Theory BijectionRel

(*  Title:      HOL/Old_Number_Theory/BijectionRel.thy
    Author:     Thomas M. Rasmussen
    Copyright   2000  University of Cambridge
*)

section ‹Bijections between sets›

theory BijectionRel
imports Main
begin

text ‹
  Inductive definitions of bijections between two different sets and
  between the same set.  Theorem for relating the two definitions.

  \bigskip
›

inductive_set
  bijR :: "('a => 'b => bool) => ('a set * 'b set) set"
  for P :: "'a => 'b => bool"
where
  empty [simp]: "({}, {}) ∈ bijR P"
| insert: "P a b ==> a ∉ A ==> b ∉ B ==> (A, B) ∈ bijR P
    ==> (insert a A, insert b B) ∈ bijR P"

text ‹
  Add extra condition to @{term insert}: @{term "∀b ∈ B. ¬ P a b"}
  (and similar for @{term A}).
›

definition
  bijP :: "('a => 'a => bool) => 'a set => bool" where
  "bijP P F = (∀a b. a ∈ F ∧ P a b --> b ∈ F)"

definition
  uniqP :: "('a => 'a => bool) => bool" where
  "uniqP P = (∀a b c d. P a b ∧ P c d --> (a = c) = (b = d))"

definition
  symP :: "('a => 'a => bool) => bool" where
  "symP P = (∀a b. P a b = P b a)"

inductive_set
  bijER :: "('a => 'a => bool) => 'a set set"
  for P :: "'a => 'a => bool"
where
  empty [simp]: "{} ∈ bijER P"
| insert1: "P a a ==> a ∉ A ==> A ∈ bijER P ==> insert a A ∈ bijER P"
| insert2: "P a b ==> a ≠ b ==> a ∉ A ==> b ∉ A ==> A ∈ bijER P
    ==> insert a (insert b A) ∈ bijER P"


text ‹\medskip @{term bijR}›

lemma fin_bijRl: "(A, B) ∈ bijR P ==> finite A"
  apply (erule bijR.induct)
  apply auto
  done

lemma fin_bijRr: "(A, B) ∈ bijR P ==> finite B"
  apply (erule bijR.induct)
  apply auto
  done

lemma aux_induct:
  assumes major: "finite F"
    and subs: "F ⊆ A"
    and cases: "P {}"
      "!!F a. F ⊆ A ==> a ∈ A ==> a ∉ F ==> P F ==> P (insert a F)"
  shows "P F"
  using major subs
  apply (induct set: finite)
   apply (blast intro: cases)+
  done


lemma inj_func_bijR_aux1:
    "A ⊆ B ==> a ∉ A ==> a ∈ B ==> inj_on f B ==> f a ∉ f ` A"
  apply (unfold inj_on_def)
  apply auto
  done

lemma inj_func_bijR_aux2:
  "∀a. a ∈ A --> P a (f a) ==> inj_on f A ==> finite A ==> F <= A
    ==> (F, f ` F) ∈ bijR P"
  apply (rule_tac F = F and A = A in aux_induct)
     apply (rule finite_subset)
      apply auto
  apply (rule bijR.insert)
     apply (rule_tac [3] inj_func_bijR_aux1)
        apply auto
  done

lemma inj_func_bijR:
  "∀a. a ∈ A --> P a (f a) ==> inj_on f A ==> finite A
    ==> (A, f ` A) ∈ bijR P"
  apply (rule inj_func_bijR_aux2)
     apply auto
  done


text ‹\medskip @{term bijER}›

lemma fin_bijER: "A ∈ bijER P ==> finite A"
  apply (erule bijER.induct)
    apply auto
  done

lemma aux1:
  "a ∉ A ==> a ∉ B ==> F ⊆ insert a A ==> F ⊆ insert a B ==> a ∈ F
    ==> ∃C. F = insert a C ∧ a ∉ C ∧ C <= A ∧ C <= B"
  apply (rule_tac x = "F - {a}" in exI)
  apply auto
  done

lemma aux2: "a ≠ b ==> a ∉ A ==> b ∉ B ==> a ∈ F ==> b ∈ F
    ==> F ⊆ insert a A ==> F ⊆ insert b B
    ==> ∃C. F = insert a (insert b C) ∧ a ∉ C ∧ b ∉ C ∧ C ⊆ A ∧ C ⊆ B"
  apply (rule_tac x = "F - {a, b}" in exI)
  apply auto
  done

lemma aux_uniq: "uniqP P ==> P a b ==> P c d ==> (a = c) = (b = d)"
  apply (unfold uniqP_def)
  apply auto
  done

lemma aux_sym: "symP P ==> P a b = P b a"
  apply (unfold symP_def)
  apply auto
  done

lemma aux_in1:
    "uniqP P ==> b ∉ C ==> P b b ==> bijP P (insert b C) ==> bijP P C"
  apply (unfold bijP_def)
  apply auto
  apply (subgoal_tac "b ≠ a")
   prefer 2
   apply clarify
  apply (simp add: aux_uniq)
  apply auto
  done

lemma aux_in2:
  "symP P ==> uniqP P ==> a ∉ C ==> b ∉ C ==> a ≠ b ==> P a b
    ==> bijP P (insert a (insert b C)) ==> bijP P C"
  apply (unfold bijP_def)
  apply auto
  apply (subgoal_tac "aa ≠ a")
   prefer 2
   apply clarify
  apply (subgoal_tac "aa ≠ b")
   prefer 2
   apply clarify
  apply (simp add: aux_uniq)
  apply (subgoal_tac "ba ≠ a")
   apply auto
  apply (subgoal_tac "P a aa")
   prefer 2
   apply (simp add: aux_sym)
  apply (subgoal_tac "b = aa")
   apply (rule_tac [2] iffD1)
    apply (rule_tac [2] a = a and c = a and P = P in aux_uniq)
      apply auto
  done

lemma aux_foo: "∀a b. Q a ∧ P a b --> R b ==> P a b ==> Q a ==> R b"
  apply auto
  done

lemma aux_bij: "bijP P F ==> symP P ==> P a b ==> (a ∈ F) = (b ∈ F)"
  apply (unfold bijP_def)
  apply (rule iffI)
  apply (erule_tac [!] aux_foo)
      apply simp_all
  apply (rule iffD2)
   apply (rule_tac P = P in aux_sym)
   apply simp_all
  done


lemma aux_bijRER:
  "(A, B) ∈ bijR P ==> uniqP P ==> symP P
    ==> ∀F. bijP P F ∧ F ⊆ A ∧ F ⊆ B --> F ∈ bijER P"
  apply (erule bijR.induct)
   apply simp
  apply (case_tac "a = b")
   apply clarify
   apply (case_tac "b ∈ F")
    prefer 2
    apply (simp add: subset_insert)
   apply (cut_tac F = F and a = b and A = A and B = B in aux1)
        prefer 6
        apply clarify
        apply (rule bijER.insert1)
          apply simp_all
   apply (subgoal_tac "bijP P C")
    apply simp
   apply (rule aux_in1)
      apply simp_all
  apply clarify
  apply (case_tac "a ∈ F")
   apply (case_tac [!] "b ∈ F")
     apply (cut_tac F = F and a = a and b = b and A = A and B = B
       in aux2)
            apply (simp_all add: subset_insert)
    apply clarify
    apply (rule bijER.insert2)
        apply simp_all
    apply (subgoal_tac "bijP P C")
     apply simp
    apply (rule aux_in2)
          apply simp_all
   apply (subgoal_tac "b ∈ F")
    apply (rule_tac [2] iffD1)
     apply (rule_tac [2] a = a and F = F and P = P in aux_bij)
       apply (simp_all (no_asm_simp))
   apply (subgoal_tac [2] "a ∈ F")
    apply (rule_tac [3] iffD2)
     apply (rule_tac [3] b = b and F = F and P = P in aux_bij)
       apply auto
  done

lemma bijR_bijER:
  "(A, A) ∈ bijR P ==>
    bijP P A ==> uniqP P ==> symP P ==> A ∈ bijER P"
  apply (cut_tac A = A and B = A and P = P in aux_bijRER)
     apply auto
  done

end