author  haftmann 
Wed, 14 Jul 2010 16:13:14 +0200  
changeset 37826  4c0a5e35931a 
parent 37792  ba0bc31b90d7 
child 37828  9e1758c7ff06 
permissions  rwrr 
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(* Title: HOL/Imperative_HOL/ex/Linked_Lists.thy 
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Author: Lukas Bulwahn, TU Muenchen 
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*) 
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header {* Linked Lists by ML references *} 
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theory Linked_Lists 
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imports Imperative_HOL Code_Integer 
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begin 
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section {* Definition of Linked Lists *} 
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setup {* Sign.add_const_constraint (@{const_name Ref}, SOME @{typ "nat \<Rightarrow> 'a\<Colon>type ref"}) *} 
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datatype 'a node = Empty  Node 'a "('a node) ref" 
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primrec 
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node_encode :: "'a\<Colon>countable node \<Rightarrow> nat" 
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where 
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"node_encode Empty = 0" 
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 "node_encode (Node x r) = Suc (to_nat (x, r))" 
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instance node :: (countable) countable 
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proof (rule countable_classI [of "node_encode"]) 
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fix x y :: "'a\<Colon>countable node" 
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show "node_encode x = node_encode y \<Longrightarrow> x = y" 
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by (induct x, auto, induct y, auto, induct y, auto) 
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qed 
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instance node :: (heap) heap .. 
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primrec make_llist :: "'a\<Colon>heap list \<Rightarrow> 'a node Heap" 
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where 
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[simp del]: "make_llist [] = return Empty" 
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 "make_llist (x#xs) = do { tl \<leftarrow> make_llist xs; 
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next \<leftarrow> ref tl; 

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return (Node x next) 

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}" 

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text {* define traverse using the MREC combinator *} 
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definition 
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traverse :: "'a\<Colon>heap node \<Rightarrow> 'a list Heap" 
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where 
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[code del]: "traverse = MREC (\<lambda>n. case n of Empty \<Rightarrow> return (Inl []) 
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 Node x r \<Rightarrow> do { tl \<leftarrow> Ref.lookup r; 
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return (Inr tl) }) 

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(\<lambda>n tl xs. case n of Empty \<Rightarrow> undefined 
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 Node x r \<Rightarrow> return (x # xs))" 
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lemma traverse_simps[code, simp]: 
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"traverse Empty = return []" 
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"traverse (Node x r) = do { tl \<leftarrow> Ref.lookup r; 
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xs \<leftarrow> traverse tl; 

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return (x#xs) 

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}" 

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unfolding traverse_def 
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by (auto simp: traverse_def MREC_rule) 
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section {* Proving correctness with relational abstraction *} 
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subsection {* Definition of list_of, list_of', refs_of and refs_of' *} 
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primrec list_of :: "heap \<Rightarrow> ('a::heap) node \<Rightarrow> 'a list \<Rightarrow> bool" 
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where 
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"list_of h r [] = (r = Empty)" 
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 "list_of h r (a#as) = (case r of Empty \<Rightarrow> False  Node b bs \<Rightarrow> (a = b \<and> list_of h (Ref.get h bs) as))" 
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definition list_of' :: "heap \<Rightarrow> ('a::heap) node ref \<Rightarrow> 'a list \<Rightarrow> bool" 
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where 
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"list_of' h r xs = list_of h (Ref.get h r) xs" 
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primrec refs_of :: "heap \<Rightarrow> ('a::heap) node \<Rightarrow> 'a node ref list \<Rightarrow> bool" 
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where 
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"refs_of h r [] = (r = Empty)" 
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 "refs_of h r (x#xs) = (case r of Empty \<Rightarrow> False  Node b bs \<Rightarrow> (x = bs) \<and> refs_of h (Ref.get h bs) xs)" 
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primrec refs_of' :: "heap \<Rightarrow> ('a::heap) node ref \<Rightarrow> 'a node ref list \<Rightarrow> bool" 
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where 
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"refs_of' h r [] = False" 
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 "refs_of' h r (x#xs) = ((x = r) \<and> refs_of h (Ref.get h x) xs)" 
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subsection {* Properties of these definitions *} 
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lemma list_of_Empty[simp]: "list_of h Empty xs = (xs = [])" 
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by (cases xs, auto) 
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lemma list_of_Node[simp]: "list_of h (Node x ps) xs = (\<exists>xs'. (xs = x # xs') \<and> list_of h (Ref.get h ps) xs')" 
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by (cases xs, auto) 
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lemma list_of'_Empty[simp]: "Ref.get h q = Empty \<Longrightarrow> list_of' h q xs = (xs = [])" 
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unfolding list_of'_def by simp 
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lemma list_of'_Node[simp]: "Ref.get h q = Node x ps \<Longrightarrow> list_of' h q xs = (\<exists>xs'. (xs = x # xs') \<and> list_of' h ps xs')" 
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unfolding list_of'_def by simp 
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lemma list_of'_Nil: "list_of' h q [] \<Longrightarrow> Ref.get h q = Empty" 
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unfolding list_of'_def by simp 
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lemma list_of'_Cons: 
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assumes "list_of' h q (x#xs)" 
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obtains n where "Ref.get h q = Node x n" and "list_of' h n xs" 
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using assms unfolding list_of'_def by (auto split: node.split_asm) 
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lemma refs_of_Empty[simp] : "refs_of h Empty xs = (xs = [])" 
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by (cases xs, auto) 
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lemma refs_of_Node[simp]: "refs_of h (Node x ps) xs = (\<exists>prs. xs = ps # prs \<and> refs_of h (Ref.get h ps) prs)" 
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by (cases xs, auto) 
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lemma refs_of'_def': "refs_of' h p ps = (\<exists>prs. (ps = (p # prs)) \<and> refs_of h (Ref.get h p) prs)" 
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by (cases ps, auto) 
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lemma refs_of'_Node: 
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assumes "refs_of' h p xs" 
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assumes "Ref.get h p = Node x pn" 
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obtains pnrs 
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where "xs = p # pnrs" and "refs_of' h pn pnrs" 
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using assms 
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unfolding refs_of'_def' by auto 
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lemma list_of_is_fun: "\<lbrakk> list_of h n xs; list_of h n ys\<rbrakk> \<Longrightarrow> xs = ys" 
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proof (induct xs arbitrary: ys n) 
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case Nil thus ?case by auto 
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next 
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case (Cons x xs') 
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thus ?case 
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131 
by (cases ys, auto split: node.split_asm) 
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132 
qed 
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133 

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134 
lemma refs_of_is_fun: "\<lbrakk> refs_of h n xs; refs_of h n ys\<rbrakk> \<Longrightarrow> xs = ys" 
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135 
proof (induct xs arbitrary: ys n) 
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136 
case Nil thus ?case by auto 
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137 
next 
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138 
case (Cons x xs') 
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139 
thus ?case 
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140 
by (cases ys, auto split: node.split_asm) 
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141 
qed 
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142 

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143 
lemma refs_of'_is_fun: "\<lbrakk> refs_of' h p as; refs_of' h p bs \<rbrakk> \<Longrightarrow> as = bs" 
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144 
unfolding refs_of'_def' by (auto dest: refs_of_is_fun) 
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145 

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146 

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147 
lemma list_of_refs_of_HOL: 
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148 
assumes "list_of h r xs" 
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149 
shows "\<exists>rs. refs_of h r rs" 
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150 
using assms 
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151 
proof (induct xs arbitrary: r) 
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152 
case Nil thus ?case by auto 
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153 
next 
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154 
case (Cons x xs') 
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155 
thus ?case 
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156 
by (cases r, auto) 
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157 
qed 
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158 

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159 
lemma list_of_refs_of: 
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160 
assumes "list_of h r xs" 
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161 
obtains rs where "refs_of h r rs" 
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162 
using list_of_refs_of_HOL[OF assms] 
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163 
by auto 
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164 

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165 
lemma list_of'_refs_of'_HOL: 
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166 
assumes "list_of' h r xs" 
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167 
shows "\<exists>rs. refs_of' h r rs" 
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168 
proof  
37725  169 
from assms obtain rs' where "refs_of h (Ref.get h r) rs'" 
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170 
unfolding list_of'_def by (rule list_of_refs_of) 
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171 
thus ?thesis unfolding refs_of'_def' by auto 
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172 
qed 
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173 

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174 
lemma list_of'_refs_of': 
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175 
assumes "list_of' h r xs" 
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176 
obtains rs where "refs_of' h r rs" 
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177 
using list_of'_refs_of'_HOL[OF assms] 
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178 
by auto 
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179 

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180 
lemma refs_of_list_of_HOL: 
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181 
assumes "refs_of h r rs" 
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182 
shows "\<exists>xs. list_of h r xs" 
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183 
using assms 
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184 
proof (induct rs arbitrary: r) 
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185 
case Nil thus ?case by auto 
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186 
next 
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187 
case (Cons r rs') 
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188 
thus ?case 
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189 
by (cases r, auto) 
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190 
qed 
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191 

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192 
lemma refs_of_list_of: 
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193 
assumes "refs_of h r rs" 
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194 
obtains xs where "list_of h r xs" 
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195 
using refs_of_list_of_HOL[OF assms] 
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196 
by auto 
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197 

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198 
lemma refs_of'_list_of'_HOL: 
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199 
assumes "refs_of' h r rs" 
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200 
shows "\<exists>xs. list_of' h r xs" 
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201 
using assms 
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202 
unfolding list_of'_def refs_of'_def' 
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203 
by (auto intro: refs_of_list_of) 
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204 

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205 

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206 
lemma refs_of'_list_of': 
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207 
assumes "refs_of' h r rs" 
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208 
obtains xs where "list_of' h r xs" 
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209 
using refs_of'_list_of'_HOL[OF assms] 
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210 
by auto 
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211 

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212 
lemma refs_of'E: "refs_of' h q rs \<Longrightarrow> q \<in> set rs" 
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213 
unfolding refs_of'_def' by auto 
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214 

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215 
lemma list_of'_refs_of'2: 
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216 
assumes "list_of' h r xs" 
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217 
shows "\<exists>rs'. refs_of' h r (r#rs')" 
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218 
proof  
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219 
from assms obtain rs where "refs_of' h r rs" by (rule list_of'_refs_of') 
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220 
thus ?thesis by (auto simp add: refs_of'_def') 
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221 
qed 
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222 

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223 
subsection {* More complicated properties of these predicates *} 
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224 

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225 
lemma list_of_append: 
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226 
"list_of h n (as @ bs) \<Longrightarrow> \<exists>m. list_of h m bs" 
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227 
apply (induct as arbitrary: n) 
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228 
apply auto 
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229 
apply (case_tac n) 
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230 
apply auto 
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231 
done 
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232 

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233 
lemma refs_of_append: "refs_of h n (as @ bs) \<Longrightarrow> \<exists>m. refs_of h m bs" 
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234 
apply (induct as arbitrary: n) 
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235 
apply auto 
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236 
apply (case_tac n) 
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237 
apply auto 
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238 
done 
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239 

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240 
lemma refs_of_next: 
37725  241 
assumes "refs_of h (Ref.get h p) rs" 
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242 
shows "p \<notin> set rs" 
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243 
proof (rule ccontr) 
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244 
assume a: "\<not> (p \<notin> set rs)" 
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245 
from this obtain as bs where split:"rs = as @ p # bs" by (fastsimp dest: split_list) 
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246 
with assms obtain q where "refs_of h q (p # bs)" by (fast dest: refs_of_append) 
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247 
with assms split show "False" 
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248 
by (cases q,auto dest: refs_of_is_fun) 
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249 
qed 
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250 

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251 
lemma refs_of_distinct: "refs_of h p rs \<Longrightarrow> distinct rs" 
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252 
proof (induct rs arbitrary: p) 
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253 
case Nil thus ?case by simp 
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254 
next 
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255 
case (Cons r rs') 
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256 
thus ?case 
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257 
by (cases p, auto simp add: refs_of_next) 
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258 
qed 
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259 

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260 
lemma refs_of'_distinct: "refs_of' h p rs \<Longrightarrow> distinct rs" 
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261 
unfolding refs_of'_def' 
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262 
by (fastsimp simp add: refs_of_distinct refs_of_next) 
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263 

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264 

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265 
subsection {* Interaction of these predicates with our heap transitions *} 
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266 

37725  267 
lemma list_of_set_ref: "refs_of h q rs \<Longrightarrow> p \<notin> set rs \<Longrightarrow> list_of (Ref.set p v h) q as = list_of h q as" 
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268 
using assms 
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269 
proof (induct as arbitrary: q rs) 
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270 
case Nil thus ?case by simp 
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271 
next 
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272 
case (Cons x xs) 
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273 
thus ?case 
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274 
proof (cases q) 
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275 
case Empty thus ?thesis by auto 
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276 
next 
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277 
case (Node a ref) 
37725  278 
from Cons(2) Node obtain rs' where 1: "refs_of h (Ref.get h ref) rs'" and rs_rs': "rs = ref # rs'" by auto 
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279 
from Cons(3) rs_rs' have "ref \<noteq> p" by fastsimp 
37725  280 
hence ref_eq: "Ref.get (Ref.set p v h) ref = (Ref.get h ref)" by (auto simp add: Ref.get_set_neq) 
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281 
from rs_rs' Cons(3) have 2: "p \<notin> set rs'" by simp 
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282 
from Cons.hyps[OF 1 2] Node ref_eq show ?thesis by simp 
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283 
qed 
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284 
qed 
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285 

37725  286 
lemma refs_of_set_ref: "refs_of h q rs \<Longrightarrow> p \<notin> set rs \<Longrightarrow> refs_of (Ref.set p v h) q as = refs_of h q as" 
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287 
proof (induct as arbitrary: q rs) 
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288 
case Nil thus ?case by simp 
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289 
next 
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290 
case (Cons x xs) 
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291 
thus ?case 
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292 
proof (cases q) 
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293 
case Empty thus ?thesis by auto 
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294 
next 
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295 
case (Node a ref) 
37725  296 
from Cons(2) Node obtain rs' where 1: "refs_of h (Ref.get h ref) rs'" and rs_rs': "rs = ref # rs'" by auto 
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297 
from Cons(3) rs_rs' have "ref \<noteq> p" by fastsimp 
37725  298 
hence ref_eq: "Ref.get (Ref.set p v h) ref = (Ref.get h ref)" by (auto simp add: Ref.get_set_neq) 
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299 
from rs_rs' Cons(3) have 2: "p \<notin> set rs'" by simp 
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300 
from Cons.hyps[OF 1 2] Node ref_eq show ?thesis by auto 
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301 
qed 
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302 
qed 
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303 

37725  304 
lemma refs_of_set_ref2: "refs_of (Ref.set p v h) q rs \<Longrightarrow> p \<notin> set rs \<Longrightarrow> refs_of (Ref.set p v h) q rs = refs_of h q rs" 
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305 
proof (induct rs arbitrary: q) 
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306 
case Nil thus ?case by simp 
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307 
next 
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308 
case (Cons x xs) 
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309 
thus ?case 
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310 
proof (cases q) 
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311 
case Empty thus ?thesis by auto 
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312 
next 
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313 
case (Node a ref) 
37725  314 
from Cons(2) Node have 1:"refs_of (Ref.set p v h) (Ref.get (Ref.set p v h) ref) xs" and x_ref: "x = ref" by auto 
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315 
from Cons(3) this have "ref \<noteq> p" by fastsimp 
37725  316 
hence ref_eq: "Ref.get (Ref.set p v h) ref = (Ref.get h ref)" by (auto simp add: Ref.get_set_neq) 
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317 
from Cons(3) have 2: "p \<notin> set xs" by simp 
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318 
with Cons.hyps 1 2 Node ref_eq show ?thesis 
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319 
by simp 
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320 
qed 
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321 
qed 
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322 

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323 
lemma list_of'_set_ref: 
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324 
assumes "refs_of' h q rs" 
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325 
assumes "p \<notin> set rs" 
37725  326 
shows "list_of' (Ref.set p v h) q as = list_of' h q as" 
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327 
proof  
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328 
from assms have "q \<noteq> p" by (auto simp only: dest!: refs_of'E) 
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329 
with assms show ?thesis 
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330 
unfolding list_of'_def refs_of'_def' 
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331 
by (auto simp add: list_of_set_ref) 
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332 
qed 
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333 

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334 
lemma list_of'_set_next_ref_Node[simp]: 
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335 
assumes "list_of' h r xs" 
37725  336 
assumes "Ref.get h p = Node x r'" 
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337 
assumes "refs_of' h r rs" 
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338 
assumes "p \<notin> set rs" 
37725  339 
shows "list_of' (Ref.set p (Node x r) h) p (x#xs) = list_of' h r xs" 
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340 
using assms 
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341 
unfolding list_of'_def refs_of'_def' 
37725  342 
by (auto simp add: list_of_set_ref Ref.noteq_sym) 
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343 

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344 
lemma refs_of'_set_ref: 
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345 
assumes "refs_of' h q rs" 
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346 
assumes "p \<notin> set rs" 
37725  347 
shows "refs_of' (Ref.set p v h) q as = refs_of' h q as" 
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348 
using assms 
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349 
proof  
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350 
from assms have "q \<noteq> p" by (auto simp only: dest!: refs_of'E) 
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351 
with assms show ?thesis 
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352 
unfolding refs_of'_def' 
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353 
by (auto simp add: refs_of_set_ref) 
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354 
qed 
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355 

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356 
lemma refs_of'_set_ref2: 
37725  357 
assumes "refs_of' (Ref.set p v h) q rs" 
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358 
assumes "p \<notin> set rs" 
37725  359 
shows "refs_of' (Ref.set p v h) q as = refs_of' h q as" 
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360 
using assms 
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361 
proof  
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362 
from assms have "q \<noteq> p" by (auto simp only: dest!: refs_of'E) 
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363 
with assms show ?thesis 
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364 
unfolding refs_of'_def' 
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365 
apply auto 
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366 
apply (subgoal_tac "prs = prsa") 
37725  367 
apply (insert refs_of_set_ref2[of p v h "Ref.get h q"]) 
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368 
apply (erule_tac x="prs" in meta_allE) 
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369 
apply auto 
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370 
apply (auto dest: refs_of_is_fun) 
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371 
done 
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372 
qed 
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changeset

373 

1a82e2e29d67
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374 
lemma refs_of'_set_next_ref: 
37725  375 
assumes "Ref.get h1 p = Node x pn" 
376 
assumes "refs_of' (Ref.set p (Node x r1) h1) p rs" 

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377 
obtains r1s where "rs = (p#r1s)" and "refs_of' h1 r1 r1s" 
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changeset

378 
using assms 
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changeset

379 
proof  
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changeset

380 
from assms refs_of'_distinct[OF assms(2)] have "\<exists> r1s. rs = (p # r1s) \<and> refs_of' h1 r1 r1s" 
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changeset

381 
apply  
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changeset

382 
unfolding refs_of'_def'[of _ p] 
37725  383 
apply (auto, frule refs_of_set_ref2) by (auto dest: Ref.noteq_sym) 
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changeset

384 
with prems show thesis by auto 
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changeset

385 
qed 
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diff
changeset

386 

1a82e2e29d67
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changeset

387 
section {* Proving make_llist and traverse correct *} 
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changeset

388 

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changeset

389 
lemma refs_of_invariant: 
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changeset

390 
assumes "refs_of h (r::('a::heap) node) xs" 
37725  391 
assumes "\<forall>refs. refs_of h r refs \<longrightarrow> (\<forall>ref \<in> set refs. Ref.present h ref \<and> Ref.present h' ref \<and> Ref.get h ref = Ref.get h' ref)" 
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changeset

392 
shows "refs_of h' r xs" 
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changeset

393 
using assms 
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changeset

394 
proof (induct xs arbitrary: r) 
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changeset

395 
case Nil thus ?case by simp 
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changeset

396 
next 
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diff
changeset

397 
case (Cons x xs') 
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changeset

398 
from Cons(2) obtain v where Node: "r = Node v x" by (cases r, auto) 
37725  399 
from Cons(2) Node have refs_of_next: "refs_of h (Ref.get h x) xs'" by simp 
400 
from Cons(23) Node have ref_eq: "Ref.get h x = Ref.get h' x" by auto 

401 
from ref_eq refs_of_next have 1: "refs_of h (Ref.get h' x) xs'" by simp 

402 
from Cons(2) Cons(3) have "\<forall>ref \<in> set xs'. Ref.present h ref \<and> Ref.present h' ref \<and> Ref.get h ref = Ref.get h' ref" 

34051
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changeset

403 
by fastsimp 
37725  404 
with Cons(3) 1 have 2: "\<forall>refs. refs_of h (Ref.get h' x) refs \<longrightarrow> (\<forall>ref \<in> set refs. Ref.present h ref \<and> Ref.present h' ref \<and> Ref.get h ref = Ref.get h' ref)" 
34051
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parents:
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changeset

405 
by (fastsimp dest: refs_of_is_fun) 
37725  406 
from Cons.hyps[OF 1 2] have "refs_of h' (Ref.get h' x) xs'" . 
34051
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changeset

407 
with Node show ?case by simp 
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diff
changeset

408 
qed 
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bulwahn
parents:
diff
changeset

409 

1a82e2e29d67
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diff
changeset

410 
lemma refs_of'_invariant: 
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changeset

411 
assumes "refs_of' h r xs" 
37725  412 
assumes "\<forall>refs. refs_of' h r refs \<longrightarrow> (\<forall>ref \<in> set refs. Ref.present h ref \<and> Ref.present h' ref \<and> Ref.get h ref = Ref.get h' ref)" 
34051
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diff
changeset

413 
shows "refs_of' h' r xs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

414 
using assms 
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bulwahn
parents:
diff
changeset

415 
proof  
37725  416 
from assms obtain prs where refs:"refs_of h (Ref.get h r) prs" and xs_def: "xs = r # prs" 
34051
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changeset

417 
unfolding refs_of'_def' by auto 
37725  418 
from xs_def assms have x_eq: "Ref.get h r = Ref.get h' r" by fastsimp 
419 
from refs assms xs_def have 2: "\<forall>refs. refs_of h (Ref.get h r) refs \<longrightarrow> 

420 
(\<forall>ref\<in>set refs. Ref.present h ref \<and> Ref.present h' ref \<and> Ref.get h ref = Ref.get h' ref)" 

34051
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changeset

421 
by (fastsimp dest: refs_of_is_fun) 
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changeset

422 
from refs_of_invariant [OF refs 2] xs_def x_eq show ?thesis 
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changeset

423 
unfolding refs_of'_def' by auto 
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changeset

424 
qed 
1a82e2e29d67
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bulwahn
parents:
diff
changeset

425 

1a82e2e29d67
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changeset

426 
lemma list_of_invariant: 
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changeset

427 
assumes "list_of h (r::('a::heap) node) xs" 
37725  428 
assumes "\<forall>refs. refs_of h r refs \<longrightarrow> (\<forall>ref \<in> set refs. Ref.present h ref \<and> Ref.present h' ref \<and> Ref.get h ref = Ref.get h' ref)" 
34051
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changeset

429 
shows "list_of h' r xs" 
1a82e2e29d67
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bulwahn
parents:
diff
changeset

430 
using assms 
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changeset

431 
proof (induct xs arbitrary: r) 
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changeset

432 
case Nil thus ?case by simp 
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changeset

433 
next 
1a82e2e29d67
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changeset

434 
case (Cons x xs') 
1a82e2e29d67
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bulwahn
parents:
diff
changeset

435 

1a82e2e29d67
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diff
changeset

436 
from Cons(2) obtain ref where Node: "r = Node x ref" 
1a82e2e29d67
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bulwahn
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diff
changeset

437 
by (cases r, auto) 
1a82e2e29d67
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diff
changeset

438 
from Cons(2) obtain rs where rs_def: "refs_of h r rs" by (rule list_of_refs_of) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

439 
from Node rs_def obtain rss where refs_of: "refs_of h r (ref#rss)" and rss_def: "rs = ref#rss" by auto 
37725  440 
from Cons(3) Node refs_of have ref_eq: "Ref.get h ref = Ref.get h' ref" 
34051
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diff
changeset

441 
by auto 
37725  442 
from Cons(2) ref_eq Node have 1: "list_of h (Ref.get h' ref) xs'" by simp 
443 
from refs_of Node ref_eq have refs_of_ref: "refs_of h (Ref.get h' ref) rss" by simp 

444 
from Cons(3) rs_def have rs_heap_eq: "\<forall>ref\<in>set rs. Ref.present h ref \<and> Ref.present h' ref \<and> Ref.get h ref = Ref.get h' ref" by simp 

445 
from refs_of_ref rs_heap_eq rss_def have 2: "\<forall>refs. refs_of h (Ref.get h' ref) refs \<longrightarrow> 

446 
(\<forall>ref\<in>set refs. Ref.present h ref \<and> Ref.present h' ref \<and> Ref.get h ref = Ref.get h' ref)" 

34051
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diff
changeset

447 
by (auto dest: refs_of_is_fun) 
1a82e2e29d67
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bulwahn
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diff
changeset

448 
from Cons(1)[OF 1 2] 
37725  449 
have "list_of h' (Ref.get h' ref) xs'" . 
34051
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changeset

450 
with Node show ?case 
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diff
changeset

451 
unfolding list_of'_def 
1a82e2e29d67
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diff
changeset

452 
by simp 
1a82e2e29d67
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bulwahn
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diff
changeset

453 
qed 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

454 

37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

455 
lemma crel_ref: 
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

456 
assumes "crel (ref v) h h' x" 
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

457 
obtains "Ref.get h' x = v" 
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

458 
and "\<not> Ref.present h x" 
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

459 
and "Ref.present h' x" 
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

460 
and "\<forall>y. Ref.present h y \<longrightarrow> Ref.get h y = Ref.get h' y" 
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

461 
(* and "lim h' = Suc (lim h)" *) 
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

462 
and "\<forall>y. Ref.present h y \<longrightarrow> Ref.present h' y" 
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

463 
using assms 
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

464 
unfolding Ref.ref_def 
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

465 
apply (elim crel_heapE) 
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

466 
unfolding Ref.alloc_def 
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

467 
apply (simp add: Let_def) 
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

468 
unfolding Ref.present_def 
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

469 
apply auto 
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

470 
unfolding Ref.get_def Ref.set_def 
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

471 
apply auto 
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

472 
done 
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

473 

34051
1a82e2e29d67
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bulwahn
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diff
changeset

474 
lemma make_llist: 
1a82e2e29d67
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bulwahn
parents:
diff
changeset

475 
assumes "crel (make_llist xs) h h' r" 
37725  476 
shows "list_of h' r xs \<and> (\<forall>rs. refs_of h' r rs \<longrightarrow> (\<forall>ref \<in> (set rs). Ref.present h' ref))" 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

477 
using assms 
1a82e2e29d67
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bulwahn
parents:
diff
changeset

478 
proof (induct xs arbitrary: h h' r) 
37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

479 
case Nil thus ?case by (auto elim: crel_returnE simp add: make_llist.simps) 
34051
1a82e2e29d67
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bulwahn
parents:
diff
changeset

480 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

481 
case (Cons x xs') 
1a82e2e29d67
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bulwahn
parents:
diff
changeset

482 
from Cons.prems obtain h1 r1 r' where make_llist: "crel (make_llist xs') h h1 r1" 
37725  483 
and crel_refnew:"crel (ref r1) h1 h' r'" and Node: "r = Node x r'" 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

484 
unfolding make_llist.simps 
37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

485 
by (auto elim!: crel_bindE crel_returnE) 
34051
1a82e2e29d67
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bulwahn
parents:
diff
changeset

486 
from Cons.hyps[OF make_llist] have list_of_h1: "list_of h1 r1 xs'" .. 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

487 
from Cons.hyps[OF make_llist] obtain rs' where rs'_def: "refs_of h1 r1 rs'" by (auto intro: list_of_refs_of) 
37725  488 
from Cons.hyps[OF make_llist] rs'_def have refs_present: "\<forall>ref\<in>set rs'. Ref.present h1 ref" by simp 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

489 
from crel_refnew rs'_def refs_present have refs_unchanged: "\<forall>refs. refs_of h1 r1 refs \<longrightarrow> 
37725  490 
(\<forall>ref\<in>set refs. Ref.present h1 ref \<and> Ref.present h' ref \<and> Ref.get h1 ref = Ref.get h' ref)" 
491 
by (auto elim!: crel_ref dest: refs_of_is_fun) 

34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

492 
with list_of_invariant[OF list_of_h1 refs_unchanged] Node crel_refnew have fstgoal: "list_of h' r (x # xs')" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

493 
unfolding list_of.simps 
37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

494 
by (auto elim!: crel_refE) 
37725  495 
from refs_unchanged rs'_def have refs_still_present: "\<forall>ref\<in>set rs'. Ref.present h' ref" by auto 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

496 
from refs_of_invariant[OF rs'_def refs_unchanged] refs_unchanged Node crel_refnew refs_still_present 
37725  497 
have sndgoal: "\<forall>rs. refs_of h' r rs \<longrightarrow> (\<forall>ref\<in>set rs. Ref.present h' ref)" 
37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

498 
by (fastsimp elim!: crel_refE dest: refs_of_is_fun) 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

499 
from fstgoal sndgoal show ?case .. 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

500 
qed 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

501 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

502 
lemma traverse: "list_of h n r \<Longrightarrow> crel (traverse n) h h r" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

503 
proof (induct r arbitrary: n) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

504 
case Nil 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

505 
thus ?case 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

506 
by (auto intro: crel_returnI) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

507 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

508 
case (Cons x xs) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

509 
thus ?case 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

510 
apply (cases n, auto) 
37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

511 
by (auto intro!: crel_bindI crel_returnI crel_lookupI) 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

512 
qed 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

513 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

514 
lemma traverse_make_llist': 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

515 
assumes crel: "crel (make_llist xs \<guillemotright>= traverse) h h' r" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

516 
shows "r = xs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

517 
proof  
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

518 
from crel obtain h1 r1 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

519 
where makell: "crel (make_llist xs) h h1 r1" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

520 
and trav: "crel (traverse r1) h1 h' r" 
37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

521 
by (auto elim!: crel_bindE) 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

522 
from make_llist[OF makell] have "list_of h1 r1 xs" .. 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

523 
from traverse [OF this] trav show ?thesis 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

524 
using crel_deterministic by fastsimp 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

525 
qed 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

526 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

527 
section {* Proving correctness of inplace reversal *} 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

528 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

529 
subsection {* Definition of inplace reversal *} 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

530 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

531 
definition rev' :: "(('a::heap) node ref \<times> 'a node ref) \<Rightarrow> 'a node ref Heap" 
37792  532 
where "rev' = MREC (\<lambda>(q, p). do { v \<leftarrow> !p; (case v of Empty \<Rightarrow> (return (Inl q)) 
533 
 Node x next \<Rightarrow> do { 

34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

534 
p := Node x q; 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

535 
return (Inr (p, next)) 
37792  536 
})}) 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

537 
(\<lambda>x s z. return z)" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

538 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

539 
lemma rev'_simps [code]: 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

540 
"rev' (q, p) = 
37792  541 
do { 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

542 
v \<leftarrow> !p; 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

543 
(case v of 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

544 
Empty \<Rightarrow> return q 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

545 
 Node x next \<Rightarrow> 
37792  546 
do { 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

547 
p := Node x q; 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

548 
rev' (p, next) 
37792  549 
}) 
550 
}" 

34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

551 
unfolding rev'_def MREC_rule[of _ _ "(q, p)"] unfolding rev'_def[symmetric] 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

552 
thm arg_cong2 
37709  553 
by (auto simp add: expand_fun_eq intro: arg_cong2[where f = "op \<guillemotright>="] split: node.split) 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

554 

37725  555 
primrec rev :: "('a:: heap) node \<Rightarrow> 'a node Heap" 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

556 
where 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

557 
"rev Empty = return Empty" 
37792  558 
 "rev (Node x n) = do { q \<leftarrow> ref Empty; p \<leftarrow> ref (Node x n); v \<leftarrow> rev' (q, p); !v }" 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

559 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

560 
subsection {* Correctness Proof *} 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

561 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

562 
lemma rev'_invariant: 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

563 
assumes "crel (rev' (q, p)) h h' v" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

564 
assumes "list_of' h q qs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

565 
assumes "list_of' h p ps" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

566 
assumes "\<forall>qrs prs. refs_of' h q qrs \<and> refs_of' h p prs \<longrightarrow> set prs \<inter> set qrs = {}" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

567 
shows "\<exists>vs. list_of' h' v vs \<and> vs = (List.rev ps) @ qs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

568 
using assms 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

569 
proof (induct ps arbitrary: qs p q h) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

570 
case Nil 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

571 
thus ?case 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

572 
unfolding rev'_simps[of q p] list_of'_def 
37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

573 
by (auto elim!: crel_bindE crel_lookupE crel_returnE) 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

574 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

575 
case (Cons x xs) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

576 
(*"LinkedList.list_of h' (get_ref v h') (List.rev xs @ x # qsa)"*) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

577 
from Cons(4) obtain ref where 
37725  578 
p_is_Node: "Ref.get h p = Node x ref" 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

579 
(*and "ref_present ref h"*) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

580 
and list_of'_ref: "list_of' h ref xs" 
37725  581 
unfolding list_of'_def by (cases "Ref.get h p", auto) 
582 
from p_is_Node Cons(2) have crel_rev': "crel (rev' (p, ref)) (Ref.set p (Node x q) h) h' v" 

37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

583 
by (auto simp add: rev'_simps [of q p] elim!: crel_bindE crel_lookupE crel_updateE) 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

584 
from Cons(3) obtain qrs where qrs_def: "refs_of' h q qrs" by (elim list_of'_refs_of') 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

585 
from Cons(4) obtain prs where prs_def: "refs_of' h p prs" by (elim list_of'_refs_of') 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

586 
from qrs_def prs_def Cons(5) have distinct_pointers: "set qrs \<inter> set prs = {}" by fastsimp 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

587 
from qrs_def prs_def distinct_pointers refs_of'E have p_notin_qrs: "p \<notin> set qrs" by fastsimp 
37725  588 
from Cons(3) qrs_def this have 1: "list_of' (Ref.set p (Node x q) h) p (x#qs)" 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

589 
unfolding list_of'_def 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

590 
apply (simp) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

591 
unfolding list_of'_def[symmetric] 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

592 
by (simp add: list_of'_set_ref) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

593 
from list_of'_refs_of'2[OF Cons(4)] p_is_Node prs_def obtain refs where refs_def: "refs_of' h ref refs" and prs_refs: "prs = p # refs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

594 
unfolding refs_of'_def' by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

595 
from prs_refs prs_def have p_not_in_refs: "p \<notin> set refs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

596 
by (fastsimp dest!: refs_of'_distinct) 
37725  597 
with refs_def p_is_Node list_of'_ref have 2: "list_of' (Ref.set p (Node x q) h) ref xs" 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

598 
by (auto simp add: list_of'_set_ref) 
37725  599 
from p_notin_qrs qrs_def have refs_of1: "refs_of' (Ref.set p (Node x q) h) p (p#qrs)" 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

600 
unfolding refs_of'_def' 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

601 
apply (simp) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

602 
unfolding refs_of'_def'[symmetric] 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

603 
by (simp add: refs_of'_set_ref) 
37725  604 
from p_not_in_refs p_is_Node refs_def have refs_of2: "refs_of' (Ref.set p (Node x q) h) ref refs" 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

605 
by (simp add: refs_of'_set_ref) 
37725  606 
from p_not_in_refs refs_of1 refs_of2 distinct_pointers prs_refs have 3: "\<forall>qrs prs. refs_of' (Ref.set p (Node x q) h) p qrs \<and> refs_of' (Ref.set p (Node x q) h) ref prs \<longrightarrow> set prs \<inter> set qrs = {}" 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

607 
apply  apply (rule allI)+ apply (rule impI) apply (erule conjE) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

608 
apply (drule refs_of'_is_fun) back back apply assumption 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

609 
apply (drule refs_of'_is_fun) back back apply assumption 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

610 
apply auto done 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

611 
from Cons.hyps [OF crel_rev' 1 2 3] show ?case by simp 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

612 
qed 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

613 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

614 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

615 
lemma rev_correctness: 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

616 
assumes list_of_h: "list_of h r xs" 
37725  617 
assumes validHeap: "\<forall>refs. refs_of h r refs \<longrightarrow> (\<forall>r \<in> set refs. Ref.present h r)" 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

618 
assumes crel_rev: "crel (rev r) h h' r'" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

619 
shows "list_of h' r' (List.rev xs)" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

620 
using assms 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

621 
proof (cases r) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

622 
case Empty 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

623 
with list_of_h crel_rev show ?thesis 
37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

624 
by (auto simp add: list_of_Empty elim!: crel_returnE) 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

625 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

626 
case (Node x ps) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

627 
with crel_rev obtain p q h1 h2 h3 v where 
37725  628 
init: "crel (ref Empty) h h1 q" 
629 
"crel (ref (Node x ps)) h1 h2 p" 

34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

630 
and crel_rev':"crel (rev' (q, p)) h2 h3 v" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

631 
and lookup: "crel (!v) h3 h' r'" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

632 
using rev.simps 
37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

633 
by (auto elim!: crel_bindE) 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

634 
from init have a1:"list_of' h2 q []" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

635 
unfolding list_of'_def 
37725  636 
by (auto elim!: crel_ref) 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

637 
from list_of_h obtain refs where refs_def: "refs_of h r refs" by (rule list_of_refs_of) 
37725  638 
from validHeap init refs_def have heap_eq: "\<forall>refs. refs_of h r refs \<longrightarrow> (\<forall>ref\<in>set refs. Ref.present h ref \<and> Ref.present h2 ref \<and> Ref.get h ref = Ref.get h2 ref)" 
639 
by (fastsimp elim!: crel_ref dest: refs_of_is_fun) 

34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

640 
from list_of_invariant[OF list_of_h heap_eq] have "list_of h2 r xs" . 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

641 
from init this Node have a2: "list_of' h2 p xs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

642 
apply  
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

643 
unfolding list_of'_def 
37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

644 
apply (auto elim!: crel_refE) 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

645 
done 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

646 
from init have refs_of_q: "refs_of' h2 q [q]" 
37725  647 
by (auto elim!: crel_ref) 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

648 
from refs_def Node have refs_of'_ps: "refs_of' h ps refs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

649 
by (auto simp add: refs_of'_def'[symmetric]) 
37725  650 
from validHeap refs_def have all_ref_present: "\<forall>r\<in>set refs. Ref.present h r" by simp 
651 
from init refs_of'_ps Node this have heap_eq: "\<forall>refs. refs_of' h ps refs \<longrightarrow> (\<forall>ref\<in>set refs. Ref.present h ref \<and> Ref.present h2 ref \<and> Ref.get h ref = Ref.get h2 ref)" 

652 
by (fastsimp elim!: crel_ref dest: refs_of'_is_fun) 

34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

653 
from refs_of'_invariant[OF refs_of'_ps this] have "refs_of' h2 ps refs" . 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

654 
with init have refs_of_p: "refs_of' h2 p (p#refs)" 
37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

655 
by (auto elim!: crel_refE simp add: refs_of'_def') 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

656 
with init all_ref_present have q_is_new: "q \<notin> set (p#refs)" 
37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

657 
by (auto elim!: crel_refE intro!: Ref.noteq_I) 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

658 
from refs_of_p refs_of_q q_is_new have a3: "\<forall>qrs prs. refs_of' h2 q qrs \<and> refs_of' h2 p prs \<longrightarrow> set prs \<inter> set qrs = {}" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

659 
by (fastsimp simp only: set.simps dest: refs_of'_is_fun) 
37725  660 
from rev'_invariant [OF crel_rev' a1 a2 a3] have "list_of h3 (Ref.get h3 v) (List.rev xs)" 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

661 
unfolding list_of'_def by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

662 
with lookup show ?thesis 
37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

663 
by (auto elim: crel_lookupE) 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

664 
qed 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

665 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

666 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

667 
section {* The merge function on Linked Lists *} 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

668 
text {* We also prove merge correct *} 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

669 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

670 
text{* First, we define merge on lists in a natural way. *} 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

671 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

672 
fun Lmerge :: "('a::ord) list \<Rightarrow> 'a list \<Rightarrow> 'a list" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

673 
where 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

674 
"Lmerge (x#xs) (y#ys) = 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

675 
(if x \<le> y then x # Lmerge xs (y#ys) else y # Lmerge (x#xs) ys)" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

676 
 "Lmerge [] ys = ys" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

677 
 "Lmerge xs [] = xs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

678 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

679 
subsection {* Definition of merge function *} 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

680 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

681 
definition merge' :: "(('a::{heap, ord}) node ref * ('a::{heap, ord})) * ('a::{heap, ord}) node ref * ('a::{heap, ord}) node ref \<Rightarrow> ('a::{heap, ord}) node ref Heap" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

682 
where 
37792  683 
"merge' = MREC (\<lambda>(_, p, q). do { v \<leftarrow> !p; w \<leftarrow> !q; 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

684 
(case v of Empty \<Rightarrow> return (Inl q) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

685 
 Node valp np \<Rightarrow> 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

686 
(case w of Empty \<Rightarrow> return (Inl p) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

687 
 Node valq nq \<Rightarrow> 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

688 
if (valp \<le> valq) then 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

689 
return (Inr ((p, valp), np, q)) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

690 
else 
37792  691 
return (Inr ((q, valq), p, nq)))) }) 
692 
(\<lambda> _ ((n, v), _, _) r. do { n := Node v r; return n })" 

34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

693 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

694 
definition merge where "merge p q = merge' (undefined, p, q)" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

695 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

696 
lemma if_return: "(if P then return x else return y) = return (if P then x else y)" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

697 
by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

698 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

699 
lemma if_distrib_App: "(if P then f else g) x = (if P then f x else g x)" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

700 
by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

701 
lemma redundant_if: "(if P then (if P then x else z) else y) = (if P then x else y)" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

702 
"(if P then x else (if P then z else y)) = (if P then x else y)" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

703 
by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

704 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

705 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

706 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

707 
lemma sum_distrib: "sum_case fl fr (case x of Empty \<Rightarrow> y  Node v n \<Rightarrow> (z v n)) = (case x of Empty \<Rightarrow> sum_case fl fr y  Node v n \<Rightarrow> sum_case fl fr (z v n))" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

708 
by (cases x) auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

709 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

710 
lemma merge: "merge' (x, p, q) = merge p q" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

711 
unfolding merge'_def merge_def 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

712 
apply (simp add: MREC_rule) done 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

713 
term "Ref.change" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

714 
lemma merge_simps [code]: 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

715 
shows "merge p q = 
37792  716 
do { v \<leftarrow> !p; 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

717 
w \<leftarrow> !q; 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

718 
(case v of node.Empty \<Rightarrow> return q 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

719 
 Node valp np \<Rightarrow> 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

720 
case w of node.Empty \<Rightarrow> return p 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

721 
 Node valq nq \<Rightarrow> 
37792  722 
if valp \<le> valq then do { r \<leftarrow> merge np q; 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

723 
p := (Node valp r); 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

724 
return p 
37792  725 
} 
726 
else do { r \<leftarrow> merge p nq; 

34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

727 
q := (Node valq r); 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

728 
return q 
37792  729 
}) 
730 
}" 

34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

731 
proof  
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

732 
{fix v x y 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

733 
have case_return: "(case v of Empty \<Rightarrow> return x  Node v n \<Rightarrow> return (y v n)) = return (case v of Empty \<Rightarrow> x  Node v n \<Rightarrow> y v n)" by (cases v) auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

734 
} note case_return = this 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

735 
show ?thesis 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

736 
unfolding merge_def[of p q] merge'_def 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

737 
apply (simp add: MREC_rule[of _ _ "(undefined, p, q)"]) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

738 
unfolding bind_bind return_bind 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

739 
unfolding merge'_def[symmetric] 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

740 
unfolding if_return case_return bind_bind return_bind sum_distrib sum.cases 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

741 
unfolding if_distrib[symmetric, where f="Inr"] 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

742 
unfolding sum.cases 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

743 
unfolding if_distrib 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

744 
unfolding split_beta fst_conv snd_conv 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

745 
unfolding if_distrib_App redundant_if merge 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

746 
.. 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

747 
qed 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

748 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

749 
subsection {* Induction refinement by applying the abstraction function to our induct rule *} 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

750 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

751 
text {* From our original induction rule Lmerge.induct, we derive a new rule with our list_of' predicate *} 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

752 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

753 
lemma merge_induct2: 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

754 
assumes "list_of' h (p::'a::{heap, ord} node ref) xs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

755 
assumes "list_of' h q ys" 
37725  756 
assumes "\<And> ys p q. \<lbrakk> list_of' h p []; list_of' h q ys; Ref.get h p = Empty \<rbrakk> \<Longrightarrow> P p q [] ys" 
757 
assumes "\<And> x xs' p q pn. \<lbrakk> list_of' h p (x#xs'); list_of' h q []; Ref.get h p = Node x pn; Ref.get h q = Empty \<rbrakk> \<Longrightarrow> P p q (x#xs') []" 

34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

758 
assumes "\<And> x xs' y ys' p q pn qn. 
37725  759 
\<lbrakk> list_of' h p (x#xs'); list_of' h q (y#ys'); Ref.get h p = Node x pn; Ref.get h q = Node y qn; 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

760 
x \<le> y; P pn q xs' (y#ys') \<rbrakk> 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

761 
\<Longrightarrow> P p q (x#xs') (y#ys')" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

762 
assumes "\<And> x xs' y ys' p q pn qn. 
37725  763 
\<lbrakk> list_of' h p (x#xs'); list_of' h q (y#ys'); Ref.get h p = Node x pn; Ref.get h q = Node y qn; 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

764 
\<not> x \<le> y; P p qn (x#xs') ys'\<rbrakk> 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

765 
\<Longrightarrow> P p q (x#xs') (y#ys')" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

766 
shows "P p q xs ys" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

767 
using assms(12) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

768 
proof (induct xs ys arbitrary: p q rule: Lmerge.induct) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

769 
case (2 ys) 
37725  770 
from 2(1) have "Ref.get h p = Empty" unfolding list_of'_def by simp 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

771 
with 2(12) assms(3) show ?case by blast 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

772 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

773 
case (3 x xs') 
37725  774 
from 3(1) obtain pn where Node: "Ref.get h p = Node x pn" by (rule list_of'_Cons) 
775 
from 3(2) have "Ref.get h q = Empty" unfolding list_of'_def by simp 

34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

776 
with Node 3(12) assms(4) show ?case by blast 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

777 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

778 
case (1 x xs' y ys') 
37725  779 
from 1(3) obtain pn where pNode:"Ref.get h p = Node x pn" 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

780 
and list_of'_pn: "list_of' h pn xs'" by (rule list_of'_Cons) 
37725  781 
from 1(4) obtain qn where qNode:"Ref.get h q = Node y qn" 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

782 
and list_of'_qn: "list_of' h qn ys'" by (rule list_of'_Cons) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

783 
show ?case 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

784 
proof (cases "x \<le> y") 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

785 
case True 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

786 
from 1(1)[OF True list_of'_pn 1(4)] assms(5) 1(34) pNode qNode True 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

787 
show ?thesis by blast 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

788 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

789 
case False 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

790 
from 1(2)[OF False 1(3) list_of'_qn] assms(6) 1(34) pNode qNode False 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

791 
show ?thesis by blast 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

792 
qed 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

793 
qed 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

794 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

795 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

796 
text {* secondly, we add the crel statement in the premise, and derive the crel statements for the single cases which we then eliminate with our crel elim rules. *} 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

797 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

798 
lemma merge_induct3: 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

799 
assumes "list_of' h p xs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

800 
assumes "list_of' h q ys" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

801 
assumes "crel (merge p q) h h' r" 
37725  802 
assumes "\<And> ys p q. \<lbrakk> list_of' h p []; list_of' h q ys; Ref.get h p = Empty \<rbrakk> \<Longrightarrow> P p q h h q [] ys" 
803 
assumes "\<And> x xs' p q pn. \<lbrakk> list_of' h p (x#xs'); list_of' h q []; Ref.get h p = Node x pn; Ref.get h q = Empty \<rbrakk> \<Longrightarrow> P p q h h p (x#xs') []" 

34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

804 
assumes "\<And> x xs' y ys' p q pn qn h1 r1 h'. 
37725  805 
\<lbrakk> list_of' h p (x#xs'); list_of' h q (y#ys');Ref.get h p = Node x pn; Ref.get h q = Node y qn; 
806 
x \<le> y; crel (merge pn q) h h1 r1 ; P pn q h h1 r1 xs' (y#ys'); h' = Ref.set p (Node x r1) h1 \<rbrakk> 

34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

807 
\<Longrightarrow> P p q h h' p (x#xs') (y#ys')" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

808 
assumes "\<And> x xs' y ys' p q pn qn h1 r1 h'. 
37725  809 
\<lbrakk> list_of' h p (x#xs'); list_of' h q (y#ys'); Ref.get h p = Node x pn; Ref.get h q = Node y qn; 
810 
\<not> x \<le> y; crel (merge p qn) h h1 r1; P p qn h h1 r1 (x#xs') ys'; h' = Ref.set q (Node y r1) h1 \<rbrakk> 

34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

811 
\<Longrightarrow> P p q h h' q (x#xs') (y#ys')" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

812 
shows "P p q h h' r xs ys" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

813 
using assms(3) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

814 
proof (induct arbitrary: h' r rule: merge_induct2[OF assms(1) assms(2)]) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

815 
case (1 ys p q) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

816 
from 1(34) have "h = h' \<and> r = q" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

817 
unfolding merge_simps[of p q] 
37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

818 
by (auto elim!: crel_lookupE crel_bindE crel_returnE) 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

819 
with assms(4)[OF 1(1) 1(2) 1(3)] show ?case by simp 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

820 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

821 
case (2 x xs' p q pn) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

822 
from 2(35) have "h = h' \<and> r = p" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

823 
unfolding merge_simps[of p q] 
37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

824 
by (auto elim!: crel_lookupE crel_bindE crel_returnE) 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

825 
with assms(5)[OF 2(14)] show ?case by simp 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

826 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

827 
case (3 x xs' y ys' p q pn qn) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

828 
from 3(35) 3(7) obtain h1 r1 where 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

829 
1: "crel (merge pn q) h h1 r1" 
37725  830 
and 2: "h' = Ref.set p (Node x r1) h1 \<and> r = p" 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

831 
unfolding merge_simps[of p q] 
37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

832 
by (auto elim!: crel_lookupE crel_bindE crel_returnE crel_ifE crel_updateE) 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

833 
from 3(6)[OF 1] assms(6) [OF 3(15)] 1 2 show ?case by simp 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

834 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

835 
case (4 x xs' y ys' p q pn qn) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

836 
from 4(35) 4(7) obtain h1 r1 where 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

837 
1: "crel (merge p qn) h h1 r1" 
37725  838 
and 2: "h' = Ref.set q (Node y r1) h1 \<and> r = q" 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

839 
unfolding merge_simps[of p q] 
37771
1bec64044b5e
spelt out relational framework in a consistent way
haftmann
parents:
37765
diff
changeset

840 
by (auto elim!: crel_lookupE crel_bindE crel_returnE crel_ifE crel_updateE) 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

841 
from 4(6)[OF 1] assms(7) [OF 4(15)] 1 2 show ?case by simp 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

842 
qed 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

843 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

844 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

845 
subsection {* Proving merge correct *} 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

846 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

847 
text {* As many parts of the following three proofs are identical, we could actually move the 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

848 
same reasoning into an extended induction rule *} 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

849 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

850 
lemma merge_unchanged: 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

851 
assumes "refs_of' h p xs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

852 
assumes "refs_of' h q ys" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

853 
assumes "crel (merge p q) h h' r'" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

854 
assumes "set xs \<inter> set ys = {}" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

855 
assumes "r \<notin> set xs \<union> set ys" 
37725  856 
shows "Ref.get h r = Ref.get h' r" 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

857 
proof  
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

858 
from assms(1) obtain ps where ps_def: "list_of' h p ps" by (rule refs_of'_list_of') 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

859 
from assms(2) obtain qs where qs_def: "list_of' h q qs" by (rule refs_of'_list_of') 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

860 
show ?thesis using assms(1) assms(2) assms(4) assms(5) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

861 
proof (induct arbitrary: xs ys r rule: merge_induct3[OF ps_def qs_def assms(3)]) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

862 
case 1 thus ?case by simp 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

863 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

864 
case 2 thus ?case by simp 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

865 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

866 
case (3 x xs' y ys' p q pn qn h1 r1 h' xs ys r) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

867 
from 3(9) 3(3) obtain pnrs 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

868 
where pnrs_def: "xs = p#pnrs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

869 
and refs_of'_pn: "refs_of' h pn pnrs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

870 
by (rule refs_of'_Node) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

871 
with 3(12) have r_in: "r \<notin> set pnrs \<union> set ys" by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

872 
from pnrs_def 3(12) have "r \<noteq> p" by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

873 
with 3(11) 3(12) pnrs_def refs_of'_distinct[OF 3(9)] have p_in: "p \<notin> set pnrs \<union> set ys" by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

874 
from 3(11) pnrs_def have no_inter: "set pnrs \<inter> set ys = {}" by auto 
37725  875 
from 3(7)[OF refs_of'_pn 3(10) this p_in] 3(3) have p_is_Node: "Ref.get h1 p = Node x pn" 
876 
by simp 

34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

877 
from 3(7)[OF refs_of'_pn 3(10) no_inter r_in] 3(8) `r \<noteq> p` show ?case 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

878 
by simp 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

879 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

880 
case (4 x xs' y ys' p q pn qn h1 r1 h' xs ys r) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

881 
from 4(10) 4(4) obtain qnrs 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

882 
where qnrs_def: "ys = q#qnrs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

883 
and refs_of'_qn: "refs_of' h qn qnrs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

884 
by (rule refs_of'_Node) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

885 
with 4(12) have r_in: "r \<notin> set xs \<union> set qnrs" by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

886 
from qnrs_def 4(12) have "r \<noteq> q" by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

887 
with 4(11) 4(12) qnrs_def refs_of'_distinct[OF 4(10)] have q_in: "q \<notin> set xs \<union> set qnrs" by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

888 
from 4(11) qnrs_def have no_inter: "set xs \<inter> set qnrs = {}" by auto 
37725  889 
from 4(7)[OF 4(9) refs_of'_qn this q_in] 4(4) have q_is_Node: "Ref.get h1 q = Node y qn" by simp 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

890 
from 4(7)[OF 4(9) refs_of'_qn no_inter r_in] 4(8) `r \<noteq> q` show ?case 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

891 
by simp 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

892 
qed 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

893 
qed 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

894 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

895 
lemma refs_of'_merge: 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

896 
assumes "refs_of' h p xs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

897 
assumes "refs_of' h q ys" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

898 
assumes "crel (merge p q) h h' r" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

899 
assumes "set xs \<inter> set ys = {}" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

900 
assumes "refs_of' h' r rs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

901 
shows "set rs \<subseteq> set xs \<union> set ys" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

902 
proof  
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

903 
from assms(1) obtain ps where ps_def: "list_of' h p ps" by (rule refs_of'_list_of') 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

904 
from assms(2) obtain qs where qs_def: "list_of' h q qs" by (rule refs_of'_list_of') 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

905 
show ?thesis using assms(1) assms(2) assms(4) assms(5) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

906 
proof (induct arbitrary: xs ys rs rule: merge_induct3[OF ps_def qs_def assms(3)]) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

907 
case 1 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

908 
from 1(5) 1(7) have "rs = ys" by (fastsimp simp add: refs_of'_is_fun) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

909 
thus ?case by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

910 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

911 
case 2 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

912 
from 2(5) 2(8) have "rs = xs" by (auto simp add: refs_of'_is_fun) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

913 
thus ?case by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

914 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

915 
case (3 x xs' y ys' p q pn qn h1 r1 h' xs ys rs) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

916 
from 3(9) 3(3) obtain pnrs 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

917 
where pnrs_def: "xs = p#pnrs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

918 
and refs_of'_pn: "refs_of' h pn pnrs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

919 
by (rule refs_of'_Node) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

920 
from 3(10) 3(9) 3(11) pnrs_def refs_of'_distinct[OF 3(9)] have p_in: "p \<notin> set pnrs \<union> set ys" by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

921 
from 3(11) pnrs_def have no_inter: "set pnrs \<inter> set ys = {}" by auto 
37725  922 
from merge_unchanged[OF refs_of'_pn 3(10) 3(6) no_inter p_in] have p_stays: "Ref.get h1 p = Ref.get h p" .. 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

923 
from 3 p_stays obtain r1s 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

924 
where rs_def: "rs = p#r1s" and refs_of'_r1:"refs_of' h1 r1 r1s" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

925 
by (auto elim: refs_of'_set_next_ref) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

926 
from 3(7)[OF refs_of'_pn 3(10) no_inter refs_of'_r1] rs_def pnrs_def show ?case by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

927 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

928 
case (4 x xs' y ys' p q pn qn h1 r1 h' xs ys rs) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

929 
from 4(10) 4(4) obtain qnrs 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

930 
where qnrs_def: "ys = q#qnrs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

931 
and refs_of'_qn: "refs_of' h qn qnrs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

932 
by (rule refs_of'_Node) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

933 
from 4(10) 4(9) 4(11) qnrs_def refs_of'_distinct[OF 4(10)] have q_in: "q \<notin> set xs \<union> set qnrs" by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

934 
from 4(11) qnrs_def have no_inter: "set xs \<inter> set qnrs = {}" by auto 
37725  935 
from merge_unchanged[OF 4(9) refs_of'_qn 4(6) no_inter q_in] have q_stays: "Ref.get h1 q = Ref.get h q" .. 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

936 
from 4 q_stays obtain r1s 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

937 
where rs_def: "rs = q#r1s" and refs_of'_r1:"refs_of' h1 r1 r1s" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

938 
by (auto elim: refs_of'_set_next_ref) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

939 
from 4(7)[OF 4(9) refs_of'_qn no_inter refs_of'_r1] rs_def qnrs_def show ?case by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

940 
qed 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

941 
qed 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

942 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

943 
lemma 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

944 
assumes "list_of' h p xs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

945 
assumes "list_of' h q ys" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

946 
assumes "crel (merge p q) h h' r" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

947 
assumes "\<forall>qrs prs. refs_of' h q qrs \<and> refs_of' h p prs \<longrightarrow> set prs \<inter> set qrs = {}" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

948 
shows "list_of' h' r (Lmerge xs ys)" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

949 
using assms(4) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

950 
proof (induct rule: merge_induct3[OF assms(13)]) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

951 
case 1 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

952 
thus ?case by simp 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

953 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

954 
case 2 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

955 
thus ?case by simp 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

956 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

957 
case (3 x xs' y ys' p q pn qn h1 r1 h') 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

958 
from 3(1) obtain prs where prs_def: "refs_of' h p prs" by (rule list_of'_refs_of') 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

959 
from 3(2) obtain qrs where qrs_def: "refs_of' h q qrs" by (rule list_of'_refs_of') 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

960 
from prs_def 3(3) obtain pnrs 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

961 
where pnrs_def: "prs = p#pnrs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

962 
and refs_of'_pn: "refs_of' h pn pnrs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

963 
by (rule refs_of'_Node) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

964 
from prs_def qrs_def 3(9) pnrs_def refs_of'_distinct[OF prs_def] have p_in: "p \<notin> set pnrs \<union> set qrs" by fastsimp 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

965 
from prs_def qrs_def 3(9) pnrs_def have no_inter: "set pnrs \<inter> set qrs = {}" by fastsimp 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

966 
from no_inter refs_of'_pn qrs_def have no_inter2: "\<forall>qrs prs. refs_of' h q qrs \<and> refs_of' h pn prs \<longrightarrow> set prs \<inter> set qrs = {}" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

967 
by (fastsimp dest: refs_of'_is_fun) 
37725  968 
from merge_unchanged[OF refs_of'_pn qrs_def 3(6) no_inter p_in] have p_stays: "Ref.get h1 p = Ref.get h p" .. 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

969 
from 3(7)[OF no_inter2] obtain rs where rs_def: "refs_of' h1 r1 rs" by (rule list_of'_refs_of') 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

970 
from refs_of'_merge[OF refs_of'_pn qrs_def 3(6) no_inter this] p_in have p_rs: "p \<notin> set rs" by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

971 
with 3(7)[OF no_inter2] 3(15) 3(8) p_rs rs_def p_stays 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

972 
show ?case by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

973 
next 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

974 
case (4 x xs' y ys' p q pn qn h1 r1 h') 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

975 
from 4(1) obtain prs where prs_def: "refs_of' h p prs" by (rule list_of'_refs_of') 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

976 
from 4(2) obtain qrs where qrs_def: "refs_of' h q qrs" by (rule list_of'_refs_of') 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

977 
from qrs_def 4(4) obtain qnrs 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

978 
where qnrs_def: "qrs = q#qnrs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

979 
and refs_of'_qn: "refs_of' h qn qnrs" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

980 
by (rule refs_of'_Node) 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

981 
from prs_def qrs_def 4(9) qnrs_def refs_of'_distinct[OF qrs_def] have q_in: "q \<notin> set prs \<union> set qnrs" by fastsimp 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

982 
from prs_def qrs_def 4(9) qnrs_def have no_inter: "set prs \<inter> set qnrs = {}" by fastsimp 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

983 
from no_inter refs_of'_qn prs_def have no_inter2: "\<forall>qrs prs. refs_of' h qn qrs \<and> refs_of' h p prs \<longrightarrow> set prs \<inter> set qrs = {}" 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

984 
by (fastsimp dest: refs_of'_is_fun) 
37725  985 
from merge_unchanged[OF prs_def refs_of'_qn 4(6) no_inter q_in] have q_stays: "Ref.get h1 q = Ref.get h q" .. 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

986 
from 4(7)[OF no_inter2] obtain rs where rs_def: "refs_of' h1 r1 rs" by (rule list_of'_refs_of') 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

987 
from refs_of'_merge[OF prs_def refs_of'_qn 4(6) no_inter this] q_in have q_rs: "q \<notin> set rs" by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

988 
with 4(7)[OF no_inter2] 4(15) 4(8) q_rs rs_def q_stays 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

989 
show ?case by auto 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

990 
qed 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

991 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

992 
section {* Code generation *} 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

993 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

994 
text {* A simple example program *} 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

995 

37792  996 
definition test_1 where "test_1 = (do { ll_xs < make_llist [1..(15::int)]; xs < traverse ll_xs; return xs })" 
997 
definition test_2 where "test_2 = (do { ll_xs < make_llist [1..(15::int)]; ll_ys < rev ll_xs; ys < traverse ll_ys; return ys })" 

34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

998 

1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

999 
definition test_3 where "test_3 = 
37792  1000 
(do { 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

1001 
ll_xs \<leftarrow> make_llist (filter (%n. n mod 2 = 0) [2..8]); 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

1002 
ll_ys \<leftarrow> make_llist (filter (%n. n mod 2 = 1) [5..11]); 
37725  1003 
r \<leftarrow> ref ll_xs; 
1004 
q \<leftarrow> ref ll_ys; 

34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

1005 
p \<leftarrow> merge r q; 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

1006 
ll_zs \<leftarrow> !p; 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

1007 
zs \<leftarrow> traverse ll_zs; 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

1008 
return zs 
37792  1009 
})" 
34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

1010 

35041
6eb917794a5c
avoid upto in generated code (is infix operator in library.ML)
haftmann
parents:
34051
diff
changeset

1011 
code_reserved SML upto 
6eb917794a5c
avoid upto in generated code (is infix operator in library.ML)
haftmann
parents:
34051
diff
changeset

1012 

34051
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

1013 
ML {* @{code test_1} () *} 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

1014 
ML {* @{code test_2} () *} 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

1015 
ML {* @{code test_3} () *} 
1a82e2e29d67
added Imperative_HOL examples; added tailrecursive combinator for monadic heap functions; adopted code generation of references; added lemmas
bulwahn
parents:
diff
changeset

1016 

37826  1017 
export_code test_1 test_2 test_3 checking SML SML_imp OCaml? OCaml_imp? Haskell? 
37750
82e0fe8b07eb
dropped ancient inplace compilation of SML; more tests
haftmann
parents:
37725
diff
changeset

1018 

37725  1019 
end 