src/HOLCF/IOA/meta_theory/ShortExecutions.ML

GPLed;

(*  Title:      HOLCF/IOA/meta_theory/ShortExecutions.thy
    ID:         $Id$
    Author:     Olaf Müller
    License:    GPL (GNU GENERAL PUBLIC LICENSE)

Some properties about (Cut ex), defined as follows:

For every execution ex there is another shorter execution (Cut ex) 
that has the same trace as ex, but its schedule ends with an external action.
*) 


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 thy 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";