\import Algebra.Group
\import Algebra.Group.Category
\import Algebra.Monoid
\import Algebra.Ordered
\import Arith.Real
\import Function.Meta
\import Logic
\import Logic.Meta
\import Meta
\import Order.LinearOrder
\import Order.PartialOrder
\import Order.StrictOrder
\import Paths
\import Paths.Meta
\import Set.Subset
\import Topology.CoverSpace
\import Topology.MetricSpace
\import Topology.TopAbGroup
\import Topology.TopAbGroup.Complete
\import Topology.UniformSpace
\import Topology.UniformSpace.Product

\class PseudoNormedAbGroup \extends PseudoMetricSpace, TopAbGroup {
| norm : E -> Real
| norm_zro : norm zro = (0 : Real)
| norm_negative {x : E} : norm (negative x) = norm x
| norm_+ {x y : E} : norm (x + y) <= norm x RealAbGroup.+ norm y
| norm-dist {x y : E} : dist x y = norm (x - y)

| +-cont => \new UniformMap (ProductUniformSpace \this \this) \this {
| func s => s.1 + s.2
| func-uniform Eu => \case dist-uniform.1 Eu \with {
| inP (eps,eps>0,h) => inP (_, makeUniform (RealAbGroup.half>0 eps>0), _, makeUniform (RealAbGroup.half>0 eps>0),
\lam (inP (_, inP (x,idp), _, inP (y,idp), p)) => \case h (x + y) \with {
| inP (V,EV,g) => inP (_, inP (V, EV, idp), rewrite p \lam {(x',y')} (xx'<eps/2,yy'<eps/2) =>
\have lem : x - x' + (y - y') = x + y - (x' + y') => +-assoc *> pmap (x +) (+-comm *> +-assoc *> pmap (y +) (inv negative_+)) *> inv +-assoc
\in g $transport (_ <) RealAbGroup.half+half$ later (repeat {3} (rewrite norm-dist) $transport (norm __ <= _) lem norm_+) <∘r OrderedAddMonoid.<_+ xx'<eps/2 yy'<eps/2) }) } } | negative-cont => \new MetricMap { | func-dist => rewrite (norm-dist, norm-dist, inv norm_negative, +-comm)$ simplify <=-refl
}
| neighborhood-uniform =>
\have lem {x} {y} : dist 0 (x - y) = dist x y => norm-dist *> inv norm_negative *> pmap norm simplify *> inv norm-dist
\in (\lam Cu => \case dist-uniform.1 Cu \with {
| inP (eps,eps>0,h) => inP (OBall eps 0, OBall-open, OBall-center eps>0, \lam x => \case h x \with {
| inP (U,CU,g) => inP (U, CU, \lam d => g $transport (< _) lem d) }) }, \lam (inP (U,Uo,U0,h)) => \case dist_open.1 Uo U0 \with { | inP (eps,eps>0,p) => dist-uniform.2$ inP (eps, eps>0, \lam x => \case h x \with {
| inP (U,CU,g) => inP (U, CU, \lam d => g $p$ transportInv (< _) lem d)
})
})
| open-top => defaultImpl PrecoverSpace open-top
| open-inter {U} {V} => defaultImpl PrecoverSpace open-inter {_} {U} {V}
| open-Union {S} => defaultImpl PrecoverSpace open-Union {_} {S}
| dist-refl => norm-dist *> pmap norm negative-right *> norm_zro
| dist-symm => norm-dist *> simplify *> norm_negative *> inv norm-dist
| dist-triang => repeat {3} (rewrite norm-dist) $transport (norm __ <= _) simplify norm_+ \default dist x y : Real => norm (x - y) \default norm-dist \as norm-dist-impl {x} {y} : dist x y = norm (x - y) => idp \func IsUnbounded : \Prop => \Pi (B : Nat) -> ∃ (x : E) (B RealAbGroup.< norm x) \lemma norm_BigSum {l : Array E} : norm (BigSum l) <= RealAbGroup.BigSum \lam j => norm (l j) \elim l | nil => transportInv (<= _) norm_zro <=-refl | x :: l => norm_+ <=∘ LinearlyOrderedAbMonoid.<=_+ <=-refl norm_BigSum } \class NormedAbGroup \extends PseudoNormedAbGroup, MetricSpace | norm-ext {x : E} : norm x = (0 : Real) -> x = zro | dist-ext p => fromZero$ norm-ext $inv norm-dist *> p \lemma norm_dist {X : PseudoNormedAbGroup} {x : X} : norm x = dist 0 x => inv norm_negative *> simplify *> inv norm-dist \lemma norm>=0 {X : PseudoNormedAbGroup} {x : X} : 0 <= norm x => rewrite norm_dist dist>=0 \lemma norm_- {X : PseudoNormedAbGroup} {x y : X} : norm (x - y) = norm (y - x) => pmap norm (inv X.negative-isInv) *> norm_negative *> simplify \lemma norm_-left {X : PseudoNormedAbGroup} {x y : X} : norm x - norm y <= norm (x - y) => transport (_ <=) (+-assoc *> pmap (_ +) negative-right *> zro-right)$ LinearlyOrderedAbMonoid.<=_+ (transport (norm __ <= _) (+-assoc *> pmap (x +) negative-left *> zro-right) norm_+) <=-refl

\lemma norm_-right {X : PseudoNormedAbGroup} {x y : X} : norm y - norm x <= norm (x - y)
=> transport (_ <=) norm_- norm_-left

\record UniformNormedAbGroupMap \extends UniformMetricMap, TopAbGroupMap {
\override Dom : PseudoNormedAbGroup
\override Cod : PseudoNormedAbGroup

| func-norm-uniform : ∀ {eps : Real} (0 < eps) ∃ (delta : Real) (0 < delta) ∀ {x : Dom} (norm x < delta -> norm (func x) < eps)
| func-dist-uniform eps>0 => \case func-norm-uniform eps>0 \with {
| inP (delta,delta>0,h) => inP (delta, delta>0, unfold \lam d => rewrite (norm-dist, inv AddGroupHom.func-minus) $h$ rewriteI norm-dist d)
}

\default func-norm-uniform eps>0 => \case dist_open.1 (func-cont OBall-open) $rewrite func-zro$ OBall-center eps>0 \with {
| inP (delta,delta>0,h) => inP (delta, delta>0, \lam d => rewrite (norm_dist, inv func-zro) $h$ rewrite norm_dist in d)
}
}

\record NormedAbGroupMap \extends UniformNormedAbGroupMap, MetricMap {
\override Dom : PseudoNormedAbGroup
\override Cod : PseudoNormedAbGroup

| func-norm {x : Dom} : norm (func x) <= norm x
| func-norm-uniform {eps} eps>0 => inP (eps, eps>0, \lam d => func-norm <∘r d)
| func-dist => unfold \$ rewrite (norm-dist, norm-dist, inv AddGroupHom.func-minus) func-norm
}

\record NormedIsometricMap \extends NormedAbGroupMap, IsometricMap {
\override Dom : PseudoNormedAbGroup
\override Cod : PseudoNormedAbGroup

| func-norm-isometry {x : Dom} : norm (func x) = norm x
| func-norm => Preorder.=_<= func-norm-isometry
| func-isometry => norm-dist *> pmap norm (inv func-minus) *> func-norm-isometry *> inv norm-dist
}

\class CompleteNormedAbGroup \extends NormedAbGroup, CompleteMetricSpace, CompleteTopAbGroup
`