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module Structure.Function.Proofs where
import Lvl
open import Functional
open import Structure.Function
open import Structure.Operator
open import Structure.Operator.Properties
open import Structure.Setoid
open import Syntax.Transitivity
open import Type
private variable ℓ ℓₑ₁ ℓₑ₂ ℓₑ₃ ℓₑ₄ : Lvl.Level
private var... | {
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module PredicateLifting where
import Level
open import Data.Empty
open import Data.Unit as Unit
open import Data.Nat
open import Data.List as List renaming ([] to Ø; [_] to [_]L)
open import NonEmptyList as NList
open import Data.Fin hiding (_+_)
open import Data.Vec as Vec renaming ([_] to [_]V; _++_ to _++V_)
open i... | {
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{-# OPTIONS --without-K #-}
-- Universes
open import Agda.Primitive
module main where
-- I don't really understand exactly which equality gets imported in agda if
-- I try to import the builtin, so define the sane MLTT one here.
infixl 4 _≡_
data _≡_ {n : Level} {A : Set n} : A → A → Set n where
refl : {a : A} → ... | {
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{-# OPTIONS --without-K --rewriting #-}
open import HoTT
module stash.modalities.FiberedPushout {i j k} {X : Type i} {Y : Type j} where
{-
X ---glue-left--- P ---glue-right--- Y
-}
module _ (P : X → Y → Type k) where
fibered-pushout-span : Span
fibered-pushout-span = record
{ A = Σ X λ x → ... | {
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------------------------------------------------------------------------
-- The Agda standard library
--
-- A Categorical view of the Sum type (Right-biased)
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Level
module Data.Sum.Categorical.Right... | {
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module MJ.Syntax.Scoped where
open import Prelude
open import Data.Maybe as Maybe using (Maybe; just)
open import Data.Maybe.All as MayAll
open import Data.Vec as Vec hiding (_∈_)
open import Data.Star.Indexed
open import Data.List
open import Data.List.Properties.Extra
open import Data.List.Prefix
open import Data.Li... | {
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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
-}
import LibraBFT.Base.KVMap as Map
open import... | {
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{-# OPTIONS --allow-unsolved-metas #-} -- FIXME
open import Everything
module Test.Test1 where
test-functor-transextensionality : ∀
{𝔬₁ 𝔯₁ ℓ₁ 𝔬₂ 𝔯₂ ℓ₂}
⦃ functor : Functor 𝔬₁ 𝔯₁ ℓ₁ 𝔬₂ 𝔯₂ ℓ₂ ⦄
(open Functor functor)
→ {!Transextensionality!.type _∼₁_ _∼̇₁_!}
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Ways to give instances of certain structures where some fields can
-- be given in terms of others
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open... | {
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{-# OPTIONS --cubical --safe #-}
module Data.Unit.Properties where
open import Data.Unit
open import Level
open import HLevels
isProp⊤ : isProp ⊤
isProp⊤ _ _ i = tt
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module ExtendedLambdaCase where
data Bool : Set where
true false : Bool
data Void : Set where
foo : Bool -> Bool -> Bool -> Bool
foo = λ { x → λ { y z → {!!} } }
data Bar : (Bool -> Bool) -> Set where
baz : (t : Void) -> Bar λ { x → {!!} }
-- with hidden argument
data Bar' : (Bool -> Bool) -> Set where
baz' ... | {
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{-# OPTIONS --without-K #-}
-- Expose all the core types and theorems.
module hott.core.theorems where
open import hott.core public
open import hott.core.equality.theorems public
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Bounded vectors, basic types and operations
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.Vec.Bounded.Base where
open import Level usin... | {
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{-# OPTIONS --erased-cubical --no-main --save-metas #-}
open import Agda.Builtin.Bool
open import Erased-cubical-Cubical
postulate
f : Not-compiled → Bool
-- It is at the time of writing not possible to give a COMPILE GHC
-- pragma for f, because Not-compiled is not compiled.
{-# COMPILE GHC f = \_ -> True #-}
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{-# OPTIONS --copatterns --sized-types --without-K #-}
module PolyFinalCoalg where
open import Data.Product
open import Data.Nat
open import Data.Fin
open import Data.Unit as Unit
open import Data.Empty
open import Data.Vec hiding (_∈_; [_])
open import Relation.Binary.PropositionalEquality
open import Function
open i... | {
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------------------------------------------------------------------------------
-- Properties related with lists (using induction on the FOTC lists type)
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
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module Thesis.BigStepSILR2 where
open import Data.Empty
open import Data.Unit.Base hiding (_≤_)
open import Data.Product
open import Relation.Binary.PropositionalEquality
open import Relation.Binary hiding (_⇒_)
open import Data.Nat -- using (ℕ; zero; suc; decTotalOrder; _<_; _≤_)
open import Data.Nat.Properties
open ... | {
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{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Groups.Homomorphisms.Definition
open import Groups.Definition
open import Setoids.Setoids
open import Sets.EquivalenceRelations
open import Rings.Definition
open import Rings.Homomorphisms.Definition
open import Groups.Homomorph... | {
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------------------------------------------------------------------------------
-- Propositional equality
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OP... | {
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{-
Importing and re-exporting this module allows for (constrained) natural number
and negative integer literals for any type (e.g. Int, ℕ₋₁, ℕ₋₂, ℕ₊₁).
-}
{-# OPTIONS --cubical --no-import-sorts --no-exact-split --safe #-}
module Cubical.Data.Nat.Literals where
open import Agda.Builtin.FromNat public
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{-# OPTIONS --without-K --rewriting #-}
open import HoTT
open import homotopy.LoopSpaceCircle
-- This file is temporarily put in the cw/ directory for
-- development, but it actually belongs to somewhere else.
module homotopy.SphereEndomorphism where
⊙SphereS-endo-out : ∀ n
→ Trunc 0 (⊙Sphere (S n) ⊙→ ⊙Sphere... | {
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open import Common.Prelude
open import Common.Reflect
open import Common.Equality
postulate
trustme : ∀ {a} {A : Set a} {x y : A} → x ≡ y
magic : Term → Term
magic _ = def (quote trustme) []
id : ∀ {a} {A : Set a} → A → A
id x = x
science : Term → Term
science _ = def (quote id) []
by-magic : ∀ n → n + 4 ≡ 3
by... | {
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-- 2010-09-06 Andreas
module IrrelevantApplication where
-- an unknown function that does not use its second argument
postulate
f : {A B : Set} -> A -> .B -> A
data _==_ {A : Set}(a : A) : A -> Set where
refl : a == a
-- the second argument is irrelevant for equality
proofIrr : {A : Set}{x y z : A} -> f x y =... | {
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-- Andreas, 2017-10-08, issue #2594, reported by gallais
-- Incomplete pattern matching should be reported as such,
-- rather than crashing on a split error.
-- {-# OPTIONS -v tc.cover:10 #-}
open import Agda.Builtin.Bool
dispatch : Bool → Set
dispatch true = Bool
dispatch false = Bool
argh : (b : Bool) → dispatc... | {
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{-# OPTIONS --without-K --safe #-}
open import Definition.Typed.EqualityRelation
module Definition.LogicalRelation.Substitution.Introductions.Pi {{eqrel : EqRelSet}} where
open EqRelSet {{...}}
open import Definition.Untyped as U hiding (wk)
open import Definition.Untyped.Properties
open import Definition.Typed
open... | {
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{-# OPTIONS --safe --warning=error --without-K #-}
open import Groups.Abelian.Definition
open import Groups.Definition
open import Setoids.Setoids
open import Sets.EquivalenceRelations
open import Lists.Lists
module Groups.Polynomials.Group {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} (G : Group S ... | {
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{-# OPTIONS --two-level #-}
open import Agda.Primitive
data Unit : Set where tt : Unit
data Unitω : Setω where tt : Unitω
data SUnit : SSet where tt : SUnit
data SUnitω : SSetω where tt : SUnitω
-- Cannot eliminate fibrant type Unit
-- unless target type is also fibrant
-- f1 : Unit → SUnit
-- f1 tt = tt
-- Cannot ... | {
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open import Common.Prelude
infixr 9 _∘_
_∘_ : {A B C : Set} → (B → C) → (A → B) → (A → C)
f ∘ g = λ x → f (g x)
test : Nat → Nat
test = _* 5 ∘ 6 +_ ∘ 2 ∸_
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{-# OPTIONS --erased-cubical --no-subtyping #-}
module Agda.Primitive.Cubical where
{-# BUILTIN CUBEINTERVALUNIV IUniv #-} -- IUniv : SSet₁
{-# BUILTIN INTERVAL I #-} -- I : IUniv
{-# BUILTIN IZERO i0 #-}
{-# BUILTIN IONE i1 #-}
-- I is treated as the type of booleans.
{-# COMPILE JS i0 = false #-}
{-# CO... | {
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module *-example where
import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl)
open Eq.≡-Reasoning using (begin_; _≡⟨⟩_; _∎)
open import Naturals using (_*_)
*-example : 3 * 4 ≡ 12
*-example =
begin
3 * 4
≡⟨⟩
4 + (2 * 4)
≡⟨⟩
4 + (4 + (1 * 4))
≡⟨⟩
4 + (4 + (4 + (0 * 4)))
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open import x1-Base
module x3-PropositionalEquality where
data _≡_ {X} : Rel X where
refl : ∀ {x} → x ≡ x
{-
What propositional equality means.
DEFINITIONAL EQUALITY
To decide if two types/terms are "the same", it reduces them to their normal form,
then compares them syntactically, plus some additional laws.
Eve... | {
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------------------------------------------------------------------------
-- A library of parser combinators
------------------------------------------------------------------------
-- This module also provides examples of parsers for which the indices
-- cannot be inferred.
module StructurallyRecursiveDescentParsing.... | {
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{-# OPTIONS --without-K --safe #-}
open import Axiom.Extensionality.Propositional using (Extensionality)
module Cats.Category.Sets.Facts.Exponential
(funext : ∀ {a b} → Extensionality a b)
where
open import Data.Product using (_×_ ; _,_ ; proj₁ ; proj₂)
open import Relation.Binary.PropositionalEquality using (_≡_... | {
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{-# OPTIONS --without-K --rewriting #-}
open import lib.Basics hiding (take; drop; take-drop-split)
open import lib.types.Pointed
open import lib.types.Paths
module lib.types.FunctionSeq where
infixr 80 _◃∘_
data FunctionSeq {i} : Type i → Type i → Type (lsucc i) where
idf-seq : {A : Type i} → FunctionSeq A A
_◃... | {
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------------------------------------------------------------------------
-- The Agda standard library
--
-- The Colist type and some operations
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe --sized-types #-}
module Codata.Colist where
open import Size
open im... | {
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-- Copyright: (c) 2016 Ertugrul Söylemez
-- License: BSD3
-- Maintainer: Ertugrul Söylemez <esz@posteo.de>
module Algebra.Group where
open import Algebra.Group.Group public
open import Algebra.Group.Monoid public
open import Algebra.Group.Semigroup public
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{-# OPTIONS --without-K #-}
module Agda.Builtin.Coinduction where
infix 1000 ♯_
{-# BUILTIN INFINITY ∞ #-}
{-# BUILTIN SHARP ♯_ #-}
{-# BUILTIN FLAT ♭ #-}
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{-# OPTIONS --without-K --safe #-}
open import Categories.Category
{-
Various combinators for working with Isomorphisms in the
context of morphism equalities
both for Category (Switch) and IsGroupoid (GroupoidR)
-}
module Categories.Morphism.Reasoning.Iso {o ℓ e} (C : Category o ℓ e) where
open import Level
op... | {
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties of operations on floats
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.Float.Properties where
open import Data.Bool.Base as B... | {
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open import Agda.Builtin.Equality
cong : ∀ {A B : Set} → (f : A → B) → ∀ {x y} → x ≡ y → f x ≡ f y
cong f refl = refl
data ⊥ : Set where
data N : Set where
ze : N
su : (⊥ → N) → N
foo : N
foo = su (\ ())
postulate
ext : ∀ {A : Set} → (f g : ⊥ → A) → f ≡ g
foo-suc : foo ≡ su (\ _ → foo)
foo-suc = cong su (e... | {
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module Automaton.Deterministic.Oper where
open import Automaton.Deterministic
open import Logic.Propositional
import Lvl
open import Data.Boolean
import Data.Boolean.Operators
open Data.Boolean.Operators.Programming
open import Data.List renaming (∅ to [])
open import Data.Tuple as Tuple using (_,_) r... | {
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open import Agda.Builtin.Equality
postulate
A : Set
id : A → A
id x = x
mutual
?Y : A → A → A
?Y = _
?X : A → A → A
?X = _
-- This tries to solve ?X x x := ?Y x y, which fails but prunes
-- ?Y x y := ?Z x. Failing ?X := ?Y we then try the other direction
-- without realising that ?Y has been insta... | {
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open import Prelude hiding (_<_)
module Implicits.Resolution.GenericFinite.TerminationCondition
where
open import Induction.WellFounded
open import Implicits.Syntax
record TerminationCondition : Set₁ where
field
TCtx : Set
_<_ : TCtx → TCtx → Set
_<?_ : (x y : TCtx) → Dec (x < y)
step : ∀ {ν} →... | {
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{-# OPTIONS --without-K --safe #-}
module Categories.Category.Finite where
open import Level
open import Data.Product using (Σ; _,_; ∃₂)
open import Function.Equality using (Π; _⟶_)
open import Function.Inverse
open import Data.Nat as ℕ hiding (_⊔_)
import Data.Fin as Fin
open import Data.Vec as Vec
open import Relat... | {
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Some properties imply others
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Relation.Unary.Consequences where
open import Relation.Unary
open... | {
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{-# OPTIONS --without-K --safe #-}
module Cats.Category.Presheaves where
open import Level using (_⊔_ ; suc)
open import Cats.Category
open import Cats.Category.Fun using (Fun)
open import Cats.Category.Op using (_ᵒᵖ)
open import Cats.Category.Setoids using (Setoids)
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{-# OPTIONS --without-K #-}
infixr 1 _‘→’_
record ⊤ : Set where
constructor tt
data ⊥ : Set where
mutual
data Type : Set where
_‘→’_ : Type → Type → Type
‘□’ : Type → Type
‘⊤’ : Type
‘⊥’ : Type
data □ : Type → Set where
Lӧb : ∀ {X} → □ (‘□’ X ‘→’ X) → □ X
‘tt’ : □ ‘⊤’
mutual
⌞_⌟ : T... | {
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------------------------------------------------------------------------
-- Truncation, defined as a HIT
------------------------------------------------------------------------
{-# OPTIONS --erased-cubical --safe #-}
-- The beginning of this module follows the HoTT book rather closely.
-- The module is parametrised... | {
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open import Prelude
module Implicits.Resolution.Termination where
open import Implicits.Resolution.Termination.SizeMeasures public
open import Implicits.Resolution.Termination.Stack public
open import Implicits.Resolution.Termination.Lemmas public
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-- This combination should not be allowed:
{-# OPTIONS --guardedness --sized-types --safe #-}
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module Relation.Ternary.Separation.Morphisms where
open import Level
open import Function
open import Relation.Unary
open import Relation.Binary.PropositionalEquality
open import Relation.Ternary.Separation
open import Data.Product
open import Function using (_∘_)
record Morphism {a b} (A : Set a) (B : Set b)
{{r :... | {
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------------------------------------------------------------------------------
-- Common (interactive and automatic) properties using the induction principle
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types ... | {
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module Proof where
-- we give an example of a structure that satisfies the monad laws, but does not have a join operation.
open import Relation.Binary.PropositionalEquality
F : Set → Set₁
F x = Set → x
return : ∀ {x} → x → F x
return = λ z _ → z
_>>=_ : ∀ {x y} -> F x → (x -> F y) → F y
a >>= f = λ z → f (a z) z
-... | {
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module ExplicitImplicitConversion where
open import ExpressionChanging
open import Data.Nat
open import Data.List
open import ParseTree
open import ScopeState
open import Data.Vec using (Vec ; _∷_)
open import Data.Fin
open import Data.List.NonEmpty
open import AgdaHelperFunctions
open import Data.Bool
open import Pars... | {
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{-# OPTIONS --with-K #-}
open import Agda.Builtin.Equality
open import Agda.Builtin.Nat
single : {m n : Nat} → suc m ≡ suc n → n ≡ m
single p with refl ← p = refl
double : {m n p : Nat} → suc m ≡ n → suc n ≡ 2 + p → m ≡ p
double p q with refl ← p | refl ← q = refl
_∋_ : (A : Set) → A → A
A ∋ a = a
-- The second eq... | {
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open import Prelude
open import dynamics-core
module lemmas-matching where
-- arrow matching produces unique answers
▸arr-unicity : ∀{τ τ2 τ3} →
τ ▸arr τ2 →
τ ▸arr τ3 →
τ2 == τ3
▸arr-unicity MAHole MAHole = refl
▸arr-unicity MAArr MAArr = refl
-- only con... | {
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module Haskell.Prim.Ord where
open import Agda.Builtin.Nat as Nat hiding (_==_; _<_)
open import Agda.Builtin.Char
open import Haskell.Prim
open import Haskell.Prim.Eq
open import Haskell.Prim.Bool
open import Haskell.Prim.Int
open import Haskell.Prim.Word
open import Haskell.Prim.Integer
open import Haskell.Prim.Do... | {
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{-# OPTIONS --safe --experimental-lossy-unification #-}
module Cubical.Algebra.Polynomials.UnivariateHIT.Polyn-nPoly where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.HLevels
open import Cubical.Data.Nat rena... | {
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{-# OPTIONS --without-K --safe #-}
module Data.Unit where
open import Agda.Builtin.Unit public
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{-# OPTIONS --experimental-irrelevance #-}
-- Andreas, 2011-04-15
-- {-# OPTIONS -v tc.data:20 #-}
module IrrelevantDataParameter where
postulate
A : Set
data K .(a : A) : Set where
c : K a
postulate
a : A
data K' .(b : A) : Set where
c : K' a
-- ok, since parameter irrelevant
-- 2011-09-09
postulate
_... | {
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module Structure.Operator.Functions where
import Lvl
open import Logic
open import Structure.Setoid
open import Structure.Operator.Properties
open import Type
private variable ℓ ℓ₁ ℓ₂ ℓ₃ ℓₑ ℓₑ₁ ℓₑ₂ ℓₑ₃ : Lvl.Level
module _ {A : Type{ℓ₁}} {X : Type{ℓ₂}} ⦃ equiv-X : Equiv{ℓₑ₁}(X) ⦄ {Y : Type{ℓ₃}} ⦃ equiv-Y : Equi... | {
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open import Relation.Binary.Core
module Heapsort.Impl1.Correctness.Order {A : Set}
(_≤_ : A → A → Set)
(tot≤ : Total _≤_)
(trans≤ : Transitive _≤_) where
open import Data.List
open import Function using (_∘_)
open import Heapsort.Impl1 _≤_ tot≤ trans≤
open impor... | {
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module Data.Option.Functions where
import Lvl
open import Data
open import Data.Boolean
open import Data.Either as Either using (_‖_)
open import Data.Option
open import Data.Tuple as Tuple using (_⨯_ ; _,_)
open import Type
private variable ℓ : Lvl.Level
private variable T A B T₁ T₂ T₃ : Type{ℓ}
-- Applies a f... | {
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------------------------------------------------------------------------
-- Partiality algebra categories
------------------------------------------------------------------------
{-# OPTIONS --cubical --safe #-}
module Partiality-algebra.Category where
open import Equality.Propositional.Cubical
open import Logical-e... | {
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module _ where
open import Common.Prelude hiding (_>>=_)
open import Common.Reflection
open import Agda.Builtin.Sigma
infix -100 This:_ this:_
data This:_ {a} {A : Set a} : A → Set where
this:_ : ∀ x → This: x
macro
runT : Tactic → Tactic
runT m = m
evalT : ∀ {a} {A : Set a} → TC A → Tactic
evalT m hole ... | {
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{-# OPTIONS --without-K --safe #-}
-- The 'original' version of Lawvere Theory, based on
-- Nat^op and IOO functors. Contrast with the weak version at
-- https://ncatlab.org/nlab/show/Lawvere+theory
-- Unfortunately, many results on the weak version are not in
-- the literature, so doing that development would be new ... | {
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{-# OPTIONS --without-K --safe #-}
module Magma where
open import Magma.Bundles public
open import Magma.Structures public
open import Magma.Definitions public
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-- Andreas, 2015-02-24, issue reported by g.x.allais
-- {-# OPTIONS -v interaction.give:100 #-}
record R : Set1 where
field
-v : Set
goal : R
goal = {!!}
-- refine here
-- WAS: error due to rendering or record as
-- record {-v = ?}
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{-
ℤ is a Commutative Ring (using QuoInt)
-}
{-# OPTIONS --safe #-}
module Cubical.Algebra.CommRing.Instances.QuoInt where
open import Cubical.Foundations.Prelude
open import Cubical.Algebra.CommRing
open import Cubical.Data.Nat using (ℕ ; zero ; suc)
open import Cubical.Data.Bool using (not)
open import Cubical.D... | {
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-- Intuitionistic propositional calculus.
-- Common syntax.
module IPC.Syntax.Common where
open import Common.Context public
-- Types, or propositions.
infixl 9 _∧_
infixl 8 _∨_
infixr 7 _▻_
data Ty : Set where
α_ : Atom → Ty
_▻_ : Ty → Ty → Ty
_∧_ : Ty → Ty → Ty
⊤ : Ty
⊥ : Ty
_∨_ : Ty → Ty → Ty
-... | {
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{-# OPTIONS --sized-types #-}
module CoSig where
open import Data.Product as Prod
open import Data.Fin
open import Data.Unit
open import Data.Empty
open import Data.Sum
open import Data.Nat
open import Size
open import Function
open import Relation.Binary.PropositionalEquality using (_≡_; refl; subst)
---- We don't ... | {
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{-# OPTIONS --without-K --safe #-}
open import Categories.Category
open import Categories.Functor.Bifunctor
module Categories.Diagram.End {o ℓ e o′ ℓ′ e′} {C : Category o ℓ e} {D : Category o′ ℓ′ e′}
(F : Bifunctor (Category.op C) C D) where
private
module C = Category C
module D = Category D
open D
open H... | {
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{- Byzantine Fault Tolerant Consensus Verification in Agda, version 0.9.
Copyright (c) 2021, Oracle and/or its affiliates.
Licensed under the Universal Permissive License v 1.0 as shown at https://opensource.oracle.com/licenses/upl
-}
open import LibraBFT.ImplShared.Consensus.Types
module LibraBFT.Impl.IO.OBM.... | {
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{- PROOF OF BOUNDING THEOREM -}
open import Preliminaries
open import Source
open import Complexity
open import Translation
open import Bounding-Lemmas
module Bounding where
boundingRec : ∀ {τ} (v : [] Source.|- nat) (val-v : val v)
(e0 : [] Source.|- τ)
(e1 : (nat :: su... | {
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------------------------------------------------------------------------------
-- Properties related with lists of natural numbers
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-univer... | {
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module Issue847 where
data ⊥ : Set where
bad : ⊥
bad = bad′
where
abstract
bad′ : ⊥
bad′ = bad
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open import Logic.Classical
open import Structure.Real
module Structure.Real.Abs {ℓ₁ ℓ₂ ℓₑ} {R} ⦃ R-equiv ⦄ (_+_) (_⋅_) (_≤_) ⦃ classical : ∀{ℓ}{P} → Classical{ℓ}(P) ⦄ ⦃ reals : RealTheory{ℓ₁}{ℓₑ}{ℓ₂} {R} ⦃ R-equiv ⦄ (_+_)(_⋅_)(_≤_) ⦄ where
open RealTheory(reals)
open import Data.Boolean
import Lvl
open import F... | {
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{-# OPTIONS --without-K --safe #-}
open import Categories.Category
open import Categories.Object.Zero
module Categories.Object.Kernel.Properties {o ℓ e} {𝒞 : Category o ℓ e} (𝒞-Zero : Zero 𝒞) where
open import Function using (_$_)
open import Categories.Diagram.Equalizer 𝒞
open import Categories.Diagram.Pullbac... | {
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-- Basic intuitionistic modal logic S4, without ∨, ⊥, or ◇.
-- Non-canonical model equipment for Kripke-style semantics.
module BasicIS4.Equipment.KripkeDyadicNonCanonical where
open import BasicIS4.Syntax.Common public
module Syntax
(_⊢_ : Cx² Ty Ty → Ty → Set)
(mono²⊢ : ∀ {A Π Π′} → Π ⊆² Π′ → Π ⊢ A →... | {
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{-# OPTIONS --without-K --safe #-}
open import Categories.Category
{-
Helper routines most often used in reasoning with commutative squares,
at the level of arrows in categories.
Basic : reasoning about identity
Pulls : use a ∘ b ≈ c as left-to-right rewrite
Pushes : use c ≈ a ∘ b as a left-to-right rewri... | {
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{-# OPTIONS --cubical --safe #-}
module TreeFold.Indexed where
open import Prelude
open import Data.Binary using (𝔹; 0ᵇ; 1ᵇ_; 2ᵇ_; ⟦_⇓⟧; ⟦_⇑⟧; inc)
open import Data.Binary.Isomorphism
open import Data.Nat
private
variable
n m : ℕ
t : Level
N : ℕ → Type t
ns : 𝔹
double : ℕ → ℕ
double n = n * 2
2... | {
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module Computability.Recursive where
open import Data.Empty using (⊥; ⊥-elim)
open import Data.Unit using (⊤)
open import Data.Bool using (Bool; false; true)
open import Data.Nat using (ℕ; zero; suc; _+_; _*_; pred)
open import Data.Product using (_×_)
open import Data.Fin using (Fin; zero; suc; punchIn; punchOut)
ope... | {
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module Experimental.Finite where
data ℕ : Set where
Z : ℕ
S : ℕ → ℕ
data Fin : ℕ → Set where
Fin-S : ∀ {n} → Fin n → Fin (S n)
Fin-Z : ∀ {n} → Fin (S n)
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{- Byzantine Fault Tolerant Consensus Verification in Agda, version 0.9.
Copyright (c) 2021, Oracle and/or its affiliates.
Licensed under the Universal Permissive License v 1.0 as shown at https://opensource.oracle.com/licenses/upl
-}
open import LibraBFT.Concrete.System
open import LibraBFT.Concrete.System.Par... | {
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open import Agda.Primitive using (lzero; lsuc; _⊔_)
open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans; cong; subst; setoid)
open import Data.Product using (_×_; Σ; _,_; proj₁; proj₂; zip; map; <_,_>; swap)
import Function.Equality
open import Relation.Binary using (Setoid)
import Relation.B... | {
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module Tactic.Monoid.Exp where
open import Prelude
open import Tactic.Reflection.Quote
open import Tactic.Deriving.Quotable
data Exp : Set where
var : Nat → Exp
ε : Exp
_⊕_ : Exp → Exp → Exp
unquoteDecl QuoteExp = deriveQuotable QuoteExp (quote Exp)
flatten : Exp → List Nat
flatten (var x) = x ∷ []
flatten ε... | {
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module cry.gfp where
-- open import Agda.Primitive using (lsuc; _⊔_)
open import Level
-- import Agda.Builtin.FromNat
open import Agda.Builtin.Bool
open import Agda.Builtin.Equality
open import Algebra.FunctionProperties.Core using (Op₁; Op₂)
open import Relation.Nullary
-- open import Relation.Binary.Core using (Rel... | {
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{-# OPTIONS --without-K --safe #-}
module Fragment.Examples.Semigroup.Types where
open import Fragment.Prelude
open import Level using (zero)
open import Function.Related
open import Function.Related.TypeIsomorphisms
using (×-isSemigroup; ⊎-isSemigroup)
open import Function.Inverse using (_↔_)
open import Data.P... | {
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module QuoteTerm where
open import Common.Reflection
open import Common.Prelude renaming (Nat to ℕ; module Nat to ℕ)
data _≡_ {a}{A : Set a}(x : A) : A → Set where
refl : x ≡ x
test₁ : quoteTerm (λ {A : Set} (x : A) → x)
≡ lam hidden (abs "A" (lam visible (abs "x" (var 0 []))))
test₁ = refl
-- Local var... | {
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record ⊤ : Set where
instance constructor tt
data ⊥ : Set where
data Inf : Set where
-∞ ∞ : Inf
Less : Inf → Inf → Set
Less ∞ _ = ⊥
Less _ ∞ = ⊤
Less _ _ = ⊥
data Bounded : Inf → Set where
bound : ∀ {b} {{lt : Less -∞ b}} → Bounded b
-- The first time around the target type is Less -∞ _b which results in no... | {
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Instantiates indexed binary structures at an index to the equivalent
-- non-indexed structures.
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module... | {
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{-# OPTIONS --cubical-compatible --safe #-}
data E (@0 A : Set) : Set where
c₁ c₂ : A → E A
@0 c₃ : A → E A
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module Relation.Ternary.Separation.Monad.Update where
open import Level hiding (Lift)
open import Function using (_∘_; case_of_)
open import Relation.Binary.PropositionalEquality using (refl)
open import Relation.Unary
open import Relation.Unary.PredicateTransformer hiding (_⊔_; [_])
open import Relation.Ternary.Separ... | {
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module Scratch.Data.List.Relation.Helpers where
open import Relation.Nullary
open import Relation.Nullary.Decidable
open import Relation.Unary
open import Relation.Binary hiding (Decidable)
open import Data.Product
open import Data.List.Base
open import Data.List.Relation.Unary.All as All
open import Data.List.Rela... | {
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Categories.Functor.Base where
open import Cubical.Foundations.Prelude
open import Cubical.Data.Sigma
open import Cubical.Categories.Category
private
variable
ℓC ℓC' ℓD ℓD' : Level
record Functor (C : Precategory ℓC ℓC') (D : Precategory ℓD ℓD')... | {
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module 14-implicitConfigurations where
postulate
Integral : Set → Set
add : ∀ {A} {{ intA : Integral A }} → A → A → A
mul : ∀ {A} {{ intA : Integral A }} → A → A → A
mod : ∀ {A} {{ intA : Integral A }} → A → A → A
N : Set
zero one two three : N
nInt : Integral N
private postulate Token : Set
record Mo... | {
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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
-}
-----------------------------------------------------------------------------... | {
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module Base.Ascription where
-- Encode infix ascription.
as' : ∀ {ℓ} (A : Set ℓ) (a : A) → A
as' _ a = a
syntax as' A a = a as A
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module UnifyTermL (A : Set) where
open import Data.Fin using (Fin; suc; zero)
open import Data.Nat using (ℕ; suc; zero)
open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong₂; cong; sym; trans)
open import Function using (_∘_)
open import Relation.Nullary using (¬_; Dec; yes; no)
open import Data.P... | {
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module Numeral.Integer.Proofs where
import Data.Either as Either
open import Data.Tuple as Tuple using (_,_)
open import Logic
import Lvl
open import Functional
open import Numeral.Integer
open import Numeral.Integer.Oper
open import Numeral.Integer.Sign
open import Numeral.Natural.Induction
open import Num... | {
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