--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/stream.ML Wed Jan 19 17:35:01 1994 +0100
@@ -0,0 +1,405 @@
+(* 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)
+ ]);
+
+