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xs Pattern matching infixr 5 v data View a Set Set where v n View a n v n x a xs BoundedVec a n View a suc n abstract view a n BoundedVec a n View a n view bVec Vec v view bVec Vec x xs x v bVec xs Increasing the bound abstract a n BoundedVec a n BoundedVec a suc n a a bVec m m

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

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y where open Setoid A renaming to open Setoid B renaming to The set of all injections between two setoids record Injection f f t t From Setoid f f To Setoid t t Set f f t t where field to From To injective Injective to The set of all injections from one set to another infix 3 f t Set f Set t Set From To Injection P

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

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for equational reasoning using a partial order open import Relation Binary module Relation Binary PartialOrderReasoning p p p P Poset p p p where open Poset P import Relation Binary

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

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Consequences module Relation Binary Consequences Core where open import Relation Binary Core open import Data Product subst resp a p A Set a Rel A P Rel A p Substitutive p P Respects subst resp P subst λ x y

Original URL path: https://nicolaspouillard.fr/publis/NomPa.agda/Relation.Binary.Consequences.Core.html (2015-10-11)

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Set a B Set b REL A B Set InjectiveREL x y z x z y z x y InjectiveRel a A Set a Rel A Set InjectiveRel A InjectiveREL A A SurjectiveREL a b A Set a B Set b REL A B Set SurjectiveREL A B InjectiveREL B A flip SurjectiveRel a A Set a Rel A Set SurjectiveRel A SurjectiveREL A A record BijectiveREL a b A

Original URL path: https://nicolaspouillard.fr/publis/NomPa.agda/Relation.Binary.Bijection.html (2015-10-11)

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LeftInverse as Left hiding id Bijective functions record Bijective f f t t From Setoid f f To Setoid t t to From To Set f f t t where field injective Injective to surjective Surjective to open Surjective surjective public left inverse of from LeftInverseOf to left inverse of x injective right inverse of to x The set of all bijections between two setoids record Bijection f f t t From Setoid f f To Setoid t t Set f f t t where field to From To bijective Bijective to open Bijective bijective public injection Injection From To injection record to to injective injective surjection Surjection From To surjection record to to surjective surjective open Surjection surjection public using equivalence right inverse left inverse LeftInverse From To left inverse record to to from from left inverse of left inverse of Identity and composition Note that these proofs are superfluous given that Bijection is equivalent to Function Inverse Inverse id s s S Setoid s s Bijection S S id S S record to F id bijective record injective Injection injective Inj id S S surjective Surjection surjective Surj id S S infixr 9 f f m m t

Original URL path: https://nicolaspouillard.fr/publis/NomPa.agda/Function.Bijection.html (2015-10-11)

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Setoid f f To Setoid t t To From From To Set f RightInverseOf g g LeftInverseOf f The set of all left inverses between two setoids record LeftInverse f f t t From Setoid f f To Setoid t t Set f f t t where field to From To from To From left inverse of from LeftInverseOf to open Setoid From open EqReasoning From injective Injective to injective x y eq begin x sym left inverse of x from to x F cong from eq from to y left inverse of y y injection Injection From To injection record to to injective injective equivalence Equivalence From To equivalence record to to from from The set of all right inverses between two setoids RightInverse f f t t From Setoid f f To Setoid t t Set RightInverse From To LeftInverse To From The set of all left inverses from one set to another Read A B as surjection from B to A infix 3 f t Set f Set t Set From To LeftInverse P setoid From P setoid To Identity and composition id s s S Setoid s s LeftInverse S S id S S record to

Original URL path: https://nicolaspouillard.fr/publis/NomPa.agda/Function.LeftInverse.html (2015-10-11)

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f t t where field from To From right inverse of from RightInverseOf to The set of all surjections from one setoid to another record Surjection f f t t From Setoid f f To Setoid t t Set f f t t where field to From To surjective Surjective to open Surjective surjective public right inverse RightInverse From To right inverse record to from from to left inverse of right inverse of injective Injective from injective LeftInverse injective right inverse injection Injection To From injection LeftInverse injection right inverse equivalence Equivalence From To equivalence record to to from from Right inverses can be turned into surjections fromRightInverse f f t t From Setoid f f To Setoid t t RightInverse From To Surjection From To fromRightInverse r record to from surjective record from to right inverse of left inverse of where open LeftInverse r The set of all surjections from one set to another infix 3 f t Set f Set t Set From To Surjection P setoid From P setoid To Identity and composition id s s S Setoid s s Surjection S S id S S record to F id surjective record from LeftInverse to id right

Original URL path: https://nicolaspouillard.fr/publis/NomPa.agda/Function.Surjection.html (2015-10-11)

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