author paulson Mon, 17 Feb 2020 11:07:27 +0000 changeset 71665 7b8a6840e85f parent 71663 fb08117a106b (current diff) parent 71664 9edb7fb69bc2 (diff) child 71667 35b2d407e558
merged
```--- a/src/HOL/Library/Ramsey.thy	Mon Feb 17 11:17:09 2020 +0100
+++ b/src/HOL/Library/Ramsey.thy	Mon Feb 17 11:07:27 2020 +0000
@@ -21,6 +21,12 @@
lemma nsets_2_eq: "nsets A 2 = (\<Union>x\<in>A. \<Union>y\<in>A - {x}. {{x, y}})"
by (auto simp: nsets_def card_2_iff)

+lemma nsets_doubleton_2_eq [simp]: "[{x, y}]\<^bsup>2\<^esup> = (if x=y then {} else {{x, y}})"
+  by (auto simp: nsets_2_eq)
+
+lemma doubleton_in_nsets_2 [simp]: "{x,y} \<in> [A]\<^bsup>2\<^esup> \<longleftrightarrow> x \<in> A \<and> y \<in> A \<and> x \<noteq> y"
+  by (auto simp: nsets_2_eq Set.doubleton_eq_iff)
+
lemma binomial_eq_nsets: "n choose k = card (nsets {0..<n} k)"
by (meson subset_eq_atLeast0_lessThan_finite)
@@ -41,7 +47,7 @@
using that card_mono leD by auto
qed

-lemma nsets_eq_empty: "n < r \<Longrightarrow> nsets {..<n} r = {}"
+lemma nsets_eq_empty: "\<lbrakk>finite A; card A < r\<rbrakk> \<Longrightarrow> nsets A r = {}"

lemma nsets_empty_iff: "nsets {} r = (if r=0 then {{}} else {})"
@@ -79,6 +85,29 @@
by (auto simp: bij_betw_def inj_on_nsets)
qed

+lemma nset_image_obtains:
+  assumes "X \<in> [f`A]\<^bsup>k\<^esup>" "inj_on f A"
+  obtains Y where "Y \<in> [A]\<^bsup>k\<^esup>" "X = f ` Y"
+  using assms
+  apply (clarsimp simp add: nsets_def subset_image_iff)
+  by (metis card_image finite_imageD inj_on_subset)
+
+lemma nsets_image_funcset:
+  assumes "g \<in> S \<rightarrow> T" and "inj_on g S"
+  shows "(\<lambda>X. g ` X) \<in> [S]\<^bsup>k\<^esup> \<rightarrow> [T]\<^bsup>k\<^esup>"
+    using assms
+    by (fastforce simp add: nsets_def card_image inj_on_subset subset_iff simp flip: image_subset_iff_funcset)
+
+lemma nsets_compose_image_funcset:
+  assumes f: "f \<in> [T]\<^bsup>k\<^esup> \<rightarrow> D" and "g \<in> S \<rightarrow> T" and "inj_on g S"
+  shows "f \<circ> (\<lambda>X. g ` X) \<in> [S]\<^bsup>k\<^esup> \<rightarrow> D"
+proof -
+  have "(\<lambda>X. g ` X) \<in> [S]\<^bsup>k\<^esup> \<rightarrow> [T]\<^bsup>k\<^esup>"
+    using assms by (simp add: nsets_image_funcset)
+  then show ?thesis
+    using f by fastforce
+qed
+
subsubsection \<open>Partition predicates\<close>

definition partn :: "'a set \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b set \<Rightarrow> bool"```
```--- a/src/HOL/Num.thy	Mon Feb 17 11:17:09 2020 +0100
+++ b/src/HOL/Num.thy	Mon Feb 17 11:07:27 2020 +0000
@@ -1107,8 +1107,11 @@
lemma less_2_cases: "n < 2 \<Longrightarrow> n = 0 \<or> n = Suc 0"