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src/HOL/Library/Diagonal_Subsequence.thy

author | immler |

Thu Nov 15 10:49:58 2012 +0100 (2012-11-15) | |

changeset 50087 | 635d73673b5e |

child 51526 | 155263089e7b |

permissions | -rw-r--r-- |

regularity of measures, therefore:

characterization of closure with infimum distance;

characterize of compact sets as totally bounded;

added Diagonal_Subsequence to Library;

introduced (enumerable) topological basis;

rational boxes as basis of ordered euclidean space;

moved some lemmas upwards

characterization of closure with infimum distance;

characterize of compact sets as totally bounded;

added Diagonal_Subsequence to Library;

introduced (enumerable) topological basis;

rational boxes as basis of ordered euclidean space;

moved some lemmas upwards

1 (* Author: Fabian Immler, TUM *)

3 header {* Sequence of Properties on Subsequences *}

5 theory Diagonal_Subsequence

6 imports SEQ

7 begin

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

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'))"

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

43 lemma subseq_seqseq:

44 shows "subseq (seqseq n)" using seqseq_ex[OF assms] by auto

46 definition reducer where "reducer n = (SOME r'. subseq r' \<and> P n (seqseq n o r'))"

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])

52 lemma seqseq_reducer[simp]:

53 "seqseq (Suc n) = seqseq n o reducer n"

54 by (simp add: reducer_def)

56 declare seqseq.simps(2)[simp del]

58 definition diagseq where "diagseq i = seqseq i i"

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)

64 lemma subseq_diagseq: "subseq diagseq"

65 using diagseq_mono by (simp add: subseq_Suc_iff diagseq_def)

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)"

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)

76 lemma ex_subseq_reduce_index: "seqseq (n + k) = seqseq n o fold_reduce n k"

77 by (induct k) simp_all

79 lemma seqseq_fold_reduce: "seqseq n = fold_reduce 0 n"

80 by (induct n) (simp_all)

82 lemma diagseq_fold_reduce: "diagseq n = fold_reduce 0 n n"

83 using seqseq_fold_reduce by (simp add: diagseq_def)

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

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

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

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)

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)

109 end

111 end