mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁶ : Semiring R
inst✝⁵ : LinearOrder R
inst✝⁴ : ExistsAddOfLE R
inst✝³ : PosMulStrictMono R
inst✝² : MulPosStrictMono R
inst✝¹ : AddLeftStrictMono R
inst✝ : AddLeftReflectLT R
a : R
⊢ 0 < a * a ↔ a ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : MulPosReflectLT G₀
a b c : G₀
hc : 0 < c
⊢ b * c⁻¹ < a ↔ b < a * c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : MulPosReflectLT G₀
a b c : G₀
hc : 0 < c
⊢ a ≤ b / c ↔ a * c ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁴ : Semiring R
inst✝³ : LinearOrder R
inst✝² : ExistsAddOfLE R
inst✝¹ : PosMulMono R
inst✝ : AddLeftMono R
a : R
ha : a < 0
b : R
hab : 0 = a + b
hb : 0 < b
⊢ b ^ 2 = b ^ 2 + a * (a + b)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : MulPosReflectLT G₀
a b c : G₀
hc : 0 < c
⊢ b / c ≤ a ↔ b ≤ a * c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : MulPosReflectLT G₀
a b c : G₀
hc : 0 < c
⊢ a < b / c ↔ a * c < b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁴ : Semiring R
inst✝³ : LinearOrder R
inst✝² : ExistsAddOfLE R
inst✝¹ : PosMulMono R
inst✝ : AddLeftMono R
a : R
ha : a < 0
b : R
hab : 0 = a + b
hb : 0 < b
⊢ b ^ 2 + a * (a + b) = a ^ 2 + (a + b) * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : MulPosReflectLT G₀
a b c : G₀
hc : 0 < c
⊢ b / c < a ↔ b < a * c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁴ : Semiring R
inst✝³ : LinearOrder R
inst✝² : ExistsAddOfLE R
inst✝¹ : PosMulMono R
inst✝ : AddLeftMono R
a : R
ha : a < 0
b : R
hab : 0 = a + b
hb : 0 < b
⊢ a ^ 2 + (a + b) * b = a ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁴ : Semiring R
inst✝³ : LinearOrder R
inst✝² : ExistsAddOfLE R
inst✝¹ : PosMulMono R
inst✝ : AddLeftMono R
a : R
⊢ 0 ≤ a ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : MulPosReflectLT G₀
a b c : G₀
hc : 0 < c
⊢ a / c ≤ b / c ↔ a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : MulPosReflectLT G₀
a b c : G₀
hc : 0 < c
⊢ a / c < b / c ↔ a < b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : MulPosReflectLT G₀
a b : G₀
ha : 0 < a
⊢ a⁻¹ ≤ b ↔ 1 ≤ b * a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : MulPosReflectLT G₀
a b : G₀
ha : 0 < a
⊢ a⁻¹ < b ↔ 1 < b * a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁵ : Semiring R
inst✝⁴ : LinearOrder R
inst✝³ : ExistsAddOfLE R
inst✝² : PosMulMono R
inst✝¹ : AddLeftMono R
inst✝ : NoZeroDivisors R
r : R
⊢ r ^ 2 ≤ 0 ↔ r = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : MulPosReflectLT G₀
a b : G₀
hb : 0 < b
⊢ 1 ≤ a / b ↔ b ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁴ : Semiring R
inst✝³ : LinearOrder R
inst✝² : ExistsAddOfLE R
inst✝¹ : PosMulMono R
inst✝ : AddLeftMono R
a : R
⊢ 0 ≤ a * a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
α : Type u_1
inst✝⁴ : Semiring R
inst✝³ : LinearOrder R
a b c : R
inst✝² : ExistsAddOfLE R
inst✝¹ : PosMulMono R
inst✝ : AddLeftMono R
⊢ 0 ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : MulPosReflectLT G₀
a b : G₀
hb : 0 < b
⊢ 1 < a / b ↔ b < a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : MulPosReflectLT G₀
a b : G₀
hb : 0 < b
⊢ a / b ≤ 1 ↔ a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁵ : Semiring R
inst✝⁴ : LinearOrder R
a b : R
inst✝³ : NoZeroDivisors R
inst✝² : ExistsAddOfLE R
inst✝¹ : PosMulMono R
inst✝ : AddLeftMono R
⊢ a * a + b * b = 0 ↔ a = 0 ∧ b = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : MulPosReflectLT G₀
a b : G₀
hb : 0 < b
⊢ a / b < 1 ↔ a < b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : MulPosReflectLT G₀
a b c : G₀
hb : 0 ≤ b
hc : 0 ≤ c
h : a ≤ b * c⁻¹
⊢ a * c ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁵ : Semiring R
inst✝⁴ : LinearOrder R
a b c : R
inst✝³ : ExistsAddOfLE R
inst✝² : PosMulMono R
inst✝¹ : AddRightMono R
inst✝ : AddRightReflectLE R
h : a * b < a * c
hbc : b ≤ c
⊢ 0 < a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : MulPosReflectLT G₀
a b c : G₀
hb : 0 ≤ b
hc : 0 ≤ c
h : a ≤ c * b
⊢ a * b⁻¹ ≤ c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : PartialOrder G₀
inst✝¹ : MulPosReflectLT G₀
a b : G₀
inst✝ : ZeroLEOneClass G₀
h : a ≤ b
hb : 0 ≤ b
⊢ a ≤ 1 * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁵ : Semiring R
inst✝⁴ : LinearOrder R
a b c : R
inst✝³ : ExistsAddOfLE R
inst✝² : MulPosMono R
inst✝¹ : AddLeftMono R
inst✝ : AddRightReflectLE R
h : b * a < c * a
hbc : b ≤ c
⊢ 0 < a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : MulPosReflectLT G₀
a b c : G₀
hab : a ≤ b
hc : 0 ≤ c
⊢ a / c ≤ b / c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝³ : CommSemiring R
inst✝² : LinearOrder R
a d : R
inst✝¹ : PosMulMono R
inst✝ : MulPosMono R
b c : R
ha : 0 ≤ a
hd : 0 ≤ d
⊢ b * a ≤ max d b * max c a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : MulPosReflectLT G₀
a b c : G₀
h : a < b
hc : 0 < c
⊢ a / c < b / c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : PartialOrder G₀
inst✝¹ : PosMulReflectLT G₀
inst✝ : MulPosReflectLT G₀
a b : G₀
ha : 0 < a
hb : 0 < b
⊢ a⁻¹ ≤ b⁻¹ ↔ b ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝³ : CommSemiring R
inst✝² : LinearOrder R
a d : R
inst✝¹ : PosMulMono R
inst✝ : MulPosMono R
b c : R
ha : 0 ≤ a
hd : 0 ≤ d
ba : b * a ≤ max d b * max c a
cd : c * d ≤ max a c * max b d
⊢ a * b ≤ max a c * max d b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : PartialOrder G₀
inst✝¹ : PosMulReflectLT G₀
inst✝ : MulPosReflectLT G₀
a b : G₀
ha : 0 < a
hb : 0 < b
⊢ a⁻¹ < b⁻¹ ↔ b < a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : PartialOrder G₀
inst✝¹ : PosMulReflectLT G₀
inst✝ : MulPosReflectLT G₀
a b : G₀
ha : 0 < a
hb : 0 < b
⊢ a⁻¹ ≤ b ↔ b⁻¹ ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : PartialOrder G₀
inst✝¹ : PosMulReflectLT G₀
inst✝ : MulPosReflectLT G₀
a b : G₀
ha : 0 < a
hb : 0 < b
⊢ a⁻¹ < b ↔ b⁻¹ < a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝³ : CommSemiring R
inst✝² : LinearOrder R
a d : R
inst✝¹ : PosMulMono R
inst✝ : MulPosMono R
b c : R
ha : 0 ≤ a
hd : 0 ≤ d
ba : b * a ≤ max d b * max c a
cd : c * d ≤ max a c * max b d
⊢ d * c ≤ max a c * max d b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : PartialOrder G₀
inst✝¹ : PosMulReflectLT G₀
inst✝ : MulPosReflectLT G₀
a b : G₀
ha : 0 < a
hb : 0 < b
⊢ a ≤ b⁻¹ ↔ b ≤ a⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁵ : CommSemiring R
inst✝⁴ : LinearOrder R
inst✝³ : ExistsAddOfLE R
inst✝² : MulPosStrictMono R
inst✝¹ : AddLeftReflectLE R
inst✝ : AddLeftMono R
a b : R
⊢ 2 * a * b ≤ a ^ 2 + b ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : PartialOrder G₀
inst✝¹ : PosMulReflectLT G₀
inst✝ : MulPosReflectLT G₀
a b : G₀
ha : 0 < a
hb : 0 < b
⊢ a < b⁻¹ ↔ b < a⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : PartialOrder G₀
inst✝¹ : PosMulReflectLT G₀
inst✝ : MulPosReflectLT G₀
a b c : G₀
ha : 0 < a
hb : 0 < b
hc : 0 < c
⊢ a / b ≤ a / c ↔ c ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : PartialOrder G₀
inst✝¹ : PosMulReflectLT G₀
inst✝ : MulPosReflectLT G₀
a b c : G₀
ha : 0 ≤ a
hc : 0 < c
h : c ≤ b
⊢ a / b ≤ a / c

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁵ : CommSemiring R
inst✝⁴ : LinearOrder R
inst✝³ : ExistsAddOfLE R
inst✝² : MulPosStrictMono R
inst✝¹ : AddLeftReflectLE R
inst✝ : AddLeftMono R
a b : R
⊢ 4 * a * b = 2 * a * b + 2 * a * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁵ : CommSemiring R
inst✝⁴ : LinearOrder R
inst✝³ : ExistsAddOfLE R
inst✝² : MulPosStrictMono R
inst✝¹ : AddLeftReflectLE R
inst✝ : AddLeftMono R
a b : R
⊢ 2 * a * b + 2 * a * b ≤ a ^ 2 + b ^ 2 + 2 * a * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁵ : CommSemiring R
inst✝⁴ : LinearOrder R
inst✝³ : ExistsAddOfLE R
inst✝² : MulPosStrictMono R
inst✝¹ : AddLeftReflectLE R
inst✝ : AddLeftMono R
a b : R
⊢ a ^ 2 + b ^ 2 + 2 * a * b = a ^ 2 + 2 * a * b + b ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : PartialOrder G₀
inst✝¹ : PosMulReflectLT G₀
inst✝ : MulPosReflectLT G₀
a b c d : G₀
hc : 0 ≤ c
hac : a ≤ c
hd : 0 < d
hdb : d ≤ b
⊢ a / b ≤ c / d

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁵ : CommSemiring R
inst✝⁴ : LinearOrder R
inst✝³ : ExistsAddOfLE R
inst✝² : MulPosStrictMono R
inst✝¹ : AddLeftReflectLE R
inst✝ : AddLeftMono R
a b : R
⊢ 4 * a * b ≤ (a + b) ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : PartialOrder G₀
inst✝¹ : PosMulReflectLT G₀
inst✝ : MulPosReflectLT G₀
a b c d : G₀
hac : a < c
hdb : d ≤ b
hc : 0 ≤ c
hd : 0 < d
⊢ b⁻¹ ≤ d⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : PartialOrder G₀
inst✝¹ : PosMulReflectLT G₀
inst✝ : MulPosReflectLT G₀
a b c d : G₀
hac : a < c
hdb : d ≤ b
hc : 0 ≤ c
hd : 0 < d
⊢ a / b < c / d

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : PartialOrder G₀
inst✝¹ : PosMulReflectLT G₀
inst✝ : MulPosReflectLT G₀
a b c d : G₀
hac : a ≤ c
hdb : d < b
hc : 0 < c
hd : 0 < d
⊢ a / b < c / d

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : LinearOrder G₀
a : G₀
inst✝ : PosMulMono G₀
⊢ a⁻¹ < 0 ↔ a < 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : LinearOrder G₀
a : G₀
inst✝ : PosMulMono G₀
⊢ a⁻¹ ≤ 0 ↔ a ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁶ : CommSemiring R
inst✝⁵ : LinearOrder R
inst✝⁴ : ExistsAddOfLE R
inst✝³ : MulPosStrictMono R
inst✝² : PosMulStrictMono R
inst✝¹ : AddLeftReflectLE R
inst✝ : AddLeftMono R
a b r : R
ha : 0 ≤ a
hb : 0 ≤ b
ht : r ^ 2 ≤ a * b
⊢ 2 * r ≤ a + b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : LinearOrder G₀
a b : G₀
inst✝ : PosMulMono G₀
ha : 0 ≤ a
hb : b ≤ 0
⊢ a / b ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : LinearOrder G₀
a : G₀
inst✝¹ : PosMulStrictMono G₀
inst✝ : ZeroLEOneClass G₀
⊢ a⁻¹ < 1 ↔ a ≤ 0 ∨ 1 < a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : LinearOrder G₀
a : G₀
inst✝¹ : PosMulStrictMono G₀
inst✝ : ZeroLEOneClass G₀
⊢ a⁻¹ ≤ 1 ↔ a ≤ 0 ∨ 1 ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : LinearOrder G₀
a : G₀
inst✝¹ : PosMulStrictMono G₀
inst✝ : ZeroLEOneClass G₀
ha₀ : 0 < a
ha₁ : a ≠ 1
⊢ Injective fun n ↦ a ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Order.RelIso.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
r : α → α → Prop
s : β → β → Prop
F : Type u_5
inst✝¹ : FunLike F α β
inst✝ : RelHomClass F r s
f : F
a : α
⊢ Acc s (f a) → Acc r a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : LinearOrder G₀
a : G₀
inst✝¹ : PosMulStrictMono G₀
inst✝ : ZeroLEOneClass G₀
ha₀ : 0 ≤ a
ha₁ : a ≠ 1
n : ℤ
⊢ a ^ n = 1 ↔ n = 0

Tactic:
by

Goal state after:
Module: Mathlib.Order.RelIso.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
u : δ → δ → Prop
f g : r →r s
h : (fun o ↦ o.toFun) f = (fun o ↦ o.toFun) g
⊢ f = g

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : LinearOrder G₀
inst✝¹ : PosMulStrictMono G₀
inst✝ : MulPosMono G₀
n : ℕ
x✝ : ↑(n + 1) ≠ 0
⊢ Set.InjOn (fun a ↦ a ^ ↑(n + 1)) {a | 0 ≤ a}

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝³ : GroupWithZero G₀
inst✝² : LinearOrder G₀
inst✝¹ : PosMulStrictMono G₀
inst✝ : MulPosMono G₀
n : ℕ
x✝ : Int.negSucc n ≠ 0
⊢ Set.InjOn (fun a ↦ a ^ Int.negSucc n) {a | 0 ≤ a}

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁵ : Ring R
inst✝⁴ : LinearOrder R
a b : R
inst✝³ : PosMulStrictMono R
inst✝² : MulPosStrictMono R
inst✝¹ : AddLeftReflectLT R
inst✝ : AddLeftStrictMono R
⊢ a * b < 0 ↔ 0 < a ∧ b < 0 ∨ a < 0 ∧ 0 < b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝¹ : CommGroupWithZero G₀
inst✝ : Preorder G₀
a b c : G₀
h : 0 ≤ a / b
⊢ c * a / (c * b) ≤ a / b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝¹ : CommGroupWithZero G₀
inst✝ : Preorder G₀
a b c : G₀
h : a / b ≤ 0
⊢ a / b ≤ c * a / (c * b)

Tactic:
by

Goal state after:
Module: Mathlib.Order.RelIso.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
r : α → α → Prop
s : β → β → Prop
inst✝¹ : Std.Trichotomous r
inst✝ : Std.Irrefl s
f : α → β
hf : ∀ {x y : α}, r x y → s (f x) (f y)
⊢ Injective f

Tactic:
by

Goal state after:
Module: Mathlib.Order.RelIso.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
r : α → α → Prop
s : β → β → Prop
f : α → β
hf : Surjective f
o : ∀ {a b : α}, r a b ↔ s (f a) (f b)
a✝ b✝ : β
h : s a✝ b✝
⊢ r (surjInv hf a✝) (surjInv hf b✝)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : CommGroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : PosMulReflectLT G₀
a b c : G₀
hc : 0 < c
⊢ a ≤ c⁻¹ * b ↔ a * c ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁵ : Ring R
inst✝⁴ : LinearOrder R
a b : R
inst✝³ : MulPosStrictMono R
inst✝² : PosMulStrictMono R
inst✝¹ : AddLeftReflectLE R
inst✝ : AddLeftMono R
⊢ a * b ≤ 0 ↔ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : CommGroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : PosMulReflectLT G₀
a b c : G₀
hc : 0 < c
⊢ c⁻¹ * b ≤ a ↔ b ≤ a * c

Tactic:
by

Goal state after:
Module: Mathlib.Order.RelIso.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
u : δ → δ → Prop
f g : r ↪r s
h : (fun x ↦ x.toFun) f = (fun x ↦ x.toFun) g
⊢ f = g

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : CommGroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : PosMulReflectLT G₀
a b c : G₀
hc : 0 < c
⊢ a ≤ b * c⁻¹ ↔ c * a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Order.RelIso.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
r : α → α → Prop
s : β → β → Prop
⊢ Injective toEmbedding

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : CommGroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : PosMulReflectLT G₀
a b c : G₀
hc : 0 < c
⊢ b * c⁻¹ ≤ a ↔ b ≤ c * a

Tactic:
by

Goal state after:
Module: Mathlib.Order.RelIso.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
u : δ → δ → Prop
f : r ↪r s
g : s ↪r t
⊢ ∀ {a b : α}, t ((f.trans g.toEmbedding) a) ((f.trans g.toEmbedding) b) ↔ r a b

Tactic:
by

Goal state after:
Module: Mathlib.Order.RelIso.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
r : α → α → Prop
s : β → β → Prop
f : r ↪r s
⊢ r = ⇑f ⁻¹'o s

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : CommGroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : PosMulReflectLT G₀
a b c d : G₀
hb : 0 < b
hd : 0 < d
⊢ a / b ≤ c / d ↔ a * d ≤ c * b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁵ : Ring R
inst✝⁴ : LinearOrder R
a b : R
inst✝³ : PosMulStrictMono R
inst✝² : MulPosStrictMono R
inst✝¹ : AddLeftMono R
inst✝ : AddLeftReflectLE R
⊢ 0 ≤ a * b ↔ (a < 0 → b ≤ 0) ∧ (b < 0 → a ≤ 0)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : CommGroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : PosMulReflectLT G₀
a b c : G₀
hc : 0 < c
⊢ a ≤ b / c ↔ c * a ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Order.RelIso.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
r : α → α → Prop
s : β → β → Prop
f : r ↪r s
a : α
⊢ Acc s (f a) → Acc r a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : CommGroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : PosMulReflectLT G₀
a b c : G₀
hc : 0 < c
⊢ b / c ≤ a ↔ b ≤ c * a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : CommGroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : PosMulReflectLT G₀
a b c : G₀
ha : 0 < a
hc : 0 < c
⊢ a ≤ b / c ↔ c ≤ b / a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁵ : Ring R
inst✝⁴ : LinearOrder R
a b : R
inst✝³ : PosMulStrictMono R
inst✝² : MulPosStrictMono R
inst✝¹ : AddLeftMono R
inst✝ : AddLeftReflectLE R
⊢ a * b ≤ 0 ↔ (0 < a → b ≤ 0) ∧ (b < 0 → 0 ≤ a)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : CommGroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : PosMulReflectLT G₀
a b c : G₀
hb : 0 < b
hc : 0 < c
⊢ a / b ≤ c ↔ a / c ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : CommGroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : PosMulReflectLT G₀
a b c : G₀
hc : 0 < c
⊢ a < c⁻¹ * b ↔ a * c < b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : CommGroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : PosMulReflectLT G₀
a b c : G₀
hc : 0 < c
⊢ c⁻¹ * b < a ↔ b < a * c

Tactic:
by

Goal state after:
Module: Mathlib.Order.RelIso.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
r✝ : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
u : δ → δ → Prop
x✝ : Setoid α
r : α → α → Prop
H : ∀ (a₁ b₁ a₂ b₂ : α), a₁ ≈ a₂ → b₁ ≈ b₂ → r a₁ b₁ = r a₂ b₂
x y : Quotient x✝
⊢ r ((Embedding.quotientOut α) x) ((Embedding.quotientOut α) y) ↔ Quotient.lift₂ r H x y

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : CommGroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : PosMulReflectLT G₀
a b c : G₀
hc : 0 < c
⊢ a < b * c⁻¹ ↔ c * a < b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.Unbundled.Basic

Goal state before:
pp: R : Type u
inst✝⁵ : Ring R
inst✝⁴ : LinearOrder R
a b : R
inst✝³ : PosMulStrictMono R
inst✝² : MulPosStrictMono R
inst✝¹ : AddLeftMono R
inst✝ : AddLeftReflectLE R
⊢ a * b ≤ 0 ↔ (a < 0 → 0 ≤ b) ∧ (0 < b → a ≤ 0)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : CommGroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : PosMulReflectLT G₀
a b c : G₀
hc : 0 < c
⊢ b * c⁻¹ < a ↔ b < c * a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.GroupWithZero.Basic

Goal state before:
pp: G₀ : Type u_3
inst✝² : CommGroupWithZero G₀
inst✝¹ : PartialOrder G₀
inst✝ : PosMulReflectLT G₀
a b c d : G₀
hb : 0 < b
hd : 0 < d
⊢ a / b < c / d ↔ a * d < c * b

Tactic:
by

Goal state after: