(*  Title:      HOL/ex/Set_Comprehension_Pointfree_Tests.thy

    Author:     Lukas Bulwahn, Rafal Kolanski

    Copyright   2012 TU Muenchen

*)

header {* Tests for the set comprehension to pointfree simproc *}

theory Set_Comprehension_Pointfree_Tests

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 file -

export_code union common_subsets products in Haskell file -

end