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### Definitions File

### Template File

### Check File

theory Defs imports Main begin end

theory Submission imports Defs begin lemma induct_pcpl: "\<lbrakk>P []; \<And>x. P [x]; \<And>x y zs. P zs \<Longrightarrow> P (x # y # zs)\<rbrakk> \<Longrightarrow> P xs" by induction_schema (pat_completeness, lexicographic_order) lemma split_splice: "\<exists>ys zs. xs = splice ys zs \<and> length ys \<ge> (length xs) div 2 \<and> length zs \<ge> (length xs) div 2" proof(induction xs rule: induct_pcpl) case 1 show ?case sorry next case (2 x) show ?case sorry next case (3 y z xs) show ?case sorry qed lemma eq_identiy: fixes f :: "nat \<Rightarrow> nat" assumes "\<forall>n. f(Suc n) = f(n)^2" shows "f(n) = f(0)^(2^n)" sorry end

theory Check imports Submission begin lemma split_splice: "\<exists>ys zs. xs = splice ys zs \<and> length ys \<ge> (length xs) div 2 \<and> length zs \<ge> (length xs) div 2" by(rule split_splice) lemma eq_identiy: fixes f :: "nat \<Rightarrow> nat" assumes "\<forall>n. f(Suc n) = f(n)^2" shows "f(n) = f(0)^(2^n)" by(rule eq_identiy[OF assms]) end

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