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Example.lean
194
Example.lean
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variable (p q r : Prop)
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variable (p q r : Prop)
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-- commutativity of ∧ and ∨
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-- commutativity of ∧ and ∨
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example : p ∧ q ↔ q ∧ p := by
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example : p ∧ q ↔ q ∧ p :=
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apply Iff.intro
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Iff.intro (λ ⟨p,q⟩ => ⟨q,p⟩) (λ ⟨q,p⟩ => ⟨p,q⟩)
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· intro pandq
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apply And.intro
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· apply And.right pandq
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· apply And.left pandq
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· intro qandp
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apply And.intro
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· apply And.right qandp
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· apply And.left qandp
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example : p ∨ q ↔ q ∨ p := by
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example : p ∨ q ↔ q ∨ p :=
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apply Iff.intro
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Iff.intro
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· intro porq
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(λ
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cases porq with
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| Or.inl p => Or.inr p
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| inl p => exact Or.inr p
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| Or.inr q => Or.inl q)
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| inr q => exact Or.inl q
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(λ
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· intro qorp
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| Or.inl q => Or.inr q
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cases qorp with
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| Or.inr p => Or.inl p)
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| inl q => exact Or.inr q
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| inr p => exact Or.inl p
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-- associativity of ∧ and ∨
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-- associativity of ∧ and ∨
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example : (p ∧ q) ∧ r ↔ p ∧ (q ∧ r) := by
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example : (p ∧ q) ∧ r ↔ p ∧ (q ∧ r) :=
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apply Iff.intro
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Iff.intro
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· intro pqandr
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(λ ⟨⟨p,q⟩,r⟩ => ⟨p,⟨q,r⟩⟩)
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apply And.intro
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(λ ⟨p,⟨q,r⟩⟩ => ⟨⟨p,q⟩,r⟩)
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· exact And.left (And.left pqandr)
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· apply And.intro
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· exact And.right (And.left pqandr)
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· exact And.right pqandr
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· intro pandqr
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apply And.intro
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· apply And.intro
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· exact And.left pandqr
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· exact And.left (And.right pandqr)
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· exact And.right (And.right pandqr)
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example : (p ∨ q) ∨ r ↔ p ∨ (q ∨ r) := by
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example : (p ∨ q) ∨ r ↔ p ∨ (q ∨ r) :=
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apply Iff.intro
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Iff.intro
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· intro pqorr
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(λ
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cases pqorr with
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| Or.inr r => Or.inr (Or.inr r)
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| inr r => apply Or.inr (Or.inr r)
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| Or.inl (Or.inr q) => Or.inr (Or.inl q)
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| inl porq => cases porq with
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| Or.inl (Or.inl p) => Or.inl p)
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| inl p => apply Or.inl p
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(λ
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| inr q => apply Or.inr (Or.inl q)
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| Or.inl p => Or.inl (Or.inl p)
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· intro porqr
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| Or.inr (Or.inl q) => Or.inl (Or.inr q)
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cases porqr with
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| Or.inr (Or.inr r) => Or.inr r)
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| inl p => apply Or.inl (Or.inl p)
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| inr qorr => cases qorr with
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| inr r => apply Or.inr r
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| inl q => apply Or.inl (Or.inr q)
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-- distributivity
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-- distributivity
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example : p ∧ (q ∨ r) ↔ (p ∧ q) ∨ (p ∧ r) := by
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example : p ∧ (q ∨ r) ↔ (p ∧ q) ∨ (p ∧ r) :=
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apply Iff.intro
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Iff.intro
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· intro pandqorr
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(λ
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cases pandqorr with
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| ⟨p, Or.inl q⟩ => Or.inl ⟨p,q⟩
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| intro p qor => cases qor with
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| ⟨p, Or.inr r⟩ => Or.inr ⟨p,r⟩)
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| inl q => apply Or.inl (And.intro p q)
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(λ
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| inr r => apply Or.inr (And.intro p r)
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| Or.inl ⟨p,q⟩ => ⟨p,Or.inl q⟩
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· intro pandqorpandr
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| Or.inr ⟨p,r⟩ => ⟨p,Or.inr r⟩)
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cases pandqorpandr with
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| inl pandq => apply And.intro (And.left pandq) (Or.inl (And.right pandq))
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| inr pandr => apply And.intro (And.left pandr) (Or.inr (And.right pandr))
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example : p ∨ (q ∧ r) ↔ (p ∨ q) ∧ (p ∨ r) := by
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example : p ∨ (q ∧ r) ↔ (p ∨ q) ∧ (p ∨ r) :=
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apply Iff.intro
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Iff.intro
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· intro porqandr
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(λ
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cases porqandr with
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| Or.inl p => ⟨Or.inl p, Or.inl p⟩
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| inl p => apply And.intro (Or.inl p) (Or.inl p)
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| Or.inr ⟨q,r⟩ => ⟨Or.inr q, Or.inr r⟩)
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| inr qandr => apply And.intro (Or.inr (And.left qandr)) (Or.inr (And.right qandr))
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(λ
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· intro porqandporr
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| ⟨Or.inr q, Or.inr r⟩ => Or.inr ⟨q,r⟩
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cases And.left porqandporr with
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| ⟨Or.inl p, _⟩ => Or.inl p
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| inl p => apply Or.inl p
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| ⟨ _, Or.inl p⟩ => Or.inl p)
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| inr q => cases And.right porqandporr with
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| inl p => apply Or.inl p
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| inr r => apply Or.inr (And.intro q r)
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-- other properties
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-- other properties
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example : (p → (q → r)) ↔ (p ∧ q → r) := by
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example : (p → (q → r)) ↔ (p ∧ q → r) :=
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apply Iff.intro
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Iff.intro
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· intro ptoqtor
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(λ f ⟨p,q⟩ => f p q)
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intro pandq
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(λ f p q => f ⟨p,q⟩)
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exact ptoqtor (And.left pandq) (And.right pandq)
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· intro pandqtor
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intro p
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intro q
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exact pandqtor (And.intro p q)
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example : ((p ∨ q) → r) ↔ (p → r) ∧ (q → r) := by
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example : ((p ∨ q) → r) ↔ (p → r) ∧ (q → r) :=
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apply Iff.intro
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Iff.intro
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· intro porqtor
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(λ porqtor => And.intro (λ p => porqtor (Or.inl p)) (λ q => porqtor (Or.inr q)))
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apply And.intro
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(λ
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· intro p
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|⟨f, _⟩, Or.inl p => f p
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exact porqtor (Or.inl p)
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|⟨_, g⟩, Or.inr q => g q)
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· intro q
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exact porqtor (Or.inr q)
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· intro ptorandqtor
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intro porq
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cases porq with
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| inl p => exact (And.left ptorandqtor) p
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| inr q => exact (And.right ptorandqtor) q
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example : ¬(p ∨ q) ↔ ¬p ∧ ¬q := by
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example : ¬(p ∨ q) ↔ ¬p ∧ ¬q :=
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apply Iff.intro
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Iff.intro
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· intro nporq
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(λ nporq => And.intro (λ p => nporq (Or.inl p)) (λ q => nporq (Or.inr q)))
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apply And.intro
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(λ
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· intro p
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| ⟨np, _⟩, Or.inl p => np p
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exact nporq (Or.inl p)
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| ⟨_ ,nq⟩, Or.inr q => nq q)
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· intro q
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exact nporq (Or.inr q)
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· intro npandnq
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intro porq
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cases porq with
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| inl p => exact (And.left npandnq) p
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| inr q => exact (And.right npandnq) q
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example : ¬p ∨ ¬q → ¬(p ∧ q) := by
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example : ¬p ∨ ¬q → ¬(p ∧ q)
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intro npornq
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| Or.inl np, ⟨p,_⟩ => np p
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cases npornq with
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| Or.inr nq, ⟨_,q⟩ => nq q
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| inl np => intro pandq; exact np (And.left pandq)
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| inr nq => intro pandq; exact nq (And.right pandq)
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example : ¬(p ∧ ¬p) := by
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example : ¬(p ∧ ¬p) :=
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intro pandnp
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λ ⟨p,np⟩ => np p
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exact (And.right pandnp) (And.left pandnp)
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example : p ∧ ¬q → ¬(p → q) :=
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example : p ∧ ¬q → ¬(p → q) :=
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λ ⟨p,nq⟩ ptoq => nq (ptoq p)
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λ ⟨p,nq⟩ ptoq => nq (ptoq p)
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@ -139,21 +87,11 @@ example : (¬p ∨ q) → (p → q)
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| (Or.inl np), p => absurd p np
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| (Or.inl np), p => absurd p np
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| (Or.inr q), _ => q
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| (Or.inr q), _ => q
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example : p ∨ False ↔ p := by
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example : p ∨ False ↔ p :=
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apply Iff.intro
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Iff.intro (λ porf => Or.elim porf id False.elim) (λ p => Or.inl p)
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· intro porf
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apply Or.elim porf id False.elim
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· intro p
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apply Or.inl p
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example : p ∧ False ↔ False := by
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apply Iff.intro
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· intro ⟨p,f⟩
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exact f
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· intro f
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apply False.elim f
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example : p ∧ False ↔ False :=
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Iff.intro (λ ⟨_,f⟩ => f) (λ f => False.elim f)
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example : (p → q) → (¬q → ¬p) :=
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example : (p → q) → (¬q → ¬p) :=
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λ ptoq nq p => absurd (ptoq p) nq
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λ ptoq nq p => absurd (ptoq p) nq
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