113 lines
3.7 KiB
Plaintext
113 lines
3.7 KiB
Plaintext
(*
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* Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
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*
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* SPDX-License-Identifier: BSD-2-Clause
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*)
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section \<open>Lemmas on list operations\<close>
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theory Even_More_List
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imports Main
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begin
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lemma upt_add_eq_append':
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assumes "i \<le> j" and "j \<le> k"
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shows "[i..<k] = [i..<j] @ [j..<k]"
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using assms le_Suc_ex upt_add_eq_append by blast
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lemma map_idem_upt_eq:
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\<open>map f [m..<n] = [m..<n]\<close> if \<open>\<And>q. m \<le> q \<Longrightarrow> q < n \<Longrightarrow> f q = q\<close>
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proof (cases \<open>n \<ge> m\<close>)
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case False
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then show ?thesis
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by simp
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next
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case True
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moreover define r where \<open>r = n - m\<close>
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ultimately have \<open>n = m + r\<close>
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by simp
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with that have \<open>\<And>q. m \<le> q \<Longrightarrow> q < m + r \<Longrightarrow> f q = q\<close>
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by simp
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then have \<open>map f [m..<m + r] = [m..<m + r]\<close>
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by (induction r) simp_all
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with \<open>n = m + r\<close> show ?thesis
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by simp
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qed
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lemma upt_zero_numeral_unfold:
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\<open>[0..<numeral n] = [0..<pred_numeral n] @ [pred_numeral n]\<close>
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by (simp add: numeral_eq_Suc)
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lemma length_takeWhile_less:
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"\<exists>x\<in>set xs. \<not> P x \<Longrightarrow> length (takeWhile P xs) < length xs"
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by (induct xs) (auto split: if_splits)
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lemma Min_eq_length_takeWhile:
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\<open>Min {m. P m} = length (takeWhile (Not \<circ> P) ([0..<n]))\<close>
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if *: \<open>\<And>m. P m \<Longrightarrow> m < n\<close> and \<open>\<exists>m. P m\<close>
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proof -
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from \<open>\<exists>m. P m\<close> obtain r where \<open>P r\<close> ..
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have \<open>Min {m. P m} = q + length (takeWhile (Not \<circ> P) ([q..<n]))\<close>
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if \<open>q \<le> n\<close> \<open>\<And>m. P m \<Longrightarrow> q \<le> m \<and> m < n\<close> for q
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using that proof (induction rule: inc_induct)
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case base
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from base [of r] \<open>P r\<close> show ?case
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by simp
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next
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case (step q)
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from \<open>q < n\<close> have *: \<open>[q..<n] = q # [Suc q..<n]\<close>
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by (simp add: upt_rec)
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show ?case
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proof (cases \<open>P q\<close>)
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case False
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then have \<open>Suc q \<le> m \<and> m < n\<close> if \<open>P m\<close> for m
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using that step.prems [of m] by (auto simp add: Suc_le_eq less_le)
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with \<open>\<not> P q\<close> show ?thesis
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by (simp add: * step.IH)
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next
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case True
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have \<open>{a. P a} \<subseteq> {0..n}\<close>
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using step.prems by (auto simp add: less_imp_le_nat)
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moreover have \<open>finite {0..n}\<close>
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by simp
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ultimately have \<open>finite {a. P a}\<close>
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by (rule finite_subset)
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with \<open>P q\<close> step.prems show ?thesis
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by (auto intro: Min_eqI simp add: *)
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qed
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qed
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from this [of 0] and that show ?thesis
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by simp
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qed
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lemma Max_eq_length_takeWhile:
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\<open>Max {m. P m} = n - Suc (length (takeWhile (Not \<circ> P) (rev [0..<n])))\<close>
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if *: \<open>\<And>m. P m \<Longrightarrow> m < n\<close> and \<open>\<exists>m. P m\<close>
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using * proof (induction n)
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case 0
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with \<open>\<exists>m. P m\<close> show ?case
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by auto
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next
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case (Suc n)
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show ?case
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proof (cases \<open>P n\<close>)
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case False
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then have \<open>m < n\<close> if \<open>P m\<close> for m
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using that Suc.prems [of m] by (auto simp add: less_le)
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with Suc.IH show ?thesis
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by auto
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next
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case True
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have \<open>{a. P a} \<subseteq> {0..n}\<close>
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using Suc.prems by (auto simp add: less_Suc_eq_le)
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moreover have \<open>finite {0..n}\<close>
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by simp
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ultimately have \<open>finite {a. P a}\<close>
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by (rule finite_subset)
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with \<open>P n\<close> Suc.prems show ?thesis
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by (auto intro: Max_eqI simp add: less_Suc_eq_le)
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qed
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qed
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end
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