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To start, she wanted to examine the difference between π(n+1) and π(n) and show that it either grows by one or stays the same.

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### Definitions File

### Template File

### Check File

theory Defs imports Main begin definition prime :: "nat \<Rightarrow> bool" where "prime n = (1 < n \<and> (\<forall>m. m dvd n \<longrightarrow> m = 1 \<or> m = n))" definition "\<pi> n = card { i. prime i \<and> i \<le> n }" end

theory Submission imports Defs begin lemma pi: "\<pi> (Suc n) - \<pi> n = (if prime (Suc n) then 1 else 0)" sorry end

theory Check imports Submission begin lemma "\<pi> (Suc n) - \<pi> n = (if prime (Suc n) then 1 else 0)" by (rule Submission.pi) end

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### Definitions File

### Template File

### Check File

theory Defs imports Main begin definition prime :: "nat \<Rightarrow> bool" where "prime n = (1 < n \<and> (\<forall>m. m dvd n \<longrightarrow> m = 1 \<or> m = n))" definition "\<pi> n = card { i. prime i \<and> i \<le> n }" end

theory Submission imports Defs begin lemma pi: "\<pi> (Suc n) - \<pi> n = (if prime (Suc n) then 1 else 0)" sorry end

theory Check imports Submission begin lemma "\<pi> (Suc n) - \<pi> n = (if prime (Suc n) then 1 else 0)" by (rule Submission.pi) end

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