34 lines
No EOL
879 B
Text
34 lines
No EOL
879 B
Text
import macros
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scope
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theorem ReflIf (A : Type)
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(R : A → A → Bool)
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(Symm : ∀ x y, R x y → R y x)
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(Trans : ∀ x y z, R x y → R y z → R x z)
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(Linked : ∀ x, ∃ y, R x y)
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:
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∀ x, R x x :=
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take x,
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obtain (w : A) (H : R x w), from (Linked x),
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let L1 : R w x := Symm x w H
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in Trans x w x H L1
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pop_scope
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scope
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-- Same example again.
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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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axiom Linked (x : A) : ∃ y, R x y
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theorem ReflIf (x : A) : R x x :=
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obtain (w : A) (H : R x w), from (Linked x),
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let L1 : R w x := Symm H
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in Trans H L1
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end
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-- Display the last two theorems
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print environment 2 |