@@ -142,8 +142,8 @@ Class ExtMetricSpaceClass (X : Type) `{MetricSpaceBall X} : Prop := {
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mspc_radius_proper : Proper ((=) ==> (≡) ==> (≡) ==> iff) ball;
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mspc_inf: ∀ x y, ball Qinf.infinite x y;
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mspc_negative: ∀ (e: Q), e < 0 → ∀ x y, ~ ball e x y;
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- mspc_refl:> ∀ e : Q, 0 ≤ e → Reflexive (ball e);
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- mspc_symm:> ∀ e, Symmetric (ball e);
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+ mspc_refl:: ∀ e : Q, 0 ≤ e → Reflexive (ball e);
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+ mspc_symm:: ∀ e, Symmetric (ball e);
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mspc_triangle: ∀ (e1 e2: Q) (a b c: X),
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ball e1 a b → ball e2 b c → ball (e1 + e2) a c;
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mspc_closed: ∀ (e: Q) (a b: X),
@@ -437,7 +437,7 @@ Definition restrict (f : X -> Y) (x : X) (r : Q) : sig (ball r x) -> Y :=
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IsUniformlyContinuous and IsLocallyUniformlyContinuous *)
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Class IsLocallyUniformlyContinuous (f : X -> Y) (lmu : X -> Q -> Q -> Qinf) :=
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- luc_prf :> forall (x : X) (r : Q), IsUniformlyContinuous (restrict f x r) (lmu x r).
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+ luc_prf :: forall (x : X) (r : Q), IsUniformlyContinuous (restrict f x r) (lmu x r).
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Global Arguments luc_prf f lmu {_} x r.
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@@ -580,7 +580,7 @@ particular, integral_lipschitz in AbstractIntegration.v defines [L] as
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[λ a r, abs (f a) + L' a r * r]. *)
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Class IsLocallyLipschitz (f : X -> Y) (L : X -> Q -> Q) :=
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- llip_prf :> forall (x : X) (r : Q), PropHolds (0 ≤ r) -> IsLipschitz (restrict f x r) (L x r).
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+ llip_prf :: forall (x : X) (r : Q), PropHolds (0 ≤ r) -> IsLipschitz (restrict f x r) (L x r).
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Global Arguments llip_prf f L {_} x r _.
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@@ -625,7 +625,7 @@ Section Contractions.
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Context `{MetricSpaceBall X, MetricSpaceBall Y}.
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Class IsContraction (f : X -> Y) (q : Q) := {
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- contr_prf :> IsLipschitz f q;
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+ contr_prf :: IsLipschitz f q;
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contr_lt_1 : q < 1
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}.
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Class Limit := lim : RegularFunction -> X.
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- Class CompleteMetricSpaceClass `{Limit} := cmspc :> Surjective reg_unit (inv := lim).
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+ Class CompleteMetricSpaceClass `{Limit} := cmspc :: Surjective reg_unit (inv := lim).
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Definition tends_to (f : RegularFunction) (l : X) :=
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forall e : Q, 0 < e -> ball e (f e) l.
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