src/HOL/Library/Diagonal_Subsequence.thy
 author Andreas Lochbihler Fri Sep 20 10:09:16 2013 +0200 (2013-09-20) changeset 53745 788730ab7da4 parent 52681 8cc7f76b827a child 57862 8f074e6e22fc permissions -rw-r--r--
prefer Code.abort over code_abort
```     1 (* Author: Fabian Immler, TUM *)
```
```     2
```
```     3 header {* Sequence of Properties on Subsequences *}
```
```     4
```
```     5 theory Diagonal_Subsequence
```
```     6 imports Complex_Main
```
```     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 definition reduce where "reduce s n = (SOME r'. subseq r' \<and> P n (s o r'))"
```
```    15
```
```    16 lemma subseq_reduce[intro, simp]:
```
```    17   "subseq s \<Longrightarrow> subseq (reduce s n)"
```
```    18   unfolding reduce_def by (rule someI2_ex[OF ex_subseq]) auto
```
```    19
```
```    20 lemma reduce_holds:
```
```    21   "subseq s \<Longrightarrow> P n (s o reduce s n)"
```
```    22   unfolding reduce_def by (rule someI2_ex[OF ex_subseq]) (auto simp: o_def)
```
```    23
```
```    24 primrec seqseq where
```
```    25   "seqseq 0 = id"
```
```    26 | "seqseq (Suc n) = seqseq n o reduce (seqseq n) n"
```
```    27
```
```    28 lemma subseq_seqseq[intro, simp]: "subseq (seqseq n)"
```
```    29 proof (induct n)
```
```    30   case (Suc n) thus ?case by (subst seqseq.simps) (auto simp: subseq_reduce intro!: subseq_o)
```
```    31 qed (simp add: subseq_def)
```
```    32
```
```    33 lemma seqseq_holds:
```
```    34   "P n (seqseq (Suc n))"
```
```    35 proof -
```
```    36   have "P n (seqseq n o reduce (seqseq n) n)"
```
```    37     by (intro reduce_holds subseq_seqseq)
```
```    38   thus ?thesis by simp
```
```    39 qed
```
```    40
```
```    41 definition diagseq where "diagseq i = seqseq i i"
```
```    42
```
```    43 lemma subseq_mono: "subseq f \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
```
```    44   by (metis le_eq_less_or_eq subseq_mono)
```
```    45
```
```    46 lemma subseq_strict_mono: "subseq f \<Longrightarrow> a < b \<Longrightarrow> f a < f b"
```
```    47   by (simp add: subseq_def)
```
```    48
```
```    49 lemma diagseq_mono: "diagseq n < diagseq (Suc n)"
```
```    50 proof -
```
```    51   have "diagseq n < seqseq n (Suc n)"
```
```    52     using subseq_seqseq[of n] by (simp add: diagseq_def subseq_def)
```
```    53   also have "\<dots> \<le> seqseq n (reduce (seqseq n) n (Suc n))"
```
```    54     by (auto intro: subseq_mono seq_suble)
```
```    55   also have "\<dots> = diagseq (Suc n)" by (simp add: diagseq_def)
```
```    56   finally show ?thesis .
```
```    57 qed
```
```    58
```
```    59 lemma subseq_diagseq: "subseq diagseq"
```
```    60   using diagseq_mono by (simp add: subseq_Suc_iff diagseq_def)
```
```    61
```
```    62 primrec fold_reduce where
```
```    63   "fold_reduce n 0 = id"
```
```    64 | "fold_reduce n (Suc k) = fold_reduce n k o reduce (seqseq (n + k)) (n + k)"
```
```    65
```
```    66 lemma subseq_fold_reduce[intro, simp]: "subseq (fold_reduce n k)"
```
```    67 proof (induct k)
```
```    68   case (Suc k) from subseq_o[OF this subseq_reduce] show ?case by (simp add: o_def)
```
```    69 qed (simp add: subseq_def)
```
```    70
```
```    71 lemma ex_subseq_reduce_index: "seqseq (n + k) = seqseq n o fold_reduce n k"
```
```    72   by (induct k) simp_all
```
```    73
```
```    74 lemma seqseq_fold_reduce: "seqseq n = fold_reduce 0 n"
```
```    75   by (induct n) (simp_all)
```
```    76
```
```    77 lemma diagseq_fold_reduce: "diagseq n = fold_reduce 0 n n"
```
```    78   using seqseq_fold_reduce by (simp add: diagseq_def)
```
```    79
```
```    80 lemma fold_reduce_add: "fold_reduce 0 (m + n) = fold_reduce 0 m o fold_reduce m n"
```
```    81   by (induct n) simp_all
```
```    82
```
```    83 lemma diagseq_add: "diagseq (k + n) = (seqseq k o (fold_reduce k n)) (k + n)"
```
```    84 proof -
```
```    85   have "diagseq (k + n) = fold_reduce 0 (k + n) (k + n)"
```
```    86     by (simp add: diagseq_fold_reduce)
```
```    87   also have "\<dots> = (seqseq k o fold_reduce k n) (k + n)"
```
```    88     unfolding fold_reduce_add seqseq_fold_reduce ..
```
```    89   finally show ?thesis .
```
```    90 qed
```
```    91
```
```    92 lemma diagseq_sub:
```
```    93   assumes "m \<le> n" shows "diagseq n = (seqseq m o (fold_reduce m (n - m))) n"
```
```    94   using diagseq_add[of m "n - m"] assms by simp
```
```    95
```
```    96 lemma subseq_diagonal_rest: "subseq (\<lambda>x. fold_reduce k x (k + x))"
```
```    97   unfolding subseq_Suc_iff fold_reduce.simps o_def
```
```    98 proof
```
```    99   fix n
```
```   100   have "fold_reduce k n (k + n) < fold_reduce k n (k + Suc n)" (is "?lhs < _")
```
```   101     by (auto intro: subseq_strict_mono)
```
```   102   also have "\<dots> \<le> fold_reduce k n (reduce (seqseq (k + n)) (k + n) (k + Suc n))"
```
```   103     by (rule subseq_mono) (auto intro!: seq_suble subseq_mono)
```
```   104   finally show "?lhs < \<dots>" .
```
```   105 qed
```
```   106
```
```   107 lemma diagseq_seqseq: "diagseq o (op + k) = (seqseq k o (\<lambda>x. fold_reduce k x (k + x)))"
```
```   108   by (auto simp: o_def diagseq_add)
```
```   109
```
```   110 lemma diagseq_holds:
```
```   111   assumes subseq_stable: "\<And>r s n. subseq r \<Longrightarrow> P n s \<Longrightarrow> P n (s o r)"
```
```   112   shows "P k (diagseq o (op + (Suc k)))"
```
```   113   unfolding diagseq_seqseq by (intro subseq_stable subseq_diagonal_rest seqseq_holds)
```
```   114
```
```   115 end
```
```   116
```
```   117 end
```