(* Title: HOLCF/IOA/meta_theory/Abstraction.thy
ID: $Id$
Author: Olaf M"uller
Copyright 1997 TU Muenchen
Abstraction Theory -- tailored for I/O automata
*)
section "cex_abs";
(* ---------------------------------------------------------------- *)
(* cex_abs *)
(* ---------------------------------------------------------------- *)
Goal "cex_abs f (s,UU) = (f s, UU)";
by (simp_tac (simpset() addsimps [cex_abs_def]) 1);
qed"cex_abs_UU";
Goal "cex_abs f (s,nil) = (f s, nil)";
by (simp_tac (simpset() addsimps [cex_abs_def]) 1);
qed"cex_abs_nil";
Goal "cex_abs f (s,(a,t)>>ex) = (f s, (a,f t) >> (snd (cex_abs f (t,ex))))";
by (simp_tac (simpset() addsimps [cex_abs_def]) 1);
qed"cex_abs_cons";
Addsimps [cex_abs_UU, cex_abs_nil, cex_abs_cons];
section "lemmas";
(* ---------------------------------------------------------------- *)
(* Lemmas *)
(* ---------------------------------------------------------------- *)
Goal "temp_weakening Q P h = (! ex. (ex |== P) --> (cex_abs h ex |== Q))";
by (simp_tac (simpset() addsimps [temp_weakening_def,temp_strengthening_def,
NOT_def,temp_sat_def,satisfies_def]) 1);
by Auto_tac;
qed"temp_weakening_def2";
Goal "state_weakening Q P h = (! s t a. P (s,a,t) --> Q (h(s),a,h(t)))";
by (simp_tac (simpset() addsimps [state_weakening_def,state_strengthening_def,
NOT_def]) 1);
by Auto_tac;
qed"state_weakening_def2";
section "Abstraction Rules for Properties";
(* ---------------------------------------------------------------- *)
(* Abstraction Rules for Properties *)
(* ---------------------------------------------------------------- *)
Goalw [cex_abs_def]
"[| is_abstraction h C A |] ==>\
\ !s. reachable C s & is_exec_frag C (s,xs) \
\ --> is_exec_frag A (cex_abs h (s,xs))";
by (Asm_full_simp_tac 1);
by (pair_induct_tac "xs" [is_exec_frag_def] 1);
(* main case *)
by (safe_tac set_cs);
by (asm_full_simp_tac (simpset() addsimps [is_abstraction_def])1);
by (forward_tac [reachable.reachable_n] 1);
by (assume_tac 1);
by (Asm_full_simp_tac 1);
qed_spec_mp"exec_frag_abstraction";
Goal "is_abstraction h C A ==> weakeningIOA A C h";
by (asm_full_simp_tac (simpset() addsimps [weakeningIOA_def])1);
by Auto_tac;
by (asm_full_simp_tac (simpset() addsimps [executions_def]) 1);
(* start state *)
by (rtac conjI 1);
by (asm_full_simp_tac (simpset() addsimps [is_abstraction_def,cex_abs_def]) 1);
(* is-execution-fragment *)
by (etac exec_frag_abstraction 1);
by (asm_full_simp_tac (simpset() addsimps [reachable.reachable_0]) 1);
qed"abs_is_weakening";
Goal "[|is_abstraction h C A; validIOA A Q; temp_strengthening Q P h |] \
\ ==> validIOA C P";
by (dtac abs_is_weakening 1);
by (asm_full_simp_tac (simpset() addsimps [weakeningIOA_def,
validIOA_def, temp_strengthening_def])1);
by (safe_tac set_cs);
by (pair_tac "ex" 1);
qed"AbsRuleT1";
(* FIX: Nach TLS.ML *)
Goal "(ex |== P .--> Q) = ((ex |== P) --> (ex |== Q))";
by (simp_tac (simpset() addsimps [IMPLIES_def,temp_sat_def, satisfies_def])1);
qed"IMPLIES_temp_sat";
Goal "(ex |== P .& Q) = ((ex |== P) & (ex |== Q))";
by (simp_tac (simpset() addsimps [AND_def,temp_sat_def, satisfies_def])1);
qed"AND_temp_sat";
Goal "(ex |== P .| Q) = ((ex |== P) | (ex |== Q))";
by (simp_tac (simpset() addsimps [OR_def,temp_sat_def, satisfies_def])1);
qed"OR_temp_sat";
Goal "(ex |== .~ P) = (~ (ex |== P))";
by (simp_tac (simpset() addsimps [NOT_def,temp_sat_def, satisfies_def])1);
qed"NOT_temp_sat";
Addsimps [IMPLIES_temp_sat,AND_temp_sat,OR_temp_sat,NOT_temp_sat];
Goalw [is_live_abstraction_def]
"[|is_live_abstraction h (C,L) (A,M); \
\ validLIOA (A,M) Q; temp_strengthening Q P h |] \
\ ==> validLIOA (C,L) P";
by Auto_tac;
by (dtac abs_is_weakening 1);
by (asm_full_simp_tac (simpset() addsimps [weakeningIOA_def, temp_weakening_def2,
validLIOA_def, validIOA_def, temp_strengthening_def])1);
by (safe_tac set_cs);
by (pair_tac "ex" 1);
qed"AbsRuleT2";
Goalw [is_live_abstraction_def]
"[|is_live_abstraction h (C,L) (A,M); \
\ validLIOA (A,M) (H1 .--> Q); temp_strengthening Q P h; \
\ temp_weakening H1 H2 h; validLIOA (C,L) H2 |] \
\ ==> validLIOA (C,L) P";
by Auto_tac;
by (dtac abs_is_weakening 1);
by (asm_full_simp_tac (simpset() addsimps [weakeningIOA_def, temp_weakening_def2,
validLIOA_def, validIOA_def, temp_strengthening_def])1);
by (safe_tac set_cs);
by (pair_tac "ex" 1);
qed"AbsRuleTImprove";
section "Correctness of safe abstraction";
(* ---------------------------------------------------------------- *)
(* Correctness of safe abstraction *)
(* ---------------------------------------------------------------- *)
Goalw [is_abstraction_def,is_ref_map_def]
"is_abstraction h C A ==> is_ref_map h C A";
by (safe_tac set_cs);
by (res_inst_tac[("x","(a,h t)>>nil")] exI 1);
by (asm_full_simp_tac (simpset() addsimps [move_def])1);
qed"abstraction_is_ref_map";
Goal "[| inp(C)=inp(A); out(C)=out(A); \
\ is_abstraction h C A |] \
\ ==> C =<| A";
by (asm_full_simp_tac (simpset() addsimps [ioa_implements_def]) 1);
by (rtac trace_inclusion 1);
by (simp_tac (simpset() addsimps [externals_def])1);
by (SELECT_GOAL (auto_tac (claset(),simpset()))1);
by (etac abstraction_is_ref_map 1);
qed"abs_safety";
section "Correctness of life abstraction";
(* ---------------------------------------------------------------- *)
(* Correctness of life abstraction *)
(* ---------------------------------------------------------------- *)
(* Reduces to Filter (Map fst x) = Filter (Map fst (Map (%(a,t). (a,x)) x),
that is to special Map Lemma *)
Goalw [cex_abs_def,mk_trace_def,filter_act_def]
"ext C = ext A \
\ ==> mk_trace C$xs = mk_trace A$(snd (cex_abs f (s,xs)))";
by (Asm_full_simp_tac 1);
by (pair_induct_tac "xs" [] 1);
qed"traces_coincide_abs";
(* Does not work with abstraction_is_ref_map as proof of abs_safety, because
is_live_abstraction includes temp_strengthening which is necessarily based
on cex_abs and not on corresp_ex. Thus, the proof is redoone in a more specific
way for cex_abs *)
Goal "[| inp(C)=inp(A); out(C)=out(A); \
\ is_live_abstraction h (C,M) (A,L) |] \
\ ==> live_implements (C,M) (A,L)";
by (asm_full_simp_tac (simpset() addsimps [is_live_abstraction_def, live_implements_def,
livetraces_def,liveexecutions_def]) 1);
by (safe_tac set_cs);
by (res_inst_tac[("x","cex_abs h ex")] exI 1);
by (safe_tac set_cs);
(* Traces coincide *)
by (pair_tac "ex" 1);
by (rtac traces_coincide_abs 1);
by (simp_tac (simpset() addsimps [externals_def])1);
by (SELECT_GOAL (auto_tac (claset(),simpset()))1);
(* cex_abs is execution *)
by (pair_tac "ex" 1);
by (asm_full_simp_tac (simpset() addsimps [executions_def]) 1);
(* start state *)
by (rtac conjI 1);
by (asm_full_simp_tac (simpset() addsimps [is_abstraction_def,cex_abs_def]) 1);
(* is-execution-fragment *)
by (etac exec_frag_abstraction 1);
by (asm_full_simp_tac (simpset() addsimps [reachable.reachable_0]) 1);
(* Liveness *)
by (asm_full_simp_tac (simpset() addsimps [temp_weakening_def2]) 1);
by (pair_tac "ex" 1);
qed"abs_liveness";
(* FIX: NAch Traces.ML bringen *)
Goalw [ioa_implements_def]
"[| A =<| B; B =<| C|] ==> A =<| C";
by Auto_tac;
qed"implements_trans";
section "Abstraction Rules for Automata";
(* ---------------------------------------------------------------- *)
(* Abstraction Rules for Automata *)
(* ---------------------------------------------------------------- *)
Goal "[| inp(C)=inp(A); out(C)=out(A); \
\ inp(Q)=inp(P); out(Q)=out(P); \
\ is_abstraction h1 C A; \
\ A =<| Q ; \
\ is_abstraction h2 Q P |] \
\ ==> C =<| P";
by (dtac abs_safety 1);
by (REPEAT (atac 1));
by (dtac abs_safety 1);
by (REPEAT (atac 1));
by (etac implements_trans 1);
by (etac implements_trans 1);
by (assume_tac 1);
qed"AbsRuleA1";
Goal "[| inp(C)=inp(A); out(C)=out(A); \
\ inp(Q)=inp(P); out(Q)=out(P); \
\ is_live_abstraction h1 (C,LC) (A,LA); \
\ live_implements (A,LA) (Q,LQ) ; \
\ is_live_abstraction h2 (Q,LQ) (P,LP) |] \
\ ==> live_implements (C,LC) (P,LP)";
by (dtac abs_liveness 1);
by (REPEAT (atac 1));
by (dtac abs_liveness 1);
by (REPEAT (atac 1));
by (etac live_implements_trans 1);
by (etac live_implements_trans 1);
by (assume_tac 1);
qed"AbsRuleA2";
Delsimps [split_paired_All];
section "Localizing Temporal Strengthenings and Weakenings";
(* ---------------------------------------------------------------- *)
(* Localizing Temproal Strengthenings - 1 *)
(* ---------------------------------------------------------------- *)
Goalw [temp_strengthening_def]
"[| temp_strengthening P1 Q1 h; \
\ temp_strengthening P2 Q2 h |] \
\ ==> temp_strengthening (P1 .& P2) (Q1 .& Q2) h";
by Auto_tac;
qed"strength_AND";
Goalw [temp_strengthening_def]
"[| temp_strengthening P1 Q1 h; \
\ temp_strengthening P2 Q2 h |] \
\ ==> temp_strengthening (P1 .| P2) (Q1 .| Q2) h";
by Auto_tac;
qed"strength_OR";
Goalw [temp_strengthening_def]
"[| temp_weakening P Q h |] \
\ ==> temp_strengthening (.~ P) (.~ Q) h";
by (asm_full_simp_tac (simpset() addsimps [temp_weakening_def2])1);
by Auto_tac;
qed"strength_NOT";
Goalw [temp_strengthening_def]
"[| temp_weakening P1 Q1 h; \
\ temp_strengthening P2 Q2 h |] \
\ ==> temp_strengthening (P1 .--> P2) (Q1 .--> Q2) h";
by (asm_full_simp_tac (simpset() addsimps [temp_weakening_def2])1);
qed"strength_IMPLIES";
(* ---------------------------------------------------------------- *)
(* Localizing Temproal Weakenings - Part 1 *)
(* ---------------------------------------------------------------- *)
Goal
"[| temp_weakening P1 Q1 h; \
\ temp_weakening P2 Q2 h |] \
\ ==> temp_weakening (P1 .& P2) (Q1 .& Q2) h";
by (asm_full_simp_tac (simpset() addsimps [temp_weakening_def2])1);
qed"weak_AND";
Goal
"[| temp_weakening P1 Q1 h; \
\ temp_weakening P2 Q2 h |] \
\ ==> temp_weakening (P1 .| P2) (Q1 .| Q2) h";
by (asm_full_simp_tac (simpset() addsimps [temp_weakening_def2])1);
qed"weak_OR";
Goalw [temp_strengthening_def]
"[| temp_strengthening P Q h |] \
\ ==> temp_weakening (.~ P) (.~ Q) h";
by (asm_full_simp_tac (simpset() addsimps [temp_weakening_def2])1);
by Auto_tac;
qed"weak_NOT";
Goalw [temp_strengthening_def]
"[| temp_strengthening P1 Q1 h; \
\ temp_weakening P2 Q2 h |] \
\ ==> temp_weakening (P1 .--> P2) (Q1 .--> Q2) h";
by (asm_full_simp_tac (simpset() addsimps [temp_weakening_def2])1);
qed"weak_IMPLIES";
(* ---------------------------------------------------------------- *)
(* Localizing Temproal Strengthenings - 2 *)
(* ---------------------------------------------------------------- *)
(* ------------------ Box ----------------------------*)
(* FIX: should be same as nil_is_Conc2 when all nils are turned to right side !! *)
Goal "(UU = x @@ y) = (((x::'a Seq)= UU) | (x=nil & y=UU))";
by (Seq_case_simp_tac "x" 1);
by Auto_tac;
qed"UU_is_Conc";
Goal
"Finite s1 --> \
\ (! ex. (s~=nil & s~=UU & ex2seq ex = s1 @@ s) --> (? ex'. s = ex2seq ex'))";
by (rtac impI 1);
by (Seq_Finite_induct_tac 1);
by (Blast_tac 1);
(* main case *)
by (clarify_tac set_cs 1);
by (pair_tac "ex" 1);
by (Seq_case_simp_tac "y" 1);
(* UU case *)
by (asm_full_simp_tac (simpset() addsimps [nil_is_Conc])1);
(* nil case *)
by (asm_full_simp_tac (simpset() addsimps [nil_is_Conc])1);
(* cons case *)
by (pair_tac "aa" 1);
by Auto_tac;
qed_spec_mp"ex2seqConc";
(* important property of ex2seq: can be shiftet, as defined "pointwise" *)
Goalw [tsuffix_def,suffix_def]
"tsuffix s (ex2seq ex) ==> ? ex'. s = (ex2seq ex')";
by Auto_tac;
by (dtac ex2seqConc 1);
by Auto_tac;
qed"ex2seq_tsuffix";
(* FIX: NAch Sequence.ML bringen *)
Goal "(Map f$s = nil) = (s=nil)";
by (Seq_case_simp_tac "s" 1);
qed"Mapnil";
Goal "(Map f$s = UU) = (s=UU)";
by (Seq_case_simp_tac "s" 1);
qed"MapUU";
(* important property of cex_absSeq: As it is a 1to1 correspondence,
properties carry over *)
Goalw [tsuffix_def,suffix_def,cex_absSeq_def]
"tsuffix s t ==> tsuffix (cex_absSeq h s) (cex_absSeq h t)";
by Auto_tac;
by (asm_full_simp_tac (simpset() addsimps [Mapnil])1);
by (asm_full_simp_tac (simpset() addsimps [MapUU])1);
by (res_inst_tac [("x","Map (%(s,a,t). (h s,a, h t))$s1")] exI 1);
by (asm_full_simp_tac (simpset() addsimps [Map2Finite,MapConc])1);
qed"cex_absSeq_tsuffix";
Goalw [temp_strengthening_def,state_strengthening_def, temp_sat_def,
satisfies_def,Box_def]
"[| temp_strengthening P Q h |]\
\ ==> temp_strengthening ([] P) ([] Q) h";
by (clarify_tac set_cs 1);
by (ftac ex2seq_tsuffix 1);
by (clarify_tac set_cs 1);
by (dres_inst_tac [("h","h")] cex_absSeq_tsuffix 1);
by (asm_full_simp_tac (simpset() addsimps [ex2seq_abs_cex])1);
qed"strength_Box";
(* ------------------ Init ----------------------------*)
Goalw [temp_strengthening_def,state_strengthening_def,
temp_sat_def,satisfies_def,Init_def,unlift_def]
"[| state_strengthening P Q h |]\
\ ==> temp_strengthening (Init P) (Init Q) h";
by (safe_tac set_cs);
by (pair_tac "ex" 1);
by (Seq_case_simp_tac "y" 1);
by (pair_tac "a" 1);
qed"strength_Init";
(* ------------------ Next ----------------------------*)
Goal
"(TL$(ex2seq (cex_abs h ex))=UU) = (TL$(ex2seq ex)=UU)";
by (pair_tac "ex" 1);
by (Seq_case_simp_tac "y" 1);
by (pair_tac "a" 1);
by (Seq_case_simp_tac "s" 1);
by (pair_tac "a" 1);
qed"TL_ex2seq_UU";
Goal
"(TL$(ex2seq (cex_abs h ex))=nil) = (TL$(ex2seq ex)=nil)";
by (pair_tac "ex" 1);
by (Seq_case_simp_tac "y" 1);
by (pair_tac "a" 1);
by (Seq_case_simp_tac "s" 1);
by (pair_tac "a" 1);
qed"TL_ex2seq_nil";
(* FIX: put to Sequence Lemmas *)
Goal "Map f$(TL$s) = TL$(Map f$s)";
by (Seq_induct_tac "s" [] 1);
qed"MapTL";
(* important property of cex_absSeq: As it is a 1to1 correspondence,
properties carry over *)
Goalw [cex_absSeq_def]
"cex_absSeq h (TL$s) = (TL$(cex_absSeq h s))";
by (simp_tac (simpset() addsimps [MapTL]) 1);
qed"cex_absSeq_TL";
(* important property of ex2seq: can be shiftet, as defined "pointwise" *)
Goal "[| (snd ex)~=UU ; (snd ex)~=nil |] ==> (? ex'. TL$(ex2seq ex) = ex2seq ex')";
by (pair_tac "ex" 1);
by (Seq_case_simp_tac "y" 1);
by (pair_tac "a" 1);
by Auto_tac;
qed"TLex2seq";
Goal "(TL$(ex2seq ex)~=nil) = ((snd ex)~=nil & (snd ex)~=UU)";
by (pair_tac "ex" 1);
by (Seq_case_simp_tac "y" 1);
by (pair_tac "a" 1);
by (Seq_case_simp_tac "s" 1);
by (pair_tac "a" 1);
qed"ex2seqnilTL";
Goalw [temp_strengthening_def,state_strengthening_def,
temp_sat_def, satisfies_def,Next_def]
"[| temp_strengthening P Q h |]\
\ ==> temp_strengthening (Next P) (Next Q) h";
by (Asm_full_simp_tac 1);
by (safe_tac set_cs);
by (asm_full_simp_tac (simpset() addsimps [TL_ex2seq_nil,TL_ex2seq_UU]) 1);
by (asm_full_simp_tac (simpset() addsimps [TL_ex2seq_nil,TL_ex2seq_UU]) 1);
by (asm_full_simp_tac (simpset() addsimps [TL_ex2seq_nil,TL_ex2seq_UU]) 1);
by (asm_full_simp_tac (simpset() addsimps [TL_ex2seq_nil,TL_ex2seq_UU]) 1);
(* cons case *)
by (asm_full_simp_tac (simpset() addsimps [TL_ex2seq_nil,TL_ex2seq_UU,
ex2seq_abs_cex,cex_absSeq_TL RS sym, ex2seqnilTL])1);
by (etac conjE 1);
by (dtac TLex2seq 1);
by (assume_tac 1);
by Auto_tac;
qed"strength_Next";
(* ---------------------------------------------------------------- *)
(* Localizing Temporal Weakenings - 2 *)
(* ---------------------------------------------------------------- *)
Goal
"[| state_weakening P Q h |]\
\ ==> temp_weakening (Init P) (Init Q) h";
by (asm_full_simp_tac (simpset() addsimps [temp_weakening_def2,
state_weakening_def2, temp_sat_def,satisfies_def,Init_def,unlift_def])1);
by (safe_tac set_cs);
by (pair_tac "ex" 1);
by (Seq_case_simp_tac "y" 1);
by (pair_tac "a" 1);
qed"weak_Init";
(* ---------------------------------------------------------------- *)
(* Localizing Temproal Strengthenings - 3 *)
(* ---------------------------------------------------------------- *)
Goalw [Diamond_def]
"[| temp_strengthening P Q h |]\
\ ==> temp_strengthening (<> P) (<> Q) h";
by (rtac strength_NOT 1);
by (rtac weak_Box 1);
by (etac weak_NOT 1);
qed"strength_Diamond";
Goalw [Leadsto_def]
"[| temp_weakening P1 P2 h;\
\ temp_strengthening Q1 Q2 h |]\
\ ==> temp_strengthening (P1 ~> Q1) (P2 ~> Q2) h";
by (rtac strength_Box 1);
by (etac strength_IMPLIES 1);
by (etac strength_Diamond 1);
qed"strength_Leadsto";
(* ---------------------------------------------------------------- *)
(* Localizing Temporal Weakenings - 3 *)
(* ---------------------------------------------------------------- *)
Goalw [Diamond_def]
"[| temp_weakening P Q h |]\
\ ==> temp_weakening (<> P) (<> Q) h";
by (rtac weak_NOT 1);
by (rtac strength_Box 1);
by (etac strength_NOT 1);
qed"weak_Diamond";
Goalw [Leadsto_def]
"[| temp_strengthening P1 P2 h;\
\ temp_weakening Q1 Q2 h |]\
\ ==> temp_weakening (P1 ~> Q1) (P2 ~> Q2) h";
by (rtac weak_Box 1);
by (etac weak_IMPLIES 1);
by (etac weak_Diamond 1);
qed"weak_Leadsto";
Goalw [WF_def]
" !!A. [| !! s. Enabled A acts (h s) ==> Enabled C acts s|] \
\ ==> temp_weakening (WF A acts) (WF C acts) h";
by (rtac weak_IMPLIES 1);
by (rtac strength_Diamond 1);
by (rtac strength_Box 1);
by (rtac strength_Init 1);
by (rtac weak_Box 2);
by (rtac weak_Diamond 2);
by (rtac weak_Init 2);
by (auto_tac (claset(),
simpset() addsimps [state_weakening_def,state_strengthening_def,
xt2_def,plift_def,option_lift_def,NOT_def]));
qed"weak_WF";
Goalw [SF_def]
" !!A. [| !! s. Enabled A acts (h s) ==> Enabled C acts s|] \
\ ==> temp_weakening (SF A acts) (SF C acts) h";
by (rtac weak_IMPLIES 1);
by (rtac strength_Box 1);
by (rtac strength_Diamond 1);
by (rtac strength_Init 1);
by (rtac weak_Box 2);
by (rtac weak_Diamond 2);
by (rtac weak_Init 2);
by (auto_tac (claset(),
simpset() addsimps [state_weakening_def,state_strengthening_def,
xt2_def,plift_def,option_lift_def,NOT_def]));
qed"weak_SF";
val weak_strength_lemmas =
[weak_OR,weak_AND,weak_NOT,weak_IMPLIES,weak_Box,weak_Next,weak_Init,
weak_Diamond,weak_Leadsto,strength_OR,strength_AND,strength_NOT,
strength_IMPLIES,strength_Box,strength_Next,strength_Init,
strength_Diamond,strength_Leadsto,weak_WF,weak_SF];
fun abstraction_tac i =
SELECT_GOAL (auto_tac (claset() addSIs weak_strength_lemmas,
simpset() addsimps [state_strengthening_def,state_weakening_def])) i;