# Theory Set_Comprehension_Pointfree_Examples

theory Set_Comprehension_Pointfree_Examples
imports Main
```(*  Title:      HOL/ex/Set_Comprehension_Pointfree_Examples.thy
Author:     Lukas Bulwahn, Rafal Kolanski
*)

section ‹Examples for the set comprehension to pointfree simproc›

theory Set_Comprehension_Pointfree_Examples
imports Main
begin

lemma
"finite (UNIV::'a set) ⟹ finite {p. ∃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'. ∃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. ∃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 (λ(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 (λ(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 (λ(x, y). g x y) ⟹ inj (λ(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_goal (* 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

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 ∨ 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. ∃a b. a ∈ A ∧ b ∈ B ∧ c = a * b}"