(* Title: CCL/fix
ID: $Id$
Author: Martin Coen, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
For fix.thy.
*)
open Fix;
(*** Fixed Point Induction ***)
val [base,step,incl] = goalw Fix.thy [INCL_def]
"[| P(bot); !!x.P(x) ==> P(f(x)); INCL(P) |] ==> P(fix(f))";
br (incl RS spec RS mp) 1;
by (rtac (Nat_ind RS ballI) 1 THEN atac 1);
by (ALLGOALS (simp_tac term_ss));
by (REPEAT (ares_tac [base,step] 1));
val fix_ind = result();
(*** Inclusive Predicates ***)
val prems = goalw Fix.thy [INCL_def]
"INCL(P) <-> (ALL f. (ALL n:Nat. P(f ^ n ` bot)) --> P(fix(f)))";
br iff_refl 1;
val inclXH = result();
val prems = goal Fix.thy
"[| !!f.ALL n:Nat.P(f^n`bot) ==> P(fix(f)) |] ==> INCL(%x.P(x))";
by (fast_tac (term_cs addIs (prems @ [XH_to_I inclXH])) 1);
val inclI = result();
val incl::prems = goal Fix.thy
"[| INCL(P); !!n.n:Nat ==> P(f^n`bot) |] ==> P(fix(f))";
by (fast_tac (term_cs addIs ([ballI RS (incl RS (XH_to_D inclXH) RS spec RS mp)]
@ prems)) 1);
val inclD = result();
val incl::prems = goal Fix.thy
"[| INCL(P); (ALL n:Nat.P(f^n`bot))-->P(fix(f)) ==> R |] ==> R";
by (fast_tac (term_cs addIs ([incl RS inclD] @ prems)) 1);
val inclE = result();
(*** Lemmas for Inclusive Predicates ***)
goal Fix.thy "INCL(%x.~ a(x) [= t)";
br inclI 1;
bd bspec 1;
br zeroT 1;
be contrapos 1;
br po_trans 1;
ba 2;
br (napplyBzero RS ssubst) 1;
by (rtac po_cong 1 THEN rtac po_bot 1);
val npo_INCL = result();
val prems = goal Fix.thy "[| INCL(P); INCL(Q) |] ==> INCL(%x.P(x) & Q(x))";
by (fast_tac (set_cs addSIs ([inclI] @ (prems RL [inclD]))) 1);;
val conj_INCL = result();
val prems = goal Fix.thy "[| !!a.INCL(P(a)) |] ==> INCL(%x.ALL a.P(a,x))";
by (fast_tac (set_cs addSIs ([inclI] @ (prems RL [inclD]))) 1);;
val all_INCL = result();
val prems = goal Fix.thy "[| !!a.a:A ==> INCL(P(a)) |] ==> INCL(%x.ALL a:A.P(a,x))";
by (fast_tac (set_cs addSIs ([inclI] @ (prems RL [inclD]))) 1);;
val ball_INCL = result();
goal Fix.thy "INCL(%x.a(x) = b(x)::'a::prog)";
by (simp_tac (term_ss addsimps [eq_iff]) 1);
by (REPEAT (resolve_tac [conj_INCL,po_INCL] 1));
val eq_INCL = result();
(*** Derivation of Reachability Condition ***)
(* Fixed points of idgen *)
goal Fix.thy "idgen(fix(idgen)) = fix(idgen)";
br (fixB RS sym) 1;
val fix_idgenfp = result();
goalw Fix.thy [idgen_def] "idgen(lam x.x) = lam x.x";
by (simp_tac term_ss 1);
br (term_case RS allI) 1;
by (ALLGOALS (simp_tac term_ss));
val id_idgenfp = result();
(* All fixed points are lam-expressions *)
val [prem] = goal Fix.thy "idgen(d) = d ==> d = lam x.?f(x)";
br (prem RS subst) 1;
bw idgen_def;
br refl 1;
val idgenfp_lam = result();
(* Lemmas for rewriting fixed points of idgen *)
val prems = goalw Fix.thy [idgen_def]
"[| a = b; a ` t = u |] ==> b ` t = u";
by (simp_tac (term_ss addsimps (prems RL [sym])) 1);
val l_lemma= result();
val idgen_lemmas =
let fun mk_thm s = prove_goalw Fix.thy [idgen_def] s
(fn [prem] => [rtac (prem RS l_lemma) 1,simp_tac term_ss 1])
in map mk_thm
[ "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)"]
end;
(* Proof of Reachability law - show that fix and lam x.x both give LEAST fixed points
of idgen and hence are they same *)
val [p1,p2,p3] = goal CCL.thy
"[| ALL x.t ` x [= u ` x; EX f.t=lam x.f(x); EX f.u=lam x.f(x) |] ==> t [= u";
br (p2 RS cond_eta RS ssubst) 1;
br (p3 RS cond_eta RS ssubst) 1;
br (p1 RS (po_lam RS iffD2)) 1;
val po_eta = result();
val [prem] = goalw Fix.thy [idgen_def] "idgen(d) = d ==> d = lam x.?f(x)";
br (prem RS subst) 1;
br refl 1;
val po_eta_lemma = result();
val [prem] = goal Fix.thy
"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)})";
by (REPEAT (step_tac term_cs 1));
by (term_case_tac "t" 1);
by (ALLGOALS (simp_tac (term_ss addsimps (POgenXH::([prem,fix_idgenfp] RL idgen_lemmas)))));
by (ALLGOALS (fast_tac set_cs));
val lemma1 = result();
val [prem] = goal Fix.thy
"idgen(d) = d ==> fix(idgen) [= d";
br (allI RS po_eta) 1;
br (lemma1 RSN(2,po_coinduct)) 1;
by (ALLGOALS (fast_tac (term_cs addIs [prem,po_eta_lemma,fix_idgenfp])));
val fix_least_idgen = result();
val [prem] = goal Fix.thy
"idgen(d) = d ==> \
\ {p.EX a b.p=<a,b> & b = d ` a} <= POgen({p.EX a b.p=<a,b> & b = d ` a})";
by (REPEAT (step_tac term_cs 1));
by (term_case_tac "a" 1);
by (ALLGOALS (simp_tac (term_ss addsimps (POgenXH::([prem] RL idgen_lemmas)))));
by (ALLGOALS (fast_tac set_cs));
val lemma2 = result();
val [prem] = goal Fix.thy
"idgen(d) = d ==> lam x.x [= d";
br (allI RS po_eta) 1;
br (lemma2 RSN(2,po_coinduct)) 1;
by (simp_tac term_ss 1);
by (ALLGOALS (fast_tac (term_cs addIs [prem,po_eta_lemma,fix_idgenfp])));
val id_least_idgen = result();
goal Fix.thy "fix(idgen) = lam x.x";
by (fast_tac (term_cs addIs [eq_iff RS iffD2,
id_idgenfp RS fix_least_idgen,
fix_idgenfp RS id_least_idgen]) 1);
val reachability = result();
(********)
val [prem] = goal Fix.thy "f = lam x.x ==> f`t = t";
br (prem RS sym RS subst) 1;
br applyB 1;
val id_apply = result();
val prems = goal Fix.thy
"[| P(bot); P(true); P(false); \
\ !!x y.[| P(x); P(y) |] ==> P(<x,y>); \
\ !!u.(!!x.P(u(x))) ==> P(lam x.u(x)); INCL(P) |] ==> \
\ P(t)";
br (reachability RS id_apply RS subst) 1;
by (res_inst_tac [("x","t")] spec 1);
br fix_ind 1;
bw idgen_def;
by (REPEAT_SOME (ares_tac [allI]));
br (applyBbot RS ssubst) 1;
brs prems 1;
br (applyB RS ssubst )1;
by (res_inst_tac [("t","xa")] term_case 1);
by (ALLGOALS (simp_tac term_ss));
by (ALLGOALS (fast_tac (term_cs addIs ([all_INCL,INCL_subst] @ prems))));
val term_ind = result();