(* Title: HOLCF/IOA/meta_theory/CompoTraces.thy
Author: Olaf Müller
*)
header {* Compositionality on Trace level *}
theory CompoTraces
imports CompoScheds ShortExecutions
begin
consts
mksch ::"('a,'s)ioa => ('a,'t)ioa => 'a Seq -> 'a Seq -> 'a Seq -> 'a Seq"
par_traces ::"['a trace_module,'a trace_module] => 'a trace_module"
defs
mksch_def:
"mksch A B == (fix$(LAM h tr schA schB. case tr of
nil => nil
| x##xs =>
(case x of
UU => UU
| Def y =>
(if y:act A then
(if y:act B then
((Takewhile (%a. a:int A)$schA)
@@ (Takewhile (%a. a:int B)$schB)
@@ (y>>(h$xs
$(TL$(Dropwhile (%a. a:int A)$schA))
$(TL$(Dropwhile (%a. a:int B)$schB))
)))
else
((Takewhile (%a. a:int A)$schA)
@@ (y>>(h$xs
$(TL$(Dropwhile (%a. a:int A)$schA))
$schB)))
)
else
(if y:act B then
((Takewhile (%a. a:int B)$schB)
@@ (y>>(h$xs
$schA
$(TL$(Dropwhile (%a. a:int B)$schB))
)))
else
UU
)
)
)))"
par_traces_def:
"par_traces TracesA TracesB ==
let trA = fst TracesA; sigA = snd TracesA;
trB = fst TracesB; sigB = snd TracesB
in
( {tr. Filter (%a. a:actions sigA)$tr : trA}
Int {tr. Filter (%a. a:actions sigB)$tr : trB}
Int {tr. Forall (%x. x:(externals sigA Un externals sigB)) tr},
asig_comp sigA sigB)"
axioms
finiteR_mksch:
"Finite (mksch A B$tr$x$y) --> Finite tr"
declaration {* fn _ => Simplifier.map_ss (fn ss => ss setmksym (K NONE)) *}
subsection "mksch rewrite rules"
lemma mksch_unfold:
"mksch A B = (LAM tr schA schB. case tr of
nil => nil
| x##xs =>
(case x of
UU => UU
| Def y =>
(if y:act A then
(if y:act B then
((Takewhile (%a. a:int A)$schA)
@@(Takewhile (%a. a:int B)$schB)
@@(y>>(mksch A B$xs
$(TL$(Dropwhile (%a. a:int A)$schA))
$(TL$(Dropwhile (%a. a:int B)$schB))
)))
else
((Takewhile (%a. a:int A)$schA)
@@ (y>>(mksch A B$xs
$(TL$(Dropwhile (%a. a:int A)$schA))
$schB)))
)
else
(if y:act B then
((Takewhile (%a. a:int B)$schB)
@@ (y>>(mksch A B$xs
$schA
$(TL$(Dropwhile (%a. a:int B)$schB))
)))
else
UU
)
)
))"
apply (rule trans)
apply (rule fix_eq2)
apply (rule mksch_def)
apply (rule beta_cfun)
apply simp
done
lemma mksch_UU: "mksch A B$UU$schA$schB = UU"
apply (subst mksch_unfold)
apply simp
done
lemma mksch_nil: "mksch A B$nil$schA$schB = nil"
apply (subst mksch_unfold)
apply simp
done
lemma mksch_cons1: "[|x:act A;x~:act B|]
==> mksch A B$(x>>tr)$schA$schB =
(Takewhile (%a. a:int A)$schA)
@@ (x>>(mksch A B$tr$(TL$(Dropwhile (%a. a:int A)$schA))
$schB))"
apply (rule trans)
apply (subst mksch_unfold)
apply (simp add: Consq_def If_and_if)
apply (simp add: Consq_def)
done
lemma mksch_cons2: "[|x~:act A;x:act B|]
==> mksch A B$(x>>tr)$schA$schB =
(Takewhile (%a. a:int B)$schB)
@@ (x>>(mksch A B$tr$schA$(TL$(Dropwhile (%a. a:int B)$schB))
))"
apply (rule trans)
apply (subst mksch_unfold)
apply (simp add: Consq_def If_and_if)
apply (simp add: Consq_def)
done
lemma mksch_cons3: "[|x:act A;x:act B|]
==> mksch A B$(x>>tr)$schA$schB =
(Takewhile (%a. a:int A)$schA)
@@ ((Takewhile (%a. a:int B)$schB)
@@ (x>>(mksch A B$tr$(TL$(Dropwhile (%a. a:int A)$schA))
$(TL$(Dropwhile (%a. a:int B)$schB))))
)"
apply (rule trans)
apply (subst mksch_unfold)
apply (simp add: Consq_def If_and_if)
apply (simp add: Consq_def)
done
lemmas compotr_simps = mksch_UU mksch_nil mksch_cons1 mksch_cons2 mksch_cons3
declare compotr_simps [simp]
subsection {* COMPOSITIONALITY on TRACE Level *}
subsubsection "Lemmata for ==>"
(* Consequence out of ext1_ext2_is_not_act1(2), which in turn are consequences out of
the compatibility of IOA, in particular out of the condition that internals are
really hidden. *)
lemma compatibility_consequence1: "(eB & ~eA --> ~A) -->
(A & (eA | eB)) = (eA & A)"
apply fast
done
(* very similar to above, only the commutativity of | is used to make a slight change *)
lemma compatibility_consequence2: "(eB & ~eA --> ~A) -->
(A & (eB | eA)) = (eA & A)"
apply fast
done
subsubsection "Lemmata for <=="
(* Lemma for substitution of looping assumption in another specific assumption *)
lemma subst_lemma1: "[| f << (g x) ; x=(h x) |] ==> f << g (h x)"
by (erule subst)
(* Lemma for substitution of looping assumption in another specific assumption *)
lemma subst_lemma2: "[| (f x) = y >> g; x=(h x) |] ==> (f (h x)) = y >> g"
by (erule subst)
lemma ForallAorB_mksch [rule_format]:
"!!A B. compatible A B ==>
! schA schB. Forall (%x. x:act (A||B)) tr
--> Forall (%x. x:act (A||B)) (mksch A B$tr$schA$schB)"
apply (tactic {* Seq_induct_tac @{context} "tr"
[@{thm Forall_def}, @{thm sforall_def}, @{thm mksch_def}] 1 *})
apply auto
apply (simp add: actions_of_par)
apply (case_tac "a:act A")
apply (case_tac "a:act B")
(* a:A, a:B *)
apply simp
apply (rule Forall_Conc_impl [THEN mp])
apply (simp add: intA_is_not_actB int_is_act)
apply (rule Forall_Conc_impl [THEN mp])
apply (simp add: intA_is_not_actB int_is_act)
(* a:A,a~:B *)
apply simp
apply (rule Forall_Conc_impl [THEN mp])
apply (simp add: intA_is_not_actB int_is_act)
apply (case_tac "a:act B")
(* a~:A, a:B *)
apply simp
apply (rule Forall_Conc_impl [THEN mp])
apply (simp add: intA_is_not_actB int_is_act)
(* a~:A,a~:B *)
apply auto
done
lemma ForallBnAmksch [rule_format (no_asm)]: "!!A B. compatible B A ==>
! schA schB. (Forall (%x. x:act B & x~:act A) tr
--> Forall (%x. x:act B & x~:act A) (mksch A B$tr$schA$schB))"
apply (tactic {* Seq_induct_tac @{context} "tr"
[@{thm Forall_def}, @{thm sforall_def}, @{thm mksch_def}] 1 *})
apply auto
apply (rule Forall_Conc_impl [THEN mp])
apply (simp add: intA_is_not_actB int_is_act)
done
lemma ForallAnBmksch [rule_format (no_asm)]: "!!A B. compatible A B ==>
! schA schB. (Forall (%x. x:act A & x~:act B) tr
--> Forall (%x. x:act A & x~:act B) (mksch A B$tr$schA$schB))"
apply (tactic {* Seq_induct_tac @{context} "tr"
[@{thm Forall_def}, @{thm sforall_def}, @{thm mksch_def}] 1 *})
apply auto
apply (rule Forall_Conc_impl [THEN mp])
apply (simp add: intA_is_not_actB int_is_act)
done
(* safe-tac makes too many case distinctions with this lemma in the next proof *)
declare FiniteConc [simp del]
lemma FiniteL_mksch [rule_format (no_asm)]: "[| Finite tr; is_asig(asig_of A); is_asig(asig_of B) |] ==>
! x y. Forall (%x. x:act A) x & Forall (%x. x:act B) y &
Filter (%a. a:ext A)$x = Filter (%a. a:act A)$tr &
Filter (%a. a:ext B)$y = Filter (%a. a:act B)$tr &
Forall (%x. x:ext (A||B)) tr
--> Finite (mksch A B$tr$x$y)"
apply (erule Seq_Finite_ind)
apply simp
(* main case *)
apply simp
apply auto
(* a: act A; a: act B *)
apply (frule sym [THEN antisym_less_inverse, THEN conjunct1, THEN divide_Seq])
apply (frule sym [THEN antisym_less_inverse, THEN conjunct1, THEN divide_Seq])
back
apply (erule conjE)+
(* Finite (tw iA x) and Finite (tw iB y) *)
apply (simp add: not_ext_is_int_or_not_act FiniteConc)
(* now for conclusion IH applicable, but assumptions have to be transformed *)
apply (drule_tac x = "x" and g = "Filter (%a. a:act A) $s" in subst_lemma2)
apply assumption
apply (drule_tac x = "y" and g = "Filter (%a. a:act B) $s" in subst_lemma2)
apply assumption
(* IH *)
apply (simp add: not_ext_is_int_or_not_act ForallTL ForallDropwhile)
(* a: act B; a~: act A *)
apply (frule sym [THEN antisym_less_inverse, THEN conjunct1, THEN divide_Seq])
apply (erule conjE)+
(* Finite (tw iB y) *)
apply (simp add: not_ext_is_int_or_not_act FiniteConc)
(* now for conclusion IH applicable, but assumptions have to be transformed *)
apply (drule_tac x = "y" and g = "Filter (%a. a:act B) $s" in subst_lemma2)
apply assumption
(* IH *)
apply (simp add: not_ext_is_int_or_not_act ForallTL ForallDropwhile)
(* a~: act B; a: act A *)
apply (frule sym [THEN antisym_less_inverse, THEN conjunct1, THEN divide_Seq])
apply (erule conjE)+
(* Finite (tw iA x) *)
apply (simp add: not_ext_is_int_or_not_act FiniteConc)
(* now for conclusion IH applicable, but assumptions have to be transformed *)
apply (drule_tac x = "x" and g = "Filter (%a. a:act A) $s" in subst_lemma2)
apply assumption
(* IH *)
apply (simp add: not_ext_is_int_or_not_act ForallTL ForallDropwhile)
(* a~: act B; a~: act A *)
apply (fastsimp intro!: ext_is_act simp: externals_of_par)
done
declare FiniteConc [simp]
declare FilterConc [simp del]
lemma reduceA_mksch1 [rule_format (no_asm)]: " [| Finite bs; is_asig(asig_of A); is_asig(asig_of B);compatible A B|] ==>
! y. Forall (%x. x:act B) y & Forall (%x. x:act B & x~:act A) bs &
Filter (%a. a:ext B)$y = Filter (%a. a:act B)$(bs @@ z)
--> (? y1 y2. (mksch A B$(bs @@ z)$x$y) = (y1 @@ (mksch A B$z$x$y2)) &
Forall (%x. x:act B & x~:act A) y1 &
Finite y1 & y = (y1 @@ y2) &
Filter (%a. a:ext B)$y1 = bs)"
apply (frule_tac A1 = "A" in compat_commute [THEN iffD1])
apply (erule Seq_Finite_ind)
apply (rule allI)+
apply (rule impI)
apply (rule_tac x = "nil" in exI)
apply (rule_tac x = "y" in exI)
apply simp
(* main case *)
apply (rule allI)+
apply (rule impI)
apply simp
apply (erule conjE)+
apply simp
(* divide_Seq on s *)
apply (frule sym [THEN antisym_less_inverse, THEN conjunct1, THEN divide_Seq])
apply (erule conjE)+
(* transform assumption f eB y = f B (s@z) *)
apply (drule_tac x = "y" and g = "Filter (%a. a:act B) $ (s@@z) " in subst_lemma2)
apply assumption
apply (simp add: not_ext_is_int_or_not_act FilterConc)
(* apply IH *)
apply (erule_tac x = "TL$ (Dropwhile (%a. a:int B) $y) " in allE)
apply (simp add: ForallTL ForallDropwhile FilterConc)
apply (erule exE)+
apply (erule conjE)+
apply (simp add: FilterConc)
(* for replacing IH in conclusion *)
apply (rotate_tac -2)
(* instantiate y1a and y2a *)
apply (rule_tac x = "Takewhile (%a. a:int B) $y @@ a>>y1" in exI)
apply (rule_tac x = "y2" in exI)
(* elminate all obligations up to two depending on Conc_assoc *)
apply (simp add: intA_is_not_actB int_is_act int_is_not_ext FilterConc)
apply (simp (no_asm) add: Conc_assoc FilterConc)
done
lemmas reduceA_mksch = conjI [THEN [6] conjI [THEN [5] reduceA_mksch1]]
lemma reduceB_mksch1 [rule_format]:
" [| Finite a_s; is_asig(asig_of A); is_asig(asig_of B);compatible A B|] ==>
! x. Forall (%x. x:act A) x & Forall (%x. x:act A & x~:act B) a_s &
Filter (%a. a:ext A)$x = Filter (%a. a:act A)$(a_s @@ z)
--> (? x1 x2. (mksch A B$(a_s @@ z)$x$y) = (x1 @@ (mksch A B$z$x2$y)) &
Forall (%x. x:act A & x~:act B) x1 &
Finite x1 & x = (x1 @@ x2) &
Filter (%a. a:ext A)$x1 = a_s)"
apply (frule_tac A1 = "A" in compat_commute [THEN iffD1])
apply (erule Seq_Finite_ind)
apply (rule allI)+
apply (rule impI)
apply (rule_tac x = "nil" in exI)
apply (rule_tac x = "x" in exI)
apply simp
(* main case *)
apply (rule allI)+
apply (rule impI)
apply simp
apply (erule conjE)+
apply simp
(* divide_Seq on s *)
apply (frule sym [THEN antisym_less_inverse, THEN conjunct1, THEN divide_Seq])
apply (erule conjE)+
(* transform assumption f eA x = f A (s@z) *)
apply (drule_tac x = "x" and g = "Filter (%a. a:act A) $ (s@@z) " in subst_lemma2)
apply assumption
apply (simp add: not_ext_is_int_or_not_act FilterConc)
(* apply IH *)
apply (erule_tac x = "TL$ (Dropwhile (%a. a:int A) $x) " in allE)
apply (simp add: ForallTL ForallDropwhile FilterConc)
apply (erule exE)+
apply (erule conjE)+
apply (simp add: FilterConc)
(* for replacing IH in conclusion *)
apply (rotate_tac -2)
(* instantiate y1a and y2a *)
apply (rule_tac x = "Takewhile (%a. a:int A) $x @@ a>>x1" in exI)
apply (rule_tac x = "x2" in exI)
(* elminate all obligations up to two depending on Conc_assoc *)
apply (simp add: intA_is_not_actB int_is_act int_is_not_ext FilterConc)
apply (simp (no_asm) add: Conc_assoc FilterConc)
done
lemmas reduceB_mksch = conjI [THEN [6] conjI [THEN [5] reduceB_mksch1]]
declare FilterConc [simp]
subsection "Filtering external actions out of mksch(tr,schA,schB) yields the oracle tr"
lemma FilterA_mksch_is_tr:
"!! A B. [| compatible A B; compatible B A;
is_asig(asig_of A); is_asig(asig_of B) |] ==>
! schA schB. Forall (%x. x:act A) schA & Forall (%x. x:act B) schB &
Forall (%x. x:ext (A||B)) tr &
Filter (%a. a:act A)$tr << Filter (%a. a:ext A)$schA &
Filter (%a. a:act B)$tr << Filter (%a. a:ext B)$schB
--> Filter (%a. a:ext (A||B))$(mksch A B$tr$schA$schB) = tr"
apply (tactic {* Seq_induct_tac @{context} "tr"
[@{thm Forall_def}, @{thm sforall_def}, @{thm mksch_def}] 1 *})
(* main case *)
(* splitting into 4 cases according to a:A, a:B *)
apply auto
(* Case a:A, a:B *)
apply (frule divide_Seq)
apply (frule divide_Seq)
back
apply (erule conjE)+
(* filtering internals of A in schA and of B in schB is nil *)
apply (simp add: not_ext_is_int_or_not_act externals_of_par intA_is_not_extB int_is_not_ext)
(* conclusion of IH ok, but assumptions of IH have to be transformed *)
apply (drule_tac x = "schA" in subst_lemma1)
apply assumption
apply (drule_tac x = "schB" in subst_lemma1)
apply assumption
(* IH *)
apply (simp add: not_ext_is_int_or_not_act ForallTL ForallDropwhile)
(* Case a:A, a~:B *)
apply (frule divide_Seq)
apply (erule conjE)+
(* filtering internals of A is nil *)
apply (simp add: not_ext_is_int_or_not_act externals_of_par intA_is_not_extB int_is_not_ext)
apply (drule_tac x = "schA" in subst_lemma1)
apply assumption
apply (simp add: not_ext_is_int_or_not_act ForallTL ForallDropwhile)
(* Case a:B, a~:A *)
apply (frule divide_Seq)
apply (erule conjE)+
(* filtering internals of A is nil *)
apply (simp add: not_ext_is_int_or_not_act externals_of_par intA_is_not_extB int_is_not_ext)
apply (drule_tac x = "schB" in subst_lemma1)
back
apply assumption
apply (simp add: not_ext_is_int_or_not_act ForallTL ForallDropwhile)
(* Case a~:A, a~:B *)
apply (fastsimp intro!: ext_is_act simp: externals_of_par)
done
subsection" Filter of mksch(tr,schA,schB) to A is schA -- take lemma proof"
lemma FilterAmksch_is_schA: "!! A B. [| compatible A B; compatible B A;
is_asig(asig_of A); is_asig(asig_of B) |] ==>
Forall (%x. x:ext (A||B)) tr &
Forall (%x. x:act A) schA & Forall (%x. x:act B) schB &
Filter (%a. a:ext A)$schA = Filter (%a. a:act A)$tr &
Filter (%a. a:ext B)$schB = Filter (%a. a:act B)$tr &
LastActExtsch A schA & LastActExtsch B schB
--> Filter (%a. a:act A)$(mksch A B$tr$schA$schB) = schA"
apply (intro strip)
apply (rule seq.take_lemmas)
apply (rule mp)
prefer 2 apply assumption
back back back back
apply (rule_tac x = "schA" in spec)
apply (rule_tac x = "schB" in spec)
apply (rule_tac x = "tr" in spec)
apply (tactic "thin_tac' 5 1")
apply (rule nat_less_induct)
apply (rule allI)+
apply (rename_tac tr schB schA)
apply (intro strip)
apply (erule conjE)+
apply (case_tac "Forall (%x. x:act B & x~:act A) tr")
apply (rule seq_take_lemma [THEN iffD2, THEN spec])
apply (tactic "thin_tac' 5 1")
apply (case_tac "Finite tr")
(* both sides of this equation are nil *)
apply (subgoal_tac "schA=nil")
apply (simp (no_asm_simp))
(* first side: mksch = nil *)
apply (auto intro!: ForallQFilterPnil ForallBnAmksch FiniteL_mksch)[1]
(* second side: schA = nil *)
apply (erule_tac A = "A" in LastActExtimplnil)
apply (simp (no_asm_simp))
apply (erule_tac Q = "%x. x:act B & x~:act A" in ForallQFilterPnil)
apply assumption
apply fast
(* case ~ Finite s *)
(* both sides of this equation are UU *)
apply (subgoal_tac "schA=UU")
apply (simp (no_asm_simp))
(* first side: mksch = UU *)
apply (auto intro!: ForallQFilterPUU finiteR_mksch [THEN mp, COMP rev_contrapos] ForallBnAmksch)[1]
(* schA = UU *)
apply (erule_tac A = "A" in LastActExtimplUU)
apply (simp (no_asm_simp))
apply (erule_tac Q = "%x. x:act B & x~:act A" in ForallQFilterPUU)
apply assumption
apply fast
(* case" ~ Forall (%x.x:act B & x~:act A) s" *)
apply (drule divide_Seq3)
apply (erule exE)+
apply (erule conjE)+
apply hypsubst
(* bring in lemma reduceA_mksch *)
apply (frule_tac x = "schA" and y = "schB" and A = "A" and B = "B" in reduceA_mksch)
apply assumption+
apply (erule exE)+
apply (erule conjE)+
(* use reduceA_mksch to rewrite conclusion *)
apply hypsubst
apply simp
(* eliminate the B-only prefix *)
apply (subgoal_tac " (Filter (%a. a :act A) $y1) = nil")
apply (erule_tac [2] ForallQFilterPnil)
prefer 2 apply assumption
prefer 2 apply fast
(* Now real recursive step follows (in y) *)
apply simp
apply (case_tac "x:act A")
apply (case_tac "x~:act B")
apply (rotate_tac -2)
apply simp
apply (subgoal_tac "Filter (%a. a:act A & a:ext B) $y1=nil")
apply (rotate_tac -1)
apply simp
(* eliminate introduced subgoal 2 *)
apply (erule_tac [2] ForallQFilterPnil)
prefer 2 apply assumption
prefer 2 apply fast
(* bring in divide Seq for s *)
apply (frule sym [THEN antisym_less_inverse, THEN conjunct1, THEN divide_Seq])
apply (erule conjE)+
(* subst divide_Seq in conclusion, but only at the righest occurence *)
apply (rule_tac t = "schA" in ssubst)
back
back
back
apply assumption
(* reduce trace_takes from n to strictly smaller k *)
apply (rule take_reduction)
(* f A (tw iA) = tw ~eA *)
apply (simp add: int_is_act not_ext_is_int_or_not_act)
apply (rule refl)
apply (simp add: int_is_act not_ext_is_int_or_not_act)
apply (rotate_tac -11)
(* now conclusion fulfills induction hypothesis, but assumptions are not ready *)
(* assumption Forall tr *)
(* assumption schB *)
apply (simp add: ext_and_act)
(* assumption schA *)
apply (drule_tac x = "schA" and g = "Filter (%a. a:act A) $rs" in subst_lemma2)
apply assumption
apply (simp add: int_is_not_ext)
(* assumptions concerning LastActExtsch, cannot be rewritten, as LastActExtsmalli are looping *)
apply (drule_tac sch = "schA" and P = "%a. a:int A" in LastActExtsmall1)
apply (frule_tac ?sch1.0 = "y1" in LastActExtsmall2)
apply assumption
(* assumption Forall schA *)
apply (drule_tac s = "schA" and P = "Forall (%x. x:act A) " in subst)
apply assumption
apply (simp add: int_is_act)
(* case x:actions(asig_of A) & x: actions(asig_of B) *)
apply (rotate_tac -2)
apply simp
apply (subgoal_tac "Filter (%a. a:act A & a:ext B) $y1=nil")
apply (rotate_tac -1)
apply simp
(* eliminate introduced subgoal 2 *)
apply (erule_tac [2] ForallQFilterPnil)
prefer 2 apply (assumption)
prefer 2 apply (fast)
(* bring in divide Seq for s *)
apply (frule sym [THEN antisym_less_inverse, THEN conjunct1, THEN divide_Seq])
apply (erule conjE)+
(* subst divide_Seq in conclusion, but only at the righest occurence *)
apply (rule_tac t = "schA" in ssubst)
back
back
back
apply assumption
(* f A (tw iA) = tw ~eA *)
apply (simp add: int_is_act not_ext_is_int_or_not_act)
(* rewrite assumption forall and schB *)
apply (rotate_tac 13)
apply (simp add: ext_and_act)
(* divide_Seq for schB2 *)
apply (frule_tac y = "y2" in sym [THEN antisym_less_inverse, THEN conjunct1, THEN divide_Seq])
apply (erule conjE)+
(* assumption schA *)
apply (drule_tac x = "schA" and g = "Filter (%a. a:act A) $rs" in subst_lemma2)
apply assumption
apply (simp add: int_is_not_ext)
(* f A (tw iB schB2) = nil *)
apply (simp add: int_is_not_ext not_ext_is_int_or_not_act intA_is_not_actB)
(* reduce trace_takes from n to strictly smaller k *)
apply (rule take_reduction)
apply (rule refl)
apply (rule refl)
(* now conclusion fulfills induction hypothesis, but assumptions are not all ready *)
(* assumption schB *)
apply (drule_tac x = "y2" and g = "Filter (%a. a:act B) $rs" in subst_lemma2)
apply assumption
apply (simp add: intA_is_not_actB int_is_not_ext)
(* conclusions concerning LastActExtsch, cannot be rewritten, as LastActExtsmalli are looping *)
apply (drule_tac sch = "schA" and P = "%a. a:int A" in LastActExtsmall1)
apply (frule_tac ?sch1.0 = "y1" in LastActExtsmall2)
apply assumption
apply (drule_tac sch = "y2" and P = "%a. a:int B" in LastActExtsmall1)
(* assumption Forall schA, and Forall schA are performed by ForallTL,ForallDropwhile *)
apply (simp add: ForallTL ForallDropwhile)
(* case x~:A & x:B *)
(* cannot occur, as just this case has been scheduled out before as the B-only prefix *)
apply (case_tac "x:act B")
apply fast
(* case x~:A & x~:B *)
(* cannot occur because of assumption: Forall (a:ext A | a:ext B) *)
apply (rotate_tac -9)
(* reduce forall assumption from tr to (x>>rs) *)
apply (simp add: externals_of_par)
apply (fast intro!: ext_is_act)
done
subsection" Filter of mksch(tr,schA,schB) to B is schB -- take lemma proof"
lemma FilterBmksch_is_schB: "!! A B. [| compatible A B; compatible B A;
is_asig(asig_of A); is_asig(asig_of B) |] ==>
Forall (%x. x:ext (A||B)) tr &
Forall (%x. x:act A) schA & Forall (%x. x:act B) schB &
Filter (%a. a:ext A)$schA = Filter (%a. a:act A)$tr &
Filter (%a. a:ext B)$schB = Filter (%a. a:act B)$tr &
LastActExtsch A schA & LastActExtsch B schB
--> Filter (%a. a:act B)$(mksch A B$tr$schA$schB) = schB"
apply (intro strip)
apply (rule seq.take_lemmas)
apply (rule mp)
prefer 2 apply assumption
back back back back
apply (rule_tac x = "schA" in spec)
apply (rule_tac x = "schB" in spec)
apply (rule_tac x = "tr" in spec)
apply (tactic "thin_tac' 5 1")
apply (rule nat_less_induct)
apply (rule allI)+
apply (rename_tac tr schB schA)
apply (intro strip)
apply (erule conjE)+
apply (case_tac "Forall (%x. x:act A & x~:act B) tr")
apply (rule seq_take_lemma [THEN iffD2, THEN spec])
apply (tactic "thin_tac' 5 1")
apply (case_tac "Finite tr")
(* both sides of this equation are nil *)
apply (subgoal_tac "schB=nil")
apply (simp (no_asm_simp))
(* first side: mksch = nil *)
apply (auto intro!: ForallQFilterPnil ForallAnBmksch FiniteL_mksch)[1]
(* second side: schA = nil *)
apply (erule_tac A = "B" in LastActExtimplnil)
apply (simp (no_asm_simp))
apply (erule_tac Q = "%x. x:act A & x~:act B" in ForallQFilterPnil)
apply assumption
apply fast
(* case ~ Finite tr *)
(* both sides of this equation are UU *)
apply (subgoal_tac "schB=UU")
apply (simp (no_asm_simp))
(* first side: mksch = UU *)
apply (force intro!: ForallQFilterPUU finiteR_mksch [THEN mp, COMP rev_contrapos] ForallAnBmksch)
(* schA = UU *)
apply (erule_tac A = "B" in LastActExtimplUU)
apply (simp (no_asm_simp))
apply (erule_tac Q = "%x. x:act A & x~:act B" in ForallQFilterPUU)
apply assumption
apply fast
(* case" ~ Forall (%x.x:act B & x~:act A) s" *)
apply (drule divide_Seq3)
apply (erule exE)+
apply (erule conjE)+
apply hypsubst
(* bring in lemma reduceB_mksch *)
apply (frule_tac y = "schB" and x = "schA" and A = "A" and B = "B" in reduceB_mksch)
apply assumption+
apply (erule exE)+
apply (erule conjE)+
(* use reduceB_mksch to rewrite conclusion *)
apply hypsubst
apply simp
(* eliminate the A-only prefix *)
apply (subgoal_tac "(Filter (%a. a :act B) $x1) = nil")
apply (erule_tac [2] ForallQFilterPnil)
prefer 2 apply (assumption)
prefer 2 apply (fast)
(* Now real recursive step follows (in x) *)
apply simp
apply (case_tac "x:act B")
apply (case_tac "x~:act A")
apply (rotate_tac -2)
apply simp
apply (subgoal_tac "Filter (%a. a:act B & a:ext A) $x1=nil")
apply (rotate_tac -1)
apply simp
(* eliminate introduced subgoal 2 *)
apply (erule_tac [2] ForallQFilterPnil)
prefer 2 apply (assumption)
prefer 2 apply (fast)
(* bring in divide Seq for s *)
apply (frule sym [THEN antisym_less_inverse, THEN conjunct1, THEN divide_Seq])
apply (erule conjE)+
(* subst divide_Seq in conclusion, but only at the righest occurence *)
apply (rule_tac t = "schB" in ssubst)
back
back
back
apply assumption
(* reduce trace_takes from n to strictly smaller k *)
apply (rule take_reduction)
(* f B (tw iB) = tw ~eB *)
apply (simp add: int_is_act not_ext_is_int_or_not_act)
apply (rule refl)
apply (simp add: int_is_act not_ext_is_int_or_not_act)
apply (rotate_tac -11)
(* now conclusion fulfills induction hypothesis, but assumptions are not ready *)
(* assumption schA *)
apply (simp add: ext_and_act)
(* assumption schB *)
apply (drule_tac x = "schB" and g = "Filter (%a. a:act B) $rs" in subst_lemma2)
apply assumption
apply (simp add: int_is_not_ext)
(* assumptions concerning LastActExtsch, cannot be rewritten, as LastActExtsmalli are looping *)
apply (drule_tac sch = "schB" and P = "%a. a:int B" in LastActExtsmall1)
apply (frule_tac ?sch1.0 = "x1" in LastActExtsmall2)
apply assumption
(* assumption Forall schB *)
apply (drule_tac s = "schB" and P = "Forall (%x. x:act B) " in subst)
apply assumption
apply (simp add: int_is_act)
(* case x:actions(asig_of A) & x: actions(asig_of B) *)
apply (rotate_tac -2)
apply simp
apply (subgoal_tac "Filter (%a. a:act B & a:ext A) $x1=nil")
apply (rotate_tac -1)
apply simp
(* eliminate introduced subgoal 2 *)
apply (erule_tac [2] ForallQFilterPnil)
prefer 2 apply (assumption)
prefer 2 apply (fast)
(* bring in divide Seq for s *)
apply (frule sym [THEN antisym_less_inverse, THEN conjunct1, THEN divide_Seq])
apply (erule conjE)+
(* subst divide_Seq in conclusion, but only at the righest occurence *)
apply (rule_tac t = "schB" in ssubst)
back
back
back
apply assumption
(* f B (tw iB) = tw ~eB *)
apply (simp add: int_is_act not_ext_is_int_or_not_act)
(* rewrite assumption forall and schB *)
apply (rotate_tac 13)
apply (simp add: ext_and_act)
(* divide_Seq for schB2 *)
apply (frule_tac y = "x2" in sym [THEN antisym_less_inverse, THEN conjunct1, THEN divide_Seq])
apply (erule conjE)+
(* assumption schA *)
apply (drule_tac x = "schB" and g = "Filter (%a. a:act B) $rs" in subst_lemma2)
apply assumption
apply (simp add: int_is_not_ext)
(* f B (tw iA schA2) = nil *)
apply (simp add: int_is_not_ext not_ext_is_int_or_not_act intA_is_not_actB)
(* reduce trace_takes from n to strictly smaller k *)
apply (rule take_reduction)
apply (rule refl)
apply (rule refl)
(* now conclusion fulfills induction hypothesis, but assumptions are not all ready *)
(* assumption schA *)
apply (drule_tac x = "x2" and g = "Filter (%a. a:act A) $rs" in subst_lemma2)
apply assumption
apply (simp add: intA_is_not_actB int_is_not_ext)
(* conclusions concerning LastActExtsch, cannot be rewritten, as LastActExtsmalli are looping *)
apply (drule_tac sch = "schB" and P = "%a. a:int B" in LastActExtsmall1)
apply (frule_tac ?sch1.0 = "x1" in LastActExtsmall2)
apply assumption
apply (drule_tac sch = "x2" and P = "%a. a:int A" in LastActExtsmall1)
(* assumption Forall schA, and Forall schA are performed by ForallTL,ForallDropwhile *)
apply (simp add: ForallTL ForallDropwhile)
(* case x~:B & x:A *)
(* cannot occur, as just this case has been scheduled out before as the B-only prefix *)
apply (case_tac "x:act A")
apply fast
(* case x~:B & x~:A *)
(* cannot occur because of assumption: Forall (a:ext A | a:ext B) *)
apply (rotate_tac -9)
(* reduce forall assumption from tr to (x>>rs) *)
apply (simp add: externals_of_par)
apply (fast intro!: ext_is_act)
done
subsection "COMPOSITIONALITY on TRACE Level -- Main Theorem"
lemma compositionality_tr:
"!! A B. [| is_trans_of A; is_trans_of B; compatible A B; compatible B A;
is_asig(asig_of A); is_asig(asig_of B)|]
==> (tr: traces(A||B)) =
(Filter (%a. a:act A)$tr : traces A &
Filter (%a. a:act B)$tr : traces B &
Forall (%x. x:ext(A||B)) tr)"
apply (simp (no_asm) add: traces_def has_trace_def)
apply auto
(* ==> *)
(* There is a schedule of A *)
apply (rule_tac x = "Filter (%a. a:act A) $sch" in bexI)
prefer 2
apply (simp add: compositionality_sch)
apply (simp add: compatibility_consequence1 externals_of_par ext1_ext2_is_not_act1)
(* There is a schedule of B *)
apply (rule_tac x = "Filter (%a. a:act B) $sch" in bexI)
prefer 2
apply (simp add: compositionality_sch)
apply (simp add: compatibility_consequence2 externals_of_par ext1_ext2_is_not_act2)
(* Traces of A||B have only external actions from A or B *)
apply (rule ForallPFilterP)
(* <== *)
(* replace schA and schB by Cut(schA) and Cut(schB) *)
apply (drule exists_LastActExtsch)
apply assumption
apply (drule exists_LastActExtsch)
apply assumption
apply (erule exE)+
apply (erule conjE)+
(* Schedules of A(B) have only actions of A(B) *)
apply (drule scheds_in_sig)
apply assumption
apply (drule scheds_in_sig)
apply assumption
apply (rename_tac h1 h2 schA schB)
(* mksch is exactly the construction of trA||B out of schA, schB, and the oracle tr,
we need here *)
apply (rule_tac x = "mksch A B$tr$schA$schB" in bexI)
(* External actions of mksch are just the oracle *)
apply (simp add: FilterA_mksch_is_tr)
(* mksch is a schedule -- use compositionality on sch-level *)
apply (simp add: compositionality_sch)
apply (simp add: FilterAmksch_is_schA FilterBmksch_is_schB)
apply (erule ForallAorB_mksch)
apply (erule ForallPForallQ)
apply (erule ext_is_act)
done
subsection {* COMPOSITIONALITY on TRACE Level -- for Modules *}
lemma compositionality_tr_modules:
"!! A B. [| is_trans_of A; is_trans_of B; compatible A B; compatible B A;
is_asig(asig_of A); is_asig(asig_of B)|]
==> Traces (A||B) = par_traces (Traces A) (Traces B)"
apply (unfold Traces_def par_traces_def)
apply (simp add: asig_of_par)
apply (rule set_ext)
apply (simp add: compositionality_tr externals_of_par)
done
declaration {* fn _ => Simplifier.map_ss (fn ss => ss setmksym (SOME o symmetric_fun)) *}
end