-- Left and right unitors.
module _ {o ℓ e o′ ℓ′ e′}
{C : BraidedMonoidalCategory o ℓ e}
{D : BraidedMonoidalCategory o′ ℓ′ e′}
{F : BraidedMonoidalFunctor C D} where
-- NOTE: Again, manual expansion seems necessary to type check in
-- reasonable time.
unitorˡ : idFˡ D ∘Fˡ F ≃ F
unitorˡ = record
{ U = LU.U
; F⇒G-isMonoidal = record
{ ε-compat = LU.ε-compat
; ⊗-homo-compat = LU.⊗-homo-compat
}
}
where module LU = U.MonoidalNaturalIsomorphism (U.unitorˡ {F = MF F})
unitorʳ : F ∘Fˡ idFˡ C ≃ F
unitorʳ = record
{ U = RU.U
; F⇒G-isMonoidal = record
{ ε-compat = RU.ε-compat
; ⊗-homo-compat = RU.⊗-homo-compat
}
}
where module RU = U.MonoidalNaturalIsomorphism (U.unitorʳ {F = MF F})
-- Associator.
module _ {o ℓ e o′ ℓ′ e′ o″ ℓ″ e″ o‴ ℓ‴ e‴}
{B : BraidedMonoidalCategory o ℓ e}
{C : BraidedMonoidalCategory o′ ℓ′ e′}
{D : BraidedMonoidalCategory o″ ℓ″ e″}
{E : BraidedMonoidalCategory o‴ ℓ‴ e‴}
{F : BraidedMonoidalFunctor B C} {G : BraidedMonoidalFunctor C D}
{H : BraidedMonoidalFunctor D E} where
-- NOTE: Again, manual expansion seems necessary to type check in
-- reasonable time.
associator : (H ∘Fˡ G) ∘Fˡ F ≃ H ∘Fˡ (G ∘Fˡ F)
associator = record
{ U = AU.U
; F⇒G-isMonoidal = record
{ ε-compat = AU.ε-compat
; ⊗-homo-compat = AU.⊗-homo-compat
}
}
where
module AU =
U.MonoidalNaturalIsomorphism (U.associator {F = MF F} {MF G} {MF H})
module Strong where
open BMF.Strong using (BraidedMonoidalFunctor)
open MNI.Strong using (IsMonoidalNaturalIsomorphism)
open BraidedMonoidalFunctor using () renaming
( F to UF
; monoidalFunctor to MF
; laxBraidedMonoidalFunctor to laxBMF
)
private module U = MNI.Strong
module _ {o ℓ e o′ ℓ′ e′}
{C : BraidedMonoidalCategory o ℓ e}
{D : BraidedMonoidalCategory o′ ℓ′ e′} where
-- Monoidal natural isomorphisms between strong braided monoidal functors.
record BraidedMonoidalNaturalIsomorphism
(F G : BraidedMonoidalFunctor C D) : Set (o ⊔ ℓ ⊔ ℓ′ ⊔ e′) where
field
U : NaturalIsomorphism (UF F) (UF G)
F⇒G-isMonoidal : IsMonoidalNaturalIsomorphism (MF F) (MF G) U
⌊_⌋ : U.MonoidalNaturalIsomorphism (MF F) (MF G)
⌊_⌋ = record { U = U ; F⇒G-isMonoidal = F⇒G-isMonoidal }
laxBNI : Lax.BraidedMonoidalNaturalIsomorphism (laxBMF F) (laxBMF G)
laxBNI = record { U = U ; F⇒G-isMonoidal = F⇒G-isMonoidal }
open Lax.BraidedMonoidalNaturalIsomorphism laxBNI public
hiding (U; F⇒G-isMonoidal; ⌊_⌋)
infix 4 _≃_
_≃_ = BraidedMonoidalNaturalIsomorphism
-- "Strengthening"
⌈_⌉ : {F G : BraidedMonoidalFunctor C D} →
U.MonoidalNaturalIsomorphism (MF F) (MF G) → F ≃ G
⌈ α ⌉ = record { U = U ; F⇒G-isMonoidal = F⇒G-isMonoidal }
where open U.MonoidalNaturalIsomorphism α
open BraidedMonoidalNaturalIsomorphism
-- Identity and compositions
infixr 9 _ⓘᵥ_
id : {F : BraidedMonoidalFunctor C D} → F ≃ F
id = ⌈ U.id ⌉
_ⓘᵥ_ : {F G H : BraidedMonoidalFunctor C D} → G ≃ H → F ≃ G → F ≃ H
α ⓘᵥ β = ⌈ ⌊ α ⌋ U.ⓘᵥ ⌊ β ⌋ ⌉
isEquivalence : IsEquivalence _≃_
isEquivalence = record
{ refl = id
; sym = λ α → record
{ U = NI.sym (U α)
; F⇒G-isMonoidal = F⇐G-isMonoidal α
}
; trans = λ α β → β ⓘᵥ α
}
where
open BraidedMonoidalNaturalIsomorphism
module _ {o ℓ e o′ ℓ′ e′ o″ ℓ″ e″}
{C : BraidedMonoidalCategory o ℓ e}
{D : BraidedMonoidalCategory o′ ℓ′ e′}
{E : BraidedMonoidalCategory o″ ℓ″ e″} where
infixr 9 _ⓘₕ_ _ⓘˡ_ _ⓘʳ_
_ⓘₕ_ : {F G : BraidedMonoidalFunctor C D}
{H I : BraidedMonoidalFunctor D E} →
H ≃ I → F ≃ G → (H ∘Fˢ F) ≃ (I ∘Fˢ G)
-- NOTE: this definition is clearly equivalent to
--
-- α ⓘₕ β = ⌈ ⌊ α ⌋ U.ⓘₕ ⌊ β ⌋ ⌉
--
-- but the latter takes an unreasonably long time to typecheck,
-- while the unfolded version typechecks almost immediately.
α ⓘₕ β = record
{ U = C.U
; F⇒G-isMonoidal = record
{ ε-compat = C.ε-compat
; ⊗-homo-compat = C.⊗-homo-compat }
}
where module C = U.MonoidalNaturalIsomorphism (⌊ α ⌋ U.ⓘₕ ⌊ β ⌋)
_ⓘˡ_ : {F G : BraidedMonoidalFunctor C D}
(H : BraidedMonoidalFunctor D E) → F ≃ G → (H ∘Fˢ F) ≃ (H ∘Fˢ G)
H ⓘˡ α = id {F = H} ⓘₕ α
_ⓘʳ_ : {G H : BraidedMonoidalFunctor D E} →
G ≃ H → (F : BraidedMonoidalFunctor C D) → (G ∘Fˢ F) ≃ (H ∘Fˢ F)
α ⓘʳ F = α ⓘₕ id {F = F}
-- Left and right unitors.
module _ {o ℓ e o′ ℓ′ e′}
{C : BraidedMonoidalCategory o ℓ e}
{D : BraidedMonoidalCategory o′ ℓ′ e′}
{F : BraidedMonoidalFunctor C D} where
-- NOTE: Again, manual expansion seems necessary to type check in
-- reasonable time.
unitorˡ : idFˢ D ∘Fˢ F ≃ F
unitorˡ = record
{ U = LU.U
; F⇒G-isMonoidal = record
{ ε-compat = LU.ε-compat
; ⊗-homo-compat = LU.⊗-homo-compat
}
}
where module LU = U.MonoidalNaturalIsomorphism (U.unitorˡ {F = MF F})
unitorʳ : F ∘Fˢ idFˢ C ≃ F
unitorʳ = record
{ U = RU.U
; F⇒G-isMonoidal = record
{ ε-compat = RU.ε-compat
; ⊗-homo-compat = RU.⊗-homo-compat
}
}
where module RU = U.MonoidalNaturalIsomorphism (U.unitorʳ {F = MF F})
-- Associator.
module _ {o ℓ e o′ ℓ′ e′ o″ ℓ″ e″ o‴ ℓ‴ e‴}
{B : BraidedMonoidalCategory o ℓ e}
{C : BraidedMonoidalCategory o′ ℓ′ e′}
{D : BraidedMonoidalCategory o″ ℓ″ e″}
{E : BraidedMonoidalCategory o‴ ℓ‴ e‴}
{F : BraidedMonoidalFunctor B C} {G : BraidedMonoidalFunctor C D}
{H : BraidedMonoidalFunctor D E} where
-- NOTE: Again, manual expansion seems necessary to type check in
-- reasonable time.
associator : (H ∘Fˢ G) ∘Fˢ F ≃ H ∘Fˢ (G ∘Fˢ F)
associator = record
{ U = AU.U
; F⇒G-isMonoidal = record
{ ε-compat = AU.ε-compat
; ⊗-homo-compat = AU.⊗-homo-compat
}
}
where
module AU =
U.MonoidalNaturalIsomorphism (U.associator {F = MF F} {MF G} {MF H})
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module Bool where
data Bool : Set where
true : Bool
false : Bool
{-# BUILTIN BOOL Bool #-}
{-# BUILTIN TRUE true #-}
{-# BUILTIN FALSE false #-}
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{-# OPTIONS --sized-types #-}
open import Relation.Binary.Core
module SelectSort {A : Set}
(_≤_ : A → A → Set)
(tot≤ : Total _≤_) where
open import Data.List
open import Data.Product
open import Data.Sum
open import Size
open import SList
open import SList.Order _≤_
select : {ι : Size} → A → SList A {ι} → A × SList A {ι}
select x snil = (x , snil)
select x (y ∙ ys)
with tot≤ x y
... | inj₁ x≤y
with select x ys
select x (y ∙ ys) | inj₁ x≤y | (z , zs) = (z , y ∙ zs)
select x (y ∙ ys) | inj₂ y≤x
with select y ys
select x (y ∙ ys) | inj₂ y≤x | (z , zs) = (z , x ∙ zs)
selectSort : {ι : Size} → SList A {ι} → SList A {ι}
selectSort snil = snil
selectSort (x ∙ xs)
with select x xs
... | (y , ys) = y ∙ (selectSort ys)
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{-# OPTIONS --without-K --safe #-}
module Categories.Category.Instance.Monoidals where
open import Level
open import Categories.Category
open import Categories.Category.Helper
open import Categories.Category.Monoidal
open import Categories.Functor.Monoidal
open import Categories.Functor.Monoidal.Properties
open import Categories.NaturalTransformation.NaturalIsomorphism
module _ o ℓ e where
Monoidals : Category (suc (o ⊔ ℓ ⊔ e)) (o ⊔ ℓ ⊔ e) (o ⊔ ℓ ⊔ e)
Monoidals = categoryHelper record
{ Obj = MonoidalCategory o ℓ e
; _⇒_ = MonoidalFunctor
; _≈_ = λ F G → M.F F ≃ M.F G
; id = idF-Monoidal _
; _∘_ = ∘-Monoidal
; assoc = λ {_ _ _ _ F G H} → associator (M.F F) (M.F G) (M.F H)
; identityˡ = unitorˡ
; identityʳ = unitorʳ
; equiv = record
{ refl = ≃.refl
; sym = ≃.sym
; trans = ≃.trans
}
; ∘-resp-≈ = _ⓘₕ_
}
where module M = MonoidalFunctor
StrongMonoidals : Category (suc (o ⊔ ℓ ⊔ e)) (o ⊔ ℓ ⊔ e) (o ⊔ ℓ ⊔ e)
StrongMonoidals = categoryHelper record
{ Obj = MonoidalCategory o ℓ e
; _⇒_ = StrongMonoidalFunctor
; _≈_ = λ F G → M.F F ≃ M.F G
; id = idF-StrongMonoidal _
; _∘_ = ∘-StrongMonoidal
; assoc = λ {_ _ _ _ F G H} → associator (M.F F) (M.F G) (M.F H)
; identityˡ = unitorˡ
; identityʳ = unitorʳ
; equiv = record
{ refl = ≃.refl
; sym = ≃.sym
; trans = ≃.trans
}
; ∘-resp-≈ = _ⓘₕ_
}
where module M = StrongMonoidalFunctor
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{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
module LogicalFramework.Disjunction where
module LF where
postulate
_∨_ : Set → Set → Set
inj₁ : {A B : Set} → A → A ∨ B
inj₂ : {A B : Set} → B → A ∨ B
case : {A B C : Set} → (A → C) → (B → C) → A ∨ B → C
∨-comm : {A B : Set} → A ∨ B → B ∨ A
∨-comm = case inj₂ inj₁
module Inductive where
open import Common.FOL.FOL
∨-comm-el : {A B : Set} → A ∨ B → B ∨ A
∨-comm-el = case inj₂ inj₁
∨-comm : {A B : Set} → A ∨ B → B ∨ A
∨-comm (inj₁ a) = inj₂ a
∨-comm (inj₂ b) = inj₁ b
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------------------------------------------------------------------------
-- A type soundness result
------------------------------------------------------------------------
{-# OPTIONS --sized-types #-}
module Lambda.Delay-monad.Type-soundness where
open import Equality.Propositional
open import Prelude
open import Prelude.Size
open import Function-universe equality-with-J
open import Maybe equality-with-J
open import Monad equality-with-J
open import Vec.Function equality-with-J
open import Delay-monad.Always
open import Delay-monad.Bisimilarity
open import Delay-monad.Monad
open import Lambda.Delay-monad.Interpreter
open import Lambda.Syntax
open Closure Tm
-- If we can prove □ ∞ (WF-MV σ) (run x), then x does not "go wrong".
does-not-go-wrong : ∀ {σ} {x : M ∞ Value} →
□ ∞ (WF-MV σ) (run x) → ¬ x ≈M fail
does-not-go-wrong (now {x = nothing} ())
does-not-go-wrong (now {x = just x} x-wf) ()
does-not-go-wrong (later x-wf) (laterˡ x↯) =
does-not-go-wrong (force x-wf) x↯
-- A "constructor" for □ i ∘ WF-MV.
_>>=-wf_ :
∀ {i σ τ} {x : M ∞ Value} {f : Value → M ∞ Value} →
□ i (WF-MV σ) (run x) →
(∀ {v} → WF-Value σ v → □ i (WF-MV τ) (run (f v))) →
□ i (WF-MV τ) (MaybeT.run (x >>= f))
x-wf >>=-wf f-wf =
□->>= x-wf λ { {nothing} ()
; {just v} v-wf → f-wf v-wf
}
-- Well-typed programs do not "go wrong".
mutual
⟦⟧-wf : ∀ {i n Γ} (t : Tm n) {σ} → Γ ⊢ t ∈ σ →
∀ {ρ} → WF-Env Γ ρ →
□ i (WF-MV σ) (run (⟦ t ⟧ ρ))
⟦⟧-wf (con i) con ρ-wf = now con
⟦⟧-wf (var x) var ρ-wf = now (ρ-wf x)
⟦⟧-wf (ƛ t) (ƛ t∈) ρ-wf = now (ƛ t∈ ρ-wf)
⟦⟧-wf (t₁ · t₂) (t₁∈ · t₂∈) {ρ} ρ-wf =
⟦⟧-wf t₁ t₁∈ ρ-wf >>=-wf λ f-wf →
⟦⟧-wf t₂ t₂∈ ρ-wf >>=-wf λ v-wf →
∙-wf f-wf v-wf
∙-wf : ∀ {i σ τ f v} →
WF-Value (σ ⇾ τ) f → WF-Value (force σ) v →
□ i (WF-MV (force τ)) (run (f ∙ v))
∙-wf (ƛ t₁∈ ρ₁-wf) v₂-wf =
later λ { .force → ⟦⟧-wf _ t₁∈ (cons-wf v₂-wf ρ₁-wf) }
type-soundness : ∀ {t : Tm 0} {σ} →
nil ⊢ t ∈ σ → ¬ ⟦ t ⟧ nil ≈M fail
type-soundness {t} {σ} =
nil ⊢ t ∈ σ ↝⟨ (λ t∈ → ⟦⟧-wf _ t∈ nil-wf) ⟩
□ ∞ (WF-MV σ) (run (⟦ t ⟧ nil)) ↝⟨ does-not-go-wrong ⟩□
¬ ⟦ t ⟧ nil ≈M fail □
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module _ where
module Test₁ where
postulate
id : {X : Set} → X → X
A : Set
x : A
record S : Set where
field
a : A
postulate
B : S → Set
record T : Set where
field
s : S
b : B s
-- Agda hangs here
t : T
t = λ
{ .T.s .S.a → x
; .T.b → id {!!}
}
module Test₂ where
postulate
id : (X : Set) → X → X
A : Set
B : A → Set
x : A
record T : Set where
field
a : A
b : B a
-- Agda hangs here
t : T
t = λ
{ .T.a → x
; .T.b → id {!!} {!!}
}
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{-# OPTIONS --cubical --safe #-}
module Data.Maybe.Sugar where
open import Prelude
open import Data.Maybe
_>>=_ : Maybe A → (A → Maybe B) → Maybe B
nothing >>= f = nothing
just x >>= f = f x
pure : A → Maybe A
pure = just
_<*>_ : Maybe (A → B) → Maybe A → Maybe B
nothing <*> xs = nothing
just f <*> nothing = nothing
just f <*> just x = just (f x)
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Automatic solvers for equations over booleans
------------------------------------------------------------------------
-- See README.Nat for examples of how to use similar solvers
{-# OPTIONS --without-K --safe #-}
module Data.Bool.Solver where
import Algebra.Solver.Ring.Simple as Solver
import Algebra.Solver.Ring.AlmostCommutativeRing as ACR
open import Data.Bool using (_≟_)
open import Data.Bool.Properties
------------------------------------------------------------------------
-- A module for automatically solving propositional equivalences
-- containing _∨_ and _∧_
module ∨-∧-Solver =
Solver (ACR.fromCommutativeSemiring ∨-∧-commutativeSemiring) _≟_
------------------------------------------------------------------------
-- A module for automatically solving propositional equivalences
-- containing _xor_ and _∧_
module xor-∧-Solver =
Solver (ACR.fromCommutativeRing xor-∧-commutativeRing) _≟_
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module WithInParModule (A : Set) where
data Nat : Set where
zero : Nat
suc : Nat -> Nat
data Bool : Set where
true : Bool
false : Bool
isZero : Nat -> Bool
isZero zero = true
isZero (suc _) = false
f : Nat -> Nat
f n with isZero n
f n | true = zero
f n | false = suc zero
g : Nat -> Nat
g zero = zero
g (suc n) with g n
g (suc n) | zero = n
g (suc n) | suc _ = n
data T : Set where
tt : T
module A (x : T) where
h : T
h with x
h | y = y
postulate
C : T -> Set
test : C (A.h tt) -> C tt
test x = x
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module Issue348 where
import Common.Irrelevance
data _==_ {A : Set1}(a : A) : A -> Set where
refl : a == a
record R : Set1 where
constructor mkR
field
.fromR : Set
reflR : (r : R) -> r == r
reflR r = refl {a = _}
-- issue: unsolved metavars resolved 2010-10-15 by making eta-expansion
-- more lazy (do not eta expand all meta variable listeners, see MetaVars.hs | {
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module Cats.End where
open import Level using (_⊔_)
open import Cats.Category
open import Cats.Category.Wedges using (Wedge ; Wedges)
open import Cats.Profunctor
module _ {lo la l≈ lo′ la′ l≈′}
{C : Category lo la l≈} {D : Category lo′ la′ l≈′}
where
IsEnd : {F : Profunctor C C D} → Wedge F → Set (lo ⊔ la ⊔ lo′ ⊔ la′ ⊔ l≈′)
IsEnd {F} = Wdg.IsTerminal
where
module Wdg = Category (Wedges F)
record End (F : Profunctor C C D) : Set (lo ⊔ la ⊔ lo′ ⊔ la′ ⊔ l≈′) where
field
wedge : Wedge F
isEnd : IsEnd wedge
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------------------------------------------------------------------------
-- The Agda standard library
--
-- An example of how Algebra.IdempotentCommutativeMonoidSolver can be
-- used
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Algebra.Solver.IdempotentCommutativeMonoid.Example where
open import Relation.Binary.PropositionalEquality using (_≡_)
open import Data.Bool.Base using (_∨_)
open import Data.Bool.Properties using (∨-idempotentCommutativeMonoid)
open import Data.Fin using (zero; suc)
open import Data.Vec using ([]; _∷_)
open import Algebra.Solver.IdempotentCommutativeMonoid
∨-idempotentCommutativeMonoid
test : ∀ x y z → (x ∨ y) ∨ (x ∨ z) ≡ (z ∨ y) ∨ x
test a b c = let _∨_ = _⊕_ in
prove 3 ((x ∨ y) ∨ (x ∨ z)) ((z ∨ y) ∨ x) (a ∷ b ∷ c ∷ [])
where
x = var zero
y = var (suc zero)
z = var (suc (suc zero))
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{-# OPTIONS --allow-unsolved-metas #-}
open import Agda.Builtin.Equality
open import Agda.Builtin.List
postulate
A : Set
nilA : A
consA : A → List A → A
w/e : {x y : A} → x ≡ y
data D : List A → Set where
nil : D []
cons : (x : A) (xs : List A) → D (x ∷ xs)
foo : ∀ {xs} (d : D xs)
(let f : D xs → A
f = λ where nil → nilA
(cons y ys) → consA y ys) →
f d ≡ nilA
foo nil = {!!}
foo (cons _ _) = w/e
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{-# OPTIONS --without-K --exact-split --safe #-}
module mwe where
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_) public
--------------------------------------------------------------------------------
data ℕ : Set lzero where
zero-ℕ : ℕ
succ-ℕ : ℕ → ℕ
add-ℕ : ℕ → ℕ → ℕ
add-ℕ x zero-ℕ = x
add-ℕ x (succ-ℕ y) = succ-ℕ (add-ℕ x y)
{- The following function returns the triangular numbers. In other words,
test-with n = (1/2) * n * (n+1) -}
test-with-1 : ℕ → ℕ
test-with-1 x with x
test-with-1 zero-ℕ | y = y
test-with-1 (succ-ℕ x) | y = add-ℕ y (test-with-1 x)
{- Now we construct a function with manual with-abstraction, of which the
definition *looks* the same. But actually we will have
test-with-2 n = n * (n+1). -}
cases-test-with-2 : ℕ → ℕ → ℕ
cases-test-with-2 zero-ℕ y = y
cases-test-with-2 (succ-ℕ x) y = add-ℕ y (cases-test-with-2 x y)
test-with-2 : ℕ → ℕ
test-with-2 x = cases-test-with-2 x x
{- The following function test-with-3 is again the same as test-with-1. -}
cases-test-with-3 : ℕ → ℕ → ℕ
cases-test-with-3 zero-ℕ y = y
cases-test-with-3 (succ-ℕ x) zero-ℕ = zero-ℕ
cases-test-with-3 (succ-ℕ x) (succ-ℕ y) =
add-ℕ (succ-ℕ y) (cases-test-with-3 x y)
test-with-3 : ℕ → ℕ
test-with-3 x = cases-test-with-3 x x
-- test-with-1 (succ-ℕ (succ-ℕ (succ-ℕ (succ-ℕ zero-ℕ))))
-- test-with-2 (succ-ℕ (succ-ℕ (succ-ℕ (succ-ℕ zero-ℕ))))
-- test-with-3 (succ-ℕ (succ-ℕ (succ-ℕ (succ-ℕ zero-ℕ))))
-- We do some basic setup, introducing only what we need. All else is removed.
id : {i : Level} {A : Set i} → A → A
id a = a
data empty : Set lzero where
ex-falso : {i : Level} {A : Set i} → empty → A
ex-falso ()
¬ : {i : Level} → Set i → Set i
¬ A = A → empty
data unit : Set lzero where
star : unit
data coprod {i j : Level} (A : Set i) (B : Set j) : Set (i ⊔ j) where
inl : A → coprod A B
inr : B → coprod A B
is-decidable : {l : Level} (A : Set l) → Set l
is-decidable A = coprod A (¬ A)
data Id {i : Level} {A : Set i} (x : A) : A → Set i where
refl : Id x x
_∙_ :
{i : Level} {A : Set i} {x y z : A} → Id x y → Id y z → Id x z
refl ∙ q = q
inv :
{i : Level} {A : Set i} {x y : A} → Id x y → Id y x
inv refl = refl
ap :
{i j : Level} {A : Set i} {B : Set j} (f : A → B) {x y : A} (p : Id x y) →
Id (f x) (f y)
ap f refl = refl
--------------------------------------------------------------------------------
-- We introduce the finite types
Fin : ℕ → Set lzero
Fin zero-ℕ = empty
Fin (succ-ℕ k) = coprod (Fin k) unit
Eq-Fin : (k : ℕ) → Fin k → Fin k → Set lzero
Eq-Fin (succ-ℕ k) (inl x) (inl y) = Eq-Fin k x y
Eq-Fin (succ-ℕ k) (inl x) (inr y) = empty
Eq-Fin (succ-ℕ k) (inr x) (inl y) = empty
Eq-Fin (succ-ℕ k) (inr x) (inr y) = unit
eq-Eq-Fin :
{k : ℕ} {x y : Fin k} → Eq-Fin k x y → Id x y
eq-Eq-Fin {succ-ℕ k} {inl x} {inl y} e = ap inl (eq-Eq-Fin e)
eq-Eq-Fin {succ-ℕ k} {inr star} {inr star} star = refl
is-decidable-Eq-Fin :
(k : ℕ) (x y : Fin k) → is-decidable (Eq-Fin k x y)
is-decidable-Eq-Fin (succ-ℕ k) (inl x) (inl y) = is-decidable-Eq-Fin k x y
is-decidable-Eq-Fin (succ-ℕ k) (inl x) (inr y) = inr id
is-decidable-Eq-Fin (succ-ℕ k) (inr x) (inl y) = inr id
is-decidable-Eq-Fin (succ-ℕ k) (inr x) (inr y) = inl star
--------------------------------------------------------------------------------
-- We now study the successor and predecessor functions on Fin k
zero-Fin : {k : ℕ} → Fin (succ-ℕ k)
zero-Fin {zero-ℕ} = inr star
zero-Fin {succ-ℕ k} = inl zero-Fin
neg-one-Fin : {k : ℕ} → Fin (succ-ℕ k)
neg-one-Fin {k} = inr star
succ-Fin : {k : ℕ} → Fin k → Fin k
succ-Fin {succ-ℕ zero-ℕ} x = x
succ-Fin {succ-ℕ (succ-ℕ k)} (inl (inl x)) = inl (succ-Fin (inl x))
succ-Fin {succ-ℕ (succ-ℕ k)} (inl (inr x)) = neg-one-Fin
succ-Fin {succ-ℕ (succ-ℕ k)} (inr x) = zero-Fin
succ-neg-one-Fin : {k : ℕ} → Id (succ-Fin (neg-one-Fin {k})) zero-Fin
succ-neg-one-Fin {zero-ℕ} = refl
succ-neg-one-Fin {succ-ℕ k} = refl
pred-Fin' : {k : ℕ} → Fin k → Fin k
pred-Fin' {succ-ℕ k} x with (is-decidable-Eq-Fin (succ-ℕ k) x zero-Fin)
pred-Fin' {succ-ℕ zero-ℕ} (inr star) | d = zero-Fin
pred-Fin' {succ-ℕ (succ-ℕ k)} (inl x) | inl e = neg-one-Fin
pred-Fin' {succ-ℕ (succ-ℕ k)} (inl x) | inr f = inl (pred-Fin' {succ-ℕ k} x)
pred-Fin' {succ-ℕ (succ-ℕ k)} (inr x) | inr f = inl neg-one-Fin
succ-pred-Fin' :
{k : ℕ} (x : Fin k) → Id (succ-Fin (pred-Fin' x)) x
succ-pred-Fin' {succ-ℕ k} x with is-decidable-Eq-Fin (succ-ℕ k) x zero-Fin
succ-pred-Fin' {succ-ℕ zero-ℕ} (inr star) | d = refl
succ-pred-Fin' {succ-ℕ (succ-ℕ k)} (inl x) | inl e =
succ-neg-one-Fin ∙ inv (eq-Eq-Fin e)
succ-pred-Fin' {succ-ℕ (succ-ℕ zero-ℕ)} (inl (inr star)) | inr f =
ex-falso (f star)
succ-pred-Fin' {succ-ℕ (succ-ℕ (succ-ℕ k))} (inl (inl x)) | inr f =
{!!} ∙ ap inl (succ-pred-Fin' {succ-ℕ (succ-ℕ k)} (inl x))
succ-pred-Fin' {succ-ℕ (succ-ℕ (succ-ℕ k))} (inl (inr star)) | inr f = refl
succ-pred-Fin' {succ-ℕ (succ-ℕ k)} (inr star) | inr f = refl
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-- Andreas, 2012-09-13
-- (The signature on the previous line does not apply to all of the
-- text in this file.)
module RelevanceSubtyping where
-- this naturally type-checks:
one : {A B : Set} → (.A → B) → A → B
one f x = f x
-- Subtyping is no longer supported for irrelevance, so the following
-- code is no longer accepted.
-- -- this type-checks because of subtyping
-- one' : {A B : Set} → (.A → B) → A → B
-- one' f = f
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{-# OPTIONS --without-K --safe #-}
module Definition.Typed.EqualityRelation where
open import Definition.Untyped
open import Definition.Typed
open import Definition.Typed.Weakening using (_∷_⊆_)
-- Generic equality relation used with the logical relation
record EqRelSet : Set₁ where
constructor eqRel
field
---------------
-- Relations --
---------------
-- Equality of types
_⊢_≅_ : Con Term → (A B : Term) → Set
-- Equality of terms
_⊢_≅_∷_ : Con Term → (t u A : Term) → Set
-- Equality of neutral terms
_⊢_~_∷_ : Con Term → (t u A : Term) → Set
----------------
-- Properties --
----------------
-- Generic equality compatibility
~-to-≅ₜ : ∀ {k l A Γ}
→ Γ ⊢ k ~ l ∷ A
→ Γ ⊢ k ≅ l ∷ A
-- Judgmental conversion compatibility
≅-eq : ∀ {A B Γ}
→ Γ ⊢ A ≅ B
→ Γ ⊢ A ≡ B
≅ₜ-eq : ∀ {t u A Γ}
→ Γ ⊢ t ≅ u ∷ A
→ Γ ⊢ t ≡ u ∷ A
-- Universe
≅-univ : ∀ {A B Γ}
→ Γ ⊢ A ≅ B ∷ U
→ Γ ⊢ A ≅ B
-- Symmetry
≅-sym : ∀ {A B Γ} → Γ ⊢ A ≅ B → Γ ⊢ B ≅ A
≅ₜ-sym : ∀ {t u A Γ} → Γ ⊢ t ≅ u ∷ A → Γ ⊢ u ≅ t ∷ A
~-sym : ∀ {k l A Γ} → Γ ⊢ k ~ l ∷ A → Γ ⊢ l ~ k ∷ A
-- Transitivity
≅-trans : ∀ {A B C Γ} → Γ ⊢ A ≅ B → Γ ⊢ B ≅ C → Γ ⊢ A ≅ C
≅ₜ-trans : ∀ {t u v A Γ} → Γ ⊢ t ≅ u ∷ A → Γ ⊢ u ≅ v ∷ A → Γ ⊢ t ≅ v ∷ A
~-trans : ∀ {k l m A Γ} → Γ ⊢ k ~ l ∷ A → Γ ⊢ l ~ m ∷ A → Γ ⊢ k ~ m ∷ A
-- Conversion
≅-conv : ∀ {t u A B Γ} → Γ ⊢ t ≅ u ∷ A → Γ ⊢ A ≡ B → Γ ⊢ t ≅ u ∷ B
~-conv : ∀ {k l A B Γ} → Γ ⊢ k ~ l ∷ A → Γ ⊢ A ≡ B → Γ ⊢ k ~ l ∷ B
-- Weakening
≅-wk : ∀ {A B ρ Γ Δ}
→ ρ ∷ Δ ⊆ Γ
→ ⊢ Δ
→ Γ ⊢ A ≅ B
→ Δ ⊢ wk ρ A ≅ wk ρ B
≅ₜ-wk : ∀ {t u A ρ Γ Δ}
→ ρ ∷ Δ ⊆ Γ
→ ⊢ Δ
→ Γ ⊢ t ≅ u ∷ A
→ Δ ⊢ wk ρ t ≅ wk ρ u ∷ wk ρ A
~-wk : ∀ {k l A ρ Γ Δ}
→ ρ ∷ Δ ⊆ Γ
→ ⊢ Δ
→ Γ ⊢ k ~ l ∷ A
→ Δ ⊢ wk ρ k ~ wk ρ l ∷ wk ρ A
-- Weak head expansion
≅-red : ∀ {A A′ B B′ Γ}
→ Γ ⊢ A ⇒* A′
→ Γ ⊢ B ⇒* B′
→ Whnf A′
→ Whnf B′
→ Γ ⊢ A′ ≅ B′
→ Γ ⊢ A ≅ B
≅ₜ-red : ∀ {a a′ b b′ A B Γ}
→ Γ ⊢ A ⇒* B
→ Γ ⊢ a ⇒* a′ ∷ B
→ Γ ⊢ b ⇒* b′ ∷ B
→ Whnf B
→ Whnf a′
→ Whnf b′
→ Γ ⊢ a′ ≅ b′ ∷ B
→ Γ ⊢ a ≅ b ∷ A
-- Universe type reflexivity
≅-Urefl : ∀ {Γ} → ⊢ Γ → Γ ⊢ U ≅ U
-- Natural number type reflexivity
≅-ℕrefl : ∀ {Γ} → ⊢ Γ → Γ ⊢ ℕ ≅ ℕ
≅ₜ-ℕrefl : ∀ {Γ} → ⊢ Γ → Γ ⊢ ℕ ≅ ℕ ∷ U
-- Π-congurence
≅-Π-cong : ∀ {F G H E Γ}
→ Γ ⊢ F
→ Γ ⊢ F ≅ H
→ Γ ∙ F ⊢ G ≅ E
→ Γ ⊢ Π F ▹ G ≅ Π H ▹ E
≅ₜ-Π-cong : ∀ {F G H E Γ}
→ Γ ⊢ F
→ Γ ⊢ F ≅ H ∷ U
→ Γ ∙ F ⊢ G ≅ E ∷ U
→ Γ ⊢ Π F ▹ G ≅ Π H ▹ E ∷ U
-- Zero reflexivity
≅ₜ-zerorefl : ∀ {Γ} → ⊢ Γ → Γ ⊢ zero ≅ zero ∷ ℕ
-- Successor congurence
≅-suc-cong : ∀ {m n Γ} → Γ ⊢ m ≅ n ∷ ℕ → Γ ⊢ suc m ≅ suc n ∷ ℕ
-- η-equality
≅-η-eq : ∀ {f g F G Γ}
→ Γ ⊢ F
→ Γ ⊢ f ∷ Π F ▹ G
→ Γ ⊢ g ∷ Π F ▹ G
→ Function f
→ Function g
→ Γ ∙ F ⊢ wk1 f ∘ var 0 ≅ wk1 g ∘ var 0 ∷ G
→ Γ ⊢ f ≅ g ∷ Π F ▹ G
-- Variable reflexivity
~-var : ∀ {x A Γ} → Γ ⊢ var x ∷ A → Γ ⊢ var x ~ var x ∷ A
-- Application congurence
~-app : ∀ {a b f g F G Γ}
→ Γ ⊢ f ~ g ∷ Π F ▹ G
→ Γ ⊢ a ≅ b ∷ F
→ Γ ⊢ f ∘ a ~ g ∘ b ∷ G [ a ]
-- Natural recursion congurence
~-natrec : ∀ {z z′ s s′ n n′ F F′ Γ}
→ Γ ∙ ℕ ⊢ F ≅ F′
→ Γ ⊢ z ≅ z′ ∷ F [ zero ]
→ Γ ⊢ s ≅ s′ ∷ Π ℕ ▹ (F ▹▹ F [ suc (var 0) ]↑)
→ Γ ⊢ n ~ n′ ∷ ℕ
→ Γ ⊢ natrec F z s n ~ natrec F′ z′ s′ n′ ∷ F [ n ]
-- Composition of universe and generic equality compatibility
~-to-≅ : ∀ {k l Γ} → Γ ⊢ k ~ l ∷ U → Γ ⊢ k ≅ l
~-to-≅ k~l = ≅-univ (~-to-≅ₜ k~l)
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module Agda.Builtin.FromNeg where
open import Agda.Primitive
open import Agda.Builtin.Nat
record Negative {a} (A : Set a) : Set (lsuc a) where
field
Constraint : Nat → Set a
fromNeg : ∀ n → {{_ : Constraint n}} → A
open Negative {{...}} public using (fromNeg)
{-# BUILTIN FROMNEG fromNeg #-}
{-# DISPLAY Negative.fromNeg _ n = fromNeg n #-}
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