src/HOLCF/explicit_domains/Coind.ML

Updated version and date

(*  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),
        (stac from_def2 1),
        (Simp_tac 1),
        (rtac refl 1)
        ]);


val from1 = prove_goal Coind.thy "from`UU = UU"
 (fn prems =>
        [
        (rtac trans 1),
        (stac from 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),
        (rewtac stream_bisim_def),
        (strip_tac 1),
        (etac exE 1),
        (case_tac "n=UU" 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),
        (case_tac "n=UU" 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),
        (stac coind_lemma1 1),
        (stac from 1),
        (asm_simp_tac (!simpset addsimps stream_rews) 1),
        (res_inst_tac [("x","dsucc`n")] exI 1),
        (rtac conjI 1),
        (rtac trans 1),
        (stac coind_lemma1 1),
        (asm_simp_tac (!simpset addsimps stream_rews) 1),
        (rtac refl 1),
        (rtac trans 1),
        (stac from 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)
        ]);