# File Trancl.ML

    (*  Title:      CCL/Trancl.ML
        ID:         $Id: Trancl.ML,v 1.7 2005/09/17 15:35:29 wenzelm Exp $
    *)

    (** Natural deduction for trans(r) **)

    val prems = goalw (the_context ()) [trans_def]
        "(!! x y z. [| <x,y>:r;  <y,z>:r |] ==> <x,z>:r) ==> trans(r)";
    by (REPEAT (ares_tac (prems@[allI,impI]) 1));
    qed "transI";

    val major::prems = goalw (the_context ()) [trans_def]
        "[| trans(r);  <a,b>:r;  <b,c>:r |] ==> <a,c>:r";
    by (cut_facts_tac [major] 1);
    by (fast_tac (FOL_cs addIs prems) 1);
    qed "transD";

    (** Identity relation **)

    Goalw [id_def] "<a,a> : id";
    by (rtac CollectI 1);
    by (rtac exI 1);
    by (rtac refl 1);
    qed "idI";

    val major::prems = goalw (the_context ()) [id_def]
        "[| p: id;  !!x.[| p = <x,x> |] ==> P  \
    \    |] ==>  P";
    by (rtac (major RS CollectE) 1);
    by (etac exE 1);
    by (eresolve_tac prems 1);
    qed "idE";

    (** Composition of two relations **)

    val prems = goalw (the_context ()) [comp_def]
        "[| <a,b>:s; <b,c>:r |] ==> <a,c> : r O s";
    by (fast_tac (set_cs addIs prems) 1);
    qed "compI";

    (*proof requires higher-level assumptions or a delaying of hyp_subst_tac*)
    val prems = goalw (the_context ()) [comp_def]
        "[| xz : r O s;  \
    \       !!x y z. [| xz = <x,z>;  <x,y>:s;  <y,z>:r |] ==> P \
    \    |] ==> P";
    by (cut_facts_tac prems 1);
    by (REPEAT (eresolve_tac [CollectE, exE, conjE] 1 ORELSE ares_tac prems 1));
    qed "compE";

    val prems = goal (the_context ())
        "[| <a,c> : r O s;  \
    \       !!y. [| <a,y>:s;  <y,c>:r |] ==> P \
    \    |] ==> P";
    by (rtac compE 1);
    by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [pair_inject,ssubst] 1));
    qed "compEpair";

    val comp_cs = set_cs addIs [compI,idI]
                           addEs [compE,idE]
                           addSEs [pair_inject];

    val prems = goal (the_context ())
        "[| r'<=r; s'<=s |] ==> (r' O s') <= (r O s)";
    by (cut_facts_tac prems 1);
    by (fast_tac comp_cs 1);
    qed "comp_mono";

    (** The relation rtrancl **)

    Goal "mono(%s. id Un (r O s))";
    by (rtac monoI 1);
    by (REPEAT (ares_tac [monoI, subset_refl, comp_mono, Un_mono] 1));
    qed "rtrancl_fun_mono";

    val rtrancl_unfold = rtrancl_fun_mono RS (rtrancl_def RS def_lfp_Tarski);

    (*Reflexivity of rtrancl*)
    Goal "<a,a> : r^*";
    by (stac rtrancl_unfold 1);
    by (fast_tac comp_cs 1);
    qed "rtrancl_refl";

    (*Closure under composition with r*)
    val prems = goal (the_context ())
        "[| <a,b> : r^*;  <b,c> : r |] ==> <a,c> : r^*";
    by (stac rtrancl_unfold 1);
    by (fast_tac (comp_cs addIs prems) 1);
    qed "rtrancl_into_rtrancl";

    (*rtrancl of r contains r*)
    val [prem] = goal (the_context ()) "[| <a,b> : r |] ==> <a,b> : r^*";
    by (rtac (rtrancl_refl RS rtrancl_into_rtrancl) 1);
    by (rtac prem 1);
    qed "r_into_rtrancl";

    (** standard induction rule **)

    val major::prems = goal (the_context ())
      "[| <a,b> : r^*; \
    \     !!x. P(<x,x>); \
    \     !!x y z.[| P(<x,y>); <x,y>: r^*; <y,z>: r |]  ==>  P(<x,z>) |] \
    \  ==>  P(<a,b>)";
    by (rtac (major RS (rtrancl_def RS def_induct)) 1);
    by (rtac rtrancl_fun_mono 1);
    by (fast_tac (comp_cs addIs prems) 1);
    qed "rtrancl_full_induct";

    (*nice induction rule*)
    val major::prems = goal (the_context ())
        "[| <a,b> : r^*;    \
    \       P(a); \
    \       !!y z.[| <a,y> : r^*;  <y,z> : r;  P(y) |] ==> P(z) |]  \
    \     ==> P(b)";
    (*by induction on this formula*)
    by (subgoal_tac "ALL y. <a,b> = <a,y> --> P(y)" 1);
    (*now solve first subgoal: this formula is sufficient*)
    by (fast_tac FOL_cs 1);
    (*now do the induction*)
    by (resolve_tac [major RS rtrancl_full_induct] 1);
    by (fast_tac (comp_cs addIs prems) 1);
    by (fast_tac (comp_cs addIs prems) 1);
    qed "rtrancl_induct";

    (*transitivity of transitive closure!! -- by induction.*)
    Goal "trans(r^*)";
    by (rtac transI 1);
    by (res_inst_tac [("b","z")] rtrancl_induct 1);
    by (DEPTH_SOLVE (eresolve_tac [asm_rl, rtrancl_into_rtrancl] 1));
    qed "trans_rtrancl";

    (*elimination of rtrancl -- by induction on a special formula*)
    val major::prems = goal (the_context ())
        "[| <a,b> : r^*;  (a = b) ==> P; \
    \       !!y.[| <a,y> : r^*; <y,b> : r |] ==> P |] \
    \    ==> P";
    by (subgoal_tac "a = b  | (EX y. <a,y> : r^* & <y,b> : r)" 1);
    by (rtac (major RS rtrancl_induct) 2);
    by (fast_tac (set_cs addIs prems) 2);
    by (fast_tac (set_cs addIs prems) 2);
    by (REPEAT (eresolve_tac ([asm_rl,exE,disjE,conjE]@prems) 1));
    qed "rtranclE";

    (**** The relation trancl ****)

    (** Conversions between trancl and rtrancl **)

    val [major] = goalw (the_context ()) [trancl_def]
        "[| <a,b> : r^+ |] ==> <a,b> : r^*";
    by (resolve_tac [major RS compEpair] 1);
    by (REPEAT (ares_tac [rtrancl_into_rtrancl] 1));
    qed "trancl_into_rtrancl";

    (*r^+ contains r*)
    val [prem] = goalw (the_context ()) [trancl_def]
       "[| <a,b> : r |] ==> <a,b> : r^+";
    by (REPEAT (ares_tac [prem,compI,rtrancl_refl] 1));
    qed "r_into_trancl";

    (*intro rule by definition: from rtrancl and r*)
    val prems = goalw (the_context ()) [trancl_def]
        "[| <a,b> : r^*;  <b,c> : r |]   ==>  <a,c> : r^+";
    by (REPEAT (resolve_tac ([compI]@prems) 1));
    qed "rtrancl_into_trancl1";

    (*intro rule from r and rtrancl*)
    val prems = goal (the_context ())
        "[| <a,b> : r;  <b,c> : r^* |]   ==>  <a,c> : r^+";
    by (resolve_tac (prems RL [rtranclE]) 1);
    by (etac subst 1);
    by (resolve_tac (prems RL [r_into_trancl]) 1);
    by (rtac (trans_rtrancl RS transD RS rtrancl_into_trancl1) 1);
    by (REPEAT (ares_tac (prems@[r_into_rtrancl]) 1));
    qed "rtrancl_into_trancl2";

    (*elimination of r^+ -- NOT an induction rule*)
    val major::prems = goal (the_context ())
        "[| <a,b> : r^+;  \
    \       <a,b> : r ==> P; \
    \       !!y.[| <a,y> : r^+;  <y,b> : r |] ==> P  \
    \    |] ==> P";
    by (subgoal_tac "<a,b> : r | (EX y. <a,y> : r^+  &  <y,b> : r)" 1);
    by (REPEAT (eresolve_tac ([asm_rl,disjE,exE,conjE]@prems) 1));
    by (rtac (rewrite_rule [trancl_def] major RS compEpair) 1);
    by (etac rtranclE 1);
    by (fast_tac comp_cs 1);
    by (fast_tac (comp_cs addSIs [rtrancl_into_trancl1]) 1);
    qed "tranclE";

    (*Transitivity of r^+.
      Proved by unfolding since it uses transitivity of rtrancl. *)
    Goalw [trancl_def] "trans(r^+)";
    by (rtac transI 1);
    by (REPEAT (etac compEpair 1));
    by (rtac (rtrancl_into_rtrancl RS (trans_rtrancl RS transD RS compI)) 1);
    by (REPEAT (assume_tac 1));
    qed "trans_trancl";

    val prems = goal (the_context ())
        "[| <a,b> : r;  <b,c> : r^+ |]   ==>  <a,c> : r^+";
    by (rtac (r_into_trancl RS (trans_trancl RS transD)) 1);
    by (resolve_tac prems 1);
    by (resolve_tac prems 1);
    qed "trancl_into_trancl2";