--- a/src/HOL/Library/Diagonal_Subsequence.thy Tue Jul 16 23:34:33 2013 +0200
+++ b/src/HOL/Library/Diagonal_Subsequence.thy Wed Jul 17 13:34:21 2013 +0200
@@ -11,66 +11,61 @@
assumes ex_subseq: "\<And>n s. subseq s \<Longrightarrow> \<exists>r'. subseq r' \<and> P n (s o r')"
begin
+definition reduce where "reduce s n = (SOME r'. subseq r' \<and> P n (s o r'))"
+
+lemma subseq_reduce[intro, simp]:
+ "subseq s \<Longrightarrow> subseq (reduce s n)"
+ unfolding reduce_def by (rule someI2_ex[OF ex_subseq]) auto
+
+lemma reduce_holds:
+ "subseq s \<Longrightarrow> P n (s o reduce s n)"
+ unfolding reduce_def by (rule someI2_ex[OF ex_subseq]) (auto simp: o_def)
+
primrec seqseq where
"seqseq 0 = id"
-| "seqseq (Suc n) = seqseq n o (SOME r'. subseq r' \<and> P n (seqseq n o r'))"
+| "seqseq (Suc n) = seqseq n o reduce (seqseq n) n"
-lemma seqseq_ex:
- shows "subseq (seqseq n) \<and>
- (\<exists>r'. seqseq (Suc n) = seqseq n o r' \<and> subseq r' \<and> P n (seqseq n o r'))"
+lemma subseq_seqseq[intro, simp]: "subseq (seqseq n)"
proof (induct n)
- case 0
- let ?P = "\<lambda>r'. subseq r' \<and> P 0 r'"
- let ?r = "Eps ?P"
- have "?P ?r" using ex_subseq[of id 0] by (intro someI_ex[of ?P]) (auto simp: subseq_def)
- thus ?case by (auto simp: subseq_def)
-next
- case (Suc n)
- then obtain r' where
- Suc': "seqseq (Suc n) = seqseq n \<circ> r'" "subseq (seqseq n)" "subseq r'"
- "P n (seqseq n o r')"
- by blast
- let ?P = "\<lambda>r'a. subseq (r'a ) \<and> P (Suc n) (seqseq n o r' o r'a)"
- let ?r = "Eps ?P"
- have "?P ?r" using ex_subseq[of "seqseq n o r'" "Suc n"] Suc'
- by (intro someI_ex[of ?P]) (auto intro: subseq_o simp: o_assoc)
- moreover have "seqseq (Suc (Suc n)) = seqseq n \<circ> r' \<circ> ?r"
- by (subst seqseq.simps) (simp only: Suc' o_assoc)
- moreover note subseq_o[OF `subseq (seqseq n)` `subseq r'`]
- ultimately show ?case unfolding Suc' by (auto simp: o_def)
+ case (Suc n) thus ?case by (subst seqseq.simps) (auto simp: subseq_reduce intro!: subseq_o)
+qed (simp add: subseq_def)
+
+lemma seqseq_holds:
+ "P n (seqseq (Suc n))"
+proof -
+ have "P n (seqseq n o reduce (seqseq n) n)"
+ by (intro reduce_holds subseq_seqseq)
+ thus ?thesis by simp
qed
-lemma subseq_seqseq:
- shows "subseq (seqseq n)" using seqseq_ex[OF assms] by auto
-
-definition reducer where "reducer n = (SOME r'. subseq r' \<and> P n (seqseq n o r'))"
-
-lemma subseq_reducer: "subseq (reducer n)" and reducer_reduces: "P n (seqseq n o reducer n)"
- unfolding atomize_conj unfolding reducer_def using subseq_seqseq
- by (rule someI_ex[OF ex_subseq])
-
-lemma seqseq_reducer[simp]:
- "seqseq (Suc n) = seqseq n o reducer n"
- by (simp add: reducer_def)
-
-declare seqseq.simps(2)[simp del]
-
definition diagseq where "diagseq i = seqseq i i"
+lemma subseq_mono: "subseq f \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
+ by (metis le_eq_less_or_eq subseq_mono)
+
+lemma subseq_strict_mono: "subseq f \<Longrightarrow> a < b \<Longrightarrow> f a < f b"
+ by (simp add: subseq_def)
+
lemma diagseq_mono: "diagseq n < diagseq (Suc n)"
- unfolding diagseq_def seqseq_reducer o_def
- by (metis subseq_mono[OF subseq_seqseq] less_le_trans lessI seq_suble subseq_reducer)
+proof -
+ have "diagseq n < seqseq n (Suc n)"
+ using subseq_seqseq[of n] by (simp add: diagseq_def subseq_def)
+ also have "\<dots> \<le> seqseq n (reduce (seqseq n) n (Suc n))"
+ by (auto intro: subseq_mono seq_suble)
+ also have "\<dots> = diagseq (Suc n)" by (simp add: diagseq_def)
+ finally show ?thesis .
+qed
lemma subseq_diagseq: "subseq diagseq"
using diagseq_mono by (simp add: subseq_Suc_iff diagseq_def)
primrec fold_reduce where
"fold_reduce n 0 = id"
-| "fold_reduce n (Suc k) = fold_reduce n k o reducer (n + k)"
+| "fold_reduce n (Suc k) = fold_reduce n k o reduce (seqseq (n + k)) (n + k)"
-lemma subseq_fold_reduce: "subseq (fold_reduce n k)"
+lemma subseq_fold_reduce[intro, simp]: "subseq (fold_reduce n k)"
proof (induct k)
- case (Suc k) from subseq_o[OF this subseq_reducer] show ?case by (simp add: o_def)
+ case (Suc k) from subseq_o[OF this subseq_reduce] show ?case by (simp add: o_def)
qed (simp add: subseq_def)
lemma ex_subseq_reduce_index: "seqseq (n + k) = seqseq n o fold_reduce n k"
@@ -100,12 +95,23 @@
lemma subseq_diagonal_rest: "subseq (\<lambda>x. fold_reduce k x (k + x))"
unfolding subseq_Suc_iff fold_reduce.simps o_def
- by (metis subseq_mono[OF subseq_fold_reduce] less_le_trans lessI add_Suc_right seq_suble
- subseq_reducer)
+proof
+ fix n
+ have "fold_reduce k n (k + n) < fold_reduce k n (k + Suc n)" (is "?lhs < _")
+ by (auto intro: subseq_strict_mono)
+ also have "\<dots> \<le> fold_reduce k n (reduce (seqseq (k + n)) (k + n) (k + Suc n))"
+ by (rule subseq_mono) (auto intro!: seq_suble subseq_mono)
+ finally show "?lhs < \<dots>" .
+qed
lemma diagseq_seqseq: "diagseq o (op + k) = (seqseq k o (\<lambda>x. fold_reduce k x (k + x)))"
by (auto simp: o_def diagseq_add)
+lemma diagseq_holds:
+ assumes subseq_stable: "\<And>r s n. subseq r \<Longrightarrow> P n s \<Longrightarrow> P n (s o r)"
+ shows "P k (diagseq o (op + (Suc k)))"
+ unfolding diagseq_seqseq by (intro subseq_stable subseq_diagonal_rest seqseq_holds)
+
end
end
--- a/src/HOL/Probability/Projective_Limit.thy Tue Jul 16 23:34:33 2013 +0200
+++ b/src/HOL/Probability/Projective_Limit.thy Wed Jul 17 13:34:21 2013 +0200
@@ -77,25 +77,20 @@
lemma (in finmap_seqs_into_compact) diagonal_tendsto: "\<exists>l. (\<lambda>i. (f (diagseq i))\<^isub>F n) ----> l"
proof -
- have "\<And>i n0. (f o seqseq i) i = f (diagseq i)" unfolding diagseq_def by simp
- from reducer_reduces obtain l where l: "(\<lambda>i. ((f \<circ> seqseq (Suc n)) i)\<^isub>F n) ----> l"
- unfolding seqseq_reducer
- by auto
- have "(\<lambda>i. (f (diagseq (i + Suc n)))\<^isub>F n) =
- (\<lambda>i. ((f o (diagseq o (op + (Suc n)))) i)\<^isub>F n)" by (simp add: add_commute)
- also have "\<dots> =
- (\<lambda>i. ((f o ((seqseq (Suc n) o (\<lambda>x. fold_reduce (Suc n) x (Suc n + x))))) i)\<^isub>F n)"
- unfolding diagseq_seqseq by simp
- also have "\<dots> = (\<lambda>i. ((f o ((seqseq (Suc n)))) i)\<^isub>F n) o (\<lambda>x. fold_reduce (Suc n) x (Suc n + x))"
- by (simp add: o_def)
- also have "\<dots> ----> l"
- proof (rule LIMSEQ_subseq_LIMSEQ[OF _ subseq_diagonal_rest], rule tendstoI)
- fix e::real assume "0 < e"
- from tendstoD[OF l `0 < e`]
- show "eventually (\<lambda>x. dist (((f \<circ> seqseq (Suc n)) x)\<^isub>F n) l < e)
- sequentially" .
+ obtain l where "(\<lambda>i. ((f o (diagseq o op + (Suc n))) i)\<^isub>F n) ----> l"
+ proof (atomize_elim, rule diagseq_holds)
+ fix r s n
+ assume "subseq r"
+ assume "\<exists>l. (\<lambda>i. ((f \<circ> s) i)\<^isub>F n) ----> l"
+ then obtain l where "((\<lambda>i. (f i)\<^isub>F n) o s) ----> l"
+ by (auto simp: o_def)
+ hence "((\<lambda>i. (f i)\<^isub>F n) o s o r) ----> l" using `subseq r`
+ by (rule LIMSEQ_subseq_LIMSEQ)
+ thus "\<exists>l. (\<lambda>i. ((f \<circ> (s \<circ> r)) i)\<^isub>F n) ----> l" by (auto simp add: o_def)
qed
- finally show ?thesis by (intro exI) (rule LIMSEQ_offset)
+ hence "(\<lambda>i. ((f (diagseq (i + Suc n))))\<^isub>F n) ----> l" by (simp add: ac_simps)
+ hence "(\<lambda>i. (f (diagseq i))\<^isub>F n) ----> l" by (rule LIMSEQ_offset)
+ thus ?thesis ..
qed
subsection {* Daniell-Kolmogorov Theorem *}