(* Title: 92/CCL/coinduction
ID: $Id$
Author: Martin Coen, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Lemmas and tactics for using the rule coinduct3 on [= and =.
*)
val [mono,prem] = goal Lfp.thy "[| mono(f); a : f(lfp(f)) |] ==> a : lfp(f)";
by (stac (mono RS lfp_Tarski) 1);
by (rtac prem 1);
qed "lfpI";
val prems = goal CCL.thy "[| a=a'; a' : A |] ==> a : A";
by (simp_tac (term_ss addsimps prems) 1);
qed "ssubst_single";
val prems = goal CCL.thy "[| a=a'; b=b'; <a',b'> : A |] ==> <a,b> : A";
by (simp_tac (term_ss addsimps prems) 1);
qed "ssubst_pair";
(***)
local
fun mk_thm s = prove_goal Term.thy s (fn mono::prems =>
[fast_tac (term_cs addIs ((mono RS coinduct3_mono_lemma RS lfpI)::prems)) 1]);
in
val ci3_RI = mk_thm "[| mono(Agen); a : R |] ==> a : lfp(%x. Agen(x) Un R Un A)";
val ci3_AgenI = mk_thm "[| mono(Agen); a : Agen(lfp(%x. Agen(x) Un R Un A)) |] ==> \
\ a : lfp(%x. Agen(x) Un R Un A)";
val ci3_AI = mk_thm "[| mono(Agen); a : A |] ==> a : lfp(%x. Agen(x) Un R Un A)";
end;
fun mk_genIs thy defs genXH gen_mono s = prove_goalw thy defs s
(fn prems => [rtac (genXH RS iffD2) 1,
(simp_tac term_ss 1),
TRY (fast_tac (term_cs addIs
([genXH RS iffD2,gen_mono RS coinduct3_mono_lemma RS lfpI]
@ prems)) 1)]);
(** POgen **)
goal Term.thy "<a,a> : PO";
by (rtac (po_refl RS (XH_to_D PO_iff)) 1);
qed "PO_refl";
val POgenIs = map (mk_genIs Term.thy data_defs POgenXH POgen_mono)
["<true,true> : POgen(R)",
"<false,false> : POgen(R)",
"[| <a,a'> : R; <b,b'> : R |] ==> <<a,b>,<a',b'>> : POgen(R)",
"[|!!x. <b(x),b'(x)> : R |] ==><lam x. b(x),lam x. b'(x)> : POgen(R)",
"<one,one> : POgen(R)",
"<a,a'> : lfp(%x. POgen(x) Un R Un PO) ==> \
\ <inl(a),inl(a')> : POgen(lfp(%x. POgen(x) Un R Un PO))",
"<b,b'> : lfp(%x. POgen(x) Un R Un PO) ==> \
\ <inr(b),inr(b')> : POgen(lfp(%x. POgen(x) Un R Un PO))",
"<zero,zero> : POgen(lfp(%x. POgen(x) Un R Un PO))",
"<n,n'> : lfp(%x. POgen(x) Un R Un PO) ==> \
\ <succ(n),succ(n')> : POgen(lfp(%x. POgen(x) Un R Un PO))",
"<[],[]> : POgen(lfp(%x. POgen(x) Un R Un PO))",
"[| <h,h'> : lfp(%x. POgen(x) Un R Un PO); \
\ <t,t'> : lfp(%x. POgen(x) Un R Un PO) |] ==> \
\ <h$t,h'$t'> : POgen(lfp(%x. POgen(x) Un R Un PO))"];
fun POgen_tac (rla,rlb) i =
SELECT_GOAL (safe_tac ccl_cs) i THEN
rtac (rlb RS (rla RS ssubst_pair)) i THEN
(REPEAT (resolve_tac (POgenIs @ [PO_refl RS (POgen_mono RS ci3_AI)] @
(POgenIs RL [POgen_mono RS ci3_AgenI]) @ [POgen_mono RS ci3_RI]) i));
(** EQgen **)
goal Term.thy "<a,a> : EQ";
by (rtac (refl RS (EQ_iff RS iffD1)) 1);
qed "EQ_refl";
val EQgenIs = map (mk_genIs Term.thy data_defs EQgenXH EQgen_mono)
["<true,true> : EQgen(R)",
"<false,false> : EQgen(R)",
"[| <a,a'> : R; <b,b'> : R |] ==> <<a,b>,<a',b'>> : EQgen(R)",
"[|!!x. <b(x),b'(x)> : R |] ==> <lam x. b(x),lam x. b'(x)> : EQgen(R)",
"<one,one> : EQgen(R)",
"<a,a'> : lfp(%x. EQgen(x) Un R Un EQ) ==> \
\ <inl(a),inl(a')> : EQgen(lfp(%x. EQgen(x) Un R Un EQ))",
"<b,b'> : lfp(%x. EQgen(x) Un R Un EQ) ==> \
\ <inr(b),inr(b')> : EQgen(lfp(%x. EQgen(x) Un R Un EQ))",
"<zero,zero> : EQgen(lfp(%x. EQgen(x) Un R Un EQ))",
"<n,n'> : lfp(%x. EQgen(x) Un R Un EQ) ==> \
\ <succ(n),succ(n')> : EQgen(lfp(%x. EQgen(x) Un R Un EQ))",
"<[],[]> : EQgen(lfp(%x. EQgen(x) Un R Un EQ))",
"[| <h,h'> : lfp(%x. EQgen(x) Un R Un EQ); \
\ <t,t'> : lfp(%x. EQgen(x) Un R Un EQ) |] ==> \
\ <h$t,h'$t'> : EQgen(lfp(%x. EQgen(x) Un R Un EQ))"];
fun EQgen_raw_tac i =
(REPEAT (resolve_tac (EQgenIs @ [EQ_refl RS (EQgen_mono RS ci3_AI)] @
(EQgenIs RL [EQgen_mono RS ci3_AgenI]) @ [EQgen_mono RS ci3_RI]) i));
(* Goals of the form R <= EQgen(R) - rewrite elements <a,b> : EQgen(R) using rews and *)
(* then reduce this to a goal <a',b'> : R (hopefully?) *)
(* rews are rewrite rules that would cause looping in the simpifier *)
fun EQgen_tac simp_set rews i =
SELECT_GOAL
(TRY (safe_tac ccl_cs) THEN
resolve_tac ((rews@[refl]) RL ((rews@[refl]) RL [ssubst_pair])) i THEN
ALLGOALS (simp_tac simp_set) THEN
ALLGOALS EQgen_raw_tac) i;