75 lines
2.4 KiB
Text
75 lines
2.4 KiB
Text
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import data.nat
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theorem tst2 (A B C D : Type) : (A × B) × (C × D) → C × B × A :=
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assume p : (A × B) × (C × D),
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obtain [a b] [c d], from p,
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(c, b, a)
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theorem tst (a b c d : Prop) : (a ∧ b) ∧ (c ∧ d) → c ∧ b ∧ a :=
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assume H,
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obtain [Ha Hb] Hc Hd, from H,
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and.intro Hc (and.intro Hb Ha)
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theorem tst22 (A B C D : Type) : (A × B) × (C × D) → C × B × A :=
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assume p,
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obtain [a b] [c d], from p,
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(c, b, a)
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theorem tst3 (A B C D : Type) : A × B × C × D → C × B × A :=
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assume p,
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obtain [[a b] c] d, from p,
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(c, b, a)
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example (p : nat → nat → Prop) : (∃ x, p x x) → ∃ x y, p x y :=
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assume ex,
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obtain x pxx, from ex,
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exists.intro x (exists.intro x pxx)
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example (p q : nat → nat → Prop) : (∃ x y, p x y ∧ q x y ∧ q y x) → ∃ x y, p x y :=
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assume ex,
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obtain x y pxy qxy qyx, from ex,
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exists.intro x (exists.intro y pxy)
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example (p : nat → nat → Type): (Σ x, p x x) → (Σ x y, p x y) :=
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assume sig,
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obtain x pxx, from sig,
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⟨x, x, pxx⟩
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example (p q : nat → nat → Type) : (Σ x y, p x y × q x y × q y x) → Σ x y, p x y :=
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assume ex : Σ x y, p x y × q x y × q y x,
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obtain x y [[pxy qxy] qyx], from ex,
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⟨x, y, pxy⟩
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example (p q : nat → nat → Type) : (Σ x y, p x y × q x y × q y x) → Σ x y, p x y :=
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assume ex,
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have ex1 : Σ x y, p x y × q x y × q y x, from ex,
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obtain x y [[pxy qxy] qyx], from ex1,
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⟨x, y, pxy⟩
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example (p q : nat → nat → Type) : (Σ x y, p x y × q x y × q y x) → Σ x y, p x y :=
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assume ex,
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obtain x y [[pxy qxy] qyx], from ex,
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⟨x, y, pxy⟩
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open nat
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definition even (a : nat) := ∃ x, a = 2*x
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example (a b : nat) (H₁ : even a) (H₂ : even b) : even (a+b) :=
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obtain x (Hx : a = 2*x), from H₁,
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obtain y (Hy : b = 2*y), from H₂,
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exists.intro
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(x+y)
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(calc a+b = 2*x + 2*y : by rewrite [Hx, Hy]
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... = 2*(x+y) : by rewrite mul.left_distrib)
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theorem dvd_of_dvd_add_left {m n₁ n₂ : ℕ} (H₁ : m ∣ n₁ + n₂) (H₂ : m ∣ n₁) : m ∣ n₂ :=
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obtain (c₁ : nat) (Hc₁ : n₁ + n₂ = m * c₁), from H₁,
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obtain (c₂ : nat) (Hc₂ : n₁ = m * c₂), from H₂,
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have aux : m * (c₁ - c₂) = n₂, from calc
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m * (c₁ - c₂) = m * c₁ - m * c₂ : by rewrite mul_sub_left_distrib
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... = n₁ + n₂ - m * c₂ : Hc₁
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... = n₁ + n₂ - n₁ : Hc₂
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... = n₂ : add_sub_cancel_left,
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dvd.intro aux
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