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------------------------------------------------------------------------
-- The Agda standard library
--
-- A categorical view of Stream
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe --sized-types #-}
module Codata.Stream.Categorical where
open import Data.Pr... | {
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module Issue470 where
data Bool : Set where
true false : Bool
_and_ : Bool → Bool → Bool
true and x = x
false and x = false
infixr 5 _∷_
data Foo : Bool → Set where
[] : Foo true
_∷_ : ∀ {b} (x : Bool) → Foo b → Foo (x and b)
Baz : Bool → Set
Baz true = Bool
Baz false = Foo false
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module Data.Permutation where
open import Prelude
open import Data.Fin as Fin hiding (_==_; _<_)
open import Data.Nat
open import Data.Vec
open import Logic.Identity
open import Logic.Base
import Logic.ChainReasoning
-- What is a permutation?
-- Answer 1: A bijection between Fin n and itself
data Permutation (n : N... | {
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{-# OPTIONS --without-K --rewriting #-}
open import HoTT
open import cohomology.Theory
module cw.cohomology.TopGrid {i} (OT : OrdinaryTheory i)
(n : ℤ) {X Y Z : Ptd i} (f : X ⊙→ Y) (g : Y ⊙→ Z) where
{-
X --> Y ---> Z
| | this |
v v one v
1 -> Y/X -> Z/X
-}
open OrdinaryTheory OT
open import ... | {
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module my-nat where
open import product
open import bool
open import eq
data ℕ : Set where
zero : ℕ
suc : ℕ → ℕ
{-# BUILTIN NATURAL ℕ #-}
infixl 10 _*_
infixl 9 _+_
--infixl 8 _<_ _=ℕ_ _≤_ _>_ _≥_
_+_ : ℕ → ℕ → ℕ
zero + n = n
suc m + n = suc (m + n)
0+ : ∀ (x : ℕ) → 0 + x ≡ x
0+ x = refl
+0 : ∀ (x : ℕ) → x +... | {
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module Common where
∗ : Set₁
∗ = Set
_∘_ : {A B C : ∗} → (B → C) → (A → B) → (A → C)
f ∘ g = λ a → f (g a)
infixr 2 _×_
infixr 1 _⊎_
data _×_ (A B : ∗) : ∗ where
_,_ : A → B → A × B
proj₁ : ∀ {A B} → A × B → A
proj₁ (a , b) = a
proj₂ : ∀ {A B} → A × B → B
proj₂ (a , b) = b
uncurry : {A... | {
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open import Agda.Builtin.Equality
open import Agda.Builtin.Size
postulate
P : (A : Set₁) → A → Set₁
p : (i : Size) (f : {_ : Size< i} → Set) (x : _) →
P ({_ : Size< i} → Set) f
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Foundations.Path where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.GroupoidLaws
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Transport
open import Cubical.F... | {
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-- Andreas, 2019-09-10, issue #4065, reported by nad
-- If the set of filtered unsolved constraints is empty, Agda
-- should print at least something like "Unsolved constraints"
postulate
A : Set
record R (F : A → Set) : Set where
module M (r : (F : A → Set) → R F) where
module T (F : _) = R (r (λ ℓ → F ℓ))
-... | {
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module _ (A : Set) where
id : A → A
id x = x
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{-# OPTIONS --without-K --safe --exact-split #-}
module Constructive.Axiom.Properties where
open import Constructive.Axiom.Properties.Base public
open import Constructive.Axiom.Properties.Transport public
open import Constructive.Axiom.Properties.Bool public
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module TemporalOps.OtherOps where
open import CategoryTheory.Instances.Reactive
open import CategoryTheory.Functor
open import CategoryTheory.Monad
open import CategoryTheory.Comonad
open import CategoryTheory.NatTrans
open import CategoryTheory.BCCCs
open import CategoryTheory.CartesianStrength
open import Temporal... | {
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primitive
primSetOmega : _
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-- Basic intuitionistic propositional calculus, without ∨ or ⊥.
-- Gentzen-style formalisation of syntax.
-- Normal forms, neutrals, and spines.
module BasicIPC.Syntax.GentzenSpinalNormalForm where
open import BasicIPC.Syntax.Gentzen public
-- Derivations.
mutual
-- Normal forms, or introductions.
infix 3 _⊢ⁿᶠ... | {
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module Agda.Builtin.FromNat where
open import Agda.Primitive
open import Agda.Builtin.Nat
record Number {a} (A : Set a) : Set (lsuc a) where
field
Constraint : Nat → Set a
fromNat : ∀ n → {{_ : Constraint n}} → A
open Number {{...}} public using (fromNat)
{-# BUILTIN FROMNAT fromNat #-}
{-# DISPLAY Numbe... | {
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open import Everything
module Test.Factsurj3 where
module Test0
{𝔵 𝔭 𝔯 ℓ} {𝔛 : Ø 𝔵}
(𝔓 : π̂ 𝔭 𝔛)
(_≈_ : ∀̇ π̂² ℓ 𝔓)
(_≈'_ : ∀̇ π̂² ℓ 𝔓)
(ℜ : π̂² 𝔯 𝔛)
(ε : Reflexivity.type ℜ)
(_◃_ : Smaphomarrow!.type ℜ 𝔓)
(_◃'_ : Smaphomarrow!.type ℜ 𝔓)
where
test-class' : ⦃ _ : Leftunit,smaphomarr... | {
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module Haskell.Prim.Foldable where
open import Haskell.Prim
open import Haskell.Prim.Num
open import Haskell.Prim.Eq
open import Haskell.Prim.List
open import Haskell.Prim.Int
open import Haskell.Prim.Bool
open import Haskell.Prim.Maybe
open import Haskell.Prim.Either
open import Haskell.Prim.Tuple
open import Haskel... | {
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{-
This file defines cohomology of a type with
coefficients in a dependent spectrum over it.
* Cohom and Cohom' are two versions of the cohomology groups
- the only difference is the carrier type.
* 'commDegreeΩ' commutes Ωs with degree shifts of the spectrum
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open import Agda.Primitive using (lzero; lsuc; _⊔_)
open import Relation.Binary.PropositionalEquality using (_≡_; refl)
open import SingleSorted.Substitution
open import Data.Nat using (ℕ; zero; suc)
import MultiSorted.Context as Context
module MultiSorted.Group where
data 𝒜 : Set₀ where
A : 𝒜
single-sort : ∀ (... | {
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Induction over _<_ for ℕᵇ.
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.Nat.Binary.Induction where
open import Data.Nat.Binary.Base
op... | {
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-- Andreas, 2018-05-28, issue #3095, cannot make hidden shadowing variable visible
data Nat : Set where
suc : {n : Nat} → Nat
data IsSuc : Nat → Set where
isSuc : ∀{n} → IsSuc (suc {n})
test : ∀{n} → IsSuc n → Set
test p = aux p
where
aux : ∀{n} → IsSuc n → Set
aux isSuc = {!n!} -- Split on n here
-- Con... | {
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{-# OPTIONS --cubical #-}
module PathWith where
open import Agda.Builtin.Cubical.Path
open import Agda.Primitive.Cubical
open import Agda.Builtin.Nat
pred : Nat → Nat
pred (suc n) = n
pred 0 = 0
f : Nat → Nat
f n with pred n
... | zero = zero
... | suc m = suc m
module _ (qqq : Nat) where
test1 : f ≡ pred
test... | {
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-- a few random examples
module Test where
import Agda.Builtin.Bool
data Nat : Set where zero : Nat -- Comment which gets eaten
suc : Nat -> Nat --Comment which is preserved
plus {- preserved comment {- which may be nested -} -} :
{- comment after Colon, also preserved-}
{-comments are essentia... | {
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module Structure.Category.NaturalTransformation.Equiv where
import Function.Equals
open Function.Equals.Dependent
import Lvl
open import Logic
open import Logic.Predicate
open import Structure.Category
open import Structure.Category.Functor
open import Structure.Category.NaturalTransformation
open imp... | {
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{-# OPTIONS --safe --warning=error --without-K #-}
open import Functions.Definition
open import Sets.CantorBijection.Proofs
module Sets.CantorBijection.CantorBijection where
open Sets.CantorBijection.Proofs using (cantorInverse ; cantorInverseLemma) public
cantorBijection : Bijection cantorInverse
Bijection.inj can... | {
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module UniDB.Basic where
open import UniDB.Spec
open import UniDB.Subst
open import Function
module _
{T : STX} {{vrT : Vr T}} {{wkT : Wk T}} {{apTT : Ap T T}} {{apVrT : ApVr T}} {{apWkTT : ApWk T T}}
{Ξ : MOR} {{upΞ : Up Ξ}} {{compΞ : Comp Ξ}} {{lkTΞ : Lk T Ξ}} {{lkUpTΞ : LkUp T Ξ}}
{{lkCompTΞ : LkCompAp T ... | {
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{-# OPTIONS --without-K --safe #-}
open import Algebra
open import Relation.Unary
open import Relation.Binary hiding (Decidable)
module Data.FingerTree.Split
{r m}
(ℳ : Monoid r m)
{s}
{ℙ : Pred (Monoid.Carrier ℳ) s}
(ℙ-resp : ℙ Respects (Monoid._≈_ ℳ))
(ℙ? : Decidable ℙ)
where
open import Relation.Nul... | {
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open import prelude renaming (_≟Nat_ to _≟_)
infixl 5 _$_
-- Copied pretty much verbatim
Var = Nat
data Term : Set where
var : Var → Term
fun : Term → Term
_$_ : Term → Term → Term
-- Doing full reduction, so values are either of the form
-- (fun v) or (x v1 ... vN)
data VarHeaded : Term → Set
data Value : ... | {
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module Data.Graph where
open import Data.List as List using (List; []; _∷_)
open import Data.Nat hiding (_⊔_)
open import Data.Product
open import Data.Vec as Vec using (Vec; []; _∷_)
open import Data.Vec.Properties
open import Level as ℓ using (Level; _⊔_)
open import Finite
open import Finite.Pigeonhole
open import ... | {
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{-# OPTIONS --safe #-}
module Cubical.Algebra.CommAlgebra.Instances.Unit where
open import Cubical.Foundations.Prelude
open import Cubical.Data.Unit
open import Cubical.Algebra.Ring
open import Cubical.Algebra.CommRing
open import Cubical.Algebra.CommRing.Instances.Unit
open import Cubical.Algebra.CommAlgebra
priva... | {
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factorial : ℕ → ℕ
factorial zero = 1
factorial (suc n) = suc n * factorial n
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{-# OPTIONS --cubical --no-import-sorts --no-exact-split --safe #-}
module Cubical.Data.Queue.Truncated2List where
open import Cubical.Foundations.Everything
open import Cubical.Foundations.SIP
open import Cubical.Structures.Queue
open import Cubical.Data.Maybe
open import Cubical.Data.List
open import Cubical.Data.... | {
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module Soundness where
{-
open import Relation.Binary.PropositionalEquality
open import Data.Nat hiding (_>_)
-}
open import StdLibStuff
open import Syntax
open import STT
open import FSC
open import DerivedProps
open import HelpDefs
mutual
sound : {n : ℕ} {Γ-t : Ctx n} (Γ : FSC-Ctx n Γ-t) (F : Form Γ-t $o) →
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{-# OPTIONS --show-implicit #-}
-- {-# OPTIONS -v tc.polarity:10 #-}
module Issue44 where
data Ty : Set where
ι : Ty
_⇒_ : Ty -> Ty -> Ty
data Con : Set where
ε : Con
_<_ : Con -> Ty -> Con
data Var : Con -> Ty -> Set where
vZ : forall {Γ σ} -> Var (Γ < σ) σ
vS : forall {Γ σ}{τ : Ty} -> Var Γ σ -> Var (Γ... | {
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open import Agda.Builtin.Nat
open import Agda.Builtin.FromNat
open import Data.Unit using (⊤)
open import Function using (const)
module Data.Nat.FromNat where
instance
NumberNat : Number Nat
NumberNat = record { Constraint = const ⊤ ; fromNat = λ n → n }
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module _ where
open import Agda.Builtin.IO
open import Agda.Builtin.String
open import Agda.Builtin.Unit
data D (A : Set) : Set where
L : A → D A
R : A → D A
data T : Set where
Var : (s : D String) → T
test : T → String
test (Var (L "abc")) = ""
test (Var (L s)) = ""
test (Var (R s)) = ""
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import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl)
open import Data.Nat using (ℕ; zero; suc)
open import Relation.Nullary using (Dec; yes; no)
data _≤_ : ℕ → ℕ → Set where
z≤n : ∀ {n : ℕ} → zero ≤ n
s≤s : ∀ {m n : ℕ} → m ≤ n → suc m ≤ suc n
_≤?_ : ∀ (m n : ℕ) → Dec (m ≤ n)
zero ≤? n = ye... | {
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module Matrix where
open import Data.Nat
open import Data.Vec
Matrix : (A : Set) → ℕ → ℕ → Set
Matrix A m n = Vec (Vec A m) n
transpose : ∀ {A m n} → Matrix A m n → Matrix A n m
transpose [] = replicate []
transpose (xs ∷ xss) = zipWith _∷_ xs (transpose xss)
a = (1 ∷ 2 ∷ 3 ∷ []) ∷ (4 ∷ 5 ∷ 6 ∷ []) ∷ []
b = transpo... | {
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-- Andreas, 2015-07-01, Issue 1599:
-- abstract data needs to be positivity checked!
-- {-# OPTIONS -v tc.pos:80 #-}
data ⊥ : Set where
abstract
T : Set → Set
T X = X → ⊥
data D : Set where
lam : T D → D
app : D → D → ⊥
app (lam f) = f
omega : D
omega = lam λ x → app x x
Omega : ⊥
Omega = ... | {
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-- submitted by Nisse, 2011-06-21
-- {-# OPTIONS -v tc.lhs.unify:15 --show-implicit #-}
module Issue423 where
-- import Common.Level
------------------------------------------------------------------------
-- Prelude
data _≡_ {A : Set} (x : A) : A → Set where
refl : x ≡ x
record Σ (A : Set) (B : A → Set) : Set ... | {
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{-# OPTIONS --cubical --safe --postfix-projections #-}
module Relation.Nullary.Decidable.Base where
open import Level
open import Data.Bool
open import Data.Empty
data Reflects {a} (A : Type a) : Bool → Type a where
ofʸ : (p : A) → Reflects A true
ofⁿ : (¬p : ¬ A) → Reflects A false
record Dec {a} (A :... | {
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{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Numbers.Naturals.Definition
open import Numbers.Naturals.Order
open import Numbers.Naturals.Naturals -- for length
open import Lists.Lists
open import Orders.Partial.Definition
open import Orders.Total.Definition
open import Fun... | {
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-- Andreas, 2017-11-12, issue #2803
-- Problem: names of hidden variable patterns
-- can get lost during case splitting.
-- They actually get lost already during lhs type checking,
-- but it is noticed only when printed back to the user
-- during case splitting.
-- {-# OPTIONS -v tc.lhs:40 #-}
record HFun (A B : Se... | {
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{-# OPTIONS --injective-type-constructors #-}
module InjectiveTypeConstructors where
data D (A : Set) : Set where
data _==_ (A : Set) : Set → Set where
refl : A == A
D' = D
injD : ∀ {A B} → D A == D' B → A == B
injD refl = refl
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------------------------------------------------------------------------
-- A quotient inductive-inductive definition of the lifting
-- construction on ω-cpos
------------------------------------------------------------------------
-- The code in this module is based on a suggestion from Paolo
-- Capriotti.
{-# OPTIO... | {
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{-# OPTIONS --without-K #-}
module Explore.Subset where
open import Relation.Binary.PropositionalEquality
-- TODO
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{-# OPTIONS --safe #-}
module Cubical.HITs.FreeAbGroup where
open import Cubical.HITs.FreeAbGroup.Base public
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-- Inductive types
-- See Chapter 15 of "Practical Foundations for Programming Languages" by Robert Harper
module ITypes where
open import Data.Vec
open import Data.Nat
open import Data.List
open import Data.Fin.Base
-- auxiliary
data Bool : Set where
True False : Bool
_if_else_ : {A : Set} → A → Bool → A → A
A i... | {
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module Numeric.Nat.BinarySearch where
open import Prelude
open import Numeric.Nat.Properties
open import Tactic.Nat
open import Control.WellFounded
open import Numeric.Nat.DivMod
data SearchResult! {a} (P : Nat → Set a) (lo hi : Nat) : Set a where
here : ∀ k (!pk : ¬ P k) (psk : P (suc k)) (lo≤k : lo ≤ k) (k<hi : ... | {
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-- Andreas, 2011-10-04
-- I'd like to infer the type of a even though it is irrelevant
-- E.g. when pressing C-c C-.
module InferIrrelevant where
f : (A : Set)(g : .A → A).(a : A) → A
f A g a = {!a!} | {
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{-
Copyright 2019 Lisandra Silva
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writin... | {
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module LecSigma where
open import CS410-Prelude
open import CS410-Nat
open import CS410-Functor
-- look in the Prelude
-- talk about *
-- talk about +
_N>_ : Nat -> Nat -> Set
m N> n = m N>= suc n
Fin : Nat -> Set
Fin n = Sg Nat \ m -> n N> m
foo : Fin 7
foo = 5 , <>
-- command-response systems
_<|_ : (C : Set)... | {
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open import Oscar.Class.Similarity
open import Oscar.Class.Reflexivity
open import Oscar.Class.Surjection
open import Oscar.Class.Leftunit
open import Oscar.Class.Smap
open import Oscar.Class.Symmetrical
open import Oscar.Class.Unit
open import Oscar.Data.Surjcollation
open import Oscar.Prelude
module Oscar.Class.Sma... | {
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module list-test where
open import bool
open import list
open import nat
open import list-to-string
open import bool-to-string
open import eq
test1 : 𝕃 𝔹
test1 = tt :: tt :: tt :: []
test2 : 𝕃 𝔹
test2 = ff :: ff :: []
test3 = test1 ++ test2
test-lem : test3 ≡ tt :: tt :: tt :: ff :: ff :: []
test-lem = refl
t... | {
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{-# OPTIONS --without-K --rewriting #-}
open import lib.Basics
module lib.types.Bool where
{-
data Bool : Type₀ where
true : Bool
false : Bool
{-# BUILTIN BOOL Bool #-}
{-# BUILTIN FALSE false #-}
{-# BUILTIN TRUE true #-}
-}
Bool = ⊤ ⊔ ⊤
pattern true = inl unit
pattern false = inr unit
⊙Bool : Ptd₀
⊙Bool = ... | {
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open import Coinduction using ( ∞ ; ♯_ ; ♭ )
open import Data.Product using ( _×_ ; _,_ ; proj₁ )
open import FRP.LTL.Time.Bound using
( Time∞ ; _≼_ ; _≺_ ; fin ; +∞ ; +∞-top
; ≼-refl ; _≼-trans_ ; ≡-impl-≽ ; ≺-impl-≼ ; ≺-impl-⋡ ; t≺+∞
; _≼-case_ ; lt ; eq ; gt )
open import FRP.LTL.Time.Interval using ( [_⟩ ;... | {
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module UselessAbstractPrimitive where
postulate Int : Set
{-# BUILTIN INTEGER Int #-}
abstract
primitive
primIntegerPlus : Int -> Int -> Int
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{-# OPTIONS --omega-in-omega --no-termination-check --overlapping-instances #-}
-- This library is licensed under 0BSD. You can read more about it at its GitHub repository: https://github.com/Zambonifofex/lightlib
module index where
-- The `Light.Libary` file exports modules referencing interface files and other mis... | {
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open import Everything
module Test.Test7 where
𝓅rop-id : ∀
{𝔵} {𝔛 : Ø 𝔵}
{𝔞} {𝔄 : 𝔛 → Ø 𝔞}
{𝔟} {𝔅 : 𝔛 → Ø 𝔟}
(let _∼_ = Arrow 𝔄 𝔅)
{ℓ̇} {_∼̇_ : ∀ {x y} → x ∼ y → x ∼ y → Ø ℓ̇}
⦃ _ : Transitivity.class _∼_ ⦄
⦃ _ : Reflexivity.class _∼_ ⦄
⦃ _ : Transleftidentity!.class _... | {
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{-
Types Summer School 2007
Bertinoro
Aug 19 - 31, 2007
Agda
Ulf Norell
-}
-- Now we're getting somewhere! Inductive families of datatypes.
module Families where
-- You can import modules defined in other files.
-- More details later...... | {
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{-# OPTIONS -Wall #-}
module noinfixdecl where
-- warning acts on data constructors
data #2 : Set where
one : #2
two : #2
neg_ : #2 → #2
-- warning acts on definitions
infixl 3 _⊓_
_⊔_ : #2 → #2 → #2
_⊓_ : #2 → #2 → #2
one ⊔ n = n
two ⊔ n = two
(neg v) ⊔ n = neg (v ⊓ neg n)
one ⊓ n = one
two ... | {
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{-# OPTIONS --without-K #-}
open import lib.Basics
open import lib.NType2
open import lib.types.Nat
open import lib.types.TLevel
open import lib.types.Empty
open import lib.types.Pi
open import lib.types.Sigma
open import lib.types.Truncation
open import lib.types.Pointed
open import lib.types.Group
open import lib.ty... | {
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module Human.Test where
open import Human.Humanity
-- sum 7
-- 0 + 1 + 2 + 3 + 4 + 5 + 6
-- function sum(x) {
-- var result = 0;
-- for (var i = 0; i < 10; ++i) {
-- result = result + i;
-- }
-- return result;
-- }
sum : Nat → Nat
sum i =
init 0
for i from 0 to 10 do:
λ result → result + 1
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------------------------------------------------------------------------
-- A small library of derived parser combinators
------------------------------------------------------------------------
module TotalParserCombinators.Lib where
open import Category.Monad
open import Codata.Musical.Notation
open import Function... | {
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{-
Definition of various kinds of categories.
This library follows the UniMath terminology, that is:
Concept Ob C Hom C Univalence
Precategory Type Type No
Category Type Set No
Univalent Category Type Set Yes
The most useful notion is Category and t... | {
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{-# OPTIONS --without-K --safe #-}
open import Algebra
open import Relation.Unary
open import Relation.Binary hiding (Decidable)
module Data.FingerTree.Split.StoredPredicate
{r m}
(ℳ : Monoid r m)
{s}
{ℙ : Pred (Monoid.Carrier ℳ) s}
(ℙ-resp : ℙ Respects (Monoid._≈_ ℳ))
(ℙ? : Decidable ℙ)
where
open Mon... | {
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------------------------------------------------------------------------
-- Commutative semirings with some additional structure ("almost"
-- commutative rings), used by the ring solver
------------------------------------------------------------------------
module Algebra.RingSolver.AlmostCommutativeRing where
open ... | {
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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 $≡... | {
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Keys for AVL trees -- the original key type extended with a new
-- minimum and maximum.
-----------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Rel... | {
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open import Data.Nat
ten : ℕ
ten = 10
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module Data.Collection.Inclusion where
open import Data.Collection.Equivalence
open import Data.Collection.Core
open import Function using (id; _∘_)
open import Function.Equivalence using (equivalence)
open import Level using (Level; suc; zero)
open import Relation.Unary hiding (_⇒_)
open import Relation.Binary
open ... | {
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{-# OPTIONS --without-K #-}
module well-typed-syntax-pre-interpreter where
open import common public
open import well-typed-syntax
open import well-typed-syntax-helpers
max-level : Level
max-level = lsuc lzero
module inner
(context-pick-if' : ∀ ℓ (P : Context → Set ℓ)
(Γ : Context)
... | {
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-- Andreas, 2016-06-20
-- Issue #191, reported 2009-12-02 by Nisse
data T (A : Set) : Set → Set1 where
c₁ : {B : Set} → T A B
c₂ : T A A
foo : {A B C : Set} → T A (B → C) → T A B → Set₁
foo c₁ y = Set
foo x y = {!y!}
-- WAS:
-- Perform the indicated case split. Agda will happily replace the
-- last ... | {
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------------------------------------------------------------------------
-- Equivalence relations
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Equality
module Equivalence-relation
{e⁺} (eq : ∀ {a p} → Equality-with-J a p e⁺) where
open Der... | {
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module Types4Crib where
open import Basics public
_<=_ : Nat -> Nat -> Set
ze <= y = One
su x <= ze = Zero
su x <= su y = x <= y
cmp : (x y : Nat) -> (x <= y) + (y <= x)
cmp ze y = inl <>
cmp (su x) ze = inr <>
cmp (su x) (su y) = cmp x y
data Bnd : Set where
bot : Bnd
# : Nat -> Bnd
top : Bnd
_<B=_ : Bnd ->... | {
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{-# OPTIONS --safe #-}
module Cubical.Algebra.AbGroup.Instances.Direct-Sum where
open import Cubical.Foundations.Prelude
open import Cubical.Algebra.AbGroup
open import Cubical.Algebra.Direct-Sum.Base
open import Cubical.Algebra.Direct-Sum.Properties
open import Cubical.Algebra.Polynomials.Multivariate.Base
privat... | {
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-- Testing the version option when used without type-check the file.
--
-- N.B. It is necessary to change the Issue1244b.out file when using
-- different versions of Agda.
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-- Andreas, 2019-04-30, issue #3731
-- Compiler backend: Do not look for a main function if not the main module.
open import Agda.Builtin.Nat
module Issue3731 where
module M where
module main where
record R : Set where
field
main : Nat
data Main : Set where
main : Main
module N where
data mai... | {
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module Function.Multi.Functions where
open import Data
import Data.Option.Functions as Option
open import Data.Tuple renaming (curry to curry₁ ; uncurry to uncurry₁) using (_⨯_ ; _,_)
open import Data.Tuple.Raise
import Data.Tuple.Raiseᵣ.Functions as Raise
open import Data.Tuple.RaiseTypeᵣ
open import Data.T... | {
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open import Mockingbird.Forest using (Forest)
-- Gödel's Forest
module Mockingbird.Problems.Chapter17 {b ℓ} (forest : Forest {b} {ℓ}) where
open import Data.Empty using (⊥-elim)
open import Data.Product using (_×_; _,_; proj₁; proj₂; ∃-syntax)
open import Data.Sum using (_⊎_; inj₁; inj₂)
open import Function using (_... | {
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------------------------------------------------------------------------
-- A definitional interpreter
------------------------------------------------------------------------
{-# OPTIONS --sized-types #-}
module Lambda.Delay-monad.Interpreter where
open import Equality.Propositional
open import Prelude
open import ... | {
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{-
This file contains:
Properties of FreeGroupoid:
- Induction principle for the FreeGroupoid on hProps
- ∥freeGroupoid∥₂ is a Group
- FreeGroup A ≡ ∥ FreeGroupoid A ∥₂
-}
{-# OPTIONS --safe #-}
module Cubical.HITs.FreeGroupoid.Properties where
open import Cubical.HITs.FreeGroupoid.Base
open import Cubical.HITs.Fr... | {
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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
-}
-- This is a selection of useful functions and definitions
-- from the ... | {
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module Data.List.Primitive where
-- In Agda 2.2.10 and below, there's no FFI binding for the stdlib
-- List type, so we have to roll our own. This will change.
data #List (X : Set) : Set where
[] : #List X
_∷_ : X → #List X → #List X
{-# COMPILED_DATA #List [] [] (:) #-}
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{-# OPTIONS --safe --experimental-lossy-unification #-}
-- This file could be proven using the file Sn
-- However the proofs are easier than in Sn
-- And so kept for pedagologic reasons
module Cubical.ZCohomology.CohomologyRings.S1 where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Functio... | {
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{-# OPTIONS --type-in-type
--no-termination-check
--no-positivity-check #-}
module IMDesc where
--********************************************
-- Prelude
--********************************************
-- Some preliminary stuffs, to avoid relying on the stdlib
--****************
-- Sigma and ... | {
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module Lib.Fin where
open import Lib.Nat
open import Lib.Bool
open import Lib.Id
data Fin : Nat -> Set where
zero : {n : Nat} -> Fin (suc n)
suc : {n : Nat} -> Fin n -> Fin (suc n)
fromNat : (n : Nat) -> Fin (suc n)
fromNat zero = zero
fromNat (suc n) = suc (fromNat n)
toNat : {n : Nat} -> Fin n -> Nat
toN... | {
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------------------------------------------------------------------------
-- Integer division
------------------------------------------------------------------------
module Data.Nat.DivMod where
open import Data.Nat
open import Data.Nat.Properties
open SemiringSolver
open import Data.Fin as Fin using (Fin; zero; suc;... | {
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{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
-- {-# OPTIONS --without-K #-}
-- [1] Hofmann, Martin and Thomas Streicher (1998). “The groupoid
-- interpretation on type theory”. In: Twenty-five Years of
-- Constru... | {
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-- Jesper, 2019-05-20: When checking confluence of two rewrite rules,
-- we disable all reductions during unification of the left-hand
-- sides. However, we should not disable reductions at the type-level,
-- as shown by this (non-confluent) example.
{-# OPTIONS --rewriting --confluence-check #-}
open import Agda.Bui... | {
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module UniDB.Morph.Weaken where
open import UniDB.Spec
--------------------------------------------------------------------------------
data Weaken : MOR where
weaken : {γ : Dom} (δ : Dom) → Weaken γ (γ ∪ δ)
instance
iLkWeaken : {T : STX} {{vrT : Vr T}} → Lk T Weaken
lk {{iLkWeaken}} (weaken δ) i = vr (wk δ ... | {
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-- Jesper, 2017-01-23: when instantiating a variable during unification,
-- we should check that the type of the variable is equal to the type
-- of the equation (and not just a subtype of it). See Issue 2407.
open import Agda.Builtin.Equality
open import Agda.Builtin.Size
data D : Size → Set where
J= : ∀ {ℓ} {s : ... | {
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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: ???
open import AISEC.Utils
open import Data.Real as ℝ using (ℝ)
open import Data.List
module MyTestModule where
f : Tensor ℝ (... | {
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module Numeral.Natural.Oper.Proofs.Structure where
open import Logic.Predicate
open import Numeral.Natural
open import Numeral.Natural.Oper.Proofs
open import Numeral.Natural.Oper
open import Relator.Equals
open import Relator.Equals.Proofs
open import Structure.Operator.Monoid
instance
[+]-monoid : Monoid(_+_)
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{-# OPTIONS --without-K #-}
module Computability.Function where
open import Computability.Prelude
import Function
variable
l₀ l₁ : Level
Injective : {A : Set l₀}{B : Set l₁} → (A → B) → Set _
Injective = Function.Injective _≡_ _≡_
Surjective : {A : Set l₀}{B : Set l₁} → (A → B) → Set _
Surjective {A = A} {B = B} ... | {
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{-# OPTIONS --safe #-}
module Cubical.HITs.FreeGroup where
open import Cubical.HITs.FreeGroup.Base public
open import Cubical.HITs.FreeGroup.Properties public
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mutual
data ℕμ : Set where
zero : ℕμ
succ : ℕμ → ℕμ
delay : ℕν → ℕμ
record ℕν : Set where
coinductive
field step : ℕμ
open ℕν public
mutual
data _≤μ_ : ℕμ → ℕμ → Set where
-- Standard part of the order
le₀ : (y : ℕμ) → zero ≤μ y
leₛ : {x y : ℕμ} → x ≤μ y → succ x ≤μ succ y
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module Issue329b where
abstract
infixl 0 D Undeclared
data D : Set where
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{-# OPTIONS --without-K --safe #-}
open import Level hiding (zero)
open import Relation.Binary
module SDG.Extra.OrderedAlgebra.Structures {a ℓ} {A : Set a} (_≈_ : Rel A ℓ) where
open import Algebra.Structures
open import Algebra.FunctionProperties
record IsOrderedCommutativeRing
(_<_ : Rel A ℓ)
(_+_ _*_ : ... | {
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{-# OPTIONS --no-termination-check #-}
module PiPath where
open import Level
open import Data.Empty
open import Data.Unit
open import Data.Sum
open import Data.Product
open import Relation.Binary.PropositionalEquality
------------------------------------------------------------------------------
-- Universe of types... | {
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