bigcode/the-stack-smol-xl · Datasets at Hugging Face

0d079ed13b1e7fca6bf68f676cd4c541fa26225a

253

agda

Agda

test/Fail/Issue998d.agda

alex-mckenna/agda

78b62cd24bbd570271a7153e44ad280e52ef3e29

[ "BSD-3-Clause" ]

3

2015-03-28T14:51:03.000Z

2015-12-07T20:14:00.000Z

test/Fail/Issue998d.agda

andersk/agda

56928ff709dcb931cb9a48c4790e5ed3739e3032

[ "BSD-3-Clause" ]

null

test/Fail/Issue998d.agda

andersk/agda

56928ff709dcb931cb9a48c4790e5ed3739e3032

[ "BSD-3-Clause" ]

1

2022-03-12T11:35:18.000Z

open import Common.Level postulate ℓ : Level f : (l : Level) (A : Set l) → Set ℓ f ℓ A = A -- Expected error: -- ℓ != ℓ of type Level -- (because one is a variable and one a defined identifier) -- when checking that the expression A has type Set ℓ

19.461538

59

0.664032

edf922c966d7704dafcdd9f95425a4e95130fd06

83

agda

Agda

src/higher.agda

pcapriotti/agda-base

bbbc3bfb2f80ad08c8e608cccfa14b83ea3d258c

[ "BSD-3-Clause" ]

20

2015-06-12T12:20:17.000Z

2022-02-01T11:25:54.000Z

src/higher.agda

pcapriotti/agda-base

bbbc3bfb2f80ad08c8e608cccfa14b83ea3d258c

[ "BSD-3-Clause" ]

4

2015-02-02T14:32:16.000Z

2016-10-26T11:57:26.000Z

src/higher.agda

pcapriotti/agda-base

bbbc3bfb2f80ad08c8e608cccfa14b83ea3d258c

[ "BSD-3-Clause" ]

4

2015-02-02T12:17:00.000Z

2019-05-04T19:31:00.000Z

{-# OPTIONS --without-K #-} module higher where open import higher.circle public

13.833333

32

0.722892

fd2cf2f2f8e017fdd4a7db289691b45b1e18e363

29,166

agda

Agda

prototyping/Properties/StrictMode.agda

saga/luau

5bb9f379b07e378db0a170e7c4030e3a943b2f14

[ "MIT" ]

null

prototyping/Properties/StrictMode.agda

saga/luau

5bb9f379b07e378db0a170e7c4030e3a943b2f14

[ "MIT" ]

null

prototyping/Properties/StrictMode.agda

saga/luau

5bb9f379b07e378db0a170e7c4030e3a943b2f14

[ "MIT" ]

null

{-# OPTIONS --rewriting #-} module Properties.StrictMode where import Agda.Builtin.Equality.Rewrite open import Agda.Builtin.Equality using (_≡_; refl) open import FFI.Data.Either using (Either; Left; Right; mapL; mapR; mapLR; swapLR; cond) open import FFI.Data.Maybe using (Maybe; just; nothing) open import Luau.Heap using (Heap; Object; function_is_end; defn; alloc; ok; next; lookup-not-allocated) renaming (_≡_⊕_↦_ to _≡ᴴ_⊕_↦_; _[_] to _[_]ᴴ; ∅ to ∅ᴴ) open import Luau.StrictMode using (Warningᴱ; Warningᴮ; Warningᴼ; Warningᴴ; UnallocatedAddress; UnboundVariable; FunctionCallMismatch; app₁; app₂; BinOpMismatch₁; BinOpMismatch₂; bin₁; bin₂; BlockMismatch; block₁; return; LocalVarMismatch; local₁; local₂; FunctionDefnMismatch; function₁; function₂; heap; expr; block; addr) open import Luau.Substitution using (_[_/_]ᴮ; _[_/_]ᴱ; _[_/_]ᴮunless_; var_[_/_]ᴱwhenever_) open import Luau.Subtyping using (_≮:_; witness; unknown; never; scalar; function; scalar-function; scalar-function-ok; scalar-function-err; scalar-scalar; function-scalar; function-ok; function-err; left; right; _,_; Tree; Language; ¬Language) open import Luau.Syntax using (Expr; yes; var; val; var_∈_; _⟨_⟩∈_; _$_; addr; number; bool; string; binexp; nil; function_is_end; block_is_end; done; return; local_←_; _∙_; fun; arg; name; ==; ~=) open import Luau.Type using (Type; nil; number; boolean; string; _⇒_; never; unknown; _∩_; _∪_; src; tgt; _≡ᵀ_; _≡ᴹᵀ_) open import Luau.TypeCheck using (_⊢ᴮ_∈_; _⊢ᴱ_∈_; _⊢ᴴᴮ_▷_∈_; _⊢ᴴᴱ_▷_∈_; nil; var; addr; app; function; block; done; return; local; orUnknown; srcBinOp; tgtBinOp) open import Luau.Var using (_≡ⱽ_) open import Luau.Addr using (_≡ᴬ_) open import Luau.VarCtxt using (VarCtxt; ∅; _⋒_; _↦_; _⊕_↦_; _⊝_; ⊕-lookup-miss; ⊕-swap; ⊕-over) renaming (_[_] to _[_]ⱽ) open import Luau.VarCtxt using (VarCtxt; ∅) open import Properties.Remember using (remember; _,_) open import Properties.Equality using (_≢_; sym; cong; trans; subst₁) open import Properties.Dec using (Dec; yes; no) open import Properties.Contradiction using (CONTRADICTION; ¬) open import Properties.Functions using (_∘_) open import Properties.Subtyping using (unknown-≮:; ≡-trans-≮:; ≮:-trans-≡; never-tgt-≮:; tgt-never-≮:; src-unknown-≮:; unknown-src-≮:; ≮:-trans; ≮:-refl; scalar-≢-impl-≮:; function-≮:-scalar; scalar-≮:-function; function-≮:-never; unknown-≮:-scalar; scalar-≮:-never; unknown-≮:-never) open import Properties.TypeCheck using (typeOfᴼ; typeOfᴹᴼ; typeOfⱽ; typeOfᴱ; typeOfᴮ; typeCheckᴱ; typeCheckᴮ; typeCheckᴼ; typeCheckᴴ) open import Luau.OpSem using (_⟦_⟧_⟶_; _⊢_⟶*_⊣_; _⊢_⟶ᴮ_⊣_; _⊢_⟶ᴱ_⊣_; app₁; app₂; function; beta; return; block; done; local; subst; binOp₀; binOp₁; binOp₂; refl; step; +; -; *; /; <; >; ==; ~=; <=; >=; ··) open import Luau.RuntimeError using (BinOpError; RuntimeErrorᴱ; RuntimeErrorᴮ; FunctionMismatch; BinOpMismatch₁; BinOpMismatch₂; UnboundVariable; SEGV; app₁; app₂; bin₁; bin₂; block; local; return; +; -; *; /; <; >; <=; >=; ··) open import Luau.RuntimeType using (RuntimeType; valueType; number; string; boolean; nil; function) data _⊑_ (H : Heap yes) : Heap yes → Set where refl : (H ⊑ H) snoc : ∀ {H′ a O} → (H′ ≡ᴴ H ⊕ a ↦ O) → (H ⊑ H′) rednᴱ⊑ : ∀ {H H′ M M′} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → (H ⊑ H′) rednᴮ⊑ : ∀ {H H′ B B′} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → (H ⊑ H′) rednᴱ⊑ (function a p) = snoc p rednᴱ⊑ (app₁ s) = rednᴱ⊑ s rednᴱ⊑ (app₂ p s) = rednᴱ⊑ s rednᴱ⊑ (beta O v p q) = refl rednᴱ⊑ (block s) = rednᴮ⊑ s rednᴱ⊑ (return v) = refl rednᴱ⊑ done = refl rednᴱ⊑ (binOp₀ p) = refl rednᴱ⊑ (binOp₁ s) = rednᴱ⊑ s rednᴱ⊑ (binOp₂ s) = rednᴱ⊑ s rednᴮ⊑ (local s) = rednᴱ⊑ s rednᴮ⊑ (subst v) = refl rednᴮ⊑ (function a p) = snoc p rednᴮ⊑ (return s) = rednᴱ⊑ s data LookupResult (H : Heap yes) a V : Set where just : (H [ a ]ᴴ ≡ just V) → LookupResult H a V nothing : (H [ a ]ᴴ ≡ nothing) → LookupResult H a V lookup-⊑-nothing : ∀ {H H′} a → (H ⊑ H′) → (H′ [ a ]ᴴ ≡ nothing) → (H [ a ]ᴴ ≡ nothing) lookup-⊑-nothing {H} a refl p = p lookup-⊑-nothing {H} a (snoc defn) p with a ≡ᴬ next H lookup-⊑-nothing {H} a (snoc defn) p | yes refl = refl lookup-⊑-nothing {H} a (snoc o) p | no q = trans (lookup-not-allocated o q) p heap-weakeningᴱ : ∀ Γ H M {H′ U} → (H ⊑ H′) → (typeOfᴱ H′ Γ M ≮: U) → (typeOfᴱ H Γ M ≮: U) heap-weakeningᴱ Γ H (var x) h p = p heap-weakeningᴱ Γ H (val nil) h p = p heap-weakeningᴱ Γ H (val (addr a)) refl p = p heap-weakeningᴱ Γ H (val (addr a)) (snoc {a = b} q) p with a ≡ᴬ b heap-weakeningᴱ Γ H (val (addr a)) (snoc {a = a} defn) p | yes refl = unknown-≮: p heap-weakeningᴱ Γ H (val (addr a)) (snoc {a = b} q) p | no r = ≡-trans-≮: (cong orUnknown (cong typeOfᴹᴼ (lookup-not-allocated q r))) p heap-weakeningᴱ Γ H (val (number x)) h p = p heap-weakeningᴱ Γ H (val (bool x)) h p = p heap-weakeningᴱ Γ H (val (string x)) h p = p heap-weakeningᴱ Γ H (M $ N) h p = never-tgt-≮: (heap-weakeningᴱ Γ H M h (tgt-never-≮: p)) heap-weakeningᴱ Γ H (function f ⟨ var x ∈ T ⟩∈ U is B end) h p = p heap-weakeningᴱ Γ H (block var b ∈ T is B end) h p = p heap-weakeningᴱ Γ H (binexp M op N) h p = p heap-weakeningᴮ : ∀ Γ H B {H′ U} → (H ⊑ H′) → (typeOfᴮ H′ Γ B ≮: U) → (typeOfᴮ H Γ B ≮: U) heap-weakeningᴮ Γ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) h p = heap-weakeningᴮ (Γ ⊕ f ↦ (T ⇒ U)) H B h p heap-weakeningᴮ Γ H (local var x ∈ T ← M ∙ B) h p = heap-weakeningᴮ (Γ ⊕ x ↦ T) H B h p heap-weakeningᴮ Γ H (return M ∙ B) h p = heap-weakeningᴱ Γ H M h p heap-weakeningᴮ Γ H done h p = p substitutivityᴱ : ∀ {Γ T U} H M v x → (typeOfᴱ H Γ (M [ v / x ]ᴱ) ≮: U) → Either (typeOfᴱ H (Γ ⊕ x ↦ T) M ≮: U) (typeOfᴱ H ∅ (val v) ≮: T) substitutivityᴱ-whenever : ∀ {Γ T U} H v x y (r : Dec(x ≡ y)) → (typeOfᴱ H Γ (var y [ v / x ]ᴱwhenever r) ≮: U) → Either (typeOfᴱ H (Γ ⊕ x ↦ T) (var y) ≮: U) (typeOfᴱ H ∅ (val v) ≮: T) substitutivityᴮ : ∀ {Γ T U} H B v x → (typeOfᴮ H Γ (B [ v / x ]ᴮ) ≮: U) → Either (typeOfᴮ H (Γ ⊕ x ↦ T) B ≮: U) (typeOfᴱ H ∅ (val v) ≮: T) substitutivityᴮ-unless : ∀ {Γ T U V} H B v x y (r : Dec(x ≡ y)) → (typeOfᴮ H (Γ ⊕ y ↦ U) (B [ v / x ]ᴮunless r) ≮: V) → Either (typeOfᴮ H ((Γ ⊕ x ↦ T) ⊕ y ↦ U) B ≮: V) (typeOfᴱ H ∅ (val v) ≮: T) substitutivityᴮ-unless-yes : ∀ {Γ Γ′ T V} H B v x y (r : x ≡ y) → (Γ′ ≡ Γ) → (typeOfᴮ H Γ (B [ v / x ]ᴮunless yes r) ≮: V) → Either (typeOfᴮ H Γ′ B ≮: V) (typeOfᴱ H ∅ (val v) ≮: T) substitutivityᴮ-unless-no : ∀ {Γ Γ′ T V} H B v x y (r : x ≢ y) → (Γ′ ≡ Γ ⊕ x ↦ T) → (typeOfᴮ H Γ (B [ v / x ]ᴮunless no r) ≮: V) → Either (typeOfᴮ H Γ′ B ≮: V) (typeOfᴱ H ∅ (val v) ≮: T) substitutivityᴱ H (var y) v x p = substitutivityᴱ-whenever H v x y (x ≡ⱽ y) p substitutivityᴱ H (val w) v x p = Left p substitutivityᴱ H (binexp M op N) v x p = Left p substitutivityᴱ H (M $ N) v x p = mapL never-tgt-≮: (substitutivityᴱ H M v x (tgt-never-≮: p)) substitutivityᴱ H (function f ⟨ var y ∈ T ⟩∈ U is B end) v x p = Left p substitutivityᴱ H (block var b ∈ T is B end) v x p = Left p substitutivityᴱ-whenever H v x x (yes refl) q = swapLR (≮:-trans q) substitutivityᴱ-whenever H v x y (no p) q = Left (≡-trans-≮: (cong orUnknown (sym (⊕-lookup-miss x y _ _ p))) q) substitutivityᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x p = substitutivityᴮ-unless H B v x f (x ≡ⱽ f) p substitutivityᴮ H (local var y ∈ T ← M ∙ B) v x p = substitutivityᴮ-unless H B v x y (x ≡ⱽ y) p substitutivityᴮ H (return M ∙ B) v x p = substitutivityᴱ H M v x p substitutivityᴮ H done v x p = Left p substitutivityᴮ-unless H B v x y (yes p) q = substitutivityᴮ-unless-yes H B v x y p (⊕-over p) q substitutivityᴮ-unless H B v x y (no p) q = substitutivityᴮ-unless-no H B v x y p (⊕-swap p) q substitutivityᴮ-unless-yes H B v x y refl refl p = Left p substitutivityᴮ-unless-no H B v x y p refl q = substitutivityᴮ H B v x q binOpPreservation : ∀ H {op v w x} → (v ⟦ op ⟧ w ⟶ x) → (tgtBinOp op ≡ typeOfᴱ H ∅ (val x)) binOpPreservation H (+ m n) = refl binOpPreservation H (- m n) = refl binOpPreservation H (/ m n) = refl binOpPreservation H (* m n) = refl binOpPreservation H (< m n) = refl binOpPreservation H (> m n) = refl binOpPreservation H (<= m n) = refl binOpPreservation H (>= m n) = refl binOpPreservation H (== v w) = refl binOpPreservation H (~= v w) = refl binOpPreservation H (·· v w) = refl reflect-subtypingᴱ : ∀ H M {H′ M′ T} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → (typeOfᴱ H′ ∅ M′ ≮: T) → Either (typeOfᴱ H ∅ M ≮: T) (Warningᴱ H (typeCheckᴱ H ∅ M)) reflect-subtypingᴮ : ∀ H B {H′ B′ T} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → (typeOfᴮ H′ ∅ B′ ≮: T) → Either (typeOfᴮ H ∅ B ≮: T) (Warningᴮ H (typeCheckᴮ H ∅ B)) reflect-subtypingᴱ H (M $ N) (app₁ s) p = mapLR never-tgt-≮: app₁ (reflect-subtypingᴱ H M s (tgt-never-≮: p)) reflect-subtypingᴱ H (M $ N) (app₂ v s) p = Left (never-tgt-≮: (heap-weakeningᴱ ∅ H M (rednᴱ⊑ s) (tgt-never-≮: p))) reflect-subtypingᴱ H (M $ N) (beta (function f ⟨ var y ∈ T ⟩∈ U is B end) v refl q) p = Left (≡-trans-≮: (cong tgt (cong orUnknown (cong typeOfᴹᴼ q))) p) reflect-subtypingᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a defn) p = Left p reflect-subtypingᴱ H (block var b ∈ T is B end) (block s) p = Left p reflect-subtypingᴱ H (block var b ∈ T is return (val v) ∙ B end) (return v) p = mapR BlockMismatch (swapLR (≮:-trans p)) reflect-subtypingᴱ H (block var b ∈ T is done end) done p = mapR BlockMismatch (swapLR (≮:-trans p)) reflect-subtypingᴱ H (binexp M op N) (binOp₀ s) p = Left (≡-trans-≮: (binOpPreservation H s) p) reflect-subtypingᴱ H (binexp M op N) (binOp₁ s) p = Left p reflect-subtypingᴱ H (binexp M op N) (binOp₂ s) p = Left p reflect-subtypingᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) (function a defn) p = mapLR (heap-weakeningᴮ _ _ B (snoc defn)) (CONTRADICTION ∘ ≮:-refl) (substitutivityᴮ _ B (addr a) f p) reflect-subtypingᴮ H (local var x ∈ T ← M ∙ B) (local s) p = Left (heap-weakeningᴮ (x ↦ T) H B (rednᴱ⊑ s) p) reflect-subtypingᴮ H (local var x ∈ T ← M ∙ B) (subst v) p = mapR LocalVarMismatch (substitutivityᴮ H B v x p) reflect-subtypingᴮ H (return M ∙ B) (return s) p = mapR return (reflect-subtypingᴱ H M s p) reflect-substitutionᴱ : ∀ {Γ T} H M v x → Warningᴱ H (typeCheckᴱ H Γ (M [ v / x ]ᴱ)) → Either (Warningᴱ H (typeCheckᴱ H (Γ ⊕ x ↦ T) M)) (Either (Warningᴱ H (typeCheckᴱ H ∅ (val v))) (typeOfᴱ H ∅ (val v) ≮: T)) reflect-substitutionᴱ-whenever : ∀ {Γ T} H v x y (p : Dec(x ≡ y)) → Warningᴱ H (typeCheckᴱ H Γ (var y [ v / x ]ᴱwhenever p)) → Either (Warningᴱ H (typeCheckᴱ H (Γ ⊕ x ↦ T) (var y))) (Either (Warningᴱ H (typeCheckᴱ H ∅ (val v))) (typeOfᴱ H ∅ (val v) ≮: T)) reflect-substitutionᴮ : ∀ {Γ T} H B v x → Warningᴮ H (typeCheckᴮ H Γ (B [ v / x ]ᴮ)) → Either (Warningᴮ H (typeCheckᴮ H (Γ ⊕ x ↦ T) B)) (Either (Warningᴱ H (typeCheckᴱ H ∅ (val v))) (typeOfᴱ H ∅ (val v) ≮: T)) reflect-substitutionᴮ-unless : ∀ {Γ T U} H B v x y (r : Dec(x ≡ y)) → Warningᴮ H (typeCheckᴮ H (Γ ⊕ y ↦ U) (B [ v / x ]ᴮunless r)) → Either (Warningᴮ H (typeCheckᴮ H ((Γ ⊕ x ↦ T) ⊕ y ↦ U) B)) (Either (Warningᴱ H (typeCheckᴱ H ∅ (val v))) (typeOfᴱ H ∅ (val v) ≮: T)) reflect-substitutionᴮ-unless-yes : ∀ {Γ Γ′ T} H B v x y (r : x ≡ y) → (Γ′ ≡ Γ) → Warningᴮ H (typeCheckᴮ H Γ (B [ v / x ]ᴮunless yes r)) → Either (Warningᴮ H (typeCheckᴮ H Γ′ B)) (Either (Warningᴱ H (typeCheckᴱ H ∅ (val v))) (typeOfᴱ H ∅ (val v) ≮: T)) reflect-substitutionᴮ-unless-no : ∀ {Γ Γ′ T} H B v x y (r : x ≢ y) → (Γ′ ≡ Γ ⊕ x ↦ T) → Warningᴮ H (typeCheckᴮ H Γ (B [ v / x ]ᴮunless no r)) → Either (Warningᴮ H (typeCheckᴮ H Γ′ B)) (Either (Warningᴱ H (typeCheckᴱ H ∅ (val v))) (typeOfᴱ H ∅ (val v) ≮: T)) reflect-substitutionᴱ H (var y) v x W = reflect-substitutionᴱ-whenever H v x y (x ≡ⱽ y) W reflect-substitutionᴱ H (val (addr a)) v x (UnallocatedAddress r) = Left (UnallocatedAddress r) reflect-substitutionᴱ H (M $ N) v x (FunctionCallMismatch p) with substitutivityᴱ H N v x p reflect-substitutionᴱ H (M $ N) v x (FunctionCallMismatch p) | Right W = Right (Right W) reflect-substitutionᴱ H (M $ N) v x (FunctionCallMismatch p) | Left q with substitutivityᴱ H M v x (src-unknown-≮: q) reflect-substitutionᴱ H (M $ N) v x (FunctionCallMismatch p) | Left q | Left r = Left ((FunctionCallMismatch ∘ unknown-src-≮: q) r) reflect-substitutionᴱ H (M $ N) v x (FunctionCallMismatch p) | Left q | Right W = Right (Right W) reflect-substitutionᴱ H (M $ N) v x (app₁ W) = mapL app₁ (reflect-substitutionᴱ H M v x W) reflect-substitutionᴱ H (M $ N) v x (app₂ W) = mapL app₂ (reflect-substitutionᴱ H N v x W) reflect-substitutionᴱ H (function f ⟨ var y ∈ T ⟩∈ U is B end) v x (FunctionDefnMismatch q) = mapLR FunctionDefnMismatch Right (substitutivityᴮ-unless H B v x y (x ≡ⱽ y) q) reflect-substitutionᴱ H (function f ⟨ var y ∈ T ⟩∈ U is B end) v x (function₁ W) = mapL function₁ (reflect-substitutionᴮ-unless H B v x y (x ≡ⱽ y) W) reflect-substitutionᴱ H (block var b ∈ T is B end) v x (BlockMismatch q) = mapLR BlockMismatch Right (substitutivityᴮ H B v x q) reflect-substitutionᴱ H (block var b ∈ T is B end) v x (block₁ W′) = mapL block₁ (reflect-substitutionᴮ H B v x W′) reflect-substitutionᴱ H (binexp M op N) v x (BinOpMismatch₁ q) = mapLR BinOpMismatch₁ Right (substitutivityᴱ H M v x q) reflect-substitutionᴱ H (binexp M op N) v x (BinOpMismatch₂ q) = mapLR BinOpMismatch₂ Right (substitutivityᴱ H N v x q) reflect-substitutionᴱ H (binexp M op N) v x (bin₁ W) = mapL bin₁ (reflect-substitutionᴱ H M v x W) reflect-substitutionᴱ H (binexp M op N) v x (bin₂ W) = mapL bin₂ (reflect-substitutionᴱ H N v x W) reflect-substitutionᴱ-whenever H a x x (yes refl) (UnallocatedAddress p) = Right (Left (UnallocatedAddress p)) reflect-substitutionᴱ-whenever H v x y (no p) (UnboundVariable q) = Left (UnboundVariable (trans (sym (⊕-lookup-miss x y _ _ p)) q)) reflect-substitutionᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x (FunctionDefnMismatch q) = mapLR FunctionDefnMismatch Right (substitutivityᴮ-unless H C v x y (x ≡ⱽ y) q) reflect-substitutionᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x (function₁ W) = mapL function₁ (reflect-substitutionᴮ-unless H C v x y (x ≡ⱽ y) W) reflect-substitutionᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) v x (function₂ W) = mapL function₂ (reflect-substitutionᴮ-unless H B v x f (x ≡ⱽ f) W) reflect-substitutionᴮ H (local var y ∈ T ← M ∙ B) v x (LocalVarMismatch q) = mapLR LocalVarMismatch Right (substitutivityᴱ H M v x q) reflect-substitutionᴮ H (local var y ∈ T ← M ∙ B) v x (local₁ W) = mapL local₁ (reflect-substitutionᴱ H M v x W) reflect-substitutionᴮ H (local var y ∈ T ← M ∙ B) v x (local₂ W) = mapL local₂ (reflect-substitutionᴮ-unless H B v x y (x ≡ⱽ y) W) reflect-substitutionᴮ H (return M ∙ B) v x (return W) = mapL return (reflect-substitutionᴱ H M v x W) reflect-substitutionᴮ-unless H B v x y (yes p) W = reflect-substitutionᴮ-unless-yes H B v x y p (⊕-over p) W reflect-substitutionᴮ-unless H B v x y (no p) W = reflect-substitutionᴮ-unless-no H B v x y p (⊕-swap p) W reflect-substitutionᴮ-unless-yes H B v x x refl refl W = Left W reflect-substitutionᴮ-unless-no H B v x y p refl W = reflect-substitutionᴮ H B v x W reflect-weakeningᴱ : ∀ Γ H M {H′} → (H ⊑ H′) → Warningᴱ H′ (typeCheckᴱ H′ Γ M) → Warningᴱ H (typeCheckᴱ H Γ M) reflect-weakeningᴮ : ∀ Γ H B {H′} → (H ⊑ H′) → Warningᴮ H′ (typeCheckᴮ H′ Γ B) → Warningᴮ H (typeCheckᴮ H Γ B) reflect-weakeningᴱ Γ H (var x) h (UnboundVariable p) = (UnboundVariable p) reflect-weakeningᴱ Γ H (val (addr a)) h (UnallocatedAddress p) = UnallocatedAddress (lookup-⊑-nothing a h p) reflect-weakeningᴱ Γ H (M $ N) h (FunctionCallMismatch p) = FunctionCallMismatch (heap-weakeningᴱ Γ H N h (unknown-src-≮: p (heap-weakeningᴱ Γ H M h (src-unknown-≮: p)))) reflect-weakeningᴱ Γ H (M $ N) h (app₁ W) = app₁ (reflect-weakeningᴱ Γ H M h W) reflect-weakeningᴱ Γ H (M $ N) h (app₂ W) = app₂ (reflect-weakeningᴱ Γ H N h W) reflect-weakeningᴱ Γ H (binexp M op N) h (BinOpMismatch₁ p) = BinOpMismatch₁ (heap-weakeningᴱ Γ H M h p) reflect-weakeningᴱ Γ H (binexp M op N) h (BinOpMismatch₂ p) = BinOpMismatch₂ (heap-weakeningᴱ Γ H N h p) reflect-weakeningᴱ Γ H (binexp M op N) h (bin₁ W′) = bin₁ (reflect-weakeningᴱ Γ H M h W′) reflect-weakeningᴱ Γ H (binexp M op N) h (bin₂ W′) = bin₂ (reflect-weakeningᴱ Γ H N h W′) reflect-weakeningᴱ Γ H (function f ⟨ var y ∈ T ⟩∈ U is B end) h (FunctionDefnMismatch p) = FunctionDefnMismatch (heap-weakeningᴮ (Γ ⊕ y ↦ T) H B h p) reflect-weakeningᴱ Γ H (function f ⟨ var y ∈ T ⟩∈ U is B end) h (function₁ W) = function₁ (reflect-weakeningᴮ (Γ ⊕ y ↦ T) H B h W) reflect-weakeningᴱ Γ H (block var b ∈ T is B end) h (BlockMismatch p) = BlockMismatch (heap-weakeningᴮ Γ H B h p) reflect-weakeningᴱ Γ H (block var b ∈ T is B end) h (block₁ W) = block₁ (reflect-weakeningᴮ Γ H B h W) reflect-weakeningᴮ Γ H (return M ∙ B) h (return W) = return (reflect-weakeningᴱ Γ H M h W) reflect-weakeningᴮ Γ H (local var y ∈ T ← M ∙ B) h (LocalVarMismatch p) = LocalVarMismatch (heap-weakeningᴱ Γ H M h p) reflect-weakeningᴮ Γ H (local var y ∈ T ← M ∙ B) h (local₁ W) = local₁ (reflect-weakeningᴱ Γ H M h W) reflect-weakeningᴮ Γ H (local var y ∈ T ← M ∙ B) h (local₂ W) = local₂ (reflect-weakeningᴮ (Γ ⊕ y ↦ T) H B h W) reflect-weakeningᴮ Γ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) h (FunctionDefnMismatch p) = FunctionDefnMismatch (heap-weakeningᴮ (Γ ⊕ x ↦ T) H C h p) reflect-weakeningᴮ Γ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) h (function₁ W) = function₁ (reflect-weakeningᴮ (Γ ⊕ x ↦ T) H C h W) reflect-weakeningᴮ Γ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) h (function₂ W) = function₂ (reflect-weakeningᴮ (Γ ⊕ f ↦ (T ⇒ U)) H B h W) reflect-weakeningᴼ : ∀ H O {H′} → (H ⊑ H′) → Warningᴼ H′ (typeCheckᴼ H′ O) → Warningᴼ H (typeCheckᴼ H O) reflect-weakeningᴼ H (just function f ⟨ var x ∈ T ⟩∈ U is B end) h (FunctionDefnMismatch p) = FunctionDefnMismatch (heap-weakeningᴮ (x ↦ T) H B h p) reflect-weakeningᴼ H (just function f ⟨ var x ∈ T ⟩∈ U is B end) h (function₁ W) = function₁ (reflect-weakeningᴮ (x ↦ T) H B h W) reflectᴱ : ∀ H M {H′ M′} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → Warningᴱ H′ (typeCheckᴱ H′ ∅ M′) → Either (Warningᴱ H (typeCheckᴱ H ∅ M)) (Warningᴴ H (typeCheckᴴ H)) reflectᴮ : ∀ H B {H′ B′} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → Warningᴮ H′ (typeCheckᴮ H′ ∅ B′) → Either (Warningᴮ H (typeCheckᴮ H ∅ B)) (Warningᴴ H (typeCheckᴴ H)) reflectᴱ H (M $ N) (app₁ s) (FunctionCallMismatch p) = cond (Left ∘ FunctionCallMismatch ∘ heap-weakeningᴱ ∅ H N (rednᴱ⊑ s) ∘ unknown-src-≮: p) (Left ∘ app₁) (reflect-subtypingᴱ H M s (src-unknown-≮: p)) reflectᴱ H (M $ N) (app₁ s) (app₁ W′) = mapL app₁ (reflectᴱ H M s W′) reflectᴱ H (M $ N) (app₁ s) (app₂ W′) = Left (app₂ (reflect-weakeningᴱ ∅ H N (rednᴱ⊑ s) W′)) reflectᴱ H (M $ N) (app₂ p s) (FunctionCallMismatch q) = cond (λ r → Left (FunctionCallMismatch (unknown-src-≮: r (heap-weakeningᴱ ∅ H M (rednᴱ⊑ s) (src-unknown-≮: r))))) (Left ∘ app₂) (reflect-subtypingᴱ H N s q) reflectᴱ H (M $ N) (app₂ p s) (app₁ W′) = Left (app₁ (reflect-weakeningᴱ ∅ H M (rednᴱ⊑ s) W′)) reflectᴱ H (M $ N) (app₂ p s) (app₂ W′) = mapL app₂ (reflectᴱ H N s W′) reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (BlockMismatch q) with substitutivityᴮ H B v x q reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (BlockMismatch q) | Left r = Right (addr a p (FunctionDefnMismatch r)) reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (BlockMismatch q) | Right r = Left (FunctionCallMismatch (≮:-trans-≡ r ((cong src (cong orUnknown (cong typeOfᴹᴼ (sym p))))))) reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (block₁ W′) with reflect-substitutionᴮ _ B v x W′ reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (block₁ W′) | Left W = Right (addr a p (function₁ W)) reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (block₁ W′) | Right (Left W) = Left (app₂ W) reflectᴱ H (val (addr a) $ N) (beta (function f ⟨ var x ∈ T ⟩∈ U is B end) v refl p) (block₁ W′) | Right (Right q) = Left (FunctionCallMismatch (≮:-trans-≡ q (cong src (cong orUnknown (cong typeOfᴹᴼ (sym p)))))) reflectᴱ H (block var b ∈ T is B end) (block s) (BlockMismatch p) = Left (cond BlockMismatch block₁ (reflect-subtypingᴮ H B s p)) reflectᴱ H (block var b ∈ T is B end) (block s) (block₁ W′) = mapL block₁ (reflectᴮ H B s W′) reflectᴱ H (block var b ∈ T is B end) (return v) W′ = Left (block₁ (return W′)) reflectᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a defn) (UnallocatedAddress ()) reflectᴱ H (binexp M op N) (binOp₀ ()) (UnallocatedAddress p) reflectᴱ H (binexp M op N) (binOp₁ s) (BinOpMismatch₁ p) = Left (cond BinOpMismatch₁ bin₁ (reflect-subtypingᴱ H M s p)) reflectᴱ H (binexp M op N) (binOp₁ s) (BinOpMismatch₂ p) = Left (BinOpMismatch₂ (heap-weakeningᴱ ∅ H N (rednᴱ⊑ s) p)) reflectᴱ H (binexp M op N) (binOp₁ s) (bin₁ W′) = mapL bin₁ (reflectᴱ H M s W′) reflectᴱ H (binexp M op N) (binOp₁ s) (bin₂ W′) = Left (bin₂ (reflect-weakeningᴱ ∅ H N (rednᴱ⊑ s) W′)) reflectᴱ H (binexp M op N) (binOp₂ s) (BinOpMismatch₁ p) = Left (BinOpMismatch₁ (heap-weakeningᴱ ∅ H M (rednᴱ⊑ s) p)) reflectᴱ H (binexp M op N) (binOp₂ s) (BinOpMismatch₂ p) = Left (cond BinOpMismatch₂ bin₂ (reflect-subtypingᴱ H N s p)) reflectᴱ H (binexp M op N) (binOp₂ s) (bin₁ W′) = Left (bin₁ (reflect-weakeningᴱ ∅ H M (rednᴱ⊑ s) W′)) reflectᴱ H (binexp M op N) (binOp₂ s) (bin₂ W′) = mapL bin₂ (reflectᴱ H N s W′) reflectᴮ H (local var x ∈ T ← M ∙ B) (local s) (LocalVarMismatch p) = Left (cond LocalVarMismatch local₁ (reflect-subtypingᴱ H M s p)) reflectᴮ H (local var x ∈ T ← M ∙ B) (local s) (local₁ W′) = mapL local₁ (reflectᴱ H M s W′) reflectᴮ H (local var x ∈ T ← M ∙ B) (local s) (local₂ W′) = Left (local₂ (reflect-weakeningᴮ (x ↦ T) H B (rednᴱ⊑ s) W′)) reflectᴮ H (local var x ∈ T ← M ∙ B) (subst v) W′ = Left (cond local₂ (cond local₁ LocalVarMismatch) (reflect-substitutionᴮ H B v x W′)) reflectᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) (function a defn) W′ with reflect-substitutionᴮ _ B (addr a) f W′ reflectᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) (function a defn) W′ | Left W = Left (function₂ (reflect-weakeningᴮ (f ↦ (T ⇒ U)) H B (snoc defn) W)) reflectᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) (function a defn) W′ | Right (Left (UnallocatedAddress ())) reflectᴮ H (function f ⟨ var y ∈ T ⟩∈ U is C end ∙ B) (function a defn) W′ | Right (Right p) = CONTRADICTION (≮:-refl p) reflectᴮ H (return M ∙ B) (return s) (return W′) = mapL return (reflectᴱ H M s W′) reflectᴴᴱ : ∀ H M {H′ M′} → (H ⊢ M ⟶ᴱ M′ ⊣ H′) → Warningᴴ H′ (typeCheckᴴ H′) → Either (Warningᴱ H (typeCheckᴱ H ∅ M)) (Warningᴴ H (typeCheckᴴ H)) reflectᴴᴮ : ∀ H B {H′ B′} → (H ⊢ B ⟶ᴮ B′ ⊣ H′) → Warningᴴ H′ (typeCheckᴴ H′) → Either (Warningᴮ H (typeCheckᴮ H ∅ B)) (Warningᴴ H (typeCheckᴴ H)) reflectᴴᴱ H (M $ N) (app₁ s) W = mapL app₁ (reflectᴴᴱ H M s W) reflectᴴᴱ H (M $ N) (app₂ v s) W = mapL app₂ (reflectᴴᴱ H N s W) reflectᴴᴱ H (M $ N) (beta O v refl p) W = Right W reflectᴴᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a p) (addr b refl W) with b ≡ᴬ a reflectᴴᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a defn) (addr b refl (FunctionDefnMismatch p)) | yes refl = Left (FunctionDefnMismatch (heap-weakeningᴮ (x ↦ T) H B (snoc defn) p)) reflectᴴᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a defn) (addr b refl (function₁ W)) | yes refl = Left (function₁ (reflect-weakeningᴮ (x ↦ T) H B (snoc defn) W)) reflectᴴᴱ H (function f ⟨ var x ∈ T ⟩∈ U is B end) (function a p) (addr b refl W) | no q = Right (addr b (lookup-not-allocated p q) (reflect-weakeningᴼ H _ (snoc p) W)) reflectᴴᴱ H (block var b ∈ T is B end) (block s) W = mapL block₁ (reflectᴴᴮ H B s W) reflectᴴᴱ H (block var b ∈ T is return (val v) ∙ B end) (return v) W = Right W reflectᴴᴱ H (block var b ∈ T is done end) done W = Right W reflectᴴᴱ H (binexp M op N) (binOp₀ s) W = Right W reflectᴴᴱ H (binexp M op N) (binOp₁ s) W = mapL bin₁ (reflectᴴᴱ H M s W) reflectᴴᴱ H (binexp M op N) (binOp₂ s) W = mapL bin₂ (reflectᴴᴱ H N s W) reflectᴴᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) (function a p) (addr b refl W) with b ≡ᴬ a reflectᴴᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) (function a defn) (addr b refl (FunctionDefnMismatch p)) | yes refl = Left (FunctionDefnMismatch (heap-weakeningᴮ (x ↦ T) H C (snoc defn) p)) reflectᴴᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) (function a defn) (addr b refl (function₁ W)) | yes refl = Left (function₁ (reflect-weakeningᴮ (x ↦ T) H C (snoc defn) W)) reflectᴴᴮ H (function f ⟨ var x ∈ T ⟩∈ U is C end ∙ B) (function a p) (addr b refl W) | no q = Right (addr b (lookup-not-allocated p q) (reflect-weakeningᴼ H _ (snoc p) W)) reflectᴴᴮ H (local var x ∈ T ← M ∙ B) (local s) W = mapL local₁ (reflectᴴᴱ H M s W) reflectᴴᴮ H (local var x ∈ T ← M ∙ B) (subst v) W = Right W reflectᴴᴮ H (return M ∙ B) (return s) W = mapL return (reflectᴴᴱ H M s W) reflect* : ∀ H B {H′ B′} → (H ⊢ B ⟶* B′ ⊣ H′) → Either (Warningᴮ H′ (typeCheckᴮ H′ ∅ B′)) (Warningᴴ H′ (typeCheckᴴ H′)) → Either (Warningᴮ H (typeCheckᴮ H ∅ B)) (Warningᴴ H (typeCheckᴴ H)) reflect* H B refl W = W reflect* H B (step s t) W = cond (reflectᴮ H B s) (reflectᴴᴮ H B s) (reflect* _ _ t W) isntNumber : ∀ H v → (valueType v ≢ number) → (typeOfᴱ H ∅ (val v) ≮: number) isntNumber H nil p = scalar-≢-impl-≮: nil number (λ ()) isntNumber H (addr a) p with remember (H [ a ]ᴴ) isntNumber H (addr a) p | (just (function f ⟨ var x ∈ T ⟩∈ U is B end) , q) = ≡-trans-≮: (cong orUnknown (cong typeOfᴹᴼ q)) (function-≮:-scalar number) isntNumber H (addr a) p | (nothing , q) = ≡-trans-≮: (cong orUnknown (cong typeOfᴹᴼ q)) (unknown-≮:-scalar number) isntNumber H (number x) p = CONTRADICTION (p refl) isntNumber H (bool x) p = scalar-≢-impl-≮: boolean number (λ ()) isntNumber H (string x) p = scalar-≢-impl-≮: string number (λ ()) isntString : ∀ H v → (valueType v ≢ string) → (typeOfᴱ H ∅ (val v) ≮: string) isntString H nil p = scalar-≢-impl-≮: nil string (λ ()) isntString H (addr a) p with remember (H [ a ]ᴴ) isntString H (addr a) p | (just (function f ⟨ var x ∈ T ⟩∈ U is B end) , q) = ≡-trans-≮: (cong orUnknown (cong typeOfᴹᴼ q)) (function-≮:-scalar string) isntString H (addr a) p | (nothing , q) = ≡-trans-≮: (cong orUnknown (cong typeOfᴹᴼ q)) (unknown-≮:-scalar string) isntString H (number x) p = scalar-≢-impl-≮: number string (λ ()) isntString H (bool x) p = scalar-≢-impl-≮: boolean string (λ ()) isntString H (string x) p = CONTRADICTION (p refl) isntFunction : ∀ H v {T U} → (valueType v ≢ function) → (typeOfᴱ H ∅ (val v) ≮: (T ⇒ U)) isntFunction H nil p = scalar-≮:-function nil isntFunction H (addr a) p = CONTRADICTION (p refl) isntFunction H (number x) p = scalar-≮:-function number isntFunction H (bool x) p = scalar-≮:-function boolean isntFunction H (string x) p = scalar-≮:-function string isntEmpty : ∀ H v → (typeOfᴱ H ∅ (val v) ≮: never) isntEmpty H nil = scalar-≮:-never nil isntEmpty H (addr a) with remember (H [ a ]ᴴ) isntEmpty H (addr a) | (just (function f ⟨ var x ∈ T ⟩∈ U is B end) , p) = ≡-trans-≮: (cong orUnknown (cong typeOfᴹᴼ p)) function-≮:-never isntEmpty H (addr a) | (nothing , p) = ≡-trans-≮: (cong orUnknown (cong typeOfᴹᴼ p)) unknown-≮:-never isntEmpty H (number x) = scalar-≮:-never number isntEmpty H (bool x) = scalar-≮:-never boolean isntEmpty H (string x) = scalar-≮:-never string runtimeBinOpWarning : ∀ H {op} v → BinOpError op (valueType v) → (typeOfᴱ H ∅ (val v) ≮: srcBinOp op) runtimeBinOpWarning H v (+ p) = isntNumber H v p runtimeBinOpWarning H v (- p) = isntNumber H v p runtimeBinOpWarning H v (* p) = isntNumber H v p runtimeBinOpWarning H v (/ p) = isntNumber H v p runtimeBinOpWarning H v (< p) = isntNumber H v p runtimeBinOpWarning H v (> p) = isntNumber H v p runtimeBinOpWarning H v (<= p) = isntNumber H v p runtimeBinOpWarning H v (>= p) = isntNumber H v p runtimeBinOpWarning H v (·· p) = isntString H v p runtimeWarningᴱ : ∀ H M → RuntimeErrorᴱ H M → Warningᴱ H (typeCheckᴱ H ∅ M) runtimeWarningᴮ : ∀ H B → RuntimeErrorᴮ H B → Warningᴮ H (typeCheckᴮ H ∅ B) runtimeWarningᴱ H (var x) UnboundVariable = UnboundVariable refl runtimeWarningᴱ H (val (addr a)) (SEGV p) = UnallocatedAddress p runtimeWarningᴱ H (M $ N) (FunctionMismatch v w p) = FunctionCallMismatch (unknown-src-≮: (isntEmpty H w) (isntFunction H v p)) runtimeWarningᴱ H (M $ N) (app₁ err) = app₁ (runtimeWarningᴱ H M err) runtimeWarningᴱ H (M $ N) (app₂ err) = app₂ (runtimeWarningᴱ H N err) runtimeWarningᴱ H (block var b ∈ T is B end) (block err) = block₁ (runtimeWarningᴮ H B err) runtimeWarningᴱ H (binexp M op N) (BinOpMismatch₁ v w p) = BinOpMismatch₁ (runtimeBinOpWarning H v p) runtimeWarningᴱ H (binexp M op N) (BinOpMismatch₂ v w p) = BinOpMismatch₂ (runtimeBinOpWarning H w p) runtimeWarningᴱ H (binexp M op N) (bin₁ err) = bin₁ (runtimeWarningᴱ H M err) runtimeWarningᴱ H (binexp M op N) (bin₂ err) = bin₂ (runtimeWarningᴱ H N err) runtimeWarningᴮ H (local var x ∈ T ← M ∙ B) (local err) = local₁ (runtimeWarningᴱ H M err) runtimeWarningᴮ H (return M ∙ B) (return err) = return (runtimeWarningᴱ H M err) wellTypedProgramsDontGoWrong : ∀ H′ B B′ → (∅ᴴ ⊢ B ⟶* B′ ⊣ H′) → (RuntimeErrorᴮ H′ B′) → Warningᴮ ∅ᴴ (typeCheckᴮ ∅ᴴ ∅ B) wellTypedProgramsDontGoWrong H′ B B′ t err with reflect* ∅ᴴ B t (Left (runtimeWarningᴮ H′ B′ err)) wellTypedProgramsDontGoWrong H′ B B′ t err | Right (addr a refl ()) wellTypedProgramsDontGoWrong H′ B B′ t err | Left W = W

84.051873

323

0.653021

736df6d3ac4af6d0a26e7007197ce5a6ef47ad5c

2,287

agda

Agda

Structure/Category/Functor/Proofs.agda

Lolirofle/stuff-in-agda

70f4fba849f2fd779c5aaa5af122ccb6a5b271ba

[ "MIT" ]

6

2020-04-07T17:58:13.000Z

2022-02-05T06:53:22.000Z

Structure/Category/Functor/Proofs.agda

Lolirofle/stuff-in-agda

70f4fba849f2fd779c5aaa5af122ccb6a5b271ba

[ "MIT" ]

null

Structure/Category/Functor/Proofs.agda

Lolirofle/stuff-in-agda

70f4fba849f2fd779c5aaa5af122ccb6a5b271ba

[ "MIT" ]

null

module Structure.Category.Functor.Proofs where open import Data.Tuple as Tuple using (_,_) open import Functional using (_$_) open import Logic.Predicate import Lvl open import Structure.Category open import Structure.Categorical.Properties open import Structure.Category.Functor open import Structure.Category.Functor.Equiv open import Structure.Function open import Structure.Operator open import Structure.Relator.Equivalence open import Structure.Relator.Properties open import Structure.Setoid open import Syntax.Transitivity open import Type private variable ℓ ℓₗₑ ℓᵣₑ : Lvl.Level private variable Obj Obj₁ Obj₂ Obj₃ : Type{ℓ} private variable Morphism Morphism₁ Morphism₂ Morphism₃ : Obj → Obj → Type{ℓ} module _ ⦃ morphism-equiv₁ : ∀{x y} → Equiv{ℓₗₑ}(Morphism₁ x y) ⦄ ⦃ morphism-equiv₂ : ∀{x y} → Equiv{ℓᵣₑ}(Morphism₂ x y) ⦄ {cat₁ : Category(Morphism₁)} {cat₂ : Category(Morphism₂)} (F : Obj₁ → Obj₂) ⦃ functor : Functor(cat₁)(cat₂)(F) ⦄ where open Category.ArrowNotation ⦃ … ⦄ open Category ⦃ … ⦄ open Functor(functor) private open module MorphismEquivₗ {x}{y} = Equiv(morphism-equiv₁{x}{y}) using () renaming (_≡_ to _≡ₗₘ_) private open module MorphismEquivᵣ {x}{y} = Equiv(morphism-equiv₂{x}{y}) using () renaming (_≡_ to _≡ᵣₘ_) private instance _ = cat₁ private instance _ = cat₂ private variable x y : Obj₁ isomorphism-preserving : ∀{f : x ⟶ y} → Morphism.Isomorphism ⦃ \{x y} → morphism-equiv₁ {x}{y} ⦄ (_∘_)(id)(f) → Morphism.Isomorphism ⦃ \{x y} → morphism-equiv₂ {x}{y} ⦄ (_∘_)(id)(map f) ∃.witness (isomorphism-preserving ([∃]-intro g)) = map g ∃.proof (isomorphism-preserving {f = f} iso@([∃]-intro g)) = (Morphism.intro $ map g ∘ map f 🝖-[ op-preserving ]-sym map(g ∘ f) 🝖-[ congruence₁(map) (inverseₗ(f)(g)) ] map id 🝖-[ id-preserving ] id 🝖-end ) , (Morphism.intro $ map f ∘ map g 🝖-[ op-preserving ]-sym map(f ∘ g) 🝖-[ congruence₁(map) (inverseᵣ(f)(g)) ] map id 🝖-[ id-preserving ] id 🝖-end ) where open Morphism.OperModule (\{x : Obj₁} → _∘_ {x = x}) open Morphism.IdModule (\{x : Obj₁} → _∘_ {x = x})(id) open Morphism.Isomorphism(\{x : Obj₁} → _∘_ {x = x})(id)(f) instance _ = iso

37.491803

187

0.651509

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agda

Agda

src/Categories/Category/Duality.agda

jaykru/agda-categories

a4053cf700bcefdf73b857c3352f1eae29382a60

[ "MIT" ]

5

2020-10-07T12:07:53.000Z

2020-10-10T21:41:32.000Z

src/Categories/Category/Duality.agda

jaykru/agda-categories

a4053cf700bcefdf73b857c3352f1eae29382a60

[ "MIT" ]

null

src/Categories/Category/Duality.agda

jaykru/agda-categories

a4053cf700bcefdf73b857c3352f1eae29382a60

[ "MIT" ]

1

2021-11-04T06:54:45.000Z

{-# OPTIONS --without-K --safe #-} open import Categories.Category module Categories.Category.Duality {o ℓ e} (C : Category o ℓ e) where open import Relation.Binary.PropositionalEquality using (_≡_; refl) open import Categories.Category.Cartesian open import Categories.Category.Cocartesian open import Categories.Category.Complete open import Categories.Category.Complete.Finitely open import Categories.Category.Cocomplete open import Categories.Category.Cocomplete.Finitely open import Categories.Object.Duality open import Categories.Diagram.Duality open import Categories.Functor private module C = Category C open C coCartesian⇒Cocartesian : Cartesian C.op → Cocartesian C coCartesian⇒Cocartesian Car = record { initial = op⊤⇒⊥ C terminal ; coproducts = record { coproduct = coProduct⇒Coproduct C product } } where open Cartesian Car Cocartesian⇒coCartesian : Cocartesian C → Cartesian C.op Cocartesian⇒coCartesian Co = record { terminal = ⊥⇒op⊤ C initial ; products = record { product = Coproduct⇒coProduct C coproduct } } where open Cocartesian Co coComplete⇒Cocomplete : ∀ {o′ ℓ′ e′} → Complete o′ ℓ′ e′ C.op → Cocomplete o′ ℓ′ e′ C coComplete⇒Cocomplete Com F = coLimit⇒Colimit C (Com F.op) where module F = Functor F Cocomplete⇒coComplete : ∀ {o′ ℓ′ e′} → Cocomplete o′ ℓ′ e′ C → Complete o′ ℓ′ e′ C.op Cocomplete⇒coComplete Coc F = Colimit⇒coLimit C (Coc F.op) where module F = Functor F coFinitelyComplete⇒FinitelyCocomplete : FinitelyComplete C.op → FinitelyCocomplete C coFinitelyComplete⇒FinitelyCocomplete FC = record { cocartesian = coCartesian⇒Cocartesian cartesian ; coequalizer = λ f g → coEqualizer⇒Coequalizer C (equalizer f g) } where open FinitelyComplete FC FinitelyCocomplete⇒coFinitelyComplete : FinitelyCocomplete C → FinitelyComplete C.op FinitelyCocomplete⇒coFinitelyComplete FC = record { cartesian = Cocartesian⇒coCartesian cocartesian ; equalizer = λ f g → Coequalizer⇒coEqualizer C (coequalizer f g) } where open FinitelyCocomplete FC module DualityConversionProperties where private op-involutive : Category.op C.op ≡ C op-involutive = refl coCartesian⇔Cocartesian : ∀(coCartesian : Cartesian C.op) → Cocartesian⇒coCartesian (coCartesian⇒Cocartesian coCartesian) ≡ coCartesian coCartesian⇔Cocartesian _ = refl coFinitelyComplete⇔FinitelyCocomplete : ∀(coFinComplete : FinitelyComplete C.op) → FinitelyCocomplete⇒coFinitelyComplete (coFinitelyComplete⇒FinitelyCocomplete coFinComplete) ≡ coFinComplete coFinitelyComplete⇔FinitelyCocomplete _ = refl

32.876543

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agda

Agda

test/succeed/SubtermTermination.agda

asr/agda-kanso

aa10ae6a29dc79964fe9dec2de07b9df28b61ed5

[ "MIT" ]

1

2019-11-27T07:26:06.000Z

test/succeed/SubtermTermination.agda

np/agda-git-experiment

20596e9dd9867166a64470dd24ea68925ff380ce

[ "MIT" ]

null

test/succeed/SubtermTermination.agda

np/agda-git-experiment

20596e9dd9867166a64470dd24ea68925ff380ce

[ "MIT" ]

null

-- Check that the termination checker can handle recursive -- calls on subterms which aren't simply variables. module SubtermTermination where data N : Set where zero : N suc : N → N f : N → N f (suc zero) = f zero f _ = zero data One? : N → Set where one : One? (suc zero) other : ∀ {n} → One? n -- Should work for dot patterns as well f′ : (n : N) → One? n → N f′ (suc .zero) one = f′ zero other f′ _ _ = zero f″ : (n : N) → One? n → N f″ ._ one = f″ zero other f″ _ _ = zero data D : Set where c₁ : D c₂ : D → D c₃ : D → D → D g : D → D g (c₃ (c₂ x) y) = g (c₂ x) g _ = c₁ {- Andreas, 2011-07-07 subterm is not complete does not work with postulates or definitions postulate i : {A : Set} → A → A data NAT : N → Set where Zero : NAT zero Suc : ∀ n → NAT (i n) → NAT (suc (i n)) h : (n : N) -> NAT n -> Set h .zero Zero = N h .(suc (i n)) (Suc n m) = h (i n) (i m) -}

19.354167

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0.547901

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Agda

test/MonoidTactic.agda

t-more/agda-prelude

da4fca7744d317b8843f2bc80a923972f65548d3

[ "MIT" ]

111

2015-01-05T11:28:15.000Z

2022-02-12T23:29:26.000Z

test/MonoidTactic.agda

t-more/agda-prelude

da4fca7744d317b8843f2bc80a923972f65548d3

[ "MIT" ]

59

2016-02-09T05:36:44.000Z

2022-01-14T07:32:36.000Z

test/MonoidTactic.agda

t-more/agda-prelude

da4fca7744d317b8843f2bc80a923972f65548d3

[ "MIT" ]

24

2015-03-12T18:03:45.000Z

2021-04-22T06:10:41.000Z

module MonoidTactic where open import Prelude open import Container.Traversable open import Tactic.Monoid open import Tactic.Reflection SemigroupAnd : Semigroup Bool _<>_ {{SemigroupAnd}} = _&&_ MonoidAnd : Monoid Bool Monoid.super MonoidAnd = SemigroupAnd mempty {{MonoidAnd}} = true Monoid/LawsAnd : Monoid/Laws Bool Monoid/Laws.super Monoid/LawsAnd = MonoidAnd left-identity {{Monoid/LawsAnd}} x = refl right-identity {{Monoid/LawsAnd}} true = refl right-identity {{Monoid/LawsAnd}} false = refl monoid-assoc {{Monoid/LawsAnd}} true y z = refl monoid-assoc {{Monoid/LawsAnd}} false y z = refl test₁ : (a b : Bool) → (a && (b && a && true)) ≡ ((a && b) && a) test₁ a b = auto-monoid {{Laws = Monoid/LawsAnd}} test₂ : ∀ {a} {A : Set a} {{Laws : Monoid/Laws A}} → (x y : A) → x <> (y <> x <> mempty) ≡ (x <> y) <> x test₂ x y = auto-monoid test₃ : ∀ {a} {A : Set a} (xs ys zs : List A) → xs ++ ys ++ zs ≡ (xs ++ []) ++ (ys ++ []) ++ zs test₃ xs ys zs = runT monoidTactic

29.176471

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0.640121

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326

agda

Agda

Categories/Presheaves.agda

copumpkin/categories

36f4181d751e2ecb54db219911d8c69afe8ba892

[ "BSD-3-Clause" ]

98

2015-04-15T14:57:33.000Z

2022-03-08T05:20:36.000Z

Categories/Presheaves.agda

p-pavel/categories

e41aef56324a9f1f8cf3cd30b2db2f73e01066f2

[ "BSD-3-Clause" ]

19

2015-05-23T06:47:10.000Z

2019-08-09T16:31:40.000Z

Categories/Presheaves.agda

p-pavel/categories

e41aef56324a9f1f8cf3cd30b2db2f73e01066f2

[ "BSD-3-Clause" ]

23

2015-02-05T13:03:09.000Z

2021-11-11T13:50:56.000Z

{-# OPTIONS --universe-polymorphism #-} module Categories.Presheaves where open import Level open import Categories.Category open import Categories.Agda open import Categories.FunctorCategory Presheaves : ∀ {o ℓ e : Level} → Category o ℓ e → Category _ _ _ Presheaves {o} {ℓ} {e} C = Functors (Category.op C) (ISetoids ℓ e)

29.636364

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Agda

test/Succeed/RewritingGlobalConfluenceWithClauses.agda

cagix/agda

cc026a6a97a3e517bb94bafa9d49233b067c7559

[ "BSD-2-Clause" ]

1,989

2015-01-09T23:51:16.000Z

2022-03-30T18:20:48.000Z

test/Succeed/RewritingGlobalConfluenceWithClauses.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

4,066

2015-01-10T11:24:51.000Z

2022-03-31T21:14:49.000Z

test/Succeed/RewritingGlobalConfluenceWithClauses.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

371

2015-01-03T14:04:08.000Z

2022-03-30T19:00:30.000Z

{-# OPTIONS --rewriting --confluence-check #-} open import Agda.Builtin.List open import Agda.Builtin.Nat open import Agda.Builtin.Equality open import Agda.Builtin.Equality.Rewrite variable A B : Set x y z : A xs ys zs : List A f : A → B m n : Nat cong : (f : A → B) → x ≡ y → f x ≡ f y cong f refl = refl trans : x ≡ y → y ≡ z → x ≡ z trans refl refl = refl +zero : m + zero ≡ m +zero {zero} = refl +zero {suc m} = cong suc +zero suc+zero : suc m + zero ≡ suc m suc+zero = +zero +suc : m + (suc n) ≡ suc (m + n) +suc {zero} = refl +suc {suc m} = cong suc +suc zero+suc : zero + (suc n) ≡ suc n zero+suc = refl suc+suc : (suc m) + (suc n) ≡ suc (suc (m + n)) suc+suc = cong suc +suc {-# REWRITE +zero +suc suc+zero zero+suc suc+suc #-} map : (A → B) → List A → List B map f [] = [] map f (x ∷ xs) = (f x) ∷ (map f xs) _++_ : List A → List A → List A [] ++ ys = ys (x ∷ xs) ++ ys = x ∷ (xs ++ ys) ++-[] : xs ++ [] ≡ xs ++-[] {xs = []} = refl ++-[] {xs = x ∷ xs} = cong (_∷_ x) ++-[] ∷-++-[] : (x ∷ xs) ++ [] ≡ x ∷ xs ∷-++-[] = ++-[] map-id : map (λ x → x) xs ≡ xs map-id {xs = []} = refl map-id {xs = x ∷ xs} = cong (_∷_ x) map-id map-id-∷ : map (λ x → x) (x ∷ xs) ≡ x ∷ xs map-id-∷ = map-id {-# REWRITE ++-[] ∷-++-[] #-} {-# REWRITE map-id map-id-∷ #-}

20.015625

52

0.50039

119d2d9465bb903f679cbd153a4a098adb7f0813

864

agda

Agda

test/Succeed/BlockOnFreshMeta.agda

redfish64/autonomic-agda

c0ae7d20728b15d7da4efff6ffadae6fe4590016

[ "BSD-3-Clause" ]

null

test/Succeed/BlockOnFreshMeta.agda

redfish64/autonomic-agda

c0ae7d20728b15d7da4efff6ffadae6fe4590016

[ "BSD-3-Clause" ]

null

test/Succeed/BlockOnFreshMeta.agda

redfish64/autonomic-agda

c0ae7d20728b15d7da4efff6ffadae6fe4590016

[ "BSD-3-Clause" ]

null

module _ where open import Common.Prelude hiding (_>>=_) open import Common.Reflection open import Common.Equality infix 0 case_of_ case_of_ : ∀ {a b} {A : Set a} {B : Set b} → A → (A → B) → B case x of f = f x blockOnFresh : TC ⊤ blockOnFresh = checkType unknown unknown >>= λ { (meta m _) → blockOnMeta m ; _ → typeError (strErr "impossible" ∷ []) } macro weirdButShouldWork : Tactic weirdButShouldWork hole = inferType hole >>= λ goal → case goal of λ { (meta _ _) → blockOnFresh ; _ → unify hole (lit (nat 42)) } -- When the goal is a meta the tactic will block on a different, fresh, meta. -- That's silly, but should still work. Once the goal is resolved the tactic -- doesn't block any more so everything should be fine. thing : _ solves : Nat thing = weirdButShouldWork solves = thing check : thing ≡ 42 check = refl

23.351351

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0.668981

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agda

Agda

Cubical/HITs/Ints/QuoInt.agda

limemloh/cubical

df4ef7edffd1c1deb3d4ff342c7178e9901c44f1

[ "MIT" ]

1

2020-03-23T23:52:11.000Z

Cubical/HITs/Ints/QuoInt.agda

limemloh/cubical

df4ef7edffd1c1deb3d4ff342c7178e9901c44f1

[ "MIT" ]

null

Cubical/HITs/Ints/QuoInt.agda

limemloh/cubical

df4ef7edffd1c1deb3d4ff342c7178e9901c44f1

[ "MIT" ]

null

{-# OPTIONS --cubical --safe #-} module Cubical.HITs.Ints.QuoInt where open import Cubical.HITs.Ints.QuoInt.Base public -- open import Cubical.HITs.Ints.QuoInt.Properties public

25.714286

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agda

Agda

src/Human/List.agda

MaisaMilena/JuiceMaker

b509eb4c4014605facfb4ee5c807cd07753d4477

[ "MIT" ]

6

2019-03-29T17:35:20.000Z

2020-11-28T05:46:27.000Z

src/Human/List.agda

MaisaMilena/JuiceMaker

b509eb4c4014605facfb4ee5c807cd07753d4477

[ "MIT" ]

null

src/Human/List.agda

MaisaMilena/JuiceMaker

b509eb4c4014605facfb4ee5c807cd07753d4477

[ "MIT" ]

null

module Human.List where open import Human.Nat infixr 5 _,_ data List {a} (A : Set a) : Set a where end : List A _,_ : (x : A) (xs : List A) → List A {-# BUILTIN LIST List #-} {-# COMPILE JS List = function(x,v) { if (x.length < 1) { return v["[]"](); } else { return v["_∷_"](x[0], x.slice(1)); } } #-} {-# COMPILE JS end = Array() #-} {-# COMPILE JS _,_ = function (x) { return function(y) { return Array(x).concat(y); }; } #-} foldr : ∀ {A : Set} {B : Set} → (A → B → B) → B → List A → B foldr c n end = n foldr c n (x , xs) = c x (foldr c n xs) length : ∀ {A : Set} → List A → Nat length = foldr (λ a n → suc n) zero -- TODO -- -- filter -- reduce -- Receives a function that transforms each element of A, a function A and a new list, B. map : ∀ {A : Set} {B : Set} → (f : A → B) → List A → List B map f end = end map f (x , xs) = (f x) , (map f xs) -- f transforms element x, return map to do a new transformation -- Sum all numbers in a list sum : List Nat → Nat sum end = zero sum (x , l) = x + (sum l) remove-last : ∀ {A : Set} → List A → List A remove-last end = end remove-last (x , l) = l

28.325

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0.551633

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agda

Agda

src/fot/FOTC/Induction/WF.agda

asr/fotc

2fc9f2b81052a2e0822669f02036c5750371b72d

[ "MIT" ]

11

2015-09-03T20:53:42.000Z

2021-09-12T16:09:54.000Z

src/fot/FOTC/Induction/WF.agda

asr/fotc

2fc9f2b81052a2e0822669f02036c5750371b72d

[ "MIT" ]

2

2016-10-12T17:28:16.000Z

2017-01-01T14:34:26.000Z

src/fot/FOTC/Induction/WF.agda

asr/fotc

2fc9f2b81052a2e0822669f02036c5750371b72d

[ "MIT" ]

3

2016-09-19T14:18:30.000Z

2018-03-14T08:50:00.000Z

------------------------------------------------------------------------------ -- Well-founded induction on natural numbers ------------------------------------------------------------------------------ {-# OPTIONS --exact-split #-} {-# OPTIONS --no-sized-types #-} {-# OPTIONS --no-universe-polymorphism #-} {-# OPTIONS --without-K #-} -- Adapted from -- http://www.iis.sinica.edu.tw/~scm/2008/well-founded-recursion-and-accessibility/ -- and the Agda standard library 0.8.1. module FOTC.Induction.WF where open import Common.Relation.Unary open import FOTC.Base ------------------------------------------------------------------------------ -- The accessibility predicate: x is accessible if everything which is -- smaller than x is also accessible (inductively). data Acc (P : D → Set)(_<_ : D → D → Set)(x : D) : Set where acc : (∀ {y} → P y → y < x → Acc P _<_ y) → Acc P _<_ x accFold : {P Q : D → Set}(_<_ : D → D → Set) → (∀ {x} → Q x → (∀ {y} → Q y → y < x → P y) → P x) → ∀ {x} → Q x → Acc Q _<_ x → P x accFold _<_ f Qx (acc h) = f Qx (λ Qy y<x → accFold _<_ f Qy (h Qy y<x)) -- The accessibility predicate encodes what it means to be -- well-founded; if all elements are accessible, then _<_ is -- well-founded. WellFounded : {P : D → Set} → (D → D → Set) → Set WellFounded {P} _<_ = ∀ {x} → P x → Acc P _<_ x WellFoundedInduction : {P Q : D → Set} {_<_ : D → D → Set} → WellFounded _<_ → (∀ {x} → Q x → (∀ {y} → Q y → y < x → P y) → P x) → ∀ {x} → Q x → P x WellFoundedInduction {_<_ = _<_} wf f Qx = accFold _<_ f Qx (wf Qx) module Subrelation {P : D → Set} {_<_ _<'_ : D → D → Set} (<⇒<' : ∀ {x y} → P x → x < y → x <' y) where accessible : Acc P _<'_ ⊆ Acc P _<_ accessible (acc h) = acc (λ Py y<x → accessible (h Py (<⇒<' Py y<x))) well-founded : WellFounded _<'_ → WellFounded _<_ well-founded wf = λ Px → accessible (wf Px) module InverseImage {P Q : D → Set} {_<_ : D → D → Set} {f : D → D} (f-Q : ∀ {x} → P x → Q (f x)) where accessible : ∀ {x} → P x → Acc Q _<_ (f x) → Acc P (λ x' y' → f x' < f y') x accessible Px (acc h) = acc (λ {y} Py fy<fx → accessible Py (h (f-Q Py) fy<fx)) wellFounded : WellFounded _<_ → WellFounded (λ x y → f x < f y) wellFounded wf = λ Px → accessible Px (wf (f-Q Px))

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agda

Agda

examples/ATPAxiomDataConstructors.agda

asr/apia

a66c5ddca2ab470539fd68c42c4fbd45f720d682

[ "MIT" ]

10

2015-09-03T20:54:16.000Z

2019-12-03T13:44:25.000Z

examples/ATPAxiomDataConstructors.agda

asr/apia

a66c5ddca2ab470539fd68c42c4fbd45f720d682

[ "MIT" ]

121

2015-01-25T13:22:12.000Z

2018-04-22T06:01:44.000Z

examples/ATPAxiomDataConstructors.agda

asr/apia

a66c5ddca2ab470539fd68c42c4fbd45f720d682

[ "MIT" ]

4

2016-05-10T23:06:19.000Z

2016-08-03T03:54:55.000Z

-- The ATP pragma with the role <axiom> can be used with data constructors. module ATPAxiomDataConstructors where postulate D : Set zero : D succ : D → D data N : D → Set where zN : N zero sN : ∀ {n} → N n → N (succ n) {-# ATP axiom zN #-}

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agda

Agda

src/Data/PropFormula/Theorems.agda

jonaprieto/agda-prop

a1730062a6aaced2bb74878c1071db06477044ae

[ "MIT" ]

13

2017-05-01T16:45:41.000Z

2022-01-17T03:33:12.000Z

src/Data/PropFormula/Theorems.agda

jonaprieto/agda-prop

a1730062a6aaced2bb74878c1071db06477044ae

[ "MIT" ]

18

2017-03-08T14:33:10.000Z

2017-12-18T16:34:21.000Z

src/Data/PropFormula/Theorems.agda

jonaprieto/agda-prop

a1730062a6aaced2bb74878c1071db06477044ae

[ "MIT" ]

2

2017-03-30T16:41:56.000Z

2017-12-01T17:01:25.000Z

------------------------------------------------------------------------------ -- Agda-Prop Library. -- A compilation of theorems in Propositional Logic ------------------------------------------------------------------------------ open import Data.Nat using ( ℕ ) module Data.PropFormula.Theorems ( n : ℕ ) where ------------------------------------------------------------------------------ open import Data.PropFormula.Theorems.Biimplication n public open import Data.PropFormula.Theorems.Classical n public open import Data.PropFormula.Theorems.Conjunction n public open import Data.PropFormula.Theorems.Disjunction n public open import Data.PropFormula.Theorems.Implication n public open import Data.PropFormula.Theorems.Mixies n public open import Data.PropFormula.Theorems.Negation n public open import Data.PropFormula.Theorems.Weakening n public ------------------------------------------------------------------------------

43.863636

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agda

Agda

test/Fail/Issue2127.agda

mdimjasevic/agda

8fb548356b275c7a1e79b768b64511ae937c738b

[ "BSD-3-Clause" ]

1,989

2015-01-09T23:51:16.000Z

2022-03-30T18:20:48.000Z

test/Fail/Issue2127.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

4,066

2015-01-10T11:24:51.000Z

2022-03-31T21:14:49.000Z

test/Fail/Issue2127.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

371

2015-01-03T14:04:08.000Z

2022-03-30T19:00:30.000Z

-- Andreas, 2016-08-02 issue #2127 reported by petercommand data Test : Set₁ where field A : Set B : Set -- second field necessary to trigger internal error -- WAS: internal error -- Should give proper error

20.181818

64

0.698198

7ceb9cda8b223740655c664c41bb153b85bc76cc

329

agda

Agda

test/Fail/Issue314.agda

mdimjasevic/agda

8fb548356b275c7a1e79b768b64511ae937c738b

[ "BSD-3-Clause" ]

1,989

2015-01-09T23:51:16.000Z

2022-03-30T18:20:48.000Z

test/Fail/Issue314.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

4,066

2015-01-10T11:24:51.000Z

2022-03-31T21:14:49.000Z

test/Fail/Issue314.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

371

2015-01-03T14:04:08.000Z

2022-03-30T19:00:30.000Z

module Issue314 where postulate A : Set data _≡_ (x : A) : A → Set where refl : x ≡ x postulate lemma : (x y : A) → x ≡ y Foo : A → Set₁ Foo x with lemma x _ Foo x | refl = Set -- Bug.agda:12,9-13 -- Failed to solve the following constraints: -- x == _23 x : A -- when checking that the pattern refl has type x ≡ _23 x

17.315789

57

0.620061

113b5a1fc1f10318a0ba8c4867231ae6fa9e0150

102,926

agda

Agda

src/templates.agda

xoltar/cedille

acf691e37210607d028f4b19f98ec26c4353bfb5

[ "MIT" ]

null

src/templates.agda

xoltar/cedille

acf691e37210607d028f4b19f98ec26c4353bfb5

[ "MIT" ]

null

src/templates.agda

xoltar/cedille

acf691e37210607d028f4b19f98ec26c4353bfb5

[ "MIT" ]

null

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(Tkk (KndArrow (KndParens "2735" (KndTpArrow (TpVar "2736" "Indices") (Star "2746")) "2748") (KndTpArrow (TpVar "2751" "Indices") (Star "2761")))) (KndTpArrow (TpApp (TpVar "2768" "Functor") (TpVar "2778" "F")) (KndPi "2786" "2788" "indices" (Tkt (TpVar "2798" "Indices")) (KndTpArrow (TpAppt (TpApp (TpVar "2811" "FixM") (TpVar "2818" "F")) (Var "2820" "indices")) (Star "2834"))))) (TpLambda "2840" "2842" "F" (Tkk (KndArrow (KndParens "2846" (KndTpArrow (TpVar "2847" "Indices") (Star "2857")) "2859") (KndTpArrow (TpVar "2862" "Indices") (Star "2872")))) (TpLambda "2875" "2877" "fmap" (Tkt (TpApp (TpVar "2884" "Functor") (TpVar "2894" "F"))) (TpLambda "2901" "2903" "indices" (Tkt (TpVar "2913" "Indices")) (TpLambda "2922" "2924" "x" (Tkt (TpAppt (TpApp (TpVar "2928" "FixM") (TpVar "2935" "F")) (Var "2937" "indices"))) (Abs "2952" Erased "2954" "Q" (Tkk (KndPi "2958" "2960" "indices" (Tkt (TpVar "2970" "Indices")) (KndTpArrow (TpAppt (TpApp (TpVar "2979" "FixM") (TpVar "2986" "F")) (Var "2988" "indices")) (Star "2998")))) (TpArrow (TpAppt (TpApp (TpApp (TpAppt (TpApp (TpVar "3007" "PrfAlgM") (TpVar "3017" "F")) (Var "3019" "fmap")) (TpParens "3026" (TpApp (TpVar "3027" "FixM") (TpVar "3034" "F")) "3036")) (TpVar "3039" "Q")) (Parens "3041" (AppTp (Var "3042" "inFixM") (TpVar "3051" "F")) "3053")) NotErased (TpAppt (TpAppt (TpVar "3056" "Q") (Var "3058" "indices")) (Var "3066" "x"))))))))) "3068") (CmdsNext (DefTermOrType OpacTrans (DefType "3070" "FixIndM" (KndPi "3080" "3082" "F" (Tkk (KndArrow (KndParens "3086" (KndTpArrow (TpVar "3087" "Indices") (Star "3097")) "3099") (KndTpArrow (TpVar "3102" "Indices") (Star "3112")))) (KndTpArrow (TpApp (TpVar "3115" "Functor") (TpVar "3125" "F")) (KndTpArrow (TpVar "3129" "Indices") (Star "3139")))) (TpLambda "3145" "3147" "F" (Tkk (KndArrow (KndParens "3151" (KndTpArrow (TpVar "3152" "Indices") (Star "3162")) "3164") (KndTpArrow (TpVar "3167" "Indices") (Star "3177")))) (TpLambda "3180" "3182" "fmap" (Tkt (TpApp (TpVar "3189" 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(TpVar "3419" "F")) (Var "3421" "fmap")) (Var "3426" "indices"))))))) (Lam "3438" Erased "3440" "F" NoClass (Lam "3443" Erased "3445" "fmap" NoClass (Lam "3451" Erased "3453" "indices" NoClass (Lam "3462" NotErased "3464" "v" NoClass (Let "3471" (DefTerm "3472" "outInd" (SomeType (TpApp (TpApp (TpVar "3481" "Cast") (TpParens "3488" (TpAppt (TpApp (TpVar "3489" "FixIndM") (TpVar "3499" "F")) (Var "3501" "fmap")) "3506")) (TpParens "3509" (TpApp (TpVar "3510" "FixM") (TpVar "3517" "F")) "3519"))) (IotaPair "3522" (Lam "3523" Erased "3525" "indices" NoClass (Lam "3534" NotErased "3536" "x" NoClass (IotaProj (Var "3539" "x") "1" "3542"))) (Beta "3544" NoTerm NoTerm) NoGuide "3546")) (IotaPair "3554" (App (App (AppTp (Var "3555" "inFixM") (TpVar "3564" "F")) Erased (Var "3567" "indices")) NotErased (Parens "3575" (App (App (App (Var "3576" "cast") Erased (Parens "3582" (App (Var "3583" "fmap") Erased (Var "3589" "outInd")) "3596")) Erased (Var "3598" "indices")) NotErased (Var "3606" "v")) 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(KndTpArrow (TpAppt (TpApp (TpVar "3986" "FixM") (TpVar "3993" "F")) (Var "3995" "indices")) (Star "4009")))))) (TpLambda "4015" "4017" "F" (Tkk (KndArrow (KndParens "4021" (KndTpArrow (TpVar "4022" "Indices") (Star "4032")) "4034") (KndTpArrow (TpVar "4037" "Indices") (Star "4047")))) (TpLambda "4050" "4052" "fmap" (Tkt (TpApp (TpVar "4059" "Functor") (TpVar "4069" "F"))) (TpLambda "4076" "4078" "Q" (Tkk (KndPi "4082" "4084" "indices" (Tkt (TpVar "4094" "Indices")) (KndTpArrow (TpAppt (TpAppt (TpApp (TpVar "4103" "FixIndM") (TpVar "4113" "F")) (Var "4115" "fmap")) (Var "4120" "indices")) (Star "4130")))) (TpLambda "4139" "4141" "indices" (Tkt (TpVar "4151" "Indices")) (TpLambda "4160" "4162" "e" (Tkt (TpAppt (TpApp (TpVar "4166" "FixM") (TpVar "4173" "F")) (Var "4175" "indices"))) (TpAppt (TpApp (TpApp (TpApp (TpVar "4192" "WithWitness") (TpParens "4206" (TpAppt (TpAppt (TpApp (TpVar "4207" "FixIndM") (TpVar "4217" "F")) (Var "4219" "fmap")) (Var "4224" "indices")) "4232")) (TpParens "4235" (TpAppt (TpApp (TpVar "4236" "FixM") (TpVar "4243" "F")) (Var "4245" "indices")) "4253")) (TpParens "4256" (TpAppt (TpVar "4257" "Q") (Var "4259" "indices")) "4267")) (Var "4268" "e")))))))) "4270") (CmdsNext (DefTermOrType OpacTrans (DefTerm "4272" "LiftProp1" (SomeType (Abs "4288" Erased "4290" "F" (Tkk (KndArrow (KndParens "4294" (KndTpArrow (TpVar "4295" "Indices") (Star "4305")) "4307") (KndTpArrow (TpVar "4310" "Indices") (Star "4320")))) (Abs "4323" Erased "4325" "fmap" (Tkt (TpApp (TpVar "4332" "Functor") (TpVar "4342" "F"))) (Abs "4349" Erased "4351" "Q" (Tkk (KndPi "4355" "4357" "indices" (Tkt (TpVar "4367" "Indices")) (KndTpArrow (TpAppt (TpAppt (TpApp (TpVar "4376" "FixIndM") (TpVar "4386" "F")) (Var "4388" "fmap")) (Var "4393" "indices")) (Star "4403")))) (Abs "4410" Erased "4412" "indices" (Tkt (TpVar "4422" "Indices")) (Abs "4431" Erased "4433" "e" (Tkt (TpAppt (TpAppt (TpApp (TpVar "4437" "FixIndM") (TpVar "4447" "F")) (Var "4449" "fmap")) (Var "4454" "indices"))) (TpArrow (TpAppt (TpAppt (TpApp (TpAppt (TpApp (TpVar "4467" "Lift") (TpVar "4474" "F")) (Var "4476" "fmap")) (TpVar "4483" "Q")) (Var "4485" "indices")) (IotaProj (Var "4493" "e") "1" "4496")) NotErased (TpAppt (TpAppt (TpVar "4499" "Q") (Var "4501" "indices")) (Var "4509" "e"))))))))) (Lam "4515" Erased "4517" "F" NoClass (Lam "4520" Erased "4522" "fmap" NoClass (Lam "4528" Erased "4530" "Q" NoClass (Lam "4533" Erased "4535" "indices" NoClass (Lam "4544" Erased "4546" "e" NoClass (Lam "4549" NotErased "4551" "pr" NoClass (Rho "4555" RhoPlain NoNums (IotaProj (Parens "4557" (App (Var "4558" "fst") NotErased (Var "4562" "pr")) "4565") "2" "4567") NoGuide (App (Var "4570" "snd") NotErased (Var "4574" "pr")))))))))) "4577") (CmdsNext (DefTermOrType OpacTrans (DefTerm "4579" "LiftProp2" (SomeType (Abs "4595" Erased "4597" "F" (Tkk (KndArrow (KndParens "4601" (KndTpArrow (TpVar "4602" "Indices") (Star "4612")) "4614") (KndTpArrow (TpVar "4617" "Indices") (Star "4627")))) (Abs 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"6096" "FixM") (TpVar "6103" "F")) (Var "6105" "indices"))) (Abs "6114" Erased "6116" "p" (Tkt (TpAppt (TpAppt (TpVar "6120" "Y") (Var "6122" "indices")) (Var "6130" "e"))) (TpEq "6133" (App (Var "6134" "Yprop3") NotErased (Var "6141" "p")) (Var "6145" "e") "6147"))))) (TpArrow (TpAppt (TpApp (TpApp (TpAppt (TpApp (TpVar "6150" "PrfAlgM") (TpVar "6160" "F")) (Var "6162" "fmap")) (TpParens "6169" (TpAppt (TpApp (TpVar "6170" "FixIndM") (TpVar "6180" "F")) (Var "6182" "fmap")) "6187")) (TpVar "6190" "Q")) (Parens "6192" (App (AppTp (Var "6193" "inFixIndM") (TpVar "6205" "F")) Erased (Var "6208" "fmap")) "6213")) NotErased (TpAppt (TpApp (TpApp (TpAppt (TpApp (TpVar "6217" "PrfAlgM") (TpVar "6227" "F")) (Var "6229" "fmap")) (TpParens "6236" (TpApp (TpVar "6237" "FixM") (TpVar "6244" "F")) "6246")) (TpVar "6249" "Y")) (Parens "6251" (AppTp (Var "6252" "inFixM") (TpVar "6261" "F")) "6263")))))))))))) (Lam "6268" Erased "6270" "F" NoClass (Lam "6273" Erased "6275" "fmap" NoClass (Lam "6281" 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NotErased (Var "6463" "r")) "6465")) Erased (Parens "6467" (App (App (Var "6468" "ih") Erased (Var "6472" "indices")) NotErased (Var "6480" "r")) "6482")) "6483") (Parens "6494" (App (App (App (Var "6495" "qp1") Erased (Var "6500" "indices")) Erased (Parens "6509" (App (App (App (Var "6510" "cast") Erased (Var "6516" "cst")) Erased (Var "6521" "indices")) NotErased (Var "6529" "r")) "6531")) NotErased (Parens "6532" (App (App (Var "6533" "ih") Erased (Var "6537" "indices")) NotErased (Var "6545" "r")) "6547")) "6548") (Var "6550" "r") "6552"))) (Beta "6554" NoTerm NoTerm) NoGuide "6556")) (App (App (App (Var "6564" "qp4") Erased (Var "6569" "indices")) NotErased (Parens "6577" (App (App (App (Var "6578" "inFixIndM") Erased (Var "6589" "fmap")) Erased (Var "6595" "indices")) NotErased (Parens "6603" (App (App (App (Var "6604" "cast") Erased (Parens "6610" (App (Var "6611" "fmap") Erased (Var "6617" "cst2")) "6622")) Erased (Var "6624" "indices")) NotErased (Var "6632" "gr")) "6635")) "6636")) NotErased (Parens "6643" (App (App (App (App (Var "6644" "q") Erased (Var "6647" "cst2")) NotErased (Parens "6652" (Lam "6653" Erased "6655" "indices" NoClass (Lam "6664" NotErased "6666" "r" NoClass (App (App (App (Var "6669" "qp3") Erased (Var "6674" "indices")) Erased (Parens "6683" (App (App (App (Var "6684" "cast") Erased (Var "6690" "cst2")) Erased (Var "6696" "indices")) NotErased (Var "6704" "r")) "6706")) NotErased (Parens "6707" (App (App (Var "6708" "ih") Erased (Var "6712" "indices")) NotErased (Var "6720" "r")) "6722")))) "6723")) Erased (Var "6733" "indices")) NotErased (Var "6741" "gr")) "6744")))))))))))))))))) "6745") (CmdsNext (DefTermOrType OpacTrans (DefTerm "6747" "indFixIndM" (SomeType (Abs "6760" Erased "6762" "F" (Tkk (KndArrow (KndParens "6766" (KndTpArrow (TpVar "6767" "Indices") (Star "6777")) "6779") (KndTpArrow (TpVar "6782" "Indices") (Star "6792")))) (Abs "6795" Erased "6797" "fmap" (Tkt (TpApp (TpVar "6804" "Functor") (TpVar "6814" "F"))) (Abs "6830" Erased "6832" "indices" (Tkt (TpVar "6842" "Indices")) (Abs "6851" NotErased "6853" "e" (Tkt (TpAppt (TpAppt (TpApp (TpVar "6857" "FixIndM") (TpVar "6867" "F")) (Var "6869" "fmap")) (Var "6874" "indices"))) (Abs "6896" Erased "6898" "Q" (Tkk (KndPi "6902" "6904" "indices" (Tkt (TpVar "6914" "Indices")) (KndTpArrow (TpAppt (TpAppt (TpApp (TpVar "6923" "FixIndM") (TpVar "6933" "F")) (Var "6935" "fmap")) (Var "6940" "indices")) (Star "6950")))) (TpArrow (TpAppt (TpApp (TpApp (TpAppt (TpApp (TpVar "6966" "PrfAlgM") (TpVar "6976" "F")) (Var "6978" "fmap")) (TpParens "6985" (TpAppt (TpApp (TpVar "6986" "FixIndM") (TpVar "6996" "F")) (Var "6998" "fmap")) "7003")) (TpVar "7006" "Q")) (Parens "7008" (App (AppTp (Var "7009" "inFixIndM") (TpVar "7021" "F")) Erased (Var "7024" "fmap")) "7029")) NotErased (TpAppt (TpAppt (TpVar "7045" "Q") (Var "7047" "indices")) (Var "7055" "e"))))))))) (Lam "7062" Erased "7064" "F" NoClass (Lam "7067" Erased "7069" "fmap" NoClass (Lam "7075" Erased "7077" "indices" NoClass (Lam "7086" NotErased "7088" "e" NoClass (Lam "7091" Erased "7093" "Q" NoClass (Lam "7096" NotErased "7098" "q" NoClass (App (App (App (AppTp (App (Var "7101" "LiftProp1") Erased (Var "7112" "fmap")) (TpVar "7119" "Q")) Erased (Var "7122" "indices")) Erased (Var "7131" "e")) NotErased (Parens "7137" (App (IotaProj (Var "7138" "e") "2" "7141") NotErased (Parens "7142" (App (App (App (App (App (App (Var "7143" "convIH") Erased (Var "7151" "fmap")) NotErased (Parens "7162" (AppTp (App (Var "7163" "LiftProp1") Erased (Var "7174" "fmap")) (TpVar "7181" "Q")) "7183")) NotErased (Parens "7190" (AppTp (App (Var "7191" "LiftProp2") Erased (Var "7202" "fmap")) (TpVar "7209" "Q")) "7211")) NotErased (Parens "7218" (AppTp (App (Var "7219" "LiftProp3") Erased (Var "7230" "fmap")) (TpVar "7237" "Q")) "7239")) NotErased (Parens "7246" (AppTp (App (Var "7247" "LiftProp4") Erased (Var "7258" "fmap")) (TpVar "7265" "Q")) "7267")) NotErased (Var "7274" "q")) "7276")) "7277"))))))))) 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"inFixIndM") (TpVar "7528" "F")) Erased (Var "7531" "fmap")) "7536"))))) (Lam "7541" Erased "7543" "F" NoClass (Lam "7546" Erased "7548" "fmap" NoClass (Lam "7554" Erased "7556" "R" NoClass (Lam "7559" Erased "7561" "c" NoClass (Lam "7564" NotErased "7566" "x" NoClass (App (Var "7569" "cast") Erased (Parens "7575" (App (Var "7576" "fmap") Erased (Var "7582" "c")) "7584")))))))) "7585") (CmdsNext (DefTermOrType OpacTrans (DefTerm "7587" "outFixIndM" (SomeType (Abs "7604" Erased "7606" "F" (Tkk (KndArrow (KndParens "7610" (KndTpArrow (TpVar "7611" "Indices") (Star "7621")) "7623") (KndTpArrow (TpVar "7626" "Indices") (Star "7636")))) (Abs "7639" Erased "7641" "fmap" (Tkt (TpApp (TpVar "7648" "Functor") (TpVar "7658" "F"))) (Abs "7661" Erased "7663" "indices" (Tkt (TpVar "7673" "Indices")) (TpArrow (TpAppt (TpAppt (TpApp (TpVar "7686" "FixIndM") (TpVar "7696" "F")) (Var "7698" "fmap")) (Var "7703" "indices")) NotErased (TpAppt (TpApp (TpVar "7713" "F") (TpParens "7717" (TpAppt (TpApp 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"8363"))) (TpArrow (TpParens "8366" (Abs "8367" Erased "8369" "indices" (Tkt (TpVar "8379" "Indices")) (TpArrow (TpAppt (TpVar "8388" "R") (Var "8390" "indices")) NotErased (TpAppt (TpVar "8400" "A") (Var "8402" "indices")))) "8410") NotErased (TpArrow (TpAppt (TpApp (TpVar "8413" "F") (TpVar "8417" "R")) (Var "8419" "indices")) NotErased (TpVar "8429" "Y")))) "8431") NotErased (TpVar "8438" "Y"))))))) "8440") (CmdsNext (DefTermOrType OpacTrans (DefTerm "8442" "intrCoend" (SomeType (Abs "8458" Erased "8460" "F" (Tkk (KndArrow (KndParens "8464" (KndTpArrow (TpVar "8465" "Indices") (Star "8475")) "8477") (KndTpArrow (TpVar "8480" "Indices") (Star "8490")))) (Abs "8497" Erased "8499" "C" (Tkk (KndTpArrow (TpVar "8503" "Indices") (Star "8513"))) (Abs "8520" Erased "8522" "R" (Tkk (KndTpArrow (TpVar "8526" "Indices") (Star "8536"))) (Abs "8543" Erased "8545" "indices" (Tkt (TpVar "8555" "Indices")) (TpArrow (TpParens "8568" (Abs "8569" Erased "8571" "indices" (Tkt (TpVar "8581" "Indices")) (TpArrow (TpAppt (TpVar "8590" "R") (Var "8592" "indices")) NotErased (TpAppt (TpVar "8602" "C") (Var "8604" "indices")))) "8612") NotErased (TpArrow (TpAppt (TpApp (TpVar "8619" "F") (TpVar "8623" "R")) (Var "8625" "indices")) NotErased (TpAppt (TpApp (TpApp (TpVar "8639" "Coend") (TpVar "8647" "F")) (TpVar "8651" "C")) (Var "8653" "indices"))))))))) (Lam "8665" Erased "8667" "F" NoClass (Lam "8670" Erased "8672" "C" NoClass (Lam "8675" Erased "8677" "R" NoClass (Lam "8680" Erased "8682" "indices" NoClass (Lam "8691" NotErased "8693" "ac" NoClass (Lam "8697" NotErased "8699" "ga" NoClass (Lam "8703" Erased "8705" "Y" NoClass (Lam "8708" NotErased "8710" "q" NoClass (App (App (Var "8713" "q") NotErased (Var "8715" "ac")) NotErased (Var "8718" "ga"))))))))))) "8721") (CmdsNext (DefTermOrType OpacTrans (DefTerm "8723" "elimCoend" (SomeType (Abs "8739" Erased "8741" "F" (Tkk (KndArrow (KndParens "8745" (KndTpArrow (TpVar "8746" "Indices") (Star "8756")) "8758") (KndTpArrow (TpVar "8761" "Indices") (Star "8771")))) (Abs "8778" Erased "8780" "A" (Tkk (KndTpArrow (TpVar "8784" "Indices") (Star "8794"))) (Abs "8801" Erased "8803" "D" (Tkk (Star "8807")) (Abs "8814" Erased "8816" "indices" (Tkt (TpVar "8826" "Indices")) (TpArrow (TpParens "8839" (Abs "8840" Erased "8842" "R" (Tkk (KndTpArrow (TpVar "8846" "Indices") (Star "8856"))) (TpArrow (TpParens "8859" (Abs "8860" Erased "8862" "indices" (Tkt (TpVar "8872" "Indices")) (TpArrow (TpAppt (TpVar "8881" "R") (Var "8883" "indices")) NotErased (TpAppt (TpVar "8893" "A") (Var "8895" "indices")))) "8903") NotErased (TpArrow (TpAppt (TpApp (TpVar "8906" "F") (TpVar "8910" "R")) (Var "8912" "indices")) NotErased (TpVar "8922" "D")))) "8924") NotErased (TpArrow (TpAppt (TpApp (TpApp (TpVar "8931" "Coend") (TpVar "8939" "F")) (TpVar "8943" "A")) (Var "8945" "indices")) NotErased (TpVar "8959" "D")))))))) (Lam "8965" Erased "8967" "F" NoClass (Lam "8970" Erased "8972" "A" NoClass (Lam "8975" Erased "8977" "D" NoClass (Lam "8980" Erased "8982" "indices" NoClass (Lam "8991" NotErased "8993" "phi" NoClass (Lam "8998" NotErased "9000" "e" NoClass (App (Var "9003" "e") NotErased (Var "9005" "phi"))))))))) "9009") (CmdsNext (DefTermOrType OpacTrans (DefType "9011" "CoendInductive" (KndPi "9032" "9034" "F" (Tkk (KndArrow (KndParens "9038" (KndTpArrow (TpVar "9039" "Indices") (Star "9049")) "9051") (KndTpArrow (TpVar "9054" "Indices") (Star "9064")))) (KndPi "9067" "9069" "C" (Tkk (KndTpArrow (TpVar "9073" "Indices") (Star "9083"))) (KndPi "9090" "9092" "indices" (Tkt (TpVar "9102" "Indices")) (KndTpArrow (TpAppt (TpApp (TpApp (TpVar "9111" "Coend") (TpVar "9119" "F")) (TpVar "9123" "C")) (Var "9125" "indices")) (Star "9135"))))) (TpLambda "9141" "9143" "F" (Tkk (KndArrow (KndParens "9147" (KndTpArrow (TpVar "9148" "Indices") (Star "9158")) "9160") (KndTpArrow (TpVar "9163" "Indices") (Star "9173")))) (TpLambda "9176" "9178" "C" (Tkk (KndTpArrow (TpVar "9182" "Indices") (Star "9192"))) (TpLambda "9199" "9201" "indices" (Tkt (TpVar "9211" "Indices")) (TpLambda "9220" "9222" "e" (Tkt (TpAppt (TpApp (TpApp (TpVar "9226" "Coend") (TpVar "9234" "F")) (TpVar "9238" "C")) (Var "9240" "indices"))) (Abs "9255" Erased "9257" "Q" (Tkk (KndPi "9261" "9263" "indices" (Tkt (TpVar "9273" "Indices")) (KndTpArrow (TpAppt (TpApp (TpApp (TpVar "9282" "Coend") (TpVar "9290" "F")) (TpVar "9294" "C")) (Var "9296" "indices")) (Star "9306")))) (TpArrow (TpParens "9315" (Abs "9316" Erased "9318" "R" (Tkk (KndTpArrow (TpVar "9322" "Indices") (Star "9332"))) (Abs "9335" Erased "9337" "indices" (Tkt (TpVar "9347" "Indices")) (Abs "9356" NotErased "9358" "c" (Tkt (TpApp (TpApp (TpVar "9362" "Cast") (TpVar "9369" "R")) (TpVar "9373" "C"))) (Abs "9384" NotErased "9386" "gr" (Tkt (TpAppt (TpApp (TpVar "9391" "F") (TpVar "9395" "R")) (Var "9397" "indices"))) (TpAppt (TpAppt (TpVar "9406" "Q") (Var "9408" "indices")) (Parens "9416" (App (App (App (AppTp (AppTp (AppTp (Var "9417" "intrCoend") (TpVar "9429" "F")) (TpVar "9433" "C")) (TpVar "9437" "R")) Erased (Var "9440" "indices")) NotErased (Parens "9448" (App (Var "9449" "cast") Erased (Var "9455" "c")) "9457")) NotErased (Var "9458" "gr")) "9461")))))) "9462") NotErased (TpAppt (TpAppt (TpVar "9471" "Q") (Var "9473" "indices")) (Var "9481" "e"))))))))) "9483") (CmdsNext (DefTermOrType OpacTrans (DefType "9485" "CoendInd" (KndArrow (KndParens "9496" (KndArrow (KndParens "9497" (KndTpArrow (TpVar "9498" "Indices") (Star "9508")) "9510") (KndTpArrow (TpVar "9513" "Indices") (Star "9523"))) "9525") (KndArrow (KndParens "9528" (KndTpArrow (TpVar "9529" "Indices") (Star "9539")) "9541") (KndTpArrow (TpVar "9544" "Indices") (Star "9554")))) (TpLambda "9560" "9562" "G" (Tkk (KndArrow (KndParens "9566" (KndTpArrow (TpVar "9567" "Indices") (Star "9577")) "9579") (KndTpArrow (TpVar "9582" "Indices") (Star "9592")))) (TpLambda "9595" "9597" "C" (Tkk (KndTpArrow (TpVar "9601" "Indices") (Star "9611"))) (TpLambda "9614" "9616" "indices" (Tkt (TpVar "9626" "Indices")) (Iota "9639" "9641" "x" (TpAppt (TpApp (TpApp (TpVar "9645" "Coend") (TpVar "9653" "G")) (TpVar "9657" "C")) (Var "9659" "indices")) (TpAppt (TpAppt (TpApp (TpApp (TpVar "9668" "CoendInductive") (TpVar "9686" "G")) (TpVar "9690" "C")) (Var "9692" "indices")) (Var "9700" "x"))))))) "9702") (CmdsNext (DefTermOrType OpacTrans (DefTerm "9705" "intrCoendInd" (SomeType (Abs "9724" Erased "9726" "F" (Tkk (KndArrow (KndParens "9730" (KndTpArrow (TpVar "9731" "Indices") (Star "9741")) "9743") (KndTpArrow (TpVar "9746" "Indices") (Star "9756")))) (Abs "9763" Erased "9765" "C" (Tkk (KndTpArrow (TpVar "9769" "Indices") (Star "9779"))) (Abs "9782" Erased "9784" "R" (Tkk (KndTpArrow (TpVar "9788" "Indices") (Star "9798"))) (Abs "9801" Erased "9803" "indices" (Tkt (TpVar "9813" "Indices")) (TpArrow (TpApp (TpApp (TpVar "9826" "Cast") (TpVar "9833" "R")) (TpVar "9837" "C")) Erased (TpArrow (TpAppt (TpApp (TpVar "9841" "F") (TpVar "9845" "R")) (Var "9847" "indices")) NotErased (TpAppt (TpApp (TpApp (TpVar "9857" "CoendInd") (TpVar "9868" "F")) (TpVar "9872" "C")) (Var "9874" "indices"))))))))) (Lam "9886" Erased "9888" "F" NoClass (Lam "9891" Erased "9893" "C" NoClass (Lam "9896" Erased "9898" "R" NoClass (Lam "9901" Erased "9903" "indices" NoClass (Lam "9912" Erased "9914" "f" NoClass (Lam "9917" NotErased "9919" "gr" NoClass (IotaPair "9927" (App (App (App (AppTp (AppTp (AppTp (Var "9928" "intrCoend") (TpVar "9940" "F")) (TpVar "9944" "C")) (TpVar "9948" "R")) Erased (Var "9951" "indices")) NotErased (Parens "9959" (App (Var "9960" "cast") Erased (Var "9966" "f")) "9968")) NotErased (Var "9969" "gr")) (Lam "9973" Erased "9975" "Q" NoClass (Lam "9978" NotErased "9980" "q" NoClass (App (App (App (AppTp (Var "9983" "q") (TpVar "9987" "R")) Erased (Var "9990" "indices")) NotErased (IotaPair "9998" (App (Var "9999" "cast") Erased (Var "10005" "f")) (Beta "10008" NoTerm NoTerm) NoGuide "10010")) NotErased (Var "10011" "gr")))) NoGuide "10014")))))))) "10015") (CmdsNext (DefTermOrType OpacTrans (DefTerm "10018" "indCoend'" (SomeType (Abs "10034" Erased "10036" "F" (Tkk (KndArrow (KndParens "10040" (KndTpArrow (TpVar "10041" "Indices") (Star "10051")) "10053") (KndTpArrow (TpVar "10056" "Indices") (Star "10066")))) (Abs "10069" Erased "10071" "C" (Tkk (KndTpArrow (TpVar "10075" "Indices") (Star "10085"))) (Abs "10088" Erased "10090" "indices" (Tkt (TpVar "10100" "Indices")) (Abs "10113" NotErased "10115" "e" (Tkt (TpAppt (TpApp (TpApp (TpVar "10119" "CoendInd") (TpVar "10130" "F")) (TpVar "10134" "C")) (Var "10136" "indices"))) (Abs "10149" Erased "10151" "Q" (Tkk (KndPi "10155" "10157" "indices" (Tkt (TpVar "10167" "Indices")) (KndTpArrow (TpAppt (TpApp (TpApp (TpVar "10176" "CoendInd") (TpVar "10187" "F")) (TpVar "10191" "C")) (Var "10193" "indices")) (Star "10203")))) (TpArrow (TpParens "10210" (Abs "10211" Erased "10213" "R" (Tkk (KndTpArrow (TpVar "10217" "Indices") (Star "10227"))) (Abs "10230" Erased "10232" "indices" (Tkt (TpVar "10242" "Indices")) (Abs "10251" Erased "10253" "c" (Tkt (TpApp (TpApp (TpVar "10257" "Cast") (TpVar "10264" "R")) (TpVar "10268" "C"))) (Abs "10277" NotErased "10279" "gr" (Tkt (TpAppt (TpApp (TpVar "10284" "F") (TpVar "10288" "R")) (Var "10290" "indices"))) (TpAppt (TpAppt (TpVar "10299" "Q") (Var "10301" "indices")) (Parens "10309" (App (App (App (AppTp (AppTp (AppTp (Var "10310" "intrCoendInd") (TpVar "10325" "F")) (TpVar "10329" "C")) (TpVar "10333" "R")) Erased (Var "10336" "indices")) Erased (Var "10345" "c")) NotErased (Var "10347" "gr")) "10350")))))) "10351") NotErased (Abs "10358" Erased "10360" "X" (Tkk (KndPi "10364" "10366" "indices" (Tkt (TpVar "10376" "Indices")) (KndTpArrow (TpAppt (TpApp (TpApp (TpVar "10385" "CoendInd") (TpVar "10396" "F")) (TpVar "10400" "C")) (Var "10402" "indices")) (Star "10412")))) (TpArrow (TpParens "10419" (Abs "10420" Erased "10422" "indices" (Tkt (TpVar "10432" "Indices")) (Abs "10441" Erased "10443" "x'" (Tkt (TpAppt (TpApp (TpApp (TpVar "10448" "CoendInd") (TpVar "10459" "F")) (TpVar "10463" "C")) (Var "10465" "indices"))) (TpArrow (TpAppt (TpAppt (TpVar "10474" "Q") (Var "10476" "indices")) (Var "10484" "x'")) NotErased (TpAppt (TpAppt (TpVar "10489" "X") (Var "10491" "indices")) (Var "10499" "x'"))))) "10502") NotErased (TpAppt (TpAppt (TpVar "10509" "X") (Var "10511" "indices")) (Var "10519" "e"))))))))))) (Lam "10525" Erased "10527" "F" NoClass (Lam "10530" Erased "10532" "C" NoClass (Lam "10535" Erased "10537" "indices" NoClass (Lam "10546" NotErased "10548" "e" NoClass (Lam "10551" Erased "10553" "Q" NoClass (Lam "10556" NotErased "10558" "q" NoClass (Theta "10561" (AbstractVars (VarsNext "indices" (VarsStart "e"))) (IotaProj (Var "10574" "e") "2" "10577") (LtermsCons NotErased (Parens "10582" (Lam "10583" Erased "10585" "R" NoClass (Lam "10588" Erased "10590" "indices" NoClass (Lam "10599" NotErased "10601" "ar" NoClass (Lam "10605" NotErased "10607" "gr" NoClass (Lam "10611" Erased "10613" "X" NoClass (Lam "10616" NotErased "10618" "qq" NoClass (App (App (App (Var "10622" "qq") Erased (Var "10626" "indices")) Erased (Parens "10641" (App (App (App (AppTp (AppTp (AppTp (Var "10642" "intrCoendInd") (TpVar "10657" "F")) (TpVar "10661" "C")) (TpVar "10665" "R")) Erased (Var "10668" "indices")) Erased (Var "10677" "ar")) NotErased (Var "10680" "gr")) "10683")) NotErased (Parens "10684" (App (App (App (AppTp (Var "10685" "q") (TpVar "10689" "R")) Erased (Var "10692" "indices")) Erased (Var "10701" "ar")) NotErased (Var "10704" "gr")) "10707")))))))) "10708") (LtermsNil "10707")))))))))) "10709") (CmdsNext (DefTermOrType OpacTrans (DefTerm "10712" "indCoend" (SomeType (Abs "10727" Erased "10729" "F" (Tkk (KndArrow (KndParens "10733" (KndTpArrow (TpVar "10734" "Indices") (Star "10744")) "10746") (KndTpArrow (TpVar "10749" "Indices") (Star "10759")))) (Abs "10762" Erased "10764" "C" (Tkk (KndTpArrow (TpVar "10768" "Indices") (Star "10778"))) (Abs "10781" Erased "10783" "indices" (Tkt (TpVar "10793" "Indices")) (Abs "10806" NotErased "10808" "e" (Tkt (TpAppt (TpApp (TpApp (TpVar "10812" "CoendInd") (TpVar "10823" "F")) (TpVar "10827" "C")) (Var "10829" "indices"))) (Abs "10842" Erased "10844" "Q" (Tkk (KndPi "10848" "10850" "indices" (Tkt (TpVar "10860" "Indices")) (KndTpArrow (TpAppt (TpApp (TpApp (TpVar "10869" "CoendInd") (TpVar "10880" "F")) (TpVar "10884" "C")) (Var "10886" "indices")) (Star "10896")))) (TpArrow (TpParens "10903" (Abs "10904" Erased "10906" "R" (Tkk (KndTpArrow (TpVar "10910" "Indices") (Star "10920"))) (Abs "10923" Erased "10925" "indices" (Tkt (TpVar "10935" "Indices")) (Abs "10944" Erased "10946" "c" (Tkt (TpApp (TpApp (TpVar "10950" "Cast") (TpVar "10957" "R")) (TpVar "10961" "C"))) (Abs "10970" NotErased "10972" "gr" (Tkt (TpAppt (TpApp (TpVar "10977" "F") (TpVar "10981" "R")) (Var "10983" "indices"))) (TpAppt (TpAppt (TpVar "10992" "Q") (Var "10994" "indices")) (Parens "11002" (App (App (App (AppTp (AppTp (AppTp (Var "11003" "intrCoendInd") (TpVar 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OpacTrans (DefTerm "11174" "fmapCoend" (SomeType (Abs "11186" Erased "11188" "F" (Tkk (KndArrow (KndParens "11192" (KndTpArrow (TpVar "11193" "Indices") (Star "11203")) "11205") (KndTpArrow (TpVar "11208" "Indices") (Star "11218")))) (TpApp (TpVar "11221" "Functor") (TpParens "11231" (TpApp (TpVar "11232" "CoendInd") (TpVar "11243" "F")) "11245")))) (Lam "11250" Erased "11252" "F" NoClass (Lam "11255" Erased "11257" "A" NoClass (Lam "11260" Erased "11262" "B" NoClass (Lam "11265" Erased "11267" "f" NoClass (IotaPair "11274" (Lam "11275" Erased "11277" "indices" NoClass (Lam "11286" NotErased "11288" "c" NoClass (Phi "11297" (Parens "11299" (Theta "11300" (AbstractVars (VarsNext "indices" (VarsStart "c"))) (Parens "11313" (App (App (AppTp (AppTp (Var "11314" "indCoend") (TpVar "11325" "F")) (TpVar "11329" "A")) Erased (Var "11332" "indices")) NotErased (Var "11340" "c")) "11342") (LtermsCons NotErased (Parens "11343" (Lam "11344" Erased "11346" "R" NoClass (Lam "11349" Erased "11351" 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NotErased (Parens "11550" (App (App (App (Var "11551" "cast") Erased (Var "11557" "i")) Erased (Var "11560" "indices")) NotErased (Var "11568" "r")) "11570")))) (Beta "11572" NoTerm NoTerm) NoGuide "11574")) NotErased (Var "11575" "gr")))))) "11578") (LtermsNil "11578"))) (Var "11580" "c") "11582"))) (Beta "11584" NoTerm NoTerm) NoGuide "11586")))))) "11587") (CmdsNext (DefTermOrType OpacTrans (DefType "11590" "AlgCVM" (KndArrow (KndParens "11599" (KndArrow (KndParens "11600" (KndTpArrow (TpVar "11601" "Indices") (Star "11611")) "11613") (KndTpArrow (TpVar "11616" "Indices") (Star "11626"))) "11628") (KndArrow (Star "11631") (KndTpArrow (TpVar "11635" "Indices") (Star "11645")))) (TpLambda "11651" "11653" "F" (Tkk (KndArrow (KndParens "11657" (KndTpArrow (TpVar "11658" "Indices") (Star "11668")) "11670") (KndTpArrow (TpVar "11673" "Indices") (Star "11683")))) (TpLambda "11686" "11688" "X" (Tkk (Star "11692")) (TpLambda "11695" "11697" "indices" (Tkt (TpVar "11707" "Indices")) (Abs "11720" Erased "11722" "R" (Tkk (KndTpArrow (TpVar "11726" "Indices") (Star "11736"))) (TpArrow (TpParens "11739" (TpArrow (TpAppt (TpVar "11740" "R") (Var "11742" "indices")) NotErased (TpAppt (TpApp (TpVar "11752" "F") (TpVar "11756" "R")) (Var "11758" "indices"))) "11766") NotErased (TpArrow (TpParens "11769" (TpArrow (TpAppt (TpVar "11770" "R") (Var "11772" "indices")) NotErased (TpVar "11782" "X")) "11784") NotErased (TpArrow (TpAppt (TpApp (TpVar "11787" "F") (TpVar "11791" "R")) (Var "11793" "indices")) NotErased (TpVar "11803" "X"))))))))) "11805") (CmdsNext (DefTermOrType OpacTrans (DefType "11808" "FixCVM" (KndArrow (KndParens "11817" (KndArrow (KndParens "11818" (KndTpArrow (TpVar "11819" "Indices") (Star "11829")) "11831") (KndTpArrow (TpVar "11834" "Indices") (Star "11844"))) "11846") (KndTpArrow (TpVar "11849" "Indices") (Star "11859"))) (TpLambda "11865" "11867" "F" (Tkk (KndArrow (KndParens "11871" (KndTpArrow (TpVar "11872" "Indices") (Star "11882")) "11884") (KndTpArrow (TpVar "11887" "Indices") (Star "11897")))) (TpLambda "11900" "11902" "indices" (Tkt (TpVar "11912" "Indices")) (Abs "11925" Erased "11927" "R" (Tkk (Star "11931")) (TpArrow (TpAppt (TpApp (TpApp (TpVar "11934" "AlgCVM") (TpVar "11943" "F")) (TpVar "11947" "R")) (Var "11949" "indices")) NotErased (TpVar "11959" "R")))))) "11961") (CmdsNext (DefTermOrType OpacTrans (DefTerm "11963" "foldCVM" (SomeType (Abs "11977" Erased "11979" "F" (Tkk (KndArrow (KndParens "11983" (KndTpArrow (TpVar "11984" "Indices") (Star "11994")) "11996") (KndTpArrow (TpVar "11999" "Indices") (Star "12009")))) (Abs "12012" Erased "12014" "A" (Tkk (Star "12018")) (Abs "12021" Erased "12023" "indices" (Tkt (TpVar "12033" "Indices")) (TpArrow (TpAppt (TpApp (TpApp (TpVar "12046" "AlgCVM") (TpVar "12055" "F")) (TpVar "12059" "A")) (Var "12061" "indices")) NotErased (TpArrow (TpAppt (TpApp (TpVar "12071" "FixCVM") (TpVar "12080" "F")) (Var "12082" "indices")) NotErased (TpVar "12092" "A"))))))) (Lam "12098" Erased "12100" "F" NoClass (Lam "12103" Erased "12105" "A" NoClass (Lam "12108" Erased "12110" "indices" NoClass (Lam "12119" NotErased "12121" "alg" NoClass (Lam "12126" NotErased "12128" "fix" NoClass (App (Var "12133" "fix") NotErased (Var "12137" "alg")))))))) "12141") (CmdsNext (DefTermOrType OpacTrans (DefType "12143" "CVProduct'" (KndArrow (KndParens "12157" (KndArrow (KndParens "12158" (KndTpArrow (TpVar "12159" "Indices") (Star "12169")) "12171") (KndTpArrow (TpVar "12174" "Indices") (Star "12184"))) "12186") (KndArrow (KndParens "12189" (KndTpArrow (TpVar "12190" "Indices") (Star "12200")) "12202") (KndTpArrow (TpVar "12205" "Indices") (Star "12215")))) (TpLambda "12221" "12223" "F" (Tkk (KndArrow (KndParens "12227" (KndTpArrow (TpVar "12228" "Indices") (Star "12238")) "12240") (KndTpArrow (TpVar "12243" "Indices") (Star "12253")))) (TpLambda "12256" "12258" "X" (Tkk (KndTpArrow (TpVar "12262" "Indices") (Star "12272"))) (TpLambda "12275" "12277" "indices" (Tkt (TpVar "12287" "Indices")) (TpApp (TpApp (TpVar "12300" "Sigma") (TpParens "12308" (TpAppt (TpApp (TpVar "12309" "F") (TpVar "12313" "X")) (Var "12315" "indices")) "12323")) (TpParens "12326" (TpLambda "12327" "12329" "_" (Tkt (TpAppt (TpApp (TpVar "12333" "F") (TpVar "12337" "X")) (Var "12339" "indices"))) (TpArrow (TpAppt (TpVar "12348" "X") (Var "12350" "indices")) NotErased (TpAppt (TpApp (TpVar "12360" "F") (TpVar "12364" "X")) (Var "12366" "indices")))) "12374")))))) "12375") (CmdsNext (DefTermOrType OpacTrans (DefType "12376" "CVF'" (KndArrow (KndParens "12389" (KndArrow (KndParens "12390" (KndTpArrow (TpVar "12391" "Indices") (Star "12401")) "12403") (KndTpArrow (TpVar "12406" "Indices") (Star "12416"))) "12418") (KndArrow (KndParens "12421" (KndTpArrow (TpVar "12422" "Indices") (Star "12432")) "12434") (KndTpArrow (TpVar "12437" "Indices") (Star "12447")))) (TpLambda "12453" "12455" "F" (Tkk (KndArrow (KndParens "12459" (KndTpArrow (TpVar "12460" "Indices") (Star "12470")) "12472") 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(TpAppt (TpApp (TpVar "13135" "F") (TpParens "13139" (TpApp (TpVar "13140" "CVFixIndM'") (TpVar "13153" "F")) "13155")) (Var "13156" "indices")))) "13164")) NotErased (Parens "13166" (Lam "13167" Erased "13169" "R" NoClass (Lam "13172" Erased "13174" "indices" NoClass (Lam "13183" Erased "13185" "c" NoClass (Lam "13188" NotErased "13190" "v" NoClass (App (App (App (Var "13193" "cast") Erased (Parens "13199" (App (Var "13200" "fmap") Erased (Var "13206" "c")) "13208")) Erased (Var "13210" "indices")) NotErased (Parens "13218" (App (Var "13219" "fst") NotErased (Var "13223" "v")) "13225")))))) "13226"))))))) "13227") (CmdsNext (DefTermOrType OpacTrans (DefType "13229" "CVProduct" (KndArrow (KndParens "13241" (KndArrow (KndParens "13242" (KndTpArrow (TpVar "13243" "Indices") (Star "13253")) "13255") (KndTpArrow (TpVar "13258" "Indices") (Star "13268"))) "13270") (KndArrow (KndParens "13273" (KndTpArrow (TpVar "13274" "Indices") (Star "13284")) "13286") (KndTpArrow (TpVar "13289" 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"13930" "F")) (Var "13932" "fmap")) (Var "13937" "indices")) NotErased (TpAppt (TpApp (TpVar "13947" "F") (TpParens "13951" (TpAppt (TpApp (TpVar "13952" "CVFixIndM") (TpVar "13964" "F")) (Var "13966" "fmap")) "13971")) (Var "13972" "indices"))))))) (Lam "13984" Erased "13986" "F" NoClass (Lam "13989" Erased "13991" "fmap" NoClass (Lam "13997" Erased "13999" "indices" NoClass (Lam "14008" NotErased "14010" "v" NoClass (App (AppTp (App (App (AppTp (AppTp (Var "14013" "indCoend") (TpParens "14028" (TpApp (TpVar "14029" "CVProduct") (TpVar "14041" "F")) "14043")) (TpParens "14050" (TpAppt (TpApp (TpVar "14051" "FixIndM") (TpParens "14061" (TpApp (TpVar "14062" "CoendInd") (TpParens "14073" (TpApp (TpVar "14074" "CVProduct") (TpVar "14086" "F")) "14088")) "14089")) (Parens "14090" (AppTp (Var "14091" "fmapCoend") (TpParens "14103" (TpApp (TpVar "14104" "CVProduct") (TpVar "14116" "F")) "14118")) "14119")) "14120")) Erased (Var "14126" "indices")) NotErased (Parens "14138" (App (App (App (Var "14139" "outFixIndM") Erased (Parens "14151" (AppTp (Var "14152" "fmapCoend") (TpParens "14164" (TpApp (TpVar "14165" "CVProduct") (TpVar "14177" "F")) "14179")) "14180")) Erased (Var "14182" "indices")) NotErased (Var "14190" "v")) "14192")) (TpParens "14199" (TpLambda "14200" "14202" "indices" (Tkt (TpVar "14212" "Indices")) (TpLambda "14221" "14223" "_" (Tkt (TpAppt (TpApp (TpApp (TpVar "14227" "CoendInd") (TpParens "14238" (TpApp (TpVar "14239" "CVProduct") (TpVar "14251" "F")) "14253")) (TpParens "14256" (TpAppt (TpApp (TpVar "14257" "CVFixIndM") (TpVar "14269" "F")) (Var "14271" "fmap")) "14276")) (Var "14277" "indices"))) (TpAppt (TpApp (TpVar "14294" "F") (TpParens "14298" (TpAppt (TpApp (TpVar "14299" "CVFixIndM") (TpVar "14311" "F")) (Var "14313" "fmap")) "14318")) (Var "14319" "indices")))) "14327")) NotErased (Parens "14332" (Lam "14333" Erased "14335" "R" NoClass (Lam "14338" Erased "14340" "indices" NoClass (Lam "14349" Erased "14351" "c" NoClass (Lam "14354" 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"14967" "14969" "indices" (Tkt (TpVar "14979" "Indices")) (KndTpArrow (TpAppt (TpAppt (TpApp (TpVar "14988" "CVFixIndM") (TpVar "15000" "F")) (Var "15002" "fmap")) (Var "15007" "indices")) (Star "15017")))) (TpArrow (TpParens "15024" (Abs "15025" Erased "15027" "R" (Tkk (KndTpArrow (TpVar "15031" "Indices") (Star "15041"))) (Abs "15044" Erased "15046" "c" (Tkt (TpApp (TpApp (TpVar "15050" "Cast") (TpVar "15057" "R")) (TpParens "15061" (TpAppt (TpApp (TpVar "15062" "CVFixIndM") (TpVar "15074" "F")) (Var "15076" "fmap")) "15081"))) (Abs "15088" Erased "15090" "out" (Tkt (Abs "15096" Erased "15098" "indices" (Tkt (TpVar "15108" "Indices")) (TpArrow (TpAppt (TpVar "15117" "R") (Var "15119" "indices")) NotErased (TpAppt (TpApp (TpVar "15129" "F") (TpVar "15133" "R")) (Var "15135" "indices"))))) (TpArrow (TpEq "15144" (Var "15145" "out") (Var "15151" "cvOutFixIndM") "15164") Erased (TpArrow (TpParens "15172" (Abs "15173" Erased "15175" "indices" (Tkt (TpVar "15185" "Indices")) (Abs "15194" NotErased "15196" "r" (Tkt (TpAppt (TpVar "15200" "R") (Var "15202" "indices"))) (TpAppt (TpAppt (TpVar "15211" "Q") (Var "15213" "indices")) (Parens "15221" (App (App (App (Var "15222" "cast") Erased (Var "15228" "c")) Erased (Var "15231" "indices")) NotErased (Var "15239" "r")) "15241")))) "15242") NotErased (Abs "15250" Erased "15252" "indices" (Tkt (TpVar "15262" "Indices")) (Abs "15271" NotErased "15273" "gr" (Tkt (TpAppt (TpApp (TpVar "15278" "F") (TpVar "15282" "R")) (Var "15284" "indices"))) (TpAppt (TpAppt (TpVar "15298" "Q") (Var "15300" "indices")) (Parens "15308" (App (App (App (Var "15309" "cvInFixIndM") Erased (Var "15322" "fmap")) Erased (Var "15328" "indices")) NotErased (Parens "15336" (App (App (App (Var "15337" "cast") Erased (Parens "15343" (App (Var "15344" "fmap") Erased (Var "15350" "c")) "15352")) Erased (Var "15354" "indices")) NotErased (Var "15362" "gr")) "15365")) "15366"))))))))) "15367") NotErased (TpAppt (TpAppt (TpVar "15374" "Q") (Var "15376" 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"indices" (Tkt (TpVar "632" "Indices")) (Abs "645" Erased "647" "X" (Tkk (KndTpArrow (TpVar "651" "Indices") (Star "661"))) (TpArrow (TpApp (TpApp (TpVar "664" "Cast") (TpParens "671" (TpApp (TpVar "672" "F") (TpVar "676" "X")) "678")) (TpVar "681" "X")) Erased (TpAppt (TpVar "685" "X") (Var "687" "indices"))))))) "695") (CmdsNext (DefTermOrType OpacTrans (DefTerm "698" "recCast" (SomeType (Abs "708" Erased "710" "F" (Tkk (KndArrow (KndParens "714" (KndTpArrow (TpVar "715" "Indices") (Star "725")) "727") (KndTpArrow (TpVar "730" "Indices") (Star "740")))) (Abs "743" Erased "745" "X" (Tkk (KndTpArrow (TpVar "749" "Indices") (Star "759"))) (TpArrow (TpApp (TpApp (TpVar "766" "Cast") (TpParens "773" (TpApp (TpVar "774" "F") (TpVar "778" "X")) "780")) (TpVar "783" "X")) Erased (TpApp (TpApp (TpVar "787" "Cast") (TpParens "794" (TpApp (TpVar "795" "Rec") (TpVar "801" "F")) "803")) (TpVar "806" "X")))))) (Lam "812" Erased "814" "F" NoClass (Lam "817" Erased "819" "X" NoClass (Lam "822" Erased "824" "c" NoClass (IotaPair "827" (Lam "828" Erased "830" "indices" NoClass (Lam "839" NotErased "841" "a" NoClass (App (AppTp (Var "844" "a") (TpVar "848" "X")) Erased (Var "851" "c")))) (Beta "854" NoTerm NoTerm) NoGuide "856"))))) "857") (CmdsNext (DefTermOrType OpacTrans (DefTerm "859" "recIn" (SomeType (Abs "867" Erased "869" "F" (Tkk (KndArrow (KndParens "873" (KndTpArrow (TpVar "874" "Indices") (Star "884")) "886") (KndTpArrow (TpVar "889" "Indices") (Star "899")))) (TpArrow (TpApp (TpVar "902" "RecFunctor") (TpVar "915" "F")) Erased (TpApp (TpApp (TpVar "923" "Cast") (TpParens "930" (TpApp (TpVar "931" "F") (TpParens "935" (TpApp (TpVar "936" "Rec") (TpVar "942" "F")) "944")) "945")) (TpParens "948" (TpApp (TpVar "949" "Rec") (TpVar "955" "F")) "957"))))) (Lam "962" Erased "964" "F" NoClass (Lam "967" Erased "969" "fmap" NoClass (IotaPair "979" (Lam "980" Erased "982" "indices" NoClass (Lam "991" NotErased "993" "a" NoClass (Lam "996" Erased "998" "X" NoClass (Lam "1001" 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NotErased "1639" "d" NoClass (App (Var "1642" "d") NotErased (Parens "1644" (Lam "1645" NotErased "1647" "e" NoClass (Lam "1650" NotErased "1652" "p" NoClass (Lam "1655" NotErased "1657" "d" NoClass (Var "1660" "d")))) "1662"))) "1663")) "1664") NotErased (TpArrow (TpParens "1676" (Iota "1677" "1679" "ih" (Abs "1684" Erased "1686" "indices" (Tkt (TpVar "1696" "Indices")) (Abs "1705" NotErased "1707" "r" (Tkt (TpAppt (TpVar "1711" "R") (Var "1713" "indices"))) (TpAppt (TpAppt (TpVar "1722" "X") (Beta "1724" NoTerm (SomeTerm (Var "1726" "r") "1728"))) (Var "1729" "indices")))) (TpEq "1738" (Var "1739" "ih") (Lam "1744" NotErased "1746" "d" NoClass (App (Var "1749" "d") NotErased (Var "1751" "alg"))) "1755")) "1756") NotErased (Abs "1768" Erased "1770" "indices" (Tkt (TpVar "1780" "Indices")) (Abs "1798" NotErased "1800" "x" (Tkt (TpAppt (TpApp (TpVar "1804" "F") (TpVar "1808" "R")) (Var "1810" "indices"))) (TpAppt (TpAppt (TpVar "1828" "X") (Beta "1830" NoTerm (SomeTerm (Lam "1832" NotErased "1834" "alg" NoClass (App (App (App (Var "1839" "alg") NotErased (Parens "1843" (Lam "1844" NotErased "1846" "d" NoClass (App (Var "1849" "d") NotErased (Parens "1851" (Lam "1852" NotErased "1854" "p" NoClass (Lam "1857" NotErased "1859" "e" NoClass (Lam "1862" NotErased "1864" "d" NoClass (Var "1867" "d")))) "1869"))) "1870")) NotErased (Parens "1871" (Lam "1872" NotErased "1874" "d" NoClass (App (Var "1877" "d") NotErased (Var "1879" "alg"))) "1883")) NotErased (Var "1884" "x"))) "1886"))) (Var "1887" "indices"))))))))) "1895") NotErased (TpAppt (TpAppt (TpVar "1906" "X") (Var "1908" "x")) (Var "1910" "indices")))))))))) "1918") (CmdsNext (DefTermOrType OpacTrans (DefType "1920" "FixM" (KndPi "1927" "1929" "F" (Tkk (KndArrow (KndParens "1933" (KndTpArrow (TpVar "1934" "Indices") (Star "1944")) "1946") (KndTpArrow (TpVar "1949" "Indices") (Star "1959")))) (KndTpArrow (TpApp (TpVar "1962" "RecFunctor") (TpVar "1975" "F")) (KndTpArrow (TpVar "1979" "Indices") (Star "1989")))) (TpLambda "1995" "1997" "F" (Tkk (KndArrow (KndParens "2001" (KndTpArrow (TpVar "2002" "Indices") (Star "2012")) "2014") (KndTpArrow (TpVar "2017" "Indices") (Star "2027")))) (TpLambda "2030" "2032" "fm" (Tkt (TpApp (TpVar "2037" "RecFunctor") (TpVar "2050" "F"))) (TpApp (TpVar "2053" "Rec") (TpParens "2059" (TpAppt (TpApp (TpVar "2060" "FixMF") (TpVar "2068" "F")) (Var "2070" "fm")) "2073"))))) "2074") (CmdsNext (DefTermOrType OpacTrans (DefTerm "2076" "FixFmap" (SomeType (Abs "2090" Erased "2092" "F" (Tkk (KndArrow (KndParens "2096" (KndTpArrow (TpVar "2097" "Indices") (Star "2107")) "2109") (KndTpArrow (TpVar "2112" "Indices") (Star "2122")))) (Abs "2125" Erased "2127" "fm" (Tkt (TpApp (TpVar "2132" "RecFunctor") (TpVar "2145" "F"))) (TpApp (TpVar "2148" "RecFunctor") (TpParens "2161" (TpAppt (TpApp (TpVar "2162" "FixMF") (TpVar "2170" "F")) (Var "2172" "fm")) "2175"))))) (Lam "2180" Erased "2182" "F" NoClass (Lam "2185" Erased "2187" "fm" NoClass (Lam "2191" Erased "2193" "D" 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Agda

src/Everything.agda

andreasabel/cubical

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[ "MIT" ]

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src/Everything.agda

andreasabel/cubical

914f655c7c0417754c2ffe494d3f6ea7a357b1c3

[ "MIT" ]

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src/Everything.agda

andreasabel/cubical

914f655c7c0417754c2ffe494d3f6ea7a357b1c3

[ "MIT" ]

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module Everything where import Control.Category import Control.Category.Functor import Control.Category.Product import Control.Category.SetsAndFunctions import Control.Category.Slice import Control.Decoration import Control.Functor import Control.Functor.NaturalTransformation import Control.Functor.Product import Control.Kleisli import Control.Lens import Control.Monad import Control.Monad.Error import Control.Monad.KleisliTriple import Control.Comonad import Control.Comonad.Store import Dimension.PartialWeakening import Dimension.PartialWeakening.Model -- import Dimension.PartialWeakening.Soundness -- still broken

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Agda

test/Succeed/ProjectionLikeAndModules1.agda

mdimjasevic/agda

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[ "BSD-3-Clause" ]

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2015-01-09T23:51:16.000Z

2022-03-30T18:20:48.000Z

test/Succeed/ProjectionLikeAndModules1.agda

Seanpm2001-languages/agda

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2022-03-31T21:14:49.000Z

test/Succeed/ProjectionLikeAndModules1.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

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2015-01-03T14:04:08.000Z

2022-03-30T19:00:30.000Z

-- {-# OPTIONS -v tc.proj.like:10 #-} {-# OPTIONS -v tc.conv:10 #-} import Common.Level module ProjectionLikeAndModules1 (A : Set) (a : A) where record ⊤ : Set where constructor tt data Wrap (W : Set) : Set where wrap : W → Wrap W data Bool : Set where true false : Bool -- `or' should be projection like in the module parameters if : Bool → {B : Set} → B → B → B if true a b = a if false a b = b postulate u v : ⊤ P : Wrap ⊤ -> Set test : (y : Bool) -> P (if y (wrap u) (wrap tt)) -> P (if y (wrap tt) (wrap v)) test y h = h -- Error: -- u != tt of type Set -- when checking that the expression h has type -- P (if y (wrap tt) (wrap v))

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Agda

src/Data/Vec/Membership/Propositional/Distinct/Properties.agda

tizmd/agda-distinct-disjoint

d4cd2a3442a9b58e6139499d16a2b31268f27f80

[ "MIT" ]

null

src/Data/Vec/Membership/Propositional/Distinct/Properties.agda

tizmd/agda-distinct-disjoint

d4cd2a3442a9b58e6139499d16a2b31268f27f80

[ "MIT" ]

null

src/Data/Vec/Membership/Propositional/Distinct/Properties.agda

tizmd/agda-distinct-disjoint

d4cd2a3442a9b58e6139499d16a2b31268f27f80

[ "MIT" ]

null

module Data.Vec.Membership.Propositional.Distinct.Properties where open import Data.Fin as Fin open import Relation.Binary.PropositionalEquality as P open import Data.Vec as Vec using (Vec; [] ; _∷_ ; _++_) open import Data.Vec.Any open import Data.Vec.Membership.Propositional.Distinct open import Data.Vec.Membership.Propositional.Disjoint renaming (Disjoint to _⋈_) open import Data.Vec.Membership.Propositional.Properties open import Data.Vec.Membership.Propositional open import Data.Product open import Data.Empty using (⊥-elim) open import Function using (_∘_) open import Function.Equivalence using (_⇔_; equivalence) distinct-++ˡ : ∀ {a}{A : Set a}{m n} (xs : Vec A m){ys : Vec A n} → Distinct (xs ++ ys) → Distinct xs distinct-++ˡ [] dis = distinct-[] distinct-++ˡ (x ∷ xs) (.x distinct-∷ dis by x∉xsys) = x distinct-∷ distinct-++ˡ xs dis by λ x∈xs → x∉xsys (∈-++⁺ˡ x∈xs) distinct-++ʳ : ∀ {a}{A : Set a}{m n} (xs : Vec A m) {ys : Vec A n} → Distinct (xs ++ ys) → Distinct ys distinct-++ʳ [] dys = dys distinct-++ʳ (x ∷ xs) (.x distinct-∷ dxsys by _) = distinct-++ʳ xs dxsys distinct-++→disjoint : ∀ {a}{A : Set a}{m n} (xs : Vec A m) {ys : Vec A n} → Distinct (xs ++ ys) → xs ⋈ ys distinct-++→disjoint [] dxsys {z} () z∈ys distinct-++→disjoint (x ∷ xs) (.x distinct-∷ dxsys by x∉xsys) {.x} (here refl) x∈ys = x∉xsys (∈-++⁺ʳ xs x∈ys) distinct-++→disjoint (x ∷ xs) (.x distinct-∷ dxsys by x₁) {z} (there z∈xs) z∈ys = distinct-++→disjoint xs dxsys z∈xs z∈ys ⋈→distinct-++ : ∀ {a}{A : Set a}{m n}{xs : Vec A m}{ys : Vec A n} → Distinct xs → Distinct ys → xs ⋈ ys → Distinct (xs ++ ys) ⋈→distinct-++ {xs = []} _ dys _ = dys ⋈→distinct-++ {xs = x ∷ xs} (.x distinct-∷ dxs by x∉xs) dys xxs⋈ys = x distinct-∷ ⋈→distinct-++ dxs dys (xxs⋈ys ∘ there) by λ x∈xs++ys → xxs⋈ys (here P.refl) (x∈xs++ys→x∉xs→x∈ys xs x∈xs++ys x∉xs) where x∈xs++ys→x∉xs→x∈ys : ∀ {a} {A : Set a} {m n} (xs : Vec A m){ys : Vec A n} → ∀ {x} → x ∈ xs ++ ys → x ∉ xs → x ∈ ys x∈xs++ys→x∉xs→x∈ys [] x∈ys _ = x∈ys x∈xs++ys→x∉xs→x∈ys (x ∷ xs) (here refl) x∉xs = ⊥-elim (x∉xs (here refl)) x∈xs++ys→x∉xs→x∈ys (x ∷ xs) (there x∈xsys) x∉xs = x∈xs++ys→x∉xs→x∈ys xs x∈xsys (x∉xs ∘ there) distinct-++⇔⋈ : ∀ {a}{A : Set a}{m n} {xs : Vec A m}{ys : Vec A n} → Distinct (xs ++ ys) ⇔ (Distinct xs × Distinct ys × xs ⋈ ys) distinct-++⇔⋈ = equivalence to from where open import Data.Nat.Properties to : ∀ {a}{A : Set a} {m n} {xs : Vec A m}{ys : Vec A n} → Distinct (xs ++ ys) → (Distinct xs × Distinct ys × xs ⋈ ys) to {xs = xs} dxsys = distinct-++ˡ xs dxsys , distinct-++ʳ xs dxsys , distinct-++→disjoint xs dxsys from : ∀ {a}{A : Set a} {m n} {xs : Vec A m}{ ys : Vec A n} → (Distinct xs × Distinct ys × xs ⋈ ys) → Distinct (xs ++ ys) from (dxs , dys , xs⋈ys) = ⋈→distinct-++ dxs dys xs⋈ys private lookup-∈ : ∀ {a n}{A : Set a} i (xs : Vec A n) → Vec.lookup i xs ∈ xs lookup-∈ () [] lookup-∈ zero (x ∷ xs) = here P.refl lookup-∈ (suc i) (x ∷ xs) = there (lookup-∈ i xs) lookup-injective : ∀ {a n}{A : Set a} {xs : Vec A n}{i j} → Distinct xs → Vec.lookup i xs ≡ Vec.lookup j xs → i ≡ j lookup-injective {i = ()} {j} distinct-[] _ lookup-injective {i = zero} {zero} (x distinct-∷ dxs by x∉xs) eq = P.refl lookup-injective {i = suc i} {suc j} (x distinct-∷ dxs by x∉xs) eq = P.cong Fin.suc (lookup-injective dxs eq) lookup-injective {xs = _ ∷ xs} {i = zero} {suc j} (x distinct-∷ dxs by x∉xs) eq rewrite eq = ⊥-elim (x∉xs (lookup-∈ j xs)) lookup-injective {xs = _ ∷ xs} {i = suc i} {zero} (x distinct-∷ dxs by x∉xs) eq rewrite P.sym eq = ⊥-elim (x∉xs (lookup-∈ i xs))

54.38806

129

0.589737

03d89ebde2ae6216930409c5aa8c7f0e2bdd2a26

28,596

agda

Agda

archive/agda-3/src/AgdaFeaturePitfallInstanceResolution.agda

m0davis/oscar

52e1cdbdee54d9a8eaee04ee518a0d7f61d25afb

[ "RSA-MD" ]

null

archive/agda-3/src/AgdaFeaturePitfallInstanceResolution.agda

m0davis/oscar

52e1cdbdee54d9a8eaee04ee518a0d7f61d25afb

[ "RSA-MD" ]

1

2019-04-29T00:35:04.000Z

2019-05-11T23:33:04.000Z

archive/agda-3/src/AgdaFeaturePitfallInstanceResolution.agda

m0davis/oscar

52e1cdbdee54d9a8eaee04ee518a0d7f61d25afb

[ "RSA-MD" ]

null

{-# OPTIONS --allow-unsolved-metas #-} {- The moral of the story is best told by comparing RegularVsConstructedMoreSimpler and RegularVsConstructed-EnhancedReg: * aliased type constructors can lose information about their dependencies, leading to some inconvenience when using a function which takes those dependencies implicitly * expressing those constructors as records (instead of as aliases) averts the above inconvenience * the loss of information happens when the resultant type is made from projections on the dependencies, where only a proper subset of all the possible projections are used TODO: what if instead of projections, we use a function? (try one that's abstract, and one that case splits on arguments) - see ProjectedMorality ... so far it looks like it doesn't matter --- not sure why TODO: what if the argument type (the one that's losing information) were data instead of record? - see DataMorality ... weirdness! -} module AgdaFeaturePitfallInstanceResolution where record Symmetry {B : Set₁} (_∼_ : B → B → Set) : Set₁ where field symmetry : ∀ {x y} → x ∼ y → y ∼ x Property : Set → Set₁ Property A = A → Set Extension : {A : Set} → Property A → Set Extension P = ∀ f → P f postulate PropertyEquivalence : ∀ {P : Set} → Property P → Property P → Set record Regular : Set where no-eta-equality infixr 5 _,_ record Σ (𝔒 : Set₁) (𝔓 : 𝔒 → Set) : Set₁ where constructor _,_ field π₀ : 𝔒 π₁ : 𝔓 π₀ open Σ public ExtensionProperty : ∀ (𝔒 : Set) → Set₁ ExtensionProperty 𝔒 = Σ (Property 𝔒) Extension _≈_ : {𝔒 : Set} → ExtensionProperty 𝔒 → ExtensionProperty 𝔒 → Set _≈_ P Q = PropertyEquivalence (π₀ P) (π₀ Q) record Instance : Set where no-eta-equality postulate instance _ : ∀ {𝔒 : Set} → Symmetry (_≈_ {𝔒 = 𝔒}) open Symmetry ⦃ … ⦄ module Test {𝔒 : Set} where test1-fails : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test1-fails P≈Q = symmetry P≈Q test2-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test2-works {P} {Q} P≈Q = symmetry {x = P} {y = Q} P≈Q test3-fails : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test3-fails {P} {Q} P≈Q = symmetry {x = _ , _} {y = _ , _} P≈Q test4-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test4-works {P} {Q} P≈Q = symmetry {x = _ , π₁ P} {y = _ , π₁ Q} P≈Q record Function : Set where no-eta-equality postulate symmetry : ∀ {𝔒} {x y : ExtensionProperty 𝔒} → x ≈ y → y ≈ x -- normalises to : {𝔒 : Set} {x y : Σ (𝔒 → Set) (λ P → (f : 𝔒) → P f)} → PropertyEquivalence (π₀ x) (π₀ y) → PropertyEquivalence (π₀ y) (π₀ x) module Test {𝔒 : Set} where test1-fails : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test1-fails P≈Q = symmetry P≈Q test2-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test2-works {P} {Q} P≈Q = symmetry {x = P} {y = Q} P≈Q test3-fails : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test3-fails {P} {Q} P≈Q = symmetry {x = _ , _} {y = _ , _} P≈Q test4-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test4-works {P} {Q} P≈Q = symmetry {x = _ , π₁ P} {y = _ , π₁ Q} P≈Q record Revamped : Set where no-eta-equality record ExtensionProperty (𝔒 : Set) : Set₁ where constructor _,_ field π₀ : Property 𝔒 π₁ : Extension π₀ open ExtensionProperty _≈_ : {𝔒 : Set} → ExtensionProperty 𝔒 → ExtensionProperty 𝔒 → Set _≈_ P Q = PropertyEquivalence (π₀ P) (π₀ Q) record Instance : Set where no-eta-equality postulate instance _ : ∀ {𝔒 : Set} → Symmetry (_≈_ {𝔒 = 𝔒}) open Symmetry ⦃ … ⦄ module Test {𝔒 : Set} where test1-fails : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test1-fails P≈Q = symmetry P≈Q test2-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test2-works {P} {Q} P≈Q = symmetry {x = P} {y = Q} P≈Q test3-fails : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test3-fails {P} {Q} P≈Q = symmetry {x = _ , _} {y = _ , _} P≈Q test4-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test4-works {P} {Q} P≈Q = symmetry {x = _ , π₁ P} {y = _ , π₁ Q} P≈Q record Function : Set where no-eta-equality postulate symmetry : ∀ {𝔒} {x y : ExtensionProperty 𝔒} → x ≈ y → y ≈ x -- normalises to : ∀ {𝔒} {x y : ExtensionProperty 𝔒} → PropertyEquivalence (π₀ x) (π₀ y) → PropertyEquivalence (π₀ y) (π₀ x) module Test {𝔒 : Set} where test1-fails : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test1-fails P≈Q = symmetry P≈Q test2-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test2-works {P} {Q} P≈Q = symmetry {x = P} {y = Q} P≈Q test3-fails : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test3-fails {P} {Q} P≈Q = symmetry {x = _ , _} {y = _ , _} P≈Q test4-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test4-works {P} {Q} P≈Q = symmetry {x = _ , π₁ P} {y = _ , π₁ Q} P≈Q record PostulatedExtensionProperty : Set where no-eta-equality postulate ExtensionProperty : Set → Set₁ π₀ : {𝔒 : Set} → ExtensionProperty 𝔒 → Property 𝔒 π₁ : {𝔒 : Set} → (P : ExtensionProperty 𝔒) → Extension (π₀ P) _,_ : {𝔒 : Set} → (π₀ : Property 𝔒) → Extension π₀ → ExtensionProperty 𝔒 _≈_ : {𝔒 : Set} → ExtensionProperty 𝔒 → ExtensionProperty 𝔒 → Set _≈_ P Q = PropertyEquivalence (π₀ P) (π₀ Q) record Instance : Set where no-eta-equality postulate instance _ : ∀ {𝔒 : Set} → Symmetry (_≈_ {𝔒 = 𝔒}) open Symmetry ⦃ … ⦄ module Test {𝔒 : Set} where test1-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test1-works P≈Q = symmetry P≈Q test2-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test2-works {P} {Q} P≈Q = symmetry {x = P} {y = Q} P≈Q test3-inexpressible : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test3-inexpressible {P} {Q} P≈Q = {!!} -- symmetry {x = _ , _} {y = _ , _} P≈Q test4-inexpressible : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test4-inexpressible {P} {Q} P≈Q = {!!} -- symmetry {x = _ , π₁ P} {y = _ , π₁ Q} P≈Q record Function : Set where no-eta-equality postulate symmetry : ∀ {𝔒} {x y : ExtensionProperty 𝔒} → x ≈ y → y ≈ x -- normalises to : ∀ {𝔒} {x y : ExtensionProperty 𝔒} → PropertyEquivalence (π₀ x) (π₀ y) → PropertyEquivalence (π₀ y) (π₀ x) module Test {𝔒 : Set} where test1-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test1-works P≈Q = symmetry P≈Q test2-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test2-works {P} {Q} P≈Q = symmetry {x = P} {y = Q} P≈Q test3-inexpressible : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test3-inexpressible {P} {Q} P≈Q = {!!} -- symmetry {x = _ , _} {y = _ , _} P≈Q test4-inexpressible : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test4-inexpressible {P} {Q} P≈Q = {!!} -- symmetry {x = _ , π₁ P} {y = _ , π₁ Q} P≈Q record Constructed : Set where no-eta-equality infixr 5 _,_ record Σ (𝔒 : Set₁) (𝔓 : 𝔒 → Set) : Set₁ where constructor _,_ field π₀ : 𝔒 π₁ : 𝔓 π₀ open Σ public ExtensionProperty : Set → Set₁ ExtensionProperty 𝔒 = Σ (Property 𝔒) Extension record _≈_ {𝔒 : Set} (P Q : ExtensionProperty 𝔒) : Set where constructor ∁ field π₀ : PropertyEquivalence (π₀ P) (π₀ Q) record Instance : Set where no-eta-equality postulate instance _ : {𝔒 : Set} → Symmetry (_≈_ {𝔒 = 𝔒}) open Symmetry ⦃ … ⦄ module Test {𝔒 : Set} where test1-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test1-works P≈Q = symmetry P≈Q test2-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test2-works {P} {Q} P≈Q = symmetry {x = P} {y = Q} P≈Q test3-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test3-works {P} {Q} P≈Q = symmetry {x = _ , _} {y = _ , _} P≈Q test4-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test4-works {P} {Q} P≈Q = symmetry {x = _ , π₁ P} {y = _ , π₁ Q} P≈Q record Function : Set where no-eta-equality postulate symmetry : ∀ {𝔒} {x y : ExtensionProperty 𝔒} → x ≈ y → y ≈ x -- normalises to : {𝔒 : Set} {x y : Σ (𝔒 → Set) (λ P → (f : 𝔒) → P f)} → x ≈ y → y ≈ x module Test {𝔒 : Set} where test1-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test1-works P≈Q = symmetry P≈Q test2-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test2-works {P} {Q} P≈Q = symmetry {x = P} {y = Q} P≈Q test3-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test3-works {P} {Q} P≈Q = symmetry {x = _ , _} {y = _ , _} P≈Q test4-works : {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test4-works {P} {Q} P≈Q = symmetry {x = _ , π₁ P} {y = _ , π₁ Q} P≈Q record RegularVsConstructed : Set where no-eta-equality infixr 5 _,_ record Σ (𝔒 : Set₁) (𝔓 : 𝔒 → Set) : Set₁ where constructor _,_ field π₀ : 𝔒 π₁ : 𝔓 π₀ open Σ public ExtensionProperty : Set → Set₁ ExtensionProperty 𝔒 = Σ (Property 𝔒) Extension record _≈R_ {𝔒 : Set} (P Q : ExtensionProperty 𝔒) : Set where constructor ∁ field π₀ : PropertyEquivalence (π₀ P) (π₀ Q) _≈F_ : {𝔒 : Set} → ExtensionProperty 𝔒 → ExtensionProperty 𝔒 → Set _≈F_ P Q = PropertyEquivalence (π₀ P) (π₀ Q) record Instance : Set where no-eta-equality postulate instance _ : {𝔒 : Set} → Symmetry (_≈R_ {𝔒 = 𝔒}) postulate instance _ : {𝔒 : Set} → Symmetry (_≈F_ {𝔒 = 𝔒}) open Symmetry ⦃ … ⦄ module Test {𝔒 : Set} where test1-worksR : {P Q : ExtensionProperty 𝔒} → P ≈R Q → Q ≈R P test1-worksR P≈Q = symmetry P≈Q test2-worksR : {P Q : ExtensionProperty 𝔒} → P ≈R Q → Q ≈R P test2-worksR {P} {Q} P≈Q = symmetry {x = P} {y = Q} P≈Q test3-worksR : {P Q : ExtensionProperty 𝔒} → P ≈R Q → Q ≈R P test3-worksR {P} {Q} P≈Q = symmetry {x = _ , _} {y = _ , _} P≈Q test4-worksR : {P Q : ExtensionProperty 𝔒} → P ≈R Q → Q ≈R P test4-worksR {P} {Q} P≈Q = symmetry {x = _ , π₁ P} {y = _ , π₁ Q} P≈Q test1-failsF : {P Q : ExtensionProperty 𝔒} → P ≈F Q → Q ≈F P test1-failsF P≈Q = symmetry P≈Q test2-worksF : {P Q : ExtensionProperty 𝔒} → P ≈F Q → Q ≈F P test2-worksF {P} {Q} P≈Q = symmetry {x = P} {y = Q} P≈Q test3-failsF : {P Q : ExtensionProperty 𝔒} → P ≈F Q → Q ≈F P test3-failsF {P} {Q} P≈Q = symmetry {x = _ , _} {y = _ , _} P≈Q test4-worksF : {P Q : ExtensionProperty 𝔒} → P ≈F Q → Q ≈F P test4-worksF {P} {Q} P≈Q = symmetry {x = _ , π₁ P} {y = _ , π₁ Q} P≈Q record Function : Set where no-eta-equality postulate symmetryR : ∀ {𝔒} {x y : ExtensionProperty 𝔒} → x ≈R y → y ≈R x postulate symmetryF : ∀ {𝔒} {x y : ExtensionProperty 𝔒} → x ≈F y → y ≈F x module Test {𝔒 : Set} where test1-worksR : {P Q : ExtensionProperty 𝔒} → P ≈R Q → Q ≈R P test1-worksR P≈Q = symmetryR P≈Q test2-worksR : {P Q : ExtensionProperty 𝔒} → P ≈R Q → Q ≈R P test2-worksR {P} {Q} P≈Q = symmetryR {x = P} {y = Q} P≈Q test3-worksR : {P Q : ExtensionProperty 𝔒} → P ≈R Q → Q ≈R P test3-worksR {P} {Q} P≈Q = symmetryR {x = _ , _} {y = _ , _} P≈Q test4-worksR : {P Q : ExtensionProperty 𝔒} → P ≈R Q → Q ≈R P test4-worksR {P} {Q} P≈Q = symmetryR {x = _ , π₁ P} {y = _ , π₁ Q} P≈Q test1-failsF : {P Q : ExtensionProperty 𝔒} → P ≈F Q → Q ≈F P test1-failsF P≈Q = symmetryF P≈Q test2-worksF : {P Q : ExtensionProperty 𝔒} → P ≈F Q → Q ≈F P test2-worksF {P} {Q} P≈Q = symmetryF {x = P} {y = Q} P≈Q test3-failsF : {P Q : ExtensionProperty 𝔒} → P ≈F Q → Q ≈F P test3-failsF {P} {Q} P≈Q = symmetryF {x = _ , _} {y = _ , _} P≈Q test4-worksF : {P Q : ExtensionProperty 𝔒} → P ≈F Q → Q ≈F P test4-worksF {P} {Q} P≈Q = symmetryF {x = _ , π₁ P} {y = _ , π₁ Q} P≈Q record RegularVsConstructedSimpler : Set where no-eta-equality infixr 5 _,_ record Σ (𝔒 : Set₁) (𝔓 : 𝔒 → Set) : Set₁ where constructor _,_ field π₀ : 𝔒 π₁ : 𝔓 π₀ open Σ public postulate Prop : Set₁ postulate Ext : Prop → Set postulate PropEq : Prop → Set ExtProp : Set₁ ExtProp = Σ Prop Ext record ≈C_ (P : ExtProp) : Set where constructor ∁ field π₀ : PropEq (π₀ P) ≈R_ : ExtProp → Set ≈R_ P = PropEq (π₀ P) record Instance : Set where no-eta-equality record Class {B : Set₁} (∼_ : B → Set) : Set₁ where field foo : ∀ {x} → ∼ x → Set open Class ⦃ … ⦄ postulate instance _ : Class ≈C_ postulate instance _ : Class ≈R_ module Test where test1-worksC : {P : ExtProp} → ≈C P → Set test1-worksC P≈Q = foo P≈Q test2-worksC : {P : ExtProp} → ≈C P → Set test2-worksC {P} P≈Q = foo {x = P} P≈Q test3-worksC : {P : ExtProp} → ≈C P → Set test3-worksC {P} P≈Q = foo {x = _ , _} P≈Q test4-worksC : {P : ExtProp} → ≈C P → Set test4-worksC {P} P≈Q = foo {x = _ , π₁ P} P≈Q test1-failsR : {P : ExtProp} → ≈R P → Set test1-failsR P≈Q = foo P≈Q test2-worksR : {P : ExtProp} → ≈R P → Set test2-worksR {P} P≈Q = foo {x = P} P≈Q test3-failsR : {P : ExtProp} → ≈R P → Set test3-failsR {P} P≈Q = foo {x = _ , _} P≈Q test4-worksR : {P : ExtProp} → ≈R P → Set test4-worksR {P} P≈Q = foo {x = _ , π₁ P} P≈Q record Function : Set where no-eta-equality postulate fooC : {x : ExtProp} → ≈C x → Set postulate fooR : {x : ExtProp} → ≈R x → Set module Test where test1-worksC : {P : ExtProp} → ≈C P → Set test1-worksC P≈Q = fooC P≈Q test2-worksC : {P : ExtProp} → ≈C P → Set test2-worksC {P} P≈Q = fooC {x = P} P≈Q test3-worksC : {P : ExtProp} → ≈C P → Set test3-worksC {P} P≈Q = fooC {x = _ , _} P≈Q test4-worksC : {P : ExtProp} → ≈C P → Set test4-worksC {P} P≈Q = fooC {x = _ , π₁ P} P≈Q test1-failsR : {P : ExtProp} → ≈R P → Set test1-failsR P≈Q = fooR P≈Q test2-worksR : {P : ExtProp} → ≈R P → Set test2-worksR {P} P≈Q = fooR {x = P} P≈Q test3-failsR : {P : ExtProp} → ≈R P → Set test3-failsR {P} P≈Q = fooR {x = _ , _} P≈Q test4-worksR : {P : ExtProp} → ≈R P → Set test4-worksR {P} P≈Q = fooR {x = _ , π₁ P} P≈Q record RegularVsConstructedMoreSimpler : Set where no-eta-equality infixr 5 _,_ record Σ (𝔒 : Set₁) (𝔓 : 𝔒 → Set) : Set₁ where constructor _,_ field π₀ : 𝔒 π₁ : Set open Σ postulate Prop : Set₁ postulate Ext : Prop → Set postulate PropEq : Prop → Set ExtProp : Set₁ ExtProp = Σ Prop Ext Reg : ExtProp → Set Reg P = PropEq (π₀ P) record Con (P : ExtProp) : Set where constructor ∁ field π₀ : Reg P module Instance where record Class {B : Set₁} (F : B → Set) : Set₁ where field foo : ∀ {x} → F x → Set open Class ⦃ … ⦄ postulate instance _ : Class Reg postulate instance _ : Class Con postulate instance _ : Class Ext postulate instance _ : Class PropEq test1-failsR : {P : ExtProp} → Reg P → Set test1-failsR P≈Q = foo P≈Q test2-worksR : {P : ExtProp} → Reg P → Set test2-worksR {P} P≈Q = foo {x = P} P≈Q test3-failsR : {P : ExtProp} → Reg P → Set test3-failsR {P} P≈Q = foo {x = _ , _} P≈Q test4-worksR : {P : ExtProp} → Reg P → Set test4-worksR {P} P≈Q = foo {x = _ , π₁ P} P≈Q test1-worksC : {P : ExtProp} → Con P → Set test1-worksC P≈Q = foo P≈Q test2-worksC : {P : ExtProp} → Con P → Set test2-worksC {P} P≈Q = foo {x = P} P≈Q test3-worksC : {P : ExtProp} → Con P → Set test3-worksC {P} P≈Q = foo {x = _ , _} P≈Q test4-worksC : {P : ExtProp} → Con P → Set test4-worksC {P} P≈Q = foo {x = _ , π₁ P} P≈Q module Function where postulate fooR : {x : ExtProp} → Reg x → Set postulate fooC : {x : ExtProp} → Con x → Set test1-failsR : {P : ExtProp} → Reg P → Set test1-failsR P≈Q = fooR P≈Q test2-worksR : {P : ExtProp} → Reg P → Set test2-worksR {P} P≈Q = fooR {x = P} P≈Q test3-failsR : {P : ExtProp} → Reg P → Set test3-failsR {P} P≈Q = fooR {x = _ , _} P≈Q test4-worksR : {P : ExtProp} → Reg P → Set test4-worksR {P} P≈Q = fooR {x = _ , π₁ P} P≈Q test1-worksC : {P : ExtProp} → Con P → Set test1-worksC P≈Q = fooC P≈Q test2-worksC : {P : ExtProp} → Con P → Set test2-worksC {P} P≈Q = fooC {x = P} P≈Q test3-worksC : {P : ExtProp} → Con P → Set test3-worksC {P} P≈Q = fooC {x = _ , _} P≈Q test4-worksC : {P : ExtProp} → Con P → Set test4-worksC {P} P≈Q = fooC {x = _ , π₁ P} P≈Q module RegularVsConstructed-EnhancedReg where infixr 5 _,_ record Σ (𝔒 : Set₁) (𝔓 : 𝔒 → Set) : Set₁ where constructor _,_ field π₀ : 𝔒 π₁ : Set open Σ postulate Prop : Set₁ postulate Ext : Prop → Set postulate PropEq : Prop → Set → Set ExtProp : Set₁ ExtProp = Σ Prop Ext Reg : ExtProp → Set Reg P = PropEq (π₀ P) (π₁ P) record Con (P : ExtProp) : Set where constructor ∁ field π₀ : Reg P module Instance where record Class {B : Set₁} (F : B → Set) : Set₁ where field foo : ∀ {x} → F x → Set open Class ⦃ … ⦄ postulate instance _ : Class Reg postulate instance _ : Class Con postulate instance _ : Class Ext test1-failsR : {P : ExtProp} → Reg P → Set test1-failsR P≈Q = foo P≈Q test2-worksR : {P : ExtProp} → Reg P → Set test2-worksR {P} P≈Q = foo {x = P} P≈Q test3-failsR : {P : ExtProp} → Reg P → Set test3-failsR {P} P≈Q = foo {x = _ , _} P≈Q test4-worksR : {P : ExtProp} → Reg P → Set test4-worksR {P} P≈Q = foo {x = _ , π₁ P} P≈Q test1-worksC : {P : ExtProp} → Con P → Set test1-worksC P≈Q = foo P≈Q test2-worksC : {P : ExtProp} → Con P → Set test2-worksC {P} P≈Q = foo {x = P} P≈Q test3-worksC : {P : ExtProp} → Con P → Set test3-worksC {P} P≈Q = foo {x = _ , _} P≈Q test4-worksC : {P : ExtProp} → Con P → Set test4-worksC {P} P≈Q = foo {x = _ , π₁ P} P≈Q module Function where postulate fooR : {x : ExtProp} → Reg x → Set postulate fooC : {x : ExtProp} → Con x → Set test1-failsR : {P : ExtProp} → Reg P → Set test1-failsR P≈Q = fooR P≈Q test2-worksR : {P : ExtProp} → Reg P → Set test2-worksR {P} P≈Q = fooR {x = P} P≈Q test3-failsR : {P : ExtProp} → Reg P → Set test3-failsR {P} P≈Q = fooR {x = _ , _} P≈Q test4-worksR : {P : ExtProp} → Reg P → Set test4-worksR {P} P≈Q = fooR {x = _ , π₁ P} P≈Q test1-worksC : {P : ExtProp} → Con P → Set test1-worksC P≈Q = fooC P≈Q test2-worksC : {P : ExtProp} → Con P → Set test2-worksC {P} P≈Q = fooC {x = P} P≈Q test3-worksC : {P : ExtProp} → Con P → Set test3-worksC {P} P≈Q = fooC {x = _ , _} P≈Q test4-worksC : {P : ExtProp} → Con P → Set test4-worksC {P} P≈Q = fooC {x = _ , π₁ P} P≈Q record ProjectedMorality : Set where no-eta-equality infixr 5 _,_ record Σ (𝔒 : Set₁) (𝔓 : 𝔒 → Set) : Set₁ where constructor _,_ field π₀ : 𝔒 π₁ : Set open Σ postulate Prop : Set₁ postulate Ext : Prop → Set postulate PropEq : Prop → Set Reg : Σ Prop Ext → Set Reg P = PropEq (π₀ P) postulate bar : ∀ {𝔒 : Set₁} → 𝔒 → 𝔒 postulate qux : ∀ {𝔒} {𝔓 : 𝔒 → Set} → Σ 𝔒 𝔓 → Σ 𝔒 𝔓 postulate fake-π₀ : ∀ {𝔒} {𝔓 : 𝔒 → Set} → Σ 𝔒 𝔓 → 𝔒 abstract abstracted-π₀ : ∀ {𝔒} {𝔓 : 𝔒 → Set} → Σ 𝔒 𝔓 → 𝔒 abstracted-π₀ x = π₀ x Reg-using-abstracted-projection : Σ Prop Ext → Set Reg-using-abstracted-projection (P0 , P1) = PropEq (abstracted-π₀ {𝔒 = Prop} {𝔓 = Ext} (P0 , P1)) Reg-using-q : Σ Prop Ext → Set Reg-using-q x = PropEq (π₀ (qux x)) Reg-using-fake-π₀ : Σ Prop Ext → Set Reg-using-fake-π₀ x = PropEq (fake-π₀ x) record Con (P : Σ Prop Ext) : Set where constructor ∁ field π₀ : Reg P record Con-using-abstracted-projection (P : Σ Prop Ext) : Set where constructor ∁ field π₀ : Reg-using-abstracted-projection P record Con-using-q (P : Σ Prop Ext) : Set where constructor ∁ field π₀ : Reg-using-q P record Con-using-fake-π₀ (P : Σ Prop Ext) : Set where constructor ∁ field π₀ : Reg-using-fake-π₀ P record Class {B : Set₁} (F : B → Set) : Set₁ where field foo : ∀ {x} → F x → Set open Class ⦃ … ⦄ postulate instance _ : Class Reg postulate instance _ : Class Reg-using-abstracted-projection postulate instance _ : Class Reg-using-q postulate instance _ : Class Reg-using-fake-π₀ postulate instance _ : Class Con postulate instance _ : Class Con-using-abstracted-projection postulate instance _ : Class Con-using-q postulate instance _ : Class Con-using-fake-π₀ test1-failsR : ∀ {P} → Reg P → Set test1-failsR = foo test1-failsRap : ∀ {P} → Reg-using-abstracted-projection P → Set test1-failsRap = foo test1-failsRq : ∀ {P} → Reg-using-q P → Set test1-failsRq = foo test1-failsRf : ∀ {P} → Reg-using-fake-π₀ P → Set test1-failsRf = foo test1-worksC : ∀ {P} → Con P → Set test1-worksC = foo test1-worksCap : ∀ {P} → Con-using-abstracted-projection P → Set test1-worksCap = foo test1-worksCq : ∀ {P} → Con-using-q P → Set test1-worksCq = foo test1-worksCf : ∀ {P} → Con-using-fake-π₀ P → Set test1-worksCf = foo record DataMorality : Set where no-eta-equality module _ (𝔒 : Set₁) (𝔓 : 𝔒 → Set) where data Σ : Set₁ where _,_ : 𝔒 → Set → Σ module _ {𝔒 : Set₁} {𝔓 : 𝔒 → Set} where dπ₀ : Σ _ 𝔓 → 𝔒 dπ₀ (x , _) = x dπ₁ : Σ _ 𝔓 → Set dπ₁ (_ , y) = y postulate Prop : Set₁ postulate Ext : Prop → Set postulate PropEq : Prop → Set Reg : Σ Prop Ext → Set Reg P = PropEq (dπ₀ P) postulate bar : ∀ {𝔒 : Set₁} → 𝔒 → 𝔒 postulate qux : ∀ {𝔒} {𝔓 : 𝔒 → Set} → Σ 𝔒 𝔓 → Σ 𝔒 𝔓 postulate fake-π₀ : ∀ {𝔒} {𝔓 : 𝔒 → Set} → Σ 𝔒 𝔓 → 𝔒 abstract abstracted-π₀ : ∀ {𝔒} {𝔓 : 𝔒 → Set} → Σ 𝔒 𝔓 → 𝔒 abstracted-π₀ x = dπ₀ x Reg-using-abstracted-projection : Σ Prop Ext → Set Reg-using-abstracted-projection (P0 , P1) = PropEq (abstracted-π₀ {𝔒 = Prop} {𝔓 = Ext} (P0 , P1)) Reg-using-q : Σ Prop Ext → Set Reg-using-q x = PropEq (dπ₀ (qux x)) Reg-using-fake-π₀ : Σ Prop Ext → Set Reg-using-fake-π₀ x = PropEq (fake-π₀ x) record Con (P : Σ Prop Ext) : Set where constructor ∁ field π₀ : Reg P record Con-using-abstracted-projection (P : Σ Prop Ext) : Set where constructor ∁ field π₀ : Reg-using-abstracted-projection P record Con-using-q (P : Σ Prop Ext) : Set where constructor ∁ field π₀ : Reg-using-q P record Con-using-fake-π₀ (P : Σ Prop Ext) : Set where constructor ∁ field π₀ : Reg-using-fake-π₀ P record Class {B : Set₁} (F : B → Set) : Set₁ where field foo : ∀ {x} → F x → Set open Class ⦃ … ⦄ postulate instance _ : Class Reg postulate instance _ : Class Reg-using-abstracted-projection postulate instance _ : Class Reg-using-q postulate instance _ : Class Reg-using-fake-π₀ postulate instance _ : Class Con postulate instance _ : Class Con-using-abstracted-projection postulate instance _ : Class Con-using-q postulate instance _ : Class Con-using-fake-π₀ test1-failsR : ∀ {P} → Reg P → Set test1-failsR = foo test1-failsRap : ∀ {P} → Reg-using-abstracted-projection P → Set test1-failsRap = foo -- woah, it actually works. why? test1-failsRq : ∀ {P} → Reg-using-q P → Set test1-failsRq = foo -- NB this doesn't fail if instance of Class Reg is excluded test1-failsRf : ∀ {P} → Reg-using-fake-π₀ P → Set test1-failsRf = foo -- NB this doesn't fail if instance of Class Reg is excluded test1-worksC : ∀ {P} → Con P → Set test1-worksC = foo test1-worksCap : ∀ {P} → Con-using-abstracted-projection P → Set test1-worksCap = foo test1-worksCq : ∀ {P} → Con-using-q P → Set test1-worksCq = foo test1-worksCf : ∀ {P} → Con-using-fake-π₀ P → Set test1-worksCf = foo module RevampedSimpleFailure where record ExtensionProperty (𝔒 : Set) : Set₁ where field π₀ : Property 𝔒 π₁ : Extension π₀ open ExtensionProperty _≈_ : {𝔒 : Set} → ExtensionProperty 𝔒 → ExtensionProperty 𝔒 → Set _≈_ P Q = PropertyEquivalence (π₀ P) (π₀ Q) postulate symmetry : ∀ {𝔒} {x y : ExtensionProperty 𝔒} → x ≈ y → y ≈ x -- normalises to : ∀ {𝔒} {x y : ExtensionProperty 𝔒} → PropertyEquivalence (π₀ x) (π₀ y) → PropertyEquivalence (π₀ y) (π₀ x) test-fails : {𝔒 : Set} {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test-fails P≈Q = symmetry P≈Q module PostulatedExtensionPropertySimpleSuccess where postulate ExtensionProperty : Set → Set₁ π₀ : {𝔒 : Set} → ExtensionProperty 𝔒 → Property 𝔒 _≈_ : {𝔒 : Set} → ExtensionProperty 𝔒 → ExtensionProperty 𝔒 → Set _≈_ P Q = PropertyEquivalence (π₀ P) (π₀ Q) postulate symmetry : ∀ {𝔒} {x y : ExtensionProperty 𝔒} → x ≈ y → y ≈ x -- normalises to : ∀ {𝔒} {x y : ExtensionProperty 𝔒} → PropertyEquivalence (π₀ {𝔒} x) (π₀ {𝔒} y) → PropertyEquivalence (π₀ {𝔒} y) (π₀ {𝔒} x) test-works : {𝔒 : Set} {P Q : ExtensionProperty 𝔒} → P ≈ Q → Q ≈ P test-works P≈Q = symmetry P≈Q module RevampedVerySimpleFailure where -- was PropertyEquivalence : ∀ {P : Set} → Property P → Property P → Set postulate _∼_ : Set → Set → Set record ExtensionProperty : Set₁ where field π₀ : Set -- was Property 𝔒 π₁ : Set -- was Extension π₀ open ExtensionProperty postulate symmetry : ∀ {x y : ExtensionProperty} → π₀ x ∼ π₀ y → π₀ y ∼ π₀ x postulate x y : ExtensionProperty test-fails : π₀ x ∼ π₀ y → π₀ y ∼ π₀ x test-fails = symmetry module PostulatedExtensionPropertyVerySimpleSuccess where postulate _∼_ : Set → Set → Set postulate ExtensionProperty : Set₁ π₀ : ExtensionProperty → Set postulate symmetry : ∀ {x y : ExtensionProperty} → π₀ x ∼ π₀ y → π₀ y ∼ π₀ x postulate x y : ExtensionProperty test-works : π₀ x ∼ π₀ y → π₀ y ∼ π₀ x test-works P≈Q = symmetry P≈Q module RevampedEvenSimplerFailure where -- was _∼_, which was PropertyEquivalence postulate F : Set → Set record ExtensionProperty : Set₁ where field π₀ : Set π₁ : Set open ExtensionProperty postulate symmetry : ∀ {x : ExtensionProperty} → F (π₀ x) → Set postulate x : ExtensionProperty postulate Fpx : F (π₀ x) test-fails1 : Set test-fails1 = symmetry Fpx test-fails2 : Set test-fails2 = symmetry {x = record { π₀ = π₀ x ; π₁ = _}} Fpx test-works-arbitrarily : Set test-works-arbitrarily = symmetry {x = record { π₀ = π₀ x ; π₁ = F (F (π₁ x)) }} Fpx module PostulatedExtensionPropertyEvenSimplerSuccess where postulate F : Set → Set postulate ExtensionProperty : Set₁ π₀ : ExtensionProperty → Set postulate symmetry : ∀ {x : ExtensionProperty} → F (π₀ x) → Set postulate x : ExtensionProperty postulate Fpx : F (π₀ x) test-works1 : Set test-works1 = symmetry Fpx test-works2 : Set test-works2 = symmetry {x = x} Fpx module RevampedEvenSimplerFailureClassified where postulate F : Set → Set record ExtensionProperty : Set₁ where field π₀ : Set π₁ : Set open ExtensionProperty record Symmetry' (R : Set → Set) : Set₁ where field symmetry : ∀ {x : ExtensionProperty} → R (π₀ x) → Set open Symmetry' ⦃ … ⦄ postulate instance _ : Symmetry' F postulate x : ExtensionProperty postulate Fpx : F (π₀ x) test-fails1 : Set test-fails1 = symmetry Fpx test-fails2 : Set test-fails2 = symmetry {x = record { π₀ = π₀ x ; π₁ = _}} Fpx test-works-arbitrarily : Set test-works-arbitrarily = symmetry {x = record { π₀ = π₀ x ; π₁ = F (F (π₁ x)) }} Fpx module PostulatedExtensionPropertyEvenSimplerSuccessClassified where postulate F : Set → Set postulate ExtensionProperty : Set₁ π₀ : ExtensionProperty → Set record Symmetry' (R : Set → Set) : Set₁ where field symmetry : ∀ {x : ExtensionProperty} → R (π₀ x) → Set open Symmetry' ⦃ … ⦄ postulate instance _ : Symmetry' F postulate x : ExtensionProperty postulate Fpx : F (π₀ x) test-works1 : Set test-works1 = symmetry Fpx test-works2 : Set test-works2 = symmetry {x = x} Fpx module RevampedVsPostulated where record R : Set₁ where field f1 : Set f2 : Set open R postulate fooR : ∀ {x : R} → f1 x → Set postulate r : R postulate f1r : f1 r test-fails1 : Set test-fails1 = fooR f1r postulate S : Set₁ g1 : S → Set postulate fooS : ∀ {x : S} → g1 x → Set postulate s : S postulate g1s : g1 s test-works1 : Set test-works1 = fooS g1s

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agda

Agda

Cubical/Foundations/Equiv/Reasoning.agda

kiana-S/univalent-foundations

8bdb0766260489f9c89a14f4c0f2ad26e5190dec

[ "MIT" ]

null

Cubical/Foundations/Equiv/Reasoning.agda

kiana-S/univalent-foundations

8bdb0766260489f9c89a14f4c0f2ad26e5190dec

[ "MIT" ]

null

Cubical/Foundations/Equiv/Reasoning.agda

kiana-S/univalent-foundations

8bdb0766260489f9c89a14f4c0f2ad26e5190dec

[ "MIT" ]

null

{-# OPTIONS --cubical --no-import-sorts --safe #-} module Cubical.Foundations.Equiv.Reasoning where open import Cubical.Foundations.Prelude using (refl; sym) open import Cubical.Relation.Binary -- Properties of equivalence ≃-reflexive : Reflexive _≃_ ≃-reflexive = ?

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Agda

test/Succeed/Issue3109.agda

mdimjasevic/agda

8fb548356b275c7a1e79b768b64511ae937c738b

[ "BSD-3-Clause" ]

1,989

2015-01-09T23:51:16.000Z

2022-03-30T18:20:48.000Z

test/Succeed/Issue3109.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

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2022-03-31T21:14:49.000Z

test/Succeed/Issue3109.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

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2015-01-03T14:04:08.000Z

2022-03-30T19:00:30.000Z

{-# OPTIONS --allow-unsolved-metas #-} postulate Nat : Set Fin : Nat → Set Foo : (n : Nat) → Fin n → Set Bar : ∀ {n m} → Foo n m → Set variable n : Nat m : Fin _ k : Foo _ m l : Foo n m open import Agda.Builtin.Equality postulate goal-type-error : Bar k foo : Bar _ foo = goal-type-error {_} {_}

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agda

Agda

currypp/.cpm/packages/currycheck/examples/withVerification/PROOF-appendAddLengths.agda

phlummox/curry-tools

7905bc4f625a94a725f9f6d8a2de1140bea5e471

[ "BSD-3-Clause" ]

null

currypp/.cpm/packages/currycheck/examples/withVerification/PROOF-appendAddLengths.agda

phlummox/curry-tools

7905bc4f625a94a725f9f6d8a2de1140bea5e471

[ "BSD-3-Clause" ]

null

currypp/.cpm/packages/currycheck/examples/withVerification/PROOF-appendAddLengths.agda

phlummox/curry-tools

7905bc4f625a94a725f9f6d8a2de1140bea5e471

[ "BSD-3-Clause" ]

null

-- Agda program using the Iowa Agda library open import bool module PROOF-appendAddLengths (Choice : Set) (choose : Choice → 𝔹) (lchoice : Choice → Choice) (rchoice : Choice → Choice) where open import eq open import nat open import list open import maybe --------------------------------------------------------------------------- -- Translated Curry operations: ++ : {a : Set} → 𝕃 a → 𝕃 a → 𝕃 a ++ [] x = x ++ (y :: z) u = y :: (++ z u) append : {a : Set} → 𝕃 a → 𝕃 a → 𝕃 a append x y = ++ x y --------------------------------------------------------------------------- appendAddLengths : {a : Set} → (x : 𝕃 a) → (y : 𝕃 a) → ((length x) + (length y)) ≡ (length (append x y)) appendAddLengths [] y = refl appendAddLengths (x :: xs) y rewrite appendAddLengths xs y = refl ---------------------------------------------------------------------------

25.257143

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agda

Agda

test/Succeed/Issue919.agda

mdimjasevic/agda

8fb548356b275c7a1e79b768b64511ae937c738b

[ "BSD-3-Clause" ]

1,989

2015-01-09T23:51:16.000Z

2022-03-30T18:20:48.000Z

test/Succeed/Issue919.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

4,066

2015-01-10T11:24:51.000Z

2022-03-31T21:14:49.000Z

test/Succeed/Issue919.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

371

2015-01-03T14:04:08.000Z

2022-03-30T19:00:30.000Z

module Issue919 where open import Common.Prelude Zero : Nat → Set Zero 0 = ⊤ Zero (suc _) = ⊥ test : (n : Nat) {p : Zero n} → Set → Set test 0 A = A test (suc _) {()} -- Horrible error for first clause: -- Cannot eliminate type Set with pattern {(implicit)} (did you supply -- too many arguments?) -- when checking that the pattern zero has type Nat -- Caused by trailing implicit insertion (see Rules/Def.hs). -- With trailing implicit insertion switched off, this should work now.

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Agda

src/Tactic/Reflection/Substitute.agda

t-more/agda-prelude

da4fca7744d317b8843f2bc80a923972f65548d3

[ "MIT" ]

null

src/Tactic/Reflection/Substitute.agda

t-more/agda-prelude

da4fca7744d317b8843f2bc80a923972f65548d3

[ "MIT" ]

null

src/Tactic/Reflection/Substitute.agda

t-more/agda-prelude

da4fca7744d317b8843f2bc80a923972f65548d3

[ "MIT" ]

null

module Tactic.Reflection.Substitute where open import Prelude hiding (abs) open import Builtin.Reflection open import Tactic.Reflection.DeBruijn IsSafe : Term → Set IsSafe (lam _ _) = ⊥ IsSafe _ = ⊤ data SafeTerm : Set where safe : (v : Term) (p : IsSafe v) → SafeTerm maybeSafe : Term → Maybe SafeTerm maybeSafe (var x args) = just (safe (var x args) _) maybeSafe (con c args) = just (safe (con c args) _) maybeSafe (def f args) = just (safe (def f args) _) maybeSafe (meta x args) = just (safe (meta x args) _) maybeSafe (lam v t) = nothing maybeSafe (pat-lam cs args) = just (safe (pat-lam cs args) _) maybeSafe (pi a b) = just (safe (pi a b) _) maybeSafe (agda-sort s) = just (safe (agda-sort s) _) maybeSafe (lit l) = just (safe (lit l) _) maybeSafe unknown = just (safe unknown _) instance DeBruijnSafeTerm : DeBruijn SafeTerm strengthenFrom {{DeBruijnSafeTerm}} k n (safe v _) = do -- Strengthening or weakening safe terms always results in safe terms, -- but proving that is a bit of a bother, thus maybeSafe. v₁ ← strengthenFrom k n v maybeSafe v₁ weakenFrom {{DeBruijnSafeTerm}} k n (safe v p) = maybe (safe unknown _) id (maybeSafe (weakenFrom k n v)) safe-term : SafeTerm → Term safe-term (safe v _) = v applyTerm : SafeTerm → List (Arg Term) → Term applyTerm v [] = safe-term v applyTerm (safe (var x args) _) args₁ = var x (args ++ args₁) applyTerm (safe (con c args) _) args₁ = con c (args ++ args₁) applyTerm (safe (def f args) _) args₁ = def f (args ++ args₁) applyTerm (safe (meta x args) _) args₁ = meta x (args ++ args₁) applyTerm (safe (lam v t) ()) args applyTerm (safe (pat-lam cs args) _) args₁ = pat-lam cs (args ++ args₁) applyTerm (safe (pi a b) _) _ = pi a b applyTerm (safe (agda-sort s) _) _ = agda-sort s applyTerm (safe (lit l) _) _ = lit l applyTerm (safe unknown _) _ = unknown Subst : Set → Set Subst A = List SafeTerm → A → A substTerm : Subst Term substArgs : Subst (List (Arg Term)) substArg : Subst (Arg Term) substAbs : Subst (Abs Term) substSort : Subst Sort substClauses : Subst (List Clause) substClause : Subst Clause substTerm σ (var x args) = case index σ x of λ { nothing → var (x - length σ) (substArgs σ args) ; (just v) → applyTerm v (substArgs σ args) } substTerm σ (con c args) = con c (substArgs σ args) substTerm σ (def f args) = def f (substArgs σ args) substTerm σ (meta x args) = meta x (substArgs σ args) substTerm σ (lam v b) = lam v (substAbs σ b) substTerm σ (pat-lam cs args) = pat-lam (substClauses σ cs) (substArgs σ args) substTerm σ (pi a b) = pi (substArg σ a) (substAbs σ b) substTerm σ (agda-sort s) = agda-sort (substSort σ s) substTerm σ (lit l) = lit l substTerm σ unknown = unknown substSort σ (set t) = set (substTerm σ t) substSort σ (lit n) = lit n substSort σ unknown = unknown substClauses σ [] = [] substClauses σ (c ∷ cs) = substClause σ c ∷ substClauses σ cs substClause σ (clause tel ps b) = case length tel of λ { zero → clause tel ps (substTerm σ b) ; (suc n) → clause tel ps (substTerm (reverse (map (λ i → safe (var i []) _) (from 0 to n)) ++ weaken (suc n) σ) b) } substClause σ (absurd-clause tel ps) = absurd-clause tel ps substArgs σ [] = [] substArgs σ (x ∷ args) = substArg σ x ∷ substArgs σ args substArg σ (arg i x) = arg i (substTerm σ x) substAbs σ (abs x v) = abs x $ substTerm (safe (var 0 []) _ ∷ weaken 1 σ) v private toArgs : Nat → List (Arg SafeTerm) → List (Arg Term) toArgs k = map (λ x → weaken k (fmap safe-term x)) SafeApplyType : Set → Set SafeApplyType A = List SafeTerm → Nat → A → List (Arg SafeTerm) → A safeApplyAbs : SafeApplyType (Abs Term) safeApplyArg : SafeApplyType (Arg Term) safeApplySort : SafeApplyType Sort -- safeApply′ env |Θ| v args = v′ -- where Γ, Δ, Θ ⊢ v -- Γ ⊢ env : Δ -- Γ ⊢ args -- Γ, Θ ⊢ v′ safeApply′ : List SafeTerm → Nat → Term → List (Arg SafeTerm) → Term safeApply′ env k (var x args) args₁ = if x <? k then var x (args ++ toArgs k args₁) else case index env (x - k) of λ { nothing → var (x - length env) (args ++ toArgs k args₁) ; (just v) → applyTerm v (args ++ toArgs k args₁) } safeApply′ env k (con c args) args₁ = con c (args ++ toArgs k args₁) safeApply′ env k (def f args) args₁ = def f (args ++ toArgs k args₁) safeApply′ env k (lam v t) (a ∷ args₁) = safeApply′ (unArg a ∷ env) k (unAbs t) args₁ safeApply′ env k (lam v b) [] = lam v $ safeApplyAbs env k b [] safeApply′ env k (pat-lam cs args) args₁ = pat-lam cs (args ++ toArgs k args₁) -- not right if applying to constructors safeApply′ env k (pi a b) _ = pi (safeApplyArg env k a []) (safeApplyAbs env k b []) safeApply′ env k (agda-sort s) args₁ = agda-sort (safeApplySort env k s []) safeApply′ env k (lit l) args₁ = lit l safeApply′ env k (meta x args) args₁ = meta x (args ++ toArgs k args₁) safeApply′ env k unknown args₁ = unknown safeApplyAbs env k (abs x b) _ = abs x (safeApply′ env (suc k) b []) safeApplyArg env k (arg i v) args₁ = arg i (safeApply′ env k v args₁) safeApplySort env k (set t) _ = set (safeApply′ env k t []) safeApplySort env k (lit n) _ = lit n safeApplySort env k unknown _ = unknown safeApply : Term → List (Arg SafeTerm) → Term safeApply v args = safeApply′ [] 0 v args

39.15

117

0.632549

118bae1011168434f4ec61f010953cd45bd7f90d

443

agda

Agda

test/Succeed/Issue175.agda

pthariensflame/agda

222c4c64b2ccf8e0fc2498492731c15e8fef32d4

[ "BSD-3-Clause" ]

3

2015-03-28T14:51:03.000Z

2015-12-07T20:14:00.000Z

test/succeed/Issue175.agda

np/agda-git-experiment

20596e9dd9867166a64470dd24ea68925ff380ce

[ "MIT" ]

null

test/succeed/Issue175.agda

np/agda-git-experiment

20596e9dd9867166a64470dd24ea68925ff380ce

[ "MIT" ]

1

2019-03-05T20:02:38.000Z

module Issue175 where data List (A : Set) : Set where [] : List A _∷_ : A → List A → List A {-# BUILTIN LIST List #-} {-# BUILTIN NIL [] #-} {-# BUILTIN CONS _∷_ #-} data _≡_ {A : Set} (x : A) : A → Set where refl : x ≡ x postulate Char : Set String : Set {-# BUILTIN CHAR Char #-} {-# BUILTIN STRING String #-} primitive primStringToList : String → List Char lemma : primStringToList "0" ≡ ('0' ∷ []) lemma = refl

17.72

47

0.568849

21dcf9bda3f9c76e9d7d6b8d5a69f2b462bd7f19

2,470

agda

Agda

Cubical/HITs/Rationals/QuoQ/Base.agda

maxdore/cubical

ef62b84397396d48135d73ba7400b71c721ddc94

[ "MIT" ]

null

Cubical/HITs/Rationals/QuoQ/Base.agda

maxdore/cubical

ef62b84397396d48135d73ba7400b71c721ddc94

[ "MIT" ]

null

Cubical/HITs/Rationals/QuoQ/Base.agda

maxdore/cubical

ef62b84397396d48135d73ba7400b71c721ddc94

[ "MIT" ]

1

2021-03-12T20:08:45.000Z

{-# OPTIONS --safe #-} module Cubical.HITs.Rationals.QuoQ.Base where open import Cubical.Foundations.Prelude open import Cubical.Foundations.Equiv open import Cubical.Data.Nat as ℕ using (discreteℕ) open import Cubical.Data.NatPlusOne open import Cubical.Data.Sigma open import Cubical.HITs.Ints.QuoInt open import Cubical.HITs.SetQuotients as SetQuotient using ([_]; eq/; discreteSetQuotients) renaming (_/_ to _//_) public open import Cubical.Relation.Nullary open import Cubical.Relation.Binary.Base open BinaryRelation ℕ₊₁→ℤ : ℕ₊₁ → ℤ ℕ₊₁→ℤ n = pos (ℕ₊₁→ℕ n) private ℕ₊₁→ℤ-hom : ∀ m n → ℕ₊₁→ℤ (m ·₊₁ n) ≡ ℕ₊₁→ℤ m · ℕ₊₁→ℤ n ℕ₊₁→ℤ-hom _ _ = refl -- ℚ as a set quotient of ℤ × ℕ₊₁ (as in the HoTT book) _∼_ : ℤ × ℕ₊₁ → ℤ × ℕ₊₁ → Type₀ (a , b) ∼ (c , d) = a · ℕ₊₁→ℤ d ≡ c · ℕ₊₁→ℤ b ℚ : Type₀ ℚ = (ℤ × ℕ₊₁) // _∼_ isSetℚ : isSet ℚ isSetℚ = SetQuotient.squash/ [_/_] : ℤ → ℕ₊₁ → ℚ [ a / b ] = [ a , b ] isEquivRel∼ : isEquivRel _∼_ isEquivRel.reflexive isEquivRel∼ (a , b) = refl isEquivRel.symmetric isEquivRel∼ (a , b) (c , d) = sym isEquivRel.transitive isEquivRel∼ (a , b) (c , d) (e , f) p q = ·-injʳ _ _ _ r where r = (a · ℕ₊₁→ℤ f) · ℕ₊₁→ℤ d ≡[ i ]⟨ ·-comm a (ℕ₊₁→ℤ f) i · ℕ₊₁→ℤ d ⟩ (ℕ₊₁→ℤ f · a) · ℕ₊₁→ℤ d ≡⟨ sym (·-assoc (ℕ₊₁→ℤ f) a (ℕ₊₁→ℤ d)) ⟩ ℕ₊₁→ℤ f · (a · ℕ₊₁→ℤ d) ≡[ i ]⟨ ℕ₊₁→ℤ f · p i ⟩ ℕ₊₁→ℤ f · (c · ℕ₊₁→ℤ b) ≡⟨ ·-assoc (ℕ₊₁→ℤ f) c (ℕ₊₁→ℤ b) ⟩ (ℕ₊₁→ℤ f · c) · ℕ₊₁→ℤ b ≡[ i ]⟨ ·-comm (ℕ₊₁→ℤ f) c i · ℕ₊₁→ℤ b ⟩ (c · ℕ₊₁→ℤ f) · ℕ₊₁→ℤ b ≡[ i ]⟨ q i · ℕ₊₁→ℤ b ⟩ (e · ℕ₊₁→ℤ d) · ℕ₊₁→ℤ b ≡⟨ sym (·-assoc e (ℕ₊₁→ℤ d) (ℕ₊₁→ℤ b)) ⟩ e · (ℕ₊₁→ℤ d · ℕ₊₁→ℤ b) ≡[ i ]⟨ e · ·-comm (ℕ₊₁→ℤ d) (ℕ₊₁→ℤ b) i ⟩ e · (ℕ₊₁→ℤ b · ℕ₊₁→ℤ d) ≡⟨ ·-assoc e (ℕ₊₁→ℤ b) (ℕ₊₁→ℤ d) ⟩ (e · ℕ₊₁→ℤ b) · ℕ₊₁→ℤ d ∎ eq/⁻¹ : ∀ x y → Path ℚ [ x ] [ y ] → x ∼ y eq/⁻¹ = SetQuotient.effective (λ _ _ → isSetℤ _ _) isEquivRel∼ discreteℚ : Discrete ℚ discreteℚ = discreteSetQuotients (discreteΣ discreteℤ (λ _ → subst Discrete 1+Path discreteℕ)) (λ _ _ → isSetℤ _ _) isEquivRel∼ (λ _ _ → discreteℤ _ _) -- Natural number and negative integer literals for ℚ open import Cubical.Data.Nat.Literals public instance fromNatℚ : HasFromNat ℚ fromNatℚ = record { Constraint = λ _ → Unit ; fromNat = λ n → [ pos n / 1 ] } instance fromNegℚ : HasFromNeg ℚ fromNegℚ = record { Constraint = λ _ → Unit ; fromNeg = λ n → [ neg n / 1 ] }

32.077922

94

0.554251

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196

agda

Agda

test/Fail/NoPatternMatching.agda

mdimjasevic/agda

8fb548356b275c7a1e79b768b64511ae937c738b

[ "BSD-3-Clause" ]

1,989

2015-01-09T23:51:16.000Z

2022-03-30T18:20:48.000Z

test/Fail/NoPatternMatching.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

4,066

2015-01-10T11:24:51.000Z

2022-03-31T21:14:49.000Z

test/Fail/NoPatternMatching.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

371

2015-01-03T14:04:08.000Z

2022-03-30T19:00:30.000Z

{-# OPTIONS --no-pattern-matching #-} id : {A : Set} (x : A) → A id x = x data Unit : Set where unit : Unit fail : Unit → Set fail unit = Unit -- Expected error: Pattern matching is disabled

16.333333

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4,046

agda

Agda

Cubical/Codata/Stream/Properties.agda

kiana-S/univalent-foundations

8bdb0766260489f9c89a14f4c0f2ad26e5190dec

[ "MIT" ]

null

Cubical/Codata/Stream/Properties.agda

kiana-S/univalent-foundations

8bdb0766260489f9c89a14f4c0f2ad26e5190dec

[ "MIT" ]

1

2022-01-27T02:07:48.000Z

Cubical/Codata/Stream/Properties.agda

kiana-S/univalent-foundations

8bdb0766260489f9c89a14f4c0f2ad26e5190dec

[ "MIT" ]

1

2021-11-22T02:02:01.000Z

{-# OPTIONS --cubical --no-import-sorts --safe --guardedness #-} module Cubical.Codata.Stream.Properties where open import Cubical.Core.Everything open import Cubical.Data.Nat open import Cubical.Foundations.Prelude open import Cubical.Foundations.Equiv open import Cubical.Foundations.Isomorphism open import Cubical.Foundations.Univalence open import Cubical.Codata.Stream.Base open Stream mapS : ∀ {A B} → (A → B) → Stream A → Stream B head (mapS f xs) = f (head xs) tail (mapS f xs) = mapS f (tail xs) even : ∀ {A} → Stream A → Stream A head (even a) = head a tail (even a) = even (tail (tail a)) odd : ∀ {A} → Stream A → Stream A head (odd a) = head (tail a) tail (odd a) = odd (tail (tail a)) merge : ∀ {A} → Stream A → Stream A → Stream A head (merge a _) = head a head (tail (merge _ b)) = head b tail (tail (merge a b)) = merge (tail a) (tail b) mapS-id : ∀ {A} {xs : Stream A} → mapS (λ x → x) xs ≡ xs head (mapS-id {xs = xs} i) = head xs tail (mapS-id {xs = xs} i) = mapS-id {xs = tail xs} i Stream-η : ∀ {A} {xs : Stream A} → xs ≡ (head xs , tail xs) head (Stream-η {A} {xs} i) = head xs tail (Stream-η {A} {xs} i) = tail xs elimS : ∀ {A} (P : Stream A → Type₀) (c : ∀ x xs → P (x , xs)) (xs : Stream A) → P xs elimS P c xs = transp (λ i → P (Stream-η {xs = xs} (~ i))) i0 (c (head xs) (tail xs)) odd≡even∘tail : ∀ {A} → (a : Stream A) → odd a ≡ even (tail a) head (odd≡even∘tail a i) = head (tail a) tail (odd≡even∘tail a i) = odd≡even∘tail (tail (tail a)) i mergeEvenOdd≡id : ∀ {A} → (a : Stream A) → merge (even a) (odd a) ≡ a head (mergeEvenOdd≡id a i) = head a head (tail (mergeEvenOdd≡id a i)) = head (tail a) tail (tail (mergeEvenOdd≡id a i)) = mergeEvenOdd≡id (tail (tail a)) i module Equality≅Bisimulation where -- Bisimulation record _≈_ {A : Type₀} (x y : Stream A) : Type₀ where coinductive field ≈head : head x ≡ head y ≈tail : tail x ≈ tail y open _≈_ bisim : {A : Type₀} → {x y : Stream A} → x ≈ y → x ≡ y head (bisim x≈y i) = ≈head x≈y i tail (bisim x≈y i) = bisim (≈tail x≈y) i misib : {A : Type₀} → {x y : Stream A} → x ≡ y → x ≈ y ≈head (misib p) = λ i → head (p i) ≈tail (misib p) = misib (λ i → tail (p i)) iso1 : {A : Type₀} → {x y : Stream A} → (p : x ≡ y) → bisim (misib p) ≡ p head (iso1 p i j) = head (p j) tail (iso1 p i j) = iso1 (λ i → tail (p i)) i j iso2 : {A : Type₀} → {x y : Stream A} → (p : x ≈ y) → misib (bisim p) ≡ p ≈head (iso2 p i) = ≈head p ≈tail (iso2 p i) = iso2 (≈tail p) i path≃bisim : {A : Type₀} → {x y : Stream A} → (x ≡ y) ≃ (x ≈ y) path≃bisim = isoToEquiv (iso misib bisim iso2 iso1) path≡bisim : {A : Type₀} → {x y : Stream A} → (x ≡ y) ≡ (x ≈ y) path≡bisim = ua path≃bisim -- misib can be implemented by transport as well. refl≈ : {A : Type₀} {x : Stream A} → x ≈ x ≈head refl≈ = refl ≈tail refl≈ = refl≈ cast : ∀ {A : Type₀} {x y : Stream A} (p : x ≡ y) → x ≈ y cast {x = x} p = transport (λ i → x ≈ p i) refl≈ misib-refl : ∀ {A : Type₀} {x : Stream A} → misib {x = x} refl ≡ refl≈ ≈head (misib-refl i) = refl ≈tail (misib-refl i) = misib-refl i misibTransp : ∀ {A : Type₀} {x y : Stream A} (p : x ≡ y) → cast p ≡ misib p misibTransp p = J (λ _ p → cast p ≡ misib p) ((transportRefl refl≈) ∙ (sym misib-refl)) p module Stream≅Nat→ {A : Type₀} where lookup : Stream A → ℕ → A lookup xs zero = head xs lookup xs (suc n) = lookup (tail xs) n tabulate : (ℕ → A) → Stream A head (tabulate f) = f zero tail (tabulate f) = tabulate (λ n → f (suc n)) lookup∘tabulate : (λ (x : _ → A) → lookup (tabulate x)) ≡ (λ x → x) lookup∘tabulate i f zero = f zero lookup∘tabulate i f (suc n) = lookup∘tabulate i (λ n → f (suc n)) n tabulate∘lookup : (λ (x : Stream A) → tabulate (lookup x)) ≡ (λ x → x) head (tabulate∘lookup i xs) = head xs tail (tabulate∘lookup i xs) = tabulate∘lookup i (tail xs) Stream≡Nat→ : Stream A ≡ (ℕ → A) Stream≡Nat→ = isoToPath (iso lookup tabulate (λ f i → lookup∘tabulate i f) (λ xs i → tabulate∘lookup i xs))

33.163934

109

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agda

Agda

src/Prelude/Nat/Properties.agda

t-more/agda-prelude

da4fca7744d317b8843f2bc80a923972f65548d3

[ "MIT" ]

111

2015-01-05T11:28:15.000Z

2022-02-12T23:29:26.000Z

src/Prelude/Nat/Properties.agda

t-more/agda-prelude

da4fca7744d317b8843f2bc80a923972f65548d3

[ "MIT" ]

59

2016-02-09T05:36:44.000Z

2022-01-14T07:32:36.000Z

src/Prelude/Nat/Properties.agda

t-more/agda-prelude

da4fca7744d317b8843f2bc80a923972f65548d3

[ "MIT" ]

24

2015-03-12T18:03:45.000Z

2021-04-22T06:10:41.000Z

module Prelude.Nat.Properties where open import Prelude.Bool open import Prelude.Nat.Core open import Prelude.Equality open import Prelude.Semiring suc-inj : ∀ {n m} → suc n ≡ suc m → n ≡ m suc-inj refl = refl --- Addition --- add-zero-r : (n : Nat) → n + 0 ≡ n add-zero-r zero = refl add-zero-r (suc n) = suc $≡ add-zero-r n add-suc-r : (n m : Nat) → n + suc m ≡ suc (n + m) add-suc-r zero m = refl add-suc-r (suc n) m = suc $≡ add-suc-r n m add-commute : (a b : Nat) → a + b ≡ b + a add-commute zero b = sym (add-zero-r _) add-commute (suc a) b = suc $≡ add-commute a b ⟨≡⟩ʳ add-suc-r b _ add-assoc : (a b c : Nat) → a + (b + c) ≡ a + b + c add-assoc zero b c = refl add-assoc (suc a) b c = suc $≡ add-assoc a b c add-inj₂ : (a b c : Nat) → a + b ≡ a + c → b ≡ c add-inj₂ zero b c eq = eq add-inj₂ (suc a) b c eq = add-inj₂ a b c (suc-inj eq) add-inj₁ : (a b c : Nat) → a + c ≡ b + c → a ≡ b add-inj₁ a b c eq = add-inj₂ c a b (add-commute c a ⟨≡⟩ eq ⟨≡⟩ add-commute b c) --- Subtraction --- --- Multiplication --- mul-one-r : (x : Nat) → x * 1 ≡ x mul-one-r zero = refl mul-one-r (suc x) = suc $≡ mul-one-r x mul-zero-r : (x : Nat) → x * 0 ≡ 0 mul-zero-r zero = refl mul-zero-r (suc x) = mul-zero-r x mul-distr-r : (x y z : Nat) → (x + y) * z ≡ x * z + y * z mul-distr-r zero y z = refl mul-distr-r (suc x) y z = z +_ $≡ mul-distr-r x y z ⟨≡⟩ add-assoc z _ _ private shuffle : (a b c d : Nat) → a + b + (c + d) ≡ a + c + (b + d) shuffle a b c d = add-assoc a _ _ ʳ⟨≡⟩ a +_ $≡ (add-assoc b c d ⟨≡⟩ _+ d $≡ add-commute b c ⟨≡⟩ʳ add-assoc c b d) ⟨≡⟩ add-assoc a _ _ mul-distr-l : (x y z : Nat) → x * (y + z) ≡ x * y + x * z mul-distr-l zero y z = refl mul-distr-l (suc x) y z = y + z +_ $≡ mul-distr-l x y z ⟨≡⟩ shuffle y z (x * y) (x * z) mul-assoc : (x y z : Nat) → x * (y * z) ≡ x * y * z mul-assoc zero y z = refl mul-assoc (suc x) y z = y * z +_ $≡ mul-assoc x y z ⟨≡⟩ʳ mul-distr-r y (x * y) z mul-commute : (x y : Nat) → x * y ≡ y * x mul-commute x zero = mul-zero-r x mul-commute x (suc y) = mul-distr-l x 1 y ⟨≡⟩ _+ x * y $≡ mul-one-r x ⟨≡⟩ x +_ $≡ mul-commute x y mul-inj₁ : (x y z : Nat) {{_ : NonZero z}} → x * z ≡ y * z → x ≡ y mul-inj₁ x y zero {{}} mul-inj₁ zero zero (suc z) eq = refl mul-inj₁ zero (suc y) (suc z) () mul-inj₁ (suc x) zero (suc z) () mul-inj₁ (suc x) (suc y) (suc z) eq = suc $≡ mul-inj₁ x y (suc z) (add-inj₂ z _ _ (suc-inj eq)) mul-inj₂ : (x y z : Nat) {{_ : NonZero x}} → x * y ≡ x * z → y ≡ z mul-inj₂ x y z eq = mul-inj₁ y z x (mul-commute y x ⟨≡⟩ eq ⟨≡⟩ mul-commute x z)

32.8875

98

0.522995

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500

agda

Agda

test/Succeed/Issue557.agda

mdimjasevic/agda

8fb548356b275c7a1e79b768b64511ae937c738b

[ "BSD-3-Clause" ]

1,989

2015-01-09T23:51:16.000Z

2022-03-30T18:20:48.000Z

test/Succeed/Issue557.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

4,066

2015-01-10T11:24:51.000Z

2022-03-31T21:14:49.000Z

test/Succeed/Issue557.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

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2015-01-03T14:04:08.000Z

2022-03-30T19:00:30.000Z

-- Andreas, 2012-01-30, bug reported by Nisse -- {-# OPTIONS -v tc.term.absurd:50 -v tc.signature:30 -v tc.conv.atom:30 -v tc.conv.elim:50 #-} module Issue557 where data ⊥ : Set where postulate A : Set a : (⊥ → ⊥) → A F : A → Set f : (a : A) → F a module M (I : Set → Set) where x : A x = a (λ ()) y : A y = M.x (λ A → A) z : F y z = f y -- cause was absurd lambda in a module, i.e., under a telescope (I : Set -> Set) -- (λ ()) must be replaced by (absurd I) not just by (absurd)

19.230769

96

0.562

6574976a0382a4cf9e2173f8d15f7d53001874d6

1,469

agda

Agda

Rings/Orders/Partial/Bounded.agda

Smaug123/agdaproofs

0f4230011039092f58f673abcad8fb0652e6b562

[ "MIT" ]

4

2019-08-08T12:44:19.000Z

2022-01-28T06:04:15.000Z

Rings/Orders/Partial/Bounded.agda

Smaug123/agdaproofs

0f4230011039092f58f673abcad8fb0652e6b562

[ "MIT" ]

14

2019-01-06T21:11:59.000Z

2020-04-11T11:03:39.000Z

Rings/Orders/Partial/Bounded.agda

Smaug123/agdaproofs

0f4230011039092f58f673abcad8fb0652e6b562

[ "MIT" ]

1

2021-11-29T13:23:07.000Z

{-# OPTIONS --safe --warning=error --without-K --guardedness #-} open import Agda.Primitive using (Level; lzero; lsuc; _⊔_) open import Setoids.Setoids open import Rings.Definition open import Rings.Orders.Partial.Definition open import Sets.EquivalenceRelations open import Sequences open import Setoids.Orders.Partial.Definition open import Functions.Definition open import LogicalFormulae open import Numbers.Naturals.Semiring open import Groups.Definition module Rings.Orders.Partial.Bounded {m n o : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {m} {o} A} {pOrder : SetoidPartialOrder S _<_} {R : Ring S _+_ _*_} (pRing : PartiallyOrderedRing R pOrder) where open Group (Ring.additiveGroup R) open import Groups.Lemmas (Ring.additiveGroup R) open Setoid S open Equivalence eq open SetoidPartialOrder pOrder BoundedAbove : Sequence A → Set (m ⊔ o) BoundedAbove x = Sg A (λ K → (n : ℕ) → index x n < K) BoundedBelow : Sequence A → Set (m ⊔ o) BoundedBelow x = Sg A (λ K → (n : ℕ) → K < index x n) Bounded : Sequence A → Set (m ⊔ o) Bounded x = Sg A (λ K → (n : ℕ) → ((Group.inverse (Ring.additiveGroup R) K) < index x n) && (index x n < K)) boundNonzero : {s : Sequence A} → (b : Bounded s) → underlying b ∼ 0G → False boundNonzero {s} (a , b) isEq with b 0 ... | bad1 ,, bad2 = irreflexive (<Transitive bad1 (<WellDefined reflexive (transitive isEq (symmetric (transitive (inverseWellDefined isEq) invIdent))) bad2))

40.805556

243

0.699796

d1da78a7b73e8a7a3d26da9a73584def9982a6b6

5,589

agda

Agda

core/lib/types/CommutingSquare.agda

mikeshulman/HoTT-Agda

e7d663b63d89f380ab772ecb8d51c38c26952dbb

[ "MIT" ]

null

core/lib/types/CommutingSquare.agda

mikeshulman/HoTT-Agda

e7d663b63d89f380ab772ecb8d51c38c26952dbb

[ "MIT" ]

null

core/lib/types/CommutingSquare.agda

mikeshulman/HoTT-Agda

e7d663b63d89f380ab772ecb8d51c38c26952dbb

[ "MIT" ]

1

2018-12-26T21:31:57.000Z

{-# OPTIONS --without-K --rewriting #-} open import lib.Basics open import lib.types.Sigma open import lib.types.Paths module lib.types.CommutingSquare where {- maps between two functions -} infix 0 _□$_ _□$_ = CommSquare.commutes CommSquare-∘v : ∀ {i₀ i₁ i₂ j₀ j₁ j₂} {A₀ : Type i₀} {A₁ : Type i₁} {A₂ : Type i₂} {B₀ : Type j₀} {B₁ : Type j₁} {B₂ : Type j₂} {f₀ : A₀ → B₀} {f₁ : A₁ → B₁} {f₂ : A₂ → B₂} {hA : A₀ → A₁} {hB : B₀ → B₁} {kA : A₁ → A₂} {kB : B₁ → B₂} → CommSquare f₁ f₂ kA kB → CommSquare f₀ f₁ hA hB → CommSquare f₀ f₂ (kA ∘ hA) (kB ∘ hB) CommSquare-∘v {hA = hA} {kB = kB} (comm-sqr □₁₂) (comm-sqr □₀₁) = comm-sqr λ a₀ → ap kB (□₀₁ a₀) ∙ □₁₂ (hA a₀) CommSquare-inverse-v : ∀ {i₀ i₁ j₀ j₁} {A₀ : Type i₀} {A₁ : Type i₁} {B₀ : Type j₀} {B₁ : Type j₁} {f₀ : A₀ → B₀} {f₁ : A₁ → B₁} {hA : A₀ → A₁} {hB : B₀ → B₁} → CommSquare f₀ f₁ hA hB → (hA-ise : is-equiv hA) (hB-ise : is-equiv hB) → CommSquare f₁ f₀ (is-equiv.g hA-ise) (is-equiv.g hB-ise) CommSquare-inverse-v {f₀ = f₀} {f₁} {hA} {hB} (comm-sqr □) hA-ise hB-ise = comm-sqr λ a₁ → ap hB.g (! (□ (hA.g a₁) ∙ ap f₁ (hA.f-g a₁))) ∙ hB.g-f (f₀ (hA.g a₁)) where module hA = is-equiv hA-ise module hB = is-equiv hB-ise abstract -- 'r' with respect to '∘v' CommSquare-inverse-inv-r : ∀ {i₀ i₁ j₀ j₁} {A₀ : Type i₀} {A₁ : Type i₁} {B₀ : Type j₀} {B₁ : Type j₁} {f₀ : A₀ → B₀} {f₁ : A₁ → B₁} {hA : A₀ → A₁} {hB : B₀ → B₁} (cs : CommSquare f₀ f₁ hA hB) (hA-ise : is-equiv hA) (hB-ise : is-equiv hB) → ∀ a₁ → (CommSquare-∘v cs (CommSquare-inverse-v cs hA-ise hB-ise) □$ a₁) == is-equiv.f-g hB-ise (f₁ a₁) ∙ ! (ap f₁ (is-equiv.f-g hA-ise a₁)) CommSquare-inverse-inv-r {f₀ = f₀} {f₁} {hA} {hB} (comm-sqr □) hA-ise hB-ise a₁ = ap hB ( ap hB.g (! (□ (hA.g a₁) ∙ ap f₁ (hA.f-g a₁))) ∙ hB.g-f (f₀ (hA.g a₁))) ∙ □ (hA.g a₁) =⟨ ap-∙ hB (ap hB.g (! (□ (hA.g a₁) ∙ ap f₁ (hA.f-g a₁)))) (hB.g-f (f₀ (hA.g a₁))) |in-ctx _∙ □ (hA.g a₁) ⟩ ( ap hB (ap hB.g (! (□ (hA.g a₁) ∙ ap f₁ (hA.f-g a₁)))) ∙ ap hB (hB.g-f (f₀ (hA.g a₁)))) ∙ □ (hA.g a₁) =⟨ ap2 _∙_ (∘-ap hB hB.g (! (□ (hA.g a₁) ∙ ap f₁ (hA.f-g a₁)))) (hB.adj (f₀ (hA.g a₁))) |in-ctx _∙ □ (hA.g a₁) ⟩ ( ap (hB ∘ hB.g) (! (□ (hA.g a₁) ∙ ap f₁ (hA.f-g a₁))) ∙ hB.f-g (hB (f₀ (hA.g a₁)))) ∙ □ (hA.g a₁) =⟨ ! (↓-app=idf-out $ apd hB.f-g (! (□ (hA.g a₁) ∙ ap f₁ (hA.f-g a₁)))) |in-ctx _∙ □ (hA.g a₁) ⟩ ( hB.f-g (f₁ a₁) ∙' (! (□ (hA.g a₁) ∙ ap f₁ (hA.f-g a₁)))) ∙ □ (hA.g a₁) =⟨ lemma (hB.f-g (f₁ a₁)) (□ (hA.g a₁)) (ap f₁ (hA.f-g a₁)) ⟩ hB.f-g (f₁ a₁) ∙ ! (ap f₁ (hA.f-g a₁)) =∎ where module hA = is-equiv hA-ise module hB = is-equiv hB-ise lemma : ∀ {i} {A : Type i} {a₀ a₁ a₂ a₃ : A} (p₀ : a₀ == a₁) (p₁ : a₃ == a₂) (p₂ : a₂ == a₁) → (p₀ ∙' (! (p₁ ∙ p₂))) ∙ p₁ == p₀ ∙ ! p₂ lemma idp idp idp = idp -- 'l' with respect to '∘v' CommSquare-inverse-inv-l : ∀ {i₀ i₁ j₀ j₁} {A₀ : Type i₀} {A₁ : Type i₁} {B₀ : Type j₀} {B₁ : Type j₁} {f₀ : A₀ → B₀} {f₁ : A₁ → B₁} {hA : A₀ → A₁} {hB : B₀ → B₁} (cs : CommSquare f₀ f₁ hA hB) (hA-ise : is-equiv hA) (hB-ise : is-equiv hB) → ∀ a₀ → (CommSquare-∘v (CommSquare-inverse-v cs hA-ise hB-ise) cs □$ a₀) == is-equiv.g-f hB-ise (f₀ a₀) ∙ ! (ap f₀ (is-equiv.g-f hA-ise a₀)) CommSquare-inverse-inv-l {f₀ = f₀} {f₁} {hA} {hB} (comm-sqr □) hA-ise hB-ise a₀ = ap hB.g (□ a₀) ∙ ( ap hB.g (! (□ (hA.g (hA a₀)) ∙ ap f₁ (hA.f-g (hA a₀)))) ∙ hB.g-f (f₀ (hA.g (hA a₀)))) =⟨ ! (hA.adj a₀) |in-ctx ap f₁ |in-ctx □ (hA.g (hA a₀)) ∙_ |in-ctx ! |in-ctx ap hB.g |in-ctx _∙ hB.g-f (f₀ (hA.g (hA a₀))) |in-ctx ap hB.g (□ a₀) ∙_ ⟩ ap hB.g (□ a₀) ∙ ( ap hB.g (! (□ (hA.g (hA a₀)) ∙ ap f₁ (ap hA (hA.g-f a₀)))) ∙ hB.g-f (f₀ (hA.g (hA a₀)))) =⟨ ∘-ap f₁ hA (hA.g-f a₀) |in-ctx □ (hA.g (hA a₀)) ∙_ |in-ctx ! |in-ctx ap hB.g |in-ctx _∙ hB.g-f (f₀ (hA.g (hA a₀))) |in-ctx ap hB.g (□ a₀) ∙_ ⟩ ap hB.g (□ a₀) ∙ ( ap hB.g (! (□ (hA.g (hA a₀)) ∙ ap (f₁ ∘ hA) (hA.g-f a₀))) ∙ hB.g-f (f₀ (hA.g (hA a₀)))) =⟨ ↓-='-out' (apd □ (hA.g-f a₀)) |in-ctx ! |in-ctx ap hB.g |in-ctx _∙ hB.g-f (f₀ (hA.g (hA a₀))) |in-ctx ap hB.g (□ a₀) ∙_ ⟩ ap hB.g (□ a₀) ∙ ( ap hB.g (! (ap (hB ∘ f₀) (hA.g-f a₀) ∙' □ a₀)) ∙ hB.g-f (f₀ (hA.g (hA a₀)))) =⟨ lemma hB.g (□ a₀) (ap (hB ∘ f₀) (hA.g-f a₀)) (hB.g-f (f₀ (hA.g (hA a₀)))) ⟩ ! (ap hB.g (ap (hB ∘ f₀) (hA.g-f a₀))) ∙' hB.g-f (f₀ (hA.g (hA a₀))) =⟨ ∘-ap hB.g (hB ∘ f₀) (hA.g-f a₀) |in-ctx ! |in-ctx _∙' hB.g-f (f₀ (hA.g (hA a₀))) ⟩ ! (ap (hB.g ∘ hB ∘ f₀) (hA.g-f a₀)) ∙' hB.g-f (f₀ (hA.g (hA a₀))) =⟨ !-ap (hB.g ∘ hB ∘ f₀) (hA.g-f a₀) |in-ctx _∙' hB.g-f (f₀ (hA.g (hA a₀))) ⟩ ap (hB.g ∘ hB ∘ f₀) (! (hA.g-f a₀)) ∙' hB.g-f (f₀ (hA.g (hA a₀))) =⟨ ! (↓-='-out' (apd (hB.g-f ∘ f₀) (! (hA.g-f a₀)))) ⟩ hB.g-f (f₀ a₀) ∙ ap f₀ (! (hA.g-f a₀)) =⟨ ap-! f₀ (hA.g-f a₀) |in-ctx hB.g-f (f₀ a₀) ∙_ ⟩ hB.g-f (f₀ a₀) ∙ ! (ap f₀ (hA.g-f a₀)) =∎ where module hA = is-equiv hA-ise module hB = is-equiv hB-ise lemma : ∀ {i j} {A : Type i} {B : Type j} (f : A → B) {a₀ a₁ a₂ : A} {b : B} (p₀ : a₀ == a₁) (p₁ : a₂ == a₀) (q₀ : f a₂ == b) → ap f p₀ ∙ (ap f (! (p₁ ∙' p₀)) ∙ q₀) == ! (ap f p₁) ∙' q₀ lemma f idp idp idp = idp

41.708955

88

0.449633

0d879ef9f4cc7b9c693e7f6c93da6c096ec3db39

14,523

agda

Agda

Cubical/Foundations/Path.agda

howsiyu/cubical

1b9c97a2140fe96fe636f4c66beedfd7b8096e8f

[ "MIT" ]

null

Cubical/Foundations/Path.agda

howsiyu/cubical

1b9c97a2140fe96fe636f4c66beedfd7b8096e8f

[ "MIT" ]

null

Cubical/Foundations/Path.agda

howsiyu/cubical

1b9c97a2140fe96fe636f4c66beedfd7b8096e8f

[ "MIT" ]

null

{-# OPTIONS --safe #-} module Cubical.Foundations.Path where open import Cubical.Foundations.Prelude open import Cubical.Foundations.Function open import Cubical.Foundations.GroupoidLaws open import Cubical.Foundations.Equiv open import Cubical.Foundations.Isomorphism open import Cubical.Foundations.Transport open import Cubical.Foundations.Univalence open import Cubical.Reflection.StrictEquiv private variable ℓ ℓ' : Level A : Type ℓ -- Less polymorphic version of `cong`, to avoid some unresolved metas cong′ : ∀ {B : Type ℓ'} (f : A → B) {x y : A} (p : x ≡ y) → Path B (f x) (f y) cong′ f = cong f {-# INLINE cong′ #-} module _ {A : I → Type ℓ} {x : A i0} {y : A i1} where toPathP⁻ : x ≡ transport⁻ (λ i → A i) y → PathP A x y toPathP⁻ p = symP (toPathP (sym p)) fromPathP⁻ : PathP A x y → x ≡ transport⁻ (λ i → A i) y fromPathP⁻ p = sym (fromPathP {A = λ i → A (~ i)} (symP p)) PathP≡Path : ∀ (P : I → Type ℓ) (p : P i0) (q : P i1) → PathP P p q ≡ Path (P i1) (transport (λ i → P i) p) q PathP≡Path P p q i = PathP (λ j → P (i ∨ j)) (transport-filler (λ j → P j) p i) q PathP≡Path⁻ : ∀ (P : I → Type ℓ) (p : P i0) (q : P i1) → PathP P p q ≡ Path (P i0) p (transport⁻ (λ i → P i) q) PathP≡Path⁻ P p q i = PathP (λ j → P (~ i ∧ j)) p (transport⁻-filler (λ j → P j) q i) PathPIsoPath : ∀ (A : I → Type ℓ) (x : A i0) (y : A i1) → Iso (PathP A x y) (transport (λ i → A i) x ≡ y) PathPIsoPath A x y .Iso.fun = fromPathP PathPIsoPath A x y .Iso.inv = toPathP PathPIsoPath A x y .Iso.rightInv q k i = hcomp (λ j → λ { (i = i0) → slide (j ∨ ~ k) ; (i = i1) → q j ; (k = i0) → transp (λ l → A (i ∨ l)) i (fromPathPFiller j) ; (k = i1) → ∧∨Square i j }) (transp (λ l → A (i ∨ ~ k ∨ l)) (i ∨ ~ k) (transp (λ l → (A (i ∨ (~ k ∧ l)))) (k ∨ i) (transp (λ l → A (i ∧ l)) (~ i) x))) where fromPathPFiller : _ fromPathPFiller = hfill (λ j → λ { (i = i0) → x ; (i = i1) → q j }) (inS (transp (λ j → A (i ∧ j)) (~ i) x)) slide : I → _ slide i = transp (λ l → A (i ∨ l)) i (transp (λ l → A (i ∧ l)) (~ i) x) ∧∨Square : I → I → _ ∧∨Square i j = hcomp (λ l → λ { (i = i0) → slide j ; (i = i1) → q (j ∧ l) ; (j = i0) → slide i ; (j = i1) → q (i ∧ l) }) (slide (i ∨ j)) PathPIsoPath A x y .Iso.leftInv q k i = outS (hcomp-unique (λ j → λ { (i = i0) → x ; (i = i1) → transp (λ l → A (j ∨ l)) j (q j) }) (inS (transp (λ l → A (i ∧ l)) (~ i) x)) (λ j → inS (transp (λ l → A (i ∧ (j ∨ l))) (~ i ∨ j) (q (i ∧ j))))) k PathP≃Path : (A : I → Type ℓ) (x : A i0) (y : A i1) → PathP A x y ≃ (transport (λ i → A i) x ≡ y) PathP≃Path A x y = isoToEquiv (PathPIsoPath A x y) PathP≡compPath : ∀ {A : Type ℓ} {x y z : A} (p : x ≡ y) (q : y ≡ z) (r : x ≡ z) → (PathP (λ i → x ≡ q i) p r) ≡ (p ∙ q ≡ r) PathP≡compPath p q r k = PathP (λ i → p i0 ≡ q (i ∨ k)) (λ j → compPath-filler p q k j) r -- a quick corollary for 3-constant functions 3-ConstantCompChar : {A : Type ℓ} {B : Type ℓ'} (f : A → B) (link : 2-Constant f) → (∀ x y z → link x y ∙ link y z ≡ link x z) → 3-Constant f 3-Constant.link (3-ConstantCompChar f link coh₂) = link 3-Constant.coh₁ (3-ConstantCompChar f link coh₂) _ _ _ = transport⁻ (PathP≡compPath _ _ _) (coh₂ _ _ _) PathP≡doubleCompPathˡ : ∀ {A : Type ℓ} {w x y z : A} (p : w ≡ y) (q : w ≡ x) (r : y ≡ z) (s : x ≡ z) → (PathP (λ i → p i ≡ s i) q r) ≡ (p ⁻¹ ∙∙ q ∙∙ s ≡ r) PathP≡doubleCompPathˡ p q r s k = PathP (λ i → p (i ∨ k) ≡ s (i ∨ k)) (λ j → doubleCompPath-filler (p ⁻¹) q s k j) r PathP≡doubleCompPathʳ : ∀ {A : Type ℓ} {w x y z : A} (p : w ≡ y) (q : w ≡ x) (r : y ≡ z) (s : x ≡ z) → (PathP (λ i → p i ≡ s i) q r) ≡ (q ≡ p ∙∙ r ∙∙ s ⁻¹) PathP≡doubleCompPathʳ p q r s k = PathP (λ i → p (i ∧ (~ k)) ≡ s (i ∧ (~ k))) q (λ j → doubleCompPath-filler p r (s ⁻¹) k j) compPathl-cancel : ∀ {ℓ} {A : Type ℓ} {x y z : A} (p : x ≡ y) (q : x ≡ z) → p ∙ (sym p ∙ q) ≡ q compPathl-cancel p q = p ∙ (sym p ∙ q) ≡⟨ assoc p (sym p) q ⟩ (p ∙ sym p) ∙ q ≡⟨ cong (_∙ q) (rCancel p) ⟩ refl ∙ q ≡⟨ sym (lUnit q) ⟩ q ∎ compPathr-cancel : ∀ {ℓ} {A : Type ℓ} {x y z : A} (p : z ≡ y) (q : x ≡ y) → (q ∙ sym p) ∙ p ≡ q compPathr-cancel {x = x} p q i j = hcomp-equivFiller (doubleComp-faces (λ _ → x) (sym p) j) (inS (q j)) (~ i) compPathl-isEquiv : {x y z : A} (p : x ≡ y) → isEquiv (λ (q : y ≡ z) → p ∙ q) compPathl-isEquiv p = isoToIsEquiv (iso (p ∙_) (sym p ∙_) (compPathl-cancel p) (compPathl-cancel (sym p))) compPathlEquiv : {x y z : A} (p : x ≡ y) → (y ≡ z) ≃ (x ≡ z) compPathlEquiv p = (p ∙_) , compPathl-isEquiv p compPathr-isEquiv : {x y z : A} (p : y ≡ z) → isEquiv (λ (q : x ≡ y) → q ∙ p) compPathr-isEquiv p = isoToIsEquiv (iso (_∙ p) (_∙ sym p) (compPathr-cancel p) (compPathr-cancel (sym p))) compPathrEquiv : {x y z : A} (p : y ≡ z) → (x ≡ y) ≃ (x ≡ z) compPathrEquiv p = (_∙ p) , compPathr-isEquiv p -- Variations of isProp→isSet for PathP isProp→SquareP : ∀ {B : I → I → Type ℓ} → ((i j : I) → isProp (B i j)) → {a : B i0 i0} {b : B i0 i1} {c : B i1 i0} {d : B i1 i1} → (r : PathP (λ j → B j i0) a c) (s : PathP (λ j → B j i1) b d) → (t : PathP (λ j → B i0 j) a b) (u : PathP (λ j → B i1 j) c d) → SquareP B t u r s isProp→SquareP {B = B} isPropB {a = a} r s t u i j = hcomp (λ { k (i = i0) → isPropB i0 j (base i0 j) (t j) k ; k (i = i1) → isPropB i1 j (base i1 j) (u j) k ; k (j = i0) → isPropB i i0 (base i i0) (r i) k ; k (j = i1) → isPropB i i1 (base i i1) (s i) k }) (base i j) where base : (i j : I) → B i j base i j = transport (λ k → B (i ∧ k) (j ∧ k)) a isProp→isPropPathP : ∀ {ℓ} {B : I → Type ℓ} → ((i : I) → isProp (B i)) → (b0 : B i0) (b1 : B i1) → isProp (PathP (λ i → B i) b0 b1) isProp→isPropPathP {B = B} hB b0 b1 = isProp→SquareP (λ _ → hB) refl refl isProp→isContrPathP : {A : I → Type ℓ} → (∀ i → isProp (A i)) → (x : A i0) (y : A i1) → isContr (PathP A x y) isProp→isContrPathP h x y = isProp→PathP h x y , isProp→isPropPathP h x y _ -- Flipping a square along its diagonal flipSquare : {a₀₀ a₀₁ : A} {a₀₋ : a₀₀ ≡ a₀₁} {a₁₀ a₁₁ : A} {a₁₋ : a₁₀ ≡ a₁₁} {a₋₀ : a₀₀ ≡ a₁₀} {a₋₁ : a₀₁ ≡ a₁₁} → Square a₀₋ a₁₋ a₋₀ a₋₁ → Square a₋₀ a₋₁ a₀₋ a₁₋ flipSquare sq i j = sq j i module _ {a₀₀ a₀₁ : A} {a₀₋ : a₀₀ ≡ a₀₁} {a₁₀ a₁₁ : A} {a₁₋ : a₁₀ ≡ a₁₁} {a₋₀ : a₀₀ ≡ a₁₀} {a₋₁ : a₀₁ ≡ a₁₁} where flipSquareEquiv : Square a₀₋ a₁₋ a₋₀ a₋₁ ≃ Square a₋₀ a₋₁ a₀₋ a₁₋ unquoteDef flipSquareEquiv = defStrictEquiv flipSquareEquiv flipSquare flipSquare flipSquarePath : Square a₀₋ a₁₋ a₋₀ a₋₁ ≡ Square a₋₀ a₋₁ a₀₋ a₁₋ flipSquarePath = ua flipSquareEquiv module _ {a₀₀ a₁₁ : A} {a₋ : a₀₀ ≡ a₁₁} {a₁₀ : A} {a₁₋ : a₁₀ ≡ a₁₁} {a₋₀ : a₀₀ ≡ a₁₀} where slideSquareFaces : (i j k : I) → Partial (i ∨ ~ i ∨ j ∨ ~ j) A slideSquareFaces i j k (i = i0) = a₋ (j ∧ ~ k) slideSquareFaces i j k (i = i1) = a₁₋ j slideSquareFaces i j k (j = i0) = a₋₀ i slideSquareFaces i j k (j = i1) = a₋ (i ∨ ~ k) slideSquare : Square a₋ a₁₋ a₋₀ refl → Square refl a₁₋ a₋₀ a₋ slideSquare sq i j = hcomp (slideSquareFaces i j) (sq i j) slideSquareEquiv : (Square a₋ a₁₋ a₋₀ refl) ≃ (Square refl a₁₋ a₋₀ a₋) slideSquareEquiv = isoToEquiv (iso slideSquare slideSquareInv fillerTo fillerFrom) where slideSquareInv : Square refl a₁₋ a₋₀ a₋ → Square a₋ a₁₋ a₋₀ refl slideSquareInv sq i j = hcomp (λ k → slideSquareFaces i j (~ k)) (sq i j) fillerTo : ∀ p → slideSquare (slideSquareInv p) ≡ p fillerTo p k i j = hcomp-equivFiller (λ k → slideSquareFaces i j (~ k)) (inS (p i j)) (~ k) fillerFrom : ∀ p → slideSquareInv (slideSquare p) ≡ p fillerFrom p k i j = hcomp-equivFiller (slideSquareFaces i j) (inS (p i j)) (~ k) -- The type of fillers of a square is equivalent to the double composition identites Square≃doubleComp : {a₀₀ a₀₁ a₁₀ a₁₁ : A} (a₀₋ : a₀₀ ≡ a₀₁) (a₁₋ : a₁₀ ≡ a₁₁) (a₋₀ : a₀₀ ≡ a₁₀) (a₋₁ : a₀₁ ≡ a₁₁) → Square a₀₋ a₁₋ a₋₀ a₋₁ ≃ (a₋₀ ⁻¹ ∙∙ a₀₋ ∙∙ a₋₁ ≡ a₁₋) Square≃doubleComp a₀₋ a₁₋ a₋₀ a₋₁ = transportEquiv (PathP≡doubleCompPathˡ a₋₀ a₀₋ a₁₋ a₋₁) -- Flipping a square in Ω²A is the same as inverting it sym≡flipSquare : {x : A} (P : Square (refl {x = x}) refl refl refl) → sym P ≡ flipSquare P sym≡flipSquare {x = x} P = sym (main refl P) where B : (q : x ≡ x) → I → Type _ B q i = PathP (λ j → x ≡ q (i ∨ j)) (λ k → q (i ∧ k)) refl main : (q : x ≡ x) (p : refl ≡ q) → PathP (λ i → B q i) (λ i j → p j i) (sym p) main q = J (λ q p → PathP (λ i → B q i) (λ i j → p j i) (sym p)) refl -- Inverting both interval arguments of a square in Ω²A is the same as doing nothing sym-cong-sym≡id : {x : A} (P : Square (refl {x = x}) refl refl refl) → P ≡ λ i j → P (~ i) (~ j) sym-cong-sym≡id {x = x} P = sym (main refl P) where B : (q : x ≡ x) → I → Type _ B q i = Path (x ≡ q i) (λ j → q (i ∨ ~ j)) λ j → q (i ∧ j) main : (q : x ≡ x) (p : refl ≡ q) → PathP (λ i → B q i) (λ i j → p (~ i) (~ j)) p main q = J (λ q p → PathP (λ i → B q i) (λ i j → p (~ i) (~ j)) p) refl -- Applying cong sym is the same as flipping a square in Ω²A flipSquare≡cong-sym : ∀ {ℓ} {A : Type ℓ} {x : A} (P : Square (refl {x = x}) refl refl refl) → flipSquare P ≡ λ i j → P i (~ j) flipSquare≡cong-sym P = sym (sym≡flipSquare P) ∙ sym (sym-cong-sym≡id (cong sym P)) -- Applying cong sym is the same as inverting a square in Ω²A sym≡cong-sym : ∀ {ℓ} {A : Type ℓ} {x : A} (P : Square (refl {x = x}) refl refl refl) → sym P ≡ cong sym P sym≡cong-sym P = sym-cong-sym≡id (sym P) -- sym induces an equivalence on identity types of paths symIso : {a b : A} → Iso (a ≡ b) (b ≡ a) symIso = iso sym sym (λ _ → refl) λ _ → refl -- J is an equivalence Jequiv : {x : A} (P : ∀ y → x ≡ y → Type ℓ') → P x refl ≃ (∀ {y} (p : x ≡ y) → P y p) Jequiv P = isoToEquiv isom where isom : Iso _ _ Iso.fun isom = J P Iso.inv isom f = f refl Iso.rightInv isom f = implicitFunExt λ {_} → funExt λ t → J (λ _ t → J P (f refl) t ≡ f t) (JRefl P (f refl)) t Iso.leftInv isom = JRefl P -- Action of PathP on equivalences (without relying on univalence) congPathIso : ∀ {ℓ ℓ'} {A : I → Type ℓ} {B : I → Type ℓ'} (e : ∀ i → A i ≃ B i) {a₀ : A i0} {a₁ : A i1} → Iso (PathP A a₀ a₁) (PathP B (e i0 .fst a₀) (e i1 .fst a₁)) congPathIso {A = A} {B} e {a₀} {a₁} .Iso.fun p i = e i .fst (p i) congPathIso {A = A} {B} e {a₀} {a₁} .Iso.inv q i = hcomp (λ j → λ { (i = i0) → retEq (e i0) a₀ j ; (i = i1) → retEq (e i1) a₁ j }) (invEq (e i) (q i)) congPathIso {A = A} {B} e {a₀} {a₁} .Iso.rightInv q k i = hcomp (λ j → λ { (i = i0) → commSqIsEq (e i0 .snd) a₀ j k ; (i = i1) → commSqIsEq (e i1 .snd) a₁ j k ; (k = i0) → e i .fst (hfill (λ j → λ { (i = i0) → retEq (e i0) a₀ j ; (i = i1) → retEq (e i1) a₁ j }) (inS (invEq (e i) (q i))) j) ; (k = i1) → q i }) (secEq (e i) (q i) k) where b = commSqIsEq congPathIso {A = A} {B} e {a₀} {a₁} .Iso.leftInv p k i = hcomp (λ j → λ { (i = i0) → retEq (e i0) a₀ (j ∨ k) ; (i = i1) → retEq (e i1) a₁ (j ∨ k) ; (k = i1) → p i }) (retEq (e i) (p i) k) congPathEquiv : ∀ {ℓ ℓ'} {A : I → Type ℓ} {B : I → Type ℓ'} (e : ∀ i → A i ≃ B i) {a₀ : A i0} {a₁ : A i1} → PathP A a₀ a₁ ≃ PathP B (e i0 .fst a₀) (e i1 .fst a₁) congPathEquiv e = isoToEquiv (congPathIso e) -- Characterizations of dependent paths in path types doubleCompPath-filler∙ : {a b c d : A} (p : a ≡ b) (q : b ≡ c) (r : c ≡ d) → PathP (λ i → p i ≡ r (~ i)) (p ∙ q ∙ r) q doubleCompPath-filler∙ {A = A} {b = b} p q r j i = hcomp (λ k → λ { (i = i0) → p j ; (i = i1) → side j k ; (j = i1) → q (i ∧ k)}) (p (j ∨ i)) where side : I → I → A side i j = hcomp (λ k → λ { (i = i1) → q j ; (j = i0) → b ; (j = i1) → r (~ i ∧ k)}) (q j) PathP→compPathL : {a b c d : A} {p : a ≡ c} {q : b ≡ d} {r : a ≡ b} {s : c ≡ d} → PathP (λ i → p i ≡ q i) r s → sym p ∙ r ∙ q ≡ s PathP→compPathL {p = p} {q = q} {r = r} {s = s} P j i = hcomp (λ k → λ { (i = i0) → p (j ∨ k) ; (i = i1) → q (j ∨ k) ; (j = i0) → doubleCompPath-filler∙ (sym p) r q (~ k) i ; (j = i1) → s i }) (P j i) PathP→compPathR : {a b c d : A} {p : a ≡ c} {q : b ≡ d} {r : a ≡ b} {s : c ≡ d} → PathP (λ i → p i ≡ q i) r s → r ≡ p ∙ s ∙ sym q PathP→compPathR {p = p} {q = q} {r = r} {s = s} P j i = hcomp (λ k → λ { (i = i0) → p (j ∧ (~ k)) ; (i = i1) → q (j ∧ (~ k)) ; (j = i0) → r i ; (j = i1) → doubleCompPath-filler∙ p s (sym q) (~ k) i}) (P j i) -- Other direction compPathL→PathP : {a b c d : A} {p : a ≡ c} {q : b ≡ d} {r : a ≡ b} {s : c ≡ d} → sym p ∙ r ∙ q ≡ s → PathP (λ i → p i ≡ q i) r s compPathL→PathP {p = p} {q = q} {r = r} {s = s} P j i = hcomp (λ k → λ { (i = i0) → p (~ k ∨ j) ; (i = i1) → q (~ k ∨ j) ; (j = i0) → doubleCompPath-filler∙ (sym p) r q k i ; (j = i1) → s i}) (P j i) compPathR→PathP : {a b c d : A} {p : a ≡ c} {q : b ≡ d} {r : a ≡ b} {s : c ≡ d} → r ≡ p ∙ s ∙ sym q → PathP (λ i → p i ≡ q i) r s compPathR→PathP {p = p} {q = q} {r = r} {s = s} P j i = hcomp (λ k → λ { (i = i0) → p (k ∧ j) ; (i = i1) → q (k ∧ j) ; (j = i0) → r i ; (j = i1) → doubleCompPath-filler∙ p s (sym q) k i}) (P j i) compPathR→PathP∙∙ : {a b c d : A} {p : a ≡ c} {q : b ≡ d} {r : a ≡ b} {s : c ≡ d} → r ≡ p ∙∙ s ∙∙ sym q → PathP (λ i → p i ≡ q i) r s compPathR→PathP∙∙ {p = p} {q = q} {r = r} {s = s} P j i = hcomp (λ k → λ { (i = i0) → p (k ∧ j) ; (i = i1) → q (k ∧ j) ; (j = i0) → r i ; (j = i1) → doubleCompPath-filler p s (sym q) (~ k) i}) (P j i)

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2021-11-04T06:54:45.000Z

------------------------------------------------------------------------ -- The Agda standard library -- -- This module is DEPRECATED. Please use `Data.Vec.Functional` instead. ------------------------------------------------------------------------ {-# OPTIONS --without-K --safe #-} -- Disabled to prevent warnings from other Table modules {-# OPTIONS --warn=noUserWarning #-} module Data.Table.Properties where {-# WARNING_ON_IMPORT "Data.Table.Properties was deprecated in v1.2. Use Data.Vec.Functional.Properties instead." #-} open import Data.Table open import Data.Table.Relation.Binary.Equality open import Data.Bool.Base using (true; false; if_then_else_) open import Data.Nat.Base using (zero; suc) open import Data.Empty using (⊥-elim) open import Data.Fin using (Fin; suc; zero; _≟_; punchIn) import Data.Fin.Properties as FP open import Data.Fin.Permutation as Perm using (Permutation; _⟨$⟩ʳ_; _⟨$⟩ˡ_) open import Data.List.Base as L using (List; _∷_; []) open import Data.List.Relation.Unary.Any using (here; there; index) open import Data.List.Membership.Propositional using (_∈_) open import Data.Product as Product using (Σ; ∃; _,_; proj₁; proj₂) open import Data.Vec.Base as V using (Vec; _∷_; []) import Data.Vec.Properties as VP open import Level using (Level) open import Function.Base using (_∘_; flip) open import Function.Inverse using (Inverse) open import Relation.Binary.PropositionalEquality as P using (_≡_; _≢_; refl; sym; cong) open import Relation.Nullary using (does) open import Relation.Nullary.Decidable using (dec-true; dec-false) open import Relation.Nullary.Negation using (contradiction) private variable a : Level A : Set a ------------------------------------------------------------------------ -- select -- Selecting from any table is the same as selecting from a constant table. select-const : ∀ {n} (z : A) (i : Fin n) t → select z i t ≗ select z i (replicate (lookup t i)) select-const z i t j with does (j ≟ i) ... | true = refl ... | false = refl -- Selecting an element from a table then looking it up is the same as looking -- up the index in the original table select-lookup : ∀ {n x i} (t : Table A n) → lookup (select x i t) i ≡ lookup t i select-lookup {i = i} t rewrite dec-true (i ≟ i) refl = refl -- Selecting an element from a table then removing the same element produces a -- constant table select-remove : ∀ {n x} i (t : Table A (suc n)) → remove i (select x i t) ≗ replicate {n = n} x select-remove i t j rewrite dec-false (punchIn i j ≟ i) (FP.punchInᵢ≢i _ _) = refl ------------------------------------------------------------------------ -- permute -- Removing an index 'i' from a table permuted with 'π' is the same as -- removing the element, then permuting with 'π' minus 'i'. remove-permute : ∀ {m n} (π : Permutation (suc m) (suc n)) i (t : Table A (suc n)) → remove (π ⟨$⟩ˡ i) (permute π t) ≗ permute (Perm.remove (π ⟨$⟩ˡ i) π) (remove i t) remove-permute π i t j = P.cong (lookup t) (Perm.punchIn-permute′ π i j) ------------------------------------------------------------------------ -- fromList fromList-∈ : ∀ {xs : List A} (i : Fin (L.length xs)) → lookup (fromList xs) i ∈ xs fromList-∈ {xs = x ∷ xs} zero = here refl fromList-∈ {xs = x ∷ xs} (suc i) = there (fromList-∈ i) index-fromList-∈ : ∀ {xs : List A} {i} → index (fromList-∈ {xs = xs} i) ≡ i index-fromList-∈ {xs = x ∷ xs} {zero} = refl index-fromList-∈ {xs = x ∷ xs} {suc i} = cong suc index-fromList-∈ fromList-index : ∀ {xs} {x : A} (x∈xs : x ∈ xs) → lookup (fromList xs) (index x∈xs) ≡ x fromList-index (here px) = sym px fromList-index (there x∈xs) = fromList-index x∈xs ------------------------------------------------------------------------ -- There exists an isomorphism between tables and vectors. ↔Vec : ∀ {n} → Inverse (≡-setoid A n) (P.setoid (Vec A n)) ↔Vec = record { to = record { _⟨$⟩_ = toVec ; cong = VP.tabulate-cong } ; from = P.→-to-⟶ fromVec ; inverse-of = record { left-inverse-of = VP.lookup∘tabulate ∘ lookup ; right-inverse-of = VP.tabulate∘lookup } } ------------------------------------------------------------------------ -- Other lookup∈ : ∀ {xs : List A} (i : Fin (L.length xs)) → ∃ λ x → x ∈ xs lookup∈ i = _ , fromList-∈ i

36.533333

87

0.573905

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541

agda

Agda

test/Fail/Issue300.agda

mdimjasevic/agda

8fb548356b275c7a1e79b768b64511ae937c738b

[ "BSD-3-Clause" ]

3

2015-03-28T14:51:03.000Z

2015-12-07T20:14:00.000Z

test/Fail/Issue300.agda

mdimjasevic/agda

8fb548356b275c7a1e79b768b64511ae937c738b

[ "BSD-3-Clause" ]

3

2018-11-14T15:31:44.000Z

2019-04-01T19:39:26.000Z

test/Fail/Issue300.agda

mdimjasevic/agda

8fb548356b275c7a1e79b768b64511ae937c738b

[ "BSD-3-Clause" ]

1

2015-09-15T14:36:15.000Z

-- {-# OPTIONS -v tc.size.solve:60 #-} module Issue300 where open import Common.Size data Nat : Size → Set where zero : (i : Size) → Nat (↑ i) suc : (i : Size) → Nat i → Nat (↑ i) -- Size meta used in a different context than the one created in A : Set₁ A = (Id : (i : Size) → Nat _ → Set) (k : Size) (m : Nat (↑ k)) (p : Id k m) → (j : Size) (n : Nat j ) → Id j n -- should solve _ with ↑ i -- 1) Id,k,m |- ↑ 1 ≤ X 1 ==> ↑ 4 ≤ X 4 -- 2) Id,k,m,p,j,n |- 1 ≤ X 1 -- Unfixed by fix for #1914 (Andreas, 2016-04-08).

23.521739

64

0.51756

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790

agda

Agda

test/Succeed/Issue3364.agda

strake/agda

c8a3cfa002e77acc5ae1993bae413fde42d4f93b

[ "BSD-3-Clause" ]

2

2019-10-29T09:40:30.000Z

2020-09-20T00:28:57.000Z

test/Succeed/Issue3364.agda

vikfret/agda

49ad0b3f0d39c01bc35123478b857e702b29fb9d

[ "BSD-3-Clause" ]

1

2020-01-26T18:22:08.000Z

test/Succeed/Issue3364.agda

vikfret/agda

49ad0b3f0d39c01bc35123478b857e702b29fb9d

[ "BSD-3-Clause" ]

1

2021-04-01T18:30:09.000Z

-- Andreas, 2018-11-03, issue #3364 -- Andreas, 2019-02-23, issue #3457 -- -- Better error when trying to import with new qualified module name. open import Agda.Builtin.Nat as Builtin.Nat -- WAS: Error: -- Not in scope: -- as at ... -- when scope checking as -- NOW: Warning -- `as' must be followed by an identifier; a qualified name is not allowed here -- when scope checking the declaration -- open import Agda.Builtin.Nat as Builtin.Nat import Agda.Builtin.Sigma as .as -- `as' must be followed by an identifier -- when scope checking the declaration -- import Agda.Builtin.Sigma as .as import Agda.Builtin.String as _ -- `as' must be followed by an identifier; an underscore is not allowed here -- when scope checking the declaration -- import Agda.Builtin.String as _

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agda

Agda

examples/examplesPaperJFP/Sized.agda

agda/ooAgda

7cc45e0148a4a508d20ed67e791544c30fecd795

[ "MIT" ]

23

2016-06-19T12:57:55.000Z

2020-10-12T23:15:25.000Z

examples/examplesPaperJFP/Sized.agda

agda/ooAgda

7cc45e0148a4a508d20ed67e791544c30fecd795

[ "MIT" ]

null

examples/examplesPaperJFP/Sized.agda

agda/ooAgda

7cc45e0148a4a508d20ed67e791544c30fecd795

[ "MIT" ]

2

2018-09-01T15:02:37.000Z

2022-03-12T11:41:00.000Z

module examplesPaperJFP.Sized where open import Data.Product using (_×_; _,_) open import Data.String open import Function using (case_of_) open import Size open import examplesPaperJFP.NativeIOSafe open import examplesPaperJFP.BasicIO using (IOInterface; Command; Response) open import examplesPaperJFP.ConsoleInterface open import examplesPaperJFP.Console using (translateIOConsoleLocal) open import examplesPaperJFP.Object using (Interface; Method; Result; cellJ; CellMethod; get; put; CellResult) module UnfoldF where open import examplesPaperJFP.Coalgebra using (F; mapF) record νF (i : Size) : Set where coinductive constructor delay field force : ∀(j : Size< i) → F (νF j) open νF using (force) unfoldF : ∀{S} (t : S → F S) → ∀ i → (S → νF i) force (unfoldF t i s) j = mapF (unfoldF t j) (t s) mutual record IO (Iᵢₒ : IOInterface) (i : Size) (A : Set) : Set where coinductive constructor delay field force : {j : Size< i} → IO′ Iᵢₒ j A data IO′ (Iᵢₒ : IOInterface) (i : Size) (A : Set) : Set where exec′ : (c : Command Iᵢₒ) (f : Response Iᵢₒ c → IO Iᵢₒ i A) → IO′ Iᵢₒ i A return′ : (a : A) → IO′ Iᵢₒ i A module NestedRecursion (Iᵢₒ : IOInterface) (A : Set) where data F (X : Set) : Set where exec′ : (c : Command Iᵢₒ) (f : Response Iᵢₒ c → X) → F X return′ : (a : A) → F X record νF (i : Size) : Set where coinductive constructor delay field force : {j : Size< i} → F (νF j) open IO public module _ {Iᵢₒ : IOInterface } (let C = Command Iᵢₒ) (let R = Response Iᵢₒ) where infixl 2 _>>=_ exec : ∀ {i A} (c : C) (f : R c → IO Iᵢₒ i A) → IO Iᵢₒ i A return : ∀ {i A} (a : A) → IO Iᵢₒ i A _>>=_ : ∀ {i A B} (m : IO Iᵢₒ i A) (k : A → IO Iᵢₒ i B) → IO Iᵢₒ i B force (exec c f) = exec′ c f force (return a) = return′ a force (_>>=_ {i} m k) {j} with force m {j} ... | exec′ c f = exec′ c λ r → _>>=_ {j} (f r) k ... | return′ a = force (k a) {j} {-# NON_TERMINATING #-} translateIO : ∀{A : Set} → (translateLocal : (c : C) → NativeIO (R c)) → IO Iᵢₒ ∞ A → NativeIO A translateIO translateLocal m = case (force m) of λ{ (exec′ c f) → (translateLocal c) native>>= λ r → translateIO translateLocal (f r) ; (return′ a) → nativeReturn a } record IOObject (Iᵢₒ : IOInterface) (I : Interface) (i : Size) : Set where coinductive field method : ∀{j : Size< i} (m : Method I) → IO Iᵢₒ ∞ (Result I m × IOObject Iᵢₒ I j) open IOObject public CellC : (i : Size) → Set CellC = IOObject ConsoleInterface (cellJ String) simpleCell : ∀{i} (s : String) → CellC i force (method (simpleCell {i} s) {j} get) = exec′ (putStrLn ("getting (" ++ s ++ ")")) λ _ → return (s , simpleCell {j} s) force (method (simpleCell _) (put s)) = exec′ (putStrLn ("putting (" ++ s ++ ")")) λ _ → return (unit , simpleCell s) program : ∀{i} → IO ConsoleInterface i Unit force program = let c₁ = simpleCell "Start" in exec′ getLine λ{ nothing → return unit; (just s) → method c₁ (put s) >>= λ{ (_ , c₂) → method c₂ get >>= λ{ (s′ , c₃) → exec (putStrLn s′) λ _ → program }}} main : NativeIO Unit main = translateIO translateIOConsoleLocal program

30.107143

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13,543

agda

Agda

Agda/18-circle.agda

UlrikBuchholtz/HoTT-Intro

1e1f8def50f9359928e52ebb2ee53ed1166487d9

[ "CC-BY-4.0" ]

333

2018-09-26T08:33:30.000Z

2022-03-22T23:50:15.000Z

Agda/18-circle.agda

UlrikBuchholtz/HoTT-Intro

1e1f8def50f9359928e52ebb2ee53ed1166487d9

[ "CC-BY-4.0" ]

8

2019-06-18T04:16:04.000Z

2020-10-16T15:27:01.000Z

Agda/18-circle.agda

UlrikBuchholtz/HoTT-Intro

1e1f8def50f9359928e52ebb2ee53ed1166487d9

[ "CC-BY-4.0" ]

30

2018-09-26T09:08:57.000Z

2022-03-16T00:33:50.000Z

{-# OPTIONS --without-K --exact-split #-} module 18-circle where import 17-number-theory open 17-number-theory public {- Section 11.1 The induction principle of the circle -} free-loops : { l1 : Level} (X : UU l1) → UU l1 free-loops X = Σ X (λ x → Id x x) base-free-loop : { l1 : Level} {X : UU l1} → free-loops X → X base-free-loop = pr1 loop-free-loop : { l1 : Level} {X : UU l1} (l : free-loops X) → Id (base-free-loop l) (base-free-loop l) loop-free-loop = pr2 -- Now we characterize the identity types of free loops Eq-free-loops : { l1 : Level} {X : UU l1} (l l' : free-loops X) → UU l1 Eq-free-loops (pair x l) l' = Σ (Id x (pr1 l')) (λ p → Id (l ∙ p) (p ∙ (pr2 l'))) reflexive-Eq-free-loops : { l1 : Level} {X : UU l1} (l : free-loops X) → Eq-free-loops l l reflexive-Eq-free-loops (pair x l) = pair refl right-unit Eq-free-loops-eq : { l1 : Level} {X : UU l1} (l l' : free-loops X) → Id l l' → Eq-free-loops l l' Eq-free-loops-eq l .l refl = reflexive-Eq-free-loops l abstract is-contr-total-Eq-free-loops : { l1 : Level} {X : UU l1} (l : free-loops X) → is-contr (Σ (free-loops X) (Eq-free-loops l)) is-contr-total-Eq-free-loops (pair x l) = is-contr-total-Eq-structure ( λ x l' p → Id (l ∙ p) (p ∙ l')) ( is-contr-total-path x) ( pair x refl) ( is-contr-is-equiv' ( Σ (Id x x) (λ l' → Id l l')) ( tot (λ l' α → right-unit ∙ α)) ( is-equiv-tot-is-fiberwise-equiv ( λ l' → is-equiv-concat right-unit l')) ( is-contr-total-path l)) abstract is-equiv-Eq-free-loops-eq : { l1 : Level} {X : UU l1} (l l' : free-loops X) → is-equiv (Eq-free-loops-eq l l') is-equiv-Eq-free-loops-eq l = fundamental-theorem-id l ( reflexive-Eq-free-loops l) ( is-contr-total-Eq-free-loops l) ( Eq-free-loops-eq l) {- We introduce dependent free loops, which are used in the induction principle of the circle. -} dependent-free-loops : { l1 l2 : Level} {X : UU l1} (l : free-loops X) (P : X → UU l2) → UU l2 dependent-free-loops l P = Σ ( P (base-free-loop l)) ( λ p₀ → Id (tr P (loop-free-loop l) p₀) p₀) Eq-dependent-free-loops : { l1 l2 : Level} {X : UU l1} (l : free-loops X) (P : X → UU l2) → ( p p' : dependent-free-loops l P) → UU l2 Eq-dependent-free-loops (pair x l) P (pair y p) p' = Σ ( Id y (pr1 p')) ( λ q → Id (p ∙ q) ((ap (tr P l) q) ∙ (pr2 p'))) reflexive-Eq-dependent-free-loops : { l1 l2 : Level} {X : UU l1} (l : free-loops X) (P : X → UU l2) → ( p : dependent-free-loops l P) → Eq-dependent-free-loops l P p p reflexive-Eq-dependent-free-loops (pair x l) P (pair y p) = pair refl right-unit Eq-dependent-free-loops-eq : { l1 l2 : Level} {X : UU l1} (l : free-loops X) (P : X → UU l2) → ( p p' : dependent-free-loops l P) → Id p p' → Eq-dependent-free-loops l P p p' Eq-dependent-free-loops-eq l P p .p refl = reflexive-Eq-dependent-free-loops l P p abstract is-contr-total-Eq-dependent-free-loops : { l1 l2 : Level} {X : UU l1} (l : free-loops X) (P : X → UU l2) → ( p : dependent-free-loops l P) → is-contr (Σ (dependent-free-loops l P) (Eq-dependent-free-loops l P p)) is-contr-total-Eq-dependent-free-loops (pair x l) P (pair y p) = is-contr-total-Eq-structure ( λ y' p' q → Id (p ∙ q) ((ap (tr P l) q) ∙ p')) ( is-contr-total-path y) ( pair y refl) ( is-contr-is-equiv' ( Σ (Id (tr P l y) y) (λ p' → Id p p')) ( tot (λ p' α → right-unit ∙ α)) ( is-equiv-tot-is-fiberwise-equiv ( λ p' → is-equiv-concat right-unit p')) ( is-contr-total-path p)) abstract is-equiv-Eq-dependent-free-loops-eq : { l1 l2 : Level} {X : UU l1} (l : free-loops X) (P : X → UU l2) ( p p' : dependent-free-loops l P) → is-equiv (Eq-dependent-free-loops-eq l P p p') is-equiv-Eq-dependent-free-loops-eq l P p = fundamental-theorem-id p ( reflexive-Eq-dependent-free-loops l P p) ( is-contr-total-Eq-dependent-free-loops l P p) ( Eq-dependent-free-loops-eq l P p) eq-Eq-dependent-free-loops : { l1 l2 : Level} {X : UU l1} (l : free-loops X) (P : X → UU l2) ( p p' : dependent-free-loops l P) → Eq-dependent-free-loops l P p p' → Id p p' eq-Eq-dependent-free-loops l P p p' = inv-is-equiv (is-equiv-Eq-dependent-free-loops-eq l P p p') {- We now define the induction principle of the circle. -} ev-free-loop' : { l1 l2 : Level} {X : UU l1} (l : free-loops X) (P : X → UU l2) → ( (x : X) → P x) → dependent-free-loops l P ev-free-loop' (pair x₀ p) P f = pair (f x₀) (apd f p) induction-principle-circle : { l1 : Level} (l2 : Level) {X : UU l1} (l : free-loops X) → UU ((lsuc l2) ⊔ l1) induction-principle-circle l2 {X} l = (P : X → UU l2) → sec (ev-free-loop' l P) {- Section 11.2 The universal property of the circle -} {- We first state the universal property of the circle -} ev-free-loop : { l1 l2 : Level} {X : UU l1} (l : free-loops X) (Y : UU l2) → ( X → Y) → free-loops Y ev-free-loop l Y f = pair (f (pr1 l)) (ap f (pr2 l)) universal-property-circle : { l1 : Level} (l2 : Level) {X : UU l1} (l : free-loops X) → UU _ universal-property-circle l2 l = ( Y : UU l2) → is-equiv (ev-free-loop l Y) {- A fairly straightforward proof of the universal property of the circle factors through the dependent universal property of the circle. -} dependent-universal-property-circle : { l1 : Level} (l2 : Level) {X : UU l1} (l : free-loops X) → UU ((lsuc l2) ⊔ l1) dependent-universal-property-circle l2 {X} l = ( P : X → UU l2) → is-equiv (ev-free-loop' l P) {- We first prove that the dependent universal property of the circle follows from the induction principle of the circle. To show this, we have to show that the section of ev-free-loop' is also a retraction. This construction is also by the induction principle of the circle, but it requires (a minimal amount of) preparations. -} Eq-subst : { l1 l2 : Level} {X : UU l1} {P : X → UU l2} (f g : (x : X) → P x) → X → UU _ Eq-subst f g x = Id (f x) (g x) tr-Eq-subst : { l1 l2 : Level} {X : UU l1} {P : X → UU l2} (f g : (x : X) → P x) { x y : X} (p : Id x y) (q : Id (f x) (g x)) (r : Id (f y) (g y))→ ( Id ((apd f p) ∙ r) ((ap (tr P p) q) ∙ (apd g p))) → ( Id (tr (Eq-subst f g) p q) r) tr-Eq-subst f g refl q .((ap id q) ∙ refl) refl = inv (right-unit ∙ (ap-id q)) dependent-free-loops-htpy : {l1 l2 : Level} {X : UU l1} {l : free-loops X} {P : X → UU l2} {f g : (x : X) → P x} → ( Eq-dependent-free-loops l P (ev-free-loop' l P f) (ev-free-loop' l P g)) → ( dependent-free-loops l (λ x → Id (f x) (g x))) dependent-free-loops-htpy {l = (pair x l)} (pair p q) = pair p (tr-Eq-subst _ _ l p p q) isretr-ind-circle : { l1 l2 : Level} {X : UU l1} (l : free-loops X) → ( ind-circle : induction-principle-circle l2 l) (P : X → UU l2) → ( (pr1 (ind-circle P)) ∘ (ev-free-loop' l P)) ~ id isretr-ind-circle l ind-circle P f = eq-htpy ( pr1 ( ind-circle ( λ t → Id (pr1 (ind-circle P) (ev-free-loop' l P f) t) (f t))) ( dependent-free-loops-htpy ( Eq-dependent-free-loops-eq l P _ _ ( pr2 (ind-circle P) (ev-free-loop' l P f))))) abstract dependent-universal-property-induction-principle-circle : { l1 l2 : Level} {X : UU l1} (l : free-loops X) → induction-principle-circle l2 l → dependent-universal-property-circle l2 l dependent-universal-property-induction-principle-circle l ind-circle P = is-equiv-has-inverse ( pr1 (ind-circle P)) ( pr2 (ind-circle P)) ( isretr-ind-circle l ind-circle P) {- We use the dependent universal property to derive a uniqeness property of dependent functions on the circle. -} dependent-uniqueness-circle : { l1 l2 : Level} {X : UU l1} (l : free-loops X) → dependent-universal-property-circle l2 l → { P : X → UU l2} (k : dependent-free-loops l P) → is-contr ( Σ ( (x : X) → P x) ( λ h → Eq-dependent-free-loops l P (ev-free-loop' l P h) k)) dependent-uniqueness-circle l dup-circle {P} k = is-contr-is-equiv' ( fib (ev-free-loop' l P) k) ( tot (λ h → Eq-dependent-free-loops-eq l P (ev-free-loop' l P h) k)) ( is-equiv-tot-is-fiberwise-equiv (λ h → is-equiv-Eq-dependent-free-loops-eq l P (ev-free-loop' l P h) k)) ( is-contr-map-is-equiv (dup-circle P) k) {- Now that we have established the dependent universal property, we can reduce the (non-dependent) universal property to the dependent case. We do so by constructing a commuting triangle relating ev-free-loop to ev-free-loop' via a comparison equivalence. -} tr-const : {i j : Level} {A : UU i} {B : UU j} {x y : A} (p : Id x y) (b : B) → Id (tr (λ (a : A) → B) p b) b tr-const refl b = refl apd-const : {i j : Level} {A : UU i} {B : UU j} (f : A → B) {x y : A} (p : Id x y) → Id (apd f p) ((tr-const p (f x)) ∙ (ap f p)) apd-const f refl = refl comparison-free-loops : { l1 l2 : Level} {X : UU l1} (l : free-loops X) (Y : UU l2) → free-loops Y → dependent-free-loops l (λ x → Y) comparison-free-loops l Y = tot (λ y l' → (tr-const (pr2 l) y) ∙ l') abstract is-equiv-comparison-free-loops : { l1 l2 : Level} {X : UU l1} (l : free-loops X) (Y : UU l2) → is-equiv (comparison-free-loops l Y) is-equiv-comparison-free-loops l Y = is-equiv-tot-is-fiberwise-equiv ( λ y → is-equiv-concat (tr-const (pr2 l) y) y) triangle-comparison-free-loops : { l1 l2 : Level} {X : UU l1} (l : free-loops X) (Y : UU l2) → ( (comparison-free-loops l Y) ∘ (ev-free-loop l Y)) ~ ( ev-free-loop' l (λ x → Y)) triangle-comparison-free-loops (pair x l) Y f = eq-Eq-dependent-free-loops ( pair x l) ( λ x → Y) ( comparison-free-loops (pair x l) Y (ev-free-loop (pair x l) Y f)) ( ev-free-loop' (pair x l) (λ x → Y) f) ( pair refl (right-unit ∙ (inv (apd-const f l)))) abstract universal-property-dependent-universal-property-circle : { l1 l2 : Level} {X : UU l1} (l : free-loops X) → ( dependent-universal-property-circle l2 l) → ( universal-property-circle l2 l) universal-property-dependent-universal-property-circle l dup-circle Y = is-equiv-right-factor ( ev-free-loop' l (λ x → Y)) ( comparison-free-loops l Y) ( ev-free-loop l Y) ( htpy-inv (triangle-comparison-free-loops l Y)) ( is-equiv-comparison-free-loops l Y) ( dup-circle (λ x → Y)) {- Now we get the universal property of the circle from the induction principle of the circle by composing the earlier two proofs. -} abstract universal-property-induction-principle-circle : { l1 l2 : Level} {X : UU l1} (l : free-loops X) → induction-principle-circle l2 l → universal-property-circle l2 l universal-property-induction-principle-circle l = ( universal-property-dependent-universal-property-circle l) ∘ ( dependent-universal-property-induction-principle-circle l) unique-mapping-property-circle : { l1 : Level} (l2 : Level) {X : UU l1} (l : free-loops X) → UU (l1 ⊔ (lsuc l2)) unique-mapping-property-circle l2 {X} l = ( Y : UU l2) (l' : free-loops Y) → is-contr (Σ (X → Y) (λ f → Eq-free-loops (ev-free-loop l Y f) l')) abstract unique-mapping-property-universal-property-circle : { l1 l2 : Level} {X : UU l1} (l : free-loops X) → universal-property-circle l2 l → unique-mapping-property-circle l2 l unique-mapping-property-universal-property-circle l up-circle Y l' = is-contr-is-equiv' ( fib (ev-free-loop l Y) l') ( tot (λ f → Eq-free-loops-eq (ev-free-loop l Y f) l')) ( is-equiv-tot-is-fiberwise-equiv ( λ f → is-equiv-Eq-free-loops-eq (ev-free-loop l Y f) l')) ( is-contr-map-is-equiv (up-circle Y) l') {- We assume that we have a circle. -} postulate 𝕊¹ : UU lzero postulate base-𝕊¹ : 𝕊¹ postulate loop-𝕊¹ : Id base-𝕊¹ base-𝕊¹ free-loop-𝕊¹ : free-loops 𝕊¹ free-loop-𝕊¹ = pair base-𝕊¹ loop-𝕊¹ postulate ind-𝕊¹ : {l : Level} → induction-principle-circle l free-loop-𝕊¹ dependent-universal-property-𝕊¹ : {l : Level} → dependent-universal-property-circle l free-loop-𝕊¹ dependent-universal-property-𝕊¹ = dependent-universal-property-induction-principle-circle free-loop-𝕊¹ ind-𝕊¹ dependent-uniqueness-𝕊¹ : {l : Level} {P : 𝕊¹ → UU l} (k : dependent-free-loops free-loop-𝕊¹ P) → is-contr (Σ ((x : 𝕊¹) → P x) (λ h → Eq-dependent-free-loops free-loop-𝕊¹ P (ev-free-loop' free-loop-𝕊¹ P h) k)) dependent-uniqueness-𝕊¹ {l} {P} k = dependent-uniqueness-circle free-loop-𝕊¹ dependent-universal-property-𝕊¹ k universal-property-𝕊¹ : {l : Level} → universal-property-circle l free-loop-𝕊¹ universal-property-𝕊¹ = universal-property-dependent-universal-property-circle free-loop-𝕊¹ dependent-universal-property-𝕊¹ -- Section 14.3 Multiplication on the circle {- Exercises -} -- Exercise 11.1 {- The dependent universal property of the circle (and hence also the induction principle of the circle, implies that the circle is connected in the sense that for any family of propositions parametrized by the circle, if the proposition at the base holds, then it holds for any x : circle. -} abstract is-connected-circle' : { l1 l2 : Level} {X : UU l1} (l : free-loops X) → ( dup-circle : dependent-universal-property-circle l2 l) ( P : X → UU l2) (is-prop-P : (x : X) → is-prop (P x)) → P (base-free-loop l) → (x : X) → P x is-connected-circle' l dup-circle P is-prop-P p = inv-is-equiv ( dup-circle P) ( pair p (center (is-prop-P _ (tr P (pr2 l) p) p)))

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Agda

Definition/LogicalRelation/Substitution/Properties.agda

fhlkfy/logrel-mltt

ea83fc4f618d1527d64ecac82d7d17e2f18ac391

[ "MIT" ]

30

2017-05-20T03:05:21.000Z

2022-03-30T18:01:07.000Z

Definition/LogicalRelation/Substitution/Properties.agda

fhlkfy/logrel-mltt

ea83fc4f618d1527d64ecac82d7d17e2f18ac391

[ "MIT" ]

4

2017-06-22T12:49:23.000Z

2021-02-22T10:37:24.000Z

Definition/LogicalRelation/Substitution/Properties.agda

fhlkfy/logrel-mltt

ea83fc4f618d1527d64ecac82d7d17e2f18ac391

[ "MIT" ]

8

2017-10-18T14:18:20.000Z

2021-11-27T15:58:33.000Z

{-# OPTIONS --without-K --safe #-} open import Definition.Typed.EqualityRelation module Definition.LogicalRelation.Substitution.Properties {{eqrel : EqRelSet}} where open EqRelSet {{...}} open import Definition.Untyped open import Definition.Untyped.Properties open import Definition.Typed open import Definition.Typed.Weakening open import Definition.LogicalRelation open import Definition.LogicalRelation.Substitution open import Definition.LogicalRelation.Substitution.Irrelevance using (irrelevanceSubst′) open import Definition.LogicalRelation.Irrelevance open import Definition.LogicalRelation.Properties import Definition.LogicalRelation.Weakening as LR open import Tools.Fin open import Tools.Nat open import Tools.Unit open import Tools.Product import Tools.PropositionalEquality as PE private variable k m n : Nat Γ : Con Term n σ σ′ : Subst m n ρ : Wk k n -- Valid substitutions are well-formed wellformedSubst : ∀ {Γ Δ} ([Γ] : ⊩ᵛ Γ) (⊢Δ : ⊢ Δ) → Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ → Δ ⊢ˢ σ ∷ Γ wellformedSubst ε ⊢Δ [σ] = id wellformedSubst ([Γ] ∙ [A]) ⊢Δ ([tailσ] , [headσ]) = wellformedSubst [Γ] ⊢Δ [tailσ] , escapeTerm (proj₁ ([A] ⊢Δ [tailσ])) [headσ] -- Valid substitution equality is well-formed wellformedSubstEq : ∀ {Γ Δ} ([Γ] : ⊩ᵛ Γ) (⊢Δ : ⊢ Δ) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ) → Δ ⊩ˢ σ ≡ σ′ ∷ Γ / [Γ] / ⊢Δ / [σ] → Δ ⊢ˢ σ ≡ σ′ ∷ Γ wellformedSubstEq ε ⊢Δ [σ] [σ≡σ′] = id wellformedSubstEq ([Γ] ∙ [A]) ⊢Δ ([tailσ] , [headσ]) ([tailσ≡σ′] , [headσ≡σ′]) = wellformedSubstEq [Γ] ⊢Δ [tailσ] [tailσ≡σ′] , ≅ₜ-eq (escapeTermEq (proj₁ ([A] ⊢Δ [tailσ])) [headσ≡σ′]) -- Extend a valid substitution with a term consSubstS : ∀ {l t A Γ Δ} ([Γ] : ⊩ᵛ Γ) (⊢Δ : ⊢ Δ) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ) ([A] : Γ ⊩ᵛ⟨ l ⟩ A / [Γ]) ([t] : Δ ⊩⟨ l ⟩ t ∷ subst σ A / proj₁ ([A] ⊢Δ [σ])) → Δ ⊩ˢ consSubst σ t ∷ Γ ∙ A / [Γ] ∙ [A] / ⊢Δ consSubstS [Γ] ⊢Δ [σ] [A] [t] = [σ] , [t] -- Extend a valid substitution equality with a term consSubstSEq : ∀ {l t A Γ Δ} ([Γ] : ⊩ᵛ Γ) (⊢Δ : ⊢ Δ) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ) ([σ≡σ′] : Δ ⊩ˢ σ ≡ σ′ ∷ Γ / [Γ] / ⊢Δ / [σ]) ([A] : Γ ⊩ᵛ⟨ l ⟩ A / [Γ]) ([t] : Δ ⊩⟨ l ⟩ t ∷ subst σ A / proj₁ ([A] ⊢Δ [σ])) → Δ ⊩ˢ consSubst σ t ≡ consSubst σ′ t ∷ Γ ∙ A / [Γ] ∙ [A] / ⊢Δ / consSubstS {t = t} {A = A} [Γ] ⊢Δ [σ] [A] [t] consSubstSEq [Γ] ⊢Δ [σ] [σ≡σ′] [A] [t] = [σ≡σ′] , reflEqTerm (proj₁ ([A] ⊢Δ [σ])) [t] -- Weakening of valid substitutions wkSubstS : ∀ {Γ Δ Δ′} ([Γ] : ⊩ᵛ Γ) (⊢Δ : ⊢ Δ) (⊢Δ′ : ⊢ Δ′) ([ρ] : ρ ∷ Δ′ ⊆ Δ) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ) → Δ′ ⊩ˢ ρ •ₛ σ ∷ Γ / [Γ] / ⊢Δ′ wkSubstS ε ⊢Δ ⊢Δ′ ρ [σ] = tt wkSubstS {σ = σ} {Γ = Γ ∙ A} ([Γ] ∙ x) ⊢Δ ⊢Δ′ ρ [σ] = let [tailσ] = wkSubstS [Γ] ⊢Δ ⊢Δ′ ρ (proj₁ [σ]) in [tailσ] , irrelevanceTerm′ (wk-subst A) (LR.wk ρ ⊢Δ′ (proj₁ (x ⊢Δ (proj₁ [σ])))) (proj₁ (x ⊢Δ′ [tailσ])) (LR.wkTerm ρ ⊢Δ′ (proj₁ (x ⊢Δ (proj₁ [σ]))) (proj₂ [σ])) -- Weakening of valid substitution equality wkSubstSEq : ∀ {Γ Δ Δ′} ([Γ] : ⊩ᵛ Γ) (⊢Δ : ⊢ Δ) (⊢Δ′ : ⊢ Δ′) ([ρ] : ρ ∷ Δ′ ⊆ Δ) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ) ([σ≡σ′] : Δ ⊩ˢ σ ≡ σ′ ∷ Γ / [Γ] / ⊢Δ / [σ]) → Δ′ ⊩ˢ ρ •ₛ σ ≡ ρ •ₛ σ′ ∷ Γ / [Γ] / ⊢Δ′ / wkSubstS [Γ] ⊢Δ ⊢Δ′ [ρ] [σ] wkSubstSEq ε ⊢Δ ⊢Δ′ ρ [σ] [σ≡σ′] = tt wkSubstSEq {Γ = Γ ∙ A} ([Γ] ∙ x) ⊢Δ ⊢Δ′ ρ [σ] [σ≡σ′] = wkSubstSEq [Γ] ⊢Δ ⊢Δ′ ρ (proj₁ [σ]) (proj₁ [σ≡σ′]) , irrelevanceEqTerm′ (wk-subst A) (LR.wk ρ ⊢Δ′ (proj₁ (x ⊢Δ (proj₁ [σ])))) (proj₁ (x ⊢Δ′ (wkSubstS [Γ] ⊢Δ ⊢Δ′ ρ (proj₁ [σ])))) (LR.wkEqTerm ρ ⊢Δ′ (proj₁ (x ⊢Δ (proj₁ [σ]))) (proj₂ [σ≡σ′])) -- Weaken a valid substitution by one type wk1SubstS : ∀ {F Γ Δ} ([Γ] : ⊩ᵛ Γ) (⊢Δ : ⊢ Δ) (⊢F : Δ ⊢ F) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ) → (Δ ∙ F) ⊩ˢ wk1Subst σ ∷ Γ / [Γ] / (⊢Δ ∙ ⊢F) wk1SubstS {F} {σ} {Γ} {Δ} [Γ] ⊢Δ ⊢F [σ] = wkSubstS [Γ] ⊢Δ (⊢Δ ∙ ⊢F) (step id) [σ] -- Weaken a valid substitution equality by one type wk1SubstSEq : ∀ {F Γ Δ} ([Γ] : ⊩ᵛ Γ) (⊢Δ : ⊢ Δ) (⊢F : Δ ⊢ F) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ) ([σ≡σ′] : Δ ⊩ˢ σ ≡ σ′ ∷ Γ / [Γ] / ⊢Δ / [σ]) → (Δ ∙ F) ⊩ˢ wk1Subst σ ≡ wk1Subst σ′ ∷ Γ / [Γ] / (⊢Δ ∙ ⊢F) / wk1SubstS [Γ] ⊢Δ ⊢F [σ] wk1SubstSEq {l} {F} {σ} {Γ} {Δ} [Γ] ⊢Δ ⊢F [σ] [σ≡σ′] = wkSubstSEq [Γ] ⊢Δ (⊢Δ ∙ ⊢F) (step id) [σ] [σ≡σ′] -- Lift a valid substitution liftSubstS : ∀ {l F Γ Δ} ([Γ] : ⊩ᵛ Γ) (⊢Δ : ⊢ Δ) ([F] : Γ ⊩ᵛ⟨ l ⟩ F / [Γ]) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ) → (Δ ∙ subst σ F) ⊩ˢ liftSubst σ ∷ Γ ∙ F / [Γ] ∙ [F] / (⊢Δ ∙ escape (proj₁ ([F] ⊢Δ [σ]))) liftSubstS {σ = σ} {F = F} {Δ = Δ} [Γ] ⊢Δ [F] [σ] = let ⊢F = escape (proj₁ ([F] ⊢Δ [σ])) [tailσ] = wk1SubstS {F = subst σ F} [Γ] ⊢Δ (escape (proj₁ ([F] ⊢Δ [σ]))) [σ] var0 = var (⊢Δ ∙ ⊢F) (PE.subst (λ x → x0 ∷ x ∈ (Δ ∙ subst σ F)) (wk-subst F) here) in [tailσ] , neuTerm (proj₁ ([F] (⊢Δ ∙ ⊢F) [tailσ])) (var x0) var0 (~-var var0) -- Lift a valid substitution equality liftSubstSEq : ∀ {l F Γ Δ} ([Γ] : ⊩ᵛ Γ) (⊢Δ : ⊢ Δ) ([F] : Γ ⊩ᵛ⟨ l ⟩ F / [Γ]) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ) ([σ≡σ′] : Δ ⊩ˢ σ ≡ σ′ ∷ Γ / [Γ] / ⊢Δ / [σ]) → (Δ ∙ subst σ F) ⊩ˢ liftSubst σ ≡ liftSubst σ′ ∷ Γ ∙ F / [Γ] ∙ [F] / (⊢Δ ∙ escape (proj₁ ([F] ⊢Δ [σ]))) / liftSubstS {F = F} [Γ] ⊢Δ [F] [σ] liftSubstSEq {σ = σ} {σ′ = σ′} {F = F} {Δ = Δ} [Γ] ⊢Δ [F] [σ] [σ≡σ′] = let ⊢F = escape (proj₁ ([F] ⊢Δ [σ])) [tailσ] = wk1SubstS {F = subst σ F} [Γ] ⊢Δ (escape (proj₁ ([F] ⊢Δ [σ]))) [σ] [tailσ≡σ′] = wk1SubstSEq [Γ] ⊢Δ (escape (proj₁ ([F] ⊢Δ [σ]))) [σ] [σ≡σ′] var0 = var (⊢Δ ∙ ⊢F) (PE.subst (λ x → x0 ∷ x ∈ (Δ ∙ subst σ F)) (wk-subst F) here) in [tailσ≡σ′] , neuEqTerm (proj₁ ([F] (⊢Δ ∙ ⊢F) [tailσ])) (var x0) (var x0) var0 var0 (~-var var0) mutual -- Valid contexts are well-formed soundContext : ⊩ᵛ Γ → ⊢ Γ soundContext ε = ε soundContext (x ∙ x₁) = soundContext x ∙ escape (irrelevance′ (subst-id _) (proj₁ (x₁ (soundContext x) (idSubstS x)))) -- From a valid context we can constuct a valid identity substitution idSubstS : ([Γ] : ⊩ᵛ Γ) → Γ ⊩ˢ idSubst ∷ Γ / [Γ] / soundContext [Γ] idSubstS ε = tt idSubstS {Γ = Γ ∙ A} ([Γ] ∙ [A]) = let ⊢Γ = soundContext [Γ] ⊢Γ∙A = soundContext ([Γ] ∙ [A]) ⊢Γ∙A′ = ⊢Γ ∙ escape (proj₁ ([A] ⊢Γ (idSubstS [Γ]))) [A]′ = wk1SubstS {F = subst idSubst A} [Γ] ⊢Γ (escape (proj₁ ([A] (soundContext [Γ]) (idSubstS [Γ])))) (idSubstS [Γ]) [tailσ] = irrelevanceSubst′ (PE.cong (_∙_ Γ) (subst-id A)) [Γ] [Γ] ⊢Γ∙A′ ⊢Γ∙A [A]′ var0 = var ⊢Γ∙A (PE.subst (λ x → x0 ∷ x ∈ (Γ ∙ A)) (wk-subst A) (PE.subst (λ x → x0 ∷ wk1 (subst idSubst A) ∈ (Γ ∙ x)) (subst-id A) here)) in [tailσ] , neuTerm (proj₁ ([A] ⊢Γ∙A [tailσ])) (var x0) var0 (~-var var0) -- Reflexivity valid substitutions reflSubst : ∀ {Γ Δ} ([Γ] : ⊩ᵛ Γ) (⊢Δ : ⊢ Δ) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ) → Δ ⊩ˢ σ ≡ σ ∷ Γ / [Γ] / ⊢Δ / [σ] reflSubst ε ⊢Δ [σ] = tt reflSubst ([Γ] ∙ x) ⊢Δ [σ] = reflSubst [Γ] ⊢Δ (proj₁ [σ]) , reflEqTerm (proj₁ (x ⊢Δ (proj₁ [σ]))) (proj₂ [σ]) -- Reflexivity of valid identity substitution reflIdSubst : ([Γ] : ⊩ᵛ Γ) → Γ ⊩ˢ idSubst ≡ idSubst ∷ Γ / [Γ] / soundContext [Γ] / idSubstS [Γ] reflIdSubst [Γ] = reflSubst [Γ] (soundContext [Γ]) (idSubstS [Γ]) -- Symmetry of valid substitution symS : ∀ {Γ Δ} ([Γ] : ⊩ᵛ Γ) (⊢Δ : ⊢ Δ) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ) ([σ′] : Δ ⊩ˢ σ′ ∷ Γ / [Γ] / ⊢Δ) → Δ ⊩ˢ σ ≡ σ′ ∷ Γ / [Γ] / ⊢Δ / [σ] → Δ ⊩ˢ σ′ ≡ σ ∷ Γ / [Γ] / ⊢Δ / [σ′] symS ε ⊢Δ [σ] [σ′] [σ≡σ′] = tt symS ([Γ] ∙ x) ⊢Δ [σ] [σ′] [σ≡σ′] = symS [Γ] ⊢Δ (proj₁ [σ]) (proj₁ [σ′]) (proj₁ [σ≡σ′]) , let [σA] = proj₁ (x ⊢Δ (proj₁ [σ])) [σ′A] = proj₁ (x ⊢Δ (proj₁ [σ′])) [σA≡σ′A] = (proj₂ (x ⊢Δ (proj₁ [σ]))) (proj₁ [σ′]) (proj₁ [σ≡σ′]) [headσ′≡headσ] = symEqTerm [σA] (proj₂ [σ≡σ′]) in convEqTerm₁ [σA] [σ′A] [σA≡σ′A] [headσ′≡headσ] -- Transitivity of valid substitution transS : ∀ {σ″ Γ Δ} ([Γ] : ⊩ᵛ Γ) (⊢Δ : ⊢ Δ) ([σ] : Δ ⊩ˢ σ ∷ Γ / [Γ] / ⊢Δ) ([σ′] : Δ ⊩ˢ σ′ ∷ Γ / [Γ] / ⊢Δ) ([σ″] : Δ ⊩ˢ σ″ ∷ Γ / [Γ] / ⊢Δ) → Δ ⊩ˢ σ ≡ σ′ ∷ Γ / [Γ] / ⊢Δ / [σ] → Δ ⊩ˢ σ′ ≡ σ″ ∷ Γ / [Γ] / ⊢Δ / [σ′] → Δ ⊩ˢ σ ≡ σ″ ∷ Γ / [Γ] / ⊢Δ / [σ] transS ε ⊢Δ [σ] [σ′] [σ″] [σ≡σ′] [σ′≡σ″] = tt transS ([Γ] ∙ x) ⊢Δ [σ] [σ′] [σ″] [σ≡σ′] [σ′≡σ″] = transS [Γ] ⊢Δ (proj₁ [σ]) (proj₁ [σ′]) (proj₁ [σ″]) (proj₁ [σ≡σ′]) (proj₁ [σ′≡σ″]) , let [σA] = proj₁ (x ⊢Δ (proj₁ [σ])) [σ′A] = proj₁ (x ⊢Δ (proj₁ [σ′])) [σ″A] = proj₁ (x ⊢Δ (proj₁ [σ″])) [σ′≡σ″]′ = convEqTerm₂ [σA] [σ′A] ((proj₂ (x ⊢Δ (proj₁ [σ]))) (proj₁ [σ′]) (proj₁ [σ≡σ′])) (proj₂ [σ′≡σ″]) in transEqTerm [σA] (proj₂ [σ≡σ′]) [σ′≡σ″]′

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65025ff805f6a03cb774bffb15ff827c1f55573e

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agda

Agda

Cubical/Data/FinSet/DecidablePredicate.agda

howsiyu/cubical

1b9c97a2140fe96fe636f4c66beedfd7b8096e8f

[ "MIT" ]

null

Cubical/Data/FinSet/DecidablePredicate.agda

howsiyu/cubical

1b9c97a2140fe96fe636f4c66beedfd7b8096e8f

[ "MIT" ]

null

Cubical/Data/FinSet/DecidablePredicate.agda

howsiyu/cubical

1b9c97a2140fe96fe636f4c66beedfd7b8096e8f

[ "MIT" ]

null

{- This files contains: - Lots of useful properties about (this) decidable predicates on finite sets. (P.S. We use the alternative definition of decidability for computational effectivity.) -} {-# OPTIONS --safe #-} module Cubical.Data.FinSet.DecidablePredicate where open import Cubical.Foundations.Prelude open import Cubical.Foundations.Function open import Cubical.Foundations.HLevels open import Cubical.Foundations.Equiv renaming (_∙ₑ_ to _⋆_) open import Cubical.Foundations.Equiv.Properties open import Cubical.HITs.PropositionalTruncation as Prop open import Cubical.Data.Bool open import Cubical.Data.Empty as Empty open import Cubical.Data.Sigma open import Cubical.Data.Fin open import Cubical.Data.SumFin renaming (Fin to SumFin) open import Cubical.Data.FinSet.Base open import Cubical.Data.FinSet.Properties open import Cubical.Relation.Nullary open import Cubical.Relation.Nullary.DecidablePropositions hiding (DecProp) renaming (DecProp' to DecProp) private variable ℓ ℓ' ℓ'' ℓ''' : Level module _ (X : Type ℓ)(p : isFinOrd X) where isDecProp¬' : isDecProp (¬ X) isDecProp¬' = _ , invEquiv (preCompEquiv (p .snd)) ⋆ SumFin¬ _ isDecProp∥∥' : isDecProp ∥ X ∥ isDecProp∥∥' = _ , propTrunc≃ (p .snd) ⋆ SumFin∥∥DecProp _ module _ (X : Type ℓ )(p : isFinOrd X) (P : X → Type ℓ') (dec : (x : X) → isDecProp (P x)) where private e = p .snd isFinOrdSub : isFinOrd (Σ X P) isFinOrdSub = _ , Σ-cong-equiv {B' = λ x → P (invEq e x)} e (transpFamily p) ⋆ Σ-cong-equiv-snd (λ x → dec (invEq e x) .snd) ⋆ SumFinSub≃ _ (fst ∘ dec ∘ invEq e) isDecProp∃' : isDecProp ∥ Σ X P ∥ isDecProp∃' = _ , Prop.propTrunc≃ ( Σ-cong-equiv {B' = λ x → P (invEq e x)} e (transpFamily p) ⋆ Σ-cong-equiv-snd (λ x → dec (invEq e x) .snd)) ⋆ SumFin∃≃ _ (fst ∘ dec ∘ invEq e) isDecProp∀' : isDecProp ((x : X) → P x) isDecProp∀' = _ , equivΠ {B' = λ x → P (invEq e x)} e (transpFamily p) ⋆ equivΠCod (λ x → dec (invEq e x) .snd) ⋆ SumFin∀≃ _ (fst ∘ dec ∘ invEq e) module _ (X : Type ℓ )(p : isFinOrd X) (a b : X) where private e = p .snd isDecProp≡' : isDecProp (a ≡ b) isDecProp≡' .fst = SumFin≡ _ (e .fst a) (e .fst b) isDecProp≡' .snd = congEquiv e ⋆ SumFin≡≃ _ _ _ module _ (X : FinSet ℓ) (P : X .fst → DecProp ℓ') where isFinSetSub : isFinSet (Σ (X .fst) (λ x → P x .fst)) isFinSetSub = Prop.rec isPropIsFinSet (λ p → isFinOrd→isFinSet (isFinOrdSub (X .fst) (_ , p) (λ x → P x .fst) (λ x → P x .snd))) (X .snd .snd) isDecProp∃ : isDecProp ∥ Σ (X .fst) (λ x → P x .fst) ∥ isDecProp∃ = Prop.rec isPropIsDecProp (λ p → isDecProp∃' (X .fst) (_ , p) (λ x → P x .fst) (λ x → P x .snd)) (X .snd .snd) isDecProp∀ : isDecProp ((x : X .fst) → P x .fst) isDecProp∀ = Prop.rec isPropIsDecProp (λ p → isDecProp∀' (X .fst) (_ , p) (λ x → P x .fst) (λ x → P x .snd)) (X .snd .snd) module _ (X : FinSet ℓ) (Y : X .fst → FinSet ℓ') (Z : (x : X .fst) → Y x .fst → DecProp ℓ'') where isDecProp∀2 : isDecProp ((x : X .fst) → (y : Y x .fst) → Z x y .fst) isDecProp∀2 = isDecProp∀ X (λ x → _ , isDecProp∀ (Y x) (Z x)) module _ (X : FinSet ℓ) (Y : X .fst → FinSet ℓ') (Z : (x : X .fst) → Y x .fst → FinSet ℓ'') (W : (x : X .fst) → (y : Y x .fst) → Z x y .fst → DecProp ℓ''') where isDecProp∀3 : isDecProp ((x : X .fst) → (y : Y x .fst) → (z : Z x y .fst) → W x y z .fst) isDecProp∀3 = isDecProp∀ X (λ x → _ , isDecProp∀2 (Y x) (Z x) (W x)) module _ (X : FinSet ℓ) where isDecProp≡ : (a b : X .fst) → isDecProp (a ≡ b) isDecProp≡ a b = Prop.rec isPropIsDecProp (λ p → isDecProp≡' (X .fst) (_ , p) a b) (X .snd .snd) module _ (P : DecProp ℓ ) (Q : DecProp ℓ') where isDecProp× : isDecProp (P .fst × Q .fst) isDecProp× .fst = P .snd .fst and Q .snd .fst isDecProp× .snd = Σ-cong-equiv (P .snd .snd) (λ _ → Q .snd .snd) ⋆ Bool→Type×≃ _ _ module _ (X : FinSet ℓ) where isDecProp¬ : isDecProp (¬ (X .fst)) isDecProp¬ = Prop.rec isPropIsDecProp (λ p → isDecProp¬' (X .fst) (_ , p)) (X .snd .snd) isDecProp∥∥ : isDecProp ∥ X .fst ∥ isDecProp∥∥ = Prop.rec isPropIsDecProp (λ p → isDecProp∥∥' (X .fst) (_ , p)) (X .snd .snd)

29.475524

94

0.604033

4ec640f4c01742f138752f0c8121b96f06be4ad9

814

agda

Agda

Definition/LogicalRelation/Substitution/Reflexivity.agda

CoqHott/logrel-mltt

e0eeebc4aa5ed791ce3e7c0dc9531bd113dfcc04

[ "MIT" ]

2

2018-06-21T08:39:01.000Z

2022-01-17T16:13:53.000Z

Definition/LogicalRelation/Substitution/Reflexivity.agda

CoqHott/logrel-mltt

e0eeebc4aa5ed791ce3e7c0dc9531bd113dfcc04

[ "MIT" ]

null

Definition/LogicalRelation/Substitution/Reflexivity.agda

CoqHott/logrel-mltt

e0eeebc4aa5ed791ce3e7c0dc9531bd113dfcc04

[ "MIT" ]

2

2022-01-26T14:55:51.000Z

2022-02-15T19:42:19.000Z

{-# OPTIONS --safe #-} open import Definition.Typed.EqualityRelation module Definition.LogicalRelation.Substitution.Reflexivity {{eqrel : EqRelSet}} where open EqRelSet {{...}} open import Definition.LogicalRelation.Properties open import Definition.LogicalRelation.Substitution open import Tools.Product -- Reflexivity of valid types. reflᵛ : ∀ {A Γ rA l} ([Γ] : ⊩ᵛ Γ) ([A] : Γ ⊩ᵛ⟨ l ⟩ A ^ rA / [Γ]) → Γ ⊩ᵛ⟨ l ⟩ A ≡ A ^ rA / [Γ] / [A] reflᵛ [Γ] [A] ⊢Δ [σ] = reflEq (proj₁ ([A] ⊢Δ [σ])) -- Reflexivity of valid terms. reflᵗᵛ : ∀ {A t Γ rA l} ([Γ] : ⊩ᵛ Γ) ([A] : Γ ⊩ᵛ⟨ l ⟩ A ^ rA / [Γ]) ([t] : Γ ⊩ᵛ⟨ l ⟩ t ∷ A ^ rA / [Γ] / [A]) → Γ ⊩ᵛ⟨ l ⟩ t ≡ t ∷ A ^ rA / [Γ] / [A] reflᵗᵛ [Γ] [A] [t] ⊢Δ [σ] = reflEqTerm (proj₁ ([A] ⊢Δ [σ])) (proj₁ ([t] ⊢Δ [σ]))

27.133333

85

0.527027

7cb2347f541a78bb17c0a2c5f11421015de94b5b

354

agda

Agda

data/declaration/Open2.agda

msuperdock/agda-unused

f327f9aab8dcb07022b857736d8201906bba02e9

[ "MIT" ]

6

2020-10-29T09:38:43.000Z

2022-03-01T16:38:05.000Z

data/declaration/Open2.agda

msuperdock/agda-unused

f327f9aab8dcb07022b857736d8201906bba02e9

[ "MIT" ]

null

data/declaration/Open2.agda

msuperdock/agda-unused

f327f9aab8dcb07022b857736d8201906bba02e9

[ "MIT" ]

1

2022-03-01T16:38:14.000Z

module Open2 where data ⊤ : Set where tt : ⊤ data ⊤' (x : ⊤) : Set where tt : ⊤' x record R : Set where field x : ⊤ y : ⊤ record S : Set₁ where field x : R open R x public renaming (x to y; y to z) postulate s : S open S s using (y) postulate p : ⊤' y

6.436364

29

0.432203

03927ba96a43a877c174398651aff853ca361d67

168

agda

Agda

test/Succeed/WarningOnImport/Impo.agda

mdimjasevic/agda

8fb548356b275c7a1e79b768b64511ae937c738b

[ "BSD-3-Clause" ]

1,989

2015-01-09T23:51:16.000Z

2022-03-30T18:20:48.000Z

test/Succeed/WarningOnImport/Impo.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

4,066

2015-01-10T11:24:51.000Z

2022-03-31T21:14:49.000Z

test/Succeed/WarningOnImport/Impo.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

371

2015-01-03T14:04:08.000Z

2022-03-30T19:00:30.000Z

module WarningOnImport.Impo where B = Set A = B {-# WARNING_ON_USAGE A "Deprecated: Use B instead" #-} {-# WARNING_ON_IMPORT "Deprecated: Use Impossible instead" #-}

21

62

0.720238

7c450c192c99171cf1c0fe5aac6c9b299be97503

1,405

agda

Agda

Cubical/Data/Nat/Base.agda

AkermanRydbeck/cubical

038bcaff93d278c627ccdcec34a4f6df2b56ad5a

[ "MIT" ]

null

Cubical/Data/Nat/Base.agda

AkermanRydbeck/cubical

038bcaff93d278c627ccdcec34a4f6df2b56ad5a

[ "MIT" ]

null

Cubical/Data/Nat/Base.agda

AkermanRydbeck/cubical

038bcaff93d278c627ccdcec34a4f6df2b56ad5a

[ "MIT" ]

null

{-# OPTIONS --no-exact-split --safe #-} module Cubical.Data.Nat.Base where open import Cubical.Core.Primitives open import Agda.Builtin.Nat public using (zero; suc; _+_) renaming (Nat to ℕ; _-_ to _∸_; _*_ to _·_) open import Cubical.Data.Nat.Literals public open import Cubical.Data.Bool.Base open import Cubical.Data.Sum.Base hiding (elim) open import Cubical.Data.Empty.Base hiding (elim) open import Cubical.Data.Unit.Base predℕ : ℕ → ℕ predℕ zero = zero predℕ (suc n) = n caseNat : ∀ {ℓ} → {A : Type ℓ} → (a0 aS : A) → ℕ → A caseNat a0 aS zero = a0 caseNat a0 aS (suc n) = aS doubleℕ : ℕ → ℕ doubleℕ zero = zero doubleℕ (suc x) = suc (suc (doubleℕ x)) -- doublesℕ n m = 2^n · m doublesℕ : ℕ → ℕ → ℕ doublesℕ zero m = m doublesℕ (suc n) m = doublesℕ n (doubleℕ m) -- iterate iter : ∀ {ℓ} {A : Type ℓ} → ℕ → (A → A) → A → A iter zero f z = z iter (suc n) f z = f (iter n f z) elim : ∀ {ℓ} {A : ℕ → Type ℓ} → A zero → ((n : ℕ) → A n → A (suc n)) → (n : ℕ) → A n elim a₀ _ zero = a₀ elim a₀ f (suc n) = f n (elim a₀ f n) isEven isOdd : ℕ → Bool isEven zero = true isEven (suc n) = isOdd n isOdd zero = false isOdd (suc n) = isEven n --Typed version private toType : Bool → Type toType false = ⊥ toType true = Unit isEvenT : ℕ → Type isEvenT n = toType (isEven n) isOddT : ℕ → Type isOddT n = isEvenT (suc n) isZero : ℕ → Bool isZero zero = true isZero (suc n) = false

21.287879

52

0.627046

6474f6c404607a6b2f486a6e58fd8e5ae0722fbf

4,692

agda

Agda

examples/instance-arguments/07-subclasses.agda

larrytheliquid/agda

477c8c37f948e6038b773409358fd8f38395f827

[ "MIT" ]

1

2018-10-10T17:08:44.000Z

examples/instance-arguments/07-subclasses.agda

masondesu/agda

70c8a575c46f6a568c7518150a1a64fcd03aa437

[ "MIT" ]

null

examples/instance-arguments/07-subclasses.agda

masondesu/agda

70c8a575c46f6a568c7518150a1a64fcd03aa437

[ "MIT" ]

1

2022-03-12T11:35:18.000Z

-- {-# OPTIONS --verbose tc.records.ifs:15 #-} -- {-# OPTIONS --verbose tc.constr.findInScope:15 #-} -- {-# OPTIONS --verbose tc.term.args.ifs:15 #-} -- {-# OPTIONS --verbose cta.record.ifs:15 #-} -- {-# OPTIONS --verbose tc.section.apply:25 #-} -- {-# OPTIONS --verbose tc.mod.apply:100 #-} -- {-# OPTIONS --verbose scope.rec:15 #-} -- {-# OPTIONS --verbose tc.rec.def:15 #-} module 07-subclasses where module Imports where module L where open import Agda.Primitive public using (Level; _⊔_) renaming (lzero to zero; lsuc to suc) -- extract from Function id : ∀ {a} {A : Set a} → A → A id x = x _$_ : ∀ {a b} {A : Set a} {B : A → Set b} → ((x : A) → B x) → ((x : A) → B x) f $ x = f x _∘_ : ∀ {a b c} {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) -- extract from Data.Bool infixr 5 _∨_ data Bool : Set where true : Bool false : Bool not : Bool → Bool not true = false not false = true _∨_ : Bool → Bool → Bool true ∨ b = true false ∨ b = b -- extract from Relation.Nullary.Decidable and friends infix 3 ¬_ data ⊥ : Set where ¬_ : ∀ {ℓ} → Set ℓ → Set ℓ ¬ P = P → ⊥ data Dec {p} (P : Set p) : Set p where yes : ( p : P) → Dec P no : (¬p : ¬ P) → Dec P ⌊_⌋ : ∀ {p} {P : Set p} → Dec P → Bool ⌊ yes _ ⌋ = true ⌊ no _ ⌋ = false -- extract from Relation.Binary.PropositionalEquality data _≡_ {a} {A : Set a} (x : A) : A → Set a where refl : x ≡ x cong : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) {x y} → x ≡ y → f x ≡ f y cong f refl = refl -- extract from Data.Nat data ℕ : Set where zero : ℕ suc : (n : ℕ) → ℕ {-# BUILTIN NATURAL ℕ #-} pred : ℕ → ℕ pred zero = zero pred (suc n) = n _≟_ : (x y : ℕ) → Dec (x ≡ y) zero ≟ zero = yes refl suc m ≟ suc n with m ≟ n suc m ≟ suc .m | yes refl = yes refl suc m ≟ suc n | no prf = no (prf ∘ cong pred) zero ≟ suc n = no λ() suc m ≟ zero = no λ() open Imports -- Begin of actual example! record Eq (A : Set) : Set where field eq : A → A → Bool primEqBool : Bool → Bool → Bool primEqBool true = id primEqBool false = not eqBool : Eq Bool eqBool = record { eq = primEqBool } primEqNat : ℕ → ℕ → Bool primEqNat a b = ⌊ a ≟ b ⌋ primLtNat : ℕ → ℕ → Bool primLtNat 0 _ = true primLtNat (suc a) (suc b) = primLtNat a b primLtNat _ _ = false neq : {t : Set} → {{eqT : Eq t}} → t → t → Bool neq a b = not $ eq a b where open Eq {{...}} record Ord₁ (A : Set) : Set where field _<_ : A → A → Bool eqA : Eq A ord₁Nat : Ord₁ ℕ ord₁Nat = record { _<_ = primLtNat; eqA = eqNat } where eqNat : Eq ℕ eqNat = record { eq = primEqNat } record Ord₂ {A : Set} (eqA : Eq A) : Set where field _<_ : A → A → Bool ord₂Nat : Ord₂ (record { eq = primEqNat }) ord₂Nat = record { _<_ = primLtNat } record Ord₃ (A : Set) : Set where field _<_ : A → A → Bool eqA : Eq A open Eq eqA public ord₃Nat : Ord₃ ℕ ord₃Nat = record { _<_ = primLtNat; eqA = eqNat } where eqNat : Eq ℕ eqNat = record { eq = primEqNat } record Ord₄ {A : Set} (eqA : Eq A) : Set where field _<_ : A → A → Bool open Eq eqA public ord₄Nat : Ord₄ (record { eq = primEqNat }) ord₄Nat = record { _<_ = primLtNat } module test₁ where open Ord₁ {{...}} open Eq {{...}} eqNat : Eq _ eqNat = eqA test₁ = 5 < 3 test₂ = eq 5 3 test₃ = eq true false test₄ : {A : Set} → {{ ordA : Ord₁ A }} → A → A → Bool test₄ a b = a < b ∨ eq a b where eqA' : Eq _ eqA' = eqA module test₂ where open Ord₂ {{...}} open Eq {{...}} eqNat : Eq ℕ eqNat = record { eq = primEqNat } test₁ = 5 < 3 test₂ = eq 5 3 test₃ = eq true false test₄ : {A : Set} → {eqA : Eq A} → {{ ordA : Ord₂ eqA }} → A → A → Bool test₄ {eqA = _} a b = a < b ∨ eq a b module test₃ where open Ord₃ {{...}} open Eq {{...}} renaming (eq to eq') test₁ = 5 < 3 test₂ = eq 5 3 test₃ = eq' true false test₄ : {A : Set} → {{ ordA : Ord₃ A }} → A → A → Bool test₄ a b = a < b ∨ eq a b module test₄ where open Ord₄ {{...}} open Eq {{...}} renaming (eq to eq') test₁ = 5 < 3 test₂ = eq 5 3 test₃ = eq' true false test₄ : {A : Set} → {eqA : Eq A} → {{ ordA : Ord₄ eqA }} → A → A → Bool test₄ a b = a < b ∨ eq a b module test₄′ where open Ord₄ {{...}} hiding (eq) open Eq {{...}} eqNat : Eq ℕ eqNat = record { eq = primEqNat } test₁ = 5 < 3 test₂ = eq 5 3 test₃ = eq true false test₄ : {A : Set} → {eqA : Eq A} → {{ ordA : Ord₄ eqA }} → A → A → Bool test₄ {eqA = _} a b = a < b ∨ eq a b

22.132075

73

0.521739

116a9d74eb27a9da5b8db966536e91480a937f3d

3,736

agda

Agda

RMonads/REM.agda

jmchapman/Relative-Monads

74707d3538bf494f4bd30263d2f5515a84733865

[ "MIT" ]

21

2015-07-30T01:25:12.000Z

2021-02-13T18:02:18.000Z

RMonads/REM.agda

jmchapman/Relative-Monads

74707d3538bf494f4bd30263d2f5515a84733865

[ "MIT" ]

3

2019-01-13T13:12:33.000Z

2019-05-29T09:50:26.000Z

RMonads/REM.agda

jmchapman/Relative-Monads

74707d3538bf494f4bd30263d2f5515a84733865

[ "MIT" ]

1

2019-11-04T21:33:13.000Z

open import Categories open import Functors open import RMonads module RMonads.REM {a b c d}{C : Cat {a}{b}}{D : Cat {c}{d}}{J : Fun C D} (M : RMonad J) where open import Library open RMonad M open Fun record RAlg : Set (a ⊔ c ⊔ d) where constructor ralg open Cat D field acar : Obj astr : ∀ {Z} → Hom (OMap J Z) acar → Hom (T Z) acar alaw1 : ∀ {Z}{f : Hom (OMap J Z) acar} → f ≅ comp (astr f) η alaw2 : ∀{Z}{W}{k : Hom (OMap J Z) (T W)} {f : Hom (OMap J W) acar} → astr (comp (astr f) k) ≅ comp (astr f) (bind k) AlgEq : {X Y : RAlg} → RAlg.acar X ≅ RAlg.acar Y → ((λ {Z} → RAlg.astr X {Z}) ≅ (λ {Z} → RAlg.astr Y {Z})) → X ≅ Y AlgEq {ralg acar astr _ _}{ralg .acar .astr _ _} refl refl = let open Cat in cong₂ (ralg acar astr) (iext λ _ → iext λ _ → ir _ _) (iext λ _ → iext λ _ → iext λ _ → iext λ _ → ir _ _) astrnat : ∀(alg : RAlg){X Y} (f : Cat.Hom C X Y) → (g : Cat.Hom D (OMap J X) (RAlg.acar alg)) (g' : Cat.Hom D (OMap J Y) (RAlg.acar alg)) → Cat.comp D g' (HMap J f) ≅ g → Cat.comp D (RAlg.astr alg g') (RMonad.bind M (Cat.comp D (RMonad.η M) (HMap J f))) ≅ RAlg.astr alg g astrnat alg f g g' p = let open RAlg alg; open Cat D in proof comp (astr g') (bind (comp η (HMap J f))) ≅⟨ sym alaw2 ⟩ astr (comp (astr g') (comp η (HMap J f))) ≅⟨ cong astr (sym ass) ⟩ astr (comp (comp (astr g') η) (HMap J f)) ≅⟨ cong (λ g₁ → astr (comp g₁ (HMap J f))) (sym alaw1) ⟩ astr (comp g' (HMap J f)) ≅⟨ cong astr p ⟩ astr g ∎ record RAlgMorph (A B : RAlg) : Set (a ⊔ c ⊔ d) where constructor ralgmorph open Cat D open RAlg field amor : Hom (acar A) (acar B) ahom : ∀{Z}{f : Hom (OMap J Z) (acar A)} → comp amor (astr A f) ≅ astr B (comp amor f) open RAlgMorph RAlgMorphEq : ∀{X Y : RAlg}{f g : RAlgMorph X Y} → amor f ≅ amor g → f ≅ g RAlgMorphEq {X}{Y}{ralgmorph amor _}{ralgmorph .amor _} refl = cong (ralgmorph amor) (iext λ _ → iext λ _ → ir _ _) lemZ : ∀{X X' Y Y' : RAlg} {f : RAlgMorph X Y}{g : RAlgMorph X' Y'} → X ≅ X' → Y ≅ Y' → amor f ≅ amor g → f ≅ g lemZ refl refl = RAlgMorphEq IdMorph : ∀{A : RAlg} → RAlgMorph A A IdMorph {A} = let open Cat D; open RAlg A in record { amor = iden; ahom = λ {_ f} → proof comp iden (astr f) ≅⟨ idl ⟩ astr f ≅⟨ cong astr (sym idl) ⟩ astr (comp iden f) ∎} CompMorph : ∀{X Y Z : RAlg} → RAlgMorph Y Z → RAlgMorph X Y → RAlgMorph X Z CompMorph {X}{Y}{Z} f g = let open Cat D; open RAlg in record { amor = comp (amor f) (amor g); ahom = λ {_ f'} → proof comp (comp (amor f) (amor g)) (astr X f') ≅⟨ ass ⟩ comp (amor f) (comp (amor g) (astr X f')) ≅⟨ cong (comp (amor f)) (ahom g) ⟩ comp (amor f) (astr Y (comp (amor g) f')) ≅⟨ ahom f ⟩ astr Z (comp (amor f) (comp (amor g) f')) ≅⟨ cong (astr Z) (sym ass) ⟩ astr Z (comp (comp (amor f) (amor g)) f') ∎} idlMorph : {X Y : RAlg}{f : RAlgMorph X Y} → CompMorph IdMorph f ≅ f idlMorph = RAlgMorphEq (Cat.idl D) idrMorph : ∀{X Y : RAlg}{f : RAlgMorph X Y} → CompMorph f IdMorph ≅ f idrMorph = RAlgMorphEq (Cat.idr D) assMorph : ∀{W X Y Z : RAlg} {f : RAlgMorph Y Z}{g : RAlgMorph X Y}{h : RAlgMorph W X} → CompMorph (CompMorph f g) h ≅ CompMorph f (CompMorph g h) assMorph = RAlgMorphEq (Cat.ass D) EM : Cat EM = record{ Obj = RAlg; Hom = RAlgMorph; iden = IdMorph; comp = CompMorph; idl = idlMorph; idr = idrMorph; ass = λ{_ _ _ _ f g h} → assMorph {f = f}{g}{h}}

30.129032

76

0.523019

3f2b9915ed3d88213be87cdbd75c76cd3566d9ed

276

agda

Agda

test/Common/Irrelevance.agda

larrytheliquid/agda

477c8c37f948e6038b773409358fd8f38395f827

[ "MIT" ]

1

2019-11-27T07:26:06.000Z

test/Common/Irrelevance.agda

np/agda-git-experiment

20596e9dd9867166a64470dd24ea68925ff380ce

[ "MIT" ]

null

test/Common/Irrelevance.agda

np/agda-git-experiment

20596e9dd9867166a64470dd24ea68925ff380ce

[ "MIT" ]

1

2022-03-12T11:35:18.000Z

-- Andreas, 2012-01-12 module Common.Irrelevance where open import Common.Level postulate .irrAxiom : ∀ {a}{A : Set a} → .A → A {-# BUILTIN IRRAXIOM irrAxiom #-} record Squash {a}(A : Set a) : Set a where constructor squash field .unsquash : A open Squash public

18.4

42

0.673913

4ef7e78516337a81789d9d81a5ead72033fee44f

2,512

agda

Agda

DataAndCodata.agda

nad/codata

1b90445566df0d3b4ba6e31bd0bac417b4c0eb0e

[ "MIT" ]

1

2021-02-13T14:48:45.000Z

DataAndCodata.agda

nad/codata

1b90445566df0d3b4ba6e31bd0bac417b4c0eb0e

[ "MIT" ]

null

DataAndCodata.agda

nad/codata

1b90445566df0d3b4ba6e31bd0bac417b4c0eb0e

[ "MIT" ]

null

------------------------------------------------------------------------ -- Data and codata can sometimes be "unified" ------------------------------------------------------------------------ -- In Haskell one can define the partial list type once, and define -- map once and for all for this type. In Agda one typically defines -- the (finite) list type + map and separately the (potentially -- infinite) colist type + map. This is not strictly necessary, -- though: the two types can be unified. The gain may be small, -- though. module DataAndCodata where open import Codata.Musical.Notation open import Function open import Relation.Binary.PropositionalEquality ------------------------------------------------------------------------ -- Conditional coinduction data Rec : Set where μ : Rec ν : Rec ∞? : Rec → Set → Set ∞? μ = id ∞? ν = ∞ ♯? : ∀ (r : Rec) {A} → A → ∞? r A ♯? μ x = x ♯? ν x = ♯ x ♭? : ∀ (r : Rec) {A} → ∞? r A → A ♭? μ = id ♭? ν = ♭ ------------------------------------------------------------------------ -- A type for definitely finite or potentially infinite lists -- If the Rec parameter is μ, then the type contains finite lists, and -- otherwise it contains potentially infinite lists. infixr 5 _∷_ data List∞? (r : Rec) (A : Set) : Set where [] : List∞? r A _∷_ : A → ∞? r (List∞? r A) → List∞? r A -- List equality. infix 4 _≈_ data _≈_ {r A} : List∞? r A → List∞? r A → Set where [] : [] ≈ [] _∷_ : ∀ {x y xs ys} → x ≡ y → ∞? r (♭? r xs ≈ ♭? r ys) → x ∷ xs ≈ y ∷ ys -- μ-lists can be seen as ν-lists. lift : ∀ {A} → List∞? μ A → List∞? ν A lift [] = [] lift (x ∷ xs) = x ∷ ♯ lift xs ------------------------------------------------------------------------ -- The map function -- Maps over any list. The definition contains separate cases for _∷_ -- depending on whether the Rec index is μ or ν, though. map : ∀ {r A B} → (A → B) → List∞? r A → List∞? r B map f [] = [] map {μ} f (x ∷ xs) = f x ∷ map f xs -- Structural recursion -- (guarded). map {ν} f (x ∷ xs) = f x ∷ ♯ map f (♭ xs) -- Guarded corecursion. -- In Haskell the map function is automatically (in effect) parametric -- in the Rec parameter. Here this property is not automatic, so I -- have proved it manually: map-parametric : ∀ {A B} (f : A → B) (xs : List∞? μ A) → map f (lift xs) ≈ lift (map f xs) map-parametric f [] = [] map-parametric f (x ∷ xs) = refl ∷ ♯ map-parametric f xs

29.904762

72

0.496815

3666f9520a710b3afbf1c2e71c5e51dd3f793232

41

agda

Agda

test/interaction/Issue2959.agda

mdimjasevic/agda

8fb548356b275c7a1e79b768b64511ae937c738b

[ "BSD-3-Clause" ]

1,989

2015-01-09T23:51:16.000Z

2022-03-30T18:20:48.000Z

test/interaction/Issue2959.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

4,066

2015-01-10T11:24:51.000Z

2022-03-31T21:14:49.000Z

test/interaction/Issue2959.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

371

2015-01-03T14:04:08.000Z

2022-03-30T19:00:30.000Z

open import Issue2959.M Set r : R r = ?

8.2

27

0.634146

37dc2b48aaf7fc1991538b52a4278a69097ca189

4,125

agda

Agda

Categories/Object/BinaryProducts/N-ary.agda

copumpkin/categories

36f4181d751e2ecb54db219911d8c69afe8ba892

[ "BSD-3-Clause" ]

98

2015-04-15T14:57:33.000Z

2022-03-08T05:20:36.000Z

Categories/Object/BinaryProducts/N-ary.agda

copumpkin/categories

36f4181d751e2ecb54db219911d8c69afe8ba892

[ "BSD-3-Clause" ]

19

2015-05-23T06:47:10.000Z

2019-08-09T16:31:40.000Z

Categories/Object/BinaryProducts/N-ary.agda

copumpkin/categories

36f4181d751e2ecb54db219911d8c69afe8ba892

[ "BSD-3-Clause" ]

23

2015-02-05T13:03:09.000Z

2021-11-11T13:50:56.000Z

{-# OPTIONS --universe-polymorphism #-} open import Categories.Category open import Categories.Object.BinaryProducts module Categories.Object.BinaryProducts.N-ary {o ℓ e} (C : Category o ℓ e) (BP : BinaryProducts C) where open Category C open BinaryProducts BP open Equiv import Categories.Object.Product open Categories.Object.Product C open import Data.Nat using (ℕ; zero; suc) open import Data.Vec open import Data.Product.N-ary hiding ([]) Prod : {n : ℕ} → Vec Obj (suc n) → Obj Prod { zero} (A ∷ []) = A Prod {suc n} (A ∷ As) = A × Prod {n} As πˡ : {n m : ℕ} → (As : Vec Obj (suc n)) → (Bs : Vec Obj (suc m)) → Prod (As ++ Bs) ⇒ Prod As πˡ { zero} (A ∷ []) Bs = π₁ πˡ {suc n} (A ∷ As) Bs = ⟨ π₁ , πˡ {n} As Bs ∘ π₂ ⟩ πʳ : {n m : ℕ} → (As : Vec Obj (suc n)) → (Bs : Vec Obj (suc m)) → Prod (As ++ Bs) ⇒ Prod Bs πʳ { zero} (A ∷ []) Bs = π₂ πʳ {suc n} (A ∷ As) Bs = πʳ {n} As Bs ∘ π₂ glue : {n m : ℕ}{X : Obj} → (As : Vec Obj (suc n)) → (Bs : Vec Obj (suc m)) → (f : X ⇒ Prod As) → (g : X ⇒ Prod Bs) → X ⇒ Prod (As ++ Bs) glue { zero}{m} (A ∷ []) Bs f g = ⟨ f , g ⟩ glue {suc n}{m} (A ∷ As) Bs f g = ⟨ π₁ ∘ f , glue As Bs (π₂ ∘ f) g ⟩ open HomReasoning .commuteˡ : {n m : ℕ}{X : Obj} → (As : Vec Obj (suc n)) → (Bs : Vec Obj (suc m)) → {f : X ⇒ Prod As} → {g : X ⇒ Prod Bs} → πˡ As Bs ∘ glue As Bs f g ≡ f commuteˡ { zero} (A ∷ []) Bs {f}{g} = commute₁ commuteˡ {suc n} (A ∷ As) Bs {f}{g} = begin ⟨ π₁ , πˡ As Bs ∘ π₂ ⟩ ∘ ⟨ π₁ ∘ f , glue As Bs (π₂ ∘ f) g ⟩ ↓⟨ ⟨⟩∘ ⟩ ⟨ π₁ ∘ ⟨ π₁ ∘ f , glue As Bs (π₂ ∘ f) g ⟩ , (πˡ As Bs ∘ π₂) ∘ ⟨ π₁ ∘ f , glue As Bs (π₂ ∘ f) g ⟩ ⟩ ↓⟨ ⟨⟩-cong₂ commute₁ assoc ⟩ ⟨ π₁ ∘ f , πˡ As Bs ∘ π₂ ∘ ⟨ π₁ ∘ f , glue As Bs (π₂ ∘ f) g ⟩ ⟩ ↓⟨ ⟨⟩-congʳ (refl ⟩∘⟨ commute₂) ⟩ ⟨ π₁ ∘ f , πˡ As Bs ∘ glue As Bs (π₂ ∘ f) g ⟩ ↓⟨ ⟨⟩-congʳ (commuteˡ As Bs) ⟩ ⟨ π₁ ∘ f , π₂ ∘ f ⟩ ↓⟨ g-η ⟩ f ∎ .commuteʳ : {n m : ℕ}{X : Obj} → (As : Vec Obj (suc n)) → (Bs : Vec Obj (suc m)) → {f : X ⇒ Prod As} → {g : X ⇒ Prod Bs} → πʳ As Bs ∘ glue As Bs f g ≡ g commuteʳ { zero} (A ∷ []) Bs {f}{g} = commute₂ commuteʳ {suc n} (A ∷ As) Bs {f}{g} = begin (πʳ As Bs ∘ π₂) ∘ ⟨ π₁ ∘ f , glue As Bs (π₂ ∘ f) g ⟩ ↓⟨ assoc ⟩ πʳ As Bs ∘ π₂ ∘ ⟨ π₁ ∘ f , glue As Bs (π₂ ∘ f) g ⟩ ↓⟨ refl ⟩∘⟨ commute₂ ⟩ πʳ As Bs ∘ glue As Bs (π₂ ∘ f) g ↓⟨ commuteʳ As Bs ⟩ g ∎ .N-universal : {n m : ℕ}{X : Obj} → (As : Vec Obj (suc n)) → (Bs : Vec Obj (suc m)) → {f : X ⇒ Prod As} → {g : X ⇒ Prod Bs} → {h : X ⇒ Prod (As ++ Bs) } → πˡ As Bs ∘ h ≡ f → πʳ As Bs ∘ h ≡ g → glue As Bs f g ≡ h N-universal { zero} (A ∷ []) Bs {f}{g}{h} h-commuteˡ h-commuteʳ = universal h-commuteˡ h-commuteʳ N-universal {suc n} (A ∷ As) Bs {f}{g}{h} h-commuteˡ h-commuteʳ = begin ⟨ π₁ ∘ f , glue As Bs (π₂ ∘ f) g ⟩ ↓⟨ ⟨⟩-congʳ (N-universal As Bs π₂∘h-commuteˡ π₂∘h-commuteʳ) ⟩ ⟨ π₁ ∘ f , π₂ ∘ h ⟩ ↑⟨ ⟨⟩-congˡ π₁∘h-commuteˡ ⟩ ⟨ π₁ ∘ h , π₂ ∘ h ⟩ ↓⟨ g-η ⟩ h ∎ where -- h-commuteˡ : ⟨ π₁ , πˡ As Bs ∘ π₂ ⟩ ∘ h ≡ f -- h-commuteʳ : (πʳ As Bs ∘ π₂) ∘ h ≡ g π₁∘h-commuteˡ : π₁ ∘ h ≡ π₁ ∘ f π₁∘h-commuteˡ = begin π₁ ∘ h ↑⟨ commute₁ ⟩∘⟨ refl ⟩ (π₁ ∘ ⟨ π₁ , πˡ As Bs ∘ π₂ ⟩) ∘ h ↓⟨ assoc ⟩ π₁ ∘ ⟨ π₁ , πˡ As Bs ∘ π₂ ⟩ ∘ h ↓⟨ refl ⟩∘⟨ h-commuteˡ ⟩ π₁ ∘ f ∎ π₂∘h-commuteˡ : πˡ As Bs ∘ π₂ ∘ h ≡ π₂ ∘ f π₂∘h-commuteˡ = begin πˡ As Bs ∘ π₂ ∘ h ↑⟨ assoc ⟩ (πˡ As Bs ∘ π₂) ∘ h ↑⟨ commute₂ ⟩∘⟨ refl ⟩ (π₂ ∘ ⟨ π₁ , πˡ As Bs ∘ π₂ ⟩) ∘ h ↓⟨ assoc ⟩ π₂ ∘ ⟨ π₁ , πˡ As Bs ∘ π₂ ⟩ ∘ h ↓⟨ refl ⟩∘⟨ h-commuteˡ ⟩ π₂ ∘ f ∎ π₂∘h-commuteʳ : πʳ As Bs ∘ π₂ ∘ h ≡ g π₂∘h-commuteʳ = trans (sym assoc) h-commuteʳ isProduct : {n m : ℕ} → (As : Vec Obj (suc n)) → (Bs : Vec Obj (suc m)) → Product (Prod As) (Prod Bs) isProduct {n}{m} As Bs = record { A×B = Prod (As ++ Bs) ; π₁ = πˡ As Bs ; π₂ = πʳ As Bs ; ⟨_,_⟩ = glue As Bs ; commute₁ = commuteˡ As Bs ; commute₂ = commuteʳ As Bs ; universal = N-universal As Bs }

25.78125

97

0.475152

116ee691a0428b2b36171620969d3f7378fffd66

1,569

agda

Agda

src/Categories/Category/Helper.agda

jaykru/agda-categories

a4053cf700bcefdf73b857c3352f1eae29382a60

[ "MIT" ]

279

2019-06-01T14:36:40.000Z

2022-03-22T00:40:14.000Z

src/Categories/Category/Helper.agda

seanpm2001/agda-categories

d9e4f578b126313058d105c61707d8c8ae987fa8

[ "MIT" ]

236

2019-06-01T14:53:54.000Z

2022-03-28T14:31:43.000Z

src/Categories/Category/Helper.agda

seanpm2001/agda-categories

d9e4f578b126313058d105c61707d8c8ae987fa8

[ "MIT" ]

64

2019-06-02T16:58:15.000Z

2022-03-14T02:00:59.000Z

{-# OPTIONS --without-K --safe #-} module Categories.Category.Helper where open import Level open import Relation.Binary using (Rel; IsEquivalence) open import Categories.Category.Core using (Category) -- Since we add extra proofs in the definition of `Category` (i.e. `sym-assoc` and -- `identity²`), we might still want to construct a `Category` in its originally -- easier manner. Thus, this redundant definition is here to ease the construction. private record CategoryHelper (o ℓ e : Level) : Set (suc (o ⊔ ℓ ⊔ e)) where infix 4 _≈_ _⇒_ infixr 9 _∘_ field Obj : Set o _⇒_ : 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) identityˡ : ∀ {A B} {f : A ⇒ B} → id ∘ f ≈ f identityʳ : ∀ {A B} {f : A ⇒ B} → f ∘ id ≈ f 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 categoryHelper : ∀ {o ℓ e} → CategoryHelper o ℓ e → Category o ℓ e categoryHelper C = record { Obj = Obj ; _⇒_ = _⇒_ ; _≈_ = _≈_ ; id = id ; _∘_ = _∘_ ; assoc = assoc ; sym-assoc = sym assoc ; identityˡ = identityˡ ; identityʳ = identityʳ ; identity² = identityˡ ; equiv = equiv ; ∘-resp-≈ = ∘-resp-≈ } where open CategoryHelper C module _ {A B} where open IsEquivalence (equiv {A} {B}) public

31.38

93

0.525813

4e5f90ee81f2910c5485e0a2d7514d31ee852b52

189

agda

Agda

Cubical/HITs/AssocList.agda

kiana-S/univalent-foundations

8bdb0766260489f9c89a14f4c0f2ad26e5190dec

[ "MIT" ]

null

Cubical/HITs/AssocList.agda

kiana-S/univalent-foundations

8bdb0766260489f9c89a14f4c0f2ad26e5190dec

[ "MIT" ]

1

2022-01-27T02:07:48.000Z

Cubical/HITs/AssocList.agda

kiana-S/univalent-foundations

8bdb0766260489f9c89a14f4c0f2ad26e5190dec

[ "MIT" ]

1

2021-11-22T02:02:01.000Z

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

27

52

0.783069

1b7ae68b5774d544b03ab3346ad592ec477a0e6b

5,660

agda

Agda

Cubical/HITs/SetQuotients/Properties.agda

Rotsor/cubical

d55cd4834ca1f171f58b4a0c46b804ea6d18191f

[ "MIT" ]

null

Cubical/HITs/SetQuotients/Properties.agda

Rotsor/cubical

d55cd4834ca1f171f58b4a0c46b804ea6d18191f

[ "MIT" ]

null

Cubical/HITs/SetQuotients/Properties.agda

Rotsor/cubical

d55cd4834ca1f171f58b4a0c46b804ea6d18191f

[ "MIT" ]

null

{- Set quotients: -} {-# OPTIONS --cubical --safe #-} module Cubical.HITs.SetQuotients.Properties where open import Cubical.HITs.SetQuotients.Base open import Cubical.Core.Everything open import Cubical.Foundations.Prelude open import Cubical.Foundations.Isomorphism open import Cubical.Foundations.Equiv open import Cubical.Foundations.HLevels open import Cubical.Foundations.HAEquiv open import Cubical.Foundations.Univalence open import Cubical.Data.Nat open import Cubical.Data.Sigma open import Cubical.Relation.Nullary open import Cubical.Relation.Binary.Base open import Cubical.HITs.PropositionalTruncation open import Cubical.HITs.SetTruncation -- Type quotients private variable ℓ ℓ' ℓ'' : Level A : Type ℓ R : A → A → Type ℓ' B : A / R → Type ℓ'' elimEq/ : (Bprop : (x : A / R ) → isProp (B x)) {x y : A / R} (eq : x ≡ y) (bx : B x) (by : B y) → PathP (λ i → B (eq i)) bx by elimEq/ {B = B} Bprop {x = x} = J (λ y eq → ∀ bx by → PathP (λ i → B (eq i)) bx by) (λ bx by → Bprop x bx by) elimSetQuotientsProp : ((x : A / R ) → isProp (B x)) → (f : (a : A) → B ( [ a ])) → (x : A / R) → B x elimSetQuotientsProp Bprop f [ x ] = f x elimSetQuotientsProp Bprop f (squash/ x y p q i j) = isOfHLevel→isOfHLevelDep {n = 2} (λ x → isProp→isSet (Bprop x)) (g x) (g y) (cong g p) (cong g q) (squash/ x y p q) i j where g = elimSetQuotientsProp Bprop f elimSetQuotientsProp Bprop f (eq/ a b r i) = elimEq/ Bprop (eq/ a b r) (f a) (f b) i -- lemma 6.10.2 in hott book -- TODO: defined truncated Sigma as ∃ []surjective : (x : A / R) → ∥ Σ[ a ∈ A ] [ a ] ≡ x ∥ []surjective = elimSetQuotientsProp (λ x → squash) (λ a → ∣ a , refl ∣) elimSetQuotients : {B : A / R → Type ℓ} → (Bset : (x : A / R) → isSet (B x)) → (f : (a : A) → (B [ a ])) → (feq : (a b : A) (r : R a b) → PathP (λ i → B (eq/ a b r i)) (f a) (f b)) → (x : A / R) → B x elimSetQuotients Bset f feq [ a ] = f a elimSetQuotients Bset f feq (eq/ a b r i) = feq a b r i elimSetQuotients Bset f feq (squash/ x y p q i j) = isOfHLevel→isOfHLevelDep {n = 2} Bset (g x) (g y) (cong g p) (cong g q) (squash/ x y p q) i j where g = elimSetQuotients Bset f feq setQuotUniversal : {B : Type ℓ} (Bset : isSet B) → (A / R → B) ≃ (Σ[ f ∈ (A → B) ] ((a b : A) → R a b → f a ≡ f b)) setQuotUniversal Bset = isoToEquiv (iso intro elim elimRightInv elimLeftInv) where intro = λ g → (λ a → g [ a ]) , λ a b r i → g (eq/ a b r i) elim = λ h → elimSetQuotients (λ x → Bset) (fst h) (snd h) elimRightInv : ∀ h → intro (elim h) ≡ h elimRightInv h = refl elimLeftInv : ∀ g → elim (intro g) ≡ g elimLeftInv = λ g → funExt (λ x → elimPropTrunc {P = λ sur → elim (intro g) x ≡ g x} (λ sur → Bset (elim (intro g) x) (g x)) (λ sur → cong (elim (intro g)) (sym (snd sur)) ∙ (cong g (snd sur))) ([]surjective x) ) open BinaryRelation effective : (Rprop : isPropValued R) (Requiv : isEquivRel R) (a b : A) → [ a ] ≡ [ b ] → R a b effective {A = A} {R = R} Rprop (EquivRel R/refl R/sym R/trans) a b p = transport aa≡ab (R/refl _) where helper : A / R → hProp helper = elimSetQuotients (λ _ → isSetHProp) (λ c → (R a c , Rprop a c)) (λ c d cd → ΣProp≡ (λ _ → isPropIsProp) (ua (PropEquiv→Equiv (Rprop a c) (Rprop a d) (λ ac → R/trans _ _ _ ac cd) (λ ad → R/trans _ _ _ ad (R/sym _ _ cd))))) aa≡ab : R a a ≡ R a b aa≡ab i = fst (helper (p i)) isEquivRel→isEffective : isPropValued R → isEquivRel R → isEffective R isEquivRel→isEffective {R = R} Rprop Req a b = isoToEquiv (iso intro elim intro-elim elim-intro) where intro : [ a ] ≡ [ b ] → R a b intro = effective Rprop Req a b elim : R a b → [ a ] ≡ [ b ] elim = eq/ a b intro-elim : ∀ x → intro (elim x) ≡ x intro-elim ab = Rprop a b _ _ elim-intro : ∀ x → elim (intro x) ≡ x elim-intro eq = squash/ _ _ _ _ discreteSetQuotients : Discrete A → isPropValued R → isEquivRel R → (∀ a₀ a₁ → Dec (R a₀ a₁)) → Discrete (A / R) discreteSetQuotients {A = A} {R = R} Adis Rprop Req Rdec = elimSetQuotients ((λ a₀ → isSetPi (λ a₁ → isProp→isSet (isPropDec (squash/ a₀ a₁))))) discreteSetQuotients' discreteSetQuotients'-eq where discreteSetQuotients' : (a : A) (y : A / R) → Dec ([ a ] ≡ y) discreteSetQuotients' a₀ = elimSetQuotients ((λ a₁ → isProp→isSet (isPropDec (squash/ [ a₀ ] a₁)))) dis dis-eq where dis : (a₁ : A) → Dec ([ a₀ ] ≡ [ a₁ ]) dis a₁ with Rdec a₀ a₁ ... | (yes p) = yes (eq/ a₀ a₁ p) ... | (no ¬p) = no λ eq → ¬p (effective Rprop Req a₀ a₁ eq ) dis-eq : (a b : A) (r : R a b) → PathP (λ i → Dec ([ a₀ ] ≡ eq/ a b r i)) (dis a) (dis b) dis-eq a b ab = J (λ b ab → ∀ k → PathP (λ i → Dec ([ a₀ ] ≡ ab i)) (dis a) k) (λ k → isPropDec (squash/ _ _) _ _) (eq/ a b ab) (dis b) discreteSetQuotients'-eq : (a b : A) (r : R a b) → PathP (λ i → (y : A / R) → Dec (eq/ a b r i ≡ y)) (discreteSetQuotients' a) (discreteSetQuotients' b) discreteSetQuotients'-eq a b ab = J (λ b ab → ∀ k → PathP (λ i → (y : A / R) → Dec (ab i ≡ y)) (discreteSetQuotients' a) k) (λ k → funExt (λ x → isPropDec (squash/ _ _) _ _)) (eq/ a b ab) (discreteSetQuotients' b)

37.733333

142

0.536219

cc3891dd3329870ef6bc3d534ace5a802e3cf579

164

agda

Agda

test/Fail/UnknownImplicitInstance.agda

mdimjasevic/agda

8fb548356b275c7a1e79b768b64511ae937c738b

[ "BSD-3-Clause" ]

1,989

2015-01-09T23:51:16.000Z

2022-03-30T18:20:48.000Z

test/Fail/UnknownImplicitInstance.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

4,066

2015-01-10T11:24:51.000Z

2022-03-31T21:14:49.000Z

test/Fail/UnknownImplicitInstance.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

371

2015-01-03T14:04:08.000Z

2022-03-30T19:00:30.000Z

module UnknownImplicitInstance where ⟨⟩ : {A : Set} {{a : A}} → A ⟨⟩ {{a}} = a postulate B : Set instance b : B f : {A : Set₁} {{a : A}} → A x : Set x = f

12.615385

36

0.493902

4ea49ed65934cac2c59e698ca56b4ab067e09706

5,061

agda

Agda

vendor/stdlib/src/Data/Fin/Dec.agda

isabella232/Lemmachine

8ef786b40e4a9ab274c6103dc697dcb658cf3db3

[ "MIT" ]

56

2015-01-20T02:11:42.000Z

2021-12-21T17:02:19.000Z

vendor/stdlib/src/Data/Fin/Dec.agda

larrytheliquid/Lemmachine

8ef786b40e4a9ab274c6103dc697dcb658cf3db3

[ "MIT" ]

1

2022-03-12T12:17:51.000Z

vendor/stdlib/src/Data/Fin/Dec.agda

isabella232/Lemmachine

8ef786b40e4a9ab274c6103dc697dcb658cf3db3

[ "MIT" ]

3

2015-07-21T16:37:58.000Z

2022-03-12T11:54:10.000Z

------------------------------------------------------------------------ -- Decision procedures for finite sets and subsets of finite sets ------------------------------------------------------------------------ module Data.Fin.Dec where open import Data.Function open import Data.Nat hiding (_<_) open import Data.Vec hiding (_∈_) open import Data.Fin open import Data.Fin.Subset open import Data.Fin.Subset.Props open import Data.Product as Prod open import Data.Empty open import Relation.Nullary open import Relation.Unary using (Pred) infix 4 _∈?_ _∈?_ : ∀ {n} x (p : Subset n) → Dec (x ∈ p) zero ∈? inside ∷ p = yes here zero ∈? outside ∷ p = no λ() suc n ∈? s ∷ p with n ∈? p ... | yes n∈p = yes (there n∈p) ... | no n∉p = no (n∉p ∘ drop-there) private restrictP : ∀ {n} → (Fin (suc n) → Set) → (Fin n → Set) restrictP P f = P (suc f) restrict : ∀ {n} {P : Fin (suc n) → Set} → (∀ f → Dec (P f)) → (∀ f → Dec (restrictP P f)) restrict dec f = dec (suc f) any? : ∀ {n} {P : Fin n → Set} → ((f : Fin n) → Dec (P f)) → Dec (∃ P) any? {zero} {P} dec = no ((¬ Fin 0 ∶ λ()) ∘ proj₁) any? {suc n} {P} dec with dec zero | any? (restrict dec) ... | yes p | _ = yes (_ , p) ... | _ | yes (_ , p') = yes (_ , p') ... | no ¬p | no ¬p' = no helper where helper : ∄ P helper (zero , p) = ¬p p helper (suc f , p') = ¬p' (_ , p') nonempty? : ∀ {n} (p : Subset n) → Dec (Nonempty p) nonempty? p = any? (λ x → x ∈? p) private restrict∈ : ∀ {n} P {Q : Fin (suc n) → Set} → (∀ {f} → Q f → Dec (P f)) → (∀ {f} → restrictP Q f → Dec (restrictP P f)) restrict∈ _ dec {f} Qf = dec {suc f} Qf decFinSubset : ∀ {n} {P Q : Fin n → Set} → (∀ f → Dec (Q f)) → (∀ {f} → Q f → Dec (P f)) → Dec (∀ {f} → Q f → P f) decFinSubset {zero} _ _ = yes λ{} decFinSubset {suc n} {P} {Q} decQ decP = helper where helper : Dec (∀ {f} → Q f → P f) helper with decFinSubset (restrict decQ) (restrict∈ P decP) helper | no ¬q⟶p = no (λ q⟶p → ¬q⟶p (λ {f} q → q⟶p {suc f} q)) helper | yes q⟶p with decQ zero helper | yes q⟶p | yes q₀ with decP q₀ helper | yes q⟶p | yes q₀ | no ¬p₀ = no (λ q⟶p → ¬p₀ (q⟶p {zero} q₀)) helper | yes q⟶p | yes q₀ | yes p₀ = yes (λ {_} → hlpr _) where hlpr : ∀ f → Q f → P f hlpr zero _ = p₀ hlpr (suc f) qf = q⟶p qf helper | yes q⟶p | no ¬q₀ = yes (λ {_} → hlpr _) where hlpr : ∀ f → Q f → P f hlpr zero q₀ = ⊥-elim (¬q₀ q₀) hlpr (suc f) qf = q⟶p qf all∈? : ∀ {n} {P : Fin n → Set} {q} → (∀ {f} → f ∈ q → Dec (P f)) → Dec (∀ {f} → f ∈ q → P f) all∈? {q = q} dec = decFinSubset (λ f → f ∈? q) dec all? : ∀ {n} {P : Fin n → Set} → (∀ f → Dec (P f)) → Dec (∀ f → P f) all? dec with all∈? {q = all inside} (λ {f} _ → dec f) ... | yes ∀p = yes (λ f → ∀p (allInside f)) ... | no ¬∀p = no (λ ∀p → ¬∀p (λ {f} _ → ∀p f)) decLift : ∀ {n} {P : Fin n → Set} → (∀ x → Dec (P x)) → (∀ p → Dec (Lift P p)) decLift dec p = all∈? (λ {x} _ → dec x) private restrictSP : ∀ {n} → Side → (Subset (suc n) → Set) → (Subset n → Set) restrictSP s P p = P (s ∷ p) restrictS : ∀ {n} {P : Subset (suc n) → Set} → (s : Side) → (∀ p → Dec (P p)) → (∀ p → Dec (restrictSP s P p)) restrictS s dec p = dec (s ∷ p) anySubset? : ∀ {n} {P : Subset n → Set} → (∀ s → Dec (P s)) → Dec (∃ P) anySubset? {zero} {P} dec with dec [] ... | yes P[] = yes (_ , P[]) ... | no ¬P[] = no helper where helper : ∄ P helper ([] , P[]) = ¬P[] P[] anySubset? {suc n} {P} dec with anySubset? (restrictS inside dec) | anySubset? (restrictS outside dec) ... | yes (_ , Pp) | _ = yes (_ , Pp) ... | _ | yes (_ , Pp) = yes (_ , Pp) ... | no ¬Pp | no ¬Pp' = no helper where helper : ∄ P helper (inside ∷ p , Pp) = ¬Pp (_ , Pp) helper (outside ∷ p , Pp') = ¬Pp' (_ , Pp') -- If a decidable predicate P over a finite set is sometimes false, -- then we can find the smallest value for which this is the case. ¬∀⟶∃¬-smallest : ∀ n (P : Pred (Fin n)) → (∀ i → Dec (P i)) → ¬ (∀ i → P i) → ∃ λ i → ¬ P i × ((j : Fin′ i) → P (inject j)) ¬∀⟶∃¬-smallest zero P dec ¬∀iPi = ⊥-elim (¬∀iPi (λ())) ¬∀⟶∃¬-smallest (suc n) P dec ¬∀iPi with dec zero ¬∀⟶∃¬-smallest (suc n) P dec ¬∀iPi | no ¬P0 = (zero , ¬P0 , λ ()) ¬∀⟶∃¬-smallest (suc n) P dec ¬∀iPi | yes P0 = Prod.map suc (Prod.map id extend′) $ ¬∀⟶∃¬-smallest n (λ n → P (suc n)) (dec ∘ suc) (¬∀iPi ∘ extend) where extend : (∀ i → P (suc i)) → (∀ i → P i) extend ∀iP[1+i] zero = P0 extend ∀iP[1+i] (suc i) = ∀iP[1+i] i extend′ : ∀ {i : Fin n} → ((j : Fin′ i) → P (suc (inject j))) → ((j : Fin′ (suc i)) → P (inject j)) extend′ g zero = P0 extend′ g (suc j) = g j

33.078431

72

0.447935

21416802f97d42f092e6f877fc67ea78d96c19af

600

agda

Agda

src/Dodo/Unary/Unique.agda

sourcedennis/agda-dodo

376f0ccee1e1aa31470890e494bcb534324f598a

[ "BSD-3-Clause" ]

null

src/Dodo/Unary/Unique.agda

sourcedennis/agda-dodo

376f0ccee1e1aa31470890e494bcb534324f598a

[ "BSD-3-Clause" ]

null

src/Dodo/Unary/Unique.agda

sourcedennis/agda-dodo

376f0ccee1e1aa31470890e494bcb534324f598a

[ "BSD-3-Clause" ]

null

{-# OPTIONS --without-K --safe #-} module Dodo.Unary.Unique where -- Stdlib imports open import Level using (Level) open import Relation.Unary using (Pred) open import Relation.Binary using (Rel) -- Local imports open import Dodo.Nullary.Unique -- # Definitions # -- | At most one element satisfies the predicate Unique₁ : ∀ {a ℓ₁ ℓ₂ : Level} {A : Set a} → Rel A ℓ₁ → Pred A ℓ₂ → Set _ Unique₁ _≈_ P = ∀ {x y} → P x → P y → x ≈ y -- | For every `x`, there exists at most one inhabitant of `P x`. UniquePred : ∀ {a ℓ : Level} {A : Set a} → Pred A ℓ → Set _ UniquePred P = ∀ x → Unique (P x)

25

65

0.643333

1198c5b1ba1cee854f6c36f3c35416b9bf5d68dd

589

agda

Agda

Utils/Monoid.agda

AndrasKovacs/SemanticsWithApplications

05200d60b4a4b2c6fa37806ced9247055d24db94

[ "MIT" ]

8

2016-09-12T04:25:39.000Z

2020-02-02T10:01:52.000Z

Utils/Monoid.agda

AndrasKovacs/SemanticsWithApplications

05200d60b4a4b2c6fa37806ced9247055d24db94

[ "MIT" ]

null

Utils/Monoid.agda

AndrasKovacs/SemanticsWithApplications

05200d60b4a4b2c6fa37806ced9247055d24db94

[ "MIT" ]

null

-- Ad-hoc monoid typeclass module module Utils.Monoid where open import Data.List open import Data.String record Monoid {α}(A : Set α) : Set α where constructor rec field append : A → A → A identity : A infixr 5 _<>_ _<>_ : ∀ {α}{A : Set α} ⦃ _ : Monoid A ⦄ → A → A → A _<>_ ⦃ dict ⦄ a b = Monoid.append dict a b mempty : ∀ {α}{A : Set α} ⦃ _ : Monoid A ⦄ → A mempty ⦃ dict ⦄ = Monoid.identity dict instance MonoidList : ∀ {α A} → Monoid {α} (List A) MonoidList = rec Data.List._++_ [] MonoidString : Monoid String MonoidString = rec Data.String._++_ ""

20.310345

52

0.609508

36fc57850ccf0cc58409240cbb15e384ab07ad73

1,686

agda

Agda

src/Bisimilarity/Equational-reasoning-instances.agda

nad/up-to

b936ff85411baf3401ad85ce85d5ff2e9aa0ca14

[ "MIT" ]

null

src/Bisimilarity/Equational-reasoning-instances.agda

nad/up-to

b936ff85411baf3401ad85ce85d5ff2e9aa0ca14

[ "MIT" ]

null

src/Bisimilarity/Equational-reasoning-instances.agda

nad/up-to

b936ff85411baf3401ad85ce85d5ff2e9aa0ca14

[ "MIT" ]

null

------------------------------------------------------------------------ -- "Equational" reasoning combinator setup ------------------------------------------------------------------------ {-# OPTIONS --sized-types #-} open import Prelude.Size open import Labelled-transition-system module Bisimilarity.Equational-reasoning-instances {ℓ} {lts : LTS ℓ} {i : Size} where open import Prelude open import Bisimilarity lts open import Equational-reasoning instance reflexive∼ : Reflexive [ i ]_∼_ reflexive∼ = is-reflexive reflexive-∼ reflexive∼′ : Reflexive [ i ]_∼′_ reflexive∼′ = is-reflexive reflexive-∼′ symmetric∼ : Symmetric [ i ]_∼_ symmetric∼ = is-symmetric symmetric-∼ symmetric∼′ : Symmetric [ i ]_∼′_ symmetric∼′ = is-symmetric symmetric-∼′ trans∼∼ : Transitive [ i ]_∼_ [ i ]_∼_ trans∼∼ = is-transitive transitive-∼ trans∼′∼ : Transitive _∼′_ [ i ]_∼_ trans∼′∼ = is-transitive λ p∼′q → transitive (force p∼′q) trans∼′∼′ : Transitive [ i ]_∼′_ [ i ]_∼′_ trans∼′∼′ = is-transitive transitive-∼′ trans∼∼′ : Transitive [ i ]_∼_ [ i ]_∼′_ trans∼∼′ = is-transitive lemma where lemma : ∀ {p q r} → [ i ] p ∼ q → [ i ] q ∼′ r → [ i ] p ∼′ r force (lemma p∼q q∼′r) = transitive-∼ p∼q (force q∼′r) convert∼∼ : Convertible [ i ]_∼_ [ i ]_∼_ convert∼∼ = is-convertible id convert∼′∼ : Convertible _∼′_ [ i ]_∼_ convert∼′∼ = is-convertible λ p∼′q → force p∼′q convert∼∼′ : Convertible [ i ]_∼_ [ i ]_∼′_ convert∼∼′ = is-convertible lemma where lemma : ∀ {p q} → [ i ] p ∼ q → [ i ] p ∼′ q force (lemma p∼q) = p∼q convert∼′∼′ : Convertible [ i ]_∼′_ [ i ]_∼′_ convert∼′∼′ = is-convertible id

27.193548

72

0.549229

19b37df11c2a09096145088644e54b7e353e07b0

200

agda

Agda

Cubical/HITs/TypeQuotients.agda

Edlyr/cubical

5de11df25b79ee49d5c084fbbe6dfc66e4147a2e

[ "MIT" ]

null

Cubical/HITs/TypeQuotients.agda

Edlyr/cubical

5de11df25b79ee49d5c084fbbe6dfc66e4147a2e

[ "MIT" ]

null

Cubical/HITs/TypeQuotients.agda

Edlyr/cubical

5de11df25b79ee49d5c084fbbe6dfc66e4147a2e

[ "MIT" ]

null

{-# 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

33.333333

56

0.8

d1860b77087e5a7c0887b820d2c05d925633a6ca

2,013

agda

Agda

JamesContractibility.agda

guillaumebrunerie/JamesConstruction

89fbc29473d2d1ed1a45c3c0e56288cdcf77050b

[ "MIT" ]

5

2016-12-07T04:34:52.000Z

2018-11-16T22:10:16.000Z

JamesContractibility.agda

guillaumebrunerie/JamesConstruction

89fbc29473d2d1ed1a45c3c0e56288cdcf77050b

[ "MIT" ]

null

JamesContractibility.agda

guillaumebrunerie/JamesConstruction

89fbc29473d2d1ed1a45c3c0e56288cdcf77050b

[ "MIT" ]

null

{-# OPTIONS --without-K --rewriting #-} open import PathInduction open import Pushout module JamesContractibility {i} (A : Type i) (⋆A : A) where open import JamesTwoMaps A ⋆A public -- We do not prove the flattening lemma here, we only prove that the following pushout is contractible T : Type i T = Pushout (span JA JA (A × JA) snd (uncurry αJ)) ⋆T : T ⋆T = inl εJ T-contr-inl-ε : inl εJ == ⋆T T-contr-inl-ε = idp T-contr-inl-α : (a : A) (x : JA) → inl x == ⋆T → inl (αJ a x) == ⋆T T-contr-inl-α a x y = push (⋆A , αJ a x) ∙ ! (ap inr (δJ (αJ a x))) ∙ ! (push (a , x)) ∙ y T-contr-inl-δ : (x : JA) (y : inl x == ⋆T) → Square (ap inl (δJ x)) y (T-contr-inl-α ⋆A x y) idp T-contr-inl-δ x y = & coh (ap-square inr (& cη (ηJ x))) (natural-square (λ z → push (⋆A , z)) (δJ x) idp (ap-∘ inr (αJ ⋆A) (δJ x))) where coh : Coh ({A : Type i} {a b : A} {p : a == b} {c : A} {q q' : c == b} (q= : Square q q' idp idp) {d : A} {r : d == c} {e : A} {s : d == e} {t : d == a} (sq : Square r t q p) → Square t s (p ∙ ! q' ∙ ! r ∙ s) idp) coh = path-induction cη : Coh ({A : Type i} {a b : A} {p : a == b} {c : A} {q : a == c} {r : b == c} (ηJ : ! p ∙ q == r) → Square p q r idp) cη = path-induction T-contr-inl : (x : JA) → inl x == ⋆T T-contr-inl = JA-elim T-contr-inl-ε T-contr-inl-α (λ x y → ↓-='-from-square idp (ap-cst ⋆T (δJ x)) (square-symmetry (T-contr-inl-δ x y))) T-contr-inr : (x : JA) → inr x == ⋆T T-contr-inr x = ap inr (δJ x) ∙ ! (push (⋆A , x)) ∙ T-contr-inl x T-contr-push : (a : A) (x : JA) → Square (T-contr-inl x) (push (a , x)) idp (T-contr-inr (αJ a x)) T-contr-push a x = & coh where coh : Coh ({A : Type i} {a b : A} {r : a == b} {c : A} {p : c == b} {d : A} {q : d == c} {e : A} {y : d == e} → Square y q idp (p ∙ ! r ∙ (r ∙ ! p ∙ ! q ∙ y))) coh = path-induction T-contr : (x : T) → x == ⋆T T-contr = Pushout-elim T-contr-inl T-contr-inr (λ {(a , x) → ↓-='-from-square (ap-idf (push (a , x))) (ap-cst ⋆T (push (a , x))) (T-contr-push a x)})

40.26

149

0.506706

adeac7d48649b135372e2ff56bc0dc682884b4ac

1,298

agda

Agda

src/Categories/Category/Monoidal/Instance/Cats.agda

yourboynico/agda-categories

6a087c592dbe58fc4bd9d02e1be9b94a9e138aca

[ "MIT" ]

279

2019-06-01T14:36:40.000Z

2022-03-22T00:40:14.000Z

src/Categories/Category/Monoidal/Instance/Cats.agda

seanpm2001/agda-categories

d9e4f578b126313058d105c61707d8c8ae987fa8

[ "MIT" ]

236

2019-06-01T14:53:54.000Z

2022-03-28T14:31:43.000Z

src/Categories/Category/Monoidal/Instance/Cats.agda

seanpm2001/agda-categories

d9e4f578b126313058d105c61707d8c8ae987fa8

[ "MIT" ]

64

2019-06-02T16:58:15.000Z

2022-03-14T02:00:59.000Z

{-# OPTIONS --without-K --safe #-} -- The category of Cats is Monoidal module Categories.Category.Monoidal.Instance.Cats where open import Level open import Categories.Category.BinaryProducts using (BinaryProducts) open import Categories.Category.Cartesian using (Cartesian) open import Categories.Category.Cartesian.Monoidal using (module CartesianMonoidal) open import Categories.Category.Instance.Cats using (Cats) open import Categories.Category.Instance.One using (One-⊤) open import Categories.Category.Monoidal using (Monoidal) open import Categories.Category.Product using (Product; πˡ; πʳ; _※_) open import Categories.Category.Product.Properties using (project₁; project₂; unique) -- Cats is a Monoidal Category with Product as Bifunctor module Product {o ℓ e : Level} where private C = Cats o ℓ e Cats-has-all : BinaryProducts C Cats-has-all = record { product = λ {A} {B} → record { A×B = Product A B ; π₁ = πˡ ; π₂ = πʳ ; ⟨_,_⟩ = _※_ ; project₁ = λ {_} {h} {i} → project₁ {i = h} {j = i} ; project₂ = λ {_} {h} {i} → project₂ {i = h} {j = i} ; unique = unique } } Cats-is : Cartesian C Cats-is = record { terminal = One-⊤ ; products = Cats-has-all } Cats-Monoidal : Monoidal C Cats-Monoidal = CartesianMonoidal.monoidal Cats-is

33.282051

85

0.702619

ede41404e31a6adad20409ece6505b20e297dd07

2,994

agda

Agda

src/Relation/Binary/Construct/Closure/SymmetricTransitive.agda

MirceaS/agda-categories

58e5ec015781be5413bdf968f7ec4fdae0ab4b21

[ "MIT" ]

null

src/Relation/Binary/Construct/Closure/SymmetricTransitive.agda

MirceaS/agda-categories

58e5ec015781be5413bdf968f7ec4fdae0ab4b21

[ "MIT" ]

null

src/Relation/Binary/Construct/Closure/SymmetricTransitive.agda

MirceaS/agda-categories

58e5ec015781be5413bdf968f7ec4fdae0ab4b21

[ "MIT" ]

null

{-# OPTIONS --without-K --safe #-} module Relation.Binary.Construct.Closure.SymmetricTransitive where open import Level open import Relation.Binary private variable a ℓ ℓ′ : Level A B : Set a module _ {A : Set a} (_≤_ : Rel A ℓ) where private variable x y z : A data Plus⇔ : Rel A (a ⊔ ℓ) where forth : x ≤ y → Plus⇔ x y back : y ≤ x → Plus⇔ x y forth⁺ : x ≤ y → Plus⇔ y z → Plus⇔ x z back⁺ : y ≤ x → Plus⇔ y z → Plus⇔ x z module _ (_∼_ : Rel A ℓ) where trans : Transitive (Plus⇔ _∼_) trans (forth r) rel′ = forth⁺ r rel′ trans (back r) rel′ = back⁺ r rel′ trans (forth⁺ r rel) rel′ = forth⁺ r (trans rel rel′) trans (back⁺ r rel) rel′ = back⁺ r (trans rel rel′) sym : Symmetric (Plus⇔ _∼_) sym (forth r) = back r sym (back r) = forth r sym (forth⁺ r rel) = trans (sym rel) (back r) sym (back⁺ r rel) = trans (sym rel) (forth r) isPartialEquivalence : IsPartialEquivalence (Plus⇔ _∼_) isPartialEquivalence = record { sym = sym ; trans = trans } partialSetoid : PartialSetoid _ _ partialSetoid = record { Carrier = A ; _≈_ = Plus⇔ _∼_ ; isPartialEquivalence = isPartialEquivalence } module _ (refl : Reflexive _∼_) where isEquivalence : IsEquivalence (Plus⇔ _∼_) isEquivalence = record { refl = forth refl ; sym = sym ; trans = trans } setoid : Setoid _ _ setoid = record { Carrier = A ; _≈_ = Plus⇔ _∼_ ; isEquivalence = isEquivalence } module _ {c e} (S : Setoid c e) where private module S = Setoid S minimal : (f : A → Setoid.Carrier S) → _∼_ =[ f ]⇒ Setoid._≈_ S → Plus⇔ _∼_ =[ f ]⇒ Setoid._≈_ S minimal f inj (forth r) = inj r minimal f inj (back r) = S.sym (inj r) minimal f inj (forth⁺ r rel) = S.trans (inj r) (minimal f inj rel) minimal f inj (back⁺ r rel) = S.trans (S.sym (inj r)) (minimal f inj rel) module Plus⇔Reasoning (_≤_ : Rel A ℓ) where infix 3 forth-synax back-syntax infixr 2 forth⁺-syntax back⁺-syntax forth-synax : ∀ x y → x ≤ y → Plus⇔ _≤_ x y forth-synax _ _ = forth syntax forth-synax x y x≤y = x ⇒⟨ x≤y ⟩∎ y ∎ back-syntax : ∀ x y → y ≤ x → Plus⇔ _≤_ x y back-syntax _ _ = back syntax back-syntax x y y≤x = x ⇐⟨ y≤x ⟩∎ y ∎ forth⁺-syntax : ∀ x {y z} → x ≤ y → Plus⇔ _≤_ y z → Plus⇔ _≤_ x z forth⁺-syntax _ = forth⁺ syntax forth⁺-syntax x x≤y y⇔z = x ⇒⟨ x≤y ⟩ y⇔z back⁺-syntax : ∀ x {y z} → y ≤ x → Plus⇔ _≤_ y z → Plus⇔ _≤_ x z back⁺-syntax _ = back⁺ syntax back⁺-syntax x y≤x y⇔z = x ⇐⟨ y≤x ⟩ y⇔z module _ {_≤_ : Rel A ℓ} {_≼_ : Rel B ℓ′} (f : A → B) (inj : _≤_ =[ f ]⇒ _≼_) where map : ∀ {x y} → Plus⇔ _≤_ x y → Plus⇔ _≼_ (f x) (f y) map (forth r) = forth (inj r) map (back r) = back (inj r) map (forth⁺ r rel) = forth⁺ (inj r) (map rel) map (back⁺ r rel) = back⁺ (inj r) (map rel)

27.218182

83

0.542418

73462b03187cbf1b42f6df6187412e515c05c67a

579

agda

Agda

agda-stdlib/src/Data/Vec/Relation/Pointwise/Inductive.agda

DreamLinuxer/popl21-artifact

fb380f2e67dcb4a94f353dbaec91624fcb5b8933

[ "MIT" ]

5

2020-10-07T12:07:53.000Z

2020-10-10T21:41:32.000Z

agda-stdlib/src/Data/Vec/Relation/Pointwise/Inductive.agda

DreamLinuxer/popl21-artifact

fb380f2e67dcb4a94f353dbaec91624fcb5b8933

[ "MIT" ]

null

agda-stdlib/src/Data/Vec/Relation/Pointwise/Inductive.agda

DreamLinuxer/popl21-artifact

fb380f2e67dcb4a94f353dbaec91624fcb5b8933

[ "MIT" ]

1

2021-11-04T06:54:45.000Z

------------------------------------------------------------------------ -- The Agda standard library -- -- This module is DEPRECATED. Please use -- Data.Vec.Relation.Binary.Pointwise.Inductive directly. ------------------------------------------------------------------------ {-# OPTIONS --without-K --safe #-} module Data.Vec.Relation.Pointwise.Inductive where open import Data.Vec.Relation.Binary.Pointwise.Inductive public {-# WARNING_ON_IMPORT "Data.Vec.Relation.Pointwise.Inductive was deprecated in v1.0. Use Data.Vec.Relation.Binary.Pointwise.Inductive instead." #-}

32.166667

72

0.578584

64e2bfd3d64965fe328aba7f8adb681ea4d68bb4

3,644

agda

Agda

examples/ISWIM.agda

mdimjasevic/agda

8fb548356b275c7a1e79b768b64511ae937c738b

[ "BSD-3-Clause" ]

1,989

2015-01-09T23:51:16.000Z

2022-03-30T18:20:48.000Z

examples/ISWIM.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

4,066

2015-01-10T11:24:51.000Z

2022-03-31T21:14:49.000Z

examples/ISWIM.agda

Seanpm2001-languages/agda

9911f73061e21a87fad76c662463257afe02c861

[ "BSD-2-Clause" ]

371

2015-01-03T14:04:08.000Z

2022-03-30T19:00:30.000Z

-- A Typed version of a subset of Landin's ISWIM from "The Next 700 Programming -- Languages" module ISWIM where data Nat : Set where zero : Nat suc : Nat -> Nat _+_ : Nat -> Nat -> Nat zero + m = m suc n + m = suc (n + m) {-# BUILTIN NATURAL Nat #-} {-# BUILTIN NATPLUS _+_ #-} data Bool : Set where true : Bool false : Bool module Syntax where infixl 100 _∙_ infixl 80 _WHERE_ _PP_ infixr 60 _─→_ infixl 40 _,_ data Type : Set where nat : Type bool : Type _─→_ : Type -> Type -> Type data Context : Set where ε : Context _,_ : Context -> Type -> Context data Var : Context -> Type -> Set where vz : {Γ : Context}{τ : Type} -> Var (Γ , τ) τ vs : {Γ : Context}{σ τ : Type} -> Var Γ τ -> Var (Γ , σ) τ data Expr (Γ : Context) : Type -> Set where var : {τ : Type} -> Var Γ τ -> Expr Γ τ litNat : Nat -> Expr Γ nat litBool : Bool -> Expr Γ bool plus : Expr Γ (nat ─→ nat ─→ nat) if : {τ : Type} -> Expr Γ (bool ─→ τ ─→ τ ─→ τ) _∙_ : {σ τ : Type} -> Expr Γ (σ ─→ τ) -> Expr Γ σ -> Expr Γ τ _WHERE_ : {σ τ ρ : Type} -> Expr (Γ , σ ─→ τ) ρ -> Expr (Γ , σ) τ -> Expr Γ ρ _PP_ : {σ τ ρ : Type} -> Expr (Γ , σ ─→ τ) ρ -> Expr (Γ , σ) ρ -> Expr Γ ρ -- ƛ x. e = f where f x = e ƛ : {Γ : Context}{σ τ : Type} -> Expr (Γ , σ) τ -> Expr Γ (σ ─→ τ) ƛ e = var vz WHERE e module Cont (R : Set) where C : Set -> Set C a = (a -> R) -> R callcc : {a : Set} -> (({b : Set} -> a -> C b) -> C a) -> C a callcc {a} g = \k -> g (\x _ -> k x) k return : {a : Set} -> a -> C a return x = \k -> k x infixr 10 _>>=_ _>>=_ : {a b : Set} -> C a -> (a -> C b) -> C b (m >>= k) ret = m \x -> k x ret module Semantics (R : Set) where open module C = Cont R open Syntax infix 60 _!_ infixl 40 _||_ ⟦_⟧type : Type -> Set ⟦_⟧type' : Type -> Set ⟦ nat ⟧type' = Nat ⟦ bool ⟧type' = Bool ⟦ σ ─→ τ ⟧type' = ⟦ σ ⟧type' -> ⟦ τ ⟧type ⟦ τ ⟧type = C ⟦ τ ⟧type' data ⟦_⟧ctx : Context -> Set where ★ : ⟦ ε ⟧ctx _||_ : {Γ : Context}{τ : Type} -> ⟦ Γ ⟧ctx -> ⟦ τ ⟧type' -> ⟦ Γ , τ ⟧ctx _!_ : {Γ : Context}{τ : Type} -> ⟦ Γ ⟧ctx -> Var Γ τ -> ⟦ τ ⟧type' ★ ! () (ρ || v) ! vz = v (ρ || v) ! vs x = ρ ! x ⟦_⟧ : {Γ : Context}{τ : Type} -> Expr Γ τ -> ⟦ Γ ⟧ctx -> ⟦ τ ⟧type ⟦ var x ⟧ ρ = return (ρ ! x) ⟦ litNat n ⟧ ρ = return n ⟦ litBool b ⟧ ρ = return b ⟦ plus ⟧ ρ = return \n -> return \m -> return (n + m) ⟦ f ∙ e ⟧ ρ = ⟦ e ⟧ ρ >>= \v -> ⟦ f ⟧ ρ >>= \w -> w v ⟦ e WHERE f ⟧ ρ = ⟦ e ⟧ (ρ || (\x -> ⟦ f ⟧ (ρ || x))) ⟦ e PP f ⟧ ρ = callcc \k -> let throw = \x -> ⟦ f ⟧ (ρ || x) >>= k in ⟦ e ⟧ (ρ || throw) ⟦ if ⟧ ρ = return \x -> return \y -> return \z -> return (iff x y z) where iff : {A : Set} -> Bool -> A -> A -> A iff true x y = x iff false x y = y module Test where open Syntax open module C = Cont Nat open module S = Semantics Nat run : Expr ε nat -> Nat run e = ⟦ e ⟧ ★ \x -> x -- 1 + 1 two : Expr ε nat two = plus ∙ litNat 1 ∙ litNat 1 -- f 1 + f 2 where f x = x three : Expr ε nat three = plus ∙ (var vz ∙ litNat 1) ∙ (var vz ∙ litNat 2) WHERE var vz -- 1 + f 1 where pp f x = x one : Expr ε nat one = plus ∙ litNat 1 ∙ (var vz ∙ litNat 1) PP var vz open Test data _==_ {a : Set}(x : a) : a -> Set where refl : x == x twoOK : run two == 2 twoOK = refl threeOK : run three == 3 threeOK = refl oneOK : run one == 1 oneOK = refl open Cont open Syntax open Semantics

23.509677

81

0.464599

657edd55bfd3cea96c0074a89859b160c8aaf47c

3,768

agda

Agda

src/Container/List.agda

L-TChen/agda-prelude

158d299b1b365e186f00d8ef5b8c6844235ee267

[ "MIT" ]

111

2015-01-05T11:28:15.000Z

2022-02-12T23:29:26.000Z

src/Container/List.agda

L-TChen/agda-prelude

158d299b1b365e186f00d8ef5b8c6844235ee267

[ "MIT" ]

59

2016-02-09T05:36:44.000Z

2022-01-14T07:32:36.000Z

src/Container/List.agda

L-TChen/agda-prelude

158d299b1b365e186f00d8ef5b8c6844235ee267

[ "MIT" ]

24

2015-03-12T18:03:45.000Z

2021-04-22T06:10:41.000Z

module Container.List where open import Prelude infixr 5 _∷_ data All {a b} {A : Set a} (P : A → Set b) : List A → Set (a ⊔ b) where [] : All P [] _∷_ : ∀ {x xs} (p : P x) (ps : All P xs) → All P (x ∷ xs) data Any {a b} {A : Set a} (P : A → Set b) : List A → Set (a ⊔ b) where zero : ∀ {x xs} (p : P x) → Any P (x ∷ xs) suc : ∀ {x xs} (i : Any P xs) → Any P (x ∷ xs) pattern zero! = zero refl -- Literal overloading for Any module _ {a b} {A : Set a} {P : A → Set b} where private AnyConstraint : List A → Nat → Set (a ⊔ b) AnyConstraint [] _ = ⊥′ AnyConstraint (x ∷ _) zero = ⊤′ {a} × P x -- hack to line up levels AnyConstraint (_ ∷ xs) (suc i) = AnyConstraint xs i natToIx : ∀ (xs : List A) n → {{_ : AnyConstraint xs n}} → Any P xs natToIx [] n {{}} natToIx (x ∷ xs) zero {{_ , px}} = zero px natToIx (x ∷ xs) (suc n) = suc (natToIx xs n) instance NumberAny : ∀ {xs} → Number (Any P xs) Number.Constraint (NumberAny {xs}) = AnyConstraint xs fromNat {{NumberAny {xs}}} = natToIx xs infix 3 _∈_ _∈_ : ∀ {a} {A : Set a} → A → List A → Set a x ∈ xs = Any (_≡_ x) xs forgetAny : ∀ {a p} {A : Set a} {P : A → Set p} {xs : List A} → Any P xs → Nat forgetAny (zero _) = zero forgetAny (suc i) = suc (forgetAny i) lookupAny : ∀ {a b} {A : Set a} {P Q : A → Set b} {xs} → All P xs → Any Q xs → Σ A (λ x → P x × Q x) lookupAny (p ∷ ps) (zero q) = _ , p , q lookupAny (p ∷ ps) (suc i) = lookupAny ps i lookup∈ : ∀ {a b} {A : Set a} {P : A → Set b} {xs x} → All P xs → x ∈ xs → P x lookup∈ (p ∷ ps) (zero refl) = p lookup∈ (p ∷ ps) (suc i) = lookup∈ ps i module _ {a b} {A : Set a} {P Q : A → Set b} (f : ∀ {x} → P x → Q x) where mapAll : ∀ {xs} → All P xs → All Q xs mapAll [] = [] mapAll (x ∷ xs) = f x ∷ mapAll xs mapAny : ∀ {xs} → Any P xs → Any Q xs mapAny (zero x) = zero (f x) mapAny (suc i) = suc (mapAny i) traverseAll : ∀ {a b} {A : Set a} {B : A → Set a} {F : Set a → Set b} {{AppF : Applicative F}} → (∀ x → F (B x)) → (xs : List A) → F (All B xs) traverseAll f [] = pure [] traverseAll f (x ∷ xs) = ⦇ f x ∷ traverseAll f xs ⦈ module _ {a b} {A : Set a} {P : A → Set b} where -- Append infixr 5 _++All_ _++All_ : ∀ {xs ys} → All P xs → All P ys → All P (xs ++ ys) [] ++All qs = qs (p ∷ ps) ++All qs = p ∷ ps ++All qs -- Delete deleteIx : ∀ xs → Any P xs → List A deleteIx (_ ∷ xs) (zero _) = xs deleteIx (x ∷ xs) (suc i) = x ∷ deleteIx xs i deleteAllIx : ∀ {c} {Q : A → Set c} {xs} → All Q xs → (i : Any P xs) → All Q (deleteIx xs i) deleteAllIx (q ∷ qs) (zero _) = qs deleteAllIx (q ∷ qs) (suc i) = q ∷ deleteAllIx qs i -- Equality -- module _ {a b} {A : Set a} {P : A → Set b} {{EqP : ∀ {x} → Eq (P x)}} where private z : ∀ {x xs} → P x → Any P (x ∷ xs) z = zero zero-inj : ∀ {x} {xs : List A} {p q : P x} → Any.zero {xs = xs} p ≡ z q → p ≡ q zero-inj refl = refl sucAny-inj : ∀ {x xs} {i j : Any P xs} → Any.suc {x = x} i ≡ Any.suc {x = x} j → i ≡ j sucAny-inj refl = refl cons-inj₁ : ∀ {x xs} {p q : P x} {ps qs : All P xs} → p All.∷ ps ≡ q ∷ qs → p ≡ q cons-inj₁ refl = refl cons-inj₂ : ∀ {x xs} {p q : P x} {ps qs : All P xs} → p All.∷ ps ≡ q ∷ qs → ps ≡ qs cons-inj₂ refl = refl instance EqAny : ∀ {xs} → Eq (Any P xs) _==_ {{EqAny}} (zero p) (zero q) = decEq₁ zero-inj (p == q) _==_ {{EqAny}} (suc i) (suc j) = decEq₁ sucAny-inj (i == j) _==_ {{EqAny}} (zero _) (suc _) = no λ () _==_ {{EqAny}} (suc _) (zero _) = no λ () EqAll : ∀ {xs} → Eq (All P xs) _==_ {{EqAll}} [] [] = yes refl _==_ {{EqAll}} (x ∷ xs) (y ∷ ys) = decEq₂ cons-inj₁ cons-inj₂ (x == y) (xs == ys)

32.765217

100

0.49018

3f162ffdc2a81b89598f01c88a059eab5a64870f

994

agda

Agda

test/asset/agda-stdlib-1.0/Data/Table/Relation/Binary/Equality.agda

omega12345/agda-mode

0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71

[ "MIT" ]

null

test/asset/agda-stdlib-1.0/Data/Table/Relation/Binary/Equality.agda

omega12345/agda-mode

0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71

[ "MIT" ]

null

test/asset/agda-stdlib-1.0/Data/Table/Relation/Binary/Equality.agda

omega12345/agda-mode

0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71

[ "MIT" ]

null

------------------------------------------------------------------------ -- The Agda standard library -- -- Pointwise table equality ------------------------------------------------------------------------ {-# OPTIONS --without-K --safe #-} module Data.Table.Relation.Binary.Equality where open import Relation.Binary using (Setoid) open import Data.Table.Base open import Data.Nat using (ℕ) open import Function using (_∘_) open import Relation.Binary.PropositionalEquality as P using (_≡_; _→-setoid_) setoid : ∀ {c p} → Setoid c p → ℕ → Setoid _ _ setoid S n = record { Carrier = Table Carrier n ; _≈_ = λ t t′ → ∀ i → lookup t i ≈ lookup t′ i ; isEquivalence = record { refl = λ i → refl ; sym = λ p → sym ∘ p ; trans = λ p q i → trans (p i) (q i) } } where open Setoid S ≡-setoid : ∀ {a} → Set a → ℕ → Setoid _ _ ≡-setoid A = setoid (P.setoid A) module _ {a} {A : Set a} {n} where open Setoid (≡-setoid A n) public using () renaming (_≈_ to _≗_)

27.611111

72

0.533199

5ed40cc0c6fe4192ec6e78cf7eb48672f235aa57

2,520

agda

Agda

Cubical/Foundations/Pointed/Base.agda

hyleIndex/cubical

ce5c2820ecb2e0fd8dce74fb0247856cdbf034c4

[ "MIT" ]

301

2018-10-17T18:00:24.000Z

2022-03-24T02:10:47.000Z

Cubical/Foundations/Pointed/Base.agda

hyleIndex/cubical

ce5c2820ecb2e0fd8dce74fb0247856cdbf034c4

[ "MIT" ]

584

2018-10-15T09:49:02.000Z

2022-03-30T12:09:17.000Z

Cubical/Foundations/Pointed/Base.agda

hyleIndex/cubical

ce5c2820ecb2e0fd8dce74fb0247856cdbf034c4

[ "MIT" ]

134

2018-11-16T06:11:03.000Z

2022-03-23T16:22:13.000Z

{-# OPTIONS --safe #-} module Cubical.Foundations.Pointed.Base where open import Cubical.Foundations.Prelude open import Cubical.Foundations.Equiv open import Cubical.Foundations.Structure open import Cubical.Foundations.Structure using (typ) public open import Cubical.Foundations.GroupoidLaws open import Cubical.Foundations.Isomorphism open import Cubical.Foundations.Univalence Pointed : (ℓ : Level) → Type (ℓ-suc ℓ) Pointed ℓ = TypeWithStr ℓ (λ x → x) pt : ∀ {ℓ} (A∙ : Pointed ℓ) → typ A∙ pt = str Pointed₀ = Pointed ℓ-zero {- Pointed functions -} _→∙_ : ∀{ℓ ℓ'} → (A : Pointed ℓ) (B : Pointed ℓ') → Type (ℓ-max ℓ ℓ') (A , a) →∙ (B , b) = Σ[ f ∈ (A → B) ] f a ≡ b _→∙_∙ : ∀{ℓ ℓ'} → (A : Pointed ℓ) (B : Pointed ℓ') → Pointed (ℓ-max ℓ ℓ') (A →∙ B ∙) .fst = A →∙ B (A →∙ B ∙) .snd .fst x = pt B (A →∙ B ∙) .snd .snd = refl idfun∙ : ∀ {ℓ} (A : Pointed ℓ) → A →∙ A idfun∙ A .fst x = x idfun∙ A .snd = refl ua∙ : ∀ {ℓ} {A B : Pointed ℓ} (e : fst A ≃ fst B) → fst e (snd A) ≡ snd B → A ≡ B fst (ua∙ e p i) = ua e i snd (ua∙ {A = A} e p i) = ua-gluePath e p i {- HIT allowing for pattern matching on pointed types -} data Pointer {ℓ} (A : Pointed ℓ) : Type ℓ where pt₀ : Pointer A ⌊_⌋ : typ A → Pointer A id : ⌊ pt A ⌋ ≡ pt₀ IsoPointedPointer : ∀ {ℓ} {A : Pointed ℓ} → Iso (typ A) (Pointer A) Iso.fun IsoPointedPointer = ⌊_⌋ Iso.inv (IsoPointedPointer {A = A}) pt₀ = pt A Iso.inv IsoPointedPointer ⌊ x ⌋ = x Iso.inv (IsoPointedPointer {A = A}) (id i) = pt A Iso.rightInv IsoPointedPointer pt₀ = id Iso.rightInv IsoPointedPointer ⌊ x ⌋ = refl Iso.rightInv IsoPointedPointer (id i) j = id (i ∧ j) Iso.leftInv IsoPointedPointer x = refl Pointed≡Pointer : ∀ {ℓ} {A : Pointed ℓ} → typ A ≡ Pointer A Pointed≡Pointer = isoToPath IsoPointedPointer Pointer∙ : ∀ {ℓ} (A : Pointed ℓ) → Pointed ℓ Pointer∙ A .fst = Pointer A Pointer∙ A .snd = pt₀ Pointed≡∙Pointer : ∀ {ℓ} {A : Pointed ℓ} → A ≡ (Pointer A , pt₀) Pointed≡∙Pointer {A = A} i = (Pointed≡Pointer {A = A} i) , helper i where helper : PathP (λ i → Pointed≡Pointer {A = A} i) (pt A) pt₀ helper = ua-gluePath (isoToEquiv (IsoPointedPointer {A = A})) id pointerFun : ∀ {ℓ ℓ'} {A : Pointed ℓ} {B : Pointed ℓ'} (f : A →∙ B) → Pointer A → Pointer B pointerFun f pt₀ = pt₀ pointerFun f ⌊ x ⌋ = ⌊ fst f x ⌋ pointerFun f (id i) = (cong ⌊_⌋ (snd f) ∙ id) i pointerFun∙ : ∀ {ℓ ℓ'} {A : Pointed ℓ} {B : Pointed ℓ'} (f : A →∙ B) → Pointer∙ A →∙ Pointer∙ B pointerFun∙ f .fst = pointerFun f pointerFun∙ f .snd = refl

32.307692

73

0.611111

fb1f0d29f01e70f4b5bf592ed117835230dd51ee

2,239

agda

Agda

agda-stdlib/src/Data/List/Relation/Binary/Permutation/Homogeneous.agda

DreamLinuxer/popl21-artifact

fb380f2e67dcb4a94f353dbaec91624fcb5b8933

[ "MIT" ]

5

2020-10-07T12:07:53.000Z

2020-10-10T21:41:32.000Z

agda-stdlib/src/Data/List/Relation/Binary/Permutation/Homogeneous.agda

DreamLinuxer/popl21-artifact

fb380f2e67dcb4a94f353dbaec91624fcb5b8933

[ "MIT" ]

null

agda-stdlib/src/Data/List/Relation/Binary/Permutation/Homogeneous.agda

DreamLinuxer/popl21-artifact

fb380f2e67dcb4a94f353dbaec91624fcb5b8933

[ "MIT" ]

1

2021-11-04T06:54:45.000Z

------------------------------------------------------------------------ -- The Agda standard library -- -- A definition for the permutation relation using setoid equality ------------------------------------------------------------------------ {-# OPTIONS --without-K --safe #-} module Data.List.Relation.Binary.Permutation.Homogeneous where open import Data.List.Base using (List; _∷_) open import Data.List.Relation.Binary.Pointwise as Pointwise using (Pointwise) open import Level using (Level; _⊔_) open import Relation.Binary private variable a r s : Level A : Set a data Permutation {A : Set a} (R : Rel A r) : Rel (List A) (a ⊔ r) where refl : ∀ {xs ys} → Pointwise R xs ys → Permutation R xs ys prep : ∀ {xs ys x y} (eq : R x y) → Permutation R xs ys → Permutation R (x ∷ xs) (y ∷ ys) swap : ∀ {xs ys x y x' y'} (eq₁ : R x x') (eq₂ : R y y') → Permutation R xs ys → Permutation R (x ∷ y ∷ xs) (y' ∷ x' ∷ ys) trans : ∀ {xs ys zs} → Permutation R xs ys → Permutation R ys zs → Permutation R xs zs ------------------------------------------------------------------------ -- The Permutation relation is an equivalence module _ {R : Rel A r} where sym : Symmetric R → Symmetric (Permutation R) sym R-sym (refl xs∼ys) = refl (Pointwise.symmetric R-sym xs∼ys) sym R-sym (prep x∼x' xs↭ys) = prep (R-sym x∼x') (sym R-sym xs↭ys) sym R-sym (swap x∼x' y∼y' xs↭ys) = swap (R-sym y∼y') (R-sym x∼x') (sym R-sym xs↭ys) sym R-sym (trans xs↭ys ys↭zs) = trans (sym R-sym ys↭zs) (sym R-sym xs↭ys) isEquivalence : Reflexive R → Symmetric R → IsEquivalence (Permutation R) isEquivalence R-refl R-sym = record { refl = refl (Pointwise.refl R-refl) ; sym = sym R-sym ; trans = trans } setoid : Reflexive R → Symmetric R → Setoid _ _ setoid R-refl R-sym = record { isEquivalence = isEquivalence R-refl R-sym } map : ∀ {R : Rel A r} {S : Rel A s} → (R ⇒ S) → (Permutation R ⇒ Permutation S) map R⇒S (refl xs∼ys) = refl (Pointwise.map R⇒S xs∼ys) map R⇒S (prep e xs∼ys) = prep (R⇒S e) (map R⇒S xs∼ys) map R⇒S (swap e₁ e₂ xs∼ys) = swap (R⇒S e₁) (R⇒S e₂) (map R⇒S xs∼ys) map R⇒S (trans xs∼ys ys∼zs) = trans (map R⇒S xs∼ys) (map R⇒S ys∼zs)

39.280702

125

0.557392

3f7b5752591469cff8e4fbffeb045a91f9c78519

1,277

agda

Agda

theorems/cw/cohomology/TipAndAugment.agda

mikeshulman/HoTT-Agda

e7d663b63d89f380ab772ecb8d51c38c26952dbb

[ "MIT" ]

null

theorems/cw/cohomology/TipAndAugment.agda

mikeshulman/HoTT-Agda

e7d663b63d89f380ab772ecb8d51c38c26952dbb

[ "MIT" ]

null

theorems/cw/cohomology/TipAndAugment.agda

mikeshulman/HoTT-Agda

e7d663b63d89f380ab772ecb8d51c38c26952dbb

[ "MIT" ]

1

2018-12-26T21:31:57.000Z

{-# OPTIONS --without-K --rewriting #-} open import HoTT open import cohomology.Theory open import cw.CW module cw.cohomology.TipAndAugment {i} (OT : OrdinaryTheory i) (⊙skel : ⊙Skeleton {i} 0) where open OrdinaryTheory OT open import homotopy.DisjointlyPointedSet open import cohomology.DisjointlyPointedSet OT module _ (m : ℤ) where CX₀ : Group i CX₀ = C m (⊙cw-head ⊙skel) CX₀-is-abelian : is-abelian CX₀ CX₀-is-abelian = C-is-abelian m (⊙cw-head ⊙skel) C2×CX₀ : Group i C2×CX₀ = C2 m ×ᴳ CX₀ abstract C2×CX₀-is-abelian : is-abelian C2×CX₀ C2×CX₀-is-abelian = ×ᴳ-is-abelian (C2 m) CX₀ (C2-is-abelian m) CX₀-is-abelian C2×CX₀-abgroup : AbGroup i C2×CX₀-abgroup = C2×CX₀ , C2×CX₀-is-abelian CX₀-β : ⊙has-cells-with-choice 0 ⊙skel i → CX₀ ≃ᴳ Πᴳ (MinusPoint (⊙cw-head ⊙skel)) (λ _ → C2 m) CX₀-β ac = C-set m (⊙cw-head ⊙skel) (snd (⊙Skeleton.skel ⊙skel)) (⊙Skeleton.pt-dec ⊙skel) ac abstract CX₀-≠-is-trivial : ∀ {m} (m≠0 : m ≠ 0) → ⊙has-cells-with-choice 0 ⊙skel i → is-trivialᴳ (CX₀ m) CX₀-≠-is-trivial {m} m≠0 ac = iso-preserves'-trivial (CX₀-β m ac) $ Πᴳ-is-trivial (MinusPoint (⊙cw-head ⊙skel)) (λ _ → C2 m) (λ _ → C-dimension m≠0) cw-coε : C2 0 →ᴳ C2×CX₀ 0 cw-coε = ×ᴳ-inl {G = C2 0} {H = CX₀ 0}

27.170213

94

0.635865

5931adde7d83016d1b98ebba83cb4dbd1da6d7d1

2,389

agda

Agda

Lec8.agda

clarkdm/CS410

523a8749f49c914bcd28402116dcbe79a78dbbf4

[ "CC0-1.0" ]

null

Lec8.agda

clarkdm/CS410

523a8749f49c914bcd28402116dcbe79a78dbbf4

[ "CC0-1.0" ]

null

Lec8.agda

clarkdm/CS410

523a8749f49c914bcd28402116dcbe79a78dbbf4

[ "CC0-1.0" ]

null

module Lec8 where open import CS410-Prelude open import CS410-Functor open import CS410-Monoid open import CS410-Nat data Maybe (X : Set) : Set where yes : X -> Maybe X no : Maybe X maybeFunctor : Functor Maybe maybeFunctor = record { map = \ { f (yes x) -> yes (f x) ; f no -> no } ; mapid = \ { (yes x) -> refl ; no -> refl } ; mapcomp = \ { f g (yes x) -> refl ; f g no -> refl } } open Functor maybeFunctor data List (X : Set) : Set where -- X scopes over the whole declaration... [] : List X -- ...so you can use it here... _::_ : X -> List X -> List X -- ...and here. infixr 3 _::_ data Hutton : Set where val : Nat -> Hutton _+H_ : Hutton -> Hutton -> Hutton hif_then_else_ : Hutton -> Hutton -> Hutton -> Hutton fail : Hutton maybeApplicative : Applicative Maybe maybeApplicative = record { pure = yes ; _<*>_ = \ { no mx -> no ; (yes f) no -> no ; (yes f) (yes x) -> yes (f x) } ; identity = \ {(yes x) -> refl ; no -> refl} ; composition = \ { (yes f) (yes g) (yes x) -> refl ; (yes f) (yes g) no -> refl ; (yes x) no mx -> refl ; no mg mx -> refl } ; homomorphism = \ f x -> refl ; interchange = \ { (yes f) y -> refl ; no y → refl } } open Applicative maybeApplicative public cond : Nat -> Nat -> Nat -> Nat cond zero t e = e cond (suc c) t e = t _>>=_ : forall {X Y} -> Maybe X -> (X -> Maybe Y) -> Maybe Y yes x >>= x2my = x2my x no >>= x2my = no eval : Hutton -> Maybe Nat eval (val x) = pure x eval (h +H h') = pure _+N_ <*> eval h <*> eval h' eval (hif c then t else e) -- = pure cond <*> eval c <*> eval t <*> eval e -- oops = eval c >>= \ { zero -> eval e ; (suc _) -> eval t} eval fail = no foo : Hutton foo = hif val 1 then (val 2 +H val 3) else (hif val 0 then val 4 else val 6) goo : Hutton goo = hif val 1 then (val 2 +H val 3) else (hif val 0 then fail else val 6) ap : forall {X Y} -> Maybe (X -> Y) -> Maybe X -> Maybe Y ap mf mx = mf >>= \ f -> mx >>= \ x -> yes (f x)

30.628205

76

0.462955

368d67c6016e6f964cf2d58162826c8bf75df96e

15,166

agda

Agda

contexts.agda

hazelgrove/hazel-palette-agda

c3225acc3c94c56376c6842b82b8b5d76912df2a

[ "MIT" ]

4

2020-10-04T06:45:06.000Z

2021-12-19T15:38:31.000Z

contexts.agda

hazelgrove/hazel-palette-agda

c3225acc3c94c56376c6842b82b8b5d76912df2a

[ "MIT" ]

9

2020-09-30T20:27:56.000Z

2020-10-20T20:44:13.000Z

contexts.agda

hazelgrove/hazelnut-livelits-agda

c3225acc3c94c56376c6842b82b8b5d76912df2a

[ "MIT" ]

null

open import Prelude open import Nat module contexts where -- variables are named with naturals in ė. therefore we represent -- contexts as functions from names for variables (nats) to possible -- bindings. _ctx : Set → Set A ctx = Nat → Maybe A -- convenient shorthand for the (unique up to fun. ext.) empty context ∅ : {A : Set} → A ctx ∅ _ = None infixr 100 ■_ -- the domain of a context is those naturals which cuase it to emit some τ dom : {A : Set} → A ctx → Nat → Set dom {A} Γ x = Σ[ τ ∈ A ] (Γ x == Some τ) -- membership, or presence, in a context _∈_ : {A : Set} (p : Nat × A) → (Γ : A ctx) → Set (x , y) ∈ Γ = (Γ x) == Some y -- this packages up an appeal to context memebership into a form that -- lets us retain more information ctxindirect : {A : Set} (Γ : A ctx) (n : Nat) → Σ[ a ∈ A ] (Γ n == Some a) + Γ n == None ctxindirect Γ n with Γ n ctxindirect Γ n | Some x = Inl (x , refl) ctxindirect Γ n | None = Inr refl -- apartness for contexts _#_ : {A : Set} (n : Nat) → (Γ : A ctx) → Set x # Γ = (Γ x) == None -- disjoint contexts are those which share no mappings _##_ : {A : Set} → A ctx → A ctx → Set _##_ {A} Γ Γ' = ((n : Nat) → dom Γ n → n # Γ') × ((n : Nat) → dom Γ' n → n # Γ) -- contexts give at most one binding for each variable ctxunicity : {A : Set} → {Γ : A ctx} {n : Nat} {t t' : A} → (n , t) ∈ Γ → (n , t') ∈ Γ → t == t' ctxunicity {n = n} p q with natEQ n n ctxunicity p q | Inl refl = someinj (! p · q) ctxunicity _ _ | Inr x≠x = abort (x≠x refl) -- warning: this is union, but it assumes WITHOUT CHECKING that the -- domains are disjoint. this is inherently asymmetric, and that's -- reflected throughout the development that follows _∪_ : {A : Set} → A ctx → A ctx → A ctx (C1 ∪ C2) x with C1 x (C1 ∪ C2) x | Some x₁ = Some x₁ (C1 ∪ C2) x | None = C2 x -- the singleton context ■_ : {A : Set} → (Nat × A) → A ctx (■ (x , a)) y with natEQ x y (■ (x , a)) .x | Inl refl = Some a ... | Inr _ = None -- context extension _,,_ : {A : Set} → A ctx → (Nat × A) → A ctx (Γ ,, (x , t)) = Γ ∪ (■ (x , t)) infixl 10 _,,_ -- used below in proof of ∪ commutativity and associativity lem-dom-union1 : {A : Set} {C1 C2 : A ctx} {x : Nat} → C1 ## C2 → dom C1 x → (C1 ∪ C2) x == C1 x lem-dom-union1 {A} {C1} {C2} {x} (d1 , d2) D with C1 x lem-dom-union1 (d1 , d2) D | Some x₁ = refl lem-dom-union1 (d1 , d2) D | None = abort (somenotnone (! (π2 D))) lem-dom-union2 : {A : Set} {C1 C2 : A ctx} {x : Nat} → C1 ## C2 → dom C1 x → (C2 ∪ C1) x == C1 x lem-dom-union2 {A} {C1} {C2} {x} (d1 , d2) D with ctxindirect C2 x lem-dom-union2 {A} {C1} {C2} {x} (d1 , d2) D | Inl x₁ = abort (somenotnone (! (π2 x₁) · d1 x D )) lem-dom-union2 {A} {C1} {C2} {x} (d1 , d2) D | Inr x₁ with C2 x lem-dom-union2 (d1 , d2) D | Inr x₂ | Some x₁ = abort (somenotnone x₂) lem-dom-union2 (d1 , d2) D | Inr x₁ | None = refl -- if the contexts in question are disjoint, then union is commutative ∪comm : {A : Set} → (C1 C2 : A ctx) → C1 ## C2 → (C1 ∪ C2) == (C2 ∪ C1) ∪comm C1 C2 (d1 , d2)= funext guts where lem-apart-union1 : {A : Set} (C1 C2 : A ctx) (x : Nat) → x # C1 → x # C2 → x # (C1 ∪ C2) lem-apart-union1 C1 C2 x apt1 apt2 with C1 x lem-apart-union1 C1 C2 x apt1 apt2 | Some x₁ = abort (somenotnone apt1) lem-apart-union1 C1 C2 x apt1 apt2 | None = apt2 lem-apart-union2 : {A : Set} (C1 C2 : A ctx) (x : Nat) → x # C1 → x # C2 → x # (C2 ∪ C1) lem-apart-union2 C1 C2 x apt1 apt2 with C2 x lem-apart-union2 C1 C2 x apt1 apt2 | Some x₁ = abort (somenotnone apt2) lem-apart-union2 C1 C2 x apt1 apt2 | None = apt1 guts : (x : Nat) → (C1 ∪ C2) x == (C2 ∪ C1) x guts x with ctxindirect C1 x | ctxindirect C2 x guts x | Inl (π1 , π2) | Inl (π3 , π4) = abort (somenotnone (! π4 · d1 x (π1 , π2))) guts x | Inl x₁ | Inr x₂ = tr (λ qq → qq == (C2 ∪ C1) x) (! (lem-dom-union1 (d1 , d2) x₁)) (tr (λ qq → C1 x == qq) (! (lem-dom-union2 (d1 , d2) x₁)) refl) guts x | Inr x₁ | Inl x₂ = tr (λ qq → (C1 ∪ C2) x == qq) (! (lem-dom-union1 (d2 , d1) x₂)) (tr (λ qq → qq == C2 x) (! (lem-dom-union2 (d2 , d1) x₂)) refl) guts x | Inr x₁ | Inr x₂ = tr (λ qq → qq == (C2 ∪ C1) x) (! (lem-apart-union1 C1 C2 x x₁ x₂)) (tr (λ qq → None == qq) (! (lem-apart-union2 C1 C2 x x₁ x₂)) refl) -- an element in the left of a union is in the union x∈∪l : {A : Set} → (Γ Γ' : A ctx) (n : Nat) (x : A) → (n , x) ∈ Γ → (n , x) ∈ (Γ ∪ Γ') x∈∪l Γ Γ' n x xin with Γ n x∈∪l Γ Γ' n x₁ xin | Some x = xin x∈∪l Γ Γ' n x () | None -- an element in the right of a union is in the union as long as the -- contexts are disjoint; this asymmetry reflects the asymmetry in the -- definition of union x∈∪r : {A : Set} → (Γ Γ' : A ctx) (n : Nat) (x : A) → (n , x) ∈ Γ' → Γ' ## Γ → (n , x) ∈ (Γ ∪ Γ') x∈∪r Γ Γ' n x nx∈ disj = tr (λ qq → (n , x) ∈ qq) (∪comm _ _ disj) (x∈∪l Γ' Γ n x nx∈) -- an element is in the context formed with just itself x∈■ : {A : Set} (n : Nat) (a : A) → (n , a) ∈ (■ (n , a)) x∈■ n a with natEQ n n x∈■ n a | Inl refl = refl x∈■ n a | Inr x = abort (x refl) -- if an index is in the domain of a singleton context, it's the only -- index in the context lem-dom-eq : {A : Set} {y : A} {n m : Nat} → dom (■ (m , y)) n → n == m lem-dom-eq {n = n} {m = m} (π1 , π2) with natEQ m n lem-dom-eq (π1 , π2) | Inl refl = refl lem-dom-eq (π1 , π2) | Inr x = abort (somenotnone (! π2)) -- a singleton context formed with an index apart from a context is -- disjoint from that context lem-apart-sing-disj : {A : Set} {n : Nat} {a : A} {Γ : A ctx} → n # Γ → (■ (n , a)) ## Γ lem-apart-sing-disj {A} {n} {a} {Γ} apt = asd1 , asd2 where asd1 : (n₁ : Nat) → dom (■ (n , a)) n₁ → n₁ # Γ asd1 m d with lem-dom-eq d asd1 .n d | refl = apt asd2 : (n₁ : Nat) → dom Γ n₁ → n₁ # (■ (n , a)) asd2 m (π1 , π2) with natEQ n m asd2 .n (π1 , π2) | Inl refl = abort (somenotnone (! π2 · apt )) asd2 m (π1 , π2) | Inr x = refl -- the only index of a singleton context is in its domain lem-domsingle : {A : Set} (p : Nat) (q : A) → dom (■ (p , q)) p lem-domsingle p q with natEQ p p lem-domsingle p q | Inl refl = q , refl lem-domsingle p q | Inr x₁ = abort (x₁ refl) -- dual of above lem-disj-sing-apart : {A : Set} {n : Nat} {a : A} {Γ : A ctx} → (■ (n , a)) ## Γ → n # Γ lem-disj-sing-apart {A} {n} {a} {Γ} (d1 , d2) = d1 n (lem-domsingle n a) -- the singleton context can only produce one non-none result lem-insingeq : {A : Set} {x x' : Nat} {τ τ' : A} → (■ (x , τ)) x' == Some τ' → τ == τ' lem-insingeq {A} {x} {x'} {τ} {τ'} eq with lem-dom-eq (τ' , eq) lem-insingeq {A} {x} {.x} {τ} {τ'} eq | refl with natEQ x x lem-insingeq refl | refl | Inl refl = refl lem-insingeq eq | refl | Inr x₁ = abort (somenotnone (! eq)) -- if an index doesn't appear in a context, and the union of that context -- with a singleton does produce a result, it must have come from the singleton lem-apart-union-eq : {A : Set} {Γ : A ctx} {x x' : Nat} {τ τ' : A} → x' # Γ → (Γ ∪ ■ (x , τ)) x' == Some τ' → τ == τ' lem-apart-union-eq {A} {Γ} {x} {x'} {τ} {τ'} apart eq with Γ x' lem-apart-union-eq apart eq | Some x = abort (somenotnone apart) lem-apart-union-eq apart eq | None = lem-insingeq eq -- if an index not in a singleton context produces a result from that -- singleton unioned with another context, the result must have come from -- the other context lem-neq-union-eq : {A : Set} {Γ : A ctx} {x x' : Nat} {τ τ' : A} → x' ≠ x → (Γ ∪ ■ (x , τ)) x' == Some τ' → Γ x' == Some τ' lem-neq-union-eq {A} {Γ} {x} {x'} {τ} {τ'} neq eq with Γ x' lem-neq-union-eq neq eq | Some x = eq lem-neq-union-eq {A} {Γ} {x} {x'} {τ} {τ'} neq eq | None with natEQ x x' lem-neq-union-eq neq eq | None | Inl x₁ = abort ((flip neq) x₁) lem-neq-union-eq neq eq | None | Inr x₁ = abort (somenotnone (! eq)) -- extending a context with a new index produces the result paired with -- that index. ctx-top : {A : Set} → (Γ : A ctx) (n : Nat) (a : A) → (n # Γ) → (n , a) ∈ (Γ ,, (n , a)) ctx-top Γ n a apt = x∈∪r Γ (■ (n , a)) n a (x∈■ n a) (lem-apart-sing-disj apt) --- lemmas building up to a proof of associativity of ∪ ctxignore1 : {A : Set} (x : Nat) (C1 C2 : A ctx) → x # C1 → (C1 ∪ C2) x == C2 x ctxignore1 x C1 C2 apt with ctxindirect C1 x ctxignore1 x C1 C2 apt | Inl x₁ = abort (somenotnone (! (π2 x₁) · apt)) ctxignore1 x C1 C2 apt | Inr x₁ with C1 x ctxignore1 x C1 C2 apt | Inr x₂ | Some x₁ = abort (somenotnone (x₂)) ctxignore1 x C1 C2 apt | Inr x₁ | None = refl ctxignore2 : {A : Set} (x : Nat) (C1 C2 : A ctx) → x # C2 → (C1 ∪ C2) x == C1 x ctxignore2 x C1 C2 apt with ctxindirect C2 x ctxignore2 x C1 C2 apt | Inl x₁ = abort (somenotnone (! (π2 x₁) · apt)) ctxignore2 x C1 C2 apt | Inr x₁ with C1 x ctxignore2 x C1 C2 apt | Inr x₂ | Some x₁ = refl ctxignore2 x C1 C2 apt | Inr x₁ | None = x₁ ctxcollapse1 : {A : Set} → (C1 C2 C3 : A ctx) (x : Nat) → (x # C3) → (C2 ∪ C3) x == C2 x → (C1 ∪ (C2 ∪ C3)) x == (C1 ∪ C2) x ctxcollapse1 C1 C2 C3 x apt eq with C2 x ctxcollapse1 C1 C2 C3 x apt eq | Some x₁ with C1 x ctxcollapse1 C1 C2 C3 x apt eq | Some x₂ | Some x₁ = refl ctxcollapse1 C1 C2 C3 x apt eq | Some x₁ | None with C2 x ctxcollapse1 C1 C2 C3 x apt eq | Some x₂ | None | Some x₁ = refl ctxcollapse1 C1 C2 C3 x apt eq | Some x₁ | None | None = apt ctxcollapse1 C1 C2 C3 x apt eq | None with C1 x ctxcollapse1 C1 C2 C3 x apt eq | None | Some x₁ = refl ctxcollapse1 C1 C2 C3 x apt eq | None | None with C2 x ctxcollapse1 C1 C2 C3 x apt eq | None | None | Some x₁ = refl ctxcollapse1 C1 C2 C3 x apt eq | None | None | None = eq ctxcollapse2 : {A : Set} → (C1 C2 C3 : A ctx) (x : Nat) → (x # C2) → (C2 ∪ C3) x == C3 x → (C1 ∪ (C2 ∪ C3)) x == (C1 ∪ C3) x ctxcollapse2 C1 C2 C3 x apt eq with C1 x ctxcollapse2 C1 C2 C3 x apt eq | Some x₁ = refl ctxcollapse2 C1 C2 C3 x apt eq | None with C2 x ctxcollapse2 C1 C2 C3 x apt eq | None | Some x₁ = eq ctxcollapse2 C1 C2 C3 x apt eq | None | None = refl ctxcollapse3 : {A : Set} → (C1 C2 C3 : A ctx) (x : Nat) → (x # C2) → ((C1 ∪ C2) ∪ C3) x == (C1 ∪ C3) x ctxcollapse3 C1 C2 C3 x apt with C1 x ctxcollapse3 C1 C2 C3 x apt | Some x₁ = refl ctxcollapse3 C1 C2 C3 x apt | None with C2 x ctxcollapse3 C1 C2 C3 x apt | None | Some x₁ = abort (somenotnone apt) ctxcollapse3 C1 C2 C3 x apt | None | None = refl -- if a union of a singleton and a ctx produces no result, the argument -- index must be apart from the ctx and disequal to the index of the -- singleton lem-union-none : {A : Set} {Γ : A ctx} {a : A} {x x' : Nat} → (Γ ∪ ■ (x , a)) x' == None → (x ≠ x') × (x' # Γ) lem-union-none {A} {Γ} {a} {x} {x'} emp with ctxindirect Γ x' lem-union-none {A} {Γ} {a} {x} {x'} emp | Inl (π1 , π2) with Γ x' lem-union-none emp | Inl (π1 , π2) | Some x = abort (somenotnone emp) lem-union-none {A} {Γ} {a} {x} {x'} emp | Inl (π1 , π2) | None with natEQ x x' lem-union-none emp | Inl (π1 , π2) | None | Inl x₁ = abort (somenotnone (! π2)) lem-union-none emp | Inl (π1 , π2) | None | Inr x₁ = x₁ , refl lem-union-none {A} {Γ} {a} {x} {x'} emp | Inr y with Γ x' lem-union-none emp | Inr y | Some x = abort (somenotnone emp) lem-union-none {A} {Γ} {a} {x} {x'} emp | Inr y | None with natEQ x x' lem-union-none emp | Inr y | None | Inl refl = abort (somenotnone emp) lem-union-none emp | Inr y | None | Inr x₁ = x₁ , refl -- converse of lem-union-none lem-none-union : {A : Set} {Γ : A ctx} {a : A} {x x' : Nat} → (x ≠ x') → (x' # Γ) → (Γ ∪ ■ (x , a)) x' == None lem-none-union {A} {Γ} {a} {x} {x'} h₁ h₂ with ctxindirect (■ (x , a)) x' lem-none-union {A} {Γ} {a} {x} {x'} h₁ h₂ | Inl (a' , h) = abort (somenotnone (!( lem-neq-union-eq (flip h₁) (tr (λ y → y == Some a') refl h)))) lem-none-union {A} {Γ} {a} {x} {x'} h₁ h₂ | Inr h = (ctxignore1 x' Γ (■ (x , a)) h₂) · h ∪assoc : {A : Set} (C1 C2 C3 : A ctx) → (C2 ## C3) → (C1 ∪ C2) ∪ C3 == C1 ∪ (C2 ∪ C3) ∪assoc C1 C2 C3 (d1 , d2) = funext guts where case2 : (x : Nat) → x # C3 → dom C2 x → ((C1 ∪ C2) ∪ C3) x == (C1 ∪ (C2 ∪ C3)) x case2 x apt dom = (ctxignore2 x (C1 ∪ C2) C3 apt) · ! (ctxcollapse1 C1 C2 C3 x apt (lem-dom-union1 (d1 , d2) dom)) case34 : (x : Nat) → x # C2 → ((C1 ∪ C2) ∪ C3) x == (C1 ∪ (C2 ∪ C3)) x case34 x apt = ctxcollapse3 C1 C2 C3 x apt · ! (ctxcollapse2 C1 C2 C3 x apt (ctxignore1 x C2 C3 apt)) guts : (x : Nat) → ((C1 ∪ C2) ∪ C3) x == (C1 ∪ (C2 ∪ C3)) x guts x with ctxindirect C2 x | ctxindirect C3 x guts x | Inl (π1 , π2) | Inl (π3 , π4) = abort (somenotnone (! π4 · d1 x (π1 , π2))) guts x | Inl x₁ | Inr x₂ = case2 x x₂ x₁ guts x | Inr x₁ | Inl x₂ = case34 x x₁ guts x | Inr x₁ | Inr x₂ = case34 x x₁ -- if x is apart from either part of a union, the answer comes from the other one lem-dom-union-apt1 : {A : Set} {Δ1 Δ2 : A ctx} {x : Nat} {y : A} → x # Δ1 → ((Δ1 ∪ Δ2) x == Some y) → (Δ2 x == Some y) lem-dom-union-apt1 {A} {Δ1} {Δ2} {x} {y} apt xin with Δ1 x lem-dom-union-apt1 apt xin | Some x₁ = abort (somenotnone apt) lem-dom-union-apt1 apt xin | None = xin lem-dom-union-apt2 : {A : Set} {Δ1 Δ2 : A ctx} {x : Nat} {y : A} → x # Δ2 → ((Δ1 ∪ Δ2) x == Some y) → (Δ1 x == Some y) lem-dom-union-apt2 {A} {Δ1} {Δ2} {x} {y} apt xin with Δ1 x lem-dom-union-apt2 apt xin | Some x₁ = xin lem-dom-union-apt2 apt xin | None = abort (somenotnone (! xin · apt)) -- the empty context is a left and right unit for ∪ ∅∪1 : {A : Set} {Γ : A ctx} → ∅ ∪ Γ == Γ ∅∪1 {A} {Γ} = refl ∅∪2 : {A : Set} {Γ : A ctx} → Γ ∪ ∅ == Γ ∅∪2 {A} {Γ} = funext guts where guts : (x : Nat) → (Γ ∪ ∅) x == Γ x guts x with Γ x guts x | Some x₁ = refl guts x | None = refl

46.664615

166

0.509891

edd0fe3821c0428ac0b4f9cae2331bdd3edad6bd

4,801

agda

Agda

LibraBFT/Impl/Consensus/Types.agda

lisandrasilva/bft-consensus-agda-1

b7dd98dd90d98fbb934ef8cb4f3314940986790d

[ "UPL-1.0" ]

null

LibraBFT/Impl/Consensus/Types.agda

lisandrasilva/bft-consensus-agda-1

b7dd98dd90d98fbb934ef8cb4f3314940986790d

[ "UPL-1.0" ]

null

LibraBFT/Impl/Consensus/Types.agda

lisandrasilva/bft-consensus-agda-1

b7dd98dd90d98fbb934ef8cb4f3314940986790d

[ "UPL-1.0" ]

null

{- Byzantine Fault Tolerant Consensus Verification in Agda, version 0.9. Copyright (c) 2020 Oracle and/or its affiliates. Licensed under the Universal Permissive License v 1.0 as shown at https://opensource.oracle.com/licenses/upl -} {-# OPTIONS --allow-unsolved-metas #-} open import LibraBFT.Prelude open import LibraBFT.Hash open import LibraBFT.Base.PKCS open import LibraBFT.Base.Encode open import LibraBFT.Base.KVMap as KVMap open import Optics.All open import Data.String using (String) -- This module defines types for an out-of-date implementation, based -- on a previous version of LibraBFT. It will be updated to model a -- more recent version in future. -- -- One important trick here is that the EventProcessor type separayes -- types that /define/ the EpochConfig and types that /use/ the -- /EpochConfig/. The advantage of doing this separation can be seen -- in Util.Util.liftEC, where we define a lifting of a function that -- does not change the bits that define the EpochConfig into the whole -- state. This enables a more elegant approach for reasoning about -- functions that do not change parts of the state responsible for -- defining the epoch config. However, the separation is not perfect, -- so sometimes fields may be modified in EpochIndep even though there -- is no epoch change. module LibraBFT.Impl.Consensus.Types where open import LibraBFT.Impl.NetworkMsg open import LibraBFT.Impl.Consensus.Types.EpochIndep public open import LibraBFT.Impl.Consensus.Types.EpochDep public -- The parts of the state of a peer that are used to -- define the EpochConfig are the SafetyRules and ValidatorVerifier: record EventProcessorEC : Set where constructor mkEventProcessorPreEC field ₋epSafetyRules : SafetyRules ₋epValidators : ValidatorVerifier open EventProcessorEC public unquoteDecl epSafetyRules epValidators = mkLens (quote EventProcessorEC) (epSafetyRules ∷ epValidators ∷ []) epEpoch : Lens EventProcessorEC EpochId epEpoch = epSafetyRules ∙ srPersistentStorage ∙ psEpoch epLastVotedRound : Lens EventProcessorEC Round epLastVotedRound = epSafetyRules ∙ srPersistentStorage ∙ psLastVotedRound -- We need enough authors to withstand the desired number of -- byzantine failures. We enforce this with a predicate over -- 'EventProcessorEC'. EventProcessorEC-correct : EventProcessorEC → Set EventProcessorEC-correct epec = let numAuthors = kvm-size (epec ^∙ epValidators ∙ vvAddressToValidatorInfo) qsize = epec ^∙ epValidators ∙ vvQuorumVotingPower bizF = numAuthors ∸ qsize in suc (3 * bizF) ≤ numAuthors EventProcessorEC-correct-≡ : (epec1 : EventProcessorEC) → (epec2 : EventProcessorEC) → (epec1 ^∙ epValidators) ≡ (epec2 ^∙ epValidators) → EventProcessorEC-correct epec1 → EventProcessorEC-correct epec2 EventProcessorEC-correct-≡ epec1 epec2 refl = id -- Given a well-formed set of definitions that defines an EpochConfig, -- α-EC will compute this EpochConfig by abstracting away the unecessary -- pieces from EventProcessorEC. -- TODO-2: update and complete when definitions are updated to more recent version α-EC : Σ EventProcessorEC EventProcessorEC-correct → EpochConfig α-EC (epec , ok) = let numAuthors = kvm-size (epec ^∙ epValidators ∙ vvAddressToValidatorInfo) qsize = epec ^∙ epValidators ∙ vvQuorumVotingPower bizF = numAuthors ∸ qsize in (mkEpochConfig {! someHash?!} (epec ^∙ epEpoch) numAuthors {!!} {!!} {!!} {!!} {!!} {!!} {!!} {!!}) α-EC-≡ : (epec1 : EventProcessorEC) → (epec2 : EventProcessorEC) → (vals≡ : (epec1 ^∙ epValidators) ≡ (epec2 ^∙ epValidators)) → (epoch≡ : (epec1 ^∙ epEpoch) ≡ (epec2 ^∙ epEpoch)) → (epec1-corr : EventProcessorEC-correct epec1) → α-EC (epec1 , epec1-corr) ≡ α-EC (epec2 , EventProcessorEC-correct-≡ epec1 epec2 vals≡ epec1-corr) α-EC-≡ epec1 epec2 refl refl epec1-corr = refl -- Finally, the EventProcessor is split in two pieces: those -- that are used to make an EpochConfig versus those that -- use an EpochConfig. record EventProcessor : Set where constructor mkEventProcessor field ₋epMeta-Msgs : List NetworkMsg -- List of messages sent by this peer ₋epEC : EventProcessorEC ₋epEC-correct : EventProcessorEC-correct ₋epEC ₋epWithEC : EventProcessorWithEC (α-EC (₋epEC , ₋epEC-correct)) -- If we want to add pieces that neither contribute to the -- construction of the EC nor need one, they should be defined in -- EventProcessor directly open EventProcessor public

45.72381

111

0.70277

3611700f25a0f853ea94628273d4f1542ad14d57

1,040

agda

Agda

Data/List/Smart.agda

oisdk/agda-playground

97a3aab1282b2337c5f43e2cfa3fa969a94c11b7

[ "MIT" ]

6

2020-09-11T17:45:41.000Z

2021-11-16T08:11:34.000Z

Data/List/Smart.agda

oisdk/agda-playground

97a3aab1282b2337c5f43e2cfa3fa969a94c11b7

[ "MIT" ]

null

Data/List/Smart.agda

oisdk/agda-playground

97a3aab1282b2337c5f43e2cfa3fa969a94c11b7

[ "MIT" ]

1

2021-11-11T12:30:21.000Z

{-# OPTIONS --cubical --safe #-} module Data.List.Smart where open import Prelude open import Data.Nat.Properties using (_≡ᴮ_; complete-==) infixr 5 _∷′_ _++′_ data List {a} (A : Type a) : Type a where []′ : List A _∷′_ : A → List A → List A _++′_ : List A → List A → List A sz′ : List A → ℕ → ℕ sz′ []′ k = k sz′ (x ∷′ xs) k = k sz′ (xs ++′ ys) k = suc (sz′ xs (sz′ ys k)) sz : List A → ℕ sz []′ = zero sz (x ∷′ xs) = zero sz (xs ++′ ys) = sz′ xs (sz ys) _HasSize_ : List A → ℕ → Type xs HasSize n = T (sz xs ≡ᴮ n) data ListView {a} (A : Type a) : Type a where Nil : ListView A Cons : A → List A → ListView A viewˡ : List A → ListView A viewˡ xs = go xs (sz xs) (complete-== (sz xs)) where go : (xs : List A) → (n : ℕ) → xs HasSize n → ListView A go []′ n p = Nil go (x ∷′ xs) n p = Cons x xs go ((x ∷′ xs) ++′ ys) n p = Cons x (xs ++′ ys) go ([]′ ++′ ys) n p = go ys n p go ((xs ++′ ys) ++′ zs) (suc n) p = go (xs ++′ (ys ++′ zs)) n p

23.636364

66

0.480769

65b8ac3a19925f749db747719001a24f9b70f3a0

1,342

agda

Agda

src/data/lib/prim/Agda/Primitive/Cubical.agda

phadej/agda

2fa8ede09451d43647f918dbfb24ff7b27c52edc

[ "BSD-3-Clause" ]

null

src/data/lib/prim/Agda/Primitive/Cubical.agda

phadej/agda

2fa8ede09451d43647f918dbfb24ff7b27c52edc

[ "BSD-3-Clause" ]

null

src/data/lib/prim/Agda/Primitive/Cubical.agda

phadej/agda

2fa8ede09451d43647f918dbfb24ff7b27c52edc

[ "BSD-3-Clause" ]

null

{-# OPTIONS --cubical #-} module Agda.Primitive.Cubical where {-# BUILTIN INTERVAL I #-} -- I : Setω {-# BUILTIN IZERO i0 #-} {-# BUILTIN IONE i1 #-} infix 30 primINeg infixr 20 primIMin primIMax primitive primIMin : I → I → I primIMax : I → I → I primINeg : I → I {-# BUILTIN ISONE IsOne #-} -- IsOne : I → Setω postulate itIsOne : IsOne i1 IsOne1 : ∀ i j → IsOne i → IsOne (primIMax i j) IsOne2 : ∀ i j → IsOne j → IsOne (primIMax i j) {-# BUILTIN ITISONE itIsOne #-} {-# BUILTIN ISONE1 IsOne1 #-} {-# BUILTIN ISONE2 IsOne2 #-} {-# BUILTIN PARTIAL Partial #-} {-# BUILTIN PARTIALP PartialP #-} postulate isOneEmpty : ∀ {a} {A : Partial i0 (Set a)} → PartialP i0 A {-# BUILTIN ISONEEMPTY isOneEmpty #-} primitive primPOr : ∀ {a} (i j : I) {A : Partial (primIMax i j) (Set a)} → PartialP i (λ z → A (IsOne1 i j z)) → PartialP j (λ z → A (IsOne2 i j z)) → PartialP (primIMax i j) A -- Computes in terms of primHComp and primTransp primComp : ∀ {a} (A : (i : I) → Set (a i)) (φ : I) → (∀ i → Partial φ (A i)) → (a : A i0) → A i1 syntax primPOr p q u t = [ p ↦ u , q ↦ t ] primitive primTransp : ∀ {a} (A : (i : I) → Set (a i)) (φ : I) → (a : A i0) → A i1 primHComp : ∀ {a} {A : Set a} {φ : I} → (∀ i → Partial φ A) → A → A

28.553191

98

0.538003

36606c14dd617e0e112f90b292a288d122d14525

2,762

agda

Agda

Cubical/Categories/Category/Base.agda

antoinevanmuylder/cubical

5b40df813434aa11631ac240409ca2c4d849453c

[ "MIT" ]

null

Cubical/Categories/Category/Base.agda

antoinevanmuylder/cubical

5b40df813434aa11631ac240409ca2c4d849453c

[ "MIT" ]

null

Cubical/Categories/Category/Base.agda

antoinevanmuylder/cubical

5b40df813434aa11631ac240409ca2c4d849453c

[ "MIT" ]

null

{-# OPTIONS --safe #-} module Cubical.Categories.Category.Base where open import Cubical.Foundations.Prelude open import Cubical.Foundations.HLevels open import Cubical.Foundations.Equiv private variable ℓ ℓ' : Level -- Categories with hom-sets record Category ℓ ℓ' : Type (ℓ-suc (ℓ-max ℓ ℓ')) where -- no-eta-equality ; NOTE: need eta equality for `opop` field ob : Type ℓ Hom[_,_] : ob → ob → Type ℓ' id : ∀ {x} → Hom[ x , x ] _⋆_ : ∀ {x y z} (f : Hom[ x , y ]) (g : Hom[ y , z ]) → Hom[ x , z ] ⋆IdL : ∀ {x y} (f : Hom[ x , y ]) → id ⋆ f ≡ f ⋆IdR : ∀ {x y} (f : Hom[ x , y ]) → f ⋆ id ≡ f ⋆Assoc : ∀ {x y z w} (f : Hom[ x , y ]) (g : Hom[ y , z ]) (h : Hom[ z , w ]) → (f ⋆ g) ⋆ h ≡ f ⋆ (g ⋆ h) isSetHom : ∀ {x y} → isSet Hom[ x , y ] -- composition: alternative to diagramatic order _∘_ : ∀ {x y z} (g : Hom[ y , z ]) (f : Hom[ x , y ]) → Hom[ x , z ] g ∘ f = f ⋆ g infixr 9 _⋆_ infixr 9 _∘_ open Category -- Helpful syntax/notation _[_,_] : (C : Category ℓ ℓ') → (x y : C .ob) → Type ℓ' _[_,_] = Hom[_,_] -- Needed to define this in order to be able to make the subsequence syntax declaration seq' : ∀ (C : Category ℓ ℓ') {x y z} (f : C [ x , y ]) (g : C [ y , z ]) → C [ x , z ] seq' = _⋆_ infixl 15 seq' syntax seq' C f g = f ⋆⟨ C ⟩ g -- composition comp' : ∀ (C : Category ℓ ℓ') {x y z} (g : C [ y , z ]) (f : C [ x , y ]) → C [ x , z ] comp' = _∘_ infixr 16 comp' syntax comp' C g f = g ∘⟨ C ⟩ f -- Isomorphisms and paths in categories record CatIso (C : Category ℓ ℓ') (x y : C .ob) : Type ℓ' where constructor catiso field mor : C [ x , y ] inv : C [ y , x ] sec : inv ⋆⟨ C ⟩ mor ≡ C .id ret : mor ⋆⟨ C ⟩ inv ≡ C .id pathToIso : {C : Category ℓ ℓ'} {x y : C .ob} (p : x ≡ y) → CatIso C x y pathToIso {C = C} p = J (λ z _ → CatIso _ _ z) (catiso idx idx (C .⋆IdL idx) (C .⋆IdL idx)) p where idx = C .id -- Univalent Categories record isUnivalent (C : Category ℓ ℓ') : Type (ℓ-max ℓ ℓ') where field univ : (x y : C .ob) → isEquiv (pathToIso {C = C} {x = x} {y = y}) -- package up the univalence equivalence univEquiv : ∀ (x y : C .ob) → (x ≡ y) ≃ (CatIso _ x y) univEquiv x y = pathToIso , univ x y -- The function extracting paths from category-theoretic isomorphisms. CatIsoToPath : {x y : C .ob} (p : CatIso _ x y) → x ≡ y CatIsoToPath {x = x} {y = y} p = equivFun (invEquiv (univEquiv x y)) p -- Opposite category _^op : Category ℓ ℓ' → Category ℓ ℓ' ob (C ^op) = ob C Hom[_,_] (C ^op) x y = C [ y , x ] id (C ^op) = id C _⋆_ (C ^op) f g = g ⋆⟨ C ⟩ f ⋆IdL (C ^op) = C .⋆IdR ⋆IdR (C ^op) = C .⋆IdL ⋆Assoc (C ^op) f g h = sym (C .⋆Assoc _ _ _) isSetHom (C ^op) = C .isSetHom

30.351648

93

0.524258

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agda

Agda

Cubical/Algebra/Group/DirProd.agda

FernandoLarrain/cubical

9acdecfa6437ec455568be4e5ff04849cc2bc13b

[ "MIT" ]

301

2018-10-17T18:00:24.000Z

2022-03-24T02:10:47.000Z

Cubical/Algebra/Group/DirProd.agda

FernandoLarrain/cubical

9acdecfa6437ec455568be4e5ff04849cc2bc13b

[ "MIT" ]

584

2018-10-15T09:49:02.000Z

2022-03-30T12:09:17.000Z

Cubical/Algebra/Group/DirProd.agda

FernandoLarrain/cubical

9acdecfa6437ec455568be4e5ff04849cc2bc13b

[ "MIT" ]

134

2018-11-16T06:11:03.000Z

2022-03-23T16:22:13.000Z

{-# OPTIONS --safe #-} module Cubical.Algebra.Group.DirProd where open import Cubical.Foundations.Prelude open import Cubical.Foundations.HLevels open import Cubical.Data.Sigma open import Cubical.Algebra.Group.Base open import Cubical.Algebra.Monoid open import Cubical.Algebra.Semigroup open GroupStr open IsGroup hiding (rid ; lid ; invr ; invl) open IsMonoid hiding (rid ; lid) open IsSemigroup DirProd : ∀ {ℓ ℓ'} → Group ℓ → Group ℓ' → Group (ℓ-max ℓ ℓ') fst (DirProd G H) = fst G × fst H 1g (snd (DirProd G H)) = (1g (snd G)) , (1g (snd H)) _·_ (snd (DirProd G H)) x y = _·_ (snd G) (fst x) (fst y) , _·_ (snd H) (snd x) (snd y) (inv (snd (DirProd G H))) x = (inv (snd G) (fst x)) , (inv (snd H) (snd x)) is-set (isSemigroup (isMonoid (isGroup (snd (DirProd G H))))) = isSet× (is-set (snd G)) (is-set (snd H)) assoc (isSemigroup (isMonoid (isGroup (snd (DirProd G H))))) x y z i = assoc (snd G) (fst x) (fst y) (fst z) i , assoc (snd H) (snd x) (snd y) (snd z) i fst (identity (isMonoid (isGroup (snd (DirProd G H)))) x) i = rid (snd G) (fst x) i , rid (snd H) (snd x) i snd (identity (isMonoid (isGroup (snd (DirProd G H)))) x) i = lid (snd G) (fst x) i , lid (snd H) (snd x) i fst (inverse (isGroup (snd (DirProd G H))) x) i = (invr (snd G) (fst x) i) , invr (snd H) (snd x) i snd (inverse (isGroup (snd (DirProd G H))) x) i = (invl (snd G) (fst x) i) , invl (snd H) (snd x) i

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agda

Agda

theorems/cw/cohomology/reconstructed/HigherCoboundary.agda

AntoineAllioux/HoTT-Agda

1037d82edcf29b620677a311dcfd4fc2ade2faa6

[ "MIT" ]

294

2015-01-09T16:23:23.000Z

2022-03-20T13:54:45.000Z

theorems/cw/cohomology/reconstructed/HigherCoboundary.agda

AntoineAllioux/HoTT-Agda

1037d82edcf29b620677a311dcfd4fc2ade2faa6

[ "MIT" ]

31

2015-03-05T20:09:00.000Z

2021-10-03T19:15:25.000Z

theorems/cw/cohomology/reconstructed/HigherCoboundary.agda

AntoineAllioux/HoTT-Agda

1037d82edcf29b620677a311dcfd4fc2ade2faa6

[ "MIT" ]

50

2015-01-10T01:48:08.000Z

2022-02-14T03:03:25.000Z

{-# OPTIONS --without-K --rewriting #-} open import HoTT open import groups.Cokernel open import cw.WedgeOfCells open import cohomology.Theory open import cw.CW module cw.cohomology.reconstructed.HigherCoboundary {i} (OT : OrdinaryTheory i) {n} (⊙skel : ⊙Skeleton {i} (S (S n))) where open OrdinaryTheory OT open import cw.cohomology.grid.LongExactSequence cohomology-theory (ℕ-to-ℤ (S n)) (⊙cw-incl-last (⊙cw-init ⊙skel)) (⊙cw-incl-last ⊙skel) open import cw.cohomology.WedgeOfCells OT cw-co∂-last : CXₙ/Xₙ₋₁ (⊙cw-init ⊙skel) (ℕ-to-ℤ (S n)) →ᴳ CXₙ/Xₙ₋₁ ⊙skel (ℕ-to-ℤ (S (S n))) cw-co∂-last = grid-co∂ cw-∂-before-Susp : Xₙ/Xₙ₋₁ (⊙Skeleton.skel ⊙skel) → Susp (Xₙ/Xₙ₋₁ (cw-init (⊙Skeleton.skel ⊙skel))) cw-∂-before-Susp = grid-∂-before-Susp ⊙cw-∂-before-Susp : ⊙Xₙ/Xₙ₋₁ (⊙Skeleton.skel ⊙skel) ⊙→ ⊙Susp (Xₙ/Xₙ₋₁ (cw-init (⊙Skeleton.skel ⊙skel))) ⊙cw-∂-before-Susp = ⊙grid-∂-before-Susp cw-∂-before-Susp-glue-β = grid-∂-before-Susp-glue-β cw-co∂-last-β = grid-co∂-β module CokerCo∂ where grp = Coker cw-co∂-last (CXₙ/Xₙ₋₁-is-abelian ⊙skel (ℕ-to-ℤ (S (S n)))) open Group grp public CokerCo∂ = CokerCo∂.grp

32

103

0.675893

11089e25fc9714c377e6102dc8b0e36c80a1030f

4,772

agda

Agda

notes/fixed-points/LFPs/List.agda

asr/fotc

2fc9f2b81052a2e0822669f02036c5750371b72d

[ "MIT" ]

11

2015-09-03T20:53:42.000Z

2021-09-12T16:09:54.000Z

notes/fixed-points/LFPs/List.agda

asr/fotc

2fc9f2b81052a2e0822669f02036c5750371b72d

[ "MIT" ]

2

2016-10-12T17:28:16.000Z

2017-01-01T14:34:26.000Z

notes/fixed-points/LFPs/List.agda

asr/fotc

2fc9f2b81052a2e0822669f02036c5750371b72d

[ "MIT" ]

3

2016-09-19T14:18:30.000Z

2018-03-14T08:50:00.000Z

------------------------------------------------------------------------------ -- Equivalence of definitions of total lists ------------------------------------------------------------------------------ {-# OPTIONS --exact-split #-} {-# OPTIONS --no-sized-types #-} {-# OPTIONS --no-universe-polymorphism #-} {-# OPTIONS --without-K #-} module LFPs.List where open import FOTC.Base open import FOTC.Base.List open import FOTC.Data.Nat.UnaryNumbers ------------------------------------------------------------------------------ module LFP where -- List is a least fixed-point of a functor -- The functor. ListF : (D → Set) → D → Set ListF A xs = xs ≡ [] ∨ (∃[ x' ] ∃[ xs' ] xs ≡ x' ∷ xs' ∧ A xs') -- List is the least fixed-point of ListF. i.e. postulate List : D → Set -- List is a pre-fixed point of ListF, i.e. -- -- ListF List ≤ List. -- -- Peter: It corresponds to the introduction rules. List-in : ∀ {xs} → xs ≡ [] ∨ (∃[ x' ] ∃[ xs' ] xs ≡ x' ∷ xs' ∧ List xs') → List xs -- The higher-order version. List-in-ho : {xs : D} → ListF List xs → List xs -- List is the least pre-fixed point of ListF, i.e. -- -- ∀ A. ListF A ≤ A ⇒ List ≤ A. -- -- Peter: It corresponds to the elimination rule of an inductively -- defined predicate. List-ind : (A : D → Set) → (∀ {xs} → xs ≡ [] ∨ (∃[ x' ] ∃[ xs' ] xs ≡ x' ∷ xs' ∧ A xs') → A xs) → ∀ {xs} → List xs → A xs -- Higher-order version. List-ind-ho : (A : D → Set) → (∀ {xs} → ListF A xs → A xs) → ∀ {xs} → List xs → A xs ---------------------------------------------------------------------------- -- List-in and List-in-ho are equivalents List-in-fo : ∀ {xs} → xs ≡ [] ∨ (∃[ x' ] ∃[ xs' ] xs ≡ x' ∷ xs' ∧ List xs') → List xs List-in-fo = List-in-ho List-in-ho' : {xs : D} → ListF List xs → List xs List-in-ho' = List-in-ho ---------------------------------------------------------------------------- -- List-ind and List-ind-ho are equivalents List-ind-fo : (A : D → Set) → (∀ {xs} → xs ≡ [] ∨ (∃[ x' ] ∃[ xs' ] xs ≡ x' ∷ xs' ∧ A xs') → A xs) → ∀ {xs} → List xs → A xs List-ind-fo = List-ind-ho List-ind-ho' : (A : D → Set) → (∀ {xs} → ListF A xs → A xs) → ∀ {xs} → List xs → A xs List-ind-ho' = List-ind ---------------------------------------------------------------------------- -- The data constructors of List. lnil : List [] lnil = List-in (inj₁ refl) lcons : ∀ x {xs} → List xs → List (x ∷ xs) lcons x {xs} Lxs = List-in (inj₂ (x , xs , refl , Lxs)) ---------------------------------------------------------------------------- -- The type theoretical induction principle for List. List-ind' : (A : D → Set) → A [] → (∀ x {xs} → A xs → A (x ∷ xs)) → ∀ {xs} → List xs → A xs List-ind' A A[] is = List-ind A prf where prf : ∀ {xs} → xs ≡ [] ∨ (∃[ x' ] ∃[ xs' ] xs ≡ x' ∷ xs' ∧ A xs') → A xs prf (inj₁ xs≡[]) = subst A (sym xs≡[]) A[] prf (inj₂ (x' , xs' , h₁ , Axs')) = subst A (sym h₁) (is x' Axs') ---------------------------------------------------------------------------- -- Example xs : D xs = 0' ∷ true ∷ 1' ∷ false ∷ [] xs-List : List xs xs-List = lcons 0' (lcons true (lcons 1' (lcons false lnil))) ------------------------------------------------------------------------------ module Data where data List : D → Set where lnil : List [] lcons : ∀ x {xs} → List xs → List (x ∷ xs) -- Induction principle. List-ind : (A : D → Set) → A [] → (∀ x {xs} → A xs → A (x ∷ xs)) → ∀ {xs} → List xs → A xs List-ind A A[] h lnil = A[] List-ind A A[] h (lcons x Lxs) = h x (List-ind A A[] h Lxs) ---------------------------------------------------------------------------- -- List-in List-in : ∀ {xs} → xs ≡ [] ∨ (∃[ x' ] ∃[ xs' ] xs ≡ x' ∷ xs' ∧ List xs') → List xs List-in {xs} h = case prf₁ prf₂ h where prf₁ : xs ≡ [] → List xs prf₁ xs≡[] = subst List (sym xs≡[]) lnil prf₂ : ∃[ x' ] ∃[ xs' ] xs ≡ x' ∷ xs' ∧ List xs' → List xs prf₂ (x' , xs' , prf , Lxs') = subst List (sym prf) (lcons x' Lxs') ---------------------------------------------------------------------------- -- The fixed-point induction principle for List. List-ind' : (A : D → Set) → (∀ {xs} → xs ≡ [] ∨ (∃[ x' ] ∃[ xs' ] xs ≡ x' ∷ xs' ∧ A xs') → A xs) → ∀ {xs} → List xs → A xs List-ind' A h Lxs = List-ind A h₁ h₂ Lxs where h₁ : A [] h₁ = h (inj₁ refl) h₂ : ∀ y {ys} → A ys → A (y ∷ ys) h₂ y {ys} Ays = h (inj₂ (y , ys , refl , Ays))

31.189542

79

0.384325

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146

agda

Agda

Categories/Enriched.agda

copumpkin/categories

36f4181d751e2ecb54db219911d8c69afe8ba892

[ "BSD-3-Clause" ]

98

2015-04-15T14:57:33.000Z

2022-03-08T05:20:36.000Z

Categories/Enriched.agda

p-pavel/categories

e41aef56324a9f1f8cf3cd30b2db2f73e01066f2

[ "BSD-3-Clause" ]

19

2015-05-23T06:47:10.000Z

2019-08-09T16:31:40.000Z

Categories/Enriched.agda

p-pavel/categories

e41aef56324a9f1f8cf3cd30b2db2f73e01066f2

[ "BSD-3-Clause" ]

23

2015-02-05T13:03:09.000Z

2021-11-11T13:50:56.000Z

{-# OPTIONS --universe-polymorphism #-} module Categories.Enriched where open import Categories.Category open import Categories.Monoidal -- moar

20.857143

39

0.794521

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agda

Agda

src/Categories/Morphism/Isomorphism.agda

yourboynico/agda-categories

6a087c592dbe58fc4bd9d02e1be9b94a9e138aca

[ "MIT" ]

279

2019-06-01T14:36:40.000Z

2022-03-22T00:40:14.000Z

src/Categories/Morphism/Isomorphism.agda

seanpm2001/agda-categories

d9e4f578b126313058d105c61707d8c8ae987fa8

[ "MIT" ]

236

2019-06-01T14:53:54.000Z

2022-03-28T14:31:43.000Z

src/Categories/Morphism/Isomorphism.agda

seanpm2001/agda-categories

d9e4f578b126313058d105c61707d8c8ae987fa8

[ "MIT" ]

64

2019-06-02T16:58:15.000Z

2022-03-14T02:00:59.000Z

{-# OPTIONS --without-K --safe #-} open import Categories.Category -- Mainly *properties* of isomorphisms, and a lot of other things too -- TODO: split things up more semantically? module Categories.Morphism.Isomorphism {o ℓ e} (𝒞 : Category o ℓ e) where open import Level using (_⊔_) open import Function using (flip) open import Data.Product using (_,_) open import Relation.Binary using (Rel; _Preserves_⟶_; IsEquivalence) open import Relation.Binary.Construct.Closure.Transitive open import Relation.Binary.PropositionalEquality as ≡ using (_≡_) import Categories.Category.Construction.Core as Core open import Categories.Category.Groupoid using (IsGroupoid) import Categories.Category.Groupoid.Properties as GroupoidProps import Categories.Morphism as Morphism import Categories.Morphism.Properties as MorphismProps import Categories.Morphism.IsoEquiv as IsoEquiv import Categories.Category.Construction.Path as Path open Core 𝒞 using (Core; Core-isGroupoid; CoreGroupoid; module Shorthands) open Morphism 𝒞 open MorphismProps 𝒞 open Path 𝒞 import Categories.Morphism.Reasoning as MR open Category 𝒞 open Definitions 𝒞 private module MCore where open GroupoidProps CoreGroupoid public open MorphismProps Core public open Morphism Core public open Path Core public variable A B C D E F : Obj open Shorthands hiding (_≅_) CommutativeIso = IsGroupoid.CommutativeSquare Core-isGroupoid -------------------- -- Also stuff about transitive closure ∘ᵢ-tc : A [ _≅_ ]⁺ B → A ≅ B ∘ᵢ-tc = MCore.∘-tc infix 4 _≃⁺_ _≃⁺_ : Rel (A [ _≅_ ]⁺ B) e _≃⁺_ = MCore._≈⁺_ TransitiveClosure : Category o (o ⊔ ℓ ⊔ e) e TransitiveClosure = MCore.Path -------------------- -- some infrastructure setup in order to say something about morphisms and isomorphisms module _ where private data IsoPlus : A [ _⇒_ ]⁺ B → Set (o ⊔ ℓ ⊔ e) where [_] : {f : A ⇒ B} {g : B ⇒ A} → Iso f g → IsoPlus [ f ] _∼⁺⟨_⟩_ : ∀ A {f⁺ : A [ _⇒_ ]⁺ B} {g⁺ : B [ _⇒_ ]⁺ C} → IsoPlus f⁺ → IsoPlus g⁺ → IsoPlus (_ ∼⁺⟨ f⁺ ⟩ g⁺) open _≅_ ≅⁺⇒⇒⁺ : A [ _≅_ ]⁺ B → A [ _⇒_ ]⁺ B ≅⁺⇒⇒⁺ [ f ] = [ from f ] ≅⁺⇒⇒⁺ (_ ∼⁺⟨ f⁺ ⟩ f⁺′) = _ ∼⁺⟨ ≅⁺⇒⇒⁺ f⁺ ⟩ ≅⁺⇒⇒⁺ f⁺′ reverse : A [ _≅_ ]⁺ B → B [ _≅_ ]⁺ A reverse [ f ] = [ ≅.sym f ] reverse (_ ∼⁺⟨ f⁺ ⟩ f⁺′) = _ ∼⁺⟨ reverse f⁺′ ⟩ reverse f⁺ reverse⇒≅-sym : (f⁺ : A [ _≅_ ]⁺ B) → ∘ᵢ-tc (reverse f⁺) ≡ ≅.sym (∘ᵢ-tc f⁺) reverse⇒≅-sym [ f ] = ≡.refl reverse⇒≅-sym (_ ∼⁺⟨ f⁺ ⟩ f⁺′) = ≡.cong₂ (Morphism.≅.trans 𝒞) (reverse⇒≅-sym f⁺′) (reverse⇒≅-sym f⁺) TransitiveClosure-groupoid : IsGroupoid TransitiveClosure TransitiveClosure-groupoid = record { _⁻¹ = reverse ; iso = λ {_ _ f⁺} → record { isoˡ = isoˡ′ f⁺ ; isoʳ = isoʳ′ f⁺ } } where open HomReasoningᵢ isoˡ′ : (f⁺ : A [ _≅_ ]⁺ B) → ∘ᵢ-tc (reverse f⁺) ∘ᵢ ∘ᵢ-tc f⁺ ≈ᵢ ≅.refl isoˡ′ f⁺ = begin ∘ᵢ-tc (reverse f⁺) ∘ᵢ ∘ᵢ-tc f⁺ ≡⟨ ≡.cong (_∘ᵢ ∘ᵢ-tc f⁺) (reverse⇒≅-sym f⁺) ⟩ ≅.sym (∘ᵢ-tc f⁺) ∘ᵢ ∘ᵢ-tc f⁺ ≈⟨ ⁻¹-iso.isoˡ ⟩ ≅.refl ∎ isoʳ′ : (f⁺ : A [ _≅_ ]⁺ B) → ∘ᵢ-tc f⁺ ∘ᵢ ∘ᵢ-tc (reverse f⁺) ≈ᵢ ≅.refl isoʳ′ f⁺ = begin ∘ᵢ-tc f⁺ ∘ᵢ ∘ᵢ-tc (reverse f⁺) ≡⟨ ≡.cong (∘ᵢ-tc f⁺ ∘ᵢ_) (reverse⇒≅-sym f⁺) ⟩ ∘ᵢ-tc f⁺ ∘ᵢ ≅.sym (∘ᵢ-tc f⁺) ≈⟨ ⁻¹-iso.isoʳ ⟩ ≅.refl ∎ from-∘ᵢ-tc : (f⁺ : A [ _≅_ ]⁺ B) → from (∘ᵢ-tc f⁺) ≡ ∘-tc (≅⁺⇒⇒⁺ f⁺) from-∘ᵢ-tc [ f ] = ≡.refl from-∘ᵢ-tc (_ ∼⁺⟨ f⁺ ⟩ f⁺′) = ≡.cong₂ _∘_ (from-∘ᵢ-tc f⁺′) (from-∘ᵢ-tc f⁺) ≅*⇒⇒*-cong : ≅⁺⇒⇒⁺ {A} {B} Preserves _≃⁺_ ⟶ _≈⁺_ ≅*⇒⇒*-cong {_} {_} {f⁺} {g⁺} f⁺≃⁺g⁺ = begin ∘-tc (≅⁺⇒⇒⁺ f⁺) ≡˘⟨ from-∘ᵢ-tc f⁺ ⟩ from (∘ᵢ-tc f⁺) ≈⟨ from-≈ f⁺≃⁺g⁺ ⟩ from (∘ᵢ-tc g⁺) ≡⟨ from-∘ᵢ-tc g⁺ ⟩ ∘-tc (≅⁺⇒⇒⁺ g⁺) ∎ where open HomReasoning ≅-shift : ∀ {f⁺ : A [ _≅_ ]⁺ B} {g⁺ : B [ _≅_ ]⁺ C} {h⁺ : A [ _≅_ ]⁺ C} → (_ ∼⁺⟨ f⁺ ⟩ g⁺) ≃⁺ h⁺ → g⁺ ≃⁺ (_ ∼⁺⟨ reverse f⁺ ⟩ h⁺) ≅-shift {f⁺ = f⁺} {g⁺ = g⁺} {h⁺ = h⁺} eq = begin ∘ᵢ-tc g⁺ ≈⟨ introʳ (I.isoʳ f⁺) ⟩ ∘ᵢ-tc g⁺ ∘ᵢ (∘ᵢ-tc f⁺ ∘ᵢ ∘ᵢ-tc (reverse f⁺)) ≈⟨ pullˡ eq ⟩ ∘ᵢ-tc h⁺ ∘ᵢ ∘ᵢ-tc (reverse f⁺) ∎ where open HomReasoningᵢ open MR Core module I {A B} (f⁺ : A [ _≅_ ]⁺ B) = Morphism.Iso (IsGroupoid.iso TransitiveClosure-groupoid {f = f⁺}) lift : ∀ {f⁺ : A [ _⇒_ ]⁺ B} → IsoPlus f⁺ → A [ _≅_ ]⁺ B lift [ iso ] = [ record { from = _ ; to = _ ; iso = iso } ] lift (_ ∼⁺⟨ iso ⟩ iso′) = _ ∼⁺⟨ lift iso ⟩ lift iso′ reduce-lift : ∀ {f⁺ : A [ _⇒_ ]⁺ B} (f′ : IsoPlus f⁺) → from (∘ᵢ-tc (lift f′)) ≡ ∘-tc f⁺ reduce-lift [ f ] = ≡.refl reduce-lift (_ ∼⁺⟨ f′ ⟩ f″) = ≡.cong₂ _∘_ (reduce-lift f″) (reduce-lift f′) lift-cong : ∀ {f⁺ g⁺ : A [ _⇒_ ]⁺ B} (f′ : IsoPlus f⁺) (g′ : IsoPlus g⁺) → f⁺ ≈⁺ g⁺ → lift f′ ≃⁺ lift g′ lift-cong {_} {_} {f⁺} {g⁺} f′ g′ eq = ⌞ from-≈′ ⌟ where open HomReasoning from-≈′ : from (∘ᵢ-tc (lift f′)) ≈ from (∘ᵢ-tc (lift g′)) from-≈′ = begin from (∘ᵢ-tc (lift f′)) ≡⟨ reduce-lift f′ ⟩ ∘-tc f⁺ ≈⟨ eq ⟩ ∘-tc g⁺ ≡˘⟨ reduce-lift g′ ⟩ from (∘ᵢ-tc (lift g′)) ∎ lift-triangle : {f : A ⇒ B} {g : C ⇒ A} {h : C ⇒ B} {k : B ⇒ C} {i : B ⇒ A} {j : A ⇒ C} → f ∘ g ≈ h → (f′ : Iso f i) → (g′ : Iso g j) → (h′ : Iso h k) → lift (_ ∼⁺⟨ [ g′ ] ⟩ [ f′ ]) ≃⁺ lift [ h′ ] lift-triangle eq f′ g′ h′ = lift-cong (_ ∼⁺⟨ [ g′ ] ⟩ [ f′ ]) [ h′ ] eq lift-square : {f : A ⇒ B} {g : C ⇒ A} {h : D ⇒ B} {i : C ⇒ D} {j : D ⇒ C} {a : B ⇒ A} {b : A ⇒ C} {c : B ⇒ D} → f ∘ g ≈ h ∘ i → (f′ : Iso f a) → (g′ : Iso g b) → (h′ : Iso h c) → (i′ : Iso i j) → lift (_ ∼⁺⟨ [ g′ ] ⟩ [ f′ ]) ≃⁺ lift (_ ∼⁺⟨ [ i′ ] ⟩ [ h′ ]) lift-square eq f′ g′ h′ i′ = lift-cong (_ ∼⁺⟨ [ g′ ] ⟩ [ f′ ]) (_ ∼⁺⟨ [ i′ ] ⟩ [ h′ ]) eq lift-pentagon : {f : A ⇒ B} {g : C ⇒ A} {h : D ⇒ C} {i : E ⇒ B} {j : D ⇒ E} {l : E ⇒ D} {a : B ⇒ A} {b : A ⇒ C} {c : C ⇒ D} {d : B ⇒ E} → f ∘ g ∘ h ≈ i ∘ j → (f′ : Iso f a) → (g′ : Iso g b) → (h′ : Iso h c) → (i′ : Iso i d) → (j′ : Iso j l) → lift (_ ∼⁺⟨ _ ∼⁺⟨ [ h′ ] ⟩ [ g′ ] ⟩ [ f′ ]) ≃⁺ lift (_ ∼⁺⟨ [ j′ ] ⟩ [ i′ ]) lift-pentagon eq f′ g′ h′ i′ j′ = lift-cong (_ ∼⁺⟨ _ ∼⁺⟨ [ h′ ] ⟩ [ g′ ] ⟩ [ f′ ]) (_ ∼⁺⟨ [ j′ ] ⟩ [ i′ ]) eq module _ where open _≅_ -- projecting isomorphism commutations to morphism commutations project-triangle : {g : A ≅ B} {f : C ≅ A} {h : C ≅ B} → g ∘ᵢ f ≈ᵢ h → from g ∘ from f ≈ from h project-triangle = from-≈ project-square : {g : A ≅ B} {f : C ≅ A} {i : D ≅ B} {h : C ≅ D} → g ∘ᵢ f ≈ᵢ i ∘ᵢ h → from g ∘ from f ≈ from i ∘ from h project-square = from-≈ -- direct lifting from morphism commutations to isomorphism commutations lift-triangle′ : {f : A ≅ B} {g : C ≅ A} {h : C ≅ B} → from f ∘ from g ≈ from h → f ∘ᵢ g ≈ᵢ h lift-triangle′ = ⌞_⌟ lift-square′ : {f : A ≅ B} {g : C ≅ A} {h : D ≅ B} {i : C ≅ D} → from f ∘ from g ≈ from h ∘ from i → f ∘ᵢ g ≈ᵢ h ∘ᵢ i lift-square′ = ⌞_⌟ lift-pentagon′ : {f : A ≅ B} {g : C ≅ A} {h : D ≅ C} {i : E ≅ B} {j : D ≅ E} → from f ∘ from g ∘ from h ≈ from i ∘ from j → f ∘ᵢ g ∘ᵢ h ≈ᵢ i ∘ᵢ j lift-pentagon′ = ⌞_⌟ open MR Core open HomReasoningᵢ open MR.GroupoidR _ Core-isGroupoid squares×≃⇒≃ : {f f′ : A ≅ B} {g : A ≅ C} {h : B ≅ D} {i i′ : C ≅ D} → CommutativeIso f g h i → CommutativeIso f′ g h i′ → i ≈ᵢ i′ → f ≈ᵢ f′ squares×≃⇒≃ sq₁ sq₂ eq = MCore.isos×≈⇒≈ eq helper₁ (MCore.≅.sym helper₂) sq₁ sq₂ where helper₁ = record { iso = ⁻¹-iso } helper₂ = record { iso = ⁻¹-iso } -- imagine a triangle prism, if all the sides and the top face commute, the bottom face commute. triangle-prism : {i′ : A ≅ B} {f′ : C ≅ A} {h′ : C ≅ B} {i : D ≅ E} {j : D ≅ A} {k : E ≅ B} {f : F ≅ D} {g : F ≅ C} {h : F ≅ E} → i′ ∘ᵢ f′ ≈ᵢ h′ → CommutativeIso i j k i′ → CommutativeIso f g j f′ → CommutativeIso h g k h′ → i ∘ᵢ f ≈ᵢ h triangle-prism {i′ = i′} {f′} {_} {i} {_} {k} {f} {g} {_} eq sq₁ sq₂ sq₃ = squares×≃⇒≃ glued sq₃ eq where glued : CommutativeIso (i ∘ᵢ f) g k (i′ ∘ᵢ f′) glued = ⟺ (glue (⟺ sq₁) (⟺ sq₂)) elim-triangleˡ : {f : A ≅ B} {g : C ≅ A} {h : D ≅ C} {i : D ≅ B} {j : D ≅ A} → f ∘ᵢ g ∘ᵢ h ≈ᵢ i → f ∘ᵢ j ≈ᵢ i → g ∘ᵢ h ≈ᵢ j elim-triangleˡ perim tri = MCore.mono _ _ (perim ○ ⟺ tri) elim-triangleˡ′ : {f : A ≅ B} {g : C ≅ A} {h : D ≅ C} {i : D ≅ B} {j : C ≅ B} → f ∘ᵢ g ∘ᵢ h ≈ᵢ i → j ∘ᵢ h ≈ᵢ i → f ∘ᵢ g ≈ᵢ j elim-triangleˡ′ {f = f} {g} {h} {i} {j} perim tri = MCore.epi _ _ ( begin (f ∘ᵢ g) ∘ᵢ h ≈⟨ ⌞ assoc ⌟ ⟩ f ∘ᵢ g ∘ᵢ h ≈⟨ perim ⟩ i ≈˘⟨ tri ⟩ j ∘ᵢ h ∎ ) cut-squareʳ : {g : A ≅ B} {f : A ≅ C} {h : B ≅ D} {i : C ≅ D} {j : B ≅ C} → CommutativeIso g f h i → i ∘ᵢ j ≈ᵢ h → j ∘ᵢ g ≈ᵢ f cut-squareʳ {g = g} {f = f} {h = h} {i = i} {j = j} sq tri = begin j ∘ᵢ g ≈⟨ switch-fromtoˡ′ {f = i} {h = j} {k = h} tri ⟩∘⟨refl ⟩ (i ⁻¹ ∘ᵢ h) ∘ᵢ g ≈⟨ ⌞ assoc ⌟ ⟩ i ⁻¹ ∘ᵢ h ∘ᵢ g ≈˘⟨ switch-fromtoˡ′ {f = i} {h = f} {k = h ∘ᵢ g} (⟺ sq) ⟩ f ∎

38.352697

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4,419

agda

Agda

src/Partiality-monad/Coinductive/Partial-order.agda

nad/partiality-monad

f69749280969f9093e5e13884c6feb0ad2506eae

[ "MIT" ]

2

2020-05-21T22:59:18.000Z

2020-07-03T08:56:08.000Z

src/Partiality-monad/Coinductive/Partial-order.agda

nad/partiality-monad

f69749280969f9093e5e13884c6feb0ad2506eae

[ "MIT" ]

null

src/Partiality-monad/Coinductive/Partial-order.agda

nad/partiality-monad

f69749280969f9093e5e13884c6feb0ad2506eae

[ "MIT" ]

null

------------------------------------------------------------------------ -- A partial order ------------------------------------------------------------------------ {-# OPTIONS --cubical --sized-types #-} open import Prelude hiding (⊥; module W) module Partiality-monad.Coinductive.Partial-order {a} {A : Type a} where open import Equality.Propositional.Cubical open import Logical-equivalence using (_⇔_) open import Prelude.Size open import Bijection equality-with-J using (_↔_) open import Equality.Path.Isomorphisms.Univalence equality-with-paths open import H-level equality-with-J open import H-level.Closure equality-with-J open import H-level.Truncation.Propositional equality-with-paths as Trunc open import Quotient equality-with-paths as Quotient open import Univalence-axiom equality-with-J open import Delay-monad open import Delay-monad.Bisimilarity as B using (_≈_) import Delay-monad.Partial-order as PO open import Partiality-monad.Coinductive -- An ordering relation. LE : A ⊥ → A ⊥ → Proposition a LE x y = Quotient.rec (λ where .[]ʳ x → LE″ x y .[]-respects-relationʳ → left-lemma″-∥∥ y .is-setʳ → is-set) x where LE′ : Delay A ∞ → Delay A ∞ → Proposition a LE′ x y = ∥ x PO.⊑ y ∥ , truncation-is-proposition abstract is-set : Is-set (∃ λ (A : Type a) → Is-proposition A) is-set = Is-set-∃-Is-proposition ext prop-ext right-lemma : ∀ {x y z} → x ≈ y → LE′ z x ≡ LE′ z y right-lemma x≈y = _↔_.to (⇔↔≡″ ext prop-ext) (record { to = ∥∥-map (flip PO.transitive-⊑≈ x≈y) ; from = ∥∥-map (flip PO.transitive-⊑≈ (B.symmetric x≈y)) }) right-lemma-∥∥ : ∀ {x y z} → ∥ x ≈ y ∥ → LE′ z x ≡ LE′ z y right-lemma-∥∥ = Trunc.rec is-set right-lemma LE″ : Delay A ∞ → A ⊥ → Proposition a LE″ x y = Quotient.rec (λ where .[]ʳ → LE′ x .[]-respects-relationʳ → right-lemma-∥∥ .is-setʳ → is-set) y abstract left-lemma : ∀ {x y z} → x ≈ y → LE′ x z ≡ LE′ y z left-lemma x≈y = _↔_.to (⇔↔≡″ ext prop-ext) (record { to = ∥∥-map (PO.transitive-≈⊑ (B.symmetric x≈y)) ; from = ∥∥-map (PO.transitive-≈⊑ x≈y) }) left-lemma″ : ∀ {x y} z → x ≈ y → LE″ x z ≡ LE″ y z left-lemma″ {x} {y} z x≈y = Quotient.elim-prop {P = λ z → LE″ x z ≡ LE″ y z} (λ where .[]ʳ _ → left-lemma x≈y .is-propositionʳ _ → Is-set-∃-Is-proposition ext prop-ext) z left-lemma″-∥∥ : ∀ {x y} z → ∥ x ≈ y ∥ → LE″ x z ≡ LE″ y z left-lemma″-∥∥ z = Trunc.rec is-set (left-lemma″ z) infix 4 _⊑_ _⊑_ : A ⊥ → A ⊥ → Type a x ⊑ y = proj₁ (LE x y) -- _⊑_ is propositional. ⊑-propositional : ∀ x y → Is-proposition (x ⊑ y) ⊑-propositional x y = proj₂ (LE x y) -- _⊑_ is reflexive. reflexive : ∀ x → x ⊑ x reflexive = Quotient.elim-prop λ where .[]ʳ x → ∣ PO.reflexive x ∣ .is-propositionʳ x → ⊑-propositional [ x ] [ x ] -- _⊑_ is antisymmetric. antisymmetric : ∀ x y → x ⊑ y → y ⊑ x → x ≡ y antisymmetric = Quotient.elim-prop λ where .[]ʳ x → Quotient.elim-prop (λ where .[]ʳ y ∥x⊑y∥ ∥y⊑x∥ → []-respects-relation $ Trunc.rec truncation-is-proposition (λ x⊑y → ∥∥-map (PO.antisymmetric x⊑y) ∥y⊑x∥) ∥x⊑y∥ .is-propositionʳ _ → Π-closure ext 1 λ _ → Π-closure ext 1 λ _ → ⊥-is-set) .is-propositionʳ _ → Π-closure ext 1 λ _ → Π-closure ext 1 λ _ → Π-closure ext 1 λ _ → ⊥-is-set -- _⊑_ is transitive. transitive : ∀ x y z → x ⊑ y → y ⊑ z → x ⊑ z transitive = Quotient.elim-prop λ where .[]ʳ x → Quotient.elim-prop λ where .[]ʳ y → Quotient.elim-prop λ where .[]ʳ z ∥x⊑y∥ → Trunc.rec truncation-is-proposition (λ y⊑z → ∥∥-map (λ x⊑y → PO.transitive x⊑y y⊑z) ∥x⊑y∥) .is-propositionʳ _ → Π-closure ext 1 λ _ → Π-closure ext 1 λ _ → ⊑-propositional [ _ ] [ _ ] .is-propositionʳ _ → Π-closure ext 1 λ z → Π-closure ext 1 λ _ → Π-closure ext 1 λ _ → ⊑-propositional [ _ ] z .is-propositionʳ _ → Π-closure ext 1 λ _ → Π-closure ext 1 λ z → Π-closure ext 1 λ _ → Π-closure ext 1 λ _ → ⊑-propositional [ _ ] z

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agda

Agda

test/Test3.agda

mchristianl/synthetic-reals

10206b5c3eaef99ece5d18bf703c9e8b2371bde4

[ "MIT" ]

3

2020-07-31T18:15:26.000Z

2022-02-19T12:15:21.000Z

test/Test3.agda

mchristianl/synthetic-reals

10206b5c3eaef99ece5d18bf703c9e8b2371bde4

[ "MIT" ]

null

test/Test3.agda

mchristianl/synthetic-reals

10206b5c3eaef99ece5d18bf703c9e8b2371bde4

[ "MIT" ]

null

{-# OPTIONS --cubical --no-import-sorts #-} module Test3 where open import Cubical.Foundations.Everything renaming (_⁻¹ to _⁻¹ᵖ; assoc to ∙-assoc) open import Cubical.Foundations.Logic abstract !_ : ∀{ℓ} {X : Type ℓ} → X → X ! x = x !-≡ : ∀{ℓ} {X : Type ℓ} → (! X) ≡ X !-≡ = refl -- makes use of the definition of `!_` within this block !!_ : ∀{ℓ} {X : Type ℓ} → X → ! X !!_ {X = X} x = transport (sym (!-≡ {X = X})) x !!⁻¹_ : ∀{ℓ} {X : Type ℓ} → ! X → X !!⁻¹_ {X = X} x = transport (!-≡ {X = X}) x infix 1 !_ infix 1 !!_ infix 1 !!⁻¹_ -- !-≡' : ∀{ℓ} {X : Type ℓ} → (! X) ≡ X -- !-≡' = refl -- cannot make use of the definition of `!_` anymore hPropRel : ∀ {ℓ} (A B : Type ℓ) (ℓ' : Level) → Type (ℓ-max ℓ (ℓ-suc ℓ')) hPropRel A B ℓ' = A → B → hProp ℓ' module TestB {ℓ ℓ'} (X : Type ℓ) (0ˣ : X) (_+_ _·_ : X → X → X) (_<_ : hPropRel X X ℓ') (let infixl 5 _+_; _+_ = _+_) where _≤_ : hPropRel X X ℓ' x ≤ y = ¬(y < x) postulate sqrt : (x : X) → {{ ! [ 0ˣ ≤ x ] }} → X 0≤x² : ∀ x → [ 0ˣ ≤ (x · x) ] instance -- module-scope instances _ = λ {x} → !! 0≤x² x test4 : (x y z : X) → [ 0ˣ ≤ x ] → [ 0ˣ ≤ y ] → X test4 x y z 0≤x 0≤y = let instance -- let-scope instances _ = !! 0≤x _ = !! 0≤y _ = !! 0≤x² x -- preferred over the instance from module-scope in ( (sqrt x) -- works + (sqrt y) -- also works + (sqrt (z · z)) -- uses instance from module scope + (sqrt (x · x)) -- uses instance from let-scope (?) -- NOTE: see https://github.com/agda/agda/issues/4688 )

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agda

Agda

Univalence/FiniteType.agda

JacquesCarette/pi-dual

003835484facfde0b770bc2b3d781b42b76184c1

[ "BSD-2-Clause" ]

14

2015-08-18T21:40:15.000Z

2021-05-05T01:07:57.000Z

Univalence/FiniteType.agda

JacquesCarette/pi-dual

003835484facfde0b770bc2b3d781b42b76184c1

[ "BSD-2-Clause" ]

4

2018-06-07T16:27:41.000Z

2021-10-29T20:41:23.000Z

Univalence/FiniteType.agda

JacquesCarette/pi-dual

003835484facfde0b770bc2b3d781b42b76184c1

[ "BSD-2-Clause" ]

3

2016-05-29T01:56:33.000Z

2019-09-10T09:47:13.000Z

{-# OPTIONS --without-K #-} module FiniteType where open import Equiv using (_≃_) open import Data.Product using (Σ; _,_) open import Data.Nat using (ℕ) open import Data.Fin using (Fin) -------------------------------------------------------------------------- -- -- A finite type is a type which is equivalent to Fin n -- FiniteType : ∀ {ℓ} → (A : Set ℓ) → (n : ℕ) → Set ℓ FiniteType A n = A ≃ Fin n --------------------------------------------------------------------------

25.368421

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0.446058

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agda

Agda

test/Fail/Issue2467.agda

mdimjasevic/agda

8fb548356b275c7a1e79b768b64511ae937c738b

[ "BSD-3-Clause" ]

1,989

2015-01-09T23:51:16.000Z

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test/Fail/Issue2467.agda

Seanpm2001-languages/agda

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test/Fail/Issue2467.agda

Seanpm2001-languages/agda

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2015-01-03T14:04:08.000Z

2022-03-30T19:00:30.000Z

-- Andreas, 2017-02-20, issue #2467 -- Proper error on missing BUILTIN REWRITE {-# OPTIONS --rewriting #-} postulate A : Set {-# REWRITE A #-} -- Should fail with error like

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Agda

src/Generic/Lib/Data/Sets.agda

iblech/Generic

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src/Generic/Lib/Data/Sets.agda

iblech/Generic

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2017-04-06T18:58:09.000Z

2022-01-04T15:43:14.000Z

src/Generic/Lib/Data/Sets.agda

iblech/Generic

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[ "MIT" ]

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2017-07-17T07:23:39.000Z

2021-01-27T12:57:09.000Z

module Generic.Lib.Data.Sets where open import Generic.Lib.Intro open import Generic.Lib.Data.Nat open import Generic.Lib.Data.Product open import Generic.Lib.Data.Pow infixl 6 _⊔ⁿ_ _⊔ⁿ_ : ∀ {n} -> Level ^ n -> Level -> Level _⊔ⁿ_ = flip $ foldPow _ _⊔_ Sets : ∀ {n} -> (αs : Level ^ n) -> Set (mapPow lsuc αs ⊔ⁿ lzero) Sets {0} _ = ⊤ Sets {suc _} (α , αs) = Set α × Sets αs FoldSets : ∀ {n β αs} -> Sets {n} αs -> Set β -> Set (αs ⊔ⁿ β) FoldSets {0} tt B = B FoldSets {suc _} (A , As) B = A -> FoldSets As B HList : ∀ {n} {αs} -> Sets {n} αs -> Set (αs ⊔ⁿ lzero) HList {0} tt = ⊤ HList {suc _} (A , As) = A × HList As applyFoldSets : ∀ {n β αs} {As : Sets {n} αs} {B : Set β} -> FoldSets As B -> HList As -> B applyFoldSets {0} y tt = y applyFoldSets {suc n} f (x , xs) = applyFoldSets (f x) xs

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agda-stdlib/src/Codata/Stream/Bisimilarity.agda

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agda-stdlib/src/Codata/Stream/Bisimilarity.agda

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agda-stdlib/src/Codata/Stream/Bisimilarity.agda

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------------------------------------------------------------------------ -- The Agda standard library -- -- Bisimilarity for Streams ------------------------------------------------------------------------ {-# OPTIONS --without-K --safe --sized-types #-} module Codata.Stream.Bisimilarity where open import Size open import Codata.Thunk open import Codata.Stream open import Level open import Data.List.NonEmpty as List⁺ using (_∷_) open import Data.List.Relation.Binary.Pointwise using (Pointwise; []; _∷_) open import Relation.Binary open import Relation.Binary.PropositionalEquality as Eq using (_≡_) private variable a b c p q r : Level A : Set a B : Set b C : Set c i : Size data Bisim {A : Set a} {B : Set b} (R : REL A B r) i : REL (Stream A ∞) (Stream B ∞) (a ⊔ b ⊔ r) where _∷_ : ∀ {x y xs ys} → R x y → Thunk^R (Bisim R) i xs ys → Bisim R i (x ∷ xs) (y ∷ ys) module _ {R : Rel A r} where reflexive : Reflexive R → Reflexive (Bisim R i) reflexive refl^R {r ∷ rs} = refl^R ∷ λ where .force → reflexive refl^R module _ {P : REL A B p} {Q : REL B A q} where symmetric : Sym P Q → Sym (Bisim P i) (Bisim Q i) symmetric sym^PQ (p ∷ ps) = sym^PQ p ∷ λ where .force → symmetric sym^PQ (ps .force) module _ {P : REL A B p} {Q : REL B C q} {R : REL A C r} where transitive : Trans P Q R → Trans (Bisim P i) (Bisim Q i) (Bisim R i) transitive trans^PQR (p ∷ ps) (q ∷ qs) = trans^PQR p q ∷ λ where .force → transitive trans^PQR (ps .force) (qs .force) isEquivalence : {R : Rel A r} → IsEquivalence R → IsEquivalence (Bisim R i) isEquivalence equiv^R = record { refl = reflexive equiv^R.refl ; sym = symmetric equiv^R.sym ; trans = transitive equiv^R.trans } where module equiv^R = IsEquivalence equiv^R setoid : Setoid a r → Size → Setoid a (a ⊔ r) setoid S i = record { isEquivalence = isEquivalence {i = i} (Setoid.isEquivalence S) } module _ {R : REL A B r} where ++⁺ : ∀ {as bs xs ys} → Pointwise R as bs → Bisim R i xs ys → Bisim R i (as ++ xs) (bs ++ ys) ++⁺ [] rs = rs ++⁺ (r ∷ pw) rs = r ∷ λ where .force → ++⁺ pw rs ⁺++⁺ : ∀ {as bs xs ys} → Pointwise R (List⁺.toList as) (List⁺.toList bs) → Thunk^R (Bisim R) i xs ys → Bisim R i (as ⁺++ xs) (bs ⁺++ ys) ⁺++⁺ (r ∷ pw) rs = r ∷ λ where .force → ++⁺ pw (rs .force) ------------------------------------------------------------------------ -- Pointwise Equality as a Bisimilarity module _ {A : Set a} where infix 1 _⊢_≈_ _⊢_≈_ : ∀ i → Stream A ∞ → Stream A ∞ → Set a _⊢_≈_ = Bisim _≡_ refl : ∀ {i} → Reflexive (i ⊢_≈_) refl = reflexive Eq.refl sym : ∀ {i} → Symmetric (i ⊢_≈_) sym = symmetric Eq.sym trans : ∀ {i} → Transitive (i ⊢_≈_) trans = transitive Eq.trans module ≈-Reasoning {a} {A : Set a} {i} where open import Relation.Binary.Reasoning.Setoid (setoid (Eq.setoid A) i) public

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Cubical/HITs/AssocList/Properties.agda

FernandoLarrain/cubical

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[ "MIT" ]

301

2018-10-17T18:00:24.000Z

2022-03-24T02:10:47.000Z

Cubical/HITs/AssocList/Properties.agda

FernandoLarrain/cubical

9acdecfa6437ec455568be4e5ff04849cc2bc13b

[ "MIT" ]

584

2018-10-15T09:49:02.000Z

2022-03-30T12:09:17.000Z

Cubical/HITs/AssocList/Properties.agda

FernandoLarrain/cubical

9acdecfa6437ec455568be4e5ff04849cc2bc13b

[ "MIT" ]

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2018-11-16T06:11:03.000Z

2022-03-23T16:22:13.000Z

{-# OPTIONS --safe #-} module Cubical.HITs.AssocList.Properties where open import Cubical.HITs.AssocList.Base as AL open import Cubical.Foundations.Everything open import Cubical.Foundations.SIP open import Cubical.HITs.FiniteMultiset as FMS open import Cubical.Data.Nat using (ℕ; zero; suc; _+_; +-assoc; isSetℕ) open import Cubical.Structures.MultiSet open import Cubical.Relation.Nullary open import Cubical.Relation.Nullary.DecidableEq private variable ℓ : Level A : Type ℓ multiPer : (a b : A) (m n : ℕ) (xs : AssocList A) → ⟨ a , m ⟩∷ ⟨ b , n ⟩∷ xs ≡ ⟨ b , n ⟩∷ ⟨ a , m ⟩∷ xs multiPer a b zero n xs = del a (⟨ b , n ⟩∷ xs) ∙ cong (λ ys → ⟨ b , n ⟩∷ ys) (sym (del a xs)) multiPer a b (suc m) zero xs = cong (λ ys → ⟨ a , suc m ⟩∷ ys) (del b xs) ∙ sym (del b (⟨ a , suc m ⟩∷ xs)) multiPer a b (suc m) (suc n) xs = ⟨ a , suc m ⟩∷ ⟨ b , suc n ⟩∷ xs ≡⟨ sym (agg a 1 m (⟨ b , suc n ⟩∷ xs)) ⟩ ⟨ a , 1 ⟩∷ ⟨ a , m ⟩∷ ⟨ b , suc n ⟩∷ xs ≡⟨ cong (λ ys → ⟨ a , 1 ⟩∷ ys) (multiPer a b m (suc n) xs) ⟩ ⟨ a , 1 ⟩∷ ⟨ b , suc n ⟩∷ ⟨ a , m ⟩∷ xs ≡⟨ cong (λ ys → ⟨ a , 1 ⟩∷ ys) (sym (agg b 1 n (⟨ a , m ⟩∷ xs))) ⟩ ⟨ a , 1 ⟩∷ ⟨ b , 1 ⟩∷ ⟨ b , n ⟩∷ ⟨ a , m ⟩∷ xs ≡⟨ per a b (⟨ b , n ⟩∷ ⟨ a , m ⟩∷ xs) ⟩ ⟨ b , 1 ⟩∷ ⟨ a , 1 ⟩∷ ⟨ b , n ⟩∷ ⟨ a , m ⟩∷ xs ≡⟨ cong (λ ys → ⟨ b , 1 ⟩∷ ⟨ a , 1 ⟩∷ ys) (multiPer b a n m xs) ⟩ ⟨ b , 1 ⟩∷ ⟨ a , 1 ⟩∷ ⟨ a , m ⟩∷ ⟨ b , n ⟩∷ xs ≡⟨ cong (λ ys → ⟨ b , 1 ⟩∷ ys) (agg a 1 m (⟨ b , n ⟩∷ xs)) ⟩ ⟨ b , 1 ⟩∷ ⟨ a , suc m ⟩∷ ⟨ b , n ⟩∷ xs ≡⟨ cong (λ ys → ⟨ b , 1 ⟩∷ ys) (multiPer a b (suc m) n xs) ⟩ ⟨ b , 1 ⟩∷ ⟨ b , n ⟩∷ ⟨ a , suc m ⟩∷ xs ≡⟨ agg b 1 n (⟨ a , suc m ⟩∷ xs) ⟩ ⟨ b , suc n ⟩∷ ⟨ a , suc m ⟩∷ xs ∎ -- Show that association lists and finite multisets are equivalent multi-∷ : A → ℕ → FMSet A → FMSet A multi-∷ x zero xs = xs multi-∷ x (suc n) xs = x ∷ multi-∷ x n xs multi-∷-agg : (x : A) (m n : ℕ) (b : FMSet A) → multi-∷ x m (multi-∷ x n b) ≡ multi-∷ x (m + n) b multi-∷-agg x zero n b = refl multi-∷-agg x (suc m) n b i = x ∷ (multi-∷-agg x m n b i) AL→FMS : AssocList A → FMSet A AL→FMS = AL.Rec.f FMS.trunc [] multi-∷ comm multi-∷-agg λ _ _ → refl FMS→AL : FMSet A → AssocList A FMS→AL = FMS.Rec.f AL.trunc ⟨⟩ (λ x xs → ⟨ x , 1 ⟩∷ xs) per AL→FMS∘FMS→AL≡id : section {A = AssocList A} AL→FMS FMS→AL AL→FMS∘FMS→AL≡id = FMS.ElimProp.f (FMS.trunc _ _) refl (λ x p → cong (λ ys → x ∷ ys) p) -- need a little lemma for other direction multi-∷-id : (x : A) (n : ℕ) (u : FMSet A) → FMS→AL (multi-∷ x n u) ≡ ⟨ x , n ⟩∷ FMS→AL u multi-∷-id x zero u = sym (del x (FMS→AL u)) multi-∷-id x (suc n) u = FMS→AL (multi-∷ x (suc n) u) ≡⟨ cong (λ ys → ⟨ x , 1 ⟩∷ ys) (multi-∷-id x n u) ⟩ ⟨ x , 1 ⟩∷ ⟨ x , n ⟩∷ (FMS→AL u) ≡⟨ agg x 1 n (FMS→AL u) ⟩ ⟨ x , (suc n) ⟩∷ (FMS→AL u) ∎ FMS→AL∘AL→FMS≡id : retract {A = AssocList A} AL→FMS FMS→AL FMS→AL∘AL→FMS≡id = AL.ElimProp.f (AL.trunc _ _) refl (λ x n {xs} p → (multi-∷-id x n (AL→FMS xs)) ∙ cong (λ ys → ⟨ x , n ⟩∷ ys) p) AssocList≃FMSet : AssocList A ≃ FMSet A AssocList≃FMSet = isoToEquiv (iso AL→FMS FMS→AL AL→FMS∘FMS→AL≡id FMS→AL∘AL→FMS≡id) FMSet≃AssocList : FMSet A ≃ AssocList A FMSet≃AssocList = isoToEquiv (iso FMS→AL AL→FMS FMS→AL∘AL→FMS≡id AL→FMS∘FMS→AL≡id) AssocList≡FMSet : AssocList A ≡ FMSet A AssocList≡FMSet = ua AssocList≃FMSet -- We want to define a multiset structure on AssocList A, we use the recursor to define the count-function module _(discA : Discrete A) where setA = Discrete→isSet discA ALcount-⟨,⟩∷* : A → A → ℕ → ℕ → ℕ ALcount-⟨,⟩∷* a x n xs with discA a x ... | yes a≡x = n + xs ... | no a≢x = xs ALcount-per* : (a x y : A) (xs : ℕ) → ALcount-⟨,⟩∷* a x 1 (ALcount-⟨,⟩∷* a y 1 xs) ≡ ALcount-⟨,⟩∷* a y 1 (ALcount-⟨,⟩∷* a x 1 xs) ALcount-per* a x y xs with discA a x | discA a y ALcount-per* a x y xs | yes a≡x | yes a≡y = refl ALcount-per* a x y xs | yes a≡x | no a≢y = refl ALcount-per* a x y xs | no a≢x | yes a≡y = refl ALcount-per* a x y xs | no a≢x | no a≢y = refl ALcount-agg* : (a x : A) (m n xs : ℕ) → ALcount-⟨,⟩∷* a x m (ALcount-⟨,⟩∷* a x n xs) ≡ ALcount-⟨,⟩∷* a x (m + n) xs ALcount-agg* a x m n xs with discA a x ... | yes _ = +-assoc m n xs ... | no _ = refl ALcount-del* : (a x : A) (xs : ℕ) → ALcount-⟨,⟩∷* a x 0 xs ≡ xs ALcount-del* a x xs with discA a x ... | yes _ = refl ... | no _ = refl ALcount : A → AssocList A → ℕ ALcount a = AL.Rec.f isSetℕ 0 (ALcount-⟨,⟩∷* a) (ALcount-per* a) (ALcount-agg* a) (ALcount-del* a) AL-with-str : MultiSet A setA AL-with-str = (AssocList A , ⟨⟩ , ⟨_, 1 ⟩∷_ , ALcount) -- We want to show that Al-with-str ≅ FMS-with-str as multiset-structures FMS→AL-EquivStr : MultiSetEquivStr A setA (FMS-with-str discA) (AL-with-str) FMSet≃AssocList FMS→AL-EquivStr = refl , (λ a xs → refl) , φ where φ : ∀ a xs → FMScount discA a xs ≡ ALcount a (FMS→AL xs) φ a = FMS.ElimProp.f (isSetℕ _ _) refl ψ where ψ : (x : A) {xs : FMSet A} → FMScount discA a xs ≡ ALcount a (FMS→AL xs) → FMScount discA a (x ∷ xs) ≡ ALcount a (FMS→AL (x ∷ xs)) ψ x {xs} p = subst B α θ where B = λ ys → FMScount discA a (x ∷ xs) ≡ ALcount a ys α : ⟨ x , 1 ⟩∷ FMS→AL xs ≡ FMS→AL (x ∷ xs) α = sym (multi-∷-id x 1 xs) θ : FMScount discA a (x ∷ xs) ≡ ALcount a (⟨ x , 1 ⟩∷ (FMS→AL xs)) θ with discA a x ... | yes _ = cong suc p ... | no ¬p = p FMS-with-str≡AL-with-str : FMS-with-str discA ≡ AL-with-str FMS-with-str≡AL-with-str = sip (multiSetUnivalentStr A setA) (FMS-with-str discA) AL-with-str (FMSet≃AssocList , FMS→AL-EquivStr)

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test/Succeed/Issue759.agda

mdimjasevic/agda

8fb548356b275c7a1e79b768b64511ae937c738b

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test/Succeed/Issue759.agda

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test/Succeed/Issue759.agda

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2022-03-30T19:00:30.000Z

-- Andreas, 2012-11-18: abstract record values module Issue759 where import Common.Level abstract record Wrap (A : Set) : Set where field wrapped : A open Wrap public wrap : {A : Set} → A → Wrap A wrap a = record { wrapped = a } -- caused 'Not in Scope: recCon-NOT-PRINTED' -- during eta-contraction in serialization -- should work now

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test/interaction/IntroSharp.agda

asr/agda-kanso

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2019-11-27T04:41:05.000Z

test/interaction/IntroSharp.agda

np/agda-git-experiment

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test/interaction/IntroSharp.agda

np/agda-git-experiment

20596e9dd9867166a64470dd24ea68925ff380ce

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-- The "intro" command manages to refine goals of type ∞ A with the -- term ♯ ?. {-# OPTIONS --universe-polymorphism #-} module IntroSharp where postulate Level : Set zero : Level suc : (i : Level) → Level _⊔_ : Level -> Level -> Level {-# BUILTIN LEVEL Level #-} {-# BUILTIN LEVELZERO zero #-} {-# BUILTIN LEVELSUC suc #-} {-# BUILTIN LEVELMAX _⊔_ #-} postulate ∞ : ∀ {a} (A : Set a) → Set a ♯_ : ∀ {a} {A : Set a} → A → ∞ A ♭ : ∀ {a} {A : Set a} → ∞ A → A {-# BUILTIN INFINITY ∞ #-} {-# BUILTIN SHARP ♯_ #-} {-# BUILTIN FLAT ♭ #-} Foo : ∞ Set Foo = ?

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test/Succeed/Issue2248.agda

Blaisorblade/Agda

802a28aa8374f15fe9d011ceb80317fdb1ec0949

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test/Succeed/Issue2248.agda

Blaisorblade/Agda

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[ "BSD-3-Clause" ]

null

test/Succeed/Issue2248.agda

Blaisorblade/Agda

802a28aa8374f15fe9d011ceb80317fdb1ec0949

[ "BSD-3-Clause" ]

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-- Andreas, 2016-10-11, AIM XXIV -- COMPILED pragma accidentially also accepted for abstract definitions -- Ulf, 2017-02-22: We now allow COMPILE pragmas on functions, and abstract -- functions should not be an exception. The original problem, however, was -- that we expected an unused argument-version of the function to be available. -- This is not the case for COMPILE'd functions. This problem has now been -- fixed. open import Common.String data Unit : Set where unit : Unit {-# COMPILE GHC Unit = data () (()) #-} postulate IO : Set → Set doNothing : IO Unit doSomething : String → IO Unit {-# COMPILE GHC IO = type IO #-} {-# BUILTIN IO IO #-} {-# COMPILE GHC doNothing = return () #-} {-# FOREIGN GHC import qualified Data.Text.IO #-} abstract putStrLn : Unit → String → IO Unit putStrLn _ = doSomething {-# COMPILE GHC putStrLn = \ _ -> Data.Text.IO.putStrLn #-} main = putStrLn unit "Hello, world!" -- WAS: compiler produced ill-formed Haskell-code -- NOW: Error on COMPILE GHC pragma

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Cubical/HITs/FiniteMultiset.agda

FernandoLarrain/cubical

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301

2018-10-17T18:00:24.000Z

2022-03-24T02:10:47.000Z

Cubical/HITs/FiniteMultiset.agda

FernandoLarrain/cubical

9acdecfa6437ec455568be4e5ff04849cc2bc13b

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2018-10-15T09:49:02.000Z

2022-03-30T12:09:17.000Z

Cubical/HITs/FiniteMultiset.agda

FernandoLarrain/cubical

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2018-11-16T06:11:03.000Z

2022-03-23T16:22:13.000Z

{-# OPTIONS --safe #-} module Cubical.HITs.FiniteMultiset where open import Cubical.HITs.FiniteMultiset.Base public open import Cubical.HITs.FiniteMultiset.Properties public

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agda-stdlib/README/Text/Tabular.agda

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agda-stdlib/README/Text/Tabular.agda

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agda-stdlib/README/Text/Tabular.agda

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2021-11-04T06:54:45.000Z

------------------------------------------------------------------------ -- The Agda standard library -- -- Examples of printing list and vec-based tables ------------------------------------------------------------------------ {-# OPTIONS --safe --without-K #-} module README.Text.Tabular where open import Function.Base open import Relation.Binary.PropositionalEquality open import Data.List.Base open import Data.String.Base open import Data.Vec.Base open import Text.Tabular.Base import Text.Tabular.List as Tabularˡ import Text.Tabular.Vec as Tabularᵛ ------------------------------------------------------------------------ -- VEC -- -- If you have a matrix of strings, you simply need to: -- * pick a configuration (see below) -- * pick an alignment for each column -- * pass the matrix -- -- The display function will then pad each string on the left, right, -- or both to respect the alignment constraints. -- It will return a list of strings corresponding to each line in the -- table. You may then: --- * use Data.String.Base's unlines to produce a String -- * use Text.Pretty's text and vcat to produce a Doc (i.e. indentable!) ------------------------------------------------------------------------ _ : unlines (Tabularᵛ.display unicode (Right ∷ Left ∷ Center ∷ []) ( ("foo" ∷ "bar" ∷ "baz" ∷ []) ∷ ("1" ∷ "2" ∷ "3" ∷ []) ∷ ("6" ∷ "5" ∷ "4" ∷ []) ∷ [])) ≡ "┌───┬───┬───┐ \ \│foo│bar│baz│ \ \├───┼───┼───┤ \ \│ 1│2 │ 3 │ \ \├───┼───┼───┤ \ \│ 6│5 │ 4 │ \ \└───┴───┴───┘" _ = refl ------------------------------------------------------------------------ -- CONFIG -- -- Configurations allow you to change the way the table is displayed. ------------------------------------------------------------------------ -- We will use the same example throughout foobar : Vec (Vec String 2) 3 foobar = ("foo" ∷ "bar" ∷ []) ∷ ("1" ∷ "2" ∷ []) ∷ ("4" ∷ "3" ∷ []) ∷ [] ------------------------------------------------------------------------ -- Basic configurations: unicode, ascii, whitespace -- unicode _ : unlines (Tabularᵛ.display unicode (Right ∷ Left ∷ []) foobar) ≡ "┌───┬───┐ \ \│foo│bar│ \ \├───┼───┤ \ \│ 1│2 │ \ \├───┼───┤ \ \│ 4│3 │ \ \└───┴───┘" _ = refl -- ascii _ : unlines (Tabularᵛ.display ascii (Right ∷ Left ∷ []) foobar) ≡ "+-------+ \ \|foo|bar| \ \|---+---| \ \| 1|2 | \ \|---+---| \ \| 4|3 | \ \+-------+" _ = refl -- whitespace _ : unlines (Tabularᵛ.display whitespace (Right ∷ Left ∷ []) foobar) ≡ "foo bar \ \ 1 2 \ \ 4 3 " _ = refl ------------------------------------------------------------------------ -- Modifiers: altering existing configurations -- In these examples we will be using unicode as the base configuration. -- However these modifiers apply to all configurations (and can even be -- combined) -- compact: drop the horizontal line between each row _ : unlines (Tabularᵛ.display (compact unicode) (Right ∷ Left ∷ []) foobar) ≡ "┌───┬───┐ \ \│foo│bar│ \ \│ 1│2 │ \ \│ 4│3 │ \ \└───┴───┘" _ = refl -- noBorder: drop the outside borders _ : unlines (Tabularᵛ.display (noBorder unicode) (Right ∷ Left ∷ []) foobar) ≡ "foo│bar \ \───┼─── \ \ 1│2 \ \───┼─── \ \ 4│3 " _ = refl -- addSpace : add whitespace space inside cells _ : unlines (Tabularᵛ.display (addSpace unicode) (Right ∷ Left ∷ []) foobar) ≡ "┌─────┬─────┐ \ \│ foo │ bar │ \ \├─────┼─────┤ \ \│ 1 │ 2 │ \ \├─────┼─────┤ \ \│ 4 │ 3 │ \ \└─────┴─────┘" _ = refl -- compact together with addSpace _ : unlines (Tabularᵛ.display (compact (addSpace unicode)) (Right ∷ Left ∷ []) foobar) ≡ "┌─────┬─────┐ \ \│ foo │ bar │ \ \│ 1 │ 2 │ \ \│ 4 │ 3 │ \ \└─────┴─────┘" _ = refl ------------------------------------------------------------------------ -- LIST -- -- Same thing as for vectors except that if the list of lists is not -- rectangular, it is padded with empty strings to make it so. If there -- are not enough alignment directives, we arbitrarily pick Left. ------------------------------------------------------------------------ _ : unlines (Tabularˡ.display unicode (Center ∷ Right ∷ []) ( ("foo" ∷ "bar" ∷ []) ∷ ("partial" ∷ "rows" ∷ "are" ∷ "ok" ∷ []) ∷ ("3" ∷ "2" ∷ "1" ∷ "..." ∷ "surprise!" ∷ []) ∷ [])) ≡ "┌───────┬────┬───┬───┬─────────┐ \ \│ foo │ bar│ │ │ │ \ \├───────┼────┼───┼───┼─────────┤ \ \│partial│rows│are│ok │ │ \ \├───────┼────┼───┼───┼─────────┤ \ \│ 3 │ 2│1 │...│surprise!│ \ \└───────┴────┴───┴───┴─────────┘" _ = refl ------------------------------------------------------------------------ -- LIST (UNSAFE) -- -- If you know *for sure* that your data is already perfectly rectangular -- i.e. all the rows of the list of lists have the same length -- in each column, all the strings have the same width -- then you can use the unsafeDisplay function defined Text.Tabular.Base. -- -- This is what gets used internally by `Text.Tabular.Vec` and -- `Text.Tabular.List` once the potentially unsafe data has been -- processed. ------------------------------------------------------------------------ _ : unlines (unsafeDisplay (compact unicode) ( ("foo" ∷ "bar" ∷ []) ∷ (" 1" ∷ " 2" ∷ []) ∷ (" 4" ∷ " 3" ∷ []) ∷ [])) ≡ "┌───┬───┐ \ \│foo│bar│ \ \│ 1│ 2│ \ \│ 4│ 3│ \ \└───┴───┘" _ = refl

26.746479

74

0.415833

659af290ca653a205aba9307ac867e1586dbff20

4,121

agda

Agda

out/CommRing/Syntax.agda

JoeyEremondi/agda-soas

ff1a985a6be9b780d3ba2beff68e902394f0a9d8

[ "MIT" ]

39

2021-11-09T20:39:55.000Z

2022-03-19T17:33:12.000Z

out/CommRing/Syntax.agda

JoeyEremondi/agda-soas

ff1a985a6be9b780d3ba2beff68e902394f0a9d8

[ "MIT" ]

1

2021-11-21T12:19:32.000Z

out/CommRing/Syntax.agda

JoeyEremondi/agda-soas

ff1a985a6be9b780d3ba2beff68e902394f0a9d8

[ "MIT" ]

4

2021-11-09T20:39:59.000Z

2022-01-24T12:49:17.000Z

{- This second-order term syntax was created from the following second-order syntax description: syntax CommRing | CR type * : 0-ary term zero : * | 𝟘 add : * * -> * | _⊕_ l20 one : * | 𝟙 mult : * * -> * | _⊗_ l30 neg : * -> * | ⊖_ r50 theory (𝟘U⊕ᴸ) a |> add (zero, a) = a (𝟘U⊕ᴿ) a |> add (a, zero) = a (⊕A) a b c |> add (add(a, b), c) = add (a, add(b, c)) (⊕C) a b |> add(a, b) = add(b, a) (𝟙U⊗ᴸ) a |> mult (one, a) = a (𝟙U⊗ᴿ) a |> mult (a, one) = a (⊗A) a b c |> mult (mult(a, b), c) = mult (a, mult(b, c)) (⊗D⊕ᴸ) a b c |> mult (a, add (b, c)) = add (mult(a, b), mult(a, c)) (⊗D⊕ᴿ) a b c |> mult (add (a, b), c) = add (mult(a, c), mult(b, c)) (𝟘X⊗ᴸ) a |> mult (zero, a) = zero (𝟘X⊗ᴿ) a |> mult (a, zero) = zero (⊖N⊕ᴸ) a |> add (neg (a), a) = zero (⊖N⊕ᴿ) a |> add (a, neg (a)) = zero (⊗C) a b |> mult(a, b) = mult(b, a) -} module CommRing.Syntax where open import SOAS.Common open import SOAS.Context open import SOAS.Variable open import SOAS.Families.Core open import SOAS.Construction.Structure open import SOAS.ContextMaps.Inductive open import SOAS.Metatheory.Syntax open import CommRing.Signature private variable Γ Δ Π : Ctx α : *T 𝔛 : Familyₛ -- Inductive term declaration module CR:Terms (𝔛 : Familyₛ) where data CR : Familyₛ where var : ℐ ⇾̣ CR mvar : 𝔛 α Π → Sub CR Π Γ → CR α Γ 𝟘 : CR * Γ _⊕_ : CR * Γ → CR * Γ → CR * Γ 𝟙 : CR * Γ _⊗_ : CR * Γ → CR * Γ → CR * Γ ⊖_ : CR * Γ → CR * Γ infixl 20 _⊕_ infixl 30 _⊗_ infixr 50 ⊖_ open import SOAS.Metatheory.MetaAlgebra ⅀F 𝔛 CRᵃ : MetaAlg CR CRᵃ = record { 𝑎𝑙𝑔 = λ where (zeroₒ ⋮ _) → 𝟘 (addₒ ⋮ a , b) → _⊕_ a b (oneₒ ⋮ _) → 𝟙 (multₒ ⋮ a , b) → _⊗_ a b (negₒ ⋮ a) → ⊖_ a ; 𝑣𝑎𝑟 = var ; 𝑚𝑣𝑎𝑟 = λ 𝔪 mε → mvar 𝔪 (tabulate mε) } module CRᵃ = MetaAlg CRᵃ module _ {𝒜 : Familyₛ}(𝒜ᵃ : MetaAlg 𝒜) where open MetaAlg 𝒜ᵃ 𝕤𝕖𝕞 : CR ⇾̣ 𝒜 𝕊 : Sub CR Π Γ → Π ~[ 𝒜 ]↝ Γ 𝕊 (t ◂ σ) new = 𝕤𝕖𝕞 t 𝕊 (t ◂ σ) (old v) = 𝕊 σ v 𝕤𝕖𝕞 (mvar 𝔪 mε) = 𝑚𝑣𝑎𝑟 𝔪 (𝕊 mε) 𝕤𝕖𝕞 (var v) = 𝑣𝑎𝑟 v 𝕤𝕖𝕞 𝟘 = 𝑎𝑙𝑔 (zeroₒ ⋮ tt) 𝕤𝕖𝕞 (_⊕_ a b) = 𝑎𝑙𝑔 (addₒ ⋮ 𝕤𝕖𝕞 a , 𝕤𝕖𝕞 b) 𝕤𝕖𝕞 𝟙 = 𝑎𝑙𝑔 (oneₒ ⋮ tt) 𝕤𝕖𝕞 (_⊗_ a b) = 𝑎𝑙𝑔 (multₒ ⋮ 𝕤𝕖𝕞 a , 𝕤𝕖𝕞 b) 𝕤𝕖𝕞 (⊖_ a) = 𝑎𝑙𝑔 (negₒ ⋮ 𝕤𝕖𝕞 a) 𝕤𝕖𝕞ᵃ⇒ : MetaAlg⇒ CRᵃ 𝒜ᵃ 𝕤𝕖𝕞 𝕤𝕖𝕞ᵃ⇒ = record { ⟨𝑎𝑙𝑔⟩ = λ{ {t = t} → ⟨𝑎𝑙𝑔⟩ t } ; ⟨𝑣𝑎𝑟⟩ = refl ; ⟨𝑚𝑣𝑎𝑟⟩ = λ{ {𝔪 = 𝔪}{mε} → cong (𝑚𝑣𝑎𝑟 𝔪) (dext (𝕊-tab mε)) } } where open ≡-Reasoning ⟨𝑎𝑙𝑔⟩ : (t : ⅀ CR α Γ) → 𝕤𝕖𝕞 (CRᵃ.𝑎𝑙𝑔 t) ≡ 𝑎𝑙𝑔 (⅀₁ 𝕤𝕖𝕞 t) ⟨𝑎𝑙𝑔⟩ (zeroₒ ⋮ _) = refl ⟨𝑎𝑙𝑔⟩ (addₒ ⋮ _) = refl ⟨𝑎𝑙𝑔⟩ (oneₒ ⋮ _) = refl ⟨𝑎𝑙𝑔⟩ (multₒ ⋮ _) = refl ⟨𝑎𝑙𝑔⟩ (negₒ ⋮ _) = refl 𝕊-tab : (mε : Π ~[ CR ]↝ Γ)(v : ℐ α Π) → 𝕊 (tabulate mε) v ≡ 𝕤𝕖𝕞 (mε v) 𝕊-tab mε new = refl 𝕊-tab mε (old v) = 𝕊-tab (mε ∘ old) v module _ (g : CR ⇾̣ 𝒜)(gᵃ⇒ : MetaAlg⇒ CRᵃ 𝒜ᵃ g) where open MetaAlg⇒ gᵃ⇒ 𝕤𝕖𝕞! : (t : CR α Γ) → 𝕤𝕖𝕞 t ≡ g t 𝕊-ix : (mε : Sub CR Π Γ)(v : ℐ α Π) → 𝕊 mε v ≡ g (index mε v) 𝕊-ix (x ◂ mε) new = 𝕤𝕖𝕞! x 𝕊-ix (x ◂ mε) (old v) = 𝕊-ix mε v 𝕤𝕖𝕞! (mvar 𝔪 mε) rewrite cong (𝑚𝑣𝑎𝑟 𝔪) (dext (𝕊-ix mε)) = trans (sym ⟨𝑚𝑣𝑎𝑟⟩) (cong (g ∘ mvar 𝔪) (tab∘ix≈id mε)) 𝕤𝕖𝕞! (var v) = sym ⟨𝑣𝑎𝑟⟩ 𝕤𝕖𝕞! 𝟘 = sym ⟨𝑎𝑙𝑔⟩ 𝕤𝕖𝕞! (_⊕_ a b) rewrite 𝕤𝕖𝕞! a | 𝕤𝕖𝕞! b = sym ⟨𝑎𝑙𝑔⟩ 𝕤𝕖𝕞! 𝟙 = sym ⟨𝑎𝑙𝑔⟩ 𝕤𝕖𝕞! (_⊗_ a b) rewrite 𝕤𝕖𝕞! a | 𝕤𝕖𝕞! b = sym ⟨𝑎𝑙𝑔⟩ 𝕤𝕖𝕞! (⊖_ a) rewrite 𝕤𝕖𝕞! a = sym ⟨𝑎𝑙𝑔⟩ -- Syntax instance for the signature CR:Syn : Syntax CR:Syn = record { ⅀F = ⅀F ; ⅀:CS = ⅀:CompatStr ; mvarᵢ = CR:Terms.mvar ; 𝕋:Init = λ 𝔛 → let open CR:Terms 𝔛 in record { ⊥ = CR ⋉ CRᵃ ; ⊥-is-initial = record { ! = λ{ {𝒜 ⋉ 𝒜ᵃ} → 𝕤𝕖𝕞 𝒜ᵃ ⋉ 𝕤𝕖𝕞ᵃ⇒ 𝒜ᵃ } ; !-unique = λ{ {𝒜 ⋉ 𝒜ᵃ} (f ⋉ fᵃ⇒) {x = t} → 𝕤𝕖𝕞! 𝒜ᵃ f fᵃ⇒ t } } } } -- Instantiation of the syntax and metatheory open Syntax CR:Syn public open CR:Terms public open import SOAS.Families.Build public open import SOAS.Syntax.Shorthands CRᵃ public open import SOAS.Metatheory CR:Syn public

26.587097

93

0.502063

ccc3cb23b821c63c1bcd1b57dceda8d7b8b816da

3,537

agda

Agda

agda-stdlib-0.9/src/Data/Star/Properties.agda

qwe2/try-agda

9d4c43b1609d3f085636376fdca73093481ab882

[ "Apache-2.0" ]

1

2016-10-20T15:52:05.000Z

agda-stdlib-0.9/src/Data/Star/Properties.agda

qwe2/try-agda

9d4c43b1609d3f085636376fdca73093481ab882

[ "Apache-2.0" ]

null

agda-stdlib-0.9/src/Data/Star/Properties.agda

qwe2/try-agda

9d4c43b1609d3f085636376fdca73093481ab882

[ "Apache-2.0" ]

null

------------------------------------------------------------------------ -- The Agda standard library -- -- Some properties related to Data.Star ------------------------------------------------------------------------ module Data.Star.Properties where open import Data.Star open import Function open import Relation.Binary open import Relation.Binary.PropositionalEquality as PropEq using (_≡_; refl; sym; cong; cong₂) import Relation.Binary.PreorderReasoning as PreR ◅◅-assoc : ∀ {i t} {I : Set i} {T : Rel I t} {i j k l} (xs : Star T i j) (ys : Star T j k) (zs : Star T k l) → (xs ◅◅ ys) ◅◅ zs ≡ xs ◅◅ (ys ◅◅ zs) ◅◅-assoc ε ys zs = refl ◅◅-assoc (x ◅ xs) ys zs = cong (_◅_ x) (◅◅-assoc xs ys zs) gmap-id : ∀ {i t} {I : Set i} {T : Rel I t} {i j} (xs : Star T i j) → gmap id id xs ≡ xs gmap-id ε = refl gmap-id (x ◅ xs) = cong (_◅_ x) (gmap-id xs) gmap-∘ : ∀ {i t} {I : Set i} {T : Rel I t} {j u} {J : Set j} {U : Rel J u} {k v} {K : Set k} {V : Rel K v} (f : J → K) (g : U =[ f ]⇒ V) (f′ : I → J) (g′ : T =[ f′ ]⇒ U) {i j} (xs : Star T i j) → (gmap {U = V} f g ∘ gmap f′ g′) xs ≡ gmap (f ∘ f′) (g ∘ g′) xs gmap-∘ f g f′ g′ ε = refl gmap-∘ f g f′ g′ (x ◅ xs) = cong (_◅_ (g (g′ x))) (gmap-∘ f g f′ g′ xs) gmap-◅◅ : ∀ {i t j u} {I : Set i} {T : Rel I t} {J : Set j} {U : Rel J u} (f : I → J) (g : T =[ f ]⇒ U) {i j k} (xs : Star T i j) (ys : Star T j k) → gmap {U = U} f g (xs ◅◅ ys) ≡ gmap f g xs ◅◅ gmap f g ys gmap-◅◅ f g ε ys = refl gmap-◅◅ f g (x ◅ xs) ys = cong (_◅_ (g x)) (gmap-◅◅ f g xs ys) gmap-cong : ∀ {i t j u} {I : Set i} {T : Rel I t} {J : Set j} {U : Rel J u} (f : I → J) (g : T =[ f ]⇒ U) (g′ : T =[ f ]⇒ U) → (∀ {i j} (x : T i j) → g x ≡ g′ x) → ∀ {i j} (xs : Star T i j) → gmap {U = U} f g xs ≡ gmap f g′ xs gmap-cong f g g′ eq ε = refl gmap-cong f g g′ eq (x ◅ xs) = cong₂ _◅_ (eq x) (gmap-cong f g g′ eq xs) fold-◅◅ : ∀ {i p} {I : Set i} (P : Rel I p) (_⊕_ : Transitive P) (∅ : Reflexive P) → (∀ {i j} (x : P i j) → ∅ ⊕ x ≡ x) → (∀ {i j k l} (x : P i j) (y : P j k) (z : P k l) → (x ⊕ y) ⊕ z ≡ x ⊕ (y ⊕ z)) → ∀ {i j k} (xs : Star P i j) (ys : Star P j k) → fold P _⊕_ ∅ (xs ◅◅ ys) ≡ fold P _⊕_ ∅ xs ⊕ fold P _⊕_ ∅ ys fold-◅◅ P _⊕_ ∅ left-unit assoc ε ys = sym (left-unit _) fold-◅◅ P _⊕_ ∅ left-unit assoc (x ◅ xs) ys = begin x ⊕ fold P _⊕_ ∅ (xs ◅◅ ys) ≡⟨ cong (_⊕_ x) $ fold-◅◅ P _⊕_ ∅ left-unit assoc xs ys ⟩ x ⊕ (fold P _⊕_ ∅ xs ⊕ fold P _⊕_ ∅ ys) ≡⟨ sym (assoc x _ _) ⟩ (x ⊕ fold P _⊕_ ∅ xs) ⊕ fold P _⊕_ ∅ ys ∎ where open PropEq.≡-Reasoning -- Reflexive transitive closures are preorders. preorder : ∀ {i t} {I : Set i} (T : Rel I t) → Preorder _ _ _ preorder T = record { _≈_ = _≡_ ; _∼_ = Star T ; isPreorder = record { isEquivalence = PropEq.isEquivalence ; reflexive = reflexive ; trans = _◅◅_ } } where reflexive : _≡_ ⇒ Star T reflexive refl = ε -- Preorder reasoning for Star. module StarReasoning {i t} {I : Set i} (T : Rel I t) where open PreR (preorder T) public renaming (_∼⟨_⟩_ to _⟶⋆⟨_⟩_; _≈⟨_⟩_ to _≡⟨_⟩_) infixr 2 _⟶⟨_⟩_ _⟶⟨_⟩_ : ∀ x {y z} → T x y → y IsRelatedTo z → x IsRelatedTo z x ⟶⟨ x⟶y ⟩ y⟶⋆z = x ⟶⋆⟨ x⟶y ◅ ε ⟩ y⟶⋆z

36.84375

88

0.430025

3745d842ecf979acff52051a0650f96fc4ca98da

265

agda

Agda

test/fail/WithoutK5.agda

dagit/agda

4383a3d20328a6c43689161496cee8eb479aca08

[ "MIT" ]

1

2019-11-27T07:26:06.000Z

test/fail/WithoutK5.agda

np/agda-git-experiment

20596e9dd9867166a64470dd24ea68925ff380ce

[ "MIT" ]

null

test/fail/WithoutK5.agda

np/agda-git-experiment

20596e9dd9867166a64470dd24ea68925ff380ce

[ "MIT" ]

null

{-# OPTIONS --without-K --show-implicit #-} module WithoutK5 where -- Equality defined with one index. data _≡_ {A : Set} (x : A) : A → Set where refl : x ≡ x weak-K : {A : Set} {a b : A} (p q : a ≡ b) (α β : p ≡ q) → α ≡ β weak-K refl .refl refl refl = refl

22.083333

64

0.558491

523bedea1f509a50cc239b76a4b76c60ae95976c

9,284

agda

Agda

Cubical/Algebra/Magma/MorphismProperties.agda

kiana-S/univalent-foundations

8bdb0766260489f9c89a14f4c0f2ad26e5190dec

[ "MIT" ]

null

Cubical/Algebra/Magma/MorphismProperties.agda

kiana-S/univalent-foundations

8bdb0766260489f9c89a14f4c0f2ad26e5190dec

[ "MIT" ]

null

Cubical/Algebra/Magma/MorphismProperties.agda

kiana-S/univalent-foundations

8bdb0766260489f9c89a14f4c0f2ad26e5190dec

[ "MIT" ]

null

{-# OPTIONS --cubical --no-import-sorts --safe #-} module Cubical.Algebra.Magma.MorphismProperties where open import Cubical.Foundations.Prelude open import Cubical.Foundations.Isomorphism open import Cubical.Foundations.Equiv open import Cubical.Foundations.Equiv.HalfAdjoint open import Cubical.Foundations.HLevels open import Cubical.Foundations.Univalence open import Cubical.Foundations.SIP open import Cubical.Foundations.Function using (_∘_; id) open import Cubical.Foundations.GroupoidLaws open import Cubical.Functions.Embedding open import Cubical.Data.Sigma open import Cubical.Data.Prod using (isPropProd) open import Cubical.Algebra open import Cubical.Algebra.Properties open import Cubical.Algebra.Magma.Morphism open import Cubical.Structures.Axioms open import Cubical.Structures.Auto open import Cubical.Structures.Record open import Cubical.Relation.Binary.Reasoning.Equality open Iso private variable ℓ ℓ′ ℓ′′ : Level L : Magma ℓ M : Magma ℓ′ N : Magma ℓ′′ isPropIsMagmaHom : ∀ (M : Magma ℓ) (N : Magma ℓ′) f → isProp (IsMagmaHom M N f) isPropIsMagmaHom M N f = isPropHomomorphic₂ (Magma.is-set N) f (Magma._•_ M) (Magma._•_ N) isSetMagmaHom : isSet (M ⟶ᴴ N) isSetMagmaHom {M = M} {N = N} = isOfHLevelRespectEquiv 2 equiv (isSetΣ (isSetΠ λ _ → is-set N) (λ f → isProp→isSet (isPropIsMagmaHom M N f))) where open Magma equiv : (Σ[ g ∈ (⟨ M ⟩ → ⟨ N ⟩) ] IsMagmaHom M N g) ≃ MagmaHom M N equiv = isoToEquiv (iso (λ (g , m) → magmahom g m) (λ (magmahom g m) → g , m) (λ _ → refl) λ _ → refl) isMagmaHomComp : {f : ⟨ L ⟩ → ⟨ M ⟩} {g : ⟨ M ⟩ → ⟨ N ⟩} → IsMagmaHom L M f → IsMagmaHom M N g → IsMagmaHom L N (g ∘ f) isMagmaHomComp {g = g} fHom gHom _ _ = cong g (fHom _ _) ∙ gHom _ _ private isMagmaHomComp′ : (f : L ⟶ᴴ M) (g : M ⟶ᴴ N) → IsMagmaHom L N (MagmaHom.fun g ∘ MagmaHom.fun f) isMagmaHomComp′ (magmahom f fHom) (magmahom g gHom) _ _ = cong g (fHom _ _) ∙ gHom _ _ compMagmaHom : (L ⟶ᴴ M) → (M ⟶ᴴ N) → (L ⟶ᴴ N) compMagmaHom f g = magmahom _ (isMagmaHomComp′ f g) compMagmaEquiv : L ≃ᴴ M → M ≃ᴴ N → L ≃ᴴ N compMagmaEquiv f g = magmaequiv (compEquiv f.eq g.eq) (isMagmaHomComp′ f.hom g.hom) where module f = MagmaEquiv f module g = MagmaEquiv g isMagmaHomId : (M : Magma ℓ) → IsMagmaHom M M id isMagmaHomId M _ _ = refl idMagmaHom : (M : Magma ℓ) → (M ⟶ᴴ M) idMagmaHom M = record { fun = id ; isHom = isMagmaHomId M } idMagmaEquiv : (M : Magma ℓ) → M ≃ᴴ M idMagmaEquiv M = record { eq = idEquiv ⟨ M ⟩ ; isHom = isMagmaHomId M } -- Isomorphism inversion isMagmaHomInv : (eqv : M ≃ᴴ N) → IsMagmaHom N M (invEq (MagmaEquiv.eq eqv)) isMagmaHomInv {M = M} {N = N} (magmaequiv eq isHom) x y = isInj-f _ _ ( f (f⁻¹ (x •ᴺ y)) ≡⟨ retEq eq _ ⟩ x •ᴺ y ≡˘⟨ cong₂ _•ᴺ_ (retEq eq x) (retEq eq y) ⟩ f (f⁻¹ x) •ᴺ f (f⁻¹ y) ≡˘⟨ isHom (f⁻¹ x) (f⁻¹ y) ⟩ f (f⁻¹ x •ᴹ f⁻¹ y) ∎) where _•ᴹ_ = Magma._•_ M _•ᴺ_ = Magma._•_ N f = equivFun eq f⁻¹ = invEq eq isInj-f : (x y : ⟨ M ⟩) → f x ≡ f y → x ≡ y isInj-f x y = invEq (_ , isEquiv→isEmbedding (eq .snd) x y) invMagmaHom : M ≃ᴴ N → (N ⟶ᴴ M) invMagmaHom eq = record { isHom = isMagmaHomInv eq } invMagmaEquiv : (M ≃ᴴ N) → (N ≃ᴴ M) invMagmaEquiv eq = record { eq = invEquiv (MagmaEquiv.eq eq) ; isHom = isMagmaHomInv eq } magmaHomEq : {f g : M ⟶ᴴ N} → (MagmaHom.fun f ≡ MagmaHom.fun g) → f ≡ g magmaHomEq {M = M} {N = N} {magmahom f fm} {magmahom g gm} p i = magmahom (p i) (p-hom i) where p-hom : PathP (λ i → IsMagmaHom M N (p i)) fm gm p-hom = toPathP (isPropIsMagmaHom M N _ _ _) magmaEquivEq : {f g : M ≃ᴴ N} → (MagmaEquiv.eq f ≡ MagmaEquiv.eq g) → f ≡ g magmaEquivEq {M = M} {N = N} {magmaequiv f fm} {magmaequiv g gm} p i = magmaequiv (p i) (p-hom i) where p-hom : PathP (λ i → IsMagmaHom M N (p i .fst)) fm gm p-hom = toPathP (isPropIsMagmaHom M N _ _ _) module MagmaΣTheory {ℓ} where RawMagmaStructure : Type ℓ → Type ℓ RawMagmaStructure A = A → A → A RawMagmaEquivStr = AutoEquivStr RawMagmaStructure rawMagmaUnivalentStr : UnivalentStr _ RawMagmaEquivStr rawMagmaUnivalentStr = autoUnivalentStr RawMagmaStructure MagmaAxioms : (A : Type ℓ) → RawMagmaStructure A → Type ℓ MagmaAxioms A _•_ = isSet A MagmaStructure : Type ℓ → Type ℓ MagmaStructure = AxiomsStructure RawMagmaStructure MagmaAxioms MagmaΣ : Type (ℓ-suc ℓ) MagmaΣ = TypeWithStr ℓ MagmaStructure isPropMagmaAxioms : (A : Type ℓ) (_•_ : RawMagmaStructure A) → isProp (MagmaAxioms A _•_) isPropMagmaAxioms _ _ = isPropIsSet MagmaEquivStr : StrEquiv MagmaStructure ℓ MagmaEquivStr = AxiomsEquivStr RawMagmaEquivStr MagmaAxioms MagmaAxiomsIsoIsMagma : {A : Type ℓ} (_•_ : RawMagmaStructure A) → Iso (MagmaAxioms A _•_) (IsMagma A _•_) fun (MagmaAxiomsIsoIsMagma s) x = ismagma x inv (MagmaAxiomsIsoIsMagma s) (ismagma x) = x rightInv (MagmaAxiomsIsoIsMagma s) _ = refl leftInv (MagmaAxiomsIsoIsMagma s) _ = refl MagmaAxioms≡IsMagma : {A : Type ℓ} (_•_ : RawMagmaStructure A) → MagmaAxioms A _•_ ≡ IsMagma A _•_ MagmaAxioms≡IsMagma s = isoToPath (MagmaAxiomsIsoIsMagma s) Magma→MagmaΣ : Magma ℓ → MagmaΣ Magma→MagmaΣ (mkmagma A _•_ isMagma) = A , _•_ , MagmaAxiomsIsoIsMagma _ .inv isMagma MagmaΣ→Magma : MagmaΣ → Magma ℓ MagmaΣ→Magma (A , _•_ , isMagma•) = mkmagma A _•_ (MagmaAxiomsIsoIsMagma _ .fun isMagma•) MagmaIsoMagmaΣ : Iso (Magma ℓ) MagmaΣ MagmaIsoMagmaΣ = iso Magma→MagmaΣ MagmaΣ→Magma (λ _ → refl) (λ _ → refl) magmaUnivalentStr : UnivalentStr MagmaStructure MagmaEquivStr magmaUnivalentStr = axiomsUnivalentStr _ isPropMagmaAxioms rawMagmaUnivalentStr MagmaΣPath : (M N : MagmaΣ) → (M ≃[ MagmaEquivStr ] N) ≃ (M ≡ N) MagmaΣPath = SIP magmaUnivalentStr MagmaEquivΣ : (M N : Magma ℓ) → Type ℓ MagmaEquivΣ M N = Magma→MagmaΣ M ≃[ MagmaEquivStr ] Magma→MagmaΣ N MagmaIsoΣPath : {M N : Magma ℓ} → Iso (MagmaEquiv M N) (MagmaEquivΣ M N) fun MagmaIsoΣPath (magmaequiv e h) = (e , h) inv MagmaIsoΣPath (e , h) = magmaequiv e h rightInv MagmaIsoΣPath _ = refl leftInv MagmaIsoΣPath _ = refl MagmaPath : (M N : Magma ℓ) → (MagmaEquiv M N) ≃ (M ≡ N) MagmaPath M N = MagmaEquiv M N ≃⟨ isoToEquiv MagmaIsoΣPath ⟩ MagmaEquivΣ M N ≃⟨ MagmaΣPath _ _ ⟩ Magma→MagmaΣ M ≡ Magma→MagmaΣ N ≃⟨ isoToEquiv (invIso (congIso MagmaIsoMagmaΣ)) ⟩ M ≡ N ■ RawMagmaΣ : Type (ℓ-suc ℓ) RawMagmaΣ = TypeWithStr ℓ RawMagmaStructure Magma→RawMagmaΣ : Magma ℓ → RawMagmaΣ Magma→RawMagmaΣ M = (⟨ M ⟩ , Magma._•_ M) InducedMagma : (M : Magma ℓ) (N : RawMagmaΣ) (e : ⟨ M ⟩ ≃ ⟨ N ⟩) → RawMagmaEquivStr (Magma→RawMagmaΣ M) N e → Magma ℓ InducedMagma M N e r = MagmaΣ→Magma (inducedStructure rawMagmaUnivalentStr (Magma→MagmaΣ M) N (e , r)) InducedMagmaPath : (M : Magma ℓ) (N : RawMagmaΣ) (e : ⟨ M ⟩ ≃ ⟨ N ⟩) (E : RawMagmaEquivStr (Magma→RawMagmaΣ M) N e) → M ≡ InducedMagma M N e E InducedMagmaPath M N e E = MagmaPath M (InducedMagma M N e E) .fst (magmaequiv e E) open MagmaΣTheory public using (InducedMagma; InducedMagmaPath) MagmaPath : (M ≃ᴴ N) ≃ (M ≡ N) MagmaPath = MagmaΣTheory.MagmaPath _ _ open Magma uaMagma : M ≃ᴴ N → M ≡ N uaMagma = equivFun MagmaPath carac-uaMagma : {M N : Magma ℓ} (f : M ≃ᴴ N) → cong Carrier (uaMagma f) ≡ ua (MagmaEquiv.eq f) carac-uaMagma (magmaequiv f m) = (refl ∙∙ ua f ∙∙ refl) ≡˘⟨ rUnit (ua f) ⟩ ua f ∎ Magma≡ : (M N : Magma ℓ) → ( Σ[ p ∈ ⟨ M ⟩ ≡ ⟨ N ⟩ ] Σ[ q ∈ PathP (λ i → p i → p i → p i) (_•_ M) (_•_ N) ] PathP (λ i → IsMagma (p i) (q i)) (isMagma M) (isMagma N)) ≃ (M ≡ N) Magma≡ M N = isoToEquiv (iso (λ (p , q , r) i → mkmagma (p i) (q i) (r i)) (λ p → cong Carrier p , cong _•_ p , cong isMagma p) (λ _ → refl) (λ _ → refl)) caracMagma≡ : (p q : M ≡ N) → cong Carrier p ≡ cong Carrier q → p ≡ q caracMagma≡ {M = M} {N = N} p q t = cong (Magma≡ M N .fst) (Σ≡Prop (λ _ → isPropΣ (isOfHLevelPathP' 1 (isSetΠ2 λ _ _ → is-set N) _ _) λ _ → isOfHLevelPathP 1 (λ { (ismagma x) (ismagma y) → cong ismagma (isPropIsSet x y) }) _ _) t) uaMagmaId : (M : Magma ℓ) → uaMagma (idMagmaEquiv M) ≡ refl uaMagmaId M = caracMagma≡ _ _ (carac-uaMagma (idMagmaEquiv M) ∙ uaIdEquiv) uaCompMagmaEquiv : {L M N : Magma ℓ} (f : L ≃ᴴ M) (g : M ≃ᴴ N) → uaMagma (compMagmaEquiv f g) ≡ uaMagma f ∙ uaMagma g uaCompMagmaEquiv f g = caracMagma≡ _ _ ( cong Carrier (uaMagma (compMagmaEquiv f g)) ≡⟨ carac-uaMagma (compMagmaEquiv f g) ⟩ ua (eq (compMagmaEquiv f g)) ≡⟨ uaCompEquiv _ _ ⟩ ua (eq f) ∙ ua (eq g) ≡⟨ cong (_∙ ua (eq g)) (sym (carac-uaMagma f)) ⟩ cong Carrier (uaMagma f) ∙ ua (eq g) ≡⟨ cong (cong Carrier (uaMagma f) ∙_) (sym (carac-uaMagma g)) ⟩ cong Carrier (uaMagma f) ∙ cong Carrier (uaMagma g) ≡⟨ sym (cong-∙ Carrier (uaMagma f) (uaMagma g)) ⟩ cong Carrier (uaMagma f ∙ uaMagma g) ∎) where open MagmaEquiv

35.707692

97

0.621069

1a0416e472bdad83a9ded3dca03908b866b48a6c

1,258

agda

Agda

prototyping/Luau/Heap.agda

FreakingBarbarians/luau

5187e64f88953f34785ffe58acd0610ee5041f5f

[ "MIT" ]

1

2022-02-11T21:30:17.000Z

prototyping/Luau/Heap.agda

FreakingBarbarians/luau

5187e64f88953f34785ffe58acd0610ee5041f5f

[ "MIT" ]

null

prototyping/Luau/Heap.agda

FreakingBarbarians/luau

5187e64f88953f34785ffe58acd0610ee5041f5f

[ "MIT" ]

null

module Luau.Heap where open import Agda.Builtin.Equality using (_≡_) open import FFI.Data.Maybe using (Maybe; just) open import FFI.Data.Vector using (Vector; length; snoc; empty) open import Luau.Addr using (Addr) open import Luau.Var using (Var) open import Luau.Syntax using (Block; Expr; nil; addr; function⟨_⟩_end) data HeapValue : Set where function_⟨_⟩_end : Var → Var → Block → HeapValue Heap = Vector HeapValue data _≡_⊕_↦_ : Heap → Heap → Addr → HeapValue → Set where defn : ∀ {H val} → ----------------------------------- (snoc H val) ≡ H ⊕ (length H) ↦ val lookup : Heap → Addr → Maybe HeapValue lookup = FFI.Data.Vector.lookup emp : Heap emp = empty data AllocResult (H : Heap) (V : HeapValue) : Set where ok : ∀ a H′ → (H′ ≡ H ⊕ a ↦ V) → AllocResult H V alloc : ∀ H V → AllocResult H V alloc H V = ok (length H) (snoc H V) defn next : Heap → Addr next = length allocated : Heap → HeapValue → Heap allocated = snoc -- next-emp : (length empty ≡ 0) next-emp = FFI.Data.Vector.length-empty -- lookup-next : ∀ V H → (lookup (allocated H V) (next H) ≡ just V) lookup-next = FFI.Data.Vector.lookup-snoc -- lookup-next-emp : ∀ V → (lookup (allocated emp V) 0 ≡ just V) lookup-next-emp = FFI.Data.Vector.lookup-snoc-empty

25.673469

71

0.651828