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Total order on the rationals (#17)

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Patrick Stevens
2019-01-13 12:05:30 +00:00
committed by GitHub
parent 9e22ba78f5
commit 2866b11be6
9 changed files with 624 additions and 52 deletions
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22
Groups/GroupDefinition.agda Normal file
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@@ -0,0 +1,22 @@
{-# OPTIONS --safe --warning=error #-}
open import LogicalFormulae
open import Setoids.Setoids
open import Functions
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Numbers.Naturals
open import Sets.FinSet
module Groups.GroupDefinition where
record Group {lvl1 lvl2} {A : Set lvl1} (S : Setoid {lvl1} {lvl2} A) (_·_ : A → A → A) : Set (lsuc lvl1 ⊔ lvl2) where
open Setoid S
field
wellDefined : {m n x y : A} → (m ∼ x) → (n ∼ y) → (m · n) ∼ (x · y)
identity : A
inverse : A → A
multAssoc : {a b c : A} → (a · (b · c)) ∼ (a · b) · c
multIdentRight : {a : A} → (a · identity) ∼ a
multIdentLeft : {a : A} → (identity · a) ∼ a
invLeft : {a : A} → (inverse a) · a ∼ identity
invRight : {a : A} → a · (inverse a) ∼ identity

20
Groups/Groups.agda
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@@ -6,19 +6,9 @@ open import Functions
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Numbers.Naturals
open import Sets.FinSet
open import Groups.GroupDefinition
module Groups.Groups where
record Group {lvl1 lvl2} {A : Set lvl1} (S : Setoid {lvl1} {lvl2} A) (_·_ : A → A → A) : Set (lsuc lvl1 ⊔ lvl2) where
open Setoid S
field
wellDefined : {m n x y : A} → (m ∼ x) → (n ∼ y) → (m · n) ∼ (x · y)
identity : A
inverse : A → A
multAssoc : {a b c : A} → (a · (b · c)) ∼ (a · b) · c
multIdentRight : {a : A} → (a · identity) ∼ a
multIdentLeft : {a : A} → (identity · a) ∼ a
invLeft : {a : A} → (inverse a) · a ∼ identity
invRight : {a : A} → a · (inverse a) ∼ identity
reflGroupWellDefined : {lvl : _} {A : Set lvl} {m n x y : A} {op : A → A → A} → m ≡ x → n ≡ y → (op m n) ≡ (op x y)
reflGroupWellDefined {lvl} {A} {m} {n} {.m} {.n} {op} refl refl = refl
@@ -185,8 +175,8 @@ module Groups.Groups where
open Transitive transitiveEq
open Symmetric symmetricEq
invContravariant : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (x y : A) → (Setoid._∼_ S (Group.inverse G (x · y)) ((Group.inverse G y) · (Group.inverse G x)))
invContravariant {S = S} {_·_} G x y = ans
invContravariant : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {x y : A} → (Setoid._∼_ S (Group.inverse G (x · y)) ((Group.inverse G y) · (Group.inverse G x)))
invContravariant {S = S} {_·_} G {x} {y} = ans
where
open Group G renaming (inverse to _^-1) renaming (identity to e)
open Setoid S
@@ -361,7 +351,7 @@ module Groups.Groups where
g : f (Group.inverse G (m ·A Group.inverse G n)) ∼ identity
g = transitive (homRespectsInverse fHom {m ·A Group.inverse G n}) (transitive (inverseWellDefined H pr) (invIdentity H))
h : f (Group.inverse G (Group.inverse G n) ·A Group.inverse G m) ∼ identity
h = transitive (GroupHom.wellDefined fHom (Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq S)) (invContravariant G m (Group.inverse G n)))) g
h = transitive (GroupHom.wellDefined fHom (Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq S)) (invContravariant G))) g
i : f (n ·A Group.inverse G m) ∼ identity
i = transitive (GroupHom.wellDefined fHom (Group.wellDefined G (Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq S)) (invTwice G n)) (Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq S))))) h
Transitive.transitive (Equivalence.transitiveEq (Setoid.eq ansSetoid)) {m} {n} {o} prmn prno = transitive (GroupHom.wellDefined fHom (Group.wellDefined G (Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq S))) (Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq S)) (Group.multIdentLeft G)))) k
@@ -392,7 +382,7 @@ module Groups.Groups where
p6 : (f x) ·B (Group.identity H ·B f (Group.inverse G m)) ∼ (f x) ·B f (Group.inverse G m)
p7 : (f x) ·B f (Group.inverse G m) ∼ f (x ·A (Group.inverse G m))
p8 : f (x ·A (Group.inverse G m)) ∼ Group.identity H
p1 = GroupHom.wellDefined fHom (Group.wellDefined G (Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq S))) (invContravariant G m n))
p1 = GroupHom.wellDefined fHom (Group.wellDefined G (Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq S))) (invContravariant G))
p2 = GroupHom.wellDefined fHom (Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq S)) (fourWayAssoc G))
p3 = GroupHom.groupHom fHom
p4 = Group.wellDefined H (Reflexive.reflexive reflexiveEq) (GroupHom.groupHom fHom)

1
Groups/GroupsLemmas.agda
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@@ -6,6 +6,7 @@ open import Functions
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Numbers.Naturals
open import Groups.Groups
open import Groups.GroupDefinition
module Groups.GroupsLemmas where
invInv : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → {x : A} → Setoid._∼_ S (Group.inverse G (Group.inverse G x)) x