Library prosa.util.nat
(* ----------------------------------[ coqtop ]---------------------------------
Welcome to Coq 8.11.2 (June 2020)
----------------------------------------------------------------------------- *)
Require Export prosa.util.tactics prosa.util.ssromega.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div.
(* Additional lemmas about natural numbers. *)
Section NatLemmas.
Lemma subh1 :
∀ m n p,
m ≥ n →
m - n + p = m + p - n.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 23)
============================
forall m n p : nat, n <= m -> m - n + p = m + p - n
----------------------------------------------------------------------------- *)
Proof. by ins; ssromega.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Lemma subh2 :
∀ m1 m2 n1 n2,
m1 ≥ m2 →
n1 ≥ n2 →
(m1 + n1) - (m2 + n2) = m1 - m2 + (n1 - n2).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 30)
============================
forall m1 m2 n1 n2 : nat,
m2 <= m1 -> n2 <= n1 -> m1 + n1 - (m2 + n2) = m1 - m2 + (n1 - n2)
----------------------------------------------------------------------------- *)
Proof. by ins; ssromega.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Lemma subh3:
∀ m n p,
m + p ≤ n →
m ≤ n - p.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 34)
============================
forall m n p : nat, m + p <= n -> m <= n - p
----------------------------------------------------------------------------- *)
Proof.
clear.
intros.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 38)
m, n, p : nat
H : m + p <= n
============================
m <= n - p
----------------------------------------------------------------------------- *)
rewrite <- leq_add2r with (p := p).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 39)
m, n, p : nat
H : m + p <= n
============================
m + p <= n - p + p
----------------------------------------------------------------------------- *)
rewrite subh1 //.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 45)
m, n, p : nat
H : m + p <= n
============================
m + p <= n + p - p
subgoal 2 (ID 46) is:
p <= n
----------------------------------------------------------------------------- *)
- by rewrite -addnBA // subnn addn0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 46)
m, n, p : nat
H : m + p <= n
============================
p <= n
----------------------------------------------------------------------------- *)
- by apply leq_trans with (m+p); first rewrite leq_addl.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
(* Simplify [n + a - b + b - a = n] if [n >= b]. *)
Lemma subn_abba:
∀ n a b,
n ≥ b →
n + a - b + b - a = n.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 40)
============================
forall n a b : nat, b <= n -> n + a - b + b - a = n
----------------------------------------------------------------------------- *)
Proof.
move⇒ n a b le_bn.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 44)
n, a, b : nat
le_bn : b <= n
============================
n + a - b + b - a = n
----------------------------------------------------------------------------- *)
rewrite subnK;
first by rewrite -addnBA // subnn addn0 //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 50)
n, a, b : nat
le_bn : b <= n
============================
b <= n + a
----------------------------------------------------------------------------- *)
rewrite -[b]addn0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 94)
n, a, b : nat
le_bn : b <= n
============================
b + 0 <= n + a
----------------------------------------------------------------------------- *)
apply leq_trans with (n := n + 0); rewrite !addn0 //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 114)
n, a, b : nat
le_bn : b <= n
============================
n <= n + a
----------------------------------------------------------------------------- *)
apply leq_addr.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Lemma add_subC:
∀ a b c,
a ≥ c →
b ≥c →
a + (b - c ) = a - c + b.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 46)
============================
forall a b c : nat, c <= a -> c <= b -> a + (b - c) = a - c + b
----------------------------------------------------------------------------- *)
Proof.
intros× AgeC BgeC.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 51)
a, b, c : nat
AgeC : c <= a
BgeC : c <= b
============================
a + (b - c) = a - c + b
----------------------------------------------------------------------------- *)
induction b;induction c;intros;try ssromega.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
(* TODO: remove when mathcomp minimum required version becomes 1.10.0 *)
Lemma ltn_subLR:
∀ a b c,
a - c < b →
a < b + c.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 50)
============================
forall a b c : nat, a - c < b -> a < b + c
----------------------------------------------------------------------------- *)
Proof.
intros× AC.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 54)
a, b, c : nat
AC : a - c < b
============================
a < b + c
----------------------------------------------------------------------------- *)
ssromega.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
(* We can drop additive terms on the lesser side of an inequality. *)
Lemma leq_addk:
∀ m n k,
n + k ≤ m →
n ≤ m.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 54)
============================
forall m n k : nat, n + k <= m -> n <= m
----------------------------------------------------------------------------- *)
Proof.
move⇒ m n p.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 57)
m, n, p : nat
============================
n + p <= m -> n <= m
----------------------------------------------------------------------------- *)
apply leq_trans.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 58)
m, n, p : nat
============================
n <= n + p
----------------------------------------------------------------------------- *)
by apply leq_addr.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End NatLemmas.
Section Interval.
(* Trivially, points before the start of an interval, or past the end of an
interval, are not included in the interval. *)
Lemma point_not_in_interval:
∀ t1 t2 t',
t2 ≤ t' ∨ t' < t1 →
∀ t,
t1 ≤ t < t2
→ t ≠ t'.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 24)
============================
forall t1 t2 t' : nat,
t2 <= t' \/ t' < t1 -> forall t : nat, t1 <= t < t2 -> t <> t'
----------------------------------------------------------------------------- *)
Proof.
move⇒ t1 t2 t' EXCLUDED t /andP [GEQ_t1 LT_t2] EQ.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 71)
t1, t2, t' : nat
EXCLUDED : t2 <= t' \/ t' < t1
t : nat
GEQ_t1 : t1 <= t
LT_t2 : t < t2
EQ : t = t'
============================
False
----------------------------------------------------------------------------- *)
subst.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 79)
t1, t2, t' : nat
EXCLUDED : t2 <= t' \/ t' < t1
LT_t2 : t' < t2
GEQ_t1 : t1 <= t'
============================
False
----------------------------------------------------------------------------- *)
case EXCLUDED ⇒ [INEQ | INEQ];
eapply leq_ltn_trans in INEQ; eauto;
by rewrite ltnn in INEQ.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End Interval.
Section NatOrderLemmas.
(* Mimic the way implicit arguments are used in [ssreflect]. *)
Set Implicit Arguments.
Unset Strict Implicit.
(* [ltn_leq_trans]: Establish that [m < p] if [m < n] and [n <= p], to mirror the
lemma [leq_ltn_trans] in [ssrnat].
NB: There is a good reason for this lemma to be "missing" in [ssrnat] --
since [m < n] is defined as [m.+1 <= n], [ltn_leq_trans] is just
[m.+1 <= n -> n <= p -> m.+1 <= p], that is [@leq_trans n m.+1 p].
Nonetheless we introduce it here because an additional (even though
arguably redundant) lemma doesn't hurt, and for newcomers the apparent
absence of the mirror case of [leq_ltn_trans] can be somewhat confusing. *)
Lemma ltn_leq_trans n m p : m < n → n ≤ p → m < p.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 24)
n, m, p : nat
============================
m < n -> n <= p -> m < p
----------------------------------------------------------------------------- *)
Proof. exact (@leq_trans n m.+1 p).
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End NatOrderLemmas.