Module mathcomp.classical.mathcomp_extra

Require Import BinPos.
From mathcomp Require choice.
From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.
From mathcomp Require Import finset interval.


Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

MathComp 1.15 additions 

Reserved Notation "f \* g" (at level 40, left associativity).
Reserved Notation "f \- g" (at level 50, left associativity).
Reserved Notation "\- f" (at level 35, f at level 35).

Number Notation positive Pos.of_num_int Pos.to_num_uint : AC_scope.

Notation "f \min g" := (Order.min_fun f g) : function_scope.
Notation "f \max g" := (Order.max_fun f g) : function_scope.

Lemma all_sig2_cond {I : Type} {T : Type} (D : pred I)
   (P Q : I -> T -> Prop) : T ->
    (forall x : I, D x -> {y : T | P x y & Q x y}) ->
  {f : forall x : I, T | forall x : I, D x -> P x (f x) & forall x : I, D x -> Q x (f x)}.

Definition olift aT rT (f : aT -> rT) := Some \o f.

Lemma oapp_comp aT rT sT (f : aT -> rT) (g : rT -> sT) x :
  oapp (g \o f) x =1 (@oapp _ _)^~ x g \o omap f.

Lemma olift_comp aT rT sT (f : aT -> rT) (g : rT -> sT) :
  olift (g \o f) = olift g \o f.

Lemma compA {A B C D : Type} (f : B -> A) (g : C -> B) (h : D -> C) :
  f \o (g \o h) = (f \o g) \o h.

Lemma can_in_pcan [rT aT : Type] (A : {pred aT}) [f : aT -> rT] [g : rT -> aT] :
  {in A, cancel f g} -> {in A, pcancel f (fun y : rT => Some (g y))}.

Lemma pcan_in_inj [rT aT : Type] [A : {pred aT}] [f : aT -> rT] [g : rT -> option aT] :
  {in A, pcancel f g} -> {in A &, injective f}.

Definition pred_oapp T (D : {pred T}) : pred (option T) :=
  [pred x | oapp (mem D) false x].

Lemma ocan_in_comp [A B C : Type] (D : {pred B}) (D' : {pred C})
    [f : B -> option A] [h : C -> option B]
    [f' : A -> B] [h' : B -> C] :
  {homo h : x / x \in D' >-> x \in pred_oapp D} ->
  {in D, ocancel f f'} -> {in D', ocancel h h'} ->
  {in D', ocancel (obind f \o h) (h' \o f')}.

Lemma eqbRL (b1 b2 : bool) : b1 = b2 -> b2 -> b1.

Definition mul_fun T (R : ringType) (f g : T -> R) x := (f x * g x)%R.
Notation "f \* g" := (mul_fun f g) : ring_scope.
Arguments mul_fun {T R} _ _ _ /.

Definition opp_fun T (R : zmodType) (f : T -> R) x := (- f x)%R.
Notation "\- f" := (opp_fun f) : ring_scope.
Arguments opp_fun {T R} _ _ /.

Coercion pair_of_interval T (I : interval T) : itv_bound T * itv_bound T :=
  let: Interval b1 b2 := I in (b1, b2).

Import Order.TTheory GRing.Theory Num.Theory.

Section itv_simp.
Variables (d : unit) (T : porderType d).
Implicit Types (x y : T) (b : bool).

Lemma ltBSide x y b b' :
  ((BSide b x < BSide b' y) = (x < y ?<= if b && ~~ b'))%O.

Lemma leBSide x y b b' :
  ((BSide b x <= BSide b' y) = (x < y ?<= if b' ==> b))%O.

Definition lteBSide := (ltBSide, leBSide).

Let BLeft_ltE x y b : (BSide b x < BLeft y)%O = (x < y)%O.
Let BRight_leE x y b : (BSide b x <= BRight y)%O = (x <= y)%O.
Let BRight_BLeft_leE x y : (BRight x <= BLeft y)%O = (x < y)%O.
Let BLeft_BRight_ltE x y : (BLeft x < BRight y)%O = (x <= y)%O.
Let BRight_BSide_ltE x y b : (BRight x < BSide b y)%O = (x < y)%O.
Let BLeft_BSide_leE x y b : (BLeft x <= BSide b y)%O = (x <= y)%O.
Let BSide_ltE x y b : (BSide b x < BSide b y)%O = (x < y)%O.
Let BSide_leE x y b : (BSide b x <= BSide b y)%O = (x <= y)%O.
Let BInfty_leE a : (a <= BInfty T false)%O
Let BInfty_geE a : (BInfty T true <= a)%O
Let BInfty_le_eqE a : (BInfty T false <= a)%O = (a == BInfty T false).
Let BInfty_ge_eqE a : (a <= BInfty T true)%O = (a == BInfty T true).
Let BInfty_ltE a : (a < BInfty T false)%O = (a != BInfty T false).
Let BInfty_gtE a : (BInfty T true < a)%O = (a != BInfty T true).
Let BInfty_ltF a : (BInfty T false < a)%O = false.
Let BInfty_gtF a : (a < BInfty T true)%O = false.
Let BInfty_BInfty_ltE : (BInfty T true < BInfty T false)%O

Lemma ltBRight_leBLeft (a : itv_bound T) (r : T) :
  (a < BRight r)%O = (a <= BLeft r)%O.
Lemma leBRight_ltBLeft (a : itv_bound T) (r : T) :
  (BRight r <= a)%O = (BLeft r < a)%O.

Definition bnd_simp := (BLeft_ltE, BRight_leE,
  BRight_BLeft_leE, BLeft_BRight_ltE,
  BRight_BSide_ltE, BLeft_BSide_leE, BSide_ltE, BSide_leE,
  BInfty_leE, BInfty_geE, BInfty_BInfty_ltE,
  BInfty_le_eqE, BInfty_ge_eqE, BInfty_ltE, BInfty_gtE, BInfty_ltF, BInfty_gtF,
  @lexx _ T, @ltxx _ T, @eqxx T).

Lemma itv_splitU1 (a : itv_bound T) (x : T) : (a <= BLeft x)%O ->
  Interval a (BRight x) =i [predU1 x & Interval a (BLeft x)].

Lemma itv_split1U (a : itv_bound T) (x : T) : (BRight x <= a)%O ->
  Interval (BLeft x) a =i [predU1 x & Interval (BRight x) a].

End itv_simp.

Definition miditv (R : numDomainType) (i : interval R) : R :=
  match i with
  | Interval (BSide _ a) (BSide _ b) => (a + b) / 2%:R
  | Interval -oo%O (BSide _ b) => b - 1
  | Interval (BSide _ a) +oo%O => a + 1
  | Interval -oo%O +oo%O => 0
  | _ => 0
  end.

Section miditv_lemmas.
Variable R : numFieldType.
Implicit Types i : interval R.

Lemma mem_miditv i : (i.1 < i.2)%O -> miditv i \in i.

Lemma miditv_le_left i b : (i.1 < i.2)%O -> (BSide b (miditv i) <= i.2)%O.

Lemma miditv_ge_right i b : (i.1 < i.2)%O -> (i.1 <= BSide b (miditv i))%O.

End miditv_lemmas.

Section itv_porderType.
Variables (d : unit) (T : orderType d).
Implicit Types (a : itv_bound T) (x y : T) (i j : interval T) (b : bool).

Lemma predC_itvl a : [predC Interval -oo%O a] =i Interval a +oo%O.

Lemma predC_itvr a : [predC Interval a +oo%O] =i Interval -oo%O a.

Lemma predC_itv i : [predC i] =i [predU Interval -oo%O i.1 & Interval i.2 +oo%O].

End itv_porderType.

Lemma sumr_le0 (R : numDomainType) I (r : seq I) (P : pred I) (F : I -> R) :
  (forall i, P i -> F i <= 0)%R -> (\sum_(i <- r | P i) F i <= 0)%R.

Lemma enum_ord0 : enum 'I_0 = [::].

Lemma enum_ordSr n : enum 'I_n.+1 =
  rcons (map (widen_ord (leqnSn _)) (enum 'I_n)) ord_max.

Lemma obindEapp {aT rT} (f : aT -> option rT) : obind f = oapp f None.

Lemma omapEbind {aT rT} (f : aT -> rT) : omap f = obind (olift f).

Lemma omapEapp {aT rT} (f : aT -> rT) : omap f = oapp (olift f) None.

Lemma oappEmap {aT rT} (f : aT -> rT) (y0 : rT) x :
  oapp f y0 x = odflt y0 (omap f x).

Lemma omap_comp aT rT sT (f : aT -> rT) (g : rT -> sT) :
  omap (g \o f) =1 omap g \o omap f.

Lemma oapp_comp_f {aT rT sT} (f : aT -> rT) (g : rT -> sT) (x : rT) :
  oapp (g \o f) (g x) =1 g \o oapp f x.

Lemma can_in_comp [A B C : Type] (D : {pred B}) (D' : {pred C})
    [f : B -> A] [h : C -> B]
    [f' : A -> B] [h' : B -> C] :
  {homo h : x / x \in D' >-> x \in D} ->
  {in D, cancel f f'} -> {in D', cancel h h'} ->
  {in D', cancel (f \o h) (h' \o f')}.

Lemma in_inj_comp A B C (f : B -> A) (h : C -> B) (P : pred B) (Q : pred C) :
  {in P &, injective f} -> {in Q &, injective h} -> {homo h : x / Q x >-> P x} ->
  {in Q &, injective (f \o h)}.

Lemma pcan_in_comp [A B C : Type] (D : {pred B}) (D' : {pred C})
    [f : B -> A] [h : C -> B]
    [f' : A -> option B] [h' : B -> option C] :
  {homo h : x / x \in D' >-> x \in D} ->
  {in D, pcancel f f'} -> {in D', pcancel h h'} ->
  {in D', pcancel (f \o h) (obind h' \o f')}.

Lemma ocan_comp [A B C : Type] [f : B -> option A] [h : C -> option B]
    [f' : A -> B] [h' : B -> C] :
  ocancel f f' -> ocancel h h' -> ocancel (obind f \o h) (h' \o f').

Lemma eqbLR (b1 b2 : bool) : b1 = b2 -> b1 -> b2.

Lemma gtr_opp (R : numDomainType) (r : R) : (0 < r)%R -> (- r < r)%R.

Lemma le_le_trans {disp : unit} {T : porderType disp} (x y z t : T) :
  (z <= x)%O -> (y <= t)%O -> (x <= y)%O -> (z <= t)%O.

Notation eqLHS := (X in (X == _))%pattern.
Notation eqRHS := (X in (_ == X))%pattern.
Notation leLHS := (X in (X <= _)%O)%pattern.
Notation leRHS := (X in (_ <= X)%O)%pattern.
Notation ltLHS := (X in (X < _)%O)%pattern.
Notation ltRHS := (X in (_ < X)%O)%pattern.
Inductive boxed T := Box of T.

Lemma eq_bigl_supp [R : eqType] [idx : R] [op : Monoid.law idx] [I : Type]
  [r : seq I] [P1 : pred I] (P2 : pred I) (F : I -> R) :
  {in [pred x | F x != idx], P1 =1 P2} ->
  \big[op/idx]_(i <- r | P1 i) F i = \big[op/idx]_(i <- r | P2 i) F i.

Lemma perm_big_supp_cond [R : eqType] [idx : R] [op : Monoid.com_law idx] [I : eqType]
  [r s : seq I] [P : pred I] (F : I -> R) :
  perm_eq [seq i <- r | P i && (F i != idx)] [seq i <- s | P i && (F i != idx)] ->
  \big[op/idx]_(i <- r | P i) F i = \big[op/idx]_(i <- s| P i) F i.

Lemma perm_big_supp [R : eqType] [idx : R] [op : Monoid.com_law idx] [I : eqType]
  [r s : seq I] [P : pred I] (F : I -> R) :
  perm_eq [seq i <- r | (F i != idx)] [seq i <- s | (F i != idx)] ->
  \big[op/idx]_(i <- r | P i) F i = \big[op/idx]_(i <- s| P i) F i.

Local Open Scope order_scope.
Local Open Scope ring_scope.
Import GRing.Theory Order.TTheory.

Lemma mulr_ge0_gt0 (R : numDomainType) (x y : R) : 0 <= x -> 0 <= y ->
  (0 < x * y) = (0 < x) && (0 < y).

Section lt_le_gt_ge.
Variable (R : numFieldType).
Implicit Types x y z a b : R.

Lemma splitr x : x = x / 2%:R + x / 2%:R.

Lemma ler_addgt0Pr x y : reflect (forall e, e > 0 -> x <= y + e) (x <= y).

Lemma ler_addgt0Pl x y : reflect (forall e, e > 0 -> x <= e + y) (x <= y).

Lemma in_segment_addgt0Pr x y z :
  reflect (forall e, e > 0 -> y \in `[x - e, z + e]) (y \in `[x, z]).

Lemma in_segment_addgt0Pl x y z :
  reflect (forall e, e > 0 -> y \in `[(- e + x), (e + z)]) (y \in `[x, z]).

Lemma lt_le a b : (forall x, x < a -> x < b) -> a <= b.

Lemma gt_ge a b : (forall x, b < x -> a < x) -> a <= b.

End lt_le_gt_ge.

Lemma itvxx d (T : porderType d) (x : T) : `[x, x] =i pred1 x.

Lemma itvxxP d (T : porderType d) (x y : T) : reflect (y = x) (y \in `[x, x]).

Lemma subset_itv_oo_cc d (T : porderType d) (a b : T) : {subset `]a, b[ <= `[a, b]}.

End of MathComp 1.15 additions 

Reserved Notation "`1- r" (format "`1- r", at level 2).
Reserved Notation "f \^-1" (at level 3, format "f \^-1", left associativity).

Lemma natr1 (R : ringType) (n : nat) : (n%:R + 1 = n.+1%:R :> R)%R.

Lemma nat1r (R : ringType) (n : nat) : (1 + n%:R = n.+1%:R :> R)%R.


Lemma fset_nat_maximum (X : choiceType) (A : {fset X})
    (f : X -> nat) : A != fset0 ->
  (exists i, i \in A /\ forall j, j \in A -> f j <= f i)%nat.

Lemma image_nat_maximum n (f : nat -> nat) :
  (exists i, i <= n /\ forall j, j <= n -> f j <= f i)%N.

Lemma card_fset_sum1 (T : choiceType) (A : {fset T}) :
  #|` A| = (\sum_(i <- A) 1)%N.

Arguments big_rmcond {R idx op I r} P.
Arguments big_rmcond_in {R idx op I r} P.

MathComp > 1.16.0 additions 

Reserved Notation "`1- x" (format "`1- x", at level 2).

Section onem.
Variable R : numDomainType.
Implicit Types r : R.

Definition onem r := 1 - r.
Local Notation "`1- r" := (onem r).

Lemma onem0 : `1-0 = 1

Lemma onem1 : `1-1 = 0

Lemma onemK r : `1-(`1-r) = r.

Lemma add_onemK r : r + `1- r = 1.

Lemma onem_gt0 r : r < 1 -> 0 < `1-r

Lemma onem_ge0 r : r <= 1 -> 0 <= `1-r.

Lemma onem_le1 r : 0 <= r -> `1-r <= 1.

Lemma onem_lt1 r : 0 < r -> `1-r < 1.

Lemma onemX_ge0 r n : 0 <= r -> r <= 1 -> 0 <= `1-(r ^+ n).

Lemma onemX_lt1 r n : 0 < r -> `1-(r ^+ n) < 1.

Lemma onemD r s : `1-(r + s) = `1-r - s.

Lemma onemMr r s : s * `1-r = s - s * r.

Lemma onemM r s : `1-(r * s) = `1-r + `1-s - `1-r * `1-s.

End onem.
Notation "`1- r" := (onem r) : ring_scope.

Lemma onemV (F : numFieldType) (x : F) : x != 0 -> `1-(x^-1) = (x - 1) / x.

Lemma lez_abs2 (a b : int) : 0 <= a -> a <= b -> (`|a| <= `|b|)%N.

Lemma ler_gtP (R : numFieldType) (x y : R) :
  reflect (forall z, z > y -> x <= z) (x <= y).

Lemma ler_ltP (R : numFieldType) (x y : R) :
  reflect (forall z, z < x -> z <= y) (x <= y).

Definition inv_fun T (R : unitRingType) (f : T -> R) x := (f x)^-1%R.
Notation "f \^-1" := (inv_fun f) : ring_scope.
Arguments inv_fun {T R} _ _ /.

Definition bound_side d (T : porderType d) (c : bool) (x : itv_bound T) :=
  if x is BSide c' _ then c == c' else false.

Lemma real_ltr_distlC [R : numDomainType] [x y : R] (e : R) :
  x - y \is Num.real -> (`|x - y| < e) = (x - e < y < x + e).

Definition proj {I} {T : I -> Type} i (f : forall i, T i) := f i.

Section DFunWith.
Variables (I : eqType) (T : I -> Type) (f : forall i, T i).

Definition dfwith i (x : T i) (j : I) : T j :=
  if (i =P j) is ReflectT ij then ecast j (T j) ij x else f j.

Lemma dfwithin i x : dfwith x i = x.

Lemma dfwithout i (x : T i) j : i != j -> dfwith x j = f j.

Variant dfwith_spec i (x : T i) : forall j, T j -> Type :=
  | DFunWithin : dfwith_spec x x
  | DFunWithout j : i != j -> dfwith_spec x (f j).

Lemma dfwithP i (x : T i) (j : I) : dfwith_spec x (dfwith x j).

Lemma projK i (x : T i) : cancel (@dfwith i) (proj i).

End DFunWith.
Arguments dfwith {I T} f i x.

Definition swap (T1 T2 : Type) (x : T1 * T2) := (x.2, x.1).

Lemma ler_sqrt {R : rcfType} (a b : R) :
  (0 <= b -> (Num.sqrt a <= Num.sqrt b) = (a <= b))%R.

Section order_min.
Variables (d : unit) (T : orderType d).
Import Order.
Local Open Scope order_scope.

Lemma lt_min_lt (x y z : T) : (min x z < min y z)%O -> (x < y)%O.

End order_min.

MathComp 2.1 additions 

From mathcomp Require Import poly.

Definition coefE :=
  (coef0, coef1, coefC, coefX, coefXn,
   coefZ, coefMC, coefCM, coefXnM, coefMXn, coefXM, coefMX, coefMNn, coefMn,
   coefN, coefB, coefD,
   coef_cons, coef_Poly, coef_poly,
   coef_deriv, coef_nderivn, coef_derivn, coef_map, coef_sum,
   coef_comp_poly).

Module Export Pdeg2.

Module Export Field.

Section Pdeg2Field.
Variable F : fieldType.
Hypothesis nz2 : 2%:R != 0 :> F.

Variable p : {poly F}.
Hypothesis degp : size p = 3%N.

Let a := p`_2.
Let b := p`_1.
Let c := p`_0.

Let pneq0 : p != 0
Let aneq0 : a != 0.
Let a2neq0 : 2%:R * a != 0
Let sqa2neq0 : (2%:R * a) ^+ 2 != 0

Let aa4 : 4%:R * a * a = (2%:R * a)^+2.

Let splitr (x : F) : x = x / 2%:R + x / 2%:R.

Let pE : p = a *: 'X^2 + b *: 'X + c%:P.

Let delta := b ^+ 2 - 4%:R * a * c.

Lemma deg2_poly_canonical :
  p = a *: (('X + (b / (2%:R * a))%:P)^+2 - (delta / (4%:R * a ^+ 2))%:P).

Variable r : F.
Hypothesis r_sqrt_delta : r ^+ 2 = delta.

Let r1 := (- b - r) / (2%:R * a).
Let r2 := (- b + r) / (2%:R * a).

Lemma deg2_poly_factor : p = a *: ('X - r1%:P) * ('X - r2%:P).

End Pdeg2Field.
End Field.

Module Real.

Section Pdeg2Real.

Variable F : realFieldType.

Section Pdeg2RealConvex.

Variable p : {poly F}.
Hypothesis degp : size p = 3%N.

Let a := p`_2.
Let b := p`_1.
Let c := p`_0.

Hypothesis age0 : 0 <= a.

Let delta := b ^+ 2 - 4%:R * a * c.

Let pneq0 : p != 0
Let aneq0 : a != 0.
Let agt0 : 0 < a
Let a4gt0 : 0 < 4%:R * a

Lemma deg2_poly_min x : p.[- b / (2%:R * a)] <= p.[x].

Lemma deg2_poly_minE : p.[- b / (2%:R * a)] = - delta / (4%:R * a).

Lemma deg2_poly_ge0 : reflect (forall x, 0 <= p.[x]) (delta <= 0).

End Pdeg2RealConvex.

End Pdeg2Real.

Section Pdeg2RealClosed.

Variable F : rcfType.

Section Pdeg2RealClosedConvex.

Variable p : {poly F}.
Hypothesis degp : size p = 3%N.

Let a := p`_2.
Let b := p`_1.
Let c := p`_0.

Let nz2 : 2%:R != 0 :> F

Let delta := b ^+ 2 - 4%:R * a * c.

Let r1 := (- b - Num.sqrt delta) / (2%:R * a).
Let r2 := (- b + Num.sqrt delta) / (2%:R * a).

Lemma deg2_poly_factor : 0 <= delta -> p = a *: ('X - r1%:P) * ('X - r2%:P).

End Pdeg2RealClosedConvex.

End Pdeg2RealClosed.
End Real.

End Pdeg2.

Section Degle2PolyRealConvex.

Variable (F : realFieldType) (p : {poly F}).
Hypothesis degp : (size p <= 3)%N.

Let a := p`_2.
Let b := p`_1.
Let c := p`_0.

Let delta := b ^+ 2 - 4%:R * a * c.

Lemma deg_le2_poly_delta_ge0 : 0 <= a -> (forall x, 0 <= p.[x]) -> delta <= 0.

End Degle2PolyRealConvex.

Section Degle2PolyRealClosedConvex.

Variable (F : rcfType) (p : {poly F}).
Hypothesis degp : (size p <= 3)%N.

Let a := p`_2.
Let b := p`_1.
Let c := p`_0.

Let delta := b ^+ 2 - 4%:R * a * c.

Lemma deg_le2_poly_ge0 : (forall x, 0 <= p.[x]) -> delta <= 0.

End Degle2PolyRealClosedConvex.

Lemma deg_le2_ge0 (F : rcfType) (a b c : F) :
  (forall x, 0 <= a * x ^+ 2 + b * x + c)%R -> (b ^+ 2 - 4%:R * a * c <= 0)%R.

Section dependent_choice_Type.
Context X (R : X -> X -> Prop).

Lemma dependent_choice_Type : (forall x, {y | R x y}) ->
  forall x0, {f | f 0%N = x0 /\ forall n, R (f n) (f n.+1)}.
End dependent_choice_Type.

Section max_min.
Variable R : realFieldType.
Import Num.Theory.

Let nz2 : 2%:R != 0 :> R

Lemma maxr_absE (x y : R) : Num.max x y = (x + y + `|x - y|) / 2%:R.

Lemma minr_absE (x y : R) : Num.min x y = (x + y - `|x - y|) / 2%:R.

End max_min.

Notation trivial := (ltac:(done)).

Section bigmax_seq.
Context d {T : orderType d} {x : T} {I : eqType}.
Variables (r : seq I) (i0 : I) (P : pred I).

Lemma le_bigmax_seq F :
  i0 \in r -> P i0 -> (F i0 <= \big[Order.max/x]_(i <- r | P i) F i)%O.

Lemma bigmax_sup_seq (m : T) (F : I -> T) :
  i0 \in r -> P i0 -> (m <= F i0)%O ->
  (m <= \big[Order.max/x]_(i <- r | P i) F i)%O.

End bigmax_seq.
Arguments le_bigmax_seq {d T} x {I r} i0 P.

Lemma gerBl {R : numDomainType} (x y : R) : 0 <= y -> x - y <= x.

Section normr.
Variable R : realDomainType.

Definition Rnpos : qualifier 0 R := [qualify x : R | x <= 0].
Lemma nposrE x : (x \is Rnpos) = (x <= 0)

Lemma ger0_le_norm :
  {in Num.nneg &, {mono (@Num.Def.normr _ R) : x y / x <= y}}.

Lemma gtr0_le_norm :
  {in Num.pos &, {mono (@Num.Def.normr _ R) : x y / x <= y}}.

Lemma ler0_ge_norm :
  {in Rnpos &, {mono (@Num.Def.normr _ R) : x y / x <= y >-> x >= y}}.

Lemma ltr0_ge_norm :
  {in Num.neg &, {mono (@Num.Def.normr _ R) : x y / x <= y >-> x >= y}}.

End normr.

Lemma leq_ltn_expn m : exists n, (2 ^ n <= m.+1 < 2 ^ n.+1)%N.

Definition monotonous d (T : porderType d) (pT : predType T) (A : pT) (f : T -> T) :=
  {in A &, {mono f : x y / (x <= y)%O}} \/ {in A &, {mono f : x y /~ (x <= y)%O}}.

Section path_lt.
Context d {T : orderType d}.
Implicit Types (a b c : T) (s : seq T).

Lemma last_filterP a (P : pred T) s :
  P a -> P (last a [seq x <- s | P x]).

Lemma path_lt_filter0 a s : path <%O a s -> [seq x <- s | (x < a)%O] = [::].

Lemma path_lt_filterT a s : path <%O a s -> [seq x <- s | (a < x)%O] = s.

Lemma path_lt_head a b s : (a < b)%O -> path <%O b s -> path <%O a s.

Lemma path_lt_last_filter a b c s :
  (a < c)%O -> (c < b)%O -> path <%O a s -> last a s = b ->
  last c [seq x <- s | (c < x)%O] = b.

Lemma path_lt_le_last a s : path <%O a s -> (a <= last a s)%O.

End path_lt.
Arguments last_filterP {d T a} P s.