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feat(pi): prove that forall x, a = x is a mere proposition

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Floris van Doorn 2015-05-05 18:33:17 -04:00 committed by Leonardo de Moura
parent e5241f84ec
commit e9ff925e2f
1 changed files with 8 additions and 3 deletions
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11
hott/types/pi.hlean
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@ -177,13 +177,18 @@ namespace pi
definition is_trunc_eq_pi [instance] [priority 500] (n : trunc_index) (f g : Πa, B a) definition is_trunc_eq_pi [instance] [priority 500] (n : trunc_index) (f g : Πa, B a)
[H : ∀a, is_trunc n (f a = g a)] : is_trunc n (f = g) := [H : ∀a, is_trunc n (f a = g a)] : is_trunc n (f = g) :=
begin begin
apply is_trunc_equiv_closed, apply equiv.symm, apply is_trunc_equiv_closed_rev,
apply eq_equiv_homotopy apply eq_equiv_homotopy
end end
/- Symmetry of Π -/ definition is_hprop_pi_eq [instance] [priority 490] (a : A) : is_hprop (Π(a' : A), a = a') :=
is_hprop_of_imp_is_contr
( assume (f : Πa', a = a'),
assert H : is_contr A, from is_contr.mk a f,
_)
definition is_equiv_flip [instance] {P : A → A' → Type} : is_equiv (@function.flip _ _ P) := /- Symmetry of Π -/
definition is_equiv_flip [instance] {P : A → A' → Type} : is_equiv (@function.flip A A' P) :=
begin begin
fapply is_equiv.mk, fapply is_equiv.mk,
exact (@function.flip _ _ (function.flip P)), exact (@function.flip _ _ (function.flip P)),