<commit_before>(*  Title:      HOL/Extraction/QuotRem.thy
    ID:         $Id$
    Author:     <NAME>, T<NAME>
*)

header {* Quotient and remainder *}

theory QuotRem imports Main begin
text {* Derivation of quotient and remainder using program extraction. *}

lemma nat_eq_dec: "\<And>n::nat. m = n \<or> m \<noteq> n"
  apply (induct m)
  apply (case_tac n)
  apply (case_tac [3] n)
  apply (simp only: nat.simps, iprover?)+
  done

theorem division: "\<exists>r q. a = Suc b * q + r \<and> r \<le> b"
proof (induct a)
  case 0<commit_msg>Moved nat_eq_dec to Util.thy<commit_after>(*  Title:      HOL/Extraction/QuotRem.thy
    ID:         $Id$
    Author:     <NAME>, TU Muenchen
*)

header {* Quotient and remainder *}

theory QuotRem imports Util begin
text {* Derivation of quotient and remainder using program extraction. *}

theorem division: "\<exists>r q. a = Suc b * q + r \<and> r \<le> b"
proof (induct a)
  case 0
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{'dir': 'git-commits-cleaned', 'id': '100538', 'n_tokens_mistral': 360, 'n_tokens_neox': 341, 'n_words': 116}