(* Title: CCL/Fix.thy
Author: Martin Coen
Copyright 1993 University of Cambridge
*)
section \<open>Tentative attempt at including fixed point induction; justified by Smith\<close>
theory Fix
imports Type
begin
definition idgen :: "i \<Rightarrow> i"
where "idgen(f) == lam t. case(t,true,false, \<lambda>x y.<f`x, f`y>, \<lambda>u. lam x. f ` u(x))"
axiomatization INCL :: "[i\<Rightarrow>o]\<Rightarrow>o" where
INCL_def: "INCL(\<lambda>x. P(x)) == (ALL f.(ALL n:Nat. P(f^n`bot)) \<longrightarrow> P(fix(f)))" and
po_INCL: "INCL(\<lambda>x. a(x) [= b(x))" and
INCL_subst: "INCL(P) \<Longrightarrow> INCL(\<lambda>x. P((g::i\<Rightarrow>i)(x)))"
subsection \<open>Fixed Point Induction\<close>
lemma fix_ind:
assumes base: "P(bot)"
and step: "\<And>x. P(x) \<Longrightarrow> P(f(x))"
and incl: "INCL(P)"
shows "P(fix(f))"
apply (rule incl [unfolded INCL_def, rule_format])
apply (rule Nat_ind [THEN ballI], assumption)
apply simp_all
apply (rule base)
apply (erule step)
done
subsection \<open>Inclusive Predicates\<close>
lemma inclXH: "INCL(P) \<longleftrightarrow> (ALL f. (ALL n:Nat. P(f ^ n ` bot)) \<longrightarrow> P(fix(f)))"
by (simp add: INCL_def)
lemma inclI: "\<lbrakk>\<And>f. ALL n:Nat. P(f^n`bot) \<Longrightarrow> P(fix(f))\<rbrakk> \<Longrightarrow> INCL(\<lambda>x. P(x))"
unfolding inclXH by blast
lemma inclD: "\<lbrakk>INCL(P); \<And>n. n:Nat \<Longrightarrow> P(f^n`bot)\<rbrakk> \<Longrightarrow> P(fix(f))"
unfolding inclXH by blast
lemma inclE: "\<lbrakk>INCL(P); (ALL n:Nat. P(f^n`bot)) \<longrightarrow> P(fix(f)) \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
by (blast dest: inclD)
subsection \<open>Lemmas for Inclusive Predicates\<close>
lemma npo_INCL: "INCL(\<lambda>x. \<not> a(x) [= t)"
apply (rule inclI)
apply (drule bspec)
apply (rule zeroT)
apply (erule contrapos)
apply (rule po_trans)
prefer 2
apply assumption
apply (subst napplyBzero)
apply (rule po_cong, rule po_bot)
done
lemma conj_INCL: "\<lbrakk>INCL(P); INCL(Q)\<rbrakk> \<Longrightarrow> INCL(\<lambda>x. P(x) \<and> Q(x))"
by (blast intro!: inclI dest!: inclD)
lemma all_INCL: "(\<And>a. INCL(P(a))) \<Longrightarrow> INCL(\<lambda>x. ALL a. P(a,x))"
by (blast intro!: inclI dest!: inclD)
lemma ball_INCL: "(\<And>a. a:A \<Longrightarrow> INCL(P(a))) \<Longrightarrow> INCL(\<lambda>x. ALL a:A. P(a,x))"
by (blast intro!: inclI dest!: inclD)
lemma eq_INCL: "INCL(\<lambda>x. a(x) = (b(x)::'a::prog))"
apply (simp add: eq_iff)
apply (rule conj_INCL po_INCL)+
done
subsection \<open>Derivation of Reachability Condition\<close>
(* Fixed points of idgen *)
lemma fix_idgenfp: "idgen(fix(idgen)) = fix(idgen)"
apply (rule fixB [symmetric])
done
lemma id_idgenfp: "idgen(lam x. x) = lam x. x"
apply (simp add: idgen_def)
apply (rule term_case [THEN allI])
apply simp_all
done
(* All fixed points are lam-expressions *)
schematic_goal idgenfp_lam: "idgen(d) = d \<Longrightarrow> d = lam x. ?f(x)"
apply (unfold idgen_def)
apply (erule ssubst)
apply (rule refl)
done
(* Lemmas for rewriting fixed points of idgen *)
lemma l_lemma: "\<lbrakk>a = b; a ` t = u\<rbrakk> \<Longrightarrow> b ` t = u"
by (simp add: idgen_def)
lemma idgen_lemmas:
"idgen(d) = d \<Longrightarrow> d ` bot = bot"
"idgen(d) = d \<Longrightarrow> d ` true = true"
"idgen(d) = d \<Longrightarrow> d ` false = false"
"idgen(d) = d \<Longrightarrow> d ` <a,b> = <d ` a,d ` b>"
"idgen(d) = d \<Longrightarrow> d ` (lam x. f(x)) = lam x. d ` f(x)"
by (erule l_lemma, simp add: idgen_def)+
(* Proof of Reachability law - show that fix and lam x.x both give LEAST fixed points
of idgen and hence are they same *)
lemma po_eta:
"\<lbrakk>ALL x. t ` x [= u ` x; EX f. t=lam x. f(x); EX f. u=lam x. f(x)\<rbrakk> \<Longrightarrow> t [= u"
apply (drule cond_eta)+
apply (erule ssubst)
apply (erule ssubst)
apply (rule po_lam [THEN iffD2])
apply simp
done
schematic_goal po_eta_lemma: "idgen(d) = d \<Longrightarrow> d = lam x. ?f(x)"
apply (unfold idgen_def)
apply (erule sym)
done
lemma lemma1:
"idgen(d) = d \<Longrightarrow>
{p. EX a b. p=<a,b> \<and> (EX t. a=fix(idgen) ` t \<and> b = d ` t)} <=
POgen({p. EX a b. p=<a,b> \<and> (EX t. a=fix(idgen) ` t \<and> b = d ` t)})"
apply clarify
apply (rule_tac t = t in term_case)
apply (simp_all add: POgenXH idgen_lemmas idgen_lemmas [OF fix_idgenfp])
apply blast
apply fast
done
lemma fix_least_idgen: "idgen(d) = d \<Longrightarrow> fix(idgen) [= d"
apply (rule allI [THEN po_eta])
apply (rule lemma1 [THEN [2] po_coinduct])
apply (blast intro: po_eta_lemma fix_idgenfp)+
done
lemma lemma2:
"idgen(d) = d \<Longrightarrow>
{p. EX a b. p=<a,b> \<and> b = d ` a} <= POgen({p. EX a b. p=<a,b> \<and> b = d ` a})"
apply clarify
apply (rule_tac t = a in term_case)
apply (simp_all add: POgenXH idgen_lemmas)
apply fast
done
lemma id_least_idgen: "idgen(d) = d \<Longrightarrow> lam x. x [= d"
apply (rule allI [THEN po_eta])
apply (rule lemma2 [THEN [2] po_coinduct])
apply simp
apply (fast intro: po_eta_lemma fix_idgenfp)+
done
lemma reachability: "fix(idgen) = lam x. x"
apply (fast intro: eq_iff [THEN iffD2]
id_idgenfp [THEN fix_least_idgen] fix_idgenfp [THEN id_least_idgen])
done
(********)
lemma id_apply: "f = lam x. x \<Longrightarrow> f`t = t"
apply (erule ssubst)
apply (rule applyB)
done
lemma term_ind:
assumes 1: "P(bot)" and 2: "P(true)" and 3: "P(false)"
and 4: "\<And>x y. \<lbrakk>P(x); P(y)\<rbrakk> \<Longrightarrow> P(<x,y>)"
and 5: "\<And>u.(\<And>x. P(u(x))) \<Longrightarrow> P(lam x. u(x))"
and 6: "INCL(P)"
shows "P(t)"
apply (rule reachability [THEN id_apply, THEN subst])
apply (rule_tac x = t in spec)
apply (rule fix_ind)
apply (unfold idgen_def)
apply (rule allI)
apply (subst applyBbot)
apply (rule 1)
apply (rule allI)
apply (rule applyB [THEN ssubst])
apply (rule_tac t = "xa" in term_case)
apply simp_all
apply (fast intro: assms INCL_subst all_INCL)+
done
end