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agdaproofs/Setoids/Functions/Extension.agda
2019-11-13 18:35:41 +00:00

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{-# OPTIONS --safe --warning=error --without-K #-}
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Setoids.Setoids
open import Setoids.Functions.Definition
open import Sets.EquivalenceRelations
module Setoids.Functions.Extension where
ExtensionallyEqual : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {f g : A B} (fWd : WellDefined S T f) (gWd : WellDefined S T g) Set (a d)
ExtensionallyEqual {A = A} {T = T} {f = f} {g = g} fWD gWD = ( {x : A} Setoid.__ T (f x) (g x))
extensionallyEqualReflexive : {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f : A B) (fWD1 fWD2 : WellDefined S T f) ExtensionallyEqual {S = S} {T} fWD1 fWD2
extensionallyEqualReflexive S T f fWD1 _ = Equivalence.reflexive (Setoid.eq T)
extensionallyEqualSymmetric : {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f g : A B) (fWD : WellDefined S T f) (gWD : WellDefined S T g) ExtensionallyEqual {S = S} {T = T} fWD gWD ExtensionallyEqual {S = S} {T} gWD fWD
extensionallyEqualSymmetric S T f g fWD gWD pr = Equivalence.symmetric (Setoid.eq T) pr
extensionallyEqualTransitive : {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f g h : A B) (fWD : WellDefined S T f) (gWD : WellDefined S T g) (hWD : WellDefined S T h) ExtensionallyEqual {S = S} {T} fWD gWD ExtensionallyEqual {S = S} {T} gWD hWD ExtensionallyEqual {S = S} {T} fWD hWD
extensionallyEqualTransitive S T f g h fWD gWD hWD pr1 pr2 = Equivalence.transitive (Setoid.eq T) pr1 pr2