--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Set_Comprehension_Pointfree_Examples.thy Fri Feb 21 21:08:03 2014 +0100
@@ -0,0 +1,140 @@
+(* Title: HOL/ex/Set_Comprehension_Pointfree_Examples.thy
+ Author: Lukas Bulwahn, Rafal Kolanski
+ Copyright 2012 TU Muenchen
+*)
+
+header {* Examples for the set comprehension to pointfree simproc *}
+
+theory Set_Comprehension_Pointfree_Examples
+imports Main
+begin
+
+declare [[simproc add: finite_Collect]]
+
+lemma
+ "finite (UNIV::'a set) ==> finite {p. EX x::'a. p = (x, x)}"
+ by simp
+
+lemma
+ "finite A ==> finite B ==> finite {f a b| a b. a : A \<and> b : B}"
+ by simp
+
+lemma
+ "finite B ==> finite A' ==> finite {f a b| a b. a : A \<and> a : A' \<and> b : B}"
+ by simp
+
+lemma
+ "finite A ==> finite B ==> finite {f a b| a b. a : A \<and> b : B \<and> b : B'}"
+ by simp
+
+lemma
+ "finite A ==> finite B ==> finite C ==> finite {f a b c| a b c. a : A \<and> b : B \<and> c : C}"
+ by simp
+
+lemma
+ "finite A ==> finite B ==> finite C ==> finite D ==>
+ finite {f a b c d| a b c d. a : A \<and> b : B \<and> c : C \<and> d : D}"
+ by simp
+
+lemma
+ "finite A ==> finite B ==> finite C ==> finite D ==> finite E ==>
+ finite {f a b c d e | a b c d e. a : A \<and> b : B \<and> c : C \<and> d : D \<and> e : E}"
+ by simp
+
+lemma
+ "finite A ==> finite B ==> finite C ==> finite D ==> finite E \<Longrightarrow>
+ finite {f a d c b e | e b c d a. b : B \<and> a : A \<and> e : E' \<and> c : C \<and> d : D \<and> e : E \<and> b : B'}"
+ by simp
+
+lemma
+ "\<lbrakk> finite A ; finite B ; finite C ; finite D \<rbrakk>
+ \<Longrightarrow> finite ({f a b c d| a b c d. a : A \<and> b : B \<and> c : C \<and> d : D})"
+ by simp
+
+lemma
+ "finite ((\<lambda>(a,b,c,d). f a b c d) ` (A \<times> B \<times> C \<times> D))
+ \<Longrightarrow> finite ({f a b c d| a b c d. a : A \<and> b : B \<and> c : C \<and> d : D})"
+ by simp
+
+lemma
+ "finite S ==> finite {s'. EX s:S. s' = f a e s}"
+ by simp
+
+lemma
+ "finite A ==> finite B ==> finite {f a b| a b. a : A \<and> b : B \<and> a \<notin> Z}"
+ by simp
+
+lemma
+ "finite A ==> finite B ==> finite R ==> finite {f a b x y| a b x y. a : A \<and> b : B \<and> (x,y) \<in> R}"
+by simp
+
+lemma
+ "finite A ==> finite B ==> finite R ==> finite {f a b x y| a b x y. a : A \<and> (x,y) \<in> R \<and> b : B}"
+by simp
+
+lemma
+ "finite A ==> finite B ==> finite R ==> finite {f a (x, b) y| y b x a. a : A \<and> (x,y) \<in> R \<and> b : B}"
+by simp
+
+lemma
+ "finite A ==> finite AA ==> finite B ==> finite {f a b| a b. (a : A \<or> a : AA) \<and> b : B \<and> a \<notin> Z}"
+by simp
+
+lemma
+ "finite A1 ==> finite A2 ==> finite A3 ==> finite A4 ==> finite A5 ==> finite B ==>
+ finite {f a b c | a b c. ((a : A1 \<and> a : A2) \<or> (a : A3 \<and> (a : A4 \<or> a : A5))) \<and> b : B \<and> a \<notin> Z}"
+apply simp
+oops
+
+lemma "finite B ==> finite {c. EX x. x : B & c = a * x}"
+by simp
+
+lemma
+ "finite A ==> finite B ==> finite {f a * g b |a b. a : A & b : B}"
+by simp
+
+lemma
+ "finite S ==> inj (%(x, y). g x y) ==> finite {f x y| x y. g x y : S}"
+ by (auto intro: finite_vimageI)
+
+lemma
+ "finite A ==> finite S ==> inj (%(x, y). g x y) ==> finite {f x y z | x y z. g x y : S & z : A}"
+ by (auto intro: finite_vimageI)
+
+lemma
+ "finite S ==> finite A ==> inj (%(x, y). g x y) ==> inj (%(x, y). h x y)
+ ==> finite {f a b c d | a b c d. g a c : S & h b d : A}"
+ by (auto intro: finite_vimageI)
+
+lemma
+ assumes "finite S" shows "finite {(a,b,c,d). ([a, b], [c, d]) : S}"
+using assms by (auto intro!: finite_vimageI simp add: inj_on_def)
+ (* injectivity to be automated with further rules and automation *)
+
+schematic_lemma (* check interaction with schematics *)
+ "finite {x :: ?'A \<Rightarrow> ?'B \<Rightarrow> bool. \<exists>a b. x = Pair_Rep a b}
+ = finite ((\<lambda>(b :: ?'B, a:: ?'A). Pair_Rep a b) ` (UNIV \<times> UNIV))"
+ by simp
+
+declare [[simproc del: finite_Collect]]
+
+
+section {* Testing simproc in code generation *}
+
+definition union :: "nat set => nat set => nat set"
+where
+ "union A B = {x. x : A \<or> x : B}"
+
+definition common_subsets :: "nat set => nat set => nat set set"
+where
+ "common_subsets S1 S2 = {S. S \<subseteq> S1 \<and> S \<subseteq> S2}"
+
+definition products :: "nat set => nat set => nat set"
+where
+ "products A B = {c. EX a b. a : A & b : B & c = a * b}"
+
+export_code products in Haskell
+
+export_code union common_subsets products in Haskell
+
+end