27 lines
563 B
Text
27 lines
563 B
Text
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import data.nat
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open nat
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inductive lift .{l} (A : Type.{l}) : Type.{l+1} :=
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up : A → lift A
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namespace lift
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definition down {A : Type} (a : lift A) : A :=
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rec (λ a, a) a
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theorem down_up {A : Type} (a : A) : down (up a) = a :=
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rfl
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theorem induction_on [protected] {A : Type} {P : lift A → Prop} (a : lift A) (H : ∀ (a : A), P (up a)) : P a :=
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rec H a
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theorem up_down {A : Type} (a' : lift A) : up (down a') = a' :=
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induction_on a' (λ a, rfl)
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end lift
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set_option pp.universes true
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check nat
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check lift nat
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open lift
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definition one1 : lift nat := up 1
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