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693 lemma assumes A: "(x,y) \<in> r*" and B: "(y,z) \<in> r*" |
693 lemma assumes A: "(x,y) \<in> r*" and B: "(y,z) \<in> r*" |
694 shows "(x,z) \<in> r*" |
694 shows "(x,z) \<in> r*" |
695 proof - |
695 proof - |
696 from A B show ?thesis |
696 from A B show ?thesis |
697 proof induct |
697 proof induct |
698 fix x assume "(x,z) \<in> r*" -- "B[y := x]" |
698 fix x assume "(x,z) \<in> r*" -- {*@{text B}[@{text y} := @{text x}]*} |
699 thus "(x,z) \<in> r*" . |
699 thus "(x,z) \<in> r*" . |
700 next |
700 next |
701 fix x' x y |
701 fix x' x y |
702 assume step: "(x',x) \<in> r" and |
702 assume step: "(x',x) \<in> r" and |
703 IH: "(y,z) \<in> r* \<Longrightarrow> (x,z) \<in> r*" and |
703 IH: "(y,z) \<in> r* \<Longrightarrow> (x,z) \<in> r*" and |