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Met and Top lecture 1 (#99)

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Patrick Stevens
2020-02-23 12:12:53 +00:00
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{-# 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 Rings.Definition
open import Rings.Homomorphisms.Definition
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Rings.Homomorphisms.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+A_ _*A_ : A → A → A} (R : Ring S _+A_ _*A_) where
open import Groups.Homomorphisms.Lemmas2 (Ring.additiveGroup R)
imageRing : {c d : _} {B : Set c} {T : Setoid {c} {d} B} {_+B_ : B → B → 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 : 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)) → ({x y m n : B} → Setoid._∼_ T x m → Setoid._∼_ T y n → Setoid._∼_ T (x *B y) (m *B n)) → Ring T _+B_ _*B_
Ring.additiveGroup (imageRing f surj respects+ respects* +wd *wd) = imageGroup f surj respects+ +wd
Ring.*WellDefined (imageRing f surj respects+ respects* +wd *wd) = *wd
Ring.1R (imageRing f surj respects+ respects* +wd *wd) = f (Ring.1R R)
Ring.groupIsAbelian (imageRing {T = T} f record { surjective = surjective ; wellDefined = wellDefined } respects+ respects* +wd *wd) {a} {b} with surjective {a}
... | x , fx=a with surjective {b}
... | y , fy=b = transitive (+wd (symmetric fx=a) (symmetric fy=b)) (transitive (transitive (symmetric respects+) (transitive (wellDefined (Ring.groupIsAbelian R)) respects+)) (+wd fy=b fx=a))
where
open Setoid T
open Equivalence eq
Ring.*Associative (imageRing {T = T} f record { surjective = surjective ; wellDefined = wellDefined } respects+ respects* +wd *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 (Ring.*Associative R)) respects*)) (*wd (transitive respects* (*wd fx=a fy=b)) fz=c))
where
open Setoid T
open Equivalence eq
Ring.*Commutative (imageRing {T = T} f record { surjective = surjective ; wellDefined = wellDefined } respects+ respects* +wd *wd) {a} {b} with surjective {a}
... | x , fx=a with surjective {b}
... | y , fy=b = transitive (*wd (symmetric fx=a) (symmetric fy=b)) (transitive (transitive (symmetric respects*) (transitive (wellDefined (Ring.*Commutative R)) respects*)) (*wd fy=b fx=a))
where
open Setoid T
open Equivalence eq
Ring.*DistributesOver+ (imageRing {T = T} f record { surjective = surjective ; wellDefined = wellDefined } respects+ respects* +wd *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) (+wd (symmetric fy=b) (symmetric fz=c))) (transitive (transitive (transitive (*wd reflexive (symmetric respects+)) (symmetric respects*)) (transitive (transitive (wellDefined (Ring.*DistributesOver+ R)) respects+) (+wd respects* respects*))) (+wd (*wd fx=a fy=b) (*wd fx=a fz=c)))
where
open Setoid T
open Equivalence eq
Ring.identIsIdent (imageRing {T = T} f record { wellDefined = wellDefined ; surjective = surjective } respects+ respects* +wd *wd) {b} with surjective {b}
Ring.identIsIdent (imageRing {T = T} f record { wellDefined = wellDefined ; surjective = surjective } respects+ respects* +wd *wd) {b} | a , fa=b = transitive (transitive (*wd reflexive (symmetric fa=b)) (transitive (symmetric respects*) (wellDefined (Ring.identIsIdent R)))) fa=b
where
open Setoid T
open Equivalence eq
homToImageRing : {c d : _} {B : Set c} {T : Setoid {c} {d} B} {_+B_ : B → B → 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))) → (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)) → (*wd : {x y m n : B} → Setoid._∼_ T x m → Setoid._∼_ T y n → Setoid._∼_ T (x *B y) (m *B n)) → RingHom R (imageRing f surj respects+ respects* +wd *wd) f
RingHom.preserves1 (homToImageRing {T = T} f surj respects+ respects* +wd *wd) = reflexive
where
open Setoid T
open Equivalence eq
RingHom.ringHom (homToImageRing f surj respects+ respects* +wd *wd) = respects*
RingHom.groupHom (homToImageRing f surj respects+ respects* +wd *wd) = homToImageGroup f surj respects+ +wd