src/HOLCF/dnat.ML

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

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

Lemmas for dnat.thy 
*)

open Dnat;

(* ------------------------------------------------------------------------*)
(* The isomorphisms dnat_rep_iso and dnat_abs_iso are strict               *)
(* ------------------------------------------------------------------------*)

val dnat_iso_strict= dnat_rep_iso RS (dnat_abs_iso RS 
	(allI  RSN (2,allI RS iso_strict)));

val dnat_rews = [dnat_iso_strict RS conjunct1,
		dnat_iso_strict RS conjunct2];

(* ------------------------------------------------------------------------*)
(* Properties of dnat_copy                                                 *)
(* ------------------------------------------------------------------------*)

fun prover defs thm =  prove_goalw Dnat.thy defs thm
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(asm_simp_tac (HOLCF_ss addsimps 
		(dnat_rews @ [dnat_abs_iso,dnat_rep_iso])) 1)
	]);

val dnat_copy = 
	[
	prover [dnat_copy_def] "dnat_copy[f][UU]=UU",
	prover [dnat_copy_def,dzero_def] "dnat_copy[f][dzero]= dzero",
	prover [dnat_copy_def,dsucc_def] 
		"n~=UU ==> dnat_copy[f][dsucc[n]] = dsucc[f[n]]"
	];

val dnat_rews =  dnat_copy @ dnat_rews; 

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

val Exh_dnat = prove_goalw Dnat.thy [dsucc_def,dzero_def]
	"n = UU | n = dzero | (? x . x~=UU & n = dsucc[x])"
 (fn prems =>
	[
	(simp_tac HOLCF_ss  1),
	(rtac (dnat_rep_iso RS subst) 1),
	(res_inst_tac [("p","dnat_rep[n]")] ssumE 1),
	(rtac disjI1 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1),
	(rtac (disjI1 RS disjI2) 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1),
	(res_inst_tac [("p","x")] oneE 1),
	(contr_tac 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1),
	(rtac (disjI2 RS disjI2) 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1),
	(fast_tac HOL_cs 1)
	]);

val dnatE = prove_goal Dnat.thy 
 "[| n=UU ==> Q; n=dzero ==> Q; !!x.[|n=dsucc[x];x~=UU|]==>Q|]==>Q"
 (fn prems =>
	[
	(rtac (Exh_dnat RS disjE) 1),
	(eresolve_tac prems 1),
	(etac disjE 1),
	(eresolve_tac prems 1),
	(REPEAT (etac exE 1)),
	(resolve_tac prems 1),
	(fast_tac HOL_cs 1),
	(fast_tac HOL_cs 1)
	]);

(* ------------------------------------------------------------------------*)
(* Properties of dnat_when                                                 *)
(* ------------------------------------------------------------------------*)

fun prover defs thm =  prove_goalw Dnat.thy defs thm
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(asm_simp_tac (HOLCF_ss addsimps 
		(dnat_rews @ [dnat_abs_iso,dnat_rep_iso])) 1)
	]);


val dnat_when = [
	prover [dnat_when_def] "dnat_when[c][f][UU]=UU",
	prover [dnat_when_def,dzero_def] "dnat_when[c][f][dzero]=c",
	prover [dnat_when_def,dsucc_def] 
		"n~=UU ==> dnat_when[c][f][dsucc[n]]=f[n]"
	];

val dnat_rews = dnat_when @ dnat_rews;

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

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

val dnat_discsel = [
	prover [is_dzero_def] "is_dzero[UU]=UU",
	prover [is_dsucc_def] "is_dsucc[UU]=UU",
	prover [dpred_def] "dpred[UU]=UU"
	];


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

val dnat_discsel = [
	prover [is_dzero_def] "is_dzero[dzero]=TT",
	prover [is_dzero_def] "n~=UU ==>is_dzero[dsucc[n]]=FF",
	prover [is_dsucc_def] "is_dsucc[dzero]=FF",
	prover [is_dsucc_def] "n~=UU ==> is_dsucc[dsucc[n]]=TT",
	prover [dpred_def] "dpred[dzero]=UU",
	prover [dpred_def] "n~=UU ==> dpred[dsucc[n]]=n"
	] @ dnat_discsel;

val dnat_rews = dnat_discsel @ dnat_rews;

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

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

val dnat_constrdef = [
	prover "is_dzero[UU] ~= UU" "dzero~=UU",
	prover "is_dsucc[UU] ~= UU" "n~=UU ==> dsucc[n]~=UU"
	]; 


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

val dnat_constrdef = [
	prover [dsucc_def] "dsucc[UU]=UU"
	] @ dnat_constrdef;

val dnat_rews = dnat_constrdef @ dnat_rews;


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

fun prover contrfun thm = prove_goal Dnat.thy thm
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(res_inst_tac [("P1","TT << FF")] classical3 1),
	(resolve_tac dist_less_tr 1),
	(dres_inst_tac [("fo5",contrfun)] monofun_cfun_arg 1),
	(etac box_less 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1)
	]);

val dnat_dist_less = 
	[
prover "is_dzero" "n~=UU ==> ~dzero << dsucc[n]",
prover "is_dsucc" "n~=UU ==> ~dsucc[n] << dzero"
	];

fun prover contrfun thm = prove_goal Dnat.thy thm
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(res_inst_tac [("P1","TT = FF")] classical3 1),
	(resolve_tac dist_eq_tr 1),
	(dres_inst_tac [("f",contrfun)] cfun_arg_cong 1),
	(etac box_equals 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1)
	]);

val dnat_dist_eq = 
	[
prover "is_dzero" "n~=UU ==> dzero ~= dsucc[n]",
prover "is_dsucc" "n~=UU ==> dsucc[n] ~= dzero"
	];

val dnat_rews = dnat_dist_less @ dnat_dist_eq @ dnat_rews;

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

val dnat_invert = 
	[
prove_goal Dnat.thy 
"[|x1~=UU; y1~=UU; dsucc[x1] << dsucc[y1] |] ==> x1<< y1"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(dres_inst_tac [("fo5","dnat_when[c][LAM x.x]")] monofun_cfun_arg 1),
	(etac box_less 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1)
	])
	];

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

val dnat_inject = 
	[
prove_goal Dnat.thy 
"[|x1~=UU; y1~=UU; dsucc[x1] = dsucc[y1] |] ==> x1= y1"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(dres_inst_tac [("f","dnat_when[c][LAM x.x]")] cfun_arg_cong 1),
	(etac box_equals 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1)
	])
	];

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


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

val dnat_discsel_def = 
	[
	prover  "n~=UU ==> is_dzero[n]~=UU",
	prover  "n~=UU ==> is_dsucc[n]~=UU"
	];

val dnat_rews = dnat_discsel_def @ dnat_rews;

 
(* ------------------------------------------------------------------------*)
(* Properties dnat_take                                                    *)
(* ------------------------------------------------------------------------*)

val dnat_take =
	[
prove_goalw Dnat.thy [dnat_take_def] "dnat_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 dnat_rews) 1)
	]),
prove_goalw Dnat.thy [dnat_take_def] "dnat_take(0)[xs]=UU"
 (fn prems =>
	[
	(asm_simp_tac iterate_ss 1)
	])];

fun prover thm = prove_goalw Dnat.thy [dnat_take_def] thm
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(simp_tac iterate_ss 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1)
	]);

val dnat_take = [
prover "dnat_take(Suc(n))[dzero]=dzero",
prover "xs~=UU ==> dnat_take(Suc(n))[dsucc[xs]]=dsucc[dnat_take(n)[xs]]"
	] @ dnat_take;


val dnat_rews = dnat_take @ dnat_rews;

(* ------------------------------------------------------------------------*)
(* take lemma for dnats                                                  *)
(* ------------------------------------------------------------------------*)

fun prover reach defs thm  = prove_goalw Dnat.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 dnat_take_lemma = prover dnat_reach  [dnat_take_def]
	"(!!n.dnat_take(n)[s1]=dnat_take(n)[s2]) ==> s1=s2";


(* ------------------------------------------------------------------------*)
(* Co -induction for dnats                                                 *)
(* ------------------------------------------------------------------------*)

val dnat_coind_lemma = prove_goalw Dnat.thy [dnat_bisim_def] 
"dnat_bisim(R) ==> ! p q.R(p,q) --> dnat_take(n)[p]=dnat_take(n)[q]"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(nat_ind_tac "n" 1),
	(simp_tac (HOLCF_ss addsimps dnat_take) 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 dnat_take) 1),
	(etac disjE 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_take) 1),
	(etac exE 1),
	(etac exE 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_take) 1),
	(REPEAT (etac conjE 1)),
	(rtac cfun_arg_cong 1),
	(fast_tac HOL_cs 1)
	]);

val dnat_coind = prove_goal Dnat.thy "[|dnat_bisim(R);R(p,q)|] ==> p = q"
 (fn prems =>
	[
	(rtac dnat_take_lemma 1),
	(rtac (dnat_coind_lemma RS spec RS spec RS mp) 1),
	(resolve_tac prems 1),
	(resolve_tac prems 1)
	]);


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

val dnat_ind = prove_goal Dnat.thy
"[| adm(P);\
\   P(UU);\
\   P(dzero);\
\   !! s1.[|s1~=UU ; P(s1)|] ==> P(dsucc[s1])|] ==> P(s)"
 (fn prems =>
	[
	(rtac (dnat_reach RS subst) 1),
	(res_inst_tac [("x","s")] spec 1),
	(rtac fix_ind 1),
	(rtac adm_all2 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 [("n","xa")] dnatE 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_copy) 1),
	(resolve_tac prems 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_copy) 1),
	(resolve_tac prems 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_copy) 1),
	(res_inst_tac [("Q","x[xb]=UU")] classical2 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1),
	(resolve_tac prems 1),
	(eresolve_tac prems 1),
	(etac spec 1)
	]);


val dnat_flat = prove_goalw Dnat.thy [flat_def] "flat(dzero)"
 (fn prems =>
	[
	(rtac allI 1),
	(res_inst_tac [("s","x")] dnat_ind 1),
	(REPEAT (resolve_tac adm_thms 1)),
	(contX_tacR 1),
	(REPEAT (resolve_tac adm_thms 1)),
	(contX_tacR 1),
	(REPEAT (resolve_tac adm_thms 1)),
	(contX_tacR 1),
	(fast_tac HOL_cs 1),
	(rtac allI 1),
	(res_inst_tac [("n","y")] dnatE 1),
	(fast_tac (HOL_cs addSIs [UU_I]) 1),
	(asm_simp_tac HOLCF_ss 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_dist_less) 1),
	(rtac allI 1),
	(res_inst_tac [("n","y")] dnatE 1),
	(fast_tac (HOL_cs addSIs [UU_I]) 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1),
	(hyp_subst_tac 1),
	(strip_tac 1),
	(rtac disjI2 1),
	(forward_tac dnat_invert 1),
	(atac 2),
	(atac 1),
	(etac allE 1),
	(dtac mp 1),
	(atac 1),
	(etac disjE 1),
	(contr_tac 1),
	(asm_simp_tac HOLCF_ss 1)
	]);

val dnat_ind = prove_goal Dnat.thy
"[| adm(P);\
\   P(UU);\
\   P(dzero);\
\   !! s1.[|s1~=UU ; P(s1)|] ==> P(dsucc[s1])|] ==> P(s)"
 (fn prems =>
	[
	(rtac (dnat_reach RS subst) 1),
	(res_inst_tac [("x","s")] spec 1),
	(rtac fix_ind 1),
	(rtac adm_all2 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 [("n","xa")] dnatE 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_copy) 1),
	(resolve_tac prems 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_copy) 1),
	(resolve_tac prems 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_copy) 1),
	(res_inst_tac [("Q","x[xb]=UU")] classical2 1),
	(asm_simp_tac (HOLCF_ss addsimps dnat_rews) 1),
	(resolve_tac prems 1),
	(eresolve_tac prems 1),
	(etac spec 1)
	]);

val dnat_ind2 = dnat_flat RS adm_flat RS dnat_ind;
(*  "[| ?P(UU); ?P(dzero);
    !!s1. [| s1 ~= UU; ?P(s1) |] ==> ?P(dsucc[s1]) |] ==> ?P(?s)" : thm
*)