20 lines
542 B
Text
20 lines
542 B
Text
import macros
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scope
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variable A : Type
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variable R : A → A → Bool
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axiom Symm {x y : A} : R x y → R y x
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axiom Trans {x y z : A} : R x y → R y z → R x z
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theorem ReflIf (Linked : ∀ x, ∃ y, R x y) : ∀ x, R x x :=
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take x, obtain (w : A) (Hw : R x w), from Linked x,
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let lemma1 : R w x := Symm Hw
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in Trans Hw lemma1
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theorem ReflIf2 (Linked : ∀ x, ∃ y, R x y) : ∀ x, R x x :=
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λ x, exists_elim (Linked x)
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(λ w Hw,
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Trans Hw (Symm Hw))
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end
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print environment 1
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