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(* Author: Tobias Nipkow *)
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theory Def_Ass_Small imports Star Com Def_Ass_Exp
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begin
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subsection "Initialization-Sensitive Small Step Semantics"
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inductive
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small_step :: "(com \<times> state) \<Rightarrow> (com \<times> state) \<Rightarrow> bool" (infix "\<rightarrow>" 55)
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where
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Assign: "aval a s = Some i \<Longrightarrow> (x ::= a, s) \<rightarrow> (SKIP, s(x := Some i))" |
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Semi1: "(SKIP;c,s) \<rightarrow> (c,s)" |
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Semi2: "(c\<^isub>1,s) \<rightarrow> (c\<^isub>1',s') \<Longrightarrow> (c\<^isub>1;c\<^isub>2,s) \<rightarrow> (c\<^isub>1';c\<^isub>2,s')" |
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IfTrue: "bval b s = Some True \<Longrightarrow> (IF b THEN c\<^isub>1 ELSE c\<^isub>2,s) \<rightarrow> (c\<^isub>1,s)" |
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IfFalse: "bval b s = Some False \<Longrightarrow> (IF b THEN c\<^isub>1 ELSE c\<^isub>2,s) \<rightarrow> (c\<^isub>2,s)" |
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While: "(WHILE b DO c,s) \<rightarrow> (IF b THEN c; WHILE b DO c ELSE SKIP,s)"
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lemmas small_step_induct = small_step.induct[split_format(complete)]
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abbreviation small_steps :: "com * state \<Rightarrow> com * state \<Rightarrow> bool" (infix "\<rightarrow>*" 55)
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where "x \<rightarrow>* y == star small_step x y"
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end
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