(*

This file is a part of IsarMathLib -

a library of formalized mathematics for Isabelle/Isar.

Copyright (C) 2005 - 2008 Slawomir Kolodynski

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header{*\isaheader{Monoid\_ZF.thy}*}

theory Monoid_ZF imports func_ZF

begin

text{*This theory provides basic facts about monoids.*}

section{*Definition and basic properties*}

text{*In this section we talk about monoids.

The notion of a monoid is similar to the notion of a semigroup

except that we require the existence of a neutral element.

It is also similar to the notion of group except that

we don't require existence of the inverse.*}

text{*Monoid is a set $G$ with an associative operation and a neutral element.

The operation is a function on $G\times G$ with values in $G$.

In the context of ZF set theory this means that it is a set of pairs

$\langle x,y \rangle$, where $x\in G\times G$ and $y\in G$. In other words

the operation is a certain subset of $(G\times G)\times G$. We express

all this by defing a predicate @{text "IsAmonoid(G,f)"}. Here $G$ is the

''carrier'' of the group and $f$ is the binary operation on it.

*}

definition

"IsAmonoid(G,f) ≡

f {is associative on} G ∧

(∃e∈G. (∀ g∈G. ( (f`(⟨e,g⟩) = g) ∧ (f`(⟨g,e⟩) = g))))"

text{* The next locale called ''monoid0'' defines a context for theorems

that concern monoids. In this contex we assume that the pair $(G,f)$

is a monoid. We will use

the @{text "⊕"} symbol to denote the monoid operation (for

no particular reason).*}

locale monoid0 =

fixes G

fixes f

assumes monoidAsssum: "IsAmonoid(G,f)"

fixes monoper (infixl "⊕" 70)

defines monoper_def [simp]: "a ⊕ b ≡ f`⟨a,b⟩";

text{*The result of the monoid operation is in the monoid (carrier).*}

lemma (in monoid0) group0_1_L1:

assumes "a∈G" "b∈G" shows "a⊕b ∈ G";

using assms monoidAsssum IsAmonoid_def IsAssociative_def apply_funtype

by auto;

text{*There is only one neutral element in a monoid.*}

lemma (in monoid0) group0_1_L2: shows

"∃!e. e∈G ∧ (∀ g∈G. ( (e⊕g = g) ∧ g⊕e = g))"

proof

fix e y

assume "e ∈ G ∧ (∀g∈G. e ⊕ g = g ∧ g ⊕ e = g)"

and "y ∈ G ∧ (∀g∈G. y ⊕ g = g ∧ g ⊕ y = g)"

then have "y⊕e = y" "y⊕e = e" by auto;

thus "e = y" by simp;

next from monoidAsssum show

"∃e. e∈ G ∧ (∀ g∈G. e⊕g = g ∧ g⊕e = g)"

using IsAmonoid_def by auto;

qed;

text{*We could put the definition of neutral element anywhere,

but it is only usable in conjuction with the above lemma.*}

definition

"TheNeutralElement(G,f) ≡

( THE e. e∈G ∧ (∀ g∈G. f`⟨e,g⟩ = g ∧ f`⟨g,e⟩ = g))";

text{*The neutral element is neutral.*}

lemma (in monoid0) unit_is_neutral:

assumes A1: "e = TheNeutralElement(G,f)"

shows "e ∈ G ∧ (∀g∈G. e ⊕ g = g ∧ g ⊕ e = g)"

proof -;

let ?n = "THE b. b∈ G ∧ (∀ g∈G. b⊕g = g ∧ g⊕b = g)";

have "∃!b. b∈ G ∧ (∀ g∈G. b⊕g = g ∧ g⊕b = g)"

using group0_1_L2 by simp;

then have "?n∈ G ∧ (∀ g∈G. ?n⊕g = g ∧ g⊕?n = g)"

by (rule theI);

with A1 show ?thesis

using TheNeutralElement_def by simp;

qed;

text{*The monoid carrier is not empty.*}

lemma (in monoid0) group0_1_L3A: shows "G≠0"

proof -;

have "TheNeutralElement(G,f) ∈ G" using unit_is_neutral

by simp;

thus ?thesis by auto;

qed;

text{* The range of the monoid operation is the whole monoid carrier.*}

lemma (in monoid0) group0_1_L3B: shows "range(f) = G"

proof;

from monoidAsssum have "f : G×G->G"

using IsAmonoid_def IsAssociative_def by simp;

then show "range(f) ⊆ G"

using func1_1_L5B by simp;

show "G ⊆ range(f)"

proof;

fix g assume A1: "g∈G"

let ?e = "TheNeutralElement(G,f)"

from A1 have "⟨?e,g⟩ ∈ G×G" "g = f`⟨?e,g⟩"

using unit_is_neutral by auto;

with `f : G×G->G` show "g ∈ range(f)"

using func1_1_L5A by blast;

qed;

qed

text{*Another way to state that the range of the monoid operation

is the whole monoid carrier.*}

lemma (in monoid0) range_carr: shows "f``(G×G) = G"

using monoidAsssum IsAmonoid_def IsAssociative_def

group0_1_L3B range_image_domain by auto;

text{*In a monoid any neutral element is the neutral element.*}

lemma (in monoid0) group0_1_L4:

assumes A1: "e ∈ G ∧ (∀g∈G. e ⊕ g = g ∧ g ⊕ e = g)"

shows "e = TheNeutralElement(G,f)"

proof -

let ?n = "THE b. b∈ G ∧ (∀ g∈G. b⊕g = g ∧ g⊕b = g)";

have "∃!b. b∈ G ∧ (∀ g∈G. b⊕g = g ∧ g⊕b = g)"

using group0_1_L2 by simp;

moreover note A1

ultimately have "?n = e" by (rule the_equality2);

then show ?thesis using TheNeutralElement_def by simp;

qed;

text{*The next lemma shows that if the if we restrict the monoid operation to

a subset of $G$ that contains the neutral element, then the

neutral element of the monoid operation is also neutral with the

restricted operation.

*}

lemma (in monoid0) group0_1_L5:

assumes A1: "∀x∈H.∀y∈H. x⊕y ∈ H"

and A2: "H⊆G"

and A3: "e = TheNeutralElement(G,f)"

and A4: "g = restrict(f,H×H)"

and A5: "e∈H"

and A6: "h∈H"

shows "g`⟨e,h⟩ = h ∧ g`⟨h,e⟩ = h";

proof -;

from A4 A6 A5 have

"g`⟨e,h⟩ = e⊕h ∧ g`⟨h,e⟩ = h⊕e"

using restrict_if by simp;

with A3 A4 A6 A2 show

"g`⟨e,h⟩ = h ∧ g`⟨h,e⟩ = h"

using unit_is_neutral by auto;

qed

text{*The next theorem shows that if the monoid operation is closed

on a subset of $G$ then this set is a (sub)monoid (although

we do not define this notion). This fact will be

useful when we study subgroups. *}

theorem (in monoid0) group0_1_T1:

assumes A1: "H {is closed under} f"

and A2: "H⊆G"

and A3: "TheNeutralElement(G,f) ∈ H"

shows "IsAmonoid(H,restrict(f,H×H))"

proof -;

let ?g = "restrict(f,H×H)"

let ?e = "TheNeutralElement(G,f)"

from monoidAsssum have "f ∈ G×G->G"

using IsAmonoid_def IsAssociative_def by simp;

moreover from A2 have "H×H ⊆ G×G" by auto;

moreover from A1 have "∀p ∈ H×H. f`(p) ∈ H"

using IsOpClosed_def by auto;

ultimately have "?g ∈ H×H->H";

using func1_2_L4 by simp;

moreover have "∀x∈H.∀y∈H.∀z∈H.

?g`⟨?g`⟨x,y⟩ ,z⟩ = ?g`⟨x,?g`⟨y,z⟩⟩"

proof -

from A1 have "∀x∈H.∀y∈H.∀z∈H.

?g`⟨?g`⟨x,y⟩,z⟩ = x⊕y⊕z"

using IsOpClosed_def restrict_if by simp;

moreover have "∀x∈H.∀y∈H.∀z∈H. x⊕y⊕z = x⊕(y⊕z)"

proof -;

from monoidAsssum have

"∀x∈G.∀y∈G.∀z∈G. x⊕y⊕z = x⊕(y⊕z)"

using IsAmonoid_def IsAssociative_def

by simp;

with A2 show ?thesis by auto;

qed;

moreover from A1 have

"∀x∈H.∀y∈H.∀z∈H. x⊕(y⊕z) = ?g`⟨ x,?g`⟨y,z⟩ ⟩"

using IsOpClosed_def restrict_if by simp;

ultimately show ?thesis by simp;

qed;

moreover have

"∃n∈H. (∀h∈H. ?g`⟨n,h⟩ = h ∧ ?g`⟨h,n⟩ = h)"

proof -;

from A1 have "∀x∈H.∀y∈H. x⊕y ∈ H"

using IsOpClosed_def by simp;

with A2 A3 have

"∀ h∈H. ?g`⟨?e,h⟩ = h ∧ ?g`⟨h,?e⟩ = h"

using group0_1_L5 by blast;

with A3 show ?thesis by auto;

qed;

ultimately show ?thesis using IsAmonoid_def IsAssociative_def

by simp;

qed;

text{*Under the assumptions of @{text " group0_1_T1"}

the neutral element of a

submonoid is the same as that of the monoid.*}

lemma group0_1_L6:

assumes A1: "IsAmonoid(G,f)"

and A2: "H {is closed under} f"

and A3: "H⊆G"

and A4: "TheNeutralElement(G,f) ∈ H"

shows "TheNeutralElement(H,restrict(f,H×H)) = TheNeutralElement(G,f)"

proof -

let ?e = "TheNeutralElement(G,f)"

let ?g = "restrict(f,H×H)"

from assms have "monoid0(H,?g)";

using monoid0_def monoid0.group0_1_T1

by simp;

moreover have

"?e ∈ H ∧ (∀h∈H. ?g`⟨?e,h⟩ = h ∧ ?g`⟨h,?e⟩ = h)"

proof -;

{ fix h assume "h ∈ H"

with assms have

"monoid0(G,f)" "∀x∈H.∀y∈H. f`⟨x,y⟩ ∈ H"

"H⊆G" "?e = TheNeutralElement(G,f)" "?g = restrict(f,H×H)"

"?e ∈ H" "h ∈ H"

using monoid0_def IsOpClosed_def by auto;

then have "?g`⟨?e,h⟩ = h ∧ ?g`⟨h,?e⟩ = h"

by (rule monoid0.group0_1_L5);

} hence "∀h∈H. ?g`⟨?e,h⟩ = h ∧ ?g`⟨h,?e⟩ = h" by simp;

with A4 show ?thesis by simp;

qed;

ultimately have "?e = TheNeutralElement(H,?g)"

by (rule monoid0.group0_1_L4);

thus ?thesis by simp;

qed;

text{*If a sum of two elements is not zero,

then at least one has to be nonzero.*}

lemma (in monoid0) sum_nonzero_elmnt_nonzero:

assumes "a ⊕ b ≠ TheNeutralElement(G,f)"

shows "a ≠ TheNeutralElement(G,f) ∨ b ≠ TheNeutralElement(G,f)";

using assms unit_is_neutral by auto;

end