# HG changeset patch
# User paulson <lp15@cam.ac.uk>
# Date 1447868220 0
# Node ID 286c1741285c190116311a20e820622f8bc141ba
# Parent  a81dc5c4d6a917067cb020b7b3bd4b42d972246d# Parent  c4a6edbfec49395b44f5be069fbadf9fc41fcd7b
Merge

diff -r c4a6edbfec49 -r 286c1741285c src/HOL/Fun.thy
--- a/src/HOL/Fun.thy	Wed Nov 18 14:28:45 2015 +0100
+++ b/src/HOL/Fun.thy	Wed Nov 18 17:37:00 2015 +0000
@@ -120,6 +120,9 @@
 lemma fcomp_id [simp]: "f \<circ>> id = f"
   by (simp add: fcomp_def)
 
+lemma fcomp_comp: "fcomp f g = comp g f" 
+  by (simp add: ext)
+
 code_printing
   constant fcomp \<rightharpoonup> (Eval) infixl 1 "#>"
 
diff -r c4a6edbfec49 -r 286c1741285c src/HOL/Library/Permutation.thy
--- a/src/HOL/Library/Permutation.thy	Wed Nov 18 14:28:45 2015 +0100
+++ b/src/HOL/Library/Permutation.thy	Wed Nov 18 17:37:00 2015 +0000
@@ -15,25 +15,25 @@
 | Cons [intro!]: "xs <~~> ys \<Longrightarrow> z # xs <~~> z # ys"
 | trans [intro]: "xs <~~> ys \<Longrightarrow> ys <~~> zs \<Longrightarrow> xs <~~> zs"
 
-lemma perm_refl [iff]: "l <~~> l"
+proposition perm_refl [iff]: "l <~~> l"
   by (induct l) auto
 
 
 subsection \<open>Some examples of rule induction on permutations\<close>
 
-lemma xperm_empty_imp: "[] <~~> ys \<Longrightarrow> ys = []"
+proposition xperm_empty_imp: "[] <~~> ys \<Longrightarrow> ys = []"
   by (induct xs == "[] :: 'a list" ys pred: perm) simp_all
 
 
 text \<open>\medskip This more general theorem is easier to understand!\<close>
 
-lemma perm_length: "xs <~~> ys \<Longrightarrow> length xs = length ys"
+proposition perm_length: "xs <~~> ys \<Longrightarrow> length xs = length ys"
   by (induct pred: perm) simp_all
 
-lemma perm_empty_imp: "[] <~~> xs \<Longrightarrow> xs = []"
+proposition perm_empty_imp: "[] <~~> xs \<Longrightarrow> xs = []"
   by (drule perm_length) auto
 
-lemma perm_sym: "xs <~~> ys \<Longrightarrow> ys <~~> xs"
+proposition perm_sym: "xs <~~> ys \<Longrightarrow> ys <~~> xs"
   by (induct pred: perm) auto
 
 
@@ -41,78 +41,72 @@
 
 text \<open>We can insert the head anywhere in the list.\<close>
 
-lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
+proposition perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
   by (induct xs) auto
 
-lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
-  apply (induct xs)
-    apply simp_all
-  apply (blast intro: perm_append_Cons)
-  done
+proposition perm_append_swap: "xs @ ys <~~> ys @ xs"
+  by (induct xs) (auto intro: perm_append_Cons)
 
-lemma perm_append_single: "a # xs <~~> xs @ [a]"
+proposition perm_append_single: "a # xs <~~> xs @ [a]"
   by (rule perm.trans [OF _ perm_append_swap]) simp
 
-lemma perm_rev: "rev xs <~~> xs"
-  apply (induct xs)
-   apply simp_all
-  apply (blast intro!: perm_append_single intro: perm_sym)
-  done
+proposition perm_rev: "rev xs <~~> xs"
+  by (induct xs) (auto intro!: perm_append_single intro: perm_sym)
 
-lemma perm_append1: "xs <~~> ys \<Longrightarrow> l @ xs <~~> l @ ys"
+proposition perm_append1: "xs <~~> ys \<Longrightarrow> l @ xs <~~> l @ ys"
   by (induct l) auto
 
-lemma perm_append2: "xs <~~> ys \<Longrightarrow> xs @ l <~~> ys @ l"
+proposition perm_append2: "xs <~~> ys \<Longrightarrow> xs @ l <~~> ys @ l"
   by (blast intro!: perm_append_swap perm_append1)
 
 
 subsection \<open>Further results\<close>
 
-lemma perm_empty [iff]: "[] <~~> xs \<longleftrightarrow> xs = []"
+proposition perm_empty [iff]: "[] <~~> xs \<longleftrightarrow> xs = []"
   by (blast intro: perm_empty_imp)
 
-lemma perm_empty2 [iff]: "xs <~~> [] \<longleftrightarrow> xs = []"
+proposition perm_empty2 [iff]: "xs <~~> [] \<longleftrightarrow> xs = []"
   apply auto
   apply (erule perm_sym [THEN perm_empty_imp])
   done
 
-lemma perm_sing_imp: "ys <~~> xs \<Longrightarrow> xs = [y] \<Longrightarrow> ys = [y]"
+proposition perm_sing_imp: "ys <~~> xs \<Longrightarrow> xs = [y] \<Longrightarrow> ys = [y]"
   by (induct pred: perm) auto
 
-lemma perm_sing_eq [iff]: "ys <~~> [y] \<longleftrightarrow> ys = [y]"
+proposition perm_sing_eq [iff]: "ys <~~> [y] \<longleftrightarrow> ys = [y]"
   by (blast intro: perm_sing_imp)
 
-lemma perm_sing_eq2 [iff]: "[y] <~~> ys \<longleftrightarrow> ys = [y]"
+proposition perm_sing_eq2 [iff]: "[y] <~~> ys \<longleftrightarrow> ys = [y]"
   by (blast dest: perm_sym)
 
 
 subsection \<open>Removing elements\<close>
 
-lemma perm_remove: "x \<in> set ys \<Longrightarrow> ys <~~> x # remove1 x ys"
+proposition perm_remove: "x \<in> set ys \<Longrightarrow> ys <~~> x # remove1 x ys"
   by (induct ys) auto
 
 
 text \<open>\medskip Congruence rule\<close>
 
-lemma perm_remove_perm: "xs <~~> ys \<Longrightarrow> remove1 z xs <~~> remove1 z ys"
+proposition perm_remove_perm: "xs <~~> ys \<Longrightarrow> remove1 z xs <~~> remove1 z ys"
   by (induct pred: perm) auto
 
-lemma remove_hd [simp]: "remove1 z (z # xs) = xs"
+proposition remove_hd [simp]: "remove1 z (z # xs) = xs"
   by auto
 
-lemma cons_perm_imp_perm: "z # xs <~~> z # ys \<Longrightarrow> xs <~~> ys"
+proposition cons_perm_imp_perm: "z # xs <~~> z # ys \<Longrightarrow> xs <~~> ys"
   by (drule_tac z = z in perm_remove_perm) auto
 
-lemma cons_perm_eq [iff]: "z#xs <~~> z#ys \<longleftrightarrow> xs <~~> ys"
+proposition cons_perm_eq [iff]: "z#xs <~~> z#ys \<longleftrightarrow> xs <~~> ys"
   by (blast intro: cons_perm_imp_perm)
 
-lemma append_perm_imp_perm: "zs @ xs <~~> zs @ ys \<Longrightarrow> xs <~~> ys"
+proposition append_perm_imp_perm: "zs @ xs <~~> zs @ ys \<Longrightarrow> xs <~~> ys"
   by (induct zs arbitrary: xs ys rule: rev_induct) auto
 
-lemma perm_append1_eq [iff]: "zs @ xs <~~> zs @ ys \<longleftrightarrow> xs <~~> ys"
+proposition perm_append1_eq [iff]: "zs @ xs <~~> zs @ ys \<longleftrightarrow> xs <~~> ys"
   by (blast intro: append_perm_imp_perm perm_append1)
 
-lemma perm_append2_eq [iff]: "xs @ zs <~~> ys @ zs \<longleftrightarrow> xs <~~> ys"
+proposition perm_append2_eq [iff]: "xs @ zs <~~> ys @ zs \<longleftrightarrow> xs <~~> ys"
   apply (safe intro!: perm_append2)
   apply (rule append_perm_imp_perm)
   apply (rule perm_append_swap [THEN perm.trans])
@@ -120,7 +114,7 @@
   apply (blast intro: perm_append_swap)
   done
 
-lemma mset_eq_perm: "mset xs = mset ys \<longleftrightarrow> xs <~~> ys"
+theorem mset_eq_perm: "mset xs = mset ys \<longleftrightarrow> xs <~~> ys"
   apply (rule iffI)
   apply (erule_tac [2] perm.induct)
   apply (simp_all add: union_ac)
@@ -140,24 +134,24 @@
   apply simp
   done
 
-lemma mset_le_perm_append: "mset xs \<le># mset ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"
+proposition mset_le_perm_append: "mset xs \<le># mset ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"
   apply (auto simp: mset_eq_perm[THEN sym] mset_le_exists_conv)
   apply (insert surj_mset)
   apply (drule surjD)
   apply (blast intro: sym)+
   done
 
-lemma perm_set_eq: "xs <~~> ys \<Longrightarrow> set xs = set ys"
+proposition perm_set_eq: "xs <~~> ys \<Longrightarrow> set xs = set ys"
   by (metis mset_eq_perm mset_eq_setD)
 
-lemma perm_distinct_iff: "xs <~~> ys \<Longrightarrow> distinct xs = distinct ys"
+proposition perm_distinct_iff: "xs <~~> ys \<Longrightarrow> distinct xs = distinct ys"
   apply (induct pred: perm)
      apply simp_all
    apply fastforce
   apply (metis perm_set_eq)
   done
 
-lemma eq_set_perm_remdups: "set xs = set ys \<Longrightarrow> remdups xs <~~> remdups ys"
+theorem eq_set_perm_remdups: "set xs = set ys \<Longrightarrow> remdups xs <~~> remdups ys"
   apply (induct xs arbitrary: ys rule: length_induct)
   apply (case_tac "remdups xs")
    apply simp_all
@@ -182,10 +176,10 @@
    apply (rule length_remdups_leq)
   done
 
-lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y \<longleftrightarrow> set x = set y"
+proposition perm_remdups_iff_eq_set: "remdups x <~~> remdups y \<longleftrightarrow> set x = set y"
   by (metis List.set_remdups perm_set_eq eq_set_perm_remdups)
 
-lemma permutation_Ex_bij:
+theorem permutation_Ex_bij:
   assumes "xs <~~> ys"
   shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! (f i))"
   using assms
@@ -246,4 +240,21 @@
   qed
 qed
 
+proposition perm_finite: "finite {B. B <~~> A}"
+proof (rule finite_subset[where B="{xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"])
+ show "finite {xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"
+   apply (cases A, simp)
+   apply (rule card_ge_0_finite)
+   apply (auto simp: card_lists_length_le)
+   done
+next
+ show "{B. B <~~> A} \<subseteq> {xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"
+   by (clarsimp simp add: perm_length perm_set_eq)
+qed
+
+proposition perm_swap:
+    assumes "i < length xs" "j < length xs"
+    shows "xs[i := xs ! j, j := xs ! i] <~~> xs"
+  using assms by (simp add: mset_eq_perm[symmetric] mset_swap)
+
 end
diff -r c4a6edbfec49 -r 286c1741285c src/HOL/List.thy
--- a/src/HOL/List.thy	Wed Nov 18 14:28:45 2015 +0100
+++ b/src/HOL/List.thy	Wed Nov 18 17:37:00 2015 +0000
@@ -2081,6 +2081,12 @@
  apply (case_tac i, auto)
 done
 
+lemma drop_rev: "drop n (rev xs) = rev (take (length xs - n) xs)"
+  by (cases "length xs < n") (auto simp: rev_take)
+
+lemma take_rev: "take n (rev xs) = rev (drop (length xs - n) xs)"
+  by (cases "length xs < n") (auto simp: rev_drop)
+
 lemma nth_take [simp]: "i < n ==> (take n xs)!i = xs!i"
 apply (induct xs arbitrary: i n, auto)
  apply (case_tac n, blast)
diff -r c4a6edbfec49 -r 286c1741285c src/HOL/Multivariate_Analysis/Path_Connected.thy
--- a/src/HOL/Multivariate_Analysis/Path_Connected.thy	Wed Nov 18 14:28:45 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Path_Connected.thy	Wed Nov 18 17:37:00 2015 +0000
@@ -2695,6 +2695,9 @@
     shows "homotopic_with P X Y f g \<longleftrightarrow> homotopic_with P X Y g f"
   using homotopic_with_symD by blast
 
+lemma split_01: "{0..1::real} = {0..1/2} \<union> {1/2..1}"
+  by force
+
 lemma split_01_prod: "{0..1::real} \<times> X = ({0..1/2} \<times> X) \<union> ({1/2..1} \<times> X)"
   by force
 
diff -r c4a6edbfec49 -r 286c1741285c src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Wed Nov 18 14:28:45 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Wed Nov 18 17:37:00 2015 +0000
@@ -6882,7 +6882,6 @@
   fixes s :: "real set"
   assumes "continuous_on {t \<in> s. t \<le> a} f"
       and "continuous_on {t \<in> s. a \<le> t} g"
-      and "continuous_on s h"
       and "a \<in> s \<Longrightarrow> f a = g a"
     shows "continuous_on s (\<lambda>t. if t \<le> a then f(t) else g(t))"
 using assms
diff -r c4a6edbfec49 -r 286c1741285c src/HOL/Orderings.thy
--- a/src/HOL/Orderings.thy	Wed Nov 18 14:28:45 2015 +0100
+++ b/src/HOL/Orderings.thy	Wed Nov 18 17:37:00 2015 +0000
@@ -1379,8 +1379,14 @@
   fix x assume "P x" "\<forall>y. P y \<longrightarrow> x \<le> y" thus "Q x" by (rule assms(2))
 qed
 
+lemma LeastI2_wellorder_ex:
+  assumes "\<exists>x. P x"
+  and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> Q a"
+  shows "Q (Least P)"
+using assms by clarify (blast intro!: LeastI2_wellorder)
+
 lemma not_less_Least: "k < (LEAST x. P x) \<Longrightarrow> \<not> P k"
-apply (simp (no_asm_use) add: not_le [symmetric])
+apply (simp add: not_le [symmetric])
 apply (erule contrapos_nn)
 apply (erule Least_le)
 done