--- a/src/HOL/HoareParallel/Mul_Gar_Coll.thy Mon Sep 21 08:45:31 2009 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1283 +0,0 @@
-
-header {* \section{The Multi-Mutator Case} *}
-
-theory Mul_Gar_Coll imports Graph OG_Syntax begin
-
-text {* The full theory takes aprox. 18 minutes. *}
-
-record mut =
- Z :: bool
- R :: nat
- T :: nat
-
-text {* Declaration of variables: *}
-
-record mul_gar_coll_state =
- M :: nodes
- E :: edges
- bc :: "nat set"
- obc :: "nat set"
- Ma :: nodes
- ind :: nat
- k :: nat
- q :: nat
- l :: nat
- Muts :: "mut list"
-
-subsection {* The Mutators *}
-
-constdefs
- Mul_mut_init :: "mul_gar_coll_state \<Rightarrow> nat \<Rightarrow> bool"
- "Mul_mut_init \<equiv> \<guillemotleft> \<lambda>n. n=length \<acute>Muts \<and> (\<forall>i<n. R (\<acute>Muts!i)<length \<acute>E
- \<and> T (\<acute>Muts!i)<length \<acute>M) \<guillemotright>"
-
- Mul_Redirect_Edge :: "nat \<Rightarrow> nat \<Rightarrow> mul_gar_coll_state ann_com"
- "Mul_Redirect_Edge j n \<equiv>
- .{\<acute>Mul_mut_init n \<and> Z (\<acute>Muts!j)}.
- \<langle>IF T(\<acute>Muts!j) \<in> Reach \<acute>E THEN
- \<acute>E:= \<acute>E[R (\<acute>Muts!j):= (fst (\<acute>E!R(\<acute>Muts!j)), T (\<acute>Muts!j))] FI,,
- \<acute>Muts:= \<acute>Muts[j:= (\<acute>Muts!j) \<lparr>Z:=False\<rparr>]\<rangle>"
-
- Mul_Color_Target :: "nat \<Rightarrow> nat \<Rightarrow> mul_gar_coll_state ann_com"
- "Mul_Color_Target j n \<equiv>
- .{\<acute>Mul_mut_init n \<and> \<not> Z (\<acute>Muts!j)}.
- \<langle>\<acute>M:=\<acute>M[T (\<acute>Muts!j):=Black],, \<acute>Muts:=\<acute>Muts[j:= (\<acute>Muts!j) \<lparr>Z:=True\<rparr>]\<rangle>"
-
- Mul_Mutator :: "nat \<Rightarrow> nat \<Rightarrow> mul_gar_coll_state ann_com"
- "Mul_Mutator j n \<equiv>
- .{\<acute>Mul_mut_init n \<and> Z (\<acute>Muts!j)}.
- WHILE True
- INV .{\<acute>Mul_mut_init n \<and> Z (\<acute>Muts!j)}.
- DO Mul_Redirect_Edge j n ;;
- Mul_Color_Target j n
- OD"
-
-lemmas mul_mutator_defs = Mul_mut_init_def Mul_Redirect_Edge_def Mul_Color_Target_def
-
-subsubsection {* Correctness of the proof outline of one mutator *}
-
-lemma Mul_Redirect_Edge: "0\<le>j \<and> j<n \<Longrightarrow>
- \<turnstile> Mul_Redirect_Edge j n
- pre(Mul_Color_Target j n)"
-apply (unfold mul_mutator_defs)
-apply annhoare
-apply(simp_all)
-apply clarify
-apply(simp add:nth_list_update)
-done
-
-lemma Mul_Color_Target: "0\<le>j \<and> j<n \<Longrightarrow>
- \<turnstile> Mul_Color_Target j n
- .{\<acute>Mul_mut_init n \<and> Z (\<acute>Muts!j)}."
-apply (unfold mul_mutator_defs)
-apply annhoare
-apply(simp_all)
-apply clarify
-apply(simp add:nth_list_update)
-done
-
-lemma Mul_Mutator: "0\<le>j \<and> j<n \<Longrightarrow>
- \<turnstile> Mul_Mutator j n .{False}."
-apply(unfold Mul_Mutator_def)
-apply annhoare
-apply(simp_all add:Mul_Redirect_Edge Mul_Color_Target)
-apply(simp add:mul_mutator_defs Mul_Redirect_Edge_def)
-done
-
-subsubsection {* Interference freedom between mutators *}
-
-lemma Mul_interfree_Redirect_Edge_Redirect_Edge:
- "\<lbrakk>0\<le>i; i<n; 0\<le>j; j<n; i\<noteq>j\<rbrakk> \<Longrightarrow>
- interfree_aux (Some (Mul_Redirect_Edge i n),{}, Some(Mul_Redirect_Edge j n))"
-apply (unfold mul_mutator_defs)
-apply interfree_aux
-apply safe
-apply(simp_all add: nth_list_update)
-done
-
-lemma Mul_interfree_Redirect_Edge_Color_Target:
- "\<lbrakk>0\<le>i; i<n; 0\<le>j; j<n; i\<noteq>j\<rbrakk> \<Longrightarrow>
- interfree_aux (Some(Mul_Redirect_Edge i n),{},Some(Mul_Color_Target j n))"
-apply (unfold mul_mutator_defs)
-apply interfree_aux
-apply safe
-apply(simp_all add: nth_list_update)
-done
-
-lemma Mul_interfree_Color_Target_Redirect_Edge:
- "\<lbrakk>0\<le>i; i<n; 0\<le>j; j<n; i\<noteq>j\<rbrakk> \<Longrightarrow>
- interfree_aux (Some(Mul_Color_Target i n),{},Some(Mul_Redirect_Edge j n))"
-apply (unfold mul_mutator_defs)
-apply interfree_aux
-apply safe
-apply(simp_all add:nth_list_update)
-done
-
-lemma Mul_interfree_Color_Target_Color_Target:
- " \<lbrakk>0\<le>i; i<n; 0\<le>j; j<n; i\<noteq>j\<rbrakk> \<Longrightarrow>
- interfree_aux (Some(Mul_Color_Target i n),{},Some(Mul_Color_Target j n))"
-apply (unfold mul_mutator_defs)
-apply interfree_aux
-apply safe
-apply(simp_all add: nth_list_update)
-done
-
-lemmas mul_mutator_interfree =
- Mul_interfree_Redirect_Edge_Redirect_Edge Mul_interfree_Redirect_Edge_Color_Target
- Mul_interfree_Color_Target_Redirect_Edge Mul_interfree_Color_Target_Color_Target
-
-lemma Mul_interfree_Mutator_Mutator: "\<lbrakk>i < n; j < n; i \<noteq> j\<rbrakk> \<Longrightarrow>
- interfree_aux (Some (Mul_Mutator i n), {}, Some (Mul_Mutator j n))"
-apply(unfold Mul_Mutator_def)
-apply(interfree_aux)
-apply(simp_all add:mul_mutator_interfree)
-apply(simp_all add: mul_mutator_defs)
-apply(tactic {* TRYALL (interfree_aux_tac) *})
-apply(tactic {* ALLGOALS (clarify_tac @{claset}) *})
-apply (simp_all add:nth_list_update)
-done
-
-subsubsection {* Modular Parameterized Mutators *}
-
-lemma Mul_Parameterized_Mutators: "0<n \<Longrightarrow>
- \<parallel>- .{\<acute>Mul_mut_init n \<and> (\<forall>i<n. Z (\<acute>Muts!i))}.
- COBEGIN
- SCHEME [0\<le> j< n]
- Mul_Mutator j n
- .{False}.
- COEND
- .{False}."
-apply oghoare
-apply(force simp add:Mul_Mutator_def mul_mutator_defs nth_list_update)
-apply(erule Mul_Mutator)
-apply(simp add:Mul_interfree_Mutator_Mutator)
-apply(force simp add:Mul_Mutator_def mul_mutator_defs nth_list_update)
-done
-
-subsection {* The Collector *}
-
-constdefs
- Queue :: "mul_gar_coll_state \<Rightarrow> nat"
- "Queue \<equiv> \<guillemotleft> length (filter (\<lambda>i. \<not> Z i \<and> \<acute>M!(T i) \<noteq> Black) \<acute>Muts) \<guillemotright>"
-
-consts M_init :: nodes
-
-constdefs
- Proper_M_init :: "mul_gar_coll_state \<Rightarrow> bool"
- "Proper_M_init \<equiv> \<guillemotleft> Blacks M_init=Roots \<and> length M_init=length \<acute>M \<guillemotright>"
-
- Mul_Proper :: "mul_gar_coll_state \<Rightarrow> nat \<Rightarrow> bool"
- "Mul_Proper \<equiv> \<guillemotleft> \<lambda>n. Proper_Roots \<acute>M \<and> Proper_Edges (\<acute>M, \<acute>E) \<and> \<acute>Proper_M_init \<and> n=length \<acute>Muts \<guillemotright>"
-
- Safe :: "mul_gar_coll_state \<Rightarrow> bool"
- "Safe \<equiv> \<guillemotleft> Reach \<acute>E \<subseteq> Blacks \<acute>M \<guillemotright>"
-
-lemmas mul_collector_defs = Proper_M_init_def Mul_Proper_def Safe_def
-
-subsubsection {* Blackening Roots *}
-
-constdefs
- Mul_Blacken_Roots :: "nat \<Rightarrow> mul_gar_coll_state ann_com"
- "Mul_Blacken_Roots n \<equiv>
- .{\<acute>Mul_Proper n}.
- \<acute>ind:=0;;
- .{\<acute>Mul_Proper n \<and> \<acute>ind=0}.
- WHILE \<acute>ind<length \<acute>M
- INV .{\<acute>Mul_Proper n \<and> (\<forall>i<\<acute>ind. i\<in>Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind\<le>length \<acute>M}.
- DO .{\<acute>Mul_Proper n \<and> (\<forall>i<\<acute>ind. i\<in>Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind<length \<acute>M}.
- IF \<acute>ind\<in>Roots THEN
- .{\<acute>Mul_Proper n \<and> (\<forall>i<\<acute>ind. i\<in>Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind<length \<acute>M \<and> \<acute>ind\<in>Roots}.
- \<acute>M:=\<acute>M[\<acute>ind:=Black] FI;;
- .{\<acute>Mul_Proper n \<and> (\<forall>i<\<acute>ind+1. i\<in>Roots \<longrightarrow> \<acute>M!i=Black) \<and> \<acute>ind<length \<acute>M}.
- \<acute>ind:=\<acute>ind+1
- OD"
-
-lemma Mul_Blacken_Roots:
- "\<turnstile> Mul_Blacken_Roots n
- .{\<acute>Mul_Proper n \<and> Roots \<subseteq> Blacks \<acute>M}."
-apply (unfold Mul_Blacken_Roots_def)
-apply annhoare
-apply(simp_all add:mul_collector_defs Graph_defs)
-apply safe
-apply(simp_all add:nth_list_update)
- apply (erule less_SucE)
- apply simp+
- apply force
-apply force
-done
-
-subsubsection {* Propagating Black *}
-
-constdefs
- Mul_PBInv :: "mul_gar_coll_state \<Rightarrow> bool"
- "Mul_PBInv \<equiv> \<guillemotleft>\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>M \<or> \<acute>l<\<acute>Queue
- \<or> (\<forall>i<\<acute>ind. \<not>BtoW(\<acute>E!i,\<acute>M)) \<and> \<acute>l\<le>\<acute>Queue\<guillemotright>"
-
- Mul_Auxk :: "mul_gar_coll_state \<Rightarrow> bool"
- "Mul_Auxk \<equiv> \<guillemotleft>\<acute>l<\<acute>Queue \<or> \<acute>M!\<acute>k\<noteq>Black \<or> \<not>BtoW(\<acute>E!\<acute>ind, \<acute>M) \<or> \<acute>obc\<subset>Blacks \<acute>M\<guillemotright>"
-
-constdefs
- Mul_Propagate_Black :: "nat \<Rightarrow> mul_gar_coll_state ann_com"
- "Mul_Propagate_Black n \<equiv>
- .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> (\<acute>Safe \<or> \<acute>l\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M)}.
- \<acute>ind:=0;;
- .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M
- \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> Blacks \<acute>M\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> (\<acute>Safe \<or> \<acute>l\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M) \<and> \<acute>ind=0}.
- WHILE \<acute>ind<length \<acute>E
- INV .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M
- \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> \<acute>Mul_PBInv \<and> \<acute>ind\<le>length \<acute>E}.
- DO .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M
- \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> \<acute>Mul_PBInv \<and> \<acute>ind<length \<acute>E}.
- IF \<acute>M!(fst (\<acute>E!\<acute>ind))=Black THEN
- .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M
- \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> \<acute>Mul_PBInv \<and> (\<acute>M!fst(\<acute>E!\<acute>ind))=Black \<and> \<acute>ind<length \<acute>E}.
- \<acute>k:=snd(\<acute>E!\<acute>ind);;
- .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M
- \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>M \<or> \<acute>l<\<acute>Queue \<or> (\<forall>i<\<acute>ind. \<not>BtoW(\<acute>E!i,\<acute>M))
- \<and> \<acute>l\<le>\<acute>Queue \<and> \<acute>Mul_Auxk ) \<and> \<acute>k<length \<acute>M \<and> \<acute>M!fst(\<acute>E!\<acute>ind)=Black
- \<and> \<acute>ind<length \<acute>E}.
- \<langle>\<acute>M:=\<acute>M[\<acute>k:=Black],,\<acute>ind:=\<acute>ind+1\<rangle>
- ELSE .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M
- \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> \<acute>Mul_PBInv \<and> \<acute>ind<length \<acute>E}.
- \<langle>IF \<acute>M!(fst (\<acute>E!\<acute>ind))\<noteq>Black THEN \<acute>ind:=\<acute>ind+1 FI\<rangle> FI
- OD"
-
-lemma Mul_Propagate_Black:
- "\<turnstile> Mul_Propagate_Black n
- .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>M \<or> \<acute>l<\<acute>Queue \<and> (\<acute>l\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M))}."
-apply(unfold Mul_Propagate_Black_def)
-apply annhoare
-apply(simp_all add:Mul_PBInv_def mul_collector_defs Mul_Auxk_def Graph6 Graph7 Graph8 Graph12 mul_collector_defs Queue_def)
---{* 8 subgoals left *}
-apply force
-apply force
-apply force
-apply(force simp add:BtoW_def Graph_defs)
---{* 4 subgoals left *}
-apply clarify
-apply(simp add: mul_collector_defs Graph12 Graph6 Graph7 Graph8)
-apply(disjE_tac)
- apply(simp_all add:Graph12 Graph13)
- apply(case_tac "M x! k x=Black")
- apply(simp add: Graph10)
- apply(rule disjI2, rule disjI1, erule subset_psubset_trans, erule Graph11, force)
-apply(case_tac "M x! k x=Black")
- apply(simp add: Graph10 BtoW_def)
- apply(rule disjI2, clarify, erule less_SucE, force)
- apply(case_tac "M x!snd(E x! ind x)=Black")
- apply(force)
- apply(force)
-apply(rule disjI2, rule disjI1, erule subset_psubset_trans, erule Graph11, force)
---{* 3 subgoals left *}
-apply force
---{* 2 subgoals left *}
-apply clarify
-apply(conjI_tac)
-apply(disjE_tac)
- apply (simp_all)
-apply clarify
-apply(erule less_SucE)
- apply force
-apply (simp add:BtoW_def)
---{* 1 subgoal left *}
-apply clarify
-apply simp
-apply(disjE_tac)
-apply (simp_all)
-apply(rule disjI1 , rule Graph1)
- apply simp_all
-done
-
-subsubsection {* Counting Black Nodes *}
-
-constdefs
- Mul_CountInv :: "mul_gar_coll_state \<Rightarrow> nat \<Rightarrow> bool"
- "Mul_CountInv \<equiv> \<guillemotleft> \<lambda>ind. {i. i<ind \<and> \<acute>Ma!i=Black}\<subseteq>\<acute>bc \<guillemotright>"
-
- Mul_Count :: "nat \<Rightarrow> mul_gar_coll_state ann_com"
- "Mul_Count n \<equiv>
- .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M
- \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> length \<acute>Ma=length \<acute>M
- \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M) )
- \<and> \<acute>q<n+1 \<and> \<acute>bc={}}.
- \<acute>ind:=0;;
- .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M
- \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> length \<acute>Ma=length \<acute>M
- \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M) )
- \<and> \<acute>q<n+1 \<and> \<acute>bc={} \<and> \<acute>ind=0}.
- WHILE \<acute>ind<length \<acute>M
- INV .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M
- \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> length \<acute>Ma=length \<acute>M \<and> \<acute>Mul_CountInv \<acute>ind
- \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M))
- \<and> \<acute>q<n+1 \<and> \<acute>ind\<le>length \<acute>M}.
- DO .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M
- \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> length \<acute>Ma=length \<acute>M \<and> \<acute>Mul_CountInv \<acute>ind
- \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M))
- \<and> \<acute>q<n+1 \<and> \<acute>ind<length \<acute>M}.
- IF \<acute>M!\<acute>ind=Black
- THEN .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M
- \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> length \<acute>Ma=length \<acute>M \<and> \<acute>Mul_CountInv \<acute>ind
- \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M))
- \<and> \<acute>q<n+1 \<and> \<acute>ind<length \<acute>M \<and> \<acute>M!\<acute>ind=Black}.
- \<acute>bc:=insert \<acute>ind \<acute>bc
- FI;;
- .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M
- \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> length \<acute>Ma=length \<acute>M \<and> \<acute>Mul_CountInv (\<acute>ind+1)
- \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M))
- \<and> \<acute>q<n+1 \<and> \<acute>ind<length \<acute>M}.
- \<acute>ind:=\<acute>ind+1
- OD"
-
-lemma Mul_Count:
- "\<turnstile> Mul_Count n
- .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M
- \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> length \<acute>Ma=length \<acute>M \<and> Blacks \<acute>Ma\<subseteq>\<acute>bc
- \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M))
- \<and> \<acute>q<n+1}."
-apply (unfold Mul_Count_def)
-apply annhoare
-apply(simp_all add:Mul_CountInv_def mul_collector_defs Mul_Auxk_def Graph6 Graph7 Graph8 Graph12 mul_collector_defs Queue_def)
---{* 7 subgoals left *}
-apply force
-apply force
-apply force
---{* 4 subgoals left *}
-apply clarify
-apply(conjI_tac)
-apply(disjE_tac)
- apply simp_all
-apply(simp add:Blacks_def)
-apply clarify
-apply(erule less_SucE)
- back
- apply force
-apply force
---{* 3 subgoals left *}
-apply clarify
-apply(conjI_tac)
-apply(disjE_tac)
- apply simp_all
-apply clarify
-apply(erule less_SucE)
- back
- apply force
-apply simp
-apply(rotate_tac -1)
-apply (force simp add:Blacks_def)
---{* 2 subgoals left *}
-apply force
---{* 1 subgoal left *}
-apply clarify
-apply(drule_tac x = "ind x" in le_imp_less_or_eq)
-apply (simp_all add:Blacks_def)
-done
-
-subsubsection {* Appending garbage nodes to the free list *}
-
-consts Append_to_free :: "nat \<times> edges \<Rightarrow> edges"
-
-axioms
- Append_to_free0: "length (Append_to_free (i, e)) = length e"
- Append_to_free1: "Proper_Edges (m, e)
- \<Longrightarrow> Proper_Edges (m, Append_to_free(i, e))"
- Append_to_free2: "i \<notin> Reach e
- \<Longrightarrow> n \<in> Reach (Append_to_free(i, e)) = ( n = i \<or> n \<in> Reach e)"
-
-constdefs
- Mul_AppendInv :: "mul_gar_coll_state \<Rightarrow> nat \<Rightarrow> bool"
- "Mul_AppendInv \<equiv> \<guillemotleft> \<lambda>ind. (\<forall>i. ind\<le>i \<longrightarrow> i<length \<acute>M \<longrightarrow> i\<in>Reach \<acute>E \<longrightarrow> \<acute>M!i=Black)\<guillemotright>"
-
- Mul_Append :: "nat \<Rightarrow> mul_gar_coll_state ann_com"
- "Mul_Append n \<equiv>
- .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>Safe}.
- \<acute>ind:=0;;
- .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>Safe \<and> \<acute>ind=0}.
- WHILE \<acute>ind<length \<acute>M
- INV .{\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv \<acute>ind \<and> \<acute>ind\<le>length \<acute>M}.
- DO .{\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv \<acute>ind \<and> \<acute>ind<length \<acute>M}.
- IF \<acute>M!\<acute>ind=Black THEN
- .{\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv \<acute>ind \<and> \<acute>ind<length \<acute>M \<and> \<acute>M!\<acute>ind=Black}.
- \<acute>M:=\<acute>M[\<acute>ind:=White]
- ELSE
- .{\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv \<acute>ind \<and> \<acute>ind<length \<acute>M \<and> \<acute>ind\<notin>Reach \<acute>E}.
- \<acute>E:=Append_to_free(\<acute>ind,\<acute>E)
- FI;;
- .{\<acute>Mul_Proper n \<and> \<acute>Mul_AppendInv (\<acute>ind+1) \<and> \<acute>ind<length \<acute>M}.
- \<acute>ind:=\<acute>ind+1
- OD"
-
-lemma Mul_Append:
- "\<turnstile> Mul_Append n
- .{\<acute>Mul_Proper n}."
-apply(unfold Mul_Append_def)
-apply annhoare
-apply(simp_all add: mul_collector_defs Mul_AppendInv_def
- Graph6 Graph7 Graph8 Append_to_free0 Append_to_free1 Graph12)
-apply(force simp add:Blacks_def)
-apply(force simp add:Blacks_def)
-apply(force simp add:Blacks_def)
-apply(force simp add:Graph_defs)
-apply force
-apply(force simp add:Append_to_free1 Append_to_free2)
-apply force
-apply force
-done
-
-subsubsection {* Collector *}
-
-constdefs
- Mul_Collector :: "nat \<Rightarrow> mul_gar_coll_state ann_com"
- "Mul_Collector n \<equiv>
-.{\<acute>Mul_Proper n}.
-WHILE True INV .{\<acute>Mul_Proper n}.
-DO
-Mul_Blacken_Roots n ;;
-.{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M}.
- \<acute>obc:={};;
-.{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc={}}.
- \<acute>bc:=Roots;;
-.{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc={} \<and> \<acute>bc=Roots}.
- \<acute>l:=0;;
-.{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>obc={} \<and> \<acute>bc=Roots \<and> \<acute>l=0}.
- WHILE \<acute>l<n+1
- INV .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M \<and>
- (\<acute>Safe \<or> (\<acute>l\<le>\<acute>Queue \<or> \<acute>bc\<subset>Blacks \<acute>M) \<and> \<acute>l<n+1)}.
- DO .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> (\<acute>Safe \<or> \<acute>l\<le>\<acute>Queue \<or> \<acute>bc\<subset>Blacks \<acute>M)}.
- \<acute>obc:=\<acute>bc;;
- Mul_Propagate_Black n;;
- .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M
- \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>M \<or> \<acute>l<\<acute>Queue
- \<and> (\<acute>l\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M))}.
- \<acute>bc:={};;
- .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M
- \<and> \<acute>obc\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>M \<or> \<acute>l<\<acute>Queue
- \<and> (\<acute>l\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M)) \<and> \<acute>bc={}}.
- \<langle> \<acute>Ma:=\<acute>M,, \<acute>q:=\<acute>Queue \<rangle>;;
- Mul_Count n;;
- .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M
- \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> length \<acute>Ma=length \<acute>M \<and> Blacks \<acute>Ma\<subseteq>\<acute>bc
- \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M))
- \<and> \<acute>q<n+1}.
- IF \<acute>obc=\<acute>bc THEN
- .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M
- \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> length \<acute>Ma=length \<acute>M \<and> Blacks \<acute>Ma\<subseteq>\<acute>bc
- \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M))
- \<and> \<acute>q<n+1 \<and> \<acute>obc=\<acute>bc}.
- \<acute>l:=\<acute>l+1
- ELSE .{\<acute>Mul_Proper n \<and> Roots\<subseteq>Blacks \<acute>M
- \<and> \<acute>obc\<subseteq>Blacks \<acute>Ma \<and> Blacks \<acute>Ma\<subseteq>Blacks \<acute>M \<and> \<acute>bc\<subseteq>Blacks \<acute>M
- \<and> length \<acute>Ma=length \<acute>M \<and> Blacks \<acute>Ma\<subseteq>\<acute>bc
- \<and> (\<acute>Safe \<or> \<acute>obc\<subset>Blacks \<acute>Ma \<or> \<acute>l<\<acute>q \<and> (\<acute>q\<le>\<acute>Queue \<or> \<acute>obc\<subset>Blacks \<acute>M))
- \<and> \<acute>q<n+1 \<and> \<acute>obc\<noteq>\<acute>bc}.
- \<acute>l:=0 FI
- OD;;
- Mul_Append n
-OD"
-
-lemmas mul_modules = Mul_Redirect_Edge_def Mul_Color_Target_def
- Mul_Blacken_Roots_def Mul_Propagate_Black_def
- Mul_Count_def Mul_Append_def
-
-lemma Mul_Collector:
- "\<turnstile> Mul_Collector n
- .{False}."
-apply(unfold Mul_Collector_def)
-apply annhoare
-apply(simp_all only:pre.simps Mul_Blacken_Roots
- Mul_Propagate_Black Mul_Count Mul_Append)
-apply(simp_all add:mul_modules)
-apply(simp_all add:mul_collector_defs Queue_def)
-apply force
-apply force
-apply force
-apply (force simp add: less_Suc_eq_le)
-apply force
-apply (force dest:subset_antisym)
-apply force
-apply force
-apply force
-done
-
-subsection {* Interference Freedom *}
-
-lemma le_length_filter_update[rule_format]:
- "\<forall>i. (\<not>P (list!i) \<or> P j) \<and> i<length list
- \<longrightarrow> length(filter P list) \<le> length(filter P (list[i:=j]))"
-apply(induct_tac "list")
- apply(simp)
-apply(clarify)
-apply(case_tac i)
- apply(simp)
-apply(simp)
-done
-
-lemma less_length_filter_update [rule_format]:
- "\<forall>i. P j \<and> \<not>(P (list!i)) \<and> i<length list
- \<longrightarrow> length(filter P list) < length(filter P (list[i:=j]))"
-apply(induct_tac "list")
- apply(simp)
-apply(clarify)
-apply(case_tac i)
- apply(simp)
-apply(simp)
-done
-
-lemma Mul_interfree_Blacken_Roots_Redirect_Edge: "\<lbrakk>0\<le>j; j<n\<rbrakk> \<Longrightarrow>
- interfree_aux (Some(Mul_Blacken_Roots n),{},Some(Mul_Redirect_Edge j n))"
-apply (unfold mul_modules)
-apply interfree_aux
-apply safe
-apply(simp_all add:Graph6 Graph9 Graph12 nth_list_update mul_mutator_defs mul_collector_defs)
-done
-
-lemma Mul_interfree_Redirect_Edge_Blacken_Roots: "\<lbrakk>0\<le>j; j<n\<rbrakk>\<Longrightarrow>
- interfree_aux (Some(Mul_Redirect_Edge j n ),{},Some (Mul_Blacken_Roots n))"
-apply (unfold mul_modules)
-apply interfree_aux
-apply safe
-apply(simp_all add:mul_mutator_defs nth_list_update)
-done
-
-lemma Mul_interfree_Blacken_Roots_Color_Target: "\<lbrakk>0\<le>j; j<n\<rbrakk>\<Longrightarrow>
- interfree_aux (Some(Mul_Blacken_Roots n),{},Some (Mul_Color_Target j n ))"
-apply (unfold mul_modules)
-apply interfree_aux
-apply safe
-apply(simp_all add:mul_mutator_defs mul_collector_defs nth_list_update Graph7 Graph8 Graph9 Graph12)
-done
-
-lemma Mul_interfree_Color_Target_Blacken_Roots: "\<lbrakk>0\<le>j; j<n\<rbrakk>\<Longrightarrow>
- interfree_aux (Some(Mul_Color_Target j n ),{},Some (Mul_Blacken_Roots n ))"
-apply (unfold mul_modules)
-apply interfree_aux
-apply safe
-apply(simp_all add:mul_mutator_defs nth_list_update)
-done
-
-lemma Mul_interfree_Propagate_Black_Redirect_Edge: "\<lbrakk>0\<le>j; j<n\<rbrakk>\<Longrightarrow>
- interfree_aux (Some(Mul_Propagate_Black n),{},Some (Mul_Redirect_Edge j n ))"
-apply (unfold mul_modules)
-apply interfree_aux
-apply(simp_all add:mul_mutator_defs mul_collector_defs Mul_PBInv_def nth_list_update Graph6)
---{* 7 subgoals left *}
-apply clarify
-apply(disjE_tac)
- apply(simp_all add:Graph6)
- apply(rule impI,rule disjI1,rule subset_trans,erule Graph3,simp,simp)
-apply(rule conjI)
- apply(rule impI,rule disjI2,rule disjI1,erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(rule impI,rule disjI2,rule disjI1,erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
---{* 6 subgoals left *}
-apply clarify
-apply(disjE_tac)
- apply(simp_all add:Graph6)
- apply(rule impI,rule disjI1,rule subset_trans,erule Graph3,simp,simp)
-apply(rule conjI)
- apply(rule impI,rule disjI2,rule disjI1,erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(rule impI,rule disjI2,rule disjI1,erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
---{* 5 subgoals left *}
-apply clarify
-apply(disjE_tac)
- apply(simp_all add:Graph6)
- apply(rule impI,rule disjI1,rule subset_trans,erule Graph3,simp,simp)
-apply(rule conjI)
- apply(rule impI,rule disjI2,rule disjI2,rule disjI1,erule less_le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(rule impI,rule disjI2,rule disjI2,rule disjI1,erule less_le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(erule conjE)
-apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(rule conjI)
- apply(rule impI,(rule disjI2)+,rule conjI)
- apply clarify
- apply(case_tac "R (Muts x! j)=i")
- apply (force simp add: nth_list_update BtoW_def)
- apply (force simp add: nth_list_update)
- apply(erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
- apply(rule impI,(rule disjI2)+, erule le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(rule conjI)
- apply(rule impI,rule disjI2,rule disjI2,rule disjI1, erule le_less_trans)
- apply(force simp add:Queue_def less_Suc_eq_le less_length_filter_update)
-apply(rule impI,rule disjI2,rule disjI2,rule disjI1, erule le_less_trans)
-apply(force simp add:Queue_def less_Suc_eq_le less_length_filter_update)
---{* 4 subgoals left *}
-apply clarify
-apply(disjE_tac)
- apply(simp_all add:Graph6)
- apply(rule impI,rule disjI1,rule subset_trans,erule Graph3,simp,simp)
-apply(rule conjI)
- apply(rule impI,rule disjI2,rule disjI2,rule disjI1,erule less_le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(rule impI,rule disjI2,rule disjI2,rule disjI1,erule less_le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(erule conjE)
-apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(rule conjI)
- apply(rule impI,(rule disjI2)+,rule conjI)
- apply clarify
- apply(case_tac "R (Muts x! j)=i")
- apply (force simp add: nth_list_update BtoW_def)
- apply (force simp add: nth_list_update)
- apply(erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
- apply(rule impI,(rule disjI2)+, erule le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(rule conjI)
- apply(rule impI,rule disjI2,rule disjI2,rule disjI1, erule le_less_trans)
- apply(force simp add:Queue_def less_Suc_eq_le less_length_filter_update)
-apply(rule impI,rule disjI2,rule disjI2,rule disjI1, erule le_less_trans)
-apply(force simp add:Queue_def less_Suc_eq_le less_length_filter_update)
---{* 3 subgoals left *}
-apply clarify
-apply(disjE_tac)
- apply(simp_all add:Graph6)
- apply (rule impI)
- apply(rule conjI)
- apply(rule disjI1,rule subset_trans,erule Graph3,simp,simp)
- apply(case_tac "R (Muts x ! j)= ind x")
- apply(simp add:nth_list_update)
- apply(simp add:nth_list_update)
- apply(case_tac "R (Muts x ! j)= ind x")
- apply(simp add:nth_list_update)
- apply(simp add:nth_list_update)
- apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(rule conjI)
- apply(rule impI)
- apply(rule conjI)
- apply(rule disjI2,rule disjI2,rule disjI1, erule less_le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
- apply(case_tac "R (Muts x ! j)= ind x")
- apply(simp add:nth_list_update)
- apply(simp add:nth_list_update)
- apply(rule impI)
- apply(rule disjI2,rule disjI2,rule disjI1, erule less_le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
- apply(rule conjI)
- apply(rule impI)
- apply(rule conjI)
- apply(rule disjI2,rule disjI2,rule disjI1, erule less_le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
- apply(case_tac "R (Muts x ! j)= ind x")
- apply(simp add:nth_list_update)
- apply(simp add:nth_list_update)
- apply(rule impI)
- apply(rule disjI2,rule disjI2,rule disjI1, erule less_le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
- apply(erule conjE)
- apply(rule conjI)
- apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(rule impI,rule conjI,(rule disjI2)+,rule conjI)
- apply clarify
- apply(case_tac "R (Muts x! j)=i")
- apply (force simp add: nth_list_update BtoW_def)
- apply (force simp add: nth_list_update)
- apply(erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
- apply(case_tac "R (Muts x ! j)= ind x")
- apply(simp add:nth_list_update)
- apply(simp add:nth_list_update)
- apply(rule impI,rule conjI)
- apply(rule disjI2,rule disjI2,rule disjI1, erule le_less_trans)
- apply(force simp add:Queue_def less_Suc_eq_le less_length_filter_update)
- apply(case_tac "R (Muts x! j)=ind x")
- apply (force simp add: nth_list_update)
- apply (force simp add: nth_list_update)
-apply(rule impI, (rule disjI2)+, erule le_trans)
-apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
---{* 2 subgoals left *}
-apply clarify
-apply(rule conjI)
- apply(disjE_tac)
- apply(simp_all add:Mul_Auxk_def Graph6)
- apply (rule impI)
- apply(rule conjI)
- apply(rule disjI1,rule subset_trans,erule Graph3,simp,simp)
- apply(case_tac "R (Muts x ! j)= ind x")
- apply(simp add:nth_list_update)
- apply(simp add:nth_list_update)
- apply(case_tac "R (Muts x ! j)= ind x")
- apply(simp add:nth_list_update)
- apply(simp add:nth_list_update)
- apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(rule impI)
- apply(rule conjI)
- apply(rule disjI2,rule disjI2,rule disjI1, erule less_le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
- apply(case_tac "R (Muts x ! j)= ind x")
- apply(simp add:nth_list_update)
- apply(simp add:nth_list_update)
- apply(rule impI)
- apply(rule conjI)
- apply(rule disjI2,rule disjI2,rule disjI1, erule less_le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
- apply(case_tac "R (Muts x ! j)= ind x")
- apply(simp add:nth_list_update)
- apply(simp add:nth_list_update)
-apply(rule impI)
-apply(rule conjI)
- apply(erule conjE)+
- apply(case_tac "M x!(T (Muts x!j))=Black")
- apply((rule disjI2)+,rule conjI)
- apply clarify
- apply(case_tac "R (Muts x! j)=i")
- apply (force simp add: nth_list_update BtoW_def)
- apply (force simp add: nth_list_update)
- apply(rule conjI)
- apply(erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
- apply(rule impI)
- apply(case_tac "R (Muts x ! j)= ind x")
- apply(simp add:nth_list_update BtoW_def)
- apply (simp add:nth_list_update)
- apply(rule impI)
- apply simp
- apply(disjE_tac)
- apply(rule disjI1, erule less_le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
- apply force
- apply(rule disjI2,rule disjI2,rule disjI1, erule le_less_trans)
- apply(force simp add:Queue_def less_Suc_eq_le less_length_filter_update)
- apply(case_tac "R (Muts x ! j)= ind x")
- apply(simp add:nth_list_update)
- apply(simp add:nth_list_update)
-apply(disjE_tac)
-apply simp_all
-apply(conjI_tac)
- apply(rule impI)
- apply(rule disjI2,rule disjI2,rule disjI1, erule less_le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(erule conjE)+
-apply(rule impI,(rule disjI2)+,rule conjI)
- apply(erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(rule impI)+
-apply simp
-apply(disjE_tac)
- apply(rule disjI1, erule less_le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply force
---{* 1 subgoal left *}
-apply clarify
-apply(disjE_tac)
- apply(simp_all add:Graph6)
- apply(rule impI,rule disjI1,rule subset_trans,erule Graph3,simp,simp)
-apply(rule conjI)
- apply(rule impI,rule disjI2,rule disjI2,rule disjI1,erule less_le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(rule impI,rule disjI2,rule disjI2,rule disjI1,erule less_le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(erule conjE)
-apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(rule conjI)
- apply(rule impI,(rule disjI2)+,rule conjI)
- apply clarify
- apply(case_tac "R (Muts x! j)=i")
- apply (force simp add: nth_list_update BtoW_def)
- apply (force simp add: nth_list_update)
- apply(erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
- apply(rule impI,(rule disjI2)+, erule le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(rule conjI)
- apply(rule impI,rule disjI2,rule disjI2,rule disjI1, erule le_less_trans)
- apply(force simp add:Queue_def less_Suc_eq_le less_length_filter_update)
-apply(rule impI,rule disjI2,rule disjI2,rule disjI1, erule le_less_trans)
-apply(force simp add:Queue_def less_Suc_eq_le less_length_filter_update)
-done
-
-lemma Mul_interfree_Redirect_Edge_Propagate_Black: "\<lbrakk>0\<le>j; j<n\<rbrakk>\<Longrightarrow>
- interfree_aux (Some(Mul_Redirect_Edge j n ),{},Some (Mul_Propagate_Black n))"
-apply (unfold mul_modules)
-apply interfree_aux
-apply safe
-apply(simp_all add:mul_mutator_defs nth_list_update)
-done
-
-lemma Mul_interfree_Propagate_Black_Color_Target: "\<lbrakk>0\<le>j; j<n\<rbrakk>\<Longrightarrow>
- interfree_aux (Some(Mul_Propagate_Black n),{},Some (Mul_Color_Target j n ))"
-apply (unfold mul_modules)
-apply interfree_aux
-apply(simp_all add: mul_collector_defs mul_mutator_defs)
---{* 7 subgoals left *}
-apply clarify
-apply (simp add:Graph7 Graph8 Graph12)
-apply(disjE_tac)
- apply(simp add:Graph7 Graph8 Graph12)
- apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(rule disjI2,rule disjI1, erule le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
- apply((rule disjI2)+,erule subset_psubset_trans, erule Graph11, simp)
-apply((rule disjI2)+,erule psubset_subset_trans, simp add: Graph9)
---{* 6 subgoals left *}
-apply clarify
-apply (simp add:Graph7 Graph8 Graph12)
-apply(disjE_tac)
- apply(simp add:Graph7 Graph8 Graph12)
- apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(rule disjI2,rule disjI1, erule le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
- apply((rule disjI2)+,erule subset_psubset_trans, erule Graph11, simp)
-apply((rule disjI2)+,erule psubset_subset_trans, simp add: Graph9)
---{* 5 subgoals left *}
-apply clarify
-apply (simp add:mul_collector_defs Mul_PBInv_def Graph7 Graph8 Graph12)
-apply(disjE_tac)
- apply(simp add:Graph7 Graph8 Graph12)
- apply(rule disjI2,rule disjI1, erule psubset_subset_trans,simp add:Graph9)
- apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(rule disjI2,rule disjI2,rule disjI1, erule less_le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
- apply(rule disjI2,rule disjI1,erule subset_psubset_trans, erule Graph11, simp)
-apply(erule conjE)
-apply(case_tac "M x!(T (Muts x!j))=Black")
- apply((rule disjI2)+)
- apply (rule conjI)
- apply(simp add:Graph10)
- apply(erule le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
-apply(rule disjI2,rule disjI1,erule subset_psubset_trans, erule Graph11, simp)
---{* 4 subgoals left *}
-apply clarify
-apply (simp add:mul_collector_defs Mul_PBInv_def Graph7 Graph8 Graph12)
-apply(disjE_tac)
- apply(simp add:Graph7 Graph8 Graph12)
- apply(rule disjI2,rule disjI1, erule psubset_subset_trans,simp add:Graph9)
- apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(rule disjI2,rule disjI2,rule disjI1, erule less_le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
- apply(rule disjI2,rule disjI1,erule subset_psubset_trans, erule Graph11, simp)
-apply(erule conjE)
-apply(case_tac "M x!(T (Muts x!j))=Black")
- apply((rule disjI2)+)
- apply (rule conjI)
- apply(simp add:Graph10)
- apply(erule le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
-apply(rule disjI2,rule disjI1,erule subset_psubset_trans, erule Graph11, simp)
---{* 3 subgoals left *}
-apply clarify
-apply (simp add:mul_collector_defs Mul_PBInv_def Graph7 Graph8 Graph12)
-apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(simp add:Graph10)
- apply(disjE_tac)
- apply simp_all
- apply(rule disjI2, rule disjI2, rule disjI1,erule less_le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
- apply(erule conjE)
- apply((rule disjI2)+,erule le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
-apply(rule conjI)
- apply(rule disjI2,rule disjI1, erule subset_psubset_trans,simp add:Graph11)
-apply (force simp add:nth_list_update)
---{* 2 subgoals left *}
-apply clarify
-apply(simp add:Mul_Auxk_def Graph7 Graph8 Graph12)
-apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(simp add:Graph10)
- apply(disjE_tac)
- apply simp_all
- apply(rule disjI2, rule disjI2, rule disjI1,erule less_le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
- apply(erule conjE)+
- apply((rule disjI2)+,rule conjI, erule le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
- apply((rule impI)+)
- apply simp
- apply(erule disjE)
- apply(rule disjI1, erule less_le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
- apply force
-apply(rule conjI)
- apply(rule disjI2,rule disjI1, erule subset_psubset_trans,simp add:Graph11)
-apply (force simp add:nth_list_update)
---{* 1 subgoal left *}
-apply clarify
-apply (simp add:mul_collector_defs Mul_PBInv_def Graph7 Graph8 Graph12)
-apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(simp add:Graph10)
- apply(disjE_tac)
- apply simp_all
- apply(rule disjI2, rule disjI2, rule disjI1,erule less_le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
- apply(erule conjE)
- apply((rule disjI2)+,erule le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
-apply(rule disjI2,rule disjI1, erule subset_psubset_trans,simp add:Graph11)
-done
-
-lemma Mul_interfree_Color_Target_Propagate_Black: "\<lbrakk>0\<le>j; j<n\<rbrakk>\<Longrightarrow>
- interfree_aux (Some(Mul_Color_Target j n),{},Some(Mul_Propagate_Black n ))"
-apply (unfold mul_modules)
-apply interfree_aux
-apply safe
-apply(simp_all add:mul_mutator_defs nth_list_update)
-done
-
-lemma Mul_interfree_Count_Redirect_Edge: "\<lbrakk>0\<le>j; j<n\<rbrakk>\<Longrightarrow>
- interfree_aux (Some(Mul_Count n ),{},Some(Mul_Redirect_Edge j n))"
-apply (unfold mul_modules)
-apply interfree_aux
---{* 9 subgoals left *}
-apply(simp add:mul_mutator_defs mul_collector_defs Mul_CountInv_def Graph6)
-apply clarify
-apply disjE_tac
- apply(simp add:Graph6)
- apply(rule impI,rule disjI1,rule subset_trans,erule Graph3,simp,simp)
- apply(simp add:Graph6)
-apply clarify
-apply disjE_tac
- apply(simp add:Graph6)
- apply(rule conjI)
- apply(rule impI,rule disjI2,rule disjI2,rule disjI1,erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
- apply(rule impI,rule disjI2,rule disjI2,rule disjI1,erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(simp add:Graph6)
---{* 8 subgoals left *}
-apply(simp add:mul_mutator_defs nth_list_update)
---{* 7 subgoals left *}
-apply(simp add:mul_mutator_defs mul_collector_defs)
-apply clarify
-apply disjE_tac
- apply(simp add:Graph6)
- apply(rule impI,rule disjI1,rule subset_trans,erule Graph3,simp,simp)
- apply(simp add:Graph6)
-apply clarify
-apply disjE_tac
- apply(simp add:Graph6)
- apply(rule conjI)
- apply(rule impI,rule disjI2,rule disjI2,rule disjI1,erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
- apply(rule impI,rule disjI2,rule disjI2,rule disjI1,erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(simp add:Graph6)
---{* 6 subgoals left *}
-apply(simp add:mul_mutator_defs mul_collector_defs Mul_CountInv_def)
-apply clarify
-apply disjE_tac
- apply(simp add:Graph6 Queue_def)
- apply(rule impI,rule disjI1,rule subset_trans,erule Graph3,simp,simp)
- apply(simp add:Graph6)
-apply clarify
-apply disjE_tac
- apply(simp add:Graph6)
- apply(rule conjI)
- apply(rule impI,rule disjI2,rule disjI2,rule disjI1,erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
- apply(rule impI,rule disjI2,rule disjI2,rule disjI1,erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(simp add:Graph6)
---{* 5 subgoals left *}
-apply(simp add:mul_mutator_defs mul_collector_defs Mul_CountInv_def)
-apply clarify
-apply disjE_tac
- apply(simp add:Graph6)
- apply(rule impI,rule disjI1,rule subset_trans,erule Graph3,simp,simp)
- apply(simp add:Graph6)
-apply clarify
-apply disjE_tac
- apply(simp add:Graph6)
- apply(rule conjI)
- apply(rule impI,rule disjI2,rule disjI2,rule disjI1,erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
- apply(rule impI,rule disjI2,rule disjI2,rule disjI1,erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(simp add:Graph6)
---{* 4 subgoals left *}
-apply(simp add:mul_mutator_defs mul_collector_defs Mul_CountInv_def)
-apply clarify
-apply disjE_tac
- apply(simp add:Graph6)
- apply(rule impI,rule disjI1,rule subset_trans,erule Graph3,simp,simp)
- apply(simp add:Graph6)
-apply clarify
-apply disjE_tac
- apply(simp add:Graph6)
- apply(rule conjI)
- apply(rule impI,rule disjI2,rule disjI2,rule disjI1,erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
- apply(rule impI,rule disjI2,rule disjI2,rule disjI1,erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(simp add:Graph6)
---{* 3 subgoals left *}
-apply(simp add:mul_mutator_defs nth_list_update)
---{* 2 subgoals left *}
-apply(simp add:mul_mutator_defs mul_collector_defs Mul_CountInv_def)
-apply clarify
-apply disjE_tac
- apply(simp add:Graph6)
- apply(rule impI,rule disjI1,rule subset_trans,erule Graph3,simp,simp)
- apply(simp add:Graph6)
-apply clarify
-apply disjE_tac
- apply(simp add:Graph6)
- apply(rule conjI)
- apply(rule impI,rule disjI2,rule disjI2,rule disjI1,erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
- apply(rule impI,rule disjI2,rule disjI2,rule disjI1,erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(simp add:Graph6)
---{* 1 subgoal left *}
-apply(simp add:mul_mutator_defs nth_list_update)
-done
-
-lemma Mul_interfree_Redirect_Edge_Count: "\<lbrakk>0\<le>j; j<n\<rbrakk>\<Longrightarrow>
- interfree_aux (Some(Mul_Redirect_Edge j n),{},Some(Mul_Count n ))"
-apply (unfold mul_modules)
-apply interfree_aux
-apply safe
-apply(simp_all add:mul_mutator_defs nth_list_update)
-done
-
-lemma Mul_interfree_Count_Color_Target: "\<lbrakk>0\<le>j; j<n\<rbrakk>\<Longrightarrow>
- interfree_aux (Some(Mul_Count n ),{},Some(Mul_Color_Target j n))"
-apply (unfold mul_modules)
-apply interfree_aux
-apply(simp_all add:mul_collector_defs mul_mutator_defs Mul_CountInv_def)
---{* 6 subgoals left *}
-apply clarify
-apply disjE_tac
- apply (simp add: Graph7 Graph8 Graph12)
- apply (simp add: Graph7 Graph8 Graph12)
-apply clarify
-apply disjE_tac
- apply (simp add: Graph7 Graph8 Graph12)
- apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(rule disjI2,rule disjI2, rule disjI1, erule le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
- apply((rule disjI2)+,(erule subset_psubset_trans)+, simp add: Graph11)
-apply (simp add: Graph7 Graph8 Graph12)
-apply((rule disjI2)+,erule psubset_subset_trans, simp add: Graph9)
---{* 5 subgoals left *}
-apply clarify
-apply disjE_tac
- apply (simp add: Graph7 Graph8 Graph12)
- apply (simp add: Graph7 Graph8 Graph12)
-apply clarify
-apply disjE_tac
- apply (simp add: Graph7 Graph8 Graph12)
- apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(rule disjI2,rule disjI2, rule disjI1, erule le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
- apply((rule disjI2)+,(erule subset_psubset_trans)+, simp add: Graph11)
-apply (simp add: Graph7 Graph8 Graph12)
-apply((rule disjI2)+,erule psubset_subset_trans, simp add: Graph9)
---{* 4 subgoals left *}
-apply clarify
-apply disjE_tac
- apply (simp add: Graph7 Graph8 Graph12)
- apply (simp add: Graph7 Graph8 Graph12)
-apply clarify
-apply disjE_tac
- apply (simp add: Graph7 Graph8 Graph12)
- apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(rule disjI2,rule disjI2, rule disjI1, erule le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
- apply((rule disjI2)+,(erule subset_psubset_trans)+, simp add: Graph11)
-apply (simp add: Graph7 Graph8 Graph12)
-apply((rule disjI2)+,erule psubset_subset_trans, simp add: Graph9)
---{* 3 subgoals left *}
-apply clarify
-apply disjE_tac
- apply (simp add: Graph7 Graph8 Graph12)
- apply (simp add: Graph7 Graph8 Graph12)
-apply clarify
-apply disjE_tac
- apply (simp add: Graph7 Graph8 Graph12)
- apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(rule disjI2,rule disjI2, rule disjI1, erule le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
- apply((rule disjI2)+,(erule subset_psubset_trans)+, simp add: Graph11)
-apply (simp add: Graph7 Graph8 Graph12)
-apply((rule disjI2)+,erule psubset_subset_trans, simp add: Graph9)
---{* 2 subgoals left *}
-apply clarify
-apply disjE_tac
- apply (simp add: Graph7 Graph8 Graph12 nth_list_update)
- apply (simp add: Graph7 Graph8 Graph12 nth_list_update)
-apply clarify
-apply disjE_tac
- apply (simp add: Graph7 Graph8 Graph12)
- apply(rule conjI)
- apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(rule disjI2,rule disjI2, rule disjI1, erule le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
- apply((rule disjI2)+,(erule subset_psubset_trans)+, simp add: Graph11)
- apply (simp add: nth_list_update)
-apply (simp add: Graph7 Graph8 Graph12)
-apply(rule conjI)
- apply((rule disjI2)+,erule psubset_subset_trans, simp add: Graph9)
-apply (simp add: nth_list_update)
---{* 1 subgoal left *}
-apply clarify
-apply disjE_tac
- apply (simp add: Graph7 Graph8 Graph12)
- apply (simp add: Graph7 Graph8 Graph12)
-apply clarify
-apply disjE_tac
- apply (simp add: Graph7 Graph8 Graph12)
- apply(case_tac "M x!(T (Muts x!j))=Black")
- apply(rule disjI2,rule disjI2, rule disjI1, erule le_trans)
- apply(force simp add:Queue_def less_Suc_eq_le le_length_filter_update Graph10)
- apply((rule disjI2)+,(erule subset_psubset_trans)+, simp add: Graph11)
-apply (simp add: Graph7 Graph8 Graph12)
-apply((rule disjI2)+,erule psubset_subset_trans, simp add: Graph9)
-done
-
-lemma Mul_interfree_Color_Target_Count: "\<lbrakk>0\<le>j; j<n\<rbrakk>\<Longrightarrow>
- interfree_aux (Some(Mul_Color_Target j n),{}, Some(Mul_Count n ))"
-apply (unfold mul_modules)
-apply interfree_aux
-apply safe
-apply(simp_all add:mul_mutator_defs nth_list_update)
-done
-
-lemma Mul_interfree_Append_Redirect_Edge: "\<lbrakk>0\<le>j; j<n\<rbrakk>\<Longrightarrow>
- interfree_aux (Some(Mul_Append n),{}, Some(Mul_Redirect_Edge j n))"
-apply (unfold mul_modules)
-apply interfree_aux
-apply(tactic {* ALLGOALS (clarify_tac @{claset}) *})
-apply(simp_all add:Graph6 Append_to_free0 Append_to_free1 mul_collector_defs mul_mutator_defs Mul_AppendInv_def)
-apply(erule_tac x=j in allE, force dest:Graph3)+
-done
-
-lemma Mul_interfree_Redirect_Edge_Append: "\<lbrakk>0\<le>j; j<n\<rbrakk>\<Longrightarrow>
- interfree_aux (Some(Mul_Redirect_Edge j n),{},Some(Mul_Append n))"
-apply (unfold mul_modules)
-apply interfree_aux
-apply(tactic {* ALLGOALS (clarify_tac @{claset}) *})
-apply(simp_all add:mul_collector_defs Append_to_free0 Mul_AppendInv_def mul_mutator_defs nth_list_update)
-done
-
-lemma Mul_interfree_Append_Color_Target: "\<lbrakk>0\<le>j; j<n\<rbrakk>\<Longrightarrow>
- interfree_aux (Some(Mul_Append n),{}, Some(Mul_Color_Target j n))"
-apply (unfold mul_modules)
-apply interfree_aux
-apply(tactic {* ALLGOALS (clarify_tac @{claset}) *})
-apply(simp_all add:mul_mutator_defs mul_collector_defs Mul_AppendInv_def Graph7 Graph8 Append_to_free0 Append_to_free1
- Graph12 nth_list_update)
-done
-
-lemma Mul_interfree_Color_Target_Append: "\<lbrakk>0\<le>j; j<n\<rbrakk>\<Longrightarrow>
- interfree_aux (Some(Mul_Color_Target j n),{}, Some(Mul_Append n))"
-apply (unfold mul_modules)
-apply interfree_aux
-apply(tactic {* ALLGOALS (clarify_tac @{claset}) *})
-apply(simp_all add: mul_mutator_defs nth_list_update)
-apply(simp add:Mul_AppendInv_def Append_to_free0)
-done
-
-subsubsection {* Interference freedom Collector-Mutator *}
-
-lemmas mul_collector_mutator_interfree =
- Mul_interfree_Blacken_Roots_Redirect_Edge Mul_interfree_Blacken_Roots_Color_Target
- Mul_interfree_Propagate_Black_Redirect_Edge Mul_interfree_Propagate_Black_Color_Target
- Mul_interfree_Count_Redirect_Edge Mul_interfree_Count_Color_Target
- Mul_interfree_Append_Redirect_Edge Mul_interfree_Append_Color_Target
- Mul_interfree_Redirect_Edge_Blacken_Roots Mul_interfree_Color_Target_Blacken_Roots
- Mul_interfree_Redirect_Edge_Propagate_Black Mul_interfree_Color_Target_Propagate_Black
- Mul_interfree_Redirect_Edge_Count Mul_interfree_Color_Target_Count
- Mul_interfree_Redirect_Edge_Append Mul_interfree_Color_Target_Append
-
-lemma Mul_interfree_Collector_Mutator: "j<n \<Longrightarrow>
- interfree_aux (Some (Mul_Collector n), {}, Some (Mul_Mutator j n))"
-apply(unfold Mul_Collector_def Mul_Mutator_def)
-apply interfree_aux
-apply(simp_all add:mul_collector_mutator_interfree)
-apply(unfold mul_modules mul_collector_defs mul_mutator_defs)
-apply(tactic {* TRYALL (interfree_aux_tac) *})
---{* 42 subgoals left *}
-apply (clarify,simp add:Graph6 Graph7 Graph8 Append_to_free0 Append_to_free1 Graph12)+
---{* 24 subgoals left *}
-apply(simp_all add:Graph6 Graph7 Graph8 Append_to_free0 Append_to_free1 Graph12)
---{* 14 subgoals left *}
-apply(tactic {* TRYALL (clarify_tac @{claset}) *})
-apply(simp_all add:Graph6 Graph7 Graph8 Append_to_free0 Append_to_free1 Graph12)
-apply(tactic {* TRYALL (rtac conjI) *})
-apply(tactic {* TRYALL (rtac impI) *})
-apply(tactic {* TRYALL (etac disjE) *})
-apply(tactic {* TRYALL (etac conjE) *})
-apply(tactic {* TRYALL (etac disjE) *})
-apply(tactic {* TRYALL (etac disjE) *})
---{* 72 subgoals left *}
-apply(simp_all add:Graph6 Graph7 Graph8 Append_to_free0 Append_to_free1 Graph12)
---{* 35 subgoals left *}
-apply(tactic {* TRYALL(EVERY'[rtac disjI1,rtac subset_trans,etac @{thm Graph3},force_tac @{clasimpset}, assume_tac]) *})
---{* 28 subgoals left *}
-apply(tactic {* TRYALL (etac conjE) *})
-apply(tactic {* TRYALL (etac disjE) *})
---{* 34 subgoals left *}
-apply(rule disjI2,rule disjI1,erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(rule disjI2,rule disjI1,erule le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update)
-apply(case_tac [!] "M x!(T (Muts x ! j))=Black")
-apply(simp_all add:Graph10)
---{* 47 subgoals left *}
-apply(tactic {* TRYALL(EVERY'[REPEAT o (rtac disjI2),etac (thm "subset_psubset_trans"),etac (thm "Graph11"),force_tac @{clasimpset}]) *})
---{* 41 subgoals left *}
-apply(tactic {* TRYALL(EVERY'[rtac disjI2, rtac disjI1, etac @{thm le_trans}, force_tac (@{claset},@{simpset} addsimps [@{thm Queue_def}, @{thm less_Suc_eq_le}, @{thm le_length_filter_update}])]) *})
---{* 35 subgoals left *}
-apply(tactic {* TRYALL(EVERY'[rtac disjI2,rtac disjI1,etac (thm "psubset_subset_trans"),rtac (thm "Graph9"),force_tac @{clasimpset}]) *})
---{* 31 subgoals left *}
-apply(tactic {* TRYALL(EVERY'[rtac disjI2,rtac disjI1,etac (thm "subset_psubset_trans"),etac (thm "Graph11"),force_tac @{clasimpset}]) *})
---{* 29 subgoals left *}
-apply(tactic {* TRYALL(EVERY'[REPEAT o (rtac disjI2),etac (thm "subset_psubset_trans"),etac (thm "subset_psubset_trans"),etac (thm "Graph11"),force_tac @{clasimpset}]) *})
---{* 25 subgoals left *}
-apply(tactic {* TRYALL(EVERY'[rtac disjI2, rtac disjI2, rtac disjI1, etac @{thm le_trans}, force_tac (@{claset},@{simpset} addsimps [@{thm Queue_def}, @{thm less_Suc_eq_le}, @{thm le_length_filter_update}])]) *})
---{* 10 subgoals left *}
-apply(rule disjI2,rule disjI2,rule conjI,erule less_le_trans,force simp add:Queue_def less_Suc_eq_le le_length_filter_update, rule disjI1, rule less_imp_le, erule less_le_trans, force simp add:Queue_def less_Suc_eq_le le_length_filter_update)+
-done
-
-subsubsection {* Interference freedom Mutator-Collector *}
-
-lemma Mul_interfree_Mutator_Collector: " j < n \<Longrightarrow>
- interfree_aux (Some (Mul_Mutator j n), {}, Some (Mul_Collector n))"
-apply(unfold Mul_Collector_def Mul_Mutator_def)
-apply interfree_aux
-apply(simp_all add:mul_collector_mutator_interfree)
-apply(unfold mul_modules mul_collector_defs mul_mutator_defs)
-apply(tactic {* TRYALL (interfree_aux_tac) *})
---{* 76 subgoals left *}
-apply (clarify,simp add: nth_list_update)+
---{* 56 subgoals left *}
-apply(clarify,simp add:Mul_AppendInv_def Append_to_free0 nth_list_update)+
-done
-
-subsubsection {* The Multi-Mutator Garbage Collection Algorithm *}
-
-text {* The total number of verification conditions is 328 *}
-
-lemma Mul_Gar_Coll:
- "\<parallel>- .{\<acute>Mul_Proper n \<and> \<acute>Mul_mut_init n \<and> (\<forall>i<n. Z (\<acute>Muts!i))}.
- COBEGIN
- Mul_Collector n
- .{False}.
- \<parallel>
- SCHEME [0\<le> j< n]
- Mul_Mutator j n
- .{False}.
- COEND
- .{False}."
-apply oghoare
---{* Strengthening the precondition *}
-apply(rule Int_greatest)
- apply (case_tac n)
- apply(force simp add: Mul_Collector_def mul_mutator_defs mul_collector_defs nth_append)
- apply(simp add: Mul_Mutator_def mul_collector_defs mul_mutator_defs nth_append)
- apply force
-apply clarify
-apply(case_tac i)
- apply(simp add:Mul_Collector_def mul_mutator_defs mul_collector_defs nth_append)
-apply(simp add: Mul_Mutator_def mul_mutator_defs mul_collector_defs nth_append nth_map_upt)
---{* Collector *}
-apply(rule Mul_Collector)
---{* Mutator *}
-apply(erule Mul_Mutator)
---{* Interference freedom *}
-apply(simp add:Mul_interfree_Collector_Mutator)
-apply(simp add:Mul_interfree_Mutator_Collector)
-apply(simp add:Mul_interfree_Mutator_Mutator)
---{* Weakening of the postcondition *}
-apply(case_tac n)
- apply(simp add:Mul_Collector_def mul_mutator_defs mul_collector_defs nth_append)
-apply(simp add:Mul_Mutator_def mul_mutator_defs mul_collector_defs nth_append)
-done
-
-end