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Set Asymmetric Patterns.
Section eq.
Let A := { X : Type & X }.
Let B := (fun x : A => projT1 x).
Let T := (fun (a' : A) (b' : B a') => projT2 a' = b').
Let T' := T.
Let t1T := (fun _ : A => unit).
Let f1 := (fun x (_ : t1T x) => projT2 x).
Let t1 := (fun x (y : t1T x) => @eq_refl (projT1 x) (projT2 x)).
Let t1T' := t1T.
Let f1' := f1.
Let t1' := t1.
Theorem eq_matches_commute
a' b' (t' : T a' b')
(rta : forall b'', T' a' b'' -> A)
(rtb : forall b'' t'', B (rta b'' t''))
(rt1 : forall y, T _ (rtb (f1' a' y) (@t1' a' y)))
(R : forall (b : B (rta b' t')), T _ b -> Type)
(r1 : forall y, R (f1 _ y) (@t1 _ y))
: match
match t' as t0' in (@eq _ _ b0') return T (rta b0' t0') (rtb b0' t0') with
| eq_refl => rt1 tt
end
as t0 in (@eq _ _ b0)
return R b0 t0
with
| eq_refl => r1 tt
end
=
match t'
as t0' in (@eq _ _ b0')
return (forall (R : forall (b : B (rta b0' t0')), T _ b -> Type)
(r1 : forall y, R (f1 _ y) (@t1 _ y)),
R _ (match t0' as t0'0 in (@eq _ _ b0'0) return T (rta b0'0 t0'0) (rtb b0'0 t0'0) with
| eq_refl => rt1 tt
end))
with
| eq_refl => fun _ r1 =>
match rt1 tt with
| eq_refl => r1 tt
end
end R r1.
Proof.
destruct t'; reflexivity.
Defined.
Theorem eq_match_beta2
a b (t : T a b)
X
(R : forall b' (t' : T a b'), X b' -> Type)
(r1 : forall y x, R _ (@t1 _ y) x)
x
: match t as t' in (@eq _ _ b') return forall x, R b' t' x with
| eq_refl => r1 tt
end (x b)
=
match t as t' in (@eq _ _ b') return R b' t' (x b') with
| eq_refl => r1 tt (x _)
end.
Proof.
destruct t; reflexivity.
Defined.
End eq.
Definition typeof {T} (_ : T) := T.
Eval compute in (eq_sym (eq_sym _)).
Goal forall T (x y : T) (p : x = y), True.
intros.
pose proof
(@eq_matches_commute
(existT (fun T => T) T x) y p
(fun b'' _ => existT (fun T => T) T b'')
(fun _ _ => x)
(fun _ => eq_refl)
(fun x' _ => x' = y)
(fun _ => eq_refl)
) as H0.
compute in H0.
change (fun (x' : T) (_ : y = x') => x' = y) with ((fun y => fun (x' : T) (_ : y = x') => x' = y) y) in H0.
Fail pose proof (fun k => @eq_trans _ _ _ k H0).
Abort.
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