(* Title: HOLCF/coind.ML
ID: $Id$
Author: Franz Regensburger
Copyright 1993 Technische Universitaet Muenchen
*)
open Coind;
(* ------------------------------------------------------------------------- *)
(* expand fixed point properties *)
(* ------------------------------------------------------------------------- *)
val nats_def2 = fix_prover2 Coind.thy nats_def
"nats = scons`dzero`(smap`dsucc`nats)";
val from_def2 = fix_prover2 Coind.thy from_def
"from = (LAM n.scons`n`(from`(dsucc`n)))";
(* ------------------------------------------------------------------------- *)
(* recursive properties *)
(* ------------------------------------------------------------------------- *)
val from = prove_goal Coind.thy "from`n = scons`n`(from`(dsucc`n))"
(fn prems =>
[
(rtac trans 1),
(rtac (from_def2 RS ssubst) 1),
(Simp_tac 1),
(rtac refl 1)
]);
val from1 = prove_goal Coind.thy "from`UU = UU"
(fn prems =>
[
(rtac trans 1),
(rtac (from RS ssubst) 1),
(resolve_tac stream_constrdef 1),
(rtac refl 1)
]);
val coind_rews =
[iterator1, iterator2, iterator3, smap1, smap2,from1];
(* ------------------------------------------------------------------------- *)
(* the example *)
(* prove: nats = from`dzero *)
(* ------------------------------------------------------------------------- *)
val coind_lemma1 = prove_goal Coind.thy "iterator`n`(smap`dsucc)`nats =\
\ scons`n`(iterator`(dsucc`n)`(smap`dsucc)`nats)"
(fn prems =>
[
(res_inst_tac [("s","n")] dnat_ind 1),
(simp_tac (!simpset addsimps (coind_rews @ stream_rews)) 1),
(simp_tac (!simpset addsimps (coind_rews @ stream_rews)) 1),
(rtac trans 1),
(rtac nats_def2 1),
(simp_tac (!simpset addsimps (coind_rews @ dnat_rews)) 1),
(rtac trans 1),
(etac iterator3 1),
(rtac trans 1),
(Asm_simp_tac 1),
(rtac trans 1),
(etac smap2 1),
(rtac cfun_arg_cong 1),
(asm_simp_tac (!simpset addsimps ([iterator3 RS sym] @ dnat_rews)) 1)
]);
val nats_eq_from = prove_goal Coind.thy "nats = from`dzero"
(fn prems =>
[
(res_inst_tac [("R",
"% p q.? n. p = iterator`n`(smap`dsucc)`nats & q = from`n")] stream_coind 1),
(res_inst_tac [("x","dzero")] exI 2),
(asm_simp_tac (!simpset addsimps coind_rews) 2),
(rewrite_goals_tac [stream_bisim_def]),
(strip_tac 1),
(etac exE 1),
(res_inst_tac [("Q","n=UU")] classical2 1),
(rtac disjI1 1),
(asm_simp_tac (!simpset addsimps coind_rews) 1),
(rtac disjI2 1),
(etac conjE 1),
(hyp_subst_tac 1),
(res_inst_tac [("x","n")] exI 1),
(res_inst_tac [("x","iterator`(dsucc`n)`(smap`dsucc)`nats")] exI 1),
(res_inst_tac [("x","from`(dsucc`n)")] exI 1),
(etac conjI 1),
(rtac conjI 1),
(rtac coind_lemma1 1),
(rtac conjI 1),
(rtac from 1),
(res_inst_tac [("x","dsucc`n")] exI 1),
(fast_tac HOL_cs 1)
]);
(* another proof using stream_coind_lemma2 *)
val nats_eq_from = prove_goal Coind.thy "nats = from`dzero"
(fn prems =>
[
(res_inst_tac [("R","% p q.? n. p = \
\ iterator`n`(smap`dsucc)`nats & q = from`n")] stream_coind 1),
(rtac stream_coind_lemma2 1),
(strip_tac 1),
(etac exE 1),
(res_inst_tac [("Q","n=UU")] classical2 1),
(asm_simp_tac (!simpset addsimps coind_rews) 1),
(res_inst_tac [("x","UU::dnat")] exI 1),
(simp_tac (!simpset addsimps coind_rews addsimps stream_rews) 1),
(etac conjE 1),
(hyp_subst_tac 1),
(rtac conjI 1),
(rtac (coind_lemma1 RS ssubst) 1),
(rtac (from RS ssubst) 1),
(asm_simp_tac (!simpset addsimps stream_rews) 1),
(res_inst_tac [("x","dsucc`n")] exI 1),
(rtac conjI 1),
(rtac trans 1),
(rtac (coind_lemma1 RS ssubst) 1),
(asm_simp_tac (!simpset addsimps stream_rews) 1),
(rtac refl 1),
(rtac trans 1),
(rtac (from RS ssubst) 1),
(asm_simp_tac (!simpset addsimps stream_rews) 1),
(rtac refl 1),
(res_inst_tac [("x","dzero")] exI 1),
(asm_simp_tac (!simpset addsimps coind_rews) 1)
]);