Theory Brouwer

(*  Title:      ZF/Induct/Brouwer.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1994  University of Cambridge
*)

section ‹Infinite branching datatype definitions›

theory Brouwer imports ZFC begin

subsection ‹The Brouwer ordinals›

consts
  brouwer :: i

datatype ⊆ "Vfrom(0, csucc(nat))"
    "brouwer" = Zero | Suc ("b ∈ brouwer") | Lim ("h ∈ nat -> brouwer")
  monos Pi_mono
  type_intros inf_datatype_intros

lemma brouwer_unfold: "brouwer = {0} + brouwer + (nat -> brouwer)"
  by (fast intro!: brouwer.intros [unfolded brouwer.con_defs]
    elim: brouwer.cases [unfolded brouwer.con_defs])

lemma brouwer_induct2 [consumes 1, case_names Zero Suc Lim]:
  assumes b: "b ∈ brouwer"
    and cases:
      "P(Zero)"
      "⋀b. ⟦b ∈ brouwer;  P(b)⟧ ⟹ P(Suc(b))"
      "⋀h. ⟦h ∈ nat -> brouwer;  ∀i ∈ nat. P(h`i)⟧ ⟹ P(Lim(h))"
  shows "P(b)"
  ― ‹A nicer induction rule than the standard one.›
  using b
  apply induct
    apply (rule cases(1))
   apply (erule (1) cases(2))
  apply (rule cases(3))
   apply (fast elim: fun_weaken_type)
  apply (fast dest: apply_type)
  done


subsection ‹The Martin-Löf wellordering type›

consts
  Well :: "[i, i ⇒ i] ⇒ i"

datatype ⊆ "Vfrom(A ∪ (⋃x ∈ A. B(x)), csucc(nat ∪ |⋃x ∈ A. B(x)|))"
    ― ‹The union with ‹nat› ensures that the cardinal is infinite.›
  "Well(A, B)" = Sup ("a ∈ A", "f ∈ B(a) -> Well(A, B)")
  monos Pi_mono
  type_intros le_trans [OF UN_upper_cardinal le_nat_Un_cardinal] inf_datatype_intros

lemma Well_unfold: "Well(A, B) = (∑x ∈ A. B(x) -> Well(A, B))"
  by (fast intro!: Well.intros [unfolded Well.con_defs]
    elim: Well.cases [unfolded Well.con_defs])


lemma Well_induct2 [consumes 1, case_names step]:
  assumes w: "w ∈ Well(A, B)"
    and step: "⋀a f. ⟦a ∈ A;  f ∈ B(a) -> Well(A,B);  ∀y ∈ B(a). P(f`y)⟧ ⟹ P(Sup(a,f))"
  shows "P(w)"
  ― ‹A nicer induction rule than the standard one.›
  using w
  apply induct
  apply (assumption | rule step)+
   apply (fast elim: fun_weaken_type)
  apply (fast dest: apply_type)
  done

lemma Well_bool_unfold: "Well(bool, λx. x) = 1 + (1 -> Well(bool, λx. x))"
  ― ‹In fact it's isomorphic to ‹nat›, but we need a recursion operator›
  ― ‹for ‹Well› to prove this.›
  apply (rule Well_unfold [THEN trans])
  apply (simp add: Sigma_bool succ_def)
  done

end