src/HOLCF/domain/theorems.ML
author paulson
Thu Sep 26 15:14:23 1996 +0200 (1996-09-26)
changeset 2033 639de962ded4
parent 1834 c780a4f39454
child 2267 b2326aefecbc
permissions -rw-r--r--
Ran expandshort; used stac instead of ssubst
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 (* theorems.ML
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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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   Updated: 26-Oct-95 debugging and enhancement of proofs for take_apps, ind
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   Updated: 16-Feb-96 bug concerning  domain Triv = triv  fixed
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   Updated: 01-Mar-96 when functional strictified, copy_def based on when_def
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   Copyright 1995, 1996 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 
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                           ([pre_tm],propT));
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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(tac st) of
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                                  None        => Sequence.null
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                                | Some(st',_) => drep st')
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in 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 _ => [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 =      case_tac (v^"=UU") 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 quant_ss = HOL_ss addsimps (map (fn s => prove_goal HOL.thy s (fn _ =>[
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                fast_tac HOL_cs 1]))["(!x. P x & Q)=((!x. P x) & Q)",
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                                     "(!x. P & Q x) = (P & (!x. Q x))"]);
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val all2E = prove_goal HOL.thy "[| !x y . P x y; P x y ==> R |] ==> R"
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 (fn prems =>[
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                                resolve_tac prems 1,
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                                cut_facts_tac prems 1,
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                                fast_tac HOL_cs 1]);
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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 Porder.thy "~ x << y ==> x ~= y" (fn prems => [
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                                rtac swap3 1,
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                                etac (antisym_less_inverse RS conjunct1) 1,
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                                resolve_tac prems 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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(*
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infixr 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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(* ----- 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, 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, 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_rest_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_rest_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_rest_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,
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                                                            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 length cons = 1 andalso 
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                                   length (snd(hd cons)) = 1 andalso 
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                                   not(is_lazy(hd(snd(hd 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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  fun bind_fun t = foldr mk_All (when_funs cons,t);
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  fun bound_fun i _ = Bound (length cons - i);
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  val when_app  = foldl (op `) (%%(dname^"_when"), mapn bound_fun 1 cons);
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  val when_appl = pg [ax_when_def] (bind_fun(mk_trp(when_app`%x_name ===
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             when_body cons (fn (m,n)=> bound_fun (n-m) 0)`(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 [] (bind_fun(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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                (bind_fun (lift_defined % (nonlazy args, 
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                mk_trp(when_app`(con_app con args) ===
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                       mk_cfapp(bound_fun n 0,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 1 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_rews = let
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  val dis_stricts = map (fn (con,_) => pg axs_dis_def (mk_trp(
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                             strict(%%(dis_name con)))) [
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                                simp_tac (HOLCF_ss addsimps 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,
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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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in dis_stricts @ dis_defins @ dis_apps end;
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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, 
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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) === 
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                                (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[("fo",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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   280
                        (args2~~variantlist(map vname args2,map vname args1))));
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   281
        in [dist arg1 arg2, dist arg2 arg1] end;
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   282
    fun distincts []      = []
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   283
    |   distincts (c::cs) = (map (distinct c) cs) :: distincts cs;
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   284
in distincts cons end;
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   285
val dists_le = flat (flat distincts_le);
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   286
val dists_eq = let
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   287
    fun distinct (_,args1) ((_,args2),leqs) = let
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        val (le1,le2) = (hd leqs, hd(tl leqs));
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   289
        val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI) in
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   290
        if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else
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   291
        if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else
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   292
                                        [eq1, eq2] end;
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   293
    fun distincts []      = []
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   294
    |   distincts ((c,leqs)::cs) = flat(map (distinct c) ((map fst cs)~~leqs)) @
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   295
                                   distincts cs;
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   296
    in distincts (cons~~distincts_le) end;
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   297
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   298
local 
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   299
  fun pgterm rel con args = let
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   300
                fun append s = upd_vname(fn v => v^s);
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   301
                val (largs,rargs) = (args, map (append "'") args);
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   302
                in pg [] (mk_trp (rel(con_app con largs,con_app con rargs)) ===>
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   303
                      lift_defined % ((nonlazy largs),lift_defined % ((nonlazy rargs),
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   304
                            mk_trp (foldr' mk_conj 
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   305
                                (map rel (map %# largs ~~ map %# rargs)))))) end;
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   306
  val cons' = filter (fn (_,args) => args<>[]) cons;
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   307
in
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   308
val inverts = map (fn (con,args) => 
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   309
                pgterm (op <<) con args (flat(map (fn arg => [
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   310
                                TRY(rtac conjI 1),
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   311
                                dres_inst_tac [("fo",sel_of arg)] monofun_cfun_arg 1,
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   312
                                asm_full_simp_tac (HOLCF_ss addsimps sel_apps) 1]
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   313
                                                      ) args))) cons';
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   314
val injects = map (fn ((con,args),inv_thm) => 
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   315
                           pgterm (op ===) con args [
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   316
                                etac (antisym_less_inverse RS conjE) 1,
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   317
                                dtac inv_thm 1, REPEAT(atac 1),
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   318
                                dtac inv_thm 1, REPEAT(atac 1),
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   319
                                TRY(safe_tac HOL_cs),
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   320
                                REPEAT(rtac antisym_less 1 ORELSE atac 1)] )
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   321
                  (cons'~~inverts);
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   322
end;
regensbu@1274
   323
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   324
(* ----- theorems concerning one induction step ----------------------------- *)
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   325
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   326
val copy_strict = pg[ax_copy_def](mk_trp(strict(dc_copy`%"f"))) [
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   327
                   asm_simp_tac(HOLCF_ss addsimps [abs_strict, when_strict,
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   328
                                                   cfst_strict,csnd_strict]) 1];
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   329
val copy_apps = map (fn (con,args) => pg [ax_copy_def]
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   330
                    (lift_defined % (nonlazy_rec args,
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   331
                        mk_trp(dc_copy`%"f"`(con_app con args) ===
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   332
                (con_app2 con (app_rec_arg (cproj (%"f") (length eqs))) args))))
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   333
                        (map (case_UU_tac (abs_strict::when_strict::con_stricts)
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   334
                                 1 o vname)
paulson@2033
   335
                         (filter (fn a => not (is_rec a orelse is_lazy a)) args)
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   336
                        @[asm_simp_tac (HOLCF_ss addsimps when_apps) 1,
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   337
                          simp_tac (HOLCF_ss addsimps axs_con_def) 1]))cons;
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   338
val copy_stricts = map (fn (con,args) => pg [] (mk_trp(dc_copy`UU`
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   339
                                        (con_app con args) ===UU))
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   340
     (let val rews = cfst_strict::csnd_strict::copy_strict::copy_apps@con_rews
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   341
                         in map (case_UU_tac rews 1) (nonlazy args) @ [
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   342
                             asm_simp_tac (HOLCF_ss addsimps rews) 1] end))
paulson@2033
   343
                        (filter (fn (_,args)=>exists is_nonlazy_rec args) cons);
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   344
val copy_rews = copy_strict::copy_apps @ copy_stricts;
regensbu@1274
   345
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   346
in     (iso_rews, exhaust, cases, when_rews,
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   347
        con_rews, sel_rews, dis_rews, dists_le, dists_eq, inverts, injects,
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   348
        copy_rews)
regensbu@1274
   349
end; (* let *)
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   350
regensbu@1274
   351
regensbu@1274
   352
fun comp_theorems thy (comp_dname, eqs: eq list, casess, con_rews, copy_rews) =
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   353
let
regensbu@1274
   354
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   355
val dummy = writeln("Proving induction properties of domain "^comp_dname^"...");
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   356
val pg = pg' thy;
regensbu@1274
   357
regensbu@1274
   358
val dnames = map (fst o fst) eqs;
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   359
val conss  = map  snd        eqs;
regensbu@1274
   360
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   361
(* ----- getting the composite axiom and definitions ------------------------ *)
regensbu@1274
   362
regensbu@1274
   363
local val ga = get_axiom thy in
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   364
val axs_reach      = map (fn dn => ga (dn ^  "_reach"   )) dnames;
regensbu@1274
   365
val axs_take_def   = map (fn dn => ga (dn ^  "_take_def")) dnames;
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   366
val axs_finite_def = map (fn dn => ga (dn ^"_finite_def")) dnames;
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   367
val ax_copy2_def   = ga (comp_dname^ "_copy_def");
regensbu@1274
   368
val ax_bisim_def   = ga (comp_dname^"_bisim_def");
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   369
end; (* local *)
regensbu@1274
   370
regensbu@1274
   371
fun dc_take dn = %%(dn^"_take");
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   372
val x_name = idx_name dnames "x"; 
regensbu@1274
   373
val P_name = idx_name dnames "P";
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   374
val n_eqs = length eqs;
oheimb@1637
   375
oheimb@1637
   376
(* ----- theorems concerning finite approximation and finite induction ------ *)
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   377
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   378
local
oheimb@1637
   379
  val iterate_Cprod_ss = simpset_of "Fix"
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   380
                         addsimps [cfst_strict, csnd_strict]addsimps Cprod_rews;
regensbu@1274
   381
  val copy_con_rews  = copy_rews @ con_rews;
oheimb@1637
   382
  val copy_take_defs =(if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
oheimb@1637
   383
  val take_stricts=pg copy_take_defs(mk_trp(foldr' mk_conj(map(fn((dn,args),_)=>
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   384
            (dc_take dn $ %"n")`UU === mk_constrain(Type(dn,args),UU)) eqs)))([
paulson@2033
   385
                        nat_ind_tac "n" 1,
paulson@2033
   386
                        simp_tac iterate_Cprod_ss 1,
paulson@2033
   387
                        asm_simp_tac (iterate_Cprod_ss addsimps copy_rews)1]);
regensbu@1274
   388
  val take_stricts' = rewrite_rule copy_take_defs take_stricts;
oheimb@1637
   389
  val take_0s = mapn(fn n=> fn dn => pg axs_take_def(mk_trp((dc_take dn $ %%"0")
paulson@2033
   390
                                                        `%x_name n === UU))[
paulson@2033
   391
                                simp_tac iterate_Cprod_ss 1]) 1 dnames;
oheimb@1637
   392
  val c_UU_tac = case_UU_tac (take_stricts'::copy_con_rews) 1;
regensbu@1274
   393
  val take_apps = pg copy_take_defs (mk_trp(foldr' mk_conj 
paulson@2033
   394
            (flat(map (fn ((dn,_),cons) => map (fn (con,args) => foldr mk_all 
paulson@2033
   395
        (map vname args,(dc_take dn $ (%%"Suc" $ %"n"))`(con_app con args) ===
paulson@2033
   396
         con_app2 con (app_rec_arg (fn n=>dc_take (nth_elem(n,dnames))$ %"n"))
paulson@2033
   397
                              args)) cons) eqs)))) ([
paulson@2033
   398
                                simp_tac iterate_Cprod_ss 1,
paulson@2033
   399
                                nat_ind_tac "n" 1,
paulson@2033
   400
                            simp_tac(iterate_Cprod_ss addsimps copy_con_rews) 1,
paulson@2033
   401
                                asm_full_simp_tac (HOLCF_ss addsimps 
paulson@2033
   402
                                      (filter (has_fewer_prems 1) copy_rews)) 1,
paulson@2033
   403
                                TRY(safe_tac HOL_cs)] @
paulson@2033
   404
                        (flat(map (fn ((dn,_),cons) => map (fn (con,args) => 
paulson@2033
   405
                                if nonlazy_rec args = [] then all_tac else
paulson@2033
   406
                                EVERY(map c_UU_tac (nonlazy_rec args)) THEN
paulson@2033
   407
                                asm_full_simp_tac (HOLCF_ss addsimps copy_rews)1
paulson@2033
   408
                                                           ) cons) eqs)));
regensbu@1274
   409
in
regensbu@1274
   410
val take_rews = atomize take_stricts @ take_0s @ atomize take_apps;
regensbu@1274
   411
end; (* local *)
regensbu@1274
   412
regensbu@1274
   413
local
regensbu@1274
   414
  fun one_con p (con,args) = foldr mk_All (map vname args,
paulson@2033
   415
        lift_defined (bound_arg (map vname args)) (nonlazy args,
paulson@2033
   416
        lift (fn arg => %(P_name (1+rec_of arg)) $ bound_arg args arg)
oheimb@1637
   417
         (filter is_rec args,mk_trp(%p $ con_app2 con (bound_arg args) args))));
regensbu@1274
   418
  fun one_eq ((p,cons),concl) = (mk_trp(%p $ UU) ===> 
paulson@2033
   419
                           foldr (op ===>) (map (one_con p) cons,concl));
oheimb@1637
   420
  fun ind_term concf = foldr one_eq (mapn (fn n => fn x => (P_name n, x))1conss,
paulson@2033
   421
                        mk_trp(foldr' mk_conj (mapn concf 1 dnames)));
regensbu@1274
   422
  val take_ss = HOL_ss addsimps take_rews;
oheimb@1637
   423
  fun quant_tac i = EVERY(mapn(fn n=> fn _=> res_inst_tac[("x",x_name n)]spec i)
paulson@2033
   424
                               1 dnames);
oheimb@1637
   425
  fun ind_prems_tac prems = EVERY(flat (map (fn cons => (
paulson@2033
   426
                                     resolve_tac prems 1 ::
paulson@2033
   427
                                     flat (map (fn (_,args) => 
paulson@2033
   428
                                       resolve_tac prems 1 ::
paulson@2033
   429
                                       map (K(atac 1)) (nonlazy args) @
paulson@2033
   430
                                       map (K(atac 1)) (filter is_rec args))
paulson@2033
   431
                                     cons))) conss));
regensbu@1274
   432
  local 
oheimb@1637
   433
    (* check whether every/exists constructor of the n-th part of the equation:
oheimb@1637
   434
       it has a possibly indirectly recursive argument that isn't/is possibly 
oheimb@1637
   435
       indirectly lazy *)
oheimb@1637
   436
    fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => 
paulson@2033
   437
          is_rec arg andalso not(rec_of arg mem ns) andalso
paulson@2033
   438
          ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse 
paulson@2033
   439
            rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) 
paulson@2033
   440
              (lazy_rec orelse is_lazy arg) (n, (nth_elem(rec_of arg,conss))))
paulson@2033
   441
          ) o snd) cons;
oheimb@1637
   442
    fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
oheimb@1637
   443
    fun warn (n,cons)  = if all_rec_to [] false (n,cons) then (writeln 
oheimb@1637
   444
        ("WARNING: domain "^nth_elem(n,dnames)^" is empty!"); true) else false;
oheimb@1637
   445
    fun lazy_rec_to ns = rec_to exists Id  lazy_rec_to ns;
oheimb@1637
   446
oheimb@1637
   447
  in val n__eqs     = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
oheimb@1637
   448
     val is_emptys = map warn n__eqs;
oheimb@1637
   449
     val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
regensbu@1274
   450
  end;
oheimb@1637
   451
in (* local *)
oheimb@1637
   452
val finite_ind = pg'' thy [] (ind_term (fn n => fn dn => %(P_name n)$
paulson@2033
   453
                             (dc_take dn $ %"n" `%(x_name n)))) (fn prems => [
paulson@2033
   454
                                quant_tac 1,
paulson@2033
   455
                                simp_tac quant_ss 1,
paulson@2033
   456
                                nat_ind_tac "n" 1,
paulson@2033
   457
                                simp_tac (take_ss addsimps prems) 1,
paulson@2033
   458
                                TRY(safe_tac HOL_cs)]
paulson@2033
   459
                                @ flat(map (fn (cons,cases) => [
paulson@2033
   460
                                 res_inst_tac [("x","x")] cases 1,
paulson@2033
   461
                                 asm_simp_tac (take_ss addsimps prems) 1]
paulson@2033
   462
                                 @ flat(map (fn (con,args) => 
paulson@2033
   463
                                  asm_simp_tac take_ss 1 ::
paulson@2033
   464
                                  map (fn arg =>
paulson@2033
   465
                                   case_UU_tac (prems@con_rews) 1 (
paulson@2033
   466
                           nth_elem(rec_of arg,dnames)^"_take n1`"^vname arg))
paulson@2033
   467
                                  (filter is_nonlazy_rec args) @ [
paulson@2033
   468
                                  resolve_tac prems 1] @
paulson@2033
   469
                                  map (K (atac 1))      (nonlazy args) @
paulson@2033
   470
                                  map (K (etac spec 1)) (filter is_rec args)) 
paulson@2033
   471
                                 cons))
paulson@2033
   472
                                (conss~~casess)));
oheimb@1637
   473
oheimb@1637
   474
val take_lemmas =mapn(fn n=> fn(dn,ax_reach)=> pg'' thy axs_take_def(mk_All("n",
paulson@2033
   475
                mk_trp(dc_take dn $ Bound 0 `%(x_name n) === 
paulson@2033
   476
                       dc_take dn $ Bound 0 `%(x_name n^"'")))
paulson@2033
   477
           ===> mk_trp(%(x_name n) === %(x_name n^"'"))) (fn prems => [
paulson@2033
   478
                        res_inst_tac[("t",x_name n    )](ax_reach RS subst) 1,
paulson@2033
   479
                        res_inst_tac[("t",x_name n^"'")](ax_reach RS subst) 1,
paulson@2033
   480
                                stac fix_def2 1,
paulson@2033
   481
                                REPEAT(CHANGED(rtac(contlub_cfun_arg RS ssubst)1
paulson@2033
   482
                                               THEN chain_tac 1)),
paulson@2033
   483
                                stac contlub_cfun_fun 1,
paulson@2033
   484
                                stac contlub_cfun_fun 2,
paulson@2033
   485
                                rtac lub_equal 3,
paulson@2033
   486
                                chain_tac 1,
paulson@2033
   487
                                rtac allI 1,
paulson@2033
   488
                                resolve_tac prems 1])) 1 (dnames~~axs_reach);
oheimb@1637
   489
oheimb@1637
   490
(* ----- theorems concerning finiteness and induction ----------------------- *)
regensbu@1274
   491
regensbu@1274
   492
val (finites,ind) = if is_finite then
oheimb@1637
   493
  let 
oheimb@1637
   494
    fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %"x" === %"x");
oheimb@1637
   495
    val finite_lemmas1a = map (fn dn => pg [] (mk_trp(defined (%"x")) ===> 
paulson@2033
   496
        mk_trp(mk_disj(mk_all("n",dc_take dn $ Bound 0 ` %"x" === UU),
paulson@2033
   497
        take_enough dn)) ===> mk_trp(take_enough dn)) [
paulson@2033
   498
                                etac disjE 1,
paulson@2033
   499
                                etac notE 1,
paulson@2033
   500
                                resolve_tac take_lemmas 1,
paulson@2033
   501
                                asm_simp_tac take_ss 1,
paulson@2033
   502
                                atac 1]) dnames;
oheimb@1637
   503
    val finite_lemma1b = pg [] (mk_trp (mk_all("n",foldr' mk_conj (mapn 
paulson@2033
   504
        (fn n => fn ((dn,args),_) => mk_constrainall(x_name n,Type(dn,args),
paulson@2033
   505
         mk_disj(dc_take dn $ Bound 1 ` Bound 0 === UU,
paulson@2033
   506
                 dc_take dn $ Bound 1 ` Bound 0 === Bound 0))) 1 eqs)))) ([
paulson@2033
   507
                                rtac allI 1,
paulson@2033
   508
                                nat_ind_tac "n" 1,
paulson@2033
   509
                                simp_tac take_ss 1,
paulson@2033
   510
                        TRY(safe_tac(empty_cs addSEs[conjE] addSIs[conjI]))] @
paulson@2033
   511
                                flat(mapn (fn n => fn (cons,cases) => [
paulson@2033
   512
                                  simp_tac take_ss 1,
paulson@2033
   513
                                  rtac allI 1,
paulson@2033
   514
                                  res_inst_tac [("x",x_name n)] cases 1,
paulson@2033
   515
                                  asm_simp_tac take_ss 1] @ 
paulson@2033
   516
                                  flat(map (fn (con,args) => 
paulson@2033
   517
                                    asm_simp_tac take_ss 1 ::
paulson@2033
   518
                                    flat(map (fn vn => [
paulson@2033
   519
                                      eres_inst_tac [("x",vn)] all_dupE 1,
paulson@2033
   520
                                      etac disjE 1,
paulson@2033
   521
                                      asm_simp_tac (HOL_ss addsimps con_rews) 1,
paulson@2033
   522
                                      asm_simp_tac take_ss 1])
paulson@2033
   523
                                    (nonlazy_rec args)))
paulson@2033
   524
                                  cons))
paulson@2033
   525
                                1 (conss~~casess))) handle ERROR => raise ERROR;
oheimb@1637
   526
    val finites = map (fn (dn,l1b) => pg axs_finite_def (mk_trp(
paulson@2033
   527
                                                %%(dn^"_finite") $ %"x"))[
paulson@2033
   528
                                case_UU_tac take_rews 1 "x",
paulson@2033
   529
                                eresolve_tac finite_lemmas1a 1,
paulson@2033
   530
                                step_tac HOL_cs 1,
paulson@2033
   531
                                step_tac HOL_cs 1,
paulson@2033
   532
                                cut_facts_tac [l1b] 1,
paulson@2033
   533
                        fast_tac HOL_cs 1]) (dnames~~atomize finite_lemma1b);
oheimb@1637
   534
  in
oheimb@1637
   535
  (finites,
oheimb@1637
   536
   pg'' thy[](ind_term (fn n => fn dn => %(P_name n) $ %(x_name n)))(fn prems =>
paulson@2033
   537
                                TRY(safe_tac HOL_cs) ::
paulson@2033
   538
                         flat (map (fn (finite,fin_ind) => [
paulson@2033
   539
                               rtac(rewrite_rule axs_finite_def finite RS exE)1,
paulson@2033
   540
                                etac subst 1,
paulson@2033
   541
                                rtac fin_ind 1,
paulson@2033
   542
                                ind_prems_tac prems]) 
paulson@2033
   543
                                   (finites~~(atomize finite_ind)) ))
regensbu@1274
   544
) end (* let *) else
oheimb@1637
   545
  (mapn (fn n => fn dn => read_instantiate_sg (sign_of thy) 
paulson@2033
   546
                    [("P",dn^"_finite "^x_name n)] excluded_middle) 1 dnames,
oheimb@1637
   547
   pg'' thy [] (foldr (op ===>) (mapn (fn n => K(mk_trp(%%"adm" $ %(P_name n))))
paulson@2033
   548
               1 dnames, ind_term (fn n => fn dn => %(P_name n) $ %(x_name n))))
paulson@2033
   549
                   (fn prems => map (fn ax_reach => rtac (ax_reach RS subst) 1) 
paulson@2033
   550
                                    axs_reach @ [
paulson@2033
   551
                                quant_tac 1,
paulson@2033
   552
                                rtac (adm_impl_admw RS wfix_ind) 1,
paulson@2033
   553
                                REPEAT_DETERM(rtac adm_all2 1),
paulson@2033
   554
                                REPEAT_DETERM(TRY(rtac adm_conj 1) THEN 
paulson@2033
   555
                                                  rtac adm_subst 1 THEN 
paulson@2033
   556
                                        cont_tacR 1 THEN resolve_tac prems 1),
paulson@2033
   557
                                strip_tac 1,
paulson@2033
   558
                                rtac (rewrite_rule axs_take_def finite_ind) 1,
paulson@2033
   559
                                ind_prems_tac prems])
regensbu@1274
   560
)
regensbu@1274
   561
end; (* local *)
regensbu@1274
   562
oheimb@1637
   563
(* ----- theorem concerning coinduction ------------------------------------- *)
oheimb@1637
   564
regensbu@1274
   565
local
regensbu@1274
   566
  val xs = mapn (fn n => K (x_name n)) 1 dnames;
oheimb@1637
   567
  fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
regensbu@1274
   568
  val take_ss = HOL_ss addsimps take_rews;
oheimb@1637
   569
  val sproj   = prj (fn s => "fst("^s^")") (fn s => "snd("^s^")");
oheimb@1637
   570
  val coind_lemma=pg[ax_bisim_def](mk_trp(mk_imp(%%(comp_dname^"_bisim") $ %"R",
paulson@2033
   571
                foldr (fn (x,t)=> mk_all(x,mk_all(x^"'",t))) (xs,
paulson@2033
   572
                  foldr mk_imp (mapn (fn n => K(proj (%"R") n_eqs n $ 
paulson@2033
   573
                                      bnd_arg n 0 $ bnd_arg n 1)) 0 dnames,
paulson@2033
   574
                    foldr' mk_conj (mapn (fn n => fn dn => 
paulson@2033
   575
                                (dc_take dn $ %"n" `bnd_arg n 0 === 
paulson@2033
   576
                                (dc_take dn $ %"n" `bnd_arg n 1)))0 dnames))))))
paulson@2033
   577
                             ([ rtac impI 1,
paulson@2033
   578
                                nat_ind_tac "n" 1,
paulson@2033
   579
                                simp_tac take_ss 1,
paulson@2033
   580
                                safe_tac HOL_cs] @
paulson@2033
   581
                                flat(mapn (fn n => fn x => [
paulson@2033
   582
                                  rotate_tac (n+1) 1,
paulson@2033
   583
                                  etac all2E 1,
paulson@2033
   584
                                  eres_inst_tac [("P1", sproj "R" n_eqs n^
paulson@2033
   585
                                        " "^x^" "^x^"'")](mp RS disjE) 1,
paulson@2033
   586
                                  TRY(safe_tac HOL_cs),
paulson@2033
   587
                                  REPEAT(CHANGED(asm_simp_tac take_ss 1))]) 
paulson@2033
   588
                                0 xs));
regensbu@1274
   589
in
regensbu@1274
   590
val coind = pg [] (mk_trp(%%(comp_dname^"_bisim") $ %"R") ===>
paulson@2033
   591
                foldr (op ===>) (mapn (fn n => fn x => 
paulson@2033
   592
                  mk_trp(proj (%"R") n_eqs n $ %x $ %(x^"'"))) 0 xs,
paulson@2033
   593
                  mk_trp(foldr' mk_conj (map (fn x => %x === %(x^"'")) xs)))) ([
paulson@2033
   594
                                TRY(safe_tac HOL_cs)] @
paulson@2033
   595
                                flat(map (fn take_lemma => [
paulson@2033
   596
                                  rtac take_lemma 1,
paulson@2033
   597
                                  cut_facts_tac [coind_lemma] 1,
paulson@2033
   598
                                  fast_tac HOL_cs 1])
paulson@2033
   599
                                take_lemmas));
regensbu@1274
   600
end; (* local *)
regensbu@1274
   601
regensbu@1274
   602
regensbu@1274
   603
in (take_rews, take_lemmas, finites, finite_ind, ind, coind)
regensbu@1274
   604
regensbu@1274
   605
end; (* let *)
regensbu@1274
   606
end; (* local *)
regensbu@1274
   607
end; (* struct *)