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{- reported by Guillaume Brunerie on 2015-09-17 -} {-# OPTIONS --rewriting #-} data _==_ {A : Set} (a : A) : A → Set where idp : a == a {-# BUILTIN REWRITE _==_ #-} postulate A B : Set f g : A → B module M (x : A) where postulate rx : f x == g x {-# REWRITE rx #-} -- This shouldn't work test : (...
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module NF {A : Set} where open import Relation.Binary.PropositionalEquality record Sing (a : A) : Set where field unpack : A eq : unpack ≡ a {-# INLINE Sing.unpack #-} record NF (v : A) : Set where field !! : Sing v {-# INLINE NF.!! #-} open NF {{...}} open import Function.Reasoning nf : (v : A){{nf...
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{-# OPTIONS --without-K --safe #-} module Categories.Functor.Construction.Constant where open import Level open import Categories.Category open import Categories.Category.Instance.One open import Categories.Category.Product open import Categories.Functor renaming (id to idF) open import Categories.NaturalTransformati...
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module Structure.Relator.Equivalence where import Lvl open import Logic open import Logic.Propositional open import Structure.Relator.Properties hiding (reflexivity ; symmetry ; transitivity) open import Type private variable ℓ₁ ℓ₂ : Lvl.Level -- An equivalence relation is a reflexive, symmetric and transitiv...
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{-# OPTIONS --without-K --rewriting #-} open import HoTT open import groups.Exactness open import homotopy.CofiberSequence open import cohomology.Theory {- Finite additivity is provable (and in a stronger form) without using - the additivity axiom. We have Cⁿ(X ∨ Y) == Cⁿ(X) × Cⁿ(Y) - and over this path ...
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-- Test case extracted from The Agda standard library -- Properties related to Any {-# OPTIONS --show-implicit #-} module ProjectionNotNormalized where open import Common.Level renaming (lsuc to suc) record Setoid c ℓ : Set (suc (c ⊔ ℓ)) where infix 4 _≈_ field Carrier : Set c _≈_ : Carr...
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{- Byzantine Fault Tolerant Consensus Verification in Agda, version 0.9. Copyright (c) 2020, 2021, Oracle and/or its affiliates. Licensed under the Universal Permissive License v 1.0 as shown at https://opensource.oracle.com/licenses/upl -} open import LibraBFT.ImplShared.Consensus.Types.EpochIndep open import ...
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{-# OPTIONS --without-K --safe #-} open import Categories.Category module Categories.Diagram.Pullback.Properties {o ℓ e} (C : Category o ℓ e) where open import Function using (_$_) open import Categories.Category.BinaryProducts C open import Categories.Category.Cartesian C open import Categories.Diagram.Pullback C ...
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module Sandbox.Pred where open import Data.Num open import Data.Num.Properties open import Data.Nat open import Data.Fin using (Fin; suc; zero; #_) open import Data.Vec open import Relation.Nullary.Decidable using (True; fromWitness) open import Relation.Binary.PropositionalEquality hiding ([_]) -- open import Relati...
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{-# OPTIONS --show-implicit #-} {-# OPTIONS --sized-types #-} module Issue298 where postulate Size : Set ↑_ : Size → Size ∞ : Size {-# BUILTIN SIZE Size #-} {-# BUILTIN SIZESUC ↑_ #-} {-# BUILTIN SIZEINF ∞ #-} data BTree : {i : Size} → Set where leaf : ∀ {i} → BTree {↑ i} node : ∀ {i} → BT...
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{-# OPTIONS --without-K --safe #-} open import Categories.Category -- Reasoning facilities about morphism equivalences (not necessarily 'squares') module Categories.Morphism.Reasoning {o ℓ e} (C : Category o ℓ e) where -- some items are defined in sub-modules open import Categories.Morphism.Reasoning.Core C public ...
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module Prelude where data Nat : Set where zero : Nat suc : Nat -> Nat {-# BUILTIN NATURAL Nat #-} data _==_ {A : Set} (x : A) : A → Set where refl : x == x data Vec (A : Set) : Nat -> Set where nil : Vec A 0 cons : forall {n} -> A -> Vec A n -> Vec A (suc n) data List (A : Set) : Set where nil : List A...
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{-# OPTIONS --cubical --safe #-} module Cubical.Data.Everything where open import Cubical.Data.BinNat public open import Cubical.Data.Bool public open import Cubical.Data.Empty public open import Cubical.Data.Fin public open import Cubical.Data.Nat public open import Cubical.Data.Int public renaming (_+_ to _+Int_ ; +...
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{- This second-order term syntax was created from the following second-order syntax description: syntax Inception | IA type L : 0-ary P : 0-ary A : 0-ary term rec : L P -> A inc : L.A P.A -> A theory (S) p : P a : P.A |> inc (l. rec (l, p[]), x. a[x]) = a[p[]] (E) a : L.A |> k : L |- inc (l. ...
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-- The unit type; also used as proposition ``Truth''. {-# OPTIONS --without-K --safe #-} module Tools.Unit where -- We reexport Agda's built-in unit type. open import Agda.Builtin.Unit public using (⊤; tt) open import Agda.Primitive record ⊤′ : Setω where instance constructor tt
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import Lvl open import Structure.Category open import Structure.Setoid open import Type module Structure.Category.Morphism.Transport {ℓₒ ℓₘ ℓₑ : Lvl.Level} (cat : CategoryObject{ℓₒ}{ℓₘ}{ℓₑ}) where import Functional.Dependent as Fn import Function.Equals open Function.Equals.Dependent open ...
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module ModuleArityMismatch where module M (A : Set) where postulate A : Set module M′ = M A A
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module Prelude where -- Function composition _∘_ : ∀ {x y z : Set} -> (y -> z) -> (x -> y) -> (x -> z) (f ∘ g) x = f (g x) -- Product type data _×_ (A B : Set) : Set where _,_ : A -> B -> A × B uncurry : ∀ {x y z : Set} -> (x -> y -> z) -> (x × y) -> z uncurry f (a , b) = f a b -- Lists data List ...
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module Issue49 where module Dummy {A : Set1} where postulate D : Set T : Set T = Dummy.D {Set} T' : Set T' = Dummy.D {A = Set}
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module NatCat where open import Relation.Binary.PropositionalEquality -- If you can show that a relation only ever has one inhabitant -- you get the category laws for free module EasyCategory (obj : Set) (_⟶_ : obj → obj → Set) (_∘_ : ∀ {x y z} → x ⟶ y → y ⟶ z → x ⟶ z) (id : ∀ x → x ⟶ x) (single-inhabitan...
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module Structure.Relator.Properties where open import Functional import Lvl open import Lang.Instance open import Logic import Structure.Relator.Names as Names open import Type private variable ℓ ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Lvl.Level private variable T A B C D E : Type{ℓ} -- Definition of a reflexive binary relation mod...
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module Issue2486.ImportB where open import Issue2486.Haskell {-# FOREIGN GHC data BBool = BTrue | BFalse #-} data BBool : Set where BTrue BFalse : BBool {-# COMPILE GHC BBool = data BBool ( BTrue | BFalse ) #-} postulate BList : Set → Set {-# FOREIGN GHC import MAlonzo.Code.Issue2486.Haskell (MyList) #-} {-# COMP...
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{- This is mostly for convenience, when working with ideals (which are defined for general rings) in a commutative ring. -} {-# OPTIONS --safe #-} module Cubical.Algebra.CommRing.Ideal where open import Cubical.Foundations.Prelude open import Cubical.Foundations.Function open import Cubical.Foundations.HLevels ope...
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------------------------------------------------------------------------------ -- Equality reasoning on inductive PA ------------------------------------------------------------------------------ {-# OPTIONS --exact-split #-} {-# OPTIONS --no-sized-types #-} {-# OPTIONS --no-universe-polymorphis...
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------------------------------------------------------------------------ -- One can construct combinators which do not preserve equality ------------------------------------------------------------------------ module TotalParserCombinators.NotACongruence where open import Codata.Musical.Notation open import Data.Bool...
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{-# OPTIONS --cubical --no-import-sorts --safe #-} module Cubical.HITs.AssocList.Base where open import Cubical.Foundations.Prelude open import Cubical.Foundations.HLevels open import Cubical.Data.Nat using (ℕ; _+_) private variable ℓ : Level A : Type ℓ infixr 5 ⟨_,_⟩∷_ data AssocList (A : Type ℓ) : Ty...
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{-# OPTIONS --without-K --safe #-} module Categories.Diagram.Finite where open import Level open import Categories.Category open import Categories.Category.Finite renaming (Finite to FiniteC) open import Categories.Functor private variable o ℓ e : Level J C : Category o ℓ e record Finite (F : Functor J C...
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-- Exercises for session 2 -- -- If unsure which exercises to do start with those marked with * -- {-# OPTIONS --cubical --allow-unsolved-metas #-} module ExerciseSession2 where open import Part1 open import Part2 open import ExerciseSession1 open import Cubical.Foundations.Equiv -- Exercises about Part 2: -- Exerc...
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{-# OPTIONS --cubical --safe #-} module Cubical.Homotopy.PointedFibration where open import Cubical.Core.Everything open import Cubical.Data.Nat open import Cubical.Foundations.Prelude open import Cubical.Foundations.Pointed open import Cubical.Foundations.Isomorphism open import Cubical.Foundations.GroupoidLaws op...
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{-# OPTIONS --without-K #-} module Wow-It-is-FV where open import Agda.Builtin.Nat open import Agda.Builtin.Equality open import Agda.Primitive levelEq : lzero ≡ lzero levelEq = refl levelEq′ : lsuc lzero ≡ lsuc lzero levelEq′ = refl trans : {l : Level} {Q : Set l} {a b c : Q} → a ≡ b → b ≡ c → a ≡ c trans re...
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open import Nat open import Prelude open import List open import contexts open import core module lemmas-general where Coerce-unicity : ∀{r ex1 ex2} → Coerce r := ex1 → Coerce r := ex2 → ex1 == ex2 Coerce-unicity CoerceUnit CoerceUnit = refl Coerce-...
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------------------------------------------------------------------------ -- The Agda standard library -- -- Lexicographic ordering of lists ------------------------------------------------------------------------ {-# OPTIONS --without-K --safe #-} module Data.List.Relation.Binary.Lex.Core where open import Data.Empt...
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-- ---------------------------------------------------------------------- -- The Agda σ-library -- -- Substitutions -- ---------------------------------------------------------------------- -- A substitution on T is defined as a mapping from -- indices to T (w/ explicit bounds). -- -- Since the domain is bounded, we...
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{-# OPTIONS --cubical --no-import-sorts --safe #-} module Cubical.Relation.Binary where open import Cubical.Relation.Binary.Base public open import Cubical.Relation.Binary.Properties public open import Cubical.Relation.Binary.Fiberwise public
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module RewriteAndWhere where open import Common.Equality data ℕ : Set where zero : ℕ good : (a b : ℕ) → a ≡ b → b ≡ a good a b eq with a | eq ... | .b | refl = foo where foo : b ≡ b foo = refl mutual aux : (a b : ℕ)(w : ℕ) → w ≡ b → b ≡ w aux a b .b refl = foo where foo : b ≡ b foo = r...
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{-# OPTIONS --cubical --safe #-} module Strict.Properties where open import Path open import Level open import Strict open import Agda.Builtin.Strict open import Function $!-≡ : {A : Type a} {B : A → Type b} → (f : ∀ x → B x) → ∀ x → (f $! x) ≡ f x $!-≡ f x = builtin-eq-to-path (primForceLemma x f) $!≡$ : {A : Type...
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------------------------------------------------------------------------ -- The Agda standard library -- -- Some derivable properties ------------------------------------------------------------------------ {-# OPTIONS --without-K --safe #-} open import Algebra module Algebra.Properties.Group {g₁ g₂} (G : Group g₁ g...
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------------------------------------------------------------------------ -- The Agda standard library -- -- Properties related to All ------------------------------------------------------------------------ {-# OPTIONS --without-K --safe #-} module Data.Vec.Relation.Unary.All.Properties where open import Data.List u...
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{- Byzantine Fault Tolerant Consensus Verification in Agda, version 0.9. Copyright (c) 2021, Oracle and/or its affiliates. Licensed under the Universal Permissive License v 1.0 as shown at https://opensource.oracle.com/licenses/upl -} module Haskell.Modules.RWS.Lens where open import Haskell.Modules.RWS open i...
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module Nat where open import Base data Nat : Set where zero : Nat suc : Nat -> Nat _=N_ : Nat -> Nat -> Set zero =N zero = True zero =N suc _ = False suc _ =N zero = False suc n =N suc m = n =N m refN : Refl _=N_ refN {zero} = T refN {suc n} = refN {n} symN : Sym _=N_ symN {zero}{zero} p = p symN {suc...
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{-# OPTIONS --without-K --safe #-} open import Categories.Category -- Power Functors, Exponentials over a Category C -- Mainly categories where the objects are functions (Fin n -> Obj) considered pointwise -- and then upgraded to Functors. module Categories.Functor.Power {o ℓ e} (C : Category o ℓ e) where open Ca...
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------------------------------------------------------------------------ -- Possibly infinite binary trees ------------------------------------------------------------------------ module Tree where open import Codata.Musical.Notation import Relation.Binary.PropositionalEquality as PropEq open PropEq using (_≡_) data...
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------------------------------------------------------------------------ -- The Agda standard library -- -- N-ary products ------------------------------------------------------------------------ -- Vectors (as in Data.Vec) also represent n-ary products, so what is -- the point of this module? The n-ary products below...
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module Properties.Equality where open import Agda.Builtin.Equality using (_≡_; refl) open import Properties.Contradiction using (¬) sym : ∀ {A : Set} {a b : A} → (a ≡ b) → (b ≡ a) sym refl = refl trans : ∀ {A : Set} {a b c : A} → (a ≡ b) → (b ≡ c) → (a ≡ c) trans refl refl = refl cong : ∀ {A B : Set} {a b : A} (f :...
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{-# OPTIONS --without-K #-} {- Ribbon is an explicit covering space construction. This construction is given by Daniel Grayson, Favonia and Guillaume Brunerie together. -} open import HoTT -- A is the pointed base space. -- El is intended to be a (group-)set, module homotopy.RibbonCover {i : ULevel} where ...
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{-# OPTIONS --cubical #-} module Equality where import Algebra.FunctionProperties import Algebra.Structures open import Cubical.Core.Everything public hiding (module Σ) open import Data.Empty open import Data.Nat as ℕ open import Data.Nat.Properties open import Data.Product as Σ open import Data.Unit open import Func...
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open import Level module ordinal where open import logic open import nat open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) open import Data.Empty open import Relation.Binary.PropositionalEquality open import Relation.Binary.Definitions open import Data.Nat.Properties open import R...
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{-# OPTIONS --sized-types #-} -- {-# OPTIONS -v tc.size.solve:20 -v tc.decl.ax:10 #-} module SizeUnsolvedConstraintsInTypeSignature where open import Common.Size data Nat : (i : Size) -> Set where Z : {i : Size} -> Nat (↑ i) S : {i : Size} -> Nat i → Nat (↑ i) one1 : (i : Size) → Nat (↑ (↑ i)) one1 i = S Z one2...
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module Section3 where open import Section1 public -- 3. The calculus of proof trees -- ============================== -- -- We define the set of proof trees of implicational logic in the ordinary style à la Church, -- except that we use explicit substitutions. -- 3.1. Definition of types -- -----------------------...
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-- Andreas, 2015-07-07 Let _ pattern be instantiable by an inaccessible pattern module _ where open import Common.Equality -- These used to work and should still work: transV1 : ∀{A : Set} (a b c : A) → a ≡ b → b ≡ c → a ≡ c transV1 _ ._ ._ refl refl = refl transV2 : ∀{A : Set} (a b c : A) → a ≡ b → b ≡ c → a ≡ c ...
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module PatternShadowsConstructor2 where module A where data A (X : Set) : Set where c : A X → A X x : A X open A using (A; c) f : ∀ {X} → A X → A X → A X f (c y) x = x f A.x _ = A.x
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------------------------------------------------------------------------ -- All library modules, along with short descriptions ------------------------------------------------------------------------ -- Note that core modules are not included. module Everything where -- Definitions of algebraic structures like monoi...
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{-# OPTIONS --sized-types #-} open import Agda.Builtin.Equality open import Agda.Builtin.Size mutual data D (i : Size) : Set where c : D′ i → D i record D′ (i : Size) : Set where coinductive field force : (j : Size< i) → D j data R (i : Size) : D ∞ → D ∞ → Set where c₁ : ∀ x y → R i (c x) (...
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{-# OPTIONS --allow-unsolved-metas #-} -- Andreas, 2014-05-17 reported by Fabien Renaud open import Common.Equality data Nat : Set where zero : Nat suc : Nat → Nat {-# BUILTIN NATURAL Nat #-} data List {a} (A : Set a) : Set a where [] : List A _∷_ : A → List A → List A length : ∀{a}{A : Set a} → List A →...
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module _ (A : Set) where data A where
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------------------------------------------------------------------------ -- The classical definition of (strong) bisimilarity ------------------------------------------------------------------------ -- This module is largely based on "Enhancements of the bisimulation -- proof method" by Pous and Sangiorgi. {-# OPTION...
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{-# OPTIONS --without-K --safe #-} module Data.Binary.Definitions where open import Function open import Data.Binary.Bits public open import Data.List using (_∷_) renaming ([] to 1ᵇ) public open import Data.Maybe open import Data.Product 𝔹⁺ : Set 𝔹⁺ = Data.List.List Bit 𝔹 : Set 𝔹 = Maybe 𝔹⁺ infixr 5 0<_ patte...
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------------------------------------------------------------------------ -- The delay monad quotiented by weak bisimilarity ------------------------------------------------------------------------ {-# OPTIONS --erased-cubical --sized-types #-} module Partiality-monad.Coinductive where open import Equality.Propositio...
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module _ where record R : Set₁ where constructor c open R {{...}}
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-- Andreas, 2019-10-01, continuing issue #3855 (erasure modality @0) -- Test case by Nisse at https://github.com/agda/agda/issues/3855#issuecomment-527164352 -- Occurs check needs to take erasure status of definitions -- (here: postulates) into account. postulate P : Set → Set p : (A : Set) → P A @0 A : S...
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{-# OPTIONS --without-K --safe #-} open import Relation.Binary using (Rel; Setoid; IsEquivalence) module Loop.Structures {a ℓ} {A : Set a} -- The underlying set (_≈_ : Rel A ℓ) -- The underlying equality relation where open import Algebra.Core open import Level using (_⊔_) open import Data.Product using (_,...
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-- Andreas, 2015-07-21 Issue 1612 -- Error "D is not strictly positive" should appear immediately. -- (There was a performance problem due to the use of Utils.Graph.....allPaths). {-# NON_TERMINATING #-} mutual data D : Set where c0 : A0 → D c1 : A1 → D c2 : A2 → D c3 : A3 → D c4 : A4 → D c...
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open import Relation.Nullary using (Dec; yes; no; ¬_) open import Relation.Nullary.Decidable using (from-yes) open import Relation.Nullary.Negation using (contradiction) open import Relation.Binary using (Tri) open Tri open import Relation.Binary.PropositionalEquality using (_≡_; _≢_; refl; sym; trans) open import Dat...
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-- Basic intuitionistic modal logic S4, without ∨, ⊥, or ◇. -- Common syntax. module BasicIS4.Syntax.Common where open import Common.ContextPair public -- Types, or propositions. infixr 10 □_ infixl 9 _∧_ infixr 7 _▻_ data Ty : Set where α_ : Atom → Ty _▻_ : Ty → Ty → Ty □_ : Ty → Ty _∧_ : Ty → Ty → Ty ...
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-- 2011-09-15 posted by Nisse -- {-# OPTIONS --show-implicit -v tc.lhs.unify:15 #-} module Issue292-16 where data _≡_ {A : Set} (x : A) : A → Set where refl : x ≡ x postulate A : Set f : A → A data C : A → Set where c : ∀ x → C (f x) record Box : Set where constructor box field a : A b : C a te...
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{-# OPTIONS --without-K --safe --overlapping-instances #-} module Interpreter where open import Data.Char hiding (_≤_) open import Data.Bool hiding (_≤_) open import Data.Nat hiding (_≤_) open import Data.Unit import Data.Nat as N open import Data.Product open import Data.Sum open import Relation.Binary.PropositionalE...
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-- Andreas, 2016-10-23 issue #2280: -- Solver throws away meta arguments. -- (Issue discovered by code review.) open import Common.Equality open import Common.Product data Bool : Set where true false : Bool postulate not : Bool → Bool data D (f : Bool → Bool) : Set where c : ∀ x → x ≡ not (f true) → D f test...
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module Lib.Prelude where infixr 90 _∘_ infixr 1 _,_ id : {A : Set} -> A -> A id x = x _∘_ : {A : Set}{B : A -> Set}{C : {x : A} -> B x -> Set} (f : {x : A}(y : B x) -> C y)(g : (x : A) -> B x)(x : A) -> C (g x) (f ∘ g) x = f (g x) data Unit : Set where unit : Unit {-# COMPILED_DATA Unit () () #-} p...
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------------------------------------------------------------------------ -- Up-to techniques for the delay monad and the alternative -- coinductive definition of weak bisimilarity ------------------------------------------------------------------------ {-# OPTIONS --sized-types #-} open import Prelude module Bisimil...
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-- TODO -- Gentzen-style formalisation of syntax. -- Simple terms. module BasicT.Syntax.Gentzen where open import BasicT.Syntax.Common public -- Derivations. infix 3 _⊢_ data _⊢_ (Γ : Cx Ty) : Ty → Set where var : ∀ {A} → A ∈ Γ → Γ ⊢ A lam : ∀ {A B} → Γ , A ⊢ B → Γ ⊢ A ▻ B app : ∀ {A B} → Γ ⊢ A ▻ B →...
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-- Andreas, 2015-05-01 -- With clauses for functions with flexible arity. -- {-# OPTIONS -v tc.with:40 #-} open import Common.Prelude open import Common.Equality mutual even : Nat → Bool even 0 = true even (suc n) = odd n odd : Nat → Bool odd 0 = false odd (suc n) = even n NPred : Nat → Set NPre...
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module par-swap.dpg where open import par-swap open import par-swap.properties open import par-swap.confluent open import par-swap.dpg-pot open import par-swap.dpg-e-view open import noetherian using (noetherian ; ∥_∥s) open import Esterel.Lang open import Esterel.Lang.Properties open import Esterel.Lang.Binding open ...
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module UnSizedIO.Console where open import NativeIO open import UnSizedIO.Base hiding (main) data ConsoleCommand : Set where putStrLn : String → ConsoleCommand getLine : ConsoleCommand ConsoleResponse : ConsoleCommand → Set ConsoleResponse (putStrLn s) = Unit ConsoleResponse getLine = String ConsoleInterf...
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module Base.Free.Instance.Maybe.Properties where open import Relation.Binary.PropositionalEquality using (refl; cong) open import Base.Free using (Free; pure; impure; _>>=_) open import Base.Free.Instance.Maybe using (Just; Nothing) renaming (Maybe to MaybeF) open...
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-- MIT License -- Copyright (c) 2021 Luca Ciccone and Luca Padovani -- Permission is hereby granted, free of charge, to any person -- obtaining a copy of this software and associated documentation -- files (the "Software"), to deal in the Software without -- restriction, including without limitation the rights to use...
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{-# OPTIONS --sized-types --show-implicit #-} module WrongSizeAssignment where postulate Size : Set _^ : Size -> Size ∞ : Size {-# BUILTIN SIZE Size #-} {-# BUILTIN SIZESUC _^ #-} {-# BUILTIN SIZEINF ∞ #-} data Empty : Set where data N : {_ : Size} -> Set where zero : N {∞} suc : forall {i} -> N ...
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{-# OPTIONS --cubical --no-import-sorts #-} open import Cubical.Foundations.Everything renaming (_⁻¹ to _⁻¹ᵖ; assoc to ∙-assoc) open import Function.Base using (_∋_; _$_) import Cubical.Algebra.Monoid as Std open import MorePropAlgebra.Bundles module MorePropAlgebra.Properties.Monoid {ℓ} (assumptions : Monoid {ℓ}) w...
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module _ (C Dummy : Set) where data Maybe : Set where nothing : Maybe IfJust : Maybe → Set → Set IfJust nothing P = P postulate qCtx : C inferRes : Maybe -- v-- Implicit is important here inferResBest : IfJust inferRes ({G : Maybe} → C) inferResBest with inferRes inferResBest...
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{-# OPTIONS --allow-unsolved-metas #-} {-# OPTIONS --no-double-check #-} -- {-# OPTIONS -v impossible:70 #-} -- {-# OPTIONS -v tc.interaction:30 #-} -- {-# OPTIONS -v tc.check.internal:20 #-} open import Agda.Builtin.Sigma record R1 (A : Set) : Set where instance prodR1 : {A : Set} {B : A → Set} → ⦃ {a : A} → R1 ...
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module DeclSN where open import Data.Sum open import Library open import Terms open import Substitution open import TermShape open import SN open import Reduction -- SN as accessibility data sn {Γ} {a} (t : Tm Γ a) : Set where acc : (∀ {t'} → t ⇒β t' → sn t') → sn t sn⇒β : ∀ {Γ} {a} {t t' : Tm Γ a} → sn t → t ⇒β...
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{-# OPTIONS --safe #-} module Definition.Typed.Consequences.TypeUnicity where open import Definition.Untyped hiding (U≢ℕ; U≢Π; U≢ne; ℕ≢Π; ℕ≢ne; Π≢ne; U≢Empty; ℕ≢Empty; Empty≢Π; Empty≢ne) open import Definition.Untyped.Properties using (subst-Univ-either) open import Definition.Typed open import Definition.Typed.Prope...
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import Data.Empty import Data.Unit import Data.Bool import Data.Nat import Data.Fin module SecondOrder.Arity where -- A notion of arity is given by a set of possible arities, and a mapping which to each arity assings a set of -- argument positions. record Arity : Set₁ where field arity : Set -- the s...
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------------------------------------------------------------------------------ -- Paradoxical combinator properties ------------------------------------------------------------------------------ {-# OPTIONS --exact-split #-} {-# OPTIONS --no-sized-types #-} {-# OPTIONS --no-universe-polymorphism...
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-------------------------------------------------------------------------------- -- Strings with markers, used for parsing -------------------------------------------------------------------------------- {-# OPTIONS --type-in-type #-} module Parse.MarkedString where open import Data.String using (fromList; fromChar;...
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{-# OPTIONS --without-K #-} module well-typed-syntax where infixl 2 _▻_ infixl 3 _‘’_ infixl 3 _‘’₁_ infixl 3 _‘’₂_ infixl 3 _‘’₃_ infixl 3 _‘’ₐ_ infixr 1 _‘→’_ infixl 3 _‘‘’’_ infixl 3 _w‘‘’’_ infixr 1 _‘‘→'’’_ infixr 1 _w‘‘→'’’_ mutual data Context : Set where ε : Context _▻_ : (Γ : Context) → Typ Γ → Con...
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{-# OPTIONS --safe #-} module Cubical.Algebra.CommRing.QuotientRing where open import Cubical.Foundations.Prelude open import Cubical.Data.Nat open import Cubical.Data.FinData open import Cubical.HITs.SetQuotients as SQ renaming (_/_ to _/sq_) open import Cubical.HITs.PropositionalTruncation as PT open import Cubic...
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{-# OPTIONS --cubical --safe #-} module Cubical.Foundations.Logic where import Cubical.Data.Empty as D import Cubical.Data.Prod as D import Cubical.Data.Sum as D import Cubical.Data.Unit as D open import Cubical.Foundations.Prelude open import Cubical.HITs.PropositionalTruncation open import Cubical.Foundation...
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------------------------------------------------------------------------ -- Propositional equality ------------------------------------------------------------------------ {-# OPTIONS --without-K --safe #-} module Equality.Propositional where open import Equality open import Logical-equivalence hiding (_∘_) open imp...
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-- Andreas, 2011-04-11 -- taken from test/succeed/HereditarilySingletonRecord.agda module FrozenMVar where data _≡_ {A : Set} (x : A) : A → Set where refl : x ≡ x data ℕ : Set where zero : ℕ suc : (n : ℕ) → ℕ one : ℕ one = _ force : one ≡ suc zero force = refl -- this tries to instantiate the frozen metavar...
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{-# OPTIONS --without-K --rewriting #-} open import HoTT module groups.KernelCstImage {i j k} {G : Group i} {H : Group j} (K : Group k) (φ : G →ᴳ H) (H-ab : is-abelian H) where open import groups.KernelImage {K = K} cst-hom φ H-ab open import groups.Cokernel φ H-ab Ker-cst-quot-Im : Ker/Im ≃ᴳ Coker Ker-...
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{- Pointed structure: X ↦ X -} {-# OPTIONS --cubical --no-import-sorts --safe #-} module Cubical.Structures.Relational.Pointed where open import Cubical.Foundations.Prelude open import Cubical.Foundations.Equiv open import Cubical.Foundations.Structure open import Cubical.Foundations.RelationalStructure open import ...
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open import Oscar.Prelude open import Oscar.Class open import Oscar.Data.𝟙 module Oscar.Class.Unit where module Unit {𝔞} (𝔄 : Ø 𝔞) = ℭLASS 𝟙 𝔄 module _ {𝔞} {𝔄 : Ø 𝔞} where ‼ = Unit.method 𝔄
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------------------------------------------------------------------------ -- A proof of univalence ------------------------------------------------------------------------ -- The code is based on code by Anders Mörtberg from Agda's reference -- manual or the cubical library. {-# OPTIONS --cubical --safe #-} module Eq...
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-- WARNING: This file was generated automatically by Vehicle -- and should not be modified manually! -- Metadata -- - Agda version: 2.6.2 -- - AISEC version: 0.1.0.1 -- - Time generated: ??? {-# OPTIONS --allow-exec #-} open import Vehicle open import Data.Product open import Data.Integer as ℤ using (ℤ) open impor...
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{-# OPTIONS --without-K #-} module VecHelpers where open import Data.Nat import Data.Fin as F open import Data.Vec open import Function renaming (_∘_ to _○_) open import Relation.Binary.PropositionalEquality open ≡-Reasoning infixl 10 _∘̬_ -- vector composition ------------------------------------------------...
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open import Data.Unit open import Data.Product open import Relation.Binary.PropositionalEquality module Spire.Examples.ComputationalDesc where ---------------------------------------------------------------------- data Desc (I : Set) : Set₁ where `⊤ : Desc I `X : (i : I) → Desc I `Σ `Π : (A : Set) (B : A → Desc...
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{-# OPTIONS --sized-types #-} open import Relation.Binary.Core module BBHeap.Height.Log {A : Set} (_≤_ : A → A → Set) (tot≤ : Total _≤_ ) where open import BBHeap _≤_ hiding (#) open import BBHeap.Height _≤_ open import BBHeap.Properties _≤_ open import Bound.Lower A open import...
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{-# OPTIONS --without-K --rewriting #-} open import lib.Basics open import lib.types.Sigma open import lib.types.Pi open import lib.types.Paths open import lib.types.Unit open import lib.types.Empty module lib.Equivalence2 where {- Pre- and post- composition with equivalences are equivalences -} module _ {i j k} {A ...
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-- This file defines instances for natural numbers and integers to -- overload various operations. {-# OPTIONS --without-K --safe #-} module Instances where -- imports from stdlib. import Data.Nat as Nat import Data.Integer as Int import Data.Integer.DivMod as IDM import Data.Nat.DivMod as NDM -- imports from local...
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{-# OPTIONS --safe #-} module Cubical.Algebra.Group.Instances.Unit where open import Cubical.Foundations.Prelude open import Cubical.Foundations.Isomorphism open import Cubical.Foundations.Structure open import Cubical.Foundations.HLevels open import Cubical.Foundations.Equiv open import Cubical.Data.Unit renaming (Un...
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