<commit_before>
(*<*)
theory Tutorial = Main:
(*>*)

chapter {* Introduction *}

section {* Step-by-step examples *}

subsection {* Summing natural numbers *}

theorem "2 * (\<Sum>i < n + 1. i) = n * (n + 1)"  (is "?P n")
proof (induct n)
  case 0
  then show "?P 0" by simp
next
  case (Suc n)
  let "?S n = _" = "?P n"
  have "?S (n + 1) = ?S n + 2 * (n + 1)" by simp
  also have "?S n = n * (n + 1)" .
  also have "\<dots> + 2 * (n + 1) = (n + 1) * (n + 2)" by simp
  finally show "?P (Suc n)" by simp<commit_msg>tuned;<commit_after>
(*<*)
theory Tutorial = Main:
(*>*)

chapter {* Introduction *}

chapter {* Technical issues *}

section {* Source texts *}

section {* Interaction and debugging *}

section {* Step-by-step examples *}

subsection {* Summing natural numbers *}

theorem "2 * (\<Sum>i < n + 1. i) = n * (n + 1)"  (is "?P n")
proof (induct n)
  case 0
  then show "?P 0" by simp
next
  case (Suc n)
  let "?S n = _" = "?P n"
  have "?S (n + 1) = ?S n + 2 * (n + 1)" by simp
  also have "?S n = n * (n + 1)" .
  also have "\<dots> + 2 * (n + 1) = (n + 1) * (n + 2)" by simp
  finally show "?P (Suc n)" by simp
{'fraction_non_alphanumeric': 0.2069271758436945, 'fraction_numerical': 0.023090586145648313, 'mean_word_length': 2.7072368421052633, 'pattern_counts': {'":': 0, '<': 11, '<?xml version=': 0, '>': 9, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'dir': 'git-commits-cleaned', 'id': '1672146', 'n_tokens_mistral': 515, 'n_tokens_neox': 483, 'n_words': 173}