src/HOL/Library/Diagonal_Subsequence.thy
author Andreas Lochbihler
Wed Feb 27 10:33:30 2013 +0100 (2013-02-27)
changeset 51288 be7e9a675ec9
parent 50087 635d73673b5e
child 51526 155263089e7b
permissions -rw-r--r--
add wellorder instance for Numeral_Type (suggested by Jesus Aransay)
     1 (* Author: Fabian Immler, TUM *)
     2 
     3 header {* Sequence of Properties on Subsequences *}
     4 
     5 theory Diagonal_Subsequence
     6 imports SEQ
     7 begin
     8 
     9 locale subseqs =
    10   fixes P::"nat\<Rightarrow>(nat\<Rightarrow>nat)\<Rightarrow>bool"
    11   assumes ex_subseq: "\<And>n s. subseq s \<Longrightarrow> \<exists>r'. subseq r' \<and> P n (s o r')"
    12 begin
    13 
    14 primrec seqseq where
    15   "seqseq 0 = id"
    16 | "seqseq (Suc n) = seqseq n o (SOME r'. subseq r' \<and> P n (seqseq n o r'))"
    17 
    18 lemma seqseq_ex:
    19   shows "subseq (seqseq n) \<and>
    20   (\<exists>r'. seqseq (Suc n) = seqseq n o r' \<and> subseq r' \<and> P n (seqseq n o r'))"
    21 proof (induct n)
    22   case 0
    23   let ?P = "\<lambda>r'. subseq r' \<and> P 0 r'"
    24   let ?r = "Eps ?P"
    25   have "?P ?r" using ex_subseq[of id 0] by (intro someI_ex[of ?P]) (auto simp: subseq_def)
    26   thus ?case by (auto simp: subseq_def)
    27 next
    28   case (Suc n)
    29   then obtain r' where
    30     Suc': "seqseq (Suc n) = seqseq n \<circ> r'" "subseq (seqseq n)" "subseq r'"
    31       "P n (seqseq n o r')"
    32     by blast
    33   let ?P = "\<lambda>r'a. subseq (r'a ) \<and> P (Suc n) (seqseq n o r' o r'a)"
    34   let ?r = "Eps ?P"
    35   have "?P ?r" using ex_subseq[of "seqseq n o r'" "Suc n"] Suc'
    36     by (intro someI_ex[of ?P]) (auto intro: subseq_o simp: o_assoc)
    37   moreover have "seqseq (Suc (Suc n)) = seqseq n \<circ> r' \<circ> ?r"
    38     by (subst seqseq.simps) (simp only: Suc' o_assoc)
    39   moreover note subseq_o[OF `subseq (seqseq n)` `subseq r'`]
    40   ultimately show ?case unfolding Suc' by (auto simp: o_def)
    41 qed
    42 
    43 lemma subseq_seqseq:
    44   shows "subseq (seqseq n)" using seqseq_ex[OF assms] by auto
    45 
    46 definition reducer where "reducer n = (SOME r'. subseq r' \<and> P n (seqseq n o r'))"
    47 
    48 lemma subseq_reducer: "subseq (reducer n)" and reducer_reduces: "P n (seqseq n o reducer n)"
    49   unfolding atomize_conj unfolding reducer_def using subseq_seqseq
    50   by (rule someI_ex[OF ex_subseq])
    51 
    52 lemma seqseq_reducer[simp]:
    53   "seqseq (Suc n) = seqseq n o reducer n"
    54   by (simp add: reducer_def)
    55 
    56 declare seqseq.simps(2)[simp del]
    57 
    58 definition diagseq where "diagseq i = seqseq i i"
    59 
    60 lemma diagseq_mono: "diagseq n < diagseq (Suc n)"
    61   unfolding diagseq_def seqseq_reducer o_def
    62   by (metis subseq_mono[OF subseq_seqseq] less_le_trans lessI seq_suble subseq_reducer)
    63 
    64 lemma subseq_diagseq: "subseq diagseq"
    65   using diagseq_mono by (simp add: subseq_Suc_iff diagseq_def)
    66 
    67 primrec fold_reduce where
    68   "fold_reduce n 0 = id"
    69 | "fold_reduce n (Suc k) = fold_reduce n k o reducer (n + k)"
    70 
    71 lemma subseq_fold_reduce: "subseq (fold_reduce n k)"
    72 proof (induct k)
    73   case (Suc k) from subseq_o[OF this subseq_reducer] show ?case by (simp add: o_def)
    74 qed (simp add: subseq_def)
    75 
    76 lemma ex_subseq_reduce_index: "seqseq (n + k) = seqseq n o fold_reduce n k"
    77   by (induct k) simp_all
    78 
    79 lemma seqseq_fold_reduce: "seqseq n = fold_reduce 0 n"
    80   by (induct n) (simp_all)
    81 
    82 lemma diagseq_fold_reduce: "diagseq n = fold_reduce 0 n n"
    83   using seqseq_fold_reduce by (simp add: diagseq_def)
    84 
    85 lemma fold_reduce_add: "fold_reduce 0 (m + n) = fold_reduce 0 m o fold_reduce m n"
    86   by (induct n) simp_all
    87 
    88 lemma diagseq_add: "diagseq (k + n) = (seqseq k o (fold_reduce k n)) (k + n)"
    89 proof -
    90   have "diagseq (k + n) = fold_reduce 0 (k + n) (k + n)"
    91     by (simp add: diagseq_fold_reduce)
    92   also have "\<dots> = (seqseq k o fold_reduce k n) (k + n)"
    93     unfolding fold_reduce_add seqseq_fold_reduce ..
    94   finally show ?thesis .
    95 qed
    96 
    97 lemma diagseq_sub:
    98   assumes "m \<le> n" shows "diagseq n = (seqseq m o (fold_reduce m (n - m))) n"
    99   using diagseq_add[of m "n - m"] assms by simp
   100 
   101 lemma subseq_diagonal_rest: "subseq (\<lambda>x. fold_reduce k x (k + x))"
   102   unfolding subseq_Suc_iff fold_reduce.simps o_def
   103   by (metis subseq_mono[OF subseq_fold_reduce] less_le_trans lessI add_Suc_right seq_suble
   104       subseq_reducer)
   105 
   106 lemma diagseq_seqseq: "diagseq o (op + k) = (seqseq k o (\<lambda>x. fold_reduce k x (k + x)))"
   107   by (auto simp: o_def diagseq_add)
   108 
   109 end
   110 
   111 end