Require Import
  HoTT.Classes.implementations.peano_naturals
  HoTT.Classes.implementations.natpair_integers
  HoTT.Classes.interfaces.abstract_algebra
  HoTT.Classes.interfaces.naturals
  HoTT.Classes.interfaces.rationals
  HoTT.Classes.interfaces.orders
  HoTT.Classes.theory.groups
  HoTT.Classes.theory.integers
  HoTT.Classes.theory.dec_fields
  HoTT.Classes.orders.sum
  HoTT.Classes.orders.dec_fields
  HoTT.Classes.orders.lattices
  HoTT.Classes.theory.additional_operations
  HoTT.Classes.tactics.ring_quote
  HoTT.Classes.tactics.ring_tac.
[Loading ML file number_string_notation_plugin.cmxs (using legacy method) ... done]

Import Quoting.Instances.
Import NatPair.Instances.
Local Set Universe Minimization ToSet.

Section contents.
Context `{Funext} `{Univalence}.
Universe UQ.
Context {Q : Type@{UQ} } {Qap : Apart@{UQ UQ} Q}
  {Qplus : Plus Q} {Qmult : Mult Q}
  {Qzero : Zero Q} {Qone : One Q} {Qneg : Negate Q} {Qrecip : DecRecip Q}
  {Qle : Le@{UQ UQ} Q} {Qlt : Lt@{UQ UQ} Q}
  {QtoField : RationalsToField@{UQ UQ UQ UQ} Q}
  {Qrats : Rationals@{UQ UQ UQ UQ UQ UQ UQ UQ UQ UQ} Q}
  {Qtrivialapart : TrivialApart Q} {Qdec : DecidablePaths Q}
  {Qmeet : Meet Q} {Qjoin : Join Q} {Qlattice : LatticeOrder Qle}
  {Qle_total : TotalRelation (@le Q _)}
  {Qabs : Abs Q}.

#[export] Instance rational_1_neq_0 : PropHolds (@apart Q _ 1 0).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
PropHolds (1 ≶ 0)
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
PropHolds (1 ≶ 0)
red.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
1 ≶ 0 apply trivial_apart.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
1 <> 0 solve_propholds.
Qed.

Record Qpos@{} : Type@{UQ} := mkQpos { pos : Q; is_pos : 0 < pos }.
Notation "Q+" := Qpos.

#[export] Instance Qpos_Q@{} : Cast Qpos Q := pos.
Arguments Qpos_Q /.

Lemma Qpos_plus_pr@{} : forall a b : Qpos, 0 < 'a + 'b.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b : Q+, 0 < ' a + ' b
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b : Q+, 0 < ' a + ' b
intros.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
0 < ' a + ' b
apply semirings.pos_plus_compat;apply is_pos.
Qed.

#[export] Instance Qpos_plus@{} : Plus Qpos := fun a b => mkQpos _ (Qpos_plus_pr a b).

#[export] Instance pos_is_pos@{} : forall q : Q+, PropHolds (0 < ' q)
  := is_pos.

Lemma pos_eq@{} : forall a b : Q+, @paths Q (' a) (' b) -> a = b.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b : Q+, ' a = ' b -> a = b
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b : Q+, ' a = ' b -> a = b
intros [a Ea] [b Eb] E.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a: Q
Ea: 0 < a
b: Q
Eb: 0 < b
E: ' {| pos := a; is_pos := Ea |} =
' {| pos := b; is_pos := Eb |}
{| pos := a; is_pos := Ea |} =
{| pos := b; is_pos := Eb |}
change (a = b) in E.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a: Q
Ea: 0 < a
b: Q
Eb: 0 < b
E: a = b
{| pos := a; is_pos := Ea |} =
{| pos := b; is_pos := Eb |}
destruct E;apply ap;apply path_ishprop.
Qed.

#[export] Instance Qpos_isset : IsHSet Q+.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
IsHSet Q+
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
IsHSet Q+
apply (@HSet.ishset_hrel_subpaths _ (fun e d => ' e = ' d)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Reflexive (fun e d : Q+ => ' e = ' d)
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x y : Q+, IsHProp (' x = ' y)
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x y : Q+, ' x = ' y -> x = y
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Reflexive (fun e d : Q+ => ' e = ' d) intros e; reflexivity.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x y : Q+, IsHProp (' x = ' y) exact _.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x y : Q+, ' x = ' y -> x = y exact pos_eq.
Qed.

#[export] Instance Qpos_one@{} : One Q+.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
One Q+
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
One Q+
exists 1.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
0 < 1 exact lt_0_1.
Defined.

#[export] Instance Qpos_mult@{} : Mult Q+.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Mult Q+
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Mult Q+
intros a b;exists (' a * ' b).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
0 < ' a * ' b
solve_propholds.
Defined.

#[export] Instance qpos_plus_comm@{} : Commutative (@plus Q+ _).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Commutative plus
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Commutative plus
hnf.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x y : Q+, x + y = y + x intros.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q+
x + y = y + x
apply pos_eq.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q+
' (x + y) = ' (y + x) change (' x + ' y = ' y + ' x).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q+
' x + ' y = ' y + ' x
apply plus_comm.
Qed.

#[export] Instance qpos_mult_comm@{} : Commutative (@mult Q+ _).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Commutative mult
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Commutative mult
hnf;intros;apply pos_eq,mult_comm.
Qed.

#[export] Instance pos_recip@{} : DecRecip Q+.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
DecRecip Q+
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
DecRecip Q+
intros e.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
Q+ exists (/ ' e).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
0 < / ' e
apply pos_dec_recip_compat.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
PropHolds (0 < ' e)
solve_propholds.
Defined.

#[export] Instance pos_of_nat@{} : Cast nat Q+.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Cast nat Q+
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Cast nat Q+
intros n.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
n: nat
Q+ destruct n as [|k].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Q+
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
k: nat
Q+
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Q+ exists 1;exact lt_0_1.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
k: nat
Q+ exists (naturals_to_semiring nat Q (S k)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
k: nat
0 < naturals_to_semiring nat Q k.+1%nat
  induction k as [|k Ik].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
0 < naturals_to_semiring nat Q 1%nat
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
k: nat
Ik: 0 < naturals_to_semiring nat Q k.+1%nat
0 < naturals_to_semiring nat Q k.+2%nat
  +
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
0 < naturals_to_semiring nat Q 1%nat change (0 < 1).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
0 < 1 exact lt_0_1.
  +
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
k: nat
Ik: 0 < naturals_to_semiring nat Q k.+1%nat
0 < naturals_to_semiring nat Q k.+2%nat change (0 < 1 + naturals_to_semiring nat Q (S k)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
k: nat
Ik: 0 < naturals_to_semiring nat Q k.+1%nat
0 < 1 + naturals_to_semiring nat Q k.+1%nat
    set (K := naturals_to_semiring nat Q (S k)) in *;clearbody K.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
k: nat
K: Q
Ik: 0 < K
0 < 1 + K
    apply pos_plus_compat.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
k: nat
K: Q
Ik: 0 < K
PropHolds (0 < 1)
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
k: nat
K: Q
Ik: 0 < K
PropHolds (0 < K)
    *
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
k: nat
K: Q
Ik: 0 < K
PropHolds (0 < 1) exact lt_0_1.
    *
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
k: nat
K: Q
Ik: 0 < K
PropHolds (0 < K) trivial.
Defined.

Lemma pos_recip_r@{} : forall e : Q+, e / e = 1.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall e : Q+, e / e = 1
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall e : Q+, e / e = 1
intros;apply pos_eq.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
' (e / e) = ' 1
unfold dec_recip,cast,pos_recip;simpl.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
' e *
' {|
    pos := / ' e;
    is_pos :=
      pos_dec_recip_compat (' e) (pos_is_pos e)
  |} = 1
change (' e / ' e = 1).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
' e / ' e = 1 apply dec_recip_inverse.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
' e <> 0
apply lt_ne_flip.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
0 < ' e solve_propholds.
Qed.

Lemma pos_recip_r'@{} : forall e : Q+, @paths Q (' e / ' e) 1.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall e : Q+, ' e / ' e = 1
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall e : Q+, ' e / ' e = 1
intros.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
' e / ' e = 1 change (' (e / e) = 1).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
' (e / e) = 1 rewrite pos_recip_r.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
' 1 = 1 reflexivity.
Qed.

Lemma pos_mult_1_r@{} : forall e : Q+, e * 1 = e.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall e : Q+, e * 1 = e
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall e : Q+, e * 1 = e
intros;apply pos_eq.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
' (e * 1) = ' e apply mult_1_r.
Qed.

Lemma pos_split2@{} : forall e : Q+, e = e / 2 + e / 2.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall e : Q+, e = e / 2 + e / 2
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall e : Q+, e = e / 2 + e / 2
intros.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
e = e / 2 + e / 2
path_via (e * (2 / 2)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
e = e * (2 / 2)
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
e * (2 / 2) = e / 2 + e / 2
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
e = e * (2 / 2) rewrite pos_recip_r,pos_mult_1_r;reflexivity.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
e * (2 / 2) = e / 2 + e / 2 apply pos_eq.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
' (e * (2 / 2)) = ' (e / 2 + e / 2) change (' e * (2 / 2) = ' e / 2 + ' e / 2).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
' e * (2 / 2) = ' e / 2 + ' e / 2
  ring_tac.ring_with_nat.
Qed.

Lemma pos_split3@{} : forall e : Q+, e = e / 3 + e / 3 + e / 3.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall e : Q+, e = e / 3 + e / 3 + e / 3
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall e : Q+, e = e / 3 + e / 3 + e / 3
intros.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
e = e / 3 + e / 3 + e / 3
path_via (e * (3 / 3)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
e = e * (3 / 3)
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
e * (3 / 3) = e / 3 + e / 3 + e / 3
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
e = e * (3 / 3) rewrite pos_recip_r,pos_mult_1_r;reflexivity.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
e * (3 / 3) = e / 3 + e / 3 + e / 3 apply pos_eq.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
' (e * (3 / 3)) = ' (e / 3 + e / 3 + e / 3) change (' e * (3 / 3) = ' e / 3 + ' e / 3 + ' e / 3).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
' e * (3 / 3) = ' e / 3 + ' e / 3 + ' e / 3
  ring_tac.ring_with_nat.
Qed.

#[export] Instance Qpos_mult_assoc@{} : Associative (@mult Q+ _).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Associative mult
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Associative mult
hnf.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x y z : Q+, x * (y * z) = x * y * z
intros;apply pos_eq.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y, z: Q+
' (x * (y * z)) = ' (x * y * z)
apply mult_assoc.
Qed.

#[export] Instance Qpos_plus_assoc@{} : Associative (@plus Q+ _).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Associative plus
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Associative plus
hnf.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x y z : Q+, x + (y + z) = x + y + z
intros;apply pos_eq.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y, z: Q+
' (x + (y + z)) = ' (x + y + z)
apply plus_assoc.
Qed.

#[export] Instance Qpos_mult_1_l@{} : LeftIdentity (@mult Q+ _) 1.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
LeftIdentity mult 1
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
LeftIdentity mult 1
hnf;intros;apply pos_eq;apply mult_1_l.
Qed.

#[export] Instance Qpos_mult_1_r@{} : RightIdentity (@mult Q+ _) 1.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
RightIdentity mult 1
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
RightIdentity mult 1
hnf;intros;apply pos_eq;apply mult_1_r.
Qed.

Lemma pos_recip_through_plus@{} : forall a b c : Q+,
  a + b = c * (a / c + b / c).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b c : Q+, a + b = c * (a / c + b / c)
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b c : Q+, a + b = c * (a / c + b / c)
intros.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c: Q+
a + b = c * (a / c + b / c) path_via ((a + b) * (c / c)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c: Q+
a + b = (a + b) * (c / c)
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c: Q+
(a + b) * (c / c) = c * (a / c + b / c)
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c: Q+
a + b = (a + b) * (c / c) rewrite pos_recip_r;apply pos_eq,symmetry,mult_1_r.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c: Q+
(a + b) * (c / c) = c * (a / c + b / c) apply pos_eq;ring_tac.ring_with_nat.
Qed.

Lemma pos_unconjugate@{} : forall a b : Q+, a * b / a = b.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b : Q+, (a * b) / a = b
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b : Q+, (a * b) / a = b
intros.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
(a * b) / a = b path_via (a / a * b).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
(a * b) / a = a / a * b
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
a / a * b = b
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
(a * b) / a = a / a * b apply pos_eq;ring_tac.ring_with_nat.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
a / a * b = b rewrite pos_recip_r;apply Qpos_mult_1_l.
Qed.

Lemma Qpos_recip_1 : / 1 = 1 :> Q+.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
/ 1 = 1
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
/ 1 = 1
apply pos_eq.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
' (/ 1) = ' 1 exact dec_recip_1.
Qed.

Lemma Qpos_plus_mult_distr_l : @LeftDistribute Q+ mult plus.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
LeftDistribute mult plus
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
LeftDistribute mult plus
hnf.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b c : Q+, a * (b + c) = a * b + a * c intros;apply pos_eq,plus_mult_distr_l.
Qed.

#[export] Instance Qpos_meet@{} : Meet Q+.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Meet Q+
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Meet Q+
intros a b.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
Q+ exists (meet (' a) (' b)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
0 < ' a ⊓ ' b
apply not_le_lt_flip.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
~ (' a ⊓ ' b ≤ 0) intros E.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ⊓ ' b ≤ 0
Empty
destruct (total_meet_either (' a) (' b)) as [E1|E1];
rewrite E1 in E;(eapply le_iff_not_lt_flip;[exact E|]);
solve_propholds.
Defined.

#[export] Instance Qpos_join@{} : Join Q+.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Join Q+
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Join Q+
intros a b.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
Q+ exists (join (' a) (' b)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
0 < ' a ⊔ ' b
apply not_le_lt_flip.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
~ (' a ⊔ ' b ≤ 0) intros E.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ⊔ ' b ≤ 0
Empty
destruct (total_join_either (' a) (' b)) as [E1|E1];
rewrite E1 in E;(eapply le_iff_not_lt_flip;[exact E|]);
solve_propholds.
Defined.

Lemma Q_sum_eq_join_meet@{} : forall a b c d : Q, a + b = c + d ->
  a + b = join a c + meet b d.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b c d : Q,
a + b = c + d -> a + b = a ⊔ c + (b ⊓ d)
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b c d : Q,
a + b = c + d -> a + b = a ⊔ c + (b ⊓ d)
intros ???? E.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, d: Q
E: a + b = c + d
a + b = a ⊔ c + (b ⊓ d)
destruct (total le a c) as [E1|E1].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, d: Q
E: a + b = c + d
E1: a ≤ c
a + b = a ⊔ c + (b ⊓ d)
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, d: Q
E: a + b = c + d
E1: c ≤ a
a + b = a ⊔ c + (b ⊓ d)
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, d: Q
E: a + b = c + d
E1: a ≤ c
a + b = a ⊔ c + (b ⊓ d) rewrite (join_r _ _ E1).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, d: Q
E: a + b = c + d
E1: a ≤ c
a + b = c + (b ⊓ d) rewrite meet_r;trivial.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, d: Q
E: a + b = c + d
E1: a ≤ c
d ≤ b
  apply (order_preserving (+ b)) in E1.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, d: Q
E: a + b = c + d
E1: a + b ≤ c + b
d ≤ b
  rewrite E in E1.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, d: Q
E: a + b = c + d
E1: c + d ≤ c + b
d ≤ b apply (order_reflecting (c +)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, d: Q
E: a + b = c + d
E1: c + d ≤ c + b
c + d ≤ c + b trivial.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, d: Q
E: a + b = c + d
E1: c ≤ a
a + b = a ⊔ c + (b ⊓ d) rewrite (join_l _ _ E1).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, d: Q
E: a + b = c + d
E1: c ≤ a
a + b = a + (b ⊓ d)
  rewrite meet_l;trivial.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, d: Q
E: a + b = c + d
E1: c ≤ a
b ≤ d
  apply (order_reflecting (a +)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, d: Q
E: a + b = c + d
E1: c ≤ a
a + b ≤ a + d rewrite E.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, d: Q
E: a + b = c + d
E1: c ≤ a
c + d ≤ a + d apply (order_preserving (+ d)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, d: Q
E: a + b = c + d
E1: c ≤ a
c ≤ a
  trivial.
Qed.

Lemma Qpos_sum_eq_join_meet@{} : forall a b c d : Q+, a + b = c + d ->
  a + b = join a c + meet b d.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b c d : Q+,
a + b = c + d -> a + b = a ⊔ c + (b ⊓ d)
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b c d : Q+,
a + b = c + d -> a + b = a ⊔ c + (b ⊓ d)
intros ???? E.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, d: Q+
E: a + b = c + d
a + b = a ⊔ c + (b ⊓ d)
apply pos_eq;apply Q_sum_eq_join_meet.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, d: Q+
E: a + b = c + d
' a + ' b = ' c + ' d
change (' a + ' b) with (' (a + b)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, d: Q+
E: a + b = c + d
' (a + b) = ' c + ' d rewrite E;reflexivity.
Qed.

Lemma Qpos_le_lt_min : forall a b : Q+, ' a <= ' b ->
  exists c ca cb, a = c + ca /\ b = c + cb.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b : Q+,
' a ≤ ' b ->
{c : Q+ &
{ca : Q+ &
{cb : Q+ & ((a = plus c ca) * (b = plus c cb))%type}}}
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b : Q+,
' a ≤ ' b ->
{c : Q+ &
{ca : Q+ &
{cb : Q+ & ((a = plus c ca) * (b = plus c cb))%type}}}
intros a b E.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
{c : Q+ &
{ca : Q+ &
{cb : Q+ & ((a = plus c ca) * (b = plus c cb))%type}}} exists (a/2),(a/2).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
{cb : Q+ &
((a = plus (a / 2) (a / 2)) * (b = plus (a / 2) cb))%type}
simple refine (exist _ _ _);simpl.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
Q+
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
((a = plus (a / 2) (a / 2)) * (b = plus (a / 2) ?Goal))%type
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
Q+ exists (' (a / 2) + (' b - ' a)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
0 < ' (a / 2) + (' b - ' a)
  apply nonneg_plus_lt_compat_r.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
0 ≤ ' b - ' a
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
0 < ' (a / 2)
  +
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
0 ≤ ' b - ' a apply (snd (flip_nonneg_minus _ _)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
' a ≤ ' b trivial.
  +
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
0 < ' (a / 2) solve_propholds.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
((a = plus (a / 2) (a / 2)) *
 (b =
  plus (a / 2)
    {|
      pos := plus (' (a / 2)) (' b - ' a);
      is_pos :=
        nonneg_plus_lt_compat_r 0 (' (a / 2))
          (' b - ' a)
          (snd (flip_nonneg_minus (' a) (' b)) E)
          (pos_is_pos (a / 2))
    |}))%type split.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
a = a / 2 + a / 2
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
b =
a / 2 +
{|
  pos := ' (a / 2) + (' b - ' a);
  is_pos :=
    nonneg_plus_lt_compat_r 0 
      (' (a / 2)) (' b - ' a)
      (snd (flip_nonneg_minus (' a) (' b)) E)
      (pos_is_pos (a / 2))
|}
  +
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
a = a / 2 + a / 2 apply pos_split2.
  +
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
b =
a / 2 +
{|
  pos := ' (a / 2) + (' b - ' a);
  is_pos :=
    nonneg_plus_lt_compat_r 0 (' (a / 2)) (' b - ' a)
      (snd (flip_nonneg_minus (' a) (' b)) E)
      (pos_is_pos (a / 2))
|} apply pos_eq.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
' b =
' (a / 2 +
   {|
     pos := ' (a / 2) + (' b - ' a);
     is_pos :=
       nonneg_plus_lt_compat_r 0 (' (a / 2))
         (' b - ' a)
         (snd (flip_nonneg_minus (' a) (' b)) E)
         (pos_is_pos (a / 2))
   |}) unfold cast at 2;simpl.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
' b =
' (a / 2) +
' {|
    pos := ' (a / 2) + (' b - ' a);
    is_pos :=
      nonneg_plus_lt_compat_r 0 (' (a / 2))
        (' b - ' a)
        (snd (flip_nonneg_minus (' a) (' b)) E)
        (pos_mult_compat (' a) (' (/ 2))
           (pos_is_pos a)
           (pos_dec_recip_compat (' 2)
              (Qpos_plus_pr 1 1)))
  |}
    unfold cast at 3;simpl.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
' b = ' (a / 2) + (' (a / 2) + (' b - ' a))
    set (a':=a/2);rewrite (pos_split2 a);unfold a';clear a'.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
' b =
' (a / 2) + (' (a / 2) + (' b - ' (a / 2 + a / 2)))
    ring_tac.ring_with_integers (NatPair.Z nat).
Qed.

Lemma Qpos_lt_min@{} : forall a b : Q+, exists c ca cb : Q+,
  a = c + ca /\ b = c + cb.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b : Q+,
{c : Q+ &
{ca : Q+ &
{cb : Q+ & ((a = plus c ca) * (b = plus c cb))%type}}}
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b : Q+,
{c : Q+ &
{ca : Q+ &
{cb : Q+ & ((a = plus c ca) * (b = plus c cb))%type}}}
intros.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
{c : Q+ &
{ca : Q+ &
{cb : Q+ & ((a = plus c ca) * (b = plus c cb))%type}}}
destruct (total le (' a) (' b)) as [E|E].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
{c : Q+ &
{ca : Q+ &
{cb : Q+ & ((a = plus c ca) * (b = plus c cb))%type}}}
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' b ≤ ' a
{c : Q+ &
{ca : Q+ &
{cb : Q+ & ((a = plus c ca) * (b = plus c cb))%type}}}
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' a ≤ ' b
{c : Q+ &
{ca : Q+ &
{cb : Q+ & ((a = plus c ca) * (b = plus c cb))%type}}} apply Qpos_le_lt_min;trivial.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: ' b ≤ ' a
{c : Q+ &
{ca : Q+ &
{cb : Q+ & ((a = plus c ca) * (b = plus c cb))%type}}} apply Qpos_le_lt_min in E.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q+
E: {c : Q+ &
{ca : Q+ &
{cb : Q+ &
((b = plus c ca) * (a = plus c cb))%type}}}
{c : Q+ &
{ca : Q+ &
{cb : Q+ & ((a = plus c ca) * (b = plus c cb))%type}}} destruct E as [c [cb [ca [E1 E2]]]].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c, cb, ca: Q+
E1: b = c + cb
E2: a = c + ca
{c : Q+ &
{ca : Q+ &
{cb : Q+ & ((a = plus c ca) * (b = plus c cb))%type}}}
  exists c,ca,cb;auto.
Qed.

Definition Qpos_diff : forall q r : Q, q < r -> Q+.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q r : Q, q < r -> Q+
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q r : Q, q < r -> Q+
intros q r E;exists (r-q).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
0 < r - q
exact (snd (flip_pos_minus _ _) E).
Defined.

Lemma Qpos_diff_pr@{} : forall q r E, r = q + ' (Qpos_diff q r E).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall (q r : Q) (E : q < r),
r = q + ' Qpos_diff q r E
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall (q r : Q) (E : q < r),
r = q + ' Qpos_diff q r E
intros q r E.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
r = q + ' Qpos_diff q r E change (r = q + (r - q)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
r = q + (r - q)
abstract ring_tac.ring_with_integers (NatPair.Z nat).
Qed.


Lemma Qmeet_plus_l : forall a b c : Q, meet (a + b) (a + c) = a + meet b c.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b c : Q, (a + b) ⊓ (a + c) = a + (b ⊓ c)
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b c : Q, (a + b) ⊓ (a + c) = a + (b ⊓ c)
intros.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c: Q
(a + b) ⊓ (a + c) = a + (b ⊓ c) destruct (total le b c) as [E|E].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c: Q
E: b ≤ c
(a + b) ⊓ (a + c) = a + (b ⊓ c)
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c: Q
E: c ≤ b
(a + b) ⊓ (a + c) = a + (b ⊓ c)
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c: Q
E: b ≤ c
(a + b) ⊓ (a + c) = a + (b ⊓ c) rewrite (meet_l _ _ E).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c: Q
E: b ≤ c
(a + b) ⊓ (a + c) = a + b apply meet_l.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c: Q
E: b ≤ c
a + b ≤ a + c
  apply (order_preserving (a +)),E.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c: Q
E: c ≤ b
(a + b) ⊓ (a + c) = a + (b ⊓ c) rewrite (meet_r _ _ E).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c: Q
E: c ≤ b
(a + b) ⊓ (a + c) = a + c apply meet_r.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c: Q
E: c ≤ b
a + c ≤ a + b
  apply (order_preserving (a +)),E.
Qed.

Lemma Qabs_nonneg@{} : forall q : Q, 0 <= abs q.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q : Q, 0 ≤ abs q
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q : Q, 0 ≤ abs q
intros q;destruct (total_abs_either q) as [E|E];destruct E as [E1 E2];rewrite E2.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E1: 0 ≤ q
E2: abs q = q
0 ≤ q
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E1: q ≤ 0
E2: abs q = - q
0 ≤ - q
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E1: 0 ≤ q
E2: abs q = q
0 ≤ q trivial.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E1: q ≤ 0
E2: abs q = - q
0 ≤ - q apply flip_nonneg_negate.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E1: q ≤ 0
E2: abs q = - q
- - q ≤ 0
 rewrite involutive;trivial.
Qed.

Lemma Qabs_nonpos_0@{} : forall q : Q, abs q <= 0 -> q = 0.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q : Q, abs q ≤ 0 -> q = 0
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q : Q, abs q ≤ 0 -> q = 0
intros q E.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E: abs q ≤ 0
q = 0 pose proof (antisymmetry le _ _ E (Qabs_nonneg _)) as E1.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E: abs q ≤ 0
E1: abs q = 0
q = 0
destruct (total_abs_either q) as [[E2 E3]|[E2 E3]];rewrite E3 in E1.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E: abs q ≤ 0
E1: q = 0
E2: 0 ≤ q
E3: abs q = q
q = 0
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E: abs q ≤ 0
E1: - q = 0
E2: q ≤ 0
E3: abs q = - q
q = 0
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E: abs q ≤ 0
E1: q = 0
E2: 0 ≤ q
E3: abs q = q
q = 0 trivial.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E: abs q ≤ 0
E1: - q = 0
E2: q ≤ 0
E3: abs q = - q
q = 0 apply (injective (-)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E: abs q ≤ 0
E1: - q = 0
E2: q ≤ 0
E3: abs q = - q
- q = - 0 rewrite negate_0.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E: abs q ≤ 0
E1: - q = 0
E2: q ≤ 0
E3: abs q = - q
- q = 0 trivial.
Qed.

Lemma Qabs_0_or_pos : forall q : Q, q = 0 |_| 0 < abs q.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q : Q, (q = 0) + (0 < abs q)
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q : Q, (q = 0) + (0 < abs q)
intros q.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
((q = 0) + (0 < abs q))%type destruct (le_or_lt (abs q) 0) as [E|E].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E: abs q ≤ 0
((q = 0) + (0 < abs q))%type
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E: 0 < abs q
((q = 0) + (0 < abs q))%type
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E: abs q ≤ 0
((q = 0) + (0 < abs q))%type left.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E: abs q ≤ 0
q = 0 apply Qabs_nonpos_0.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E: abs q ≤ 0
abs q ≤ 0 trivial.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E: 0 < abs q
((q = 0) + (0 < abs q))%type right.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E: 0 < abs q
0 < abs q trivial.
Qed.

Lemma Qabs_of_nonneg@{} : forall q : Q, 0 <= q -> abs q = q.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q : Q, 0 ≤ q -> abs q = q
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q : Q, 0 ≤ q -> abs q = q
intro;apply ((abs_sig _).2).
Qed.

Lemma Qabs_of_nonpos : forall q : Q, q <= 0 -> abs q = - q.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q : Q, q ≤ 0 -> abs q = - q
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q : Q, q ≤ 0 -> abs q = - q
intro;apply ((abs_sig _).2).
Qed.

Lemma Qabs_le_raw@{} : forall x : Q, x <= abs x.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x : Q, x ≤ abs x
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x : Q, x ≤ abs x
intros x;destruct (total_abs_either x) as [[E1 E2]|[E1 E2]].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
E1: 0 ≤ x
E2: abs x = x
x ≤ abs x
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
E1: x ≤ 0
E2: abs x = - x
x ≤ abs x
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
E1: 0 ≤ x
E2: abs x = x
x ≤ abs x rewrite E2;reflexivity.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
E1: x ≤ 0
E2: abs x = - x
x ≤ abs x transitivity (0:Q);trivial.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
E1: x ≤ 0
E2: abs x = - x
(0 : Q) ≤ abs x
  rewrite E2.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
E1: x ≤ 0
E2: abs x = - x
0 ≤ - x apply flip_nonpos_negate.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
E1: x ≤ 0
E2: abs x = - x
x ≤ 0 trivial.
Qed.

Lemma Qabs_neg@{} : forall x : Q, abs (- x) = abs x.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x : Q, abs (- x) = abs x
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x : Q, abs (- x) = abs x
intros x.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
abs (- x) = abs x destruct (total_abs_either x) as [[E1 E2]|[E1 E2]].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
E1: 0 ≤ x
E2: abs x = x
abs (- x) = abs x
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
E1: x ≤ 0
E2: abs x = - x
abs (- x) = abs x
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
E1: 0 ≤ x
E2: abs x = x
abs (- x) = abs x rewrite E2.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
E1: 0 ≤ x
E2: abs x = x
abs (- x) = x path_via (- - x);[|rewrite involutive;trivial].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
E1: 0 ≤ x
E2: abs x = x
abs (- x) = - - x
  apply ((abs_sig (- x)).2).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
E1: 0 ≤ x
E2: abs x = x
- x ≤ 0 apply flip_nonneg_negate;trivial.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
E1: x ≤ 0
E2: abs x = - x
abs (- x) = abs x rewrite E2.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
E1: x ≤ 0
E2: abs x = - x
abs (- x) = - x apply ((abs_sig (- x)).2).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
E1: x ≤ 0
E2: abs x = - x
0 ≤ - x apply flip_nonpos_negate;trivial.
Qed.

Lemma Qabs_le_neg_raw : forall x : Q, - x <= abs x.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x : Q, - x ≤ abs x
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x : Q, - x ≤ abs x
intros x.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
- x ≤ abs x rewrite <-Qabs_neg.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x: Q
- x ≤ abs (- x) apply Qabs_le_raw.
Qed.

Lemma Q_abs_le_pr@{} : forall x y : Q, abs x <= y <-> - y <= x /\ x <= y.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x y : Q, abs x ≤ y <-> - y ≤ x ≤ y
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x y : Q, abs x ≤ y <-> - y ≤ x ≤ y
intros x y;split.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q
abs x ≤ y -> - y ≤ x ≤ y
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q
- y ≤ x ≤ y -> abs x ≤ y
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q
abs x ≤ y -> - y ≤ x ≤ y intros E.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q
E: abs x ≤ y
- y ≤ x ≤ y split.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q
E: abs x ≤ y
- y ≤ x
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q
E: abs x ≤ y
x ≤ y
  +
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q
E: abs x ≤ y
- y ≤ x apply flip_le_negate.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q
E: abs x ≤ y
- x ≤ - - y rewrite involutive.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q
E: abs x ≤ y
- x ≤ y transitivity (abs x);trivial.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q
E: abs x ≤ y
- x ≤ abs x
    apply Qabs_le_neg_raw.
  +
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q
E: abs x ≤ y
x ≤ y transitivity (abs x);trivial.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q
E: abs x ≤ y
x ≤ abs x
    apply Qabs_le_raw.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q
- y ≤ x ≤ y -> abs x ≤ y intros [E1 E2].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q
E1: - y ≤ x
E2: x ≤ y
abs x ≤ y
  destruct (total_abs_either x) as [[E3 E4]|[E3 E4]];rewrite E4.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q
E1: - y ≤ x
E2: x ≤ y
E3: 0 ≤ x
E4: abs x = x
x ≤ y
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q
E1: - y ≤ x
E2: x ≤ y
E3: x ≤ 0
E4: abs x = - x
- x ≤ y
  +
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q
E1: - y ≤ x
E2: x ≤ y
E3: 0 ≤ x
E4: abs x = x
x ≤ y trivial.
  +
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
x, y: Q
E1: - y ≤ x
E2: x ≤ y
E3: x ≤ 0
E4: abs x = - x
- x ≤ y apply flip_le_negate;rewrite involutive;trivial.
Qed.

Lemma Qabs_is_join@{} : forall q : Q, abs q = join (- q) q.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q : Q, abs q = - q ⊔ q
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q : Q, abs q = - q ⊔ q
intros q.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
abs q = - q ⊔ q symmetry.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
- q ⊔ q = abs q
destruct (total_abs_either q) as [[E1 E2]|[E1 E2]];rewrite E2.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E1: 0 ≤ q
E2: abs q = q
- q ⊔ q = q
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E1: q ≤ 0
E2: abs q = - q
- q ⊔ q = - q
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E1: 0 ≤ q
E2: abs q = q
- q ⊔ q = q apply join_r.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E1: 0 ≤ q
E2: abs q = q
- q ≤ q transitivity (0:Q);trivial.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E1: 0 ≤ q
E2: abs q = q
- q ≤ (0 : Q)
  apply flip_nonneg_negate;trivial.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E1: q ≤ 0
E2: abs q = - q
- q ⊔ q = - q apply join_l.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E1: q ≤ 0
E2: abs q = - q
q ≤ - q transitivity (0:Q);trivial.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
E1: q ≤ 0
E2: abs q = - q
(0 : Q) ≤ - q
  apply flip_nonpos_negate;trivial.
Qed.

Lemma Qlt_join : forall a b c : Q, a < c -> b < c ->
  join a b < c.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b c : Q, a < c -> b < c -> a ⊔ b < c
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b c : Q, a < c -> b < c -> a ⊔ b < c
intros a b c E1 E2.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b, c: Q
E1: a < c
E2: b < c
a ⊔ b < c
destruct (total le a b) as [E3|E3];rewrite ?(join_r _ _ E3),?(join_l _ _ E3);
trivial.
Qed.

Lemma Q_average_between@{} : forall q r : Q, q < r -> q < (q + r) / 2 < r.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q r : Q, q < r -> q < (q + r) / 2 < r
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q r : Q, q < r -> q < (q + r) / 2 < r
intros q r E.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
q < (q + r) / 2 < r
split.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
q < (q + r) / 2
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
(q + r) / 2 < r
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
q < (q + r) / 2 apply flip_pos_minus.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
0 < (q + r) / 2 - q
  assert (Hrw : (q + r) / 2 - q = (r - q) / 2);[|rewrite Hrw;clear Hrw].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
(q + r) / 2 - q = (r - q) / 2
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
0 < (r - q) / 2
  {
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
(q + r) / 2 - q = (r - q) / 2 path_via ((q + r) / 2 - 2 / 2 * q).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
(q + r) / 2 - q = (q + r) / 2 - 2 / 2 * q
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
(q + r) / 2 - 2 / 2 * q = (r - q) / 2
    {
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
(q + r) / 2 - q = (q + r) / 2 - 2 / 2 * q rewrite dec_recip_inverse;[|solve_propholds].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
(q + r) / 2 - q = (q + r) / 2 - 1 * q
      rewrite mult_1_l;trivial.
    }
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
(q + r) / 2 - 2 / 2 * q = (r - q) / 2
    ring_tac.ring_with_integers (NatPair.Z nat).
  }
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
0 < (r - q) / 2
  apply pos_mult_compat;[|exact _].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
PropHolds (0 < r - q)
  red.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
0 < r - q apply (snd (flip_pos_minus _ _)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
q < r trivial.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
(q + r) / 2 < r apply flip_pos_minus.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
0 < r - (q + r) / 2
  assert (Hrw : r - (q + r) / 2 = (r - q) / 2);[|rewrite Hrw;clear Hrw].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
r - (q + r) / 2 = (r - q) / 2
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
0 < (r - q) / 2
  {
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
r - (q + r) / 2 = (r - q) / 2 path_via (2 / 2 * r - (q + r) / 2).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
r - (q + r) / 2 = 2 / 2 * r - (q + r) / 2
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
2 / 2 * r - (q + r) / 2 = (r - q) / 2
    {
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
r - (q + r) / 2 = 2 / 2 * r - (q + r) / 2 rewrite dec_recip_inverse;[|solve_propholds].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
r - (q + r) / 2 = 1 * r - (q + r) / 2
      rewrite mult_1_l;trivial.
    }
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
2 / 2 * r - (q + r) / 2 = (r - q) / 2
    ring_tac.ring_with_integers (NatPair.Z nat).
  }
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
0 < (r - q) / 2
  apply pos_mult_compat;[|exact _].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
PropHolds (0 < r - q)
  red.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
0 < r - q apply (snd (flip_pos_minus _ _)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
E: q < r
q < r trivial.
Qed.

Lemma path_avg_split_diff_l (q r : Q) : q + ((r - q) / 2) = (r + q) / 2.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
q + (r - q) / 2 = (r + q) / 2
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
q + (r - q) / 2 = (r + q) / 2
  pattern q at 1.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(fun q0 : Q => q0 + (r - q) / 2 = (r + q) / 2) q
  rewrite <- (mult_1_r q).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
q * 1 + (r - q) / 2 = (r + q) / 2
  pattern (1 : Q) at 1.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(fun q0 : Q => q * q0 + (r - q) / 2 = (r + q) / 2)
  (1 : Q)
  rewrite <- (dec_recip_inverse 2) by solve_propholds.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
q * (2 / 2) + (r - q) / 2 = (r + q) / 2
  rewrite (associativity q 2 (/2)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(q * 2) / 2 + (r - q) / 2 = (r + q) / 2
  rewrite <- (distribute_r (q*2) (r-q) (/2)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(q * 2 + (r - q)) / 2 = (r + q) / 2
  rewrite (distribute_l q 1 1).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(q * 1 + q * 1 + (r - q)) / 2 = (r + q) / 2
  rewrite (mult_1_r q).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(q + q + (r - q)) / 2 = (r + q) / 2
  rewrite (commutativity (q+q) (r-q)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(r - q + (q + q)) / 2 = (r + q) / 2
  rewrite <- (associativity r (-q) (q+q)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(r + (- q + (q + q))) / 2 = (r + q) / 2
  rewrite (associativity (-q) q q).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(r + (- q + q + q)) / 2 = (r + q) / 2
  rewrite (plus_negate_l q).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(r + (0 + q)) / 2 = (r + q) / 2
  rewrite (plus_0_l q).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(r + q) / 2 = (r + q) / 2
  reflexivity.
Qed.

Lemma path_avg_split_diff_r (q r : Q) : r - ((r - q) / 2) = (r + q) / 2.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
r - (r - q) / 2 = (r + q) / 2
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
r - (r - q) / 2 = (r + q) / 2
  pattern r at 1.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(fun q0 : Q => q0 - (r - q) / 2 = (r + q) / 2) r
  rewrite <- (mult_1_r r).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
r * 1 - (r - q) / 2 = (r + q) / 2
  pattern (1 : Q) at 1.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(fun q0 : Q => r * q0 - (r - q) / 2 = (r + q) / 2)
  (1 : Q)
  rewrite <- (dec_recip_inverse 2) by solve_propholds.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
r * (2 / 2) - (r - q) / 2 = (r + q) / 2
  rewrite (associativity r 2 (/2)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(r * 2) / 2 - (r - q) / 2 = (r + q) / 2
  rewrite negate_mult_distr_l.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(r * 2) / 2 + - (r - q) / 2 = (r + q) / 2
  rewrite <- (distribute_r (r*2) (-(r-q)) (/2)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(r * 2 - (r - q)) / 2 = (r + q) / 2
  rewrite (distribute_l r 1 1).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(r * 1 + r * 1 - (r - q)) / 2 = (r + q) / 2
  rewrite (mult_1_r r).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(r + r - (r - q)) / 2 = (r + q) / 2
  rewrite (commutativity (r+r) (-(r-q))).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(- (r - q) + (r + r)) / 2 = (r + q) / 2
  rewrite <- negate_swap_r.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(q - r + (r + r)) / 2 = (r + q) / 2
  rewrite <- (associativity q (-r) (r+r)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(q + (- r + (r + r))) / 2 = (r + q) / 2
  rewrite (associativity (-r) r r).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(q + (- r + r + r)) / 2 = (r + q) / 2
  rewrite (plus_negate_l r).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(q + (0 + r)) / 2 = (r + q) / 2
  rewrite (plus_0_l r).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(q + r) / 2 = (r + q) / 2
  rewrite (plus_comm q r).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
(r + q) / 2 = (r + q) / 2
  reflexivity.
Qed.

Lemma pos_gt_both : forall a b : Q, forall e, a < ' e -> b < ' e ->
  exists d d', a < ' d /\ b < ' d /\ e = d + d'.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall (a b : Q) (e : Q+),
a < ' e ->
b < ' e ->
{d : Q+ &
{d' : Q+ &
((a < ' d) * ((b < ' d) * (e = plus d d')))%type}}
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall (a b : Q) (e : Q+),
a < ' e ->
b < ' e ->
{d : Q+ &
{d' : Q+ &
((a < ' d) * ((b < ' d) * (e = plus d d')))%type}}
assert (Haux : forall a b : Q, a <= b -> forall e, a < ' e -> b < ' e ->
  exists d d', a < ' d /\ b < ' d /\ e = d + d').
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b : Q,
a ≤ b ->
forall e : Q+,
a < ' e ->
b < ' e ->
{d : Q+ &
{d' : Q+ &
((a < ' d) * ((b < ' d) * (e = plus d d')))%type}}
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Haux: forall a b : Q,
a ≤ b ->
forall e : Q+,
a < ' e ->
b < ' e ->
{d : Q+ &
{d' : Q+ &
((a < ' d) * ((b < ' d) * (e = plus d d')))%type}}
forall (a b : Q) (e : Q+),
a < ' e ->
b < ' e ->
{d : Q+ &
{d' : Q+ &
((a < ' d) * ((b < ' d) * (e = plus d d')))%type}}
{
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b : Q,
a ≤ b ->
forall e : Q+,
a < ' e ->
b < ' e ->
{d : Q+ &
{d' : Q+ &
((a < ' d) * ((b < ' d) * (e = plus d d')))%type}} intros a b E e E1 E2.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
E: a ≤ b
e: Q+
E1: a < ' e
E2: b < ' e
{d : Q+ &
{d' : Q+ &
((a < ' d) * ((b < ' d) * (e = plus d d')))%type}}
  pose proof (Q_average_between _ _ (Qlt_join _ 0 _ E2 prop_holds)) as [E3 E4].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
E: a ≤ b
e: Q+
E1: a < ' e
E2: b < ' e
E3: b ⊔ 0 < (b ⊔ 0 + ' e) / 2
E4: (b ⊔ 0 + ' e) / 2 < ' e
{d : Q+ &
{d' : Q+ &
((a < ' d) * ((b < ' d) * (e = plus d d')))%type}}
  exists (mkQpos _ (le_lt_trans _ _ _ (join_ub_r _ _) E3)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
E: a ≤ b
e: Q+
E1: a < ' e
E2: b < ' e
E3: b ⊔ 0 < (b ⊔ 0 + ' e) / 2
E4: (b ⊔ 0 + ' e) / 2 < ' e
{d' : Q+ &
((a <
  ' {|
      pos := plus (b ⊔ 0) (' e) / 2;
      is_pos :=
        le_lt_trans 0 (b ⊔ 0) (plus (b ⊔ 0) (' e) / 2)
          (join_ub_r b 0) E3
    |}) *
 ((b <
   ' {|
       pos := plus (b ⊔ 0) (' e) / 2;
       is_pos :=
         le_lt_trans 0 (b ⊔ 0)
           (plus (b ⊔ 0) (' e) / 2) (join_ub_r b 0) E3
     |}) *
  (e =
   plus
     {|
       pos := plus (b ⊔ 0) (' e) / 2;
       is_pos :=
         le_lt_trans 0 (b ⊔ 0)
           (plus (b ⊔ 0) (' e) / 2) (join_ub_r b 0) E3
     |} d')))%type}
  unfold cast at 1 4;simpl.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
E: a ≤ b
e: Q+
E1: a < ' e
E2: b < ' e
E3: b ⊔ 0 < (b ⊔ 0 + ' e) / 2
E4: (b ⊔ 0 + ' e) / 2 < ' e
{d' : Q+ &
((a < plus (b ⊔ 0) (' e) / 2) *
 ((b < plus (b ⊔ 0) (' e) / 2) *
  (e =
   plus
     {|
       pos := plus (b ⊔ 0) (' e) / 2;
       is_pos :=
         le_lt_trans 0 (b ⊔ 0)
           (plus (b ⊔ 0) (' e) / 2) (join_ub_r b 0) E3
     |} d')))%type}
  exists (Qpos_diff _ _ E4).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
E: a ≤ b
e: Q+
E1: a < ' e
E2: b < ' e
E3: b ⊔ 0 < (b ⊔ 0 + ' e) / 2
E4: (b ⊔ 0 + ' e) / 2 < ' e
((a < plus (b ⊔ 0) (' e) / 2) *
 ((b < plus (b ⊔ 0) (' e) / 2) *
  (e =
   plus
     {|
       pos := plus (b ⊔ 0) (' e) / 2;
       is_pos :=
         le_lt_trans 0 (b ⊔ 0)
           (plus (b ⊔ 0) (' e) / 2) (join_ub_r b 0) E3
     |} (Qpos_diff (plus (b ⊔ 0) (' e) / 2) (' e) E4))))%type
  repeat split.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
E: a ≤ b
e: Q+
E1: a < ' e
E2: b < ' e
E3: b ⊔ 0 < (b ⊔ 0 + ' e) / 2
E4: (b ⊔ 0 + ' e) / 2 < ' e
a < (b ⊔ 0 + ' e) / 2
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
E: a ≤ b
e: Q+
E1: a < ' e
E2: b < ' e
E3: b ⊔ 0 < (b ⊔ 0 + ' e) / 2
E4: (b ⊔ 0 + ' e) / 2 < ' e
b < (b ⊔ 0 + ' e) / 2
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
E: a ≤ b
e: Q+
E1: a < ' e
E2: b < ' e
E3: b ⊔ 0 < (b ⊔ 0 + ' e) / 2
E4: (b ⊔ 0 + ' e) / 2 < ' e
e =
{|
  pos := (b ⊔ 0 + ' e) / 2;
  is_pos :=
    le_lt_trans 0 (b ⊔ 0) 
      ((b ⊔ 0 + ' e) / 2) 
      (join_ub_r b 0) E3
|} + Qpos_diff ((b ⊔ 0 + ' e) / 2) (' e) E4
  -
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
E: a ≤ b
e: Q+
E1: a < ' e
E2: b < ' e
E3: b ⊔ 0 < (b ⊔ 0 + ' e) / 2
E4: (b ⊔ 0 + ' e) / 2 < ' e
a < (b ⊔ 0 + ' e) / 2 apply le_lt_trans with b;trivial.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
E: a ≤ b
e: Q+
E1: a < ' e
E2: b < ' e
E3: b ⊔ 0 < (b ⊔ 0 + ' e) / 2
E4: (b ⊔ 0 + ' e) / 2 < ' e
b < (b ⊔ 0 + ' e) / 2
    apply le_lt_trans with (join b 0);trivial.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
E: a ≤ b
e: Q+
E1: a < ' e
E2: b < ' e
E3: b ⊔ 0 < (b ⊔ 0 + ' e) / 2
E4: (b ⊔ 0 + ' e) / 2 < ' e
b ≤ b ⊔ 0
    apply join_ub_l.
  -
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
E: a ≤ b
e: Q+
E1: a < ' e
E2: b < ' e
E3: b ⊔ 0 < (b ⊔ 0 + ' e) / 2
E4: (b ⊔ 0 + ' e) / 2 < ' e
b < (b ⊔ 0 + ' e) / 2 apply le_lt_trans with (join b 0);trivial.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
E: a ≤ b
e: Q+
E1: a < ' e
E2: b < ' e
E3: b ⊔ 0 < (b ⊔ 0 + ' e) / 2
E4: (b ⊔ 0 + ' e) / 2 < ' e
b ≤ b ⊔ 0 apply join_ub_l.
  -
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
E: a ≤ b
e: Q+
E1: a < ' e
E2: b < ' e
E3: b ⊔ 0 < (b ⊔ 0 + ' e) / 2
E4: (b ⊔ 0 + ' e) / 2 < ' e
e =
{|
  pos := (b ⊔ 0 + ' e) / 2;
  is_pos :=
    le_lt_trans 0 (b ⊔ 0) ((b ⊔ 0 + ' e) / 2)
      (join_ub_r b 0) E3
|} + Qpos_diff ((b ⊔ 0 + ' e) / 2) (' e) E4 apply pos_eq.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
E: a ≤ b
e: Q+
E1: a < ' e
E2: b < ' e
E3: b ⊔ 0 < (b ⊔ 0 + ' e) / 2
E4: (b ⊔ 0 + ' e) / 2 < ' e
' e =
' ({|
     pos := (b ⊔ 0 + ' e) / 2;
     is_pos :=
       le_lt_trans 0 (b ⊔ 0) ((b ⊔ 0 + ' e) / 2)
         (join_ub_r b 0) E3
   |} + Qpos_diff ((b ⊔ 0 + ' e) / 2) (' e) E4) unfold cast at 2;simpl.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
E: a ≤ b
e: Q+
E1: a < ' e
E2: b < ' e
E3: b ⊔ 0 < (b ⊔ 0 + ' e) / 2
E4: (b ⊔ 0 + ' e) / 2 < ' e
' e =
' {|
    pos := (b ⊔ 0 + ' e) / 2;
    is_pos :=
      le_lt_trans 0 (b ⊔ 0) ((b ⊔ 0 + ' e) / 2)
        (join_ub_r b 0) E3
  |} + ' Qpos_diff ((b ⊔ 0 + ' e) / 2) (' e) E4 unfold cast at 2;simpl.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
E: a ≤ b
e: Q+
E1: a < ' e
E2: b < ' e
E3: b ⊔ 0 < (b ⊔ 0 + ' e) / 2
E4: (b ⊔ 0 + ' e) / 2 < ' e
' e =
(b ⊔ 0 + ' e) / 2 +
' Qpos_diff ((b ⊔ 0 + ' e) / 2) (' e) E4
    unfold cast at 3;simpl.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
E: a ≤ b
e: Q+
E1: a < ' e
E2: b < ' e
E3: b ⊔ 0 < (b ⊔ 0 + ' e) / 2
E4: (b ⊔ 0 + ' e) / 2 < ' e
' e = (b ⊔ 0 + ' e) / 2 + (' e - (b ⊔ 0 + ' e) / 2)
    abstract ring_tac.ring_with_integers (NatPair.Z nat).
}
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Haux: forall a b : Q,
a ≤ b ->
forall e : Q+,
a < ' e ->
b < ' e ->
{d : Q+ &
{d' : Q+ &
((a < ' d) * ((b < ' d) * (e = plus d d')))%type}}
forall (a b : Q) (e : Q+),
a < ' e ->
b < ' e ->
{d : Q+ &
{d' : Q+ &
((a < ' d) * ((b < ' d) * (e = plus d d')))%type}}
intros a b e E1 E2.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Haux: forall a b : Q,
a ≤ b ->
forall e : Q+,
a < ' e ->
b < ' e ->
{d : Q+ &
{d' : Q+ &
((a < ' d) * ((b < ' d) * (e = plus d d')))%type}}
a, b: Q
e: Q+
E1: a < ' e
E2: b < ' e
{d : Q+ &
{d' : Q+ &
((a < ' d) * ((b < ' d) * (e = plus d d')))%type}} destruct (total le a b) as [E|E];auto.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Haux: forall a b : Q,
a ≤ b ->
forall e : Q+,
a < ' e ->
b < ' e ->
{d : Q+ &
{d' : Q+ &
((a < ' d) * ((b < ' d) * (e = plus d d')))%type}}
a, b: Q
e: Q+
E1: a < ' e
E2: b < ' e
E: b ≤ a
{d : Q+ &
{d' : Q+ &
((a < ' d) * ((b < ' d) * (e = plus d d')))%type}}
destruct (Haux _ _ E e) as [d [d' [E3 [E4 E5]]]];trivial.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Haux: forall a b : Q,
a ≤ b ->
forall e : Q+,
a < ' e ->
b < ' e ->
{d : Q+ &
{d' : Q+ &
((a < ' d) * ((b < ' d) * (e = plus d d')))%type}}
a, b: Q
e: Q+
E1: a < ' e
E2: b < ' e
E: b ≤ a
d, d': Q+
E3: b < ' d
E4: a < ' d
E5: e = d + d'
{d : Q+ &
{d' : Q+ &
((a < ' d) * ((b < ' d) * (e = plus d d')))%type}}
eauto.
Qed.

Lemma two_fourth_is_one_half@{} : 2/4 =  1/2 :> Q+.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
2 / 4 = 1 / 2
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
2 / 4 = 1 / 2
assert (Hrw : 4 = 2 * 2 :> Q) by ring_tac.ring_with_nat.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Hrw: 4 = 2 * 2
2 / 4 = 1 / 2
apply pos_eq.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Hrw: 4 = 2 * 2
' (2 / 4) = ' (1 / 2) repeat (unfold cast;simpl).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
Hrw: 4 = 2 * 2
2 / 4 = 1 / 2
rewrite Hrw;clear Hrw.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
2 / (2 * 2) = 1 / 2
rewrite dec_recip_distr.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
2 * (/ 2 / 2) = 1 / 2
rewrite mult_assoc.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
(2 / 2) / 2 = 1 / 2 rewrite dec_recip_inverse;[|solve_propholds].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
1 / 2 = 1 / 2
reflexivity.
Unshelve.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
X: IsSemiCRing Q
Unit -> Q exact (fun _ => 1). (* <- wtf *)
Qed.

Lemma Q_triangle_le : forall q r : Q, abs (q + r) <= abs q + abs r.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q r : Q, abs (q + r) ≤ abs q + abs r
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q r : Q, abs (q + r) ≤ abs q + abs r
intros.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
abs (q + r) ≤ abs q + abs r rewrite (Qabs_is_join (q + r)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
- (q + r) ⊔ (q + r) ≤ abs q + abs r
apply join_le.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
- (q + r) ≤ abs q + abs r
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
q + r ≤ abs q + abs r
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
- (q + r) ≤ abs q + abs r rewrite negate_plus_distr.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
- q - r ≤ abs q + abs r
  apply plus_le_compat;apply Qabs_le_neg_raw.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
q + r ≤ abs q + abs r apply plus_le_compat;apply Qabs_le_raw.
Qed.

Lemma Qabs_triangle_alt_aux : forall x y : Q, abs x - abs y <= abs (x - y).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x y : Q, abs x - abs y ≤ abs (x - y)
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x y : Q, abs x - abs y ≤ abs (x - y)
intros q r.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
abs q - abs r ≤ abs (q - r)
apply (order_reflecting (+ (abs r))).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
abs q - abs r + abs r ≤ abs (q - r) + abs r
assert (Hrw : abs q - abs r + abs r = abs q)
  by ring_tac.ring_with_integers (NatPair.Z nat);
rewrite Hrw;clear Hrw.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
abs q ≤ abs (q - r) + abs r
etransitivity;[|apply Q_triangle_le].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
abs q ≤ abs (q - r + r)
assert (Hrw : q - r + r = q)
  by ring_tac.ring_with_integers (NatPair.Z nat);
rewrite Hrw;clear Hrw.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
abs q ≤ abs q
reflexivity.
Qed.

Lemma Qabs_triangle_alt : forall x y : Q, abs (abs x - abs y) <= abs (x - y).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x y : Q, abs (abs x - abs y) ≤ abs (x - y)
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall x y : Q, abs (abs x - abs y) ≤ abs (x - y)
intros q r.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
abs (abs q - abs r) ≤ abs (q - r)
rewrite (Qabs_is_join (abs q - abs r)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
- (abs q - abs r) ⊔ (abs q - abs r) ≤ abs (q - r)
apply join_le.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
- (abs q - abs r) ≤ abs (q - r)
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
abs q - abs r ≤ abs (q - r)
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
- (abs q - abs r) ≤ abs (q - r) rewrite <-(Qabs_neg (q - r)),<-!negate_swap_r.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
abs r - abs q ≤ abs (r - q)
  apply Qabs_triangle_alt_aux.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
abs q - abs r ≤ abs (q - r) apply Qabs_triangle_alt_aux.
Qed.

Lemma Q_dense@{} : forall q r : Q, q < r -> exists s, q < s < r.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q r : Q, q < r -> {s : Q & q < s < r}
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q r : Q, q < r -> {s : Q & q < s < r}
intros q r E;econstructor;apply Q_average_between,E.
Qed.

Lemma Qabs_neg_flip@{} : forall a b : Q, abs (a - b) = abs (b - a).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b : Q, abs (a - b) = abs (b - a)
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b : Q, abs (a - b) = abs (b - a)
intros a b.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
abs (a - b) = abs (b - a)
rewrite <-Qabs_neg.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
abs (- (a - b)) = abs (b - a) rewrite <-negate_swap_r.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
abs (b - a) = abs (b - a) trivial.
Qed.

Definition pos_of_Q : Q -> Q+
  := fun q => {| pos := abs q + 1;
    is_pos := le_lt_trans _ _ _ (Qabs_nonneg q)
                (fst (pos_plus_lt_compat_r _ _) lt_0_1) |}.

Lemma Q_abs_plus_1_bounds@{} : forall q : Q,
  - ' (pos_of_Q q) ≤ q
  ≤ ' (pos_of_Q q).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q : Q, - ' pos_of_Q q ≤ q ≤ ' pos_of_Q q
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q : Q, - ' pos_of_Q q ≤ q ≤ ' pos_of_Q q
intros.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
- ' pos_of_Q q ≤ q ≤ ' pos_of_Q q change (- (abs q + 1) ≤ q ≤ (abs q + 1)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
- (abs q + 1) ≤ q ≤ abs q + 1 split.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
- (abs q + 1) ≤ q
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
q ≤ abs q + 1
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
- (abs q + 1) ≤ q apply flip_le_negate.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
- q ≤ - - (abs q + 1) rewrite involutive.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
- q ≤ abs q + 1
  transitivity (abs q).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
- q ≤ abs q
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
abs q ≤ abs q + 1
  +
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
- q ≤ abs q apply Qabs_le_neg_raw.
  +
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
abs q ≤ abs q + 1 apply nonneg_plus_le_compat_r.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
0 ≤ 1 solve_propholds.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
q ≤ abs q + 1 transitivity (abs q).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
q ≤ abs q
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
abs q ≤ abs q + 1
  +
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
q ≤ abs q apply Qabs_le_raw.
  +
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
abs q ≤ abs q + 1 apply nonneg_plus_le_compat_r.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q: Q
0 ≤ 1 solve_propholds.
Qed.

Lemma Qabs_mult@{} : forall a b : Q, abs (a * b) = abs a * abs b.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b : Q, abs (a * b) = abs a * abs b
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a b : Q, abs (a * b) = abs a * abs b
intros a b.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
abs (a * b) = abs a * abs b
destruct (total_abs_either a) as [Ea|Ea];destruct Ea as [Ea1 Ea2];rewrite Ea2;
destruct (total_abs_either b) as [Eb|Eb];destruct Eb as [Eb1 Eb2];rewrite Eb2.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
Ea1: 0 ≤ a
Ea2: abs a = a
Eb1: 0 ≤ b
Eb2: abs b = b
abs (a * b) = a * b
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
Ea1: 0 ≤ a
Ea2: abs a = a
Eb1: b ≤ 0
Eb2: abs b = - b
abs (a * b) = a * - b
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
Ea1: a ≤ 0
Ea2: abs a = - a
Eb1: 0 ≤ b
Eb2: abs b = b
abs (a * b) = - a * b
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
Ea1: a ≤ 0
Ea2: abs a = - a
Eb1: b ≤ 0
Eb2: abs b = - b
abs (a * b) = - a * - b
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
Ea1: 0 ≤ a
Ea2: abs a = a
Eb1: 0 ≤ b
Eb2: abs b = b
abs (a * b) = a * b apply ((abs_sig (a*b)).2).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
Ea1: 0 ≤ a
Ea2: abs a = a
Eb1: 0 ≤ b
Eb2: abs b = b
0 ≤ a * b apply nonneg_mult_compat;trivial.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
Ea1: 0 ≤ a
Ea2: abs a = a
Eb1: b ≤ 0
Eb2: abs b = - b
abs (a * b) = a * - b rewrite <-negate_mult_distr_r.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
Ea1: 0 ≤ a
Ea2: abs a = a
Eb1: b ≤ 0
Eb2: abs b = - b
abs (a * b) = - (a * b)
  apply ((abs_sig (a*b)).2).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
Ea1: 0 ≤ a
Ea2: abs a = a
Eb1: b ≤ 0
Eb2: abs b = - b
a * b ≤ 0 apply nonneg_nonpos_mult;trivial.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
Ea1: a ≤ 0
Ea2: abs a = - a
Eb1: 0 ≤ b
Eb2: abs b = b
abs (a * b) = - a * b rewrite <-negate_mult_distr_l.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
Ea1: a ≤ 0
Ea2: abs a = - a
Eb1: 0 ≤ b
Eb2: abs b = b
abs (a * b) = - (a * b)
  apply ((abs_sig (a*b)).2).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
Ea1: a ≤ 0
Ea2: abs a = - a
Eb1: 0 ≤ b
Eb2: abs b = b
a * b ≤ 0 apply nonpos_nonneg_mult;trivial.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
Ea1: a ≤ 0
Ea2: abs a = - a
Eb1: b ≤ 0
Eb2: abs b = - b
abs (a * b) = - a * - b rewrite negate_mult_negate.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
Ea1: a ≤ 0
Ea2: abs a = - a
Eb1: b ≤ 0
Eb2: abs b = - b
abs (a * b) = a * b apply ((abs_sig (a*b)).2).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
a, b: Q
Ea1: a ≤ 0
Ea2: abs a = - a
Eb1: b ≤ 0
Eb2: abs b = - b
0 ≤ a * b
  apply nonpos_mult;trivial.
Qed.

Lemma Qpos_neg_le@{} : forall a : Q+, - ' a <= ' a.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a : Q+, - ' a ≤ ' a
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall a : Q+, - ' a ≤ ' a
intros a;apply between_nonneg;solve_propholds.
Qed.

Definition Qpos_upper (e : Q+) := exists x : Q, ' e <= x.

Definition Qpos_upper_inject e : Q -> Qpos_upper e.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
Q -> Qpos_upper e
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
Q -> Qpos_upper e
intros x.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
x: Q
Qpos_upper e exists (join x (' e)).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
e: Q+
x: Q
' e ≤ x ⊔ ' e apply join_ub_r.
Defined.

#[export] Instance QLe_dec : forall q r : Q, Decidable (q <= r).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q r : Q, Decidable (q ≤ r)
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q r : Q, Decidable (q ≤ r)
intros q r;destruct (le_or_lt q r).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
l: q ≤ r
Decidable (q ≤ r)
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
l: r < q
Decidable (q ≤ r)
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
l: q ≤ r
Decidable (q ≤ r) left;trivial.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
l: r < q
Decidable (q ≤ r) right;intros ?.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
l: r < q
X: q ≤ r
Empty apply (irreflexivity lt q).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
l: r < q
X: q ≤ r
q < q
  apply le_lt_trans with r;trivial.
Qed.

#[export] Instance QLt_dec : forall q r : Q, Decidable (q < r).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q r : Q, Decidable (q < r)
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
forall q r : Q, Decidable (q < r)
intros q r;destruct (le_or_lt r q).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
l: r ≤ q
Decidable (q < r)
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
l: q < r
Decidable (q < r)
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
l: r ≤ q
Decidable (q < r) right;intros ?.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
l: r ≤ q
X: q < r
Empty apply (irreflexivity lt q).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
l: r ≤ q
X: q < r
q < q
  apply lt_le_trans with r;trivial.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
q, r: Q
l: q < r
Decidable (q < r) left;trivial.
Qed.

Section enumerable.
Context `{Enumerable Q}.

Definition Qpos_enumerator : nat -> Q+.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
nat -> Q+
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
nat -> Q+
intros n.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
n: nat
Q+
destruct (le_or_lt (enumerator Q n) 0) as [E|E].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
n: nat
E: enumerator Q n ≤ 0
Q+
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
n: nat
E: 0 < enumerator Q n
Q+
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
n: nat
E: enumerator Q n ≤ 0
Q+ exact 1.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
n: nat
E: 0 < enumerator Q n
Q+ exists (enumerator Q n);trivial.
Defined.

Lemma Qpos_is_enumerator :
  IsSurjection@{UQ} Qpos_enumerator.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
ReflectiveSubuniverse.IsConnMap (Tr (-1))
  Qpos_enumerator
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
ReflectiveSubuniverse.IsConnMap (Tr (-1))
  Qpos_enumerator
apply BuildIsSurjection.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
forall b : Q+, merely (hfiber Qpos_enumerator b)
unfold hfiber.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
forall b : Q+,
merely {x : nat & Qpos_enumerator x = b}
intros e;generalize (@center _ (enumerator_issurj Q (' e))).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
e: Q+
Tr (-1) (hfiber (enumerator Q) (' e)) ->
merely {x : nat & Qpos_enumerator x = e} apply (Trunc_ind _).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
e: Q+
hfiber (enumerator Q) (' e) ->
merely {x : nat & Qpos_enumerator x = e}
intros [n E].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
e: Q+
n: nat
E: enumerator Q n = ' e
merely {x : nat & Qpos_enumerator x = e} apply tr;exists n.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
e: Q+
n: nat
E: enumerator Q n = ' e
Qpos_enumerator n = e
unfold Qpos_enumerator.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
e: Q+
n: nat
E: enumerator Q n = ' e
match le_or_lt (enumerator Q n) 0 with
| inl _ => 1
| inr l => {| pos := enumerator Q n; is_pos := l |}
end = e destruct (le_or_lt (enumerator Q n) 0) as [E1|E1].
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
e: Q+
n: nat
E: enumerator Q n = ' e
E1: enumerator Q n ≤ 0
1 = e
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
e: Q+
n: nat
E: enumerator Q n = ' e
E1: 0 < enumerator Q n
{| pos := enumerator Q n; is_pos := E1 |} = e
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
e: Q+
n: nat
E: enumerator Q n = ' e
E1: enumerator Q n ≤ 0
1 = e destruct (irreflexivity lt 0).
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
e: Q+
n: nat
E: enumerator Q n = ' e
E1: enumerator Q n ≤ 0
0 < 0 apply lt_le_trans with (enumerator Q n);trivial.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
e: Q+
n: nat
E: enumerator Q n = ' e
E1: enumerator Q n ≤ 0
0 < enumerator Q n
  rewrite E;solve_propholds.
-
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
e: Q+
n: nat
E: enumerator Q n = ' e
E1: 0 < enumerator Q n
{| pos := enumerator Q n; is_pos := E1 |} = e apply pos_eq,E.
Qed.

#[export] Instance Qpos_enumerable : Enumerable Q+.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
Enumerable Q+
Proof.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
Enumerable Q+
  exists Qpos_enumerator.
H: Funext
H0: Univalence
Q: Type
Qap: Apart Q
Qplus: Plus Q
Qmult: Mult Q
Qzero: Zero Q
Qone: One Q
Qneg: Negate Q
Qrecip: DecRecip Q
Qle: Le Q
Qlt: Lt Q
QtoField: RationalsToField Q
Qrats: Rationals Q
Qtrivialapart: TrivialApart Q
Qdec: DecidablePaths Q
Qmeet: Meet Q
Qjoin: Join Q
Qlattice: LatticeOrder Qle
Qle_total: TotalRelation le
Qabs: Abs Q
H1: Enumerable Q
ReflectiveSubuniverse.IsConnMap (Tr (-1))
  Qpos_enumerator
  first [exact Qpos_is_enumerator@{Uhuge Ularge}|
         exact Qpos_is_enumerator@{}].
Qed.

End enumerable.

End contents.

Arguments Qpos Q {_ _}.