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src/HOL/Library/Diagonal_Subsequence.thy

changeset 50087 | 635d73673b5e |

child 51526 | 155263089e7b |

--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Library/Diagonal_Subsequence.thy Thu Nov 15 10:49:58 2012 +0100 @@ -0,0 +1,111 @@ +(* Author: Fabian Immler, TUM *) + +header {* Sequence of Properties on Subsequences *} + +theory Diagonal_Subsequence +imports SEQ +begin + +locale subseqs = + fixes P::"nat\<Rightarrow>(nat\<Rightarrow>nat)\<Rightarrow>bool" + assumes ex_subseq: "\<And>n s. subseq s \<Longrightarrow> \<exists>r'. subseq r' \<and> P n (s o r')" +begin + +primrec seqseq where + "seqseq 0 = id" +| "seqseq (Suc n) = seqseq n o (SOME r'. subseq r' \<and> P n (seqseq n o r'))" + +lemma seqseq_ex: + shows "subseq (seqseq n) \<and> + (\<exists>r'. seqseq (Suc n) = seqseq n o r' \<and> subseq r' \<and> P n (seqseq n o r'))" +proof (induct n) + case 0 + let ?P = "\<lambda>r'. subseq r' \<and> P 0 r'" + let ?r = "Eps ?P" + have "?P ?r" using ex_subseq[of id 0] by (intro someI_ex[of ?P]) (auto simp: subseq_def) + thus ?case by (auto simp: subseq_def) +next + case (Suc n) + then obtain r' where + Suc': "seqseq (Suc n) = seqseq n \<circ> r'" "subseq (seqseq n)" "subseq r'" + "P n (seqseq n o r')" + by blast + let ?P = "\<lambda>r'a. subseq (r'a ) \<and> P (Suc n) (seqseq n o r' o r'a)" + let ?r = "Eps ?P" + have "?P ?r" using ex_subseq[of "seqseq n o r'" "Suc n"] Suc' + by (intro someI_ex[of ?P]) (auto intro: subseq_o simp: o_assoc) + moreover have "seqseq (Suc (Suc n)) = seqseq n \<circ> r' \<circ> ?r" + by (subst seqseq.simps) (simp only: Suc' o_assoc) + moreover note subseq_o[OF `subseq (seqseq n)` `subseq r'`] + ultimately show ?case unfolding Suc' by (auto simp: o_def) +qed + +lemma subseq_seqseq: + shows "subseq (seqseq n)" using seqseq_ex[OF assms] by auto + +definition reducer where "reducer n = (SOME r'. subseq r' \<and> P n (seqseq n o r'))" + +lemma subseq_reducer: "subseq (reducer n)" and reducer_reduces: "P n (seqseq n o reducer n)" + unfolding atomize_conj unfolding reducer_def using subseq_seqseq + by (rule someI_ex[OF ex_subseq]) + +lemma seqseq_reducer[simp]: + "seqseq (Suc n) = seqseq n o reducer n" + by (simp add: reducer_def) + +declare seqseq.simps(2)[simp del] + +definition diagseq where "diagseq i = seqseq i i" + +lemma diagseq_mono: "diagseq n < diagseq (Suc n)" + unfolding diagseq_def seqseq_reducer o_def + by (metis subseq_mono[OF subseq_seqseq] less_le_trans lessI seq_suble subseq_reducer) + +lemma subseq_diagseq: "subseq diagseq" + using diagseq_mono by (simp add: subseq_Suc_iff diagseq_def) + +primrec fold_reduce where + "fold_reduce n 0 = id" +| "fold_reduce n (Suc k) = fold_reduce n k o reducer (n + k)" + +lemma subseq_fold_reduce: "subseq (fold_reduce n k)" +proof (induct k) + case (Suc k) from subseq_o[OF this subseq_reducer] show ?case by (simp add: o_def) +qed (simp add: subseq_def) + +lemma ex_subseq_reduce_index: "seqseq (n + k) = seqseq n o fold_reduce n k" + by (induct k) simp_all + +lemma seqseq_fold_reduce: "seqseq n = fold_reduce 0 n" + by (induct n) (simp_all) + +lemma diagseq_fold_reduce: "diagseq n = fold_reduce 0 n n" + using seqseq_fold_reduce by (simp add: diagseq_def) + +lemma fold_reduce_add: "fold_reduce 0 (m + n) = fold_reduce 0 m o fold_reduce m n" + by (induct n) simp_all + +lemma diagseq_add: "diagseq (k + n) = (seqseq k o (fold_reduce k n)) (k + n)" +proof - + have "diagseq (k + n) = fold_reduce 0 (k + n) (k + n)" + by (simp add: diagseq_fold_reduce) + also have "\<dots> = (seqseq k o fold_reduce k n) (k + n)" + unfolding fold_reduce_add seqseq_fold_reduce .. + finally show ?thesis . +qed + +lemma diagseq_sub: + assumes "m \<le> n" shows "diagseq n = (seqseq m o (fold_reduce m (n - m))) n" + using diagseq_add[of m "n - m"] assms by simp + +lemma subseq_diagonal_rest: "subseq (\<lambda>x. fold_reduce k x (k + x))" + unfolding subseq_Suc_iff fold_reduce.simps o_def + by (metis subseq_mono[OF subseq_fold_reduce] less_le_trans lessI add_Suc_right seq_suble + subseq_reducer) + +lemma diagseq_seqseq: "diagseq o (op + k) = (seqseq k o (\<lambda>x. fold_reduce k x (k + x)))" + by (auto simp: o_def diagseq_add) + +end + +end