Theory Order_ZF_1

theory Order_ZF_1
imports Order ZF1
(*   This file is a part of IsarMathLib - 
    a library of formalized mathematics for Isabelle/Isar.

    Copyright (C) 2005, 2006, 2007  Slawomir Kolodynski

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

theory Order_ZF_1 imports Order ZF1

begin

text{*In @{text "Order_ZF"} we define some notions related to order relations
  based on the nonstrict orders ($\leq $ type). Some people however prefer to talk
  about these notions in terms of the strict order relation ($<$ type). 
  This is the case for the standard Isabelle @{text "Order.thy"} and also for 
  Metamath. In this theory file we repeat some developments from @{text "Order_ZF"}
  using the strict order relation as a basis.This is mostly useful for Metamath 
  translation, but is also of some general interest. The names of theorems are 
  copied from Metamath.*}


section{*Definitions and basic properties*}

text{*In this section we introduce some definitions taken from Metamath and relate 
  them to the ones used by the standard Isabelle @{text "Order.thy"}.
  *}

text{* The next definition is the strict version of the linear order.
  What we write as @{text "R Orders A"} is written $R Ord A$ in Metamath.
  *}

definition
StrictOrder (infix "Orders" 65) where
  "R Orders A ≡ ∀x y z. (x∈A ∧ y∈A ∧ z∈A) --> 
  (⟨x,y⟩ ∈ R <-> ¬(x=y ∨ ⟨y,x⟩ ∈ R)) ∧ 
  (⟨x,y⟩ ∈ R ∧ ⟨y,z⟩ ∈ R --> ⟨x,z⟩ ∈ R)"

text{*The definition of supremum for a (strict) linear order.*}

definition
  "Sup(B,A,R) ≡ 
  \<Union> {x ∈ A. (∀y∈B. ⟨x,y⟩ ∉ R) ∧ 
  (∀y∈A. ⟨y,x⟩ ∈ R --> (∃z∈B. ⟨y,z⟩ ∈ R))}"

text{*Definition of infimum for a linear order. 
  It is defined in terms of supremum.*}

definition
  "Infim(B,A,R) ≡ Sup(B,A,converse(R))"

text{*If relation $R$ orders a set $A$, (in Metamath sense) then $R$ 
  is irreflexive, transitive and linear therefore is a total order on $A$ 
  (in Isabelle sense).*}

lemma orders_imp_tot_ord: assumes A1: "R Orders A"
  shows 
  "irrefl(A,R)"
  "trans[A](R)"
  "part_ord(A,R)"
  "linear(A,R)"
  "tot_ord(A,R)"
proof -
  from A1 have I:
    "∀x y z. (x∈A ∧ y∈A ∧ z∈A) --> 
    (⟨x,y⟩ ∈ R <-> ¬(x=y ∨ ⟨y,x⟩ ∈ R)) ∧ 
    (⟨x,y⟩ ∈ R ∧ ⟨y,z⟩ ∈ R --> ⟨x,z⟩ ∈ R)"
    unfolding StrictOrder_def by simp
  then have "∀x∈A. ⟨x,x⟩ ∉ R" by blast
  then show "irrefl(A,R)" using irrefl_def by simp
  moreover
  from I have 
    "∀x∈A. ∀y∈A. ∀z∈A. ⟨x,y⟩ ∈ R --> ⟨y,z⟩ ∈ R --> ⟨x,z⟩ ∈ R"
    by blast
  then show "trans[A](R)" unfolding trans_on_def by blast
  ultimately show "part_ord(A,R)" using part_ord_def
    by simp
  moreover
  from I have
    "∀x∈A. ∀y∈A. ⟨x,y⟩ ∈ R ∨ x=y ∨ ⟨y,x⟩ ∈ R"
    by blast
  then show "linear(A,R)" unfolding linear_def by blast
  ultimately show "tot_ord(A,R)" using tot_ord_def
    by simp
qed

text{*A converse of @{text "orders_imp_tot_ord"}. Together with that
  theorem this shows that Metamath's notion of an order relation is equivalent to
  Isabelles @{text "tot_ord"} predicate. *}

lemma tot_ord_imp_orders: assumes A1: "tot_ord(A,R)"
  shows "R Orders A"
proof -
  from A1 have 
    I: "linear(A,R)" and  
    II: "irrefl(A,R)" and 
    III: "trans[A](R)" and
    IV: "part_ord(A,R)"
    using tot_ord_def part_ord_def by auto
  from IV have "asym(R ∩ A×A)"
    using part_ord_Imp_asym by simp
  then have V: "∀x y. ⟨x,y⟩ ∈ (R ∩ A×A) --> ¬(⟨y,x⟩ ∈ (R ∩ A×A))"
    unfolding asym_def by blast
  from I have VI: "∀x∈A. ∀y∈A. ⟨x,y⟩ ∈ R ∨ x=y ∨ ⟨y,x⟩ ∈ R"
    unfolding linear_def by blast
  from III have VII:
    "∀x∈A. ∀y∈A. ∀z∈A. ⟨x,y⟩ ∈ R --> ⟨y,z⟩ ∈ R --> ⟨x,z⟩ ∈ R"
    unfolding trans_on_def by blast
  { fix x y z
    assume T: "x∈A" "y∈A" "z∈A"
    have "⟨x,y⟩ ∈ R <-> ¬(x=y ∨ ⟨y,x⟩ ∈ R)"
    proof
      assume A2: "⟨x,y⟩ ∈ R"
      with V T have  "¬(⟨y,x⟩ ∈ R)" by blast
      moreover from II T A2 have "x≠y" using irrefl_def
	by auto
      ultimately show "¬(x=y ∨ ⟨y,x⟩ ∈ R)" by simp
    next assume  "¬(x=y ∨ ⟨y,x⟩ ∈ R)"
      with VI T show "⟨x,y⟩ ∈ R" by auto
    qed
    moreover from VII T have
      "⟨x,y⟩ ∈ R ∧ ⟨y,z⟩ ∈ R --> ⟨x,z⟩ ∈ R"
      by blast
    ultimately have "(⟨x,y⟩ ∈ R <-> ¬(x=y ∨ ⟨y,x⟩ ∈ R)) ∧ 
      (⟨x,y⟩ ∈ R ∧ ⟨y,z⟩ ∈ R --> ⟨x,z⟩ ∈ R)"
      by simp
  } then have "∀x y z. (x∈A ∧ y∈A ∧ z∈A) --> 
      (⟨x,y⟩ ∈ R <-> ¬(x=y ∨ ⟨y,x⟩ ∈ R)) ∧ 
      (⟨x,y⟩ ∈ R ∧ ⟨y,z⟩ ∈ R --> ⟨x,z⟩ ∈ R)"
    by auto
  then show "R Orders A" using StrictOrder_def by simp
qed
    
section{*Properties of (strict) total orders *}

text{*In this section we discuss the properties of strict order relations. 
  This continues the development contained in the standard Isabelle's 
  @{text "Order.thy"} with a view towards using the theorems 
  translated from Metamath.*}

text{*A relation orders a set iff the converse relation orders a set. Going
  one way we can use the the lemma @{text "tot_od_converse"} from the standard 
  Isabelle's @{text "Order.thy"}.The other way is a bit more complicated (note that
  in Isabelle for @{text "converse(converse(r)) = r"} one needs $r$ to consist
  of ordered pairs, which does not follow from the @{text "StrictOrder"} 
  definition above).*}

lemma cnvso: shows "R Orders A <-> converse(R) Orders A"
proof
  let ?r = "converse(R)"
  assume "R Orders A"
  then have "tot_ord(A,?r)" using orders_imp_tot_ord tot_ord_converse
    by simp
  then show "?r Orders A" using tot_ord_imp_orders
    by simp
next
  let ?r = "converse(R)"
  assume "?r Orders A"
  then have A2: "∀x y z. (x∈A ∧ y∈A ∧ z∈A) --> 
    (⟨x,y⟩ ∈ ?r <-> ¬(x=y ∨ ⟨y,x⟩ ∈ ?r)) ∧ 
    (⟨x,y⟩ ∈ ?r ∧ ⟨y,z⟩ ∈ ?r --> ⟨x,z⟩ ∈ ?r)"
    using StrictOrder_def by simp
  { fix x y z
    assume "x∈A ∧ y∈A ∧ z∈A"
    with A2 have
      I: "⟨y,x⟩ ∈ ?r <-> ¬(x=y ∨ ⟨x,y⟩ ∈ ?r)" and
      II: "⟨y,x⟩ ∈ ?r ∧ ⟨z,y⟩ ∈ ?r --> ⟨z,x⟩ ∈ ?r"
      by auto
    from I have "⟨x,y⟩ ∈ R <-> ¬(x=y ∨ ⟨y,x⟩ ∈ R)"
      by auto
    moreover from II have "⟨x,y⟩ ∈ R ∧ ⟨y,z⟩ ∈ R --> ⟨x,z⟩ ∈ R"
      by auto
    ultimately have "(⟨x,y⟩ ∈ R <-> ¬(x=y ∨ ⟨y,x⟩ ∈ R)) ∧ 
      (⟨x,y⟩ ∈ R ∧ ⟨y,z⟩ ∈ R --> ⟨x,z⟩ ∈ R)" by simp
  } then have  "∀x y z. (x∈A ∧ y∈A ∧ z∈A) --> 
      (⟨x,y⟩ ∈ R <-> ¬(x=y ∨ ⟨y,x⟩ ∈ R)) ∧ 
      (⟨x,y⟩ ∈ R ∧ ⟨y,z⟩ ∈ R --> ⟨x,z⟩ ∈ R)"
    by auto
  then show "R Orders A" using StrictOrder_def by simp
qed

text{*Supremum is unique, if it exists.*}

lemma supeu: assumes A1: "R Orders A" and A2: "x∈A" and
  A3: "∀y∈B. ⟨x,y⟩ ∉ R" and A4: "∀y∈A. ⟨y,x⟩ ∈ R  --> ( ∃z∈B. ⟨y,z⟩ ∈ R)"
   shows 
  "∃!x. x∈A∧(∀y∈B. ⟨x,y⟩ ∉ R) ∧ (∀y∈A. ⟨y,x⟩ ∈ R --> ( ∃z∈B. ⟨y,z⟩ ∈ R))"
proof
  from A2 A3 A4 show
    "∃ x. x∈A∧(∀y∈B. ⟨x,y⟩ ∉ R) ∧ (∀y∈A. ⟨y,x⟩ ∈ R --> ( ∃z∈B. ⟨y,z⟩ ∈ R))"
    by auto
next fix x1 x2
  assume A5:
    "x1 ∈ A ∧ (∀y∈B. ⟨x1,y⟩ ∉ R) ∧ (∀y∈A. ⟨y,x1⟩ ∈ R --> ( ∃z∈B. ⟨y,z⟩ ∈ R))"
    "x2 ∈ A ∧ (∀y∈B. ⟨x2,y⟩ ∉ R) ∧ (∀y∈A. ⟨y,x2⟩ ∈ R --> ( ∃z∈B. ⟨y,z⟩ ∈ R))"
  from A1 have "linear(A,R)" using orders_imp_tot_ord tot_ord_def
    by simp
  then have "∀x∈A. ∀y∈A. ⟨x,y⟩ ∈ R ∨ x=y ∨ ⟨y,x⟩ ∈ R"
    unfolding linear_def by blast
  with A5 have "⟨x1,x2⟩ ∈ R ∨ x1=x2 ∨ ⟨x2,x1⟩ ∈ R" by blast
  moreover 
  { assume "⟨x1,x2⟩ ∈ R"
    with A5 obtain z where "z∈B" and "⟨x1,z⟩ ∈ R" by auto
    with A5 have False by auto }
  moreover
  { assume "⟨x2,x1⟩ ∈ R"
    with A5 obtain z where "z∈B" and "⟨x2,z⟩ ∈ R" by auto
    with A5 have False by auto }
  ultimately  show "x1 = x2" by auto
qed

text{*Supremum has expected properties if it exists.*}

lemma sup_props: assumes A1: "R Orders A" and 
  A2: "∃x∈A. (∀y∈B. ⟨x,y⟩ ∉ R) ∧ (∀y∈A. ⟨y,x⟩ ∈ R --> ( ∃z∈B. ⟨y,z⟩ ∈ R))"
  shows 
  "Sup(B,A,R) ∈ A"
  "∀y∈B. ⟨Sup(B,A,R),y⟩ ∉ R"
  "∀y∈A. ⟨y,Sup(B,A,R)⟩ ∈ R --> ( ∃z∈B. ⟨y,z⟩ ∈ R )"
proof -
  let ?S = "{x∈A. (∀y∈B. ⟨x,y⟩ ∉ R) ∧ (∀y∈A. ⟨y,x⟩ ∈ R --> ( ∃z∈B. ⟨y,z⟩ ∈ R ) ) }"
  from A2 obtain x where 
    "x∈A" and "(∀y∈B. ⟨x,y⟩ ∉ R)" and "∀y∈A. ⟨y,x⟩ ∈ R --> ( ∃z∈B. ⟨y,z⟩ ∈ R)"
    by auto
  with A1 have I:
    "∃!x. x∈A∧(∀y∈B. ⟨x,y⟩ ∉ R) ∧ (∀y∈A. ⟨y,x⟩ ∈ R --> ( ∃z∈B. ⟨y,z⟩ ∈ R))"
    using supeu by simp
  then have "( \<Union>?S ) ∈ A" by (rule ZF1_1_L9)
  then show "Sup(B,A,R) ∈ A" using Sup_def by simp
  from I have II:
    "(∀y∈B. ⟨\<Union>?S ,y⟩ ∉ R) ∧ (∀y∈A. ⟨y,\<Union>?S⟩ ∈ R --> ( ∃z∈B. ⟨y,z⟩ ∈ R))"
    by (rule ZF1_1_L9)
  hence "∀y∈B. ⟨\<Union>?S,y⟩ ∉ R" by blast
  moreover have III: "(\<Union>?S) = Sup(B,A,R)" using Sup_def by simp
  ultimately show "∀y∈B. ⟨Sup(B,A,R),y⟩ ∉ R" by simp
  from II have IV: "∀y∈A. ⟨y,\<Union>?S⟩ ∈ R --> ( ∃z∈B. ⟨y,z⟩ ∈ R)"
    by blast
  { fix y assume A3: "y∈A" and "⟨y,Sup(B,A,R)⟩ ∈ R"
    with III have "⟨y,\<Union>?S⟩ ∈ R" by simp
    with IV A3 have "∃z∈B. ⟨y,z⟩ ∈ R" by blast
  } thus  "∀y∈A. ⟨y,Sup(B,A,R)⟩ ∈ R --> ( ∃z∈B. ⟨y,z⟩ ∈ R )"
    by simp
qed

text{*Elements greater or equal than any element of $B$ are 
  greater or equal than supremum of $B$.*}

lemma supnub: assumes A1: "R Orders A" and A2: 
  "∃x∈A. (∀y∈B. ⟨x,y⟩ ∉ R) ∧ (∀y∈A. ⟨y,x⟩ ∈ R --> ( ∃z∈B. ⟨y,z⟩ ∈ R))"
  and A3: "c ∈ A" and A4: "∀z∈B. ⟨c,z⟩ ∉ R"
  shows "⟨c, Sup(B,A,R)⟩ ∉ R"
proof -
  from A1 A2 have
    "∀y∈A. ⟨y,Sup(B,A,R)⟩ ∈ R --> ( ∃z∈B. ⟨y,z⟩ ∈ R )"
    by (rule sup_props)
  with A3 A4 show "⟨c, Sup(B,A,R)⟩ ∉ R" by auto
qed

end