# File ShortExecutions.ML

    (*  Title:      HOLCF/IOA/meta_theory/ShortExecutions.thy
        ID:         $Id: ShortExecutions.ML,v 1.21 2005/09/03 20:27:06 wenzelm Exp $
        Author:     Olaf Müller
    *)

    fun thin_tac' j =
      rotate_tac (j - 1) THEN'
      etac thin_rl THEN'
      rotate_tac (~ (j - 1));

    (* ---------------------------------------------------------------- *)
                       section "oraclebuild rewrite rules";
    (* ---------------------------------------------------------------- *)

    bind_thm ("oraclebuild_unfold", fix_prover2 (the_context ()) oraclebuild_def
    "oraclebuild P = (LAM s t. case t of \
    \       nil => nil\
    \    | x##xs => \
    \      (case x of\
    \        UU => UU\
    \      | Def y => (Takewhile (%a.~ P a)$s)\
    \                  @@ (y>>(oraclebuild P$(TL$(Dropwhile (%a.~ P a)$s))$xs))\
    \      )\
    \    )");

    Goal "oraclebuild P$sch$UU = UU";
    by (stac oraclebuild_unfold 1);
    by (Simp_tac 1);
    qed"oraclebuild_UU";

    Goal "oraclebuild P$sch$nil = nil";
    by (stac oraclebuild_unfold 1);
    by (Simp_tac 1);
    qed"oraclebuild_nil";

    Goal "oraclebuild P$s$(x>>t) = \
    \         (Takewhile (%a.~ P a)$s)   \
    \          @@ (x>>(oraclebuild P$(TL$(Dropwhile (%a.~ P a)$s))$t))";
    by (rtac trans 1);
    by (stac oraclebuild_unfold 1);
    by (asm_full_simp_tac (simpset() addsimps [Consq_def]) 1);
    by (simp_tac (simpset() addsimps [Consq_def]) 1);
    qed"oraclebuild_cons";

    (* ---------------------------------------------------------------- *)
                       section "Cut rewrite rules";
    (* ---------------------------------------------------------------- *)

    Goalw [Cut_def]
    "[| Forall (%a.~ P a) s; Finite s|] \
    \           ==> Cut P s =nil";
    by (subgoal_tac "Filter P$s = nil" 1);
    by (asm_simp_tac (simpset() addsimps [oraclebuild_nil]) 1);
    by (rtac ForallQFilterPnil 1);
    by (REPEAT (atac 1));
    qed"Cut_nil";

    Goalw [Cut_def]
    "[| Forall (%a.~ P a) s; ~Finite s|] \
    \           ==> Cut P s =UU";
    by (subgoal_tac "Filter P$s= UU" 1);
    by (asm_simp_tac (simpset() addsimps [oraclebuild_UU]) 1);
    by (rtac ForallQFilterPUU 1);
    by (REPEAT (atac 1));
    qed"Cut_UU";

    Goalw [Cut_def]
    "[| P t;  Forall (%x.~ P x) ss; Finite ss|] \
    \           ==> Cut P (ss @@ (t>> rs)) \
    \                = ss @@ (t >> Cut P rs)";

    by (asm_full_simp_tac (simpset() addsimps [ForallQFilterPnil,oraclebuild_cons,
              TakewhileConc,DropwhileConc]) 1);
    qed"Cut_Cons";

    (* ---------------------------------------------------------------- *)
                       section "Cut lemmas for main theorem";
    (* ---------------------------------------------------------------- *)

    Goal "Filter P$s = Filter P$(Cut P s)";

    by (res_inst_tac [("A1","%x. True")
                     ,("Q1","%x.~ P x"), ("x1","s")]
                     (take_lemma_induct RS mp) 1);
    by (Fast_tac 3);
    by (case_tac "Finite s" 1);
    by (asm_full_simp_tac (simpset() addsimps [Cut_nil,
                 ForallQFilterPnil]) 1);
    by (asm_full_simp_tac (simpset() addsimps [Cut_UU,
                 ForallQFilterPUU]) 1);
    (* main case *)
    by (asm_full_simp_tac (simpset() addsimps [Cut_Cons,ForallQFilterPnil]) 1);
    by Auto_tac;
    qed"FilterCut";

    Goal "Cut P (Cut P s) = (Cut P s)";

    by (res_inst_tac [("A1","%x. True")
                     ,("Q1","%x.~ P x"), ("x1","s")]
                     (take_lemma_less_induct RS mp) 1);
    by (Fast_tac 3);
    by (case_tac "Finite s" 1);
    by (asm_full_simp_tac (simpset() addsimps [Cut_nil,
                 ForallQFilterPnil]) 1);
    by (asm_full_simp_tac (simpset() addsimps [Cut_UU,
                 ForallQFilterPUU]) 1);
    (* main case *)
    by (asm_full_simp_tac (simpset() addsimps [Cut_Cons,ForallQFilterPnil]) 1);
    by (rtac take_reduction 1);
    by Auto_tac;
    qed"Cut_idemp";

    Goal "Map f$(Cut (P o f) s) = Cut P (Map f$s)";

    by (res_inst_tac [("A1","%x. True")
                     ,("Q1","%x.~ P (f x)"), ("x1","s")]
                     (take_lemma_less_induct RS mp) 1);
    by (Fast_tac 3);
    by (case_tac "Finite s" 1);
    by (asm_full_simp_tac (simpset() addsimps [Cut_nil]) 1);
    by (rtac (Cut_nil RS sym) 1);
    by (asm_full_simp_tac (simpset() addsimps [ForallMap,o_def]) 1);
    by (asm_full_simp_tac (simpset() addsimps [Map2Finite]) 1);
    (* csae ~ Finite s *)
    by (asm_full_simp_tac (simpset() addsimps [Cut_UU]) 1);
    by (rtac (Cut_UU (*RS sym*)) 1);
    by (asm_full_simp_tac (simpset() addsimps [ForallMap,o_def]) 1);
    by (asm_full_simp_tac (simpset() addsimps [Map2Finite]) 1);
    (* main case *)
    by (asm_full_simp_tac (simpset() addsimps [Cut_Cons,MapConc,
              ForallMap,FiniteMap1,o_def]) 1);
    by (rtac take_reduction 1);
    by Auto_tac;
    qed"MapCut";

    Goal "~Finite s --> Cut P s << s";
    by (strip_tac 1);
    by (rtac (take_lemma_less RS iffD1) 1);
    by (strip_tac 1);
    by (rtac mp 1);
    by (assume_tac 2);
    by (thin_tac' 1 1);
    by (res_inst_tac [("x","s")] spec 1);
    by (rtac nat_less_induct 1);
    by (strip_tac 1);
    ren "na n s" 1;
    by (case_tac "Forall (%x. ~ P x) s" 1);
    by (rtac (take_lemma_less RS iffD2 RS spec) 1);
    by (asm_full_simp_tac (simpset() addsimps [Cut_UU]) 1);
    (* main case *)
    by (dtac divide_Seq3 1);
    by (REPEAT (etac exE 1));
    by (REPEAT (etac conjE 1));
    by (hyp_subst_tac 1);
    by (asm_full_simp_tac (simpset() addsimps [Cut_Cons]) 1);
    by (rtac take_reduction_less 1);
    (* auto makes also reasoning about Finiteness of parts of s ! *)
    by Auto_tac;
    qed_spec_mp"Cut_prefixcl_nFinite";

    Goal "!!ex .is_exec_frag A (s,ex) ==> is_exec_frag A (s,Cut P ex)";
    by (case_tac "Finite ex" 1);
    by (cut_inst_tac [("s","ex"),("P","P")] Cut_prefixcl_Finite 1);
    by (assume_tac 1);
    by (etac exE 1);
    by (rtac exec_prefix2closed 1);
    by (eres_inst_tac [("s","ex"),("t","Cut P ex @@ y")] subst 1);
    by (assume_tac 1);
    by (etac exec_prefixclosed 1);
    by (etac Cut_prefixcl_nFinite 1);
    qed"execThruCut";

    (* ---------------------------------------------------------------- *)
                       section "Main Cut Theorem";
    (* ---------------------------------------------------------------- *)

    Goalw [schedules_def,has_schedule_def]
     "[|sch : schedules A ; tr = Filter (%a. a:ext A)$sch|] \
    \   ==> ? sch. sch : schedules A & \
    \              tr = Filter (%a. a:ext A)$sch &\
    \              LastActExtsch A sch";

    by (safe_tac set_cs);
    by (res_inst_tac [("x","filter_act$(Cut (%a. fst a:ext A) (snd ex))")] exI 1);
    by (asm_full_simp_tac (simpset() addsimps [executions_def]) 1);
    by (pair_tac "ex" 1);
    by (safe_tac set_cs);
    by (res_inst_tac [("x","(x,Cut (%a. fst a:ext A) y)")] exI 1);
    by (Asm_simp_tac 1);

    (* Subgoal 1: Lemma:  propagation of execution through Cut *)

    by (asm_full_simp_tac (simpset() addsimps [execThruCut]) 1);

    (* Subgoal 2:  Lemma:  Filter P s = Filter P (Cut P s) *)

    by (simp_tac (simpset() addsimps [filter_act_def]) 1);
    by (subgoal_tac "Map fst$(Cut (%a. fst a: ext A) y)= Cut (%a. a:ext A) (Map fst$y)" 1);

    by (rtac (rewrite_rule [o_def] MapCut) 2);
    by (asm_full_simp_tac (simpset() addsimps [FilterCut RS sym]) 1);

    (* Subgoal 3: Lemma: Cut idempotent  *)

    by (simp_tac (simpset() addsimps [LastActExtsch_def,filter_act_def]) 1);
    by (subgoal_tac "Map fst$(Cut (%a. fst a: ext A) y)= Cut (%a. a:ext A) (Map fst$y)" 1);
    by (rtac (rewrite_rule [o_def] MapCut) 2);
    by (asm_full_simp_tac (simpset() addsimps [Cut_idemp]) 1);
    qed"exists_LastActExtsch";

    (* ---------------------------------------------------------------- *)
                       section "Further Cut lemmas";
    (* ---------------------------------------------------------------- *)

    Goalw [LastActExtsch_def]
      "[| LastActExtsch A sch; Filter (%x. x:ext A)$sch = nil |] \
    \   ==> sch=nil";
    by (dtac FilternPnilForallP 1);
    by (etac conjE 1);
    by (dtac Cut_nil 1);
    by (assume_tac 1);
    by (Asm_full_simp_tac 1);
    qed"LastActExtimplnil";

    Goalw [LastActExtsch_def]
      "[| LastActExtsch A sch; Filter (%x. x:ext A)$sch = UU |] \
    \   ==> sch=UU";
    by (dtac FilternPUUForallP 1);
    by (etac conjE 1);
    by (dtac Cut_UU 1);
    by (assume_tac 1);
    by (Asm_full_simp_tac 1);
    qed"LastActExtimplUU";