--- a/src/HOL/Probability/Infinite_Product_Measure.thy Wed Feb 12 08:35:57 2014 +0100
+++ b/src/HOL/Probability/Infinite_Product_Measure.thy Wed Feb 12 08:35:57 2014 +0100
@@ -190,13 +190,13 @@
let ?P =
"\<lambda>k wk w. w \<in> space (Pi\<^sub>M (J (Suc k)) M) \<and> restrict w (J k) = wk \<and>
(\<forall>n. ?a / 2 ^ (Suc k + 1) \<le> ?q (Suc k) n w)"
- def w \<equiv> "nat_rec w0 (\<lambda>k wk. Eps (?P k wk))"
+ def w \<equiv> "rec_nat w0 (\<lambda>k wk. Eps (?P k wk))"
{ fix k have w: "w k \<in> space (Pi\<^sub>M (J k) M) \<and>
(\<forall>n. ?a / 2 ^ (k + 1) \<le> ?q k n (w k)) \<and> (k \<noteq> 0 \<longrightarrow> restrict (w k) (J (k - 1)) = w (k - 1))"
proof (induct k)
case 0 with w0 show ?case
- unfolding w_def nat_rec_0 by auto
+ unfolding w_def rec_nat_0 by auto
next
case (Suc k)
then have wk: "w k \<in> space (Pi\<^sub>M (J k) M)" by auto
@@ -241,7 +241,7 @@
(auto split: split_merge intro!: extensional_merge_sub ext simp: space_PiM PiE_iff)
qed
then have "?P k (w k) (w (Suc k))"
- unfolding w_def nat_rec_Suc unfolding w_def[symmetric]
+ unfolding w_def rec_nat_Suc unfolding w_def[symmetric]
by (rule someI_ex)
then show ?case by auto
qed
@@ -480,10 +480,10 @@
lemma comb_seq_0: "comb_seq 0 \<omega> \<omega>' = \<omega>'"
by (auto simp add: comb_seq_def)
-lemma comb_seq_Suc: "comb_seq (Suc n) \<omega> \<omega>' = comb_seq n \<omega> (nat_case (\<omega> n) \<omega>')"
+lemma comb_seq_Suc: "comb_seq (Suc n) \<omega> \<omega>' = comb_seq n \<omega> (case_nat (\<omega> n) \<omega>')"
by (auto simp add: comb_seq_def not_less less_Suc_eq le_imp_diff_is_add intro!: ext split: nat.split)
-lemma comb_seq_Suc_0[simp]: "comb_seq (Suc 0) \<omega> = nat_case (\<omega> 0)"
+lemma comb_seq_Suc_0[simp]: "comb_seq (Suc 0) \<omega> = case_nat (\<omega> 0)"
by (intro ext) (simp add: comb_seq_Suc comb_seq_0)
lemma comb_seq_less: "i < n \<Longrightarrow> comb_seq n \<omega> \<omega>' i = \<omega> i"
@@ -492,11 +492,11 @@
lemma comb_seq_add: "comb_seq n \<omega> \<omega>' (i + n) = \<omega>' i"
by (auto split: nat.split split_comb_seq)
-lemma nat_case_comb_seq: "nat_case s' (comb_seq n \<omega> \<omega>') (i + n) = nat_case (nat_case s' \<omega> n) \<omega>' i"
+lemma case_nat_comb_seq: "case_nat s' (comb_seq n \<omega> \<omega>') (i + n) = case_nat (case_nat s' \<omega> n) \<omega>' i"
by (auto split: nat.split split_comb_seq)
-lemma nat_case_comb_seq':
- "nat_case s (comb_seq i \<omega> \<omega>') = comb_seq (Suc i) (nat_case s \<omega>) \<omega>'"
+lemma case_nat_comb_seq':
+ "case_nat s (comb_seq i \<omega> \<omega>') = comb_seq (Suc i) (case_nat s \<omega>) \<omega>'"
by (auto split: split_comb_seq nat.split)
locale sequence_space = product_prob_space "\<lambda>i. M" "UNIV :: nat set" for M
@@ -570,7 +570,7 @@
qed simp_all
lemma PiM_iter:
- "distr (M \<Otimes>\<^sub>M S) S (\<lambda>(s, \<omega>). nat_case s \<omega>) = S" (is "?D = _")
+ "distr (M \<Otimes>\<^sub>M S) S (\<lambda>(s, \<omega>). case_nat s \<omega>) = S" (is "?D = _")
proof (rule PiM_eq)
let ?I = "UNIV::nat set" and ?M = "\<lambda>n. M"
let "distr _ _ ?f" = "?D"