4dd6cead83
It was not a good idea to use heterogeneous equality as the default equality in Lean.
It creates the following problems.
- Heterogeneous equality does not propagate constraints in the elaborator.
For example, suppose that l has type (List Int), then the expression
l = nil
will not propagate the type (List Int) to nil.
- It is easy to write false. For example, suppose x has type Real, and the user
writes x = 0. This is equivalent to false, since 0 has type Nat. The elaborator cannot introduce
the coercion since x = 0 is a type correct expression.
Homogeneous equality does not suffer from the problems above.
We keep heterogeneous equality because it is useful for generating proof terms.
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
28 lines
1.1 KiB
Text
28 lines
1.1 KiB
Text
Set: pp::colors
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Set: pp::unicode
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Assumed: C
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Assumed: D
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Assumed: R
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Proved: R2
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Set: lean::pp::implicit
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Variable C {A B : Type} (H : eq::explicit Type A B) (a : A) : B
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Definition C::explicit (A B : Type) (H : A = B) (a : A) : B := C H a
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Variable D {A A' : Type} {B : A → Type} {B' : A' → Type} (H : eq::explicit Type (Π x : A, B x) (Π x : A', B' x)) :
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eq::explicit Type A A'
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Definition D::explicit (A A' : Type) (B : A → Type) (B' : A' → Type) (H : (Π x : A, B x) = (Π x : A', B' x)) : A =
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A' :=
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D H
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Variable R {A A' : Type}
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{B : A → Type}
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{B' : A' → Type}
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(H : eq::explicit Type (Π x : A, B x) (Π x : A', B' x))
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(a : A) :
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eq::explicit Type (B a) (B' (C::explicit A A' (D::explicit A A' (λ x : A, B x) (λ x : A', B' x) H) a))
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Definition R::explicit (A A' : Type)
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(B : A → Type)
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(B' : A' → Type)
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(H : (Π x : A, B x) = (Π x : A', B' x))
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(a : A) : (B a) = (B' (C (D H) a)) :=
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R H a
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Theorem R2 (A1 A2 B1 B2 : Type) (H : eq::explicit Type (A1 → B1) (A2 → B2)) (a : A1) : eq::explicit Type B1 B2 :=
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R::explicit A1 A2 (λ x : A1, B1) (λ x : A2, B2) H a
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