Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# Library Coq.QArith.QArith_base

Require Export ZArith_base.
Require Export ZArithRing.
Require Export Morphisms Setoid Bool.

# Definition of Q and basic properties

Rationals are pairs of Z and positive numbers.
Record Q : Set := Qmake {Qnum : Z; Qden : positive}.
Delimit Scope hex_Q_scope with xQ.
Delimit Scope Q_scope with Q.
Open Scope Q_scope.
Ltac simpl_mult := rewrite ?Pos2Z.inj_mul.
a#b denotes the fraction a over b.
Notation "a # b" := (Qmake a b) (at level 55, no associativity) : Q_scope.
Definition inject_Z (x : Z) := Qmake x 1.
Notation QDen p := (Zpos (Qden p)).
Definition Qeq (p q : Q) := (Qnum p * QDen q)%Z = (Qnum q * QDen p)%Z.
Definition Qle (x y : Q) := (Qnum x * QDen y <= Qnum y * QDen x)%Z.
Definition Qlt (x y : Q) := (Qnum x * QDen y < Qnum y * QDen x)%Z.
Notation Qgt a b := (Qlt b a) (only parsing).
Notation Qge a b := (Qle b a) (only parsing).
Infix "==" := Qeq (at level 70, no associativity) : Q_scope.
Infix "<" := Qlt : Q_scope.
Infix "<=" := Qle : Q_scope.
Notation "x > y" := (Qlt y x)(only parsing) : Q_scope.
Notation "x >= y" := (Qle y x)(only parsing) : Q_scope.
Notation "x <= y <= z" := (x<=y/\y<=z) : Q_scope.
injection from Z is injective.
Lemma inject_Z_injective (a b: Z): inject_Z a == inject_Z b <-> a = b.
Another approach : using Qcompare for defining order relations.
Definition Qcompare (p q : Q) := (Qnum p * QDen q ?= Qnum q * QDen p)%Z.
Notation "p ?= q" := (Qcompare p q) : Q_scope.
Lemma Qeq_alt p q : (p == q) <-> (p ?= q) = Eq.
Lemma Qlt_alt p q : (p<q) <-> (p?=q = Lt).
Lemma Qgt_alt p q : (p>q) <-> (p?=q = Gt).
Lemma Qle_alt p q : (p<=q) <-> (p?=q <> Gt).
Lemma Qge_alt p q : (p>=q) <-> (p?=q <> Lt).
#[global]
Hint Unfold Qeq Qlt Qle : qarith.
#[global]
Hint Extern 5 (?X1 <> ?X2) => intro; discriminate: qarith.
Lemma Qcompare_antisym x y : CompOpp (x ?= y) = (y ?= x).
Lemma Qcompare_spec x y : CompareSpec (x==y) (x<y) (y<x) (x ?= y).

# Properties of equality.

Theorem Qeq_refl x : x == x.
Theorem Qeq_sym x y : x == y -> y == x.
Theorem Qeq_trans x y z : x == y -> y == z -> x == z.
#[global]
Hint Immediate Qeq_sym : qarith.
#[global]
Hint Resolve Qeq_refl Qeq_trans : qarith.
In a word, Qeq is a setoid equality.
Instance Q_Setoid : Equivalence Qeq.
Furthermore, this equality is decidable:
Theorem Qeq_dec x y : {x==y} + {~ x==y}.
Definition Qeq_bool x y :=
(Zeq_bool (Qnum x * QDen y) (Qnum y * QDen x))%Z.
Definition Qle_bool x y :=
(Z.leb (Qnum x * QDen y) (Qnum y * QDen x))%Z.
Lemma Qeq_bool_iff x y : Qeq_bool x y = true <-> x == y.
Lemma Qeq_bool_eq x y : Qeq_bool x y = true -> x == y.
Lemma Qeq_eq_bool x y : x == y -> Qeq_bool x y = true.
Lemma Qeq_bool_neq x y : Qeq_bool x y = false -> ~ x == y.
Lemma Qle_bool_iff x y : Qle_bool x y = true <-> x <= y.
Lemma Qle_bool_imp_le x y : Qle_bool x y = true -> x <= y.
Theorem Qnot_eq_sym x y : ~x == y -> ~y == x.
Lemma Qeq_bool_comm x y: Qeq_bool x y = Qeq_bool y x.
Lemma Qeq_bool_refl x: Qeq_bool x x = true.
Lemma Qeq_bool_sym x y: Qeq_bool x y = true -> Qeq_bool y x = true.
Lemma Qeq_bool_trans x y z: Qeq_bool x y = true -> Qeq_bool y z = true -> Qeq_bool x z = true.
#[global]
Hint Resolve Qnot_eq_sym : qarith.

# Addition, multiplication and opposite

The addition, multiplication and opposite are defined in the straightforward way:
Definition Qplus (x y : Q) :=
(Qnum x * QDen y + Qnum y * QDen x) # (Qden x * Qden y).
Definition Qmult (x y : Q) := (Qnum x * Qnum y) # (Qden x * Qden y).
Definition Qopp (x : Q) := (- Qnum x) # (Qden x).
Definition Qminus (x y : Q) := Qplus x (Qopp y).
Definition Qinv (x : Q) :=
match Qnum x with
| Z0 => 0#1
| Zpos p => (QDen x)#p
| Zneg p => (Zneg (Qden x))#p
end.
Definition Qdiv (x y : Q) := Qmult x (Qinv y).
Infix "+" := Qplus : Q_scope.
Notation "- x" := (Qopp x) : Q_scope.
Infix "-" := Qminus : Q_scope.
Infix "*" := Qmult : Q_scope.
Notation "/ x" := (Qinv x) : Q_scope.
Infix "/" := Qdiv : Q_scope.
Number notation for constants
Inductive IZ :=
| IZpow_pos : Z -> positive -> IZ
| IZ0 : IZ
| IZpos : positive -> IZ
| IZneg : positive -> IZ.
Inductive IQ :=
| IQmake : IZ -> positive -> IQ
| IQmult : IQ -> IQ -> IQ
| IQdiv : IQ -> IQ -> IQ.
Definition IZ_of_Z z :=
match z with
| Z0 => IZ0
| Zpos e => IZpos e
| Zneg e => IZneg e
end.
Definition IZ_to_Z z :=
match z with
| IZ0 => Some Z0
| IZpos e => Some (Zpos e)
| IZneg e => Some (Zneg e)
| IZpow_pos _ _ => None
end.
Definition of_decimal (d:Decimal.decimal) : IQ :=
let '(i, f, e) :=
match d with
| Decimal.Decimal i f => (i, f, Decimal.Pos Decimal.Nil)
| Decimal.DecimalExp i f e => (i, f, e)
end in
let num := Z.of_int (Decimal.app_int i f) in
let den := Nat.iter (Decimal.nb_digits f) (Pos.mul 10) 1%positive in
let q := IQmake (IZ_of_Z num) den in
let e := Z.of_int e in
match e with
| Z0 => q
| Zpos e => IQmult q (IQmake (IZpow_pos 10 e) 1)
| Zneg e => IQdiv q (IQmake (IZpow_pos 10 e) 1)
end.
Definition IQmake_to_decimal num den :=
let num := Z.to_int num in
let (den, e_den) := Decimal.nztail (Pos.to_uint den) in
match den with
| Decimal.D1 Decimal.Nil =>
match e_den with
| O => Some (Decimal.Decimal num Decimal.Nil)
| ne =>
let ai := Decimal.abs num in
let ni := Decimal.nb_digits ai in
if Nat.ltb ne ni then
let i := Decimal.del_tail_int ne num in
let f := Decimal.del_head (Nat.sub ni ne) ai in
Some (Decimal.Decimal i f)
else
let z := match num with
| Decimal.Pos _ => Decimal.Pos (Decimal.zero)
| Decimal.Neg _ => Decimal.Neg (Decimal.zero) end in
Some (Decimal.Decimal z (Nat.iter (Nat.sub ne ni) Decimal.D0 ai))
end
| _ => None
end.
Definition IQmake_to_decimal' num den :=
match IZ_to_Z num with
| None => None
| Some num => IQmake_to_decimal num den
end.
Definition to_decimal (n : IQ) : option Decimal.decimal :=
match n with
| IQmake num den => IQmake_to_decimal' num den
| IQmult (IQmake num den) (IQmake (IZpow_pos 10 e) 1) =>
match IQmake_to_decimal' num den with
| Some (Decimal.Decimal i f) =>
Some (Decimal.DecimalExp i f (Pos.to_int e))
| _ => None
end
| IQdiv (IQmake num den) (IQmake (IZpow_pos 10 e) 1) =>
match IQmake_to_decimal' num den with
| Some (Decimal.Decimal i f) =>
Some (Decimal.DecimalExp i f (Decimal.Neg (Pos.to_uint e)))
| _ => None
end
| _ => None
end.
Definition of_hexadecimal (d:Hexadecimal.hexadecimal) : IQ :=
let '(i, f, e) :=
match d with
| Hexadecimal.Hexadecimal i f => (i, f, Decimal.Pos Decimal.Nil)
| Hexadecimal.HexadecimalExp i f e => (i, f, e)
end in
let num := Z.of_hex_int (Hexadecimal.app_int i f) in
let den := Nat.iter (Hexadecimal.nb_digits f) (Pos.mul 16) 1%positive in
let q := IQmake (IZ_of_Z num) den in
let e := Z.of_int e in
match e with
| Z0 => q
| Zpos e => IQmult q (IQmake (IZpow_pos 2 e) 1)
| Zneg e => IQdiv q (IQmake (IZpow_pos 2 e) 1)
end.
Definition IQmake_to_hexadecimal num den :=
let num := Z.to_hex_int num in
let (den, e_den) := Hexadecimal.nztail (Pos.to_hex_uint den) in
match den with
| Hexadecimal.D1 Hexadecimal.Nil =>
match e_den with
| O => Some (Hexadecimal.Hexadecimal num Hexadecimal.Nil)
| ne =>
let ai := Hexadecimal.abs num in
let ni := Hexadecimal.nb_digits ai in
if Nat.ltb ne ni then
let i := Hexadecimal.del_tail_int ne num in
let f := Hexadecimal.del_head (Nat.sub ni ne) ai in
Some (Hexadecimal.Hexadecimal i f)
else
let z := match num with
| Hexadecimal.Pos _ => Hexadecimal.Pos (Hexadecimal.zero)
| Hexadecimal.Neg _ => Hexadecimal.Neg (Hexadecimal.zero) end in
Some (Hexadecimal.Hexadecimal z (Nat.iter (Nat.sub ne ni) Hexadecimal.D0 ai))
end
| _ => None
end.
Definition IQmake_to_hexadecimal' num den :=
match IZ_to_Z num with
| None => None
| Some num => IQmake_to_hexadecimal num den
end.
Definition to_hexadecimal (n : IQ) : option Hexadecimal.hexadecimal :=
match n with
| IQmake num den => IQmake_to_hexadecimal' num den
| IQmult (IQmake num den) (IQmake (IZpow_pos 2 e) 1) =>
match IQmake_to_hexadecimal' num den with
| Some (Hexadecimal.Hexadecimal i f) =>
Some (Hexadecimal.HexadecimalExp i f (Pos.to_int e))
| _ => None
end
| IQdiv (IQmake num den) (IQmake (IZpow_pos 2 e) 1) =>
match IQmake_to_hexadecimal' num den with
| Some (Hexadecimal.Hexadecimal i f) =>
Some (Hexadecimal.HexadecimalExp i f (Decimal.Neg (Pos.to_uint e)))
| _ => None
end
| _ => None
end.
Definition of_number (n : Number.number) : IQ :=
match n with
| Number.Decimal d => of_decimal d
| Number.Hexadecimal h => of_hexadecimal h
end.
Definition to_number (q:IQ) : option Number.number :=
match to_decimal q with
| None => None
| Some q => Some (Number.Decimal q)
end.
Definition to_hex_number q :=
match to_hexadecimal q with
| None => None
| Some q => Some (Number.Hexadecimal q)
end.
A light notation for Zpos
Lemma Qmake_Qdiv a b : a#b==inject_Z a/inject_Z (Zpos b).

# Setoid compatibility results

Instance Qplus_comp : Proper (Qeq==>Qeq==>Qeq) Qplus.
Open Scope Z_scope.
Close Scope Z_scope.
Instance Qopp_comp : Proper (Qeq==>Qeq) Qopp.
Open Scope Z_scope.
Close Scope Z_scope.
Instance Qminus_comp : Proper (Qeq==>Qeq==>Qeq) Qminus.
Instance Qmult_comp : Proper (Qeq==>Qeq==>Qeq) Qmult.
Open Scope Z_scope.
Close Scope Z_scope.
Instance Qinv_comp : Proper (Qeq==>Qeq) Qinv.
Open Scope Z_scope.
Close Scope Z_scope.
Instance Qdiv_comp : Proper (Qeq==>Qeq==>Qeq) Qdiv.
Instance Qcompare_comp : Proper (Qeq==>Qeq==>eq) Qcompare.
Open Scope Z_scope.
Close Scope Z_scope.
Instance Qle_comp : Proper (Qeq==>Qeq==>iff) Qle.
Instance Qlt_compat : Proper (Qeq==>Qeq==>iff) Qlt.
Instance Qeqb_comp : Proper (Qeq==>Qeq==>eq) Qeq_bool.
Instance Qleb_comp : Proper (Qeq==>Qeq==>eq) Qle_bool.
0 and 1 are apart
Lemma Q_apart_0_1 : ~ 1 == 0.

# Properties of Qadd

Addition is associative:
Theorem Qplus_assoc : forall x y z, x+(y+z)==(x+y)+z.
0 is a neutral element for addition:
Lemma Qplus_0_l : forall x, 0+x == x.
Lemma Qplus_0_r : forall x, x+0 == x.
Commutativity of addition:
Theorem Qplus_comm : forall x y, x+y == y+x.

# Properties of Qopp

Lemma Qopp_involutive : forall q, - -q == q.
Theorem Qplus_opp_r : forall q, q+(-q) == 0.
Injectivity of addition (uses theory about Qopp above):
Lemma Qplus_inj_r (x y z: Q):
x + z == y + z <-> x == y.
Lemma Qplus_inj_l (x y z: Q):
z + x == z + y <-> x == y.

# Properties of Qmult

Multiplication is associative:
Theorem Qmult_assoc : forall n m p, n*(m*p)==(n*m)*p.
multiplication and zero
Lemma Qmult_0_l : forall x , 0*x == 0.
Lemma Qmult_0_r : forall x , x*0 == 0.
1 is a neutral element for multiplication:
Lemma Qmult_1_l : forall n, 1*n == n.
Theorem Qmult_1_r : forall n, n*1==n.
Commutativity of multiplication
Theorem Qmult_comm : forall x y, x*y==y*x.
Distributivity over Qadd
Theorem Qmult_plus_distr_r : forall x y z, x*(y+z)==(x*y)+(x*z).
Theorem Qmult_plus_distr_l : forall x y z, (x+y)*z==(x*z)+(y*z).
Integrality
Theorem Qmult_integral : forall x y, x*y==0 -> x==0 \/ y==0.
Theorem Qmult_integral_l : forall x y, ~ x == 0 -> x*y == 0 -> y == 0.

# inject_Z is a ring homomorphism:

Lemma inject_Z_plus (x y: Z): inject_Z (x + y) = inject_Z x + inject_Z y.
Lemma inject_Z_mult (x y: Z): inject_Z (x * y) = inject_Z x * inject_Z y.
Lemma inject_Z_opp (x: Z): inject_Z (- x) = - inject_Z x.

# Inverse and division.

Lemma Qinv_involutive : forall q, (/ / q) == q.
Theorem Qmult_inv_r : forall x, ~ x == 0 -> x*(/x) == 1.
Lemma Qinv_mult_distr : forall p q, / (p * q) == /p * /q.
Theorem Qdiv_mult_l : forall x y, ~ y == 0 -> (x*y)/y == x.
Theorem Qmult_div_r : forall x y, ~ y == 0 -> y*(x/y) == x.
Lemma Qinv_plus_distr : forall a b c, ((a # c) + (b # c) == (a+b) # c)%Q.
Lemma Qinv_minus_distr : forall a b c, (a # c) + - (b # c) == (a-b) # c.
Injectivity of Qmult (requires theory about Qinv above):
Lemma Qmult_inj_r (x y z: Q): ~ z == 0 -> (x * z == y * z <-> x == y).
Lemma Qmult_inj_l (x y z: Q): ~ z == 0 -> (z * x == z * y <-> x == y).

# Properties of order upon Q.

Lemma Qle_refl x : x<=x.
Lemma Qle_antisym x y : x<=y -> y<=x -> x==y.
Lemma Qle_trans : forall x y z, x<=y -> y<=z -> x<=z.
Open Scope Z_scope.
Close Scope Z_scope.
#[global]
Hint Resolve Qle_trans : qarith.
Lemma Qlt_irrefl x : ~x<x.
Lemma Qlt_not_eq x y : x<y -> ~ x==y.
Lemma Zle_Qle (x y: Z): (x <= y)%Z = (inject_Z x <= inject_Z y).
Lemma Zlt_Qlt (x y: Z): (x < y)%Z = (inject_Z x < inject_Z y).
Large = strict or equal
Lemma Qle_lteq x y : x<=y <-> x<y \/ x==y.
Lemma Qlt_le_weak x y : x<y -> x<=y.
Lemma Qle_lt_trans : forall x y z, x<=y -> y<z -> x<z.
Open Scope Z_scope.
Close Scope Z_scope.
Lemma Qlt_le_trans : forall x y z, x<y -> y<=z -> x<z.
Open Scope Z_scope.
Close Scope Z_scope.
Lemma Qlt_trans : forall x y z, x<y -> y<z -> x<z.
x<y iff ~(y<=x)
Lemma Qnot_lt_le x y : ~ x < y -> y <= x.
Lemma Qnot_le_lt x y : ~ x <= y -> y < x.
Lemma Qlt_not_le x y : x < y -> ~ y <= x.
Lemma Qle_not_lt x y : x <= y -> ~ y < x.
Lemma Qle_lt_or_eq : forall x y, x<=y -> x<y \/ x==y.
#[global]
Hint Resolve Qle_not_lt Qlt_not_le Qnot_le_lt Qnot_lt_le
Qlt_le_weak Qlt_not_eq Qle_antisym Qle_refl: qarith.
Some decidability results about orders.
Lemma Q_dec : forall x y, {x<y} + {y<x} + {x==y}.
Lemma Qlt_le_dec : forall x y, {x<y} + {y<=x}.
Lemma Qarchimedean : forall q : Q, { p : positive | q < Z.pos p # 1 }.
Compatibility of operations with respect to order.
Lemma Qopp_le_compat : forall p q, p<=q -> -q <= -p.
#[global]
Hint Resolve Qopp_le_compat : qarith.
Lemma Qle_minus_iff : forall p q, p <= q <-> 0 <= q+-p.
Lemma Qlt_minus_iff : forall p q, p < q <-> 0 < q+-p.
Lemma Qplus_le_compat :
forall x y z t, x<=y -> z<=t -> x+z <= y+t.
Open Scope Z_scope.
Close Scope Z_scope.
Lemma Qplus_lt_le_compat :
forall x y z t, x<y -> z<=t -> x+z < y+t.
Open Scope Z_scope.
Close Scope Z_scope.
Lemma Qplus_le_l (x y z: Q): x + z <= y + z <-> x <= y.
Lemma Qplus_le_r (x y z: Q): z + x <= z + y <-> x <= y.
Lemma Qplus_lt_l (x y z: Q): x + z < y + z <-> x < y.
Lemma Qplus_lt_r (x y z: Q): z + x < z + y <-> x < y.
Lemma Qmult_le_compat_r : forall x y z, x <= y -> 0 <= z -> x*z <= y*z.
Open Scope Z_scope.
Close Scope Z_scope.
Lemma Qmult_lt_0_le_reg_r : forall x y z, 0 < z -> x*z <= y*z -> x <= y.
Open Scope Z_scope.
Close Scope Z_scope.
Lemma Qmult_le_r (x y z: Q): 0 < z -> (x*z <= y*z <-> x <= y).
Lemma Qmult_le_l (x y z: Q): 0 < z -> (z*x <= z*y <-> x <= y).
Lemma Qmult_lt_compat_r : forall x y z, 0 < z -> x < y -> x*z < y*z.
Open Scope Z_scope.
Close Scope Z_scope.
Lemma Qmult_lt_r: forall x y z, 0 < z -> (x*z < y*z <-> x < y).
Open Scope Z_scope.
Close Scope Z_scope.
Lemma Qmult_lt_l (x y z: Q): 0 < z -> (z*x < z*y <-> x < y).
Lemma Qmult_le_0_compat : forall a b, 0 <= a -> 0 <= b -> 0 <= a*b.
Lemma Qinv_le_0_compat : forall a, 0 <= a -> 0 <= /a.
Lemma Qle_shift_div_l : forall a b c,
0 < c -> a*c <= b -> a <= b/c.
Lemma Qle_shift_inv_l : forall a c,
0 < c -> a*c <= 1 -> a <= /c.
Lemma Qle_shift_div_r : forall a b c,
0 < b -> a <= c*b -> a/b <= c.
Lemma Qle_shift_inv_r : forall b c,
0 < b -> 1 <= c*b -> /b <= c.
Lemma Qinv_lt_0_compat : forall a, 0 < a -> 0 < /a.
Lemma Qlt_shift_div_l : forall a b c,
0 < c -> a*c < b -> a < b/c.
Lemma Qlt_shift_inv_l : forall a c,
0 < c -> a*c < 1 -> a < /c.
Lemma Qlt_shift_div_r : forall a b c,
0 < b -> a < c*b -> a/b < c.
Lemma Qlt_shift_inv_r : forall b c,
0 < b -> 1 < c*b -> /b < c.
Lemma Qinv_lt_contravar : forall a b : Q,
0 < a -> 0 < b -> (a < b <-> /b < /a).

# Rational to the n-th power

Definition Qpower_positive : Q -> positive -> Q :=
pow_pos Qmult.
Instance Qpower_positive_comp : Proper (Qeq==>eq==>Qeq) Qpower_positive.
Definition Qpower (q:Q) (z:Z) :=
match z with
| Zpos p => Qpower_positive q p
| Z0 => 1
| Zneg p => /Qpower_positive q p
end.
Notation " q ^ z " := (Qpower q z) : Q_scope.
Instance Qpower_comp : Proper (Qeq==>eq==>Qeq) Qpower.