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{-# OPTIONS --without-K #-} module ConcretePermutation where import Level using (zero) open import Data.Nat using (ℕ; _+_; _*_) open import Data.Fin using (Fin) open import Data.Product using (proj₁; proj₂) open import Data.Vec using (tabulate) open import Algebra using (CommutativeSemiring) open import Algebra.Structures using (IsSemigroup; IsCommutativeMonoid; IsCommutativeSemiring) open import Relation.Binary using (IsEquivalence) open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; cong; cong₂; module ≡-Reasoning) open import Equiv using (_≃_; sym≃;p∘!p≡id) -- open import FinVec using (FinVec; 1C; _∘̂_; _⊎c_; _×c_; unite+; uniti+; unite+r; uniti+r; assocl+; assocr+; swap+cauchy; unite*; uniti*; unite*r; uniti*r; assocl*; assocr*; swap⋆cauchy; dist*+; factor*+; distl*+; factorl*+; right-zero*l; right-zero*r) open import FinVecProperties using (∘̂-assoc; ∘̂-lid; ∘̂-rid; ~⇒≡; 1C⊎1C≡1C; 1C×1C≡1C; 1C₀⊎x≡x; ⊎c-distrib; ×c-distrib; unite+∘̂uniti+~id; uniti+∘̂unite+~id; unite+r∘̂uniti+r~id; uniti+r∘̂unite+r~id; assocl+∘̂assocr+~id; assocr+∘̂assocl+~id; swap+-inv; unite*∘̂uniti*~id; uniti*∘̂unite*~id; unite*r∘̂uniti*r~id; uniti*r∘̂unite*r~id; assocl*∘̂assocr*~id; assocr*∘̂assocl*~id; swap*-inv; dist*+∘̂factor*+~id; factor*+∘̂dist*+~id; distl*+∘̂factorl*+~id; factorl*+∘̂distl*+~id; right-zero*l∘̂right-zero*r~id; right-zero*r∘̂right-zero*l~id) ------------------------------------------------------------------------------ -- A concrete permutation has 4 components: -- - the one-line notation for a permutation -- - the one-line notation for the inverse permutation -- - and 2 proofs that these are indeed inverses record CPerm (values : ℕ) (size : ℕ) : Set where constructor cp field π : FinVec values size πᵒ : FinVec size values αp : π ∘̂ πᵒ ≡ 1C βp : πᵒ ∘̂ π ≡ 1C ------------------------------------------------------------------------------ -- Now the goal is to prove that CPerm m n is a commutative semiring -- (including symmetry now) -- First it is an equivalence relation idp : ∀ {n} → CPerm n n idp {n} = cp 1C 1C (∘̂-rid _) (∘̂-lid _) symp : ∀ {m n} → CPerm m n → CPerm n m symp (cp p₁ p₂ α β) = cp p₂ p₁ β α transp : ∀ {m₁ m₂ m₃} → CPerm m₂ m₁ → CPerm m₃ m₂ → CPerm m₃ m₁ transp {n} (cp π πᵒ αp βp) (cp π₁ πᵒ₁ αp₁ βp₁) = cp (π ∘̂ π₁) (πᵒ₁ ∘̂ πᵒ) pf₁ pf₂ where open ≡-Reasoning pf₁ : (π ∘̂ π₁) ∘̂ (πᵒ₁ ∘̂ πᵒ) ≡ 1C pf₁ = begin ( (π ∘̂ π₁) ∘̂ (πᵒ₁ ∘̂ πᵒ) ≡⟨ ∘̂-assoc _ _ _ ⟩ ((π ∘̂ π₁) ∘̂ πᵒ₁) ∘̂ πᵒ ≡⟨ cong (λ x → x ∘̂ πᵒ) (sym (∘̂-assoc _ _ _)) ⟩ (π ∘̂ (π₁ ∘̂ πᵒ₁)) ∘̂ πᵒ ≡⟨ cong (λ x → (π ∘̂ x) ∘̂ πᵒ) (αp₁) ⟩ (π ∘̂ 1C) ∘̂ πᵒ ≡⟨ cong (λ x → x ∘̂ πᵒ) (∘̂-rid _) ⟩ π ∘̂ πᵒ ≡⟨ αp ⟩ 1C ∎) pf₂ : (πᵒ₁ ∘̂ πᵒ) ∘̂ (π ∘̂ π₁) ≡ 1C pf₂ = begin ( (πᵒ₁ ∘̂ πᵒ) ∘̂ (π ∘̂ π₁) ≡⟨ ∘̂-assoc _ _ _ ⟩ ((πᵒ₁ ∘̂ πᵒ) ∘̂ π) ∘̂ π₁ ≡⟨ cong (λ x → x ∘̂ π₁) (sym (∘̂-assoc _ _ _)) ⟩ (πᵒ₁ ∘̂ (πᵒ ∘̂ π)) ∘̂ π₁ ≡⟨ cong (λ x → (πᵒ₁ ∘̂ x) ∘̂ π₁) βp ⟩ (πᵒ₁ ∘̂ 1C) ∘̂ π₁ ≡⟨ cong (λ x → x ∘̂ π₁) (∘̂-rid _) ⟩ πᵒ₁ ∘̂ π₁ ≡⟨ βp₁ ⟩ 1C ∎) -- units -- zero permutation 0p : CPerm 0 0 0p = idp {0} -- Additive monoid _⊎p_ : ∀ {m₁ m₂ n₁ n₂} → CPerm m₁ m₂ → CPerm n₁ n₂ → CPerm (m₁ + n₁) (m₂ + n₂) _⊎p_ {m₁} {m₂} {n₁} {n₂} π₀ π₁ = cp ((π π₀) ⊎c (π π₁)) ((πᵒ π₀) ⊎c (πᵒ π₁)) pf₁ pf₂ where open CPerm open ≡-Reasoning pf₁ : (π π₀ ⊎c π π₁) ∘̂ (πᵒ π₀ ⊎c πᵒ π₁) ≡ 1C pf₁ = begin ( (π π₀ ⊎c π π₁) ∘̂ (πᵒ π₀ ⊎c πᵒ π₁) ≡⟨ ⊎c-distrib {p₁ = π π₀} ⟩ (π π₀ ∘̂ πᵒ π₀) ⊎c (π π₁ ∘̂ πᵒ π₁) ≡⟨ cong₂ _⊎c_ (αp π₀) (αp π₁) ⟩ 1C {m₂} ⊎c 1C {n₂} ≡⟨ 1C⊎1C≡1C {m₂} ⟩ 1C ∎) pf₂ : (πᵒ π₀ ⊎c πᵒ π₁) ∘̂ (π π₀ ⊎c π π₁) ≡ 1C pf₂ = begin ( (πᵒ π₀ ⊎c πᵒ π₁) ∘̂ (π π₀ ⊎c π π₁) ≡⟨ ⊎c-distrib {p₁ = πᵒ π₀} ⟩ (πᵒ π₀ ∘̂ π π₀) ⊎c (πᵒ π₁ ∘̂ π π₁) ≡⟨ cong₂ _⊎c_ (βp π₀) (βp π₁) ⟩ 1C {m₁} ⊎c 1C {n₁} ≡⟨ 1C⊎1C≡1C {m₁} ⟩ 1C ∎ ) -- For the rest of the permutations, it is convenient to lift things from -- FinVec in one go mkPerm : {m n : ℕ} → (Fin m ≃ Fin n) → CPerm m n mkPerm {m} {n} eq = cp p q p∘̂q≡1 q∘̂p≡1 where f = proj₁ eq g = proj₁ (sym≃ eq) p = tabulate g -- note the flip! q = tabulate f q∘̂p≡1 = ~⇒≡ {f = g} {g = f} (p∘!p≡id {p = eq}) p∘̂q≡1 = ~⇒≡ {f = f} {g = g} (p∘!p≡id {p = sym≃ eq}) -- units unite+p : {m : ℕ} → CPerm m (0 + m) unite+p {m} = cp (unite+ {m}) (uniti+ {m}) (unite+∘̂uniti+~id {m}) (uniti+∘̂unite+~id {m}) uniti+p : {m : ℕ} → CPerm (0 + m) m uniti+p {m} = symp (unite+p {m}) unite+rp : {m : ℕ} → CPerm m (m + 0) unite+rp {m} = cp (unite+r {m}) (uniti+r) (unite+r∘̂uniti+r~id) (uniti+r∘̂unite+r~id) uniti+rp : {m : ℕ} → CPerm (m + 0) m uniti+rp {m} = symp (unite+rp {m}) -- commutativity swap+p : {m n : ℕ} → CPerm (n + m) (m + n) swap+p {m} {n} = cp (swap+cauchy m n) (swap+cauchy n m) (swap+-inv {m}) (swap+-inv {n}) -- associativity assocl+p : {m n o : ℕ} → CPerm ((m + n) + o) (m + (n + o)) assocl+p {m} = cp (assocl+ {m}) (assocr+ {m}) (assocl+∘̂assocr+~id {m}) (assocr+∘̂assocl+~id {m}) assocr+p : {m n o : ℕ} → CPerm (m + (n + o)) ((m + n) + o) assocr+p {m} = symp (assocl+p {m}) -- Multiplicative monoid _×p_ : ∀ {m₁ m₂ n₁ n₂} → CPerm m₁ m₂ → CPerm n₁ n₂ → CPerm (m₁ * n₁) (m₂ * n₂) _×p_ {m₁} {m₂} {n₁} {n₂} π₀ π₁ = cp ((π π₀) ×c (π π₁)) ((πᵒ π₀) ×c (πᵒ π₁)) pf₁ pf₂ where open CPerm open ≡-Reasoning pf₁ : (π π₀ ×c π π₁) ∘̂ (πᵒ π₀ ×c πᵒ π₁) ≡ 1C pf₁ = begin ( (π π₀ ×c π π₁) ∘̂ (πᵒ π₀ ×c πᵒ π₁) ≡⟨ ×c-distrib {p₁ = π π₀} ⟩ (π π₀ ∘̂ πᵒ π₀) ×c (π π₁ ∘̂ πᵒ π₁) ≡⟨ cong₂ _×c_ (αp π₀) (αp π₁) ⟩ 1C ×c 1C ≡⟨ 1C×1C≡1C ⟩ 1C ∎) pf₂ : (πᵒ π₀ ×c πᵒ π₁) ∘̂ (π π₀ ×c π π₁) ≡ 1C pf₂ = begin ( (πᵒ π₀ ×c πᵒ π₁) ∘̂ (π π₀ ×c π π₁) ≡⟨ ×c-distrib {p₁ = πᵒ π₀} ⟩ (πᵒ π₀ ∘̂ π π₀) ×c (πᵒ π₁ ∘̂ π π₁) ≡⟨ cong₂ _×c_ (βp π₀) (βp π₁) ⟩ 1C ×c 1C ≡⟨ 1C×1C≡1C ⟩ 1C ∎) -- units unite*p : {m : ℕ} → CPerm m (1 * m) unite*p {m} = cp (unite* {m}) (uniti* {m}) (unite*∘̂uniti*~id {m}) (uniti*∘̂unite*~id {m}) uniti*p : {m : ℕ} → CPerm (1 * m) m uniti*p {m} = symp (unite*p {m}) unite*rp : {m : ℕ} → CPerm m (m * 1) unite*rp {m} = cp (unite*r {m}) (uniti*r {m}) (unite*r∘̂uniti*r~id {m}) (uniti*r∘̂unite*r~id {m}) uniti*rp : {m : ℕ} → CPerm (m * 1) m uniti*rp {m} = symp (unite*rp {m}) -- commutativity swap*p : {m n : ℕ} → CPerm (n * m) (m * n) swap*p {m} {n} = cp (swap⋆cauchy m n) (swap⋆cauchy n m) (swap*-inv {m}) (swap*-inv {n}) -- associativity assocl*p : {m n o : ℕ} → CPerm ((m * n) * o) (m * (n * o)) assocl*p {m} = cp (assocl* {m}) (assocr* {m}) (assocl*∘̂assocr*~id {m}) (assocr*∘̂assocl*~id {m}) assocr*p : {m n o : ℕ} → CPerm (m * (n * o)) ((m * n) * o) assocr*p {m} = symp (assocl*p {m}) -- Distributivity -- right-zero absorbing permutation 0pr : ∀ {n} → CPerm 0 (n * 0) 0pr {n} = cp (right-zero*l {n}) (right-zero*r {n}) (right-zero*l∘̂right-zero*r~id {n}) (right-zero*r∘̂right-zero*l~id {n}) -- and its symmetric version 0pl : ∀ {n} → CPerm (n * 0) 0 0pl {n} = symp (0pr {n}) distp : {m n o : ℕ} → CPerm (m * o + n * o) ((m + n) * o) distp {m} {n} {o} = cp (dist*+ {m}) (factor*+ {m}) (dist*+∘̂factor*+~id {m}) (factor*+∘̂dist*+~id {m}) factorp : {m n o : ℕ} → CPerm ((m + n) * o) (m * o + n * o) factorp {m} = symp (distp {m}) distlp : {m n o : ℕ} → CPerm (m * n + m * o) (m * (n + o)) distlp {m} {n} {o} = cp (distl*+ {m}) (factorl*+ {m}) (distl*+∘̂factorl*+~id {m}) (factorl*+∘̂distl*+~id {m}) factorlp : {m n o : ℕ} → CPerm (m * (n + o)) (m * n + m * o) factorlp {m} = symp (distlp {m}) ------------------------------------------------------------------------------ -- Commutative semiring structure cpermIsEquiv : IsEquivalence {Level.zero} {Level.zero} {ℕ} CPerm cpermIsEquiv = record { refl = idp; sym = symp; trans = λ p q → transp q p } cpermPlusIsSG : IsSemigroup {Level.zero} {Level.zero} {ℕ} CPerm _+_ cpermPlusIsSG = record { isEquivalence = cpermIsEquiv ; assoc = λ m n o → assocl+p {m} {n} {o} ; ∙-cong = _⊎p_ } cpermTimesIsSG : IsSemigroup {Level.zero} {Level.zero} {ℕ} CPerm _*_ cpermTimesIsSG = record { isEquivalence = cpermIsEquiv ; assoc = λ m n o → assocl*p {m} {n} {o} ; ∙-cong = _×p_ } cpermPlusIsCM : IsCommutativeMonoid CPerm _+_ 0 cpermPlusIsCM = record { isSemigroup = cpermPlusIsSG ; identityˡ = λ m → idp ; comm = λ m n → swap+p {n} {m} } cpermTimesIsCM : IsCommutativeMonoid CPerm _*_ 1 cpermTimesIsCM = record { isSemigroup = cpermTimesIsSG ; identityˡ = λ m → uniti*p {m} ; comm = λ m n → swap*p {n} {m} } cpermIsCSR : IsCommutativeSemiring CPerm _+_ _*_ 0 1 cpermIsCSR = record { +-isCommutativeMonoid = cpermPlusIsCM ; *-isCommutativeMonoid = cpermTimesIsCM ; distribʳ = λ o m n → factorp {m} {n} {o} ; zeroˡ = λ m → 0p } cpermCSR : CommutativeSemiring Level.zero Level.zero cpermCSR = record { Carrier = ℕ ; _≈_ = CPerm; _+_ = _+_ ; _*_ = _*_ ; 0# = 0 ; 1# = 1 ; isCommutativeSemiring = cpermIsCSR } ------------------------------------------------------------------------------

{-# OPTIONS --omega-in-omega --no-termination-check --overlapping-instances #-} open import Light.Library.Data.Natural as ℕ using (ℕ) open import Light.Package using (Package) module Light.Literals.Natural ⦃ package : Package record { ℕ } ⦄ where open import Light.Literals.Definition.Natural using (FromNatural) open FromNatural using (convert) instance from‐natural : FromNatural ℕ convert from‐natural n = n

------------------------------------------------------------------------ -- The Agda standard library -- -- Decorated star-lists ------------------------------------------------------------------------ {-# OPTIONS --with-K --safe #-} module Data.Star.Decoration where open import Data.Unit open import Function open import Level open import Relation.Binary open import Relation.Binary.Construct.Closure.ReflexiveTransitive -- A predicate on relation "edges" (think of the relation as a graph). EdgePred : {ℓ r : Level} (p : Level) {I : Set ℓ} → Rel I r → Set (suc p ⊔ ℓ ⊔ r) EdgePred p T = ∀ {i j} → T i j → Set p data NonEmptyEdgePred {ℓ r p : Level} {I : Set ℓ} (T : Rel I r) (P : EdgePred p T) : Set (ℓ ⊔ r ⊔ p) where nonEmptyEdgePred : ∀ {i j} {x : T i j} (p : P x) → NonEmptyEdgePred T P -- Decorating an edge with more information. data DecoratedWith {ℓ r p : Level} {I : Set ℓ} {T : Rel I r} (P : EdgePred p T) : Rel (NonEmpty (Star T)) p where ↦ : ∀ {i j k} {x : T i j} {xs : Star T j k} (p : P x) → DecoratedWith P (nonEmpty (x ◅ xs)) (nonEmpty xs) module _ {ℓ r p : Level} {I : Set ℓ} {T : Rel I r} {P : EdgePred p T} where edge : ∀ {i j} → DecoratedWith {T = T} P i j → NonEmpty T edge (↦ {x = x} p) = nonEmpty x decoration : ∀ {i j} → (d : DecoratedWith {T = T} P i j) → P (NonEmpty.proof (edge d)) decoration (↦ p) = p -- Star-lists decorated with extra information. All P xs means that -- all edges in xs satisfy P. All : ∀ {ℓ r p} {I : Set ℓ} {T : Rel I r} → EdgePred p T → EdgePred (ℓ ⊔ (r ⊔ p)) (Star T) All P {j = j} xs = Star (DecoratedWith P) (nonEmpty xs) (nonEmpty {y = j} ε) -- We can map over decorated vectors. gmapAll : ∀ {ℓ ℓ′ r p q} {I : Set ℓ} {T : Rel I r} {P : EdgePred p T} {J : Set ℓ′} {U : Rel J r} {Q : EdgePred q U} {i j} {xs : Star T i j} (f : I → J) (g : T =[ f ]⇒ U) → (∀ {i j} {x : T i j} → P x → Q (g x)) → All P xs → All {T = U} Q (gmap f g xs) gmapAll f g h ε = ε gmapAll f g h (↦ x ◅ xs) = ↦ (h x) ◅ gmapAll f g h xs -- Since we don't automatically have gmap id id xs ≡ xs it is easier -- to implement mapAll in terms of map than in terms of gmapAll. mapAll : ∀ {ℓ r p q} {I : Set ℓ} {T : Rel I r} {P : EdgePred p T} {Q : EdgePred q T} {i j} {xs : Star T i j} → (∀ {i j} {x : T i j} → P x → Q x) → All P xs → All Q xs mapAll {P = P} {Q} f ps = map F ps where F : DecoratedWith P ⇒ DecoratedWith Q F (↦ x) = ↦ (f x) -- We can decorate star-lists with universally true predicates. decorate : ∀ {ℓ r p} {I : Set ℓ} {T : Rel I r} {P : EdgePred p T} {i j} → (∀ {i j} (x : T i j) → P x) → (xs : Star T i j) → All P xs decorate f ε = ε decorate f (x ◅ xs) = ↦ (f x) ◅ decorate f xs -- We can append Alls. Unfortunately _◅◅_ does not quite work. infixr 5 _◅◅◅_ _▻▻▻_ _◅◅◅_ : ∀ {ℓ r p} {I : Set ℓ} {T : Rel I r} {P : EdgePred p T} {i j k} {xs : Star T i j} {ys : Star T j k} → All P xs → All P ys → All P (xs ◅◅ ys) ε ◅◅◅ ys = ys (↦ x ◅ xs) ◅◅◅ ys = ↦ x ◅ xs ◅◅◅ ys _▻▻▻_ : ∀ {ℓ r p} {I : Set ℓ} {T : Rel I r} {P : EdgePred p T} {i j k} {xs : Star T j k} {ys : Star T i j} → All P xs → All P ys → All P (xs ▻▻ ys) _▻▻▻_ = flip _◅◅◅_

module Category.Fibration where open import Data.Product open import Category.Category open import Category.Subcategory open import Category.Funct

-- 2010-10-04 -- termination checker no longer counts stripping off a record constructor -- as decrease module Issue334 where data Unit : Set where unit : Unit record E : Set where inductive constructor mkE field fromE : E spam : Unit f : E -> Set f (mkE e unit) = f e -- the record pattern translation does not apply to f -- still, should not termination check, because -- f (mkE e u) = f e -- also does not!

open import Relation.Binary.PropositionalEquality open import Data.Unit using (⊤ ; tt) open import Data.Product renaming (_×_ to _∧_ ; proj₁ to fst ; proj₂ to snd) open import Data.Sum renaming (_⊎_ to _∨_ ; inj₁ to left ; inj₂ to right) open import Data.Nat using (ℕ ; zero ; suc) open import Data.Product renaming (proj₁ to fst ; proj₂ to snd) open import Data.List hiding (length ; head) {- --- 1. Order on natural numbers --- -} data _≤_ : ℕ → ℕ → Set where z≤n : {n : ℕ} → zero ≤ n s≤s : {m n : ℕ} (m≤n : m ≤ n) → suc m ≤ suc n {- 1.1 Compatibility with successor -} ≤-pred : {m n : ℕ} → (suc m ≤ suc n) → m ≤ n ≤-pred (s≤s i) = i ≤-suc : {m n : ℕ} → (m ≤ n) → suc m ≤ suc n ≤-suc i = s≤s i {- 1.2 Reflexivity -} ≤-refl : (n : ℕ) → n ≤ n ≤-refl zero = z≤n ≤-refl (suc n) = s≤s (≤-refl n) {- 1.3 Transitivity -} ≤-trans : {m n p : ℕ} → (m ≤ n) → (n ≤ p) → (m ≤ p) ≤-trans z≤n i2 = z≤n ≤-trans (s≤s i1) (s≤s i2) = ≤-suc (≤-trans i1 i2) {- 1.4 Totality -} _≤?_ : (m n : ℕ) → (m ≤ n) ∨ (n ≤ m) zero ≤? n = left z≤n suc m ≤? zero = right z≤n suc m ≤? suc n with m ≤? n ... | left e = left (≤-suc e) ... | right e = right (≤-suc e) {- --- 2. Insertion sort --- -} {- 2.1 Insertion -} insert : (x : ℕ) → (l : List ℕ) → List ℕ insert x [] = x ∷ [] insert x (x₁ ∷ l) with x ≤? x₁ ... | left e = x ∷ (x₁ ∷ l) ... | right e = x₁ ∷ (insert x l) {- 2.2 Sorting -} sort : List ℕ → List ℕ sort [] = [] sort (x ∷ l) = insert x (sort l) test : List ℕ test = sort (4 ∷ 1 ∷ 45 ∷ 8 ∷ 32 ∷ 12 ∷ 1 ∷ []) {- 2.3 The bounded below predicate -} data _≤*_ : ℕ → List ℕ → Set where [] : {x : ℕ} → x ≤* [] _∷_ : {x y : ℕ} → {l : List ℕ} → (x ≤ y) → (x ≤* l) → (x ≤* (y ∷ l)) {- 2.4 The sorted predicate -} data sorted : (l : List ℕ) → Set where [] : sorted [] _∷_ : {x : ℕ} → {l : List ℕ} → (x ≤* l) → sorted l → (sorted (x ∷ l)) {- 2.5 Insert is sorting -} ≤*-trans : {x y : ℕ} → (l : List ℕ) → x ≤ y → y ≤* l → x ≤* l ≤*-trans [] x≤y y≤*l = [] ≤*-trans (x₁ ∷ l) x≤y (y≤x₁ ∷ y≤*l) = ≤-trans x≤y y≤x₁ ∷ ≤*-trans l x≤y y≤*l sorted-lemma : {x y : ℕ} → (l : List ℕ) → x ≤ y → sorted (y ∷ l) → x ≤* (y ∷ l) sorted-lemma l x≤y (y≤*l ∷ s) = x≤y ∷ (≤*-trans l x≤y y≤*l) insert-lemma : (x y : ℕ) → (l : List ℕ) → y ≤ x → y ≤* l → y ≤* insert x l insert-lemma x y [] y≤x y≤*l = y≤x ∷ [] insert-lemma x y (x₁ ∷ l) y≤x (y≤x₁ ∷ y≤*l) with x ≤? x₁ ... | right x₁≤x = y≤x₁ ∷ insert-lemma x y l y≤x y≤*l ... | left x≤x₁ = y≤x ∷ (y≤x₁ ∷ y≤*l) insert-sorting : (x : ℕ) → (l : List ℕ) → sorted l → sorted (insert x l) insert-sorting x [] s = [] ∷ [] insert-sorting x (y ∷ l) (y≤*l ∷ s) with x ≤? y ... | left x≤y = (x≤y ∷ (≤*-trans l x≤y y≤*l)) ∷ (y≤*l ∷ s) ... | right y≤x = insert-lemma x y l y≤x y≤*l ∷ (insert-sorting x l s) {- 2.6 Sort is sorting -} sort-sorting : (l : List ℕ) → sorted (sort l) sort-sorting [] = [] sort-sorting (x ∷ l) = insert-sorting x (sort l) (sort-sorting l) {- 2.7 The problem of specification -} {- Empty lists are always sorted, so always return an empty list -} f : List ℕ → List ℕ f l = [] f-sorting : (l : List ℕ) → sorted (f l) f-sorting _ = [] {- 2.8 Intrinsic approach -} mutual data Sorted : Set where nil : Sorted cons : (x : ℕ) → (l : Sorted) → x ≤ (head x l) → Sorted head : ℕ → Sorted → ℕ head d nil = d head d (cons x l x₁) = x mutual insert' : (x : ℕ) → Sorted → Sorted insert' x nil = nil insert' x (cons x₁ l x₁≤hd) with x ≤? x₁ ... | left x≤x₁ = cons x (cons x₁ l x₁≤hd) x≤x₁ ... | right x₁≤x = cons x₁ (insert' x l) (ins-lemma x x₁ l x₁≤x x₁≤hd) {- In other words, y ≤ x and y ≤* l gives us y ≤* (insert x l) -} ins-lemma : (x y : ℕ) → (l : Sorted) → y ≤ x → y ≤ head y l → y ≤ head y (insert' x l) ins-lemma x y nil y≤x y≤hd = ≤-refl y ins-lemma x y (cons x₁ l x₁≤hd) y≤x y≤hd with x ≤? x₁ ... | left x≤x₁ = y≤x ... | right x₁≤x = y≤hd sort' : (l : List ℕ) → Sorted sort' [] = nil sort' (x ∷ l) = insert' x (sort' l) {- --- 4. Merge sort --- -} {- 4.1 Splitting -} split : {A : Set} → List A → List A × List A split [] = [] , [] split (x ∷ []) = (x ∷ []) , [] split (x ∷ y ∷ l) = let (l₁ , l₂) = split l in x ∷ l₁ , y ∷ l₂ test-split : List ℕ × List ℕ test-split = split (4 ∷ 1 ∷ 45 ∷ 8 ∷ 32 ∷ 12 ∷ 1 ∷ []) {- 4.2 Merging -} merge : (l m : List ℕ) → List ℕ merge [] m = m merge l [] = l merge (x ∷ l) (y ∷ m) with x ≤? y ... | left x≤y = x ∷ (merge l (y ∷ m)) ... | right y≤x = y ∷ (merge (x ∷ l) m) {- 4.3 Sorting -} {-# TERMINATING #-} mergesort-v1 : List ℕ → List ℕ mergesort-v1 [] = [] mergesort-v1 (x ∷ []) = x ∷ [] mergesort-v1 l = let (l₁ , l₂) = split l in merge (mergesort-v1 l₁) (mergesort-v1 l₂) test-merge-v1 : List ℕ test-merge-v1 = mergesort-v1 (4 ∷ 1 ∷ 45 ∷ 8 ∷ 32 ∷ 12 ∷ 1 ∷ []) {- 4.4 Splitting is decreasing -} data _<_ : ℕ → ℕ → Set where s<s : {m n : ℕ} (m<n : suc m ≤ n) → m < n length : {A : Set} (l : List A) → ℕ length [] = zero length (x ∷ l) = suc (length l) test-length : ℕ test-length = length (4 ∷ []) {- Surprisingly we need this lemma to have a compact proof that splitting is decreasing -} ≤-to-suc : (m : ℕ) → m ≤ suc m ≤-to-suc zero = z≤n ≤-to-suc (suc m) = s≤s (≤-to-suc m) {- Splitting is strictly decreasing only for lists with at least two elements Another approach would be to add `2 ≤ length l` as an argument, but it complicates the proof and Agda's pattern matching shows problematic behavior in that case [see at the end] -} <-split₁ : {A : Set} → (x y : A) → (l : List A) → length (fst (split (x ∷ y ∷ l))) < length (x ∷ y ∷ l) <-split₁ x y [] = s<s (≤-refl 2) <-split₁ x y (x₁ ∷ []) = s<s (s≤s (s≤s (≤-refl 1))) <-split₁ x y (x₁ ∷ x₂ ∷ l) with <-split₁ x y l ... | s<s h = s<s (s≤s (s≤s (≤-trans (≤-to-suc (suc (length (fst (split l))))) h))) <-split₂ : {A : Set} → (x y : A) → (l : List A) → length (snd (split (x ∷ y ∷ l))) < length (x ∷ y ∷ l) <-split₂ x y [] = s<s (≤-refl 2) <-split₂ x y (x₁ ∷ []) = s<s (s≤s (s≤s z≤n)) <-split₂ x y (x₁ ∷ x₂ ∷ l) with <-split₂ x y l ... | s<s h = s<s (s≤s (s≤s (≤-trans (≤-to-suc (suc (length (snd (split l))))) h))) {- 4.5 The fuel definition of merge -} {- Convenience function -} <-to-≤ : {m n : ℕ} → m < n → suc m ≤ n <-to-≤ (s<s m<n) = m<n {- The key part of this definition is proving that the recursive calls are decreasing -} mergesort-fuel : (n : ℕ) → (l : List ℕ) → (length l ≤ n) → List ℕ mergesort-fuel n [] l≤n = [] mergesort-fuel n (x ∷ []) l≤n = x ∷ [] mergesort-fuel (suc n) (x ∷ y ∷ l) l≤n = let (l₁ , l₂) = split (x ∷ y ∷ l) in merge (mergesort-fuel n l₁ (≤-trans (≤-pred (<-to-≤ (<-split₁ x y l))) (≤-pred l≤n))) (mergesort-fuel n l₂ (≤-trans (≤-pred (<-to-≤ (<-split₂ x y l))) (≤-pred l≤n))) mergesort : (l : List ℕ) → List ℕ mergesort l = mergesort-fuel (length l) l (≤-refl (length l)) test-merge : List ℕ test-merge = mergesort (4 ∷ 1 ∷ 45 ∷ 8 ∷ 32 ∷ 12 ∷ 1 ∷ []) {- 4.6 Merge sort is sorting -} {- Intrinsic approach: using the Sorted type -} {- Step 1: merging two sorted lists produces a sorted list -} mutual merge' : (l m : Sorted) → Sorted merge' nil m = m merge' l nil = l merge' (cons x l x≤l) (cons y m y≤m) with x ≤? y ... | left x≤y = cons x (merge' l (cons y m y≤m)) (merge-lemma x l (cons y m y≤m) x≤l x≤y) ... | right y≤x = cons y (merge' (cons x l x≤l) m) (merge-lemma y (cons x l x≤l) m y≤x y≤m) {- Lemma: if x is a lower bound on both l and m, it's also a lower bound on the merged list -} merge-lemma : (x : ℕ) → (l m : Sorted) → x ≤ head x l → x ≤ head x m → x ≤ head x (merge' l m) merge-lemma x nil m x≤l x≤m = x≤m merge-lemma x (cons y l y≤l) nil x≤l x≤m = x≤l merge-lemma x (cons y l y≤l) (cons z m z≤m) x≤l x≤m with y ≤? z ... | left y≤z = x≤l ... | right z≤y = x≤m {- Step 2: same-approach to mergesort-fuel -} mergesort-fuel-v2 : (n : ℕ) → (l : List ℕ) → (length l ≤ n) → Sorted mergesort-fuel-v2 _ [] _ = nil mergesort-fuel-v2 _ (x ∷ []) _ = cons x nil (≤-refl x) mergesort-fuel-v2 (suc n) (x ∷ y ∷ l) l≤n = let (l₁ , l₂) = split (x ∷ y ∷ l) in merge' (mergesort-fuel-v2 n l₁ (≤-trans (≤-pred (<-to-≤ (<-split₁ x y l))) (≤-pred l≤n))) (mergesort-fuel-v2 n l₂ (≤-trans (≤-pred (<-to-≤ (<-split₂ x y l))) (≤-pred l≤n))) {- Step 3: mergesort-v3 comes to life -} mergesort-v3 : List ℕ → Sorted mergesort-v3 l = mergesort-fuel-v2 (length l) l (≤-refl (length l)) {- --- APPENDIX --- -} {- Showcase: Agda's pattern matching forces us to consider a case where 2 ≤ 1 -} <-split₁-x : {A : Set} → (l : List A) → 2 ≤ length l → length (fst (split l)) < length l <-split₁-x (x ∷ []) (s≤s ()) <-split₁-x (x ∷ y ∷ l) 2≤l' with 2 ≤? length l ... | left 2≤l = s<s (s≤s (≤-trans (<-to-≤ (<-split₁-x l 2≤l)) (≤-to-suc (length l)))) ... | right l≤2 = {!!} {- would need to refine l here -}

record R : Set₁ where X = Y -- should get error here field A : Set

------------------------------------------------------------------------ -- The Agda standard library -- -- Simple combinators working solely on and with functions ------------------------------------------------------------------------ -- The contents of this file can be accessed from `Function`. {-# OPTIONS --without-K --safe #-} module Function.Base where open import Level open import Strict private variable a b c d e : Level A : Set a B : Set b C : Set c D : Set d E : Set e ------------------------------------------------------------------------ -- Some simple functions id : A → A id x = x const : A → B → A const x = λ _ → x ------------------------------------------------------------------------ -- Operations on dependent functions -- These are functions whose output has a type that depends on the -- value of the input to the function. infixr 9 _∘_ infixl 8 _ˢ_ infixl 0 _|>_ infix 0 case_return_of_ infixr -1 _$_ _$!_ -- Composition _∘_ : ∀ {A : Set a} {B : A → Set b} {C : {x : A} → B x → Set c} → (∀ {x} (y : B x) → C y) → (g : (x : A) → B x) → ((x : A) → C (g x)) f ∘ g = λ x → f (g x) -- Flipping order of arguments flip : ∀ {A : Set a} {B : Set b} {C : A → B → Set c} → ((x : A) (y : B) → C x y) → ((y : B) (x : A) → C x y) flip f = λ y x → f x y -- Application - note that _$_ is right associative, as in Haskell. -- If you want a left associative infix application operator, use -- Category.Functor._<$>_ from Category.Monad.Identity.IdentityMonad. _$_ : ∀ {A : Set a} {B : A → Set b} → ((x : A) → B x) → ((x : A) → B x) f $ x = f x -- Strict (call-by-value) application _$!_ : ∀ {A : Set a} {B : A → Set b} → ((x : A) → B x) → ((x : A) → B x) _$!_ = flip force -- Flipped application (aka pipe-forward) _|>_ : ∀ {A : Set a} {B : A → Set b} → (a : A) → (∀ a → B a) → B a _|>_ = flip _$_ -- The S combinator - written infix as in Conor McBride's paper -- "Outrageous but Meaningful Coincidences: Dependent type-safe syntax -- and evaluation". _ˢ_ : ∀ {A : Set a} {B : A → Set b} {C : (x : A) → B x → Set c} → ((x : A) (y : B x) → C x y) → (g : (x : A) → B x) → ((x : A) → C x (g x)) f ˢ g = λ x → f x (g x) -- Converting between implicit and explicit function spaces. _$- : ∀ {A : Set a} {B : A → Set b} → ((x : A) → B x) → ({x : A} → B x) f $- = f _ λ- : ∀ {A : Set a} {B : A → Set b} → ({x : A} → B x) → ((x : A) → B x) λ- f = λ x → f -- Case expressions (to be used with pattern-matching lambdas, see -- README.Case). case_return_of_ : ∀ {A : Set a} (x : A) (B : A → Set b) → ((x : A) → B x) → B x case x return B of f = f x ------------------------------------------------------------------------ -- Non-dependent versions of dependent operations -- Any of the above operations for dependent functions will also work -- for non-dependent functions but sometimes Agda has difficulty -- inferring the non-dependency. Primed (′ = \prime) versions of the -- operations are therefore provided below that sometimes have better -- inference properties. infixr 9 _∘′_ infixl 0 _|>′_ infix 0 case_of_ infixr -1 _$′_ _$!′_ -- Composition _∘′_ : (B → C) → (A → B) → (A → C) f ∘′ g = _∘_ f g -- Application _$′_ : (A → B) → (A → B) _$′_ = _$_ -- Strict (call-by-value) application _$!′_ : (A → B) → (A → B) _$!′_ = _$!_ -- Flipped application (aka pipe-forward) _|>′_ : A → (A → B) → B _|>′_ = _|>_ -- Case expressions (to be used with pattern-matching lambdas, see -- README.Case). case_of_ : A → (A → B) → B case x of f = case x return _ of f ------------------------------------------------------------------------ -- Operations that are only defined for non-dependent functions infixr 0 _-[_]-_ infixl 1 _on_ infixl 1 _⟨_⟩_ infixl 0 _∋_ -- Binary application _⟨_⟩_ : A → (A → B → C) → B → C x ⟨ f ⟩ y = f x y -- Composition of a binary function with a unary function _on_ : (B → B → C) → (A → B) → (A → A → C) _*_ on f = λ x y → f x * f y -- Composition of three binary functions _-[_]-_ : (A → B → C) → (C → D → E) → (A → B → D) → (A → B → E) f -[ _*_ ]- g = λ x y → f x y * g x y -- In Agda you cannot annotate every subexpression with a type -- signature. This function can be used instead. _∋_ : (A : Set a) → A → A A ∋ x = x -- Conversely it is sometimes useful to be able to extract the -- type of a given expression. typeOf : {A : Set a} → A → Set a typeOf {A = A} _ = A -- Construct an element of the given type by instance search. it : {A : Set a} → {{A}} → A it {{x}} = x

open import Relation.Binary.Core module PLRTree.Insert {A : Set} (_≤_ : A → A → Set) (tot≤ : Total _≤_) where open import Data.Sum open import PLRTree {A} insert : A → PLRTree → PLRTree insert x leaf = node perfect x leaf leaf insert x (node perfect y l r) with tot≤ x y | l | r ... | inj₁ x≤y | leaf | leaf = node right x (node perfect y leaf leaf) leaf ... | inj₁ x≤y | _ | _ = node left x (insert y l) r ... | inj₂ y≤x | leaf | leaf = node right y (node perfect x leaf leaf) leaf ... | inj₂ y≤x | _ | _ = node left y (insert x l) r insert x (node left y l r) with tot≤ x y ... | inj₁ x≤y with insert y l ... | node perfect y' l' r' = node right x (node perfect y' l' r') r ... | t = node left x t r insert x (node left y l r) | inj₂ y≤x with insert x l ... | node perfect y' l' r' = node right y (node perfect y' l' r') r ... | t = node left y t r insert x (node right y l r) with tot≤ x y ... | inj₁ x≤y with insert y r ... | node perfect y' l' r' = node perfect x l (node perfect y' l' r') ... | t = node right x l t insert x (node right y l r) | inj₂ y≤x with insert x r ... | node perfect y' l' r' = node perfect y l (node perfect y' l' r') ... | t = node right y l t

open import Common.Prelude open import Common.Equality infixr 5 _∷_ data Vec (A : Set) : Nat → Set where [] : Vec A zero _∷_ : ∀ {n} → A → Vec A n → Vec A (suc n) record Eq (A : Set) : Set where field _==_ : (x y : A) → Maybe (x ≡ y) open Eq {{...}} data Σ (A : Set) (B : A → Set) : Set where _,_ : (x : A) → B x → Σ A B eqNat : ∀ n m → Maybe (n ≡ m) eqNat zero zero = just refl eqNat zero (suc m) = nothing eqNat (suc n) zero = nothing eqNat (suc n) (suc m) with eqNat n m eqNat (suc n) (suc m) | nothing = nothing eqNat (suc n) (suc .n) | just refl = just refl eqVec : ∀ {A n} {{EqA : Eq A}} (xs ys : Vec A n) → Maybe (xs ≡ ys) eqVec [] [] = just refl eqVec (x ∷ xs) ( y ∷ ys) with x == y | eqVec xs ys eqVec (x ∷ xs) ( y ∷ ys) | nothing | _ = nothing eqVec (x ∷ xs) ( y ∷ ys) | _ | nothing = nothing eqVec (x ∷ xs) (.x ∷ .xs) | just refl | just refl = just refl eqSigma : ∀ {A B} {{EqA : Eq A}} {{EqB : ∀ {x} → Eq (B x)}} (x y : Σ A B) → Maybe (x ≡ y) eqSigma (x , y) (x₁ , y₁) with x == x₁ eqSigma (x , y) (x₁ , y₁) | nothing = nothing eqSigma (x , y) (.x , y₁) | just refl with y == y₁ eqSigma (x , y) (.x , y₁) | just refl | nothing = nothing eqSigma (x , y) (.x , .y) | just refl | just refl = just refl instance EqNat : Eq Nat EqNat = record { _==_ = eqNat } EqVec : ∀ {A n} {{_ : Eq A}} → Eq (Vec A n) EqVec = record { _==_ = eqVec } EqSigma : ∀ {A B} {{_ : Eq A}} {{_ : ∀ {x} → Eq (B x)}} → Eq (Σ A B) EqSigma = record { _==_ = eqSigma } cmpSigma : (xs ys : Σ Nat (Vec Nat)) → Maybe (xs ≡ ys) cmpSigma xs ys = xs == ys

-- Quotient category {-# OPTIONS --safe #-} module Cubical.Categories.Constructions.Quotient where open import Cubical.Categories.Category.Base open import Cubical.Categories.Functor.Base open import Cubical.Categories.Limits.Terminal open import Cubical.Foundations.HLevels open import Cubical.Foundations.Prelude open import Cubical.HITs.SetQuotients renaming ([_] to ⟦_⟧) private variable ℓ ℓ' ℓq : Level module _ (C : Category ℓ ℓ') where open Category C module _ (_~_ : {x y : ob} (f g : Hom[ x , y ] ) → Type ℓq) (~refl : {x y : ob} (f : Hom[ x , y ] ) → f ~ f) (~cong : {x y z : ob} (f f' : Hom[ x , y ]) → f ~ f' → (g g' : Hom[ y , z ]) → g ~ g' → (f ⋆ g) ~ (f' ⋆ g')) where private Hom[_,_]/~ = λ (x y : ob) → Hom[ x , y ] / _~_ module _ {x y z : ob} where _⋆/~_ : Hom[ x , y ]/~ → Hom[ y , z ]/~ → Hom[ x , z ]/~ _⋆/~_ = rec2 squash/ (λ f g → ⟦ f ⋆ g ⟧) (λ f f' g f~f' → eq/ _ _ (~cong _ _ f~f' _ _ (~refl _))) (λ f g g' g~g' → eq/ _ _ (~cong _ _ (~refl _) _ _ g~g')) module _ {x y : ob} where ⋆/~IdL : (f : Hom[ x , y ]/~) → (⟦ id ⟧ ⋆/~ f) ≡ f ⋆/~IdL = elimProp (λ _ → squash/ _ _) (λ _ → cong ⟦_⟧ (⋆IdL _)) ⋆/~IdR : (f : Hom[ x , y ]/~) → (f ⋆/~ ⟦ id ⟧) ≡ f ⋆/~IdR = elimProp (λ _ → squash/ _ _) (λ _ → cong ⟦_⟧ (⋆IdR _)) module _ {x y z w : ob} where ⋆/~Assoc : (f : Hom[ x , y ]/~) (g : Hom[ y , z ]/~) (h : Hom[ z , w ]/~) → ((f ⋆/~ g) ⋆/~ h) ≡ (f ⋆/~ (g ⋆/~ h)) ⋆/~Assoc = elimProp3 (λ _ _ _ → squash/ _ _) (λ _ _ _ → cong ⟦_⟧ (⋆Assoc _ _ _)) open Category QuotientCategory : Category ℓ (ℓ-max ℓ' ℓq) QuotientCategory .ob = ob C QuotientCategory .Hom[_,_] x y = (C [ x , y ]) / _~_ QuotientCategory .id = ⟦ id C ⟧ QuotientCategory ._⋆_ = _⋆/~_ QuotientCategory .⋆IdL = ⋆/~IdL QuotientCategory .⋆IdR = ⋆/~IdR QuotientCategory .⋆Assoc = ⋆/~Assoc QuotientCategory .isSetHom = squash/ private C/~ = QuotientCategory -- Quotient map open Functor QuoFunctor : Functor C C/~ QuoFunctor .F-ob x = x QuoFunctor .F-hom = ⟦_⟧ QuoFunctor .F-id = refl QuoFunctor .F-seq f g = refl -- Quotients preserve initial / terminal objects isInitial/~ : {z : ob C} → isInitial C z → isInitial C/~ z isInitial/~ zInit x = ⟦ zInit x .fst ⟧ , elimProp (λ _ → squash/ _ _) λ f → eq/ _ _ (subst (_~ f) (sym (zInit x .snd f)) (~refl _)) isFinal/~ : {z : ob C} → isFinal C z → isFinal C/~ z isFinal/~ zFinal x = ⟦ zFinal x .fst ⟧ , elimProp (λ _ → squash/ _ _) λ f → eq/ _ _ (subst (_~ f) (sym (zFinal x .snd f)) (~refl _))

{-# OPTIONS --cubical --no-import-sorts --safe #-} module Cubical.Foundations.Pointed.Properties where open import Cubical.Foundations.Prelude open import Cubical.Foundations.Pointed.Base open import Cubical.Foundations.Function open import Cubical.Foundations.GroupoidLaws open import Cubical.Foundations.Isomorphism open import Cubical.Data.Sigma private variable ℓ ℓ' ℓA ℓB ℓC ℓD : Level -- the default pointed Π-type: A is pointed, and B has a base point in the chosen fiber Π∙ : (A : Pointed ℓ) (B : typ A → Type ℓ') (ptB : B (pt A)) → Type (ℓ-max ℓ ℓ') Π∙ A B ptB = Σ[ f ∈ ((a : typ A) → B a) ] f (pt A) ≡ ptB -- the unpointed Π-type becomes a pointed type if the fibers are all pointed Πᵘ∙ : (A : Type ℓ) (B : A → Pointed ℓ') → Pointed (ℓ-max ℓ ℓ') Πᵘ∙ A B .fst = ∀ a → typ (B a) Πᵘ∙ A B .snd a = pt (B a) -- if the base and all fibers are pointed, we have the pointed pointed Π-type Πᵖ∙ : (A : Pointed ℓ) (B : typ A → Pointed ℓ') → Pointed (ℓ-max ℓ ℓ') Πᵖ∙ A B .fst = Π∙ A (typ ∘ B) (pt (B (pt A))) Πᵖ∙ A B .snd .fst a = pt (B a) Πᵖ∙ A B .snd .snd = refl -- the default pointed Σ-type is just the Σ-type, but as a pointed type Σ∙ : (A : Pointed ℓ) (B : typ A → Type ℓ') (ptB : B (pt A)) → Pointed (ℓ-max ℓ ℓ') Σ∙ A B ptB .fst = Σ[ a ∈ typ A ] B a Σ∙ A B ptB .snd .fst = pt A Σ∙ A B ptB .snd .snd = ptB -- version if B is a family of pointed types Σᵖ∙ : (A : Pointed ℓ) (B : typ A → Pointed ℓ') → Pointed (ℓ-max ℓ ℓ') Σᵖ∙ A B = Σ∙ A (typ ∘ B) (pt (B (pt A))) _×∙_ : (A∙ : Pointed ℓ) (B∙ : Pointed ℓ') → Pointed (ℓ-max ℓ ℓ') (A∙ ×∙ B∙) .fst = (typ A∙) × (typ B∙) (A∙ ×∙ B∙) .snd .fst = pt A∙ (A∙ ×∙ B∙) .snd .snd = pt B∙ -- composition of pointed maps _∘∙_ : {A : Pointed ℓA} {B : Pointed ℓB} {C : Pointed ℓC} (g : B →∙ C) (f : A →∙ B) → (A →∙ C) ((g , g∙) ∘∙ (f , f∙)) .fst x = g (f x) ((g , g∙) ∘∙ (f , f∙)) .snd = (cong g f∙) ∙ g∙ -- pointed identity id∙ : (A : Pointed ℓA) → (A →∙ A) id∙ A .fst x = x id∙ A .snd = refl -- constant pointed map const∙ : (A : Pointed ℓA) (B : Pointed ℓB) → (A →∙ B) const∙ _ B .fst _ = B .snd const∙ _ B .snd = refl -- left identity law for pointed maps ∘∙-idˡ : {A : Pointed ℓA} {B : Pointed ℓB} (f : A →∙ B) → f ∘∙ id∙ A ≡ f ∘∙-idˡ f = ΣPathP ( refl , (lUnit (f .snd)) ⁻¹ ) -- right identity law for pointed maps ∘∙-idʳ : {A : Pointed ℓA} {B : Pointed ℓB} (f : A →∙ B) → id∙ B ∘∙ f ≡ f ∘∙-idʳ f = ΣPathP ( refl , (rUnit (f .snd)) ⁻¹ ) -- associativity for composition of pointed maps ∘∙-assoc : {A : Pointed ℓA} {B : Pointed ℓB} {C : Pointed ℓC} {D : Pointed ℓD} (h : C →∙ D) (g : B →∙ C) (f : A →∙ B) → (h ∘∙ g) ∘∙ f ≡ h ∘∙ (g ∘∙ f) ∘∙-assoc (h , h∙) (g , g∙) (f , f∙) = ΣPathP (refl , q) where q : (cong (h ∘ g) f∙) ∙ (cong h g∙ ∙ h∙) ≡ cong h (cong g f∙ ∙ g∙) ∙ h∙ q = ( (cong (h ∘ g) f∙) ∙ (cong h g∙ ∙ h∙) ≡⟨ refl ⟩ (cong h (cong g f∙)) ∙ (cong h g∙ ∙ h∙) ≡⟨ assoc (cong h (cong g f∙)) (cong h g∙) h∙ ⟩ (cong h (cong g f∙) ∙ cong h g∙) ∙ h∙ ≡⟨ cong (λ p → p ∙ h∙) ((cong-∙ h (cong g f∙) g∙) ⁻¹) ⟩ (cong h (cong g f∙ ∙ g∙) ∙ h∙) ∎ )

{-# OPTIONS --safe #-} module Definition.Typed.EqRelInstance where open import Definition.Untyped open import Definition.Typed open import Definition.Typed.Properties open import Definition.Typed.Weakening open import Definition.Typed.Properties open import Definition.Typed.Reduction open import Definition.Typed.EqualityRelation open import Tools.Function Urefl : ∀ {r l Γ} → ⊢ Γ → Γ ⊢ (Univ r l) ≡ (Univ r l) ^ [ ! , next l ] Urefl {l = ⁰} ⊢Γ = refl (univ (univ 0<1 ⊢Γ)) Urefl {l = ¹} ⊢Γ = refl (Uⱼ ⊢Γ) instance eqRelInstance : EqRelSet eqRelInstance = eqRel _⊢_≡_^_ _⊢_≡_∷_^_ _⊢_≡_∷_^_ idᶠ idᶠ idᶠ univ un-univ≡ sym genSym genSym trans genTrans genTrans conv conv wkEq wkEqTerm wkEqTerm reduction reductionₜ Urefl (refl ∘ᶠ univ 0<1) (refl ∘ᶠ ℕⱼ) (refl ∘ᶠ Emptyⱼ) Π-cong ∃-cong (refl ∘ᶠ zeroⱼ) suc-cong (λ lF lG x x₁ x₂ x₃ x₄ x₅ → η-eq lF lG x x₁ x₂ x₅) genVar app-cong natrec-cong Emptyrec-cong Id-cong (λ ⊢Γ → Id-cong (refl (ℕⱼ ⊢Γ))) (λ ⊢Γ → Id-cong (refl (ℕⱼ ⊢Γ)) (refl (zeroⱼ ⊢Γ))) (λ ⊢Γ t → Id-cong (refl (ℕⱼ ⊢Γ)) (suc-cong t)) (λ ⊢Γ → Id-cong (refl (univ 0<1 ⊢Γ))) (λ ⊢Γ → Id-cong (refl (univ 0<1 ⊢Γ)) (refl (ℕⱼ ⊢Γ))) (λ ⊢A B → Id-cong (refl (univ 0<1 (wfEq (univ ⊢A)))) ⊢A B) cast-cong (λ ⊢Γ → cast-cong (refl (ℕⱼ ⊢Γ))) (λ ⊢Γ → cast-cong (refl (ℕⱼ ⊢Γ)) (refl (ℕⱼ ⊢Γ))) cast-cong (λ ⊢A ⊢P P → cast-cong (refl (ℕⱼ (wf (univ ⊢A)))) P) (λ ⊢A ⊢P P → cast-cong P (refl (ℕⱼ (wf (univ ⊢A))))) (λ ⊢A ⊢P P ⊢A' ⊢P' P' → cast-cong P P') (λ ⊢A ⊢P P ⊢A' ⊢P' P' → cast-cong P P') proof-irrelevance

{- 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.Abstract.Types open import LibraBFT.Abstract.Types.EpochConfig open import Util.Lemmas open import Util.Prelude open WithAbsVote -- For each desired property (VotesOnce and PreferredRoundRule), we have a -- module containing a Type that defines a property that an implementation -- should prove, and a proof that it implies the corresponding rule used by -- the abstract proofs. Then, we use those proofs to instantiate thmS5, -- and the use thmS5 to prove a number of correctness conditions. -- -- TODO-1: refactor this file to separate the definitions and proofs of -- VotesOnce and PreferredRoundRule from their use in proving the correctness -- properties. module LibraBFT.Abstract.Properties (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.RecordChain UID _≟UID_ NodeId 𝓔 𝓥 open import LibraBFT.Abstract.RecordChain.Assumptions UID _≟UID_ NodeId 𝓔 𝓥 open import LibraBFT.Abstract.System UID _≟UID_ NodeId 𝓔 𝓥 open import LibraBFT.Abstract.RecordChain.Properties UID _≟UID_ NodeId 𝓔 𝓥 open EpochConfig 𝓔 module WithAssumptions {ℓ} (InSys : Record → Set ℓ) (no-collisions-InSys : NoCollisions InSys) (votes-once : VotesOnlyOnceRule InSys) (preferred-round : PreferredRoundRule InSys) where open All-InSys-props InSys CommitsDoNotConflict : ∀{q q'} → {rc : RecordChain (Q q)} → All-InSys rc → {rc' : RecordChain (Q q')} → All-InSys rc' → {b b' : Block} → CommitRule rc b → CommitRule rc' b' → (B b) ∈RC rc' ⊎ (B b') ∈RC rc CommitsDoNotConflict ais ais' cr cr' with WithInvariants.thmS5 InSys votes-once preferred-round ais ais' cr cr' -- We use the implementation-provided evidence that Block ids are injective among -- Block actually in the system to dismiss the first possibility ...| inj₁ ((_ , neq , h≡) , (is1 , is2)) = ⊥-elim (neq (no-collisions-InSys is1 is2 h≡)) ...| inj₂ corr = corr -- When we are dealing with a /Complete/ InSys predicate, we can go a few steps -- further and prove that commits do not conflict even if we have only partial -- knowledge about Records represented in the system. module _ (∈QC⇒AllSent : Complete InSys) where -- For a /complete/ system (i.e., one in which peers vote for a Block only if -- they know of a RecordChain up to that Block whose Records are all InSys), we -- can prove that CommitRules based on RecordChainFroms similarly do not -- conflict, provided all of the Records in the RecordChainFroms are InSys. -- This enables peers not participating in consensus to confirm commits even if -- they are sent only a "commit certificate" that contains enough of a -- RecordChain to confirm the CommitRule. Note that it is this "sending" that -- justfies the assumption that the RecordChainFroms on which the CommitRules -- are based are All-InSys. CommitsDoNotConflict' : ∀{o o' q q'} → {rcf : RecordChainFrom o (Q q)} → All-InSys rcf → {rcf' : RecordChainFrom o' (Q q')} → All-InSys rcf' → {b b' : Block} → CommitRuleFrom rcf b → CommitRuleFrom rcf' b' → Σ (RecordChain (Q q')) ((B b) ∈RC_) ⊎ Σ (RecordChain (Q q)) ((B b') ∈RC_) CommitsDoNotConflict' {cb} {q = q} {q'} {rcf} rcfAll∈sys {rcf'} rcf'All∈sys crf crf' with bft-property (qVotes-C1 q) (qVotes-C1 q') ...| α , α∈qmem , α∈q'mem , hα with Any-sym (Any-map⁻ α∈qmem) | Any-sym (Any-map⁻ α∈q'mem) ...| α∈q | α∈q' with ∈QC⇒AllSent {q = q} hα α∈q (rcfAll∈sys here) | ∈QC⇒AllSent {q = q'} hα α∈q' (rcf'All∈sys here) ...| ab , (arc , ais) , ab←q | ab' , (arc' , ais') , ab←q' with crf⇒cr rcf (step arc ab←q) crf | crf⇒cr rcf' (step arc' ab←q') crf' ...| inj₁ ((_ , neq , h≡) , (is1 , is2)) | _ = ⊥-elim (neq (no-collisions-InSys (rcfAll∈sys is1) (ais (∈RC-simple-¬here arc ab←q (λ ()) is2)) h≡)) ...| inj₂ _ | inj₁ ((_ , neq , h≡) , (is1 , is2)) = ⊥-elim (neq (no-collisions-InSys (rcf'All∈sys is1) (ais' (∈RC-simple-¬here arc' ab←q' (λ ()) is2)) h≡)) ...| inj₂ cr | inj₂ cr' with CommitsDoNotConflict (All-InSys-step ais ab←q (rcfAll∈sys here)) (All-InSys-step ais' ab←q' (rcf'All∈sys here)) cr cr' ...| inj₁ b∈arc' = inj₁ (step arc' ab←q' , b∈arc') ...| inj₂ b'∈arc = inj₂ (step arc ab←q , b'∈arc)

module Data.List.First.Properties {ℓ}{A : Set ℓ} where open import Data.Product open import Data.List open import Data.List.Any open import Relation.Binary.PropositionalEquality open import Function open import Data.Empty open import Data.List.First open import Data.List.Membership.Propositional first⟶∈ : ∀ {B : A → Set} {x l} → First B x l → (x ∈ l × B x) first⟶∈ (here {x = x} p) = here refl , p first⟶∈ (there x' ¬px f) with (first⟶∈ f) first⟶∈ (there x' ¬px f) | x∈l , p = there x∈l , p first-unique : ∀ {P : A → Set}{x y v} → First P x v → First P y v → x ≡ y first-unique (here x) (here y) = refl first-unique (here {x = x} px) (there .x ¬px r) = ⊥-elim (¬px px) first-unique (there x ¬px l) (here {x = .x} px) = ⊥-elim (¬px px) first-unique (there x' _ l) (there .x' _ r) = first-unique l r

{-# OPTIONS --without-K --safe #-} open import Categories.Category module Categories.Category.Construction.Spans {o ℓ e} (𝒞 : Category o ℓ e) where open import Level open import Categories.Category.Diagram.Span 𝒞 open import Categories.Morphism.Reasoning 𝒞 open Category 𝒞 open HomReasoning open Equiv open Span open Span⇒ Spans : Obj → Obj → Category (o ⊔ ℓ) (o ⊔ ℓ ⊔ e) e Spans X Y = record { Obj = Span X Y ; _⇒_ = Span⇒ ; _≈_ = λ f g → arr f ≈ arr g ; id = id-span⇒ ; _∘_ = _∘ₛ_ ; assoc = assoc ; sym-assoc = sym-assoc ; identityˡ = identityˡ ; identityʳ = identityʳ ; identity² = identity² ; equiv = record { refl = refl ; sym = sym ; trans = trans } ; ∘-resp-≈ = ∘-resp-≈ }

{- Definition of join for ◇ and associated lemmas. -} module TemporalOps.Diamond.JoinLemmas where open import CategoryTheory.Categories open import CategoryTheory.Instances.Reactive open import CategoryTheory.Functor open import CategoryTheory.NatTrans open import CategoryTheory.Monad open import TemporalOps.Common open import TemporalOps.Next open import TemporalOps.Delay open import TemporalOps.Diamond.Functor open import TemporalOps.Diamond.Join import Relation.Binary.PropositionalEquality as ≡ open import Data.Product open import Relation.Binary.HeterogeneousEquality as ≅ using (_≅_ ; ≅-to-≡ ; ≡-to-≅ ; cong₂) open import Data.Nat.Properties using (+-identityʳ ; +-comm ; +-suc ; +-assoc) open import Holes.Term using (⌞_⌟) open import Holes.Cong.Propositional -- | Auxiliary lemmas -- Equality of two delayed values ◇-≅ : ∀{A : τ}{n n′ k k′ : ℕ} -> {v : delay A by k at n} {v′ : delay A by k′ at n′} -> (pk : k ≡ k′) (pn : n ≡ n′) -> v ≅ v′ -> _≅_ {A = ◇ A at n} (k , v) {B = ◇ A at n′} (k′ , v′) ◇-≅ refl refl ≅.refl = ≅.refl -- | Lemmas involving μ -- Two consecutive shifts can be composed μ-shift-comp : ∀{A} {n k l : ℕ} {a : ◇ A at n} -> μ-shift l (k + n) (μ-shift {A} k n a) ≅ μ-shift {A} (l + k) n a μ-shift-comp {A} {n} {k} {l} {j , v} = begin μ-shift l (k + n) (μ-shift k n (j , v)) ≡⟨⟩ μ-shift l (k + n) (k + j , v′) ≡⟨⟩ l + (k + j) , rew (sym (delay-+ l (k + j) (k + n))) v′ ≅⟨ ◇-≅ (sym (+-assoc l k j)) ((sym (+-assoc l k n))) pr ⟩ (l + k) + j , rew (sym (delay-+ (l + k) j n)) v ≡⟨⟩ μ-shift (l + k) n (j , v) ∎ where open ≅.≅-Reasoning v′ : delay A by (k + j) at (k + n) v′ = rew (sym (delay-+ k j n)) v v≅v′ : v ≅ v′ v≅v′ = rew-to-≅ (sym (delay-+ k j n)) pr : rew (sym (delay-+ l (k + j) (k + n))) v′ ≅ rew (sym (delay-+ (l + k) j n)) v pr = delay-assoc-sym l k j n v′ v (≅.sym v≅v′) private module μ = _⟹_ μ-◇ open ≅.≅-Reasoning -- Shift and multiplication can be interchanged μ-interchange : ∀{A}{n k : ℕ}{a : ◇ ◇ A at n} -> μ.at A (k + n) (μ-shift k n a) ≅ μ-shift k n (μ.at A n a) μ-interchange {A} {n} {k} {l , y} with inspect (compareLeq l n) μ-interchange {A} {.(l + m)} {k} {l , v} | snd==[ .l + m ] with≡ pf = begin μ.at A (k + (l + m)) (μ-shift k (l + m) (l , v)) ≡⟨⟩ μ.at A (k + (l + m)) (k + l , rew (sym (delay-+ k l (l + m))) v) ≡⟨⟩ μ-compare A (k + (l + m)) (k + l) v′ (compareLeq (k + l) (k + (l + m))) ≅⟨ ≅-cong₃ (λ x y z → μ-compare A x (k + l) y z) (≡-to-≅ (sym (+-assoc k l m))) v′≅v″ (compare-snd-+-assoc l m k pf) ⟩ μ-compare A ((k + l) + m) (k + l) v″ (snd==[ (k + l) + m ]) ≡⟨⟩ μ-shift (k + l) m ⌞ rew (delay-+-left0 (k + l) m) v″ ⌟ ≡⟨ cong! (pr k l m v″ v v″≅v) ⟩ μ-shift (k + l) m (rew (delay-+-left0 l m) v) ≅⟨ ≅.sym ( μ-shift-comp {A} {m} {l} {k} {(rew (delay-+-left0 l m) v)} ) ⟩ μ-shift k (l + m) (μ-shift l m (rew (delay-+-left0 l m) v)) ≡⟨⟩ μ-shift k (l + m) (μ-compare A (l + m) l v ⌞ snd==[ l + m ] ⌟) ≡⟨ cong! (sym pf) ⟩ μ-shift k (l + m) (μ-compare A (l + m) l v (compareLeq l (l + m))) ≡⟨⟩ μ-shift k (l + m) (μ.at A (l + m) (l , v)) ∎ where v′ : delay ◇ A by (k + l) at (k + (l + m)) v′ = rew (sym (delay-+ k l (l + m))) v v≅v′ : v ≅ v′ v≅v′ = rew-to-≅ (sym (delay-+ k l (l + m))) lemma-assoc : ∀{A : τ} -> (a b c : ℕ) -> delay ◇ A by (a + b) at (a + (b + c)) ≡ delay ◇ A by (a + b) at ((a + b) + c) lemma-assoc a b c rewrite sym (+-assoc a b c) = refl v″ : delay ◇ A by (k + l) at ((k + l) + m) v″ = (rew (lemma-assoc k l m) v′) v′≅v″ : v′ ≅ v″ v′≅v″ = rew-to-≅ (lemma-assoc k l m) v″≅v : v″ ≅ v v″≅v = ≅.trans (≅.sym v′≅v″) (≅.sym v≅v′) pr : ∀{A} (k l m : ℕ) -> Proof-≡ (delay-+-left0 {A} (k + l) m) (delay-+-left0 {A} l m) pr zero l m v .v ≅.refl = refl pr (suc k) l m = pr k l m -- l = n + suc j μ-interchange {A} {.n} {k} {.(n + suc l) , v} | fst==suc[ n + l ] with≡ pf = begin μ.at A (k + n) (μ-shift k n (n + suc l , v)) ≡⟨⟩ μ.at A (k + n) (k + (n + suc l) , rew (sym (delay-+ k (n + suc l) n)) v) ≡⟨⟩ μ-compare A (k + n) (k + (n + suc l)) v′ (compareLeq (k + (n + suc l)) (k + n)) ≅⟨ ≅-cong₃ ((λ x y z → μ-compare A (k + n) x y z)) (≡-to-≅ (sym (+-assoc k n (suc l)))) v′≅v″ (compare-fst-+-assoc n l k pf) ⟩ μ-compare A (k + n) ((k + n) + suc l) v″ fst==suc[ (k + n) + l ] ≡⟨⟩ (k + n) + suc l , rew (delay-⊤ (k + n) l) top.tt ≅⟨ ◇-≅ (+-assoc k n (suc l)) refl (pr n k l) ⟩ k + (n + suc l) , rew (sym (delay-+ k (n + suc l) n)) (rew (delay-⊤ n l) top.tt) ≡⟨⟩ μ-shift k n (n + suc l , rew (delay-⊤ n l) top.tt) ≡⟨⟩ μ-shift k n (μ-compare A n (n + suc l) v ⌞ fst==suc[ n + l ] ⌟) ≡⟨ cong! (sym pf) ⟩ μ-shift k n (μ-compare A n (n + suc l) v (compareLeq (n + suc l) n)) ≡⟨⟩ μ-shift k n (μ.at A n (n + suc l , v)) ∎ where v′ : delay ◇ A by (k + (n + suc l)) at (k + n) v′ = (rew (sym (delay-+ k (n + suc l) n)) v) lemma-assoc : ∀{A : τ} -> (a b c : ℕ) -> delay ◇ A by (a + (b + c)) at (a + b) ≡ delay ◇ A by ((a + b) + c) at (a + b) lemma-assoc {A} a b c rewrite sym (+-assoc a b c) = refl v″ : delay ◇ A by ((k + n) + suc l) at (k + n) v″ = (rew (lemma-assoc k n (suc l)) v′) v′≅v″ : v′ ≅ v″ v′≅v″ = rew-to-≅ (lemma-assoc k n (suc l)) pr : ∀{A} (n k l : ℕ) -> rew (delay-⊤ {A} (k + n) l) top.tt ≅ rew (sym (delay-+ {A} k (n + suc l) n)) (rew (delay-⊤ n l) top.tt) pr n zero l = ≅.refl pr n (suc k) l = pr n k l

{-# OPTIONS --omega-in-omega --no-termination-check --overlapping-instances #-} module Light.Subtyping where open import Light.Level using (_⊔_ ; ++_) open import Light.Variable.Levels open import Light.Variable.Sets record DirectSubtyping (𝕒 : Set aℓ) (𝕓 : Set bℓ) : Set (aℓ ⊔ bℓ) where constructor #_ field cast₁ : 𝕒 → 𝕓 open DirectSubtyping ⦃ ... ⦄ public record StrictSubtyping (𝕒 : Set aℓ) (𝕓 : Set aℓ) : Set (++ aℓ) data Subtyping {aℓ} : ∀ (𝕒 : Set aℓ) (𝕓 : Set aℓ) → Set (++ aℓ) record StrictSubtyping {aℓ} 𝕒 𝕓 where inductive instance constructor transitivity′ field {Thing} : Set aℓ field ⦃ direct ⦄ : DirectSubtyping 𝕒 Thing field ⦃ indirect ⦄ : Subtyping Thing 𝕓 data Subtyping where instance reflexivity : Subtyping 𝕒 𝕒 instance from‐strict : ∀ ⦃ strict : StrictSubtyping 𝕒 𝕓 ⦄ → Subtyping 𝕒 𝕓 cast : ∀ ⦃ subtyping : Subtyping 𝕒 𝕓 ⦄ → 𝕒 → 𝕓 cast ⦃ subtyping = reflexivity ⦄ a = a cast ⦃ subtyping = from‐strict ⦄ a = cast (cast₁ a) -- Note: These cannot be instances, they must be written explicitly. transitivity′′ : ∀ ⦃ a‐to‐b : Subtyping 𝕒 𝕓 ⦄ ⦃ b‐to‐c : Subtyping 𝕓 𝕔 ⦄ → DirectSubtyping 𝕒 𝕔 transitivity′′ ⦃ a‐to‐b = a‐to‐b ⦄ ⦃ b‐to‐c = b‐to‐c ⦄ = # λ a → cast ⦃ subtyping = b‐to‐c ⦄ (cast ⦃ subtyping = a‐to‐b ⦄ a) transitivity : ∀ ⦃ a‐to‐b : Subtyping 𝕒 𝕓 ⦄ ⦃ b‐to‐c : Subtyping 𝕓 𝕔 ⦄ → Subtyping 𝕒 𝕔 transitivity ⦃ a‐to‐b = a‐to‐b ⦄ ⦃ b‐to‐c = b‐to‐c ⦄ = from‐strict where instance _ = transitivity′′ ⦃ a‐to‐b = a‐to‐b ⦄ ⦃ b‐to‐c = b‐to‐c ⦄ module _ ⦃ c‐to‐a : Subtyping 𝕔 𝕒 ⦄ ⦃ b‐to‐d : Subtyping 𝕓 𝕕 ⦄ where instance explicit‐variance : DirectSubtyping (𝕒 → 𝕓) (𝕔 → 𝕕) implicit‐variance : DirectSubtyping (∀ {a : 𝕒} → 𝕓) (∀ {c : 𝕔} → 𝕕) instance‐variance : DirectSubtyping (∀ ⦃ a : 𝕒 ⦄ → 𝕓) (∀ ⦃ c : 𝕔 ⦄ → 𝕕) explicit‐variance = # λ f c → cast ⦃ subtyping = b‐to‐d ⦄ (f (cast c)) implicit‐variance = # λ f {c = c} → cast ⦃ subtyping = b‐to‐d ⦄ (f {a = cast c}) instance‐variance = # λ f ⦃ c = c ⦄ → cast ⦃ subtyping = b‐to‐d ⦄ (f ⦃ a = cast c ⦄)

module Issue561 where open import Common.Char open import Common.Prelude primitive primIsDigit : Char → Bool postulate IO : Set → Set return : ∀ {A} → A → IO A {-# BUILTIN IO IO #-} main : IO Bool main = return true

{-# OPTIONS --without-K #-} module library.types.Types where open import library.Basics open import library.types.Empty public open import library.types.Unit public open import library.types.Bool public open import library.types.Nat public open import library.types.Int public open import library.types.TLevel public open import library.types.Paths public open import library.types.Sigma public open import library.types.Pi public open import library.types.Coproduct public open import library.types.Lift public open import library.types.Circle public open import library.types.Span public open import library.types.Pushout public open import library.types.PushoutFlattening public open import library.types.Suspension public open import library.types.Join public open import library.types.Torus public open import library.types.Truncation public open import library.types.Cospan public open import library.types.Pullback public open import library.types.Group public open import library.types.Groupoid public open import library.types.GroupSet public open import library.types.KG1 public open import library.types.Pointed public open import library.types.LoopSpace public open import library.types.HomotopyGroup public open import library.types.PathSet public open import library.types.FundamentalGroupoid public open import library.types.Cover public open import library.types.OneSkeleton public open import library.types.PathSeq public -- This should probably not be exported -- module Generic1HIT {i j} (A : Type i) (B : Type j) (f g : B → A) where -- open import library.types.Generic1HIT A B f g public

record Unit : Set where constructor tt postulate C : Set c : C g : C f : Unit → C f tt = c record R : Set where constructor r g = c

-- A variant of code reported by Andreas Abel (who suggested that this -- way to trigger the bug might have been due to NAD). {-# OPTIONS --guardedness --sized-types #-} open import Agda.Builtin.Sigma open import Agda.Builtin.Size data ⊥ : Set where record Delay (A : Set) : Set where coinductive constructor ♯ field ♭ : Σ A λ _ → ⊥ → Delay A -- Recursive in order to please the termination checker. open Delay data D : Size → Set where inn : ∀ i → Delay (D i) → D (↑ i) iter : ∀{i} → D i → ⊥ iter (inn i t) = iter (fst (♭ t)) -- Should be rejected by the termination checker. bla : Delay (D ∞) ♭ bla = inn ∞ bla , λ() false : ⊥ false = iter (fst (♭ bla))

test = forall _let_ → Set

module SystemF.BigStep.Types where open import Prelude -- types are indexed by the number of open tvars infixl 10 _⇒_ data Type (n : ℕ) : Set where Unit : Type n ν : (i : Fin n) → Type n _⇒_ : Type n → Type n → Type n ∀' : Type (suc n) → Type n open import Data.Fin.Substitution open import Data.Vec module App {T} (l : Lift T Type )where open Lift l hiding (var) _/_ : ∀ {m n} → Type m → Sub T m n → Type n Unit / s = Unit ν i / s = lift (lookup i s) (a ⇒ b) / s = (a / s) ⇒ (b / s) ∀' x / s = ∀' (x / (s ↑)) open Application (record { _/_ = _/_ }) using (_/✶_) open import Data.Star Unit-/✶-↑✶ : ∀ k {m n} (ρs : Subs T m n) → Unit /✶ ρs ↑✶ k ≡ Unit Unit-/✶-↑✶ k ε = refl Unit-/✶-↑✶ k (x ◅ ρs) = cong₂ _/_ (Unit-/✶-↑✶ k ρs) refl ∀-/✶-↑✶ : ∀ k {m n t} (ρs : Subs T m n) → ∀' t /✶ ρs ↑✶ k ≡ ∀' (t /✶ ρs ↑✶ suc k) ∀-/✶-↑✶ k ε = refl ∀-/✶-↑✶ k (ρ ◅ ρs) = cong₂ _/_ (∀-/✶-↑✶ k ρs) refl ⇒-/✶-↑✶ : ∀ k {m n t₁ t₂} (ρs : Subs T m n) → t₁ ⇒ t₂ /✶ ρs ↑✶ k ≡ (t₁ /✶ ρs ↑✶ k) ⇒ (t₂ /✶ ρs ↑✶ k) ⇒-/✶-↑✶ k ε = refl ⇒-/✶-↑✶ k (ρ ◅ ρs) = cong₂ _/_ (⇒-/✶-↑✶ k ρs) refl tmSubst : TermSubst Type tmSubst = record { var = ν; app = App._/_ } open TermSubst tmSubst hiding (var; subst) public module Lemmas where open import Data.Fin.Substitution.Lemmas tyLemmas : TermLemmas Type tyLemmas = record { termSubst = tmSubst ; app-var = refl ; /✶-↑✶ = Lemma./✶-↑✶ } where module Lemma {T₁ T₂} {lift₁ : Lift T₁ Type} {lift₂ : Lift T₂ Type} where open Lifted lift₁ using () renaming (_↑✶_ to _↑✶₁_; _/✶_ to _/✶₁_) open Lifted lift₂ using () renaming (_↑✶_ to _↑✶₂_; _/✶_ to _/✶₂_) /✶-↑✶ : ∀ {m n} (ρs₁ : Subs T₁ m n) (ρs₂ : Subs T₂ m n) → (∀ k x → ν x /✶₁ ρs₁ ↑✶₁ k ≡ ν x /✶₂ ρs₂ ↑✶₂ k) → ∀ k t → t /✶₁ ρs₁ ↑✶₁ k ≡ t /✶₂ ρs₂ ↑✶₂ k /✶-↑✶ ρs₁ ρs₂ hyp k Unit = begin _ ≡⟨ App.Unit-/✶-↑✶ _ k ρs₁ ⟩ Unit ≡⟨ sym $ App.Unit-/✶-↑✶ _ k ρs₂ ⟩ _ ∎ /✶-↑✶ ρs₁ ρs₂ hyp k (ν i) = hyp k i /✶-↑✶ ρs₁ ρs₂ hyp k (a ⇒ b) = begin _ ≡⟨ App.⇒-/✶-↑✶ _ k ρs₁ ⟩ (a /✶₁ ρs₁ ↑✶₁ k) ⇒ (b /✶₁ ρs₁ ↑✶₁ k) ≡⟨ cong₂ _⇒_ (/✶-↑✶ ρs₁ ρs₂ hyp k a) ((/✶-↑✶ ρs₁ ρs₂ hyp k b)) ⟩ (a /✶₂ ρs₂ ↑✶₂ k) ⇒ (b /✶₂ ρs₂ ↑✶₂ k) ≡⟨ sym $ App.⇒-/✶-↑✶ _ k ρs₂ ⟩ _ ∎ /✶-↑✶ ρs₁ ρs₂ hyp k (∀' x) = begin _ ≡⟨ App.∀-/✶-↑✶ _ k ρs₁ ⟩ ∀' (x /✶₁ ρs₁ ↑✶₁ (suc k)) ≡⟨ cong ∀' (/✶-↑✶ ρs₁ ρs₂ hyp (suc k) x) ⟩ ∀' (x /✶₂ ρs₂ ↑✶₂ (suc k)) ≡⟨ sym $ App.∀-/✶-↑✶ _ k ρs₂ ⟩ _ ∎ open TermLemmas tyLemmas public hiding (var)

record R (A : Set) : Set where constructor c₂ field f : A → A open module R′ (A : Set) (r : R A) = R {A = A} r renaming (f to f′) _ : (@0 A : Set) → R A → A → A _ = λ A → f′ {A = A}

{-# OPTIONS --cubical --no-import-sorts --safe #-} module Cubical.Algebra.Magma.Properties where open import Cubical.Core.Everything open import Cubical.Foundations.Prelude open import Cubical.Foundations.HLevels open import Cubical.Algebra open import Cubical.Algebra.Magma.Morphism open import Cubical.Algebra.Magma.MorphismProperties public using (MagmaPath; uaMagma; carac-uaMagma; Magma≡; caracMagma≡) private variable ℓ : Level isPropIsMagma : {M : Type ℓ} {_•_ : Op₂ M} → isProp (IsMagma M _•_) isPropIsMagma (ismagma x) (ismagma y) = cong ismagma (isPropIsSet x y)

module Numeral.Natural.Oper.Summation.Range where import Lvl open import Data.List open import Data.List.Functions open import Numeral.Natural open import Type _‥_ : ℕ → ℕ → List(ℕ) _ ‥ 𝟎 = ∅ 𝟎 ‥ 𝐒 b = 𝟎 ⊰ map 𝐒(𝟎 ‥ b) 𝐒 a ‥ 𝐒 b = map 𝐒(a ‥ b) ‥_ : ℕ → List(ℕ) ‥ b = 𝟎 ‥ b _‥₌_ : ℕ → ℕ → List(ℕ) a ‥₌ b = a ‥ 𝐒(b) ‥₌_ : ℕ → List(ℕ) ‥₌ b = 𝟎 ‥₌ b

{-# OPTIONS --without-K --safe #-} module Dodo.Unary.Equality where -- Stdlib imports import Relation.Binary.PropositionalEquality as Eq open Eq using (_≡_; refl) open import Level using (Level; _⊔_) open import Function using (_∘_) open import Relation.Unary using (Pred) -- # Definitions infix 4 _⊆₁'_ _⊆₁_ _⇔₁_ -- Binary relation subset helper. Generally, use `_⊆₁_` (below). _⊆₁'_ : {a ℓ₁ ℓ₂ : Level} {A : Set a} → (P : Pred A ℓ₁) → (R : Pred A ℓ₂) → Set (a ⊔ ℓ₁ ⊔ ℓ₂) _⊆₁'_ {A = A} P R = ∀ (x : A) → P x → R x -- Somehow, Agda cannot infer P and R from `P ⇒ R`, and requires them explicitly passed. -- For proof convenience, wrap the proof in this structure, which explicitly conveys P and R -- to the type-checker. data _⊆₁_ {a ℓ₁ ℓ₂ : Level} {A : Set a} (P : Pred A ℓ₁) (R : Pred A ℓ₂) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where ⊆: : P ⊆₁' R → P ⊆₁ R data _⇔₁_ {a ℓ₁ ℓ₂ : Level} {A : Set a} (P : Pred A ℓ₁) (R : Pred A ℓ₂) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where ⇔: : P ⊆₁' R → R ⊆₁' P → P ⇔₁ R -- # Helpers ⇔₁-intro : {a ℓ₁ ℓ₂ : Level} {A : Set a} → {P : Pred A ℓ₁} {R : Pred A ℓ₂} → P ⊆₁ R → R ⊆₁ P → P ⇔₁ R ⇔₁-intro (⊆: P⊆R) (⊆: R⊆P) = ⇔: P⊆R R⊆P ⇔₁-compose : ∀ {a b ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level} {A : Set a} {B : Set b} {P : Pred A ℓ₁} {Q : Pred A ℓ₂} {R : Pred B ℓ₃} {S : Pred B ℓ₄} → ( P ⊆₁ Q → R ⊆₁ S ) → ( Q ⊆₁ P → S ⊆₁ R ) → P ⇔₁ Q → R ⇔₁ S ⇔₁-compose ⊆-proof ⊇-proof (⇔: P⊆Q R⊆S) = ⇔₁-intro (⊆-proof (⊆: P⊆Q)) (⊇-proof (⊆: R⊆S)) -- # Properties -- ## Properties: ⊆₁ module _ {a ℓ : Level} {A : Set a} {R : Pred A ℓ} where ⊆₁-refl : R ⊆₁ R ⊆₁-refl = ⊆: (λ _ Rx → Rx) module _ {a ℓ₁ ℓ₂ ℓ₃ : Level} {A : Set a} {P : Pred A ℓ₁} {Q : Pred A ℓ₂} {R : Pred A ℓ₃} where ⊆₁-trans : P ⊆₁ Q → Q ⊆₁ R → P ⊆₁ R ⊆₁-trans (⊆: P⊆Q) (⊆: Q⊆R) = ⊆: (λ x Qx → Q⊆R x (P⊆Q x Qx)) -- ## Properties: ⇔₁ module _ {a ℓ : Level} {A : Set a} {R : Pred A ℓ} where ⇔₁-refl : R ⇔₁ R ⇔₁-refl = ⇔₁-intro ⊆₁-refl ⊆₁-refl module _ {a ℓ₁ ℓ₂ : Level} {A : Set a} {Q : Pred A ℓ₁} {R : Pred A ℓ₂} where ⇔₁-sym : Q ⇔₁ R → R ⇔₁ Q -- WARNING: Do *NOT* use `Symmetric _⇔_`. It messes up the universe levels. ⇔₁-sym (⇔: Q⊆R R⊆Q) = ⇔: R⊆Q Q⊆R module _ {a ℓ₁ ℓ₂ ℓ₃ : Level} {A : Set a} {P : Pred A ℓ₁} {Q : Pred A ℓ₂} {R : Pred A ℓ₃} where ⇔₁-trans : P ⇔₁ Q → Q ⇔₁ R → P ⇔₁ R ⇔₁-trans (⇔: P⊆Q Q⊆P) (⇔: Q⊆R R⊆Q) = ⇔₁-intro (⊆₁-trans (⊆: P⊆Q) (⊆: Q⊆R)) (⊆₁-trans (⊆: R⊆Q) (⊆: Q⊆P)) -- # Operations -- ## Operations: ⇔₁ and ⊆₁ conversion un-⊆₁ : ∀ {a ℓ₁ ℓ₂ : Level} {A : Set a} {P : Pred A ℓ₁} {R : Pred A ℓ₂} → P ⊆₁ R → P ⊆₁' R un-⊆₁ (⊆: P⊆R) = P⊆R unlift-⊆₁ : ∀ {a b ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level} {A : Set a} {B : Set b} {P : Pred A ℓ₁} {Q : Pred A ℓ₂} {R : Pred B ℓ₃} {S : Pred B ℓ₄} → ( P ⊆₁ Q → R ⊆₁ S ) → ( P ⊆₁' Q → R ⊆₁' S ) unlift-⊆₁ f P⊆Q = un-⊆₁ (f (⊆: P⊆Q)) lift-⊆₁ : ∀ {a b ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level} {A : Set a} {B : Set b} {P : Pred A ℓ₁} {Q : Pred A ℓ₂} {R : Pred B ℓ₃} {S : Pred B ℓ₄} → ( P ⊆₁' Q → R ⊆₁' S ) → ( P ⊆₁ Q → R ⊆₁ S ) lift-⊆₁ f (⊆: P⊆Q) = ⊆: (f P⊆Q) ⇔₁-to-⊆₁ : {a ℓ₁ ℓ₂ : Level} {A : Set a} {P : Pred A ℓ₁} {Q : Pred A ℓ₂} → P ⇔₁ Q ------ → P ⊆₁ Q ⇔₁-to-⊆₁ (⇔: P⊆Q _) = ⊆: P⊆Q ⇔₁-to-⊇₁ : {a ℓ₁ ℓ₂ : Level} {A : Set a} {P : Pred A ℓ₁} {Q : Pred A ℓ₂} → P ⇔₁ Q ------ → Q ⊆₁ P ⇔₁-to-⊇₁ (⇔: _ Q⊆P) = ⊆: Q⊆P -- ## Operations: ⊆₁ ⊆₁-apply : ∀ {a ℓ₁ ℓ₂ : Level} {A : Set a} {P : Pred A ℓ₁} {Q : Pred A ℓ₂} → P ⊆₁ Q → {x : A} → P x ------- → Q x ⊆₁-apply (⊆: P⊆Q) {x} = P⊆Q x ⊆₁-substˡ : ∀ {a ℓ₁ ℓ₂ ℓ₃ : Level} {A : Set a} {P : Pred A ℓ₁} {Q : Pred A ℓ₂} {R : Pred A ℓ₃} {x : A} → P ⇔₁ Q → P ⊆₁ R ------ → Q ⊆₁ R ⊆₁-substˡ (⇔: _ Q⊆P) P⊆R = ⊆₁-trans (⊆: Q⊆P) P⊆R ⊆₁-substʳ : ∀ {a ℓ₁ ℓ₂ ℓ₃ : Level} {A : Set a} {P : Pred A ℓ₁} {Q : Pred A ℓ₂} {R : Pred A ℓ₃} → Q ⇔₁ R → P ⊆₁ Q ------ → P ⊆₁ R ⊆₁-substʳ (⇔: Q⊆R _) P⊆Q = ⊆₁-trans P⊆Q (⊆: Q⊆R) ≡-to-⊆₁ : {a b ℓ : Level} {A : Set a} {P : Pred A ℓ} {Q : Pred A ℓ} → P ≡ Q ------ → P ⊆₁ Q ≡-to-⊆₁ refl = ⊆: (λ _ Px → Px) -- ## Operations: ⇔₁ ⇔₁-apply-⊆₁ : {a ℓ₁ ℓ₂ : Level} {A : Set a} {P : Pred A ℓ₁} {Q : Pred A ℓ₂} → P ⇔₁ Q → {x : A} → P x ------- → Q x ⇔₁-apply-⊆₁ = ⊆₁-apply ∘ ⇔₁-to-⊆₁ ⇔₁-apply-⊇₁ : {a ℓ₁ ℓ₂ : Level} {A : Set a} {P : Pred A ℓ₁} {Q : Pred A ℓ₂} → P ⇔₁ Q → {x : A} → Q x ------- → P x ⇔₁-apply-⊇₁ = ⊆₁-apply ∘ ⇔₁-to-⊇₁ ≡-to-⇔₁ : {a ℓ : Level} {A : Set a} {P : Pred A ℓ} {Q : Pred A ℓ} → P ≡ Q ------ → P ⇔₁ Q ≡-to-⇔₁ refl = ⇔₁-intro ⊆₁-refl ⊆₁-refl -- # Reasoning -- ## Reasoning: ⊆₁ module ⊆₁-Reasoning {a ℓ₁ : Level} {A : Set a} where infix 3 _⊆₁∎ infixr 2 step-⊆₁ infix 1 begin⊆₁_ begin⊆₁_ : {ℓ₂ : Level} {P : Pred A ℓ₁} {Q : Pred A ℓ₂} → P ⊆₁ Q ------ → P ⊆₁ Q begin⊆₁_ P⊆Q = P⊆Q step-⊆₁ : ∀ {ℓ₂ ℓ₃ : Level} → (P : Pred A ℓ₁) → {Q : Pred A ℓ₂} → {R : Pred A ℓ₃} → Q ⊆₁ R → P ⊆₁ Q ------ → P ⊆₁ R step-⊆₁ P Q⊆R P⊆Q = ⊆₁-trans P⊆Q Q⊆R _⊆₁∎ : ∀ (P : Pred A ℓ₁) → P ⊆₁ P _⊆₁∎ _ = ⊆₁-refl syntax step-⊆₁ P Q⊆R P⊆Q = P ⊆₁⟨ P⊆Q ⟩ Q⊆R -- ## Reasoning: ⇔₁ module ⇔₁-Reasoning {a ℓ₁ : Level} {A : Set a} where infix 3 _⇔₁∎ infixr 2 _⇔₁⟨⟩_ step-⇔₁ infix 1 begin⇔₁_ begin⇔₁_ : {ℓ₂ : Level} {P : Pred A ℓ₁} {Q : Pred A ℓ₂} → P ⇔₁ Q ------ → P ⇔₁ Q begin⇔₁_ P⇔Q = P⇔Q _⇔₁⟨⟩_ : {ℓ₂ : Level} (P : Pred A ℓ₁) → {Q : Pred A ℓ₂} → P ⇔₁ Q --------------- → P ⇔₁ Q _ ⇔₁⟨⟩ x≡y = x≡y step-⇔₁ : ∀ {ℓ₂ ℓ₃ : Level} → (P : Pred A ℓ₁) → {Q : Pred A ℓ₂} → {R : Pred A ℓ₃} → Q ⇔₁ R → P ⇔₁ Q --------------- → P ⇔₁ R step-⇔₁ _ Q⇔R P⇔Q = ⇔₁-trans P⇔Q Q⇔R _⇔₁∎ : ∀ (P : Pred A ℓ₁) → P ⇔₁ P _⇔₁∎ _ = ⇔₁-refl syntax step-⇔₁ P Q⇔R P⇔Q = P ⇔₁⟨ P⇔Q ⟩ Q⇔R

{-# OPTIONS --cubical --safe --postfix-projections #-} module Data.Bool.Properties where open import Prelude open import Data.Bool open import Data.Unit.Properties T? : ∀ x → Dec (T x) T? x .does = x T? false .why = ofⁿ id T? true .why = ofʸ tt isPropT : ∀ x → isProp (T x) isPropT false = isProp⊥ isPropT true = isProp⊤ discreteBool : Discrete Bool discreteBool false y .does = not y discreteBool true y .does = y discreteBool false false .why = ofʸ refl discreteBool false true .why = ofⁿ λ p → subst (bool ⊤ ⊥) p tt discreteBool true false .why = ofⁿ λ p → subst (bool ⊥ ⊤) p tt discreteBool true true .why = ofʸ refl

{-# OPTIONS --without-K #-} module P where open import Data.Empty open import Data.Unit open import Data.Sum open import Data.Product open import Relation.Binary.PropositionalEquality ------------------------------------------------------------------------------ -- For now, a groupoid is just a set Groupoid : Set₁ Groupoid = Set mutual -- types data B : Set where ZERO : B ONE : B _+_ : B → B → B _*_ : B → B → B _~_ : {b : B} → ⟦ b ⟧ → ⟦ b ⟧ → B -- values ⟦_⟧ : B → Groupoid ⟦ ZERO ⟧ = ⊥ ⟦ ONE ⟧ = ⊤ ⟦ b₁ + b₂ ⟧ = ⟦ b₁ ⟧ ⊎ ⟦ b₂ ⟧ ⟦ b₁ * b₂ ⟧ = ⟦ b₁ ⟧ × ⟦ b₂ ⟧ ⟦ v₁ ~ v₂ ⟧ = v₁ ≡ v₂ -- pointed types data PB : Set where POINTED : Σ B (λ b → ⟦ b ⟧) → PB RECIP : PB → PB -- lift B type constructors to pointed types _++_ : PB → PB → PB (POINTED (b₁ , v₁)) ++ (POINTED (b₂ , v₂)) = POINTED (b₁ + b₂ , inj₁ v₁) _ ++ _ = {!!} _**_ : PB → PB → PB (POINTED (b₁ , v₁)) ** (POINTED (b₂ , v₂)) = POINTED (b₁ * b₂ , (v₁ , v₂)) _ ** _ = {!!} -- All the pi combinators now work on pointed types data _⟷_ : PB → PB → Set₁ where swap⋆ : {pb₁ pb₂ : PB} → (pb₁ ** pb₂) ⟷ (pb₂ ** pb₁) eta : {pb : PB} → POINTED (ONE , tt) ⟷ RECIP pb -- induction principle to reason about identities; needed ??? ind : {b : B} → (C : (v₁ v₂ : ⟦ b ⟧) → (p : ⟦ _~_ {b} v₁ v₂ ⟧) → Set) → (c : (v : ⟦ b ⟧) → C v v refl) → (v₁ v₂ : ⟦ b ⟧) → (p : ⟦ _~_ {b} v₁ v₂ ⟧) → C v₁ v₂ p ind C c v .v refl = c v -- Examples Bool : B Bool = ONE + ONE false : ⟦ Bool ⟧ false = inj₁ tt true : ⟦ Bool ⟧ true = inj₂ tt pb1 : PB pb1 = POINTED (Bool , false) pb2 : PB pb2 = POINTED (Bool , true) FalseID : B FalseID = _~_ {Bool} false false pb3 : PB pb3 = POINTED (FalseID , refl) FalseID^2 : B FalseID^2 = _~_ {FalseID} refl refl pb4 : PB pb4 = POINTED (FalseID^2 , refl) ------------------------------------------------------------------------------

-- Basic intuitionistic logic of proofs, without ∨, ⊥, or +. -- Gentzen-style formalisation of syntax with context pairs. -- Normal forms and neutrals. module BasicILP.Syntax.DyadicGentzenNormalForm where open import BasicILP.Syntax.DyadicGentzen public -- Derivations. mutual -- Normal forms, or introductions. infix 3 _⊢ⁿᶠ_ data _⊢ⁿᶠ_ : Cx² Ty Box → Ty → Set where neⁿᶠ : ∀ {A Γ Δ} → Γ ⁏ Δ ⊢ⁿᵉ A → Γ ⁏ Δ ⊢ⁿᶠ A lamⁿᶠ : ∀ {A B Γ Δ} → Γ , A ⁏ Δ ⊢ⁿᶠ B → Γ ⁏ Δ ⊢ⁿᶠ A ▻ B boxⁿᶠ : ∀ {Ψ Ω A Γ Δ} → (x : Ψ ⁏ Ω ⊢ A) → Γ ⁏ Δ ⊢ⁿᶠ [ Ψ ⁏ Ω ⊢ x ] A pairⁿᶠ : ∀ {A B Γ Δ} → Γ ⁏ Δ ⊢ⁿᶠ A → Γ ⁏ Δ ⊢ⁿᶠ B → Γ ⁏ Δ ⊢ⁿᶠ A ∧ B unitⁿᶠ : ∀ {Γ Δ} → Γ ⁏ Δ ⊢ⁿᶠ ⊤ -- Neutrals, or eliminations. infix 3 _⊢ⁿᵉ_ data _⊢ⁿᵉ_ : Cx² Ty Box → Ty → Set where varⁿᵉ : ∀ {A Γ Δ} → A ∈ Γ → Γ ⁏ Δ ⊢ⁿᵉ A appⁿᵉ : ∀ {A B Γ Δ} → Γ ⁏ Δ ⊢ⁿᵉ A ▻ B → Γ ⁏ Δ ⊢ⁿᶠ A → Γ ⁏ Δ ⊢ⁿᵉ B mvarⁿᵉ : ∀ {Ψ Ω A x Γ Δ} → [ Ψ ⁏ Ω ⊢ x ] A ∈ Δ → {{_ : Γ ⁏ Δ ⊢⋆ⁿᶠ Ψ}} → {{_ : Γ ⁏ Δ ⊢⍟ⁿᶠ Ω}} → Γ ⁏ Δ ⊢ⁿᵉ A unboxⁿᵉ : ∀ {Ψ Ω A C x Γ Δ} → Γ ⁏ Δ ⊢ⁿᵉ [ Ψ ⁏ Ω ⊢ x ] A → Γ ⁏ Δ , [ Ψ ⁏ Ω ⊢ x ] A ⊢ⁿᶠ C → Γ ⁏ Δ ⊢ⁿᵉ C fstⁿᵉ : ∀ {A B Γ Δ} → Γ ⁏ Δ ⊢ⁿᵉ A ∧ B → Γ ⁏ Δ ⊢ⁿᵉ A sndⁿᵉ : ∀ {A B Γ Δ} → Γ ⁏ Δ ⊢ⁿᵉ A ∧ B → Γ ⁏ Δ ⊢ⁿᵉ B infix 3 _⊢⋆ⁿᶠ_ _⊢⋆ⁿᶠ_ : Cx² Ty Box → Cx Ty → Set Γ ⁏ Δ ⊢⋆ⁿᶠ ∅ = 𝟙 Γ ⁏ Δ ⊢⋆ⁿᶠ Ξ , A = Γ ⁏ Δ ⊢⋆ⁿᶠ Ξ × Γ ⁏ Δ ⊢ⁿᶠ A infix 3 _⊢⍟ⁿᶠ_ _⊢⍟ⁿᶠ_ : Cx² Ty Box → Cx Box → Set Γ ⁏ Δ ⊢⍟ⁿᶠ ∅ = 𝟙 Γ ⁏ Δ ⊢⍟ⁿᶠ Ξ , [ Ψ ⁏ Ω ⊢ x ] A = Γ ⁏ Δ ⊢⍟ⁿᶠ Ξ × Γ ⁏ Δ ⊢ⁿᶠ [ Ψ ⁏ Ω ⊢ x ] A infix 3 _⊢⋆ⁿᵉ_ _⊢⋆ⁿᵉ_ : Cx² Ty Box → Cx Ty → Set Γ ⁏ Δ ⊢⋆ⁿᵉ ∅ = 𝟙 Γ ⁏ Δ ⊢⋆ⁿᵉ Ξ , A = Γ ⁏ Δ ⊢⋆ⁿᵉ Ξ × Γ ⁏ Δ ⊢ⁿᵉ A infix 3 _⊢⍟ⁿᵉ_ _⊢⍟ⁿᵉ_ : Cx² Ty Box → Cx Box → Set Γ ⁏ Δ ⊢⍟ⁿᵉ ∅ = 𝟙 Γ ⁏ Δ ⊢⍟ⁿᵉ Ξ , [ Ψ ⁏ Ω ⊢ x ] A = Γ ⁏ Δ ⊢⍟ⁿᵉ Ξ × Γ ⁏ Δ ⊢ⁿᵉ [ Ψ ⁏ Ω ⊢ x ] A -- Translation to simple terms. mutual nf→tm : ∀ {A Γ Δ} → Γ ⁏ Δ ⊢ⁿᶠ A → Γ ⁏ Δ ⊢ A nf→tm (neⁿᶠ t) = ne→tm t nf→tm (lamⁿᶠ t) = lam (nf→tm t) nf→tm (boxⁿᶠ t) = box t nf→tm (pairⁿᶠ t u) = pair (nf→tm t) (nf→tm u) nf→tm unitⁿᶠ = unit ne→tm : ∀ {A Γ Δ} → Γ ⁏ Δ ⊢ⁿᵉ A → Γ ⁏ Δ ⊢ A ne→tm (varⁿᵉ i) = var i ne→tm (appⁿᵉ t u) = app (ne→tm t) (nf→tm u) ne→tm (mvarⁿᵉ i {{ts}} {{us}}) = mvar i {{nf→tm⋆ ts}} {{nf→tm⍟ us}} ne→tm (unboxⁿᵉ t u) = unbox (ne→tm t) (nf→tm u) ne→tm (fstⁿᵉ t) = fst (ne→tm t) ne→tm (sndⁿᵉ t) = snd (ne→tm t) nf→tm⋆ : ∀ {Ξ Γ Δ} → Γ ⁏ Δ ⊢⋆ⁿᶠ Ξ → Γ ⁏ Δ ⊢⋆ Ξ nf→tm⋆ {∅} ∙ = ∙ nf→tm⋆ {Ξ , A} (ts , t) = nf→tm⋆ ts , nf→tm t nf→tm⍟ : ∀ {Ξ Γ Δ} → Γ ⁏ Δ ⊢⍟ⁿᶠ Ξ → Γ ⁏ Δ ⊢⍟ Ξ nf→tm⍟ {∅} ∙ = ∙ nf→tm⍟ {Ξ , _} (ts , t) = nf→tm⍟ ts , nf→tm t ne→tm⋆ : ∀ {Ξ Γ Δ} → Γ ⁏ Δ ⊢⋆ⁿᵉ Ξ → Γ ⁏ Δ ⊢⋆ Ξ ne→tm⋆ {∅} ∙ = ∙ ne→tm⋆ {Ξ , A} (ts , t) = ne→tm⋆ ts , ne→tm t ne→tm⍟ : ∀ {Ξ Γ Δ} → Γ ⁏ Δ ⊢⍟ⁿᵉ Ξ → Γ ⁏ Δ ⊢⍟ Ξ ne→tm⍟ {∅} ∙ = ∙ ne→tm⍟ {Ξ , _} (ts , t) = ne→tm⍟ ts , ne→tm t -- Monotonicity with respect to context inclusion. mutual mono⊢ⁿᶠ : ∀ {A Γ Γ′ Δ} → Γ ⊆ Γ′ → Γ ⁏ Δ ⊢ⁿᶠ A → Γ′ ⁏ Δ ⊢ⁿᶠ A mono⊢ⁿᶠ η (neⁿᶠ t) = neⁿᶠ (mono⊢ⁿᵉ η t) mono⊢ⁿᶠ η (lamⁿᶠ t) = lamⁿᶠ (mono⊢ⁿᶠ (keep η) t) mono⊢ⁿᶠ η (boxⁿᶠ t) = boxⁿᶠ t mono⊢ⁿᶠ η (pairⁿᶠ t u) = pairⁿᶠ (mono⊢ⁿᶠ η t) (mono⊢ⁿᶠ η u) mono⊢ⁿᶠ η unitⁿᶠ = unitⁿᶠ mono⊢ⁿᵉ : ∀ {A Γ Γ′ Δ} → Γ ⊆ Γ′ → Γ ⁏ Δ ⊢ⁿᵉ A → Γ′ ⁏ Δ ⊢ⁿᵉ A mono⊢ⁿᵉ η (varⁿᵉ i) = varⁿᵉ (mono∈ η i) mono⊢ⁿᵉ η (appⁿᵉ t u) = appⁿᵉ (mono⊢ⁿᵉ η t) (mono⊢ⁿᶠ η u) mono⊢ⁿᵉ η (mvarⁿᵉ i {{ts}} {{us}}) = mvarⁿᵉ i {{mono⊢⋆ⁿᶠ η ts}} {{mono⊢⍟ⁿᶠ η us}} mono⊢ⁿᵉ η (unboxⁿᵉ t u) = unboxⁿᵉ (mono⊢ⁿᵉ η t) (mono⊢ⁿᶠ η u) mono⊢ⁿᵉ η (fstⁿᵉ t) = fstⁿᵉ (mono⊢ⁿᵉ η t) mono⊢ⁿᵉ η (sndⁿᵉ t) = sndⁿᵉ (mono⊢ⁿᵉ η t) mono⊢⋆ⁿᶠ : ∀ {Ξ Γ Γ′ Δ} → Γ ⊆ Γ′ → Γ ⁏ Δ ⊢⋆ⁿᶠ Ξ → Γ′ ⁏ Δ ⊢⋆ⁿᶠ Ξ mono⊢⋆ⁿᶠ {∅} η ∙ = ∙ mono⊢⋆ⁿᶠ {Ξ , A} η (ts , t) = mono⊢⋆ⁿᶠ η ts , mono⊢ⁿᶠ η t mono⊢⍟ⁿᶠ : ∀ {Ξ Γ Γ′ Δ} → Γ ⊆ Γ′ → Γ ⁏ Δ ⊢⍟ⁿᶠ Ξ → Γ′ ⁏ Δ ⊢⍟ⁿᶠ Ξ mono⊢⍟ⁿᶠ {∅} η ∙ = ∙ mono⊢⍟ⁿᶠ {Ξ , _} η (ts , t) = mono⊢⍟ⁿᶠ η ts , mono⊢ⁿᶠ η t mono⊢⋆ⁿᵉ : ∀ {Ξ Γ Γ′ Δ} → Γ ⊆ Γ′ → Γ ⁏ Δ ⊢⋆ⁿᵉ Ξ → Γ′ ⁏ Δ ⊢⋆ⁿᵉ Ξ mono⊢⋆ⁿᵉ {∅} η ∙ = ∙ mono⊢⋆ⁿᵉ {Ξ , A} η (ts , t) = mono⊢⋆ⁿᵉ η ts , mono⊢ⁿᵉ η t mono⊢⍟ⁿᵉ : ∀ {Ξ Γ Γ′ Δ} → Γ ⊆ Γ′ → Γ ⁏ Δ ⊢⍟ⁿᵉ Ξ → Γ′ ⁏ Δ ⊢⍟ⁿᵉ Ξ mono⊢⍟ⁿᵉ {∅} η ∙ = ∙ mono⊢⍟ⁿᵉ {Ξ , _} η (ts , t) = mono⊢⍟ⁿᵉ η ts , mono⊢ⁿᵉ η t -- Monotonicity with respect to modal context inclusion. mutual mmono⊢ⁿᶠ : ∀ {A Γ Δ Δ′} → Δ ⊆ Δ′ → Γ ⁏ Δ ⊢ⁿᶠ A → Γ ⁏ Δ′ ⊢ⁿᶠ A mmono⊢ⁿᶠ θ (neⁿᶠ t) = neⁿᶠ (mmono⊢ⁿᵉ θ t) mmono⊢ⁿᶠ θ (lamⁿᶠ t) = lamⁿᶠ (mmono⊢ⁿᶠ θ t) mmono⊢ⁿᶠ θ (boxⁿᶠ t) = boxⁿᶠ t mmono⊢ⁿᶠ θ (pairⁿᶠ t u) = pairⁿᶠ (mmono⊢ⁿᶠ θ t) (mmono⊢ⁿᶠ θ u) mmono⊢ⁿᶠ θ unitⁿᶠ = unitⁿᶠ mmono⊢ⁿᵉ : ∀ {A Γ Δ Δ′} → Δ ⊆ Δ′ → Γ ⁏ Δ ⊢ⁿᵉ A → Γ ⁏ Δ′ ⊢ⁿᵉ A mmono⊢ⁿᵉ θ (varⁿᵉ i) = varⁿᵉ i mmono⊢ⁿᵉ θ (appⁿᵉ t u) = appⁿᵉ (mmono⊢ⁿᵉ θ t) (mmono⊢ⁿᶠ θ u) mmono⊢ⁿᵉ θ (mvarⁿᵉ i {{ts}} {{us}}) = mvarⁿᵉ (mono∈ θ i) {{mmono⊢⋆ⁿᶠ θ ts}} {{mmono⊢⍟ⁿᶠ θ us}} mmono⊢ⁿᵉ θ (unboxⁿᵉ t u) = unboxⁿᵉ (mmono⊢ⁿᵉ θ t) (mmono⊢ⁿᶠ (keep θ) u) mmono⊢ⁿᵉ θ (fstⁿᵉ t) = fstⁿᵉ (mmono⊢ⁿᵉ θ t) mmono⊢ⁿᵉ θ (sndⁿᵉ t) = sndⁿᵉ (mmono⊢ⁿᵉ θ t) mmono⊢⋆ⁿᶠ : ∀ {Ξ Γ Δ Δ′} → Δ ⊆ Δ′ → Γ ⁏ Δ ⊢⋆ⁿᶠ Ξ → Γ ⁏ Δ′ ⊢⋆ⁿᶠ Ξ mmono⊢⋆ⁿᶠ {∅} θ ∙ = ∙ mmono⊢⋆ⁿᶠ {Ξ , A} θ (ts , t) = mmono⊢⋆ⁿᶠ θ ts , mmono⊢ⁿᶠ θ t mmono⊢⍟ⁿᶠ : ∀ {Ξ Γ Δ Δ′} → Δ ⊆ Δ′ → Γ ⁏ Δ ⊢⍟ⁿᶠ Ξ → Γ ⁏ Δ′ ⊢⍟ⁿᶠ Ξ mmono⊢⍟ⁿᶠ {∅} θ ∙ = ∙ mmono⊢⍟ⁿᶠ {Ξ , _} θ (ts , t) = mmono⊢⍟ⁿᶠ θ ts , mmono⊢ⁿᶠ θ t mmono⊢⋆ⁿᵉ : ∀ {Ξ Γ Δ Δ′} → Δ ⊆ Δ′ → Γ ⁏ Δ ⊢⋆ⁿᵉ Ξ → Γ ⁏ Δ′ ⊢⋆ⁿᵉ Ξ mmono⊢⋆ⁿᵉ {∅} θ ∙ = ∙ mmono⊢⋆ⁿᵉ {Ξ , A} θ (ts , t) = mmono⊢⋆ⁿᵉ θ ts , mmono⊢ⁿᵉ θ t mmono⊢⍟ⁿᵉ : ∀ {Ξ Γ Δ Δ′} → Δ ⊆ Δ′ → Γ ⁏ Δ ⊢⍟ⁿᵉ Ξ → Γ ⁏ Δ′ ⊢⍟ⁿᵉ Ξ mmono⊢⍟ⁿᵉ {∅} θ ∙ = ∙ mmono⊢⍟ⁿᵉ {Ξ , _} θ (ts , t) = mmono⊢⍟ⁿᵉ θ ts , mmono⊢ⁿᵉ θ t -- Monotonicity using context pairs. mono²⊢ⁿᶠ : ∀ {A Π Π′} → Π ⊆² Π′ → Π ⊢ⁿᶠ A → Π′ ⊢ⁿᶠ A mono²⊢ⁿᶠ (η , θ) = mono⊢ⁿᶠ η ∘ mmono⊢ⁿᶠ θ mono²⊢ⁿᵉ : ∀ {A Π Π′} → Π ⊆² Π′ → Π ⊢ⁿᵉ A → Π′ ⊢ⁿᵉ A mono²⊢ⁿᵉ (η , θ) = mono⊢ⁿᵉ η ∘ mmono⊢ⁿᵉ θ -- Generalised reflexivity. refl⊢⋆ⁿᵉ : ∀ {Γ Ψ Δ} → {{_ : Ψ ⊆ Γ}} → Γ ⁏ Δ ⊢⋆ⁿᵉ Ψ refl⊢⋆ⁿᵉ {∅} {{done}} = ∙ refl⊢⋆ⁿᵉ {Γ , A} {{skip η}} = mono⊢⋆ⁿᵉ weak⊆ (refl⊢⋆ⁿᵉ {{η}}) refl⊢⋆ⁿᵉ {Γ , A} {{keep η}} = mono⊢⋆ⁿᵉ weak⊆ (refl⊢⋆ⁿᵉ {{η}}) , varⁿᵉ top

module EquationalTheory where open import Library open import Syntax open import RenamingAndSubstitution -- Single collapsing substitution. sub1 : ∀{Γ σ τ} → Tm Γ σ → Tm (Γ , σ) τ → Tm Γ τ sub1 {Γ}{σ}{τ} u t = sub (subId , u) t -- Typed β-η-equality. data _≡βη_ {Γ : Cxt} : ∀{σ} → Tm Γ σ → Tm Γ σ → Set where -- Axioms. beta≡ : ∀{σ τ} {t : Tm (Γ , σ) τ} {u : Tm Γ σ} → -------------------------- app (abs t) u ≡βη sub1 u t eta≡ : ∀{σ τ} (t : Tm Γ (σ ⇒ τ)) → ------------------------------------- abs (app (weak _ t) (var zero)) ≡βη t -- Congruence rules. var≡ : ∀{σ} (x : Var Γ σ) → --------------- var x ≡βη var x abs≡ : ∀{σ τ}{t t' : Tm (Γ , σ) τ} → t ≡βη t' → ---------------- abs t ≡βη abs t' app≡ : ∀{σ τ}{t t' : Tm Γ (σ ⇒ τ)}{u u' : Tm Γ σ} → t ≡βη t' → u ≡βη u' → --------------------- app t u ≡βη app t' u' -- Equivalence rules. refl≡ : ∀{a} (t {t'} : Tm Γ a) → t ≡ t' → ------- t ≡βη t' sym≡ : ∀{a}{t t' : Tm Γ a} (t'≡t : t' ≡βη t) → ----------------- t ≡βη t' trans≡ : ∀{a}{t₁ t₂ t₃ : Tm Γ a} (t₁≡t₂ : t₁ ≡βη t₂) (t₂≡t₃ : t₂ ≡βη t₃) → ---------------------------------- t₁ ≡βη t₃ -- A calculation on renamings needed for renaming of eta≡. ren-eta≡ : ∀ {Γ Δ a b} (t : Tm Γ (a ⇒ b)) (ρ : Ren Δ Γ) → ren (wkr ρ , zero) (ren (wkr renId) t) ≡ ren (wkr {σ = a} renId) (ren ρ t) ren-eta≡ t ρ = begin ren (wkr ρ , zero) (ren (wkr renId) t) ≡⟨ sym (rencomp _ _ _) ⟩ ren (renComp (wkr ρ , zero) (wkr renId)) t ≡⟨ cong (λ ρ₁ → ren ρ₁ t) (lemrr _ _ _) ⟩ ren (renComp (wkr ρ) renId) t ≡⟨ cong (λ ρ₁ → ren ρ₁ t) (ridr _) ⟩ ren (wkr ρ) t ≡⟨ cong (λ ρ₁ → ren ρ₁ t) (cong wkr (sym (lidr _))) ⟩ ren (wkr (renComp renId ρ)) t ≡⟨ cong (λ ρ₁ → ren ρ₁ t) (sym (wkrcomp _ _)) ⟩ ren (renComp (wkr renId) ρ) t ≡⟨ rencomp _ _ _ ⟩ ren (wkr renId) (ren ρ t) ∎ where open ≡-Reasoning -- Definitional equality is closed under renaming. ren≡βη : ∀{Γ a} {t : Tm Γ a}{t' : Tm Γ a} → t ≡βη t' → ∀{Δ}(ρ : Ren Δ Γ) → ren ρ t ≡βη ren ρ t' ren≡βη (beta≡ {t = t}{u = u}) ρ = trans≡ beta≡ $ refl≡ _ $ trans (subren (subId , ren ρ u) (liftr ρ) t) (trans (cong (λ xs → sub xs t) (cong₂ Sub._,_ (trans (lemsr subId (ren ρ u) ρ) (trans (sidl (ren2sub ρ)) (sym $ sidr (ren2sub ρ)))) (ren2subren ρ u))) (sym $ rensub ρ (subId , u) t)) -- TODO: factor out reasoning about renamings and substitutions ren≡βη (eta≡ {a} t) ρ rewrite ren-eta≡ t ρ = eta≡ (ren ρ t) -- OLD: -- ren≡βη (eta≡ t) ρ = trans≡ -- (abs≡ (app≡ (refl≡ _ -- (trans (sym $ rencomp (liftr ρ) (wkr renId) t) -- (trans (cong (λ xs → ren xs t) -- (trans (lemrr (wkr ρ) zero renId) -- (trans (ridr (wkr ρ)) -- (trans (cong wkr (sym (lidr ρ))) -- (sym (wkrcomp renId ρ)))))) -- (rencomp (wkr renId) ρ t)))) -- (refl≡ _))) -- (eta≡ _) ren≡βη (var≡ x) ρ = var≡ (lookr ρ x) ren≡βη (abs≡ p) ρ = abs≡ (ren≡βη p (liftr ρ)) ren≡βη (app≡ p q) ρ = app≡ (ren≡βη p ρ) (ren≡βη q ρ) ren≡βη (refl≡ _ refl) ρ = refl≡ _ refl ren≡βη (sym≡ p) ρ = sym≡ (ren≡βη p ρ) ren≡βη (trans≡ p q) ρ = trans≡ (ren≡βη p ρ) (ren≡βη q ρ)

open import Relation.Binary.Core module InsertSort.Impl2.Correctness.Order {A : Set} (_≤_ : A → A → Set) (tot≤ : Total _≤_) where open import Data.List open import Function using (_∘_) open import InsertSort.Impl2 _≤_ tot≤ open import List.Sorted _≤_ open import OList _≤_ open import OList.Properties _≤_ theorem-insertSort-sorted : (xs : List A) → Sorted (forget (insertSort xs)) theorem-insertSort-sorted = lemma-olist-sorted ∘ insertSort

{-# OPTIONS --prop #-} open import Agda.Builtin.Nat data T : Nat → Prop where To : T zero Tn : ∀ n → T n ummm : ∀ n → T n → {! !} ummm n t = {! t !}

-- A variant of code reported by Andreas Abel. {-# OPTIONS --guardedness --sized-types #-} open import Common.Coinduction renaming (∞ to Delay) open import Common.Size open import Common.Product data ⊥ : Set where record Stream (A : Set) : Set where inductive constructor delay field force : Delay (A × Stream A) open Stream head : ∀{A} → Stream A → A head s = proj₁ (♭ (force s)) -- This type should be empty, as Stream A is isomorphic to ℕ → A. data D : (i : Size) → Set where lim : ∀ i → Stream (D i) → D (↑ i) -- Emptiness witness for D. empty : ∀ i → D i → ⊥ empty .(↑ i) (lim i s) = empty i (head s) -- BAD: But we can construct an inhabitant. inh : Stream (D ∞) inh = delay (♯ (lim ∞ inh , inh)) -- Should be rejected by termination checker. absurd : ⊥ absurd = empty ∞ (lim ∞ inh)

open import Oscar.Prelude open import Oscar.Class open import Oscar.Class.IsPrecategory open import Oscar.Class.IsCategory open import Oscar.Class.HasEquivalence open import Oscar.Class.Reflexivity open import Oscar.Class.Symmetry open import Oscar.Class.Transextensionality open import Oscar.Class.Transassociativity open import Oscar.Class.Transleftidentity open import Oscar.Class.Transrightidentity open import Oscar.Class.Transitivity open import Oscar.Class.Precategory open import Oscar.Class.Category open import Oscar.Property.Setoid.Proposequality open import Oscar.Property.Setoid.Proposextensequality open import Oscar.Data.Proposequality import Oscar.Class.Reflexivity.Function import Oscar.Class.Transextensionality.Proposequality module Oscar.Property.Category.Function where module _ {𝔬 : Ł} where instance TransitivityFunction : Transitivity.class Function⟦ 𝔬 ⟧ TransitivityFunction .⋆ f g = g ∘ f module _ {𝔬 : Ł} where instance HasEquivalenceFunction : ∀ {A B : Ø 𝔬} → HasEquivalence (Function⟦ 𝔬 ⟧ A B) 𝔬 HasEquivalenceFunction .⋆ = _≡̇_ HasEquivalenceFunction .Rℭlass.⋆⋆ = ! instance TransassociativityFunction : Transassociativity.class Function⟦ 𝔬 ⟧ _≡̇_ transitivity TransassociativityFunction .⋆ _ _ _ _ = ∅ instance 𝓣ransextensionalityFunctionProposextensequality : Transextensionality.class Function⟦ 𝔬 ⟧ Proposextensequality transitivity 𝓣ransextensionalityFunctionProposextensequality .⋆ {f₂ = f₂} f₁≡̇f₂ g₁≡̇g₂ x rewrite f₁≡̇f₂ x = g₁≡̇g₂ (f₂ x) instance IsPrecategoryFunction : IsPrecategory Function⟦ 𝔬 ⟧ _≡̇_ transitivity IsPrecategoryFunction = ∁ instance TransleftidentityFunction : Transleftidentity.class Function⟦ 𝔬 ⟧ _≡̇_ ε transitivity TransleftidentityFunction .⋆ _ = ∅ TransrightidentityFunction : Transrightidentity.class Function⟦ 𝔬 ⟧ _≡̇_ ε transitivity TransrightidentityFunction .⋆ _ = ∅ instance IsCategoryFunction : IsCategory Function⟦ 𝔬 ⟧ _≡̇_ ε transitivity IsCategoryFunction = ∁ PrecategoryFunction : Precategory _ _ _ PrecategoryFunction = ∁ Function⟦ 𝔬 ⟧ Proposextensequality transitivity CategoryFunction : Category _ _ _ CategoryFunction = ∁ Function⟦ 𝔬 ⟧ Proposextensequality ε transitivity

------------------------------------------------------------------------ -- The Agda standard library -- -- Floats ------------------------------------------------------------------------ module Data.Float where open import Data.Bool hiding (_≟_) open import Relation.Nullary.Decidable open import Relation.Nullary open import Relation.Binary.PropositionalEquality as PropEq using (_≡_) open import Relation.Binary.PropositionalEquality.TrustMe open import Data.String using (String) postulate Float : Set {-# BUILTIN FLOAT Float #-} primitive primFloatEquality : Float → Float → Bool primShowFloat : Float → String show : Float → String show = primShowFloat _≟_ : (x y : Float) → Dec (x ≡ y) x ≟ y with primFloatEquality x y ... | true = yes trustMe ... | false = no whatever where postulate whatever : _

module Issue2447.Type-error where import Issue2447.M Rejected : Set Rejected = Set

{-# OPTIONS --safe --warning=error --without-K #-} open import Groups.Homomorphisms.Definition open import Groups.Definition open import Setoids.Setoids open import Rings.Definition open import Agda.Primitive using (Level; lzero; lsuc; _⊔_) module Rings.Homomorphisms.Definition where record RingHom {m n o p : _} {A : Set m} {B : Set n} {SA : Setoid {m} {o} A} {SB : Setoid {n} {p} B} {_+A_ : A → A → A} {_*A_ : A → A → A} (R : Ring SA _+A_ _*A_) {_+B_ : B → B → B} {_*B_ : B → B → B} (S : Ring SB _+B_ _*B_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where open Ring S open Group additiveGroup open Setoid SB field preserves1 : f (Ring.1R R) ∼ 1R ringHom : {r s : A} → f (r *A s) ∼ (f r) *B (f s) groupHom : GroupHom (Ring.additiveGroup R) additiveGroup f

{-# OPTIONS --cubical --no-import-sorts --safe #-} module Cubical.Relation.Binary.Base where open import Cubical.Core.Everything open import Cubical.Foundations.Prelude open import Cubical.Foundations.HLevels open import Cubical.Foundations.Isomorphism open import Cubical.Foundations.Equiv open import Cubical.Foundations.Equiv.Fiberwise open import Cubical.Data.Sigma open import Cubical.HITs.SetQuotients.Base open import Cubical.HITs.PropositionalTruncation.Base private variable ℓA ℓ≅A ℓA' ℓ≅A' : Level Rel : ∀ {ℓ} (A B : Type ℓ) (ℓ' : Level) → Type (ℓ-max ℓ (ℓ-suc ℓ')) Rel A B ℓ' = A → B → Type ℓ' PropRel : ∀ {ℓ} (A B : Type ℓ) (ℓ' : Level) → Type (ℓ-max ℓ (ℓ-suc ℓ')) PropRel A B ℓ' = Σ[ R ∈ Rel A B ℓ' ] ∀ a b → isProp (R a b) idPropRel : ∀ {ℓ} (A : Type ℓ) → PropRel A A ℓ idPropRel A .fst a a' = ∥ a ≡ a' ∥ idPropRel A .snd _ _ = squash invPropRel : ∀ {ℓ ℓ'} {A B : Type ℓ} → PropRel A B ℓ' → PropRel B A ℓ' invPropRel R .fst b a = R .fst a b invPropRel R .snd b a = R .snd a b compPropRel : ∀ {ℓ ℓ' ℓ''} {A B C : Type ℓ} → PropRel A B ℓ' → PropRel B C ℓ'' → PropRel A C (ℓ-max ℓ (ℓ-max ℓ' ℓ'')) compPropRel R S .fst a c = ∥ Σ[ b ∈ _ ] (R .fst a b × S .fst b c) ∥ compPropRel R S .snd _ _ = squash graphRel : ∀ {ℓ} {A B : Type ℓ} → (A → B) → Rel A B ℓ graphRel f a b = f a ≡ b module BinaryRelation {ℓ ℓ' : Level} {A : Type ℓ} (R : Rel A A ℓ') where isRefl : Type (ℓ-max ℓ ℓ') isRefl = (a : A) → R a a isSym : Type (ℓ-max ℓ ℓ') isSym = (a b : A) → R a b → R b a isTrans : Type (ℓ-max ℓ ℓ') isTrans = (a b c : A) → R a b → R b c → R a c record isEquivRel : Type (ℓ-max ℓ ℓ') where constructor equivRel field reflexive : isRefl symmetric : isSym transitive : isTrans isPropValued : Type (ℓ-max ℓ ℓ') isPropValued = (a b : A) → isProp (R a b) isEffective : Type (ℓ-max ℓ ℓ') isEffective = (a b : A) → isEquiv (eq/ {R = R} a b) impliesIdentity : Type _ impliesIdentity = {a a' : A} → (R a a') → (a ≡ a') -- the total space corresponding to the binary relation w.r.t. a relSinglAt : (a : A) → Type (ℓ-max ℓ ℓ') relSinglAt a = Σ[ a' ∈ A ] (R a a') -- the statement that the total space is contractible at any a contrRelSingl : Type (ℓ-max ℓ ℓ') contrRelSingl = (a : A) → isContr (relSinglAt a) isUnivalent : Type (ℓ-max ℓ ℓ') isUnivalent = (a a' : A) → (R a a') ≃ (a ≡ a') contrRelSingl→isUnivalent : isRefl → contrRelSingl → isUnivalent contrRelSingl→isUnivalent ρ c a a' = isoToEquiv i where h : isProp (relSinglAt a) h = isContr→isProp (c a) aρa : relSinglAt a aρa = a , ρ a Q : (y : A) → a ≡ y → _ Q y _ = R a y i : Iso (R a a') (a ≡ a') Iso.fun i r = cong fst (h aρa (a' , r)) Iso.inv i = J Q (ρ a) Iso.rightInv i = J (λ y p → cong fst (h aρa (y , J Q (ρ a) p)) ≡ p) (J (λ q _ → cong fst (h aρa (a , q)) ≡ refl) (J (λ α _ → cong fst α ≡ refl) refl (isContr→isProp (isProp→isContrPath h aρa aρa) refl (h aρa aρa))) (sym (JRefl Q (ρ a)))) Iso.leftInv i r = J (λ w β → J Q (ρ a) (cong fst β) ≡ snd w) (JRefl Q (ρ a)) (h aρa (a' , r)) isUnivalent→contrRelSingl : isUnivalent → contrRelSingl isUnivalent→contrRelSingl u a = q where abstract f : (x : A) → a ≡ x → R a x f x p = invEq (u a x) p t : singl a → relSinglAt a t (x , p) = x , f x p q : isContr (relSinglAt a) q = isOfHLevelRespectEquiv 0 (t , totalEquiv _ _ f λ x → invEquiv (u a x) .snd) (isContrSingl a) EquivRel : ∀ {ℓ} (A : Type ℓ) (ℓ' : Level) → Type (ℓ-max ℓ (ℓ-suc ℓ')) EquivRel A ℓ' = Σ[ R ∈ Rel A A ℓ' ] BinaryRelation.isEquivRel R EquivPropRel : ∀ {ℓ} (A : Type ℓ) (ℓ' : Level) → Type (ℓ-max ℓ (ℓ-suc ℓ')) EquivPropRel A ℓ' = Σ[ R ∈ PropRel A A ℓ' ] BinaryRelation.isEquivRel (R .fst) record RelIso {A : Type ℓA} (_≅_ : Rel A A ℓ≅A) {A' : Type ℓA'} (_≅'_ : Rel A' A' ℓ≅A') : Type (ℓ-max (ℓ-max ℓA ℓA') (ℓ-max ℓ≅A ℓ≅A')) where constructor reliso field fun : A → A' inv : A' → A rightInv : (a' : A') → fun (inv a') ≅' a' leftInv : (a : A) → inv (fun a) ≅ a open BinaryRelation RelIso→Iso : {A : Type ℓA} {A' : Type ℓA'} (_≅_ : Rel A A ℓ≅A) (_≅'_ : Rel A' A' ℓ≅A') (uni : impliesIdentity _≅_) (uni' : impliesIdentity _≅'_) (f : RelIso _≅_ _≅'_) → Iso A A' Iso.fun (RelIso→Iso _ _ _ _ f) = RelIso.fun f Iso.inv (RelIso→Iso _ _ _ _ f) = RelIso.inv f Iso.rightInv (RelIso→Iso _ _ uni uni' f) a' = uni' (RelIso.rightInv f a') Iso.leftInv (RelIso→Iso _ _ uni uni' f) a = uni (RelIso.leftInv f a)

module Thesis.SIRelBigStep.Types where open import Data.Empty open import Data.Product open import Relation.Nullary open import Relation.Binary.PropositionalEquality open import Relation.Binary hiding (_⇒_) data Type : Set where _⇒_ : (σ τ : Type) → Type pair : (σ τ : Type) → Type nat : Type infixr 20 _⇒_ open import Base.Syntax.Context Type public open import Base.Syntax.Vars Type public -- Decidable equivalence for types and contexts. Needed later for ⊕ on closures. ⇒-inj : ∀ {τ1 τ2 τ3 τ4 : Type} → _≡_ {A = Type} (τ1 ⇒ τ2) (τ3 ⇒ τ4) → τ1 ≡ τ3 × τ2 ≡ τ4 ⇒-inj refl = refl , refl pair-inj : ∀ {τ1 τ2 τ3 τ4 : Type} → _≡_ {A = Type} (pair τ1 τ2) (pair τ3 τ4) → τ1 ≡ τ3 × τ2 ≡ τ4 pair-inj refl = refl , refl _≟Type_ : (τ1 τ2 : Type) → Dec (τ1 ≡ τ2) (τ1 ⇒ τ2) ≟Type (τ3 ⇒ τ4) with τ1 ≟Type τ3 | τ2 ≟Type τ4 (τ1 ⇒ τ2) ≟Type (.τ1 ⇒ .τ2) | yes refl | yes refl = yes refl (τ1 ⇒ τ2) ≟Type (.τ1 ⇒ τ4) | yes refl | no ¬q = no (λ x → ¬q (proj₂ (⇒-inj x))) (τ1 ⇒ τ2) ≟Type (τ3 ⇒ τ4) | no ¬p | q = no (λ x → ¬p (proj₁ (⇒-inj x))) (τ1 ⇒ τ2) ≟Type pair τ3 τ4 = no (λ ()) (τ1 ⇒ τ2) ≟Type nat = no (λ ()) pair τ1 τ2 ≟Type (τ3 ⇒ τ4) = no (λ ()) pair τ1 τ2 ≟Type pair τ3 τ4 with τ1 ≟Type τ3 | τ2 ≟Type τ4 pair τ1 τ2 ≟Type pair .τ1 .τ2 | yes refl | yes refl = yes refl pair τ1 τ2 ≟Type pair τ3 τ4 | yes p | no ¬q = no (λ x → ¬q (proj₂ (pair-inj x))) pair τ1 τ2 ≟Type pair τ3 τ4 | no ¬p | q = no (λ x → ¬p (proj₁ (pair-inj x))) pair τ1 τ2 ≟Type nat = no (λ ()) nat ≟Type pair τ1 τ2 = no (λ ()) nat ≟Type (τ1 ⇒ τ2) = no (λ ()) nat ≟Type nat = yes refl •-inj : ∀ {τ1 τ2 : Type} {Γ1 Γ2 : Context} → _≡_ {A = Context} (τ1 • Γ1) (τ2 • Γ2) → τ1 ≡ τ2 × Γ1 ≡ Γ2 •-inj refl = refl , refl _≟Ctx_ : (Γ1 Γ2 : Context) → Dec (Γ1 ≡ Γ2) ∅ ≟Ctx ∅ = yes refl ∅ ≟Ctx (τ2 • Γ2) = no (λ ()) (τ1 • Γ1) ≟Ctx ∅ = no (λ ()) (τ1 • Γ1) ≟Ctx (τ2 • Γ2) with τ1 ≟Type τ2 | Γ1 ≟Ctx Γ2 (τ1 • Γ1) ≟Ctx (.τ1 • .Γ1) | yes refl | yes refl = yes refl (τ1 • Γ1) ≟Ctx (.τ1 • Γ2) | yes refl | no ¬q = no (λ x → ¬q (proj₂ (•-inj x))) (τ1 • Γ1) ≟Ctx (τ2 • Γ2) | no ¬p | q = no (λ x → ¬p (proj₁ (•-inj x))) ≟Ctx-refl : ∀ Γ → Γ ≟Ctx Γ ≡ yes refl ≟Ctx-refl Γ with Γ ≟Ctx Γ ≟Ctx-refl Γ | yes refl = refl ≟Ctx-refl Γ | no ¬p = ⊥-elim (¬p refl)

module MLib.Prelude where open import MLib.Prelude.FromStdlib public module Fin where open import MLib.Prelude.Fin public open Fin using (Fin; zero; suc) hiding (module Fin) public

open import Agda.Builtin.Nat data Vec (A : Set) : Nat → Set where [] : Vec A 0 _∷_ : ∀ {n} → A → Vec A n → Vec A (suc n) infixr 5 _∷_ _++_ _++_ : ∀ {A m n} → Vec A m → Vec A n → Vec A (m + n) [] ++ ys = ys (x ∷ xs) ++ ys = x ∷ xs ++ ys T : ∀ {A n} → Vec A n → Set T [] = Nat T (x ∷ xs) = Vec Nat 0 foo : ∀ {A} m n (xs : Vec A m) (ys : Vec A n) → T (xs ++ ys) foo m n xs ys with {m + n} | xs ++ ys ... | [] = 0 ... | z ∷ zs = []

module HelloWorld where open import Common.IO open import Common.Unit main : IO Unit main = putStr "Hello World"

open import Agda.Builtin.Equality data D (A : Set) : Set where {-# INJECTIVE D #-} test : {A₁ A₂ : Set} → D A₁ ≡ D A₂ → A₁ ≡ A₂ test refl = refl postulate f : Set → Set {-# INJECTIVE f #-} test₂ : {A₁ A₂ : Set} → f A₁ ≡ f A₂ → A₁ ≡ A₂ test₂ refl = refl

module Avionics.Bool where open import Data.Bool using (Bool; true; false; _∧_; T) open import Data.Unit using (⊤; tt) open import Data.Product using (_×_; _,_) open import Relation.Binary.PropositionalEquality using (_≡_; refl; inspect; [_]) --open import Avionics.Product using (_×_; ⟨_,_⟩) --TODO: Replace with T⇔≡ from standard library ≡→T : ∀ {b : Bool} → b ≡ true → T b ≡→T refl = tt T→≡ : ∀ {b : Bool} → T b → b ≡ true T→≡ {true} tt = refl T∧→× : ∀ {x y} → T (x ∧ y) → (T x) × (T y) T∧→× {true} {true} tt = tt , tt --TODO: Find a way to extract the function below from `T-∧` (standard library) --T∧→× {x} {y} = ? -- Equivalence.to (T-∧ {x} {y}) ×→T∧ : ∀ {x y} → (T x) × (T y) → T (x ∧ y) ×→T∧ {true} {true} (tt , tt) = tt lem∧ : {a b : Bool} → a ∧ b ≡ true → a ≡ true × b ≡ true lem∧ {true} {true} refl = refl , refl ∧≡true→×≡ : ∀ {A B : Set} {f : A → Bool} {g : B → Bool} (n : A) (m : B) → f n ∧ g m ≡ true → f n ≡ true × g m ≡ true ∧≡true→×≡ {f = f} {g = g} n m fn∧gm≡true = lem∧ {f n} {g m} fn∧gm≡true

{-# OPTIONS --universe-polymorphism #-} open import Level open import Categories.Category module Categories.Power.Functorial {o ℓ e : Level} (C : Category o ℓ e) where open import Relation.Binary.PropositionalEquality using (≡-irrelevance) open import Data.Unit using (⊤; tt) open import Function using () renaming (_∘_ to _∙_) open import Relation.Binary using (module IsEquivalence) open import Categories.Support.PropositionalEquality import Categories.Power as Power open Power C open import Categories.Functor hiding (identityˡ) renaming (_≡_ to _≡F_; _∘_ to _∘F_; id to idF) open import Categories.NaturalTransformation using (NaturalTransformation; module NaturalTransformation) open import Categories.Discrete open import Categories.Equivalence.Strong hiding (module Equiv) open Category using (Obj) open Category C using (_⇒_; _∘_; _≡_; module Equiv) open import Categories.FunctorCategory open import Categories.Lift open import Categories.NaturalIsomorphism as NI using (NaturalIsomorphism; module NaturalIsomorphism) open Functor using () renaming (F₀ to map₀; F₁ to map₁) open NaturalTransformation using (η) z : Level z = o ⊔ ℓ ⊔ e open import Categories.Categories using (Categories) import Categories.Morphisms as M open M (Categories z z e) using (_≅_) exp→functor₀ : ∀ {I} → Obj (Exp I) → Functor (Discrete I) C exp→functor₀ X = record { F₀ = X ; F₁ = my-F₁ ; identity = Equiv.refl ; homomorphism = λ {_ _ _ A≣B B≣C} → my-homomorphism A≣B B≣C ; F-resp-≡ = λ {_ _ e1 e2} _ → Equiv.reflexive (≣-cong my-F₁ (≡-irrelevance e1 e2)) } where my-F₁ : ∀ {A B} → (A ≣ B) → (X A ⇒ X B) my-F₁ ≣-refl = C.id .my-homomorphism : ∀ {A B C} (A≣B : A ≣ B) (B≣C : B ≣ C) → my-F₁ (≣-trans A≣B B≣C) ≡ my-F₁ B≣C ∘ my-F₁ A≣B my-homomorphism ≣-refl ≣-refl = Equiv.sym C.identityˡ exp→functor₁ : ∀ {I} {X Y} → Exp I [ X , Y ] → NaturalTransformation (exp→functor₀ X) (exp→functor₀ Y) exp→functor₁ {X = X} {Y} F = record { η = F; commute = my-commute } where .my-commute : ∀ {A B} (A≣B : A ≣ B) → F B ∘ map₁ (exp→functor₀ X) A≣B ≡ map₁ (exp→functor₀ Y) A≣B ∘ F A my-commute ≣-refl = Equiv.trans C.identityʳ (Equiv.sym C.identityˡ) exp→functor : ∀ {I} → Functor (Exp I) (Functors (Discrete I) C) exp→functor = record { F₀ = exp→functor₀ ; F₁ = exp→functor₁ ; identity = Equiv.refl ; homomorphism = Equiv.refl ; F-resp-≡ = λ eqs {x} → eqs x } functor→exp : ∀ {I} → Functor (Functors (Discrete I) C) (Exp I) functor→exp = record { F₀ = Functor.F₀ ; F₁ = NaturalTransformation.η ; identity = λ _ → Equiv.refl ; homomorphism = λ _ → Equiv.refl ; F-resp-≡ = λ eqs i → eqs {i} } semicanonical : ∀ {I} → (F : Functor (Discrete I) C) → F ≡F (exp→functor₀ (map₀ F)) semicanonical {I} F ≣-refl = ≡⇒∼ F.identity where module F = Functor F open Heterogeneous C exp≋functor : ∀ {I} → StrongEquivalence (Exp I) (Functors (Discrete I) C) exp≋functor {I} = record { F = exp→functor ; G = functor→exp ; weak-inverse = record { F∘G≅id = record { F⇒G = record { η = Sc.F⇐G ; commute = λ f → Equiv.trans C.identityˡ (Equiv.sym C.identityʳ) } ; F⇐G = record { η = Sc.F⇒G ; commute = λ f → Equiv.trans C.identityˡ (Equiv.sym C.identityʳ) } ; iso = λ X → record { isoˡ = C.identityˡ; isoʳ = C.identityˡ } } ; G∘F≅id = IsEquivalence.refl NI.equiv } } where FDIC = Functors (Discrete I) C module Sc (X : Obj FDIC) = NaturalIsomorphism (NI.≡→iso X (exp→functor₀ (map₀ X)) (semicanonical X)) exp≅functor : (ext : ∀ {a b} {A : Set a} {B : A → Set b} (f g : (x : A) → B x) → (∀ x → f x ≣ g x) → f ≣ g) → (id-propositionally-unique : (A : C.Obj) (f : A ⇒ A) → .(f ≡ C.id) → f ≣ C.id) → ∀ {I} → LiftC z z e (Exp I) ≅ Functors (Discrete I) C exp≅functor ext id-propositionally-unique {I} = record { f = LiftFˡ exp→functor ; g = LiftFʳ functor→exp ; iso = record { isoˡ = λ f → Heterogeneous.refl _ ; isoʳ = λ {A B} f → ir A B f } } where FDIC = Functors (Discrete I) C f∘g = LiftFˡ exp→functor ∘F LiftFʳ functor→exp squash : ∀ (A : Obj FDIC) → Obj FDIC squash A = exp→functor₀ (map₀ A) ≣-cong′ : ∀ {a b} {A : Set a} {B : Set b} {x y : A} (f : (z : A) → (y ≣ z) → B) → (eq : x ≣ y) → f x (≣-sym eq) ≣ f y ≣-refl ≣-cong′ f ≣-refl = ≣-refl squash-does-nothing : (A : Obj FDIC) → squash A ≣ A squash-does-nothing A = ≣-cong′ {B = Obj FDIC} (λ f eq → record { F₀ = map₀ A; F₁ = λ {i j : I} → f i j; identity = λ {i} → ≣-subst (λ f → f i i ≣-refl ≡ C.id) eq (Functor.identity A {i}); homomorphism = λ {i j k i≣j j≣k} → ≣-subst (λ f → f i k (≣-trans i≣j j≣k) ≡ f j k j≣k ∘ f i j i≣j) eq (Functor.homomorphism A {f = i≣j} {j≣k}); F-resp-≡ = λ {i j ij ji} → ≣-subst (λ f → ⊤ → f i j ij ≡ f i j ji) eq (Functor.F-resp-≡ A {i} {j} {ij} {ji}) }) lemma₃ where .lemma₁ : (i : I) (eq : i ≣ i) → map₁ A eq ≡ C.id lemma₁ i eq = Equiv.trans (Functor.F-resp-≡ A tt) (Functor.identity A) lemma₂ : (i j : I) (eq : i ≣ j) → map₁ (squash A) eq ≣ map₁ A eq lemma₂ i .i ≣-refl = ≣-sym (id-propositionally-unique (map₀ A i) (map₁ A ≣-refl) (lemma₁ i ≣-refl)) lemma₃ : (λ (i j : I) → map₁ (squash A) {i} {j}) ≣ (λ (i j : I) → map₁ A {i} {j}) lemma₃ = ext (λ (i j : I) → map₁ (squash A)) (λ (i j : I) → map₁ A) (λ (i : I) → ext (λ (j : I) → map₁ (squash A)) (λ (j : I) → map₁ A) (λ (j : I) → ext (map₁ (squash A)) (map₁ A) (lemma₂ i j))) ∼-subst : ∀ {o ℓ e} {C : Category o ℓ e} {A B A′ B′ : Obj C} (f : C [ A , B ]) (g : C [ A′ , B′ ]) (A′≣A : A′ ≣ A) (B′≣B : B′ ≣ B) → .(C [ ≣-subst (λ X → C [ X , B ]) A′≣A (≣-subst (λ Y → C [ A′ , Y ]) B′≣B g) ≡ f ]) → C [ g ∼ f ] ∼-subst {C = C} f g ≣-refl ≣-refl eq = ≡⇒∼ eq where open Heterogeneous C .∼-unsubst : ∀ {o ℓ e} {C : Category o ℓ e} {A B A′ B′ : Obj C} (f : C [ A , B ]) (g : C [ A′ , B′ ]) (A′≣A : A′ ≣ A) (B′≣B : B′ ≣ B) → C [ g ∼ f ] → C [ ≣-subst (λ X → C [ X , B ]) A′≣A (≣-subst (λ Y → C [ A′ , Y ]) B′≣B g) ≡ f ] ∼-unsubst f g ≣-refl ≣-refl (Heterogeneous.≡⇒∼ eq) = irr eq where open Heterogeneous C ≣-subst-irrel : ∀ {a p} {A : Set a} (P : A → Set p) {x y : A} → (eq₁ eq₂ : x ≣ y) → ∀ {it} → ≣-subst P eq₁ it ≣ ≣-subst P eq₂ it ≣-subst-irrel _ ≣-refl ≣-refl = ≣-refl ≣-subst-cong : ∀ {a a′ p} {A : Set a} {A′ : Set a′} (P′ : A′ → Set p) (f : A → A′) {x y : A} (eq : x ≣ y) → ∀ {it} → ≣-subst (P′ ∙ f) eq it ≣ ≣-subst P′ (≣-cong f eq) it ≣-subst-cong P′ f ≣-refl = ≣-refl ≣-subst-fatten : ∀ {a a′ p} {A : Set a} {A′ : Set a′} (P′ : A′ → Set p) (f : A → A′) {x y : A} (eq : x ≣ y) (eq′ : f x ≣ f y) → ∀ {it} → ≣-subst (P′ ∙ f) eq it ≣ ≣-subst P′ eq′ it ≣-subst-fatten P′ f eq eq′ = ≣-trans (≣-subst-cong P′ f eq) (≣-subst-irrel P′ (≣-cong f eq) eq′) η-under-substˡ : ∀ {o ℓ e o′ ℓ′ e′} {C : Category o ℓ e} {D : Category o′ ℓ′ e′} {F F′ G : Functor C D} (α : NaturalTransformation F′ G) (F′≣F : F′ ≣ F) (c : Obj C) → η (≣-subst (λ H → NaturalTransformation H G) F′≣F α) c ≣ (≣-subst (λ (H : Functor C D) → D [ map₀ H c , map₀ G c ]) F′≣F (η α c)) η-under-substˡ α ≣-refl c = ≣-refl η-under-substʳ : ∀ {o ℓ e o′ ℓ′ e′} {C : Category o ℓ e} {D : Category o′ ℓ′ e′} {F G G′ : Functor C D} (α : NaturalTransformation F G′) (G′≣G : G′ ≣ G) (c : Obj C) → η (≣-subst (λ H → NaturalTransformation F H) G′≣G α) c ≣ (≣-subst (λ (H : Functor C D) → D [ map₀ F c , map₀ H c ]) G′≣G (η α c)) η-under-substʳ α ≣-refl c = ≣-refl .lemma : (A B : Obj FDIC) (f : FDIC [ A , B ]) (i : I) → C [ η (≣-subst (λ X → FDIC [ X , B ]) (squash-does-nothing A) (≣-subst (λ Y → FDIC [ squash A , Y ]) (squash-does-nothing B) (map₁ f∘g f))) i ≡ η f i ] lemma A B f i = let loc X = map₀ X i in let sqdnA = squash-does-nothing A in let sqdnB = squash-does-nothing B in let MESS = ≣-subst (λ Y → FDIC [ squash A , Y ]) sqdnB (map₁ f∘g f) in C.Equiv.reflexive ( begin η (≣-subst (λ X → FDIC [ X , B ]) sqdnA MESS) i ≣⟨ η-under-substˡ MESS sqdnA i ⟩ ≣-subst (λ X → C [ loc X , loc B ]) sqdnA (η MESS i) ≣⟨ ≣-subst-fatten (λ c → C [ c , loc B ]) loc sqdnA ≣-refl ⟩ η (≣-subst (λ Y → FDIC [ squash A , Y ]) sqdnB (map₁ f∘g f)) i ≣⟨ η-under-substʳ (map₁ f∘g f) sqdnB i ⟩ ≣-subst (λ Y → C [ loc A , loc Y ]) sqdnB (η (map₁ f∘g f) i) ≣⟨ ≣-subst-fatten (λ c → C [ loc A , c ]) loc sqdnB ≣-refl ⟩ η f i ∎) where open ≣-reasoning ir : (A B : Obj FDIC) (f : FDIC [ A , B ]) → FDIC [ map₁ f∘g f ∼ f ] ir A B f = ∼-subst {C = FDIC} f (map₁ f∘g f) (squash-does-nothing A) (squash-does-nothing B) (λ {x} → lemma A B f x)

{-# OPTIONS --without-K --exact-split --allow-unsolved-metas #-} module 16-sets where import 15-number-theory open 15-number-theory public

module examplesPaperJFP.ConsoleInterface where open import examplesPaperJFP.NativeIOSafe open import examplesPaperJFP.BasicIO hiding (main) module _ where data ConsoleCommand : Set where getLine : ConsoleCommand putStrLn : String → ConsoleCommand ConsoleResponse : ConsoleCommand → Set ConsoleResponse getLine = Maybe String ConsoleResponse (putStrLn s) = Unit ConsoleInterface : IOInterface Command ConsoleInterface = ConsoleCommand Response ConsoleInterface = ConsoleResponse

------------------------------------------------------------------------ -- A variant of the set quotients from Quotient ------------------------------------------------------------------------ {-# OPTIONS --erased-cubical --safe #-} -- Partly following the HoTT book. -- -- Unlike the HoTT book, but following the cubical library (in which -- set quotients were implemented by Zesen Qian and Anders Mörtberg), -- the quotienting relations are not always required to be -- propositional. Furthermore, unlike the code in the cubical library, -- the set truncation constructor takes an argument stating that the -- quotienting relation is propositional. -- The module is parametrised by a notion of equality. The higher -- constructors of the HIT defining quotients use path equality, but -- the supplied notion of equality is used for many other things. import Equality.Path as P module Quotient.Set-truncated-if-propositional {e⁺} (eq : ∀ {a p} → P.Equality-with-paths a p e⁺) where private open module D = P.Derived-definitions-and-properties eq hiding (elim) open import Logical-equivalence using (_⇔_) open import Prelude open import Bijection equality-with-J using (_↔_) open import Equality.Decidable-UIP equality-with-J open import Equality.Path.Isomorphisms eq open import Equivalence equality-with-J as Eq using (_≃_) open import Equivalence-relation equality-with-J open import Function-universe equality-with-J as F hiding (_∘_; id) open import H-level equality-with-J as H import H-level P.equality-with-J as PH open import H-level.Closure equality-with-J import H-level.Truncation.Church equality-with-J as Trunc open import H-level.Truncation.Propositional eq as TruncP using (∥_∥; ∣_∣; Surjective; Axiom-of-countable-choice) open import Preimage equality-with-J using (_⁻¹_) import Quotient eq as Quotient import Quotient.Families-of-equivalence-classes equality-with-J as QF open import Sum equality-with-J open import Surjection equality-with-J using (_↠_) open import Univalence-axiom equality-with-J private variable a a₁ a₂ b p : Level k : Isomorphism-kind A A₁ A₂ B : Type a P : A → Type p e f r r₁ r₂ x y : A R R₁ R₂ : A → A → Type r ------------------------------------------------------------------------ -- Quotients -- The quotient type constructor. infix 5 _/_ data _/_ (A : Type a) (R : A → A → Type r) : Type (a ⊔ r) where [_] : A → A / R []-respects-relationᴾ : R x y → [ x ] P.≡ [ y ] /-is-setᴾ : (∀ x y → Is-proposition (R x y)) → P.Is-set (A / R) -- [_] respects the quotient relation. []-respects-relation : R x y → _≡_ {A = A / R} [ x ] [ y ] []-respects-relation = _↔_.from ≡↔≡ ∘ []-respects-relationᴾ -- If R is pointwise propositional, then A / R is a set. /-is-set : (∀ x y → Is-proposition (R x y)) → Is-set (A / R) /-is-set = _↔_.from (H-level↔H-level 2) ∘ /-is-setᴾ ------------------------------------------------------------------------ -- Eliminators -- An eliminator, expressed using paths. record Elimᴾ′ {A : Type a} {R : A → A → Type r} (P : A / R → Type p) : Type (a ⊔ r ⊔ p) where no-eta-equality field []ʳ : ∀ x → P [ x ] []-respects-relationʳ : (r : R x y) → P.[ (λ i → P ([]-respects-relationᴾ r i)) ] []ʳ x ≡ []ʳ y is-setʳ : (prop : ∀ x y → Is-proposition (R x y)) {eq₁ eq₂ : x P.≡ y} {p : P x} {q : P y} (eq₃ : P.[ (λ i → P (eq₁ i)) ] p ≡ q) (eq₄ : P.[ (λ i → P (eq₂ i)) ] p ≡ q) → P.[ (λ i → P.[ (λ j → P (/-is-setᴾ prop eq₁ eq₂ i j)) ] p ≡ q) ] eq₃ ≡ eq₄ open Elimᴾ′ public elimᴾ′ : Elimᴾ′ P → (x : A / R) → P x elimᴾ′ {A = A} {R = R} {P = P} e = helper where module E = Elimᴾ′ e helper : (x : A / R) → P x helper [ x ] = E.[]ʳ x helper ([]-respects-relationᴾ r i) = E.[]-respects-relationʳ r i helper (/-is-setᴾ prop p q i j) = E.is-setʳ prop (λ i → helper (p i)) (λ i → helper (q i)) i j -- A possibly more useful eliminator, expressed using paths. record Elimᴾ {A : Type a} {R : A → A → Type r} (P : A / R → Type p) : Type (a ⊔ r ⊔ p) where no-eta-equality field []ʳ : ∀ x → P [ x ] []-respects-relationʳ : (r : R x y) → P.[ (λ i → P ([]-respects-relationᴾ r i)) ] []ʳ x ≡ []ʳ y is-setʳ : (∀ x y → Is-proposition (R x y)) → ∀ x → P.Is-set (P x) open Elimᴾ public elimᴾ : Elimᴾ P → (x : A / R) → P x elimᴾ e = elimᴾ′ λ where .[]ʳ → E.[]ʳ .[]-respects-relationʳ → E.[]-respects-relationʳ .is-setʳ prop → P.heterogeneous-UIP (E.is-setʳ prop) _ where module E = Elimᴾ e private -- One can define elimᴾ′ using elimᴾ. elimᴾ′₂ : Elimᴾ′ P → (x : A / R) → P x elimᴾ′₂ {P = P} e = elimᴾ λ where .[]ʳ → E.[]ʳ .[]-respects-relationʳ → E.[]-respects-relationʳ .is-setʳ prop x {y} {z} p q → $⟨ E.is-setʳ prop p q ⟩ P.[ (λ i → P.[ (λ j → P (/-is-setᴾ prop P.refl P.refl i j)) ] y ≡ z) ] p ≡ q ↝⟨ P.subst (λ eq → P.[ (λ i → P.[ (λ j → P (eq i j)) ] y ≡ z) ] p ≡ q) (PH.mono₁ 2 (/-is-setᴾ prop) _ _) ⟩ P.[ (λ _ → P.[ (λ _ → P x) ] y ≡ z) ] p ≡ q ↔⟨⟩ p P.≡ q □ where module E = Elimᴾ′ e -- A non-dependent eliminator, expressed using paths. record Recᴾ {A : Type a} (R : A → A → Type r) (B : Type b) : Type (a ⊔ r ⊔ b) where no-eta-equality field []ʳ : A → B []-respects-relationʳ : (r : R x y) → []ʳ x P.≡ []ʳ y is-setʳ : (∀ x y → Is-proposition (R x y)) → P.Is-set B open Recᴾ public recᴾ : Recᴾ R B → A / R → B recᴾ r = elimᴾ λ where .[]ʳ → R.[]ʳ .[]-respects-relationʳ → R.[]-respects-relationʳ .is-setʳ prop _ → R.is-setʳ prop where module R = Recᴾ r -- An eliminator. record Elim {A : Type a} {R : A → A → Type r} (P : A / R → Type p) : Type (a ⊔ r ⊔ p) where no-eta-equality field []ʳ : ∀ x → P [ x ] []-respects-relationʳ : (r : R x y) → subst P ([]-respects-relation r) ([]ʳ x) ≡ []ʳ y is-setʳ : (∀ x y → Is-proposition (R x y)) → ∀ x → Is-set (P x) open Elim public elim : Elim P → (x : A / R) → P x elim e = elimᴾ λ where .[]ʳ → E.[]ʳ .[]-respects-relationʳ → subst≡→[]≡ ∘ E.[]-respects-relationʳ .is-setʳ prop → _↔_.to (H-level↔H-level 2) ∘ E.is-setʳ prop where module E = Elim e -- A computation rule. elim-[]-respects-relation : dcong (elim e) ([]-respects-relation r) ≡ e .[]-respects-relationʳ r elim-[]-respects-relation = dcong-subst≡→[]≡ (refl _) -- A non-dependent eliminator. record Rec {A : Type a} (R : A → A → Type r) (B : Type b) : Type (a ⊔ r ⊔ b) where no-eta-equality field []ʳ : A → B []-respects-relationʳ : (r : R x y) → []ʳ x ≡ []ʳ y is-setʳ : (∀ x y → Is-proposition (R x y)) → Is-set B open Rec public rec : Rec R B → A / R → B rec r = recᴾ λ where .[]ʳ → R.[]ʳ .[]-respects-relationʳ → _↔_.to ≡↔≡ ∘ R.[]-respects-relationʳ .is-setʳ → _↔_.to (H-level↔H-level 2) ∘ R.is-setʳ where module R = Rec r -- A computation rule. rec-[]-respects-relation : cong (rec {R = R} {B = B} r₁) ([]-respects-relation {x = x} {y = y} r₂) ≡ r₁ .[]-respects-relationʳ r₂ rec-[]-respects-relation = cong-≡↔≡ (refl _) -- A variant of elim that can be used if the motive composed with [_] -- is a family of propositions. -- -- I took the idea for this eliminator from Nicolai Kraus. record Elim-prop {A : Type a} {R : A → A → Type r} (P : A / R → Type p) : Type (a ⊔ r ⊔ p) where no-eta-equality field []ʳ : ∀ x → P [ x ] is-propositionʳ : ∀ x → Is-proposition (P [ x ]) open Elim-prop public elim-prop : Elim-prop P → (x : A / R) → P x elim-prop e = elim λ where .[]ʳ → E.[]ʳ .[]-respects-relationʳ _ → E.is-propositionʳ _ _ _ .is-setʳ _ → elim λ where .[]ʳ → mono₁ 1 ∘ E.is-propositionʳ .[]-respects-relationʳ _ → H-level-propositional ext 2 _ _ .is-setʳ _ _ → mono₁ 1 (H-level-propositional ext 2) where module E = Elim-prop e -- A variant of rec that can be used if the motive is a proposition. record Rec-prop (A : Type a) (B : Type b) : Type (a ⊔ b) where no-eta-equality field []ʳ : A → B is-propositionʳ : Is-proposition B open Rec-prop public rec-prop : Rec-prop A B → A / R → B rec-prop r = elim-prop λ where .[]ʳ → R.[]ʳ .is-propositionʳ _ → R.is-propositionʳ where module R = Rec-prop r ------------------------------------------------------------------------ -- Some properties -- [_] is surjective. []-surjective : Surjective ([_] {R = R}) []-surjective = elim-prop λ where .[]ʳ x → ∣ x , refl _ ∣ .is-propositionʳ _ → TruncP.truncation-is-proposition -- If the relation is a propositional equivalence relation, then it is -- equivalent to equality under [_] (assuming propositional -- extensionality). -- -- The basic structure of this proof is that of Proposition 2 in -- "Quotienting the Delay Monad by Weak Bisimilarity" by Chapman, -- Uustalu and Veltri. related≃[equal] : {R : A → A → Type r} → Propositional-extensionality r → Is-equivalence-relation R → (∀ {x y} → Is-proposition (R x y)) → ∀ {x y} → R x y ≃ _≡_ {A = A / R} [ x ] [ y ] related≃[equal] {A = A} {r = r} {R = R} prop-ext R-equiv R-prop {x = x} {y = y} = _↠_.from (Eq.≃↠⇔ R-prop (/-is-set λ _ _ → R-prop)) (record { to = []-respects-relation ; from = λ [x]≡[y] → $⟨ reflexive ⟩ proj₁ (R′ x [ x ]) ↝⟨ ≡⇒→ (cong (proj₁ ∘ R′ x) [x]≡[y]) ⟩ proj₁ (R′ x [ y ]) ↝⟨ id ⟩□ R x y □ }) where open Is-equivalence-relation R-equiv lemma : ∀ {x y z} → R y z → (R x y , R-prop) ≡ (R x z , R-prop) lemma {x} {y} {z} = R y z ↝⟨ (λ r → record { to = flip transitive r ; from = flip transitive (symmetric r) }) ⟩ R x y ⇔ R x z ↔⟨ ⇔↔≡″ ext prop-ext ⟩□ (R x y , R-prop) ≡ (R x z , R-prop) □ R′ : A → A / R → Proposition r R′ x = rec λ where .[]ʳ y → R x y , R-prop .[]-respects-relationʳ → lemma .is-setʳ _ → Is-set-∃-Is-proposition ext prop-ext -- Quotienting with equality (for a set) amounts to the same thing as -- not quotienting at all. /≡↔ : Is-set A → A / _≡_ ↔ A /≡↔ A-set = record { surjection = record { logical-equivalence = record { from = [_] ; to = rec λ where .[]ʳ → id .[]-respects-relationʳ → id .is-setʳ prop → prop _ _ } ; right-inverse-of = λ _ → refl _ } ; left-inverse-of = elim λ where .[]ʳ _ → refl _ .is-setʳ prop _ → mono₁ 2 (/-is-set prop) .[]-respects-relationʳ _ → /-is-set (λ _ _ → A-set) _ _ } -- Quotienting with a trivial, pointwise propositional relation -- amounts to the same thing as using the propositional truncation -- operator. /trivial↔∥∥ : (∀ x y → R x y) → (∀ {x y} → Is-proposition (R x y)) → A / R ↔ ∥ A ∥ /trivial↔∥∥ {A = A} {R = R} trivial prop = record { surjection = record { logical-equivalence = record { from = TruncP.rec /-prop [_] ; to = rec-prop λ where .[]ʳ → ∣_∣ .is-propositionʳ → TruncP.truncation-is-proposition } ; right-inverse-of = TruncP.elim _ (λ _ → ⇒≡ 1 TruncP.truncation-is-proposition) (λ _ → refl _) } ; left-inverse-of = elim-prop λ where .[]ʳ _ → refl _ .is-propositionʳ _ → /-is-set λ _ _ → prop } where /-prop : Is-proposition (A / R) /-prop = elim-prop λ where .[]ʳ x → elim-prop λ where .[]ʳ y → []-respects-relation (trivial x y) .is-propositionʳ _ → /-is-set λ _ _ → prop .is-propositionʳ _ → Π-closure ext 1 λ _ → /-is-set λ _ _ → prop -- The previous property gives us an alternative to -- constant-function≃∥inhabited∥⇒inhabited. constant-function↔∥inhabited∥⇒inhabited : Is-set B → (∃ λ (f : A → B) → Constant f) ↔ (∥ A ∥ → B) constant-function↔∥inhabited∥⇒inhabited {B = B} {A = A} B-set = (∃ λ (f : A → B) → Constant f) ↝⟨ record { surjection = record { logical-equivalence = record { to = λ { (f , c) → rec λ where .[]ʳ → f .[]-respects-relationʳ _ → c _ _ .is-setʳ _ → B-set } ; from = λ f → (f ∘ [_]) , (λ _ _ → cong f ([]-respects-relation _)) } ; right-inverse-of = λ f → ⟨ext⟩ $ elim λ where .[]ʳ _ → refl _ .[]-respects-relationʳ _ → B-set _ _ .is-setʳ _ _ → mono₁ 2 B-set } ; left-inverse-of = λ _ → Σ-≡,≡→≡ (refl _) ((Π-closure ext 1 λ _ → Π-closure ext 1 λ _ → B-set) _ _) } ⟩ (A / (λ _ _ → ⊤) → B) ↝⟨ →-cong₁ ext (/trivial↔∥∥ _ (mono₁ 0 ⊤-contractible)) ⟩□ (∥ A ∥ → B) □ -- The two directions of the proposition above compute in the -- "right" way. Note that (at the time of writing) an analogue of -- the second property below fails to hold definitionally for -- constant-function≃∥inhabited∥⇒inhabited. _ : {B-set : Is-set B} → _↔_.to (constant-function↔∥inhabited∥⇒inhabited B-set) f ∣ x ∣ ≡ proj₁ f x _ = refl _ _ : {B-set : Is-set B} → proj₁ (_↔_.from (constant-function↔∥inhabited∥⇒inhabited B-set) f) x ≡ f ∣ x ∣ _ = refl _ ------------------------------------------------------------------------ -- Preservation lemmas -- A preservation lemma for functions. /-map : ((∀ x y → Is-proposition (R₁ x y)) → (∀ x y → Is-proposition (R₂ x y))) → (A₁→A₂ : A₁ → A₂) → (∀ x y → R₁ x y → R₂ (A₁→A₂ x) (A₁→A₂ y)) → A₁ / R₁ → A₂ / R₂ /-map {R₁ = R₁} {R₂ = R₂} prop A₁→A₂ R₁→R₂ = rec λ where .[]ʳ → [_] ∘ A₁→A₂ .is-setʳ → /-is-set ∘ prop .[]-respects-relationʳ {x = x} {y = y} → R₁ x y ↝⟨ R₁→R₂ _ _ ⟩ R₂ (A₁→A₂ x) (A₁→A₂ y) ↝⟨ []-respects-relation ⟩□ [ A₁→A₂ x ] ≡ [ A₁→A₂ y ] □ -- A preservation lemma for logical equivalences. /-cong-⇔ : ((∀ x y → Is-proposition (R₁ x y)) ⇔ (∀ x y → Is-proposition (R₂ x y))) → (A₁⇔A₂ : A₁ ⇔ A₂) → (∀ x y → R₁ x y → R₂ (_⇔_.to A₁⇔A₂ x) (_⇔_.to A₁⇔A₂ y)) → (∀ x y → R₂ x y → R₁ (_⇔_.from A₁⇔A₂ x) (_⇔_.from A₁⇔A₂ y)) → A₁ / R₁ ⇔ A₂ / R₂ /-cong-⇔ prop A₁⇔A₂ R₁→R₂ R₂→R₁ = record { to = /-map (_⇔_.to prop) (_⇔_.to A₁⇔A₂) R₁→R₂ ; from = /-map (_⇔_.from prop) (_⇔_.from A₁⇔A₂) R₂→R₁ } -- A preservation lemma for split surjections. /-cong-↠ : ((∀ x y → Is-proposition (R₁ x y)) ⇔ (∀ x y → Is-proposition (R₂ x y))) → (A₁↠A₂ : A₁ ↠ A₂) → (∀ x y → R₁ x y ↠ R₂ (_↠_.to A₁↠A₂ x) (_↠_.to A₁↠A₂ y)) → A₁ / R₁ ↠ A₂ / R₂ /-cong-↠ {R₁ = R₁} {R₂ = R₂} prop A₁↠A₂ R₁↠R₂ = record { logical-equivalence = /-cong-⇔ prop (_↠_.logical-equivalence A₁↠A₂) (λ x y → _↠_.to (R₁↠R₂ x y)) (λ x y → R₂ x y ↝⟨ ≡⇒↝ _ (sym $ cong₂ R₂ (right-inverse-of x) (right-inverse-of y)) ⟩ R₂ (to (from x)) (to (from y)) ↝⟨ _↠_.from (R₁↠R₂ _ _) ⟩□ R₁ (from x) (from y) □) ; right-inverse-of = elim λ where .[]ʳ x → [ to (from x) ] ≡⟨ cong [_] $ right-inverse-of x ⟩∎ [ x ] ∎ .is-setʳ prop _ → mono₁ 1 (/-is-set prop) .[]-respects-relationʳ {x = x} {y = y} r → let tr = λ x y → _↠_.to (R₁↠R₂ x y) fr = λ x y → _↠_.from (R₁↠R₂ (from x) (from y)) ∘ ≡⇒↝ _ (sym (cong₂ R₂ (right-inverse-of x) (right-inverse-of y))) t = /-map (_⇔_.to prop) to tr f = /-map (_⇔_.from prop) from fr in subst (λ x → t (f x) ≡ x) ([]-respects-relation r) (cong [_] (right-inverse-of x)) ≡⟨ subst-in-terms-of-trans-and-cong ⟩ trans (sym (cong (t ∘ f) ([]-respects-relation r))) (trans (cong [_] (right-inverse-of x)) (cong id ([]-respects-relation r))) ≡⟨ cong₂ (λ eq₁ eq₂ → trans (sym eq₁) (trans (cong [_] (right-inverse-of x)) eq₂)) (sym $ cong-∘ _ _ _) (sym $ cong-id _) ⟩ trans (sym (cong t (cong f ([]-respects-relation r)))) (trans (cong [_] (right-inverse-of x)) ([]-respects-relation r)) ≡⟨ cong (λ eq → trans (sym (cong t eq)) (trans (cong [_] (right-inverse-of x)) ([]-respects-relation r))) $ rec-[]-respects-relation ⟩ trans (sym (cong t ([]-respects-relation (fr _ _ r)))) (trans (cong [_] (right-inverse-of x)) ([]-respects-relation r)) ≡⟨ cong (λ eq → trans (sym eq) (trans (cong [_] (right-inverse-of x)) ([]-respects-relation r))) $ rec-[]-respects-relation ⟩ trans (sym ([]-respects-relation (tr _ _ (fr _ _ r)))) (trans (cong [_] (right-inverse-of x)) ([]-respects-relation r)) ≡⟨ cong (λ r′ → trans (sym ([]-respects-relation r′)) (trans (cong [_] (right-inverse-of x)) ([]-respects-relation r))) $ _↠_.right-inverse-of (R₁↠R₂ _ _) _ ⟩ trans (sym ([]-respects-relation (≡⇒↝ _ (sym (cong₂ R₂ (right-inverse-of x) (right-inverse-of y))) r))) (trans (cong [_] (right-inverse-of x)) ([]-respects-relation r)) ≡⟨ lemma _ _ _ ⟩∎ cong [_] (right-inverse-of y) ∎ } where open _↠_ A₁↠A₂ lemma : ∀ _ _ _ → _ lemma x y r = elim₁ (λ eq → trans (sym ([]-respects-relation (≡⇒↝ _ (sym (cong₂ R₂ (right-inverse-of x) eq)) r))) (trans (cong [_] (right-inverse-of x)) ([]-respects-relation r)) ≡ cong [_] eq) (elim₁ (λ eq → trans (sym ([]-respects-relation (≡⇒↝ _ (sym (cong₂ R₂ eq (refl _))) r))) (trans (cong [_] eq) ([]-respects-relation r)) ≡ cong [_] (refl _)) (trans (sym ([]-respects-relation (≡⇒↝ _ (sym (cong₂ R₂ (refl _) (refl _))) r))) (trans (cong [_] (refl _)) ([]-respects-relation r)) ≡⟨ cong₂ (λ eq₁ eq₂ → trans (sym ([]-respects-relation (≡⇒↝ _ eq₁ r))) (trans eq₂ ([]-respects-relation r))) (trans (cong sym $ cong₂-refl R₂) sym-refl) (cong-refl _) ⟩ trans (sym ([]-respects-relation (≡⇒↝ _ (refl _) r))) (trans (refl _) ([]-respects-relation r)) ≡⟨ cong₂ (λ eq₁ eq₂ → trans (sym ([]-respects-relation eq₁)) eq₂) (cong (_$ r) ≡⇒↝-refl) (trans-reflˡ _) ⟩ trans (sym ([]-respects-relation r)) ([]-respects-relation r) ≡⟨ trans-symˡ _ ⟩ refl _ ≡⟨ sym $ cong-refl _ ⟩∎ cong [_] (refl _) ∎) (right-inverse-of x)) (right-inverse-of y) private -- A preservation lemma for equivalences. /-cong-≃ : ((∀ x y → Is-proposition (R₁ x y)) ⇔ (∀ x y → Is-proposition (R₂ x y))) → (A₁≃A₂ : A₁ ≃ A₂) → (∀ x y → R₁ x y ≃ R₂ (to-implication A₁≃A₂ x) (to-implication A₁≃A₂ y)) → A₁ / R₁ ≃ A₂ / R₂ /-cong-≃ {R₁ = R₁} {R₂ = R₂} prop A₁≃A₂ R₁≃R₂ = Eq.↔⇒≃ (record { surjection = /-cong-↠ prop (_≃_.surjection A₁≃A₂) λ x y → R₁ x y ↔⟨ R₁≃R₂ x y ⟩□ R₂ (to x) (to y) □ ; left-inverse-of = elim λ where .[]ʳ x → [ from (to x) ] ≡⟨ cong [_] $ left-inverse-of x ⟩∎ [ x ] ∎ .is-setʳ prop _ → mono₁ 1 (/-is-set prop) .[]-respects-relationʳ {x = x} {y = y} r → let tr = λ x y → _≃_.to (R₁≃R₂ x y) fr = λ x y → _≃_.from (R₁≃R₂ (from x) (from y)) ∘ ≡⇒↝ _ (sym (cong₂ R₂ (right-inverse-of x) (right-inverse-of y))) t = /-map (_⇔_.to prop) to tr f = /-map (_⇔_.from prop) from fr in subst (λ x → f (t x) ≡ x) ([]-respects-relation r) (cong [_] (left-inverse-of x)) ≡⟨ subst-in-terms-of-trans-and-cong ⟩ trans (sym (cong (f ∘ t) ([]-respects-relation r))) (trans (cong [_] (left-inverse-of x)) (cong id ([]-respects-relation r))) ≡⟨ cong₂ (λ eq₁ eq₂ → trans (sym eq₁) (trans (cong [_] (left-inverse-of x)) eq₂)) (sym $ cong-∘ _ _ _) (sym $ cong-id _) ⟩ trans (sym (cong f (cong t ([]-respects-relation r)))) (trans (cong [_] (left-inverse-of x)) ([]-respects-relation r)) ≡⟨ cong (λ eq → trans (sym (cong f eq)) (trans (cong [_] (left-inverse-of x)) ([]-respects-relation r))) $ rec-[]-respects-relation ⟩ trans (sym (cong f ([]-respects-relation (tr _ _ r)))) (trans (cong [_] (left-inverse-of x)) ([]-respects-relation r)) ≡⟨ cong (λ eq → trans (sym eq) (trans (cong [_] (left-inverse-of x)) ([]-respects-relation r))) $ rec-[]-respects-relation ⟩ trans (sym ([]-respects-relation (fr _ _ (tr _ _ r)))) (trans (cong [_] (left-inverse-of x)) ([]-respects-relation r)) ≡⟨ cong (λ r′ → trans (sym ([]-respects-relation r′)) (trans (cong [_] (left-inverse-of x)) ([]-respects-relation r))) $ lemma₁ _ _ _ ⟩ trans (sym ([]-respects-relation (≡⇒↝ _ (sym (cong₂ R₁ (left-inverse-of x) (left-inverse-of y))) r))) (trans (cong [_] (left-inverse-of x)) ([]-respects-relation r)) ≡⟨ lemma₂ _ _ _ ⟩∎ cong [_] (left-inverse-of y) ∎ }) where open _≃_ A₁≃A₂ lemma₀ : ∀ _ _ _ → _ lemma₀ x y r = elim₁ (λ eq → _≃_.from (R₁≃R₂ (from (to x)) _) (≡⇒↝ _ (sym (cong₂ (λ x y → R₂ (to x) (to y)) (left-inverse-of x) eq)) (_≃_.to (R₁≃R₂ x y) r)) ≡ ≡⇒↝ _ (sym (cong₂ R₁ (left-inverse-of x) eq)) r) (elim₁ (λ eq → _≃_.from (R₁≃R₂ _ _) (≡⇒↝ _ (sym (cong₂ (λ x y → R₂ (to x) (to y)) eq (refl _))) (_≃_.to (R₁≃R₂ x y) r)) ≡ ≡⇒↝ _ (sym (cong₂ R₁ eq (refl _))) r) (_≃_.from (R₁≃R₂ _ _) (≡⇒↝ _ (sym (cong₂ (λ x y → R₂ (to x) (to y)) (refl _) (refl _))) (_≃_.to (R₁≃R₂ x y) r)) ≡⟨ cong (λ eq → _≃_.from (R₁≃R₂ _ _) (≡⇒↝ _ eq (_≃_.to (R₁≃R₂ x y) r))) $ trans (cong sym $ cong₂-refl _) sym-refl ⟩ _≃_.from (R₁≃R₂ _ _) (≡⇒↝ _ (refl _) (_≃_.to (R₁≃R₂ x y) r)) ≡⟨ cong (_≃_.from (R₁≃R₂ _ _)) $ cong (_$ _≃_.to (R₁≃R₂ x y) r) ≡⇒↝-refl ⟩ _≃_.from (R₁≃R₂ _ _) (_≃_.to (R₁≃R₂ x y) r) ≡⟨ _≃_.left-inverse-of (R₁≃R₂ _ _) _ ⟩ r ≡⟨ sym $ cong (_$ r) ≡⇒↝-refl ⟩ ≡⇒↝ _ (refl _) r ≡⟨ cong (λ eq → ≡⇒↝ _ eq r) $ sym $ trans (cong sym $ cong₂-refl _) sym-refl ⟩∎ ≡⇒↝ _ (sym (cong₂ R₁ (refl _) (refl _))) r ∎) (left-inverse-of x)) (left-inverse-of y) lemma₁ : ∀ _ _ _ → _ lemma₁ x y r = _≃_.from (R₁≃R₂ (from (to x)) (from (to y))) (≡⇒↝ _ (sym (cong₂ R₂ (right-inverse-of (to x)) (right-inverse-of (to y)))) (_≃_.to (R₁≃R₂ x y) r)) ≡⟨⟩ _≃_.from (R₁≃R₂ (from (to x)) (from (to y))) (≡⇒↝ _ (sym (trans (cong (flip R₂ _) (right-inverse-of (to x))) (cong (R₂ _) (right-inverse-of (to y))))) (_≃_.to (R₁≃R₂ x y) r)) ≡⟨ cong₂ (λ eq₁ eq₂ → _≃_.from (R₁≃R₂ (from (to x)) (from (to y))) (≡⇒↝ _ (sym (trans eq₁ eq₂)) (_≃_.to (R₁≃R₂ x y) r))) (trans (cong (cong _) $ sym $ left-right-lemma _) (cong-∘ _ _ _)) (trans (cong (cong _) $ sym $ left-right-lemma _) (cong-∘ _ _ _)) ⟩ _≃_.from (R₁≃R₂ (from (to x)) (from (to y))) (≡⇒↝ _ (sym (trans (cong (flip R₂ _ ∘ to) (left-inverse-of x)) (cong (R₂ _ ∘ to) (left-inverse-of y)))) (_≃_.to (R₁≃R₂ x y) r)) ≡⟨⟩ _≃_.from (R₁≃R₂ (from (to x)) (from (to y))) (≡⇒↝ _ (sym (cong₂ (λ x y → R₂ (to x) (to y)) (left-inverse-of x) (left-inverse-of y))) (_≃_.to (R₁≃R₂ x y) r)) ≡⟨ lemma₀ _ _ _ ⟩∎ ≡⇒↝ _ (sym (cong₂ R₁ (left-inverse-of x) (left-inverse-of y))) r ∎ lemma₂ : ∀ _ _ _ → _ lemma₂ x y r = elim₁ (λ eq → trans (sym ([]-respects-relation (≡⇒↝ _ (sym (cong₂ R₁ (left-inverse-of x) eq)) r))) (trans (cong [_] (left-inverse-of x)) ([]-respects-relation r)) ≡ cong [_] eq) (elim₁ (λ eq → trans (sym ([]-respects-relation (≡⇒↝ _ (sym (cong₂ R₁ eq (refl _))) r))) (trans (cong [_] eq) ([]-respects-relation r)) ≡ cong [_] (refl _)) (trans (sym ([]-respects-relation (≡⇒↝ _ (sym (cong₂ R₁ (refl _) (refl _))) r))) (trans (cong [_] (refl _)) ([]-respects-relation r)) ≡⟨ cong₂ (λ eq₁ eq₂ → trans (sym ([]-respects-relation (≡⇒↝ _ eq₁ r))) (trans eq₂ ([]-respects-relation r))) (trans (cong sym $ cong₂-refl R₁) sym-refl) (cong-refl _) ⟩ trans (sym ([]-respects-relation (≡⇒↝ _ (refl _) r))) (trans (refl _) ([]-respects-relation r)) ≡⟨ cong₂ (λ eq₁ eq₂ → trans (sym ([]-respects-relation eq₁)) eq₂) (cong (_$ r) ≡⇒↝-refl) (trans-reflˡ _) ⟩ trans (sym ([]-respects-relation r)) ([]-respects-relation r) ≡⟨ trans-symˡ _ ⟩ refl _ ≡⟨ sym $ cong-refl _ ⟩∎ cong [_] (refl _) ∎) (left-inverse-of x)) (left-inverse-of y) -- A preservation lemma for isomorphisms. /-cong-↔ : {A₁ : Type a₁} {R₁ : A₁ → A₁ → Type r₁} {A₂ : Type a₂} {R₂ : A₂ → A₂ → Type r₂} → ((∀ x y → Is-proposition (R₁ x y)) ⇔ (∀ x y → Is-proposition (R₂ x y))) → (A₁↔A₂ : A₁ ↔[ k ] A₂) → (∀ x y → R₁ x y ↔[ k ] R₂ (to-implication A₁↔A₂ x) (to-implication A₁↔A₂ y)) → A₁ / R₁ ↔[ k ] A₂ / R₂ /-cong-↔ {k = k} {R₁ = R₁} {R₂ = R₂} prop A₁↔A₂ R₁↔R₂ = from-isomorphism $ /-cong-≃ prop A₁≃A₂ λ x y → R₁ x y ↔⟨ R₁↔R₂ x y ⟩ R₂ (to-implication A₁↔A₂ x) (to-implication A₁↔A₂ y) ↝⟨ ≡⇒↝ _ $ cong₂ (λ f g → R₂ (f x) (g y)) (to-implication∘from-isomorphism k equivalence) (to-implication∘from-isomorphism k equivalence) ⟩□ R₂ (_≃_.to A₁≃A₂ x) (_≃_.to A₁≃A₂ y) □ where A₁≃A₂ = from-isomorphism A₁↔A₂ ------------------------------------------------------------------------ -- The quotients defined here can be related to the ones defined in -- Quotient -- If the quotient relation is propositional, then the definition of -- quotients given in Quotient is equivalent to the one given here. /≃/ : (∀ {x y} → Is-proposition (R x y)) → A / R ≃ A Quotient./ R /≃/ R-prop = Eq.↔⇒≃ (record { surjection = record { logical-equivalence = record { to = rec (λ where .[]ʳ → Quotient.[_] .[]-respects-relationʳ → Quotient.[]-respects-relation .is-setʳ _ → Quotient./-is-set) ; from = Quotient.rec λ where .Quotient.Rec.[]ʳ → [_] .Quotient.Rec.[]-respects-relationʳ → []-respects-relation .Quotient.Rec.is-setʳ → /-is-set λ _ _ → R-prop } ; right-inverse-of = Quotient.elim-prop λ where .Quotient.Elim-prop.[]ʳ _ → refl _ .Quotient.Elim-prop.is-propositionʳ _ → Quotient./-is-set } ; left-inverse-of = elim-prop λ where .Elim-prop.[]ʳ _ → refl _ .Elim-prop.is-propositionʳ _ → /-is-set λ _ _ → R-prop }) ------------------------------------------------------------------------ -- The quotients defined here can be related to the ones defined in -- QF.Families-of-equivalence-classes private -- An alternative definition of the quotients from -- QF.Families-of-equivalence-classes. infix 5 _/′_ _/′_ : (A : Type a) → (A → A → Type a) → Type (lsuc a) _/′_ {a = a} A R = ∃ λ (P : A → Type a) → ∥ (∃ λ x → R x ≡ P) ∥ /↔/′ : A QF./ R ↔ A /′ R /↔/′ {A = A} {R = R} = A QF./ R ↔⟨⟩ (∃ λ (P : A → Type _) → Trunc.∥ (∃ λ x → R x ≡ P) ∥ 1 _) ↝⟨ (∃-cong λ _ → inverse $ TruncP.∥∥↔∥∥ lzero) ⟩ (∃ λ (P : A → Type _) → ∥ (∃ λ x → R x ≡ P) ∥) ↔⟨⟩ A /′ R □ [_]′ : A → A /′ R [_]′ = _↔_.to /↔/′ ∘ QF.[_] rec′ : {A : Type a} {R : A → A → Type a} → (∀ {x} → R x x) → (B : Type a) → Is-set B → (f : A → B) → (∀ {x y} → R x y → f x ≡ f y) → A /′ R → B rec′ refl B B-set f R⇒≡ = QF.rec ext refl B B-set f R⇒≡ ∘ _↔_.from /↔/′ elim-Prop′ : {A : Type a} {R : A → A → Type a} → QF.Strong-equivalence-with surjection R → (B : A /′ R → Type (lsuc a)) → (∀ x → Is-proposition (B [ x ]′)) → (f : ∀ x → B [ x ]′) → ∀ x → B x elim-Prop′ strong-equivalence B B-prop f x = subst B (_↔_.right-inverse-of /↔/′ _) $ QF.elim-Prop ext strong-equivalence (B ∘ _↔_.to /↔/′) B-prop f (_↔_.from /↔/′ x) -- If the relation is a propositional equivalence relation of a -- certain size, then the quotients defined above are isomorphic to -- families of equivalence relations, defined in a certain way -- (assuming univalence). /↔ : {A : Type a} {R : A → A → Type a} → Univalence a → Univalence lzero → Is-equivalence-relation R → (∀ {x y} → Is-proposition (R x y)) → A / R ↔ ∃ λ (P : A → Type a) → ∥ (∃ λ x → R x ≡ P) ∥ /↔ {a = a} {A = A} {R = R} univ univ₀ R-equiv R-prop = record { surjection = record { logical-equivalence = record { to = to ; from = from } ; right-inverse-of = to∘from } ; left-inverse-of = from∘to } where R-is-strong-equivalence : QF.Strong-equivalence R R-is-strong-equivalence = QF.propositional-equivalence⇒strong-equivalence ext univ R-equiv (λ _ _ → R-prop) to : A / R → A /′ R to = rec λ where .[]ʳ → [_]′ .[]-respects-relationʳ {x = x} {y = y} → R x y ↝⟨ _≃_.to (QF.related↝[equal] ext R-is-strong-equivalence) ⟩ QF.[ x ] ≡ QF.[ y ] ↝⟨ cong (_↔_.to /↔/′) ⟩□ [ x ]′ ≡ [ y ]′ □ .is-setʳ prop → $⟨ (λ {_ _} → QF.quotient's-h-level-is-1-+-relation's-h-level ext univ univ₀ 1 (λ _ _ → R-prop)) ⟩ Is-set (A QF./ R) ↝⟨ H.respects-surjection (_↔_.surjection /↔/′) 2 ⟩□ Is-set (A /′ R) □ from : A /′ R → A / R from = rec′ (Is-equivalence-relation.reflexive R-equiv) _ (/-is-set λ _ _ → R-prop) [_] []-respects-relation to∘from : ∀ x → to (from x) ≡ x to∘from = elim-Prop′ (QF.strong-equivalence⇒strong-equivalence-with R-is-strong-equivalence) _ (λ x → $⟨ (λ {_ _} → QF.quotient's-h-level-is-1-+-relation's-h-level ext univ univ₀ 1 λ _ _ → R-prop) ⟩ Is-set (A QF./ R) ↝⟨ H.respects-surjection (_↔_.surjection /↔/′) 2 ⟩ Is-set (A /′ R) ↝⟨ +⇒≡ {n = 1} ⟩□ Is-proposition (to (from [ x ]′) ≡ [ x ]′) □) (λ _ → refl _) from∘to : ∀ x → from (to x) ≡ x from∘to = elim-prop λ where .[]ʳ _ → refl _ .is-propositionʳ _ → /-is-set λ _ _ → R-prop -- If the relation is a propositional equivalence relation of a -- certain size, then the definition of quotients given in -- QF.Families-of-equivalence-classes is isomorphic to the one given -- here (assuming univalence). /↔/ : {A : Type a} {R : A → A → Type a} → Univalence a → Univalence lzero → Is-equivalence-relation R → (R-prop : ∀ {x y} → Is-proposition (R x y)) → A QF./ R ↔ A / R /↔/ {a = a} {A = A} {R = R} univ univ₀ R-equiv R-prop = A QF./ R ↔⟨⟩ (∃ λ (P : A → Type a) → Trunc.∥ (∃ λ x → R x ≡ P) ∥ 1 (lsuc a)) ↝⟨ (∃-cong λ _ → inverse $ TruncP.∥∥↔∥∥ lzero) ⟩ (∃ λ (P : A → Type a) → ∥ (∃ λ x → R x ≡ P) ∥) ↝⟨ inverse $ /↔ univ univ₀ R-equiv R-prop ⟩□ A / R □ ------------------------------------------------------------------------ -- Various type formers commute with quotients -- _⊎_ commutes with quotients if the relations are pointwise -- propositional. ⊎/-comm : (∀ {x y} → Is-proposition (R₁ x y)) → (∀ {x y} → Is-proposition (R₂ x y)) → (A₁ ⊎ A₂) / (R₁ ⊎ᴾ R₂) ↔ A₁ / R₁ ⊎ A₂ / R₂ ⊎/-comm R₁-prop R₂-prop = record { surjection = record { logical-equivalence = record { to = rec λ where .[]ʳ → ⊎-map [_] [_] .is-setʳ prop → ⊎-closure 0 is-set₁ is-set₂ .[]-respects-relationʳ {x = inj₁ _} {y = inj₁ _} → cong inj₁ ∘ []-respects-relation .[]-respects-relationʳ {x = inj₂ _} {y = inj₂ _} → cong inj₂ ∘ []-respects-relation ; from = Prelude.[ rec (λ where .[]ʳ → [_] ∘ inj₁ .[]-respects-relationʳ → []-respects-relation .is-setʳ prop → is-set₁₂) , rec (λ where .[]ʳ → [_] ∘ inj₂ .[]-respects-relationʳ → []-respects-relation .is-setʳ prop → is-set₁₂) ] } ; right-inverse-of = Prelude.[ elim-prop (λ where .[]ʳ _ → refl _ .is-propositionʳ _ → ⊎-closure 0 is-set₁ is-set₂) , elim-prop (λ where .[]ʳ _ → refl _ .is-propositionʳ _ → ⊎-closure 0 is-set₁ is-set₂) ] } ; left-inverse-of = elim-prop λ where .[]ʳ → Prelude.[ (λ _ → refl _) , (λ _ → refl _) ] .is-propositionʳ _ → is-set₁₂ } where is-set₁ = /-is-set λ _ _ → R₁-prop is-set₂ = /-is-set λ _ _ → R₂-prop is-set₁₂ = /-is-set λ x y → ⊎ᴾ-preserves-Is-proposition R₁-prop R₂-prop {x = x} {y = y} -- Maybe commutes with quotients if the relation is pointwise -- propositional. -- -- Chapman, Uustalu and Veltri mention a similar result in -- "Quotienting the Delay Monad by Weak Bisimilarity". Maybe/-comm : (∀ {x y} → Is-proposition (R x y)) → Maybe A / Maybeᴾ R ↔ Maybe (A / R) Maybe/-comm {A = A} {R = R} R-prop = Maybe A / Maybeᴾ R ↝⟨ ⊎/-comm (↑-closure 1 (mono₁ 0 ⊤-contractible)) R-prop ⟩ ⊤ / Trivial ⊎ A / R ↝⟨ /trivial↔∥∥ _ (↑-closure 1 (mono₁ 0 ⊤-contractible)) ⊎-cong F.id ⟩ ∥ ⊤ ∥ ⊎ A / R ↝⟨ TruncP.∥∥↔ (mono₁ 0 ⊤-contractible) ⊎-cong F.id ⟩□ Maybe (A / R) □ -- A simplification lemma for Maybe/-comm. Maybe/-comm-[] : {R : A → A → Type r} {R-prop : ∀ {x y} → Is-proposition (R x y)} → _↔_.to (Maybe/-comm {R = R} R-prop) ∘ [_] ≡ ⊎-map id [_] Maybe/-comm-[] {R-prop = R-prop} = _↔_.to (Maybe/-comm R-prop) ∘ [_] ≡⟨⟩ ⊎-map _ id ∘ ⊎-map _ id ∘ ⊎-map [_] [_] ≡⟨ cong (_∘ ⊎-map [_] [_]) $ sym $ ⟨ext⟩ ⊎-map-∘ ⟩ ⊎-map _ id ∘ ⊎-map [_] [_] ≡⟨ sym $ ⟨ext⟩ ⊎-map-∘ ⟩∎ ⊎-map id [_] ∎ -- The sigma type former commutes (kind of) with quotients, assuming -- that certain families are propositional. Σ/-comm : {P : A / R → Type p} → (∀ {x} → Is-proposition (P x)) → (∀ {x y} → Is-proposition (R x y)) → Σ (A / R) P ↔ Σ A (P ∘ [_]) / (R on proj₁) Σ/-comm {A = A} {R = R} {P = P} P-prop R-prop = record { surjection = record { logical-equivalence = record { to = uncurry $ elim λ where .[]ʳ → curry [_] .[]-respects-relationʳ {x = x} {y = y} r → ⟨ext⟩ λ P[y] → subst (λ x → P x → Σ A (P ∘ [_]) / (R on proj₁)) ([]-respects-relation r) (curry [_] x) P[y] ≡⟨ subst-→-domain P {f = curry [_] _} ([]-respects-relation r) ⟩ [ (x , subst P (sym $ []-respects-relation r) P[y]) ] ≡⟨ []-respects-relation r ⟩∎ [ (y , P[y]) ] ∎ .is-setʳ prop _ → Π-closure ext 2 λ _ → /-is-set λ _ _ → prop _ _ ; from = rec λ where .[]ʳ → Σ-map [_] id .is-setʳ _ → Σ-closure 2 (/-is-set λ _ _ → R-prop) (λ _ → mono₁ 1 P-prop) .[]-respects-relationʳ {x = (x₁ , x₂)} {y = (y₁ , y₂)} → R x₁ y₁ ↝⟨ []-respects-relation ⟩ [ x₁ ] ≡ [ y₁ ] ↔⟨ ignore-propositional-component P-prop ⟩ ([ x₁ ] , x₂) ≡ ([ y₁ ] , y₂) □ } ; right-inverse-of = elim-prop λ where .[]ʳ _ → refl _ .is-propositionʳ _ → /-is-set λ _ _ → R-prop } ; left-inverse-of = uncurry $ elim-prop λ where .[]ʳ _ _ → refl _ .is-propositionʳ _ → Π-closure ext 1 λ _ → Σ-closure 2 (/-is-set λ _ _ → R-prop) (λ _ → mono₁ 1 P-prop) } -- The type former λ X → ℕ → X commutes with quotients, assuming that -- the quotient relation is a propositional equivalence relation, and -- also assuming countable choice and propositional extensionality. -- -- This result is very similar to Proposition 5 in "Quotienting the -- Delay Monad by Weak Bisimilarity" by Chapman, Uustalu and Veltri. -- The forward component of the isomorphism. This component can be -- defined without any extra assumptions. ℕ→/-comm-to : (ℕ → A) / (ℕ →ᴾ R) → (ℕ → A / R) ℕ→/-comm-to {A = A} = rec λ where .[]ʳ f n → [ f n ] .[]-respects-relationʳ r → ⟨ext⟩ ([]-respects-relation ∘ r) .is-setʳ prop → Π-closure ext 2 λ _ → /-is-set (prop⇒prop prop) where prop⇒prop : ((f g : ℕ → A) → Is-proposition ((ℕ →ᴾ R) f g)) → ((x y : A) → Is-proposition (R x y)) prop⇒prop prop x y p q = cong (_$ 0) $ prop (λ _ → x) (λ _ → y) (λ _ → p) (λ _ → q) -- The isomorphism. ℕ→/-comm : {A : Type a} {R : A → A → Type r} → Axiom-of-countable-choice (a ⊔ r) → Propositional-extensionality r → Is-equivalence-relation R → (∀ {x y} → Is-proposition (R x y)) → (ℕ → A) / (ℕ →ᴾ R) ↔ (ℕ → A / R) ℕ→/-comm {A = A} {R = R} cc prop-ext R-equiv R-prop = record { surjection = record { logical-equivalence = record { to = ℕ→/-comm-to ; from = from unit } ; right-inverse-of = to∘from unit } ; left-inverse-of = from∘to unit } where [_]→ : (ℕ → A) → (ℕ → A / R) [ f ]→ n = [ f n ] -- A module that is introduced to ensure that []→-surjective is not -- in the same abstract block as the abstract definitions below. module Dummy where abstract []→-surjective : Surjective [_]→ []→-surjective f = $⟨ []-surjective ⟩ Surjective [_] ↝⟨ (λ surj → surj ∘ f) ⦂ (_ → _) ⟩ (∀ n → ∥ (∃ λ x → [ x ] ≡ f n) ∥) ↔⟨ TruncP.countable-choice-bijection cc ⟩ ∥ (∀ n → ∃ λ x → [ x ] ≡ f n) ∥ ↔⟨ TruncP.∥∥-cong ΠΣ-comm ⟩ ∥ (∃ λ g → ∀ n → [ g n ] ≡ f n) ∥ ↔⟨⟩ ∥ (∃ λ g → ∀ n → [ g ]→ n ≡ f n) ∥ ↔⟨ TruncP.∥∥-cong (∃-cong λ _ → Eq.extensionality-isomorphism ext) ⟩□ ∥ (∃ λ g → [ g ]→ ≡ f) ∥ □ open Dummy from₁ : ∀ f → [_]→ ⁻¹ f → ℕ→/-comm-to ⁻¹ f from₁ f (g , [g]→≡f) = [ g ] , (ℕ→/-comm-to [ g ] ≡⟨⟩ [ g ]→ ≡⟨ [g]→≡f ⟩∎ f ∎) from₁-constant : ∀ f → Constant (from₁ f) from₁-constant f (g₁ , [g₁]→≡f) (g₂ , [g₂]→≡f) = $⟨ (λ n → cong (_$ n) ( [ g₁ ]→ ≡⟨ [g₁]→≡f ⟩ f ≡⟨ sym [g₂]→≡f ⟩∎ [ g₂ ]→ ∎)) ⟩ (∀ n → [ g₁ ]→ n ≡ [ g₂ ]→ n) ↔⟨⟩ (∀ n → [ g₁ n ] ≡ [ g₂ n ]) ↔⟨ ∀-cong ext (λ _ → inverse $ related≃[equal] prop-ext R-equiv R-prop) ⟩ (∀ n → R (g₁ n) (g₂ n)) ↔⟨⟩ (ℕ →ᴾ R) g₁ g₂ ↝⟨ []-respects-relation ⟩ [ g₁ ] ≡ [ g₂ ] ↔⟨ ignore-propositional-component (Π-closure ext 2 λ _ → /-is-set λ _ _ → R-prop) ⟩□ ([ g₁ ] , [g₁]→≡f) ≡ ([ g₂ ] , [g₂]→≡f) □ from₂ : Unit → ∀ f → ∥ [_]→ ⁻¹ f ∥ → ℕ→/-comm-to ⁻¹ f from₂ unit f = _≃_.to (TruncP.constant-function≃∥inhabited∥⇒inhabited (Σ-closure 2 (/-is-set λ _ _ → →ᴾ-preserves-Is-proposition R ext R-prop) λ _ → mono₁ 1 (Π-closure ext 2 (λ _ → /-is-set λ _ _ → R-prop)))) (from₁ f , from₁-constant f) unblock-from₂ : ∀ x f p → from₂ x f ∣ p ∣ ≡ from₁ f p unblock-from₂ unit _ _ = refl _ abstract from₃ : Unit → (f : ℕ → A / R) → ℕ→/-comm-to ⁻¹ f from₃ x f = from₂ x f ([]→-surjective f) from : Unit → (ℕ → A / R) → (ℕ → A) / (ℕ →ᴾ R) from x f = proj₁ (from₃ x f) to∘from : ∀ x f → ℕ→/-comm-to (from x f) ≡ f to∘from x f = proj₂ (from₃ x f) from∘to : ∀ x f → from x (ℕ→/-comm-to f) ≡ f from∘to x = elim-prop λ where .[]ʳ f → from x (ℕ→/-comm-to [ f ]) ≡⟨⟩ proj₁ (from₂ x [ f ]→ ([]→-surjective [ f ]→)) ≡⟨ cong (proj₁ ∘ from₂ x [ f ]→) $ TruncP.truncation-is-proposition _ _ ⟩ proj₁ (from₂ x [ f ]→ ∣ f , refl _ ∣) ≡⟨ cong proj₁ $ unblock-from₂ x _ (f , refl _) ⟩ proj₁ (from₁ [ f ]→ (f , refl _)) ≡⟨⟩ [ f ] ∎ .is-propositionʳ _ → /-is-set λ _ _ → →ᴾ-preserves-Is-proposition R ext R-prop ------------------------------------------------------------------------ -- Quotient-like eliminators -- If there is a split surjection from a quotient type to some other -- type, then one can construct a quotient-like eliminator for the -- other type. -- -- This kind of construction is used in "Quotienting the Delay Monad -- by Weak Bisimilarity" by Chapman, Uustalu and Veltri. ↠-eliminator : (surj : A / R ↠ B) (P : B → Type p) (p-[] : ∀ x → P (_↠_.to surj [ x ])) → (∀ {x y} (r : R x y) → subst P (cong (_↠_.to surj) ([]-respects-relation r)) (p-[] x) ≡ p-[] y) → ((∀ x y → Is-proposition (R x y)) → ∀ x → Is-set (P x)) → ∀ x → P x ↠-eliminator surj P p-[] ok P-set x = subst P (_↠_.right-inverse-of surj x) p′ where p′ : P (_↠_.to surj (_↠_.from surj x)) p′ = elim (λ where .[]ʳ → p-[] .[]-respects-relationʳ {x = x} {y = y} r → subst (P ∘ _↠_.to surj) ([]-respects-relation r) (p-[] x) ≡⟨ subst-∘ P (_↠_.to surj) ([]-respects-relation r) ⟩ subst P (cong (_↠_.to surj) ([]-respects-relation r)) (p-[] x) ≡⟨ ok r ⟩∎ p-[] y ∎ .is-setʳ prop _ → P-set prop _) (_↠_.from surj x) -- The eliminator "computes" in the "right" way for elements that -- satisfy a certain property, assuming that the quotient relation is -- pointwise propositional. ↠-eliminator-[] : ∀ (surj : A / R ↠ B) (P : B → Type p) (p-[] : ∀ x → P (_↠_.to surj [ x ])) (ok : ∀ {x y} (r : R x y) → subst P (cong (_↠_.to surj) ([]-respects-relation r)) (p-[] x) ≡ p-[] y) (P-set : (∀ x y → Is-proposition (R x y)) → ∀ x → Is-set (P x)) x → (∀ {x y} → Is-proposition (R x y)) → _↠_.from surj (_↠_.to surj [ x ]) ≡ [ x ] → ↠-eliminator surj P p-[] ok P-set (_↠_.to surj [ x ]) ≡ p-[] x ↠-eliminator-[] {R = R} surj P p-[] ok P-set x R-prop hyp = subst P (_↠_.right-inverse-of surj (_↠_.to surj [ x ])) (elim e′ (_↠_.from surj (_↠_.to surj [ x ]))) ≡⟨ cong (λ p → subst P p (elim e′ _)) $ H.respects-surjection surj 2 (/-is-set λ _ _ → R-prop) (_↠_.right-inverse-of surj (_↠_.to surj [ x ])) (cong (_↠_.to surj) hyp) ⟩ subst P (cong (_↠_.to surj) hyp) (elim e′ (_↠_.from surj (_↠_.to surj [ x ]))) ≡⟨ D.elim (λ {x y} p → subst P (cong (_↠_.to surj) p) (elim e′ x) ≡ elim e′ y) (λ y → subst P (cong (_↠_.to surj) (refl _)) (elim e′ y) ≡⟨ cong (λ p → subst P p (elim e′ _)) $ cong-refl (_↠_.to surj) ⟩ subst P (refl _) (elim e′ y) ≡⟨ subst-refl P _ ⟩∎ elim e′ y ∎) hyp ⟩ elim e′ [ x ] ≡⟨⟩ p-[] x ∎ where e′ = _ -- If there is a bijection from a quotient type to some other type, -- then one can also construct a quotient-like eliminator for the -- other type. ↔-eliminator : (bij : A / R ↔ B) (P : B → Type p) (p-[] : ∀ x → P (_↔_.to bij [ x ])) → (∀ {x y} (r : R x y) → subst P (cong (_↔_.to bij) ([]-respects-relation r)) (p-[] x) ≡ p-[] y) → ((∀ x y → Is-proposition (R x y)) → ∀ x → Is-set (P x)) → ∀ x → P x ↔-eliminator bij = ↠-eliminator (_↔_.surjection bij) -- This latter eliminator always "computes" in the "right" way, -- assuming that the quotient relation is pointwise propositional. ↔-eliminator-[] : ∀ (bij : A / R ↔ B) (P : B → Type p) (p-[] : ∀ x → P (_↔_.to bij [ x ])) (ok : ∀ {x y} (r : R x y) → subst P (cong (_↔_.to bij) ([]-respects-relation r)) (p-[] x) ≡ p-[] y) (P-set : (∀ x y → Is-proposition (R x y)) → ∀ x → Is-set (P x)) x → (∀ {x y} → Is-proposition (R x y)) → ↔-eliminator bij P p-[] ok P-set (_↔_.to bij [ x ]) ≡ p-[] x ↔-eliminator-[] bij P p-[] ok P-set x R-prop = ↠-eliminator-[] (_↔_.surjection bij) P p-[] ok P-set x R-prop (_↔_.left-inverse-of bij [ x ]) -- A quotient-like eliminator for functions of type ℕ → A / R, where R -- is a propositional equivalence relation. Defined using countable -- choice and propositional extensionality. -- -- This eliminator is taken from Corollary 1 in "Quotienting the Delay -- Monad by Weak Bisimilarity" by Chapman, Uustalu and Veltri. ℕ→/-elim : {A : Type a} {R : A → A → Type r} → Axiom-of-countable-choice (a ⊔ r) → Propositional-extensionality r → Is-equivalence-relation R → (∀ {x y} → Is-proposition (R x y)) → (P : (ℕ → A / R) → Type p) (p-[] : ∀ f → P (λ n → [ f n ])) → (∀ {f g} (r : (ℕ →ᴾ R) f g) → subst P (cong ℕ→/-comm-to ([]-respects-relation r)) (p-[] f) ≡ p-[] g) → (∀ f → Is-set (P f)) → ∀ f → P f ℕ→/-elim cc prop-ext R-equiv R-prop P p-[] ok P-set = ↔-eliminator (ℕ→/-comm cc prop-ext R-equiv R-prop) P p-[] ok (λ _ → P-set) -- The eliminator "computes" in the "right" way. ℕ→/-elim-[] : ∀ {A : Type a} {R : A → A → Type r} (cc : Axiom-of-countable-choice (a ⊔ r)) (prop-ext : Propositional-extensionality r) (R-equiv : Is-equivalence-relation R) (R-prop : ∀ {x y} → Is-proposition (R x y)) (P : (ℕ → A / R) → Type p) (p-[] : ∀ f → P (λ n → [ f n ])) (ok : ∀ {f g} (r : (ℕ →ᴾ R) f g) → subst P (cong ℕ→/-comm-to ([]-respects-relation r)) (p-[] f) ≡ p-[] g) (P-set : ∀ f → Is-set (P f)) f → ℕ→/-elim cc prop-ext R-equiv R-prop P p-[] ok P-set (λ n → [ f n ]) ≡ p-[] f ℕ→/-elim-[] {R = R} cc prop-ext R-equiv R-prop P p-[] ok P-set f = ↔-eliminator-[] (ℕ→/-comm cc prop-ext R-equiv R-prop) P p-[] ok (λ _ → P-set) f (→ᴾ-preserves-Is-proposition R ext R-prop)

module List.Sorted {A : Set}(_≤_ : A → A → Set) where open import Data.List data Sorted : List A → Set where nils : Sorted [] singls : (x : A) → Sorted [ x ] conss : {x y : A}{xs : List A} → x ≤ y → Sorted (y ∷ xs) → Sorted (x ∷ y ∷ xs)

{-# OPTIONS --cubical --no-import-sorts --safe #-} module Cubical.HITs.TypeQuotients where open import Cubical.HITs.TypeQuotients.Base public open import Cubical.HITs.TypeQuotients.Properties public

{- Descriptor language for easily defining structures -} {-# OPTIONS --cubical --no-import-sorts --no-exact-split --safe #-} module Cubical.Structures.Macro where open import Cubical.Foundations.Prelude open import Cubical.Foundations.Function open import Cubical.Foundations.HLevels open import Cubical.Foundations.Equiv open import Cubical.Foundations.SIP open import Cubical.Functions.FunExtEquiv open import Cubical.Data.Sigma open import Cubical.Data.Maybe open import Cubical.Structures.Constant open import Cubical.Structures.Function open import Cubical.Structures.Maybe open import Cubical.Structures.Parameterized open import Cubical.Structures.Pointed open import Cubical.Structures.Product data TranspDesc (ℓ : Level) : Typeω where -- constant structure: X ↦ A constant : ∀ {ℓ'} (A : Type ℓ') → TranspDesc ℓ -- pointed structure: X ↦ X var : TranspDesc ℓ -- product of structures S,T : X ↦ (S X × T X) _,_ : (d₀ : TranspDesc ℓ) (d₁ : TranspDesc ℓ) → TranspDesc ℓ -- functions between structures S,T: X ↦ (S X → T X) function : (d₀ : TranspDesc ℓ) (d₁ : TranspDesc ℓ) → TranspDesc ℓ -- Maybe on a structure S: X ↦ Maybe (S X) maybe : TranspDesc ℓ → TranspDesc ℓ -- arbitrary transport structure foreign : ∀ {ℓ'} {S : Type ℓ → Type ℓ'} (α : EquivAction S) → TransportStr α → TranspDesc ℓ data Desc (ℓ : Level) : Typeω where -- constant structure: X ↦ A constant : ∀ {ℓ'} (A : Type ℓ') → Desc ℓ -- pointed structure: X ↦ X var : Desc ℓ -- product of structures S,T : X ↦ (S X × T X) _,_ : (d₀ : Desc ℓ) (d₁ : Desc ℓ) → Desc ℓ -- functions between structures S,T : X ↦ (S X → T X) function : (d₀ : Desc ℓ) (d₁ : Desc ℓ) → Desc ℓ -- functions between structures S,T where S is functorial : X ↦ (S X → T X) function+ : (d₀ : TranspDesc ℓ) (d₁ : Desc ℓ) → Desc ℓ -- Maybe on a structure S: X ↦ Maybe (S X) maybe : Desc ℓ → Desc ℓ -- univalent structure from transport structure transpDesc : TranspDesc ℓ → Desc ℓ -- arbitrary univalent notion of structure foreign : ∀ {ℓ' ℓ''} {S : Type ℓ → Type ℓ'} (ι : StrEquiv S ℓ'') → UnivalentStr S ι → Desc ℓ infixr 4 _,_ {- Transport structures -} transpMacroLevel : ∀ {ℓ} → TranspDesc ℓ → Level transpMacroLevel (constant {ℓ'} x) = ℓ' transpMacroLevel {ℓ} var = ℓ transpMacroLevel {ℓ} (d₀ , d₁) = ℓ-max (transpMacroLevel d₀) (transpMacroLevel d₁) transpMacroLevel (function d₀ d₁) = ℓ-max (transpMacroLevel d₀) (transpMacroLevel d₁) transpMacroLevel (maybe d) = transpMacroLevel d transpMacroLevel (foreign {ℓ'} α τ) = ℓ' -- Structure defined by a transport descriptor TranspMacroStructure : ∀ {ℓ} (d : TranspDesc ℓ) → Type ℓ → Type (transpMacroLevel d) TranspMacroStructure (constant A) X = A TranspMacroStructure var X = X TranspMacroStructure (d₀ , d₁) X = TranspMacroStructure d₀ X × TranspMacroStructure d₁ X TranspMacroStructure (function d₀ d₁) X = TranspMacroStructure d₀ X → TranspMacroStructure d₁ X TranspMacroStructure (maybe d) = MaybeStructure (TranspMacroStructure d) TranspMacroStructure (foreign {S = S} α τ) = S -- Action defined by a transport descriptor transpMacroAction : ∀ {ℓ} (d : TranspDesc ℓ) → EquivAction (TranspMacroStructure d) transpMacroAction (constant A) = constantEquivAction A transpMacroAction var = pointedEquivAction transpMacroAction (d₀ , d₁) = productEquivAction (transpMacroAction d₀) (transpMacroAction d₁) transpMacroAction (function d₀ d₁) = functionEquivAction (transpMacroAction d₀) (transpMacroAction d₁) transpMacroAction (maybe d) = maybeEquivAction (transpMacroAction d) transpMacroAction (foreign α _) = α -- Action defines a transport structure transpMacroTransportStr : ∀ {ℓ} (d : TranspDesc ℓ) → TransportStr (transpMacroAction d) transpMacroTransportStr (constant A) = constantTransportStr A transpMacroTransportStr var = pointedTransportStr transpMacroTransportStr (d₀ , d₁) = productTransportStr (transpMacroAction d₀) (transpMacroTransportStr d₀) (transpMacroAction d₁) (transpMacroTransportStr d₁) transpMacroTransportStr (function d₀ d₁) = functionTransportStr (transpMacroAction d₀) (transpMacroTransportStr d₀) (transpMacroAction d₁) (transpMacroTransportStr d₁) transpMacroTransportStr (maybe d) = maybeTransportStr (transpMacroAction d) (transpMacroTransportStr d) transpMacroTransportStr (foreign α τ) = τ {- General structures -} macroStrLevel : ∀ {ℓ} → Desc ℓ → Level macroStrLevel (constant {ℓ'} x) = ℓ' macroStrLevel {ℓ} var = ℓ macroStrLevel {ℓ} (d₀ , d₁) = ℓ-max (macroStrLevel d₀) (macroStrLevel d₁) macroStrLevel {ℓ} (function+ d₀ d₁) = ℓ-max (transpMacroLevel d₀) (macroStrLevel d₁) macroStrLevel (function d₀ d₁) = ℓ-max (macroStrLevel d₀) (macroStrLevel d₁) macroStrLevel (maybe d) = macroStrLevel d macroStrLevel (transpDesc d) = transpMacroLevel d macroStrLevel (foreign {ℓ'} _ _) = ℓ' macroEquivLevel : ∀ {ℓ} → Desc ℓ → Level macroEquivLevel (constant {ℓ'} x) = ℓ' macroEquivLevel {ℓ} var = ℓ macroEquivLevel (d₀ , d₁) = ℓ-max (macroEquivLevel d₀) (macroEquivLevel d₁) macroEquivLevel {ℓ} (function+ d₀ d₁) = ℓ-max (transpMacroLevel d₀) (macroEquivLevel d₁) macroEquivLevel (function d₀ d₁) = ℓ-max (macroStrLevel d₀) (ℓ-max (macroEquivLevel d₀) (macroEquivLevel d₁)) macroEquivLevel (maybe d) = macroEquivLevel d macroEquivLevel (transpDesc d) = transpMacroLevel d macroEquivLevel (foreign {ℓ'' = ℓ''} _ _) = ℓ'' -- Structure defined by a descriptor MacroStructure : ∀ {ℓ} (d : Desc ℓ) → Type ℓ → Type (macroStrLevel d) MacroStructure (constant A) X = A MacroStructure var X = X MacroStructure (d₀ , d₁) X = MacroStructure d₀ X × MacroStructure d₁ X MacroStructure (function+ d₀ d₁) X = TranspMacroStructure d₀ X → MacroStructure d₁ X MacroStructure (function d₀ d₁) X = MacroStructure d₀ X → MacroStructure d₁ X MacroStructure (maybe d) = MaybeStructure (MacroStructure d) MacroStructure (transpDesc d) = TranspMacroStructure d MacroStructure (foreign {S = S} _ _) = S -- Notion of structured equivalence defined by a descriptor MacroEquivStr : ∀ {ℓ} → (d : Desc ℓ) → StrEquiv {ℓ} (MacroStructure d) (macroEquivLevel d) MacroEquivStr (constant A) = ConstantEquivStr A MacroEquivStr var = PointedEquivStr MacroEquivStr (d₀ , d₁) = ProductEquivStr (MacroEquivStr d₀) (MacroEquivStr d₁) MacroEquivStr (function+ d₀ d₁) = FunctionEquivStr+ (transpMacroAction d₀) (MacroEquivStr d₁) MacroEquivStr (function d₀ d₁) = FunctionEquivStr (MacroEquivStr d₀) (MacroEquivStr d₁) MacroEquivStr (maybe d) = MaybeEquivStr (MacroEquivStr d) MacroEquivStr (transpDesc d) = EquivAction→StrEquiv (transpMacroAction d) MacroEquivStr (foreign ι _) = ι -- Proof that structure induced by descriptor is univalent MacroUnivalentStr : ∀ {ℓ} → (d : Desc ℓ) → UnivalentStr (MacroStructure d) (MacroEquivStr d) MacroUnivalentStr (constant A) = constantUnivalentStr A MacroUnivalentStr var = pointedUnivalentStr MacroUnivalentStr (d₀ , d₁) = productUnivalentStr (MacroEquivStr d₀) (MacroUnivalentStr d₀) (MacroEquivStr d₁) (MacroUnivalentStr d₁) MacroUnivalentStr (function+ d₀ d₁) = functionUnivalentStr+ (transpMacroAction d₀) (transpMacroTransportStr d₀) (MacroEquivStr d₁) (MacroUnivalentStr d₁) MacroUnivalentStr (function d₀ d₁) = functionUnivalentStr (MacroEquivStr d₀) (MacroUnivalentStr d₀) (MacroEquivStr d₁) (MacroUnivalentStr d₁) MacroUnivalentStr (maybe d) = maybeUnivalentStr (MacroEquivStr d) (MacroUnivalentStr d) MacroUnivalentStr (transpDesc d) = TransportStr→UnivalentStr (transpMacroAction d) (transpMacroTransportStr d) MacroUnivalentStr (foreign _ θ) = θ -- Module for easy importing module Macro ℓ (d : Desc ℓ) where structure = MacroStructure d equiv = MacroEquivStr d univalent = MacroUnivalentStr d

------------------------------------------------------------------------ -- Lists where all elements satisfy a given property ------------------------------------------------------------------------ module Data.List.All where open import Data.Function open import Data.List as List hiding (map; all) open import Data.List.Any as Any using (here; there) open Any.Membership-≡ using (_∈_; _⊆_) open import Data.Product as Prod using (_,_) open import Relation.Nullary import Relation.Nullary.Decidable as Dec open import Relation.Unary using (Pred) renaming (_⊆_ to _⋐_) open import Relation.Binary.PropositionalEquality -- All P xs means that all elements in xs satisfy P. infixr 5 _∷_ data All {A} (P : A → Set) : List A → Set where [] : All P [] _∷_ : ∀ {x xs} (px : P x) (pxs : All P xs) → All P (x ∷ xs) head : ∀ {A} {P : A → Set} {x xs} → All P (x ∷ xs) → P x head (px ∷ pxs) = px tail : ∀ {A} {P : A → Set} {x xs} → All P (x ∷ xs) → All P xs tail (px ∷ pxs) = pxs lookup : ∀ {A} {P : A → Set} {xs} → All P xs → (∀ {x} → x ∈ xs → P x) lookup [] () lookup (px ∷ pxs) (here refl) = px lookup (px ∷ pxs) (there x∈xs) = lookup pxs x∈xs tabulate : ∀ {A} {P : A → Set} {xs} → (∀ {x} → x ∈ xs → P x) → All P xs tabulate {xs = []} hyp = [] tabulate {xs = x ∷ xs} hyp = hyp (here refl) ∷ tabulate (hyp ∘ there) map : ∀ {A} {P Q : Pred A} → P ⋐ Q → All P ⋐ All Q map g [] = [] map g (px ∷ pxs) = g px ∷ map g pxs all : ∀ {A} {P : A → Set} → (∀ x → Dec (P x)) → (xs : List A) → Dec (All P xs) all p [] = yes [] all p (x ∷ xs) with p x all p (x ∷ xs) | yes px = Dec.map (_∷_ px , tail) (all p xs) all p (x ∷ xs) | no ¬px = no (¬px ∘ head)

open import Agda.Builtin.List open import Agda.Builtin.Reflection renaming (bindTC to _>>=_) open import Agda.Builtin.Bool open import Agda.Builtin.Equality open import Agda.Builtin.Nat open import Agda.Builtin.Sigma data Unit : Set where unit : Unit data Empty : Set where qq : ∀ {a} {A : Set a} → A → Term → TC _ qq t hole = withNormalisation true do `t ← quoteTC t ``t ← quoteTC `t unify hole ``t macro qU = qq {A = Unit → Set → Set} qE = qq {A = Empty → Set → Set} qA = qq {A = Set → Set} case_of_ : ∀ {a b} {A : Set a} {B : Set b} → A → (A → B) → B case x of f = f x pattern vArg x = arg (arg-info visible relevant) x pattern [_] x = x ∷ [] map : {A B : Set} → (A → B) → List A → List B map f [] = [] map f (x ∷ xs) = f x ∷ map f xs mkArgs : List Nat → List (Arg Term) mkArgs = map λ i → vArg (var i []) unit,X=>_∙_ : Nat → List Nat → Term unit,X=> n ∙ args = pat-lam [ clause [ "X" , vArg (agda-sort (lit 0)) ] (vArg (con (quote unit) []) ∷ vArg (var 0) ∷ []) (var n []) ] (mkArgs args) abs,X=>∙_ : List Nat → Term abs,X=>∙ is = pat-lam [ absurd-clause ( ("()" , vArg (def (quote Empty) [])) ∷ ("X" , vArg (agda-sort (lit 0))) ∷ []) (vArg absurd ∷ vArg (var 0) ∷ []) ] (mkArgs is) abs=>∙_ : List Nat → Term abs=>∙ is = pat-lam [ absurd-clause [ "()" , vArg (def (quote Empty) []) ] (vArg absurd ∷ []) ] (mkArgs is) _ : qU (λ { unit X → X }) ≡ unit,X=> 0 ∙ [] _ = refl _ : (B : Set) → qU (λ { unit X → B }) ≡ unit,X=> 1 ∙ [] _ = λ _ → refl _ : qE (λ { () X }) ≡ abs,X=>∙ [] _ = refl _ : (B : Set) → qE (λ { () X }) ≡ abs,X=>∙ [] _ = λ _ → refl _ : (u : Unit) → qA (case u of λ { unit X → X }) ≡ (unit,X=> 0 ∙ [ 0 ]) _ = λ _ → refl _ : (B : Set) (u : Unit) → qA (case u of λ { unit X → B }) ≡ unit,X=> 2 ∙ [ 0 ] _ = λ _ _ → refl _ : (B : Set) (e : Empty) → qA (case e of λ { () X }) ≡ abs,X=>∙ [ 0 ] _ = λ _ _ → refl _ : qE (λ ()) ≡ abs=>∙ [] _ = refl _ : (B : Set) → qE (λ ()) ≡ abs=>∙ [] _ = λ _ → refl _ : (B : Set) (e : Empty) → qA (case e of λ ()) ≡ abs=>∙ [ 0 ] _ = λ _ _ → refl module _ (A : Set) where _ : qU (λ { unit X → X }) ≡ unit,X=> 0 ∙ [] _ = refl _ : (B : Set) → qU (λ { unit X → B }) ≡ unit,X=> 1 ∙ [] _ = λ _ → refl _ : qE (λ { () X }) ≡ abs,X=>∙ [] _ = refl _ : (u : Unit) → qA (case u of λ { unit X → X }) ≡ (unit,X=> 0 ∙ [ 0 ]) _ = λ _ → refl _ : (B : Set) (u : Unit) → qA (case u of λ { unit X → B }) ≡ unit,X=> 2 ∙ [ 0 ] _ = λ _ _ → refl _ : (B : Set) (e : Empty) → qA (case e of λ { () X }) ≡ abs,X=>∙ [ 0 ] _ = λ _ _ → refl _ : qE (λ ()) ≡ abs=>∙ [] _ = refl _ : (B : Set) → qE (λ ()) ≡ abs=>∙ [] _ = λ _ → refl _ : (B : Set) (e : Empty) → qA (case e of λ ()) ≡ abs=>∙ [ 0 ] _ = λ _ _ → refl

module API.Theorems where open import Algebra open import Algebra.Theorems open import API open import Prelude open import Reasoning -- vertices [x] == vertex x vertices-vertex : ∀ {A} {x : A} -> vertices [ x ] ≡ vertex x vertices-vertex = +identity >> reflexivity -- edge x y == clique [x, y] edge-clique : ∀ {A} {x y : A} -> edge x y ≡ clique (x :: [ y ]) edge-clique = symmetry (R *right-identity) -- vertices xs ⊆ clique xs vertices-clique : ∀ {A} {xs : List A} -> vertices xs ⊆ clique xs vertices-clique {_} {[]} = ⊆reflexivity vertices-clique {a} {_ :: t} = ⊆transitivity (⊆right-monotony (vertices-clique {a} {t})) ⊆connect -- clique (xs ++ ys) == connect (clique xs) (clique ys) connect-clique : ∀ {A} {xs ys : List A} -> clique (xs ++ ys) ≡ connect (clique xs) (clique ys) connect-clique {_} {[]} = symmetry *left-identity connect-clique {a} {_ :: t} = R (connect-clique {a} {t}) >> *associativity

module LC.Reduction where open import LC.Base open import LC.Subst open import Data.Nat open import Data.Nat.Properties open import Relation.Nullary -- β-reduction infix 3 _β→_ data _β→_ : Term → Term → Set where β-ƛ-∙ : ∀ {M N} → ((ƛ M) ∙ N) β→ (M [ N ]) β-ƛ : ∀ {M N} → M β→ N → ƛ M β→ ƛ N β-∙-l : ∀ {L M N} → M β→ N → M ∙ L β→ N ∙ L β-∙-r : ∀ {L M N} → M β→ N → L ∙ M β→ L ∙ N open import Relation.Binary.Construct.Closure.ReflexiveTransitive infix 2 _β→*_ _β→*_ : Term → Term → Set _β→*_ = Star _β→_ {-# DISPLAY Star _β→_ = _β→*_ #-} open import Relation.Binary.PropositionalEquality hiding ([_]; preorder) ≡⇒β→* : ∀ {M N} → M ≡ N → M β→* N ≡⇒β→* refl = ε cong-var : ∀ {x y} → x ≡ y → var x β→* var y cong-var {x} {y} refl = ε cong-ƛ : {M N : Term} → M β→* N → ƛ M β→* ƛ N cong-ƛ = gmap _ β-ƛ cong-∙-l : {L M N : Term} → M β→* N → M ∙ L β→* N ∙ L cong-∙-l = gmap _ β-∙-l cong-∙-r : {L M N : Term} → M β→* N → L ∙ M β→* L ∙ N cong-∙-r = gmap _ β-∙-r cong-∙ : {M M' N N' : Term} → M β→* M' → N β→* N' → M ∙ N β→* M' ∙ N' cong-∙ M→M' N→N' = (cong-∙-l M→M') ◅◅ (cong-∙-r N→N') open import LC.Subst.Term cong-lift : {n i : ℕ} {M N : Term} → M β→ N → lift n i M β→* lift n i N cong-lift (β-ƛ M→N) = cong-ƛ (cong-lift M→N) cong-lift (β-ƛ-∙ {M} {N}) = β-ƛ-∙ ◅ ≡⇒β→* (lemma M N) cong-lift (β-∙-l M→N) = cong-∙-l (cong-lift M→N) cong-lift (β-∙-r M→N) = cong-∙-r (cong-lift M→N) cong-[]-r : ∀ L {M N i} → M β→ N → L [ M / i ] β→* L [ N / i ] cong-[]-r (var x) {M} {N} {i} M→N with match x i ... | Under _ = ε ... | Exact _ = cong-lift M→N ... | Above _ _ = ε cong-[]-r (ƛ L) M→N = cong-ƛ (cong-[]-r L M→N) cong-[]-r (K ∙ L) M→N = cong-∙ (cong-[]-r K M→N) (cong-[]-r L M→N) cong-[]-l : ∀ {M N L i} → M β→ N → M [ L / i ] β→* N [ L / i ] cong-[]-l {ƛ M} (β-ƛ M→N) = cong-ƛ (cong-[]-l M→N) cong-[]-l {.(ƛ K) ∙ M} {L = L} (β-ƛ-∙ {K}) = β-ƛ-∙ ◅ ≡⇒β→* (subst-lemma K M L) cong-[]-l {K ∙ M} (β-∙-l M→N) = cong-∙-l (cong-[]-l M→N) cong-[]-l {K ∙ M} (β-∙-r M→N) = cong-∙-r (cong-[]-l M→N) cong-[] : {M M' N N' : Term} → M β→* M' → N β→* N' → M [ N ] β→* M' [ N' ] cong-[] {M} ε ε = ε cong-[] {M} {N = L} {N'} ε (_◅_ {j = N} L→N N→N') = M[L]→M[N] ◅◅ M[N]→M[N'] where M[L]→M[N] : M [ L ] β→* M [ N ] M[L]→M[N] = cong-[]-r M L→N M[N]→M[N'] : M [ N ] β→* M [ N' ] M[N]→M[N'] = cong-[] {M} ε N→N' cong-[] {M} (K→M ◅ M→M') N→N' = cong-[]-l K→M ◅◅ cong-[] M→M' N→N'

-- Copyright: (c) 2016 Ertugrul Söylemez -- License: BSD3 -- Maintainer: Ertugrul Söylemez <esz@posteo.de> -- -- This module contains definitions that are fundamental and/or used -- everywhere. module Core where open import Agda.Builtin.Equality public renaming (refl to ≡-refl) using (_≡_) open import Agda.Builtin.FromNat public open import Agda.Builtin.FromNeg public open import Agda.Builtin.FromString public open import Agda.Builtin.Unit using (⊤; tt) public open import Agda.Primitive using (Level; lsuc; lzero; _⊔_) public -- An empty type (or a false hypothesis). data ⊥ : Set where -- Dependent sums (or existential quantification). record Σ {a b} (A : Set a) (B : A → Set b) : Set (a ⊔ b) where constructor _,_ field fst : A snd : B fst open Σ public infixr 4 _,_ infixr 0 _because_ _because:_ ∃ : ∀ {a b} {A : Set a} (B : A → Set b) → Set (a ⊔ b) ∃ = Σ _ _because_ : ∀ {a b} {A : Set a} {B : A → Set b} → (x : A) → B x → Σ A B _because_ = _,_ _because:_ : ∀ {a b} {A : Set a} {B : A → Set b} → (x : A) → B x → Σ A B _because:_ = _,_ _×_ : ∀ {a b} (A : Set a) (B : Set b) → Set (a ⊔ b) A × B = Σ A (λ _ → B) infixr 7 _×_ -- Tagged unions. data Either {a} {b} (A : Set a) (B : Set b) : Set (a ⊔ b) where Left : A → Either A B Right : B → Either A B -- Equivalence relations. record Equiv {a r} (A : Set a) : Set (a ⊔ lsuc r) where field _≈_ : A → A → Set r refl : ∀ {x} → x ≈ x sym : ∀ {x y} → x ≈ y → y ≈ x trans : ∀ {x y z} → x ≈ y → y ≈ z → x ≈ z infix 4 _≈_ -- Helper functions for equational reasoning. begin_ : ∀ {x y} → x ≈ y → x ≈ y begin_ p = p _≈[_]_ : ∀ x {y z} → x ≈ y → y ≈ z → x ≈ z _ ≈[ x≈y ] y≈z = trans x≈y y≈z _≈[]_ : ∀ x {y} → x ≈ y → x ≈ y _ ≈[] p = p _qed : ∀ (x : A) → x ≈ x _qed _ = refl infix 1 begin_ infixr 2 _≈[_]_ _≈[]_ infix 3 _qed PropEq : ∀ {a} → (A : Set a) → Equiv A PropEq A = record { _≈_ = _≡_; refl = ≡-refl; sym = sym'; trans = trans' } where sym' : ∀ {x y} → x ≡ y → y ≡ x sym' ≡-refl = ≡-refl trans' : ∀ {x y z} → x ≡ y → y ≡ z → x ≡ z trans' ≡-refl q = q module PropEq {a} {A : Set a} = Equiv (PropEq A) FuncEq : ∀ {a b} (A : Set a) (B : Set b) → Equiv (A → B) FuncEq A B = record { _≈_ = λ f g → ∀ x → f x ≡ g x; refl = λ _ → ≡-refl; sym = λ p x → PropEq.sym (p x); trans = λ p q x → PropEq.trans (p x) (q x) } module FuncEq {a b} {A : Set a} {B : Set b} = Equiv (FuncEq A B) cong : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) {x y} → x ≡ y → f x ≡ f y cong _ ≡-refl = ≡-refl cong2 : ∀ {a b c} {A : Set a} {B : Set b} {C : Set c} (f : A → B → C) → ∀ {x1 x2 : A} {y1 y2 : B} → x1 ≡ x2 → y1 ≡ y2 → f x1 y1 ≡ f x2 y2 cong2 _ ≡-refl ≡-refl = ≡-refl -- Partial orders. record PartialOrder {a re rl} (A : Set a) : Set (a ⊔ lsuc (re ⊔ rl)) where field Eq : Equiv {r = re} A module ≈ = Equiv Eq open ≈ public using (_≈_) field _≤_ : (x : A) → (y : A) → Set rl antisym : ∀ {x y} → x ≤ y → y ≤ x → x ≈ y refl' : ∀ {x y} → x ≈ y → x ≤ y trans : ∀ {x y z} → x ≤ y → y ≤ z → x ≤ z infix 4 _≤_ refl : ∀ {x} → x ≤ x refl = refl' ≈.refl -- Helper functions for transitivity reasoning. begin_ : ∀ {x y} → x ≤ y → x ≤ y begin_ p = p _≤[_]_ : ∀ x {y z} → x ≤ y → y ≤ z → x ≤ z _ ≤[ x≤y ] y≤z = trans x≤y y≤z _≤[]_ : ∀ x {y} → x ≤ y → x ≤ y _ ≤[] p = p _qed : ∀ (x : A) → x ≤ x _qed _ = refl infix 1 begin_ infixr 2 _≤[_]_ _≤[]_ infix 3 _qed -- Total orders. record TotalOrder {a re rl} (A : Set a) : Set (a ⊔ lsuc (re ⊔ rl)) where field partialOrder : PartialOrder {a} {re} {rl} A open PartialOrder partialOrder public field total : ∀ x y → Either (x ≤ y) (y ≤ x) -- Low-priority function application. _$_ : ∀ {a} {A : Set a} → A → A _$_ f = f infixr 0 _$_ -- Given two predicates, this is the predicate that requires both. _and_ : ∀ {a r1 r2} {A : Set a} → (A → Set r1) → (A → Set r2) → A → Set (r1 ⊔ r2) (P and Q) x = P x × Q x infixr 6 _and_ -- Use instance resolution to find a value of the target type. it : ∀ {a} {A : Set a} {{_ : A}} → A it {{x}} = x -- Not. not : ∀ {a} → Set a → Set a not A = A → ⊥ -- Given two predicates, this is the predicate that requires at least -- one of them. _or_ : ∀ {a r1 r2} {A : Set a} → (A → Set r1) → (A → Set r2) → A → Set (r1 ⊔ r2) (P or Q) x = Either (P x) (Q x) -- Values with inline type signatures. the : ∀ {a} (A : Set a) → A → A the _ x = x -- Decidable properties. data Decision {a} (P : Set a) : Set a where yes : (p : P) → Decision P no : (np : not P) → Decision P

module Oscar.Data.Term.AlphaConversion.internal {𝔣} (FunctionName : Set 𝔣) where open import Oscar.Data.Term FunctionName open import Oscar.Data.Equality open import Oscar.Data.Fin open import Oscar.Data.Vec open import Oscar.Function open import Oscar.Relation infixr 19 _◂_ _◂s_ mutual _◂_ : ∀ {m n} → m ⟨ Fin ⟩→ n → m ⟨ Term ⟩→ n _◂_ f (i 𝑥) = i (f 𝑥) _◂_ f leaf = leaf _◂_ f (τ₁ fork τ₂) = (f ◂ τ₁) fork (f ◂ τ₂) _◂_ f (function 𝑓 τs) = function 𝑓 (f ◂s τs) _◂s_ : ∀ {m n} → m ⟨ Fin ⟩→ n → ∀ {N} → m ⟨ Terms N ⟩→ n _◂s_ f [] = [] _◂s_ f (τ ∷ τs) = f ◂ τ ∷ f ◂s τs mutual ◂-identity : ∀ {m} (τ : Term m) → id ◂ τ ≡ τ ◂-identity (i _) = refl ◂-identity leaf = refl ◂-identity (τ₁ fork τ₂) rewrite ◂-identity τ₁ | ◂-identity τ₂ = refl ◂-identity (function 𝑓 τs) rewrite ◂s-identity τs = refl ◂s-identity : ∀ {N m} (τs : Terms N m) → id ◂s τs ≡ τs ◂s-identity [] = refl ◂s-identity (τ ∷ τs) rewrite ◂-identity τ | ◂s-identity τs = refl mutual ◂-associativity : ∀ {l m n} (f : l ⟨ Fin ⟩→ m) (g : m ⟨ Fin ⟩→ n) → (τ : Term l) → (g ∘ f) ◂ τ ≡ g ◂ f ◂ τ ◂-associativity _ _ (i _) = refl ◂-associativity _ _ leaf = refl ◂-associativity f g (τ₁ fork τ₂) rewrite ◂-associativity f g τ₁ | ◂-associativity f g τ₂ = refl ◂-associativity f g (function 𝑓 τs) rewrite ◂s-associativity f g τs = refl ◂s-associativity : ∀ {l m n} (f : l ⟨ Fin ⟩→ m) (g : m ⟨ Fin ⟩→ n) → ∀ {N} (x : Terms N l) → (g ∘ f) ◂s x ≡ g ◂s f ◂s x ◂s-associativity _ _ [] = refl ◂s-associativity f g (τ ∷ τs) rewrite ◂-associativity f g τ | ◂s-associativity f g τs = refl mutual ◂-extensionality : ∀ {m n} {f g : m ⟨ Fin ⟩→ n} → f ≡̇ g → f ◂_ ≡̇ g ◂_ ◂-extensionality f≡̇g (i 𝑥) rewrite f≡̇g 𝑥 = refl ◂-extensionality f≡̇g leaf = refl ◂-extensionality f≡̇g (τ₁ fork τ₂) rewrite ◂-extensionality f≡̇g τ₁ | ◂-extensionality f≡̇g τ₂ = refl ◂-extensionality f≡̇g (function 𝑓 τs) rewrite ◂s-extensionality f≡̇g τs = refl ◂s-extensionality : ∀ {m n} {f g : m ⟨ Fin ⟩→ n} → f ≡̇ g → ∀ {N} → _◂s_ f {N} ≡̇ _◂s_ g {N} ◂s-extensionality _ [] = refl ◂s-extensionality f≡̇g (τ ∷ τs) rewrite ◂-extensionality f≡̇g τ | ◂s-extensionality f≡̇g τs = refl

{-# OPTIONS --warning=error --safe --without-K #-} open import LogicalFormulae open import Numbers.Naturals.Definition module Numbers.Naturals.Addition where infix 15 _+N_ _+N_ : ℕ → ℕ → ℕ zero +N y = y succ x +N y = succ (x +N y) {-# BUILTIN NATPLUS _+N_ #-} addZeroRight : (x : ℕ) → (x +N zero) ≡ x addZeroRight zero = refl addZeroRight (succ x) rewrite addZeroRight x = refl private succExtracts : (x y : ℕ) → (x +N succ y) ≡ (succ (x +N y)) succExtracts zero y = refl succExtracts (succ x) y = applyEquality succ (succExtracts x y) succCanMove : (x y : ℕ) → (x +N succ y) ≡ (succ x +N y) succCanMove x y = transitivity (succExtracts x y) refl additionNIsCommutative : (x y : ℕ) → (x +N y) ≡ (y +N x) additionNIsCommutative zero y = equalityCommutative (addZeroRight y) additionNIsCommutative (succ x) zero = transitivity (addZeroRight (succ x)) refl additionNIsCommutative (succ x) (succ y) = transitivity refl (applyEquality succ (transitivity (succCanMove x y) (additionNIsCommutative (succ x) y))) addingPreservesEqualityRight : {a b : ℕ} (c : ℕ) → (a ≡ b) → (a +N c ≡ b +N c) addingPreservesEqualityRight {a} {b} c pr = applyEquality (λ n -> n +N c) pr addingPreservesEqualityLeft : {a b : ℕ} (c : ℕ) → (a ≡ b) → (c +N a ≡ c +N b) addingPreservesEqualityLeft {a} {b} c pr = applyEquality (λ n -> c +N n) pr additionNIsAssociative : (a b c : ℕ) → ((a +N b) +N c) ≡ (a +N (b +N c)) additionNIsAssociative zero b c = refl additionNIsAssociative (succ a) zero c = transitivity (transitivity (applyEquality (λ n → n +N c) (applyEquality succ (addZeroRight a))) refl) (transitivity refl refl) additionNIsAssociative (succ a) (succ b) c = transitivity refl (transitivity refl (transitivity (applyEquality succ (additionNIsAssociative a (succ b) c)) refl)) succIsAddOne : (a : ℕ) → succ a ≡ a +N succ zero succIsAddOne a = equalityCommutative (transitivity (additionNIsCommutative a (succ zero)) refl) canSubtractFromEqualityRight : {a b c : ℕ} → (a +N b ≡ c +N b) → a ≡ c canSubtractFromEqualityRight {a} {zero} {c} pr = transitivity (equalityCommutative (addZeroRight a)) (transitivity pr (addZeroRight c)) canSubtractFromEqualityRight {a} {succ b} {c} pr rewrite additionNIsCommutative a (succ b) | additionNIsCommutative c (succ b) | additionNIsCommutative b a | additionNIsCommutative b c = canSubtractFromEqualityRight {a} {b} {c} (succInjective pr) canSubtractFromEqualityLeft : {a b c : ℕ} → (a +N b ≡ a +N c) → b ≡ c canSubtractFromEqualityLeft {a} {b} {c} pr rewrite additionNIsCommutative a b | additionNIsCommutative a c = canSubtractFromEqualityRight {b} {a} {c} pr

module ProcessSyntax where open import Data.List open import Typing open import Syntax -- processes data Proc (Φ : TCtx) : Set where exp : (e : Expr Φ TUnit) → Proc Φ par : ∀ {Φ₁ Φ₂} → (sp : Split Φ Φ₁ Φ₂) → (P₁ : Proc Φ₁) → (P₂ : Proc Φ₂) → Proc Φ res : (s : SType) → (P : Proc (TChan (SType.force s) ∷ TChan (SType.force (dual s)) ∷ Φ)) → Proc Φ

------------------------------------------------------------------------------ -- Distributive laws on a binary operation: Task B ------------------------------------------------------------------------------ {-# OPTIONS --exact-split #-} {-# OPTIONS --no-sized-types #-} {-# OPTIONS --no-universe-polymorphism #-} {-# OPTIONS --without-K #-} module DistributiveLaws.TaskB-TopDownATP where open import DistributiveLaws.Base open import Common.FOL.Relation.Binary.EqReasoning ------------------------------------------------------------------------------ -- We found the longest chains of equalities from -- DistributiveLaws.TaskB-I which are prove by the ATPs, following a -- top-down approach. prop₂ : ∀ u x y z → (x · y · (z · u)) · (( x · y · ( z · u)) · (x · z · (y · u))) ≡ x · z · (y · u) prop₂ u x y z = -- The numbering of the proof step justifications are associated with -- the numbers used in DistributiveLaws.TaskB-I. xy·zu · (xy·zu · xz·yu) ≡⟨ j₁₋₅ ⟩ xy·zu · (xz · xu·yu · (y·zu · xz·yu)) ≡⟨ j₅₋₉ ⟩ xy·zu · (xz · xyu · (yxz · yu)) ≡⟨ j₉₋₁₄ ⟩ xz · xyu · (yz · xyu) · (xz · xyu · (y·xu · z·yu)) ≡⟨ j₁₄₋₂₀ ⟩ xz · xyu · (y·xu · (y·yu · z·yu) · (z · xu·yu · (y·xu · z·yu))) ≡⟨ j₂₀₋₂₃ ⟩ xz · xyu · (y · xu·zu · (z · xu·yu · (y·xu · z·yu))) ≡⟨ j₂₃₋₂₅ ⟩ (xz · xyu) · (y · xu·zu · (z·xu · y·xu · z·yu)) ≡⟨ j₂₅₋₃₀ ⟩ xz · xyu · (y·zy · xzu) ≡⟨ j₃₀₋₃₅ ⟩ xz·yu ∎ where -- Two variables abbreviations xz = x · z yu = y · u yz = y · z {-# ATP definitions xz yu yz #-} -- Three variables abbreviations xyu = x · y · u xyz = x · y · z xzu = x · z · u yxz = y · x · z {-# ATP definitions xyu xyz xzu yxz #-} y·xu = y · (x · u) y·yu = y · (y · u) y·zu = y · (z · u) y·zy = y · (z · y) {-# ATP definitions y·xu y·yu y·zu y·zy #-} z·xu = z · (x · u) z·yu = z · (y · u) {-# ATP definitions z·xu z·yu #-} -- Four variables abbreviations xu·yu = x · u · (y · u) xu·zu = x · u · (z · u) {-# ATP definitions xu·yu xu·zu #-} xy·zu = x · y · (z · u) {-# ATP definition xy·zu #-} xz·yu = x · z · (y · u) {-# ATP definition xz·yu #-} -- Steps justifications postulate j₁₋₅ : xy·zu · (xy·zu · xz·yu) ≡ xy·zu · (xz · xu·yu · (y·zu · xz·yu)) {-# ATP prove j₁₋₅ #-} postulate j₅₋₉ : xy·zu · (xz · xu·yu · (y·zu · xz·yu)) ≡ xy·zu · (xz · xyu · (yxz · yu)) {-# ATP prove j₅₋₉ #-} postulate j₉₋₁₄ : xy·zu · (xz · xyu · (yxz · yu)) ≡ xz · xyu · (yz · xyu) · (xz · xyu · (y·xu · z·yu)) {-# ATP prove j₉₋₁₄ #-} postulate j₁₄₋₂₀ : xz · xyu · (yz · xyu) · (xz · xyu · (y·xu · z·yu)) ≡ xz · xyu · (y·xu · (y·yu · z·yu) · (z · xu·yu · (y·xu · z·yu))) {-# ATP prove j₁₄₋₂₀ #-} postulate j₂₀₋₂₃ : xz · xyu · (y·xu · (y·yu · z·yu) · (z · xu·yu · (y·xu · z·yu))) ≡ xz · xyu · (y · xu·zu · (z · xu·yu · (y·xu · z·yu))) {-# ATP prove j₂₀₋₂₃ #-} postulate j₂₃₋₂₅ : xz · xyu · (y · xu·zu · (z · xu·yu · (y·xu · z·yu))) ≡ (xz · xyu) · (y · xu·zu · (z·xu · y·xu · z·yu)) {-# ATP prove j₂₃₋₂₅ #-} postulate j₂₅₋₃₀ : (xz · xyu) · (y · xu·zu · (z·xu · y·xu · z·yu)) ≡ xz · xyu · (y·zy · xzu) {-# ATP prove j₂₅₋₃₀ #-} postulate j₃₀₋₃₅ : xz · xyu · (y·zy · xzu) ≡ xz·yu {-# ATP prove j₃₀₋₃₅ #-}

module Sandbox.IndRec where -- Ornamental Algebras, Algebraic Ornaments, CONOR McBRIDE -- https://personal.cis.strath.ac.uk/conor.mcbride/pub/OAAO/Ornament.pdf -- A Finite Axiomtization of Inductive-Recursion definitions, Peter Dybjer, Anton Setzer -- http://www.cse.chalmers.se/~peterd/papers/Finite_IR.pdf open import Data.Product using (_×_; Σ; _,_) open import Data.Bool open import Data.Unit open import Data.Empty open import Relation.Binary.PropositionalEquality open ≡-Reasoning data Desc : Set₁ where arg : (A : Set) -- a bag of tags to choose constructors with → (A → Desc) -- given a tag, return the description of the constructor it corresponds to → Desc rec : Desc → Desc -- recursive subnode ret : Desc -- stop -- the "decoder", "interpreter" ⟦_⟧ : Desc → Set → Set ⟦ arg A D ⟧ R = Σ A (λ a → ⟦ D a ⟧ R) ⟦ rec D ⟧ R = R × ⟦ D ⟧ R ⟦ ret ⟧ R = ⊤ -- "μ" in some other literature data Data (D : Desc) : Set where ⟨_⟩ : ⟦ D ⟧ (Data D) → Data D -- or "in" out : ∀ {D} → Data D → ⟦ D ⟧ (Data D) out ⟨ x ⟩ = x map : ∀ {A B} → (d : Desc) → (A → B) → ⟦ d ⟧ A → ⟦ d ⟧ B map (arg A d) f (a , y) = a , (map (d a) f y) map (rec desc) f (a , x) = (f a) , (map desc f x) map (ret) f tt = tt -- fold : ∀ {A} → (F : Desc) → (⟦ F ⟧ A → A) → Data F → A -- fold F alg ⟨ x ⟩ = alg (map F (fold F alg) x) -- mapFold F alg x = map F (fold F alg) x -- fold F alg x = alg (mapFold F alg (out x)) -- mapFold F alg x = map F (λ y → alg (mapFold F alg (out y))) x mapFold : ∀ {A} (F G : Desc) → (⟦ G ⟧ A → A) → ⟦ F ⟧ (Data G) → ⟦ F ⟧ A mapFold (arg T decoder) G alg (t , cnstrctr) = t , (mapFold (decoder t) G alg cnstrctr) mapFold (rec F) G alg (⟨ x ⟩ , xs) = alg (mapFold G G alg x) , mapFold F G alg xs mapFold ret G alg x = tt fold : ∀ {A} → (F : Desc) → (⟦ F ⟧ A → A) → Data F → A fold F alg ⟨ x ⟩ = alg (mapFold F F alg x) -- ℕ ℕDesc : Desc ℕDesc = arg Bool λ { false → ret ; true → rec ret } ℕ : Set ℕ = Data ℕDesc zero : ℕ zero = ⟨ (false , tt) ⟩ suc : ℕ → ℕ suc n = ⟨ (true , (n , tt)) ⟩ -- induction principle on ℕ indℕ : (P : ℕ → Set) → (P zero) → ((n : ℕ) → (P n) → (P (suc n))) → (x : ℕ) → P x indℕ P base step ⟨ true , n-1 , _ ⟩ = step n-1 (indℕ P base step n-1) indℕ P base step ⟨ false , _ ⟩ = base _+_ : ℕ → ℕ → ℕ x + y = indℕ (λ _ → ℕ) y (λ n x → suc x) x -- Maybe MaybeDesc : Set → Desc MaybeDesc A = arg Bool (λ { false → ret ; true → arg A (λ x → ret) }) Maybe : Set → Set Maybe A = Data (MaybeDesc A) nothing : ∀ {A} → Maybe A nothing = ⟨ (false , tt) ⟩ just : ∀ {A} → A → Maybe A just a = ⟨ (true , (a , tt)) ⟩ mapMaybe : ∀ {A B} → (A → B) → Maybe A → Maybe B mapMaybe f ⟨ true , a , tt ⟩ = ⟨ (true , ((f a) , tt)) ⟩ mapMaybe f ⟨ false , tt ⟩ = ⟨ false , tt ⟩ -- List ListDesc : Set → Desc ListDesc A = arg Bool (λ { false → ret ; true → arg A (λ _ → rec ret )}) List : Set → Set List A = Data (ListDesc A) nil : ∀ {A} → List A nil = ⟨ (false , tt) ⟩ cons : ∀ {A} → A → List A → List A cons x xs = ⟨ (true , (x , (xs , tt))) ⟩ foldList : ∀ {A B} (f : A → B → B) → B → List A → B foldList {A} {B} f e xs = fold (ListDesc A) alg xs where alg : ⟦ ListDesc A ⟧ B → B alg (true , n , acc , tt) = f n acc alg (false , tt) = e

{-# OPTIONS --without-K --safe #-} module Categories.Category.Finite where open import Level open import Data.Nat using (ℕ) open import Data.Fin open import Categories.Adjoint.Equivalence open import Categories.Category open import Categories.Functor open import Categories.Category.Finite.Fin -- definition of a finite category -- -- the idea is to require a functor from C to a category generated from a finite shape -- is the right adjoint. -- -- Question: it seems to me right adjoint is enough, though the original plan is to -- use adjoint equivalence. intuitively, the shape category is an "overapproximation" -- of C, which is a very strong constraint. so requiring adjoint equivalence sounds an -- unnecessarily stronger constraint. is adjoint equivalence necessary? -- -- Answer: probably yes. adjoint equivalence seems necessary as the notion needs to -- show that shapes are preserved. -- -- c.f. Categories.Adjoint.Equivalence.Properties.⊣equiv-preserves-diagram record Finite {o ℓ e} (C : Category o ℓ e) : Set (o ⊔ ℓ ⊔ e) where field shape : FinCatShape open FinCatShape public renaming (size to ∣Obj∣) shapeCat : Category _ _ _ shapeCat = FinCategory shape -- -- /------------\ -- < - \ -- C | S -- \ - ^ -- \------------/ -- field ⊣equiv : ⊣Equivalence shapeCat C module ⊣equiv = ⊣Equivalence ⊣equiv open ⊣equiv public

module Generic.Lib.Data.Maybe where open import Data.Maybe.Base using (Maybe; just; nothing; maybe; maybe′; from-just; fromMaybe) public open import Generic.Lib.Intro open import Generic.Lib.Category infixr 0 _?>_ instance MaybeMonad : ∀ {α} -> RawMonad {α} Maybe MaybeMonad = record { return = just ; _>>=_ = Data.Maybe.Base._>>=_ } MaybeApplicative : ∀ {α} -> RawApplicative {α} Maybe MaybeApplicative = rawIApplicative MaybeFunctor : ∀ {α} -> RawFunctor {α} Maybe MaybeFunctor = rawFunctor _?>_ : ∀ {α} {A : Set α} -> Bool -> A -> Maybe A true ?> x = just x false ?> x = nothing

open import Type module Relator.Sets {ℓ ℓₑ ℓₛ} {E : Type{ℓₑ}} {S : Type{ℓₛ}} (_∈_ : E → S → Type{ℓ}) where open import Functional open import Logic.Propositional open import Logic.Predicate _∉_ : E → S → Type _∉_ = (¬_) ∘₂ (_∈_) _∋_ : S → E → Type _∋_ = swap(_∈_) _∌_ : S → E → Type _∌_ = (¬_) ∘₂ (_∋_) _⊆_ : S → S → Type _⊆_ L₁ L₂ = ∀ₗ(x ↦ ((_→ᶠ_) on₂ (x ∈_)) L₁ L₂) _⊇_ : S → S → Type _⊇_ L₁ L₂ = ∀ₗ(x ↦ ((_←_) on₂ (x ∈_)) L₁ L₂) _≡_ : S → S → Type _≡_ L₁ L₂ = ∀ₗ(x ↦ ((_↔_) on₂ (x ∈_)) L₁ L₂) _⊈_ : S → S → Type _⊈_ = (¬_) ∘₂ (_⊆_) _⊉_ : S → S → Type _⊉_ = (¬_) ∘₂ (_⊇_) _≢_ : S → S → Type _≢_ = (¬_) ∘₂ (_≡_)

module Issue804 where mutual {-# NO_TERMINATION_CHECK #-} Foo : Set Foo = Foo -- WAS: The pragma above doesn't apply to Foo. An informative error message -- could be helpful. -- NOW: The pragma does apply to Foo.

------------------------------------------------------------------------ -- A type used to index a combined definition of strong and weak -- bisimilarity and expansion ------------------------------------------------------------------------ {-# OPTIONS --safe #-} module Delay-monad.Bisimilarity.Kind where open import Equality.Propositional open import Prelude -- The three different relations are defined as a single relation, -- indexed by the Kind type. data Not-strong-kind : Type where weak expansion : Not-strong-kind data Kind : Type where strong : Kind other : Not-strong-kind → Kind -- Kind equality is decidable. infix 4 _≟-Kind_ _≟-Not-strong-kind_ _≟-Not-strong-kind_ : Decidable-equality Not-strong-kind weak ≟-Not-strong-kind weak = yes refl weak ≟-Not-strong-kind expansion = no λ () expansion ≟-Not-strong-kind weak = no λ () expansion ≟-Not-strong-kind expansion = yes refl _≟-Kind_ : Decidable-equality Kind _≟-Kind_ strong strong = yes refl _≟-Kind_ strong (other _) = no λ() _≟-Kind_ (other _) strong = no λ() _≟-Kind_ (other k₁) (other k₂) = ⊎-map (cong other) (λ { hyp refl → hyp refl }) (k₁ ≟-Not-strong-kind k₂)

{-# OPTIONS --safe --experimental-lossy-unification #-} module Cubical.Algebra.CommRing.BinomialThm where open import Cubical.Foundations.Prelude open import Cubical.Foundations.Function open import Cubical.Data.Nat renaming ( _+_ to _+ℕ_ ; _·_ to _·ℕ_ ; _^_ to _^ℕ_ ; +-comm to +ℕ-comm ; ·-assoc to ·ℕ-assoc ; ·-comm to ·ℕ-comm ; _choose_ to _ℕchoose_ ; snotz to ℕsnotz) open import Cubical.Data.Nat.Order open import Cubical.Data.FinData open import Cubical.Data.Empty as ⊥ open import Cubical.Algebra.Monoid.BigOp open import Cubical.Algebra.Ring.BigOps open import Cubical.Algebra.CommRing open import Cubical.Algebra.RingSolver.Reflection private variable ℓ : Level module BinomialThm (R' : CommRing ℓ) where open CommRingStr (snd R') open Exponentiation R' open CommRingTheory R' open Sum (CommRing→Ring R') private R = fst R' _choose_ : ℕ → ℕ → R n choose zero = 1r zero choose suc k = 0r suc n choose suc k = n choose (suc k) + n choose k n<k→nChooseK≡0 : ∀ n k → n < k → n choose k ≡ 0r n<k→nChooseK≡0 zero zero (m , p) = ⊥.rec (ℕsnotz (sym (+ℕ-comm m 1) ∙ p)) n<k→nChooseK≡0 zero (suc k) (m , p) = refl n<k→nChooseK≡0 (suc n) zero (m , p) = ⊥.rec (ℕsnotz {n = m +ℕ (suc n)} (sym (+-suc _ _) ∙ p)) n<k→nChooseK≡0 (suc n) (suc k) (m , p) = cong₂ (_+_) p1 p2 ∙ +Lid 0r where p1 : n choose suc k ≡ 0r p1 = n<k→nChooseK≡0 n (suc k) (suc m , sym (+-suc _ _) ∙ p) p2 : n choose k ≡ 0r p2 = n<k→nChooseK≡0 n k (m , injSuc (sym (+-suc _ _) ∙ p)) nChooseN+1 : ∀ n → n choose (suc n) ≡ 0r nChooseN+1 n = n<k→nChooseK≡0 n (suc n) (0 , refl) BinomialVec : (n : ℕ) → R → R → FinVec R (suc n) BinomialVec n x y i = (n choose (toℕ i)) · x ^ (toℕ i) · y ^ (n ∸ toℕ i) BinomialThm : ∀ (n : ℕ) (x y : R) → (x + y) ^ n ≡ ∑ (BinomialVec n x y) BinomialThm zero x y = solve R' BinomialThm (suc n) x y = (x + y) ^ suc n ≡⟨ refl ⟩ (x + y) · (x + y) ^ n ≡⟨ cong ((x + y) ·_) (BinomialThm n x y) ⟩ (x + y) · ∑ (BinomialVec n x y) ≡⟨ ·Ldist+ _ _ _ ⟩ x · ∑ (BinomialVec n x y) + y · ∑ (BinomialVec n x y) ≡⟨ cong₂ (_+_) (∑Mulrdist _ (BinomialVec n x y)) (∑Mulrdist _ (BinomialVec n x y)) ⟩ ∑ (λ i → x · BinomialVec n x y i) + ∑ (λ i → y · BinomialVec n x y i) ≡⟨ refl ⟩ ∑ {n = suc n} (λ i → x · ((n choose (toℕ i)) · x ^ (toℕ i) · y ^ (n ∸ toℕ i))) + ∑ {n = suc n} (λ i → y · ((n choose (toℕ i)) · x ^ (toℕ i) · y ^ (n ∸ toℕ i))) ≡⟨ cong₂ (_+_) (∑Ext xVecPath) (∑Ext yVecPath) ⟩ ∑ xVec + ∑ yVec ≡⟨ cong (_+ ∑ yVec) (∑Last xVec) ⟩ ∑ (xVec ∘ weakenFin) + xⁿ⁺¹ + (yⁿ⁺¹ + ∑ (yVec ∘ suc)) ≡⟨ solve3 _ _ _ _ ⟩ yⁿ⁺¹ + (∑ (xVec ∘ weakenFin) + ∑ (yVec ∘ suc)) + xⁿ⁺¹ ≡⟨ cong (λ s → yⁿ⁺¹ + s + xⁿ⁺¹) (sym (∑Split _ _)) ⟩ yⁿ⁺¹ + (∑ middleVec) + xⁿ⁺¹ ≡⟨ cong (λ s → yⁿ⁺¹ + s + xⁿ⁺¹) (∑Ext middlePath) ⟩ yⁿ⁺¹ + ∑ ((BinomialVec (suc n) x y) ∘ weakenFin ∘ suc) + xⁿ⁺¹ ≡⟨ refl ⟩ ∑ ((BinomialVec (suc n) x y) ∘ weakenFin) + xⁿ⁺¹ ≡⟨ cong (∑ ((BinomialVec (suc n) x y) ∘ weakenFin) +_) xⁿ⁺¹Path ∙ sym (∑Last (BinomialVec (suc n) x y)) ⟩ ∑ (BinomialVec (suc n) x y) ∎ where xVec : FinVec R (suc n) xVec i = (n choose (toℕ i)) · x ^ (suc (toℕ i)) · y ^ (n ∸ toℕ i) solve1 : ∀ x nci xⁱ yⁿ⁻ⁱ → x · (nci · xⁱ · yⁿ⁻ⁱ) ≡ nci · (x · xⁱ) · yⁿ⁻ⁱ solve1 = solve R' xVecPath : ∀ (i : Fin (suc n)) → x · ((n choose (toℕ i)) · x ^ (toℕ i) · y ^ (n ∸ toℕ i)) ≡ xVec i xVecPath i = solve1 _ _ _ _ yVec : FinVec R (suc n) yVec i = (n choose (toℕ i)) · x ^ (toℕ i) · y ^ (suc (n ∸ toℕ i)) solve2 : ∀ y nci xⁱ yⁿ⁻ⁱ → y · (nci · xⁱ · yⁿ⁻ⁱ) ≡ nci · xⁱ · (y · yⁿ⁻ⁱ) solve2 = solve R' yVecPath : ∀ (i : Fin (suc n)) → y · ((n choose (toℕ i)) · x ^ (toℕ i) · y ^ (n ∸ toℕ i)) ≡ yVec i yVecPath i = solve2 _ _ _ _ xⁿ⁺¹ : R xⁿ⁺¹ = xVec (fromℕ n) yⁿ⁺¹ : R yⁿ⁺¹ = yVec zero xⁿ⁺¹Path : xⁿ⁺¹ ≡ BinomialVec (suc n) x y (fromℕ (suc n)) xⁿ⁺¹Path = cong (λ m → m · (x · x ^ toℕ (fromℕ n)) · y ^ (n ∸ toℕ (fromℕ n))) (sym (+Lid _) ∙ cong (_+ (n choose toℕ (fromℕ n))) (sym (subst (λ m → (n choose suc m) ≡ 0r) (sym (toFromId n)) (nChooseN+1 n)))) solve3 : ∀ sx sy xⁿ⁺¹ yⁿ⁺¹ → sx + xⁿ⁺¹ + (yⁿ⁺¹ + sy) ≡ yⁿ⁺¹ + (sx + sy) + xⁿ⁺¹ solve3 = solve R' middleVec : FinVec R n middleVec i = xVec (weakenFin i) + yVec (suc i) middlePath : ∀ (i : Fin n) → middleVec i ≡ BinomialVec (suc n) x y (weakenFin (suc i)) middlePath i = transport (λ j → (n choose toℕ (weakenFin i)) · (x · x ^ toℕ (weakenFin i)) · y ^ (n ∸ toℕ (weakenFin i)) + (n choose suc (weakenRespToℕ i j)) · (x · x ^ (weakenRespToℕ i j)) · sym yHelper j ≡ ((n choose suc (toℕ (weakenFin i))) + (n choose toℕ (weakenFin i))) · (x · x ^ toℕ (weakenFin i)) · y ^ (n ∸ toℕ (weakenFin i))) (solve4 _ _ _ _) where yHelper : (y · y ^ (n ∸ suc (toℕ i))) ≡ y ^ (n ∸ toℕ (weakenFin i)) yHelper = cong (λ m → y · y ^ (n ∸ suc m)) (sym (weakenRespToℕ i)) ∙ cong (y ^_) (≤-∸-suc (subst (λ m → suc m ≤ n) (sym (weakenRespToℕ _)) (toℕ<n i))) solve4 : ∀ nci ncsi xxⁱ yⁿ⁻ⁱ → nci · xxⁱ · yⁿ⁻ⁱ + ncsi · xxⁱ · yⁿ⁻ⁱ ≡ (ncsi + nci) · xxⁱ · yⁿ⁻ⁱ solve4 = solve R'

postulate Bool : Set D : Bool → Set f : (a : Bool) → D a ⟦_⟧ : ∀ {a} → D a → Bool record State : Set where field bool : Bool open State {{...}} postulate guard : {S : Set} → ({{_ : S}} → Bool) → S test : State test = guard ⟦ f bool ⟧ -- doesn't work, used to work in agda 2.4

{-# OPTIONS --without-K --safe #-} module Categories.Category.Unbundled where -- This is basically identical to Category, except that the -- Obj type is a parameter rather than a field. open import Level open import Function.Base using (flip) open import Relation.Binary using (Rel; IsEquivalence) record Category {o : Level} (Obj : Set o) (ℓ e : Level) : Set (suc (o ⊔ ℓ ⊔ e)) where eta-equality infix 4 _≈_ _⇒_ infixr 9 _∘_ field _⇒_ : Rel Obj ℓ _≈_ : ∀ {A B} → Rel (A ⇒ B) e id : ∀ {A} → (A ⇒ A) _∘_ : ∀ {A B C} → (B ⇒ C) → (A ⇒ B) → (A ⇒ C) field assoc : ∀ {A B C D} {f : A ⇒ B} {g : B ⇒ C} {h : C ⇒ D} → (h ∘ g) ∘ f ≈ h ∘ (g ∘ f) -- We add a symmetric proof of associativity so that the opposite category of the -- opposite category is definitionally equal to the original category. See how -- `op` is implemented. sym-assoc : ∀ {A B C D} {f : A ⇒ B} {g : B ⇒ C} {h : C ⇒ D} → h ∘ (g ∘ f) ≈ (h ∘ g) ∘ f identityˡ : ∀ {A B} {f : A ⇒ B} → id ∘ f ≈ f identityʳ : ∀ {A B} {f : A ⇒ B} → f ∘ id ≈ f -- We add a proof of "neutral" identity proof, in order to ensure the opposite of -- constant functor is definitionally equal to itself. identity² : ∀ {A} → id ∘ id {A} ≈ id {A} equiv : ∀ {A B} → IsEquivalence (_≈_ {A} {B}) ∘-resp-≈ : ∀ {A B C} {f h : B ⇒ C} {g i : A ⇒ B} → f ≈ h → g ≈ i → f ∘ g ≈ h ∘ i

open import Agda.Builtin.Unit open import Agda.Builtin.Sigma open import Agda.Builtin.List open import Agda.Builtin.Equality open import Agda.Builtin.Reflection renaming (bindTC to infixl 4 _>>=_) macro quoteDef : Name → Term → TC ⊤ quoteDef f hole = (getDefinition f >>= quoteTC) >>= unify hole postulate A : Set B : A → Set -- foo is projection-like foo : (x : A) → B x → B x foo x y = y pattern vArg x = arg (arg-info visible relevant) x test : quoteDef foo ≡ function (clause (("x" , vArg (def (quote A) [])) ∷ ("y" , vArg (def (quote B) (vArg (var 0 []) ∷ []))) ∷ []) (vArg (var 1) ∷ vArg (var 0) ∷ []) (var 0 []) ∷ []) test = refl

-- Andreas, 2012-09-17 documenting an internal error in InternalToAbstract.hs -- {-# OPTIONS -v syntax.reify.con:30 -v tc.with.type:30 #-} module NameFirstIfHidden where Total : (D : (A : Set) → A → Set)(A : Set) → Set Total D A = forall a → D A a data D (A : Set) : (a : A) → Set where c : Total D A postulate P : {A : Set}{a : A} → D A a → Set f : (A : Set)(a : A) → P (c a) → D A a f A a p with Total ... | _ = c a -- should succeed, triggered an internal error before

{-# OPTIONS --safe --warning=error --without-K #-} open import Groups.Definition open import Setoids.Setoids open import Sets.EquivalenceRelations open import Groups.Homomorphisms.Definition open import Groups.Homomorphisms.Lemmas open import Groups.Subgroups.Definition open import Groups.Subgroups.Normal.Definition open import Groups.Lemmas module Groups.Homomorphisms.Kernel {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+G_ : A → A → A} {_+H_ : B → B → B} {G : Group S _+G_} {H : Group T _+H_} {f : A → B} (fHom : GroupHom G H f) where open Setoid T open Equivalence eq groupKernelPred : A → Set d groupKernelPred a = Setoid._∼_ T (f a) (Group.0G H) groupKernelPredWd : {x y : A} → (Setoid._∼_ S x y) → groupKernelPred x → groupKernelPred y groupKernelPredWd x=y fx=0 = transitive (GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) x=y)) fx=0 groupKernelIsSubgroup : Subgroup G groupKernelPred Subgroup.closedUnderPlus groupKernelIsSubgroup fg=0 fh=0 = transitive (transitive (GroupHom.groupHom fHom) (Group.+WellDefined H fg=0 fh=0)) (Group.identLeft H) Subgroup.containsIdentity groupKernelIsSubgroup = imageOfIdentityIsIdentity fHom Subgroup.closedUnderInverse groupKernelIsSubgroup fg=0 = transitive (homRespectsInverse fHom) (transitive (inverseWellDefined H fg=0) (invIdent H)) Subgroup.isSubset groupKernelIsSubgroup = groupKernelPredWd groupKernelIsNormalSubgroup : normalSubgroup G groupKernelIsSubgroup groupKernelIsNormalSubgroup {g} fk=0 = transitive (transitive (transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H reflexive (transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H fk=0 reflexive) (Group.identLeft H)))) (symmetric (GroupHom.groupHom fHom)))) (GroupHom.wellDefined fHom (Group.invRight G {g}))) (imageOfIdentityIsIdentity fHom)

open import Categories open import Functors open import RMonads module RMonads.CatofRAdj {a b c d}{C : Cat {a}{b}}{D : Cat {c}{d}}{J : Fun C D} (M : RMonad J) where open import Library open import RAdjunctions open Fun open Cat record ObjAdj {e f} : Set ((a ⊔ b ⊔ c ⊔ d ⊔ lsuc e ⊔ lsuc f)) where open RMonad M field E : Cat {e}{f} adj : RAdj J E open RAdj adj field law : R ○ L ≅ TFun M ηlaw : ∀{X} → left (iden E {OMap L X}) ≅ η {X} bindlaw : ∀{X Y}{f : Hom D (OMap J X) (T Y)} → HMap R (right (subst (Hom D (OMap J X)) (fcong Y (cong OMap (sym law))) f)) ≅ bind f open ObjAdj record HomAdj {e f}(A B : ObjAdj {e}{f}) : Set (a ⊔ b ⊔ c ⊔ d ⊔ lsuc (e ⊔ f)) where constructor homadj open RAdj field K : Fun (E A) (E B) Llaw : K ○ L (adj A) ≅ L (adj B) Rlaw : R (adj A) ≅ R (adj B) ○ K rightlaw : {X : Obj C}{Y : Obj (E A)} {f : Hom D (OMap J X) (OMap (R (adj A)) Y)} → HMap K (right (adj A) {X}{Y} f) ≅ right (adj B) {X} {OMap K Y} (subst (Hom D (OMap J X)) (fcong Y (cong OMap Rlaw)) f) open HomAdj HomAdjEq : ∀{a b}{A B : ObjAdj {a}{b}} (f g : HomAdj A B) → K f ≅ K g → f ≅ g HomAdjEq {A = A}{B} (homadj K _ _ _) (homadj .K _ _ _) refl = cong₃ (homadj K) (ir _ _) (ir _ _) (iext λ _ → iext λ _ → iext λ _ → hir refl) rightlawlem : ∀{e f}{E : Cat {e}{f}}(R : Fun E D)(L : Fun C E) → (p : OMap R ≅ OMap (R ○ (IdF E))) → (right : {X : Obj C} {Y : Obj E} → Hom D (OMap J X) (OMap R Y) → Hom E (OMap L X) Y) → {X : Obj C}{Y : Obj E}{f : Hom D (OMap J X) (OMap R Y)} → right f ≅ right (subst (Hom D (OMap J X)) (fcong Y p) f) rightlawlem R L refl right = refl idLlaw : ∀{e f}(X : ObjAdj {e}{f}) → IdF (E X) ○ RAdj.L (adj X) ≅ RAdj.L (adj X) idLlaw X = FunctorEq _ _ refl refl idRlaw : ∀{e f}(X : ObjAdj {e}{f}) → RAdj.R (adj X) ≅ RAdj.R (adj X) ○ IdF (E X) idRlaw X = FunctorEq _ _ refl refl idrightlaw : ∀{e f} (X : ObjAdj {e}{f}) → {X₁ : Obj C} {Y : Obj (E X)} {f : Hom D (OMap J X₁) (OMap (RAdj.R (adj X)) Y)} → HMap (IdF (E X)) (RAdj.right (adj X) f) ≅ RAdj.right (adj X) (subst (Hom D (OMap J X₁)) (fcong Y (cong OMap (idRlaw X))) f) idrightlaw X = rightlawlem (R (adj X)) (L (adj X)) (cong OMap (FunctorEq (R (adj X)) (R (adj X) ○ IdF (E X)) refl refl)) (right (adj X)) where open RAdj idHomAdj : ∀{e f}{X : ObjAdj {e}{f}} → HomAdj X X idHomAdj {X = X} = record { K = IdF (E X); Llaw = idLlaw X; Rlaw = idRlaw X; rightlaw = idrightlaw X } HMaplem : ∀{a b c d}{C : Cat {a}{b}}{D : Cat {c}{d}}{X X' Y Y' : Obj C} → X ≅ X' → Y ≅ Y' → {f : Hom C X Y}{f' : Hom C X' Y'} → f ≅ f' → (F : Fun C D) → HMap F {X}{Y} f ≅ HMap F {X'}{Y'} f' HMaplem refl refl refl F = refl rightlawlem2 : ∀{e f g h i j}{E : Cat {e}{f}}{F : Cat {g}{h}}{G : Cat {i}{j}} (J : Fun C D) (RX : Fun E D)(LX : Fun C E) (RY : Fun F D)(LY : Fun C F) (RZ : Fun G D)(LZ : Fun C G) (KA : Fun E F)(KB : Fun F G) → (rightX : {X : Obj C}{Y : Obj E} → Hom D (OMap J X) (OMap RX Y) → Hom E (OMap LX X) Y) (rightY : {X : Obj C}{Y : Obj F} → Hom D (OMap J X) (OMap RY Y) → Hom F (OMap LY X) Y) (rightZ : {X : Obj C}{Y : Obj G} → Hom D (OMap J X) (OMap RZ Y) → Hom G (OMap LZ X) Y) → (p : RY ≅ RZ ○ KB)(q : RX ≅ RY ○ KA) → (p' : KA ○ LX ≅ LY) → (r : {X : Obj C}{Y : Obj F}{f : Hom D (OMap J X) (OMap RY Y)} → HMap KB (rightY f) ≅ rightZ (subst (Hom D (OMap J X)) (fcong Y (cong OMap p)) f)) → (s : {X : Obj C}{Y : Obj E}{f : Hom D (OMap J X) (OMap RX Y)} → HMap KA (rightX f) ≅ rightY (subst (Hom D (OMap J X)) (fcong Y (cong OMap q)) f)) → ∀{A B}(h : Hom D (OMap J A) (OMap RX B)) → HMap (KB ○ KA) (rightX h) ≅ rightZ (subst (Hom D (OMap J A)) (fcong B (cong OMap (trans q (cong (λ F₁ → F₁ ○ KA) p)))) h) rightlawlem2 J .((RZ ○ KB) ○ KA) LX .(RZ ○ KB) .(KA ○ LX) RZ LZ KA KB rightX rightY rightZ refl refl refl r s h = trans (cong (HMap KB) s) r compLlaw : ∀{e f}{X Y Z : ObjAdj {e}{f}} → (f : HomAdj Y Z)(g : HomAdj X Y) → (K f ○ K g) ○ RAdj.L (adj X) ≅ RAdj.L (adj Z) compLlaw f g = FunctorEq _ _ (ext (λ A → trans (cong (OMap (K f)) (cong (λ F → OMap F A) (Llaw g))) (cong (λ F → OMap F A) (Llaw f)))) (iext λ A → iext λ B → ext λ h → trans (HMaplem (cong (λ F → OMap F A) (Llaw g)) (cong (λ F → OMap F B) (Llaw g)) (cong (λ F → HMap F h) (Llaw g)) (K f)) (cong (λ F → HMap F h) (Llaw f))) compRlaw : ∀{e f}{X Y Z : ObjAdj {e}{f}} → (f : HomAdj Y Z)(g : HomAdj X Y) → RAdj.R (adj X) ≅ RAdj.R (adj Z) ○ (K f ○ K g) compRlaw f g = FunctorEq _ _ (ext (λ A → trans (cong (λ F → OMap F A) (Rlaw g)) (cong (λ F → OMap F (OMap (K g) A)) (Rlaw f)))) (iext λ A → iext λ B → ext λ h → trans (cong (λ F → HMap F h) (Rlaw g)) (cong (λ F → HMap F (HMap (K g) h)) (Rlaw f))) comprightlaw : ∀{e f}{X Y Z : ObjAdj {e}{f}} → (f : HomAdj Y Z)(g : HomAdj X Y) → {A : Obj C} {B : Obj (E X)} {h : Hom D (OMap J A) (OMap (RAdj.R (adj X)) B)} → HMap (K f ○ K g) (RAdj.right (adj X) h) ≅ RAdj.right (adj Z) (subst (Hom D (OMap J A)) (fcong B (cong OMap (compRlaw f g))) h) comprightlaw {X = X}{Y}{Z} f g {A}{B}{h} = trans (rightlawlem2 J (R (adj X)) (L (adj X)) (R (adj Y)) (L (adj Y)) (R (adj Z)) (L (adj Z)) (K g) (K f) (right (adj X)) (right (adj Y)) (right (adj Z)) (Rlaw f) (Rlaw g) (Llaw g) (rightlaw f) (rightlaw g) h) (cong (right (adj Z)) (trans (stripsubst (Hom D (OMap J A)) h (fcong B (cong OMap (trans (Rlaw g) (cong (λ F → F ○ K g) (Rlaw f)))))) (sym (stripsubst (Hom D (OMap J A)) h (fcong B (cong OMap (compRlaw f g))))))) where open RAdj compHomAdj : ∀{e f}{X Y Z : ObjAdj {e}{f}} → HomAdj Y Z → HomAdj X Y → HomAdj X Z compHomAdj f g = record { K = K f ○ K g; Llaw = compLlaw f g; Rlaw = compRlaw f g; rightlaw = comprightlaw f g } idlHomAdj : ∀{e f}{X Y : ObjAdj {e}{f}} {f : HomAdj X Y} → compHomAdj idHomAdj f ≅ f idlHomAdj = HomAdjEq _ _ (FunctorEq _ _ refl refl) idrHomAdj : ∀{e f}{X Y : ObjAdj {e}{f}} {f : HomAdj X Y} → compHomAdj f idHomAdj ≅ f idrHomAdj = HomAdjEq _ _ (FunctorEq _ _ refl refl) assHomAdj : ∀{a b}{W X Y Z : ObjAdj {a}{b}} {f : HomAdj Y Z} {g : HomAdj X Y} {h : HomAdj W X} → compHomAdj (compHomAdj f g) h ≅ compHomAdj f (compHomAdj g h) assHomAdj = HomAdjEq _ _ (FunctorEq _ _ refl refl) CatofAdj : ∀{a b} → Cat CatofAdj {a}{b} = record{ Obj = ObjAdj {a}{b}; Hom = HomAdj; iden = idHomAdj; comp = compHomAdj; idl = idlHomAdj; idr = idrHomAdj; ass = λ{_ _ _ _ f g h} → assHomAdj {f = f}{g}{h}}

{-# OPTIONS --universe-polymorphism #-} open import Categories.Category module Categories.Functor.Core where open import Level open import Categories.Support.EqReasoning record Functor {o ℓ e o′ ℓ′ e′} (C : Category o ℓ e) (D : Category o′ ℓ′ e′) : Set (o ⊔ ℓ ⊔ e ⊔ o′ ⊔ ℓ′ ⊔ e′) where eta-equality private module C = Category C private module D = Category D field F₀ : C.Obj → D.Obj {F₁} : ∀ {A B} → C [ A , B ] → D [ F₀ A , F₀ B ] .{identity} : ∀ {A} → D [ F₁ (C.id {A}) ≡ D.id ] .{homomorphism} : ∀ {X Y Z} {f : C [ X , Y ]} {g : C [ Y , Z ]} → D [ F₁ (C [ g ∘ f ]) ≡ D [ F₁ g ∘ F₁ f ] ] .{F-resp-≡} : ∀ {A B} {F G : C [ A , B ]} → C [ F ≡ G ] → D [ F₁ F ≡ F₁ G ] op : Functor C.op D.op op = record { F₀ = F₀ ; F₁ = F₁ ; identity = identity ; homomorphism = homomorphism ; F-resp-≡ = F-resp-≡ } Endofunctor : ∀ {o ℓ e} → Category o ℓ e → Set _ Endofunctor C = Functor C C Contravariant : ∀ {o ℓ e o′ ℓ′ e′} (C : Category o ℓ e) (D : Category o′ ℓ′ e′) → Set _ Contravariant C D = Functor C.op D where module C = Category C id : ∀ {o ℓ e} {C : Category o ℓ e} → Endofunctor C id {C = C} = record { F₀ = λ x → x ; F₁ = λ x → x ; identity = refl ; homomorphism = refl ; F-resp-≡ = λ x → x } where open Category.Equiv C infixr 9 _∘_ _∘_ : ∀ {o ℓ e} {o′ ℓ′ e′} {o′′ ℓ′′ e′′} {C : Category o ℓ e} {D : Category o′ ℓ′ e′} {E : Category o′′ ℓ′′ e′′} → Functor D E → Functor C D → Functor C E _∘_ {C = C} {D = D} {E = E} F G = record { F₀ = λ x → F₀ (G₀ x) ; F₁ = λ f → F₁ (G₁ f) ; identity = identity′ ; homomorphism = homomorphism′ ; F-resp-≡ = ∘-resp-≡′ } where module C = Category C module D = Category D module E = Category E module F = Functor F module G = Functor G open F open G renaming (F₀ to G₀; F₁ to G₁; F-resp-≡ to G-resp-≡) .identity′ : ∀ {A} → E [ F₁ (G₁ (C.id {A})) ≡ E.id ] identity′ = begin F₁ (G₁ C.id) ≈⟨ F-resp-≡ G.identity ⟩ F₁ D.id ≈⟨ F.identity ⟩ E.id ∎ where open SetoidReasoning E.hom-setoid .homomorphism′ : ∀ {X Y Z} {f : C [ X , Y ]} {g : C [ Y , Z ]} → E [ F₁ (G₁ (C [ g ∘ f ])) ≡ E [ F₁ (G₁ g) ∘ F₁ (G₁ f) ] ] homomorphism′ {f = f} {g = g} = begin F₁ (G₁ (C [ g ∘ f ])) ≈⟨ F-resp-≡ G.homomorphism ⟩ F₁ (D [ G₁ g ∘ G₁ f ]) ≈⟨ F.homomorphism ⟩ (E [ F₁ (G₁ g) ∘ F₁ (G₁ f) ]) ∎ where open SetoidReasoning E.hom-setoid .∘-resp-≡′ : ∀ {A B} {F G : C [ A , B ]} → C [ F ≡ G ] → E [ F₁ (G₁ F) ≡ F₁ (G₁ G) ] ∘-resp-≡′ = λ x → F-resp-≡ (G-resp-≡ x)

module FunClause where f : {A : Set} → A → A f x = y where y = x z = x data List (A : Set) : Set where nil : List A cons : A → List A → List A snoc : {A : Set} → List A → A → List A snoc nil y = cons y nil snoc (cons x xs) y = cons x (snoc xs y)

------------------------------------------------------------------------ -- Brandt and Henglein's subterm relation ------------------------------------------------------------------------ module RecursiveTypes.Subterm where open import Algebra open import Data.Fin using (Fin; zero; suc; lift) open import Data.Nat open import Data.List using (List; []; _∷_; [_]; _++_) open import Data.List.Properties open import Data.List.Relation.Unary.Any using (here; there) open import Data.List.Relation.Unary.Any.Properties using (++↔; map-with-∈↔) open import Data.List.Membership.Propositional open import Data.List.Membership.Propositional.Properties open import Data.List.Relation.Binary.BagAndSetEquality as BSEq open import Data.List.Relation.Binary.Subset.Propositional open import Data.List.Relation.Binary.Subset.Propositional.Properties open import Data.Product open import Data.Sum open import Function.Base open import Function.Equality using (_⟨$⟩_) open import Function.Inverse using (module Inverse) open import Relation.Binary import Relation.Binary.Reasoning.Preorder as PreR import Relation.Binary.Reasoning.Setoid as EqR import Relation.Binary.PropositionalEquality as P open import Data.Fin.Substitution private open module SetM {A : Set} = CommutativeMonoid (BSEq.commutativeMonoid set A) using (_≈_) open import RecursiveTypes.Syntax open import RecursiveTypes.Substitution hiding (subst) renaming (id to idˢ) open // using (_//_) ------------------------------------------------------------------------ -- Subterms -- The subterm relation. infix 4 _⊑_ data _⊑_ {n} : Ty n → Ty n → Set where refl : ∀ {τ} → τ ⊑ τ unfold : ∀ {σ τ₁ τ₂} (σ⊑ : σ ⊑ unfold[μ τ₁ ⟶ τ₂ ]) → σ ⊑ μ τ₁ ⟶ τ₂ ⟶ˡ : ∀ {σ τ₁ τ₂} (σ⊑τ₁ : σ ⊑ τ₁) → σ ⊑ τ₁ ⟶ τ₂ ⟶ʳ : ∀ {σ τ₁ τ₂} (σ⊑τ₂ : σ ⊑ τ₂) → σ ⊑ τ₁ ⟶ τ₂ -- Some simple consequences. unfold′ : ∀ {n} {τ₁ τ₂ : Ty (suc n)} → unfold[μ τ₁ ⟶ τ₂ ] ⊑ μ τ₁ ⟶ τ₂ unfold′ = unfold refl ⟶ˡ′ : ∀ {n} {τ₁ τ₂ : Ty n} → τ₁ ⊑ τ₁ ⟶ τ₂ ⟶ˡ′ = ⟶ˡ refl ⟶ʳ′ : ∀ {n} {τ₁ τ₂ : Ty n} → τ₂ ⊑ τ₁ ⟶ τ₂ ⟶ʳ′ = ⟶ʳ refl -- The subterm relation is transitive. trans : ∀ {n} {σ τ χ : Ty n} → σ ⊑ τ → τ ⊑ χ → σ ⊑ χ trans σ⊑τ refl = σ⊑τ trans σ⊑τ (unfold τ⊑) = unfold (trans σ⊑τ τ⊑) trans σ⊑τ (⟶ˡ τ⊑δ₁) = ⟶ˡ (trans σ⊑τ τ⊑δ₁) trans σ⊑τ (⟶ʳ τ⊑δ₂) = ⟶ʳ (trans σ⊑τ τ⊑δ₂) ------------------------------------------------------------------------ -- Subterm closure -- The list τ_∗ contains exactly the subterms of τ (possibly with -- duplicates). -- -- The important property is completeness, which ensures that the -- number of distinct subterms of a given type is always finite. mutual infix 7 _∗ _∗′ _∗ : ∀ {n} → Ty n → List (Ty n) τ ∗ = τ ∷ τ ∗′ _∗′ : ∀ {n} → Ty n → List (Ty n) (τ₁ ⟶ τ₂) ∗′ = τ₁ ∗ ++ τ₂ ∗ (μ τ₁ ⟶ τ₂) ∗′ = (τ₁ ⟶ τ₂ ∷ τ₁ ∗ ++ τ₂ ∗) // sub (μ τ₁ ⟶ τ₂) τ ∗′ = [] ------------------------------------------------------------------------ -- Soundness -- _/_ is monotonous in its left argument. /-monoˡ : ∀ {m n σ τ} {ρ : Sub Ty m n} → σ ⊑ τ → σ / ρ ⊑ τ / ρ /-monoˡ refl = refl /-monoˡ (⟶ˡ σ⊑τ₁) = ⟶ˡ (/-monoˡ σ⊑τ₁) /-monoˡ (⟶ʳ σ⊑τ₂) = ⟶ʳ (/-monoˡ σ⊑τ₂) /-monoˡ {ρ = ρ} (unfold {σ} {τ₁} {τ₂} σ⊑) = unfold $ P.subst (λ χ → σ / ρ ⊑ χ) (sub-commutes (τ₁ ⟶ τ₂)) (/-monoˡ σ⊑) sub-⊑-μ : ∀ {n} {σ : Ty (suc n)} {τ₁ τ₂} → σ ⊑ τ₁ ⟶ τ₂ → σ / sub (μ τ₁ ⟶ τ₂) ⊑ μ τ₁ ⟶ τ₂ sub-⊑-μ σ⊑τ₁⟶τ₂ = unfold (/-monoˡ σ⊑τ₁⟶τ₂) -- All list elements are subterms. sound : ∀ {n σ} (τ : Ty n) → σ ∈ τ ∗ → σ ⊑ τ sound τ (here P.refl) = refl sound (τ₁ ⟶ τ₂) (there σ∈) = [ ⟶ˡ ∘ sound τ₁ , ⟶ʳ ∘ sound τ₂ ]′ (Inverse.from (++↔ {xs = τ₁ ∗}) ⟨$⟩ σ∈) sound (μ τ₁ ⟶ τ₂) (there (here P.refl)) = unfold refl sound (μ τ₁ ⟶ τ₂) (there (there σ∈)) with Inverse.from (map-∈↔ (λ σ → σ / sub (μ τ₁ ⟶ τ₂)) {xs = τ₁ ∗ ++ τ₂ ∗}) ⟨$⟩ σ∈ ... | (χ , χ∈ , P.refl) = sub-⊑-μ $ [ ⟶ˡ ∘ sound τ₁ , ⟶ʳ ∘ sound τ₂ ]′ (Inverse.from (++↔ {xs = τ₁ ∗}) ⟨$⟩ χ∈) sound ⊥ (there ()) sound ⊤ (there ()) sound (var x) (there ()) ------------------------------------------------------------------------ -- Completeness ++-lemma : ∀ {A} xs ys {zs : List A} → (xs ++ zs) ++ (ys ++ zs) ≈ (xs ++ ys) ++ zs ++-lemma xs ys {zs} = begin (xs ++ zs) ++ (ys ++ zs) ≈⟨ SetM.assoc xs zs (ys ++ zs) ⟩ xs ++ (zs ++ (ys ++ zs)) ≈⟨ SetM.∙-cong (SetM.refl {x = xs}) (begin zs ++ (ys ++ zs) ≈⟨ SetM.∙-cong (SetM.refl {x = zs}) (SetM.comm ys zs) ⟩ zs ++ (zs ++ ys) ≈⟨ SetM.sym $ SetM.assoc zs zs ys ⟩ (zs ++ zs) ++ ys ≈⟨ SetM.∙-cong (++-idempotent zs) SetM.refl ⟩ zs ++ ys ≈⟨ SetM.comm zs ys ⟩ ys ++ zs ∎) ⟩ xs ++ (ys ++ zs) ≈⟨ SetM.sym $ SetM.assoc xs ys zs ⟩ (xs ++ ys) ++ zs ∎ where open EqR ([ set ]-Equality _) open BSEq.⊆-Reasoning mutual -- Weakening "commutes" with _∗. wk-∗-commute : ∀ k {n} (σ : Ty (k + n)) → σ / wk ↑⋆ k ∗ ⊆ σ ∗ // wk ↑⋆ k wk-∗-commute k σ (here P.refl) = here P.refl wk-∗-commute k σ (there •∈•) = there $ wk-∗′-commute k σ •∈• wk-∗′-commute : ∀ k {n} (σ : Ty (k + n)) → σ / wk ↑⋆ k ∗′ ⊆ σ ∗′ // wk ↑⋆ k wk-∗′-commute k (σ₁ ⟶ σ₂) = begin σ₁ ⟶ σ₂ / wk ↑⋆ k ∗′ ≡⟨ P.refl ⟩ σ₁ / wk ↑⋆ k ∗ ++ σ₂ / wk ↑⋆ k ∗ ⊆⟨ ++⁺ (wk-∗-commute k σ₁) (wk-∗-commute k σ₂) ⟩ σ₁ ∗ // wk ↑⋆ k ++ σ₂ ∗ // wk ↑⋆ k ≡⟨ P.sym $ map-++-commute (λ σ → σ / wk ↑⋆ k) (σ₁ ∗) (σ₂ ∗) ⟩ (σ₁ ∗ ++ σ₂ ∗) // wk ↑⋆ k ≡⟨ P.refl ⟩ σ₁ ⟶ σ₂ ∗′ // wk ↑⋆ k ∎ wk-∗′-commute k (μ σ₁ ⟶ σ₂) = begin (μ σ₁ ⟶ σ₂) / wk ↑⋆ k ∗′ ≡⟨ P.refl ⟩ σ₁ ⟶ σ₂ / wk ↑⋆ (suc k) / sub (σ / wk ↑⋆ k) ∷ (σ₁ / wk ↑⋆ (suc k) ∗ ++ σ₂ / wk ↑⋆ (suc k) ∗) // sub (σ / wk ↑⋆ k) ⊆⟨ ++⁺ lem₁ lem₂ ⟩ σ₁ ⟶ σ₂ / sub σ / wk ↑⋆ k ∷ (σ₁ ∗ ++ σ₂ ∗) // sub σ // wk ↑⋆ k ≡⟨ P.refl ⟩ μ σ₁ ⟶ σ₂ ∗′ // wk ↑⋆ k ∎ where σ = μ σ₁ ⟶ σ₂ lem₁ : _ ⊆ _ lem₁ = begin [ σ₁ ⟶ σ₂ / wk ↑⋆ (suc k) / sub (σ / wk ↑⋆ k) ] ≡⟨ P.cong [_] $ P.sym $ sub-commutes (σ₁ ⟶ σ₂) ⟩ [ σ₁ ⟶ σ₂ / sub σ / wk ↑⋆ k ] ∎ lem₂ : _ ⊆ _ lem₂ = begin (σ₁ / wk ↑⋆ (suc k) ∗ ++ σ₂ / wk ↑⋆ (suc k) ∗) // sub (σ / wk ↑⋆ k) ⊆⟨ map⁺ _ (++⁺ (wk-∗-commute (suc k) σ₁) (wk-∗-commute (suc k) σ₂)) ⟩ (σ₁ ∗ // wk ↑⋆ (suc k) ++ σ₂ ∗ // wk ↑⋆ (suc k)) // sub (σ / wk ↑⋆ k) ≡⟨ P.cong (λ σs → σs // sub (σ / wk ↑⋆ k)) $ P.sym $ map-++-commute (λ σ → σ / wk ↑⋆ suc k) (σ₁ ∗) (σ₂ ∗) ⟩ (σ₁ ∗ ++ σ₂ ∗) // wk ↑⋆ (suc k) // sub (σ / wk ↑⋆ k) ≡⟨ P.sym $ //.sub-commutes (σ₁ ∗ ++ σ₂ ∗) ⟩ (σ₁ ∗ ++ σ₂ ∗) // sub σ // wk ↑⋆ k ∎ wk-∗′-commute k (var x) = begin var x / wk ↑⋆ k ∗′ ≡⟨ P.cong _∗′ (var-/-wk-↑⋆ k x) ⟩ var (lift k suc x) ∗′ ≡⟨ P.refl ⟩ [] ⊆⟨ (λ ()) ⟩ var x ∗′ // wk ↑⋆ k ∎ wk-∗′-commute k ⊥ = λ () wk-∗′-commute k ⊤ = λ () -- Substitution "commutes" with _∗. sub-∗′-commute-var : ∀ k {n} x (τ : Ty n) → var x / sub τ ↑⋆ k ∗′ ⊆ τ ∗ // wk⋆ k sub-∗′-commute-var zero zero τ = begin τ ∗′ ⊆⟨ there ⟩ τ ∗ ≡⟨ P.sym $ //.id-vanishes (τ ∗) ⟩ τ ∗ // wk⋆ zero ∎ sub-∗′-commute-var zero (suc x) τ = begin var x / idˢ ∗′ ≡⟨ P.cong _∗′ (id-vanishes (var x)) ⟩ var x ∗′ ≡⟨ P.refl ⟩ [] ⊆⟨ (λ ()) ⟩ τ ∗ // wk⋆ zero ∎ sub-∗′-commute-var (suc k) zero τ = begin [] ⊆⟨ (λ ()) ⟩ τ ∗ // wk⋆ (suc k) ∎ sub-∗′-commute-var (suc k) (suc x) τ = begin var (suc x) / sub τ ↑⋆ suc k ∗′ ≡⟨ P.cong _∗′ (suc-/-↑ {ρ = sub τ ↑⋆ k} x) ⟩ var x / sub τ ↑⋆ k / wk ∗′ ⊆⟨ wk-∗′-commute zero (var x / sub τ ↑⋆ k) ⟩ var x / sub τ ↑⋆ k ∗′ // wk ⊆⟨ map⁺ _ (sub-∗′-commute-var k x τ) ⟩ τ ∗ // wk⋆ k // wk ≡⟨ P.sym $ //./-weaken (τ ∗) ⟩ τ ∗ // wk⋆ (suc k) ∎ sub-∗-commute : ∀ k {n} σ (τ : Ty n) → σ / sub τ ↑⋆ k ∗ ⊆ σ ∗ // sub τ ↑⋆ k ++ τ ∗ // wk⋆ k sub-∗-commute k σ τ (here P.refl) = here P.refl sub-∗-commute k ⊥ τ •∈• = Inverse.to ++↔ ⟨$⟩ inj₁ •∈• sub-∗-commute k ⊤ τ •∈• = Inverse.to ++↔ ⟨$⟩ inj₁ •∈• sub-∗-commute k (var x) τ (there •∈•) = there $ sub-∗′-commute-var k x τ •∈• sub-∗-commute k (σ₁ ⟶ σ₂) τ {χ} (there •∈•) = there $ χ ∈⟨ •∈• ⟩ (σ₁ / ρ) ∗ ++ (σ₂ / ρ) ∗ ⊆⟨ ++⁺ (sub-∗-commute k σ₁ τ) (sub-∗-commute k σ₂ τ) ⟩ (σ₁ ∗ // ρ ++ τ ∗ // wk⋆ k) ++ (σ₂ ∗ // ρ ++ τ ∗ // wk⋆ k) ∼⟨ ++-lemma (σ₁ ∗ // ρ) (σ₂ ∗ // ρ) ⟩ (σ₁ ∗ // ρ ++ σ₂ ∗ // ρ) ++ τ ∗ // wk⋆ k ≡⟨ P.cong₂ _++_ (P.sym $ map-++-commute (λ σ → σ / ρ) (σ₁ ∗) (σ₂ ∗)) P.refl ⟩ (σ₁ ∗ ++ σ₂ ∗) // ρ ++ τ ∗ // wk⋆ k ∎ where ρ = sub τ ↑⋆ k sub-∗-commute k (μ σ₁ ⟶ σ₂) τ (there (here P.refl)) = there $ here $ P.sym $ sub-commutes (σ₁ ⟶ σ₂) sub-∗-commute k (μ σ₁ ⟶ σ₂) τ {χ} (there (there •∈•)) = there $ there $ χ ∈⟨ •∈• ⟩ ((σ₁ / ρ ↑) ∗ ++ (σ₂ / ρ ↑) ∗) // sub (σ / ρ) ⊆⟨ map⁺ _ (begin (σ₁ / ρ ↑) ∗ ++ (σ₂ / ρ ↑) ∗ ⊆⟨ ++⁺ (sub-∗-commute (suc k) σ₁ τ) (sub-∗-commute (suc k) σ₂ τ) ⟩ (σ₁ ∗ // ρ ↑ ++ τ ∗ // wk⋆ (suc k)) ++ (σ₂ ∗ // ρ ↑ ++ τ ∗ // wk⋆ (suc k)) ∼⟨ ++-lemma (σ₁ ∗ // ρ ↑) (σ₂ ∗ // ρ ↑) ⟩ (σ₁ ∗ // ρ ↑ ++ σ₂ ∗ // ρ ↑) ++ τ ∗ // wk⋆ (suc k) ≡⟨ P.cong₂ _++_ (P.sym $ map-++-commute (λ σ → σ / ρ ↑) (σ₁ ∗) (σ₂ ∗)) P.refl ⟩ (σ₁ ∗ ++ σ₂ ∗) // ρ ↑ ++ τ ∗ // wk⋆ (suc k) ∎) ⟩ ((σ₁ ∗ ++ σ₂ ∗) // ρ ↑ ++ τ ∗ // wk⋆ (suc k)) // sub (σ / ρ) ≡⟨ map-++-commute (λ χ → χ / sub (σ / ρ)) ((σ₁ ∗ ++ σ₂ ∗) // ρ ↑) _ ⟩ (σ₁ ∗ ++ σ₂ ∗) // ρ ↑ // sub (σ / ρ) ++ τ ∗ // wk⋆ (suc k) // sub (σ / ρ) ≡⟨ P.cong₂ _++_ (P.sym $ //.sub-commutes (σ₁ ∗ ++ σ₂ ∗)) lem ⟩ (σ₁ ∗ ++ σ₂ ∗) // sub σ // ρ ++ τ ∗ // wk⋆ k ∎ where ρ = sub τ ↑⋆ k σ = μ σ₁ ⟶ σ₂ lem = ≡R.begin τ ∗ // wk⋆ (suc k) // sub (σ / ρ) ≡R.≡⟨ P.cong₂ _//_ (//./-weaken (τ ∗)) P.refl ⟩ τ ∗ // wk⋆ k // wk // sub (σ / ρ) ≡R.≡⟨ //.wk-sub-vanishes (τ ∗ // wk⋆ k) ⟩ τ ∗ // wk⋆ k ≡R.∎ where module ≡R = P.≡-Reasoning -- The list contains all subterms. complete : ∀ {n} {σ τ : Ty n} → σ ⊑ τ → σ ∈ τ ∗ complete refl = here P.refl complete (⟶ˡ σ⊑τ₁) = there (Inverse.to ++↔ ⟨$⟩ inj₁ (complete σ⊑τ₁)) complete (⟶ʳ σ⊑τ₂) = there (Inverse.to (++↔ {P = P._≡_ _} {xs = _ ∗}) ⟨$⟩ inj₂ (complete σ⊑τ₂)) complete (unfold {σ} {τ₁} {τ₂} σ⊑) = σ ∈⟨ complete σ⊑ ⟩ unfold[μ τ₁ ⟶ τ₂ ] ∗ ⊆⟨ sub-∗-commute zero (τ₁ ⟶ τ₂) τ ⟩ τ₁ ⟶ τ₂ ∗ // sub τ ++ τ ∗ // idˢ ≡⟨ P.cong (_++_ (τ₁ ⟶ τ₂ ∗ // sub τ)) (//.id-vanishes (τ ∗)) ⟩ τ ∗′ ++ τ ∗ ⊆⟨ ++⁺ (there {x = τ} {xs = τ ∗′}) id ⟩ τ ∗ ++ τ ∗ ∼⟨ ++-idempotent (τ ∗) ⟩ τ ∗ ∎ where τ = μ τ₁ ⟶ τ₂ ------------------------------------------------------------------------ -- A wrapper function -- Pairs up subterms with proofs. subtermsOf : ∀ {n} (τ : Ty n) → List (∃ λ σ → σ ⊑ τ) subtermsOf τ = map-with-∈ (τ ∗) (-,_ ∘′ sound τ) -- subtermsOf is complete. subtermsOf-complete : ∀ {n} {σ τ : Ty n} → σ ⊑ τ → ∃ λ σ⊑τ → (σ , σ⊑τ) ∈ subtermsOf τ subtermsOf-complete {σ = σ} {τ} σ⊑τ = (-, Inverse.to (map-with-∈↔ {f = -,_ ∘′ sound τ}) ⟨$⟩ (σ , complete σ⊑τ , P.refl))

-- {-# OPTIONS -v reify:80 #-} open import Common.Prelude open import Common.Reflection open import Common.Equality module Issue1345 (A : Set) where unquoteDecl idNat = define (vArg idNat) (funDef (pi (vArg (def (quote Nat) [])) (abs "" (def (quote Nat) []))) (clause (vArg (var "") ∷ []) (var 0 []) ∷ [])) thm : ∀ n → idNat n ≡ n thm n = refl

------------------------------------------------------------------------------ -- Testing the erasing of the duplicate definitions required by a conjecture ------------------------------------------------------------------------------ {-# OPTIONS --exact-split #-} {-# OPTIONS --no-sized-types #-} {-# OPTIONS --no-universe-polymorphism #-} {-# OPTIONS --without-K #-} module DuplicateAgdaDefinitions1 where infix 7 _≈_ ------------------------------------------------------------------------------ postulate D : Set _≈_ : D → D → Set Eq : D → D → Set Eq x y = x ≈ y {-# ATP definition Eq #-} postulate Eq-trans : ∀ {x y z} → Eq x y → Eq y z → Eq x z postulate Eq-trans₂ : ∀ {w x y z} → Eq w x → Eq x y → Eq y z → Eq w z {-# ATP prove Eq-trans₂ Eq-trans #-}

{-# OPTIONS --cubical --safe --postfix-projections #-} module Cardinality.Infinite.ManifestEnumerable where open import Prelude open import Data.List.Kleene open import Data.Fin import Data.Nat as ℕ open import Data.Nat using (_+_) open import Cubical.Data.Sigma.Properties open import Cubical.Foundations.Prelude using (J) import Data.List.Kleene.Membership as Kleene open import Codata.Stream open import Cardinality.Infinite.Split open import HITs.PropositionalTruncation.Sugar private variable u : Level U : A → Type u s : Level S : ℕ → Type s -- ℰ : Type a → Type a -- ℰ A = Σ[ xs ⦂ Stream A ] Π[ x ⦂ A ] ∥ x ∈ xs ∥ -- ℰ≡ℕ↠ : ℰ A ≡ (ℕ ↠ A) -- ℰ≡ℕ↠ = refl -- _∥Σ∥_ : ℰ A → (∀ x → ℰ (U x)) → ℰ (Σ A U) -- (xs ∥Σ∥ ys) .fst = cantor (xs .fst) (fst ∘ ys) -- (xs ∥Σ∥ ys) .snd (x , y) = -- concat-∈ -- (x , y) -- (xs .fst * (fst ∘ ys)) ∥$∥ -- ⦇ (*-cover x (xs .fst) y (fst ∘ ys)) (xs .snd x) (ys x .snd y) ⦈

{-# OPTIONS --rewriting #-} open import Agda.Primitive open import Agda.Builtin.Equality open import Agda.Builtin.Equality.Rewrite variable ℓ ℓ' : Level A : Set ℓ B : Set ℓ' x y z : A sym : x ≡ y → y ≡ x sym refl = refl _∙_ : x ≡ y → y ≡ z → x ≡ z refl ∙ refl = refl leftId : (p : x ≡ y) → refl ∙ p ≡ p leftId refl = refl rightId : (p : x ≡ y) → p ∙ refl ≡ p rightId refl = refl transport : (P : A → Set ℓ') → x ≡ y → P x → P y transport _ refl px = px ap : (f : A → B) → x ≡ y → f x ≡ f y ap _ refl = refl apd : (P : A → Set ℓ') → (f : (a : A) → P a) → (p : x ≡ y) → transport P p (f x) ≡ f y apd _ _ refl = refl postulate Size : Set base : Size next : Size → Size inf : Size lim : next inf ≡ inf postulate elim : (P : Size → Set ℓ) → (pb : P base) → (pn : (s : Size) → P s → P (next s)) → (pi : P inf) → -- (elim P pb pn pi pl (next inf)) = (elim P pb pn pi pl inf) (pl : transport P lim (pn inf pi) ≡ pi) → (s : Size) → P s elim-base : (P : Size → Set ℓ) → (pb : P base) → (pn : (s : Size) → P s → P (next s)) → (pi : P inf) → (pl : transport P lim (pn inf pi) ≡ pi) → elim P pb pn pi pl base ≡ pb elim-next : (P : Size → Set ℓ) → (pb : P base) → (pn : (s : Size) → P s → P (next s)) → (pi : P inf) → (pl : transport P lim (pn inf pi) ≡ pi) → (s : Size) → elim P pb pn pi pl (next s) ≡ pn s (elim P pb pn pi pl s) elim-inf : (P : Size → Set ℓ) → (pb : P base) → (pn : (s : Size) → P s → P (next s)) → (pi : P inf) → (pl : transport P lim (pn inf pi) ≡ pi) → elim P pb pn pi pl inf ≡ pi {-# REWRITE elim-base elim-next elim-inf #-} postulate elim-lim : (P : Size → Set ℓ) → (pb : P base) → (pn : (s : Size) → P s → P (next s)) → (pi : P inf) → (pl : transport P lim (pn inf pi) ≡ pi) → apd P (elim P pb pn pi pl) lim ≡ pl {-# REWRITE elim-lim #-} -- postulate lim-refl : transport (λ s → s ≡ inf) lim lim ≡ refl absurd : ∀ {A s} → base ≡ next s → A absurd {A} p = let discr : Size → Set discr = elim (λ _ → Set) Size (λ _ _ → A) A (apd (λ _ → Set) (λ _ → A) lim) in transport discr p base inj : ∀ {s t} → next s ≡ next t → s ≡ t inj {s} {t} p = let pred : Size → Size pred = elim (λ _ → Size) base (λ s _ → s) inf (apd (λ _ → Size) (λ s → inf) lim) in ap pred p data Nat : Size → Set where zero : (s : Size) → Nat (next s) succ : (s : Size) → Nat s → Nat (next s) prev : (P : Size → Set ℓ') → P (next inf) → P inf prev P pni = transport P lim pni double' : ∀ s → Nat s → Nat inf double' _ (zero _) = prev Nat (zero inf) double' _ (succ s n) = prev Nat (succ inf (prev Nat (succ inf (double' s n)))) -- pn inf pi ≡ pi -- zero: zero = ?pi -- next: succ _ (succ _ ?pi) = ?pi double : ∀ s → Nat s → Nat inf double = elim (λ s → Nat s → Nat inf) {! !} {! !} {! !} {! !}

---------------------------------------------------------------------------- -- Well-founded induction on the relation _◁_ ---------------------------------------------------------------------------- {-# OPTIONS --exact-split #-} {-# OPTIONS --no-sized-types #-} {-# OPTIONS --no-universe-polymorphism #-} {-# OPTIONS --without-K #-} -- N.B This module does not contain combined proofs, but it imports -- modules which contain combined proofs. module FOT.FOTC.Program.McCarthy91.WF-Relation.Induction.NonAcc.WellFoundedATP where open import FOTC.Base open import FOTC.Data.Nat open import FOTC.Program.McCarthy91.McCarthy91 open import FOTC.Program.McCarthy91.WF-Relation open import FOTC.Program.McCarthy91.WF-Relation.PropertiesATP ---------------------------------------------------------------------------- -- Adapted from FOTC.Data.Nat.Induction.WellFoundedI.wfInd-LT₁. {-# TERMINATING #-} ◁-wfind : (P : D → Set) → (∀ {n} → N n → (∀ {m} → N m → m ◁ n → P m) → P n) → ∀ {n} → N n → P n ◁-wfind P h Nn = h Nn (helper Nn) where helper : ∀ {n m} → N n → N m → m ◁ n → P m helper Nn nzero 0◁n = ⊥-elim (0◁x→⊥ Nn 0◁n) -- This equation does not pass the termination check. helper nzero (nsucc Nm) Sm◁0 = h (nsucc Nm) (λ {m'} Nm' m'◁Sm → let m'◁0 : m' ◁ zero m'◁0 = ◁-trans Nm' (nsucc Nm) nzero m'◁Sm Sm◁0 in helper nzero Nm' m'◁0 ) -- Other version of the previous equation (this version neither -- pass the termination check). -- helper nzero (nsucc {m} Nm) Sm◁0 = h (nsucc Nm) -- (λ {m'} Nm' m'◁Sm → -- let m'◁m : MCR m' m -- m'◁m = x◁Sy→x◁y Nm' Nm m'◁Sm -- in helper Nm Nm' m'◁m -- ) -- Other version of the previous equation (this version neither -- pass the termination check). -- helper nzero Nm m◁0 = h Nm -- (λ {m'} Nm' m'◁m → -- let m'◁0 : MCR m' zero -- m'◁0 = ◁-trans Nm' Nm nzero m'◁m m◁0 -- in helper nzero Nm' m'◁0 -- ) -- Other version of the previous equation (this version neither -- pass the termination check). -- helper {m = m} nzero Nm m◁0 = -- h Nm (λ {m'} Nm' m'◁m → helper Nm Nm' m'◁m) helper (nsucc {n} Nn) (nsucc {m} Nm) Sm◁Sn = helper Nn (nsucc Nm) Sm◁n where Sm◁n : succ₁ m ◁ n Sm◁n = x◁Sy→x◁y (nsucc Nm) Nn Sm◁Sn

{-# OPTIONS --no-qualified-instances #-} module NoQualifiedInstances-ParameterizedImport where postulate T : Set open import NoQualifiedInstances.ParameterizedImport.A T as A postulate f : {{A.I}} → A.I test : A.I test = f

{- 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 -} module LibraBFT.ImplShared.Util.Dijkstra.All where open import Dijkstra.All public open import LibraBFT.ImplShared.LBFT public

{-# OPTIONS --allow-unsolved-metas #-} module lambda.system-d where open import Data.Nat open import Data.Fin hiding (lift) open import lambda.vec open import lambda.untyped infixr 21 _∧_ infixr 22 _⇒_ data type : Set where base : ℕ → type _⇒_ _∧_ : type → type → type context : ℕ → Set context = vec type infix 10 _⊢_∶_ data _⊢_∶_ {n : ℕ} (Γ : context n) : term n → type → Set where ax : ∀ {i} → Γ ⊢ var i ∶ lookup i Γ lam : ∀ {A B x} → Γ ▸ A ⊢ x ∶ B → Γ ⊢ lam x ∶ A ⇒ B app : ∀ {A B x y} → Γ ⊢ x ∶ (A ⇒ B) → Γ ⊢ y ∶ A → Γ ⊢ app x y ∶ B ∧ⁱ : ∀ {A B x} → Γ ⊢ x ∶ A → Γ ⊢ x ∶ B → Γ ⊢ x ∶ A ∧ B ∧ᵉˡ : ∀ {A B x} → Γ ⊢ x ∶ A ∧ B → Γ ⊢ x ∶ A ∧ᵉʳ : ∀ {A B x} → Γ ⊢ x ∶ A ∧ B → Γ ⊢ x ∶ B ⊢_∶_ : term 0 → type → Set ⊢ x ∶ t = ε ⊢ x ∶ t -- This lemma is actually false, because i did not setup the context -- inclusions properly. It's not really complicated to fix: just -- transpose the whole machinery from lambda.stack to lambda.vec. The -- only thing is that it's a bit long and boring to get the same job -- done on an indexed datatype. Substitution is hard to get right... lift-p : ∀ {n A X Γ} {x : term n} → Γ ⊢ x ∶ A → Γ ▸ X ⊢ lift x ∶ A lift-p ax = ax lift-p (lam x) = {!!} lift-p (app x y) = app (lift-p x) (lift-p y) lift-p (∧ⁱ x₁ x₂) = ∧ⁱ (lift-p x₁) (lift-p x₂) lift-p (∧ᵉˡ x) = ∧ᵉˡ (lift-p x) lift-p (∧ᵉʳ x) = ∧ᵉʳ (lift-p x) ↑-p : ∀ {n m X Γ Δ} {ρ : Fin n → term m} → ((i : Fin n) → Δ ⊢ ρ i ∶ lookup i Γ) → (i : Fin (suc n)) → (Δ ▸ X) ⊢ ↑ ρ i ∶ lookup i (Γ ▸ X) ↑-p ρT zero = ax ↑-p ρT (suc i) = lift-p (ρT i) subst-p : ∀ {n m A} {x : term n} {Γ : context n} {Δ : context m} {ρ : Fin n → term m} → Γ ⊢ x ∶ A → ((i : Fin n) → Δ ⊢ ρ i ∶ lookup i Γ) → Δ ⊢ subst x ρ ∶ A subst-p (ax {i}) ρT = ρT i subst-p (lam x) ρT = lam (subst-p x (↑-p ρT)) subst-p (app x y) ρT = app (subst-p x ρT) (subst-p y ρT) subst-p (∧ⁱ x₁ x₂) ρT = ∧ⁱ (subst-p x₁ ρT) (subst-p x₂ ρT) subst-p (∧ᵉˡ x) ρT = ∧ᵉˡ (subst-p x ρT) subst-p (∧ᵉʳ x) ρT = ∧ᵉʳ (subst-p x ρT)

module README where ------------------------------------------------------------------------ -- The Agda "standard" library, version 0.2 -- -- Author: Nils Anders Danielsson, with contributions from -- Jean-Philippe Bernardy, Samuel Bronson, Liang-Ting Chen, Dan Doel, -- Patrik Jansson, Shin-Cheng Mu, and Ulf Norell ------------------------------------------------------------------------ -- This version of the library has been tested using Agda 2.2.4. -- Note that no guarantees are currently made about forwards or -- backwards compatibility, the library is still at an experimental -- stage. -- To make use of the library, add the path to the library’s root -- directory (src) to the Agda search path, either using the -- --include-path flag or by customising the Emacs mode variable -- agda2-include-dirs (M-x customize-group RET agda2 RET). -- To compile programs using some of the IO functions you need to -- install the Haskell package utf8-string (available from Hackage). -- Contributions to this library are welcome. Please send -- contributions in the form of darcs patches (run -- darcs send --output <patch file> and attach the patch file to an -- email), and include a statement saying that you agree to release -- your library patches under the library's licence. ------------------------------------------------------------------------ -- Module hierarchy ------------------------------------------------------------------------ -- The top-level module names of the library are currently allocated -- as follows: -- -- • Algebra -- Abstract algebra (monoids, groups, rings etc.), along with -- properties needed to specify these structures (associativity, -- commutativity, etc.), and operations on and proofs about the -- structures. -- • Category -- Category theory-inspired idioms used to structure functional -- programs (functors and monads, for instance). -- • Coinduction -- Support for coinduction. -- • Data -- Data types and properties about data types. (Also some -- combinators working solely on and with functions; see -- Data.Function.) -- • Foreign -- Related to the foreign function interface. -- • Induction -- A general framework for induction (includes lexicographic and -- well-founded induction). -- • IO -- Input/output-related functions. -- • Relation -- Properties of and proofs about relations (mostly homogeneous -- binary relations). -- • Size -- Sizes used by the sized types mechanism. ------------------------------------------------------------------------ -- A selection of useful library modules ------------------------------------------------------------------------ -- Note that module names in source code are often hyperlinked to the -- corresponding module. In the Emacs mode you can follow these -- hyperlinks by typing M-. or clicking with the middle mouse button. -- • Some data types import Data.Bool -- Booleans. import Data.Char -- Characters. import Data.Empty -- The empty type. import Data.Fin -- Finite sets. import Data.List -- Lists. import Data.Maybe -- The maybe type. import Data.Nat -- Natural numbers. import Data.Product -- Products. import Data.Stream -- Streams. import Data.String -- Strings. import Data.Sum -- Disjoint sums. import Data.Unit -- The unit type. import Data.Vec -- Fixed-length vectors. -- • Some types used to structure computations import Category.Functor -- Functors. import Category.Applicative -- Applicative functors. import Category.Monad -- Monads. -- • Equality -- Propositional equality: import Relation.Binary.PropositionalEquality -- Convenient syntax for "equational reasoning" using a preorder: import Relation.Binary.PreorderReasoning -- Solver for commutative ring or semiring equalities: import Algebra.RingSolver -- • Properties of functions, sets and relations -- Monoids, rings and similar algebraic structures: import Algebra -- Negation, decidability, and similar operations on sets: import Relation.Nullary -- Properties of homogeneous binary relations: import Relation.Binary -- • Induction -- An abstraction of various forms of recursion/induction: import Induction -- Well-founded induction: import Induction.WellFounded -- Various forms of induction for natural numbers: import Induction.Nat -- • Support for coinduction import Coinduction -- • IO import IO ------------------------------------------------------------------------ -- Record hierarchies ------------------------------------------------------------------------ -- When an abstract hierarchy of some sort (for instance semigroup → -- monoid → group) is included in the library the basic approach is to -- specify the properties of every concept in terms of a record -- containing just properties, parameterised on the underlying -- operations, sets etc.: -- -- record IsSemigroup {A} (≈ : Rel A) (∙ : Op₂ A) : Set where -- open FunctionProperties ≈ -- field -- isEquivalence : IsEquivalence ≈ -- assoc : Associative ∙ -- ∙-pres-≈ : ∙ Preserves₂ ≈ ⟶ ≈ ⟶ ≈ -- -- More specific concepts are then specified in terms of the simpler -- ones: -- -- record IsMonoid {A} (≈ : Rel A) (∙ : Op₂ A) (ε : A) : Set where -- open FunctionProperties ≈ -- field -- isSemigroup : IsSemigroup ≈ ∙ -- identity : Identity ε ∙ -- -- open IsSemigroup isSemigroup public -- -- Note here that open IsSemigroup isSemigroup public ensures that the -- fields of the isSemigroup record can be accessed directly; this -- technique enables the user of an IsMonoid record to use underlying -- records without having to manually open an entire record hierarchy. -- This is not always possible, though. Consider the following definition -- of preorders: -- -- record IsPreorder {A : Set} -- (_≈_ : Rel A) -- The underlying equality. -- (_∼_ : Rel A) -- The relation. -- : Set where -- field -- isEquivalence : IsEquivalence _≈_ -- -- Reflexivity is expressed in terms of an underlying equality: -- reflexive : _≈_ ⇒ _∼_ -- trans : Transitive _∼_ -- ∼-resp-≈ : _∼_ Respects₂ _≈_ -- -- module Eq = IsEquivalence isEquivalence -- -- refl : Reflexive _∼_ -- refl = reflexive Eq.refl -- -- The Eq module in IsPreorder is not opened publicly, because it -- contains some fields which clash with fields or other definitions -- in IsPreorder. -- Records packing up properties with the corresponding operations, -- sets, etc. are sometimes also defined: -- -- record Semigroup : Set₁ where -- infixl 7 _∙_ -- infix 4 _≈_ -- field -- carrier : Set -- _≈_ : Rel carrier -- _∙_ : Op₂ carrier -- isSemigroup : IsSemigroup _≈_ _∙_ -- -- open IsSemigroup isSemigroup public -- -- setoid : Setoid -- setoid = record { isEquivalence = isEquivalence } -- -- record Monoid : Set₁ where -- infixl 7 _∙_ -- infix 4 _≈_ -- field -- carrier : Set -- _≈_ : Rel carrier -- _∙_ : Op₂ carrier -- ε : carrier -- isMonoid : IsMonoid _≈_ _∙_ ε -- -- open IsMonoid isMonoid public -- -- semigroup : Semigroup -- semigroup = record { isSemigroup = isSemigroup } -- -- open Semigroup semigroup public using (setoid) -- -- Note that the Monoid record does not include a Semigroup field. -- Instead the Monoid /module/ includes a "repackaging function" -- semigroup which converts a Monoid to a Semigroup. -- The above setup may seem a bit complicated, but we think it makes the -- library quite easy to work with, while also providing enough -- flexibility. ------------------------------------------------------------------------ -- More documentation ------------------------------------------------------------------------ -- Some examples showing where the natural numbers and some related -- operations and properties are defined, and how they can be used: import README.Nat ------------------------------------------------------------------------ -- Core modules ------------------------------------------------------------------------ -- Some modules have names ending in ".Core". These modules are -- internal, and have (mostly) been created to avoid mutual recursion -- between modules. They should not be imported directly; their -- contents are reexported by other modules. ------------------------------------------------------------------------ -- All library modules ------------------------------------------------------------------------ -- For short descriptions of every library module, see Everything: import Everything

------------------------------------------------------------------------ -- The Agda standard library -- -- Functors ------------------------------------------------------------------------ -- Note that currently the functor laws are not included here. module Category.Functor where open import Function open import Level record RawFunctor {ℓ} (F : Set ℓ → Set ℓ) : Set (suc ℓ) where infixl 4 _<$>_ _<$_ field _<$>_ : ∀ {A B} → (A → B) → F A → F B _<$_ : ∀ {A B} → A → F B → F A x <$ y = const x <$> y

------------------------------------------------------------------------------ -- Testing variable names clash ------------------------------------------------------------------------------ {-# OPTIONS --exact-split #-} {-# OPTIONS --no-sized-types #-} {-# OPTIONS --no-universe-polymorphism #-} {-# OPTIONS --without-K #-} module VariableNamesClash where postulate D : Set _≡_ : D → D → Set a b : D postulate a≡b : a ≡ b {-# ATP axiom a≡b #-} foo : (n : D) → a ≡ b foo n = prf n where -- The translation of this postulate must use two diferents -- quantified variables names e.g. ∀ x. ∀ y. a ≡ b. postulate prf : (n : D) → a ≡ b {-# ATP prove prf #-}

-- Solver for Functor {-# OPTIONS --without-K --safe #-} open import Categories.Category open import Categories.Functor renaming (id to idF) module Experiment.Categories.Solver.Functor {o ℓ e o′ ℓ′ e′} {𝒞 : Category o ℓ e} {𝒟 : Category o′ ℓ′ e′} (F : Functor 𝒞 𝒟) where open import Level open import Relation.Binary using (Rel) import Categories.Morphism.Reasoning as MR import Experiment.Categories.Solver.Category module 𝒞 = Category 𝒞 module CS = Experiment.Categories.Solver.Category 𝒞 open CS using (:id; _:∘_; ∥_∥) renaming (∥-∥ to ∥-∥′) public open Category 𝒟 open HomReasoning open MR 𝒟 open Functor F private variable A B C D E : Obj infixr 9 _:∘_ data Expr : Rel Obj (o ⊔ o′ ⊔ ℓ ⊔ ℓ′) where :id : Expr A A _:∘_ : Expr B C → Expr A B → Expr A C :F₁ : ∀ {A B} → CS.Expr A B → Expr (F₀ A) (F₀ B) ∥_∥ : A ⇒ B → Expr A B -- Semantics ⟦_⟧ : Expr A B → A ⇒ B ⟦ :id ⟧ = id ⟦ e₁ :∘ e₂ ⟧ = ⟦ e₁ ⟧ ∘ ⟦ e₂ ⟧ ⟦ :F₁ e ⟧ = F₁ CS.⟦ e ⟧ ⟦ ∥ f ∥ ⟧ = f F₁⟦_⟧N∘_ : ∀ {B C} → CS.Expr B C → A ⇒ F₀ B → A ⇒ F₀ C F₁⟦ :id ⟧N∘ g = g F₁⟦ e₁ :∘ e₂ ⟧N∘ g = F₁⟦ e₁ ⟧N∘ (F₁⟦ e₂ ⟧N∘ g) F₁⟦ ∥ f ∥ ⟧N∘ g = F₁ f ∘ g ⟦_⟧N∘_ : Expr B C → A ⇒ B → A ⇒ C ⟦ :id ⟧N∘ g = g ⟦ e₁ :∘ e₂ ⟧N∘ g = ⟦ e₁ ⟧N∘ (⟦ e₂ ⟧N∘ g) ⟦ :F₁ e ⟧N∘ g = F₁⟦ e ⟧N∘ g ⟦ ∥ f ∥ ⟧N∘ g = f ∘ g ⟦_⟧N : Expr A B → A ⇒ B ⟦ e ⟧N = ⟦ e ⟧N∘ id F₁⟦e⟧N∘f≈F₁⟦e⟧∘f : ∀ {B C} (e : CS.Expr B C) (g : A ⇒ F₀ B) → F₁⟦ e ⟧N∘ g ≈ F₁ CS.⟦ e ⟧ ∘ g F₁⟦e⟧N∘f≈F₁⟦e⟧∘f :id g = begin g ≈˘⟨ identityˡ ⟩ id ∘ g ≈˘⟨ identity ⟩∘⟨refl ⟩ F₁ 𝒞.id ∘ g ∎ F₁⟦e⟧N∘f≈F₁⟦e⟧∘f (e₁ :∘ e₂) g = begin F₁⟦ e₁ ⟧N∘ (F₁⟦ e₂ ⟧N∘ g) ≈⟨ F₁⟦e⟧N∘f≈F₁⟦e⟧∘f e₁ (F₁⟦ e₂ ⟧N∘ g) ⟩ F₁ CS.⟦ e₁ ⟧ ∘ (F₁⟦ e₂ ⟧N∘ g) ≈⟨ pushʳ (F₁⟦e⟧N∘f≈F₁⟦e⟧∘f e₂ g) ⟩ (F₁ CS.⟦ e₁ ⟧ ∘ F₁ CS.⟦ e₂ ⟧) ∘ g ≈˘⟨ homomorphism ⟩∘⟨refl ⟩ F₁ (CS.⟦ e₁ ⟧ 𝒞.∘ CS.⟦ e₂ ⟧) ∘ g ∎ F₁⟦e⟧N∘f≈F₁⟦e⟧∘f ∥ x ∥ g = refl ⟦e⟧N∘f≈⟦e⟧∘f : (e : Expr B C) (g : A ⇒ B) → ⟦ e ⟧N∘ g ≈ ⟦ e ⟧ ∘ g ⟦e⟧N∘f≈⟦e⟧∘f :id g = ⟺ identityˡ ⟦e⟧N∘f≈⟦e⟧∘f (e₁ :∘ e₂) g = begin ⟦ e₁ ⟧N∘ (⟦ e₂ ⟧N∘ g) ≈⟨ ⟦e⟧N∘f≈⟦e⟧∘f e₁ (⟦ e₂ ⟧N∘ g) ⟩ ⟦ e₁ ⟧ ∘ (⟦ e₂ ⟧N∘ g) ≈⟨ pushʳ (⟦e⟧N∘f≈⟦e⟧∘f e₂ g) ⟩ (⟦ e₁ ⟧ ∘ ⟦ e₂ ⟧) ∘ g ∎ ⟦e⟧N∘f≈⟦e⟧∘f (:F₁ e) g = F₁⟦e⟧N∘f≈F₁⟦e⟧∘f e g ⟦e⟧N∘f≈⟦e⟧∘f ∥ f ∥ g = refl ⟦e⟧N≈⟦e⟧ : (e : Expr A B) → ⟦ e ⟧N ≈ ⟦ e ⟧ ⟦e⟧N≈⟦e⟧ e = ⟦e⟧N∘f≈⟦e⟧∘f e id ○ identityʳ solve : (e₁ e₂ : Expr A B) → ⟦ e₁ ⟧N ≈ ⟦ e₂ ⟧N → ⟦ e₁ ⟧ ≈ ⟦ e₂ ⟧ solve e₁ e₂ eq = begin ⟦ e₁ ⟧ ≈˘⟨ ⟦e⟧N≈⟦e⟧ e₁ ⟩ ⟦ e₁ ⟧N ≈⟨ eq ⟩ ⟦ e₂ ⟧N ≈⟨ ⟦e⟧N≈⟦e⟧ e₂ ⟩ ⟦ e₂ ⟧ ∎ ∥-∥ : ∀ {f : A ⇒ B} → Expr A B ∥-∥ {f = f} = ∥ f ∥

module Thesis.LangChanges where open import Thesis.Changes open import Thesis.IntChanges open import Thesis.Types open import Thesis.Contexts open import Thesis.Environments open import Relation.Binary.PropositionalEquality Chτ : (τ : Type) → Set Chτ τ = ⟦ Δt τ ⟧Type ΔΓ : Context → Context ΔΓ ∅ = ∅ ΔΓ (τ • Γ) = Δt τ • τ • ΔΓ Γ ChΓ : ∀ (Γ : Context) → Set ChΓ Γ = ⟦ ΔΓ Γ ⟧Context [_]τ_from_to_ : ∀ (τ : Type) → (dv : Chτ τ) → (v1 v2 : ⟦ τ ⟧Type) → Set isCompChangeStructureτ : ∀ τ → IsCompChangeStructure ⟦ τ ⟧Type ⟦ Δt τ ⟧Type [ τ ]τ_from_to_ changeStructureτ : ∀ τ → ChangeStructure ⟦ τ ⟧Type changeStructureτ τ = record { isCompChangeStructure = isCompChangeStructureτ τ } instance ichangeStructureτ : ∀ {τ} → ChangeStructure ⟦ τ ⟧Type ichangeStructureτ {τ} = changeStructureτ τ [ σ ⇒ τ ]τ df from f1 to f2 = ∀ (da : Chτ σ) (a1 a2 : ⟦ σ ⟧Type) → [ σ ]τ da from a1 to a2 → [ τ ]τ df a1 da from f1 a1 to f2 a2 [ unit ]τ tt from tt to tt = ⊤ [ int ]τ dv from v1 to v2 = v1 + dv ≡ v2 [ pair σ τ ]τ (da , db) from (a1 , b1) to (a2 , b2) = [ σ ]τ da from a1 to a2 × [ τ ]τ db from b1 to b2 [ sum σ τ ]τ dv from v1 to v2 = sch_from_to_ dv v1 v2 isCompChangeStructureτ (σ ⇒ τ) = ChangeStructure.isCompChangeStructure (funCS {{changeStructureτ σ}} {{changeStructureτ τ}}) isCompChangeStructureτ unit = ChangeStructure.isCompChangeStructure unitCS isCompChangeStructureτ int = ChangeStructure.isCompChangeStructure intCS isCompChangeStructureτ (pair σ τ) = ChangeStructure.isCompChangeStructure (pairCS {{changeStructureτ σ}} {{changeStructureτ τ}}) isCompChangeStructureτ (sum σ τ) = ChangeStructure.isCompChangeStructure (sumCS {{changeStructureτ σ}} {{changeStructureτ τ}}) open import Data.Unit -- We can define an alternative semantics for contexts, that defines environments as nested products: ⟦_⟧Context-prod : Context → Set ⟦ ∅ ⟧Context-prod = ⊤ ⟦ τ • Γ ⟧Context-prod = ⟦ τ ⟧Type × ⟦ Γ ⟧Context-prod -- And define a change structure for such environments, reusing the change -- structure constructor for pairs. However, this change structure does not -- duplicate base values in the environment. We probably *could* define a -- different change structure for pairs that gives the right result. changeStructureΓ-old : ∀ Γ → ChangeStructure ⟦ Γ ⟧Context-prod changeStructureΓ-old ∅ = unitCS changeStructureΓ-old (τ • Γ) = pairCS {{changeStructureτ τ}} {{changeStructureΓ-old Γ}} data [_]Γ_from_to_ : ∀ Γ → ChΓ Γ → (ρ1 ρ2 : ⟦ Γ ⟧Context) → Set where v∅ : [ ∅ ]Γ ∅ from ∅ to ∅ _v•_ : ∀ {τ Γ dv v1 v2 dρ ρ1 ρ2} → (dvv : [ τ ]τ dv from v1 to v2) → (dρρ : [ Γ ]Γ dρ from ρ1 to ρ2) → [ τ • Γ ]Γ (dv • v1 • dρ) from (v1 • ρ1) to (v2 • ρ2) module _ where _e⊕_ : ∀ {Γ} → ⟦ Γ ⟧Context → ChΓ Γ → ⟦ Γ ⟧Context _e⊕_ ∅ ∅ = ∅ _e⊕_ (v • ρ) (dv • _ • dρ) = v ⊕ dv • ρ e⊕ dρ _e⊝_ : ∀ {Γ} → ⟦ Γ ⟧Context → ⟦ Γ ⟧Context → ChΓ Γ _e⊝_ ∅ ∅ = ∅ _e⊝_ (v₂ • ρ₂) (v₁ • ρ₁) = v₂ ⊝ v₁ • v₁ • ρ₂ e⊝ ρ₁ isEnvCompCS : ∀ Γ → IsCompChangeStructure ⟦ Γ ⟧Context (ChΓ Γ) [ Γ ]Γ_from_to_ efromto→⊕ : ∀ {Γ} (dρ : ChΓ Γ) (ρ1 ρ2 : ⟦ Γ ⟧Context) → [ Γ ]Γ dρ from ρ1 to ρ2 → (ρ1 e⊕ dρ) ≡ ρ2 efromto→⊕ .∅ .∅ .∅ v∅ = refl efromto→⊕ .(_ • _ • _) .(_ • _) .(_ • _) (dvv v• dρρ) = cong₂ _•_ (fromto→⊕ _ _ _ dvv) (efromto→⊕ _ _ _ dρρ) e⊝-fromto : ∀ {Γ} → (ρ1 ρ2 : ⟦ Γ ⟧Context) → [ Γ ]Γ ρ2 e⊝ ρ1 from ρ1 to ρ2 e⊝-fromto ∅ ∅ = v∅ e⊝-fromto (v1 • ρ1) (v2 • ρ2) = ⊝-fromto v1 v2 v• e⊝-fromto ρ1 ρ2 _e⊚_ : ∀ {Γ} → ChΓ Γ → ChΓ Γ → ChΓ Γ _e⊚_ {∅} ∅ ∅ = ∅ _e⊚_ {τ • Γ} (dv1 • v1 • dρ1) (dv2 • _ • dρ2) = dv1 ⊚[ ⟦ τ ⟧Type ] dv2 • v1 • dρ1 e⊚ dρ2 e⊚-fromto : ∀ Γ → (ρ1 ρ2 ρ3 : ⟦ Γ ⟧Context) (dρ1 dρ2 : ChΓ Γ) → [ Γ ]Γ dρ1 from ρ1 to ρ2 → [ Γ ]Γ dρ2 from ρ2 to ρ3 → [ Γ ]Γ (dρ1 e⊚ dρ2) from ρ1 to ρ3 e⊚-fromto ∅ ∅ ∅ ∅ ∅ ∅ v∅ v∅ = v∅ e⊚-fromto (τ • Γ) (v1 • ρ1) (v2 • ρ2) (v3 • ρ3) (dv1 • (.v1 • dρ1)) (dv2 • (.v2 • dρ2)) (dvv1 v• dρρ1) (dvv2 v• dρρ2) = ⊚-fromto v1 v2 v3 dv1 dv2 dvv1 dvv2 v• e⊚-fromto Γ ρ1 ρ2 ρ3 dρ1 dρ2 dρρ1 dρρ2 isEnvCompCS Γ = record { isChangeStructure = record { _⊕_ = _e⊕_ ; fromto→⊕ = efromto→⊕ ; _⊝_ = _e⊝_ ; ⊝-fromto = e⊝-fromto } ; _⊚_ = _e⊚_ ; ⊚-fromto = e⊚-fromto Γ } changeStructureΓ : ∀ Γ → ChangeStructure ⟦ Γ ⟧Context changeStructureΓ Γ = record { isCompChangeStructure = isEnvCompCS Γ } instance ichangeStructureΓ : ∀ {Γ} → ChangeStructure ⟦ Γ ⟧Context ichangeStructureΓ {Γ} = changeStructureΓ Γ changeStructureΓτ : ∀ Γ τ → ChangeStructure (⟦ Γ ⟧Context → ⟦ τ ⟧Type) changeStructureΓτ Γ τ = funCS instance ichangeStructureΓτ : ∀ {Γ τ} → ChangeStructure (⟦ Γ ⟧Context → ⟦ τ ⟧Type) ichangeStructureΓτ {Γ} {τ} = changeStructureΓτ Γ τ

{-# OPTIONS --cubical --guardedness --safe #-} module Data.PolyP.Universe where open import Prelude hiding (_⟨_⟩_) open import Data.Vec.Iterated open import Data.Fin.Indexed -------------------------------------------------------------------------------- -- -- The Universe of functors we're interested in. -- -------------------------------------------------------------------------------- data Functor (n : ℕ) : Type where ! : Fin n → Functor n _⊕_ : (F G : Functor n) → Functor n _⊗_ : (F G : Functor n) → Functor n μ⟨_⟩ : Functor (suc n) → Functor n ⓪ : Functor n ① : Functor n infixl 6 _⊕_ infixl 7 _⊗_ Params : ℕ → Type₁ Params = Vec Type variable n m k : ℕ F G : Functor n As Bs : Params n --------------------------------------------------------------------------------- -- -- Interpretation -- --------------------------------------------------------------------------------- mutual ⟦_⟧ : Functor n → Params n → Type ⟦ ! i ⟧ xs = xs [ i ] ⟦ F ⊕ G ⟧ xs = ⟦ F ⟧ xs ⊎ ⟦ G ⟧ xs ⟦ F ⊗ G ⟧ xs = ⟦ F ⟧ xs × ⟦ G ⟧ xs ⟦ μ⟨ F ⟩ ⟧ xs = μ F xs ⟦ ⓪ ⟧ xs = ⊥ ⟦ ① ⟧ xs = ⊤ record μ (F : Functor (suc n)) (As : Params n) : Type where pattern; inductive; constructor ⟨_⟩ field unwrap : ⟦ F ⟧ (μ F As ∷ As) open μ public

------------------------------------------------------------------------------ -- The alternating bit protocol (ABP) is correct ------------------------------------------------------------------------------ {-# OPTIONS --exact-split #-} {-# OPTIONS --no-sized-types #-} {-# OPTIONS --no-universe-polymorphism #-} {-# OPTIONS --without-K #-} -- This module proves the correctness of the ABP by simplifing the -- formalization in Dybjer and Sander (1989) using a stronger (maybe -- invalid) co-induction principle. module FOT.FOTC.Program.ABP.StrongerInductionPrinciple.CorrectnessProofI where open import FOT.FOTC.Relation.Binary.Bisimilarity.Type open import FOT.FOTC.Program.ABP.StrongerInductionPrinciple.LemmaI open import FOTC.Base open import FOTC.Base.List open import FOTC.Data.Stream.Type open import FOTC.Data.Stream.Equality.PropertiesI open import FOTC.Program.ABP.ABP hiding ( B ) open import FOTC.Program.ABP.Fair.Type open import FOTC.Program.ABP.Terms open import FOTC.Relation.Binary.Bisimilarity.Type ------------------------------------------------------------------------------ -- Main theorem. abpCorrect : ∀ {b is os₁ os₂} → Bit b → Stream is → Fair os₁ → Fair os₂ → is ≈ abpTransfer b os₁ os₂ is abpCorrect {b} {is} {os₁} {os₂} Bb Sis Fos₁ Fos₂ = ≈-stronger-coind B h refl where B : D → D → Set B xs ys = xs ≡ xs h : B is (abpTransfer b os₁ os₂ is) → ∃[ i' ] ∃[ is' ] ∃[ js' ] is ≡ i' ∷ is' ∧ abpTransfer b os₁ os₂ is ≡ i' ∷ js' ∧ B is' js' h _ with Stream-out Sis ... | i' , is' , prf₁ , Sis' with lemma Bb Fos₁ Fos₂ prf₂ where aux₁ aux₂ aux₃ aux₄ aux₅ : D aux₁ = send b aux₂ = ack b aux₃ = out b aux₄ = corrupt os₁ aux₅ = corrupt os₂ prf₂ : S b (i' ∷ is') os₁ os₂ (has aux₁ aux₂ aux₃ aux₄ aux₅ (i' ∷ is')) (hbs aux₁ aux₂ aux₃ aux₄ aux₅ (i' ∷ is')) (hcs aux₁ aux₂ aux₃ aux₄ aux₅ (i' ∷ is')) (hds aux₁ aux₂ aux₃ aux₄ aux₅ (i' ∷ is')) (abpTransfer b os₁ os₂ (i' ∷ is')) prf₂ = has-eq aux₁ aux₂ aux₃ aux₄ aux₅ (i' ∷ is') , hbs-eq aux₁ aux₂ aux₃ aux₄ aux₅ (i' ∷ is') , hcs-eq aux₁ aux₂ aux₃ aux₄ aux₅ (i' ∷ is') , hds-eq aux₁ aux₂ aux₃ aux₄ aux₅ (i' ∷ is') , trans (abpTransfer-eq b os₁ os₂ (i' ∷ is')) (transfer-eq aux₁ aux₂ aux₃ aux₄ aux₅ (i' ∷ is')) ... | js' , prf₃ = i' , is' , js' , prf₁ , subst (λ t → abpTransfer b os₁ os₂ t ≡ i' ∷ js' ) (sym prf₁) prf₃ , refl ------------------------------------------------------------------------------ -- abpTransfer produces a Stream. abpTransfer-Stream : ∀ {b is os₁ os₂} → Bit b → Stream is → Fair os₁ → Fair os₂ → Stream (abpTransfer b os₁ os₂ is) abpTransfer-Stream Bb Sis Fos₁ Fos₂ = ≈→Stream₂ (abpCorrect Bb Sis Fos₁ Fos₂) ------------------------------------------------------------------------------ -- References -- -- Dybjer, Peter and Sander, Herbert P. (1989). A Functional -- Programming Approach to the Specification and Verification of -- Concurrent Systems. Formal Aspects of Computing 1, pp. 303–319.

{-# OPTIONS --safe #-} module Cubical.Categories.Instances.Rings where open import Cubical.Foundations.Prelude open import Cubical.Foundations.Function open import Cubical.Algebra.Ring open import Cubical.Categories.Category open Category open RingHoms RingsCategory : ∀ {ℓ} → Category (ℓ-suc ℓ) ℓ ob RingsCategory = Ring _ Hom[_,_] RingsCategory = RingHom id RingsCategory {R} = idRingHom R _⋆_ RingsCategory = compRingHom ⋆IdL RingsCategory = compIdRingHom ⋆IdR RingsCategory = idCompRingHom ⋆Assoc RingsCategory = compAssocRingHom isSetHom RingsCategory = isSetRingHom _ _

-- Algorithmic equality. {-# OPTIONS --without-K --safe #-} module Definition.Conversion where open import Definition.Untyped open import Definition.Typed open import Tools.Nat open import Tools.Product import Tools.PropositionalEquality as PE infix 10 _⊢_~_↑_ infix 10 _⊢_~_↓_ infix 10 _⊢_[conv↑]_ infix 10 _⊢_[conv↓]_ infix 10 _⊢_[conv↑]_∷_ infix 10 _⊢_[conv↓]_∷_ mutual -- Neutral equality. data _⊢_~_↑_ (Γ : Con Term) : (k l A : Term) → Set where var-refl : ∀ {x y A} → Γ ⊢ var x ∷ A → x PE.≡ y → Γ ⊢ var x ~ var y ↑ A app-cong : ∀ {k l t v F G} → Γ ⊢ k ~ l ↓ Π F ▹ G → Γ ⊢ t [conv↑] v ∷ F → Γ ⊢ k ∘ t ~ l ∘ v ↑ G [ t ] fst-cong : ∀ {p r F G} → Γ ⊢ p ~ r ↓ Σ F ▹ G → Γ ⊢ fst p ~ fst r ↑ F snd-cong : ∀ {p r F G} → Γ ⊢ p ~ r ↓ Σ F ▹ G → Γ ⊢ snd p ~ snd r ↑ G [ fst p ] natrec-cong : ∀ {k l h g a₀ b₀ F G} → Γ ∙ ℕ ⊢ F [conv↑] G → Γ ⊢ a₀ [conv↑] b₀ ∷ F [ zero ] → Γ ⊢ h [conv↑] g ∷ Π ℕ ▹ (F ▹▹ F [ suc (var 0) ]↑) → Γ ⊢ k ~ l ↓ ℕ → Γ ⊢ natrec F a₀ h k ~ natrec G b₀ g l ↑ F [ k ] Emptyrec-cong : ∀ {k l F G} → Γ ⊢ F [conv↑] G → Γ ⊢ k ~ l ↓ Empty → Γ ⊢ Emptyrec F k ~ Emptyrec G l ↑ F -- Neutral equality with types in WHNF. record _⊢_~_↓_ (Γ : Con Term) (k l B : Term) : Set where inductive constructor [~] field A : Term D : Γ ⊢ A ⇒* B whnfB : Whnf B k~l : Γ ⊢ k ~ l ↑ A -- Type equality. record _⊢_[conv↑]_ (Γ : Con Term) (A B : Term) : Set where inductive constructor [↑] field A′ B′ : Term D : Γ ⊢ A ⇒* A′ D′ : Γ ⊢ B ⇒* B′ whnfA′ : Whnf A′ whnfB′ : Whnf B′ A′<>B′ : Γ ⊢ A′ [conv↓] B′ -- Type equality with types in WHNF. data _⊢_[conv↓]_ (Γ : Con Term) : (A B : Term) → Set where U-refl : ⊢ Γ → Γ ⊢ U [conv↓] U ℕ-refl : ⊢ Γ → Γ ⊢ ℕ [conv↓] ℕ Empty-refl : ⊢ Γ → Γ ⊢ Empty [conv↓] Empty Unit-refl : ⊢ Γ → Γ ⊢ Unit [conv↓] Unit ne : ∀ {K L} → Γ ⊢ K ~ L ↓ U → Γ ⊢ K [conv↓] L Π-cong : ∀ {F G H E} → Γ ⊢ F → Γ ⊢ F [conv↑] H → Γ ∙ F ⊢ G [conv↑] E → Γ ⊢ Π F ▹ G [conv↓] Π H ▹ E Σ-cong : ∀ {F G H E} → Γ ⊢ F → Γ ⊢ F [conv↑] H → Γ ∙ F ⊢ G [conv↑] E → Γ ⊢ Σ F ▹ G [conv↓] Σ H ▹ E -- Term equality. record _⊢_[conv↑]_∷_ (Γ : Con Term) (t u A : Term) : Set where inductive constructor [↑]ₜ field B t′ u′ : Term D : Γ ⊢ A ⇒* B d : Γ ⊢ t ⇒* t′ ∷ B d′ : Γ ⊢ u ⇒* u′ ∷ B whnfB : Whnf B whnft′ : Whnf t′ whnfu′ : Whnf u′ t<>u : Γ ⊢ t′ [conv↓] u′ ∷ B -- Term equality with types and terms in WHNF. data _⊢_[conv↓]_∷_ (Γ : Con Term) : (t u A : Term) → Set where ℕ-ins : ∀ {k l} → Γ ⊢ k ~ l ↓ ℕ → Γ ⊢ k [conv↓] l ∷ ℕ Empty-ins : ∀ {k l} → Γ ⊢ k ~ l ↓ Empty → Γ ⊢ k [conv↓] l ∷ Empty Unit-ins : ∀ {k l} → Γ ⊢ k ~ l ↓ Unit → Γ ⊢ k [conv↓] l ∷ Unit ne-ins : ∀ {k l M N} → Γ ⊢ k ∷ N → Γ ⊢ l ∷ N → Neutral N → Γ ⊢ k ~ l ↓ M → Γ ⊢ k [conv↓] l ∷ N univ : ∀ {A B} → Γ ⊢ A ∷ U → Γ ⊢ B ∷ U → Γ ⊢ A [conv↓] B → Γ ⊢ A [conv↓] B ∷ U zero-refl : ⊢ Γ → Γ ⊢ zero [conv↓] zero ∷ ℕ suc-cong : ∀ {m n} → Γ ⊢ m [conv↑] n ∷ ℕ → Γ ⊢ suc m [conv↓] suc n ∷ ℕ η-eq : ∀ {f g F G} → Γ ⊢ f ∷ Π F ▹ G → Γ ⊢ g ∷ Π F ▹ G → Function f → Function g → Γ ∙ F ⊢ wk1 f ∘ var 0 [conv↑] wk1 g ∘ var 0 ∷ G → Γ ⊢ f [conv↓] g ∷ Π F ▹ G Σ-η : ∀ {p r F G} → Γ ⊢ p ∷ Σ F ▹ G → Γ ⊢ r ∷ Σ F ▹ G → Product p → Product r → Γ ⊢ fst p [conv↑] fst r ∷ F → Γ ⊢ snd p [conv↑] snd r ∷ G [ fst p ] → Γ ⊢ p [conv↓] r ∷ Σ F ▹ G η-unit : ∀ {k l} → Γ ⊢ k ∷ Unit → Γ ⊢ l ∷ Unit → Whnf k → Whnf l → Γ ⊢ k [conv↓] l ∷ Unit star-refl : ∀ {Γ} → ⊢ Γ → Γ ⊢ star [conv↓] star ∷ Unit star-refl ⊢Γ = η-unit (starⱼ ⊢Γ) (starⱼ ⊢Γ) starₙ starₙ

{-# OPTIONS --without-K #-} open import HoTT open import cohomology.Theory {- Ordinary cohomology groups of the n-torus Tⁿ = (S¹)ⁿ. - We have Cᵏ(Tⁿ) == C⁰(S⁰)^(n choose' k) where _choose'_ defined as below. - This argument could give Cᵏ((Sᵐ)ⁿ) with a little more work. -} module cohomology.Torus {i} (OT : OrdinaryTheory i) where open OrdinaryTheory OT open import cohomology.Sn OT open import cohomology.SphereProduct cohomology-theory open import cohomology.Unit cohomology-theory {- Almost n choose k, but with n choose' O = 0 for any n. -} _choose'_ : ℕ → ℤ → ℕ n choose' negsucc _ = 0 n choose' pos O = 0 n choose' pos (S O) = n O choose' pos (S (S k)) = 0 S n choose' pos (S (S k)) = (n choose' (pos (S (S k)))) + (n choose' (pos (S k))) _-⊙Torus : ℕ → Ptd i O -⊙Torus = ⊙Lift ⊙Unit (S n) -⊙Torus = ⊙Lift {j = i} ⊙S¹ ⊙× (n -⊙Torus) C-nTorus : (k : ℤ) (n : ℕ) → C k (n -⊙Torus) == C 0 (⊙Lift ⊙S⁰) ^ᴳ (n choose' k) C-nTorus (negsucc k) O = C-Unit (negsucc k) C-nTorus (negsucc k) (S n) = C-Sphere× (negsucc k) 1 (n -⊙Torus) ∙ ap (λ K → K ×ᴳ (C (negsucc k) (n -⊙Torus) ×ᴳ C (negsucc k) (⊙Susp^ 1 (n -⊙Torus)))) (C-Sphere-≠ (negsucc k) 1 (ℤ-negsucc≠pos _ _)) ∙ ×ᴳ-unit-l {G = C (negsucc k) (n -⊙Torus) ×ᴳ C (negsucc k) (⊙Susp (n -⊙Torus))} ∙ ap (λ K → K ×ᴳ C (negsucc k) (⊙Susp (n -⊙Torus))) (C-nTorus (negsucc k) n) ∙ ×ᴳ-unit-l {G = C (negsucc k) (⊙Susp (n -⊙Torus))} ∙ group-ua (C-Susp (negsucc (S k)) (n -⊙Torus)) ∙ C-nTorus (negsucc (S k)) n C-nTorus (pos O) O = C-Unit (pos O) C-nTorus (pos O) (S n) = C-Sphere× (pos O) 1 (n -⊙Torus) ∙ ap (λ K → K ×ᴳ (C 0 (n -⊙Torus) ×ᴳ C 0 (⊙Susp (n -⊙Torus)))) (C-Sphere-≠ (pos O) 1 (pos-≠ (ℕ-O≠S _))) ∙ ×ᴳ-unit-l {G = C 0 (n -⊙Torus) ×ᴳ C 0 (⊙Susp (n -⊙Torus))} ∙ ap (λ K → K ×ᴳ C 0 (⊙Susp (n -⊙Torus))) (C-nTorus 0 n) ∙ ×ᴳ-unit-l {G = C 0 (⊙Susp (n -⊙Torus))} ∙ group-ua (C-Susp (negsucc O) (n -⊙Torus)) ∙ C-nTorus (negsucc O) n C-nTorus (pos (S O)) O = C-Unit 1 C-nTorus (pos (S O)) (S n) = C-Sphere× 1 1 (n -⊙Torus) ∙ ap (λ K → K ×ᴳ (C 1 (n -⊙Torus) ×ᴳ C 1 (⊙Susp (n -⊙Torus)))) (C-Sphere-diag 1) ∙ ap (λ K → C 0 (⊙Lift ⊙S⁰) ×ᴳ K) (ap2 _×ᴳ_ (C-nTorus 1 n) (group-ua (C-Susp 0 (n -⊙Torus)) ∙ C-nTorus 0 n) ∙ ×ᴳ-unit-r {G = C 0 (⊙Lift ⊙S⁰) ^ᴳ (n choose' 1)}) C-nTorus (pos (S (S k))) O = C-Unit (pos (S (S k))) C-nTorus (pos (S (S k))) (S n) = C-Sphere× (pos (S (S k))) 1 (n -⊙Torus) ∙ ap (λ K → K ×ᴳ (C (pos (S (S k))) (n -⊙Torus) ×ᴳ C (pos (S (S k))) (⊙Susp (n -⊙Torus)))) (C-Sphere-≠ (pos (S (S k))) 1 (pos-≠ (ℕ-S-≠ (ℕ-S≠O k)))) ∙ ×ᴳ-unit-l {G = (C (pos (S (S k))) (n -⊙Torus)) ×ᴳ (C (pos (S (S k))) (⊙Susp (n -⊙Torus)))} ∙ ap2 _×ᴳ_ (C-nTorus (pos (S (S k))) n) (group-ua (C-Susp (pos (S k)) (n -⊙Torus)) ∙ C-nTorus (pos (S k)) n) ∙ ^ᴳ-sum (C 0 (⊙Lift ⊙S⁰)) (n choose' pos (S (S k))) (n choose' pos (S k))

module Languages.ILL.TypeCheck where open import bool open import maybe open import Languages.ILL.TypeSyntax open import Languages.ILL.Syntax open import Utils.HaskellTypes open import Utils.HaskellFunctions open import Utils.Exception data StateT (s : Set) (m : Set → Set) (a : Set) : Set where stateT : (s → m (Prod a s)) → StateT s m a runStateT : ∀{s m a} → StateT s m a → (s → m (Prod a s)) runStateT (stateT f) = f returnSTE : ∀{s A} → A → StateT s (Either Exception) A returnSTE x = stateT (λ y → right (x , y)) infixl 20 _>>=STE_ _>>=STE_ : ∀{s A B} → StateT s (Either Exception) A → (A → StateT s (Either Exception) B) → StateT s (Either Exception) B (stateT st) >>=STE f = stateT (λ y → st y >>=E (λ p → runStateT (f (fst p)) (snd p))) infixl 20 _>>STE_ _>>STE_ : ∀{s A B} → StateT s (Either Exception) A → StateT s (Either Exception) B → StateT s (Either Exception) B c₁ >>STE c₂ = c₁ >>=STE (λ _ → c₂) throw : ∀{s A} → Exception → StateT s (Either Exception) A throw x = stateT (λ _ → error x) lift : ∀{s A} → Either Exception A → StateT s (Either Exception) A lift (Left x) = stateT (λ _ → error x) lift (Right x) = stateT (λ y → right (x , y)) get : ∀{S} → StateT S (Either Exception) S get = stateT (λ x → right (x , x)) put : ∀{S} → S → StateT S (Either Exception) Unit put s = stateT (λ x → Right (triv , s)) CTXEl : Set CTXEl = Prod String Type CTX : Set CTX = List CTXEl CTXV-eq : CTXEl → CTXEl → 𝔹 CTXV-eq (x , ty) (y , ty') = x str-eq y TC : Set → Set TC = StateT CTX (Either Exception) _isDisjointWith_ : CTX → CTX → TC Unit ctx₁ isDisjointWith ctx₂ with disjoint CTXV-eq ctx₁ ctx₂ ... | tt = returnSTE triv ... | ff = throw TypeErrorDuplicatedFreeVar getTypeCTX : String → CTX → maybe Type getTypeCTX x ctx = lookup _str-eq_ x ctx {-# TERMINATING #-} subctxFV : CTX → Term → Either Exception CTX subctxFV ctx Triv = right [] subctxFV ctx t@(FVar x) with getTypeCTX x ctx ... | just ty = right ((x , ty) :: []) ... | nothing = error VarNotInCtx subctxFV ctx (BVar _ _ _) = right [] subctxFV ctx (Let t₁ _ _ t₂) = (subctxFV ctx t₁) >>=E (λ l₁ → (subctxFV ctx t₂) >>=E λ l₂ → right (l₁ ++ l₂)) subctxFV ctx (Lam _ _ t) = subctxFV ctx t subctxFV ctx (App t₁ t₂) = (subctxFV ctx t₁) >>=E (λ l₁ → (subctxFV ctx t₂) >>=E λ l₂ → right (l₁ ++ l₂)) subctxFV ctx (Tensor t₁ t₂) = (subctxFV ctx t₁) >>=E (λ l₁ → (subctxFV ctx t₂) >>=E λ l₂ → right (l₁ ++ l₂)) subctxFV ctx (Promote ms t) = mctx₁ >>=E (λ ctx₁ → (subctxFV ctx t) >>=E (λ ctx₂ → right (ctx₁ ++ ctx₂))) where aux = (λ m r → (subctxFV ctx (fstT m)) >>=E (λ l₁ → r >>=E (λ l₂ → right (l₁ ++ l₂)))) mctx₁ = foldr aux (right []) ms subctxFV ctx (Discard t₁ t₂) = (subctxFV ctx t₁) >>=E (λ l₁ → (subctxFV ctx t₂) >>=E λ l₂ → right (l₁ ++ l₂)) subctxFV ctx (Copy t₁ _ t₂) = (subctxFV ctx t₁) >>=E (λ l₁ → (subctxFV ctx t₂) >>=E λ l₂ → right (l₁ ++ l₂)) subctxFV ctx (Derelict t) = subctxFV ctx t isTop : Type → TC Type isTop Top = returnSTE Top isTop _ = throw TypeErrorLetNotTop isTensor : Type → TC (Prod Type Type) isTensor (Tensor x y) = returnSTE (x , y) isTensor _ = throw TypeErrorLetNotTensor isBang : Type → TC Type isBang (Bang ty) = returnSTE ty isBang _ = throw TypeErrorPromoteNotBang isImp : Type → TC (Prod Type Type) isImp (Imp A B) = returnSTE (A , B) isImp _ = throw TypeErrorAppNotImp _tyEq_ : Type → Type → TC Unit _tyEq_ ty₁ ty₂ with ty₁ eq-type ty₂ ... | tt = returnSTE triv ... | ff = throw TypeErrorTypesNotEqual {-# TERMINATING #-} typeCheck' : Term → TC Type typeCheck' Triv = get >>=STE checkCTX where checkCTX : CTX → TC Type checkCTX [] = returnSTE Top checkCTX _ = throw NonEmptyCtx typeCheck' (FVar x) = get >>=STE checkCtx where checkCtx : CTX → TC Type checkCtx ((y , ty) :: []) with x str-eq y ... | tt = returnSTE ty ... | ff = throw VarNotInCtx checkCtx _ = throw NonLinearCtx typeCheck' (BVar _ _ _) = throw NonlocallyClosed typeCheck' (Let t₁ ty PTriv t₂) = get >>=STE (λ ctx → lift (subctxFV ctx t₁) >>=STE (λ ctx₁ → lift (subctxFV ctx t₂) >>=STE λ ctx₂ → (ctx₁ isDisjointWith ctx₂) >>STE ((put ctx₁) >>STE typeCheck' t₁) >>=STE (λ ty₁ → isTop ty₁ >>STE (put ctx₂ >>STE typeCheck' t₂)))) typeCheck' (Let t₁ ty (PTensor x y) t₂) = get >>=STE (λ ctx → lift (subctxFV ctx t₁) >>=STE (λ ctx₁ → lift (subctxFV ctx t₂) >>=STE (λ ctx₂ → (ctx₁ isDisjointWith ctx₂) >>STE ((put ctx₁) >>STE typeCheck' t₁) >>=STE (λ ty₁ → (isTensor ty₁) >>=STE (λ tys → let A = fst tys B = snd tys t₂' = open-t 0 y RLPV (FVar y) (open-t 0 x LLPV (FVar x) t₂) in put ((x , A) :: (y , B) :: ctx₂) >>STE typeCheck' t₂'))))) typeCheck' (Lam x ty t) = get >>=STE (λ ctx → (put ((x , ty) :: ctx)) >>STE typeCheck' (open-t 0 x BV (FVar x) t) >>=STE (λ ty₂ → returnSTE (Imp ty ty₂))) typeCheck' (App t₁ t₂) = get >>=STE (λ ctx → lift (subctxFV ctx t₁) >>=STE (λ ctx₁ → lift (subctxFV ctx t₂) >>=STE (λ ctx₂ → (ctx₁ isDisjointWith ctx₂) >>STE (put ctx₁) >>STE (typeCheck' t₁) >>=STE (λ ty₁ → isImp ty₁ >>=STE (λ tys → (put ctx₂) >>STE (typeCheck' t₂) >>=STE (λ ty₂ → ((fst tys) tyEq ty₂) >>STE returnSTE (snd tys))))))) typeCheck' (Tensor t₁ t₂) = get >>=STE (λ ctx → lift (subctxFV ctx t₁) >>=STE (λ ctx₁ → lift (subctxFV ctx t₂) >>=STE (λ ctx₂ → (ctx₁ isDisjointWith ctx₂) >>STE (put ctx₁) >>STE typeCheck' t₁ >>=STE (λ ty₁ → (put ctx₂) >>STE typeCheck' t₂ >>=STE (λ ty₂ → returnSTE (Tensor ty₁ ty₂)))))) typeCheck' (Promote ms t) = put [] >>STE getSubctxs ms >>=STE areDisjointCtxs >>STE checkVectorOpenTerm ms t >>=STE typeCheck' >>=STE (λ ty → returnSTE (Bang ty)) where getSubctxs : List (Triple Term String Type) → TC (List CTX) getSubctxs ((triple t _ _) :: rest) = get >>=STE (λ ctx → (lift (subctxFV ctx t)) >>=STE (λ ctx' → (getSubctxs rest) >>=STE (λ r → returnSTE (ctx' :: r)))) getSubctxs l = returnSTE [] areDisjointCtxs : List CTX → TC Unit areDisjointCtxs [] = returnSTE triv areDisjointCtxs (ctx₁ :: rest) = aux ctx₁ rest >>STE areDisjointCtxs rest where aux : CTX → List CTX → TC Unit aux ctx₁ (ctx₂ :: rest) = (ctx₁ isDisjointWith ctx₂) >>STE aux ctx₁ rest aux _ [] = returnSTE triv checkVectorTerm : Term → TC Type checkVectorTerm t = get >>=STE (λ ctx → lift (subctxFV ctx t) >>=STE (λ ctx₁ → (put ctx₁) >>STE typeCheck' t)) checkVectorOpenTerm : List (Triple Term String Type) → Term → TC Term checkVectorOpenTerm [] t = throw IllformedPromote checkVectorOpenTerm (triple t₁ x₁ ty₁ :: rest) t = get >>=STE (λ ctx → checkVectorTerm t₁ >>=STE (λ ty₁' → (isBang ty₁' >>STE (ty₁ tyEq ty₁')) >>STE (put ((x₁ , ty₁) :: ctx) >>STE ((checkVectorOpenTerm rest t) >>=STE (λ t' → returnSTE (open-t 0 x₁ PBV (FVar x₁) t')))))) typeCheck' (Discard t₁ t₂) = get >>=STE (λ ctx → lift (subctxFV ctx t₁) >>=STE (λ ctx₁ → lift (subctxFV ctx t₂) >>=STE (λ ctx₂ → (ctx₁ isDisjointWith ctx₂) >>STE (put ctx₁) >>STE typeCheck' t₁ >>=STE isBang >>STE (put ctx₂) >>STE typeCheck' t₂))) typeCheck' (Copy t₁ (x , y) t₂) = get >>=STE (λ ctx → lift (subctxFV ctx t₁) >>=STE (λ ctx₁ → lift (subctxFV ctx t₂) >>=STE (λ ctx₂ → (ctx₁ isDisjointWith ctx₂) >>STE (put ctx₁) >>STE typeCheck' t₁ >>=STE (λ ty₁ → isBang ty₁ >>STE (put ((x , ty₁) :: (y , ty₁) :: ctx₂) >>STE typeCheck' t₂'))))) where t₂' = open-t 0 y RCPV (FVar y) (open-t 0 x LCPV (FVar x) t₂) typeCheck' (Derelict t) = typeCheck' t >>=STE isBang typeCheck : Term → Either Exception Type typeCheck t with runStateT (typeCheck' t) [] ... | Left e = Left e ... | Right (ty , _) = right ty

module Examples.Resolution where open import Prelude open import Implicits.Syntax open import Implicits.Syntax.Type.Constructors open import Implicits.WellTyped open import Implicits.Substitutions open import Extensions.ListFirst open Rules unit = 0 -- simple implicit resolution by implication module ex₁ where r : Type zero r = ∀' (tp-weaken tnat ⇒ (TVar zero)) -- proof that the above type is a rule r-isrule : IsRule r r-isrule = ∀'-rule (rule (tp-weaken tnat) (TVar zero)) -- the context used K = tnat ∷K nil -- partial implicit resolution -- as described by Oliveira et al module ex₂ where r : Type zero r = (tnat ⇒ TC unit ⇒ TC unit) -- proof that the above type is a rule r-isrule : IsRule r r-isrule = rule tnat (TC unit ⇒ TC unit) K : Ktx 0 1 K = TC unit ∷K nil -- higher order resolution module ex₃ where r : Type zero r = ∀' ((TC unit ⇒ TVar zero) ⇒ (TVar zero ×' TVar zero)) r' : Type zero r' = (TC unit ⇒ tnat) ⇒ (tnat ×' tnat) r-isrule : IsRule r r-isrule = ∀'-rule (rule (TC unit ⇒ TVar zero) (TVar zero ×' TVar zero)) r'-isrule : IsRule r' r'-isrule = rule (TC unit ⇒ tnat) (tnat ×' tnat) K : Ktx 0 1 K = (TC unit ⇒ tnat) ∷K nil

module foldr-monoid-foldl where import Relation.Binary.PropositionalEquality as Eq open Eq using (_≡_; refl; sym; trans; cong) open Eq.≡-Reasoning open import lists using (List; []; _∷_; [_]; [_,_]; [_,_,_]; foldr; IsMonoid) open IsMonoid open import foldl using (foldl) postulate -- 外延性の公理 extensionality : ∀ {A B : Set} {f g : A → B} → (∀ (x : A) → f x ≡ g x) ----------------------- → f ≡ g -- foldr-monoidのfoldl版 -- _⊗_ と e がモノイドをなすとき、任意の値でfoldを再表現できる foldl-monoid : ∀ {A : Set} → (_⊗_ : A → A → A) (e : A) → IsMonoid _⊗_ e → ∀ (xs : List A) (y : A) → foldl _⊗_ y xs ≡ y ⊗ (foldl _⊗_ e xs) foldl-monoid _⊗_ e ⊗-monoid [] y = begin foldl _⊗_ y [] ≡⟨⟩ y ≡⟨ sym (identityʳ ⊗-monoid y) ⟩ y ⊗ e ≡⟨⟩ y ⊗ (foldl _⊗_ e []) ∎ foldl-monoid _⊗_ e ⊗-monoid (x ∷ xs) y = begin foldl _⊗_ y (x ∷ xs) ≡⟨⟩ foldl _⊗_ (y ⊗ x) xs ≡⟨ foldl-monoid _⊗_ e ⊗-monoid xs (y ⊗ x) ⟩ (y ⊗ x) ⊗ (foldl _⊗_ e xs) ≡⟨ assoc ⊗-monoid y x (foldl _⊗_ e xs) ⟩ y ⊗ (x ⊗ (foldl _⊗_ e xs)) ≡⟨ cong (y ⊗_) (sym (foldl-monoid _⊗_ e ⊗-monoid xs x)) ⟩ y ⊗ (foldl _⊗_ x xs) ≡⟨ cong (λ e⊗x → y ⊗ (foldl _⊗_ e⊗x xs)) (sym (identityˡ ⊗-monoid x)) ⟩ y ⊗ (foldl _⊗_ (e ⊗ x) xs) ≡⟨⟩ y ⊗ (foldl _⊗_ e (x ∷ xs)) ∎ -- 外延性の公理を用いた証明のための補題 lemma : ∀ {A : Set} → (_⊗_ : A → A → A) (e : A) → IsMonoid _⊗_ e → ∀ (xs : List A) → foldr _⊗_ e xs ≡ foldl _⊗_ e xs lemma _⊗_ e ⊗-monoid [] = begin foldr _⊗_ e [] ≡⟨⟩ e ≡⟨⟩ foldl _⊗_ e [] ∎ lemma _⊗_ e ⊗-monoid (x ∷ xs) = begin foldr _⊗_ e (x ∷ xs) ≡⟨⟩ x ⊗ (foldr _⊗_ e xs) ≡⟨ cong (x ⊗_) (lemma _⊗_ e ⊗-monoid xs) ⟩ x ⊗ (foldl _⊗_ e xs) ≡⟨ sym (foldl-monoid _⊗_ e ⊗-monoid xs x) ⟩ foldl _⊗_ x xs ≡⟨ cong (λ e⊗x → foldl _⊗_ e⊗x xs) (sym (identityˡ ⊗-monoid x)) ⟩ foldl _⊗_ (e ⊗ x) xs ≡⟨⟩ foldl _⊗_ e (x ∷ xs) ∎ -- _⊗_ と e がモノイドをなすとき、foldrとfoldlが等しくなることの証明 foldr-monoid-foldl : ∀ {A : Set} → (_⊗_ : A → A → A) (e : A) → IsMonoid _⊗_ e → foldr _⊗_ e ≡ foldl _⊗_ e foldr-monoid-foldl _⊗_ e ⊗-monoid = extensionality (lemma _⊗_ e ⊗-monoid)

module Human.Empty where data Empty : Set where void : ∀ {P : Set} → Empty → P void ()