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Set Implicit Arguments.
Module A.
Set Universe Polymorphism.
Set Primitive Projections.
Set Asymmetric Patterns.
Inductive paths {A} (x : A) : A -> Type := idpath : paths x x
where "x = y" := (@paths _ x y) : type_scope.
Record sigT {A : Type} (P : A -> Type) := existT { projT1 : A; projT2 : P projT1 }.
Arguments existT {A} _ _ _.
Definition transport {A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P x) : P y :=
match p with idpath => u end.
Notation "x .1" := (projT1 x) (at level 3).
Notation "x .2" := (projT2 x) (at level 3).
Notation "( x ; y )" := (existT _ x y).
Set Printing All.
Definition path_sigma_uncurried {A : Type} (P : A -> Type) (u v : sigT P)
(pq : sigT (fun p : u.1 = v.1 => transport _ p u.2 = v.2))
: u = v
:= match pq with
| existT p q =>
match u, v return (forall p0 : (u.1 = v.1), (transport P p0 u.2 = v.2) -> (u=v)) with
| (x;y), (x';y') => fun p1 (q1 : transport P p1 (existT P x y).2 = (existT P x' y').2) =>
match p1 in (_ = x'') return (forall y'', (transport _ p1 y = y'') -> (x;y)=(x'';y'')) with
| idpath => fun y' (q2 : transport _ (@idpath _ _) y = y') =>
match q2 in (_ = y'') return (x;y) = (x;y'') with
| idpath => @idpath _ _
end
end y' q1
end p q
end.
(* Toplevel input, characters 341-357:
Error:
In environment
A : Type
P : forall _ : A, Type
u : @sigT A P
v : @sigT A P
pq :
@sigT (@paths A (projT1 u) (projT1 v))
(fun p : @paths A (projT1 u) (projT1 v) =>
@paths (P (projT1 v)) (@transport A P (projT1 u) (projT1 v) p (projT2 u))
(projT2 v))
p : @paths A (projT1 u) (projT1 v)
q :
@paths (P (projT1 v)) (@transport A P (projT1 u) (projT1 v) p (projT2 u))
(projT2 v)
x : A
y : P x
x' : A
y' : P x'
p1 : @paths A (projT1 (@existT A P x y)) (projT1 (@existT A P x' y'))
The term "projT2 (@existT A P x y)" has type "P (projT1 (@existT A P x y))"
while it is expected to have type "P (projT1 (@existT A P x y))".
*)
End A.
Module B.
Set Universe Polymorphism.
Set Primitive Projections.
Set Asymmetric Patterns.
Inductive paths {A} (x : A) : A -> Type := idpath : paths x x
where "x = y" := (@paths _ x y) : type_scope.
Record sigT {A : Type} (P : A -> Type) := existT { projT1 : A; projT2 : P projT1 }.
Arguments existT {A} _ _ _.
Definition transport {A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P x) : P y :=
match p with idpath => u end.
Notation "x .1" := (projT1 x) (at level 3).
Notation "x .2" := (projT2 x) (at level 3).
Notation "( x ; y )" := (existT _ x y).
Set Printing All.
Definition path_sigma_uncurried {A : Type} (P : A -> Type) (u v : sigT P)
(pq : sigT (fun p : u.1 = v.1 => transport _ p u.2 = v.2))
: u = v.
Proof.
destruct u as [x y].
destruct v. (* Toplevel input, characters 0-11:
Error: Illegal application:
The term "transport" of type
"forall (A : Type) (P : forall _ : A, Type) (x y : A)
(_ : @paths A x y) (_ : P x), P y"
cannot be applied to the terms
"A" : "Type"
"P" : "forall _ : A, Type"
"projT1 (@existT A P x y)" : "A"
"projT1 v" : "A"
"p" : "@paths A (projT1 (@existT A P x y)) (projT1 v)"
"projT2 (@existT A P x y)" : "P (projT1 (@existT A P x y))"
The 5th term has type "@paths A (projT1 (@existT A P x y)) (projT1 v)"
which should be coercible to
"@paths A (projT1 (@existT A P x y)) (projT1 v)".
*)
Abort.
End B.
¤ Dauer der Verarbeitung: 0.18 Sekunden
(vorverarbeitet)
¤
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