Theory Enumeration_ZF

theory Enumeration_ZF
imports NatOrder_ZF FiniteSeq_ZF FinOrd_ZF
(*
    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