# File wfd.ML

    (*  Title:      CCL/Wfd.ML
        ID:         $Id: wfd.ML,v 1.6 2005/09/17 15:35:31 wenzelm Exp $
    *)

    (***********)

    val [major,prem] = goalw (the_context ()) [Wfd_def]
        "[| Wfd(R);       \
    \       !!x.[| ALL y. <y,x>: R --> P(y) |] ==> P(x) |]  ==>  \
    \    P(a)";
    by (rtac (major RS spec RS mp RS spec RS CollectD) 1);
    by (fast_tac (set_cs addSIs [prem RS CollectI]) 1);
    qed "wfd_induct";

    val [p1,p2,p3] = goal (the_context ())
        "[| !!x y.<x,y> : R ==> Q(x); \
    \       ALL x. (ALL y. <y,x> : R --> y : P) --> x : P; \
    \       !!x. Q(x) ==> x:P |] ==> a:P";
    by (rtac (p2 RS  spec  RS mp) 1);
    by (fast_tac (set_cs addSIs [p1 RS p3]) 1);
    qed "wfd_strengthen_lemma";

    fun wfd_strengthen_tac s i = res_inst_tac [("Q",s)] wfd_strengthen_lemma i THEN
                                 assume_tac (i+1);

    val wfd::prems = goal (the_context ()) "[| Wfd(r);  <a,x>:r;  <x,a>:r |] ==> P";
    by (subgoal_tac "ALL x. <a,x>:r --> <x,a>:r --> P" 1);
    by (fast_tac (FOL_cs addIs prems) 1);
    by (rtac (wfd RS  wfd_induct) 1);
    by (ALLGOALS (fast_tac (ccl_cs addSIs prems)));
    qed "wf_anti_sym";

    val prems = goal (the_context ()) "[| Wfd(r);  <a,a>: r |] ==> P";
    by (rtac wf_anti_sym 1);
    by (REPEAT (resolve_tac prems 1));
    qed "wf_anti_refl";

    (*** Irreflexive transitive closure ***)

    val [prem] = goal (the_context ()) "Wfd(R) ==> Wfd(R^+)";
    by (rewtac Wfd_def);
    by (REPEAT (ares_tac [allI,ballI,impI] 1));
    (*must retain the universal formula for later use!*)
    by (rtac allE 1 THEN assume_tac 1);
    by (etac mp 1);
    by (rtac (prem RS wfd_induct) 1);
    by (rtac (impI RS allI) 1);
    by (etac tranclE 1);
    by (fast_tac ccl_cs 1);
    by (etac (spec RS mp RS spec RS mp) 1);
    by (REPEAT (atac 1));
    qed "trancl_wf";

    (*** Lexicographic Ordering ***)

    Goalw [lex_def]
     "p : ra**rb <-> (EX a a' b b'. p = <<a,b>,<a',b'>> & (<a,a'> : ra | a=a' & <b,b'> : rb))";
    by (fast_tac ccl_cs 1);
    qed "lexXH";

    val prems = goal (the_context ())
     "<a,a'> : ra ==> <<a,b>,<a',b'>> : ra**rb";
    by (fast_tac (ccl_cs addSIs (prems @ [lexXH RS iffD2])) 1);
    qed "lexI1";

    val prems = goal (the_context ())
     "<b,b'> : rb ==> <<a,b>,<a,b'>> : ra**rb";
    by (fast_tac (ccl_cs addSIs (prems @ [lexXH RS iffD2])) 1);
    qed "lexI2";

    val major::prems = goal (the_context ())
     "[| p : ra**rb;  \
    \    !!a a' b b'.[| <a,a'> : ra; p=<<a,b>,<a',b'>> |] ==> R;  \
    \    !!a b b'.[| <b,b'> : rb;  p = <<a,b>,<a,b'>> |] ==> R  |] ==> \
    \ R";
    by (rtac (major RS (lexXH RS iffD1) RS exE) 1);
    by (REPEAT_SOME (eresolve_tac ([exE,conjE,disjE]@prems)));
    by (ALLGOALS (fast_tac ccl_cs));
    qed "lexE";

    val [major,minor] = goal (the_context ())
     "[| p : r**s;  !!a a' b b'. p = <<a,b>,<a',b'>> ==> P |] ==>P";
    by (rtac (major RS lexE) 1);
    by (ALLGOALS (fast_tac (set_cs addSEs [minor])));
    qed "lex_pair";

    val [wfa,wfb] = goal (the_context ())
     "[| Wfd(R); Wfd(S) |] ==> Wfd(R**S)";
    by (rewtac Wfd_def);
    by (safe_tac ccl_cs);
    by (wfd_strengthen_tac "%x. EX a b. x=<a,b>" 1);
    by (fast_tac (term_cs addSEs [lex_pair]) 1);
    by (subgoal_tac "ALL a b.<a,b>:P" 1);
    by (fast_tac ccl_cs 1);
    by (rtac (wfa RS wfd_induct RS allI) 1);
    by (rtac (wfb RS wfd_induct RS allI) 1);back();
    by (fast_tac (type_cs addSEs [lexE]) 1);
    qed "lex_wf";

    (*** Mapping ***)

    Goalw [wmap_def]
     "p : wmap(f,r) <-> (EX x y. p=<x,y>  &  <f(x),f(y)> : r)";
    by (fast_tac ccl_cs 1);
    qed "wmapXH";

    val prems = goal (the_context ())
     "<f(a),f(b)> : r ==> <a,b> : wmap(f,r)";
    by (fast_tac (ccl_cs addSIs (prems @ [wmapXH RS iffD2])) 1);
    qed "wmapI";

    val major::prems = goal (the_context ())
     "[| p : wmap(f,r);  !!a b.[| <f(a),f(b)> : r;  p=<a,b> |] ==> R |] ==> R";
    by (rtac (major RS (wmapXH RS iffD1) RS exE) 1);
    by (REPEAT_SOME (eresolve_tac ([exE,conjE,disjE]@prems)));
    by (ALLGOALS (fast_tac ccl_cs));
    qed "wmapE";

    val [wf] = goal (the_context ())
     "Wfd(r) ==> Wfd(wmap(f,r))";
    by (rewtac Wfd_def);
    by (safe_tac ccl_cs);
    by (subgoal_tac "ALL b. ALL a. f(a)=b-->a:P" 1);
    by (fast_tac ccl_cs 1);
    by (rtac (wf RS wfd_induct RS allI) 1);
    by (safe_tac ccl_cs);
    by (etac (spec RS mp) 1);
    by (safe_tac (ccl_cs addSEs [wmapE]));
    by (etac (spec RS mp RS spec RS mp) 1);
    by (assume_tac 1);
    by (rtac refl 1);
    qed "wmap_wf";

    (* Projections *)

    val prems = goal (the_context ()) "<xa,ya> : r ==> <<xa,xb>,<ya,yb>> : wmap(fst,r)";
    by (rtac wmapI 1);
    by (simp_tac (term_ss addsimps prems) 1);
    qed "wfstI";

    val prems = goal (the_context ()) "<xb,yb> : r ==> <<xa,xb>,<ya,yb>> : wmap(snd,r)";
    by (rtac wmapI 1);
    by (simp_tac (term_ss addsimps prems) 1);
    qed "wsndI";

    val prems = goal (the_context ()) "<xc,yc> : r ==> <<xa,<xb,xc>>,<ya,<yb,yc>>> : wmap(thd,r)";
    by (rtac wmapI 1);
    by (simp_tac (term_ss addsimps prems) 1);
    qed "wthdI";

    (*** Ground well-founded relations ***)

    val prems = goalw (the_context ()) [wf_def]
        "[| Wfd(r);  a : r |] ==> a : wf(r)";
    by (fast_tac (set_cs addSIs prems) 1);
    qed "wfI";

    val prems = goalw (the_context ()) [Wfd_def] "Wfd({})";
    by (fast_tac (set_cs addEs [EmptyXH RS iffD1 RS FalseE]) 1);
    qed "Empty_wf";

    val prems = goalw (the_context ()) [wf_def] "Wfd(wf(R))";
    by (res_inst_tac [("Q","Wfd(R)")] (excluded_middle RS disjE) 1);
    by (ALLGOALS (asm_simp_tac CCL_ss));
    by (rtac Empty_wf 1);
    qed "wf_wf";

    Goalw [NatPR_def]  "p : NatPR <-> (EX x:Nat. p=<x,succ(x)>)";
    by (fast_tac set_cs 1);
    qed "NatPRXH";

    Goalw [ListPR_def]  "p : ListPR(A) <-> (EX h:A. EX t:List(A).p=<t,h$t>)";
    by (fast_tac set_cs 1);
    qed "ListPRXH";

    val NatPRI = refl RS (bexI RS (NatPRXH RS iffD2));
    val ListPRI = refl RS (bexI RS (bexI RS (ListPRXH RS iffD2)));

    Goalw [Wfd_def]  "Wfd(NatPR)";
    by (safe_tac set_cs);
    by (wfd_strengthen_tac "%x. x:Nat" 1);
    by (fast_tac (type_cs addSEs [XH_to_E NatPRXH]) 1);
    by (etac Nat_ind 1);
    by (ALLGOALS (fast_tac (type_cs addEs [XH_to_E NatPRXH])));
    qed "NatPR_wf";

    Goalw [Wfd_def]  "Wfd(ListPR(A))";
    by (safe_tac set_cs);
    by (wfd_strengthen_tac "%x. x:List(A)" 1);
    by (fast_tac (type_cs addSEs [XH_to_E ListPRXH]) 1);
    by (etac List_ind 1);
    by (ALLGOALS (fast_tac (type_cs addEs [XH_to_E ListPRXH])));
    qed "ListPR_wf";