(* Title: CCL/Fix.thy
Author: Martin Coen
Copyright 1993 University of Cambridge
*)
header {* Tentative attempt at including fixed point induction; justified by Smith *}
theory Fix
imports Type
begin
consts
idgen :: "[i]=>i"
INCL :: "[i=>o]=>o"
defs
idgen_def:
"idgen(f) == lam t. case(t,true,false,%x y.<f`x, f`y>,%u. lam x. f ` u(x))"
axioms
INCL_def: "INCL(%x. P(x)) == (ALL f.(ALL n:Nat. P(f^n`bot)) --> P(fix(f)))"
po_INCL: "INCL(%x. a(x) [= b(x))"
INCL_subst: "INCL(P) ==> INCL(%x. P((g::i=>i)(x)))"
subsection {* Fixed Point Induction *}
lemma fix_ind:
assumes base: "P(bot)"
and step: "!!x. P(x) ==> 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 {* Inclusive Predicates *}
lemma inclXH: "INCL(P) <-> (ALL f. (ALL n:Nat. P(f ^ n ` bot)) --> P(fix(f)))"
by (simp add: INCL_def)
lemma inclI: "[| !!f. ALL n:Nat. P(f^n`bot) ==> P(fix(f)) |] ==> INCL(%x. P(x))"
unfolding inclXH by blast
lemma inclD: "[| INCL(P); !!n. n:Nat ==> P(f^n`bot) |] ==> P(fix(f))"
unfolding inclXH by blast
lemma inclE: "[| INCL(P); (ALL n:Nat. P(f^n`bot))-->P(fix(f)) ==> R |] ==> R"
by (blast dest: inclD)
subsection {* Lemmas for Inclusive Predicates *}
lemma npo_INCL: "INCL(%x.~ 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: "[| INCL(P); INCL(Q) |] ==> INCL(%x. P(x) & Q(x))"
by (blast intro!: inclI dest!: inclD)
lemma all_INCL: "[| !!a. INCL(P(a)) |] ==> INCL(%x. ALL a. P(a,x))"
by (blast intro!: inclI dest!: inclD)
lemma ball_INCL: "[| !!a. a:A ==> INCL(P(a)) |] ==> INCL(%x. ALL a:A. P(a,x))"
by (blast intro!: inclI dest!: inclD)
lemma eq_INCL: "INCL(%x. a(x) = (b(x)::'a::prog))"
apply (simp add: eq_iff)
apply (rule conj_INCL po_INCL)+
done
subsection {* Derivation of Reachability Condition *}
(* 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_lemma idgenfp_lam: "idgen(d) = d ==> 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: "[| a = b; a ` t = u |] ==> b ` t = u"
by (simp add: idgen_def)
lemma idgen_lemmas:
"idgen(d) = d ==> d ` bot = bot"
"idgen(d) = d ==> d ` true = true"
"idgen(d) = d ==> d ` false = false"
"idgen(d) = d ==> d ` <a,b> = <d ` a,d ` b>"
"idgen(d) = d ==> 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:
"[| ALL x. t ` x [= u ` x; EX f. t=lam x. f(x); EX f. u=lam x. f(x) |] ==> t [= u"
apply (drule cond_eta)+
apply (erule ssubst)
apply (erule ssubst)
apply (rule po_lam [THEN iffD2])
apply simp
done
schematic_lemma po_eta_lemma: "idgen(d) = d ==> d = lam x. ?f(x)"
apply (unfold idgen_def)
apply (erule sym)
done
lemma lemma1:
"idgen(d) = d ==>
{p. EX a b. p=<a,b> & (EX t. a=fix(idgen) ` t & b = d ` t)} <=
POgen({p. EX a b. p=<a,b> & (EX t. a=fix(idgen) ` t & 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 ==> 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 ==>
{p. EX a b. p=<a,b> & b = d ` a} <= POgen({p. EX a b. p=<a,b> & 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 ==> 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 ==> 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: "!!x y.[| P(x); P(y) |] ==> P(<x,y>)"
and 5: "!!u.(!!x. P(u(x))) ==> 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