patrick/agdaproofs
1
0
Fork 0
mirror of https://github.com/Smaug123/agdaproofs synced 2025-10-18 17:48:40 +00:00
Code Issues Packages Projects Releases Wiki Activity

Met and Top lecture 1 (#99)

Browse Source
This commit is contained in:
Patrick Stevens
2020-02-23 12:12:53 +00:00
committed by GitHub
parent f41f5226b9
commit 29216d26cb
9 changed files with 475 additions and 0 deletions
Download Patch File Download Diff File Expand all files Collapse all files

52
Groups/Homomorphisms/Lemmas2.agda Normal file
View File

@@ -0,0 +1,52 @@
{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Groups.Groups
open import Groups.Homomorphisms.Definition
open import Groups.Definition
open import Numbers.Naturals.Naturals
open import Setoids.Orders
open import Setoids.Setoids
open import Functions
open import Sets.EquivalenceRelations
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Groups.Homomorphisms.Lemmas2 {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+A_ : A → A → A} (G : Group S _+A_) where
imageGroup : {c d : _} {B : Set c} {T : Setoid {c} {d} B} {_+B_ : B → B → B} → (f : A → B) → SetoidSurjection S T f → ({x y : A} → Setoid._∼_ T (f (x +A y)) ((f x) +B (f y))) → ({x y m n : B} → Setoid._∼_ T x m → Setoid._∼_ T y n → Setoid._∼_ T (x +B y) (m +B n)) → Group T _+B_
Group.+WellDefined (imageGroup f surj respects+ wd) {m} {n} {x} {y} = wd
Group.0G (imageGroup f surj respects+ wd) = f (Group.0G G)
Group.inverse (imageGroup f record { wellDefined = wellDefined ; surjective = surjective } respects+ wd) b with surjective {b}
Group.inverse (imageGroup f record { wellDefined = wellDefined ; surjective = surjective } respects+ wd) b | a , fa=b = f (Group.inverse G a)
Group.+Associative (imageGroup {T = T} f record { wellDefined = wellDefined ; surjective = surjective } respects+ wd) {a} {b} {c} with surjective {a}
... | x , fx=a with surjective {b}
... | y , fy=b with surjective {c}
... | z , fz=c = transitive (wd (symmetric fx=a) (transitive (wd (symmetric fy=b) (symmetric fz=c)) (symmetric respects+))) (transitive (transitive (symmetric respects+) (transitive (wellDefined (Group.+Associative G)) respects+)) (wd (transitive respects+ (wd fx=a fy=b)) fz=c))
where
open Setoid T
open Equivalence eq
Group.identRight (imageGroup {T = T} f record { wellDefined = wd ; surjective = surjective } respects+ bWd) {b} with surjective {b}
... | a , fa=b = transitive (bWd (symmetric fa=b) reflexive) (transitive (symmetric respects+) (transitive (wd (Group.identRight G)) fa=b))
where
open Setoid T
open Equivalence eq
Group.identLeft (imageGroup {T = T} f record { wellDefined = wellDefined ; surjective = surjective } respects+ wd) {b} with surjective {b}
... | a , fa=b = transitive (wd reflexive (symmetric fa=b)) (transitive (symmetric respects+) (transitive (wellDefined (Group.identLeft G)) fa=b))
where
open Setoid T
open Equivalence eq
Group.invLeft (imageGroup {T = T} f record { wellDefined = wellDefined ; surjective = surjective } respects+ wd) {b} with surjective {b}
Group.invLeft (imageGroup {T = T} f record { wellDefined = wellDefined ; surjective = surjective } respects+ wd) {b} | a , fa=b = transitive (wd reflexive (symmetric fa=b)) (transitive (symmetric respects+) (wellDefined (Group.invLeft G)))
where
open Setoid T
open Equivalence eq
Group.invRight (imageGroup f record { wellDefined = wellDefined ; surjective = surjective } respects+ wd) {b} with surjective {b}
Group.invRight (imageGroup {T = T} f record { wellDefined = wellDefined ; surjective = surjective } respects+ wd) {b} | a , fa=b = transitive (wd (symmetric fa=b) reflexive) (transitive (symmetric respects+) (wellDefined (Group.invRight G)))
where
open Setoid T
open Equivalence eq
homToImageGroup : {c d : _} {B : Set c} {T : Setoid {c} {d} B} {_+B_ : B → B → B} → (f : A → B) → (surj : SetoidSurjection S T f) → (respects+ : {x y : A} → Setoid._∼_ T (f (x +A y)) ((f x) +B (f y))) → (wd : {x y m n : B} → Setoid._∼_ T x m → Setoid._∼_ T y n → Setoid._∼_ T (x +B y) (m +B n)) → GroupHom G (imageGroup f surj respects+ wd) f
GroupHom.groupHom (homToImageGroup f surj respects+ wd) = respects+
GroupHom.wellDefined (homToImageGroup f surj respects+ wd) = SetoidSurjection.wellDefined surj