
diff -r 98f0a09a33c3 -r 791e3b4c4039 src/HOL/HoareParallel/Graph.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/HoareParallel/Graph.thy	Tue Mar 05 17:11:25 2002 +0100
@@ -0,0 +1,426 @@
+
+header {* \chapter{Case Study: Single and Multi-Mutator Garbage Collection Algorithms}
+
+\section {Formalization of the Memory} *}
+
+theory Graph = Main:
+
+datatype node = Black | White
+
+types 
+  nodes = "node list"
+  edge  = "nat \<times> nat"
+  edges = "edge list"
+
+consts Roots :: "nat set"
+
+constdefs
+  Proper_Roots :: "nodes \<Rightarrow> bool"
+  "Proper_Roots M \<equiv> Roots\<noteq>{} \<and> Roots \<subseteq> {i. i<length M}"
+
+  Proper_Edges :: "(nodes \<times> edges) \<Rightarrow> bool"
+  "Proper_Edges \<equiv> (\<lambda>(M,E). \<forall>i<length E. fst(E!i)<length M \<and> snd(E!i)<length M)"
+
+  BtoW :: "(edge \<times> nodes) \<Rightarrow> bool"
+  "BtoW \<equiv> (\<lambda>(e,M). (M!fst e)=Black \<and> (M!snd e)\<noteq>Black)"
+
+  Blacks :: "nodes \<Rightarrow> nat set"
+  "Blacks M \<equiv> {i. i<length M \<and> M!i=Black}"
+
+  Reach :: "edges \<Rightarrow> nat set"
+  "Reach E \<equiv> {x. (\<exists>path. 1<length path \<and> path!(length path - 1)\<in>Roots \<and> x=path!0
+              \<and> (\<forall>i<length path - 1. (\<exists>j<length E. E!j=(path!(i+1), path!i))))
+	      \<or> x\<in>Roots}"
+
+text{* Reach: the set of reachable nodes is the set of Roots together with the
+nodes reachable from some Root by a path represented by a list of
+  nodes (at least two since we traverse at least one edge), where two
+consecutive nodes correspond to an edge in E. *}
+
+subsection {* Proofs about Graphs *}
+
+lemmas Graph_defs= Blacks_def Proper_Roots_def Proper_Edges_def BtoW_def
+declare Graph_defs [simp]
+
+subsubsection{* Graph 1. *}
+
+lemma Graph1_aux [rule_format]: 
+  "\<lbrakk> Roots\<subseteq>Blacks M; \<forall>i<length E. \<not>BtoW(E!i,M)\<rbrakk>
+  \<Longrightarrow> 1< length path \<longrightarrow> (path!(length path - 1))\<in>Roots \<longrightarrow>  
+  (\<forall>i<length path - 1. (\<exists>j. j < length E \<and> E!j=(path!(Suc i), path!i))) 
+  \<longrightarrow> M!(path!0) = Black"
+apply(induct_tac "path")
+ apply force
+apply clarify
+apply simp
+apply(case_tac "list")
+ apply force
+apply simp
+apply(rotate_tac -1)
+apply(erule_tac x = "0" in all_dupE)
+apply simp
+apply clarify
+apply(erule allE , erule (1) notE impE)
+apply simp
+apply(erule mp)
+apply(case_tac "lista")
+ apply force
+apply simp
+apply(erule mp)
+apply clarify
+apply(erule_tac x = "Suc i" in allE)
+apply force
+done
+
+lemma Graph1: 
+  "\<lbrakk>Roots\<subseteq>Blacks M; Proper_Edges(M, E); \<forall>i<length E. \<not>BtoW(E!i,M) \<rbrakk> 
+  \<Longrightarrow> Reach E\<subseteq>Blacks M"
+apply (unfold Reach_def)
+apply simp
+apply clarify
+apply(erule disjE)
+ apply clarify
+ apply(rule conjI)
+  apply(subgoal_tac "0< length path - Suc 0")
+   apply(erule allE , erule (1) notE impE)
+   apply force
+  apply simp
+ apply(rule Graph1_aux)
+apply auto
+done
+
+subsubsection{* Graph 2. *}
+
+lemma Ex_first_occurrence [rule_format]: 
+  "P (n::nat) \<longrightarrow> (\<exists>m. P m \<and> (\<forall>i. i<m \<longrightarrow> \<not> P i))";
+apply(rule nat_less_induct)
+apply clarify
+apply(case_tac "\<forall>m. m<n \<longrightarrow> \<not> P m")
+apply auto
+done
+
+lemma Compl_lemma: "(n::nat)\<le>l \<Longrightarrow> (\<exists>m. m\<le>l \<and> n=l - m)"
+apply(rule_tac x = "l - n" in exI)
+apply arith
+done
+
+lemma Ex_last_occurrence: 
+  "\<lbrakk>P (n::nat); n\<le>l\<rbrakk> \<Longrightarrow> (\<exists>m. P (l - m) \<and> (\<forall>i. i<m \<longrightarrow> \<not>P (l - i)))"
+apply(drule Compl_lemma)
+apply clarify
+apply(erule Ex_first_occurrence)
+done
+
+lemma Graph2: 
+  "\<lbrakk>T \<in> Reach E; R<length E\<rbrakk> \<Longrightarrow> T \<in> Reach (E[R:=(fst(E!R), T)])"
+apply (unfold Reach_def)
+apply clarify
+apply simp
+apply(case_tac "\<forall>z<length path. fst(E!R)\<noteq>path!z")
+ apply(rule_tac x = "path" in exI)
+ apply simp
+ apply clarify
+ apply(erule allE , erule (1) notE impE)
+ apply clarify
+ apply(rule_tac x = "j" in exI)
+ apply(case_tac "j=R")
+  apply(erule_tac x = "Suc i" in allE)
+  apply simp
+  apply arith
+ apply (force simp add:nth_list_update)
+apply simp
+apply(erule exE)
+apply(subgoal_tac "z \<le> length path - Suc 0")
+ prefer 2 apply arith
+apply(drule_tac P = "\<lambda>m. m<length path \<and> fst(E!R)=path!m" in Ex_last_occurrence)
+ apply assumption
+apply clarify
+apply simp
+apply(rule_tac x = "(path!0)#(drop (length path - Suc m) path)" in exI)
+apply simp
+apply(case_tac "length path - (length path - Suc m)")
+ apply arith
+apply simp
+apply(subgoal_tac "(length path - Suc m) + nat \<le> length path")
+ prefer 2 apply arith
+apply(drule nth_drop)
+apply simp
+apply(subgoal_tac "length path - Suc m + nat = length path - Suc 0")
+ prefer 2 apply arith 
+apply simp
+apply clarify
+apply(case_tac "i")
+ apply(force simp add: nth_list_update)
+apply simp
+apply(subgoal_tac "(length path - Suc m) + nata \<le> length path")
+ prefer 2 apply arith
+apply simp
+apply(subgoal_tac "(length path - Suc m) + (Suc nata) \<le> length path")
+ prefer 2 apply arith
+apply simp
+apply(erule_tac x = "length path - Suc m + nata" in allE)
+apply simp
+apply clarify
+apply(rule_tac x = "j" in exI)
+apply(case_tac "R=j")
+ prefer 2 apply force
+apply simp
+apply(drule_tac t = "path ! (length path - Suc m)" in sym)
+apply simp
+apply(case_tac " length path - Suc 0 < m")
+ apply(subgoal_tac "(length path - Suc m)=0")
+  prefer 2 apply arith
+ apply(rotate_tac -1)
+ apply(simp del: diff_is_0_eq)
+ apply(subgoal_tac "Suc nata\<le>nat")
+ prefer 2 apply arith
+ apply(drule_tac n = "Suc nata" in Compl_lemma)
+ apply clarify
+ apply force
+apply(drule leI)
+apply(subgoal_tac "Suc (length path - Suc m + nata)=(length path - Suc 0) - (m - Suc nata)")
+ apply(erule_tac x = "m - (Suc nata)" in allE)
+ apply(case_tac "m")
+  apply simp
+ apply simp
+ apply(subgoal_tac "natb - nata < Suc natb")
+  prefer 2 apply(erule thin_rl)+ apply arith
+ apply simp
+ apply(case_tac "length path")
+  apply force
+ apply simp
+apply(frule_tac i1 = "length path" and j1 = "length path - Suc 0" and k1 = "m" in diff_diff_right [THEN mp])
+apply(erule_tac V = "length path - Suc m + nat = length path - Suc 0" in thin_rl)
+apply simp
+apply arith
+done
+
+
+subsubsection{* Graph 3. *}
+
+lemma Graph3: 
+  "\<lbrakk> T\<in>Reach E; R<length E \<rbrakk> \<Longrightarrow> Reach(E[R:=(fst(E!R),T)]) \<subseteq> Reach E"
+apply (unfold Reach_def)
+apply clarify
+apply simp
+apply(case_tac "\<exists>i<length path - 1. (fst(E!R),T)=(path!(Suc i),path!i)")
+--{* the changed edge is part of the path *}
+ apply(erule exE)
+ apply(drule_tac P = "\<lambda>i. i<length path - 1 \<and> (fst(E!R),T)=(path!Suc i,path!i)" in Ex_first_occurrence)
+ apply clarify
+ apply(erule disjE)
+--{* T is NOT a root *}
+  apply clarify
+  apply(rule_tac x = "(take m path)@patha" in exI)
+  apply(subgoal_tac "\<not>(length path\<le>m)")
+   prefer 2 apply arith
+  apply(simp add: min_def)
+  apply(rule conjI)
+   apply(subgoal_tac "\<not>(m + length patha - 1 < m)")
+    prefer 2 apply arith
+   apply(simp add: nth_append min_def)
+  apply(rule conjI)
+   apply(case_tac "m")
+    apply force
+   apply(case_tac "path")
+    apply force
+   apply force
+  apply clarify
+  apply(case_tac "Suc i\<le>m")
+   apply(erule_tac x = "i" in allE)
+   apply simp
+   apply clarify
+   apply(rule_tac x = "j" in exI)
+   apply(case_tac "Suc i<m")
+    apply(simp add: nth_append min_def)
+    apply(case_tac "R=j")
+     apply(simp add: nth_list_update)
+     apply(case_tac "i=m")
+      apply force
+     apply(erule_tac x = "i" in allE)
+     apply force
+    apply(force simp add: nth_list_update)
+   apply(simp add: nth_append min_def)
+   apply(subgoal_tac "i=m - 1")
+    prefer 2 apply arith
+   apply(case_tac "R=j")
+    apply(erule_tac x = "m - 1" in allE)
+    apply(simp add: nth_list_update)
+   apply(force simp add: nth_list_update)
+  apply(simp add: nth_append min_def)
+  apply(rotate_tac -4)
+  apply(erule_tac x = "i - m" in allE)
+  apply(subgoal_tac "Suc (i - m)=(Suc i - m)" )
+    prefer 2 apply arith
+   apply simp
+  apply(erule mp)
+  apply arith
+--{* T is a root *}
+ apply(case_tac "m=0")
+  apply force
+ apply(rule_tac x = "take (Suc m) path" in exI)
+ apply(subgoal_tac "\<not>(length path\<le>Suc m)" )
+  prefer 2 apply arith
+ apply(simp add: min_def)
+ apply clarify
+ apply(erule_tac x = "i" in allE)
+ apply simp
+ apply clarify
+ apply(case_tac "R=j")
+  apply(force simp add: nth_list_update)
+ apply(force simp add: nth_list_update)
+--{* the changed edge is not part of the path *}
+apply(rule_tac x = "path" in exI)
+apply simp
+apply clarify
+apply(erule_tac x = "i" in allE)
+apply clarify
+apply(case_tac "R=j")
+ apply(erule_tac x = "i" in allE)
+ apply simp
+apply(force simp add: nth_list_update)
+done
+
+subsubsection{* Graph 4. *}
+
+lemma Graph4: 
+  "\<lbrakk>T \<in> Reach E; Roots\<subseteq>Blacks M; I\<le>length E; T<length M; R<length E; 
+  \<forall>i<I. \<not>BtoW(E!i,M); R<I; M!fst(E!R)=Black; M!T\<noteq>Black\<rbrakk> \<Longrightarrow> 
+  (\<exists>r. I\<le>r \<and> r<length E \<and> BtoW(E[R:=(fst(E!R),T)]!r,M))"
+apply (unfold Reach_def)
+apply simp
+apply(erule disjE)
+ prefer 2 apply force
+apply clarify
+--{* there exist a black node in the path to T *}
+apply(case_tac "\<exists>m<length path. M!(path!m)=Black")
+ apply(erule exE)
+ apply(drule_tac P = "\<lambda>m. m<length path \<and> M!(path!m)=Black" in Ex_first_occurrence)
+ apply clarify
+ apply(case_tac "ma")
+  apply force
+ apply simp
+ apply(case_tac "length path")
+  apply force
+ apply simp
+ apply(rotate_tac -5)
+ apply(erule_tac x = "nat" in allE)
+ apply simp
+ apply clarify
+ apply(erule_tac x = "nat" in allE)
+ apply simp
+ apply(case_tac "j<I")
+  apply(erule_tac x = "j" in allE)
+  apply force
+ apply(rule_tac x = "j" in exI)
+ apply(force  simp add: nth_list_update)
+apply simp
+apply(rotate_tac -1)
+apply(erule_tac x = "length path - 1" in allE)
+apply(case_tac "length path")
+ apply force
+apply force
+done
+
+subsubsection {* Graph 5. *}
+
+lemma Graph5: 
+  "\<lbrakk> T \<in> Reach E ; Roots \<subseteq> Blacks M; \<forall>i<R. \<not>BtoW(E!i,M); T<length M; 
+    R<length E; M!fst(E!R)=Black; M!snd(E!R)=Black; M!T \<noteq> Black\<rbrakk> 
+   \<Longrightarrow> (\<exists>r. R<r \<and> r<length E \<and> BtoW(E[R:=(fst(E!R),T)]!r,M))"
+apply (unfold Reach_def)
+apply simp
+apply(erule disjE)
+ prefer 2 apply force
+apply clarify
+--{* there exist a black node in the path to T*}
+apply(case_tac "\<exists>m<length path. M!(path!m)=Black")
+ apply(erule exE)
+ apply(drule_tac P = "\<lambda>m. m<length path \<and> M!(path!m)=Black" in Ex_first_occurrence)
+ apply clarify
+ apply(case_tac "ma")
+  apply force
+ apply simp
+ apply(case_tac "length path")
+  apply force
+ apply simp
+ apply(rotate_tac -5)
+ apply(erule_tac x = "nat" in allE)
+ apply simp
+ apply clarify
+ apply(erule_tac x = "nat" in allE)
+ apply simp
+ apply(case_tac "j\<le>R")
+  apply(drule le_imp_less_or_eq)
+  apply(erule disjE)
+   apply(erule allE , erule (1) notE impE)
+   apply force
+  apply force
+ apply(rule_tac x = "j" in exI)
+ apply(force  simp add: nth_list_update)
+apply simp
+apply(rotate_tac -1)
+apply(erule_tac x = "length path - 1" in allE)
+apply(case_tac "length path")
+ apply force
+apply force
+done
+
+subsubsection {* Graph 6, 7, 8. *}
+
+lemma Graph6: 
+ "\<lbrakk>Proper_Edges(M,E); R<length E ; T<length M\<rbrakk> \<Longrightarrow> Proper_Edges(M,E[R:=(fst(E!R),T)])"
+apply (unfold Proper_Edges_def)
+ apply(force  simp add: nth_list_update)
+done
+
+lemma Graph7: 
+ "\<lbrakk>Proper_Edges(M,E)\<rbrakk> \<Longrightarrow> Proper_Edges(M[T:=a],E)"
+apply (unfold Proper_Edges_def)
+apply force
+done
+
+lemma Graph8: 
+ "\<lbrakk>Proper_Roots(M)\<rbrakk> \<Longrightarrow> Proper_Roots(M[T:=a])"
+apply (unfold Proper_Roots_def)
+apply force
+done
+
+text{* Some specific lemmata for the verification of garbage collection algorithms. *}
+
+lemma Graph9: "j<length M \<Longrightarrow> Blacks M\<subseteq>Blacks (M[j := Black])"
+apply (unfold Blacks_def)
+ apply(force simp add: nth_list_update)
+done
+
+lemma Graph10 [rule_format (no_asm)]: "\<forall>i. M!i=a \<longrightarrow>M[i:=a]=M"
+apply(induct_tac "M")
+apply auto
+apply(case_tac "i")
+apply auto
+done
+
+lemma Graph11 [rule_format (no_asm)]: 
+  "\<lbrakk> M!j\<noteq>Black;j<length M\<rbrakk> \<Longrightarrow> Blacks M \<subset> Blacks (M[j := Black])"
+apply (unfold Blacks_def)
+apply(rule psubsetI)
+ apply(force simp add: nth_list_update)
+apply safe
+apply(erule_tac c = "j" in equalityCE)
+apply auto
+done
+
+lemma Graph12: "\<lbrakk>a\<subseteq>Blacks M;j<length M\<rbrakk> \<Longrightarrow> a\<subseteq>Blacks (M[j := Black])"
+apply (unfold Blacks_def)
+apply(force simp add: nth_list_update)
+done
+
+lemma Graph13: "\<lbrakk>a\<subset> Blacks M;j<length M\<rbrakk> \<Longrightarrow> a \<subset> Blacks (M[j := Black])"
+apply (unfold Blacks_def)
+apply(erule psubset_subset_trans)
+apply(force simp add: nth_list_update)
+done
+
+declare Graph_defs [simp del]
+
+end




