src/HOL/Library/Diagonal_Subsequence.thy
changeset 50087 635d73673b5e
child 51526 155263089e7b
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/HOL/Library/Diagonal_Subsequence.thy	Thu Nov 15 10:49:58 2012 +0100
     1.3 @@ -0,0 +1,111 @@
     1.4 +(* Author: Fabian Immler, TUM *)
     1.5 +
     1.6 +header {* Sequence of Properties on Subsequences *}
     1.7 +
     1.8 +theory Diagonal_Subsequence
     1.9 +imports SEQ
    1.10 +begin
    1.11 +
    1.12 +locale subseqs =
    1.13 +  fixes P::"nat\<Rightarrow>(nat\<Rightarrow>nat)\<Rightarrow>bool"
    1.14 +  assumes ex_subseq: "\<And>n s. subseq s \<Longrightarrow> \<exists>r'. subseq r' \<and> P n (s o r')"
    1.15 +begin
    1.16 +
    1.17 +primrec seqseq where
    1.18 +  "seqseq 0 = id"
    1.19 +| "seqseq (Suc n) = seqseq n o (SOME r'. subseq r' \<and> P n (seqseq n o r'))"
    1.20 +
    1.21 +lemma seqseq_ex:
    1.22 +  shows "subseq (seqseq n) \<and>
    1.23 +  (\<exists>r'. seqseq (Suc n) = seqseq n o r' \<and> subseq r' \<and> P n (seqseq n o r'))"
    1.24 +proof (induct n)
    1.25 +  case 0
    1.26 +  let ?P = "\<lambda>r'. subseq r' \<and> P 0 r'"
    1.27 +  let ?r = "Eps ?P"
    1.28 +  have "?P ?r" using ex_subseq[of id 0] by (intro someI_ex[of ?P]) (auto simp: subseq_def)
    1.29 +  thus ?case by (auto simp: subseq_def)
    1.30 +next
    1.31 +  case (Suc n)
    1.32 +  then obtain r' where
    1.33 +    Suc': "seqseq (Suc n) = seqseq n \<circ> r'" "subseq (seqseq n)" "subseq r'"
    1.34 +      "P n (seqseq n o r')"
    1.35 +    by blast
    1.36 +  let ?P = "\<lambda>r'a. subseq (r'a ) \<and> P (Suc n) (seqseq n o r' o r'a)"
    1.37 +  let ?r = "Eps ?P"
    1.38 +  have "?P ?r" using ex_subseq[of "seqseq n o r'" "Suc n"] Suc'
    1.39 +    by (intro someI_ex[of ?P]) (auto intro: subseq_o simp: o_assoc)
    1.40 +  moreover have "seqseq (Suc (Suc n)) = seqseq n \<circ> r' \<circ> ?r"
    1.41 +    by (subst seqseq.simps) (simp only: Suc' o_assoc)
    1.42 +  moreover note subseq_o[OF `subseq (seqseq n)` `subseq r'`]
    1.43 +  ultimately show ?case unfolding Suc' by (auto simp: o_def)
    1.44 +qed
    1.45 +
    1.46 +lemma subseq_seqseq:
    1.47 +  shows "subseq (seqseq n)" using seqseq_ex[OF assms] by auto
    1.48 +
    1.49 +definition reducer where "reducer n = (SOME r'. subseq r' \<and> P n (seqseq n o r'))"
    1.50 +
    1.51 +lemma subseq_reducer: "subseq (reducer n)" and reducer_reduces: "P n (seqseq n o reducer n)"
    1.52 +  unfolding atomize_conj unfolding reducer_def using subseq_seqseq
    1.53 +  by (rule someI_ex[OF ex_subseq])
    1.54 +
    1.55 +lemma seqseq_reducer[simp]:
    1.56 +  "seqseq (Suc n) = seqseq n o reducer n"
    1.57 +  by (simp add: reducer_def)
    1.58 +
    1.59 +declare seqseq.simps(2)[simp del]
    1.60 +
    1.61 +definition diagseq where "diagseq i = seqseq i i"
    1.62 +
    1.63 +lemma diagseq_mono: "diagseq n < diagseq (Suc n)"
    1.64 +  unfolding diagseq_def seqseq_reducer o_def
    1.65 +  by (metis subseq_mono[OF subseq_seqseq] less_le_trans lessI seq_suble subseq_reducer)
    1.66 +
    1.67 +lemma subseq_diagseq: "subseq diagseq"
    1.68 +  using diagseq_mono by (simp add: subseq_Suc_iff diagseq_def)
    1.69 +
    1.70 +primrec fold_reduce where
    1.71 +  "fold_reduce n 0 = id"
    1.72 +| "fold_reduce n (Suc k) = fold_reduce n k o reducer (n + k)"
    1.73 +
    1.74 +lemma subseq_fold_reduce: "subseq (fold_reduce n k)"
    1.75 +proof (induct k)
    1.76 +  case (Suc k) from subseq_o[OF this subseq_reducer] show ?case by (simp add: o_def)
    1.77 +qed (simp add: subseq_def)
    1.78 +
    1.79 +lemma ex_subseq_reduce_index: "seqseq (n + k) = seqseq n o fold_reduce n k"
    1.80 +  by (induct k) simp_all
    1.81 +
    1.82 +lemma seqseq_fold_reduce: "seqseq n = fold_reduce 0 n"
    1.83 +  by (induct n) (simp_all)
    1.84 +
    1.85 +lemma diagseq_fold_reduce: "diagseq n = fold_reduce 0 n n"
    1.86 +  using seqseq_fold_reduce by (simp add: diagseq_def)
    1.87 +
    1.88 +lemma fold_reduce_add: "fold_reduce 0 (m + n) = fold_reduce 0 m o fold_reduce m n"
    1.89 +  by (induct n) simp_all
    1.90 +
    1.91 +lemma diagseq_add: "diagseq (k + n) = (seqseq k o (fold_reduce k n)) (k + n)"
    1.92 +proof -
    1.93 +  have "diagseq (k + n) = fold_reduce 0 (k + n) (k + n)"
    1.94 +    by (simp add: diagseq_fold_reduce)
    1.95 +  also have "\<dots> = (seqseq k o fold_reduce k n) (k + n)"
    1.96 +    unfolding fold_reduce_add seqseq_fold_reduce ..
    1.97 +  finally show ?thesis .
    1.98 +qed
    1.99 +
   1.100 +lemma diagseq_sub:
   1.101 +  assumes "m \<le> n" shows "diagseq n = (seqseq m o (fold_reduce m (n - m))) n"
   1.102 +  using diagseq_add[of m "n - m"] assms by simp
   1.103 +
   1.104 +lemma subseq_diagonal_rest: "subseq (\<lambda>x. fold_reduce k x (k + x))"
   1.105 +  unfolding subseq_Suc_iff fold_reduce.simps o_def
   1.106 +  by (metis subseq_mono[OF subseq_fold_reduce] less_le_trans lessI add_Suc_right seq_suble
   1.107 +      subseq_reducer)
   1.108 +
   1.109 +lemma diagseq_seqseq: "diagseq o (op + k) = (seqseq k o (\<lambda>x. fold_reduce k x (k + x)))"
   1.110 +  by (auto simp: o_def diagseq_add)
   1.111 +
   1.112 +end
   1.113 +
   1.114 +end