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