--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/HOLCF/IOA/Abstraction.thy Thu Dec 31 12:43:09 2015 +0100
@@ -0,0 +1,614 @@
+(* Title: HOL/HOLCF/IOA/Abstraction.thy
+ Author: Olaf Müller
+*)
+
+section \<open>Abstraction Theory -- tailored for I/O automata\<close>
+
+theory Abstraction
+imports LiveIOA
+begin
+
+default_sort type
+
+definition
+ cex_abs :: "('s1 => 's2) => ('a,'s1)execution => ('a,'s2)execution" where
+ "cex_abs f ex = (f (fst ex), Map (%(a,t). (a,f t))$(snd ex))"
+definition
+ \<comment> \<open>equals cex_abs on Sequences -- after ex2seq application\<close>
+ cex_absSeq :: "('s1 => 's2) => ('a option,'s1)transition Seq => ('a option,'s2)transition Seq" where
+ "cex_absSeq f = (%s. Map (%(s,a,t). (f s,a,f t))$s)"
+
+definition
+ is_abstraction ::"[('s1=>'s2),('a,'s1)ioa,('a,'s2)ioa] => bool" where
+ "is_abstraction f C A =
+ ((!s:starts_of(C). f(s):starts_of(A)) &
+ (!s t a. reachable C s & s \<midarrow>a\<midarrow>C\<rightarrow> t
+ --> (f s) \<midarrow>a\<midarrow>A\<rightarrow> (f t)))"
+
+definition
+ weakeningIOA :: "('a,'s2)ioa => ('a,'s1)ioa => ('s1 => 's2) => bool" where
+ "weakeningIOA A C h = (!ex. ex : executions C --> cex_abs h ex : executions A)"
+definition
+ temp_strengthening :: "('a,'s2)ioa_temp => ('a,'s1)ioa_temp => ('s1 => 's2) => bool" where
+ "temp_strengthening Q P h = (!ex. (cex_abs h ex \<TTurnstile> Q) --> (ex \<TTurnstile> P))"
+definition
+ temp_weakening :: "('a,'s2)ioa_temp => ('a,'s1)ioa_temp => ('s1 => 's2) => bool" where
+ "temp_weakening Q P h = temp_strengthening (\<^bold>\<not> Q) (\<^bold>\<not> P) h"
+
+definition
+ state_strengthening :: "('a,'s2)step_pred => ('a,'s1)step_pred => ('s1 => 's2) => bool" where
+ "state_strengthening Q P h = (!s t a. Q (h(s),a,h(t)) --> P (s,a,t))"
+definition
+ state_weakening :: "('a,'s2)step_pred => ('a,'s1)step_pred => ('s1 => 's2) => bool" where
+ "state_weakening Q P h = state_strengthening (\<^bold>\<not>Q) (\<^bold>\<not>P) h"
+
+definition
+ is_live_abstraction :: "('s1 => 's2) => ('a,'s1)live_ioa => ('a,'s2)live_ioa => bool" where
+ "is_live_abstraction h CL AM =
+ (is_abstraction h (fst CL) (fst AM) &
+ temp_weakening (snd AM) (snd CL) h)"
+
+
+axiomatization where
+(* thm about ex2seq which is not provable by induction as ex2seq is not continous *)
+ex2seq_abs_cex:
+ "ex2seq (cex_abs h ex) = cex_absSeq h (ex2seq ex)"
+
+axiomatization where
+(* analog to the proved thm strength_Box - proof skipped as trivial *)
+weak_Box:
+"temp_weakening P Q h
+ ==> temp_weakening (\<box>P) (\<box>Q) h"
+
+axiomatization where
+(* analog to the proved thm strength_Next - proof skipped as trivial *)
+weak_Next:
+"temp_weakening P Q h
+ ==> temp_weakening (Next P) (Next Q) h"
+
+
+subsection "cex_abs"
+
+lemma cex_abs_UU: "cex_abs f (s,UU) = (f s, UU)"
+ by (simp add: cex_abs_def)
+
+lemma cex_abs_nil: "cex_abs f (s,nil) = (f s, nil)"
+ by (simp add: cex_abs_def)
+
+lemma cex_abs_cons: "cex_abs f (s,(a,t)\<leadsto>ex) = (f s, (a,f t) \<leadsto> (snd (cex_abs f (t,ex))))"
+ by (simp add: cex_abs_def)
+
+declare cex_abs_UU [simp] cex_abs_nil [simp] cex_abs_cons [simp]
+
+
+subsection "lemmas"
+
+lemma temp_weakening_def2: "temp_weakening Q P h = (! ex. (ex \<TTurnstile> P) --> (cex_abs h ex \<TTurnstile> Q))"
+ apply (simp add: temp_weakening_def temp_strengthening_def NOT_def temp_sat_def satisfies_def)
+ apply auto
+ done
+
+lemma state_weakening_def2: "state_weakening Q P h = (! s t a. P (s,a,t) --> Q (h(s),a,h(t)))"
+ apply (simp add: state_weakening_def state_strengthening_def NOT_def)
+ apply auto
+ done
+
+
+subsection "Abstraction Rules for Properties"
+
+lemma exec_frag_abstraction [rule_format]:
+ "[| is_abstraction h C A |] ==>
+ !s. reachable C s & is_exec_frag C (s,xs)
+ --> is_exec_frag A (cex_abs h (s,xs))"
+apply (unfold cex_abs_def)
+apply simp
+apply (tactic \<open>pair_induct_tac @{context} "xs" [@{thm is_exec_frag_def}] 1\<close>)
+txt \<open>main case\<close>
+apply (auto dest: reachable.reachable_n simp add: is_abstraction_def)
+done
+
+
+lemma abs_is_weakening: "is_abstraction h C A ==> weakeningIOA A C h"
+apply (simp add: weakeningIOA_def)
+apply auto
+apply (simp add: executions_def)
+txt \<open>start state\<close>
+apply (rule conjI)
+apply (simp add: is_abstraction_def cex_abs_def)
+txt \<open>is-execution-fragment\<close>
+apply (erule exec_frag_abstraction)
+apply (simp add: reachable.reachable_0)
+done
+
+
+lemma AbsRuleT1: "[|is_abstraction h C A; validIOA A Q; temp_strengthening Q P h |]
+ ==> validIOA C P"
+apply (drule abs_is_weakening)
+apply (simp add: weakeningIOA_def validIOA_def temp_strengthening_def)
+apply (auto simp add: split_paired_all)
+done
+
+
+(* FIX: Nach TLS.ML *)
+
+lemma IMPLIES_temp_sat: "(ex \<TTurnstile> P \<^bold>\<longrightarrow> Q) = ((ex \<TTurnstile> P) --> (ex \<TTurnstile> Q))"
+ by (simp add: IMPLIES_def temp_sat_def satisfies_def)
+
+lemma AND_temp_sat: "(ex \<TTurnstile> P \<^bold>\<and> Q) = ((ex \<TTurnstile> P) & (ex \<TTurnstile> Q))"
+ by (simp add: AND_def temp_sat_def satisfies_def)
+
+lemma OR_temp_sat: "(ex \<TTurnstile> P \<^bold>\<or> Q) = ((ex \<TTurnstile> P) | (ex \<TTurnstile> Q))"
+ by (simp add: OR_def temp_sat_def satisfies_def)
+
+lemma NOT_temp_sat: "(ex \<TTurnstile> \<^bold>\<not> P) = (~ (ex \<TTurnstile> P))"
+ by (simp add: NOT_def temp_sat_def satisfies_def)
+
+declare IMPLIES_temp_sat [simp] AND_temp_sat [simp] OR_temp_sat [simp] NOT_temp_sat [simp]
+
+
+lemma AbsRuleT2:
+ "[|is_live_abstraction h (C,L) (A,M);
+ validLIOA (A,M) Q; temp_strengthening Q P h |]
+ ==> validLIOA (C,L) P"
+apply (unfold is_live_abstraction_def)
+apply auto
+apply (drule abs_is_weakening)
+apply (simp add: weakeningIOA_def temp_weakening_def2 validLIOA_def validIOA_def temp_strengthening_def)
+apply (auto simp add: split_paired_all)
+done
+
+
+lemma AbsRuleTImprove:
+ "[|is_live_abstraction h (C,L) (A,M);
+ validLIOA (A,M) (H1 \<^bold>\<longrightarrow> Q); temp_strengthening Q P h;
+ temp_weakening H1 H2 h; validLIOA (C,L) H2 |]
+ ==> validLIOA (C,L) P"
+apply (unfold is_live_abstraction_def)
+apply auto
+apply (drule abs_is_weakening)
+apply (simp add: weakeningIOA_def temp_weakening_def2 validLIOA_def validIOA_def temp_strengthening_def)
+apply (auto simp add: split_paired_all)
+done
+
+
+subsection "Correctness of safe abstraction"
+
+lemma abstraction_is_ref_map:
+"is_abstraction h C A ==> is_ref_map h C A"
+apply (unfold is_abstraction_def is_ref_map_def)
+apply auto
+apply (rule_tac x = "(a,h t) \<leadsto>nil" in exI)
+apply (simp add: move_def)
+done
+
+
+lemma abs_safety: "[| inp(C)=inp(A); out(C)=out(A);
+ is_abstraction h C A |]
+ ==> C =<| A"
+apply (simp add: ioa_implements_def)
+apply (rule trace_inclusion)
+apply (simp (no_asm) add: externals_def)
+apply (auto)[1]
+apply (erule abstraction_is_ref_map)
+done
+
+
+subsection "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 *)
+lemma traces_coincide_abs:
+ "ext C = ext A
+ ==> mk_trace C$xs = mk_trace A$(snd (cex_abs f (s,xs)))"
+apply (unfold cex_abs_def mk_trace_def filter_act_def)
+apply simp
+apply (tactic \<open>pair_induct_tac @{context} "xs" [] 1\<close>)
+done
+
+
+(* 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 *)
+lemma abs_liveness: "[| inp(C)=inp(A); out(C)=out(A);
+ is_live_abstraction h (C,M) (A,L) |]
+ ==> live_implements (C,M) (A,L)"
+apply (simp add: is_live_abstraction_def live_implements_def livetraces_def liveexecutions_def)
+apply auto
+apply (rule_tac x = "cex_abs h ex" in exI)
+apply auto
+ (* Traces coincide *)
+ apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
+ apply (rule traces_coincide_abs)
+ apply (simp (no_asm) add: externals_def)
+ apply (auto)[1]
+
+ (* cex_abs is execution *)
+ apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
+ apply (simp add: executions_def)
+ (* start state *)
+ apply (rule conjI)
+ apply (simp add: is_abstraction_def cex_abs_def)
+ (* is-execution-fragment *)
+ apply (erule exec_frag_abstraction)
+ apply (simp add: reachable.reachable_0)
+
+ (* Liveness *)
+apply (simp add: temp_weakening_def2)
+ apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
+done
+
+(* FIX: NAch Traces.ML bringen *)
+
+lemma implements_trans:
+"[| A =<| B; B =<| C|] ==> A =<| C"
+by (auto simp add: ioa_implements_def)
+
+
+subsection "Abstraction Rules for Automata"
+
+lemma AbsRuleA1: "[| 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"
+apply (drule abs_safety)
+apply assumption+
+apply (drule abs_safety)
+apply assumption+
+apply (erule implements_trans)
+apply (erule implements_trans)
+apply assumption
+done
+
+
+lemma AbsRuleA2: "!!LC. [| 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)"
+apply (drule abs_liveness)
+apply assumption+
+apply (drule abs_liveness)
+apply assumption+
+apply (erule live_implements_trans)
+apply (erule live_implements_trans)
+apply assumption
+done
+
+
+declare split_paired_All [simp del]
+
+
+subsection "Localizing Temporal Strengthenings and Weakenings"
+
+lemma strength_AND:
+"[| temp_strengthening P1 Q1 h;
+ temp_strengthening P2 Q2 h |]
+ ==> temp_strengthening (P1 \<^bold>\<and> P2) (Q1 \<^bold>\<and> Q2) h"
+apply (unfold temp_strengthening_def)
+apply auto
+done
+
+lemma strength_OR:
+"[| temp_strengthening P1 Q1 h;
+ temp_strengthening P2 Q2 h |]
+ ==> temp_strengthening (P1 \<^bold>\<or> P2) (Q1 \<^bold>\<or> Q2) h"
+apply (unfold temp_strengthening_def)
+apply auto
+done
+
+lemma strength_NOT:
+"[| temp_weakening P Q h |]
+ ==> temp_strengthening (\<^bold>\<not> P) (\<^bold>\<not> Q) h"
+apply (unfold temp_strengthening_def)
+apply (simp add: temp_weakening_def2)
+apply auto
+done
+
+lemma strength_IMPLIES:
+"[| temp_weakening P1 Q1 h;
+ temp_strengthening P2 Q2 h |]
+ ==> temp_strengthening (P1 \<^bold>\<longrightarrow> P2) (Q1 \<^bold>\<longrightarrow> Q2) h"
+apply (unfold temp_strengthening_def)
+apply (simp add: temp_weakening_def2)
+done
+
+
+lemma weak_AND:
+"[| temp_weakening P1 Q1 h;
+ temp_weakening P2 Q2 h |]
+ ==> temp_weakening (P1 \<^bold>\<and> P2) (Q1 \<^bold>\<and> Q2) h"
+apply (simp add: temp_weakening_def2)
+done
+
+lemma weak_OR:
+"[| temp_weakening P1 Q1 h;
+ temp_weakening P2 Q2 h |]
+ ==> temp_weakening (P1 \<^bold>\<or> P2) (Q1 \<^bold>\<or> Q2) h"
+apply (simp add: temp_weakening_def2)
+done
+
+lemma weak_NOT:
+"[| temp_strengthening P Q h |]
+ ==> temp_weakening (\<^bold>\<not> P) (\<^bold>\<not> Q) h"
+apply (unfold temp_strengthening_def)
+apply (simp add: temp_weakening_def2)
+apply auto
+done
+
+lemma weak_IMPLIES:
+"[| temp_strengthening P1 Q1 h;
+ temp_weakening P2 Q2 h |]
+ ==> temp_weakening (P1 \<^bold>\<longrightarrow> P2) (Q1 \<^bold>\<longrightarrow> Q2) h"
+apply (unfold temp_strengthening_def)
+apply (simp add: temp_weakening_def2)
+done
+
+
+subsubsection \<open>Box\<close>
+
+(* FIX: should be same as nil_is_Conc2 when all nils are turned to right side !! *)
+lemma UU_is_Conc: "(UU = x @@ y) = (((x::'a Seq)= UU) | (x=nil & y=UU))"
+apply (tactic \<open>Seq_case_simp_tac @{context} "x" 1\<close>)
+done
+
+lemma ex2seqConc [rule_format]:
+"Finite s1 -->
+ (! ex. (s~=nil & s~=UU & ex2seq ex = s1 @@ s) --> (? ex'. s = ex2seq ex'))"
+apply (rule impI)
+apply (tactic \<open>Seq_Finite_induct_tac @{context} 1\<close>)
+apply blast
+(* main case *)
+apply clarify
+apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
+apply (tactic \<open>Seq_case_simp_tac @{context} "x2" 1\<close>)
+(* UU case *)
+apply (simp add: nil_is_Conc)
+(* nil case *)
+apply (simp add: nil_is_Conc)
+(* cons case *)
+apply (tactic \<open>pair_tac @{context} "aa" 1\<close>)
+apply auto
+done
+
+(* important property of ex2seq: can be shiftet, as defined "pointwise" *)
+
+lemma ex2seq_tsuffix:
+"tsuffix s (ex2seq ex) ==> ? ex'. s = (ex2seq ex')"
+apply (unfold tsuffix_def suffix_def)
+apply auto
+apply (drule ex2seqConc)
+apply auto
+done
+
+
+(* FIX: NAch Sequence.ML bringen *)
+
+lemma Mapnil: "(Map f$s = nil) = (s=nil)"
+apply (tactic \<open>Seq_case_simp_tac @{context} "s" 1\<close>)
+done
+
+lemma MapUU: "(Map f$s = UU) = (s=UU)"
+apply (tactic \<open>Seq_case_simp_tac @{context} "s" 1\<close>)
+done
+
+
+(* important property of cex_absSeq: As it is a 1to1 correspondence,
+ properties carry over *)
+
+lemma cex_absSeq_tsuffix:
+"tsuffix s t ==> tsuffix (cex_absSeq h s) (cex_absSeq h t)"
+apply (unfold tsuffix_def suffix_def cex_absSeq_def)
+apply auto
+apply (simp add: Mapnil)
+apply (simp add: MapUU)
+apply (rule_tac x = "Map (% (s,a,t) . (h s,a, h t))$s1" in exI)
+apply (simp add: Map2Finite MapConc)
+done
+
+
+lemma strength_Box:
+"[| temp_strengthening P Q h |]
+ ==> temp_strengthening (\<box>P) (\<box>Q) h"
+apply (unfold temp_strengthening_def state_strengthening_def temp_sat_def satisfies_def Box_def)
+apply clarify
+apply (frule ex2seq_tsuffix)
+apply clarify
+apply (drule_tac h = "h" in cex_absSeq_tsuffix)
+apply (simp add: ex2seq_abs_cex)
+done
+
+
+subsubsection \<open>Init\<close>
+
+lemma strength_Init:
+"[| state_strengthening P Q h |]
+ ==> temp_strengthening (Init P) (Init Q) h"
+apply (unfold temp_strengthening_def state_strengthening_def
+ temp_sat_def satisfies_def Init_def unlift_def)
+apply auto
+apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
+apply (tactic \<open>Seq_case_simp_tac @{context} "x2" 1\<close>)
+apply (tactic \<open>pair_tac @{context} "a" 1\<close>)
+done
+
+
+subsubsection \<open>Next\<close>
+
+lemma TL_ex2seq_UU:
+"(TL$(ex2seq (cex_abs h ex))=UU) = (TL$(ex2seq ex)=UU)"
+apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
+apply (tactic \<open>Seq_case_simp_tac @{context} "x2" 1\<close>)
+apply (tactic \<open>pair_tac @{context} "a" 1\<close>)
+apply (tactic \<open>Seq_case_simp_tac @{context} "s" 1\<close>)
+apply (tactic \<open>pair_tac @{context} "a" 1\<close>)
+done
+
+lemma TL_ex2seq_nil:
+"(TL$(ex2seq (cex_abs h ex))=nil) = (TL$(ex2seq ex)=nil)"
+apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
+apply (tactic \<open>Seq_case_simp_tac @{context} "x2" 1\<close>)
+apply (tactic \<open>pair_tac @{context} "a" 1\<close>)
+apply (tactic \<open>Seq_case_simp_tac @{context} "s" 1\<close>)
+apply (tactic \<open>pair_tac @{context} "a" 1\<close>)
+done
+
+(* FIX: put to Sequence Lemmas *)
+lemma MapTL: "Map f$(TL$s) = TL$(Map f$s)"
+apply (tactic \<open>Seq_induct_tac @{context} "s" [] 1\<close>)
+done
+
+(* important property of cex_absSeq: As it is a 1to1 correspondence,
+ properties carry over *)
+
+lemma cex_absSeq_TL:
+"cex_absSeq h (TL$s) = (TL$(cex_absSeq h s))"
+apply (unfold cex_absSeq_def)
+apply (simp add: MapTL)
+done
+
+(* important property of ex2seq: can be shiftet, as defined "pointwise" *)
+
+lemma TLex2seq: "[| (snd ex)~=UU ; (snd ex)~=nil |] ==> (? ex'. TL$(ex2seq ex) = ex2seq ex')"
+apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
+apply (tactic \<open>Seq_case_simp_tac @{context} "x2" 1\<close>)
+apply (tactic \<open>pair_tac @{context} "a" 1\<close>)
+apply auto
+done
+
+
+lemma ex2seqnilTL: "(TL$(ex2seq ex)~=nil) = ((snd ex)~=nil & (snd ex)~=UU)"
+apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
+apply (tactic \<open>Seq_case_simp_tac @{context} "x2" 1\<close>)
+apply (tactic \<open>pair_tac @{context} "a" 1\<close>)
+apply (tactic \<open>Seq_case_simp_tac @{context} "s" 1\<close>)
+apply (tactic \<open>pair_tac @{context} "a" 1\<close>)
+done
+
+
+lemma strength_Next:
+"[| temp_strengthening P Q h |]
+ ==> temp_strengthening (Next P) (Next Q) h"
+apply (unfold temp_strengthening_def state_strengthening_def temp_sat_def satisfies_def Next_def)
+apply simp
+apply auto
+apply (simp add: TL_ex2seq_nil TL_ex2seq_UU)
+apply (simp add: TL_ex2seq_nil TL_ex2seq_UU)
+apply (simp add: TL_ex2seq_nil TL_ex2seq_UU)
+apply (simp add: TL_ex2seq_nil TL_ex2seq_UU)
+(* cons case *)
+apply (simp add: TL_ex2seq_nil TL_ex2seq_UU ex2seq_abs_cex cex_absSeq_TL [symmetric] ex2seqnilTL)
+apply (erule conjE)
+apply (drule TLex2seq)
+apply assumption
+apply auto
+done
+
+
+text \<open>Localizing Temporal Weakenings - 2\<close>
+
+lemma weak_Init:
+"[| state_weakening P Q h |]
+ ==> temp_weakening (Init P) (Init Q) h"
+apply (simp add: temp_weakening_def2 state_weakening_def2
+ temp_sat_def satisfies_def Init_def unlift_def)
+apply auto
+apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
+apply (tactic \<open>Seq_case_simp_tac @{context} "x2" 1\<close>)
+apply (tactic \<open>pair_tac @{context} "a" 1\<close>)
+done
+
+
+text \<open>Localizing Temproal Strengthenings - 3\<close>
+
+lemma strength_Diamond:
+"[| temp_strengthening P Q h |]
+ ==> temp_strengthening (\<diamond>P) (\<diamond>Q) h"
+apply (unfold Diamond_def)
+apply (rule strength_NOT)
+apply (rule weak_Box)
+apply (erule weak_NOT)
+done
+
+lemma strength_Leadsto:
+"[| temp_weakening P1 P2 h;
+ temp_strengthening Q1 Q2 h |]
+ ==> temp_strengthening (P1 \<leadsto> Q1) (P2 \<leadsto> Q2) h"
+apply (unfold Leadsto_def)
+apply (rule strength_Box)
+apply (erule strength_IMPLIES)
+apply (erule strength_Diamond)
+done
+
+
+text \<open>Localizing Temporal Weakenings - 3\<close>
+
+lemma weak_Diamond:
+"[| temp_weakening P Q h |]
+ ==> temp_weakening (\<diamond>P) (\<diamond>Q) h"
+apply (unfold Diamond_def)
+apply (rule weak_NOT)
+apply (rule strength_Box)
+apply (erule strength_NOT)
+done
+
+lemma weak_Leadsto:
+"[| temp_strengthening P1 P2 h;
+ temp_weakening Q1 Q2 h |]
+ ==> temp_weakening (P1 \<leadsto> Q1) (P2 \<leadsto> Q2) h"
+apply (unfold Leadsto_def)
+apply (rule weak_Box)
+apply (erule weak_IMPLIES)
+apply (erule weak_Diamond)
+done
+
+lemma weak_WF:
+ " !!A. [| !! s. Enabled A acts (h s) ==> Enabled C acts s|]
+ ==> temp_weakening (WF A acts) (WF C acts) h"
+apply (unfold WF_def)
+apply (rule weak_IMPLIES)
+apply (rule strength_Diamond)
+apply (rule strength_Box)
+apply (rule strength_Init)
+apply (rule_tac [2] weak_Box)
+apply (rule_tac [2] weak_Diamond)
+apply (rule_tac [2] weak_Init)
+apply (auto simp add: state_weakening_def state_strengthening_def
+ xt2_def plift_def option_lift_def NOT_def)
+done
+
+lemma weak_SF:
+ " !!A. [| !! s. Enabled A acts (h s) ==> Enabled C acts s|]
+ ==> temp_weakening (SF A acts) (SF C acts) h"
+apply (unfold SF_def)
+apply (rule weak_IMPLIES)
+apply (rule strength_Box)
+apply (rule strength_Diamond)
+apply (rule strength_Init)
+apply (rule_tac [2] weak_Box)
+apply (rule_tac [2] weak_Diamond)
+apply (rule_tac [2] weak_Init)
+apply (auto simp add: state_weakening_def state_strengthening_def
+ xt2_def plift_def option_lift_def NOT_def)
+done
+
+
+lemmas 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
+
+ML \<open>
+fun abstraction_tac ctxt =
+ SELECT_GOAL (auto_tac
+ (ctxt addSIs @{thms weak_strength_lemmas}
+ addsimps [@{thm state_strengthening_def}, @{thm state_weakening_def}]))
+\<close>
+
+method_setup abstraction = \<open>Scan.succeed (SIMPLE_METHOD' o abstraction_tac)\<close>
+
+end