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@affeldt-aist
affeldt-aist committed Nov 10, 2024
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Showing 1 changed file with 57 additions and 79 deletions.
136 changes: 57 additions & 79 deletions theories/topology_theory/order_topology.v
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Original file line number Diff line number Diff line change
Expand Up @@ -224,97 +224,75 @@ End induced_order_topology.
Lemma min_continuous {d} {T : orderTopologicalType d} :
continuous (fun xy : T * T => Order.min xy.1 xy.2).
Proof.
case=> x y; case : (pselect (x = y)).
move=> <- U; rewrite /= min_l // => ux; exists (U, U) => //=.
case=> a b [/= ? ?]; case /orP : (le_total a b) => ?; first by rewrite min_l.
by rewrite min_r.
move=>/eqP; wlog xy : x y / (x < y)%O.
move=> WH /[dup] /lt_total/orP []; first exact: WH.
rewrite eq_sym; move=> yx yNx.
have -> : (fun (xy : T *T ) => Order.min xy.1 xy.2) =
((fun xy => Order.min xy.1 xy.2) \o @swap T T).
apply/funext; case => a b /=; have /orP [? | ?] := le_total a b.
by rewrite min_l // min_r.
by rewrite min_r // min_l.
apply: continuous_comp; first exact: swap_continuous.
by apply: WH.
move=> _ U /=; (rewrite min_l //; last exact: ltW) => Ux.
case : (pselect (exists z, x < z < y)%O).
case=> z xzy; exists (U `&` `]-oo,z[, `]z,+oo[%classic) => //=.
- split; [apply: filterI =>// |]; apply: open_nbhs_nbhs; split.
+ exact: lray_open.
+ by rewrite set_itvE; case/andP: xzy.
+ exact: rray_open.
+ by rewrite set_itvE; case/andP: xzy.
- case=> a b /= [][Ua]; rewrite ?in_itv andbT /= => az ?.
by rewrite min_l //; apply: ltW; apply: (lt_trans az).
move/forallNP => /= xNy.
exists (U `&` `]-oo,y[, `]x,+oo[%classic) => //=.
- split; [apply: filterI =>// |]; apply: open_nbhs_nbhs; split.
+ exact: lray_open.
+ by rewrite set_itvE.
+ exact: rray_open.
+ by rewrite set_itvE.
- case=> a b /= [][Ua]; rewrite ?in_itv andbT /= => ay xb.
rewrite min_l //; rewrite leNgt; have := xNy b; apply: contra_notN.
move=> ba; apply/andP; split => //.
by apply: (lt_trans ba).
case=> x y; have [<- U /=|]:= eqVneq x y.
rewrite min_l// => Ux; exists (U, U) => //= -[a b [Ua Ub]].
by have /orP[?|?]/= := le_total a b; [rewrite min_l|rewrite min_r].
wlog xy : x y / (x < y)%O.
move=> WH /[dup] /lt_total/orP[|yx /eqP/nesym/eqP yNx]; first exact: WH.
rewrite (_ : (fun _ => _) = (fun xy => Order.min xy.1 xy.2) \o @swap T T).
by apply: continuous_comp; [exact: swap_continuous|exact: WH].
apply/funext => -[a b/=]; have /orP[ab|ba] := le_total a b.
- by rewrite min_l // min_r.
- by rewrite min_r // min_l.
move=> _ U /=; rewrite (min_l (ltW xy)) => Ux.
have [[z xzy]|/forallNP/= xNy] := pselect (exists z, x < z < y)%O.
exists (U `&` `]-oo, z[, `]z, +oo[%classic) => /=.
split; [apply: filterI =>//|]; apply: open_nbhs_nbhs.
- by split; [exact: lray_open|rewrite set_itvE; case/andP: xzy].
- by split; [exact: rray_open|rewrite set_itvE; case/andP: xzy].
case=> a b /= [[Ua]]; rewrite !in_itv andbT /= => az zb.
by rewrite min_l// (ltW (lt_trans az _)).
exists (U `&` `]-oo, y[, `]x, +oo[%classic) => /=.
split; [apply: filterI => //|]; apply: open_nbhs_nbhs.
- by split; [exact: lray_open|rewrite set_itvE].
- by split; [exact: rray_open|rewrite set_itvE].
case=> a b /= [[Ua]]; rewrite !in_itv andbT /= => ay xb.
rewrite min_l// leNgt; apply: contra_notN (xNy b) => ba.
by rewrite xb (lt_trans ba).
Qed.

Lemma min_fun_continuous {d} {X : topologicalType} {T : orderTopologicalType d}
(f g : X -> T):
Lemma min_fun_continuous {d} {X : topologicalType} {T : orderTopologicalType d}
(f g : X -> T) :
continuous f -> continuous g -> continuous (f \min g).
Proof.
move=> fc gc z; apply: continuous2_cvg; first apply min_continuous.
exact: fc.
exact: gc.
move=> fc gc z; apply: continuous2_cvg; [|exact: fc|exact: gc].
by apply min_continuous.
Qed.

Lemma max_continuous {d} {T : orderTopologicalType d} :
continuous (fun xy : T * T => Order.max xy.1 xy.2).
Proof.
case=> x y; case : (pselect (x = y)).
move=> <- U; rewrite /= max_r // => ux; exists (U, U) => //=.
case=> a b [/= ? ?]; case /orP : (le_total a b) => ?; first by rewrite max_r.
by rewrite max_l.
move=>/eqP; wlog xy : x y / (x < y)%O.
move=> WH /[dup] /lt_total/orP []; first exact: WH.
rewrite eq_sym; move=> yx yNx.
have -> : (fun (xy : T *T ) => Order.max xy.1 xy.2) =
((fun xy => Order.max xy.1 xy.2) \o @swap T T).
apply/funext; case => a b /=; have /orP [? | ?] := le_total a b.
by rewrite max_r // max_l.
by rewrite max_l // max_r.
apply: continuous_comp; first exact: swap_continuous.
by apply: WH.
move=> _ U /=; (rewrite max_r //; last exact: ltW) => Ux.
case : (pselect (exists z, x < z < y)%O).
case=> z xzy; exists (`]-oo,z[%classic, U `&` `]z,+oo[%classic) => //=.
- split; [|apply: filterI =>//]; apply: open_nbhs_nbhs; split.
+ exact: lray_open.
+ by rewrite set_itvE; case/andP: xzy.
+ exact: rray_open.
+ by rewrite set_itvE; case/andP: xzy.
- case=> a b /= [] + []; rewrite ?in_itv andbT /= => az ? zb.
by rewrite max_r //; apply: ltW; apply: (lt_trans az).
move/forallNP => /= xNy.
exists (`]-oo,y[%classic, U `&` `]x,+oo[%classic) => //=.
- split; [|apply: filterI =>//]; apply: open_nbhs_nbhs; split.
+ exact: lray_open.
+ by rewrite set_itvE.
+ exact: rray_open.
+ by rewrite set_itvE.
- case=> a b /=; rewrite ?in_itv /= andbT => [/=] [ay] [?] xb.
rewrite max_r //; rewrite leNgt; have := xNy b; apply: contra_notN.
move=> ba; apply/andP; split => //.
by apply: (lt_trans ba).
case=> x y; have [<- U|] := eqVneq x y.
rewrite /= max_r // => ux; exists (U, U) => //= -[a b] [/= Ua Ub].
by have /orP[?|?]/= := le_total a b; [rewrite max_r|rewrite max_l].
wlog xy : x y / (x < y)%O.
move=> WH /[dup] /lt_total/orP[|yx /eqP/nesym/eqP yNx]; first exact: WH.
rewrite (_ : (fun _ => _) = (fun xy => Order.max xy.1 xy.2) \o @swap T T).
by apply: continuous_comp; [exact: swap_continuous|exact: WH].
apply/funext => -[a b] /=; have /orP [ab|ba] := le_total a b.
- by rewrite max_r // max_l.
- by rewrite max_l // max_r.
move=> _ U /=; rewrite (max_r (ltW xy)) => Ux.
have [[z xzy]|/forallNP /= xNy] := pselect (exists z, x < z < y)%O.
exists (`]-oo, z[%classic, U `&` `]z, +oo[) => /=.
split; [|apply: filterI =>//]; apply: open_nbhs_nbhs.
- by split; [exact: lray_open|rewrite set_itvE; case/andP: xzy].
- by split; [exact: rray_open|rewrite set_itvE; case/andP: xzy].
case=> a b /= [] + []; rewrite !in_itv andbT /= => az Ub zb.
by rewrite (max_r (ltW (lt_trans az _))).
exists (`]-oo, y[%classic, U `&` `]x, +oo[) => /=.
split; [|apply: filterI => //]; apply: open_nbhs_nbhs.
- by split; [exact: lray_open|rewrite set_itvE].
- by split; [exact: rray_open|rewrite set_itvE].
case=> a b /=; rewrite !in_itv /= andbT => [/=] [ay] [Ub] xb.
rewrite max_r// leNgt; apply: contra_notN (xNy b) => ba.
by rewrite xb (lt_trans ba).
Qed.

Lemma max_fun_continuous {d} {X : topologicalType} {T : orderTopologicalType d}
Lemma max_fun_continuous {d} {X : topologicalType} {T : orderTopologicalType d}
(f g : X -> T):
continuous f -> continuous g -> continuous (f \max g).
Proof.
move=> fc gc z; apply: continuous2_cvg; first apply max_continuous.
exact: fc.
exact: gc.
move=> fc gc z; apply: continuous2_cvg; [|exact: fc|exact: gc].
by apply max_continuous.
Qed.

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