31 lines
755 B
Text
31 lines
755 B
Text
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/-
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This example demonstrates why allowing types such as
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inductive D : Type :=
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| intro : (D → D) → D
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would make the system inconsistent
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-/
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/- If we were allowed to form the inductive type
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inductive D : Type :=
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| intro : (D → D) → D
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we would get the following
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-/
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universe l
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-- The new type A
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axiom D : Type.{l}
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-- The constructor
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axiom introD : (D → D) → D
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-- The eliminator
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axiom recD : Π {C : D → Type}, (Π (f : D → D) (r : Π d, C (f d)), C (introD f)) → (Π (d : D), C d)
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-- We would also get a computational rule for the eliminator, but we don't need it for deriving the inconsistency.
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definition id : D → D := λd, d
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definition v : D := introD id
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theorem inconsistent : false :=
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recD (λ f ih, ih v) v
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