# File CCL.ML

    (*  Title:      CCL/CCL.ML
        ID:         $Id: CCL.ML,v 1.17 2005/09/17 15:35:26 wenzelm Exp $
        Author:     Martin Coen, Cambridge University Computer Laboratory
        Copyright   1993  University of Cambridge
    *)

    val ccl_data_defs = [apply_def,fix_def];

    val CCL_ss = set_ss addsimps [po_refl];

    (*** Congruence Rules ***)

    (*similar to AP_THM in Gordon's HOL*)
    qed_goal "fun_cong" (the_context ()) "(f::'a=>'b) = g ==> f(x)=g(x)"
      (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);

    (*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
    qed_goal "arg_cong" (the_context ()) "x=y ==> f(x)=f(y)"
     (fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);

    Goal  "(ALL x. f(x) = g(x)) --> (%x. f(x)) = (%x. g(x))";
    by (simp_tac (CCL_ss addsimps [eq_iff]) 1);
    by (fast_tac (set_cs addIs [po_abstractn]) 1);
    bind_thm("abstractn", standard (allI RS (result() RS mp)));

    fun type_of_terms (Const("Trueprop",_) $
                       (Const("op =",(Type ("fun", [t,_]))) $ _ $ _)) = t;

    fun abs_prems thm =
       let fun do_abs n thm (Type ("fun", [_,t])) = do_abs n (abstractn RSN (n,thm)) t
             | do_abs n thm _                     = thm
           fun do_prems n      [] thm = thm
             | do_prems n (x::xs) thm = do_prems (n+1) xs (do_abs n thm (type_of_terms x));
       in do_prems 1 (prems_of thm) thm
       end;

    val caseBs = [caseBtrue,caseBfalse,caseBpair,caseBlam,caseBbot];

    (*** Termination and Divergence ***)

    Goalw [Trm_def,Dvg_def] "Trm(t) <-> ~ t = bot";
    by (rtac iff_refl 1);
    qed "Trm_iff";

    Goalw [Trm_def,Dvg_def] "Dvg(t) <-> t = bot";
    by (rtac iff_refl 1);
    qed "Dvg_iff";

    (*** Constructors are injective ***)

    val prems = goal (the_context ())
        "[| x=a;  y=b;  x=y |] ==> a=b";
    by  (REPEAT (SOMEGOAL (ares_tac (prems@[box_equals]))));
    qed "eq_lemma";

    fun mk_inj_rl thy rews s =
          let fun mk_inj_lemmas r = ([arg_cong] RL [(r RS (r RS eq_lemma))]);
              val inj_lemmas = List.concat (map mk_inj_lemmas rews);
              val tac = REPEAT (ares_tac [iffI,allI,conjI] 1 ORELSE
                                eresolve_tac inj_lemmas 1 ORELSE
                                asm_simp_tac (CCL_ss addsimps rews) 1)
          in prove_goal thy s (fn _ => [tac])
          end;

    val ccl_injs = map (mk_inj_rl (the_context ()) caseBs)
                   ["<a,b> = <a',b'> <-> (a=a' & b=b')",
                    "(lam x. b(x) = lam x. b'(x)) <-> ((ALL z. b(z)=b'(z)))"];

    val pair_inject = ((hd ccl_injs) RS iffD1) RS conjE;

    (*** Constructors are distinct ***)

    local
      fun pairs_of f x [] = []
        | pairs_of f x (y::ys) = (f x y) :: (f y x) :: (pairs_of f x ys);

      fun mk_combs ff [] = []
        | mk_combs ff (x::xs) = (pairs_of ff x xs) @ mk_combs ff xs;

    (* Doesn't handle binder types correctly *)
      fun saturate thy sy name =
           let fun arg_str 0 a s = s
             | arg_str 1 a s = "(" ^ a ^ "a" ^ s ^ ")"
             | arg_str n a s = arg_str (n-1) a ("," ^ a ^ (chr((ord "a")+n-1)) ^ s);
               val sg = sign_of thy;
               val T = case Sign.const_type sg (Sign.intern_const (sign_of thy) sy) of
                                NONE => error(sy^" not declared") | SOME(T) => T;
               val arity = length (fst (strip_type T));
           in sy ^ (arg_str arity name "") end;

      fun mk_thm_str thy a b = "~ " ^ (saturate thy a "a") ^ " = " ^ (saturate thy b "b");

      val lemma = prove_goal (the_context ()) "t=t' --> case(t,b,c,d,e) = case(t',b,c,d,e)"
                       (fn _ => [simp_tac CCL_ss 1]) RS mp;
      fun mk_lemma (ra,rb) = [lemma] RL [ra RS (rb RS eq_lemma)] RL
                               [distinctness RS notE,sym RS (distinctness RS notE)];
    in
      fun mk_lemmas rls = List.concat (map mk_lemma (mk_combs pair rls));
      fun mk_dstnct_rls thy xs = mk_combs (mk_thm_str thy) xs;
    end;

    val caseB_lemmas = mk_lemmas caseBs;

    val ccl_dstncts =
            let fun mk_raw_dstnct_thm rls s =
                      prove_goal (the_context ()) s (fn _=> [rtac notI 1,eresolve_tac rls 1])
            in map (mk_raw_dstnct_thm caseB_lemmas)
                    (mk_dstnct_rls (the_context ()) ["bot","true","false","pair","lambda"]) end;

    fun mk_dstnct_thms thy defs inj_rls xs =
              let fun mk_dstnct_thm rls s = prove_goalw thy defs s
                                   (fn _ => [simp_tac (CCL_ss addsimps (rls@inj_rls)) 1])
              in map (mk_dstnct_thm ccl_dstncts) (mk_dstnct_rls thy xs) end;

    fun mkall_dstnct_thms thy defs i_rls xss = List.concat (map (mk_dstnct_thms thy defs i_rls) xss);

    (*** Rewriting and Proving ***)

    fun XH_to_I rl = rl RS iffD2;
    fun XH_to_D rl = rl RS iffD1;
    val XH_to_E = make_elim o XH_to_D;
    val XH_to_Is = map XH_to_I;
    val XH_to_Ds = map XH_to_D;
    val XH_to_Es = map XH_to_E;

    val ccl_rews = caseBs @ ccl_injs @ ccl_dstncts;
    val ccl_ss = CCL_ss addsimps ccl_rews;

    val ccl_cs = set_cs addSEs (pair_inject::(ccl_dstncts RL [notE]))
                        addSDs (XH_to_Ds ccl_injs);

    (****** Facts from gfp Definition of [= and = ******)

    val major::prems = goal (the_context ()) "[| A=B;  a:B <-> P |] ==> a:A <-> P";
    by (resolve_tac (prems RL [major RS ssubst]) 1);
    qed "XHlemma1";

    Goal "(P(t,t') <-> Q) --> (<t,t'> : {p. EX t t'. p=<t,t'> &  P(t,t')} <-> Q)";
    by (fast_tac ccl_cs 1);
    bind_thm("XHlemma2", result() RS mp);

    (*** Pre-Order ***)

    Goalw [POgen_def,SIM_def]  "mono(%X. POgen(X))";
    by (rtac monoI 1);
    by (safe_tac ccl_cs);
    by (REPEAT_SOME (resolve_tac [exI,conjI,refl]));
    by (ALLGOALS (simp_tac ccl_ss));
    by (ALLGOALS (fast_tac set_cs));
    qed "POgen_mono";

    Goalw [POgen_def,SIM_def]
     "<t,t'> : POgen(R) <-> t= bot | (t=true & t'=true)  | (t=false & t'=false) | \
    \          (EX a a' b b'. t=<a,b> &  t'=<a',b'>  & <a,a'> : R & <b,b'> : R) | \
    \          (EX f f'. t=lam x. f(x) &  t'=lam x. f'(x) & (ALL x. <f(x),f'(x)> : R))";
    by (rtac (iff_refl RS XHlemma2) 1);
    qed "POgenXH";

    Goal
      "t [= t' <-> t=bot | (t=true & t'=true) | (t=false & t'=false) | \
    \                (EX a a' b b'. t=<a,b> &  t'=<a',b'>  & a [= a' & b [= b') | \
    \                (EX f f'. t=lam x. f(x) &  t'=lam x. f'(x) & (ALL x. f(x) [= f'(x)))";
    by (simp_tac (ccl_ss addsimps [PO_iff] delsimps ex_simps) 1);
    by (rtac (rewrite_rule [POgen_def,SIM_def]
                     (POgen_mono RS (PO_def RS def_gfp_Tarski) RS XHlemma1)) 1);
    by (rtac (iff_refl RS XHlemma2) 1);
    qed "poXH";

    Goal "bot [= b";
    by (rtac (poXH RS iffD2) 1);
    by (simp_tac ccl_ss 1);
    qed "po_bot";

    Goal "a [= bot --> a=bot";
    by (rtac impI 1);
    by (dtac (poXH RS iffD1) 1);
    by (etac rev_mp 1);
    by (simp_tac ccl_ss 1);
    bind_thm("bot_poleast", result() RS mp);

    Goal "<a,b> [= <a',b'> <->  a [= a' & b [= b'";
    by (rtac (poXH RS iff_trans) 1);
    by (simp_tac ccl_ss 1);
    qed "po_pair";

    Goal "lam x. f(x) [= lam x. f'(x) <-> (ALL x. f(x) [= f'(x))";
    by (rtac (poXH RS iff_trans) 1);
    by (simp_tac ccl_ss 1);
    by (REPEAT (ares_tac [iffI,allI] 1 ORELSE eresolve_tac [exE,conjE] 1));
    by (asm_simp_tac ccl_ss 1);
    by (fast_tac ccl_cs 1);
    qed "po_lam";

    val ccl_porews = [po_bot,po_pair,po_lam];

    val [p1,p2,p3,p4,p5] = goal (the_context ())
        "[| t [= t';  a [= a';  b [= b';  !!x y. c(x,y) [= c'(x,y); \
    \       !!u. d(u) [= d'(u) |] ==> case(t,a,b,c,d) [= case(t',a',b',c',d')";
    by (rtac (p1 RS po_cong RS po_trans) 1);
    by (rtac (p2 RS po_cong RS po_trans) 1);
    by (rtac (p3 RS po_cong RS po_trans) 1);
    by (rtac (p4 RS po_abstractn RS po_abstractn RS po_cong RS po_trans) 1);
    by (res_inst_tac [("f1","%d. case(t',a',b',c',d)")]
                   (p5 RS po_abstractn RS po_cong RS po_trans) 1);
    by (rtac po_refl 1);
    qed "case_pocong";

    val [p1,p2] = goalw (the_context ()) ccl_data_defs
        "[| f [= f';  a [= a' |] ==> f ` a [= f' ` a'";
    by (REPEAT (ares_tac [po_refl,case_pocong,p1,p2 RS po_cong] 1));
    qed "apply_pocong";

    val prems = goal (the_context ()) "~ lam x. b(x) [= bot";
    by (rtac notI 1);
    by (dtac bot_poleast 1);
    by (etac (distinctness RS notE) 1);
    qed "npo_lam_bot";

    val eq1::eq2::prems = goal (the_context ())
        "[| x=a;  y=b;  x[=y |] ==> a[=b";
    by (rtac (eq1 RS subst) 1);
    by (rtac (eq2 RS subst) 1);
    by (resolve_tac prems 1);
    qed "po_lemma";

    Goal "~ <a,b> [= lam x. f(x)";
    by (rtac notI 1);
    by (rtac (npo_lam_bot RS notE) 1);
    by (etac (case_pocong RS (caseBlam RS (caseBpair RS po_lemma))) 1);
    by (REPEAT (resolve_tac [po_refl,npo_lam_bot] 1));
    qed "npo_pair_lam";

    Goal "~ lam x. f(x) [= <a,b>";
    by (rtac notI 1);
    by (rtac (npo_lam_bot RS notE) 1);
    by (etac (case_pocong RS (caseBpair RS (caseBlam RS po_lemma))) 1);
    by (REPEAT (resolve_tac [po_refl,npo_lam_bot] 1));
    qed "npo_lam_pair";

    fun mk_thm s = prove_goal (the_context ()) s (fn _ =>
                              [rtac notI 1,dtac case_pocong 1,etac rev_mp 5,
                               ALLGOALS (simp_tac ccl_ss),
                               REPEAT (resolve_tac [po_refl,npo_lam_bot] 1)]);

    val npo_rls = [npo_pair_lam,npo_lam_pair] @ map mk_thm
                ["~ true [= false",          "~ false [= true",
                 "~ true [= <a,b>",          "~ <a,b> [= true",
                 "~ true [= lam x. f(x)","~ lam x. f(x) [= true",
                "~ false [= <a,b>",          "~ <a,b> [= false",
                "~ false [= lam x. f(x)","~ lam x. f(x) [= false"];

    (* Coinduction for [= *)

    val prems = goal (the_context ()) "[|  <t,u> : R;  R <= POgen(R) |] ==> t [= u";
    by (rtac (PO_def RS def_coinduct RS (PO_iff RS iffD2)) 1);
    by (REPEAT (ares_tac prems 1));
    qed "po_coinduct";

    fun po_coinduct_tac s i = res_inst_tac [("R",s)] po_coinduct i;

    (*************** EQUALITY *******************)

    Goalw [EQgen_def,SIM_def]  "mono(%X. EQgen(X))";
    by (rtac monoI 1);
    by (safe_tac set_cs);
    by (REPEAT_SOME (resolve_tac [exI,conjI,refl]));
    by (ALLGOALS (simp_tac ccl_ss));
    by (ALLGOALS (fast_tac set_cs));
    qed "EQgen_mono";

    Goalw [EQgen_def,SIM_def]
      "<t,t'> : EQgen(R) <-> (t=bot & t'=bot)  | (t=true & t'=true)  | \
    \                                            (t=false & t'=false) | \
    \                (EX a a' b b'. t=<a,b> &  t'=<a',b'>  & <a,a'> : R & <b,b'> : R) | \
    \                (EX f f'. t=lam x. f(x) &  t'=lam x. f'(x) & (ALL x.<f(x),f'(x)> : R))";
    by (rtac (iff_refl RS XHlemma2) 1);
    qed "EQgenXH";

    Goal
      "t=t' <-> (t=bot & t'=bot)  | (t=true & t'=true)  | (t=false & t'=false) | \
    \                    (EX a a' b b'. t=<a,b> &  t'=<a',b'>  & a=a' & b=b') | \
    \                    (EX f f'. t=lam x. f(x) &  t'=lam x. f'(x) & (ALL x. f(x)=f'(x)))";
    by (subgoal_tac
      "<t,t'> : EQ <-> (t=bot & t'=bot)  | (t=true & t'=true) | (t=false & t'=false) | \
    \             (EX a a' b b'. t=<a,b> &  t'=<a',b'>  & <a,a'> : EQ & <b,b'> : EQ) | \
    \             (EX f f'. t=lam x. f(x) &  t'=lam x. f'(x) & (ALL x. <f(x),f'(x)> : EQ))" 1);
    by (etac rev_mp 1);
    by (simp_tac (CCL_ss addsimps [EQ_iff RS iff_sym]) 1);
    by (rtac (rewrite_rule [EQgen_def,SIM_def]
                     (EQgen_mono RS (EQ_def RS def_gfp_Tarski) RS XHlemma1)) 1);
    by (rtac (iff_refl RS XHlemma2) 1);
    qed "eqXH";

    val prems = goal (the_context ()) "[|  <t,u> : R;  R <= EQgen(R) |] ==> t = u";
    by (rtac (EQ_def RS def_coinduct RS (EQ_iff RS iffD2)) 1);
    by (REPEAT (ares_tac prems 1));
    qed "eq_coinduct";

    val prems = goal (the_context ())
        "[|  <t,u> : R;  R <= EQgen(lfp(%x. EQgen(x) Un R Un EQ)) |] ==> t = u";
    by (rtac (EQ_def RS def_coinduct3 RS (EQ_iff RS iffD2)) 1);
    by (REPEAT (ares_tac (EQgen_mono::prems) 1));
    qed "eq_coinduct3";

    fun eq_coinduct_tac s i = res_inst_tac [("R",s)] eq_coinduct i;
    fun eq_coinduct3_tac s i = res_inst_tac [("R",s)] eq_coinduct3 i;

    (*** Untyped Case Analysis and Other Facts ***)

    Goalw [apply_def]  "(EX f. t=lam x. f(x)) --> t = lam x.(t ` x)";
    by (safe_tac ccl_cs);
    by (simp_tac ccl_ss 1);
    bind_thm("cond_eta", result() RS mp);

    Goal "(t=bot) | (t=true) | (t=false) | (EX a b. t=<a,b>) | (EX f. t=lam x. f(x))";
    by (cut_facts_tac [refl RS (eqXH RS iffD1)] 1);
    by (fast_tac set_cs 1);
    qed "exhaustion";

    val prems = goal (the_context ())
        "[| P(bot);  P(true);  P(false);  !!x y. P(<x,y>);  !!b. P(lam x. b(x)) |] ==> P(t)";
    by (cut_facts_tac [exhaustion] 1);
    by (REPEAT_SOME (ares_tac prems ORELSE' eresolve_tac [disjE,exE,ssubst]));
    qed "term_case";

    fun term_case_tac a i = res_inst_tac [("t",a)] term_case i;