Documentation

Std.Data.Int.Order

Results about the order properties of the integers, and the integers as an ordered ring. #

Order properties of the integers #

theorem Int.not_le_of_gt {a : Int} {b : Int} :
b < a¬a b

Alias of the reverse direction of Int.not_le.

theorem Int.lt_of_not_ge {a : Int} {b : Int} :
¬a bb < a

Alias of the forward direction of Int.not_le.

theorem Int.le_of_not_le {a : Int} {b : Int} :
¬a bb a
@[simp]
theorem Int.eq_negSucc_of_lt_zero {a : Int} :
a < 0∃ (n : Nat), a = Int.negSucc n
theorem Int.lt_of_add_lt_add_left {a : Int} {b : Int} {c : Int} (h : a + b < a + c) :
b < c
theorem Int.lt_of_add_lt_add_right {a : Int} {b : Int} {c : Int} (h : a + b < c + b) :
a < c
theorem Int.add_lt_add_iff_left {b : Int} {c : Int} (a : Int) :
a + b < a + c b < c
theorem Int.add_lt_add_iff_right {a : Int} {b : Int} (c : Int) :
a + c < b + c a < b
theorem Int.add_lt_add {a : Int} {b : Int} {c : Int} {d : Int} (h₁ : a < b) (h₂ : c < d) :
a + c < b + d
theorem Int.add_lt_add_of_le_of_lt {a : Int} {b : Int} {c : Int} {d : Int} (h₁ : a b) (h₂ : c < d) :
a + c < b + d
theorem Int.add_lt_add_of_lt_of_le {a : Int} {b : Int} {c : Int} {d : Int} (h₁ : a < b) (h₂ : c d) :
a + c < b + d
theorem Int.lt_add_of_pos_right (a : Int) {b : Int} (h : 0 < b) :
a < a + b
theorem Int.lt_add_of_pos_left (a : Int) {b : Int} (h : 0 < b) :
a < b + a
theorem Int.add_nonneg {a : Int} {b : Int} (ha : 0 a) (hb : 0 b) :
0 a + b
theorem Int.add_pos {a : Int} {b : Int} (ha : 0 < a) (hb : 0 < b) :
0 < a + b
theorem Int.add_pos_of_pos_of_nonneg {a : Int} {b : Int} (ha : 0 < a) (hb : 0 b) :
0 < a + b
theorem Int.add_pos_of_nonneg_of_pos {a : Int} {b : Int} (ha : 0 a) (hb : 0 < b) :
0 < a + b
theorem Int.add_nonpos {a : Int} {b : Int} (ha : a 0) (hb : b 0) :
a + b 0
theorem Int.add_neg {a : Int} {b : Int} (ha : a < 0) (hb : b < 0) :
a + b < 0
theorem Int.add_neg_of_neg_of_nonpos {a : Int} {b : Int} (ha : a < 0) (hb : b 0) :
a + b < 0
theorem Int.add_neg_of_nonpos_of_neg {a : Int} {b : Int} (ha : a 0) (hb : b < 0) :
a + b < 0
theorem Int.lt_add_of_le_of_pos {a : Int} {b : Int} {c : Int} (hbc : b c) (ha : 0 < a) :
b < c + a
theorem Int.add_one_le_iff {a : Int} {b : Int} :
a + 1 b a < b
theorem Int.lt_add_one_iff {a : Int} {b : Int} :
a < b + 1 a b
@[simp]
theorem Int.succ_ofNat_pos (n : Nat) :
0 < n + 1
theorem Int.le_add_one {a : Int} {b : Int} (h : a b) :
a b + 1
theorem Int.nonneg_of_neg_nonpos {a : Int} (h : -a 0) :
0 a
theorem Int.nonpos_of_neg_nonneg {a : Int} (h : 0 -a) :
a 0
theorem Int.lt_of_neg_lt_neg {a : Int} {b : Int} (h : -b < -a) :
a < b
theorem Int.pos_of_neg_neg {a : Int} (h : -a < 0) :
0 < a
theorem Int.neg_of_neg_pos {a : Int} (h : 0 < -a) :
a < 0
theorem Int.le_neg_of_le_neg {a : Int} {b : Int} (h : a -b) :
b -a
theorem Int.neg_le_of_neg_le {a : Int} {b : Int} (h : -a b) :
-b a
theorem Int.lt_neg_of_lt_neg {a : Int} {b : Int} (h : a < -b) :
b < -a
theorem Int.neg_lt_of_neg_lt {a : Int} {b : Int} (h : -a < b) :
-b < a
theorem Int.sub_nonpos_of_le {a : Int} {b : Int} (h : a b) :
a - b 0
theorem Int.le_of_sub_nonpos {a : Int} {b : Int} (h : a - b 0) :
a b
theorem Int.sub_neg_of_lt {a : Int} {b : Int} (h : a < b) :
a - b < 0
theorem Int.lt_of_sub_neg {a : Int} {b : Int} (h : a - b < 0) :
a < b
theorem Int.add_le_of_le_neg_add {a : Int} {b : Int} {c : Int} (h : b -a + c) :
a + b c
theorem Int.le_neg_add_of_add_le {a : Int} {b : Int} {c : Int} (h : a + b c) :
b -a + c
theorem Int.add_le_of_le_sub_left {a : Int} {b : Int} {c : Int} (h : b c - a) :
a + b c
theorem Int.le_sub_left_of_add_le {a : Int} {b : Int} {c : Int} (h : a + b c) :
b c - a
theorem Int.add_le_of_le_sub_right {a : Int} {b : Int} {c : Int} (h : a c - b) :
a + b c
theorem Int.le_sub_right_of_add_le {a : Int} {b : Int} {c : Int} (h : a + b c) :
a c - b
theorem Int.le_add_of_neg_add_le {a : Int} {b : Int} {c : Int} (h : -b + a c) :
a b + c
theorem Int.neg_add_le_of_le_add {a : Int} {b : Int} {c : Int} (h : a b + c) :
-b + a c
theorem Int.le_add_of_sub_left_le {a : Int} {b : Int} {c : Int} (h : a - b c) :
a b + c
theorem Int.le_add_of_sub_right_le {a : Int} {b : Int} {c : Int} (h : a - c b) :
a b + c
theorem Int.sub_right_le_of_le_add {a : Int} {b : Int} {c : Int} (h : a b + c) :
a - c b
theorem Int.le_add_of_neg_add_le_left {a : Int} {b : Int} {c : Int} (h : -b + a c) :
a b + c
theorem Int.neg_add_le_left_of_le_add {a : Int} {b : Int} {c : Int} (h : a b + c) :
-b + a c
theorem Int.le_add_of_neg_add_le_right {a : Int} {b : Int} {c : Int} (h : -c + a b) :
a b + c
theorem Int.neg_add_le_right_of_le_add {a : Int} {b : Int} {c : Int} (h : a b + c) :
-c + a b
theorem Int.le_add_of_neg_le_sub_left {a : Int} {b : Int} {c : Int} (h : -a b - c) :
c a + b
theorem Int.neg_le_sub_left_of_le_add {a : Int} {b : Int} {c : Int} (h : c a + b) :
-a b - c
theorem Int.le_add_of_neg_le_sub_right {a : Int} {b : Int} {c : Int} (h : -b a - c) :
c a + b
theorem Int.neg_le_sub_right_of_le_add {a : Int} {b : Int} {c : Int} (h : c a + b) :
-b a - c
theorem Int.sub_le_of_sub_le {a : Int} {b : Int} {c : Int} (h : a - b c) :
a - c b
theorem Int.sub_le_sub_left {a : Int} {b : Int} (h : a b) (c : Int) :
c - b c - a
theorem Int.sub_le_sub_right {a : Int} {b : Int} (h : a b) (c : Int) :
a - c b - c
theorem Int.sub_le_sub {a : Int} {b : Int} {c : Int} {d : Int} (hab : a b) (hcd : c d) :
a - d b - c
theorem Int.add_lt_of_lt_neg_add {a : Int} {b : Int} {c : Int} (h : b < -a + c) :
a + b < c
theorem Int.lt_neg_add_of_add_lt {a : Int} {b : Int} {c : Int} (h : a + b < c) :
b < -a + c
theorem Int.add_lt_of_lt_sub_left {a : Int} {b : Int} {c : Int} (h : b < c - a) :
a + b < c
theorem Int.lt_sub_left_of_add_lt {a : Int} {b : Int} {c : Int} (h : a + b < c) :
b < c - a
theorem Int.add_lt_of_lt_sub_right {a : Int} {b : Int} {c : Int} (h : a < c - b) :
a + b < c
theorem Int.lt_sub_right_of_add_lt {a : Int} {b : Int} {c : Int} (h : a + b < c) :
a < c - b
theorem Int.lt_add_of_neg_add_lt {a : Int} {b : Int} {c : Int} (h : -b + a < c) :
a < b + c
theorem Int.neg_add_lt_of_lt_add {a : Int} {b : Int} {c : Int} (h : a < b + c) :
-b + a < c
theorem Int.lt_add_of_sub_left_lt {a : Int} {b : Int} {c : Int} (h : a - b < c) :
a < b + c
theorem Int.sub_left_lt_of_lt_add {a : Int} {b : Int} {c : Int} (h : a < b + c) :
a - b < c
theorem Int.lt_add_of_sub_right_lt {a : Int} {b : Int} {c : Int} (h : a - c < b) :
a < b + c
theorem Int.sub_right_lt_of_lt_add {a : Int} {b : Int} {c : Int} (h : a < b + c) :
a - c < b
theorem Int.lt_add_of_neg_add_lt_left {a : Int} {b : Int} {c : Int} (h : -b + a < c) :
a < b + c
theorem Int.neg_add_lt_left_of_lt_add {a : Int} {b : Int} {c : Int} (h : a < b + c) :
-b + a < c
theorem Int.lt_add_of_neg_add_lt_right {a : Int} {b : Int} {c : Int} (h : -c + a < b) :
a < b + c
theorem Int.neg_add_lt_right_of_lt_add {a : Int} {b : Int} {c : Int} (h : a < b + c) :
-c + a < b
theorem Int.lt_add_of_neg_lt_sub_left {a : Int} {b : Int} {c : Int} (h : -a < b - c) :
c < a + b
theorem Int.neg_lt_sub_left_of_lt_add {a : Int} {b : Int} {c : Int} (h : c < a + b) :
-a < b - c
theorem Int.lt_add_of_neg_lt_sub_right {a : Int} {b : Int} {c : Int} (h : -b < a - c) :
c < a + b
theorem Int.neg_lt_sub_right_of_lt_add {a : Int} {b : Int} {c : Int} (h : c < a + b) :
-b < a - c
theorem Int.sub_lt_of_sub_lt {a : Int} {b : Int} {c : Int} (h : a - b < c) :
a - c < b
theorem Int.sub_lt_sub_left {a : Int} {b : Int} (h : a < b) (c : Int) :
c - b < c - a
theorem Int.sub_lt_sub_right {a : Int} {b : Int} (h : a < b) (c : Int) :
a - c < b - c
theorem Int.sub_lt_sub {a : Int} {b : Int} {c : Int} {d : Int} (hab : a < b) (hcd : c < d) :
a - d < b - c
theorem Int.sub_lt_sub_of_le_of_lt {a : Int} {b : Int} {c : Int} {d : Int} (hab : a b) (hcd : c < d) :
a - d < b - c
theorem Int.sub_lt_sub_of_lt_of_le {a : Int} {b : Int} {c : Int} {d : Int} (hab : a < b) (hcd : c d) :
a - d < b - c
theorem Int.add_le_add_three {a : Int} {b : Int} {c : Int} {d : Int} {e : Int} {f : Int} (h₁ : a d) (h₂ : b e) (h₃ : c f) :
a + b + c d + e + f
theorem Int.exists_eq_neg_ofNat {a : Int} (H : a 0) :
∃ (n : Nat), a = -n
theorem Int.lt_of_add_one_le {a : Int} {b : Int} (H : a + 1 b) :
a < b
theorem Int.lt_add_one_of_le {a : Int} {b : Int} (H : a b) :
a < b + 1
theorem Int.le_of_lt_add_one {a : Int} {b : Int} (H : a < b + 1) :
a b
theorem Int.sub_one_lt_of_le {a : Int} {b : Int} (H : a b) :
a - 1 < b
theorem Int.le_of_sub_one_lt {a : Int} {b : Int} (H : a - 1 < b) :
a b
theorem Int.le_sub_one_of_lt {a : Int} {b : Int} (H : a < b) :
a b - 1
theorem Int.lt_of_le_sub_one {a : Int} {b : Int} (H : a b - 1) :
a < b
theorem Int.mul_lt_mul {a : Int} {b : Int} {c : Int} {d : Int} (h₁ : a < c) (h₂ : b d) (h₃ : 0 < b) (h₄ : 0 c) :
a * b < c * d
theorem Int.mul_lt_mul' {a : Int} {b : Int} {c : Int} {d : Int} (h₁ : a c) (h₂ : b < d) (h₃ : 0 b) (h₄ : 0 < c) :
a * b < c * d
theorem Int.mul_neg_of_pos_of_neg {a : Int} {b : Int} (ha : 0 < a) (hb : b < 0) :
a * b < 0
theorem Int.mul_neg_of_neg_of_pos {a : Int} {b : Int} (ha : a < 0) (hb : 0 < b) :
a * b < 0
theorem Int.mul_nonneg_of_nonpos_of_nonpos {a : Int} {b : Int} (ha : a 0) (hb : b 0) :
0 a * b
theorem Int.mul_lt_mul_of_neg_left {a : Int} {b : Int} {c : Int} (h : b < a) (hc : c < 0) :
c * a < c * b
theorem Int.mul_lt_mul_of_neg_right {a : Int} {b : Int} {c : Int} (h : b < a) (hc : c < 0) :
a * c < b * c
theorem Int.mul_pos_of_neg_of_neg {a : Int} {b : Int} (ha : a < 0) (hb : b < 0) :
0 < a * b
theorem Int.mul_self_le_mul_self {a : Int} {b : Int} (h1 : 0 a) (h2 : a b) :
a * a b * b
theorem Int.mul_self_lt_mul_self {a : Int} {b : Int} (h1 : 0 a) (h2 : a < b) :
a * a < b * b
@[simp]
theorem Int.sign_zero :
@[simp]
theorem Int.sign_one :
@[simp]
theorem Int.sign_of_add_one (x : Nat) :
Int.sign (x + 1) = 1
@[simp]
theorem Int.sign_negSucc (x : Nat) :
theorem Int.natAbs_sign (z : Int) :
Int.natAbs (Int.sign z) = if z = 0 then 0 else 1
theorem Int.sign_ofNat_of_nonzero {n : Nat} (hn : n 0) :
Int.sign n = 1
@[simp]
theorem Int.sign_neg (z : Int) :
@[simp]
theorem Int.sign_mul (a : Int) (b : Int) :
theorem Int.sign_eq_one_of_pos {a : Int} (h : 0 < a) :
theorem Int.sign_eq_neg_one_of_neg {a : Int} (h : a < 0) :
Int.sign a = -1
theorem Int.eq_zero_of_sign_eq_zero {a : Int} :
Int.sign a = 0a = 0
theorem Int.pos_of_sign_eq_one {a : Int} :
Int.sign a = 10 < a
theorem Int.neg_of_sign_eq_neg_one {a : Int} :
Int.sign a = -1a < 0
@[simp]
theorem Int.sign_eq_zero_iff_zero (a : Int) :
Int.sign a = 0 a = 0
@[simp]
@[simp]
theorem Int.sign_nonneg {x : Int} :
theorem Int.natAbs_mul_self {a : Int} :
(Int.natAbs a * Int.natAbs a) = a * a
theorem Int.eq_nat_or_neg (a : Int) :
∃ (n : Nat), a = n a = -n
theorem Int.natAbs_mul_natAbs_eq {a : Int} {b : Int} {c : Nat} (h : a * b = c) :
@[simp]
theorem Int.natAbs_mul_self' (a : Int) :
(Int.natAbs a) * (Int.natAbs a) = a * a
theorem Int.natAbs_eq_iff {a : Int} {n : Nat} :
Int.natAbs a = n a = n a = -n
@[deprecated Int.natAbs_of_nonneg]
theorem Int.ofNat_natAbs_eq_of_nonneg {a : Int} (H : 0 a) :
(Int.natAbs a) = a

Alias of Int.natAbs_of_nonneg.

theorem Int.negSucc_eq' (m : Nat) :
Int.negSucc m = -m - 1
theorem Int.natAbs_lt_natAbs_of_nonneg_of_lt {a : Int} {b : Int} (w₁ : 0 a) (w₂ : a < b) :
theorem Int.eq_natAbs_iff_mul_eq_zero {a : Int} {n : Nat} :
Int.natAbs a = n (a - n) * (a + n) = 0

toNat #

theorem Int.mem_toNat' (a : Int) (n : Nat) :
Int.toNat' a = some n a = n