37 lines
961 B
Text
37 lines
961 B
Text
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open prod sigma
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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 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 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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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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open nat
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definition even (a : nat) := Σ 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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⟨x+y,
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calc a+b = 2*x + 2*y : by rewrite [Hx, Hy]
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... = 2*(x+y) : sorry⟩
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