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(* theorems.ML
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ID: $Id$
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Author : David von Oheimb
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Created: 06-Jun-95
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Updated: 08-Jun-95 first proof from cterms
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Updated: 26-Jun-95 proofs for exhaustion thms
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Updated: 27-Jun-95 proofs for discriminators, constructors and selectors
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Updated: 06-Jul-95 proofs for distinctness, invertibility and injectivity
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Updated: 17-Jul-95 proofs for induction rules
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Updated: 19-Jul-95 proof for co-induction rule
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Updated: 28-Aug-95 definedness theorems for selectors (completion)
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Updated: 05-Sep-95 simultaneous domain equations (main part)
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Updated: 11-Sep-95 simultaneous domain equations (coding finished)
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Updated: 13-Sep-95 simultaneous domain equations (debugging)
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Copyright 1995 TU Muenchen
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*)
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structure Domain_Theorems = struct
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local
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open Domain_Library;
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infixr 0 ===>;infixr 0 ==>;infix 0 == ;
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infix 1 ===; infix 1 ~= ; infix 1 <<; infix 1 ~<<;
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infix 9 ` ; infix 9 `% ; infix 9 `%%; infixr 9 oo;
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(* ----- general proof facilities ------------------------------------------------- *)
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fun inferT sg pre_tm = #2(Sign.infer_types sg (K None)(K None)[]true([pre_tm],propT));
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(*
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infix 0 y;
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val b=0;
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fun _ y t = by t;
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fun g defs t = let val sg = sign_of thy;
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val ct = Thm.cterm_of sg (inferT sg t);
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in goalw_cterm defs ct end;
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*)
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fun pg'' thy defs t = let val sg = sign_of thy;
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val ct = Thm.cterm_of sg (inferT sg t);
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in prove_goalw_cterm defs ct end;
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fun pg' thy defs t tacsf=pg'' thy defs t (fn [] => tacsf
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| prems=> (cut_facts_tac prems 1)::tacsf);
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fun REPEAT_DETERM_UNTIL p tac =
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let fun drep st = if p st then Sequence.single st
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else (case Sequence.pull(tapply(tac,st)) of
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None => Sequence.null
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| Some(st',_) => drep st')
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in Tactic drep end;
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val UNTIL_SOLVED = REPEAT_DETERM_UNTIL (has_fewer_prems 1);
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local val trueI2 = prove_goal HOL.thy "f~=x ==> True" (fn prems => [rtac TrueI 1]) in
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val kill_neq_tac = dtac trueI2 end;
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fun case_UU_tac rews i v = res_inst_tac [("Q",v^"=UU")] classical2 i THEN
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asm_simp_tac (HOLCF_ss addsimps rews) i;
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val chain_tac = REPEAT_DETERM o resolve_tac
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[is_chain_iterate, ch2ch_fappR, ch2ch_fappL];
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(* ----- general proofs ----------------------------------------------------------- *)
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val swap3 = prove_goal HOL.thy "[| Q ==> P; ~P |] ==> ~Q" (fn prems => [
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cut_facts_tac prems 1,
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etac swap 1,
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dtac notnotD 1,
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etac (hd prems) 1]);
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val dist_eqI = prove_goal Porder0.thy "~ x << y ==> x ~= y" (fn prems => [
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cut_facts_tac prems 1,
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etac swap 1,
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dtac notnotD 1,
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asm_simp_tac HOLCF_ss 1]);
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val cfst_strict = prove_goal Cprod3.thy "cfst`UU = UU" (fn _ => [
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(simp_tac (HOLCF_ss addsimps [inst_cprod_pcpo2]) 1)]);
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val csnd_strict = prove_goal Cprod3.thy "csnd`UU = UU" (fn _ => [
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(simp_tac (HOLCF_ss addsimps [inst_cprod_pcpo2]) 1)]);
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in
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fun theorems thy (((dname,_),cons) : eq, eqs :eq list) =
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let
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val dummy = writeln ("Proving isomorphism properties of domain "^dname^"...");
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val pg = pg' thy;
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(* ----- getting the axioms and definitions --------------------------------------- *)
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local val ga = get_axiom thy in
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val ax_abs_iso = ga (dname^"_abs_iso" );
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val ax_rep_iso = ga (dname^"_rep_iso" );
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val ax_when_def = ga (dname^"_when_def" );
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val axs_con_def = map (fn (con,_) => ga (extern_name con ^"_def")) cons;
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val axs_dis_def = map (fn (con,_) => ga ( dis_name con ^"_def")) cons;
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val axs_sel_def = flat(map (fn (_,args) =>
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map (fn arg => ga (sel_of arg ^"_def")) args) cons);
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val ax_copy_def = ga (dname^"_copy_def" );
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end; (* local *)
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(* ----- theorems concerning the isomorphism -------------------------------------- *)
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val dc_abs = %%(dname^"_abs");
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val dc_rep = %%(dname^"_rep");
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val dc_copy = %%(dname^"_copy");
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val x_name = "x";
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val (rep_strict, abs_strict) = let
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val r = ax_rep_iso RS (ax_abs_iso RS (allI RSN(2,allI RS iso_strict)))
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in (r RS conjunct1, r RS conjunct2) end;
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val abs_defin' = pg [] ((dc_abs`%x_name === UU) ==> (%x_name === UU)) [
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res_inst_tac [("t",x_name)] (ax_abs_iso RS subst) 1,
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etac ssubst 1,
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rtac rep_strict 1];
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val rep_defin' = pg [] ((dc_rep`%x_name === UU) ==> (%x_name === UU)) [
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res_inst_tac [("t",x_name)] (ax_rep_iso RS subst) 1,
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etac ssubst 1,
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rtac abs_strict 1];
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val iso_rews = [ax_abs_iso,ax_rep_iso,abs_strict,rep_strict];
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local
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val iso_swap = pg [] (dc_rep`%"x" === %"y" ==> %"x" === dc_abs`%"y") [
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dres_inst_tac [("f",dname^"_abs")] cfun_arg_cong 1,
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etac (ax_rep_iso RS subst) 1];
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fun exh foldr1 cn quant foldr2 var = let
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fun one_con (con,args) = let val vns = map vname args in
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foldr quant (vns, foldr2 ((%x_name === con_app2 con (var vns) vns)::
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map (defined o (var vns)) (nonlazy args))) end
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in foldr1 ((cn(%x_name===UU))::map one_con cons) end;
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in
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val cases = let
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fun common_tac thm = rtac thm 1 THEN contr_tac 1;
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fun unit_tac true = common_tac liftE1
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| unit_tac _ = all_tac;
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fun prod_tac [] = common_tac oneE
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| prod_tac [arg] = unit_tac (is_lazy arg)
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| prod_tac (arg::args) =
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common_tac sprodE THEN
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kill_neq_tac 1 THEN
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unit_tac (is_lazy arg) THEN
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prod_tac args;
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fun sum_one_tac p = SELECT_GOAL(EVERY[
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rtac p 1,
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rewrite_goals_tac axs_con_def,
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dtac iso_swap 1,
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simp_tac HOLCF_ss 1,
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UNTIL_SOLVED(fast_tac HOL_cs 1)]) 1;
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fun sum_tac [(_,args)] [p] =
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prod_tac args THEN sum_one_tac p
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| sum_tac ((_,args)::cons') (p::prems) = DETERM(
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common_tac ssumE THEN
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kill_neq_tac 1 THEN kill_neq_tac 2 THEN
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prod_tac args THEN sum_one_tac p) THEN
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sum_tac cons' prems
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| sum_tac _ _ = Imposs "theorems:sum_tac";
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in pg'' thy [] (exh (fn l => foldr (op ===>) (l,mk_trp(%"P")))
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(fn T => T ==> %"P") mk_All
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(fn l => foldr (op ===>) (map mk_trp l,mk_trp(%"P")))
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bound_arg)
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(fn prems => [
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cut_facts_tac [excluded_middle] 1,
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etac disjE 1,
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rtac (hd prems) 2,
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etac rep_defin' 2,
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if is_one_con_one_arg (not o is_lazy) cons
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then rtac (hd (tl prems)) 1 THEN atac 2 THEN
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rewrite_goals_tac axs_con_def THEN
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simp_tac (HOLCF_ss addsimps [ax_rep_iso]) 1
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else sum_tac cons (tl prems)])end;
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val exhaust = pg [] (mk_trp(exh (foldr' mk_disj) Id mk_ex (foldr' mk_conj) (K %))) [
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rtac cases 1,
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UNTIL_SOLVED(fast_tac HOL_cs 1)];
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end;
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local
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val when_app = foldl (op `) (%%(dname^"_when"), map % (when_funs cons));
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val when_appl = pg [ax_when_def] (mk_trp(when_app`%x_name===when_body cons
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(fn (_,n) => %(nth_elem(n-1,when_funs cons)))`(dc_rep`%x_name))) [
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simp_tac HOLCF_ss 1];
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in
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val when_strict = pg [] ((if is_one_con_one_arg (K true) cons
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then fn t => mk_trp(strict(%"f")) ===> t else Id)(mk_trp(strict when_app))) [
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simp_tac(HOLCF_ss addsimps [when_appl,rep_strict]) 1];
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val when_apps = let fun one_when n (con,args) = pg axs_con_def
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(lift_defined % (nonlazy args, mk_trp(when_app`(con_app con args) ===
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mk_cfapp(%(nth_elem(n,when_funs cons)),map %# args))))[
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asm_simp_tac (HOLCF_ss addsimps [when_appl,ax_abs_iso]) 1];
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in mapn one_when 0 cons end;
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end;
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val when_rews = when_strict::when_apps;
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(* ----- theorems concerning the constructors, discriminators and selectors ------- *)
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val dis_stricts = map (fn (con,_) => pg axs_dis_def (mk_trp(
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(if is_one_con_one_arg (K true) cons then mk_not else Id)
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(strict(%%(dis_name con))))) [
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simp_tac (HOLCF_ss addsimps (if is_one_con_one_arg (K true) cons
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then [ax_when_def] else when_rews)) 1]) cons;
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val dis_apps = let fun one_dis c (con,args)= pg (axs_dis_def)
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(lift_defined % (nonlazy args, (*(if is_one_con_one_arg is_lazy cons
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then curry (lift_defined %#) args else Id)
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#################*)
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(mk_trp((%%(dis_name c))`(con_app con args) ===
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%%(if con=c then "TT" else "FF"))))) [
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asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
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in flat(map (fn (c,_) => map (one_dis c) cons) cons) end;
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val dis_defins = map (fn (con,args) => pg [] (defined(%x_name)==>
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defined(%%(dis_name con)`%x_name)) [
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rtac cases 1,
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contr_tac 1,
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UNTIL_SOLVED (CHANGED(asm_simp_tac
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(HOLCF_ss addsimps dis_apps) 1))]) cons;
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val dis_rews = dis_stricts @ dis_defins @ dis_apps;
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val con_stricts = flat(map (fn (con,args) => map (fn vn =>
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pg (axs_con_def)
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(mk_trp(con_app2 con (fn arg => if vname arg = vn
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then UU else %# arg) args === UU))[
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asm_simp_tac (HOLCF_ss addsimps [abs_strict]) 1]
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) (nonlazy args)) cons);
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val con_defins = map (fn (con,args) => pg []
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(lift_defined % (nonlazy args,
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mk_trp(defined(con_app con args)))) ([
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rtac swap3 1] @ (if is_one_con_one_arg (K true) cons
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then [
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if is_lazy (hd args) then rtac defined_up 2
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else atac 2,
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rtac abs_defin' 1,
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asm_full_simp_tac (HOLCF_ss addsimps axs_con_def) 1]
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else [
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eres_inst_tac [("f",dis_name con)] cfun_arg_cong 1,
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asm_simp_tac (HOLCF_ss addsimps dis_rews) 1])))cons;
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val con_rews = con_stricts @ con_defins;
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val sel_stricts = let fun one_sel sel = pg axs_sel_def (mk_trp(strict(%%sel))) [
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simp_tac (HOLCF_ss addsimps when_rews) 1];
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in flat(map (fn (_,args) => map (fn arg => one_sel (sel_of arg)) args) cons) end;
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val sel_apps = let fun one_sel c n sel = map (fn (con,args) =>
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let val nlas = nonlazy args;
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val vns = map vname args;
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in pg axs_sel_def (lift_defined %
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(filter (fn v => con=c andalso (v<>nth_elem(n,vns))) nlas,
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mk_trp((%%sel)`(con_app con args) === (if con=c then %(nth_elem(n,vns)) else UU))))
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( (if con=c then []
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else map(case_UU_tac(when_rews@con_stricts)1) nlas)
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@(if con=c andalso ((nth_elem(n,vns)) mem nlas)
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then[case_UU_tac (when_rews @ con_stricts) 1
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(nth_elem(n,vns))] else [])
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@ [asm_simp_tac(HOLCF_ss addsimps when_rews)1])end) cons;
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in flat(map (fn (c,args) =>
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flat(mapn (fn n => fn arg => one_sel c n (sel_of arg)) 0 args)) cons) end;
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val sel_defins = if length cons = 1 then map (fn arg => pg [] (defined(%x_name) ==>
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defined(%%(sel_of arg)`%x_name)) [
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rtac cases 1,
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contr_tac 1,
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UNTIL_SOLVED (CHANGED(asm_simp_tac
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(HOLCF_ss addsimps sel_apps) 1))])
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(filter_out is_lazy (snd(hd cons))) else [];
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val sel_rews = sel_stricts @ sel_defins @ sel_apps;
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val distincts_le = let
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fun dist (con1, args1) (con2, args2) = pg []
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(lift_defined % ((nonlazy args1),
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(mk_trp (con_app con1 args1 ~<< con_app con2 args2))))([
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rtac swap3 1,
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eres_inst_tac [("fo5",dis_name con1)] monofun_cfun_arg 1]
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@ map (case_UU_tac (con_stricts @ dis_rews) 1) (nonlazy args2)
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@[asm_simp_tac (HOLCF_ss addsimps dis_rews) 1]);
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fun distinct (con1,args1) (con2,args2) =
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let val arg1 = (con1, args1);
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val arg2 = (con2, (map (fn (arg,vn) => upd_vname (K vn) arg)
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(args2~~variantlist(map vname args2,map vname args1))));
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in [dist arg1 arg2, dist arg2 arg1] end;
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fun distincts [] = []
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| distincts (c::cs) = (map (distinct c) cs) :: distincts cs;
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in distincts cons end;
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val dists_le = flat (flat distincts_le);
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val dists_eq = let
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fun distinct (_,args1) ((_,args2),leqs) = let
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val (le1,le2) = (hd leqs, hd(tl leqs));
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val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI) in
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if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else
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if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else
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[eq1, eq2] end;
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fun distincts [] = []
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| distincts ((c,leqs)::cs) = flat(map (distinct c) ((map fst cs)~~leqs)) @
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distincts cs;
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in distincts (cons~~distincts_le) end;
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local
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fun pgterm rel con args = let
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fun append s = upd_vname(fn v => v^s);
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val (largs,rargs) = (args, map (append "'") args);
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in pg [] (mk_trp (rel(con_app con largs,con_app con rargs)) ===>
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lift_defined % ((nonlazy largs),lift_defined % ((nonlazy rargs),
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mk_trp (foldr' mk_conj
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(map rel (map %# largs ~~ map %# rargs)))))) end;
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val cons' = filter (fn (_,args) => args<>[]) cons;
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in
|
|
302 |
val inverts = map (fn (con,args) =>
|
1461
|
303 |
pgterm (op <<) con args (flat(map (fn arg => [
|
|
304 |
TRY(rtac conjI 1),
|
|
305 |
dres_inst_tac [("fo5",sel_of arg)] monofun_cfun_arg 1,
|
|
306 |
asm_full_simp_tac (HOLCF_ss addsimps sel_apps) 1]
|
|
307 |
) args))) cons';
|
1274
|
308 |
val injects = map (fn ((con,args),inv_thm) =>
|
1461
|
309 |
pgterm (op ===) con args [
|
|
310 |
etac (antisym_less_inverse RS conjE) 1,
|
|
311 |
dtac inv_thm 1, REPEAT(atac 1),
|
|
312 |
dtac inv_thm 1, REPEAT(atac 1),
|
|
313 |
TRY(safe_tac HOL_cs),
|
|
314 |
REPEAT(rtac antisym_less 1 ORELSE atac 1)] )
|
|
315 |
(cons'~~inverts);
|
1274
|
316 |
end;
|
|
317 |
|
|
318 |
(* ----- theorems concerning one induction step ----------------------------------- *)
|
|
319 |
|
|
320 |
val copy_strict = pg [ax_copy_def] ((if is_one_con_one_arg (K true) cons then fn t =>
|
1461
|
321 |
mk_trp(strict(cproj (%"f") eqs (rec_of (hd(snd(hd cons)))))) ===> t
|
|
322 |
else Id) (mk_trp(strict(dc_copy`%"f")))) [
|
|
323 |
asm_simp_tac(HOLCF_ss addsimps [abs_strict,rep_strict,
|
|
324 |
cfst_strict,csnd_strict]) 1];
|
1274
|
325 |
val copy_apps = map (fn (con,args) => pg (ax_copy_def::axs_con_def)
|
1461
|
326 |
(lift_defined %# (filter is_nonlazy_rec args,
|
|
327 |
mk_trp(dc_copy`%"f"`(con_app con args) ===
|
|
328 |
(con_app2 con (app_rec_arg (cproj (%"f") eqs)) args))))
|
|
329 |
(map (case_UU_tac [ax_abs_iso] 1 o vname)
|
|
330 |
(filter(fn a=>not(is_rec a orelse is_lazy a))args)@
|
|
331 |
[asm_simp_tac (HOLCF_ss addsimps [ax_abs_iso]) 1])
|
|
332 |
)cons;
|
1274
|
333 |
val copy_stricts = map(fn(con,args)=>pg[](mk_trp(dc_copy`UU`(con_app con args) ===UU))
|
1461
|
334 |
(let val rews = cfst_strict::csnd_strict::copy_strict::copy_apps@con_rews
|
|
335 |
in map (case_UU_tac rews 1) (nonlazy args) @ [
|
|
336 |
asm_simp_tac (HOLCF_ss addsimps rews) 1] end))
|
|
337 |
(filter (fn (_,args)=>exists is_nonlazy_rec args) cons);
|
1274
|
338 |
val copy_rews = copy_strict::copy_apps @ copy_stricts;
|
|
339 |
|
|
340 |
in (iso_rews, exhaust, cases, when_rews,
|
1461
|
341 |
con_rews, sel_rews, dis_rews, dists_eq, dists_le, inverts, injects,
|
|
342 |
copy_rews)
|
1274
|
343 |
end; (* let *)
|
|
344 |
|
|
345 |
|
|
346 |
fun comp_theorems thy (comp_dname, eqs: eq list, casess, con_rews, copy_rews) =
|
|
347 |
let
|
|
348 |
|
|
349 |
val dummy = writeln ("Proving induction properties of domain "^comp_dname^"...");
|
|
350 |
val pg = pg' thy;
|
|
351 |
|
|
352 |
val dnames = map (fst o fst) eqs;
|
|
353 |
val conss = map snd eqs;
|
|
354 |
|
|
355 |
(* ----- getting the composite axiom and definitions ------------------------------ *)
|
|
356 |
|
|
357 |
local val ga = get_axiom thy in
|
|
358 |
val axs_reach = map (fn dn => ga (dn ^ "_reach" )) dnames;
|
|
359 |
val axs_take_def = map (fn dn => ga (dn ^ "_take_def")) dnames;
|
|
360 |
val axs_finite_def = map (fn dn => ga (dn ^"_finite_def")) dnames;
|
|
361 |
val ax_copy2_def = ga (comp_dname^ "_copy_def");
|
|
362 |
val ax_bisim_def = ga (comp_dname^"_bisim_def");
|
|
363 |
end; (* local *)
|
|
364 |
|
|
365 |
(* ----- theorems concerning finiteness and induction ----------------------------- *)
|
|
366 |
|
|
367 |
fun dc_take dn = %%(dn^"_take");
|
|
368 |
val x_name = idx_name dnames "x";
|
|
369 |
val P_name = idx_name dnames "P";
|
|
370 |
|
|
371 |
local
|
1461
|
372 |
val iterate_ss = simpset_of "Fix";
|
1274
|
373 |
val iterate_Cprod_strict_ss = iterate_ss addsimps [cfst_strict, csnd_strict];
|
|
374 |
val iterate_Cprod_ss = iterate_ss addsimps [cfst2,csnd2,csplit2];
|
|
375 |
val copy_con_rews = copy_rews @ con_rews;
|
|
376 |
val copy_take_defs = (if length dnames=1 then [] else [ax_copy2_def]) @axs_take_def;
|
|
377 |
val take_stricts = pg copy_take_defs (mk_trp(foldr' mk_conj (map (fn ((dn,args),_)=>
|
1461
|
378 |
(dc_take dn $ %"n")`UU === mk_constrain(Type(dn,args),UU)) eqs)))([
|
|
379 |
nat_ind_tac "n" 1,
|
|
380 |
simp_tac iterate_ss 1,
|
|
381 |
simp_tac iterate_Cprod_strict_ss 1,
|
|
382 |
asm_simp_tac iterate_Cprod_ss 1,
|
|
383 |
TRY(safe_tac HOL_cs)] @
|
|
384 |
map(K(asm_simp_tac (HOL_ss addsimps copy_rews)1))dnames);
|
1274
|
385 |
val take_stricts' = rewrite_rule copy_take_defs take_stricts;
|
|
386 |
val take_0s = mapn (fn n => fn dn => pg axs_take_def(mk_trp((dc_take dn $ %%"0")
|
1461
|
387 |
`%x_name n === UU))[
|
|
388 |
simp_tac iterate_Cprod_strict_ss 1]) 1 dnames;
|
1274
|
389 |
val take_apps = pg copy_take_defs (mk_trp(foldr' mk_conj
|
1461
|
390 |
(flat(map (fn ((dn,_),cons) => map (fn (con,args) => foldr mk_all
|
|
391 |
(map vname args,(dc_take dn $ (%%"Suc" $ %"n"))`(con_app con args) ===
|
|
392 |
con_app2 con (app_rec_arg (fn n=>dc_take (nth_elem(n,dnames))$ %"n"))
|
|
393 |
args)) cons) eqs)))) ([
|
|
394 |
nat_ind_tac "n" 1,
|
|
395 |
simp_tac iterate_Cprod_strict_ss 1,
|
|
396 |
simp_tac (HOLCF_ss addsimps copy_con_rews) 1,
|
|
397 |
TRY(safe_tac HOL_cs)] @
|
|
398 |
(flat(map (fn ((dn,_),cons) => map (fn (con,args) => EVERY (
|
|
399 |
asm_full_simp_tac iterate_Cprod_ss 1::
|
|
400 |
map (case_UU_tac (take_stricts'::copy_con_rews) 1)
|
|
401 |
(nonlazy args) @[
|
|
402 |
asm_full_simp_tac (HOLCF_ss addsimps copy_rews) 1])
|
|
403 |
) cons) eqs)));
|
1274
|
404 |
in
|
|
405 |
val take_rews = atomize take_stricts @ take_0s @ atomize take_apps;
|
|
406 |
end; (* local *)
|
|
407 |
|
|
408 |
val take_lemmas = mapn (fn n => fn(dn,ax_reach) => pg'' thy axs_take_def (mk_All("n",
|
1461
|
409 |
mk_trp(dc_take dn $ Bound 0 `%(x_name n) ===
|
|
410 |
dc_take dn $ Bound 0 `%(x_name n^"'")))
|
|
411 |
===> mk_trp(%(x_name n) === %(x_name n^"'"))) (fn prems => [
|
|
412 |
res_inst_tac[("t",x_name n )](ax_reach RS subst) 1,
|
|
413 |
res_inst_tac[("t",x_name n^"'")](ax_reach RS subst) 1,
|
|
414 |
rtac (fix_def2 RS ssubst) 1,
|
|
415 |
REPEAT(CHANGED(rtac (contlub_cfun_arg RS ssubst) 1
|
|
416 |
THEN chain_tac 1)),
|
|
417 |
rtac (contlub_cfun_fun RS ssubst) 1,
|
|
418 |
rtac (contlub_cfun_fun RS ssubst) 2,
|
|
419 |
rtac lub_equal 3,
|
|
420 |
chain_tac 1,
|
|
421 |
rtac allI 1,
|
|
422 |
resolve_tac prems 1])) 1 (dnames~~axs_reach);
|
1274
|
423 |
|
|
424 |
local
|
|
425 |
fun one_con p (con,args) = foldr mk_All (map vname args,
|
1461
|
426 |
lift_defined (bound_arg (map vname args)) (nonlazy args,
|
|
427 |
lift (fn arg => %(P_name (1+rec_of arg)) $ bound_arg args arg)
|
|
428 |
(filter is_rec args,mk_trp(%p $ con_app2 con (bound_arg args) args))));
|
1274
|
429 |
fun one_eq ((p,cons),concl) = (mk_trp(%p $ UU) ===>
|
1461
|
430 |
foldr (op ===>) (map (one_con p) cons,concl));
|
1274
|
431 |
fun ind_term concf = foldr one_eq (mapn (fn n => fn x => (P_name n, x)) 1 conss,
|
1461
|
432 |
mk_trp(foldr' mk_conj (mapn (fn n => concf (P_name n,x_name n)) 1 dnames)));
|
1274
|
433 |
val take_ss = HOL_ss addsimps take_rews;
|
|
434 |
fun ind_tacs tacsf thms1 thms2 prems = TRY(safe_tac HOL_cs)::
|
1461
|
435 |
flat (mapn (fn n => fn (thm1,thm2) =>
|
|
436 |
tacsf (n,prems) (thm1,thm2) @
|
|
437 |
flat (map (fn cons =>
|
|
438 |
(resolve_tac prems 1 ::
|
|
439 |
flat (map (fn (_,args) =>
|
|
440 |
resolve_tac prems 1::
|
|
441 |
map (K(atac 1)) (nonlazy args) @
|
|
442 |
map (K(atac 1)) (filter is_rec args))
|
|
443 |
cons)))
|
|
444 |
conss))
|
|
445 |
0 (thms1~~thms2));
|
1274
|
446 |
local
|
|
447 |
fun all_rec_to ns lazy_rec (n,cons) = forall (exists (fn arg =>
|
1461
|
448 |
is_rec arg andalso not(rec_of arg mem ns) andalso
|
|
449 |
((rec_of arg = n andalso not(lazy_rec orelse is_lazy arg)) orelse
|
|
450 |
rec_of arg <> n andalso all_rec_to (rec_of arg::ns)
|
|
451 |
(lazy_rec orelse is_lazy arg) (n, (nth_elem(rec_of arg,conss))))
|
|
452 |
) o snd) cons;
|
1274
|
453 |
fun warn (n,cons) = if all_rec_to [] false (n,cons) then (writeln
|
1461
|
454 |
("WARNING: domain "^nth_elem(n,dnames)^" is empty!"); true)
|
|
455 |
else false;
|
1274
|
456 |
fun lazy_rec_to ns lazy_rec (n,cons) = exists (exists (fn arg =>
|
1461
|
457 |
is_rec arg andalso not(rec_of arg mem ns) andalso
|
|
458 |
((rec_of arg = n andalso (lazy_rec orelse is_lazy arg)) orelse
|
|
459 |
rec_of arg <> n andalso lazy_rec_to (rec_of arg::ns)
|
|
460 |
(lazy_rec orelse is_lazy arg) (n, (nth_elem(rec_of arg,conss))))
|
|
461 |
) o snd) cons;
|
1274
|
462 |
in val is_emptys = map warn (mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs);
|
|
463 |
val is_finite = forall (not o lazy_rec_to [] false)
|
1461
|
464 |
(mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs)
|
1274
|
465 |
end;
|
|
466 |
in
|
|
467 |
val finite_ind = pg'' thy [] (ind_term (fn (P,x) => fn dn =>
|
1461
|
468 |
mk_all(x,%P $ (dc_take dn $ %"n" `Bound 0)))) (fn prems=> [
|
|
469 |
nat_ind_tac "n" 1,
|
|
470 |
simp_tac (take_ss addsimps prems) 1,
|
|
471 |
TRY(safe_tac HOL_cs)]
|
|
472 |
@ flat(mapn (fn n => fn (cons,cases) => [
|
|
473 |
res_inst_tac [("x",x_name n)] cases 1,
|
|
474 |
asm_simp_tac (take_ss addsimps prems) 1]
|
|
475 |
@ flat(map (fn (con,args) =>
|
|
476 |
asm_simp_tac take_ss 1 ::
|
|
477 |
map (fn arg =>
|
|
478 |
case_UU_tac (prems@con_rews) 1 (
|
|
479 |
nth_elem(rec_of arg,dnames)^"_take n1`"^vname arg))
|
|
480 |
(filter is_nonlazy_rec args) @ [
|
|
481 |
resolve_tac prems 1] @
|
|
482 |
map (K (atac 1)) (nonlazy args) @
|
|
483 |
map (K (etac spec 1)) (filter is_rec args))
|
|
484 |
cons))
|
|
485 |
1 (conss~~casess)));
|
1274
|
486 |
|
|
487 |
val (finites,ind) = if is_finite then
|
|
488 |
let
|
|
489 |
fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %"x" === %"x");
|
|
490 |
val finite_lemmas1a = map (fn dn => pg [] (mk_trp(defined (%"x")) ===>
|
1461
|
491 |
mk_trp(mk_disj(mk_all("n",dc_take dn $ Bound 0 ` %"x" === UU),
|
|
492 |
take_enough dn)) ===> mk_trp(take_enough dn)) [
|
|
493 |
etac disjE 1,
|
|
494 |
etac notE 1,
|
|
495 |
resolve_tac take_lemmas 1,
|
|
496 |
asm_simp_tac take_ss 1,
|
|
497 |
atac 1]) dnames;
|
1274
|
498 |
val finite_lemma1b = pg [] (mk_trp (mk_all("n",foldr' mk_conj (mapn
|
1461
|
499 |
(fn n => fn ((dn,args),_) => mk_constrainall(x_name n,Type(dn,args),
|
|
500 |
mk_disj(dc_take dn $ Bound 1 ` Bound 0 === UU,
|
|
501 |
dc_take dn $ Bound 1 ` Bound 0 === Bound 0))) 1 eqs)))) ([
|
|
502 |
rtac allI 1,
|
|
503 |
nat_ind_tac "n" 1,
|
|
504 |
simp_tac take_ss 1,
|
|
505 |
TRY(safe_tac(empty_cs addSEs[conjE] addSIs[conjI]))] @
|
|
506 |
flat(mapn (fn n => fn (cons,cases) => [
|
|
507 |
simp_tac take_ss 1,
|
|
508 |
rtac allI 1,
|
|
509 |
res_inst_tac [("x",x_name n)] cases 1,
|
|
510 |
asm_simp_tac take_ss 1] @
|
|
511 |
flat(map (fn (con,args) =>
|
|
512 |
asm_simp_tac take_ss 1 ::
|
|
513 |
flat(map (fn arg => [
|
|
514 |
eres_inst_tac [("x",vname arg)] all_dupE 1,
|
|
515 |
etac disjE 1,
|
|
516 |
asm_simp_tac (HOL_ss addsimps con_rews) 1,
|
|
517 |
asm_simp_tac take_ss 1])
|
|
518 |
(filter is_nonlazy_rec args)))
|
|
519 |
cons))
|
|
520 |
1 (conss~~casess))) handle ERROR => raise ERROR;
|
1274
|
521 |
val all_finite=map (fn(dn,l1b)=>pg axs_finite_def (mk_trp(%%(dn^"_finite") $ %"x"))[
|
1461
|
522 |
case_UU_tac take_rews 1 "x",
|
|
523 |
eresolve_tac finite_lemmas1a 1,
|
|
524 |
step_tac HOL_cs 1,
|
|
525 |
step_tac HOL_cs 1,
|
|
526 |
cut_facts_tac [l1b] 1,
|
|
527 |
fast_tac HOL_cs 1]) (dnames~~atomize finite_lemma1b);
|
1274
|
528 |
in
|
|
529 |
(all_finite,
|
|
530 |
pg'' thy [] (ind_term (fn (P,x) => fn dn => %P $ %x))
|
1461
|
531 |
(ind_tacs (fn _ => fn (all_fin,finite_ind) => [
|
|
532 |
rtac (rewrite_rule axs_finite_def all_fin RS exE) 1,
|
|
533 |
etac subst 1,
|
|
534 |
rtac finite_ind 1]) all_finite (atomize finite_ind))
|
1274
|
535 |
) end (* let *) else
|
|
536 |
(mapn (fn n => fn dn => read_instantiate_sg (sign_of thy)
|
1461
|
537 |
[("P",dn^"_finite "^x_name n)] excluded_middle) 1 dnames,
|
1274
|
538 |
pg'' thy [] (foldr (op ===>) (mapn (fn n =>K(mk_trp(%%"adm" $ %(P_name n))))1
|
1461
|
539 |
dnames,ind_term (fn(P,x)=>fn dn=> %P $ %x)))
|
|
540 |
(ind_tacs (fn (n,prems) => fn (ax_reach,finite_ind) =>[
|
|
541 |
rtac (ax_reach RS subst) 1,
|
|
542 |
res_inst_tac [("x",x_name n)] spec 1,
|
|
543 |
rtac wfix_ind 1,
|
|
544 |
rtac adm_impl_admw 1,
|
|
545 |
resolve_tac adm_thms 1,
|
|
546 |
rtac adm_subst 1,
|
|
547 |
cont_tacR 1,
|
|
548 |
resolve_tac prems 1,
|
|
549 |
strip_tac 1,
|
|
550 |
rtac(rewrite_rule axs_take_def finite_ind) 1])
|
|
551 |
axs_reach (atomize finite_ind))
|
1274
|
552 |
)
|
|
553 |
end; (* local *)
|
|
554 |
|
|
555 |
local
|
|
556 |
val xs = mapn (fn n => K (x_name n)) 1 dnames;
|
|
557 |
fun bnd_arg n i = Bound(2*(length dnames - n)-i-1);
|
|
558 |
val take_ss = HOL_ss addsimps take_rews;
|
|
559 |
val sproj = bin_branchr (fn s => "fst("^s^")") (fn s => "snd("^s^")");
|
|
560 |
val coind_lemma = pg [ax_bisim_def] (mk_trp(mk_imp(%%(comp_dname^"_bisim") $ %"R",
|
1461
|
561 |
foldr (fn (x,t)=> mk_all(x,mk_all(x^"'",t))) (xs,
|
|
562 |
foldr mk_imp (mapn (fn n => K(proj (%"R") dnames n $
|
|
563 |
bnd_arg n 0 $ bnd_arg n 1)) 0 dnames,
|
|
564 |
foldr' mk_conj (mapn (fn n => fn dn =>
|
|
565 |
(dc_take dn $ %"n" `bnd_arg n 0 ===
|
|
566 |
(dc_take dn $ %"n" `bnd_arg n 1))) 0 dnames)))))) ([
|
|
567 |
rtac impI 1,
|
|
568 |
nat_ind_tac "n" 1,
|
|
569 |
simp_tac take_ss 1,
|
|
570 |
safe_tac HOL_cs] @
|
|
571 |
flat(mapn (fn n => fn x => [
|
|
572 |
etac allE 1, etac allE 1,
|
|
573 |
eres_inst_tac [("P1",sproj "R" dnames n^
|
|
574 |
" "^x^" "^x^"'")](mp RS disjE) 1,
|
|
575 |
TRY(safe_tac HOL_cs),
|
|
576 |
REPEAT(CHANGED(asm_simp_tac take_ss 1))])
|
|
577 |
0 xs));
|
1274
|
578 |
in
|
|
579 |
val coind = pg [] (mk_trp(%%(comp_dname^"_bisim") $ %"R") ===>
|
1461
|
580 |
foldr (op ===>) (mapn (fn n => fn x =>
|
|
581 |
mk_trp(proj (%"R") dnames n $ %x $ %(x^"'"))) 0 xs,
|
|
582 |
mk_trp(foldr' mk_conj (map (fn x => %x === %(x^"'")) xs)))) ([
|
|
583 |
TRY(safe_tac HOL_cs)] @
|
|
584 |
flat(map (fn take_lemma => [
|
|
585 |
rtac take_lemma 1,
|
|
586 |
cut_facts_tac [coind_lemma] 1,
|
|
587 |
fast_tac HOL_cs 1])
|
|
588 |
take_lemmas));
|
1274
|
589 |
end; (* local *)
|
|
590 |
|
|
591 |
|
|
592 |
in (take_rews, take_lemmas, finites, finite_ind, ind, coind)
|
|
593 |
|
|
594 |
end; (* let *)
|
|
595 |
end; (* local *)
|
|
596 |
end; (* struct *)
|