src/HOL/Library/Diagonal_Subsequence.thy
 author haftmann Fri Mar 22 19:18:08 2019 +0000 (3 months ago) changeset 69946 494934c30f38 parent 67399 eab6ce8368fa permissions -rw-r--r--
improved code equations taken over from AFP
```     1 (* Author: Fabian Immler, TUM *)
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```     2
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```     3 section \<open>Sequence of Properties on Subsequences\<close>
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```     4
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```     5 theory Diagonal_Subsequence
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```     6 imports Complex_Main
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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. strict_mono (s::nat\<Rightarrow>nat) \<Longrightarrow> \<exists>r'. strict_mono r' \<and> P n (s \<circ> r')"
```
```    12 begin
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```    13
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```    14 definition reduce where "reduce s n = (SOME r'::nat\<Rightarrow>nat. strict_mono r' \<and> P n (s \<circ> r'))"
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```    15
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```    16 lemma subseq_reduce[intro, simp]:
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```    17   "strict_mono s \<Longrightarrow> strict_mono (reduce s n)"
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```    18   unfolding reduce_def by (rule someI2_ex[OF ex_subseq]) auto
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```    19
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```    20 lemma reduce_holds:
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```    21   "strict_mono s \<Longrightarrow> P n (s \<circ> reduce s n)"
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```    22   unfolding reduce_def by (rule someI2_ex[OF ex_subseq]) (auto simp: o_def)
```
```    23
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```    24 primrec seqseq :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
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```    25   "seqseq 0 = id"
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```    26 | "seqseq (Suc n) = seqseq n \<circ> reduce (seqseq n) n"
```
```    27
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```    28 lemma subseq_seqseq[intro, simp]: "strict_mono (seqseq n)"
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```    29 proof (induct n)
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```    30   case 0 thus ?case by (simp add: strict_mono_def)
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```    31 next
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```    32   case (Suc n) thus ?case by (subst seqseq.simps) (auto intro!: strict_mono_o)
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```    33 qed
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```    34
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```    35 lemma seqseq_holds:
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```    36   "P n (seqseq (Suc n))"
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```    37 proof -
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```    38   have "P n (seqseq n \<circ> reduce (seqseq n) n)"
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```    39     by (intro reduce_holds subseq_seqseq)
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```    40   thus ?thesis by simp
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```    41 qed
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```    42
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```    43 definition diagseq :: "nat \<Rightarrow> nat" where "diagseq i = seqseq i i"
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```    44
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```    45 lemma diagseq_mono: "diagseq n < diagseq (Suc n)"
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```    46 proof -
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```    47   have "diagseq n < seqseq n (Suc n)"
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```    48     using subseq_seqseq[of n] by (simp add: diagseq_def strict_mono_def)
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```    49   also have "\<dots> \<le> seqseq n (reduce (seqseq n) n (Suc n))"
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```    50     using strict_mono_less_eq seq_suble by blast
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```    51   also have "\<dots> = diagseq (Suc n)" by (simp add: diagseq_def)
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```    52   finally show ?thesis .
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```    53 qed
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```    54
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```    55 lemma subseq_diagseq: "strict_mono diagseq"
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```    56   using diagseq_mono by (simp add: strict_mono_Suc_iff diagseq_def)
```
```    57
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```    58 primrec fold_reduce where
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```    59   "fold_reduce n 0 = id"
```
```    60 | "fold_reduce n (Suc k) = fold_reduce n k \<circ> reduce (seqseq (n + k)) (n + k)"
```
```    61
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```    62 lemma subseq_fold_reduce[intro, simp]: "strict_mono (fold_reduce n k)"
```
```    63 proof (induct k)
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```    64   case (Suc k) from strict_mono_o[OF this subseq_reduce] show ?case by (simp add: o_def)
```
```    65 qed (simp add: strict_mono_def)
```
```    66
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```    67 lemma ex_subseq_reduce_index: "seqseq (n + k) = seqseq n \<circ> fold_reduce n k"
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```    68   by (induct k) simp_all
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```    69
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```    70 lemma seqseq_fold_reduce: "seqseq n = fold_reduce 0 n"
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```    71   by (induct n) (simp_all)
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```    72
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```    73 lemma diagseq_fold_reduce: "diagseq n = fold_reduce 0 n n"
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```    74   using seqseq_fold_reduce by (simp add: diagseq_def)
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```    75
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```    76 lemma fold_reduce_add: "fold_reduce 0 (m + n) = fold_reduce 0 m \<circ> fold_reduce m n"
```
```    77   by (induct n) simp_all
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```    78
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```    79 lemma diagseq_add: "diagseq (k + n) = (seqseq k \<circ> (fold_reduce k n)) (k + n)"
```
```    80 proof -
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```    81   have "diagseq (k + n) = fold_reduce 0 (k + n) (k + n)"
```
```    82     by (simp add: diagseq_fold_reduce)
```
```    83   also have "\<dots> = (seqseq k \<circ> fold_reduce k n) (k + n)"
```
```    84     unfolding fold_reduce_add seqseq_fold_reduce ..
```
```    85   finally show ?thesis .
```
```    86 qed
```
```    87
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```    88 lemma diagseq_sub:
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```    89   assumes "m \<le> n" shows "diagseq n = (seqseq m \<circ> (fold_reduce m (n - m))) n"
```
```    90   using diagseq_add[of m "n - m"] assms by simp
```
```    91
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```    92 lemma subseq_diagonal_rest: "strict_mono (\<lambda>x. fold_reduce k x (k + x))"
```
```    93   unfolding strict_mono_Suc_iff fold_reduce.simps o_def
```
```    94 proof
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```    95   fix n
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```    96   have "fold_reduce k n (k + n) < fold_reduce k n (k + Suc n)" (is "?lhs < _")
```
```    97     by (auto intro: strict_monoD)
```
```    98   also have "\<dots> \<le> fold_reduce k n (reduce (seqseq (k + n)) (k + n) (k + Suc n))"
```
```    99     by (auto intro: less_mono_imp_le_mono seq_suble strict_monoD)
```
```   100   finally show "?lhs < \<dots>" .
```
```   101 qed
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```   102
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```   103 lemma diagseq_seqseq: "diagseq \<circ> ((+) k) = (seqseq k \<circ> (\<lambda>x. fold_reduce k x (k + x)))"
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```   104   by (auto simp: o_def diagseq_add)
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```   105
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```   106 lemma diagseq_holds:
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```   107   assumes subseq_stable: "\<And>r s n. strict_mono r \<Longrightarrow> P n s \<Longrightarrow> P n (s \<circ> r)"
```
```   108   shows "P k (diagseq \<circ> ((+) (Suc k)))"
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```   109   unfolding diagseq_seqseq by (intro subseq_stable subseq_diagonal_rest seqseq_holds)
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```   110
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```   111 end
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```   112
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```   113 end
```