(* Title: ZF/Induct/Brouwer.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
*)
header {* Infinite branching datatype definitions *}
theory Brouwer imports Main_ZFC begin
subsection {* The Brouwer ordinals *}
consts
brouwer :: i
datatype \<subseteq> "Vfrom(0, csucc(nat))"
"brouwer" = Zero | Suc ("b \<in> brouwer") | Lim ("h \<in> 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:
"[| b \<in> brouwer;
P(Zero);
!!b. [| b \<in> brouwer; P(b) |] ==> P(Suc(b));
!!h. [| h \<in> nat -> brouwer; \<forall>i \<in> nat. P(h`i)
|] ==> P(Lim(h))
|] ==> P(b)"
-- {* A nicer induction rule than the standard one. *}
proof -
case rule_context
assume "b \<in> brouwer"
thus ?thesis
apply induct
apply (assumption | rule rule_context)+
apply (fast elim: fun_weaken_type)
apply (fast dest: apply_type)
done
qed
subsection {* The Martin-Löf wellordering type *}
consts
Well :: "[i, i => i] => i"
datatype \<subseteq> "Vfrom(A \<union> (\<Union>x \<in> A. B(x)), csucc(nat \<union> |\<Union>x \<in> A. B(x)|))"
-- {* The union with @{text nat} ensures that the cardinal is infinite. *}
"Well(A, B)" = Sup ("a \<in> A", "f \<in> 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) = (\<Sigma> x \<in> 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:
"[| w \<in> Well(A, B);
!!a f. [| a \<in> A; f \<in> B(a) -> Well(A,B); \<forall>y \<in> B(a). P(f`y)
|] ==> P(Sup(a,f))
|] ==> P(w)"
-- {* A nicer induction rule than the standard one. *}
proof -
case rule_context
assume "w \<in> Well(A, B)"
thus ?thesis
apply induct
apply (assumption | rule rule_context)+
apply (fast elim: fun_weaken_type)
apply (fast dest: apply_type)
done
qed
lemma Well_bool_unfold: "Well(bool, \<lambda>x. x) = 1 + (1 -> Well(bool, \<lambda>x. x))"
-- {* In fact it's isomorphic to @{text nat}, but we need a recursion operator *}
-- {* for @{text Well} to prove this. *}
apply (rule Well_unfold [THEN trans])
apply (simp add: Sigma_bool Pi_empty1 succ_def)
done
end