theory set
imports Main
begin
lemma "EX x X. ALL y. EX z Z. (~P(y,y) | P(x,x) | ~S(z,x)) &
(S(x,y) | ~S(y,z) | Q(Z,Z)) &
(Q(X,y) | ~Q(y,Z) | S(X,X))"
by metis
lemma "P(n::nat) ==> ~P(0) ==> n ~= 0"
by metis
sledgehammer_params [isar_proof, isar_shrink_factor = 1]
lemma
"(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))"
proof -
have F1: "∀(x\<^isub>2::'b => bool) x\<^isub>1::'b => bool. x\<^isub>1 ⊆ x\<^isub>1 ∪ x\<^isub>2" by (metis Un_commute Un_upper2)
have F2a: "∀(x\<^isub>2::'b => bool) x\<^isub>1::'b => bool. x\<^isub>1 ⊆ x\<^isub>2 --> x\<^isub>2 = x\<^isub>2 ∪ x\<^isub>1" by (metis Un_commute subset_Un_eq)
have F2: "∀(x\<^isub>2::'b => bool) x\<^isub>1::'b => bool. x\<^isub>1 ⊆ x\<^isub>2 ∧ x\<^isub>2 ⊆ x\<^isub>1 --> x\<^isub>1 = x\<^isub>2" by (metis F2a subset_Un_eq)
{ assume "¬ Z ⊆ X"
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis Un_upper2) }
moreover
{ assume AA1: "Y ∪ Z ≠ X"
{ assume "¬ Y ⊆ X"
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis F1) }
moreover
{ assume AAA1: "Y ⊆ X ∧ Y ∪ Z ≠ X"
{ assume "¬ Z ⊆ X"
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis Un_upper2) }
moreover
{ assume "(Z ⊆ X ∧ Y ⊆ X) ∧ Y ∪ Z ≠ X"
hence "Y ∪ Z ⊆ X ∧ X ≠ Y ∪ Z" by (metis Un_subset_iff)
hence "Y ∪ Z ≠ X ∧ ¬ X ⊆ Y ∪ Z" by (metis F2)
hence "∃x\<^isub>1::'a => bool. Y ⊆ x\<^isub>1 ∪ Z ∧ Y ∪ Z ≠ X ∧ ¬ X ⊆ x\<^isub>1 ∪ Z" by (metis F1)
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis Un_upper2) }
ultimately have "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis AAA1) }
ultimately have "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis AA1) }
moreover
{ assume "∃x\<^isub>1::'a => bool. (Z ⊆ x\<^isub>1 ∧ Y ⊆ x\<^isub>1) ∧ ¬ X ⊆ x\<^isub>1"
{ assume "¬ Y ⊆ X"
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis F1) }
moreover
{ assume AAA1: "Y ⊆ X ∧ Y ∪ Z ≠ X"
{ assume "¬ Z ⊆ X"
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis Un_upper2) }
moreover
{ assume "(Z ⊆ X ∧ Y ⊆ X) ∧ Y ∪ Z ≠ X"
hence "Y ∪ Z ⊆ X ∧ X ≠ Y ∪ Z" by (metis Un_subset_iff)
hence "Y ∪ Z ≠ X ∧ ¬ X ⊆ Y ∪ Z" by (metis F2)
hence "∃x\<^isub>1::'a => bool. Y ⊆ x\<^isub>1 ∪ Z ∧ Y ∪ Z ≠ X ∧ ¬ X ⊆ x\<^isub>1 ∪ Z" by (metis F1)
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis Un_upper2) }
ultimately have "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis AAA1) }
ultimately have "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by blast }
moreover
{ assume "¬ Y ⊆ X"
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis F1) }
ultimately show "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by metis
qed
sledgehammer_params [isar_proof, isar_shrink_factor = 2]
lemma
"(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))"
proof -
have F1: "∀(x\<^isub>2::'b => bool) x\<^isub>1::'b => bool. x\<^isub>1 ⊆ x\<^isub>2 ∧ x\<^isub>2 ⊆ x\<^isub>1 --> x\<^isub>1 = x\<^isub>2" by (metis Un_commute subset_Un_eq)
{ assume AA1: "∃x\<^isub>1::'a => bool. (Z ⊆ x\<^isub>1 ∧ Y ⊆ x\<^isub>1) ∧ ¬ X ⊆ x\<^isub>1"
{ assume AAA1: "Y ⊆ X ∧ Y ∪ Z ≠ X"
{ assume "¬ Z ⊆ X"
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis Un_upper2) }
moreover
{ assume "Y ∪ Z ⊆ X ∧ X ≠ Y ∪ Z"
hence "∃x\<^isub>1::'a => bool. Y ⊆ x\<^isub>1 ∪ Z ∧ Y ∪ Z ≠ X ∧ ¬ X ⊆ x\<^isub>1 ∪ Z" by (metis F1 Un_commute Un_upper2)
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis Un_upper2) }
ultimately have "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis AAA1 Un_subset_iff) }
moreover
{ assume "¬ Y ⊆ X"
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis Un_commute Un_upper2) }
ultimately have "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis AA1 Un_subset_iff) }
moreover
{ assume "¬ Z ⊆ X"
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis Un_upper2) }
moreover
{ assume "¬ Y ⊆ X"
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis Un_commute Un_upper2) }
moreover
{ assume AA1: "Y ⊆ X ∧ Y ∪ Z ≠ X"
{ assume "¬ Z ⊆ X"
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis Un_upper2) }
moreover
{ assume "Y ∪ Z ⊆ X ∧ X ≠ Y ∪ Z"
hence "∃x\<^isub>1::'a => bool. Y ⊆ x\<^isub>1 ∪ Z ∧ Y ∪ Z ≠ X ∧ ¬ X ⊆ x\<^isub>1 ∪ Z" by (metis F1 Un_commute Un_upper2)
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis Un_upper2) }
ultimately have "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis AA1 Un_subset_iff) }
ultimately show "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by metis
qed
sledgehammer_params [isar_proof, isar_shrink_factor = 3]
lemma
"(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))"
proof -
have F1a: "∀(x\<^isub>2::'b => bool) x\<^isub>1::'b => bool. x\<^isub>1 ⊆ x\<^isub>2 --> x\<^isub>2 = x\<^isub>2 ∪ x\<^isub>1" by (metis Un_commute subset_Un_eq)
have F1: "∀(x\<^isub>2::'b => bool) x\<^isub>1::'b => bool. x\<^isub>1 ⊆ x\<^isub>2 ∧ x\<^isub>2 ⊆ x\<^isub>1 --> x\<^isub>1 = x\<^isub>2" by (metis F1a subset_Un_eq)
{ assume "(Z ⊆ X ∧ Y ⊆ X) ∧ Y ∪ Z ≠ X"
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis F1 Un_commute Un_subset_iff Un_upper2) }
moreover
{ assume AA1: "∃x\<^isub>1::'a => bool. (Z ⊆ x\<^isub>1 ∧ Y ⊆ x\<^isub>1) ∧ ¬ X ⊆ x\<^isub>1"
{ assume "(Z ⊆ X ∧ Y ⊆ X) ∧ Y ∪ Z ≠ X"
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis F1 Un_commute Un_subset_iff Un_upper2) }
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis AA1 Un_commute Un_subset_iff Un_upper2) }
ultimately show "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis Un_commute Un_upper2)
qed
sledgehammer_params [isar_proof, isar_shrink_factor = 4]
lemma
"(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))"
proof -
have F1: "∀(x\<^isub>2::'b => bool) x\<^isub>1::'b => bool. x\<^isub>1 ⊆ x\<^isub>2 ∧ x\<^isub>2 ⊆ x\<^isub>1 --> x\<^isub>1 = x\<^isub>2" by (metis Un_commute subset_Un_eq)
{ assume "¬ Y ⊆ X"
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis Un_commute Un_upper2) }
moreover
{ assume AA1: "Y ⊆ X ∧ Y ∪ Z ≠ X"
{ assume "∃x\<^isub>1::'a => bool. Y ⊆ x\<^isub>1 ∪ Z ∧ Y ∪ Z ≠ X ∧ ¬ X ⊆ x\<^isub>1 ∪ Z"
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis Un_upper2) }
hence "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis AA1 F1 Un_commute Un_subset_iff Un_upper2) }
ultimately show "(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V::'a => bool. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))" by (metis Un_subset_iff Un_upper2)
qed
sledgehammer_params [isar_proof, isar_shrink_factor = 1]
lemma
"(X = Y ∪ Z) = (Y ⊆ X ∧ Z ⊆ X ∧ (∀V. Y ⊆ V ∧ Z ⊆ V --> X ⊆ V))"
by (metis Un_least Un_upper1 Un_upper2 set_eq_subset)
lemma "(X = Y ∩ Z) = (X ⊆ Y ∧ X ⊆ Z ∧ (∀V. V ⊆ Y ∧ V ⊆ Z --> V ⊆ X))"
by (metis Int_greatest Int_lower1 Int_lower2 subset_antisym)
lemma fixedpoint: "∃!x. f (g x) = x ==> ∃!y. g (f y) = y"
by metis
lemma "∃!x. f (g x) = x ==> ∃!y. g (f y) = y"
proof -
assume "∃!x::'a. f (g x) = x"
thus "∃!y::'b. g (f y) = y" by metis
qed
lemma
"∀x ∈ S. \<Union>S ⊆ x ==> ∃z. S ⊆ {z}"
by (metis Set.subsetI Union_upper insertCI set_eq_subset)
lemma
"∀x ∈ S. \<Union>S ⊆ x ==> ∃z. S ⊆ {z}"
by (metis Set.subsetI Union_upper insert_iff set_eq_subset)
lemma singleton_example_2:
"∀x ∈ S. \<Union>S ⊆ x ==> ∃z. S ⊆ {z}"
proof -
assume "∀x ∈ S. \<Union>S ⊆ x"
hence "∀x\<^isub>1. x\<^isub>1 ⊆ \<Union>S ∧ x\<^isub>1 ∈ S --> x\<^isub>1 = \<Union>S" by (metis set_eq_subset)
hence "∀x\<^isub>1. x\<^isub>1 ∈ S --> x\<^isub>1 = \<Union>S" by (metis Union_upper)
hence "∀x\<^isub>1::('a => bool) => bool. \<Union>S ∈ x\<^isub>1 --> S ⊆ x\<^isub>1" by (metis subsetI)
hence "∀x\<^isub>1::('a => bool) => bool. S ⊆ insert (\<Union>S) x\<^isub>1" by (metis insert_iff)
thus "∃z. S ⊆ {z}" by metis
qed
text {*
From W. W. Bledsoe and Guohui Feng, SET-VAR. JAR 11 (3), 1993, pages
293-314.
*}
lemma "∃B. (∀x ∈ B. x ≤ (0::int))"
"D ∈ F ==> ∃G. ∀A ∈ G. ∃B ∈ F. A ⊆ B"
"P a ==> ∃A. (∀x ∈ A. P x) ∧ (∃y. y ∈ A)"
"a < b ∧ b < (c::int) ==> ∃B. a ∉ B ∧ b ∈ B ∧ c ∉ B"
"P (f b) ==> ∃s A. (∀x ∈ A. P x) ∧ f s ∈ A"
"P (f b) ==> ∃s A. (∀x ∈ A. P x) ∧ f s ∈ A"
"∃A. a ∉ A"
"(∀C. (0, 0) ∈ C ∧ (∀x y. (x, y) ∈ C --> (Suc x, Suc y) ∈ C) --> (n, m) ∈ C) ∧ Q n --> Q m"
apply (metis all_not_in_conv)
apply (metis all_not_in_conv)
apply (metis mem_def)
apply (metis less_int_def singleton_iff)
apply (metis mem_def)
apply (metis mem_def)
apply (metis all_not_in_conv)
by (metis pair_in_Id_conv)
end