Library iris.prelude.numbers
This file collects some trivial facts on the Coq types nat and N for
natural numbers, and the type Z for integers. It also declares some useful
notations.
From Coq Require Export Eqdep PArith NArith ZArith NPeano.
From Coq Require Import QArith Qcanon.
From iris.prelude Require Export base decidable option.
Open Scope nat_scope.
Coercion Z.of_nat : nat >-> Z.
Instance comparison_eq_dec (c1 c2 : comparison) : Decision (c1 = c2).
Proof. solve_decision. Defined.
From Coq Require Import QArith Qcanon.
From iris.prelude Require Export base decidable option.
Open Scope nat_scope.
Coercion Z.of_nat : nat >-> Z.
Instance comparison_eq_dec (c1 c2 : comparison) : Decision (c1 = c2).
Proof. solve_decision. Defined.
Notations and properties of nat
Arguments minus !_ !_ /.
Reserved Notation "x ≤ y ≤ z" (at level 70, y at next level).
Reserved Notation "x ≤ y < z" (at level 70, y at next level).
Reserved Notation "x < y < z" (at level 70, y at next level).
Reserved Notation "x < y ≤ z" (at level 70, y at next level).
Reserved Notation "x ≤ y ≤ z ≤ z'"
(at level 70, y at next level, z at next level).
Infix "≤" := le : nat_scope.
Notation "x ≤ y ≤ z" := (x ≤ y ∧ y ≤ z)%nat : nat_scope.
Notation "x ≤ y < z" := (x ≤ y ∧ y < z)%nat : nat_scope.
Notation "x < y < z" := (x < y ∧ y < z)%nat : nat_scope.
Notation "x < y ≤ z" := (x < y ∧ y ≤ z)%nat : nat_scope.
Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z')%nat : nat_scope.
Notation "(≤)" := le (only parsing) : nat_scope.
Notation "(<)" := lt (only parsing) : nat_scope.
Infix "`div`" := Nat.div (at level 35) : nat_scope.
Infix "`mod`" := Nat.modulo (at level 35) : nat_scope.
Infix "`max`" := Nat.max (at level 35) : nat_scope.
Infix "`min`" := Nat.min (at level 35) : nat_scope.
Instance nat_eq_dec: ∀ x y : nat, Decision (x = y) := eq_nat_dec.
Instance nat_le_dec: ∀ x y : nat, Decision (x ≤ y) := le_dec.
Instance nat_lt_dec: ∀ x y : nat, Decision (x < y) := lt_dec.
Instance nat_inhabited: Inhabited nat := populate 0%nat.
Instance: Inj (=) (=) S.
Proof. by injection 1. Qed.
Instance: PartialOrder (≤).
Proof. repeat split; repeat intro; auto with lia. Qed.
Instance nat_le_pi: ∀ x y : nat, ProofIrrel (x ≤ y).
Proof.
assert (∀ x y (p : x ≤ y) y' (q : x ≤ y'),
y = y' → eq_dep nat (le x) y p y' q) as aux.
{ fix 3. intros x ? [|y p] ? [|y' q].
- done.
- clear nat_le_pi. intros; exfalso; auto with lia.
- clear nat_le_pi. intros; exfalso; auto with lia.
- injection 1. intros Hy. by case (nat_le_pi x y p y' q Hy). }
intros x y p q.
by apply (Eqdep_dec.eq_dep_eq_dec (λ x y, decide (x = y))), aux.
Qed.
Instance nat_lt_pi: ∀ x y : nat, ProofIrrel (x < y).
Proof. apply _. Qed.
Definition sum_list_with {A} (f : A → nat) : list A → nat :=
fix go l :=
match l with
| [] ⇒ 0
| x :: l ⇒ f x + go l
end.
Notation sum_list := (sum_list_with id).
Lemma Nat_lt_succ_succ n : n < S (S n).
Proof. auto with arith. Qed.
Lemma Nat_mul_split_l n x1 x2 y1 y2 :
x2 < n → y2 < n → x1 × n + x2 = y1 × n + y2 → x1 = y1 ∧ x2 = y2.
Proof.
intros Hx2 Hy2 E. cut (x1 = y1); [intros; subst;lia |].
revert y1 E. induction x1; simpl; intros [|?]; simpl; auto with lia.
Qed.
Lemma Nat_mul_split_r n x1 x2 y1 y2 :
x1 < n → y1 < n → x1 + x2 × n = y1 + y2 × n → x1 = y1 ∧ x2 = y2.
Proof. intros. destruct (Nat_mul_split_l n x2 x1 y2 y1); auto with lia. Qed.
Notation lcm := Nat.lcm.
Notation divide := Nat.divide.
Notation "( x | y )" := (divide x y) : nat_scope.
Instance divide_dec x y : Decision (x | y).
Proof.
refine (cast_if (decide (lcm x y = y))); by rewrite Nat.divide_lcm_iff.
Defined.
Instance: PartialOrder divide.
Proof.
repeat split; try apply _. intros ??. apply Nat.divide_antisym_nonneg; lia.
Qed.
Hint Extern 0 (_ | _) ⇒ reflexivity.
Lemma Nat_divide_ne_0 x y : (x | y) → y ≠ 0 → x ≠ 0.
Proof. intros Hxy Hy →. by apply Hy, Nat.divide_0_l. Qed.
Reserved Notation "x ≤ y ≤ z" (at level 70, y at next level).
Reserved Notation "x ≤ y < z" (at level 70, y at next level).
Reserved Notation "x < y < z" (at level 70, y at next level).
Reserved Notation "x < y ≤ z" (at level 70, y at next level).
Reserved Notation "x ≤ y ≤ z ≤ z'"
(at level 70, y at next level, z at next level).
Infix "≤" := le : nat_scope.
Notation "x ≤ y ≤ z" := (x ≤ y ∧ y ≤ z)%nat : nat_scope.
Notation "x ≤ y < z" := (x ≤ y ∧ y < z)%nat : nat_scope.
Notation "x < y < z" := (x < y ∧ y < z)%nat : nat_scope.
Notation "x < y ≤ z" := (x < y ∧ y ≤ z)%nat : nat_scope.
Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z')%nat : nat_scope.
Notation "(≤)" := le (only parsing) : nat_scope.
Notation "(<)" := lt (only parsing) : nat_scope.
Infix "`div`" := Nat.div (at level 35) : nat_scope.
Infix "`mod`" := Nat.modulo (at level 35) : nat_scope.
Infix "`max`" := Nat.max (at level 35) : nat_scope.
Infix "`min`" := Nat.min (at level 35) : nat_scope.
Instance nat_eq_dec: ∀ x y : nat, Decision (x = y) := eq_nat_dec.
Instance nat_le_dec: ∀ x y : nat, Decision (x ≤ y) := le_dec.
Instance nat_lt_dec: ∀ x y : nat, Decision (x < y) := lt_dec.
Instance nat_inhabited: Inhabited nat := populate 0%nat.
Instance: Inj (=) (=) S.
Proof. by injection 1. Qed.
Instance: PartialOrder (≤).
Proof. repeat split; repeat intro; auto with lia. Qed.
Instance nat_le_pi: ∀ x y : nat, ProofIrrel (x ≤ y).
Proof.
assert (∀ x y (p : x ≤ y) y' (q : x ≤ y'),
y = y' → eq_dep nat (le x) y p y' q) as aux.
{ fix 3. intros x ? [|y p] ? [|y' q].
- done.
- clear nat_le_pi. intros; exfalso; auto with lia.
- clear nat_le_pi. intros; exfalso; auto with lia.
- injection 1. intros Hy. by case (nat_le_pi x y p y' q Hy). }
intros x y p q.
by apply (Eqdep_dec.eq_dep_eq_dec (λ x y, decide (x = y))), aux.
Qed.
Instance nat_lt_pi: ∀ x y : nat, ProofIrrel (x < y).
Proof. apply _. Qed.
Definition sum_list_with {A} (f : A → nat) : list A → nat :=
fix go l :=
match l with
| [] ⇒ 0
| x :: l ⇒ f x + go l
end.
Notation sum_list := (sum_list_with id).
Lemma Nat_lt_succ_succ n : n < S (S n).
Proof. auto with arith. Qed.
Lemma Nat_mul_split_l n x1 x2 y1 y2 :
x2 < n → y2 < n → x1 × n + x2 = y1 × n + y2 → x1 = y1 ∧ x2 = y2.
Proof.
intros Hx2 Hy2 E. cut (x1 = y1); [intros; subst;lia |].
revert y1 E. induction x1; simpl; intros [|?]; simpl; auto with lia.
Qed.
Lemma Nat_mul_split_r n x1 x2 y1 y2 :
x1 < n → y1 < n → x1 + x2 × n = y1 + y2 × n → x1 = y1 ∧ x2 = y2.
Proof. intros. destruct (Nat_mul_split_l n x2 x1 y2 y1); auto with lia. Qed.
Notation lcm := Nat.lcm.
Notation divide := Nat.divide.
Notation "( x | y )" := (divide x y) : nat_scope.
Instance divide_dec x y : Decision (x | y).
Proof.
refine (cast_if (decide (lcm x y = y))); by rewrite Nat.divide_lcm_iff.
Defined.
Instance: PartialOrder divide.
Proof.
repeat split; try apply _. intros ??. apply Nat.divide_antisym_nonneg; lia.
Qed.
Hint Extern 0 (_ | _) ⇒ reflexivity.
Lemma Nat_divide_ne_0 x y : (x | y) → y ≠ 0 → x ≠ 0.
Proof. intros Hxy Hy →. by apply Hy, Nat.divide_0_l. Qed.
Notations and properties of positive
Open Scope positive_scope.
Infix "≤" := Pos.le : positive_scope.
Notation "x ≤ y ≤ z" := (x ≤ y ∧ y ≤ z) : positive_scope.
Notation "x ≤ y < z" := (x ≤ y ∧ y < z) : positive_scope.
Notation "x < y < z" := (x < y ∧ y < z) : positive_scope.
Notation "x < y ≤ z" := (x < y ∧ y ≤ z) : positive_scope.
Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z') : positive_scope.
Notation "(≤)" := Pos.le (only parsing) : positive_scope.
Notation "(<)" := Pos.lt (only parsing) : positive_scope.
Notation "(~0)" := xO (only parsing) : positive_scope.
Notation "(~1)" := xI (only parsing) : positive_scope.
Arguments Pos.of_nat : simpl never.
Arguments Pmult : simpl never.
Instance positive_eq_dec: ∀ x y : positive, Decision (x = y) := Pos.eq_dec.
Instance positive_inhabited: Inhabited positive := populate 1.
Instance maybe_xO : Maybe xO := λ p, match p with p~0 ⇒ Some p | _ ⇒ None end.
Instance maybe_x1 : Maybe xI := λ p, match p with p~1 ⇒ Some p | _ ⇒ None end.
Instance: Inj (=) (=) (~0).
Proof. by injection 1. Qed.
Instance: Inj (=) (=) (~1).
Proof. by injection 1. Qed.
Infix "≤" := Pos.le : positive_scope.
Notation "x ≤ y ≤ z" := (x ≤ y ∧ y ≤ z) : positive_scope.
Notation "x ≤ y < z" := (x ≤ y ∧ y < z) : positive_scope.
Notation "x < y < z" := (x < y ∧ y < z) : positive_scope.
Notation "x < y ≤ z" := (x < y ∧ y ≤ z) : positive_scope.
Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z') : positive_scope.
Notation "(≤)" := Pos.le (only parsing) : positive_scope.
Notation "(<)" := Pos.lt (only parsing) : positive_scope.
Notation "(~0)" := xO (only parsing) : positive_scope.
Notation "(~1)" := xI (only parsing) : positive_scope.
Arguments Pos.of_nat : simpl never.
Arguments Pmult : simpl never.
Instance positive_eq_dec: ∀ x y : positive, Decision (x = y) := Pos.eq_dec.
Instance positive_inhabited: Inhabited positive := populate 1.
Instance maybe_xO : Maybe xO := λ p, match p with p~0 ⇒ Some p | _ ⇒ None end.
Instance maybe_x1 : Maybe xI := λ p, match p with p~1 ⇒ Some p | _ ⇒ None end.
Instance: Inj (=) (=) (~0).
Proof. by injection 1. Qed.
Instance: Inj (=) (=) (~1).
Proof. by injection 1. Qed.
Since positive represents lists of bits, we define list operations
on it. These operations are in reverse, as positives are treated as snoc
lists instead of cons lists.
Fixpoint Papp (p1 p2 : positive) : positive :=
match p2 with
| 1 ⇒ p1
| p2~0 ⇒ (Papp p1 p2)~0
| p2~1 ⇒ (Papp p1 p2)~1
end.
Infix "++" := Papp : positive_scope.
Notation "(++)" := Papp (only parsing) : positive_scope.
Notation "( p ++)" := (Papp p) (only parsing) : positive_scope.
Notation "(++ q )" := (λ p, Papp p q) (only parsing) : positive_scope.
Fixpoint Preverse_go (p1 p2 : positive) : positive :=
match p2 with
| 1 ⇒ p1
| p2~0 ⇒ Preverse_go (p1~0) p2
| p2~1 ⇒ Preverse_go (p1~1) p2
end.
Definition Preverse : positive → positive := Preverse_go 1.
Global Instance: LeftId (=) 1 (++).
Proof. intros p. by induction p; intros; f_equal/=. Qed.
Global Instance: RightId (=) 1 (++).
Proof. done. Qed.
Global Instance: Assoc (=) (++).
Proof. intros ?? p. by induction p; intros; f_equal/=. Qed.
Global Instance: ∀ p : positive, Inj (=) (=) (++ p).
Proof. intros p ???. induction p; simplify_eq; auto. Qed.
Lemma Preverse_go_app p1 p2 p3 :
Preverse_go p1 (p2 ++ p3) = Preverse_go p1 p3 ++ Preverse_go 1 p2.
Proof.
revert p3 p1 p2.
cut (∀ p1 p2 p3, Preverse_go (p2 ++ p3) p1 = p2 ++ Preverse_go p3 p1).
{ by intros go p3; induction p3; intros p1 p2; simpl; auto; rewrite <-?go. }
intros p1; induction p1 as [p1 IH|p1 IH|]; intros p2 p3; simpl; auto.
- apply (IH _ (_~1)).
- apply (IH _ (_~0)).
Qed.
Lemma Preverse_app p1 p2 : Preverse (p1 ++ p2) = Preverse p2 ++ Preverse p1.
Proof. unfold Preverse. by rewrite Preverse_go_app. Qed.
Lemma Preverse_xO p : Preverse (p~0) = (1~0) ++ Preverse p.
Proof Preverse_app p (1~0).
Lemma Preverse_xI p : Preverse (p~1) = (1~1) ++ Preverse p.
Proof Preverse_app p (1~1).
Fixpoint Plength (p : positive) : nat :=
match p with 1 ⇒ 0%nat | p~0 | p~1 ⇒ S (Plength p) end.
Lemma Papp_length p1 p2 : Plength (p1 ++ p2) = (Plength p2 + Plength p1)%nat.
Proof. by induction p2; f_equal/=. Qed.
Close Scope positive_scope.
match p2 with
| 1 ⇒ p1
| p2~0 ⇒ (Papp p1 p2)~0
| p2~1 ⇒ (Papp p1 p2)~1
end.
Infix "++" := Papp : positive_scope.
Notation "(++)" := Papp (only parsing) : positive_scope.
Notation "( p ++)" := (Papp p) (only parsing) : positive_scope.
Notation "(++ q )" := (λ p, Papp p q) (only parsing) : positive_scope.
Fixpoint Preverse_go (p1 p2 : positive) : positive :=
match p2 with
| 1 ⇒ p1
| p2~0 ⇒ Preverse_go (p1~0) p2
| p2~1 ⇒ Preverse_go (p1~1) p2
end.
Definition Preverse : positive → positive := Preverse_go 1.
Global Instance: LeftId (=) 1 (++).
Proof. intros p. by induction p; intros; f_equal/=. Qed.
Global Instance: RightId (=) 1 (++).
Proof. done. Qed.
Global Instance: Assoc (=) (++).
Proof. intros ?? p. by induction p; intros; f_equal/=. Qed.
Global Instance: ∀ p : positive, Inj (=) (=) (++ p).
Proof. intros p ???. induction p; simplify_eq; auto. Qed.
Lemma Preverse_go_app p1 p2 p3 :
Preverse_go p1 (p2 ++ p3) = Preverse_go p1 p3 ++ Preverse_go 1 p2.
Proof.
revert p3 p1 p2.
cut (∀ p1 p2 p3, Preverse_go (p2 ++ p3) p1 = p2 ++ Preverse_go p3 p1).
{ by intros go p3; induction p3; intros p1 p2; simpl; auto; rewrite <-?go. }
intros p1; induction p1 as [p1 IH|p1 IH|]; intros p2 p3; simpl; auto.
- apply (IH _ (_~1)).
- apply (IH _ (_~0)).
Qed.
Lemma Preverse_app p1 p2 : Preverse (p1 ++ p2) = Preverse p2 ++ Preverse p1.
Proof. unfold Preverse. by rewrite Preverse_go_app. Qed.
Lemma Preverse_xO p : Preverse (p~0) = (1~0) ++ Preverse p.
Proof Preverse_app p (1~0).
Lemma Preverse_xI p : Preverse (p~1) = (1~1) ++ Preverse p.
Proof Preverse_app p (1~1).
Fixpoint Plength (p : positive) : nat :=
match p with 1 ⇒ 0%nat | p~0 | p~1 ⇒ S (Plength p) end.
Lemma Papp_length p1 p2 : Plength (p1 ++ p2) = (Plength p2 + Plength p1)%nat.
Proof. by induction p2; f_equal/=. Qed.
Close Scope positive_scope.
Notations and properties of N
Infix "≤" := N.le : N_scope.
Notation "x ≤ y ≤ z" := (x ≤ y ∧ y ≤ z)%N : N_scope.
Notation "x ≤ y < z" := (x ≤ y ∧ y < z)%N : N_scope.
Notation "x < y < z" := (x < y ∧ y < z)%N : N_scope.
Notation "x < y ≤ z" := (x < y ∧ y ≤ z)%N : N_scope.
Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z')%N : N_scope.
Notation "(≤)" := N.le (only parsing) : N_scope.
Notation "(<)" := N.lt (only parsing) : N_scope.
Infix "`div`" := N.div (at level 35) : N_scope.
Infix "`mod`" := N.modulo (at level 35) : N_scope.
Arguments N.add _ _ : simpl never.
Instance: Inj (=) (=) Npos.
Proof. by injection 1. Qed.
Instance N_eq_dec: ∀ x y : N, Decision (x = y) := N.eq_dec.
Program Instance N_le_dec (x y : N) : Decision (x ≤ y)%N :=
match Ncompare x y with Gt ⇒ right _ | _ ⇒ left _ end.
Solve Obligations with naive_solver.
Program Instance N_lt_dec (x y : N) : Decision (x < y)%N :=
match Ncompare x y with Lt ⇒ left _ | _ ⇒ right _ end.
Solve Obligations with naive_solver.
Instance N_inhabited: Inhabited N := populate 1%N.
Instance: PartialOrder (≤)%N.
Proof.
repeat split; red. apply N.le_refl. apply N.le_trans. apply N.le_antisymm.
Qed.
Hint Extern 0 (_ ≤ _)%N ⇒ reflexivity.
Notation "x ≤ y ≤ z" := (x ≤ y ∧ y ≤ z)%N : N_scope.
Notation "x ≤ y < z" := (x ≤ y ∧ y < z)%N : N_scope.
Notation "x < y < z" := (x < y ∧ y < z)%N : N_scope.
Notation "x < y ≤ z" := (x < y ∧ y ≤ z)%N : N_scope.
Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z')%N : N_scope.
Notation "(≤)" := N.le (only parsing) : N_scope.
Notation "(<)" := N.lt (only parsing) : N_scope.
Infix "`div`" := N.div (at level 35) : N_scope.
Infix "`mod`" := N.modulo (at level 35) : N_scope.
Arguments N.add _ _ : simpl never.
Instance: Inj (=) (=) Npos.
Proof. by injection 1. Qed.
Instance N_eq_dec: ∀ x y : N, Decision (x = y) := N.eq_dec.
Program Instance N_le_dec (x y : N) : Decision (x ≤ y)%N :=
match Ncompare x y with Gt ⇒ right _ | _ ⇒ left _ end.
Solve Obligations with naive_solver.
Program Instance N_lt_dec (x y : N) : Decision (x < y)%N :=
match Ncompare x y with Lt ⇒ left _ | _ ⇒ right _ end.
Solve Obligations with naive_solver.
Instance N_inhabited: Inhabited N := populate 1%N.
Instance: PartialOrder (≤)%N.
Proof.
repeat split; red. apply N.le_refl. apply N.le_trans. apply N.le_antisymm.
Qed.
Hint Extern 0 (_ ≤ _)%N ⇒ reflexivity.
Notations and properties of Z
Open Scope Z_scope.
Infix "≤" := Z.le : Z_scope.
Notation "x ≤ y ≤ z" := (x ≤ y ∧ y ≤ z) : Z_scope.
Notation "x ≤ y < z" := (x ≤ y ∧ y < z) : Z_scope.
Notation "x < y < z" := (x < y ∧ y < z) : Z_scope.
Notation "x < y ≤ z" := (x < y ∧ y ≤ z) : Z_scope.
Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z') : Z_scope.
Notation "(≤)" := Z.le (only parsing) : Z_scope.
Notation "(<)" := Z.lt (only parsing) : Z_scope.
Infix "`div`" := Z.div (at level 35) : Z_scope.
Infix "`mod`" := Z.modulo (at level 35) : Z_scope.
Infix "`quot`" := Z.quot (at level 35) : Z_scope.
Infix "`rem`" := Z.rem (at level 35) : Z_scope.
Infix "≪" := Z.shiftl (at level 35) : Z_scope.
Infix "≫" := Z.shiftr (at level 35) : Z_scope.
Instance: Inj (=) (=) Zpos.
Proof. by injection 1. Qed.
Instance: Inj (=) (=) Zneg.
Proof. by injection 1. Qed.
Instance Z_eq_dec: ∀ x y : Z, Decision (x = y) := Z.eq_dec.
Instance Z_le_dec: ∀ x y : Z, Decision (x ≤ y) := Z_le_dec.
Instance Z_lt_dec: ∀ x y : Z, Decision (x < y) := Z_lt_dec.
Instance Z_inhabited: Inhabited Z := populate 1.
Instance: PartialOrder (≤).
Proof.
repeat split; red. apply Z.le_refl. apply Z.le_trans. apply Z.le_antisymm.
Qed.
Lemma Z_pow_pred_r n m : 0 < m → n × n ^ (Z.pred m) = n ^ m.
Proof.
intros. rewrite <-Z.pow_succ_r, Z.succ_pred. done. by apply Z.lt_le_pred.
Qed.
Lemma Z_quot_range_nonneg k x y : 0 ≤ x < k → 0 < y → 0 ≤ x `quot` y < k.
Proof.
intros [??] ?.
destruct (decide (y = 1)); subst; [rewrite Z.quot_1_r; auto |].
destruct (decide (x = 0)); subst; [rewrite Z.quot_0_l; auto with lia |].
split. apply Z.quot_pos; lia. trans x; auto. apply Z.quot_lt; lia.
Qed.
Arguments Z.to_nat _ : simpl never.
Arguments Z.mul _ _ : simpl never.
Arguments Z.add _ _ : simpl never.
Arguments Z.opp _ : simpl never.
Arguments Z.pow _ _ : simpl never.
Arguments Z.div _ _ : simpl never.
Arguments Z.modulo _ _ : simpl never.
Arguments Z.quot _ _ : simpl never.
Arguments Z.rem _ _ : simpl never.
Lemma Z_to_nat_neq_0_pos x : Z.to_nat x ≠ 0%nat → 0 < x.
Proof. by destruct x. Qed.
Lemma Z_to_nat_neq_0_nonneg x : Z.to_nat x ≠ 0%nat → 0 ≤ x.
Proof. by destruct x. Qed.
Lemma Z_mod_pos x y : 0 < y → 0 ≤ x `mod` y.
Proof. apply Z.mod_pos_bound. Qed.
Hint Resolve Z.lt_le_incl : zpos.
Hint Resolve Z.add_nonneg_pos Z.add_pos_nonneg Z.add_nonneg_nonneg : zpos.
Hint Resolve Z.mul_nonneg_nonneg Z.mul_pos_pos : zpos.
Hint Resolve Z.pow_pos_nonneg Z.pow_nonneg: zpos.
Hint Resolve Z_mod_pos Z.div_pos : zpos.
Hint Extern 1000 ⇒ lia : zpos.
Lemma Z_to_nat_nonpos x : x ≤ 0 → Z.to_nat x = 0%nat.
Proof. destruct x; simpl; auto using Z2Nat.inj_neg. by intros []. Qed.
Lemma Z2Nat_inj_pow (x y : nat) : Z.of_nat (x ^ y) = x ^ y.
Proof.
induction y as [|y IH]; [by rewrite Z.pow_0_r, Nat.pow_0_r|].
by rewrite Nat.pow_succ_r, Nat2Z.inj_succ, Z.pow_succ_r,
Nat2Z.inj_mul, IH by auto with zpos.
Qed.
Lemma Nat2Z_divide n m : (Z.of_nat n | Z.of_nat m) ↔ (n | m)%nat.
Proof.
split.
- rewrite <-(Nat2Z.id m) at 2; intros [i ->]; ∃ (Z.to_nat i).
destruct (decide (0 ≤ i)%Z).
{ by rewrite Z2Nat.inj_mul, Nat2Z.id by lia. }
by rewrite !Z_to_nat_nonpos by auto using Z.mul_nonpos_nonneg with lia.
- intros [i ->]. ∃ (Z.of_nat i). by rewrite Nat2Z.inj_mul.
Qed.
Lemma Z2Nat_divide n m :
0 ≤ n → 0 ≤ m → (Z.to_nat n | Z.to_nat m)%nat ↔ (n | m).
Proof. intros. by rewrite <-Nat2Z_divide, !Z2Nat.id by done. Qed.
Lemma Z2Nat_inj_div x y : Z.of_nat (x `div` y) = x `div` y.
Proof.
destruct (decide (y = 0%nat)); [by subst; destruct x |].
apply Z.div_unique with (x `mod` y)%nat.
{ left. rewrite <-(Nat2Z.inj_le 0), <-Nat2Z.inj_lt.
apply Nat.mod_bound_pos; lia. }
by rewrite <-Nat2Z.inj_mul, <-Nat2Z.inj_add, <-Nat.div_mod.
Qed.
Lemma Z2Nat_inj_mod x y : Z.of_nat (x `mod` y) = x `mod` y.
Proof.
destruct (decide (y = 0%nat)); [by subst; destruct x |].
apply Z.mod_unique with (x `div` y)%nat.
{ left. rewrite <-(Nat2Z.inj_le 0), <-Nat2Z.inj_lt.
apply Nat.mod_bound_pos; lia. }
by rewrite <-Nat2Z.inj_mul, <-Nat2Z.inj_add, <-Nat.div_mod.
Qed.
Close Scope Z_scope.
Infix "≤" := Z.le : Z_scope.
Notation "x ≤ y ≤ z" := (x ≤ y ∧ y ≤ z) : Z_scope.
Notation "x ≤ y < z" := (x ≤ y ∧ y < z) : Z_scope.
Notation "x < y < z" := (x < y ∧ y < z) : Z_scope.
Notation "x < y ≤ z" := (x < y ∧ y ≤ z) : Z_scope.
Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z') : Z_scope.
Notation "(≤)" := Z.le (only parsing) : Z_scope.
Notation "(<)" := Z.lt (only parsing) : Z_scope.
Infix "`div`" := Z.div (at level 35) : Z_scope.
Infix "`mod`" := Z.modulo (at level 35) : Z_scope.
Infix "`quot`" := Z.quot (at level 35) : Z_scope.
Infix "`rem`" := Z.rem (at level 35) : Z_scope.
Infix "≪" := Z.shiftl (at level 35) : Z_scope.
Infix "≫" := Z.shiftr (at level 35) : Z_scope.
Instance: Inj (=) (=) Zpos.
Proof. by injection 1. Qed.
Instance: Inj (=) (=) Zneg.
Proof. by injection 1. Qed.
Instance Z_eq_dec: ∀ x y : Z, Decision (x = y) := Z.eq_dec.
Instance Z_le_dec: ∀ x y : Z, Decision (x ≤ y) := Z_le_dec.
Instance Z_lt_dec: ∀ x y : Z, Decision (x < y) := Z_lt_dec.
Instance Z_inhabited: Inhabited Z := populate 1.
Instance: PartialOrder (≤).
Proof.
repeat split; red. apply Z.le_refl. apply Z.le_trans. apply Z.le_antisymm.
Qed.
Lemma Z_pow_pred_r n m : 0 < m → n × n ^ (Z.pred m) = n ^ m.
Proof.
intros. rewrite <-Z.pow_succ_r, Z.succ_pred. done. by apply Z.lt_le_pred.
Qed.
Lemma Z_quot_range_nonneg k x y : 0 ≤ x < k → 0 < y → 0 ≤ x `quot` y < k.
Proof.
intros [??] ?.
destruct (decide (y = 1)); subst; [rewrite Z.quot_1_r; auto |].
destruct (decide (x = 0)); subst; [rewrite Z.quot_0_l; auto with lia |].
split. apply Z.quot_pos; lia. trans x; auto. apply Z.quot_lt; lia.
Qed.
Arguments Z.to_nat _ : simpl never.
Arguments Z.mul _ _ : simpl never.
Arguments Z.add _ _ : simpl never.
Arguments Z.opp _ : simpl never.
Arguments Z.pow _ _ : simpl never.
Arguments Z.div _ _ : simpl never.
Arguments Z.modulo _ _ : simpl never.
Arguments Z.quot _ _ : simpl never.
Arguments Z.rem _ _ : simpl never.
Lemma Z_to_nat_neq_0_pos x : Z.to_nat x ≠ 0%nat → 0 < x.
Proof. by destruct x. Qed.
Lemma Z_to_nat_neq_0_nonneg x : Z.to_nat x ≠ 0%nat → 0 ≤ x.
Proof. by destruct x. Qed.
Lemma Z_mod_pos x y : 0 < y → 0 ≤ x `mod` y.
Proof. apply Z.mod_pos_bound. Qed.
Hint Resolve Z.lt_le_incl : zpos.
Hint Resolve Z.add_nonneg_pos Z.add_pos_nonneg Z.add_nonneg_nonneg : zpos.
Hint Resolve Z.mul_nonneg_nonneg Z.mul_pos_pos : zpos.
Hint Resolve Z.pow_pos_nonneg Z.pow_nonneg: zpos.
Hint Resolve Z_mod_pos Z.div_pos : zpos.
Hint Extern 1000 ⇒ lia : zpos.
Lemma Z_to_nat_nonpos x : x ≤ 0 → Z.to_nat x = 0%nat.
Proof. destruct x; simpl; auto using Z2Nat.inj_neg. by intros []. Qed.
Lemma Z2Nat_inj_pow (x y : nat) : Z.of_nat (x ^ y) = x ^ y.
Proof.
induction y as [|y IH]; [by rewrite Z.pow_0_r, Nat.pow_0_r|].
by rewrite Nat.pow_succ_r, Nat2Z.inj_succ, Z.pow_succ_r,
Nat2Z.inj_mul, IH by auto with zpos.
Qed.
Lemma Nat2Z_divide n m : (Z.of_nat n | Z.of_nat m) ↔ (n | m)%nat.
Proof.
split.
- rewrite <-(Nat2Z.id m) at 2; intros [i ->]; ∃ (Z.to_nat i).
destruct (decide (0 ≤ i)%Z).
{ by rewrite Z2Nat.inj_mul, Nat2Z.id by lia. }
by rewrite !Z_to_nat_nonpos by auto using Z.mul_nonpos_nonneg with lia.
- intros [i ->]. ∃ (Z.of_nat i). by rewrite Nat2Z.inj_mul.
Qed.
Lemma Z2Nat_divide n m :
0 ≤ n → 0 ≤ m → (Z.to_nat n | Z.to_nat m)%nat ↔ (n | m).
Proof. intros. by rewrite <-Nat2Z_divide, !Z2Nat.id by done. Qed.
Lemma Z2Nat_inj_div x y : Z.of_nat (x `div` y) = x `div` y.
Proof.
destruct (decide (y = 0%nat)); [by subst; destruct x |].
apply Z.div_unique with (x `mod` y)%nat.
{ left. rewrite <-(Nat2Z.inj_le 0), <-Nat2Z.inj_lt.
apply Nat.mod_bound_pos; lia. }
by rewrite <-Nat2Z.inj_mul, <-Nat2Z.inj_add, <-Nat.div_mod.
Qed.
Lemma Z2Nat_inj_mod x y : Z.of_nat (x `mod` y) = x `mod` y.
Proof.
destruct (decide (y = 0%nat)); [by subst; destruct x |].
apply Z.mod_unique with (x `div` y)%nat.
{ left. rewrite <-(Nat2Z.inj_le 0), <-Nat2Z.inj_lt.
apply Nat.mod_bound_pos; lia. }
by rewrite <-Nat2Z.inj_mul, <-Nat2Z.inj_add, <-Nat.div_mod.
Qed.
Close Scope Z_scope.
Notations and properties of Qc
Open Scope Qc_scope.
Delimit Scope Qc_scope with Qc.
Notation "1" := (Q2Qc 1) : Qc_scope.
Notation "2" := (1+1) : Qc_scope.
Notation "- 1" := (Qcopp 1) : Qc_scope.
Notation "- 2" := (Qcopp 2) : Qc_scope.
Notation "x - y" := (x + -y) : Qc_scope.
Notation "x / y" := (x × /y) : Qc_scope.
Infix "≤" := Qcle : Qc_scope.
Notation "x ≤ y ≤ z" := (x ≤ y ∧ y ≤ z) : Qc_scope.
Notation "x ≤ y < z" := (x ≤ y ∧ y < z) : Qc_scope.
Notation "x < y < z" := (x < y ∧ y < z) : Qc_scope.
Notation "x < y ≤ z" := (x < y ∧ y ≤ z) : Qc_scope.
Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z') : Qc_scope.
Notation "(≤)" := Qcle (only parsing) : Qc_scope.
Notation "(<)" := Qclt (only parsing) : Qc_scope.
Hint Extern 1 (_ ≤ _) ⇒ reflexivity || discriminate.
Arguments Qred _ : simpl never.
Instance Qc_eq_dec: ∀ x y : Qc, Decision (x = y) := Qc_eq_dec.
Program Instance Qc_le_dec (x y : Qc) : Decision (x ≤ y) :=
if Qclt_le_dec y x then right _ else left _.
Next Obligation. intros x y; apply Qclt_not_le. Qed.
Next Obligation. done. Qed.
Program Instance Qc_lt_dec (x y : Qc) : Decision (x < y) :=
if Qclt_le_dec x y then left _ else right _.
Solve Obligations with done.
Next Obligation. intros x y; apply Qcle_not_lt. Qed.
Instance: PartialOrder (≤).
Proof.
repeat split; red. apply Qcle_refl. apply Qcle_trans. apply Qcle_antisym.
Qed.
Instance: StrictOrder (<).
Proof.
split; red. intros x Hx. by destruct (Qclt_not_eq x x). apply Qclt_trans.
Qed.
Lemma Qcmult_0_l x : 0 × x = 0.
Proof. ring. Qed.
Lemma Qcmult_0_r x : x × 0 = 0.
Proof. ring. Qed.
Lemma Qcplus_diag x : (x + x)%Qc = (2 × x)%Qc.
Proof. ring. Qed.
Lemma Qcle_ngt (x y : Qc) : x ≤ y ↔ ¬y < x.
Proof. split; auto using Qcle_not_lt, Qcnot_lt_le. Qed.
Lemma Qclt_nge (x y : Qc) : x < y ↔ ¬y ≤ x.
Proof. split; auto using Qclt_not_le, Qcnot_le_lt. Qed.
Lemma Qcplus_le_mono_l (x y z : Qc) : x ≤ y ↔ z + x ≤ z + y.
Proof.
split; intros.
- by apply Qcplus_le_compat.
- replace x with ((0 - z) + (z + x)) by ring.
replace y with ((0 - z) + (z + y)) by ring.
by apply Qcplus_le_compat.
Qed.
Lemma Qcplus_le_mono_r (x y z : Qc) : x ≤ y ↔ x + z ≤ y + z.
Proof. rewrite !(Qcplus_comm _ z). apply Qcplus_le_mono_l. Qed.
Lemma Qcplus_lt_mono_l (x y z : Qc) : x < y ↔ z + x < z + y.
Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_l. Qed.
Lemma Qcplus_lt_mono_r (x y z : Qc) : x < y ↔ x + z < y + z.
Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_r. Qed.
Instance: Inj (=) (=) Qcopp.
Proof.
intros x y H. by rewrite <-(Qcopp_involutive x), H, Qcopp_involutive.
Qed.
Instance: ∀ z, Inj (=) (=) (Qcplus z).
Proof.
intros z x y H. by apply (anti_symm (≤));
rewrite (Qcplus_le_mono_l _ _ z), H.
Qed.
Instance: ∀ z, Inj (=) (=) (λ x, x + z).
Proof.
intros z x y H. by apply (anti_symm (≤));
rewrite (Qcplus_le_mono_r _ _ z), H.
Qed.
Lemma Qcplus_pos_nonneg (x y : Qc) : 0 < x → 0 ≤ y → 0 < x + y.
Proof.
intros. apply Qclt_le_trans with (x + 0); [by rewrite Qcplus_0_r|].
by apply Qcplus_le_mono_l.
Qed.
Lemma Qcplus_nonneg_pos (x y : Qc) : 0 ≤ x → 0 < y → 0 < x + y.
Proof. rewrite (Qcplus_comm x). auto using Qcplus_pos_nonneg. Qed.
Lemma Qcplus_pos_pos (x y : Qc) : 0 < x → 0 < y → 0 < x + y.
Proof. auto using Qcplus_pos_nonneg, Qclt_le_weak. Qed.
Lemma Qcplus_nonneg_nonneg (x y : Qc) : 0 ≤ x → 0 ≤ y → 0 ≤ x + y.
Proof.
intros. trans (x + 0); [by rewrite Qcplus_0_r|].
by apply Qcplus_le_mono_l.
Qed.
Lemma Qcplus_neg_nonpos (x y : Qc) : x < 0 → y ≤ 0 → x + y < 0.
Proof.
intros. apply Qcle_lt_trans with (x + 0); [|by rewrite Qcplus_0_r].
by apply Qcplus_le_mono_l.
Qed.
Lemma Qcplus_nonpos_neg (x y : Qc) : x ≤ 0 → y < 0 → x + y < 0.
Proof. rewrite (Qcplus_comm x). auto using Qcplus_neg_nonpos. Qed.
Lemma Qcplus_neg_neg (x y : Qc) : x < 0 → y < 0 → x + y < 0.
Proof. auto using Qcplus_nonpos_neg, Qclt_le_weak. Qed.
Lemma Qcplus_nonpos_nonpos (x y : Qc) : x ≤ 0 → y ≤ 0 → x + y ≤ 0.
Proof.
intros. trans (x + 0); [|by rewrite Qcplus_0_r].
by apply Qcplus_le_mono_l.
Qed.
Lemma Qcmult_le_mono_nonneg_l x y z : 0 ≤ z → x ≤ y → z × x ≤ z × y.
Proof. intros. rewrite !(Qcmult_comm z). by apply Qcmult_le_compat_r. Qed.
Lemma Qcmult_le_mono_nonneg_r x y z : 0 ≤ z → x ≤ y → x × z ≤ y × z.
Proof. intros. by apply Qcmult_le_compat_r. Qed.
Lemma Qcmult_le_mono_pos_l x y z : 0 < z → x ≤ y ↔ z × x ≤ z × y.
Proof.
split; auto using Qcmult_le_mono_nonneg_l, Qclt_le_weak.
rewrite !Qcle_ngt, !(Qcmult_comm z).
intuition auto using Qcmult_lt_compat_r.
Qed.
Lemma Qcmult_le_mono_pos_r x y z : 0 < z → x ≤ y ↔ x × z ≤ y × z.
Proof. rewrite !(Qcmult_comm _ z). by apply Qcmult_le_mono_pos_l. Qed.
Lemma Qcmult_lt_mono_pos_l x y z : 0 < z → x < y ↔ z × x < z × y.
Proof. intros. by rewrite !Qclt_nge, <-Qcmult_le_mono_pos_l. Qed.
Lemma Qcmult_lt_mono_pos_r x y z : 0 < z → x < y ↔ x × z < y × z.
Proof. intros. by rewrite !Qclt_nge, <-Qcmult_le_mono_pos_r. Qed.
Lemma Qcmult_pos_pos x y : 0 < x → 0 < y → 0 < x × y.
Proof.
intros. apply Qcle_lt_trans with (0 × y); [by rewrite Qcmult_0_l|].
by apply Qcmult_lt_mono_pos_r.
Qed.
Lemma Qcmult_nonneg_nonneg x y : 0 ≤ x → 0 ≤ y → 0 ≤ x × y.
Proof.
intros. trans (0 × y); [by rewrite Qcmult_0_l|].
by apply Qcmult_le_mono_nonneg_r.
Qed.
Lemma inject_Z_Qred n : Qred (inject_Z n) = inject_Z n.
Proof. apply Qred_identity; auto using Z.gcd_1_r. Qed.
Coercion Qc_of_Z (n : Z) : Qc := Qcmake _ (inject_Z_Qred n).
Lemma Z2Qc_inj_0 : Qc_of_Z 0 = 0.
Proof. by apply Qc_is_canon. Qed.
Lemma Z2Qc_inj_1 : Qc_of_Z 1 = 1.
Proof. by apply Qc_is_canon. Qed.
Lemma Z2Qc_inj_2 : Qc_of_Z 2 = 2.
Proof. by apply Qc_is_canon. Qed.
Lemma Z2Qc_inj n m : Qc_of_Z n = Qc_of_Z m → n = m.
Proof. by injection 1. Qed.
Lemma Z2Qc_inj_iff n m : Qc_of_Z n = Qc_of_Z m ↔ n = m.
Proof. split. auto using Z2Qc_inj. by intros →. Qed.
Lemma Z2Qc_inj_le n m : (n ≤ m)%Z ↔ Qc_of_Z n ≤ Qc_of_Z m.
Proof. by rewrite Zle_Qle. Qed.
Lemma Z2Qc_inj_lt n m : (n < m)%Z ↔ Qc_of_Z n < Qc_of_Z m.
Proof. by rewrite Zlt_Qlt. Qed.
Lemma Z2Qc_inj_add n m : Qc_of_Z (n + m) = Qc_of_Z n + Qc_of_Z m.
Proof. apply Qc_is_canon; simpl. by rewrite Qred_correct, inject_Z_plus. Qed.
Lemma Z2Qc_inj_mul n m : Qc_of_Z (n × m) = Qc_of_Z n × Qc_of_Z m.
Proof. apply Qc_is_canon; simpl. by rewrite Qred_correct, inject_Z_mult. Qed.
Lemma Z2Qc_inj_opp n : Qc_of_Z (-n) = -Qc_of_Z n.
Proof. apply Qc_is_canon; simpl. by rewrite Qred_correct, inject_Z_opp. Qed.
Lemma Z2Qc_inj_sub n m : Qc_of_Z (n - m) = Qc_of_Z n - Qc_of_Z m.
Proof.
apply Qc_is_canon; simpl.
by rewrite !Qred_correct, <-inject_Z_opp, <-inject_Z_plus.
Qed.
Close Scope Qc_scope.
Delimit Scope Qc_scope with Qc.
Notation "1" := (Q2Qc 1) : Qc_scope.
Notation "2" := (1+1) : Qc_scope.
Notation "- 1" := (Qcopp 1) : Qc_scope.
Notation "- 2" := (Qcopp 2) : Qc_scope.
Notation "x - y" := (x + -y) : Qc_scope.
Notation "x / y" := (x × /y) : Qc_scope.
Infix "≤" := Qcle : Qc_scope.
Notation "x ≤ y ≤ z" := (x ≤ y ∧ y ≤ z) : Qc_scope.
Notation "x ≤ y < z" := (x ≤ y ∧ y < z) : Qc_scope.
Notation "x < y < z" := (x < y ∧ y < z) : Qc_scope.
Notation "x < y ≤ z" := (x < y ∧ y ≤ z) : Qc_scope.
Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z') : Qc_scope.
Notation "(≤)" := Qcle (only parsing) : Qc_scope.
Notation "(<)" := Qclt (only parsing) : Qc_scope.
Hint Extern 1 (_ ≤ _) ⇒ reflexivity || discriminate.
Arguments Qred _ : simpl never.
Instance Qc_eq_dec: ∀ x y : Qc, Decision (x = y) := Qc_eq_dec.
Program Instance Qc_le_dec (x y : Qc) : Decision (x ≤ y) :=
if Qclt_le_dec y x then right _ else left _.
Next Obligation. intros x y; apply Qclt_not_le. Qed.
Next Obligation. done. Qed.
Program Instance Qc_lt_dec (x y : Qc) : Decision (x < y) :=
if Qclt_le_dec x y then left _ else right _.
Solve Obligations with done.
Next Obligation. intros x y; apply Qcle_not_lt. Qed.
Instance: PartialOrder (≤).
Proof.
repeat split; red. apply Qcle_refl. apply Qcle_trans. apply Qcle_antisym.
Qed.
Instance: StrictOrder (<).
Proof.
split; red. intros x Hx. by destruct (Qclt_not_eq x x). apply Qclt_trans.
Qed.
Lemma Qcmult_0_l x : 0 × x = 0.
Proof. ring. Qed.
Lemma Qcmult_0_r x : x × 0 = 0.
Proof. ring. Qed.
Lemma Qcplus_diag x : (x + x)%Qc = (2 × x)%Qc.
Proof. ring. Qed.
Lemma Qcle_ngt (x y : Qc) : x ≤ y ↔ ¬y < x.
Proof. split; auto using Qcle_not_lt, Qcnot_lt_le. Qed.
Lemma Qclt_nge (x y : Qc) : x < y ↔ ¬y ≤ x.
Proof. split; auto using Qclt_not_le, Qcnot_le_lt. Qed.
Lemma Qcplus_le_mono_l (x y z : Qc) : x ≤ y ↔ z + x ≤ z + y.
Proof.
split; intros.
- by apply Qcplus_le_compat.
- replace x with ((0 - z) + (z + x)) by ring.
replace y with ((0 - z) + (z + y)) by ring.
by apply Qcplus_le_compat.
Qed.
Lemma Qcplus_le_mono_r (x y z : Qc) : x ≤ y ↔ x + z ≤ y + z.
Proof. rewrite !(Qcplus_comm _ z). apply Qcplus_le_mono_l. Qed.
Lemma Qcplus_lt_mono_l (x y z : Qc) : x < y ↔ z + x < z + y.
Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_l. Qed.
Lemma Qcplus_lt_mono_r (x y z : Qc) : x < y ↔ x + z < y + z.
Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_r. Qed.
Instance: Inj (=) (=) Qcopp.
Proof.
intros x y H. by rewrite <-(Qcopp_involutive x), H, Qcopp_involutive.
Qed.
Instance: ∀ z, Inj (=) (=) (Qcplus z).
Proof.
intros z x y H. by apply (anti_symm (≤));
rewrite (Qcplus_le_mono_l _ _ z), H.
Qed.
Instance: ∀ z, Inj (=) (=) (λ x, x + z).
Proof.
intros z x y H. by apply (anti_symm (≤));
rewrite (Qcplus_le_mono_r _ _ z), H.
Qed.
Lemma Qcplus_pos_nonneg (x y : Qc) : 0 < x → 0 ≤ y → 0 < x + y.
Proof.
intros. apply Qclt_le_trans with (x + 0); [by rewrite Qcplus_0_r|].
by apply Qcplus_le_mono_l.
Qed.
Lemma Qcplus_nonneg_pos (x y : Qc) : 0 ≤ x → 0 < y → 0 < x + y.
Proof. rewrite (Qcplus_comm x). auto using Qcplus_pos_nonneg. Qed.
Lemma Qcplus_pos_pos (x y : Qc) : 0 < x → 0 < y → 0 < x + y.
Proof. auto using Qcplus_pos_nonneg, Qclt_le_weak. Qed.
Lemma Qcplus_nonneg_nonneg (x y : Qc) : 0 ≤ x → 0 ≤ y → 0 ≤ x + y.
Proof.
intros. trans (x + 0); [by rewrite Qcplus_0_r|].
by apply Qcplus_le_mono_l.
Qed.
Lemma Qcplus_neg_nonpos (x y : Qc) : x < 0 → y ≤ 0 → x + y < 0.
Proof.
intros. apply Qcle_lt_trans with (x + 0); [|by rewrite Qcplus_0_r].
by apply Qcplus_le_mono_l.
Qed.
Lemma Qcplus_nonpos_neg (x y : Qc) : x ≤ 0 → y < 0 → x + y < 0.
Proof. rewrite (Qcplus_comm x). auto using Qcplus_neg_nonpos. Qed.
Lemma Qcplus_neg_neg (x y : Qc) : x < 0 → y < 0 → x + y < 0.
Proof. auto using Qcplus_nonpos_neg, Qclt_le_weak. Qed.
Lemma Qcplus_nonpos_nonpos (x y : Qc) : x ≤ 0 → y ≤ 0 → x + y ≤ 0.
Proof.
intros. trans (x + 0); [|by rewrite Qcplus_0_r].
by apply Qcplus_le_mono_l.
Qed.
Lemma Qcmult_le_mono_nonneg_l x y z : 0 ≤ z → x ≤ y → z × x ≤ z × y.
Proof. intros. rewrite !(Qcmult_comm z). by apply Qcmult_le_compat_r. Qed.
Lemma Qcmult_le_mono_nonneg_r x y z : 0 ≤ z → x ≤ y → x × z ≤ y × z.
Proof. intros. by apply Qcmult_le_compat_r. Qed.
Lemma Qcmult_le_mono_pos_l x y z : 0 < z → x ≤ y ↔ z × x ≤ z × y.
Proof.
split; auto using Qcmult_le_mono_nonneg_l, Qclt_le_weak.
rewrite !Qcle_ngt, !(Qcmult_comm z).
intuition auto using Qcmult_lt_compat_r.
Qed.
Lemma Qcmult_le_mono_pos_r x y z : 0 < z → x ≤ y ↔ x × z ≤ y × z.
Proof. rewrite !(Qcmult_comm _ z). by apply Qcmult_le_mono_pos_l. Qed.
Lemma Qcmult_lt_mono_pos_l x y z : 0 < z → x < y ↔ z × x < z × y.
Proof. intros. by rewrite !Qclt_nge, <-Qcmult_le_mono_pos_l. Qed.
Lemma Qcmult_lt_mono_pos_r x y z : 0 < z → x < y ↔ x × z < y × z.
Proof. intros. by rewrite !Qclt_nge, <-Qcmult_le_mono_pos_r. Qed.
Lemma Qcmult_pos_pos x y : 0 < x → 0 < y → 0 < x × y.
Proof.
intros. apply Qcle_lt_trans with (0 × y); [by rewrite Qcmult_0_l|].
by apply Qcmult_lt_mono_pos_r.
Qed.
Lemma Qcmult_nonneg_nonneg x y : 0 ≤ x → 0 ≤ y → 0 ≤ x × y.
Proof.
intros. trans (0 × y); [by rewrite Qcmult_0_l|].
by apply Qcmult_le_mono_nonneg_r.
Qed.
Lemma inject_Z_Qred n : Qred (inject_Z n) = inject_Z n.
Proof. apply Qred_identity; auto using Z.gcd_1_r. Qed.
Coercion Qc_of_Z (n : Z) : Qc := Qcmake _ (inject_Z_Qred n).
Lemma Z2Qc_inj_0 : Qc_of_Z 0 = 0.
Proof. by apply Qc_is_canon. Qed.
Lemma Z2Qc_inj_1 : Qc_of_Z 1 = 1.
Proof. by apply Qc_is_canon. Qed.
Lemma Z2Qc_inj_2 : Qc_of_Z 2 = 2.
Proof. by apply Qc_is_canon. Qed.
Lemma Z2Qc_inj n m : Qc_of_Z n = Qc_of_Z m → n = m.
Proof. by injection 1. Qed.
Lemma Z2Qc_inj_iff n m : Qc_of_Z n = Qc_of_Z m ↔ n = m.
Proof. split. auto using Z2Qc_inj. by intros →. Qed.
Lemma Z2Qc_inj_le n m : (n ≤ m)%Z ↔ Qc_of_Z n ≤ Qc_of_Z m.
Proof. by rewrite Zle_Qle. Qed.
Lemma Z2Qc_inj_lt n m : (n < m)%Z ↔ Qc_of_Z n < Qc_of_Z m.
Proof. by rewrite Zlt_Qlt. Qed.
Lemma Z2Qc_inj_add n m : Qc_of_Z (n + m) = Qc_of_Z n + Qc_of_Z m.
Proof. apply Qc_is_canon; simpl. by rewrite Qred_correct, inject_Z_plus. Qed.
Lemma Z2Qc_inj_mul n m : Qc_of_Z (n × m) = Qc_of_Z n × Qc_of_Z m.
Proof. apply Qc_is_canon; simpl. by rewrite Qred_correct, inject_Z_mult. Qed.
Lemma Z2Qc_inj_opp n : Qc_of_Z (-n) = -Qc_of_Z n.
Proof. apply Qc_is_canon; simpl. by rewrite Qred_correct, inject_Z_opp. Qed.
Lemma Z2Qc_inj_sub n m : Qc_of_Z (n - m) = Qc_of_Z n - Qc_of_Z m.
Proof.
apply Qc_is_canon; simpl.
by rewrite !Qred_correct, <-inject_Z_opp, <-inject_Z_plus.
Qed.
Close Scope Qc_scope.
Positive rationals
The theory of positive rationals is very incomplete. We merely provide some operations and theorems that are relevant for fractional permissions.
Record Qp := mk_Qp { Qp_car :> Qc ; Qp_prf : (0 < Qp_car)%Qc }.
Hint Resolve Qp_prf.
Delimit Scope Qp_scope with Qp.
Bind Scope Qp_scope with Qp.
Arguments Qp_car _%Qp.
Definition Qp_one : Qp := mk_Qp 1 eq_refl.
Program Definition Qp_plus (x y : Qp) : Qp := mk_Qp (x + y) _.
Next Obligation. by intros x y; apply Qcplus_pos_pos. Qed.
Definition Qp_minus (x y : Qp) : option Qp :=
let z := (x - y)%Qc in
match decide (0 < z)%Qc with left Hz ⇒ Some (mk_Qp z Hz) | _ ⇒ None end.
Program Definition Qp_div (x : Qp) (y : positive) : Qp := mk_Qp (x / ('y)%Z) _.
Next Obligation.
intros x y. assert (0 < ('y)%Z)%Qc.
{ apply (Z2Qc_inj_lt 0%Z (' y)), Pos2Z.is_pos. }
by rewrite (Qcmult_lt_mono_pos_r _ _ ('y)%Z), Qcmult_0_l,
<-Qcmult_assoc, Qcmult_inv_l, Qcmult_1_r.
Qed.
Notation "1" := Qp_one : Qp_scope.
Infix "+" := Qp_plus : Qp_scope.
Infix "-" := Qp_minus : Qp_scope.
Infix "/" := Qp_div : Qp_scope.
Instance Qp_inhabited : Inhabited Qp := populate 1%Qp.
Lemma Qp_eq x y : x = y ↔ Qp_car x = Qp_car y.
Proof.
split; [by intros ->|].
destruct x, y; intros; simplify_eq/=; f_equal; apply (proof_irrel _).
Qed.
Instance Qp_plus_assoc : Assoc (=) Qp_plus.
Proof. intros x y z; apply Qp_eq, Qcplus_assoc. Qed.
Instance Qp_plus_comm : Comm (=) Qp_plus.
Proof. intros x y; apply Qp_eq, Qcplus_comm. Qed.
Lemma Qp_minus_diag x : (x - x)%Qp = None.
Proof. unfold Qp_minus. by rewrite Qcplus_opp_r. Qed.
Lemma Qp_op_minus x y : ((x + y) - x)%Qp = Some y.
Proof.
unfold Qp_minus; simpl.
rewrite (Qcplus_comm x), <- Qcplus_assoc, Qcplus_opp_r, Qcplus_0_r.
destruct (decide _) as [|[]]; auto. by f_equal; apply Qp_eq.
Qed.
Lemma Qp_div_1 x : (x / 1 = x)%Qp.
Proof.
apply Qp_eq; simpl.
rewrite <-(Qcmult_div_r x 1) at 2 by done. by rewrite Qcmult_1_l.
Qed.
Lemma Qp_div_S x y : (x / (2 × y) + x / (2 × y) = x / y)%Qp.
Proof.
apply Qp_eq; simpl.
rewrite <-Qcmult_plus_distr_l, Pos2Z.inj_mul, Z2Qc_inj_mul, Z2Qc_inj_2.
rewrite Qcplus_diag. by field_simplify.
Qed.
Lemma Qp_div_2 x : (x / 2 + x / 2 = x)%Qp.
Proof.
change 2%positive with (2 × 1)%positive. by rewrite Qp_div_S, Qp_div_1.
Qed.
Hint Resolve Qp_prf.
Delimit Scope Qp_scope with Qp.
Bind Scope Qp_scope with Qp.
Arguments Qp_car _%Qp.
Definition Qp_one : Qp := mk_Qp 1 eq_refl.
Program Definition Qp_plus (x y : Qp) : Qp := mk_Qp (x + y) _.
Next Obligation. by intros x y; apply Qcplus_pos_pos. Qed.
Definition Qp_minus (x y : Qp) : option Qp :=
let z := (x - y)%Qc in
match decide (0 < z)%Qc with left Hz ⇒ Some (mk_Qp z Hz) | _ ⇒ None end.
Program Definition Qp_div (x : Qp) (y : positive) : Qp := mk_Qp (x / ('y)%Z) _.
Next Obligation.
intros x y. assert (0 < ('y)%Z)%Qc.
{ apply (Z2Qc_inj_lt 0%Z (' y)), Pos2Z.is_pos. }
by rewrite (Qcmult_lt_mono_pos_r _ _ ('y)%Z), Qcmult_0_l,
<-Qcmult_assoc, Qcmult_inv_l, Qcmult_1_r.
Qed.
Notation "1" := Qp_one : Qp_scope.
Infix "+" := Qp_plus : Qp_scope.
Infix "-" := Qp_minus : Qp_scope.
Infix "/" := Qp_div : Qp_scope.
Instance Qp_inhabited : Inhabited Qp := populate 1%Qp.
Lemma Qp_eq x y : x = y ↔ Qp_car x = Qp_car y.
Proof.
split; [by intros ->|].
destruct x, y; intros; simplify_eq/=; f_equal; apply (proof_irrel _).
Qed.
Instance Qp_plus_assoc : Assoc (=) Qp_plus.
Proof. intros x y z; apply Qp_eq, Qcplus_assoc. Qed.
Instance Qp_plus_comm : Comm (=) Qp_plus.
Proof. intros x y; apply Qp_eq, Qcplus_comm. Qed.
Lemma Qp_minus_diag x : (x - x)%Qp = None.
Proof. unfold Qp_minus. by rewrite Qcplus_opp_r. Qed.
Lemma Qp_op_minus x y : ((x + y) - x)%Qp = Some y.
Proof.
unfold Qp_minus; simpl.
rewrite (Qcplus_comm x), <- Qcplus_assoc, Qcplus_opp_r, Qcplus_0_r.
destruct (decide _) as [|[]]; auto. by f_equal; apply Qp_eq.
Qed.
Lemma Qp_div_1 x : (x / 1 = x)%Qp.
Proof.
apply Qp_eq; simpl.
rewrite <-(Qcmult_div_r x 1) at 2 by done. by rewrite Qcmult_1_l.
Qed.
Lemma Qp_div_S x y : (x / (2 × y) + x / (2 × y) = x / y)%Qp.
Proof.
apply Qp_eq; simpl.
rewrite <-Qcmult_plus_distr_l, Pos2Z.inj_mul, Z2Qc_inj_mul, Z2Qc_inj_2.
rewrite Qcplus_diag. by field_simplify.
Qed.
Lemma Qp_div_2 x : (x / 2 + x / 2 = x)%Qp.
Proof.
change 2%positive with (2 × 1)%positive. by rewrite Qp_div_S, Qp_div_1.
Qed.