--- a/src/HOLCF/IOA/meta_theory/Abstraction.ML Mon Jun 22 17:12:27 1998 +0200
+++ b/src/HOLCF/IOA/meta_theory/Abstraction.ML Mon Jun 22 17:13:09 1998 +0200
@@ -14,15 +14,15 @@
(* cex_abs *)
(* ---------------------------------------------------------------- *)
-goal thy "cex_abs f (s,UU) = (f s, UU)";
+Goal "cex_abs f (s,UU) = (f s, UU)";
by (simp_tac (simpset() addsimps [cex_abs_def]) 1);
qed"cex_abs_UU";
-goal thy "cex_abs f (s,nil) = (f s, nil)";
+Goal "cex_abs f (s,nil) = (f s, nil)";
by (simp_tac (simpset() addsimps [cex_abs_def]) 1);
qed"cex_abs_nil";
-goal thy "cex_abs f (s,(a,t)>>ex) = (f s, (a,f t) >> (snd (cex_abs f (t,ex))))";
+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";
@@ -36,13 +36,13 @@
(* Lemmas *)
(* ---------------------------------------------------------------- *)
-goal thy "temp_weakening Q P h = (! ex. (ex |== P) --> (cex_abs h ex |== Q))";
+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);
auto();
qed"temp_weakening_def2";
-goal thy "state_weakening Q P h = (! s t a. P (s,a,t) --> Q (h(s),a,h(t)))";
+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);
auto();
@@ -56,7 +56,7 @@
(* ---------------------------------------------------------------- *)
-goalw thy [cex_abs_def]
+Goalw [cex_abs_def]
"!!h.[| is_abstraction h C A |] ==>\
\ !s. reachable C s & is_exec_frag C (s,xs) \
\ --> is_exec_frag A (cex_abs h (s,xs))";
@@ -72,7 +72,7 @@
qed"exec_frag_abstraction";
-goal thy "!!A. is_abstraction h C A ==> weakeningIOA A C h";
+Goal "!!A. is_abstraction h C A ==> weakeningIOA A C h";
by (asm_full_simp_tac (simpset() addsimps [weakeningIOA_def])1);
auto();
by (asm_full_simp_tac (simpset() addsimps [executions_def]) 1);
@@ -85,7 +85,7 @@
qed"abs_is_weakening";
-goal thy "!!A. [|is_abstraction h C A; validIOA A Q; temp_strengthening Q P h |] \
+Goal "!!A. [|is_abstraction h C A; validIOA A Q; temp_strengthening Q P h |] \
\ ==> validIOA C P";
bd abs_is_weakening 1;
by (asm_full_simp_tac (simpset() addsimps [weakeningIOA_def,
@@ -97,26 +97,26 @@
(* FIX: Nach TLS.ML *)
-goal thy "(ex |== P .--> Q) = ((ex |== P) --> (ex |== Q))";
+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 thy "(ex |== P .& Q) = ((ex |== P) & (ex |== Q))";
+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 thy "(ex |== P .| Q) = ((ex |== P) | (ex |== Q))";
+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 thy "(ex |== .~ P) = (~ (ex |== P))";
+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 thy [is_live_abstraction_def]
+Goalw [is_live_abstraction_def]
"!!A. [|is_live_abstraction h (C,L) (A,M); \
\ validLIOA (A,M) Q; temp_strengthening Q P h |] \
\ ==> validLIOA (C,L) P";
@@ -129,7 +129,7 @@
qed"AbsRuleT2";
-goalw thy [is_live_abstraction_def]
+Goalw [is_live_abstraction_def]
"!!A. [|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 |] \
@@ -150,7 +150,7 @@
(* ---------------------------------------------------------------- *)
-goalw thy [is_abstraction_def,is_ref_map_def]
+Goalw [is_abstraction_def,is_ref_map_def]
"!! h. 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);
@@ -158,7 +158,7 @@
qed"abstraction_is_ref_map";
-goal thy "!! h. [| inp(C)=inp(A); out(C)=out(A); \
+Goal "!! h. [| 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);
@@ -178,7 +178,7 @@
(* Reduces to Filter (Map fst x) = Filter (Map fst (Map (%(a,t). (a,x)) x),
that is to special Map Lemma *)
-goalw thy [cex_abs_def,mk_trace_def,filter_act_def]
+Goalw [cex_abs_def,mk_trace_def,filter_act_def]
"!! f. ext C = ext A \
\ ==> mk_trace C`xs = mk_trace A`(snd (cex_abs f (s,xs)))";
by (Asm_full_simp_tac 1);
@@ -186,7 +186,7 @@
qed"traces_coincide_abs";
-goalw thy [cex_abs_def]
+Goalw [cex_abs_def]
"!!f.[| is_abstraction h C A |] ==>\
\ !s. reachable C s & is_exec_frag C (s,xs) \
\ --> is_exec_frag A (cex_abs h (s,xs))";
@@ -210,7 +210,7 @@
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 thy "!! h. [| inp(C)=inp(A); out(C)=out(A); \
+Goal "!! h. [| inp(C)=inp(A); out(C)=out(A); \
\ is_live_abstraction h (C,M) (A,L) |] \
\ ==> live_implements (C,M) (A,L)";
@@ -242,7 +242,7 @@
(* FIX: NAch Traces.ML bringen *)
-goalw thy [ioa_implements_def]
+Goalw [ioa_implements_def]
"!! A. [| A =<| B; B =<| C|] ==> A =<| C";
auto();
qed"implements_trans";
@@ -254,7 +254,7 @@
(* Abstraction Rules for Automata *)
(* ---------------------------------------------------------------- *)
-goal thy "!! C. [| inp(C)=inp(A); out(C)=out(A); \
+Goal "!! C. [| inp(C)=inp(A); out(C)=out(A); \
\ inp(Q)=inp(P); out(Q)=out(P); \
\ is_abstraction h1 C A; \
\ A =<| Q ; \
@@ -270,7 +270,7 @@
qed"AbsRuleA1";
-goal thy "!! C. [| inp(C)=inp(A); out(C)=out(A); \
+Goal "!! C. [| 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) ; \
@@ -295,28 +295,28 @@
(* Localizing Temproal Strengthenings - 1 *)
(* ---------------------------------------------------------------- *)
-goalw thy [temp_strengthening_def]
+Goalw [temp_strengthening_def]
"!! h. [| temp_strengthening P1 Q1 h; \
\ temp_strengthening P2 Q2 h |] \
\ ==> temp_strengthening (P1 .& P2) (Q1 .& Q2) h";
auto();
qed"strength_AND";
-goalw thy [temp_strengthening_def]
+Goalw [temp_strengthening_def]
"!! h. [| temp_strengthening P1 Q1 h; \
\ temp_strengthening P2 Q2 h |] \
\ ==> temp_strengthening (P1 .| P2) (Q1 .| Q2) h";
auto();
qed"strength_OR";
-goalw thy [temp_strengthening_def]
+Goalw [temp_strengthening_def]
"!! h. [| temp_weakening P Q h |] \
\ ==> temp_strengthening (.~ P) (.~ Q) h";
by (asm_full_simp_tac (simpset() addsimps [temp_weakening_def2])1);
auto();
qed"strength_NOT";
-goalw thy [temp_strengthening_def]
+Goalw [temp_strengthening_def]
"!! h. [| temp_weakening P1 Q1 h; \
\ temp_strengthening P2 Q2 h |] \
\ ==> temp_strengthening (P1 .--> P2) (Q1 .--> Q2) h";
@@ -329,28 +329,28 @@
(* Localizing Temproal Weakenings - Part 1 *)
(* ---------------------------------------------------------------- *)
-goal thy
+Goal
"!! h. [| 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 thy
+Goal
"!! h. [| 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 thy [temp_strengthening_def]
+Goalw [temp_strengthening_def]
"!! h. [| temp_strengthening P Q h |] \
\ ==> temp_weakening (.~ P) (.~ Q) h";
by (asm_full_simp_tac (simpset() addsimps [temp_weakening_def2])1);
auto();
qed"weak_NOT";
-goalw thy [temp_strengthening_def]
+Goalw [temp_strengthening_def]
"!! h. [| temp_strengthening P1 Q1 h; \
\ temp_weakening P2 Q2 h |] \
\ ==> temp_weakening (P1 .--> P2) (Q1 .--> Q2) h";
@@ -366,12 +366,12 @@
(* ------------------ Box ----------------------------*)
(* FIX: should be same as nil_is_Conc2 when all nils are turned to right side !! *)
-goal thy "(UU = x @@ y) = (((x::'a Seq)= UU) | (x=nil & y=UU))";
+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 thy
+Goal
"Finite s1 --> \
\ (! ex. (s~=nil & s~=UU & ex2seq ex = s1 @@ s) --> (? ex'. s = ex2seq ex'))";
by (rtac impI 1);
@@ -392,7 +392,7 @@
(* important property of ex2seq: can be shiftet, as defined "pointwise" *)
-goalw thy [tsuffix_def,suffix_def]
+Goalw [tsuffix_def,suffix_def]
"!!s. tsuffix s (ex2seq ex) ==> ? ex'. s = (ex2seq ex')";
auto();
bd ex2seqConc 1;
@@ -400,11 +400,11 @@
qed"ex2seq_tsuffix";
-goal thy "(Map f`s = nil) = (s=nil)";
+Goal "(Map f`s = nil) = (s=nil)";
by (Seq_case_simp_tac "s" 1);
qed"Mapnil";
-goal thy "(Map f`s = UU) = (s=UU)";
+Goal "(Map f`s = UU) = (s=UU)";
by (Seq_case_simp_tac "s" 1);
qed"MapUU";
@@ -412,7 +412,7 @@
(* important property of cex_absSeq: As it is a 1to1 correspondence,
properties carry over *)
-goalw thy [tsuffix_def,suffix_def,cex_absSeq_def]
+Goalw [tsuffix_def,suffix_def,cex_absSeq_def]
"!! s. tsuffix s t ==> tsuffix (cex_absSeq h s) (cex_absSeq h t)";
auto();
by (asm_full_simp_tac (simpset() addsimps [Mapnil])1);
@@ -422,7 +422,7 @@
qed"cex_absSeq_tsuffix";
-goalw thy [temp_strengthening_def,state_strengthening_def, temp_sat_def,
+Goalw [temp_strengthening_def,state_strengthening_def, temp_sat_def,
satisfies_def,Box_def]
"!! h. [| temp_strengthening P Q h |]\
\ ==> temp_strengthening ([] P) ([] Q) h";
@@ -436,7 +436,7 @@
(* ------------------ Init ----------------------------*)
-goalw thy [temp_strengthening_def,state_strengthening_def,
+Goalw [temp_strengthening_def,state_strengthening_def,
temp_sat_def,satisfies_def,Init_def,unlift_def]
"!! h. [| state_strengthening P Q h |]\
\ ==> temp_strengthening (Init P) (Init Q) h";
@@ -449,7 +449,7 @@
(* ------------------ Next ----------------------------*)
-goal thy
+Goal
"(TL`(ex2seq (cex_abs h ex))=UU) = (TL`(ex2seq ex)=UU)";
by (pair_tac "ex" 1);
by (Seq_case_simp_tac "y" 1);
@@ -458,7 +458,7 @@
by (pair_tac "a" 1);
qed"TL_ex2seq_UU";
-goal thy
+Goal
"(TL`(ex2seq (cex_abs h ex))=nil) = (TL`(ex2seq ex)=nil)";
by (pair_tac "ex" 1);
by (Seq_case_simp_tac "y" 1);
@@ -468,21 +468,21 @@
qed"TL_ex2seq_nil";
(* FIX: put to Sequence Lemmas *)
-goal thy "Map f`(TL`s) = TL`(Map f`s)";
+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 thy [cex_absSeq_def]
+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 thy "!!ex. [| (snd ex)~=UU ; (snd ex)~=nil |] ==> (? ex'. TL`(ex2seq ex) = ex2seq ex')";
+Goal "!!ex. [| (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);
@@ -490,7 +490,7 @@
qed"TLex2seq";
-goal thy "(TL`(ex2seq ex)~=UU) = ((snd ex)~=UU)";
+Goal "(TL`(ex2seq ex)~=UU) = ((snd ex)~=UU)";
by (pair_tac "ex" 1);
by (Seq_case_simp_tac "y" 1);
by (pair_tac "a" 1);
@@ -498,7 +498,7 @@
by (pair_tac "a" 1);
qed"ex2seqUUTL";
-goal thy "(TL`(ex2seq ex)~=nil) = ((snd ex)~=nil)";
+Goal "(TL`(ex2seq ex)~=nil) = ((snd ex)~=nil)";
by (pair_tac "ex" 1);
by (Seq_case_simp_tac "y" 1);
by (pair_tac "a" 1);
@@ -507,7 +507,7 @@
qed"ex2seqnilTL";
-goalw thy [temp_strengthening_def,state_strengthening_def,
+Goalw [temp_strengthening_def,state_strengthening_def,
temp_sat_def, satisfies_def,Next_def]
"!! h. [| temp_strengthening P Q h |]\
\ ==> temp_strengthening (Next P) (Next Q) h";
@@ -532,7 +532,7 @@
(* ---------------------------------------------------------------- *)
-goal thy
+Goal
"!! h. [| state_weakening P Q h |]\
\ ==> temp_weakening (Init P) (Init Q) h";
by (asm_full_simp_tac (simpset() addsimps [temp_weakening_def2,
@@ -548,7 +548,7 @@
(* analog to strengthening thm above, with analog lemmas used *)
-goalw thy [state_weakening_def]
+Goalw [state_weakening_def]
"!! h. [| temp_weakening P Q h |]\
\ ==> temp_weakening ([] P) ([] Q) h";
by (asm_full_simp_tac (simpset() addsimps [temp_weakening_def2,
@@ -556,7 +556,7 @@
(* analog to strengthening thm above, with analog lemmas used *)
-goalw thy [state_weakening_def]
+Goalw [state_weakening_def]
"!! h. [| temp_weakening P Q h |]\
\ ==> temp_weakening (Next P) (Next Q) h";
by (asm_full_simp_tac (simpset() addsimps [temp_weakening_def2,
@@ -569,7 +569,7 @@
(* ---------------------------------------------------------------- *)
-goalw thy [Diamond_def]
+Goalw [Diamond_def]
"!! h. [| temp_strengthening P Q h |]\
\ ==> temp_strengthening (<> P) (<> Q) h";
br strength_NOT 1;
@@ -577,7 +577,7 @@
be weak_NOT 1;
qed"strength_Diamond";
-goalw thy [Leadsto_def]
+Goalw [Leadsto_def]
"!! h. [| temp_weakening P1 P2 h;\
\ temp_strengthening Q1 Q2 h |]\
\ ==> temp_strengthening (P1 ~> Q1) (P2 ~> Q2) h";
@@ -592,7 +592,7 @@
(* ---------------------------------------------------------------- *)
-goalw thy [Diamond_def]
+Goalw [Diamond_def]
"!! h. [| temp_weakening P Q h |]\
\ ==> temp_weakening (<> P) (<> Q) h";
br weak_NOT 1;
@@ -600,7 +600,7 @@
be strength_NOT 1;
qed"weak_Diamond";
-goalw thy [Leadsto_def]
+Goalw [Leadsto_def]
"!! h. [| temp_strengthening P1 P2 h;\
\ temp_weakening Q1 Q2 h |]\
\ ==> temp_weakening (P1 ~> Q1) (P2 ~> Q2) h";
@@ -609,7 +609,7 @@
be weak_Diamond 1;
qed"weak_Leadsto";
-goalw thy [WF_def]
+Goalw [WF_def]
" !!A. [| !! s. Enabled A acts (h s) ==> Enabled C acts s|] \
\ ==> temp_weakening (WF A acts) (WF C acts) h";
br weak_IMPLIES 1;
@@ -624,7 +624,7 @@
xt2_def,plift_def,option_lift_def,NOT_def]));
qed"weak_WF";
-goalw thy [SF_def]
+Goalw [SF_def]
" !!A. [| !! s. Enabled A acts (h s) ==> Enabled C acts s|] \
\ ==> temp_weakening (SF A acts) (SF C acts) h";
br weak_IMPLIES 1;