author  wenzelm 
Tue, 16 Jan 2018 09:58:17 +0100  
changeset 67445  4311845b0412 
parent 67443  3abf6a722518 
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(* Title: ZF/UNITY/Mutex.thy 
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Author: Sidi O Ehmety, Computer Laboratory 
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Copyright 2001 University of Cambridge 

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Based on "A Family of 2Process Mutual Exclusion Algorithms" by J Misra. 
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Variables' types are introduced globally so that type verification 

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reduces to the usual ZF typechecking \<in> an illtyed expression will 

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reduce to the empty set. 

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*) 
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section\<open>Mutual Exclusion\<close> 
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theory Mutex 

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imports SubstAx 

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begin 

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text\<open>Based on "A Family of 2Process Mutual Exclusion Algorithms" by J Misra 
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Variables' types are introduced globally so that type verification reduces to 

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the usual ZF typechecking: an illtyed expressions reduce to the empty set. 

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\<close> 
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abbreviation "p == Var([0])" 
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abbreviation "m == Var([1])" 

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abbreviation "n == Var([0,0])" 

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abbreviation "u == Var([0,1])" 

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abbreviation "v == Var([1,0])" 

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axiomatization where \<comment> \<open>Type declarations\<close> 
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p_type: "type_of(p)=bool & default_val(p)=0" and 
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m_type: "type_of(m)=int & default_val(m)=#0" and 

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n_type: "type_of(n)=int & default_val(n)=#0" and 

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u_type: "type_of(u)=bool & default_val(u)=0" and 

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v_type: "type_of(v)=bool & default_val(v)=0" 
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definition 
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(** The program for process U **) 
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"U0 == {<s,t>:state*state. t = s(u:=1, m:=#1) & s`m = #0}" 
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definition 
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"U1 == {<s,t>:state*state. t = s(p:= s`v, m:=#2) & s`m = #1}" 
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definition 
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"U2 == {<s,t>:state*state. t = s(m:=#3) & s`p=0 & s`m = #2}" 

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definition 
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"U3 == {<s,t>:state*state. t=s(u:=0, m:=#4) & s`m = #3}" 

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definition 
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"U4 == {<s,t>:state*state. t = s(p:=1, m:=#0) & s`m = #4}" 

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(** The program for process V **) 
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definition 
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"V0 == {<s,t>:state*state. t = s (v:=1, n:=#1) & s`n = #0}" 

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definition 
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"V1 == {<s,t>:state*state. t = s(p:=not(s`u), n:=#2) & s`n = #1}" 

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definition 
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"V2 == {<s,t>:state*state. t = s(n:=#3) & s`p=1 & s`n = #2}" 

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definition 
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"V3 == {<s,t>:state*state. t = s (v:=0, n:=#4) & s`n = #3}" 
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definition 
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"V4 == {<s,t>:state*state. t = s (p:=0, n:=#0) & s`n = #4}" 

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definition 
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"Mutex == mk_program({s:state. s`u=0 & s`v=0 & s`m = #0 & s`n = #0}, 

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{U0, U1, U2, U3, U4, V0, V1, V2, V3, V4}, Pow(state*state))" 
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(** The correct invariants **) 

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definition 
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"IU == {s:state. (s`u = 1\<longleftrightarrow>(#1 $\<le> s`m & s`m $\<le> #3)) 
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& (s`m = #3 \<longrightarrow> s`p=0)}" 
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definition 
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"IV == {s:state. (s`v = 1 \<longleftrightarrow> (#1 $\<le> s`n & s`n $\<le> #3)) 
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& (s`n = #3 \<longrightarrow> s`p=1)}" 
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(** The faulty invariant (for U alone) **) 

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definition 
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"bad_IU == {s:state. (s`u = 1 \<longleftrightarrow> (#1 $\<le> s`m & s`m $\<le> #3))& 
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(#3 $\<le> s`m & s`m $\<le> #4 \<longrightarrow> s`p=0)}" 

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(** Variables' types **) 

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declare p_type [simp] u_type [simp] v_type [simp] m_type [simp] n_type [simp] 

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lemma u_value_type: "s \<in> state ==>s`u \<in> bool" 

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apply (unfold state_def) 

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apply (drule_tac a = u in apply_type, auto) 

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done 

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lemma v_value_type: "s \<in> state ==> s`v \<in> bool" 

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apply (unfold state_def) 

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apply (drule_tac a = v in apply_type, auto) 

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done 

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lemma p_value_type: "s \<in> state ==> s`p \<in> bool" 

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apply (unfold state_def) 

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apply (drule_tac a = p in apply_type, auto) 

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done 

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lemma m_value_type: "s \<in> state ==> s`m \<in> int" 

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apply (unfold state_def) 

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apply (drule_tac a = m in apply_type, auto) 

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done 

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lemma n_value_type: "s \<in> state ==>s`n \<in> int" 

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apply (unfold state_def) 

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apply (drule_tac a = n in apply_type, auto) 

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done 

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declare p_value_type [simp] u_value_type [simp] v_value_type [simp] 

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m_value_type [simp] n_value_type [simp] 

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declare p_value_type [TC] u_value_type [TC] v_value_type [TC] 

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m_value_type [TC] n_value_type [TC] 

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text\<open>Mutex is a program\<close> 
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lemma Mutex_in_program [simp,TC]: "Mutex \<in> program" 

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by (simp add: Mutex_def) 

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declare Mutex_def [THEN def_prg_Init, simp] 

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declare Mutex_def [program] 
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declare U0_def [THEN def_act_simp, simp] 

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declare U1_def [THEN def_act_simp, simp] 

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declare U2_def [THEN def_act_simp, simp] 

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declare U3_def [THEN def_act_simp, simp] 

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declare U4_def [THEN def_act_simp, simp] 

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declare V0_def [THEN def_act_simp, simp] 

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declare V1_def [THEN def_act_simp, simp] 

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declare V2_def [THEN def_act_simp, simp] 

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declare V3_def [THEN def_act_simp, simp] 

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declare V4_def [THEN def_act_simp, simp] 

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declare U0_def [THEN def_set_simp, simp] 

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declare U1_def [THEN def_set_simp, simp] 

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declare U2_def [THEN def_set_simp, simp] 

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declare U3_def [THEN def_set_simp, simp] 

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declare U4_def [THEN def_set_simp, simp] 

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declare V0_def [THEN def_set_simp, simp] 

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declare V1_def [THEN def_set_simp, simp] 

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declare V2_def [THEN def_set_simp, simp] 

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declare V3_def [THEN def_set_simp, simp] 

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declare V4_def [THEN def_set_simp, simp] 

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declare IU_def [THEN def_set_simp, simp] 

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declare IV_def [THEN def_set_simp, simp] 

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declare bad_IU_def [THEN def_set_simp, simp] 

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lemma IU: "Mutex \<in> Always(IU)" 

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apply (rule AlwaysI, force) 
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apply (unfold Mutex_def, safety, auto) 

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done 
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lemma IV: "Mutex \<in> Always(IV)" 

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apply (rule AlwaysI, force) 
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apply (unfold Mutex_def, safety) 

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done 
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(*The safety property: mutual exclusion*) 

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lemma mutual_exclusion: "Mutex \<in> Always({s \<in> state. ~(s`m = #3 & s`n = #3)})" 

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apply (rule Always_weaken) 
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apply (rule Always_Int_I [OF IU IV], auto) 
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done 

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(*The bad invariant FAILS in V1*) 

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lemma less_lemma: "[ x$<#1; #3 $\<le> x ] ==> P" 
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apply (drule_tac j = "#1" and k = "#3" in zless_zle_trans) 
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apply (drule_tac [2] j = x in zle_zless_trans, auto) 

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done 

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lemma "Mutex \<in> Always(bad_IU)" 

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apply (rule AlwaysI, force) 
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apply (unfold Mutex_def, safety, auto) 
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apply (subgoal_tac "#1 $\<le> #3") 
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apply (drule_tac x = "#1" and y = "#3" in zle_trans, auto) 
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apply (simp (no_asm) add: not_zless_iff_zle [THEN iff_sym]) 

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apply auto 

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(*Resulting state: n=1, p=false, m=4, u=false. 
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Execution of V1 (the command of process v guarded by n=1) sets p:=true, 
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violating the invariant!*) 

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oops 

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(*** Progress for U ***) 

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lemma U_F0: "Mutex \<in> {s \<in> state. s`m=#2} Unless {s \<in> state. s`m=#3}" 

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by (unfold op_Unless_def Mutex_def, safety) 
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lemma U_F1: 

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"Mutex \<in> {s \<in> state. s`m=#1} \<longmapsto>w {s \<in> state. s`p = s`v & s`m = #2}" 
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by (unfold Mutex_def, ensures U1) 
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lemma U_F2: "Mutex \<in> {s \<in> state. s`p =0 & s`m = #2} \<longmapsto>w {s \<in> state. s`m = #3}" 
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apply (cut_tac IU) 
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apply (unfold Mutex_def, ensures U2) 
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done 
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lemma U_F3: "Mutex \<in> {s \<in> state. s`m = #3} \<longmapsto>w {s \<in> state. s`p=1}" 
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apply (rule_tac B = "{s \<in> state. s`m = #4}" in LeadsTo_Trans) 
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apply (unfold Mutex_def) 

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apply (ensures U3) 
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apply (ensures U4) 

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done 
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lemma U_lemma2: "Mutex \<in> {s \<in> state. s`m = #2} \<longmapsto>w {s \<in> state. s`p=1}" 
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apply (rule LeadsTo_Diff [OF LeadsTo_weaken_L 
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Int_lower2 [THEN subset_imp_LeadsTo]]) 

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apply (rule LeadsTo_Trans [OF U_F2 U_F3], auto) 

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apply (auto dest!: p_value_type simp add: bool_def) 

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done 

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lemma U_lemma1: "Mutex \<in> {s \<in> state. s`m = #1} \<longmapsto>w {s \<in> state. s`p =1}" 
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by (rule LeadsTo_Trans [OF U_F1 [THEN LeadsTo_weaken_R] U_lemma2], blast) 
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lemma eq_123: "i \<in> int ==> (#1 $\<le> i & i $\<le> #3) \<longleftrightarrow> (i=#1  i=#2  i=#3)" 
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apply auto 
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apply (auto simp add: neq_iff_zless) 

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apply (drule_tac [4] j = "#3" and i = i in zle_zless_trans) 

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apply (drule_tac [2] j = i and i = "#1" in zle_zless_trans) 

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apply (drule_tac j = i and i = "#1" in zle_zless_trans, auto) 

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apply (rule zle_anti_sym) 

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apply (simp_all (no_asm_simp) add: zless_add1_iff_zle [THEN iff_sym]) 

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done 

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lemma U_lemma123: "Mutex \<in> {s \<in> state. #1 $\<le> s`m & s`m $\<le> #3} \<longmapsto>w {s \<in> state. s`p=1}" 
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by (simp add: eq_123 Collect_disj_eq LeadsTo_Un_distrib U_lemma1 U_lemma2 U_F3) 
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(*Misra's F4*) 

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lemma u_Leadsto_p: "Mutex \<in> {s \<in> state. s`u = 1} \<longmapsto>w {s \<in> state. s`p=1}" 
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by (rule Always_LeadsTo_weaken [OF IU U_lemma123], auto) 
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(*** Progress for V ***) 

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lemma V_F0: "Mutex \<in> {s \<in> state. s`n=#2} Unless {s \<in> state. s`n=#3}" 

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by (unfold op_Unless_def Mutex_def, safety) 
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lemma V_F1: "Mutex \<in> {s \<in> state. s`n=#1} \<longmapsto>w {s \<in> state. s`p = not(s`u) & s`n = #2}" 
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by (unfold Mutex_def, ensures "V1") 
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lemma V_F2: "Mutex \<in> {s \<in> state. s`p=1 & s`n = #2} \<longmapsto>w {s \<in> state. s`n = #3}" 
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apply (cut_tac IV) 
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apply (unfold Mutex_def, ensures "V2") 
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done 
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lemma V_F3: "Mutex \<in> {s \<in> state. s`n = #3} \<longmapsto>w {s \<in> state. s`p=0}" 
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apply (rule_tac B = "{s \<in> state. s`n = #4}" in LeadsTo_Trans) 
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apply (unfold Mutex_def) 

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apply (ensures V3) 
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apply (ensures V4) 

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done 
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lemma V_lemma2: "Mutex \<in> {s \<in> state. s`n = #2} \<longmapsto>w {s \<in> state. s`p=0}" 
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apply (rule LeadsTo_Diff [OF LeadsTo_weaken_L 
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Int_lower2 [THEN subset_imp_LeadsTo]]) 

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apply (rule LeadsTo_Trans [OF V_F2 V_F3], auto) 
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apply (auto dest!: p_value_type simp add: bool_def) 
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done 

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lemma V_lemma1: "Mutex \<in> {s \<in> state. s`n = #1} \<longmapsto>w {s \<in> state. s`p = 0}" 
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by (rule LeadsTo_Trans [OF V_F1 [THEN LeadsTo_weaken_R] V_lemma2], blast) 
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lemma V_lemma123: "Mutex \<in> {s \<in> state. #1 $\<le> s`n & s`n $\<le> #3} \<longmapsto>w {s \<in> state. s`p = 0}" 
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by (simp add: eq_123 Collect_disj_eq LeadsTo_Un_distrib V_lemma1 V_lemma2 V_F3) 
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(*Misra's F4*) 

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lemma v_Leadsto_not_p: "Mutex \<in> {s \<in> state. s`v = 1} \<longmapsto>w {s \<in> state. s`p = 0}" 
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by (rule Always_LeadsTo_weaken [OF IV V_lemma123], auto) 
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(** Absence of starvation **) 

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(*Misra's F6*) 

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lemma m1_Leadsto_3: "Mutex \<in> {s \<in> state. s`m = #1} \<longmapsto>w {s \<in> state. s`m = #3}" 
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apply (rule LeadsTo_cancel2 [THEN LeadsTo_Un_duplicate]) 
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apply (rule_tac [2] U_F2) 

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apply (simp add: Collect_conj_eq) 

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apply (subst Un_commute) 

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apply (rule LeadsTo_cancel2 [THEN LeadsTo_Un_duplicate]) 

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apply (rule_tac [2] PSP_Unless [OF v_Leadsto_not_p U_F0]) 

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apply (rule U_F1 [THEN LeadsTo_weaken_R], auto) 

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apply (auto dest!: v_value_type simp add: bool_def) 

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done 

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(*The same for V*) 

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lemma n1_Leadsto_3: "Mutex \<in> {s \<in> state. s`n = #1} \<longmapsto>w {s \<in> state. s`n = #3}" 
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apply (rule LeadsTo_cancel2 [THEN LeadsTo_Un_duplicate]) 
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apply (rule_tac [2] V_F2) 

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apply (simp add: Collect_conj_eq) 

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apply (subst Un_commute) 

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apply (rule LeadsTo_cancel2 [THEN LeadsTo_Un_duplicate]) 

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apply (rule_tac [2] PSP_Unless [OF u_Leadsto_p V_F0]) 

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apply (rule V_F1 [THEN LeadsTo_weaken_R], auto) 

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apply (auto dest!: u_value_type simp add: bool_def) 

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done 

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end 