patrick/agdaproofs
1
0
Fork 0
mirror of https://github.com/Smaug123/agdaproofs synced 2025-10-17 09:28:40 +00:00
Code Issues Packages Projects Releases Wiki Activity

More rings stuff (#83)

Browse Source
This commit is contained in:
Patrick Stevens
2019-11-23 13:53:54 +00:00
committed by GitHub
parent 660d7aa27c
commit 2ed7bd8044
12 changed files with 260 additions and 17 deletions
Download Patch File Download Diff File Expand all files Collapse all files

9
Setoids/Functions/Definition.agda
View File

@@ -1,10 +1,11 @@
{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Setoids.Setoids
open import Setoids.Subset
module Setoids.Functions.Definition where
WellDefined : {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f : A → B) → Set (a ⊔ c ⊔ d)
WellDefined {A = A} S T f = {x y : A} → Setoid._∼_ S x y → Setoid._∼_ T (f x) (f y)
module Setoids.Functions.Definition {a b c d : _} {A : Set a} {B : Set b} where
WellDefined : (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f : A → B) → Set (a ⊔ c ⊔ d)
WellDefined S T f = {x y : A} → Setoid._∼_ S x y → Setoid._∼_ T (f x) (f y)

19
Setoids/Functions/Lemmas.agda Normal file
View File

@@ -0,0 +1,19 @@
{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Setoids.Setoids
open import Setoids.Subset
open import Setoids.Functions.Definition
open import Sets.EquivalenceRelations
module Setoids.Functions.Lemmas {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {f : A → B} (w : WellDefined S T f) where
inverseImagePred : {e : _} → {pred : B → Set e} → (sub : subset T pred) → A → Set (b ⊔ d ⊔ e)
inverseImagePred {pred = pred} subset a = Sg B (λ b → (pred b) && (Setoid._∼_ T (f a) b))
inverseImageWellDefined : {e : _} {pred : B → Set e} → (sub : subset T pred) → subset S (inverseImagePred sub)
inverseImageWellDefined sub {x} {y} x=y (b , (predB ,, fx=b)) = f x , (sub (symmetric fx=b) predB ,, symmetric (w x=y))
where
open Setoid T
open Equivalence eq