author | berghofe |
Wed, 30 Apr 2014 15:43:44 +0200 | |
changeset 56798 | 939e88e79724 |
parent 44890 | 22f665a2e91c |
child 58130 | 5e9170812356 |
permissions | -rw-r--r-- |
41561 | 1 |
(* Title: HOL/SPARK/Examples/Liseq/Longest_Increasing_Subsequence.thy |
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Author: Stefan Berghofer |
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Copyright: secunet Security Networks AG |
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*) |
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theory Longest_Increasing_Subsequence |
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imports SPARK |
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begin |
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text {* |
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Set of all increasing subsequences in a prefix of an array |
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*} |
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definition iseq :: "(nat \<Rightarrow> 'a::linorder) \<Rightarrow> nat \<Rightarrow> nat set set" where |
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"iseq xs l = {is. (\<forall>i\<in>is. i < l) \<and> |
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(\<forall>i\<in>is. \<forall>j\<in>is. i \<le> j \<longrightarrow> xs i \<le> xs j)}" |
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text {* |
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Length of longest increasing subsequence in a prefix of an array |
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*} |
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definition liseq :: "(nat \<Rightarrow> 'a::linorder) \<Rightarrow> nat \<Rightarrow> nat" where |
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"liseq xs i = Max (card ` iseq xs i)" |
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text {* |
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Length of longest increasing subsequence ending at a particular position |
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*} |
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definition liseq' :: "(nat \<Rightarrow> 'a::linorder) \<Rightarrow> nat \<Rightarrow> nat" where |
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939e88e79724
Discontinued old spark_open; spark_open_siv is now spark_open
berghofe
parents:
44890
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"liseq' xs i = Max (card ` (iseq xs (Suc i) \<inter> {is. Max is = i}))" |
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lemma iseq_finite: "finite (iseq xs i)" |
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apply (simp add: iseq_def) |
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apply (rule finite_subset [OF _ |
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finite_Collect_subsets [of "{j. j < i}"]]) |
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apply auto |
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done |
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lemma iseq_finite': "is \<in> iseq xs i \<Longrightarrow> finite is" |
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by (auto simp add: iseq_def bounded_nat_set_is_finite) |
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lemma iseq_singleton: "i < l \<Longrightarrow> {i} \<in> iseq xs l" |
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by (simp add: iseq_def) |
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lemma iseq_trivial: "{} \<in> iseq xs i" |
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by (simp add: iseq_def) |
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lemma iseq_nonempty: "iseq xs i \<noteq> {}" |
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by (auto intro: iseq_trivial) |
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lemma liseq'_ge1: "1 \<le> liseq' xs x" |
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apply (simp add: liseq'_def) |
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apply (subgoal_tac "iseq xs (Suc x) \<inter> {is. Max is = x} \<noteq> {}") |
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apply (simp add: Max_ge_iff iseq_finite) |
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apply (rule_tac x="{x}" in bexI) |
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apply (auto intro: iseq_singleton) |
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done |
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lemma liseq_expand: |
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assumes R: "\<And>is. liseq xs i = card is \<Longrightarrow> is \<in> iseq xs i \<Longrightarrow> |
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(\<And>js. js \<in> iseq xs i \<Longrightarrow> card js \<le> card is) \<Longrightarrow> P" |
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shows "P" |
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proof - |
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have "Max (card ` iseq xs i) \<in> card ` iseq xs i" |
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by (rule Max_in) (simp_all add: iseq_finite iseq_nonempty) |
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then obtain js where js: "liseq xs i = card js" and "js \<in> iseq xs i" |
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by (rule imageE) (simp add: liseq_def) |
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moreover { |
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fix js' |
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assume "js' \<in> iseq xs i" |
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then have "card js' \<le> card js" |
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by (simp add: js [symmetric] liseq_def iseq_finite iseq_trivial) |
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} |
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ultimately show ?thesis by (rule R) |
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qed |
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lemma liseq'_expand: |
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assumes R: "\<And>is. liseq' xs i = card is \<Longrightarrow> is \<in> iseq xs (Suc i) \<Longrightarrow> |
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finite is \<Longrightarrow> Max is = i \<Longrightarrow> |
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(\<And>js. js \<in> iseq xs (Suc i) \<Longrightarrow> Max js = i \<Longrightarrow> card js \<le> card is) \<Longrightarrow> |
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is \<noteq> {} \<Longrightarrow> P" |
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shows "P" |
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proof - |
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have "Max (card ` (iseq xs (Suc i) \<inter> {is. Max is = i})) \<in> |
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card ` (iseq xs (Suc i) \<inter> {is. Max is = i})" |
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by (auto simp add: iseq_finite intro!: iseq_singleton Max_in) |
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then obtain js where js: "liseq' xs i = card js" and "js \<in> iseq xs (Suc i)" |
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and "finite js" and "Max js = i" |
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by (auto simp add: liseq'_def intro: iseq_finite') |
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moreover { |
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fix js' |
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assume "js' \<in> iseq xs (Suc i)" "Max js' = i" |
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then have "card js' \<le> card js" |
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by (auto simp add: js [symmetric] liseq'_def iseq_finite intro!: iseq_singleton) |
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} |
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note max = this |
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moreover have "card {i} \<le> card js" |
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by (rule max) (simp_all add: iseq_singleton) |
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then have "js \<noteq> {}" by auto |
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ultimately show ?thesis by (rule R) |
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qed |
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lemma liseq'_ge: |
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"j = card js \<Longrightarrow> js \<in> iseq xs (Suc i) \<Longrightarrow> Max js = i \<Longrightarrow> |
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js \<noteq> {} \<Longrightarrow> j \<le> liseq' xs i" |
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by (simp add: liseq'_def iseq_finite) |
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lemma liseq'_eq: |
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"j = card js \<Longrightarrow> js \<in> iseq xs (Suc i) \<Longrightarrow> Max js = i \<Longrightarrow> |
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js \<noteq> {} \<Longrightarrow> (\<And>js'. js' \<in> iseq xs (Suc i) \<Longrightarrow> Max js' = i \<Longrightarrow> finite js' \<Longrightarrow> |
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js' \<noteq> {} \<Longrightarrow> card js' \<le> card js) \<Longrightarrow> |
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j = liseq' xs i" |
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new fastforce replacing fastsimp - less confusing name
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by (fastforce simp add: liseq'_def iseq_finite |
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intro: Max_eqI [symmetric]) |
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lemma liseq_ge: |
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"j = card js \<Longrightarrow> js \<in> iseq xs i \<Longrightarrow> j \<le> liseq xs i" |
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by (auto simp add: liseq_def iseq_finite) |
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lemma liseq_eq: |
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"j = card js \<Longrightarrow> js \<in> iseq xs i \<Longrightarrow> |
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(\<And>js'. js' \<in> iseq xs i \<Longrightarrow> finite js' \<Longrightarrow> |
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js' \<noteq> {} \<Longrightarrow> card js' \<le> card js) \<Longrightarrow> |
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j = liseq xs i" |
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44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
41588
diff
changeset
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by (fastforce simp add: liseq_def iseq_finite |
41561 | 126 |
intro: Max_eqI [symmetric]) |
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lemma max_notin: "finite xs \<Longrightarrow> Max xs < x \<Longrightarrow> x \<notin> xs" |
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by (cases "xs = {}") auto |
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lemma iseq_insert: |
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"xs (Max is) \<le> xs i \<Longrightarrow> is \<in> iseq xs i \<Longrightarrow> |
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is \<union> {i} \<in> iseq xs (Suc i)" |
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apply (frule iseq_finite') |
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apply (cases "is = {}") |
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apply (auto simp add: iseq_def) |
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apply (rule order_trans [of _ "xs (Max is)"]) |
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apply auto |
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apply (thin_tac "\<forall>a\<in>is. a < i") |
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apply (drule_tac x=ia in bspec) |
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apply assumption |
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apply (drule_tac x="Max is" in bspec) |
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apply (auto intro: Max_in) |
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done |
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lemma iseq_diff: "is \<in> iseq xs (Suc (Max is)) \<Longrightarrow> |
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is - {Max is} \<in> iseq xs (Suc (Max (is - {Max is})))" |
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apply (frule iseq_finite') |
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apply (simp add: iseq_def less_Suc_eq_le) |
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done |
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lemma iseq_butlast: |
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assumes "js \<in> iseq xs (Suc i)" and "js \<noteq> {}" |
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and "Max js \<noteq> i" |
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shows "js \<in> iseq xs i" |
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proof - |
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from assms have fin: "finite js" |
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by (simp add: iseq_finite') |
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with assms have "Max js \<in> js" |
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by auto |
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with assms have "Max js < i" |
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by (auto simp add: iseq_def) |
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with fin assms have "\<forall>j\<in>js. j < i" |
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by simp |
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with assms show ?thesis |
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by (simp add: iseq_def) |
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qed |
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lemma iseq_mono: "is \<in> iseq xs i \<Longrightarrow> i \<le> j \<Longrightarrow> is \<in> iseq xs j" |
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by (auto simp add: iseq_def) |
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lemma diff_nonempty: |
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assumes "1 < card is" |
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shows "is - {i} \<noteq> {}" |
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proof - |
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from assms have fin: "finite is" by (auto intro: card_ge_0_finite) |
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with assms fin have "card is - 1 \<le> card (is - {i})" |
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by (simp add: card_Diff_singleton_if) |
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with assms have "0 < card (is - {i})" by simp |
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then show ?thesis by (simp add: card_gt_0_iff) |
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qed |
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lemma Max_diff: |
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assumes "1 < card is" |
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shows "Max (is - {Max is}) < Max is" |
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proof - |
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from assms have "finite is" by (auto intro: card_ge_0_finite) |
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moreover from assms have "is - {Max is} \<noteq> {}" |
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by (rule diff_nonempty) |
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ultimately show ?thesis using assms |
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apply (auto simp add: not_less) |
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apply (subgoal_tac "a \<le> Max is") |
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apply auto |
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done |
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qed |
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lemma iseq_nth: "js \<in> iseq xs l \<Longrightarrow> 1 < card js \<Longrightarrow> |
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xs (Max (js - {Max js})) \<le> xs (Max js)" |
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apply (auto simp add: iseq_def) |
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apply (subgoal_tac "Max (js - {Max js}) \<in> js") |
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apply (thin_tac "\<forall>i\<in>js. i < l") |
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apply (drule_tac x="Max (js - {Max js})" in bspec) |
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apply assumption |
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apply (drule_tac x="Max js" in bspec) |
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using card_gt_0_iff [of js] |
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apply simp |
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using Max_diff [of js] |
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apply simp |
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using Max_in [of "js - {Max js}", OF _ diff_nonempty] card_gt_0_iff [of js] |
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apply auto |
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done |
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lemma card_leq1_singleton: |
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assumes "finite xs" "xs \<noteq> {}" "card xs \<le> 1" |
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obtains x where "xs = {x}" |
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using assms |
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by induct simp_all |
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lemma longest_iseq1: |
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"liseq' xs i = |
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Max ({0} \<union> {liseq' xs j |j. j < i \<and> xs j \<le> xs i}) + 1" |
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proof - |
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have "Max ({0} \<union> {liseq' xs j |j. j < i \<and> xs j \<le> xs i}) = liseq' xs i - 1" |
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proof (rule Max_eqI) |
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fix y |
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assume "y \<in> {0} \<union> {liseq' xs j |j. j < i \<and> xs j \<le> xs i}" |
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then show "y \<le> liseq' xs i - 1" |
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proof |
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assume "y \<in> {liseq' xs j |j. j < i \<and> xs j \<le> xs i}" |
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then obtain j where j: "j < i" "xs j \<le> xs i" "y = liseq' xs j" |
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by auto |
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have "liseq' xs j + 1 \<le> liseq' xs i" |
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proof (rule liseq'_expand) |
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fix "is" |
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assume H: "liseq' xs j = card is" "is \<in> iseq xs (Suc j)" |
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"finite is" "Max is = j" "is \<noteq> {}" |
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from H j have "card is + 1 = card (is \<union> {i})" |
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by (simp add: card_insert max_notin) |
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moreover { |
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from H j have "xs (Max is) \<le> xs i" by simp |
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moreover from `j < i` have "Suc j \<le> i" by simp |
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with `is \<in> iseq xs (Suc j)` have "is \<in> iseq xs i" |
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by (rule iseq_mono) |
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ultimately have "is \<union> {i} \<in> iseq xs (Suc i)" |
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by (rule iseq_insert) |
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} moreover from H j have "Max (is \<union> {i}) = i" by simp |
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moreover have "is \<union> {i} \<noteq> {}" by simp |
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ultimately have "card is + 1 \<le> liseq' xs i" |
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by (rule liseq'_ge) |
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with H show ?thesis by simp |
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qed |
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with j show "y \<le> liseq' xs i - 1" |
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by simp |
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qed simp |
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next |
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have "liseq' xs i \<le> 1 \<or> |
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(\<exists>j. liseq' xs i - 1 = liseq' xs j \<and> j < i \<and> xs j \<le> xs i)" |
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proof (rule liseq'_expand) |
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fix "is" |
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assume H: "liseq' xs i = card is" "is \<in> iseq xs (Suc i)" |
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"finite is" "Max is = i" "is \<noteq> {}" |
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assume R: "\<And>js. js \<in> iseq xs (Suc i) \<Longrightarrow> Max js = i \<Longrightarrow> |
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card js \<le> card is" |
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show ?thesis |
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proof (cases "card is \<le> 1") |
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case True with H show ?thesis by simp |
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next |
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case False |
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then have "1 < card is" by simp |
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then have "Max (is - {Max is}) < Max is" |
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by (rule Max_diff) |
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from `is \<in> iseq xs (Suc i)` `1 < card is` |
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have "xs (Max (is - {Max is})) \<le> xs (Max is)" |
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by (rule iseq_nth) |
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have "card is - 1 = liseq' xs (Max (is - {i}))" |
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proof (rule liseq'_eq) |
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from `Max is = i` [symmetric] `finite is` `is \<noteq> {}` |
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show "card is - 1 = card (is - {i})" by simp |
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next |
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from `is \<in> iseq xs (Suc i)` `Max is = i` [symmetric] |
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show "is - {i} \<in> iseq xs (Suc (Max (is - {i})))" |
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by simp (rule iseq_diff) |
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next |
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from `1 < card is` |
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show "is - {i} \<noteq> {}" by (rule diff_nonempty) |
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next |
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fix js |
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assume "js \<in> iseq xs (Suc (Max (is - {i})))" |
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"Max js = Max (is - {i})" "finite js" "js \<noteq> {}" |
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from `xs (Max (is - {Max is})) \<le> xs (Max is)` |
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`Max js = Max (is - {i})` `Max is = i` |
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have "xs (Max js) \<le> xs i" by simp |
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moreover from `Max is = i` `Max (is - {Max is}) < Max is` |
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have "Suc (Max (is - {i})) \<le> i" |
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by simp |
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with `js \<in> iseq xs (Suc (Max (is - {i})))` |
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have "js \<in> iseq xs i" |
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by (rule iseq_mono) |
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ultimately have "js \<union> {i} \<in> iseq xs (Suc i)" |
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by (rule iseq_insert) |
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moreover from `js \<noteq> {}` `finite js` `Max js = Max (is - {i})` |
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`Max is = i` [symmetric] `Max (is - {Max is}) < Max is` |
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have "Max (js \<union> {i}) = i" |
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by simp |
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ultimately have "card (js \<union> {i}) \<le> card is" by (rule R) |
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moreover from `Max is = i` [symmetric] `finite js` |
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`Max (is - {Max is}) < Max is` `Max js = Max (is - {i})` |
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have "i \<notin> js" by (simp add: max_notin) |
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with `finite js` |
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have "card (js \<union> {i}) = card ((js \<union> {i}) - {i}) + 1" |
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by simp |
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ultimately show "card js \<le> card (is - {i})" |
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using `i \<notin> js` `Max is = i` [symmetric] `is \<noteq> {}` `finite is` |
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by simp |
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qed simp |
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with H `Max (is - {Max is}) < Max is` |
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`xs (Max (is - {Max is})) \<le> xs (Max is)` |
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show ?thesis by auto |
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qed |
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320 |
qed |
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then show "liseq' xs i - 1 \<in> {0} \<union> |
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{liseq' xs j |j. j < i \<and> xs j \<le> xs i}" by simp |
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qed simp |
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moreover have "1 \<le> liseq' xs i" by (rule liseq'_ge1) |
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ultimately show ?thesis by simp |
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qed |
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328 |
lemma longest_iseq2': "liseq xs i < liseq' xs i \<Longrightarrow> |
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liseq xs (Suc i) = liseq' xs i" |
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apply (rule_tac xs=xs and i=i in liseq'_expand) |
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apply simp |
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apply (rule liseq_eq [symmetric]) |
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apply (rule refl) |
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apply assumption |
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apply (case_tac "Max js' = i") |
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apply simp |
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apply (drule_tac js=js' in iseq_butlast) |
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apply assumption+ |
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apply (drule_tac js=js' in liseq_ge [OF refl]) |
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apply simp |
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done |
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lemma longest_iseq2: "liseq xs i < liseq' xs i \<Longrightarrow> |
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liseq xs i + 1 = liseq' xs i" |
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apply (rule_tac xs=xs and i=i in liseq'_expand) |
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apply simp |
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apply (rule_tac xs=xs and i=i in liseq_expand) |
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apply (drule_tac s="Max is" in sym) |
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apply simp |
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apply (case_tac "card is \<le> 1") |
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apply simp |
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apply (drule iseq_diff) |
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apply (drule_tac i="Suc (Max (is - {Max is}))" and j="Max is" in iseq_mono) |
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apply (simp add: less_eq_Suc_le [symmetric]) |
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apply (rule Max_diff) |
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apply simp |
|
357 |
apply (drule_tac x="is - {Max is}" in meta_spec, |
|
358 |
drule meta_mp, assumption) |
|
359 |
apply simp |
|
360 |
done |
|
361 |
||
362 |
lemma longest_iseq3: |
|
363 |
"liseq xs j = liseq' xs i \<Longrightarrow> xs i \<le> xs j \<Longrightarrow> i < j \<Longrightarrow> |
|
364 |
liseq xs (Suc j) = liseq xs j + 1" |
|
365 |
apply (rule_tac xs=xs and i=j in liseq_expand) |
|
366 |
apply simp |
|
367 |
apply (rule_tac xs=xs and i=i in liseq'_expand) |
|
368 |
apply simp |
|
369 |
apply (rule_tac js="isa \<union> {j}" in liseq_eq [symmetric]) |
|
370 |
apply (simp add: card_insert card_Diff_singleton_if max_notin) |
|
371 |
apply (rule iseq_insert) |
|
372 |
apply simp |
|
373 |
apply (erule iseq_mono) |
|
374 |
apply simp |
|
375 |
apply (case_tac "j = Max js'") |
|
376 |
apply simp |
|
377 |
apply (drule iseq_diff) |
|
378 |
apply (drule_tac x="js' - {j}" in meta_spec) |
|
379 |
apply (drule meta_mp) |
|
380 |
apply simp |
|
381 |
apply (case_tac "card js' \<le> 1") |
|
382 |
apply (erule_tac xs=js' in card_leq1_singleton) |
|
383 |
apply assumption+ |
|
384 |
apply (simp add: iseq_trivial) |
|
385 |
apply (erule iseq_mono) |
|
386 |
apply (simp add: less_eq_Suc_le [symmetric]) |
|
387 |
apply (rule Max_diff) |
|
388 |
apply simp |
|
389 |
apply (rule le_diff_iff [THEN iffD1, of 1]) |
|
390 |
apply (simp add: card_0_eq [symmetric] del: card_0_eq) |
|
391 |
apply (simp add: card_insert) |
|
392 |
apply (subgoal_tac "card (js' - {j}) = card js' - 1") |
|
393 |
apply (simp add: card_insert card_Diff_singleton_if max_notin) |
|
394 |
apply (frule_tac A=js' in Max_in) |
|
395 |
apply assumption |
|
396 |
apply (simp add: card_Diff_singleton_if) |
|
397 |
apply (drule_tac js=js' in iseq_butlast) |
|
398 |
apply assumption |
|
399 |
apply (erule not_sym) |
|
400 |
apply (drule_tac x=js' in meta_spec) |
|
401 |
apply (drule meta_mp) |
|
402 |
apply assumption |
|
403 |
apply (simp add: card_insert_disjoint max_notin) |
|
404 |
done |
|
405 |
||
406 |
lemma longest_iseq4: |
|
407 |
"liseq xs j = liseq' xs i \<Longrightarrow> xs i \<le> xs j \<Longrightarrow> i < j \<Longrightarrow> |
|
408 |
liseq' xs j = liseq' xs i + 1" |
|
409 |
apply (rule_tac xs=xs and i=j in liseq_expand) |
|
410 |
apply simp |
|
411 |
apply (rule_tac xs=xs and i=i in liseq'_expand) |
|
412 |
apply simp |
|
413 |
apply (rule_tac js="isa \<union> {j}" in liseq'_eq [symmetric]) |
|
414 |
apply (simp add: card_insert card_Diff_singleton_if max_notin) |
|
415 |
apply (rule iseq_insert) |
|
416 |
apply simp |
|
417 |
apply (erule iseq_mono) |
|
418 |
apply simp |
|
419 |
apply simp |
|
420 |
apply simp |
|
421 |
apply (drule_tac s="Max js'" in sym) |
|
422 |
apply simp |
|
423 |
apply (drule iseq_diff) |
|
424 |
apply (drule_tac x="js' - {j}" in meta_spec) |
|
425 |
apply (drule meta_mp) |
|
426 |
apply simp |
|
427 |
apply (case_tac "card js' \<le> 1") |
|
428 |
apply (erule_tac xs=js' in card_leq1_singleton) |
|
429 |
apply assumption+ |
|
430 |
apply (simp add: iseq_trivial) |
|
431 |
apply (erule iseq_mono) |
|
432 |
apply (simp add: less_eq_Suc_le [symmetric]) |
|
433 |
apply (rule Max_diff) |
|
434 |
apply simp |
|
435 |
apply (rule le_diff_iff [THEN iffD1, of 1]) |
|
436 |
apply (simp add: card_0_eq [symmetric] del: card_0_eq) |
|
437 |
apply (simp add: card_insert) |
|
438 |
apply (subgoal_tac "card (js' - {j}) = card js' - 1") |
|
439 |
apply (simp add: card_insert card_Diff_singleton_if max_notin) |
|
440 |
apply (frule_tac A=js' in Max_in) |
|
441 |
apply assumption |
|
442 |
apply (simp add: card_Diff_singleton_if) |
|
443 |
done |
|
444 |
||
445 |
lemma longest_iseq5: "liseq' xs i \<le> liseq xs i \<Longrightarrow> |
|
446 |
liseq xs (Suc i) = liseq xs i" |
|
447 |
apply (rule_tac i=i and xs=xs in liseq'_expand) |
|
448 |
apply simp |
|
449 |
apply (rule_tac xs=xs and i=i in liseq_expand) |
|
450 |
apply simp |
|
451 |
apply (rule liseq_eq [symmetric]) |
|
452 |
apply (rule refl) |
|
453 |
apply (erule iseq_mono) |
|
454 |
apply simp |
|
455 |
apply (case_tac "Max js' = i") |
|
456 |
apply (drule_tac x=js' in meta_spec) |
|
457 |
apply simp |
|
458 |
apply (drule iseq_butlast, assumption, assumption) |
|
459 |
apply simp |
|
460 |
done |
|
461 |
||
462 |
lemma liseq_empty: "liseq xs 0 = 0" |
|
463 |
apply (rule_tac js="{}" in liseq_eq [symmetric]) |
|
464 |
apply simp |
|
465 |
apply (rule iseq_trivial) |
|
466 |
apply (simp add: iseq_def) |
|
467 |
done |
|
468 |
||
469 |
lemma liseq'_singleton: "liseq' xs 0 = 1" |
|
470 |
by (simp add: longest_iseq1 [of _ 0]) |
|
471 |
||
472 |
lemma liseq_singleton: "liseq xs (Suc 0) = Suc 0" |
|
473 |
by (simp add: longest_iseq2' liseq_empty liseq'_singleton) |
|
474 |
||
475 |
lemma liseq'_Suc_unfold: |
|
476 |
"A j \<le> x \<Longrightarrow> |
|
477 |
(insert 0 {liseq' A j' |j'. j' < Suc j \<and> A j' \<le> x}) = |
|
478 |
(insert 0 {liseq' A j' |j'. j' < j \<and> A j' \<le> x}) \<union> |
|
479 |
{liseq' A j}" |
|
480 |
by (auto simp add: less_Suc_eq) |
|
481 |
||
482 |
lemma liseq'_Suc_unfold': |
|
483 |
"\<not> (A j \<le> x) \<Longrightarrow> |
|
484 |
{liseq' A j' |j'. j' < Suc j \<and> A j' \<le> x} = |
|
485 |
{liseq' A j' |j'. j' < j \<and> A j' \<le> x}" |
|
486 |
by (auto simp add: less_Suc_eq) |
|
487 |
||
488 |
lemma iseq_card_limit: |
|
489 |
assumes "is \<in> iseq A i" |
|
490 |
shows "card is \<le> i" |
|
491 |
proof - |
|
492 |
from assms have "is \<subseteq> {0..<i}" |
|
493 |
by (auto simp add: iseq_def) |
|
494 |
with finite_atLeastLessThan have "card is \<le> card {0..<i}" |
|
495 |
by (rule card_mono) |
|
496 |
with card_atLeastLessThan show ?thesis by simp |
|
497 |
qed |
|
498 |
||
499 |
lemma liseq_limit: "liseq A i \<le> i" |
|
500 |
by (rule_tac xs=A and i=i in liseq_expand) |
|
501 |
(simp add: iseq_card_limit) |
|
502 |
||
503 |
lemma liseq'_limit: "liseq' A i \<le> i + 1" |
|
504 |
by (rule_tac xs=A and i=i in liseq'_expand) |
|
505 |
(simp add: iseq_card_limit) |
|
506 |
||
507 |
definition max_ext :: "(nat \<Rightarrow> 'a::linorder) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" where |
|
508 |
"max_ext A i j = Max ({0} \<union> {liseq' A j' |j'. j' < j \<and> A j' \<le> A i})" |
|
509 |
||
510 |
lemma max_ext_limit: "max_ext A i j \<le> j" |
|
511 |
apply (auto simp add: max_ext_def) |
|
512 |
apply (drule Suc_leI) |
|
513 |
apply (cut_tac i=j' and A=A in liseq'_limit) |
|
514 |
apply simp |
|
515 |
done |
|
516 |
||
517 |
||
518 |
text {* Proof functions *} |
|
519 |
||
520 |
abbreviation (input) |
|
521 |
"arr_conv a \<equiv> (\<lambda>n. a (int n))" |
|
522 |
||
523 |
lemma idx_conv_suc: |
|
524 |
"0 \<le> i \<Longrightarrow> nat (i + 1) = nat i + 1" |
|
525 |
by simp |
|
526 |
||
527 |
abbreviation liseq_ends_at' :: "(int \<Rightarrow> 'a::linorder) \<Rightarrow> int \<Rightarrow> int" where |
|
528 |
"liseq_ends_at' A i \<equiv> int (liseq' (\<lambda>l. A (int l)) (nat i))" |
|
529 |
||
530 |
abbreviation liseq_prfx' :: "(int \<Rightarrow> 'a::linorder) \<Rightarrow> int \<Rightarrow> int" where |
|
531 |
"liseq_prfx' A i \<equiv> int (liseq (\<lambda>l. A (int l)) (nat i))" |
|
532 |
||
533 |
abbreviation max_ext' :: "(int \<Rightarrow> 'a::linorder) \<Rightarrow> int \<Rightarrow> int \<Rightarrow> int" where |
|
534 |
"max_ext' A i j \<equiv> int (max_ext (\<lambda>l. A (int l)) (nat i) (nat j))" |
|
535 |
||
536 |
spark_proof_functions |
|
537 |
liseq_ends_at = "liseq_ends_at' :: (int \<Rightarrow> int) \<Rightarrow> int \<Rightarrow> int" |
|
538 |
liseq_prfx = "liseq_prfx' :: (int \<Rightarrow> int) \<Rightarrow> int \<Rightarrow> int" |
|
539 |
max_ext = "max_ext' :: (int \<Rightarrow> int) \<Rightarrow> int \<Rightarrow> int \<Rightarrow> int" |
|
540 |
||
541 |
||
542 |
text {* The verification conditions *} |
|
543 |
||
56798
939e88e79724
Discontinued old spark_open; spark_open_siv is now spark_open
berghofe
parents:
44890
diff
changeset
|
544 |
spark_open "liseq/liseq_length" |
41561 | 545 |
|
546 |
spark_vc procedure_liseq_length_5 |
|
547 |
by (simp_all add: liseq_singleton liseq'_singleton) |
|
548 |
||
549 |
spark_vc procedure_liseq_length_6 |
|
550 |
proof - |
|
551 |
from H1 H2 H3 H4 |
|
552 |
have eq: "liseq (arr_conv a) (nat i) = |
|
553 |
liseq' (arr_conv a) (nat pmax)" |
|
554 |
by simp |
|
555 |
from H14 H3 H4 |
|
556 |
have pmax1: "arr_conv a (nat pmax) \<le> arr_conv a (nat i)" |
|
557 |
by simp |
|
558 |
from H3 H4 have pmax2: "nat pmax < nat i" |
|
559 |
by simp |
|
560 |
{ |
|
561 |
fix i2 |
|
562 |
assume i2: "0 \<le> i2" "i2 \<le> i" |
|
563 |
have "(l(i := l pmax + 1)) i2 = |
|
564 |
int (liseq' (arr_conv a) (nat i2))" |
|
565 |
proof (cases "i2 = i") |
|
566 |
case True |
|
567 |
from eq pmax1 pmax2 have "liseq' (arr_conv a) (nat i) = |
|
568 |
liseq' (arr_conv a) (nat pmax) + 1" |
|
569 |
by (rule longest_iseq4) |
|
570 |
with True H1 H3 H4 show ?thesis |
|
571 |
by simp |
|
572 |
next |
|
573 |
case False |
|
574 |
with H1 i2 show ?thesis |
|
575 |
by simp |
|
576 |
qed |
|
577 |
} |
|
578 |
then show ?C1 by simp |
|
579 |
from eq pmax1 pmax2 |
|
580 |
have "liseq (arr_conv a) (Suc (nat i)) = |
|
581 |
liseq (arr_conv a) (nat i) + 1" |
|
582 |
by (rule longest_iseq3) |
|
583 |
with H2 H3 H4 show ?C2 |
|
584 |
by (simp add: idx_conv_suc) |
|
585 |
qed |
|
586 |
||
587 |
spark_vc procedure_liseq_length_7 |
|
588 |
proof - |
|
589 |
from H1 show ?C1 |
|
590 |
by (simp add: max_ext_def longest_iseq1 [of _ "nat i"]) |
|
591 |
from H6 |
|
592 |
have m: "max_ext (arr_conv a) (nat i) (nat i) + 1 = |
|
593 |
liseq' (arr_conv a) (nat i)" |
|
594 |
by (simp add: max_ext_def longest_iseq1 [of _ "nat i"]) |
|
595 |
with H2 H18 |
|
596 |
have gt: "liseq (arr_conv a) (nat i) < liseq' (arr_conv a) (nat i)" |
|
597 |
by simp |
|
598 |
then have "liseq' (arr_conv a) (nat i) = liseq (arr_conv a) (nat i) + 1" |
|
599 |
by (rule longest_iseq2 [symmetric]) |
|
600 |
with H2 m show ?C2 by simp |
|
601 |
from gt have "liseq (arr_conv a) (Suc (nat i)) = liseq' (arr_conv a) (nat i)" |
|
602 |
by (rule longest_iseq2') |
|
603 |
with m H6 show ?C3 by (simp add: idx_conv_suc) |
|
604 |
qed |
|
605 |
||
606 |
spark_vc procedure_liseq_length_8 |
|
607 |
proof - |
|
608 |
{ |
|
609 |
fix i2 |
|
610 |
assume i2: "0 \<le> i2" "i2 \<le> i" |
|
611 |
have "(l(i := max_ext' a i i + 1)) i2 = |
|
612 |
int (liseq' (arr_conv a) (nat i2))" |
|
613 |
proof (cases "i2 = i") |
|
614 |
case True |
|
615 |
with H1 show ?thesis |
|
616 |
by (simp add: max_ext_def longest_iseq1 [of _ "nat i"]) |
|
617 |
next |
|
618 |
case False |
|
619 |
with H1 i2 show ?thesis by simp |
|
620 |
qed |
|
621 |
} |
|
622 |
then show ?C1 by simp |
|
623 |
from H2 H6 H18 |
|
624 |
have "liseq' (arr_conv a) (nat i) \<le> liseq (arr_conv a) (nat i)" |
|
625 |
by (simp add: max_ext_def longest_iseq1 [of _ "nat i"]) |
|
626 |
then have "liseq (arr_conv a) (Suc (nat i)) = liseq (arr_conv a) (nat i)" |
|
627 |
by (rule longest_iseq5) |
|
628 |
with H2 H6 show ?C2 by (simp add: idx_conv_suc) |
|
629 |
qed |
|
630 |
||
631 |
spark_vc procedure_liseq_length_12 |
|
632 |
by (simp add: max_ext_def) |
|
633 |
||
634 |
spark_vc procedure_liseq_length_13 |
|
635 |
using H1 H6 H13 H21 H22 |
|
636 |
by (simp add: max_ext_def |
|
637 |
idx_conv_suc liseq'_Suc_unfold max_def del: Max_less_iff) |
|
638 |
||
639 |
spark_vc procedure_liseq_length_14 |
|
640 |
using H1 H6 H13 H21 |
|
641 |
by (cases "a j \<le> a i") |
|
642 |
(simp_all add: max_ext_def |
|
643 |
idx_conv_suc liseq'_Suc_unfold liseq'_Suc_unfold') |
|
644 |
||
645 |
spark_vc procedure_liseq_length_19 |
|
646 |
using H3 H4 H5 H8 H9 |
|
647 |
apply (rule_tac y="int (nat i)" in order_trans) |
|
648 |
apply (cut_tac A="arr_conv a" and i="nat i" and j="nat i" in max_ext_limit) |
|
649 |
apply simp_all |
|
650 |
done |
|
651 |
||
652 |
spark_vc procedure_liseq_length_23 |
|
653 |
using H2 H3 H4 H7 H8 H11 |
|
654 |
apply (rule_tac y="int (nat i)" in order_trans) |
|
655 |
apply (cut_tac A="arr_conv a" and i="nat i" in liseq_limit) |
|
656 |
apply simp_all |
|
657 |
done |
|
658 |
||
659 |
spark_vc procedure_liseq_length_29 |
|
660 |
using H2 H3 H8 H13 |
|
661 |
by (simp add: add1_zle_eq [symmetric]) |
|
662 |
||
663 |
spark_end |
|
664 |
||
665 |
end |