src/HOLCF/Stream.ML

Franz Regensburger's Higher-Order Logic of Computable Functions embedding LCF
in HOL.

(*  Title: 	HOLCF/stream.ML
    ID:         $Id$
    Author: 	Franz Regensburger
    Copyright   1993 Technische Universitaet Muenchen

Lemmas for stream.thy
*)

open Stream;

(* ------------------------------------------------------------------------*)
(* The isomorphisms stream_rep_iso and stream_abs_iso are strict           *)
(* ------------------------------------------------------------------------*)

val stream_iso_strict= stream_rep_iso RS (stream_abs_iso RS 
	(allI  RSN (2,allI RS iso_strict)));

val stream_rews = [stream_iso_strict RS conjunct1,
		stream_iso_strict RS conjunct2];

(* ------------------------------------------------------------------------*)
(* Properties of stream_copy                                               *)
(* ------------------------------------------------------------------------*)

fun prover defs thm =  prove_goalw Stream.thy defs thm
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(asm_simp_tac (HOLCF_ss addsimps 
		(stream_rews @ [stream_abs_iso,stream_rep_iso])) 1)
	]);

val stream_copy = 
	[
	prover [stream_copy_def] "stream_copy[f][UU]=UU",
	prover [stream_copy_def,scons_def] 
	"x~=UU ==> stream_copy[f][scons[x][xs]]= scons[x][f[xs]]"
	];

val stream_rews =  stream_copy @ stream_rews; 

(* ------------------------------------------------------------------------*)
(* Exhaustion and elimination for streams                                  *)
(* ------------------------------------------------------------------------*)

val Exh_stream = prove_goalw Stream.thy [scons_def]
	"s = UU | (? x xs. x~=UU & s = scons[x][xs])"
 (fn prems =>
	[
	(simp_tac HOLCF_ss  1),
	(rtac (stream_rep_iso RS subst) 1),
	(res_inst_tac [("p","stream_rep[s]")] sprodE 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
	(asm_simp_tac HOLCF_ss  1),
	(res_inst_tac [("p","y")] liftE1 1),
	(contr_tac 1),
	(rtac disjI2 1),
	(rtac exI 1),
	(rtac exI 1),
	(etac conjI 1),
	(asm_simp_tac HOLCF_ss  1)
	]);

val streamE = prove_goal Stream.thy 
	"[| s=UU ==> Q; !!x xs.[|s=scons[x][xs];x~=UU|]==>Q|]==>Q"
 (fn prems =>
	[
	(rtac (Exh_stream RS disjE) 1),
	(eresolve_tac prems 1),
	(etac exE 1),
	(etac exE 1),
	(resolve_tac prems 1),
	(fast_tac HOL_cs 1),
	(fast_tac HOL_cs 1)
	]);

(* ------------------------------------------------------------------------*)
(* Properties of stream_when                                               *)
(* ------------------------------------------------------------------------*)

fun prover defs thm =  prove_goalw Stream.thy defs thm
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(asm_simp_tac (HOLCF_ss addsimps 
		(stream_rews @ [stream_abs_iso,stream_rep_iso])) 1)
	]);


val stream_when = [
	prover [stream_when_def] "stream_when[f][UU]=UU",
	prover [stream_when_def,scons_def] 
		"x~=UU ==> stream_when[f][scons[x][xs]]= f[x][xs]"
	];

val stream_rews = stream_when @ stream_rews;

(* ------------------------------------------------------------------------*)
(* Rewrites for  discriminators and  selectors                             *)
(* ------------------------------------------------------------------------*)

fun prover defs thm = prove_goalw Stream.thy defs thm
 (fn prems =>
	[
	(simp_tac (HOLCF_ss addsimps stream_rews) 1)
	]);

val stream_discsel = [
	prover [is_scons_def] "is_scons[UU]=UU",
	prover [shd_def] "shd[UU]=UU",
	prover [stl_def] "stl[UU]=UU"
	];

fun prover defs thm = prove_goalw Stream.thy defs thm
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1)
	]);

val stream_discsel = [
prover [is_scons_def,shd_def,stl_def] "x~=UU ==> is_scons[scons[x][xs]]=TT",
prover [is_scons_def,shd_def,stl_def] "x~=UU ==> shd[scons[x][xs]]=x",
prover [is_scons_def,shd_def,stl_def] "x~=UU ==> stl[scons[x][xs]]=xs"
	] @ stream_discsel;

val stream_rews = stream_discsel @ stream_rews;

(* ------------------------------------------------------------------------*)
(* Definedness and strictness                                              *)
(* ------------------------------------------------------------------------*)

fun prover contr thm = prove_goal Stream.thy thm
 (fn prems =>
	[
	(res_inst_tac [("P1",contr)] classical3 1),
	(simp_tac (HOLCF_ss addsimps stream_rews) 1),
	(dtac sym 1),
	(asm_simp_tac HOLCF_ss  1),
	(simp_tac (HOLCF_ss addsimps (prems @ stream_rews)) 1)
	]);

val stream_constrdef = [
	prover "is_scons[UU] ~= UU" "x~=UU ==> scons[x][xs]~=UU"
	]; 

fun prover defs thm = prove_goalw Stream.thy defs thm
 (fn prems =>
	[
	(simp_tac (HOLCF_ss addsimps stream_rews) 1)
	]);

val stream_constrdef = [
	prover [scons_def] "scons[UU][xs]=UU"
	] @ stream_constrdef;

val stream_rews = stream_constrdef @ stream_rews;


(* ------------------------------------------------------------------------*)
(* Distinctness wrt. << and =                                              *)
(* ------------------------------------------------------------------------*)


(* ------------------------------------------------------------------------*)
(* Invertibility                                                           *)
(* ------------------------------------------------------------------------*)

val stream_invert =
	[
prove_goal Stream.thy "[|x1~=UU; y1~=UU;\
\ scons[x1][x2] << scons[y1][y2]|] ==> x1<< y1 & x2 << y2"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac conjI 1),
	(dres_inst_tac [("fo5","stream_when[LAM x l.x]")] monofun_cfun_arg 1),
	(etac box_less 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_when) 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_when) 1),
	(dres_inst_tac [("fo5","stream_when[LAM x l.l]")] monofun_cfun_arg 1),
	(etac box_less 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_when) 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_when) 1)
	])
	];

(* ------------------------------------------------------------------------*)
(* Injectivity                                                             *)
(* ------------------------------------------------------------------------*)

val stream_inject = 
	[
prove_goal Stream.thy "[|x1~=UU; y1~=UU;\
\ scons[x1][x2] = scons[y1][y2]|] ==> x1= y1 & x2 = y2"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac conjI 1),
	(dres_inst_tac [("f","stream_when[LAM x l.x]")] cfun_arg_cong 1),
	(etac box_equals 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_when) 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_when) 1),
	(dres_inst_tac [("f","stream_when[LAM x l.l]")] cfun_arg_cong 1),
	(etac box_equals 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_when) 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_when) 1)
	])
	];

(* ------------------------------------------------------------------------*)
(* definedness for  discriminators and  selectors                          *)
(* ------------------------------------------------------------------------*)

fun prover thm = prove_goal Stream.thy thm
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac streamE 1),
	(contr_tac 1),
	(REPEAT (asm_simp_tac (HOLCF_ss addsimps stream_discsel) 1))
	]);

val stream_discsel_def = 
	[
	prover "s~=UU ==> is_scons[s]~=UU", 
	prover "s~=UU ==> shd[s]~=UU" 
	];

val stream_rews = stream_discsel_def @ stream_rews;

(* ------------------------------------------------------------------------*)
(* Properties stream_take                                                  *)
(* ------------------------------------------------------------------------*)

val stream_take =
	[
prove_goalw Stream.thy [stream_take_def] "stream_take(n)[UU]=UU"
 (fn prems =>
	[
	(res_inst_tac [("n","n")] natE 1),
	(asm_simp_tac iterate_ss 1),
	(asm_simp_tac iterate_ss 1),
	(simp_tac (HOLCF_ss addsimps stream_rews) 1)
	]),
prove_goalw Stream.thy [stream_take_def] "stream_take(0)[xs]=UU"
 (fn prems =>
	[
	(asm_simp_tac iterate_ss 1)
	])];

fun prover thm = prove_goalw Stream.thy [stream_take_def] thm
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(simp_tac iterate_ss 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1)
	]);

val stream_take = [
prover 
  "x~=UU ==> stream_take(Suc(n))[scons[x][xs]]=scons[x][stream_take(n)[xs]]"
	] @ stream_take;

val stream_rews = stream_take @ stream_rews;

(* ------------------------------------------------------------------------*)
(* take lemma for streams                                                  *)
(* ------------------------------------------------------------------------*)

fun prover reach defs thm  = prove_goalw Stream.thy defs thm
 (fn prems =>
	[
	(res_inst_tac [("t","s1")] (reach RS subst) 1),
	(res_inst_tac [("t","s2")] (reach RS subst) 1),
	(rtac (fix_def2 RS ssubst) 1),
	(rtac (contlub_cfun_fun RS ssubst) 1),
	(rtac is_chain_iterate 1),
	(rtac (contlub_cfun_fun RS ssubst) 1),
	(rtac is_chain_iterate 1),
	(rtac lub_equal 1),
	(rtac (is_chain_iterate RS ch2ch_fappL) 1),
	(rtac (is_chain_iterate RS ch2ch_fappL) 1),
	(rtac allI 1),
	(resolve_tac prems 1)
	]);

val stream_take_lemma = prover stream_reach  [stream_take_def]
	"(!!n.stream_take(n)[s1]=stream_take(n)[s2]) ==> s1=s2";


(* ------------------------------------------------------------------------*)
(* Co -induction for streams                                               *)
(* ------------------------------------------------------------------------*)

val stream_coind_lemma = prove_goalw Stream.thy [stream_bisim_def] 
"stream_bisim(R) ==> ! p q.R(p,q) --> stream_take(n)[p]=stream_take(n)[q]"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(nat_ind_tac "n" 1),
	(simp_tac (HOLCF_ss addsimps stream_rews) 1),
	(strip_tac 1),
	((etac allE 1) THEN (etac allE 1) THEN (etac (mp RS disjE) 1)),
	(atac 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
	(etac exE 1),
	(etac exE 1),
	(etac exE 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
	(REPEAT (etac conjE 1)),
	(rtac cfun_arg_cong 1),
	(fast_tac HOL_cs 1)
	]);

val stream_coind = prove_goal Stream.thy "[|stream_bisim(R);R(p,q)|] ==> p = q"
 (fn prems =>
	[
	(rtac stream_take_lemma 1),
	(rtac (stream_coind_lemma RS spec RS spec RS mp) 1),
	(resolve_tac prems 1),
	(resolve_tac prems 1)
	]);

(* ------------------------------------------------------------------------*)
(* structural induction for admissible predicates                          *)
(* ------------------------------------------------------------------------*)

val stream_ind = prove_goal Stream.thy
"[| adm(P);\
\  P(UU);\
\  !! x s1.[|x~=UU;P(s1)|] ==> P(scons[x][s1])\
\  |] ==> P(s)"
 (fn prems =>
	[
	(rtac (stream_reach RS subst) 1),
	(res_inst_tac [("x","s")] spec 1),
	(rtac fix_ind 1),
	(rtac (allI RS adm_all) 1),
	(rtac adm_subst 1),
	(contX_tacR 1),
	(resolve_tac prems 1),
	(simp_tac HOLCF_ss 1),
	(resolve_tac prems 1),
	(strip_tac 1),
	(res_inst_tac [("s","xa")] streamE 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
	(resolve_tac prems 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
	(resolve_tac prems 1),
	(atac 1),
	(etac spec 1)
	]);


(* ------------------------------------------------------------------------*)
(* simplify use of Co-induction                                            *)
(* ------------------------------------------------------------------------*)

val surjectiv_scons = prove_goal Stream.thy "scons[shd[s]][stl[s]]=s"
 (fn prems =>
	[
	(res_inst_tac [("s","s")] streamE 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_rews) 1)
	]);


val stream_coind_lemma2 = prove_goalw Stream.thy [stream_bisim_def]
"!s1 s2. R(s1, s2)-->shd[s1]=shd[s2] & R(stl[s1],stl[s2]) ==>stream_bisim(R)"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(strip_tac 1),
	(etac allE 1),
	(etac allE 1),
	(dtac mp 1),
	(atac 1),
	(etac conjE 1),
	(res_inst_tac [("Q","s1 = UU & s2 = UU")] classical2 1),
	(rtac disjI1 1),
	(fast_tac HOL_cs 1),
	(rtac disjI2 1),
	(rtac disjE 1),
	(etac (de_morgan2 RS ssubst) 1),
	(res_inst_tac [("x","shd[s1]")] exI 1),
	(res_inst_tac [("x","stl[s1]")] exI 1),
	(res_inst_tac [("x","stl[s2]")] exI 1),
	(rtac conjI 1),
	(eresolve_tac stream_discsel_def 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_rews addsimps [surjectiv_scons]) 1),
	(eres_inst_tac [("s","shd[s1]"),("t","shd[s2]")] subst 1),
	(simp_tac (HOLCF_ss addsimps stream_rews addsimps [surjectiv_scons]) 1),
	(res_inst_tac [("x","shd[s2]")] exI 1),
	(res_inst_tac [("x","stl[s1]")] exI 1),
	(res_inst_tac [("x","stl[s2]")] exI 1),
	(rtac conjI 1),
	(eresolve_tac stream_discsel_def 1),
	(asm_simp_tac (HOLCF_ss addsimps stream_rews addsimps [surjectiv_scons]) 1),
	(res_inst_tac [("s","shd[s1]"),("t","shd[s2]")] ssubst 1),
	(etac sym 1),
	(simp_tac (HOLCF_ss addsimps stream_rews addsimps [surjectiv_scons]) 1)
	]);