Theory Group_ZF_1

theory Group_ZF_1
imports Group_ZF
(* 
This file is a part of IsarMathLib -
a library of formalized mathematics for Isabelle/Isar.

Copyright (C) 2008 Slawomir Kolodynski

This program is free software; Redistribution and use in source and binary forms,
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*)


header{*\isaheader{Group\_ZF\_1.thy}*}

theory Group_ZF_1 imports Group_ZF

begin

text{*In this theory we consider right and left translations and odd
functions.*}


section{*Translations*}

text{*In this section we consider translations. Translations are maps
$T: G\rightarrow G$ of the form $T_g (a) = g\cdot a$ or
$T_g (a) = a\cdot g$. We also consider two-dimensional translations
$T_g : G\times G \rightarrow G\times G$, where
$T_g(a,b) = (a\cdot g, b\cdot g)$ or $T_g(a,b) = (g\cdot a, g\cdot b)$.
*}


text{*For an element $a\in G$ the right translation is defined
a function (set of pairs) such that its value (the second element
of a pair) is the value of the group operation on the first element
of the pair and $g$. This looks a bit strange in the raw set notation,
when we write a function explicitely as a set of pairs and value of
the group operation on the pair $\langle a,b \rangle$
as @{text "P`⟨a,b⟩"} instead of the usual infix $a\cdot b$
or $a + b$. *}


definition
"RightTranslation(G,P,g) ≡ {⟨ a,b⟩ ∈ G×G. P`⟨a,g⟩ = b}"

text{*A similar definition of the left translation.*}

definition
"LeftTranslation(G,P,g) ≡ {⟨a,b⟩ ∈ G×G. P`⟨g,a⟩ = b}"

text{*Translations map $G$ into $G$. Two dimensional translations map
$G\times G$ into itself.*}


lemma (in group0) group0_5_L1: assumes A1: "g∈G"
shows "RightTranslation(G,P,g) : G->G" and "LeftTranslation(G,P,g) : G->G"
proof -
from A1 have "∀a∈G. a·g ∈ G" and "∀a∈G. g·a ∈ G"
using group_oper_assocA apply_funtype by auto;
then show
"RightTranslation(G,P,g) : G->G"
"LeftTranslation(G,P,g) : G->G"
using RightTranslation_def LeftTranslation_def func1_1_L11A
by auto;
qed;

text{*The values of the translations are what we expect.*}

lemma (in group0) group0_5_L2: assumes "g∈G" "a∈G"
shows
"RightTranslation(G,P,g)`(a) = a·g"
"LeftTranslation(G,P,g)`(a) = g·a"
using assms group0_5_L1 RightTranslation_def LeftTranslation_def
func1_1_L11B by auto;

text{*Composition of left translations is a left translation by the product.*}

lemma (in group0) group0_5_L4: assumes A1: "g∈G" "h∈G" "a∈G" and
A2: "Tg = LeftTranslation(G,P,g)" "Th = LeftTranslation(G,P,h)"
shows
"Tg`(Th`(a)) = g·h·a"
"Tg`(Th`(a)) = LeftTranslation(G,P,g·h)`(a)"
proof -;
from A1 have I: "h·a∈G" "g·h∈G"
using group_oper_assocA apply_funtype by auto;
with A1 A2 show "Tg`(Th`(a)) = g·h·a"
using group0_5_L2 group_oper_assoc by simp;
with A1 A2 I show
"Tg`(Th`(a)) = LeftTranslation(G,P,g·h)`(a)"
using group0_5_L2 group_oper_assoc by simp;
qed;


text{*Composition of right translations is a right translation by
the product.*}


lemma (in group0) group0_5_L5: assumes A1: "g∈G" "h∈G" "a∈G" and
A2: "Tg = RightTranslation(G,P,g)" "Th = RightTranslation(G,P,h)"
shows
"Tg`(Th`(a)) = a·h·g"
"Tg`(Th`(a)) = RightTranslation(G,P,h·g)`(a)"
proof -
from A1 have I: "a·h∈G" "h·g ∈G"
using group_oper_assocA apply_funtype by auto;
with A1 A2 show "Tg`(Th`(a)) = a·h·g"
using group0_5_L2 group_oper_assoc by simp;
with A1 A2 I show
"Tg`(Th`(a)) = RightTranslation(G,P,h·g)`(a)"
using group0_5_L2 group_oper_assoc by simp;
qed

text{*Point free version of @{text "group0_5_L4"} and @{text "group0_5_L5"}. *}

lemma (in group0) trans_comp: assumes "g∈G" "h∈G" shows
"RightTranslation(G,P,g) O RightTranslation(G,P,h) = RightTranslation(G,P,h·g)"
"LeftTranslation(G,P,g) O LeftTranslation(G,P,h) = LeftTranslation(G,P,g·h)"
proof -
let ?Tg = "RightTranslation(G,P,g)"
let ?Th = "RightTranslation(G,P,h)"
from assms have "?Tg:G->G" and "?Th:G->G"
using group0_5_L1 by auto
then have "?Tg O ?Th:G->G" using comp_fun by simp
moreover from assms have "RightTranslation(G,P,h·g):G->G"
using group_op_closed group0_5_L1 by simp
moreover from assms `?Th:G->G` have
"∀a∈G. (?Tg O ?Th)`(a) = RightTranslation(G,P,h·g)`(a)"
using comp_fun_apply group0_5_L5 by simp
ultimately show "?Tg O ?Th = RightTranslation(G,P,h·g)"
by (rule func_eq)
next
let ?Tg = "LeftTranslation(G,P,g)"
let ?Th = "LeftTranslation(G,P,h)"
from assms have "?Tg:G->G" and "?Th:G->G"
using group0_5_L1 by auto
then have "?Tg O ?Th:G->G" using comp_fun by simp
moreover from assms have "LeftTranslation(G,P,g·h):G->G"
using group_op_closed group0_5_L1 by simp
moreover from assms `?Th:G->G` have
"∀a∈G. (?Tg O ?Th)`(a) = LeftTranslation(G,P,g·h)`(a)"
using comp_fun_apply group0_5_L4 by simp
ultimately show "?Tg O ?Th = LeftTranslation(G,P,g·h)"
by (rule func_eq)
qed

text{*The image of a set under a composition of translations is the same as
the image under translation by a product.*}


lemma (in group0) trans_comp_image: assumes A1: "g∈G" "h∈G" and
A2: "Tg = LeftTranslation(G,P,g)" "Th = LeftTranslation(G,P,h)"
shows "Tg``(Th``(A)) = LeftTranslation(G,P,g·h)``(A)"
proof -
from A2 have "Tg``(Th``(A)) = (Tg O Th)``(A)"
using image_comp by simp
with assms show ?thesis using trans_comp by simp
qed

text{*Another form of the image of a set under a composition of translations *}

lemma (in group0) group0_5_L6:
assumes A1: "g∈G" "h∈G" and A2: "A⊆G" and
A3: "Tg = RightTranslation(G,P,g)" "Th = RightTranslation(G,P,h)"
shows "Tg``(Th``(A)) = {a·h·g. a∈A}"
proof -;
from A2 have "∀a∈A. a∈G" by auto;
from A1 A3 have "Tg : G->G" "Th : G->G"
using group0_5_L1 by auto;
with assms `∀a∈A. a∈G` show
"Tg``(Th``(A)) = {a·h·g. a∈A}"
using func1_1_L15C group0_5_L5 by auto;
qed

text{*The translation by neutral element is the identity on group.*}

lemma (in group0) trans_neutral: shows
"RightTranslation(G,P,\<one>) = id(G)" and "LeftTranslation(G,P,\<one>) = id(G)"
proof -
have "RightTranslation(G,P,\<one>):G->G" and "∀a∈G. RightTranslation(G,P,\<one>)`(a) = a"
using group0_2_L2 group0_5_L1 group0_5_L2 by auto
then show "RightTranslation(G,P,\<one>) = id(G)" by (rule indentity_fun)
have "LeftTranslation(G,P,\<one>):G->G" and "∀a∈G. LeftTranslation(G,P,\<one>)`(a) = a"
using group0_2_L2 group0_5_L1 group0_5_L2 by auto
then show "LeftTranslation(G,P,\<one>) = id(G)" by (rule indentity_fun)
qed

text{*Composition of translations by an element and its inverse is identity.*}

lemma (in group0) trans_comp_id: assumes "g∈G" shows
"RightTranslation(G,P,g) O RightTranslation(G,P,g¯) = id(G)" and
"RightTranslation(G,P,g¯) O RightTranslation(G,P,g) = id(G)" and
"LeftTranslation(G,P,g) O LeftTranslation(G,P,g¯) = id(G)" and
"LeftTranslation(G,P,g¯) O LeftTranslation(G,P,g) = id(G)"
using assms inverse_in_group trans_comp group0_2_L6 trans_neutral by auto

text{*Translations are bijective.*}

lemma (in group0) trans_bij: assumes "g∈G" shows
"RightTranslation(G,P,g) ∈ bij(G,G)" and "LeftTranslation(G,P,g) ∈ bij(G,G)"
proof-
from assms have
"RightTranslation(G,P,g):G->G" and
"RightTranslation(G,P,g¯):G->G" and
"RightTranslation(G,P,g) O RightTranslation(G,P,g¯) = id(G)"
"RightTranslation(G,P,g¯) O RightTranslation(G,P,g) = id(G)"
using inverse_in_group group0_5_L1 trans_comp_id by auto
then show "RightTranslation(G,P,g) ∈ bij(G,G)" using fg_imp_bijective by simp
from assms have
"LeftTranslation(G,P,g):G->G" and
"LeftTranslation(G,P,g¯):G->G" and
"LeftTranslation(G,P,g) O LeftTranslation(G,P,g¯) = id(G)"
"LeftTranslation(G,P,g¯) O LeftTranslation(G,P,g) = id(G)"
using inverse_in_group group0_5_L1 trans_comp_id by auto
then show "LeftTranslation(G,P,g) ∈ bij(G,G)" using fg_imp_bijective by simp
qed

text{*Converse of a translation is translation by the inverse.*}

lemma (in group0) trans_conv_inv: assumes "g∈G" shows
"converse(RightTranslation(G,P,g)) = RightTranslation(G,P,g¯)" and
"converse(LeftTranslation(G,P,g)) = LeftTranslation(G,P,g¯)" and
"LeftTranslation(G,P,g) = converse(LeftTranslation(G,P,g¯))" and
"RightTranslation(G,P,g) = converse(RightTranslation(G,P,g¯))"
proof -
from assms have
"RightTranslation(G,P,g) ∈ bij(G,G)" "RightTranslation(G,P,g¯) ∈ bij(G,G)" and
"LeftTranslation(G,P,g) ∈ bij(G,G)" "LeftTranslation(G,P,g¯) ∈ bij(G,G)"
using trans_bij inverse_in_group by auto
moreover from assms have
"RightTranslation(G,P,g¯) O RightTranslation(G,P,g) = id(G)" and
"LeftTranslation(G,P,g¯) O LeftTranslation(G,P,g) = id(G)" and
"LeftTranslation(G,P,g) O LeftTranslation(G,P,g¯) = id(G)" and
"LeftTranslation(G,P,g¯) O LeftTranslation(G,P,g) = id(G)"
using trans_comp_id by auto
ultimately show
"converse(RightTranslation(G,P,g)) = RightTranslation(G,P,g¯)" and
"converse(LeftTranslation(G,P,g)) = LeftTranslation(G,P,g¯)" and
"LeftTranslation(G,P,g) = converse(LeftTranslation(G,P,g¯))" and
"RightTranslation(G,P,g) = converse(RightTranslation(G,P,g¯))"
using comp_id_conv by auto
qed

text{*The image of a set by translation is the same as the inverse image by
by the inverse element translation.*}


lemma (in group0) trans_image_vimage: assumes "g∈G" shows
"LeftTranslation(G,P,g)``(A) = LeftTranslation(G,P,g¯)-``(A)" and
"RightTranslation(G,P,g)``(A) = RightTranslation(G,P,g¯)-``(A)"
using assms trans_conv_inv vimage_converse by auto

text{*Another way of looking at translations is that they are sections
of the group operation. *}


lemma (in group0) trans_eq_section: assumes "g∈G" shows
"RightTranslation(G,P,g) = Fix2ndVar(P,g)" and
"LeftTranslation(G,P,g) = Fix1stVar(P,g)"
proof -
let ?T = "RightTranslation(G,P,g)"
let ?F = "Fix2ndVar(P,g)"
from assms have "?T: G->G" and "?F: G->G"
using group0_5_L1 group_oper_assocA fix_2nd_var_fun by auto
moreover from assms have "∀a∈G. ?T`(a) = ?F`(a)"
using group0_5_L2 group_oper_assocA fix_var_val by simp
ultimately show "?T = ?F" by (rule func_eq)
next
let ?T = "LeftTranslation(G,P,g)"
let ?F = "Fix1stVar(P,g)"
from assms have "?T: G->G" and "?F: G->G"
using group0_5_L1 group_oper_assocA fix_1st_var_fun by auto
moreover from assms have "∀a∈G. ?T`(a) = ?F`(a)"
using group0_5_L2 group_oper_assocA fix_var_val by simp
ultimately show "?T = ?F" by (rule func_eq)
qed

text{*A lemma about translating sets.*}

lemma (in group0) ltrans_image: assumes A1: "V⊆G" and A2: "x∈G"
shows "LeftTranslation(G,P,x)``(V) = {x·v. v∈V}"
proof -
from assms have "LeftTranslation(G,P,x)``(V) = {LeftTranslation(G,P,x)`(v). v∈V}"
using group0_5_L1 func_imagedef by blast
moreover from assms have "∀v∈V. LeftTranslation(G,P,x)`(v) = x·v"
using group0_5_L2 by auto
ultimately show ?thesis by auto
qed

text{*A technical lemma about solving equations with translations.*}

lemma (in group0) ltrans_inv_in: assumes A1: "V⊆G" and A2: "y∈G" and
A3: "x ∈ LeftTranslation(G,P,y)``(GroupInv(G,P)``(V))"
shows "y ∈ LeftTranslation(G,P,x)``(V)"
proof -
have "x∈G"
proof -
from A2 have "LeftTranslation(G,P,y):G->G" using group0_5_L1 by simp
then have "LeftTranslation(G,P,y)``(GroupInv(G,P)``(V)) ⊆ G"
using func1_1_L6 by simp
with A3 show "x∈G" by auto
qed
have "∃v∈V. x = y·v¯"
proof -
have "GroupInv(G,P): G->G" using groupAssum group0_2_T2
by simp
with assms obtain z where "z ∈ GroupInv(G,P)``(V)" and "x = y·z"
using func1_1_L6 ltrans_image by auto
with A1 `GroupInv(G,P): G->G` show ?thesis using func_imagedef by auto
qed
then obtain v where "v∈V" and "x = y·v¯" by auto
with A1 A2 have "y = x·v" using inv_cancel_two by auto
with assms `x∈G` `v∈V` show ?thesis using ltrans_image by auto
qed

text{*We can look at the result of interval arithmetic operation as union of
translated sets.*}


lemma (in group0) image_ltrans_union: assumes "A⊆G" "B⊆G" shows
"(P {lifted to subsets of} G)`⟨A,B⟩ = (\<Union>a∈A. LeftTranslation(G,P,a)``(B))"
proof
from assms have I: "(P {lifted to subsets of} G)`⟨A,B⟩ = {a·b . ⟨a,b⟩ ∈ A×B}"
using group_oper_assocA lift_subsets_explained by simp
{ fix c assume "c ∈ (P {lifted to subsets of} G)`⟨A,B⟩"
with I obtain a b where "c = a·b" and "a∈A" "b∈B" by auto
hence "c ∈ {a·b. b∈B}" by auto
moreover from assms `a∈A` have
"LeftTranslation(G,P,a)``(B) = {a·b. b∈B}" using ltrans_image by auto
ultimately have "c ∈ LeftTranslation(G,P,a)``(B)" by simp
with `a∈A` have "c ∈ (\<Union>a∈A. LeftTranslation(G,P,a)``(B))" by auto
} thus "(P {lifted to subsets of} G)`⟨A,B⟩ ⊆ (\<Union>a∈A. LeftTranslation(G,P,a)``(B))"
by auto
{ fix c assume "c ∈ (\<Union>a∈A. LeftTranslation(G,P,a)``(B))"
then obtain a where "a∈A" and "c ∈ LeftTranslation(G,P,a)``(B)"
by auto
moreover from assms `a∈A` have "LeftTranslation(G,P,a)``(B) = {a·b. b∈B}"
using ltrans_image by auto
ultimately obtain b where "b∈B" and "c = a·b" by auto
with I `a∈A` have "c ∈ (P {lifted to subsets of} G)`⟨A,B⟩" by auto
} thus "(\<Union>a∈A. LeftTranslation(G,P,a)``(B)) ⊆ (P {lifted to subsets of} G)`⟨A,B⟩"
by auto
qed

text{*If the neutral element belongs to a set, then an element of group belongs
the translation of that set.*}


lemma (in group0) neut_trans_elem:
assumes A1: "A⊆G" "g∈G" and A2: "\<one>∈A"
shows "g ∈ LeftTranslation(G,P,g)``(A)"
proof -
from assms have "g·\<one> ∈ LeftTranslation(G,P,g)``(A)"
using ltrans_image by auto
with A1 show ?thesis using group0_2_L2 by simp
qed

text{*The neutral element belongs to the translation of a set by the inverse
of an element that belongs to it.*}


lemma (in group0) elem_trans_neut: assumes A1: "A⊆G" and A2: "g∈A"
shows "\<one> ∈ LeftTranslation(G,P,g¯)``(A)"
proof -
from assms have "g¯ ∈ G" using inverse_in_group by auto
with assms have "g¯·g ∈ LeftTranslation(G,P,g¯)``(A)"
using ltrans_image by auto
moreover from assms have "g¯·g = \<one>" using group0_2_L6 by auto
ultimately show ?thesis by simp
qed

section{*Odd functions*}

text{*This section is about odd functions.*}

text{*Odd functions are those that commute with the group inverse:
$f(a^{-1}) = (f(a))^{-1}.$*}


definition
"IsOdd(G,P,f) ≡ (∀a∈G. f`(GroupInv(G,P)`(a)) = GroupInv(G,P)`(f`(a)) )"

text{*Let's see the definition of an odd function in a more readable
notation.*}


lemma (in group0) group0_6_L1:
shows "IsOdd(G,P,p) <-> ( ∀a∈G. p`(a¯) = (p`(a))¯ )"
using IsOdd_def by simp;

text{*We can express the definition of an odd function in two ways.*}

lemma (in group0) group0_6_L2:
assumes A1: "p : G->G"
shows
"(∀a∈G. p`(a¯) = (p`(a))¯) <-> (∀a∈G. (p`(a¯))¯ = p`(a))"
proof
assume "∀a∈G. p`(a¯) = (p`(a))¯"
with A1 show "∀a∈G. (p`(a¯))¯ = p`(a)"
using apply_funtype group_inv_of_inv by simp;
next assume A2: "∀a∈G. (p`(a¯))¯ = p`(a)"
{ fix a assume "a∈G"
with A1 A2 have
"p`(a¯) ∈ G" and "((p`(a¯))¯)¯ = (p`(a))¯"
using apply_funtype inverse_in_group by auto;
then have "p`(a¯) = (p`(a))¯"
using group_inv_of_inv by simp;
} then show "∀a∈G. p`(a¯) = (p`(a))¯" by simp;
qed;

end