# File Type.ML

    (*  Title:      CCL/Type.ML
        ID:         $Id: Type.ML,v 1.11 2005/09/17 15:35:29 wenzelm Exp $
        Author:     Martin Coen, Cambridge University Computer Laboratory
        Copyright   1992  University of Cambridge
    *)

    val simp_type_defs = [Subtype_def,Unit_def,Bool_def,Plus_def,Sigma_def,Pi_def,
                          Lift_def,Tall_def,Tex_def];
    val ind_type_defs = [Nat_def,List_def];

    val simp_data_defs = [one_def,inl_def,inr_def];
    val ind_data_defs = [zero_def,succ_def,nil_def,cons_def];

    goal (the_context ()) "A <= B <-> (ALL x. x:A --> x:B)";
    by (fast_tac set_cs 1);
    qed "subsetXH";

    (*** Exhaustion Rules ***)

    fun mk_XH_tac thy defs rls s = prove_goalw thy defs s (fn _ => [cfast_tac rls 1]);
    val XH_tac = mk_XH_tac (the_context ()) simp_type_defs [];

    val EmptyXH = XH_tac "a : {} <-> False";
    val SubtypeXH = XH_tac "a : {x:A. P(x)} <-> (a:A & P(a))";
    val UnitXH = XH_tac "a : Unit          <-> a=one";
    val BoolXH = XH_tac "a : Bool          <-> a=true | a=false";
    val PlusXH = XH_tac "a : A+B           <-> (EX x:A. a=inl(x)) | (EX x:B. a=inr(x))";
    val PiXH   = XH_tac "a : PROD x:A. B(x) <-> (EX b. a=lam x. b(x) & (ALL x:A. b(x):B(x)))";
    val SgXH   = XH_tac "a : SUM x:A. B(x)  <-> (EX x:A. EX y:B(x).a=<x,y>)";

    val XHs = [EmptyXH,SubtypeXH,UnitXH,BoolXH,PlusXH,PiXH,SgXH];

    val LiftXH = XH_tac "a : [A] <-> (a=bot | a:A)";
    val TallXH = XH_tac "a : TALL X. B(X) <-> (ALL X. a:B(X))";
    val TexXH  = XH_tac "a : TEX X. B(X) <-> (EX X. a:B(X))";

    val case_rls = XH_to_Es XHs;

    (*** Canonical Type Rules ***)

    fun mk_canT_tac thy xhs s = prove_goal thy s
                     (fn prems => [fast_tac (set_cs addIs (prems @ (xhs RL [iffD2]))) 1]);
    val canT_tac = mk_canT_tac (the_context ()) XHs;

    val oneT   = canT_tac "one : Unit";
    val trueT  = canT_tac "true : Bool";
    val falseT = canT_tac "false : Bool";
    val lamT   = canT_tac "[| !!x. x:A ==> b(x):B(x) |] ==> lam x. b(x) : Pi(A,B)";
    val pairT  = canT_tac "[| a:A; b:B(a) |] ==> <a,b>:Sigma(A,B)";
    val inlT   = canT_tac "a:A ==> inl(a) : A+B";
    val inrT   = canT_tac "b:B ==> inr(b) : A+B";

    val canTs = [oneT,trueT,falseT,pairT,lamT,inlT,inrT];

    (*** Non-Canonical Type Rules ***)

    local
    val lemma = prove_goal (the_context ()) "[| a:B(u);  u=v |] ==> a : B(v)"
                       (fn prems => [cfast_tac prems 1]);
    in
    fun mk_ncanT_tac thy defs top_crls crls s = prove_goalw thy defs s
      (fn major::prems => [(resolve_tac ([major] RL top_crls) 1),
                           (REPEAT_SOME (eresolve_tac (crls @ [exE,bexE,conjE,disjE]))),
                           (ALLGOALS (asm_simp_tac term_ss)),
                           (ALLGOALS (ares_tac (prems RL [lemma]) ORELSE'
                                      etac bspec )),
                           (safe_tac (ccl_cs addSIs prems))]);
    end;

    val ncanT_tac = mk_ncanT_tac (the_context ()) [] case_rls case_rls;

    val ifT = ncanT_tac
         "[| b:Bool; b=true ==> t:A(true); b=false ==> u:A(false) |] ==> \
    \     if b then t else u : A(b)";

    val applyT = ncanT_tac
        "[| f : Pi(A,B);  a:A |] ==> f ` a : B(a)";

    val splitT = ncanT_tac
        "[| p:Sigma(A,B); !!x y. [| x:A;  y:B(x); p=<x,y>  |] ==> c(x,y):C(<x,y>) |] ==>  \
    \     split(p,c):C(p)";

    val whenT = ncanT_tac
         "[| p:A+B; !!x.[| x:A;  p=inl(x) |] ==> a(x):C(inl(x)); \
    \               !!y.[| y:B;  p=inr(y) |] ==> b(y):C(inr(y)) |] ==> \
    \     when(p,a,b) : C(p)";

    val ncanTs = [ifT,applyT,splitT,whenT];

    (*** Subtypes ***)

    val SubtypeD1 = standard ((SubtypeXH RS iffD1) RS conjunct1);
    val SubtypeD2 = standard ((SubtypeXH RS iffD1) RS conjunct2);

    val prems = goal (the_context ())
         "[| a:A;  P(a) |] ==> a : {x:A. P(x)}";
    by (REPEAT (resolve_tac (prems@[SubtypeXH RS iffD2,conjI]) 1));
    qed "SubtypeI";

    val prems = goal (the_context ())
         "[| a : {x:A. P(x)};  [| a:A;  P(a) |] ==> Q |] ==> Q";
    by (REPEAT (resolve_tac (prems@[SubtypeD1,SubtypeD2]) 1));
    qed "SubtypeE";

    (*** Monotonicity ***)

    Goal "mono (%X. X)";
    by (REPEAT (ares_tac [monoI] 1));
    qed "idM";

    Goal "mono(%X. A)";
    by (REPEAT (ares_tac [monoI,subset_refl] 1));
    qed "constM";

    val major::prems = goal (the_context ())
        "mono(%X. A(X)) ==> mono(%X.[A(X)])";
    by (rtac (subsetI RS monoI) 1);
    by (dtac (LiftXH RS iffD1) 1);
    by (etac disjE 1);
    by (etac (disjI1 RS (LiftXH RS iffD2)) 1);
    by (rtac (disjI2 RS (LiftXH RS iffD2)) 1);
    by (etac (major RS monoD RS subsetD) 1);
    by (assume_tac 1);
    qed "LiftM";

    val prems = goal (the_context ())
        "[| mono(%X. A(X)); !!x X. x:A(X) ==> mono(%X. B(X,x)) |] ==> \
    \    mono(%X. Sigma(A(X),B(X)))";
    by (REPEAT (ares_tac ([subsetI RS monoI] @ canTs) 1 ORELSE
                eresolve_tac ([bspec,exE,conjE,disjE,bexE] @ case_rls) 1 ORELSE
                (resolve_tac (prems RL [monoD RS subsetD]) 1 THEN assume_tac 1) ORELSE
                hyp_subst_tac 1));
    qed "SgM";

    val prems = goal (the_context ())
        "[| !!x. x:A ==> mono(%X. B(X,x)) |] ==> mono(%X. Pi(A,B(X)))";
    by (REPEAT (ares_tac ([subsetI RS monoI] @ canTs) 1 ORELSE
                eresolve_tac ([bspec,exE,conjE,disjE,bexE] @ case_rls) 1 ORELSE
                (resolve_tac (prems RL [monoD RS subsetD]) 1 THEN assume_tac 1) ORELSE
                hyp_subst_tac 1));
    qed "PiM";

    val prems = goal (the_context ())
         "[| mono(%X. A(X));  mono(%X. B(X)) |] ==> mono(%X. A(X)+B(X))";
    by (REPEAT (ares_tac ([subsetI RS monoI] @ canTs) 1 ORELSE
                eresolve_tac ([bspec,exE,conjE,disjE,bexE] @ case_rls) 1 ORELSE
                (resolve_tac (prems RL [monoD RS subsetD]) 1 THEN assume_tac 1) ORELSE
                hyp_subst_tac 1));
    qed "PlusM";

    (**************** RECURSIVE TYPES ******************)

    (*** Conversion Rules for Fixed Points via monotonicity and Tarski ***)

    Goal "mono(%X. Unit+X)";
    by (REPEAT (ares_tac [PlusM,constM,idM] 1));
    qed "NatM";
    bind_thm("def_NatB", result() RS (Nat_def RS def_lfp_Tarski));

    Goal "mono(%X.(Unit+Sigma(A,%y. X)))";
    by (REPEAT (ares_tac [PlusM,SgM,constM,idM] 1));
    qed "ListM";
    bind_thm("def_ListB", result() RS (List_def RS def_lfp_Tarski));
    bind_thm("def_ListsB", result() RS (Lists_def RS def_gfp_Tarski));

    Goal "mono(%X.({} + Sigma(A,%y. X)))";
    by (REPEAT (ares_tac [PlusM,SgM,constM,idM] 1));
    qed "IListsM";
    bind_thm("def_IListsB", result() RS (ILists_def RS def_gfp_Tarski));

    val ind_type_eqs = [def_NatB,def_ListB,def_ListsB,def_IListsB];

    (*** Exhaustion Rules ***)

    fun mk_iXH_tac teqs ddefs rls s = prove_goalw (the_context ()) ddefs s
               (fn _ => [resolve_tac (teqs RL [XHlemma1]) 1,
                         fast_tac (set_cs addSIs canTs addSEs case_rls) 1]);

    val iXH_tac = mk_iXH_tac ind_type_eqs ind_data_defs [];

    val NatXH  = iXH_tac "a : Nat <-> (a=zero | (EX x:Nat. a=succ(x)))";
    val ListXH = iXH_tac "a : List(A) <-> (a=[] | (EX x:A. EX xs:List(A).a=x$xs))";
    val ListsXH = iXH_tac "a : Lists(A) <-> (a=[] | (EX x:A. EX xs:Lists(A).a=x$xs))";
    val IListsXH = iXH_tac "a : ILists(A) <-> (EX x:A. EX xs:ILists(A).a=x$xs)";

    val iXHs = [NatXH,ListXH];
    val icase_rls = XH_to_Es iXHs;

    (*** Type Rules ***)

    val icanT_tac = mk_canT_tac (the_context ()) iXHs;
    val incanT_tac = mk_ncanT_tac (the_context ()) [] icase_rls case_rls;

    val zeroT = icanT_tac "zero : Nat";
    val succT = icanT_tac "n:Nat ==> succ(n) : Nat";
    val nilT  = icanT_tac "[] : List(A)";
    val consT = icanT_tac "[| h:A;  t:List(A) |] ==> h$t : List(A)";

    val icanTs = [zeroT,succT,nilT,consT];

    val ncaseT = incanT_tac
         "[| n:Nat; n=zero ==> b:C(zero); \
    \        !!x.[| x:Nat;  n=succ(x) |] ==> c(x):C(succ(x)) |] ==>  \
    \     ncase(n,b,c) : C(n)";

    val lcaseT = incanT_tac
         "[| l:List(A); l=[] ==> b:C([]); \
    \        !!h t.[| h:A;  t:List(A); l=h$t |] ==> c(h,t):C(h$t) |] ==> \
    \     lcase(l,b,c) : C(l)";

    val incanTs = [ncaseT,lcaseT];

    (*** Induction Rules ***)

    val ind_Ms = [NatM,ListM];

    fun mk_ind_tac ddefs tdefs Ms canTs case_rls s = prove_goalw (the_context ()) ddefs s
         (fn major::prems => [resolve_tac (Ms RL ([major] RL (tdefs RL [def_induct]))) 1,
                              fast_tac (set_cs addSIs (prems @ canTs) addSEs case_rls) 1]);

    val ind_tac = mk_ind_tac ind_data_defs ind_type_defs ind_Ms canTs case_rls;

    val Nat_ind = ind_tac
         "[| n:Nat; P(zero); !!x.[| x:Nat; P(x) |] ==> P(succ(x)) |] ==>  \
    \     P(n)";

    val List_ind = ind_tac
         "[| l:List(A); P([]); \
    \        !!x xs.[| x:A;  xs:List(A); P(xs) |] ==> P(x$xs) |] ==> \
    \     P(l)";

    val inds = [Nat_ind,List_ind];

    (*** Primitive Recursive Rules ***)

    fun mk_prec_tac inds s = prove_goal (the_context ()) s
         (fn major::prems => [resolve_tac ([major] RL inds) 1,
                              ALLGOALS (simp_tac term_ss THEN'
                                        fast_tac (set_cs addSIs prems))]);
    val prec_tac = mk_prec_tac inds;

    val nrecT = prec_tac
         "[| n:Nat; b:C(zero); \
    \        !!x g.[| x:Nat; g:C(x) |] ==> c(x,g):C(succ(x)) |] ==>  \
    \     nrec(n,b,c) : C(n)";

    val lrecT = prec_tac
         "[| l:List(A); b:C([]); \
    \        !!x xs g.[| x:A;  xs:List(A); g:C(xs) |] ==> c(x,xs,g):C(x$xs) |] ==>  \
    \     lrec(l,b,c) : C(l)";

    val precTs = [nrecT,lrecT];

    (*** Theorem proving ***)

    val [major,minor] = goal (the_context ())
        "[| <a,b> : Sigma(A,B);  [| a:A;  b:B(a) |] ==> P   \
    \    |] ==> P";
    by (rtac (major RS (XH_to_E SgXH)) 1);
    by (rtac minor 1);
    by (ALLGOALS (fast_tac term_cs));
    qed "SgE2";

    (* General theorem proving ignores non-canonical term-formers,             *)
    (*         - intro rules are type rules for canonical terms                *)
    (*         - elim rules are case rules (no non-canonical terms appear)     *)

    val type_cs = term_cs addSIs (SubtypeI::(canTs @ icanTs))
                          addSEs (SubtypeE::(XH_to_Es XHs));

    (*** Infinite Data Types ***)

    val [mono] = goal (the_context ()) "mono(f) ==> lfp(f) <= gfp(f)";
    by (rtac (lfp_lowerbound RS subset_trans) 1);
    by (rtac (mono RS gfp_lemma3) 1);
    by (rtac subset_refl 1);
    qed "lfp_subset_gfp";

    val prems = goal (the_context ())
        "[| a:A;  !!x X.[| x:A;  ALL y:A. t(y):X |] ==> t(x) : B(X) |] ==> \
    \    t(a) : gfp(B)";
    by (rtac coinduct 1);
    by (res_inst_tac [("P","%x. EX y:A. x=t(y)")] CollectI 1);
    by (ALLGOALS (fast_tac (ccl_cs addSIs prems)));
    qed "gfpI";

    val rew::prem::prems = goal (the_context ())
        "[| C==gfp(B);  a:A;  !!x X.[| x:A;  ALL y:A. t(y):X |] ==> t(x) : B(X) |] ==> \
    \    t(a) : C";
    by (rewtac rew);
    by (REPEAT (ares_tac ((prem RS gfpI)::prems) 1));
    qed "def_gfpI";

    (* EG *)

    val prems = goal (the_context ())
        "letrec g x be zero$g(x) in g(bot) : Lists(Nat)";
    by (rtac (refl RS (XH_to_I UnitXH) RS (Lists_def RS def_gfpI)) 1);
    by (stac letrecB 1);
    by (rewtac cons_def);
    by (fast_tac type_cs 1);
    result();