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(* Author: Tobias Nipkow *)


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theory Def_Ass_Sound_Big imports Def_Ass Def_Ass_Big


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begin


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subsection "Soundness wrt Big Steps"


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text{* Note the special form of the induction because one of the arguments


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of the inductive predicate is not a variable but the term @{term"Some s"}: *}


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theorem Sound:


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"\<lbrakk> (c,Some s) \<Rightarrow> s'; D A c A'; A \<subseteq> dom s \<rbrakk>


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\<Longrightarrow> \<exists> t. s' = Some t \<and> A' \<subseteq> dom t"

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proof (induction c "Some s" s' arbitrary: s A A' rule:big_step_induct)

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case AssignNone thus ?case


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by auto (metis aval_Some option.simps(3) subset_trans)


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next


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case Semi thus ?case by auto metis


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next


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case IfTrue thus ?case by auto blast


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next


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case IfFalse thus ?case by auto blast


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next


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case IfNone thus ?case


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by auto (metis bval_Some option.simps(3) order_trans)


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next


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case WhileNone thus ?case


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by auto (metis bval_Some option.simps(3) order_trans)


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next


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case (WhileTrue b s c s' s'')


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from `D A (WHILE b DO c) A'` obtain A' where "D A c A'" by blast


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then obtain t' where "s' = Some t'" "A \<subseteq> dom t'"


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by (metis D_incr WhileTrue(3,7) subset_trans)


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from WhileTrue(5)[OF this(1) WhileTrue(6) this(2)] show ?case .


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qed auto


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corollary sound: "\<lbrakk> D (dom s) c A'; (c,Some s) \<Rightarrow> s' \<rbrakk> \<Longrightarrow> s' \<noteq> None"


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by (metis Sound not_Some_eq subset_refl)


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end
