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