24 lines
1.6 KiB
Text
24 lines
1.6 KiB
Text
(1, 2) : ℕ × ℕ
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⟨trivial, trivial⟩ : true ∧ true
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anc1.lean:5:0: warning: declaration '[anonymous]' uses sorry
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⟨1, _⟩ : Σ' (x : ℕ), x > 0
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show true, from true.intro : true
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Exists.intro 1 (id (λ (a : 1 = 0), nat.no_confusion a)) : ∃ (x : ℕ), 1 ≠ 0
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λ (A B C : Prop) (Ha : A) (Hb : B) (Hc : C), show B ∧ A, from ⟨Hb, Ha⟩ :
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∀ (A B C : Prop), A → B → C → B ∧ A
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λ (A B C : Prop) (Ha : A) (Hb : B) (Hc : C), show B ∧ A ∧ C ∧ A, from ⟨Hb, ⟨Ha, ⟨Hc, Ha⟩⟩⟩ :
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∀ (A B C : Prop), A → B → C → B ∧ A ∧ C ∧ A
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λ (A B C : Prop) (Ha : A) (Hb : B) (Hc : C), show B ∧ A ∧ C ∧ A, from ⟨Hb, ⟨Ha, ⟨Hc, Ha⟩⟩⟩ :
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∀ (A B C : Prop), A → B → C → B ∧ A ∧ C ∧ A
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λ (A B C : Prop) (Ha : A) (Hb : B) (Hc : C),
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show ((B ∧ true) ∧ A) ∧ C ∧ A, from ⟨⟨⟨Hb, true.intro⟩, Ha⟩, ⟨Hc, Ha⟩⟩ :
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∀ (A B C : Prop), A → B → C → ((B ∧ true) ∧ A) ∧ C ∧ A
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λ (A : Type u) (P Q : A → Prop) (a : A) (H1 : P a) (H2 : Q a),
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show ∃ (x : A), P x ∧ Q x, from Exists.intro a ⟨H1, H2⟩ :
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∀ (A : Type u) (P Q : A → Prop) (a : A), P a → Q a → (∃ (x : A), P x ∧ Q x)
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λ (A : Type u) (P Q : A → Prop) (a b : A) (H1 : P a) (H2 : Q b),
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show ∃ (x y : A), P x ∧ Q y, from Exists.intro a (Exists.intro b ⟨H1, H2⟩) :
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∀ (A : Type u) (P Q : A → Prop) (a b : A), P a → Q b → (∃ (x y : A), P x ∧ Q y)
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λ (A : Type u) (P Q : A → Prop) (a b : A) (H1 : P a) (H2 : Q b),
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show ∃ (x y : A), P x ∧ Q y, from Exists.intro a (Exists.intro b ⟨H1, H2⟩) :
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∀ (A : Type u) (P Q : A → Prop) (a b : A), P a → Q b → (∃ (x y : A), P x ∧ Q y)
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