# HG changeset patch
# User immler
# Date 1374060861 -7200
# Node ID 8cc7f76b827af0eceda2b2d9bd2a4469a2d99e54
# Parent  c16f35e5a5aa3861a5f94dc1230d8859363de879
tuned definition of seqseq; clarified usage of diagseq via diagseq_holds

diff -r c16f35e5a5aa -r 8cc7f76b827a src/HOL/Library/Diagonal_Subsequence.thy
--- 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
diff -r c16f35e5a5aa -r 8cc7f76b827a src/HOL/Probability/Projective_Limit.thy
--- 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 *}