Theory Set_Comprehension_Pointfree_Tests

theory Set_Comprehension_Pointfree_Tests
imports Main
(*  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

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 ∧ b : B}"
by simp

lemma
"finite B ==> finite A' ==> finite {f a b| a b. a : A ∧ a : A' ∧ b : B}"
by simp

lemma
"finite A ==> finite B ==> finite {f a b| a b. a : A ∧ b : B ∧ b : B'}"
by simp

lemma
"finite A ==> finite B ==> finite C ==> finite {f a b c| a b c. a : A ∧ b : B ∧ 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 ∧ b : B ∧ c : C ∧ 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 ∧ b : B ∧ c : C ∧ d : D ∧ e : E}"

by simp

lemma
"finite A ==> finite B ==> finite C ==> finite D ==> finite E ==>
finite {f a d c b e | e b c d a. b : B ∧ a : A ∧ e : E' ∧ c : C ∧ d : D ∧ e : E ∧ b : B'}"

by simp

lemma
"[| finite A ; finite B ; finite C ; finite D |]
==> finite ({f a b c d| a b c d. a : A ∧ b : B ∧ c : C ∧ d : D})"

by simp

lemma
"finite ((λ(a,b,c,d). f a b c d) ` (A × B × C × D))
==> finite ({f a b c d| a b c d. a : A ∧ b : B ∧ c : C ∧ 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 ∧ b : B ∧ a ∉ Z}"
by simp

lemma
"finite A ==> finite B ==> finite R ==> finite {f a b x y| a b x y. a : A ∧ b : B ∧ (x,y) ∈ R}"
by simp

lemma
"finite A ==> finite B ==> finite R ==> finite {f a b x y| a b x y. a : A ∧ (x,y) ∈ R ∧ b : B}"
by simp

lemma
"finite A ==> finite B ==> finite R ==> finite {f a (x, b) y| y b x a. a : A ∧ (x,y) ∈ R ∧ b : B}"
by simp

lemma
"finite A ==> finite AA ==> finite B ==> finite {f a b| a b. (a : A ∨ a : AA) ∧ b : B ∧ a ∉ 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 ∧ a : A2) ∨ (a : A3 ∧ (a : A4 ∨ a : A5))) ∧ b : B ∧ a ∉ 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 => ?'B => bool. ∃a b. x = Pair_Rep a b}
= finite ((λ(b :: ?'B, a:: ?'A). Pair_Rep a b) ` (UNIV × UNIV))"

by simp


section {* Testing simproc in code generation *}

definition union :: "nat set => nat set => nat set"
where
"union A B = {x. x : A ∨ x : B}"

definition common_subsets :: "nat set => nat set => nat set set"
where
"common_subsets S1 S2 = {S. S ⊆ S1 ∧ S ⊆ 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