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  • f t f if b then t else t cong if true refl cong if false refl if not a A Set a b t t A if b then t else t if not b then t else t if not true refl if not false refl data Bool b b Bool Set where true Bool true true false Bool false false private module Bool Internals where refl Reflexive Bool refl true true refl false false sym Symmetric Bool sym true true sym false false trans Transitive Bool trans true id trans false id subst Substitutive Bool subst true id subst false id Decidable Bool true true yes true false false yes false true false no λ false true no λ isEquivalence IsEquivalence Bool isEquivalence record refl refl sym sym trans trans isDecEquivalence IsDecEquivalence Bool isDecEquivalence record isEquivalence isEquivalence setoid Setoid setoid record Carrier Bool Bool isEquivalence isEquivalence decSetoid DecSetoid decSetoid record Carrier Bool Bool isDecEquivalence isDecEquivalence equality Equality Bool equality record isEquivalence isEquivalence subst subst module Bool Props where open Bool Internals public using subst decSetoid equality open DecSetoid decSetoid public open Equality equality public hiding subst isEquivalence refl reflexive sym trans if then else a a a A Set a a a Bool A A A if then else if then else if then else true x x if then else false x x If then else A Set B Set b Bool A B if Set b then A else B If then else If then else If then else true x x If then else false x x x y Bool Bool true true true true false false false true false false false true module where x y Bool Set x y T x y refl Reflexive refl true refl false subst Substitutive subst true true id subst false false id subst true false subst false true sym Symmetric sym x y eq subst λ y y x x y eq refl x trans Transitive trans x y z x y y z subst x y z y z x y Decidable true true yes false false yes true false no λ false true no λ isEquivalence IsEquivalence isEquivalence record refl λ x refl x sym λ x y sym x y trans λ x y z trans x y z isDecEquivalence IsDecEquivalence isDecEquivalence record isEquivalence isEquivalence setoid Setoid setoid record Carrier Bool isEquivalence isEquivalence decSetoid DecSetoid decSetoid record Carrier Bool isDecEquivalence isDecEquivalence module Bool Reasoning Setoid Reasoning Bool Props setoid open Data Bool public true b T b Bool true b true true true true false false b T not b Bool false b false true false false false T b b true T b T refl Tnot b b false T not b Tnot refl T b T b b true T true refl T false Tnot b T not b b false Tnot false refl Tnot true T b b T b T b T b b T p q from B

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Data.Bool.NP.html (2015-10-11)
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  • Set a b c C λ a λ b C a b a b A Set a B Set b Set a b A B Σ x A B a b A Set a B Set b A B A B Unique existence Parametrised on the underlying equality a b A Set a A A Set A Set b Set a b B λ x B x y B y x y Functions Sometimes the first component can be inferred a b A Set a B A Set b x B x B y y a b c A Set a B A Set b C x B x Set c f x A B x x A C f x x A Σ B x C f g x f x g x map a b p q A Set a B Set b P A Set p Q B Set q f A B x P x Q f x Σ A P Σ B Q map f g x y f x g y zip a b c p q r A Set a B Set b C Set c P A Set p Q B Set q R C Set r A B C x y P x Q y R x y Σ A P Σ B Q Σ C R zip a p b q a b p q swap a b A Set a B Set b A B B A swap x y y x a b i j A Set a B Set b A B Set i A B Set j A B Set f g f g a b c d A Set a B Set b C Set c D Set d A B C A B D

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Data.Product.html (2015-10-11)
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  • and operations are defined in Data Unit Core open import Data Unit Core public A unit type defined as a record type Note that the name of this type is top not T record Set where constructor tt record x y Set where Operations Decidable A yes refl Decidable yes total Total total inj Properties preorder Preorder preorder PropEq preorder setoid Setoid setoid PropEq setoid decTotalOrder DecTotalOrder decTotalOrder record Carrier

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Data.Unit.html (2015-10-11)
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  • i N m Fin n reduce zero i i m i reduce suc m zero reduce suc m suc i s s i m reduce i i m inject m n n inject n i Fin n Fin i Fin n inject i zero inject i suc i zero zero inject i suc i suc j suc inject j inject n i Fin suc n Fin i Fin n inject n zero i suc inject i zero inject n suc i suc zero zero inject n suc i suc suc j suc inject j inject m n Fin m Fin m N n inject n zero zero inject n suc i suc inject n i inject m Fin m Fin suc m inject zero zero inject suc i suc inject i inject m n Fin m m N n Fin n inject zero Nat s s le zero inject suc i Nat s s le suc inject i le Operations Folds fold T Set m n T n T suc n n T suc n Fin m T m fold T f x zero x fold T f x suc i f fold T f x i fold n t T Fin suc n Set t i T inject i T suc i T zero i T i fold T f x zero x fold n zero T f x suc fold n suc n T f x suc i f i fold T inject f inject x i Lifts functions lift m n k Fin m Fin n Fin k N m Fin k N n lift zero f i f i lift suc k f zero zero lift suc k f suc i suc lift k f i m n m n infixl 6 m n i Fin

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Data.Fin.html (2015-10-11)
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  • Empty The Agda standard library Empty type module Data Empty where open import Level data Set where IMPORT Data FFI COMPILED DATA Data FFI AgdaEmpty elim w Whatever Set w

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Data.Empty.html (2015-10-11)
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  • PropositionalEquality as PropEq using open import Relation Binary PropositionalEquality TrustMe Types Finite strings postulate String Set BUILTIN STRING String COMPILED TYPE String String Possibly infinite strings Costring Set Costring Colist Char Operations private primitive primStringAppend String String String primStringToList String List Char primStringFromList List Char String primStringEquality String String Bool infixr 5 String String String primStringAppend toList String List Char toList primStringToList fromList List Char String fromList primStringFromList toVec s

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Data.String.html (2015-10-11)
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  • zero B B A suc n B A A n B A Set n B Set Set A zero B B A suc n B A A n B Endo a Set a Set a Endo A A A Cmp a Set a Set a Cmp A A A Bool More properties about fold are in Data Nat NP nest a A Set a Endo Endo A TMP nest n f x fold x f n nest zero f x x nest suc n f x f nest n f x module nest Properties a A Set a f Endo A where nest nest 0 f id nest refl nest nest 1 f f nest refl nest nest 2 f f f nest refl nest nest 3 f f f f nest refl nest m n nest m n f nest m f nest n f nest zero n refl nest suc m n cong f nest m n nest m n nest m n f nest m f nest n f nest m n x cong flip x nest m n nest m n nest m n f nest m nest n f nest zero n x refl nest suc m n x nest suc m n f x refl nest n m n f x nest n m n x nest n f nest m n f x cong nest n f nest m n x nest n f nest m nest n f x refl nest n f nest m nest n f x refl nest suc m nest n f x where open Reasoning WRONG module more nest Properties a A Set a where nest f Endo Endo A g m n nest m f g nest n f g nest m n f g nest

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Function.NP.html (2015-10-11)
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  • Set a where infix 4 field isEquivalence IsEquivalence Decidable open IsEquivalence isEquivalence public record DecSetoid c Set suc c where infix 4 field Carrier Set c Rel Carrier isDecEquivalence IsDecEquivalence open IsDecEquivalence isDecEquivalence public setoid Setoid c setoid record isEquivalence isEquivalence open Setoid setoid public using preorder Partial orders record IsPartialOrder a A Set a Rel A Rel A Set a where field isPreorder IsPreorder antisym Antisymmetric open IsPreorder isPreorder public renaming resp to resp record Poset c Set suc c where infix 4 field Carrier Set c Rel Carrier Rel Carrier isPartialOrder IsPartialOrder open IsPartialOrder isPartialOrder public preorder Preorder c preorder record isPreorder isPreorder Decidable partial orders record IsDecPartialOrder a A Set a Rel A Rel A Set a where infix 4 field isPartialOrder IsPartialOrder Decidable Decidable private module PO IsPartialOrder isPartialOrder open PO public hiding module Eq module Eq where isDecEquivalence IsDecEquivalence isDecEquivalence record isEquivalence PO isEquivalence open IsDecEquivalence isDecEquivalence public record DecPoset c Set suc c where infix 4 field Carrier Set c Rel Carrier Rel Carrier isDecPartialOrder IsDecPartialOrder private module DPO IsDecPartialOrder isDecPartialOrder open DPO public hiding module Eq poset Poset c poset record isPartialOrder isPartialOrder open Poset poset public using preorder module Eq where decSetoid DecSetoid c decSetoid record isDecEquivalence DPO Eq isDecEquivalence open DecSetoid decSetoid public Strict partial orders record IsStrictPartialOrder a A Set a Rel A Rel A Set a where field isEquivalence IsEquivalence irrefl Irreflexive trans Transitive resp Respects module Eq IsEquivalence isEquivalence record StrictPartialOrder c Set suc c where infix 4 field Carrier Set c Rel Carrier Rel Carrier isStrictPartialOrder IsStrictPartialOrder open IsStrictPartialOrder isStrictPartialOrder public Total orders record IsTotalOrder a A Set a Rel A Rel A Set a where field isPartialOrder IsPartialOrder total Total open IsPartialOrder isPartialOrder public record TotalOrder c Set suc c where infix 4 field Carrier Set

    Original URL path: http://nicolaspouillard.fr/publis/lfmtp2012-talk/html/Relation.Binary.html (2015-10-11)
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