src/HOL/Library/Diagonal_Subsequence.thy
 author wenzelm Wed Jun 17 11:03:05 2015 +0200 (2015-06-17) changeset 60500 903bb1495239 parent 58881 b9556a055632 child 66447 a1f5c5c26fa6 permissions -rw-r--r--
isabelle update_cartouches;
1 (* Author: Fabian Immler, TUM *)
3 section \<open>Sequence of Properties on Subsequences\<close>
5 theory Diagonal_Subsequence
6 imports Complex_Main
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 definition reduce where "reduce s n = (SOME r'. subseq r' \<and> P n (s o r'))"
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
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)
24 primrec seqseq where
25   "seqseq 0 = id"
26 | "seqseq (Suc n) = seqseq n o reduce (seqseq n) n"
28 lemma subseq_seqseq[intro, simp]: "subseq (seqseq n)"
29 proof (induct n)
30   case 0 thus ?case by (simp add: subseq_def)
31 next
32   case (Suc n) thus ?case by (subst seqseq.simps) (auto intro!: subseq_o)
33 qed
35 lemma seqseq_holds:
36   "P n (seqseq (Suc n))"
37 proof -
38   have "P n (seqseq n o reduce (seqseq n) n)"
39     by (intro reduce_holds subseq_seqseq)
40   thus ?thesis by simp
41 qed
43 definition diagseq where "diagseq i = seqseq i i"
45 lemma subseq_mono: "subseq f \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
46   by (metis le_eq_less_or_eq subseq_mono)
48 lemma subseq_strict_mono: "subseq f \<Longrightarrow> a < b \<Longrightarrow> f a < f b"
49   by (simp add: subseq_def)
51 lemma diagseq_mono: "diagseq n < diagseq (Suc n)"
52 proof -
53   have "diagseq n < seqseq n (Suc n)"
54     using subseq_seqseq[of n] by (simp add: diagseq_def subseq_def)
55   also have "\<dots> \<le> seqseq n (reduce (seqseq n) n (Suc n))"
56     by (auto intro: subseq_mono seq_suble)
57   also have "\<dots> = diagseq (Suc n)" by (simp add: diagseq_def)
58   finally show ?thesis .
59 qed
61 lemma subseq_diagseq: "subseq diagseq"
62   using diagseq_mono by (simp add: subseq_Suc_iff diagseq_def)
64 primrec fold_reduce where
65   "fold_reduce n 0 = id"
66 | "fold_reduce n (Suc k) = fold_reduce n k o reduce (seqseq (n + k)) (n + k)"
68 lemma subseq_fold_reduce[intro, simp]: "subseq (fold_reduce n k)"
69 proof (induct k)
70   case (Suc k) from subseq_o[OF this subseq_reduce] show ?case by (simp add: o_def)
71 qed (simp add: subseq_def)
73 lemma ex_subseq_reduce_index: "seqseq (n + k) = seqseq n o fold_reduce n k"
74   by (induct k) simp_all
76 lemma seqseq_fold_reduce: "seqseq n = fold_reduce 0 n"
77   by (induct n) (simp_all)
79 lemma diagseq_fold_reduce: "diagseq n = fold_reduce 0 n n"
80   using seqseq_fold_reduce by (simp add: diagseq_def)
82 lemma fold_reduce_add: "fold_reduce 0 (m + n) = fold_reduce 0 m o fold_reduce m n"
83   by (induct n) simp_all
85 lemma diagseq_add: "diagseq (k + n) = (seqseq k o (fold_reduce k n)) (k + n)"
86 proof -
87   have "diagseq (k + n) = fold_reduce 0 (k + n) (k + n)"
88     by (simp add: diagseq_fold_reduce)
89   also have "\<dots> = (seqseq k o fold_reduce k n) (k + n)"
90     unfolding fold_reduce_add seqseq_fold_reduce ..
91   finally show ?thesis .
92 qed
94 lemma diagseq_sub:
95   assumes "m \<le> n" shows "diagseq n = (seqseq m o (fold_reduce m (n - m))) n"
96   using diagseq_add[of m "n - m"] assms by simp
98 lemma subseq_diagonal_rest: "subseq (\<lambda>x. fold_reduce k x (k + x))"
99   unfolding subseq_Suc_iff fold_reduce.simps o_def
100 proof
101   fix n
102   have "fold_reduce k n (k + n) < fold_reduce k n (k + Suc n)" (is "?lhs < _")
103     by (auto intro: subseq_strict_mono)
104   also have "\<dots> \<le> fold_reduce k n (reduce (seqseq (k + n)) (k + n) (k + Suc n))"
105     by (rule subseq_mono) (auto intro!: seq_suble subseq_mono)
106   finally show "?lhs < \<dots>" .
107 qed
109 lemma diagseq_seqseq: "diagseq o (op + k) = (seqseq k o (\<lambda>x. fold_reduce k x (k + x)))"
110   by (auto simp: o_def diagseq_add)
112 lemma diagseq_holds:
113   assumes subseq_stable: "\<And>r s n. subseq r \<Longrightarrow> P n s \<Longrightarrow> P n (s o r)"
114   shows "P k (diagseq o (op + (Suc k)))"
115   unfolding diagseq_seqseq by (intro subseq_stable subseq_diagonal_rest seqseq_holds)
117 end
119 end