67 lines
1.9 KiB
Text
67 lines
1.9 KiB
Text
import data.nat
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open algebra
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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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namespace hidden
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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 left_distrib)
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end hidden
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