--- a/src/HOL/Fun.thy Wed Nov 18 10:12:37 2015 +0100
+++ b/src/HOL/Fun.thy Wed Nov 18 15:23:34 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 "#>"
--- a/src/HOL/Library/Permutation.thy Wed Nov 18 10:12:37 2015 +0100
+++ b/src/HOL/Library/Permutation.thy Wed Nov 18 15:23:34 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
--- a/src/HOL/List.thy Wed Nov 18 10:12:37 2015 +0100
+++ b/src/HOL/List.thy Wed Nov 18 15:23:34 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)
--- a/src/HOL/Multivariate_Analysis/Path_Connected.thy Wed Nov 18 10:12:37 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Path_Connected.thy Wed Nov 18 15:23:34 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
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Wed Nov 18 10:12:37 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Wed Nov 18 15:23:34 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
--- a/src/HOL/Orderings.thy Wed Nov 18 10:12:37 2015 +0100
+++ b/src/HOL/Orderings.thy Wed Nov 18 15:23:34 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