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Set Implicit Arguments.
Require Import Logic.
(*Global Set Universe Polymorphism.*)
Global Set Asymmetric Patterns.
Local Set Primitive Projections.
Local Open Scope type_scope.
Record prod (A B : Type) : Type :=
pair { fst : A; snd : B }.
Arguments pair {A B} _ _.
Notation "x * y" := (prod x y) : type_scope.
Notation "( x , y , .. , z )" := (pair .. (pair x y) .. z) : core_scope.
Generalizable Variables X A B f g n.
Inductive paths {A : Type} (a : A) : A -> Type :=
idpath : paths a a.
Arguments idpath {A a} , [A] a.
Notation "x = y :> A" := (@paths A x y) : type_scope.
Notation "x = y" := (x = y :>_) : type_scope.
Definition transport {A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P x) : P y :=
match p with idpath => u end.
Definition transport_prod' {A : Type} {P Q : A -> Type} {a a' : A} (p : a = a')
(z : P a * Q a)
: transport (fun a => P a * Q a) p z = (transport _ p (fst z), transport _ p (snd z))
:= match p as p' return transport (fun a0 => P a0 * Q a0) p' z = (transport P p' (fst z), transport Q p' (snd z)) with
| idpath => idpath
end. (* success *)
Definition transport_prod {A : Type} {P Q : A -> Type} {a a' : A} (p : a = a')
(z : P a * Q a)
: transport (fun a => P a * Q a) p z = (transport _ p (fst z), transport _ p (snd z))
:= match p with
| idpath => idpath
end.
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