35 lines
1.1 KiB
Text
35 lines
1.1 KiB
Text
(*
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"Type casting" library.
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*)
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(*
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The cast operator allows us to cast an element of type A
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into B if we provide a proof that types A and B are equal.
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*)
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Variable cast {A B : (Type U)} : A == B → A → B.
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(*
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The CastEq axiom states that for any cast of x is equal to x.
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*)
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Axiom CastEq {A B : (Type U)} (H : A == B) (x : A) : x == cast H x.
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(*
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The CastApp axiom "propagates" the cast over application
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*)
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Axiom CastApp {A A' : (Type U)} {B : A → (Type U)} {B' : A' → (Type U)}
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(H1 : (Π x : A, B x) == (Π x : A', B' x)) (H2 : A == A')
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(f : Π x : A, B x) (x : A) :
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cast H1 f (cast H2 x) == f x.
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(*
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If two (dependent) function spaces are equal, then their domains are equal.
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*)
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Axiom DomInj {A A' : (Type U)} {B : A → (Type U)} {B' : A' → (Type U)}
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(H : (Π x : A, B x) == (Π x : A', B' x)) :
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A == A'.
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(*
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If two (dependent) function spaces are equal, then their ranges are equal.
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*)
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Axiom RanInj {A A' : (Type U)} {B : A → (Type U)} {B' : A' → (Type U)}
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(H : (Π x : A, B x) == (Π x : A', B' x)) (a : A) :
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B a == B' (cast (DomInj H) a).
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