Require Import
  HoTT.Classes.theory.apartness
  HoTT.Classes.interfaces.abstract_algebra
  HoTT.Classes.interfaces.orders
  HoTT.Classes.theory.rings.
[Loading ML file number_string_notation_plugin.cmxs (using legacy method) ... done]
Require Export
  HoTT.Classes.orders.orders
  HoTT.Classes.orders.maps.

Generalizable Variables R Rlt f.

Section semiring_order.
  Context `{SemiRingOrder R} `{!IsSemiCRing R}.
(*   Add Ring R : (stdlib_semiring_theory R). *)

  #[export] Instance plus_le_embed_l : forall (z : R), OrderEmbedding (+z).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
forall z : R, OrderEmbedding (fun y : R => y + z)
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
forall z : R, OrderEmbedding (fun y : R => y + z)
  intro.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
z: R
OrderEmbedding (fun y : R => y + z) split.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
z: R
OrderPreserving (fun y : R => y + z)
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
z: R
OrderReflecting (fun y : R => y + z)
  -
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
z: R
OrderPreserving (fun y : R => y + z) exact order_preserving_flip.
  -
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
z: R
OrderReflecting (fun y : R => y + z) exact order_reflecting_flip.
  Qed.

  #[export] Instance plus_ordered_cancel_l : forall z, LeftCancellation (+) z.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
forall z : R, LeftCancellation plus z
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
forall z : R, LeftCancellation plus z
  intros z x y E.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
z, x, y: R
E: z + x = z + y
x = y
  apply (antisymmetry (≤)); apply (order_reflecting (z+)); apply eq_le;trivial.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
z, x, y: R
E: z + x = z + y
z + y = z + x
  apply symmetry;trivial.
  Qed.

  #[export] Instance plus_ordered_cancel_r : forall z, RightCancellation (+) z.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
forall z : R, RightCancellation plus z
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
forall z : R, RightCancellation plus z
  intros.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
z: R
RightCancellation plus z apply (right_cancel_from_left (+)).
  Qed.

  Lemma nonneg_plus_le_compat_r x z : 0 ≤ z <-> x ≤ x + z.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, z: R
0 ≤ z <-> x ≤ x + z
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, z: R
0 ≤ z <-> x ≤ x + z
  pattern x at 1.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, z: R
(fun r : R => 0 ≤ z <-> r ≤ x + z) x apply (transport _ (plus_0_r x)).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, z: R
0 ≤ z <-> x + 0 ≤ x + z
  split; intros.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, z: R
X: 0 ≤ z
x + 0 ≤ x + z
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, z: R
X: x + 0 ≤ x + z
0 ≤ z
  -
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, z: R
X: 0 ≤ z
x + 0 ≤ x + z apply (order_preserving _).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, z: R
X: 0 ≤ z
0 ≤ z trivial.
  -
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, z: R
X: x + 0 ≤ x + z
0 ≤ z apply (order_reflecting (x+)).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, z: R
X: x + 0 ≤ x + z
x + 0 ≤ x + z trivial.
  Qed.

  Lemma nonneg_plus_le_compat_l x z : 0 ≤ z <-> x ≤ z + x.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, z: R
0 ≤ z <-> x ≤ z + x
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, z: R
0 ≤ z <-> x ≤ z + x
  rewrite (commutativity (f:=plus)).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, z: R
0 ≤ z <-> x ≤ x + z
  apply nonneg_plus_le_compat_r.
  Qed.

  Lemma plus_le_compat x₁ y₁ x₂ y₂ : x₁ ≤ y₁ -> x₂ ≤ y₂ -> x₁ + x₂ ≤ y₁ + y₂.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
x₁ ≤ y₁ -> x₂ ≤ y₂ -> x₁ + x₂ ≤ y₁ + y₂
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
x₁ ≤ y₁ -> x₂ ≤ y₂ -> x₁ + x₂ ≤ y₁ + y₂
  intros E1 E2.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
E1: x₁ ≤ y₁
E2: x₂ ≤ y₂
x₁ + x₂ ≤ y₁ + y₂
  transitivity (y₁ + x₂).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
E1: x₁ ≤ y₁
E2: x₂ ≤ y₂
x₁ + x₂ ≤ y₁ + x₂
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
E1: x₁ ≤ y₁
E2: x₂ ≤ y₂
y₁ + x₂ ≤ y₁ + y₂
  -
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
E1: x₁ ≤ y₁
E2: x₂ ≤ y₂
x₁ + x₂ ≤ y₁ + x₂ apply (order_preserving (+ x₂));trivial.
  -
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
E1: x₁ ≤ y₁
E2: x₂ ≤ y₂
y₁ + x₂ ≤ y₁ + y₂ apply (order_preserving (y₁ +));trivial.
  Qed.

  Lemma plus_le_compat_r x y z : 0 ≤ z -> x ≤ y -> x ≤ y + z.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
0 ≤ z -> x ≤ y -> x ≤ y + z
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
0 ≤ z -> x ≤ y -> x ≤ y + z
  intros.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
X: 0 ≤ z
X0: x ≤ y
x ≤ y + z rewrite <-(plus_0_r x).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
X: 0 ≤ z
X0: x ≤ y
x + 0 ≤ y + z
  apply plus_le_compat;trivial.
  Qed.

  Lemma plus_le_compat_l x y z : 0 ≤ z -> x ≤ y -> x ≤ z + y.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
0 ≤ z -> x ≤ y -> x ≤ z + y
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
0 ≤ z -> x ≤ y -> x ≤ z + y
  rewrite (commutativity (f:=plus)).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
0 ≤ z -> x ≤ y -> x ≤ y + z
  apply plus_le_compat_r.
  Qed.

  Lemma nonpos_plus_compat x y : x ≤ 0 -> y ≤ 0 -> x + y ≤ 0.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
x ≤ 0 -> y ≤ 0 -> x + y ≤ 0
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
x ≤ 0 -> y ≤ 0 -> x + y ≤ 0
  intros.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x ≤ 0
X0: y ≤ 0
x + y ≤ 0 rewrite <-(plus_0_r 0).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x ≤ 0
X0: y ≤ 0
x + y ≤ 0 + 0
  apply plus_le_compat;trivial.
  Qed.

  Instance nonneg_plus_compat (x y : R)
    : PropHolds (0 ≤ x) -> PropHolds (0 ≤ y) -> PropHolds (0 ≤ x + y).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
PropHolds (0 ≤ x) ->
PropHolds (0 ≤ y) -> PropHolds (0 ≤ x + y)
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
PropHolds (0 ≤ x) ->
PropHolds (0 ≤ y) -> PropHolds (0 ≤ x + y)
  intros.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
X: PropHolds (0 ≤ x)
X0: PropHolds (0 ≤ y)
PropHolds (0 ≤ x + y)
  apply plus_le_compat_l;trivial.
  Qed.

  Lemma decompose_le {x y} : x ≤ y -> exists z, 0 ≤ z /\ y = x + z.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
x ≤ y -> {z : R & ((0 ≤ z) * (y = plus x z))%type}
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
x ≤ y -> {z : R & ((0 ≤ z) * (y = plus x z))%type}
  intros E.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
E: x ≤ y
{z : R & ((0 ≤ z) * (y = plus x z))%type}
  destruct (srorder_partial_minus x y E) as [z Ez].
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
E: x ≤ y
z: R
Ez: y = x + z
{z : R & ((0 ≤ z) * (y = plus x z))%type}
  exists z.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
E: x ≤ y
z: R
Ez: y = x + z
((0 ≤ z) * (y = plus x z))%type split; [| trivial].
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
E: x ≤ y
z: R
Ez: y = x + z
0 ≤ z
  apply (order_reflecting (x+)).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
E: x ≤ y
z: R
Ez: y = x + z
x + 0 ≤ x + z
  rewrite plus_0_r, <-Ez.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
E: x ≤ y
z: R
Ez: y = x + z
x ≤ y
  trivial.
  Qed.

  Lemma compose_le x y z : 0 ≤ z -> y = x + z -> x ≤ y.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
0 ≤ z -> y = x + z -> x ≤ y
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
0 ≤ z -> y = x + z -> x ≤ y
  intros E1 E2.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
E1: 0 ≤ z
E2: y = x + z
x ≤ y
  rewrite E2.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
E1: 0 ≤ z
E2: y = x + z
x ≤ x + z
  apply nonneg_plus_le_compat_r.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
E1: 0 ≤ z
E2: y = x + z
0 ≤ z
  trivial.
  Qed.

  #[export] Instance nonneg_mult_le_l : forall (z : R), PropHolds (0 ≤ z) ->
    OrderPreserving (z *.).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (0 ≤ z) -> OrderPreserving (mult z)
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (0 ≤ z) -> OrderPreserving (mult z)
  intros z E.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
z: R
E: PropHolds (0 ≤ z)
OrderPreserving (mult z)
  repeat (split; try exact _).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
z: R
E: PropHolds (0 ≤ z)
OrderPreserving (mult z)
  intros x y F.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
z: R
E: PropHolds (0 ≤ z)
x, y: R
F: x ≤ y
z * x ≤ z * y
  destruct (decompose_le F) as [a [Ea1 Ea2]].
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
z: R
E: PropHolds (0 ≤ z)
x, y: R
F: x ≤ y
a: R
Ea1: 0 ≤ a
Ea2: y = x + a
z * x ≤ z * y
  rewrite Ea2, plus_mult_distr_l.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
z: R
E: PropHolds (0 ≤ z)
x, y: R
F: x ≤ y
a: R
Ea1: 0 ≤ a
Ea2: y = x + a
z * x ≤ z * x + z * a
  apply nonneg_plus_le_compat_r.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
z: R
E: PropHolds (0 ≤ z)
x, y: R
F: x ≤ y
a: R
Ea1: 0 ≤ a
Ea2: y = x + a
0 ≤ z * a
  apply nonneg_mult_compat;trivial.
  Qed.

  #[export] Instance nonneg_mult_le_r : forall (z : R), PropHolds (0 ≤ z) ->
    OrderPreserving (.* z).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (0 ≤ z) ->
OrderPreserving (fun y : R => y * z)
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (0 ≤ z) ->
OrderPreserving (fun y : R => y * z)
  intros.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
z: R
X: PropHolds (0 ≤ z)
OrderPreserving (fun y : R => y * z) exact order_preserving_flip.
  Qed.

  Lemma mult_le_compat x₁ y₁ x₂ y₂ :
    0 ≤ x₁ -> 0 ≤ x₂ -> x₁ ≤ y₁ -> x₂ ≤ y₂ -> x₁ * x₂ ≤ y₁ * y₂.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
0 ≤ x₁ ->
0 ≤ x₂ -> x₁ ≤ y₁ -> x₂ ≤ y₂ -> x₁ * x₂ ≤ y₁ * y₂
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
0 ≤ x₁ ->
0 ≤ x₂ -> x₁ ≤ y₁ -> x₂ ≤ y₂ -> x₁ * x₂ ≤ y₁ * y₂
  intros Ex₁ Ey₁ E1 E2.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
Ex₁: 0 ≤ x₁
Ey₁: 0 ≤ x₂
E1: x₁ ≤ y₁
E2: x₂ ≤ y₂
x₁ * x₂ ≤ y₁ * y₂
  transitivity (y₁ * x₂).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
Ex₁: 0 ≤ x₁
Ey₁: 0 ≤ x₂
E1: x₁ ≤ y₁
E2: x₂ ≤ y₂
x₁ * x₂ ≤ y₁ * x₂
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
Ex₁: 0 ≤ x₁
Ey₁: 0 ≤ x₂
E1: x₁ ≤ y₁
E2: x₂ ≤ y₂
y₁ * x₂ ≤ y₁ * y₂
  -
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
Ex₁: 0 ≤ x₁
Ey₁: 0 ≤ x₂
E1: x₁ ≤ y₁
E2: x₂ ≤ y₂
x₁ * x₂ ≤ y₁ * x₂ apply (order_preserving_flip_nonneg (.*.) x₂);trivial.
  -
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
Ex₁: 0 ≤ x₁
Ey₁: 0 ≤ x₂
E1: x₁ ≤ y₁
E2: x₂ ≤ y₂
y₁ * x₂ ≤ y₁ * y₂ apply (order_preserving_nonneg (.*.) y₁); [| trivial].
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
Ex₁: 0 ≤ x₁
Ey₁: 0 ≤ x₂
E1: x₁ ≤ y₁
E2: x₂ ≤ y₂
0 ≤ y₁
    transitivity x₁;trivial.
  Qed.

  Lemma ge_1_mult_le_compat_r x y z : 1 ≤ z -> 0 ≤ y -> x ≤ y -> x ≤ y * z.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
1 ≤ z -> 0 ≤ y -> x ≤ y -> x ≤ y * z
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
1 ≤ z -> 0 ≤ y -> x ≤ y -> x ≤ y * z
  intros.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
X: 1 ≤ z
X0: 0 ≤ y
X1: x ≤ y
x ≤ y * z
  transitivity y; [trivial |].
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
X: 1 ≤ z
X0: 0 ≤ y
X1: x ≤ y
y ≤ y * z
  pattern y at 1;apply (transport _ (mult_1_r y)).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
X: 1 ≤ z
X0: 0 ≤ y
X1: x ≤ y
y * 1 ≤ y * z
  apply (order_preserving_nonneg (.*.) y);trivial.
  Qed.

  Lemma ge_1_mult_le_compat_l x y z : 1 ≤ z -> 0 ≤ y -> x ≤ y -> x ≤ z * y.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
1 ≤ z -> 0 ≤ y -> x ≤ y -> x ≤ z * y
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
1 ≤ z -> 0 ≤ y -> x ≤ y -> x ≤ z * y
  rewrite (commutativity (f:=mult)).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
1 ≤ z -> 0 ≤ y -> x ≤ y -> x ≤ y * z
  apply ge_1_mult_le_compat_r.
  Qed.

  Lemma flip_nonpos_mult_l x y z : z ≤ 0 -> x ≤ y -> z * y ≤ z * x.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
z ≤ 0 -> x ≤ y -> z * y ≤ z * x
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
z ≤ 0 -> x ≤ y -> z * y ≤ z * x
  intros Ez Exy.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
Ez: z ≤ 0
Exy: x ≤ y
z * y ≤ z * x
  destruct (decompose_le Ez) as [a [Ea1 Ea2]], (decompose_le Exy) as [b [Eb1 Eb2]].
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
Ez: z ≤ 0
Exy: x ≤ y
a: R
Ea1: 0 ≤ a
Ea2: 0 = z + a
b: R
Eb1: 0 ≤ b
Eb2: y = x + b
z * y ≤ z * x
  rewrite Eb2.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
Ez: z ≤ 0
Exy: x ≤ y
a: R
Ea1: 0 ≤ a
Ea2: 0 = z + a
b: R
Eb1: 0 ≤ b
Eb2: y = x + b
z * (x + b) ≤ z * x
  apply compose_le with (a * b).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
Ez: z ≤ 0
Exy: x ≤ y
a: R
Ea1: 0 ≤ a
Ea2: 0 = z + a
b: R
Eb1: 0 ≤ b
Eb2: y = x + b
0 ≤ a * b
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
Ez: z ≤ 0
Exy: x ≤ y
a: R
Ea1: 0 ≤ a
Ea2: 0 = z + a
b: R
Eb1: 0 ≤ b
Eb2: y = x + b
z * x = z * (x + b) + a * b
  -
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
Ez: z ≤ 0
Exy: x ≤ y
a: R
Ea1: 0 ≤ a
Ea2: 0 = z + a
b: R
Eb1: 0 ≤ b
Eb2: y = x + b
0 ≤ a * b apply nonneg_mult_compat;trivial.
  -
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
Ez: z ≤ 0
Exy: x ≤ y
a: R
Ea1: 0 ≤ a
Ea2: 0 = z + a
b: R
Eb1: 0 ≤ b
Eb2: y = x + b
z * x = z * (x + b) + a * b transitivity (z * x + (z + a) * b).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
Ez: z ≤ 0
Exy: x ≤ y
a: R
Ea1: 0 ≤ a
Ea2: 0 = z + a
b: R
Eb1: 0 ≤ b
Eb2: y = x + b
z * x = z * x + (z + a) * b
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
Ez: z ≤ 0
Exy: x ≤ y
a: R
Ea1: 0 ≤ a
Ea2: 0 = z + a
b: R
Eb1: 0 ≤ b
Eb2: y = x + b
z * x + (z + a) * b = z * (x + b) + a * b
    +
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
Ez: z ≤ 0
Exy: x ≤ y
a: R
Ea1: 0 ≤ a
Ea2: 0 = z + a
b: R
Eb1: 0 ≤ b
Eb2: y = x + b
z * x = z * x + (z + a) * b rewrite <-Ea2.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
Ez: z ≤ 0
Exy: x ≤ y
a: R
Ea1: 0 ≤ a
Ea2: 0 = z + a
b: R
Eb1: 0 ≤ b
Eb2: y = x + b
z * x = z * x + 0 * b rewrite mult_0_l,plus_0_r.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
Ez: z ≤ 0
Exy: x ≤ y
a: R
Ea1: 0 ≤ a
Ea2: 0 = z + a
b: R
Eb1: 0 ≤ b
Eb2: y = x + b
z * x = z * x reflexivity.
    +
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
Ez: z ≤ 0
Exy: x ≤ y
a: R
Ea1: 0 ≤ a
Ea2: 0 = z + a
b: R
Eb1: 0 ≤ b
Eb2: y = x + b
z * x + (z + a) * b = z * (x + b) + a * b rewrite plus_mult_distr_r,plus_mult_distr_l.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
Ez: z ≤ 0
Exy: x ≤ y
a: R
Ea1: 0 ≤ a
Ea2: 0 = z + a
b: R
Eb1: 0 ≤ b
Eb2: y = x + b
z * x + (z * b + a * b) = z * x + z * b + a * b
      apply associativity.
  Qed.

  Lemma flip_nonpos_mult_r x y z : z ≤ 0 -> x ≤ y -> y * z ≤ x * z.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
z ≤ 0 -> x ≤ y -> y * z ≤ x * z
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
z ≤ 0 -> x ≤ y -> y * z ≤ x * z
  rewrite 2!(commutativity _ z).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y, z: R
z ≤ 0 -> x ≤ y -> z * y ≤ z * x
  apply flip_nonpos_mult_l.
  Qed.

  Lemma nonpos_mult x y : x ≤ 0 -> y ≤ 0 -> 0 ≤ x * y.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
x ≤ 0 -> y ≤ 0 -> 0 ≤ x * y
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
x ≤ 0 -> y ≤ 0 -> 0 ≤ x * y
  intros.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x ≤ 0
X0: y ≤ 0
0 ≤ x * y
  rewrite <-(mult_0_r x).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x ≤ 0
X0: y ≤ 0
x * 0 ≤ x * y
  apply flip_nonpos_mult_l;trivial.
  Qed.

  Lemma nonpos_nonneg_mult x y : x ≤ 0 -> 0 ≤ y -> x * y ≤ 0.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
x ≤ 0 -> 0 ≤ y -> x * y ≤ 0
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
x ≤ 0 -> 0 ≤ y -> x * y ≤ 0
  intros.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x ≤ 0
X0: 0 ≤ y
x * y ≤ 0
  rewrite <-(mult_0_r x).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x ≤ 0
X0: 0 ≤ y
x * y ≤ x * 0
  apply flip_nonpos_mult_l;trivial.
  Qed.

  Lemma nonneg_nonpos_mult x y : 0 ≤ x -> y ≤ 0 -> x * y ≤ 0.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
0 ≤ x -> y ≤ 0 -> x * y ≤ 0
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
0 ≤ x -> y ≤ 0 -> x * y ≤ 0
  intros.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 ≤ x
X0: y ≤ 0
x * y ≤ 0
  rewrite (commutativity (f:=mult)).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 ≤ x
X0: y ≤ 0
y * x ≤ 0
  apply nonpos_nonneg_mult;trivial.
  Qed.
End semiring_order.

(* Due to bug #2528 *)
#[export]
Hint Extern 7 (PropHolds (0 ≤ _ + _)) =>
  eapply @nonneg_plus_compat : typeclass_instances.

Section strict_semiring_order.
  Context `{IsSemiCRing R} `{!StrictSemiRingOrder Rlt}.
(*   Add Ring Rs : (stdlib_semiring_theory R). *)

  #[export] Instance plus_lt_embed : forall (z : R), StrictOrderEmbedding (+z).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
forall z : R,
StrictOrderEmbedding (fun y : R => y + z)
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
forall z : R,
StrictOrderEmbedding (fun y : R => y + z)
  intro.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
z: R
StrictOrderEmbedding (fun y : R => y + z) split.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
z: R
StrictlyOrderPreserving (fun y : R => y + z)
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
z: R
StrictlyOrderReflecting (fun y : R => y + z)
  -
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
z: R
StrictlyOrderPreserving (fun y : R => y + z) exact strictly_order_preserving_flip.
  -
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
z: R
StrictlyOrderReflecting (fun y : R => y + z) exact strictly_order_reflecting_flip.
  Qed.

  Lemma pos_plus_lt_compat_r x z : 0 < z <-> x < x + z.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, z: R
0 < z <-> x < x + z
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, z: R
0 < z <-> x < x + z
  pattern x at 1;apply (transport _ (plus_0_r x)).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, z: R
0 < z <-> x + 0 < x + z
  split; intros.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, z: R
X: 0 < z
x + 0 < x + z
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, z: R
X: x + 0 < x + z
0 < z
  -
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, z: R
X: 0 < z
x + 0 < x + z apply (strictly_order_preserving _);trivial.
  -
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, z: R
X: x + 0 < x + z
0 < z apply (strictly_order_reflecting (x+));trivial.
  Qed.

  Lemma pos_plus_lt_compat_l x z : 0 < z -> x < z + x.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, z: R
0 < z -> x < z + x
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, z: R
0 < z -> x < z + x
  rewrite (commutativity (f:=plus)).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, z: R
0 < z -> x < x + z
  apply pos_plus_lt_compat_r.
  Qed.

  Lemma plus_lt_compat x₁ y₁ x₂ y₂ : x₁ < y₁ -> x₂ < y₂ -> x₁ + x₂ < y₁ + y₂.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x₁, y₁, x₂, y₂: R
x₁ < y₁ -> x₂ < y₂ -> x₁ + x₂ < y₁ + y₂
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x₁, y₁, x₂, y₂: R
x₁ < y₁ -> x₂ < y₂ -> x₁ + x₂ < y₁ + y₂
  intros E1 E2.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x₁, y₁, x₂, y₂: R
E1: x₁ < y₁
E2: x₂ < y₂
x₁ + x₂ < y₁ + y₂
  transitivity (y₁ + x₂).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x₁, y₁, x₂, y₂: R
E1: x₁ < y₁
E2: x₂ < y₂
x₁ + x₂ < y₁ + x₂
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x₁, y₁, x₂, y₂: R
E1: x₁ < y₁
E2: x₂ < y₂
y₁ + x₂ < y₁ + y₂
  -
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x₁, y₁, x₂, y₂: R
E1: x₁ < y₁
E2: x₂ < y₂
x₁ + x₂ < y₁ + x₂ apply (strictly_order_preserving (+ x₂));trivial.
  -
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x₁, y₁, x₂, y₂: R
E1: x₁ < y₁
E2: x₂ < y₂
y₁ + x₂ < y₁ + y₂ apply (strictly_order_preserving (y₁ +));trivial.
  Qed.

  Lemma plus_lt_compat_r x y z : 0 < z -> x < y -> x < y + z.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
0 < z -> x < y -> x < y + z
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
0 < z -> x < y -> x < y + z
  intros.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
X: 0 < z
X0: x < y
x < y + z rewrite <-(plus_0_r x).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
X: 0 < z
X0: x < y
x + 0 < y + z apply plus_lt_compat;trivial.
  Qed.

  Lemma plus_lt_compat_l x y z : 0 < z -> x < y -> x < z + y.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
0 < z -> x < y -> x < z + y
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
0 < z -> x < y -> x < z + y
  rewrite (commutativity (f:=plus)).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
0 < z -> x < y -> x < y + z
  apply plus_lt_compat_r.
  Qed.

  Lemma neg_plus_compat x y : x < 0 -> y < 0 -> x + y < 0.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
x < 0 -> y < 0 -> x + y < 0
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
x < 0 -> y < 0 -> x + y < 0
  intros.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
X: x < 0
X0: y < 0
x + y < 0 rewrite <-(plus_0_r 0).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
X: x < 0
X0: y < 0
x + y < 0 + 0
  apply plus_lt_compat;trivial.
  Qed.

  Instance pos_plus_compat (x y : R)
    : PropHolds (0 < x) -> PropHolds (0 < y) -> PropHolds (0 < x + y).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
PropHolds (0 < x) ->
PropHolds (0 < y) -> PropHolds (0 < x + y)
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
PropHolds (0 < x) ->
PropHolds (0 < y) -> PropHolds (0 < x + y)
  intros.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
X: PropHolds (0 < x)
X0: PropHolds (0 < y)
PropHolds (0 < x + y) apply plus_lt_compat_l;trivial.
  Qed.

  Lemma compose_lt x y z : 0 < z -> y = x + z -> x < y.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
0 < z -> y = x + z -> x < y
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
0 < z -> y = x + z -> x < y
  intros E1 E2.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
E1: 0 < z
E2: y = x + z
x < y
  rewrite E2.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
E1: 0 < z
E2: y = x + z
x < x + z
  apply pos_plus_lt_compat_r;trivial.
  Qed.

  Lemma decompose_lt {x y} : x < y -> exists z, 0 < z /\ y = x + z.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
x < y -> {z : R & ((0 < z) * (y = plus x z))%type}
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
x < y -> {z : R & ((0 < z) * (y = plus x z))%type}
  intros E.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
E: x < y
{z : R & ((0 < z) * (y = plus x z))%type}
  destruct (strict_srorder_partial_minus x y E) as [z Ez].
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
E: x < y
z: R
Ez: y = x + z
{z : R & ((0 < z) * (y = plus x z))%type}
  exists z.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
E: x < y
z: R
Ez: y = x + z
((0 < z) * (y = plus x z))%type split; [| trivial].
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
E: x < y
z: R
Ez: y = x + z
0 < z
  apply (strictly_order_reflecting (x+)).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
E: x < y
z: R
Ez: y = x + z
x + 0 < x + z
  rewrite <-Ez, rings.plus_0_r.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
E: x < y
z: R
Ez: y = x + z
x < y trivial.
  Qed.

  #[export] Instance pos_mult_lt_l : forall (z : R), PropHolds (0 < z) ->
    StrictlyOrderPreserving (z *.).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
forall z : R,
PropHolds (0 < z) -> StrictlyOrderPreserving (mult z)
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
forall z : R,
PropHolds (0 < z) -> StrictlyOrderPreserving (mult z)
  intros z E x y F.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
z: R
E: PropHolds (0 < z)
x, y: R
F: x < y
z * x < z * y
  destruct (decompose_lt F) as [a [Ea1 Ea2]].
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
z: R
E: PropHolds (0 < z)
x, y: R
F: x < y
a: R
Ea1: 0 < a
Ea2: y = x + a
z * x < z * y
  rewrite Ea2, plus_mult_distr_l.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
z: R
E: PropHolds (0 < z)
x, y: R
F: x < y
a: R
Ea1: 0 < a
Ea2: y = x + a
z * x < z * x + z * a
  apply pos_plus_lt_compat_r.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
z: R
E: PropHolds (0 < z)
x, y: R
F: x < y
a: R
Ea1: 0 < a
Ea2: y = x + a
0 < z * a
  apply pos_mult_compat;trivial.
  Qed.

  #[export] Instance pos_mult_lt_r : forall (z : R), PropHolds (0 < z) ->
    StrictlyOrderPreserving (.* z).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
forall z : R,
PropHolds (0 < z) ->
StrictlyOrderPreserving (fun y : R => y * z)
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
forall z : R,
PropHolds (0 < z) ->
StrictlyOrderPreserving (fun y : R => y * z)
  intros.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
z: R
X: PropHolds (0 < z)
StrictlyOrderPreserving (fun y : R => y * z) exact strictly_order_preserving_flip.
  Qed.

  Lemma mult_lt_compat x₁ y₁ x₂ y₂ :
    0 < x₁ -> 0 < x₂ -> x₁ < y₁ -> x₂ < y₂ -> x₁ * x₂ < y₁ * y₂.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x₁, y₁, x₂, y₂: R
0 < x₁ ->
0 < x₂ -> x₁ < y₁ -> x₂ < y₂ -> x₁ * x₂ < y₁ * y₂
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x₁, y₁, x₂, y₂: R
0 < x₁ ->
0 < x₂ -> x₁ < y₁ -> x₂ < y₂ -> x₁ * x₂ < y₁ * y₂
  intros Ex₁ Ey₁ E1 E2.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x₁, y₁, x₂, y₂: R
Ex₁: 0 < x₁
Ey₁: 0 < x₂
E1: x₁ < y₁
E2: x₂ < y₂
x₁ * x₂ < y₁ * y₂
  transitivity (y₁ * x₂).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x₁, y₁, x₂, y₂: R
Ex₁: 0 < x₁
Ey₁: 0 < x₂
E1: x₁ < y₁
E2: x₂ < y₂
x₁ * x₂ < y₁ * x₂
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x₁, y₁, x₂, y₂: R
Ex₁: 0 < x₁
Ey₁: 0 < x₂
E1: x₁ < y₁
E2: x₂ < y₂
y₁ * x₂ < y₁ * y₂
  -
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x₁, y₁, x₂, y₂: R
Ex₁: 0 < x₁
Ey₁: 0 < x₂
E1: x₁ < y₁
E2: x₂ < y₂
x₁ * x₂ < y₁ * x₂ apply (strictly_order_preserving_flip_pos (.*.) x₂);trivial.
  -
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x₁, y₁, x₂, y₂: R
Ex₁: 0 < x₁
Ey₁: 0 < x₂
E1: x₁ < y₁
E2: x₂ < y₂
y₁ * x₂ < y₁ * y₂ apply (strictly_order_preserving_pos (.*.) y₁); [| trivial ].
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x₁, y₁, x₂, y₂: R
Ex₁: 0 < x₁
Ey₁: 0 < x₂
E1: x₁ < y₁
E2: x₂ < y₂
0 < y₁
    transitivity x₁;trivial.
  Qed.

  Lemma gt_1_mult_lt_compat_r x y z : 1 < z -> 0 < y -> x < y -> x < y * z.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
1 < z -> 0 < y -> x < y -> x < y * z
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
1 < z -> 0 < y -> x < y -> x < y * z
  intros.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
X: 1 < z
X0: 0 < y
X1: x < y
x < y * z
  transitivity y; [ trivial |].
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
X: 1 < z
X0: 0 < y
X1: x < y
y < y * z
  pattern y at 1;apply (transport _ (mult_1_r y)).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
X: 1 < z
X0: 0 < y
X1: x < y
y * 1 < y * z
  apply (strictly_order_preserving_pos (.*.) y);trivial.
  Qed.

  Lemma gt_1_mult_lt_compat_l x y z : 1 < z -> 0 < y -> x < y -> x < z * y.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
1 < z -> 0 < y -> x < y -> x < z * y
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
1 < z -> 0 < y -> x < y -> x < z * y
  rewrite (commutativity (f:=mult)).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
1 < z -> 0 < y -> x < y -> x < y * z
  apply gt_1_mult_lt_compat_r.
  Qed.

  Lemma flip_neg_mult_l x y z : z < 0 -> x < y -> z * y < z * x.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
z < 0 -> x < y -> z * y < z * x
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
z < 0 -> x < y -> z * y < z * x
  intros Ez Exy.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
Ez: z < 0
Exy: x < y
z * y < z * x
  destruct (decompose_lt Ez) as [a [Ea1 Ea2]], (decompose_lt Exy) as [b [Eb1 Eb2]].
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
Ez: z < 0
Exy: x < y
a: R
Ea1: 0 < a
Ea2: 0 = z + a
b: R
Eb1: 0 < b
Eb2: y = x + b
z * y < z * x
  rewrite Eb2.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
Ez: z < 0
Exy: x < y
a: R
Ea1: 0 < a
Ea2: 0 = z + a
b: R
Eb1: 0 < b
Eb2: y = x + b
z * (x + b) < z * x
  apply compose_lt with (a * b).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
Ez: z < 0
Exy: x < y
a: R
Ea1: 0 < a
Ea2: 0 = z + a
b: R
Eb1: 0 < b
Eb2: y = x + b
0 < a * b
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
Ez: z < 0
Exy: x < y
a: R
Ea1: 0 < a
Ea2: 0 = z + a
b: R
Eb1: 0 < b
Eb2: y = x + b
z * x = z * (x + b) + a * b
  -
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
Ez: z < 0
Exy: x < y
a: R
Ea1: 0 < a
Ea2: 0 = z + a
b: R
Eb1: 0 < b
Eb2: y = x + b
0 < a * b apply pos_mult_compat;trivial.
  -
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
Ez: z < 0
Exy: x < y
a: R
Ea1: 0 < a
Ea2: 0 = z + a
b: R
Eb1: 0 < b
Eb2: y = x + b
z * x = z * (x + b) + a * b transitivity (z * x + (z + a) * b).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
Ez: z < 0
Exy: x < y
a: R
Ea1: 0 < a
Ea2: 0 = z + a
b: R
Eb1: 0 < b
Eb2: y = x + b
z * x = z * x + (z + a) * b
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
Ez: z < 0
Exy: x < y
a: R
Ea1: 0 < a
Ea2: 0 = z + a
b: R
Eb1: 0 < b
Eb2: y = x + b
z * x + (z + a) * b = z * (x + b) + a * b
    +
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
Ez: z < 0
Exy: x < y
a: R
Ea1: 0 < a
Ea2: 0 = z + a
b: R
Eb1: 0 < b
Eb2: y = x + b
z * x = z * x + (z + a) * b rewrite <-Ea2.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
Ez: z < 0
Exy: x < y
a: R
Ea1: 0 < a
Ea2: 0 = z + a
b: R
Eb1: 0 < b
Eb2: y = x + b
z * x = z * x + 0 * b rewrite mult_0_l,plus_0_r;reflexivity.
    +
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
Ez: z < 0
Exy: x < y
a: R
Ea1: 0 < a
Ea2: 0 = z + a
b: R
Eb1: 0 < b
Eb2: y = x + b
z * x + (z + a) * b = z * (x + b) + a * b rewrite plus_mult_distr_r,plus_mult_distr_l.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
Ez: z < 0
Exy: x < y
a: R
Ea1: 0 < a
Ea2: 0 = z + a
b: R
Eb1: 0 < b
Eb2: y = x + b
z * x + (z * b + a * b) = z * x + z * b + a * b
      apply associativity.
  Qed.

  Lemma flip_neg_mult_r x y z : z < 0 -> x < y -> y * z < x * z.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
z < 0 -> x < y -> y * z < x * z
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
z < 0 -> x < y -> y * z < x * z
  rewrite 2!(commutativity _ z).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y, z: R
z < 0 -> x < y -> z * y < z * x
  apply flip_neg_mult_l.
  Qed.

  Lemma neg_mult x y : x < 0 -> y < 0 -> 0 < x * y.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
x < 0 -> y < 0 -> 0 < x * y
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
x < 0 -> y < 0 -> 0 < x * y
  intros.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
X: x < 0
X0: y < 0
0 < x * y
  rewrite <-(mult_0_r x).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
X: x < 0
X0: y < 0
x * 0 < x * y
  apply flip_neg_mult_l;trivial.
  Qed.

  Lemma pos_mult x y : 0 < x -> 0 < y -> 0 < x * y.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
0 < x -> 0 < y -> 0 < x * y
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
0 < x -> 0 < y -> 0 < x * y
    intros xpos ypos.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
xpos: 0 < x
ypos: 0 < y
0 < x * y
    rewrite <-(mult_0_r x).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
xpos: 0 < x
ypos: 0 < y
x * 0 < x * y
    apply (pos_mult_lt_l); assumption.
  Qed.

  Lemma neg_pos_mult x y : x < 0 -> 0 < y -> x * y < 0.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
x < 0 -> 0 < y -> x * y < 0
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
x < 0 -> 0 < y -> x * y < 0
  intros.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
X: x < 0
X0: 0 < y
x * y < 0
  rewrite <-(mult_0_r x).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
X: x < 0
X0: 0 < y
x * y < x * 0
  apply flip_neg_mult_l;trivial.
  Qed.

  Lemma pos_neg_mult x y : 0 < x -> y < 0 -> x * y < 0.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
0 < x -> y < 0 -> x * y < 0
  Proof.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
0 < x -> y < 0 -> x * y < 0
  intros.
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
X: 0 < x
X0: y < 0
x * y < 0 rewrite (commutativity (f:=mult)).
R: Type
Aplus: Plus R
Amult: Mult R
Azero: Zero R
Aone: One R
H: IsSemiCRing R
Rlt: Lt R
StrictSemiRingOrder0: StrictSemiRingOrder Rlt
x, y: R
X: 0 < x
X0: y < 0
y * x < 0
  apply neg_pos_mult;trivial.
  Qed.
End strict_semiring_order.

(* Due to bug #2528 *)
#[export]
Hint Extern 7 (PropHolds (0 < _ + _)) =>
  eapply @pos_plus_compat : typeclass_instances.

Section pseudo_semiring_order.
  Context `{PseudoSemiRingOrder R} `{!IsSemiCRing R}.
(*   Add Ring Rp : (stdlib_semiring_theory R). *)

  Local Existing Instance pseudo_order_apart.

  #[export] Instance pseudosrorder_strictsrorder : StrictSemiRingOrder (_ : Lt R).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
StrictSemiRingOrder (Alt : Lt R)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
StrictSemiRingOrder (Alt : Lt R)
  split; try exact _.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall x y : R, x < y -> {z : R & y = x + z}
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall x y : R,
PropHolds (0 < x) ->
PropHolds (0 < y) -> PropHolds (0 < x * y)
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall x y : R, x < y -> {z : R & y = x + z} intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x < y
{z : R & y = x + z} apply pseudo_srorder_partial_minus, lt_flip.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x < y
x < y trivial.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall x y : R,
PropHolds (0 < x) ->
PropHolds (0 < y) -> PropHolds (0 < x * y) exact pseudo_srorder_pos_mult_compat.
  Qed.

  #[export] Instance plus_strong_ext : StrongBinaryExtensionality (+).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
StrongBinaryExtensionality plus
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
StrongBinaryExtensionality plus
  assert (forall z, StrongExtensionality (z +)).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall z : R, StrongExtensionality (plus z)
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
X: forall z : R, StrongExtensionality (plus z)
StrongBinaryExtensionality plus
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall z : R, StrongExtensionality (plus z) intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
StrongExtensionality (plus z) exact pseudo_order_embedding_ext.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
X: forall z : R, StrongExtensionality (plus z)
StrongBinaryExtensionality plus exact apartness.strong_binary_setoid_morphism_commutative.
  Qed.

  #[export] Instance plus_strong_cancel_l
    : forall z, StrongLeftCancellation (+) z.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall z : R, StrongLeftCancellation plus z
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall z : R, StrongLeftCancellation plus z
  intros z x y E.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z, x, y: R
E: x ≶ y
z + x ≶ z + y
  apply apart_iff_total_lt in E;apply apart_iff_total_lt.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z, x, y: R
E: ((x < y) + (y < x))%type
((plus z x < plus z y) + (plus z y < plus z x))%type
  destruct E; [left | right]; apply (strictly_order_preserving (z +));trivial.
  Qed.

  #[export] Instance plus_strong_cancel_r
    : forall z, StrongRightCancellation (+) z.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall z : R, StrongRightCancellation plus z
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall z : R, StrongRightCancellation plus z
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
StrongRightCancellation plus z exact (strong_right_cancel_from_left (+)).
  Qed.

  Lemma neg_mult_decompose x y : x * y < 0 -> (x < 0 /\ 0 < y) |_| (0 < x /\ y < 0).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
x * y < 0 -> (x < 0 < y) + (0 < x) * (y < 0)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
x * y < 0 -> (x < 0 < y) + (0 < x) * (y < 0)
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
((x < 0 < y) + (0 < x) * (y < 0))%type
  assert (0 ≶ x) as Ex;[|assert (apart 0 y) as Ey].
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
0 ≶ x
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
Ex: 0 ≶ x
0 ≶ y
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
Ex: 0 ≶ x
Ey: 0 ≶ y
((x < 0 < y) + (0 < x) * (y < 0))%type
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
0 ≶ x apply (strong_extensionality (.* y)).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
0 * y ≶ x * y
    rewrite mult_0_l.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
0 ≶ x * y apply pseudo_order_lt_apart_flip;trivial.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
Ex: 0 ≶ x
0 ≶ y apply (strong_extensionality (x *.)).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
Ex: 0 ≶ x
x * 0 ≶ x * y
    rewrite mult_0_r.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
Ex: 0 ≶ x
0 ≶ x * y apply pseudo_order_lt_apart_flip;trivial.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
Ex: 0 ≶ x
Ey: 0 ≶ y
((x < 0 < y) + (0 < x) * (y < 0))%type apply apart_iff_total_lt in Ex;apply apart_iff_total_lt in Ey.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
Ex: ((0 < x) + (x < 0))%type
Ey: ((0 < y) + (y < 0))%type
((x < 0 < y) + (0 < x) * (y < 0))%type
    destruct Ex as [Ex|Ex], Ey as [Ey|Ey]; try auto.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
Ex: 0 < x
Ey: 0 < y
((x < 0 < y) + (0 < x) * (y < 0))%type
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
Ex: x < 0
Ey: y < 0
((x < 0 < y) + (0 < x) * (y < 0))%type
    +
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
Ex: 0 < x
Ey: 0 < y
((x < 0 < y) + (0 < x) * (y < 0))%type destruct (irreflexivity (<) 0).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
Ex: 0 < x
Ey: 0 < y
0 < 0
      transitivity (x * y); [| trivial].
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
Ex: 0 < x
Ey: 0 < y
0 < x * y
      apply pos_mult_compat;trivial.
    +
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
Ex: x < 0
Ey: y < 0
((x < 0 < y) + (0 < x) * (y < 0))%type destruct (irreflexivity (<) 0).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
Ex: x < 0
Ey: y < 0
0 < 0
      transitivity (x * y); [| trivial].
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: x * y < 0
Ex: x < 0
Ey: y < 0
0 < x * y
      apply neg_mult;trivial.
  Qed.

  Lemma pos_mult_decompose x y : 0 < x * y -> (0 < x /\ 0 < y) |_| (x < 0 /\ y < 0).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
0 < x * y -> (0 < x) * (0 < y) + (x < 0) * (y < 0)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
0 < x * y -> (0 < x) * (0 < y) + (x < 0) * (y < 0)
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
((0 < x) * (0 < y) + (x < 0) * (y < 0))%type
  assert (0 ≶ x /\ apart 0 y) as [Ex Ey];[split|].
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
0 ≶ x
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
0 ≶ y
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
Ex: 0 ≶ x
Ey: 0 ≶ y
((0 < x) * (0 < y) + (x < 0) * (y < 0))%type
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
0 ≶ x apply (strong_extensionality (.* y)).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
0 * y ≶ x * y
    rewrite mult_0_l.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
0 ≶ x * y apply pseudo_order_lt_apart;trivial.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
0 ≶ y apply (strong_extensionality (x *.)).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
x * 0 ≶ x * y
    rewrite mult_0_r.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
0 ≶ x * y apply pseudo_order_lt_apart;trivial.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
Ex: 0 ≶ x
Ey: 0 ≶ y
((0 < x) * (0 < y) + (x < 0) * (y < 0))%type apply apart_iff_total_lt in Ex;apply apart_iff_total_lt in Ey.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
Ex: ((0 < x) + (x < 0))%type
Ey: ((0 < y) + (y < 0))%type
((0 < x) * (0 < y) + (x < 0) * (y < 0))%type
    destruct Ex as [Ex|Ex], Ey as [Ey|Ey]; try auto.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
Ex: 0 < x
Ey: y < 0
((0 < x) * (0 < y) + (x < 0) * (y < 0))%type
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
Ex: x < 0
Ey: 0 < y
((0 < x) * (0 < y) + (x < 0) * (y < 0))%type
    +
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
Ex: 0 < x
Ey: y < 0
((0 < x) * (0 < y) + (x < 0) * (y < 0))%type destruct (irreflexivity (<) 0).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
Ex: 0 < x
Ey: y < 0
0 < 0
      transitivity (x * y); [trivial |].
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
Ex: 0 < x
Ey: y < 0
x * y < 0
      apply pos_neg_mult;trivial.
    +
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
Ex: x < 0
Ey: 0 < y
((0 < x) * (0 < y) + (x < 0) * (y < 0))%type destruct (irreflexivity (<) 0).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
Ex: x < 0
Ey: 0 < y
0 < 0
      transitivity (x * y); [trivial |].
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
Ex: x < 0
Ey: 0 < y
x * y < 0
      apply neg_pos_mult;trivial.
  Qed.

  #[export] Instance pos_mult_reflect_l
    : forall (z : R), PropHolds (0 < z) -> StrictlyOrderReflecting (z *.).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (0 < z) -> StrictlyOrderReflecting (mult z)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (0 < z) -> StrictlyOrderReflecting (mult z)
  intros z Ez x y E1.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: PropHolds (0 < z)
x, y: R
E1: z * x < z * y
x < y
  apply not_lt_apart_lt_flip.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: PropHolds (0 < z)
x, y: R
E1: z * x < z * y
~ (y < x)
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: PropHolds (0 < z)
x, y: R
E1: z * x < z * y
y ≶ x
  +
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: PropHolds (0 < z)
x, y: R
E1: z * x < z * y
~ (y < x) intros E2.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: PropHolds (0 < z)
x, y: R
E1: z * x < z * y
E2: y < x
Empty apply (lt_flip _ _ E1).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: PropHolds (0 < z)
x, y: R
E1: z * x < z * y
E2: y < x
z * y < z * x
    apply (strictly_order_preserving (z *.));trivial.
  +
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: PropHolds (0 < z)
x, y: R
E1: z * x < z * y
y ≶ x apply (strong_extensionality (z *.)).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: PropHolds (0 < z)
x, y: R
E1: z * x < z * y
z * y ≶ z * x
    apply pseudo_order_lt_apart_flip;trivial.
  Qed.

  #[export] Instance pos_mult_reflect_r
    : forall (z : R), PropHolds (0 < z) -> StrictlyOrderReflecting (.* z).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (0 < z) ->
StrictlyOrderReflecting (fun y : R => y * z)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (0 < z) ->
StrictlyOrderReflecting (fun y : R => y * z)
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
X: PropHolds (0 < z)
StrictlyOrderReflecting (fun y : R => y * z)
  exact strictly_order_reflecting_flip.
  Qed.

  #[export] Instance apartzero_mult_strong_cancel_l
    : forall z, PropHolds (z ≶ 0) -> StrongLeftCancellation (.*.) z.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (z ≶ 0) -> StrongLeftCancellation mult z
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (z ≶ 0) -> StrongLeftCancellation mult z
  intros z Ez x y E.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: PropHolds (z ≶ 0)
x, y: R
E: x ≶ y
z * x ≶ z * y red in Ez.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: z ≶ 0
x, y: R
E: x ≶ y
z * x ≶ z * y
  apply apart_iff_total_lt in E;apply apart_iff_total_lt in Ez;
  apply apart_iff_total_lt.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: ((z < 0) + (0 < z))%type
x, y: R
E: ((x < y) + (y < x))%type
((mult z x < mult z y) + (mult z y < mult z x))%type
  destruct E as [E|E], Ez as [Ez|Ez].
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: z < 0
x, y: R
E: x < y
((mult z x < mult z y) + (mult z y < mult z x))%type
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: 0 < z
x, y: R
E: x < y
((mult z x < mult z y) + (mult z y < mult z x))%type
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: z < 0
x, y: R
E: y < x
((mult z x < mult z y) + (mult z y < mult z x))%type
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: 0 < z
x, y: R
E: y < x
((mult z x < mult z y) + (mult z y < mult z x))%type
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: z < 0
x, y: R
E: x < y
((mult z x < mult z y) + (mult z y < mult z x))%type right.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: z < 0
x, y: R
E: x < y
z * y < z * x apply flip_neg_mult_l;trivial.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: 0 < z
x, y: R
E: x < y
((mult z x < mult z y) + (mult z y < mult z x))%type left.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: 0 < z
x, y: R
E: x < y
z * x < z * y apply (strictly_order_preserving_pos (.*.) z);trivial.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: z < 0
x, y: R
E: y < x
((mult z x < mult z y) + (mult z y < mult z x))%type left.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: z < 0
x, y: R
E: y < x
z * x < z * y apply flip_neg_mult_l;trivial.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: 0 < z
x, y: R
E: y < x
((mult z x < mult z y) + (mult z y < mult z x))%type right.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
Ez: 0 < z
x, y: R
E: y < x
z * y < z * x apply (strictly_order_preserving_pos (.*.) z);trivial.
  Qed.

  #[export] Instance apartzero_mult_strong_cancel_r
    : forall z, PropHolds (z ≶ 0) -> StrongRightCancellation (.*.) z.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (z ≶ 0) -> StrongRightCancellation mult z
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (z ≶ 0) -> StrongRightCancellation mult z
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
X: PropHolds (z ≶ 0)
StrongRightCancellation mult z
  exact (strong_right_cancel_from_left (.*.)).
  Qed.

  #[export] Instance apartzero_mult_cancel_l
    : forall z, PropHolds (z ≶ 0) -> LeftCancellation (.*.) z.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (z ≶ 0) -> LeftCancellation mult z
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (z ≶ 0) -> LeftCancellation mult z
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
X: PropHolds (z ≶ 0)
LeftCancellation mult z exact _.
  Qed.

  #[export] Instance apartzero_mult_cancel_r
    : forall z, PropHolds (z ≶ 0) -> RightCancellation (.*.) z.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (z ≶ 0) -> RightCancellation mult z
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (z ≶ 0) -> RightCancellation mult z
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
z: R
X: PropHolds (z ≶ 0)
RightCancellation mult z exact _.
  Qed.

  Lemma square_pos x : x ≶ 0 -> 0 < x * x.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x: R
x ≶ 0 -> 0 < x * x
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x: R
x ≶ 0 -> 0 < x * x
  intros E.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x ≶ 0
0 < x * x apply apart_iff_total_lt in E.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: ((x < 0) + (0 < x))%type
0 < x * x
  destruct E as [E|E].
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x < 0
0 < x * x
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: 0 < x
0 < x * x
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x < 0
0 < x * x destruct (decompose_lt E) as [z [Ez1 Ez2]].
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x < 0
z: R
Ez1: 0 < z
Ez2: 0 = x + z
0 < x * x
    apply compose_lt with (z * z).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x < 0
z: R
Ez1: 0 < z
Ez2: 0 = x + z
0 < z * z
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x < 0
z: R
Ez1: 0 < z
Ez2: 0 = x + z
x * x = 0 + z * z
    +
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x < 0
z: R
Ez1: 0 < z
Ez2: 0 = x + z
0 < z * z apply pos_mult_compat;trivial.
    +
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x < 0
z: R
Ez1: 0 < z
Ez2: 0 = x + z
x * x = 0 + z * z rewrite plus_0_l.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x < 0
z: R
Ez1: 0 < z
Ez2: 0 = x + z
x * x = z * z
      apply (left_cancellation (+) (x * z)).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x < 0
z: R
Ez1: 0 < z
Ez2: 0 = x + z
x * z + x * x = x * z + z * z
      rewrite <-plus_mult_distr_r, <-plus_mult_distr_l.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x < 0
z: R
Ez1: 0 < z
Ez2: 0 = x + z
x * (z + x) = (x + z) * z
      rewrite (commutativity (f:=plus) z x), <-!Ez2.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x < 0
z: R
Ez1: 0 < z
Ez2: 0 = x + z
x * 0 = 0 * z
      rewrite mult_0_l,mult_0_r.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x < 0
z: R
Ez1: 0 < z
Ez2: 0 = x + z
0 = 0 reflexivity.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: 0 < x
0 < x * x apply pos_mult_compat;trivial.
  Qed.

  Lemma pos_mult_rev_l x y : 0 < x * y -> 0 < y -> 0 < x.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
0 < x * y -> 0 < y -> 0 < x
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
0 < x * y -> 0 < y -> 0 < x
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
X0: 0 < y
0 < x apply (strictly_order_reflecting (.* y)).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
X0: 0 < y
0 * y < x * y
  rewrite rings.mult_0_l;trivial.
  Qed.

  Lemma pos_mult_rev_r x y : 0 < x * y -> 0 < x -> 0 < y.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
0 < x * y -> 0 < x -> 0 < y
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
0 < x * y -> 0 < x -> 0 < y
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
X0: 0 < x
0 < y apply pos_mult_rev_l with x.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
X0: 0 < x
0 < y * x
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
X0: 0 < x
0 < x
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
X0: 0 < x
0 < y * x rewrite (commutativity (f:=mult));trivial.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 < x * y
X0: 0 < x
0 < x trivial.
  Qed.

  Context `{PropHolds (1 ≶ 0)}.

  Instance lt_0_1 : PropHolds (0 < 1).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
PropHolds (0 < 1)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
PropHolds (0 < 1)
  red.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
0 < 1
  rewrite <-(mult_1_l 1).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
0 < 1 * 1
  apply square_pos;trivial.
  Qed.

  Instance lt_0_2 : PropHolds (0 < 2).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
PropHolds (0 < 2)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
PropHolds (0 < 2)
  exact _.
  Qed.

  Instance lt_0_3 : PropHolds (0 < 3).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
PropHolds (0 < 3)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
PropHolds (0 < 3)
  exact _.
  Qed.

  Instance lt_0_4 : PropHolds (0 < 4).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
PropHolds (0 < 4)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
PropHolds (0 < 4)
  exact _.
  Qed.

  Lemma lt_1_2 : 1 < 2.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
1 < 2
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
1 < 2
  apply pos_plus_lt_compat_r, lt_0_1.
  Qed.

  Lemma lt_1_3 : 1 < 3.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
1 < 3
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
1 < 3
  apply pos_plus_lt_compat_r, lt_0_2.
  Qed.

  Lemma lt_1_4 : 1 < 4.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
1 < 4
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
1 < 4
  apply pos_plus_lt_compat_r, lt_0_3.
  Qed.

  Lemma lt_2_3 : 2 < 3.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
2 < 3
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
2 < 3
  apply (strictly_order_preserving (1+)), lt_1_2.
  Qed.

  Lemma lt_2_4 : 2 < 4.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
2 < 4
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
2 < 4
  apply (strictly_order_preserving (1+)), lt_1_3.
  Qed.

  Lemma lt_3_4 : 3 < 4.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
3 < 4
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
3 < 4
  apply (strictly_order_preserving (1+)), lt_2_3.
  Qed.

  Instance apart_0_2 : PropHolds (2 ≶ 0).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
PropHolds (2 ≶ 0)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
PropHolds (2 ≶ 0)
  red.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
2 ≶ 0 apply symmetry.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Alt: Lt R
H4: PseudoSemiRingOrder Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
0 ≶ 2
  apply pseudo_order_lt_apart, lt_0_2.
  Qed.
End pseudo_semiring_order.

#[export]
Hint Extern 7 (PropHolds (0 < 1)) => eapply @lt_0_1 : typeclass_instances.
#[export]
Hint Extern 7 (PropHolds (0 < 2)) => eapply @lt_0_2 : typeclass_instances.
#[export]
Hint Extern 7 (PropHolds (0 < 3)) => eapply @lt_0_3 : typeclass_instances.
#[export]
Hint Extern 7 (PropHolds (0 < 4)) => eapply @lt_0_4 : typeclass_instances.
#[export]
Hint Extern 7 (PropHolds (2 ≶ 0)) => eapply @apart_0_2 : typeclass_instances.

Section full_pseudo_semiring_order.
  Context `{FullPseudoSemiRingOrder R} `{!IsSemiCRing R}.

(*   Add Ring Rf : (stdlib_semiring_theory R). *)

  #[export] Instance fullpseudosrorder_fullpseudoorder
    : FullPseudoOrder (_ : Le R) (_ : Lt R).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
FullPseudoOrder (Ale : Le R) (Alt : Lt R)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
FullPseudoOrder (Ale : Le R) (Alt : Lt R)
  split.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
is_mere_relation R Ale
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
PseudoOrder Alt
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
forall x y : R, x ≤ y <-> ~ (y < x)
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
is_mere_relation R Ale exact _.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
PseudoOrder Alt exact _.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
forall x y : R, x ≤ y <-> ~ (y < x) exact full_pseudo_srorder_le_iff_not_lt_flip.
  Qed.

  #[export] Instance fullpseudosrorder_srorder : SemiRingOrder (_ : Le R).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
SemiRingOrder (Ale : Le R)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
SemiRingOrder (Ale : Le R)
  split; try exact _.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
forall x y : R, x ≤ y -> {z : R & y = x + z}
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
forall z : R, OrderEmbedding (plus z)
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
forall x y : R,
PropHolds (0 ≤ x) ->
PropHolds (0 ≤ y) -> PropHolds (0 ≤ x * y)
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
forall x y : R, x ≤ y -> {z : R & y = x + z} intros x y E.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
E: x ≤ y
{z : R & y = x + z} apply le_iff_not_lt_flip in E.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
E: ~ (y < x)
{z : R & y = x + z}
    apply pseudo_srorder_partial_minus;trivial.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
forall z : R, OrderEmbedding (plus z) intros z.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
z: R
OrderEmbedding (plus z) repeat (split; try exact _).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
z: R
OrderPreserving (plus z)
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
z: R
OrderReflecting (plus z)
    +
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
z: R
OrderPreserving (plus z) intros x y E1.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
z, x, y: R
E1: x ≤ y
z + x ≤ z + y apply le_iff_not_lt_flip in E1;apply le_iff_not_lt_flip.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
z, x, y: R
E1: ~ (y < x)
~ (z + y < z + x)
      intros E2.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
z, x, y: R
E1: ~ (y < x)
E2: z + y < z + x
Empty apply E1.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
z, x, y: R
E1: ~ (y < x)
E2: z + y < z + x
y < x
      apply (strictly_order_reflecting (z+)).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
z, x, y: R
E1: ~ (y < x)
E2: z + y < z + x
z + y < z + x trivial.
    +
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
z: R
OrderReflecting (plus z) intros x y E1.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
z, x, y: R
E1: z + x ≤ z + y
x ≤ y  apply le_iff_not_lt_flip in E1;apply le_iff_not_lt_flip.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
z, x, y: R
E1: ~ (z + y < z + x)
~ (y < x)
      intros E2.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
z, x, y: R
E1: ~ (z + y < z + x)
E2: y < x
Empty apply E1.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
z, x, y: R
E1: ~ (z + y < z + x)
E2: y < x
z + y < z + x
      apply (strictly_order_preserving _);trivial.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
forall x y : R,
PropHolds (0 ≤ x) ->
PropHolds (0 ≤ y) -> PropHolds (0 ≤ x * y) intros x y Ex Ey.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
Ex: PropHolds (0 ≤ x)
Ey: PropHolds (0 ≤ y)
PropHolds (0 ≤ x * y)
    apply le_iff_not_lt_flip in Ex;apply le_iff_not_lt_flip in Ey;
    apply le_iff_not_lt_flip.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
Ex: ~ (x < 0)
Ey: ~ (y < 0)
~ (x * y < 0)
    intros E.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
Ex: ~ (x < 0)
Ey: ~ (y < 0)
E: x * y < 0
Empty
    destruct (neg_mult_decompose x y E) as [[? ?]|[? ?]];auto.
  Qed.

  #[export] Instance orderreflecting_left_mult : forall (z : R), PropHolds (0 < z) -> OrderReflecting (z *.).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (0 < z) -> OrderReflecting (mult z)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (0 < z) -> OrderReflecting (mult z)
  intros z E.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
z: R
E: PropHolds (0 < z)
OrderReflecting (mult z) exact full_pseudo_order_reflecting.
  Qed.

  #[export] Instance orderreflecting_right_mult : forall (z : R), PropHolds (0 < z) -> OrderReflecting (.* z).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (0 < z) ->
OrderReflecting (fun y : R => y * z)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
forall z : R,
PropHolds (0 < z) ->
OrderReflecting (fun y : R => y * z)
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
z: R
X: PropHolds (0 < z)
OrderReflecting (fun y : R => y * z) exact order_reflecting_flip.
  Qed.

  Lemma plus_lt_le_compat x₁ y₁ x₂ y₂ : x₁ < y₁ -> x₂ ≤ y₂ -> x₁ + x₂ < y₁ + y₂.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
x₁ < y₁ -> x₂ ≤ y₂ -> x₁ + x₂ < y₁ + y₂
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
x₁ < y₁ -> x₂ ≤ y₂ -> x₁ + x₂ < y₁ + y₂
  intros E1 E2.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
E1: x₁ < y₁
E2: x₂ ≤ y₂
x₁ + x₂ < y₁ + y₂
  apply lt_le_trans with (y₁ + x₂).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
E1: x₁ < y₁
E2: x₂ ≤ y₂
x₁ + x₂ < y₁ + x₂
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
E1: x₁ < y₁
E2: x₂ ≤ y₂
y₁ + x₂ ≤ y₁ + y₂
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
E1: x₁ < y₁
E2: x₂ ≤ y₂
x₁ + x₂ < y₁ + x₂ apply (strictly_order_preserving (+ x₂));trivial.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
E1: x₁ < y₁
E2: x₂ ≤ y₂
y₁ + x₂ ≤ y₁ + y₂ apply (order_preserving (y₁ +));trivial.
  Qed.

  Lemma plus_le_lt_compat x₁ y₁ x₂ y₂ : x₁ ≤ y₁ -> x₂ < y₂ -> x₁ + x₂ < y₁ + y₂.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
x₁ ≤ y₁ -> x₂ < y₂ -> x₁ + x₂ < y₁ + y₂
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
x₁ ≤ y₁ -> x₂ < y₂ -> x₁ + x₂ < y₁ + y₂
  intros E1 E2.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
E1: x₁ ≤ y₁
E2: x₂ < y₂
x₁ + x₂ < y₁ + y₂
  apply le_lt_trans with (y₁ + x₂).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
E1: x₁ ≤ y₁
E2: x₂ < y₂
x₁ + x₂ ≤ y₁ + x₂
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
E1: x₁ ≤ y₁
E2: x₂ < y₂
y₁ + x₂ < y₁ + y₂
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
E1: x₁ ≤ y₁
E2: x₂ < y₂
x₁ + x₂ ≤ y₁ + x₂ apply (order_preserving (+ x₂));trivial.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x₁, y₁, x₂, y₂: R
E1: x₁ ≤ y₁
E2: x₂ < y₂
y₁ + x₂ < y₁ + y₂ apply (strictly_order_preserving (y₁ +));trivial.
  Qed.

  Lemma nonneg_plus_lt_compat_r x y z : 0 ≤ z -> x < y -> x < y + z.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y, z: R
0 ≤ z -> x < y -> x < y + z
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y, z: R
0 ≤ z -> x < y -> x < y + z
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y, z: R
X: 0 ≤ z
X0: x < y
x < y + z rewrite <-(plus_0_r x).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y, z: R
X: 0 ≤ z
X0: x < y
x + 0 < y + z apply plus_lt_le_compat;trivial.
  Qed.

  Lemma nonneg_plus_lt_compat_l x y z : 0 ≤ z -> x < y -> x < z + y.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y, z: R
0 ≤ z -> x < y -> x < z + y
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y, z: R
0 ≤ z -> x < y -> x < z + y
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y, z: R
X: 0 ≤ z
X0: x < y
x < z + y rewrite (commutativity (f:=plus)).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y, z: R
X: 0 ≤ z
X0: x < y
x < y + z
  apply nonneg_plus_lt_compat_r;trivial.
  Qed.

  Lemma pos_plus_le_lt_compat_r x y z : 0 < z -> x ≤ y -> x < y + z.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y, z: R
0 < z -> x ≤ y -> x < y + z
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y, z: R
0 < z -> x ≤ y -> x < y + z
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y, z: R
X: 0 < z
X0: x ≤ y
x < y + z rewrite <-(plus_0_r x).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y, z: R
X: 0 < z
X0: x ≤ y
x + 0 < y + z apply plus_le_lt_compat;trivial.
  Qed.

  Lemma pos_plus_le_lt_compat_l x y z : 0 < z -> x ≤ y -> x < z + y.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y, z: R
0 < z -> x ≤ y -> x < z + y
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y, z: R
0 < z -> x ≤ y -> x < z + y
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y, z: R
X: 0 < z
X0: x ≤ y
x < z + y rewrite (commutativity (f:=plus)).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y, z: R
X: 0 < z
X0: x ≤ y
x < y + z
  apply pos_plus_le_lt_compat_r;trivial.
  Qed.

  Lemma square_nonneg x : 0 ≤ x * x.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x: R
0 ≤ x * x
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x: R
0 ≤ x * x
  apply not_lt_le_flip.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x: R
~ (x * x < 0) intros E.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x * x < 0
Empty
  destruct (lt_antisym (x * x) 0).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x * x < 0
x * x < 0 < x * x
  split; [trivial |].
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x * x < 0
0 < x * x
  apply square_pos.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x * x < 0
x ≶ 0
  pose proof pseudo_order_apart.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x * x < 0
X: IsApart R
x ≶ 0
  apply (strong_extensionality (x *.)).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x * x < 0
X: IsApart R
x * x ≶ x * 0
  rewrite mult_0_r.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x * x < 0
X: IsApart R
x * x ≶ 0
  apply lt_apart.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x: R
E: x * x < 0
X: IsApart R
x * x < 0 trivial.
  Qed.

  Lemma nonneg_mult_rev_l x y : 0 ≤ x * y -> 0 < y -> 0 ≤ x.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
0 ≤ x * y -> 0 < y -> 0 ≤ x
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
0 ≤ x * y -> 0 < y -> 0 ≤ x
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 ≤ x * y
X0: 0 < y
0 ≤ x apply (order_reflecting (.* y)).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 ≤ x * y
X0: 0 < y
0 * y ≤ x * y
  rewrite rings.mult_0_l.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 ≤ x * y
X0: 0 < y
0 ≤ x * y trivial.
  Qed.

  Lemma nonneg_mult_rev_r x y : 0 ≤ x * y -> 0 < x -> 0 ≤ y.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
0 ≤ x * y -> 0 < x -> 0 ≤ y
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
0 ≤ x * y -> 0 < x -> 0 ≤ y
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 ≤ x * y
X0: 0 < x
0 ≤ y apply nonneg_mult_rev_l with x.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 ≤ x * y
X0: 0 < x
0 ≤ y * x
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 ≤ x * y
X0: 0 < x
0 < x
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 ≤ x * y
X0: 0 < x
0 ≤ y * x rewrite (commutativity (f:=mult)).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 ≤ x * y
X0: 0 < x
0 ≤ x * y trivial.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 0 ≤ x * y
X0: 0 < x
0 < x trivial.
  Qed.

  Instance le_0_1 : PropHolds (0 ≤ 1).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
PropHolds (0 ≤ 1)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
PropHolds (0 ≤ 1)
  red.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
0 ≤ 1 rewrite <-(mult_1_r 1).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
0 ≤ 1 * 1 apply square_nonneg.
  Qed.

  Instance le_0_2 : PropHolds (0 ≤ 2).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
PropHolds (0 ≤ 2)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
PropHolds (0 ≤ 2) solve_propholds. Qed.

  Instance le_0_3 : PropHolds (0 ≤ 3).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
PropHolds (0 ≤ 3)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
PropHolds (0 ≤ 3) solve_propholds. Qed.

  Instance le_0_4 : PropHolds (0 ≤ 4).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
PropHolds (0 ≤ 4)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
PropHolds (0 ≤ 4) solve_propholds. Qed.

  Lemma le_1_2 : 1 ≤ 2.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
1 ≤ 2
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
1 ≤ 2
  apply nonneg_plus_le_compat_r, le_0_1.
  Qed.

  Lemma le_1_3 : 1 ≤ 3.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
1 ≤ 3
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
1 ≤ 3
  apply nonneg_plus_le_compat_r, le_0_2.
  Qed.

  Lemma le_1_4 : 1 ≤ 4.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
1 ≤ 4
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
1 ≤ 4
  apply nonneg_plus_le_compat_r, le_0_3.
  Qed.

  Lemma le_2_3 : 2 ≤ 3.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
2 ≤ 3
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
2 ≤ 3
  apply (order_preserving (1+)), le_1_2.
  Qed.

  Lemma le_2_4 : 2 ≤ 4.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
2 ≤ 4
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
2 ≤ 4
  apply (order_preserving (1+)), le_1_3.
  Qed.

  Lemma le_3_4 : 3 ≤ 4.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
3 ≤ 4
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
3 ≤ 4
  apply (order_preserving (1+)), le_2_3.
  Qed.

  Lemma ge_1_mult_compat x y : 1 ≤ x -> 1 ≤ y -> 1 ≤ x * y.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
1 ≤ x -> 1 ≤ y -> 1 ≤ x * y
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
1 ≤ x -> 1 ≤ y -> 1 ≤ x * y
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 1 ≤ x
X0: 1 ≤ y
1 ≤ x * y
  apply ge_1_mult_le_compat_r; trivial.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 1 ≤ x
X0: 1 ≤ y
0 ≤ x
  transitivity 1.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 1 ≤ x
X0: 1 ≤ y
0 ≤ 1
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 1 ≤ x
X0: 1 ≤ y
1 ≤ x
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 1 ≤ x
X0: 1 ≤ y
0 ≤ 1 exact le_0_1.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 1 ≤ x
X0: 1 ≤ y
1 ≤ x trivial.
  Qed.

  Lemma gt_1_ge_1_mult_compat x y : 1 < x -> 1 ≤ y -> 1 < x * y.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
1 < x -> 1 ≤ y -> 1 < x * y
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
1 < x -> 1 ≤ y -> 1 < x * y
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 1 < x
X0: 1 ≤ y
1 < x * y
  apply lt_le_trans with x; trivial.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 1 < x
X0: 1 ≤ y
x ≤ x * y
  apply ge_1_mult_le_compat_r;[trivial| |apply reflexivity].
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 1 < x
X0: 1 ≤ y
0 ≤ x
  transitivity 1.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 1 < x
X0: 1 ≤ y
0 ≤ 1
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 1 < x
X0: 1 ≤ y
1 ≤ x
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 1 < x
X0: 1 ≤ y
0 ≤ 1 exact le_0_1.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 1 < x
X0: 1 ≤ y
1 ≤ x apply lt_le;trivial.
  Qed.

  Lemma ge_1_gt_1_mult_compat x y : 1 ≤ x -> 1 < y -> 1 < x * y.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
1 ≤ x -> 1 < y -> 1 < x * y
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
1 ≤ x -> 1 < y -> 1 < x * y
  intros.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 1 ≤ x
X0: 1 < y
1 < x * y
  rewrite (commutativity (f:=mult)).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
x, y: R
X: 1 ≤ x
X0: 1 < y
1 < y * x
  apply gt_1_ge_1_mult_compat;trivial.
  Qed.

  Lemma pos_mult_le_lt_compat : forall a b c d, 0 <= a /\ a <= b -> 0 < b ->
    0 <= c /\ c < d ->
    a * c < b * d.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
forall a b c d : R,
0 ≤ a ≤ b -> 0 < b -> 0 ≤ c < d -> a * c < b * d
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
forall a b c d : R,
0 ≤ a ≤ b -> 0 < b -> 0 ≤ c < d -> a * c < b * d
  intros a b c d [E1 E2] E3 [E4 E5] .
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
a, b, c, d: R
E1: 0 ≤ a
E2: a ≤ b
E3: 0 < b
E4: 0 ≤ c
E5: c < d
a * c < b * d
  apply le_lt_trans with (b * c).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
a, b, c, d: R
E1: 0 ≤ a
E2: a ≤ b
E3: 0 < b
E4: 0 ≤ c
E5: c < d
a * c ≤ b * c
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
a, b, c, d: R
E1: 0 ≤ a
E2: a ≤ b
E3: 0 < b
E4: 0 ≤ c
E5: c < d
b * c < b * d
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
a, b, c, d: R
E1: 0 ≤ a
E2: a ≤ b
E3: 0 < b
E4: 0 ≤ c
E5: c < d
a * c ≤ b * c apply mult_le_compat;auto.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
a, b, c, d: R
E1: 0 ≤ a
E2: a ≤ b
E3: 0 < b
E4: 0 ≤ c
E5: c < d
b * c < b * d apply (strictly_order_preserving (b *.)).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
a, b, c, d: R
E1: 0 ≤ a
E2: a ≤ b
E3: 0 < b
E4: 0 ≤ c
E5: c < d
c < d trivial.
  Qed.

  Context `{PropHolds (1 ≶ 0)}.

  Lemma not_le_1_0 : ~(1 ≤ 0).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
~ (1 ≤ 0)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
~ (1 ≤ 0)
  apply lt_not_le_flip, lt_0_1.
  Qed.

  Lemma not_le_2_0 : ~(2 ≤ 0).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
~ (2 ≤ 0)
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
~ (2 ≤ 0)
  apply lt_not_le_flip, lt_0_2.
  Qed.

  Lemma repeat_nat_nonneg : forall n, 0 <= Core.nat_iter n (plus 1) 0.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
forall n : nat,
0
≤ (fix F (m : nat) : R :=
     match m with
     | 0%nat => 0
     | m'.+1%nat => 1 + F m'
     end) n
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
forall n : nat,
0
≤ (fix F (m : nat) : R :=
     match m with
     | 0%nat => 0
     | m'.+1%nat => 1 + F m'
     end) n
  induction n;simpl.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
0 ≤ 0
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n: nat
IHn: 0
≤ (fix F (m : nat) : R :=
     match m with
     | 0%nat => 0
     | m'.+1%nat => 1 + F m'
     end) n
0
≤ 1 +
  (fix F (m : nat) : R :=
     match m with
     | 0%nat => 0
     | m'.+1%nat => 1 + F m'
     end) n
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
0 ≤ 0 reflexivity.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n: nat
IHn: 0
≤ (fix F (m : nat) : R :=
     match m with
     | 0%nat => 0
     | m'.+1%nat => 1 + F m'
     end) n
0
≤ 1 +
  (fix F (m : nat) : R :=
     match m with
     | 0%nat => 0
     | m'.+1%nat => 1 + F m'
     end) n apply nonneg_plus_compat.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n: nat
IHn: 0
≤ (fix F (m : nat) : R :=
     match m with
     | 0%nat => 0
     | m'.+1%nat => 1 + F m'
     end) n
PropHolds (0 ≤ 1)
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n: nat
IHn: 0
≤ (fix F (m : nat) : R :=
     match m with
     | 0%nat => 0
     | m'.+1%nat => 1 + F m'
     end) n
PropHolds
  (0
   ≤ (fix F (m : nat) : R :=
        match m with
        | 0%nat => 0
        | m'.+1%nat => 1 + F m'
        end) n)
    +
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n: nat
IHn: 0
≤ (fix F (m : nat) : R :=
     match m with
     | 0%nat => 0
     | m'.+1%nat => 1 + F m'
     end) n
PropHolds (0 ≤ 1) exact _.
    +
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n: nat
IHn: 0
≤ (fix F (m : nat) : R :=
     match m with
     | 0%nat => 0
     | m'.+1%nat => 1 + F m'
     end) n
PropHolds
  (0
   ≤ (fix F (m : nat) : R :=
        match m with
        | 0%nat => 0
        | m'.+1%nat => 1 + F m'
        end) n) exact IHn.
  Qed.

  Lemma repeat_nat_pos : forall n, 0 < Core.nat_iter (S n) (plus 1) 0.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
forall n : nat,
0 <
(fix F (m : nat) : R :=
   match m with
   | 0%nat => 0
   | m'.+1%nat => 1 + F m'
   end) n.+1%nat
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
forall n : nat,
0 <
(fix F (m : nat) : R :=
   match m with
   | 0%nat => 0
   | m'.+1%nat => 1 + F m'
   end) n.+1%nat
  intros n.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n: nat
0 <
(fix F (m : nat) : R :=
   match m with
   | 0%nat => 0
   | m'.+1%nat => 1 + F m'
   end) n.+1%nat simpl.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n: nat
0 <
1 +
(fix F (m : nat) : R :=
   match m with
   | 0%nat => 0
   | m'.+1%nat => 1 + F m'
   end) n
  apply pos_plus_le_lt_compat_l.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n: nat
0 < 1
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n: nat
0
≤ (fix F (m : nat) : R :=
     match m with
     | 0%nat => 0
     | m'.+1%nat => 1 + F m'
     end) n
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n: nat
0 < 1 solve_propholds.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n: nat
0
≤ (fix F (m : nat) : R :=
     match m with
     | 0%nat => 0
     | m'.+1%nat => 1 + F m'
     end) n apply repeat_nat_nonneg.
  Qed.

  Local Existing Instance pseudo_order_apart.

  #[export] Instance ordered_characteristic_0 : FieldCharacteristic R 0.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
FieldCharacteristic R 0
  Proof.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
FieldCharacteristic R 0
  hnf.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
forall n : nat,
Core.lt 0 n ->
(forall m : nat, n <> Core.nat_mul 0 m) <->
(fix F (m : nat) : R :=
   match m with
   | 0%nat => 0
   | m'.+1%nat => 1 + F m'
   end) n ≶ 0 intros [|n] _;split.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
(forall m : nat, 0%nat <> Core.nat_mul 0 m) -> 0 ≶ 0
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
0 ≶ 0 -> forall m : nat, 0%nat <> Core.nat_mul 0 m
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n: nat
(forall m : nat, n.+1%nat <> Core.nat_mul 0 m) ->
1 +
(fix F (m : nat) : R :=
   match m with
   | 0%nat => 0
   | m'.+1%nat => 1 + F m'
   end) n ≶ 0
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n: nat
1 +
(fix F (m : nat) : R :=
   match m with
   | 0%nat => 0
   | m'.+1%nat => 1 + F m'
   end) n ≶ 0 ->
forall m : nat, n.+1%nat <> Core.nat_mul 0 m
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
(forall m : nat, 0%nat <> Core.nat_mul 0 m) -> 0 ≶ 0 intros E'.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
E': forall m : nat, 0%nat <> Core.nat_mul 0 m
0 ≶ 0 destruct (E' O).
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
E': forall m : nat, 0%nat <> Core.nat_mul 0 m
0%nat = Core.nat_mul 0 0 reflexivity.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
0 ≶ 0 -> forall m : nat, 0%nat <> Core.nat_mul 0 m intros E';apply (irreflexivity _) in E';destruct E'.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n: nat
(forall m : nat, n.+1%nat <> Core.nat_mul 0 m) ->
1 +
(fix F (m : nat) : R :=
   match m with
   | 0%nat => 0
   | m'.+1%nat => 1 + F m'
   end) n ≶ 0 intros _;apply apart_iff_total_lt;right;apply repeat_nat_pos.
  -
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n: nat
1 +
(fix F (m : nat) : R :=
   match m with
   | 0%nat => 0
   | m'.+1%nat => 1 + F m'
   end) n ≶ 0 ->
forall m : nat, n.+1%nat <> Core.nat_mul 0 m intros _ m;simpl.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n, m: nat
n.+1%nat <> 0%nat intros E.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n, m: nat
E: n.+1%nat = 0%nat
Empty
    apply (ap (fun n => match n with | S _ => Unit | _ => Empty end)) in E;
    simpl in E.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n, m: nat
E: Unit = Empty
Empty rewrite <-E.
R: Type
H: Apart R
H0: Plus R
H1: Mult R
H2: Zero R
H3: One R
Ale: Le R
Alt: Lt R
H4: FullPseudoSemiRingOrder Ale Alt
IsSemiCRing0: IsSemiCRing R
H5: PropHolds (1 ≶ 0)
n, m: nat
E: Unit = Empty
Unit trivial.
  Qed.
End full_pseudo_semiring_order.

(* Due to bug #2528 *)
#[export]
Hint Extern 7 (PropHolds (0 ≤ 1)) => eapply @le_0_1 : typeclass_instances.
#[export]
Hint Extern 7 (PropHolds (0 ≤ 2)) => eapply @le_0_2 : typeclass_instances.
#[export]
Hint Extern 7 (PropHolds (0 ≤ 3)) => eapply @le_0_3 : typeclass_instances.
#[export]
Hint Extern 7 (PropHolds (0 ≤ 4)) => eapply @le_0_4 : typeclass_instances.

Section dec_semiring_order.
  (* Maybe these assumptions can be weakened? *)
  Context `{SemiRingOrder R} `{Apart R} `{!TrivialApart R}
    `{!NoZeroDivisors R} `{!TotalRelation (≤)} `{DecidablePaths R}.

  Context `{Rlt : Lt R} `{is_mere_relation R lt}
    (lt_correct : forall x y, x < y <-> x ≤ y /\ x <> y).

  Instance dec_srorder_fullpseudo
    : FullPseudoOrder _ _ := dec_full_pseudo_order lt_correct.
  Local Existing Instance pseudo_order_apart.

  Instance dec_pseudo_srorder: PseudoSemiRingOrder (<).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
PseudoSemiRingOrder lt
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
PseudoSemiRingOrder lt
  split; try exact _.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
forall x y : R, ~ (y < x) -> {z : R & y = x + z}
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
forall z : R, StrictOrderEmbedding (plus z)
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
StrongBinaryExtensionality mult
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
forall x y : R,
PropHolds (0 < x) ->
PropHolds (0 < y) -> PropHolds (0 < x * y)
  -
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
forall x y : R, ~ (y < x) -> {z : R & y = x + z} intros x y E.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
x, y: R
E: ~ (y < x)
{z : R & y = x + z} apply srorder_partial_minus, not_lt_le_flip;trivial.
  -
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
forall z : R, StrictOrderEmbedding (plus z) intros z.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
z: R
StrictOrderEmbedding (plus z) repeat (split; try exact _).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
z: R
StrictlyOrderPreserving (plus z)
    intros x y E.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
z, x, y: R
E: x < y
z + x < z + y apply lt_correct in E;apply lt_correct.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
z, x, y: R
E: ((x ≤ y) * (x <> y))%type
((plus z x ≤ plus z y) * (plus z x <> plus z y))%type
    destruct E as [E2a E2b].
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
z, x, y: R
E2a: x ≤ y
E2b: x <> y
((plus z x ≤ plus z y) * (plus z x <> plus z y))%type split.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
z, x, y: R
E2a: x ≤ y
E2b: x <> y
z + x ≤ z + y
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
z, x, y: R
E2a: x ≤ y
E2b: x <> y
z + x <> z + y
    +
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
z, x, y: R
E2a: x ≤ y
E2b: x <> y
z + x ≤ z + y apply (order_preserving (z+));trivial.
    +
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
z, x, y: R
E2a: x ≤ y
E2b: x <> y
z + x <> z + y intros E3.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
z, x, y: R
E2a: x ≤ y
E2b: x <> y
E3: z + x = z + y
Empty apply E2b.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
z, x, y: R
E2a: x ≤ y
E2b: x <> y
E3: z + x = z + y
x = y apply (left_cancellation (+) z);trivial.
  -
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
StrongBinaryExtensionality mult exact (apartness.dec_strong_binary_morphism (.*.)).
  -
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
forall x y : R,
PropHolds (0 < x) ->
PropHolds (0 < y) -> PropHolds (0 < x * y) intros x y E1 E2.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
x, y: R
E1: PropHolds (0 < x)
E2: PropHolds (0 < y)
PropHolds (0 < x * y)
    apply lt_correct in E1;apply lt_correct in E2;apply lt_correct.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
x, y: R
E1: ((0 ≤ x) * (0 <> x))%type
E2: ((0 ≤ y) * (0 <> y))%type
((0 ≤ mult x y) * (0 <> mult x y))%type
    destruct E1 as [E1a E1b], E2 as [E2a E2b].
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
x, y: R
E1a: 0 ≤ x
E1b: 0 <> x
E2a: 0 ≤ y
E2b: 0 <> y
((0 ≤ mult x y) * (0 <> mult x y))%type split.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
x, y: R
E1a: 0 ≤ x
E1b: 0 <> x
E2a: 0 ≤ y
E2b: 0 <> y
0 ≤ x * y
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
x, y: R
E1a: 0 ≤ x
E1b: 0 <> x
E2a: 0 ≤ y
E2b: 0 <> y
0 <> x * y
    +
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
x, y: R
E1a: 0 ≤ x
E1b: 0 <> x
E2a: 0 ≤ y
E2b: 0 <> y
0 ≤ x * y apply nonneg_mult_compat;trivial.
    +
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
x, y: R
E1a: 0 ≤ x
E1b: 0 <> x
E2a: 0 ≤ y
E2b: 0 <> y
0 <> x * y apply symmetric_neq.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
x, y: R
E1a: 0 ≤ x
E1b: 0 <> x
E2a: 0 ≤ y
E2b: 0 <> y
x * y <> 0
      apply mult_ne_0; apply symmetric_neq;trivial.
  Qed.

  Instance dec_full_pseudo_srorder: FullPseudoSemiRingOrder (≤) (<).
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
FullPseudoSemiRingOrder le lt
  Proof.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
FullPseudoSemiRingOrder le lt
  split; try exact _.
R: Type
H: Plus R
H0: Mult R
H1: Zero R
H2: One R
Ale: Le R
H3: SemiRingOrder Ale
H4: Apart R
TrivialApart0: TrivialApart R
NoZeroDivisors0: NoZeroDivisors R
TotalRelation0: TotalRelation le
H5: DecidablePaths R
Rlt: Lt R
is_mere_relation0: is_mere_relation R lt
lt_correct: forall x y : R,
x < y <-> (x ≤ y) * (x <> y)
forall x y : R, x ≤ y <-> ~ (y < x)
  exact le_iff_not_lt_flip.
  Qed.
End dec_semiring_order.

Section another_semiring.
  Context `{SemiRingOrder R1}.

  Lemma projected_srorder `{IsSemiCRing R2} `{R2le : Le R2}
    `{is_mere_relation R2 R2le} (f : R2 -> R1)
    `{!IsSemiRingPreserving f} `{!IsInjective f}
    : (forall x y, x ≤ y <-> f x ≤ f y) -> (forall x y : R2, x ≤ y -> exists z, y = x + z) ->
      SemiRingOrder R2le.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
(forall x y : R2, x ≤ y <-> f x ≤ f y) ->
(forall x y : R2, x ≤ y -> {z : R2 & y = x + z}) ->
SemiRingOrder R2le
  Proof.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
(forall x y : R2, x ≤ y <-> f x ≤ f y) ->
(forall x y : R2, x ≤ y -> {z : R2 & y = x + z}) ->
SemiRingOrder R2le
  intros P.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
(forall x y : R2, x ≤ y -> {z : R2 & y = x + z}) ->
SemiRingOrder R2le pose proof (projected_partial_order f P).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
(forall x y : R2, x ≤ y -> {z : R2 & y = x + z}) ->
SemiRingOrder R2le
  repeat (split; try exact _).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
z: R2
OrderPreserving (plus z)
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
z: R2
OrderReflecting (plus z)
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
forall x y : R2,
PropHolds (0 ≤ x) ->
PropHolds (0 ≤ y) -> PropHolds (0 ≤ x * y)
  -
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
forall x y : R2, x ≤ y -> {z : R2 & y = x + z} assumption.
  -
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
z: R2
OrderPreserving (plus z) red;intros.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
z, x, y: R2
X1: x ≤ y
z + x ≤ z + y apply P.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
z, x, y: R2
X1: x ≤ y
f (z + x) ≤ f (z + y)
    rewrite 2!(preserves_plus (f:=f)).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
z, x, y: R2
X1: x ≤ y
f z + f x ≤ f z + f y apply (order_preserving _), P.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
z, x, y: R2
X1: x ≤ y
x ≤ y trivial.
  -
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
z: R2
OrderReflecting (plus z) red;intros.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
z, x, y: R2
X1: z + x ≤ z + y
x ≤ y apply P.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
z, x, y: R2
X1: z + x ≤ z + y
f x ≤ f y apply (order_reflecting (f z +)).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
z, x, y: R2
X1: z + x ≤ z + y
f z + f x ≤ f z + f y
    rewrite <-2!preserves_plus.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
z, x, y: R2
X1: z + x ≤ z + y
f (z + x) ≤ f (z + y) apply P.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
z, x, y: R2
X1: z + x ≤ z + y
z + x ≤ z + y trivial.
  -
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
forall x y : R2,
PropHolds (0 ≤ x) ->
PropHolds (0 ≤ y) -> PropHolds (0 ≤ x * y) intros.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
x, y: R2
X1: PropHolds (0 ≤ x)
X2: PropHolds (0 ≤ y)
PropHolds (0 ≤ x * y) apply P.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
x, y: R2
X1: PropHolds (0 ≤ x)
X2: PropHolds (0 ≤ y)
f 0 ≤ f (x * y) rewrite preserves_mult, preserves_0.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
R2: Type
Aplus: Plus R2
Amult: Mult R2
Azero: Zero R2
Aone: One R2
H4: IsSemiCRing R2
R2le: Le R2
is_mere_relation0: is_mere_relation R2 R2le
f: R2 -> R1
IsSemiRingPreserving0: IsSemiRingPreserving f
IsInjective0: IsInjective f
P: forall x y : R2, x ≤ y <-> f x ≤ f y
X: PartialOrder R2le
X0: forall x y : R2, x ≤ y -> {z : R2 & y = x + z}
x, y: R2
X1: PropHolds (0 ≤ x)
X2: PropHolds (0 ≤ y)
0 ≤ f x * f y
    apply nonneg_mult_compat; rewrite <-(preserves_0 (f:=f)); apply P;trivial.
  Qed.

 Context `{!IsSemiCRing R1} `{SemiRingOrder R2} `{!IsSemiCRing R2}
   `{!IsSemiRingPreserving (f : R1 -> R2)}.

  (* If a morphism agrees on the positive cone then it is order preserving *)
  Lemma preserving_preserves_nonneg : (forall x, 0 ≤ x -> 0 ≤ f x) -> OrderPreserving f.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
(forall x : R1, 0 ≤ x -> 0 ≤ f x) -> OrderPreserving f
  Proof.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
(forall x : R1, 0 ≤ x -> 0 ≤ f x) -> OrderPreserving f
  intros E.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 ≤ x -> 0 ≤ f x
OrderPreserving f
  repeat (split; try exact _).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 ≤ x -> 0 ≤ f x
OrderPreserving f
  intros x y F.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 ≤ x -> 0 ≤ f x
x, y: R1
F: x ≤ y
f x ≤ f y
  destruct (decompose_le F) as [z [Ez1 Ez2]].
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 ≤ x -> 0 ≤ f x
x, y: R1
F: x ≤ y
z: R1
Ez1: 0 ≤ z
Ez2: y = x + z
f x ≤ f y
  apply compose_le with (f z).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 ≤ x -> 0 ≤ f x
x, y: R1
F: x ≤ y
z: R1
Ez1: 0 ≤ z
Ez2: y = x + z
0 ≤ f z
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 ≤ x -> 0 ≤ f x
x, y: R1
F: x ≤ y
z: R1
Ez1: 0 ≤ z
Ez2: y = x + z
f y = f x + f z
  -
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 ≤ x -> 0 ≤ f x
x, y: R1
F: x ≤ y
z: R1
Ez1: 0 ≤ z
Ez2: y = x + z
0 ≤ f z apply E;trivial.
  -
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 ≤ x -> 0 ≤ f x
x, y: R1
F: x ≤ y
z: R1
Ez1: 0 ≤ z
Ez2: y = x + z
f y = f x + f z rewrite Ez2, (preserves_plus (f:=f)).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 ≤ x -> 0 ≤ f x
x, y: R1
F: x ≤ y
z: R1
Ez1: 0 ≤ z
Ez2: y = x + z
f x + f z = f x + f z trivial.
  Qed.

  Instance preserves_nonneg `{!OrderPreserving f} x
    : PropHolds (0 ≤ x) -> PropHolds (0 ≤ f x).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
OrderPreserving0: OrderPreserving f
x: R1
PropHolds (0 ≤ x) -> PropHolds (0 ≤ f x)
  Proof.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
OrderPreserving0: OrderPreserving f
x: R1
PropHolds (0 ≤ x) -> PropHolds (0 ≤ f x)
  intros.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
OrderPreserving0: OrderPreserving f
x: R1
X: PropHolds (0 ≤ x)
PropHolds (0 ≤ f x) rewrite <-(preserves_0 (f:=f)).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
OrderPreserving0: OrderPreserving f
x: R1
X: PropHolds (0 ≤ x)
PropHolds (f 0 ≤ f x) apply (order_preserving f);trivial.
  Qed.

  Lemma preserves_nonpos `{!OrderPreserving f} x : x ≤ 0 -> f x ≤ 0.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
OrderPreserving0: OrderPreserving f
x: R1
x ≤ 0 -> f x ≤ 0
  Proof.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
OrderPreserving0: OrderPreserving f
x: R1
x ≤ 0 -> f x ≤ 0
  intros.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
OrderPreserving0: OrderPreserving f
x: R1
X: x ≤ 0
f x ≤ 0 rewrite <-(preserves_0 (f:=f)).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
OrderPreserving0: OrderPreserving f
x: R1
X: x ≤ 0
f x ≤ f 0 apply (order_preserving f);trivial.
  Qed.

  Lemma preserves_ge_1 `{!OrderPreserving f} x : 1 ≤ x -> 1 ≤ f x.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
OrderPreserving0: OrderPreserving f
x: R1
1 ≤ x -> 1 ≤ f x
  Proof.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
OrderPreserving0: OrderPreserving f
x: R1
1 ≤ x -> 1 ≤ f x
  intros.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
OrderPreserving0: OrderPreserving f
x: R1
X: 1 ≤ x
1 ≤ f x rewrite <-(preserves_1 (f:=f)).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
OrderPreserving0: OrderPreserving f
x: R1
X: 1 ≤ x
f 1 ≤ f x apply (order_preserving f);trivial.
  Qed.

  Lemma preserves_le_1 `{!OrderPreserving f} x : x ≤ 1 -> f x ≤ 1.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
OrderPreserving0: OrderPreserving f
x: R1
x ≤ 1 -> f x ≤ 1
  Proof.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
OrderPreserving0: OrderPreserving f
x: R1
x ≤ 1 -> f x ≤ 1
  intros.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
OrderPreserving0: OrderPreserving f
x: R1
X: x ≤ 1
f x ≤ 1 rewrite <-(preserves_1 (f:=f)).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Ale: Le R1
H3: SemiRingOrder Ale
IsSemiCRing0: IsSemiCRing R1
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Ale0: Le R2
H8: SemiRingOrder Ale0
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
OrderPreserving0: OrderPreserving f
x: R1
X: x ≤ 1
f x ≤ f 1 apply (order_preserving f);trivial.
  Qed.
End another_semiring.

Section another_semiring_strict.
  Context `{StrictSemiRingOrder R1} `{StrictSemiRingOrder R2}
    `{!IsSemiCRing R1} `{!IsSemiCRing R2}
    `{!IsSemiRingPreserving (f : R1 -> R2)}.

  Lemma strictly_preserving_preserves_pos
    : (forall x, 0 < x -> 0 < f x) -> StrictlyOrderPreserving f.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
(forall x : R1, 0 < x -> 0 < f x) ->
StrictlyOrderPreserving f
  Proof.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
(forall x : R1, 0 < x -> 0 < f x) ->
StrictlyOrderPreserving f
  intros E.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 < x -> 0 < f x
StrictlyOrderPreserving f
  repeat (split; try exact _).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 < x -> 0 < f x
StrictlyOrderPreserving f
  intros x y F.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 < x -> 0 < f x
x, y: R1
F: x < y
f x < f y
  destruct (decompose_lt F) as [z [Ez1 Ez2]].
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 < x -> 0 < f x
x, y: R1
F: x < y
z: R1
Ez1: 0 < z
Ez2: y = x + z
f x < f y
  apply compose_lt with (f z).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 < x -> 0 < f x
x, y: R1
F: x < y
z: R1
Ez1: 0 < z
Ez2: y = x + z
0 < f z
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 < x -> 0 < f x
x, y: R1
F: x < y
z: R1
Ez1: 0 < z
Ez2: y = x + z
f y = f x + f z
  -
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 < x -> 0 < f x
x, y: R1
F: x < y
z: R1
Ez1: 0 < z
Ez2: y = x + z
0 < f z apply E.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 < x -> 0 < f x
x, y: R1
F: x < y
z: R1
Ez1: 0 < z
Ez2: y = x + z
0 < z trivial.
  -
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 < x -> 0 < f x
x, y: R1
F: x < y
z: R1
Ez1: 0 < z
Ez2: y = x + z
f y = f x + f z rewrite Ez2, (preserves_plus (f:=f)).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
E: forall x : R1, 0 < x -> 0 < f x
x, y: R1
F: x < y
z: R1
Ez1: 0 < z
Ez2: y = x + z
f x + f z = f x + f z trivial.
  Qed.

  Instance preserves_pos `{!StrictlyOrderPreserving f} x
   : PropHolds (0 < x) -> PropHolds (0 < f x).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
StrictlyOrderPreserving0: StrictlyOrderPreserving f
x: R1
PropHolds (0 < x) -> PropHolds (0 < f x)
  Proof.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
StrictlyOrderPreserving0: StrictlyOrderPreserving f
x: R1
PropHolds (0 < x) -> PropHolds (0 < f x)
  intros.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
StrictlyOrderPreserving0: StrictlyOrderPreserving f
x: R1
X: PropHolds (0 < x)
PropHolds (0 < f x) rewrite <-(preserves_0 (f:=f)).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
StrictlyOrderPreserving0: StrictlyOrderPreserving f
x: R1
X: PropHolds (0 < x)
PropHolds (f 0 < f x)
  apply (strictly_order_preserving f);trivial.
  Qed.

  Lemma preserves_neg `{!StrictlyOrderPreserving f} x
    : x < 0 -> f x < 0.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
StrictlyOrderPreserving0: StrictlyOrderPreserving f
x: R1
x < 0 -> f x < 0
  Proof.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
StrictlyOrderPreserving0: StrictlyOrderPreserving f
x: R1
x < 0 -> f x < 0
  intros.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
StrictlyOrderPreserving0: StrictlyOrderPreserving f
x: R1
X: x < 0
f x < 0 rewrite <-(preserves_0 (f:=f)).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
StrictlyOrderPreserving0: StrictlyOrderPreserving f
x: R1
X: x < 0
f x < f 0
  apply (strictly_order_preserving f);trivial.
  Qed.

  Lemma preserves_gt_1 `{!StrictlyOrderPreserving f} x
    : 1 < x -> 1 < f x.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
StrictlyOrderPreserving0: StrictlyOrderPreserving f
x: R1
1 < x -> 1 < f x
  Proof.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
StrictlyOrderPreserving0: StrictlyOrderPreserving f
x: R1
1 < x -> 1 < f x
  intros.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
StrictlyOrderPreserving0: StrictlyOrderPreserving f
x: R1
X: 1 < x
1 < f x rewrite <-(preserves_1 (f:=f)).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
StrictlyOrderPreserving0: StrictlyOrderPreserving f
x: R1
X: 1 < x
f 1 < f x
  apply (strictly_order_preserving f);trivial.
  Qed.

  Lemma preserves_lt_1 `{!StrictlyOrderPreserving f} x
    : x < 1 -> f x < 1.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
StrictlyOrderPreserving0: StrictlyOrderPreserving f
x: R1
x < 1 -> f x < 1
  Proof.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
StrictlyOrderPreserving0: StrictlyOrderPreserving f
x: R1
x < 1 -> f x < 1
  intros.
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
StrictlyOrderPreserving0: StrictlyOrderPreserving f
x: R1
X: x < 1
f x < 1 rewrite <-(preserves_1 (f:=f)).
R1: Type
H: Plus R1
H0: Mult R1
H1: Zero R1
H2: One R1
Alt: Lt R1
H3: StrictSemiRingOrder Alt
R2: Type
H4: Plus R2
H5: Mult R2
H6: Zero R2
H7: One R2
Alt0: Lt R2
H8: StrictSemiRingOrder Alt0
IsSemiCRing0: IsSemiCRing R1
IsSemiCRing1: IsSemiCRing R2
f: R1 -> R2
IsSemiRingPreserving0: IsSemiRingPreserving
  (f : R1 -> R2)
StrictlyOrderPreserving0: StrictlyOrderPreserving f
x: R1
X: x < 1
f x < f 1
  apply (strictly_order_preserving f);trivial.
  Qed.
End another_semiring_strict.

(* Due to bug #2528 *)
#[export]
Hint Extern 15 (PropHolds (_ ≤ _ _)) =>
  eapply @preserves_nonneg : typeclass_instances.
#[export]
Hint Extern 15 (PropHolds (_ < _ _)) =>
  eapply @preserves_pos : typeclass_instances.

(* Oddly enough, the above hints do not work for goals of the following shape? *)
#[export]
Hint Extern 15 (PropHolds (_ ≤ '_)) =>
  eapply @preserves_nonneg : typeclass_instances.
#[export]
Hint Extern 15 (PropHolds (_ < '_)) =>
  eapply @preserves_pos : typeclass_instances.