(*

This file is a part of IsarMathLib -

a library of formalized mathematics for Isabelle/Isar.

Copyright (C) 2005, 2006 Slawomir Kolodynski

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*)

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

theory Ring_ZF_1 imports Ring_ZF Group_ZF_3

begin

text{*This theory is devoted to the part of ring theory specific the

construction of real numbers in the @{text "Real_ZF_x"} series of theories.

The goal is to

show that classes of almost homomorphisms form a ring.*}

section{*The ring of classes of almost homomorphisms*}

text{*Almost homomorphisms do not form a ring as the regular homomorphisms

do because the lifted group operation is not distributive with respect to

composition -- we have $s\circ (r\cdot q) \neq s\circ r\cdot s\circ q$ in

general. However, we do have

$s\circ (r\cdot q) \approx s\circ r\cdot s\circ q$ in the sense of the

equivalence relation defined by the group of finite range

functions (that is a normal subgroup of almost homomorphisms,

if the group is abelian). This allows to define a natural ring structure

on the classes of almost homomorphisms. *}

text{*The next lemma provides a formula useful for proving that two sides

of the distributive law equation for almost homomorphisms are almost

equal.*}

lemma (in group1) Ring_ZF_1_1_L1:

assumes A1: "s∈AH" "r∈AH" "q∈AH" and A2: "n∈G"

shows

"((so(r•q))`(n))·(((sor)•(soq))`(n))¯= δ(s,⟨ r`(n),q`(n)⟩)"

"((r•q)os)`(n) = ((ros)•(qos))`(n)"

proof -;

from groupAssum isAbelian A1 have T1:

"r•q ∈ AH" "sor ∈ AH" "soq ∈ AH" "(sor)•(soq) ∈ AH"

"ros ∈ AH" "qos ∈ AH" "(ros)•(qos) ∈ AH"

using Group_ZF_3_2_L15 Group_ZF_3_4_T1 by auto;

from A1 A2 have T2: "r`(n) ∈ G" "q`(n) ∈ G" "s`(n) ∈ G"

"s`(r`(n)) ∈ G" "s`(q`(n)) ∈ G" "δ(s,⟨ r`(n),q`(n)⟩) ∈ G"

"s`(r`(n))·s`(q`(n)) ∈ G" "r`(s`(n)) ∈ G" "q`(s`(n)) ∈ G"

"r`(s`(n))·q`(s`(n)) ∈ G"

using AlmostHoms_def apply_funtype Group_ZF_3_2_L4B

group0_2_L1 monoid0.group0_1_L1 by auto;

with T1 A1 A2 isAbelian show

"((so(r•q))`(n))·(((sor)•(soq))`(n))¯= δ(s,⟨ r`(n),q`(n)⟩)"

"((r•q)os)`(n) = ((ros)•(qos))`(n)"

using Group_ZF_3_2_L12 Group_ZF_3_4_L2 Group_ZF_3_4_L1 group0_4_L6A

by auto;

qed;

text{*The sides of the distributive law equations for almost homomorphisms

are almost equal.*}

lemma (in group1) Ring_ZF_1_1_L2:

assumes A1: "s∈AH" "r∈AH" "q∈AH"

shows

"so(r•q) ≈ (sor)•(soq)"

"(r•q)os = (ros)•(qos)"

proof -

from A1 have "∀n∈G. ⟨ r`(n),q`(n)⟩ ∈ G×G"

using AlmostHoms_def apply_funtype by auto;

moreover from A1 have "{δ(s,x). x ∈ G×G} ∈ Fin(G)"

using AlmostHoms_def by simp;

ultimately have "{δ(s,⟨ r`(n),q`(n)⟩). n∈G} ∈ Fin(G)"

by (rule Finite1_L6B);

with A1 have

"{((so(r•q))`(n))·(((sor)•(soq))`(n))¯. n ∈ G} ∈ Fin(G)"

using Ring_ZF_1_1_L1 by simp;

moreover from groupAssum isAbelian A1 A1 have

"so(r•q) ∈ AH" "(sor)•(soq) ∈ AH"

using Group_ZF_3_2_L15 Group_ZF_3_4_T1 by auto;

ultimately show "so(r•q) ≈ (sor)•(soq)"

using Group_ZF_3_4_L12 by simp;

from groupAssum isAbelian A1 have

"(r•q)os : G->G" "(ros)•(qos) : G->G"

using Group_ZF_3_2_L15 Group_ZF_3_4_T1 AlmostHoms_def

by auto;

moreover from A1 have

"∀n∈G. ((r•q)os)`(n) = ((ros)•(qos))`(n)"

using Ring_ZF_1_1_L1 by simp;

ultimately show "(r•q)os = (ros)•(qos)"

using fun_extension_iff by simp;

qed;

text{*The essential condition to show the distributivity for the

operations defined on classes of almost homomorphisms.*}

lemma (in group1) Ring_ZF_1_1_L3:

assumes A1: "R = QuotientGroupRel(AH,Op1,FR)"

and A2: "a ∈ AH//R" "b ∈ AH//R" "c ∈ AH//R"

and A3: "A = ProjFun2(AH,R,Op1)" "M = ProjFun2(AH,R,Op2)"

shows "M`⟨a,A`⟨ b,c⟩⟩ = A`⟨M`⟨ a,b⟩,M`⟨ a,c⟩⟩ ∧

M`⟨A`⟨ b,c⟩,a⟩ = A`⟨M`⟨ b,a⟩,M`⟨ c,a⟩⟩"

proof;

from A2 obtain s q r where D1: "s∈AH" "r∈AH" "q∈AH"

"a = R``{s}" "b = R``{q}" "c = R``{r}"

using quotient_def by auto

from A1 have T1:"equiv(AH,R)"

using Group_ZF_3_3_L3 by simp;

with A1 A3 D1 groupAssum isAbelian have

"M`⟨ a,A`⟨ b,c⟩ ⟩ = R``{so(q•r)}"

using Group_ZF_3_3_L4 EquivClass_1_L10

Group_ZF_3_2_L15 Group_ZF_3_4_L13A by simp;

also have "R``{so(q•r)} = R``{(soq)•(sor)}"

proof -

from T1 D1 have "equiv(AH,R)" "so(q•r)≈(soq)•(sor)"

using Ring_ZF_1_1_L2 by auto;

with A1 show ?thesis using equiv_class_eq by simp;

qed;

also from A1 T1 D1 A3 have

"R``{(soq)•(sor)} = A`⟨M`⟨ a,b⟩,M`⟨ a,c⟩⟩"

using Group_ZF_3_3_L4 Group_ZF_3_4_T1 EquivClass_1_L10

Group_ZF_3_3_L3 Group_ZF_3_4_L13A EquivClass_1_L10 Group_ZF_3_4_T1

by simp;

finally show "M`⟨a,A`⟨ b,c⟩⟩ = A`⟨M`⟨ a,b⟩,M`⟨ a,c⟩⟩" by simp;

from A1 A3 T1 D1 groupAssum isAbelian show

"M`⟨A`⟨ b,c⟩,a⟩ = A`⟨M`⟨ b,a⟩,M`⟨ c,a⟩⟩"

using Group_ZF_3_3_L4 EquivClass_1_L10 Group_ZF_3_4_L13A

Group_ZF_3_2_L15 Ring_ZF_1_1_L2 Group_ZF_3_4_T1 by simp;

qed;

text{*The projection of the first group operation on almost homomorphisms

is distributive with respect to the second group operation.*}

lemma (in group1) Ring_ZF_1_1_L4:

assumes A1: "R = QuotientGroupRel(AH,Op1,FR)"

and A2: "A = ProjFun2(AH,R,Op1)" "M = ProjFun2(AH,R,Op2)"

shows "IsDistributive(AH//R,A,M)"

proof -;

from A1 A2 have "∀a∈(AH//R).∀b∈(AH//R).∀c∈(AH//R).

M`⟨a,A`⟨ b,c⟩⟩ = A`⟨M`⟨ a,b⟩, M`⟨ a,c⟩⟩ ∧

M`⟨A`⟨ b,c⟩, a⟩ = A`⟨M`⟨ b,a⟩,M`⟨ c,a⟩⟩"

using Ring_ZF_1_1_L3 by simp;

then show ?thesis using IsDistributive_def by simp;

qed;

text{*The classes of almost homomorphisms form a ring.*}

theorem (in group1) Ring_ZF_1_1_T1:

assumes "R = QuotientGroupRel(AH,Op1,FR)"

and "A = ProjFun2(AH,R,Op1)" "M = ProjFun2(AH,R,Op2)"

shows "IsAring(AH//R,A,M)"

using assms QuotientGroupOp_def Group_ZF_3_3_T1 Group_ZF_3_4_T2

Ring_ZF_1_1_L4 IsAring_def by simp;

end