(*

This file is a part of IsarMathLib -

a library of formalized mathematics for Isabelle/Isar.

Copyright (C) 2008 Slawomir Kolodynski

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*)

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

theory Enumeration_ZF imports NatOrder_ZF FiniteSeq_ZF FinOrd_ZF

begin

text{*Suppose $r$ is a linear order on a set $A$ that has $n$ elements,

where $n \in\mathbb{N}$ . In the @{text "FinOrd_ZF"}

theory we prove a theorem stating that there is a unique

order isomorphism between $n = \{0,1,..,n-1\}$ (with natural order) and

$A$. Another way of stating that is that there is

a unique way of counting the elements of $A$ in the order increasing

according to relation $r$. Yet another way of stating the same thing is

that there is a unique sorted list of elements of $A$. We will call this

list the @{text "Enumeration"} of $A$. *}

section{*Enumerations: definition and notation*}

text{*In this section we introduce the notion of enumeration

and define a proof context (a ''locale'' in Isabelle terms)

that sets up the notation for writing about enumarations.*}

text{*We define enumeration as the only order isomorphism beween

a set $A$ and the number of its elements. We are using

the formula $\bigcup \{x\} = x$ to extract the only element

from a singleton. @{text "Le"} is the (natural) order on natural

numbers, defined is @{text "Nat_ZF"} theory in the standard

Isabelle library.*}

definition

"Enumeration(A,r) ≡ \<Union> ord_iso(|A|,Le,A,r)"

text{*To set up the notation we define a locale @{text "enums"}. In this

locale we will assume that $r$ is a linear order on some set $X$.

In most applications this set will be just the set of natural numbers.

Standard Isabelle uses $\leq $ to denote the "less or equal" relation

on natural numbers. We will use the @{text "\<lsq>"} symbol to denote

the relation $r$. Those two symbols usually look the same in the presentation,

but they are different in the source.To shorten the notation the enumeration

@{text "Enumeration(A,r)"} will be denoted as $\sigma (A)$.

Similarly as in the @{text "Semigroup"} theory we will write

$a\hookleftarrow x$ for the result of appending an element $x$ to

the finite sequence (list) $a$. Finally,

$a\sqcup b$ will denote the concatenation of the lists $a$ and $b$.*}

locale enums =

fixes X r

assumes linord: "IsLinOrder(X,r)"

fixes ler (infix "\<lsq>" 70)

defines ler_def[simp]: "x \<lsq> y ≡ ⟨x,y⟩ ∈ r"

fixes σ

defines σ_def [simp]: "σ(A) ≡ Enumeration(A,r)"

fixes append (infix "\<hookleftarrow>" 72)

defines append_def[simp]: "a \<hookleftarrow> x ≡ Append(a,x)"

fixes concat (infixl "\<squnion>" 69)

defines concat_def[simp]: "a \<squnion> b ≡ Concat(a,b)";

section{*Properties of enumerations*}

text{*In this section we prove basic facts about enumerations. *}

text{*A special case of the existence and uniqueess

of the order isomorphism for finite sets

when the first set is a natural number.*}

lemma (in enums) ord_iso_nat_fin:

assumes "A ∈ FinPow(X)" and "n ∈ nat" and "A ≈ n"

shows "∃!f. f ∈ ord_iso(n,Le,A,r)"

using assms NatOrder_ZF_1_L2 linord nat_finpow_nat

fin_ord_iso_ex_uniq by simp;

text{*An enumeration is an order isomorhism, a bijection, and a list.*}

lemma (in enums) enum_props: assumes "A ∈ FinPow(X)"

shows

"σ(A) ∈ ord_iso(|A|,Le, A,r)"

"σ(A) ∈ bij(|A|,A)"

"σ(A) : |A| -> A"

proof -

from assms have

"IsLinOrder(nat,Le)" and "|A| ∈ FinPow(nat)" and "A ≈ |A|"

using NatOrder_ZF_1_L2 card_fin_is_nat nat_finpow_nat

by auto;

with assms show "σ(A) ∈ ord_iso(|A|,Le, A,r)"

using linord fin_ord_iso_ex_uniq sigleton_extract

Enumeration_def by simp;

then show "σ(A) ∈ bij(|A|,A)" and "σ(A) : |A| -> A"

using ord_iso_def bij_def surj_def

by auto;

qed

text{*A corollary from @{text "enum_props"}. Could have been attached as

another assertion, but this slows down verification of some other proofs.

*}

lemma (in enums) enum_fun: assumes "A ∈ FinPow(X)"

shows "σ(A) : |A| -> X"

proof -

from assms have "σ(A) : |A| -> A" and "A⊆X"

using enum_props FinPow_def by auto;

then show "σ(A) : |A| -> X" by (rule func1_1_L1B);

qed;

text{*If a list is an order isomorphism then it must be the enumeration.

*}

lemma (in enums) ord_iso_enum: assumes A1: "A ∈ FinPow(X)" and

A2: "n ∈ nat" and A3: "f ∈ ord_iso(n,Le,A,r)"

shows "f = σ(A)"

proof -

from A3 have "n ≈ A" using ord_iso_def eqpoll_def

by auto;

then have "A ≈ n" by (rule eqpoll_sym);

with A1 A2 have "∃!f. f ∈ ord_iso(n,Le,A,r)"

using ord_iso_nat_fin by simp;

with assms `A ≈ n` show "f = σ(A)"

using enum_props card_card by blast;

qed;

text{*What is the enumeration of the empty set?*}

lemma (in enums) empty_enum: shows "σ(0) = 0"

proof -

have

"0 ∈ FinPow(X)" and "0 ∈ nat" and "0 ∈ ord_iso(0,Le,0,r)"

using empty_in_finpow empty_ord_iso_empty

by auto;

then show "σ(0) = 0" using ord_iso_enum

by blast;

qed;

text{*Adding a new maximum to a set appends it to the enumeration.*}

lemma (in enums) enum_append:

assumes A1: "A ∈ FinPow(X)" and A2: "b ∈ X-A" and

A3: "∀a∈A. a\<lsq>b"

shows " σ(A ∪ {b}) = σ(A)\<hookleftarrow> b"

proof -

let ?f = "σ(A) ∪ {⟨|A|,b⟩}"

from A1 have "|A| ∈ nat" using card_fin_is_nat

by simp;

from A1 A2 have "A ∪ {b} ∈ FinPow(X)"

using singleton_in_finpow union_finpow by simp;

moreover from this have "|A ∪ {b}| ∈ nat"

using card_fin_is_nat by simp;

moreover have "?f ∈ ord_iso(|A ∪ {b}| , Le, A ∪ {b} ,r)"

proof -

from A1 A2 have

"σ(A) ∈ ord_iso(|A|,Le, A,r)" and

"|A| ∉ |A|" and "b ∉ A"

using enum_props mem_not_refl by auto;

moreover from `|A| ∈ nat` have

"∀k ∈ |A|. ⟨k, |A|⟩ ∈ Le"

using elem_nat_is_nat by blast;

moreover from A3 have "∀a∈A. ⟨a,b⟩ ∈ r" by simp;

moreover have "antisym(Le)" and "antisym(r)"

using linord NatOrder_ZF_1_L2 IsLinOrder_def by auto;

moreover

from A2 `|A| ∈ nat` have

"⟨|A|,|A|⟩ ∈ Le" and "⟨b,b⟩ ∈ r"

using linord NatOrder_ZF_1_L2 IsLinOrder_def

total_is_refl refl_def by auto;

hence "⟨|A|,|A|⟩ ∈ Le <-> ⟨b,b⟩ ∈ r" by simp;

ultimately have "?f ∈ ord_iso(|A| ∪ {|A|} , Le, A ∪ {b} ,r)"

by (rule ord_iso_extend);

with A1 A2 show "?f ∈ ord_iso(|A ∪ {b}| , Le, A ∪ {b} ,r)"

using card_fin_add_one by simp;

qed;

ultimately have "?f = σ(A ∪ {b})"

using ord_iso_enum by simp;

moreover have "σ(A)\<hookleftarrow> b = ?f"

proof -

have "σ(A)\<hookleftarrow> b = σ(A) ∪ {⟨domain(σ(A)),b⟩}"

using Append_def by simp;

moreover from A1 have "domain(σ(A)) = |A|"

using enum_props func1_1_L1 by blast;

ultimately show "σ(A)\<hookleftarrow> b = ?f" by simp;

qed

ultimately show "σ(A ∪ {b}) = σ(A)\<hookleftarrow> b" by simp;

qed;

text{*What is the enumeration of a singleton?*}

lemma (in enums) enum_singleton:

assumes A1: "x∈X" shows "σ({x}): 1 -> X" and "σ({x})`(0) = x"

proof -

from A1 have

"0 ∈ FinPow(X)" and "x ∈ (X - 0)" and "∀a∈0. a\<lsq>x"

using empty_in_finpow by auto;

then have "σ(0 ∪ {x}) = σ(0)\<hookleftarrow> x" by (rule enum_append);

with A1 show "σ({x}): 1 -> X" and "σ({x})`(0) = x"

using empty_enum empty_append1 by auto;

qed;

end