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theory Reverse
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imports Main
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begin
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fun T_append :: "'a list \<Rightarrow> 'a list \<Rightarrow> nat" where
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"T_append [] ys = 1" |
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"T_append (x#xs) ys = T_append xs ys + 1"
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fun T_rev :: "'a list \<Rightarrow> nat" where
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"T_rev [] = 1" |
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"T_rev (x#xs) = T_rev xs + T_append (rev xs) [x] + 1"
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lemma T_append: "T_append xs ys = length xs + 1"
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by(induction xs) auto
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lemma T_rev: "T_rev xs \<le> (length xs + 1)^2"
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by(induction xs) (auto simp: T_append power2_eq_square)
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fun itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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"itrev [] ys = ys" |
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"itrev (x#xs) ys = itrev xs (x # ys)"
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lemma itrev: "itrev xs ys = rev xs @ ys"
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by(induction xs arbitrary: ys) auto
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lemma itrev_Nil: "itrev xs [] = rev xs"
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by(simp add: itrev)
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fun T_itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> nat" where
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"T_itrev [] ys = 1" |
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"T_itrev (x#xs) ys = T_itrev xs (x # ys) + 1"
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lemma T_itrev: "T_itrev xs ys = length xs + 1"
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by(induction xs arbitrary: ys) auto
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end |