mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.GroupTheory.MonoidLocalization.Basic

Goal state before:
pp: M : Type u_1
G : Type u_2
inst✝¹ : CommMonoid M
inst✝ : CommGroup G
S : Submonoid M
f : M → G
hf : Injective f
surj : ∀ (g : G), ∃ x, ∃ y ∈ S, g = f x / f y
x✝ y✝ : M
eq : f x✝ = f y✝
⊢ ∃ c, ↑c * x✝ = ↑c * y✝

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Sub.WithTop

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : Add α
inst✝³ : LE α
inst✝² : OrderBot α
inst✝¹ : Sub α
inst✝ : OrderedSub α
⊢ OrderedSub (WithTop α)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝³ : DecidableEq α
inst✝² : MulZeroClass α
a : WithTop α
inst✝¹ : Preorder α
inst✝ : MulPosStrictMono α
h₀ : 0 < a
hinf : a ≠ ⊤
⊢ StrictMono fun x ↦ x * a

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.MonoidLocalization.Basic

Goal state before:
pp: S : AddSubmonoid ℕ
hS : S ≠ ⊥
z : ℕ
⊢ ↑z = ↑z - ↑0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : MulZeroOneClass α
inst✝ : Nontrivial α
a : α
⊢ 1 * ↑a = ↑a

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : MulZeroOneClass α
inst✝ : Nontrivial α
a : α
⊢ ↑a * 1 = ↑a

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.MonoidLocalization.Basic

Goal state before:
pp: S : AddSubmonoid ℕ
hS : S ≠ ⊥
z n : ℕ
hnS : n ∈ S
hn0 : n ≠ 0
key : z < n * (z / n + 1)
⊢ -↑z = ↑((z / n + 1) * n - z) - ↑((z / n + 1) * n)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : Monoid R
S : Submonoid R
inst✝² : OreSet S
X : Type ?u.10
inst✝¹ : MulAction R X
T : Type u_2
inst✝ : Monoid T
f : R →* T
fS : ↥S →* Tˣ
hf : ∀ (s : ↥S), f ↑s = ↑(fS s)
x y : OreLocalization S R
⊢ liftExpand (fun r s ↦ ↑(fS s)⁻¹ * f r) ⋯ (x * y) =
    liftExpand (fun r s ↦ ↑(fS s)⁻¹ * f r) ⋯ x * liftExpand (fun r s ↦ ↑(fS s)⁻¹ * f r) ⋯ y

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.MonoidLocalization.Basic

Goal state before:
pp: S : AddSubmonoid ℕ
hS : S ≠ ⊥
z : ℤ
⊢ ∃ x, ∃ y ∈ S, z = ↑x - ↑y

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
inst✝² : Monoid R
S : Submonoid R
inst✝¹ : OreSet S
T : Type u_2
inst✝ : Monoid T
f : R →* T
fS : ↥S →* Tˣ
hf : ∀ (s : ↥S), f ↑s = ↑(fS s)
r : R
⊢ (universalMulHom f fS hf) (numeratorHom r) = f r

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.MonoidLocalization.Basic

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝¹ : CommMonoid M
S : Submonoid M
inst✝ : CommMonoid N
f : S.LocalizationMap N
⊢ Injective ⇑f ↔ ∀ ⦃x : M⦄, x ∈ S → IsRegular x

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.MonoidLocalization.Basic

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f : ⊤.LocalizationMap N
⊢ Injective ⇑f ↔ IsCancelMul M

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝⁶ : DecidableEq α
R : Type u_2
S : Type u_3
inst✝⁵ : MulZeroOneClass R
inst✝⁴ : DecidableEq R
inst✝³ : Nontrivial R
inst✝² : MulZeroOneClass S
inst✝¹ : DecidableEq S
inst✝ : Nontrivial S
f : R →*₀ S
hf : Function.Injective ⇑f
this : ∀ (z : WithTop R), map (⇑f) z = 0 ↔ z = 0
x : R
hx : ↑x ≠ 0
hy : ⊤ ≠ 0
⊢ ↑(f x) ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝⁶ : DecidableEq α
R : Type u_2
S : Type u_3
inst✝⁵ : MulZeroOneClass R
inst✝⁴ : DecidableEq R
inst✝³ : Nontrivial R
inst✝² : MulZeroOneClass S
inst✝¹ : DecidableEq S
inst✝ : Nontrivial S
f : R →*₀ S
hf : Function.Injective ⇑f
x y : WithTop R
⊢ map (⇑f) (x * y) = map (⇑f) x * map (⇑f) y

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
inst✝² : Monoid R
S : Submonoid R
inst✝¹ : OreSet S
T : Type u_2
inst✝ : Monoid T
f : R →* T
fS : ↥S →* Tˣ
hf : ∀ (s : ↥S), f ↑s = ↑(fS s)
φ : OreLocalization S R →* T
huniv : ∀ (r : R), φ (numeratorHom r) = f r
⊢ φ = universalMulHom f fS hf

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : SemigroupWithZero α
inst✝ : NoZeroDivisors α
a b c : WithTop α
⊢ a * b * c = a * (b * c)

Tactic:
rcases eq_or_ne a 0 with (rfl | ha); · simp only [zero_mul]
    rcases eq_or_ne b 0 with (rfl | hb); · simp only [zero_mul, mul_zero]
    rcases eq_or_ne c 0 with (rfl | hc); · simp only [mul_zero]
    cases a with | top => simp [hb, hc] | coe a => ?_
    cases b with | top => simp [mul_top ha, top_mul hc] | coe b => ?_
    cases c with
    | top =>
      rw [mul_top hb, mul_top ha]
      rw [← coe_zero, ne_eq, coe_eq_coe] at ha hb
      simp [ha, hb]
    | coe c => simp only [← coe_mul, mul_assoc]

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : SemigroupWithZero α
inst✝ : NoZeroDivisors α
a b c : WithTop α
⊢ a * b * c = a * (b * c)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.MonoidLocalization.Basic

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝¹ : CommMonoid M
S : Submonoid M
inst✝ : CommMonoid N
f : S.LocalizationMap N
m : M
hm : IsRegular m
⊢ IsRegular (f m)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
R' : Type u_2
M : Type u_3
X : Type u_4
inst✝¹² : Monoid M
S : Submonoid M
inst✝¹¹ : OreSet S
inst✝¹⁰ : MulAction M X
inst✝⁹ : SMul R X
inst✝⁸ : SMul R M
inst✝⁷ : IsScalarTower R M M
inst✝⁶ : IsScalarTower R M X
inst✝⁵ : SMul R' X
inst✝⁴ : SMul R' M
inst✝³ : IsScalarTower R' M M
inst✝² : IsScalarTower R' M X
inst✝¹ : SMul R R'
inst✝ : IsScalarTower R R' M
c : R
r : X
t : M
s : ↥S
ht : t * ↑s ∈ S
⊢ oreNum (c • 1) s • r /ₒ oreDenom (c • 1) s = oreNum (c • 1) ⟨t * ↑s, ht⟩ • t • r /ₒ oreDenom (c • 1) ⟨t * ↑s, ht⟩

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.MonoidLocalization.Basic

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝² : CommMonoid M
S : Submonoid M
inst✝¹ : CommMonoid N
f : S.LocalizationMap N
inst✝ : IsCancelMul M
⊢ IsCancelMul N

Tactic:
simp_rw [isCancelMul_iff_forall_isRegular, Commute.isRegular_iff (Commute.all _),
    ← Commute.isRightRegular_iff (Commute.all _)]
  intro n
  have ⟨ms, eq⟩ := f.surj n
  exact (eq ▸ f.map_isRegular (isCancelMul_iff_forall_isRegular.mp ‹_› ms.1)).2.of_mul

Goal state after:
Module: Mathlib.GroupTheory.MonoidLocalization.Basic

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝² : CommMonoid M
S : Submonoid M
inst✝¹ : CommMonoid N
f : S.LocalizationMap N
inst✝ : IsCancelMul M
⊢ IsCancelMul N

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
M : Type u_3
X : Type u_4
inst✝⁶ : Monoid M
S : Submonoid M
inst✝⁵ : OreSet S
inst✝⁴ : MulAction M X
inst✝³ : SMul R X
inst✝² : SMul R M
inst✝¹ : IsScalarTower R M M
inst✝ : IsScalarTower R M X
r : R
x : X
s : ↥S
⊢ r • (x /ₒ s) = oreNum (r • 1) s • x /ₒ oreDenom (r • 1) s

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.MonoidLocalization.Basic

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝² : CommMonoid M
S : Submonoid M
inst✝¹ : CommMonoid N
f : S.LocalizationMap N
inst✝ : Subsingleton M
x y : N
⊢ x = y

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝³ : DecidableEq α
inst✝² : MonoidWithZero α
inst✝¹ : NoZeroDivisors α
inst✝ : Nontrivial α
x : WithTop α
n : ℕ
a : WithTop α
⊢ a ^ 0 = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: M : Type u_3
X : Type u_4
inst✝² : Monoid M
S : Submonoid M
inst✝¹ : OreSet S
inst✝ : MulAction M X
r : M
x : OreLocalization S X
⊢ (r /ₒ 1) • x = r • x

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
M : Type u_3
X : Type u_4
inst✝⁶ : Monoid M
S : Submonoid M
inst✝⁵ : OreSet S
inst✝⁴ : MulAction M X
inst✝³ : SMul R X
inst✝² : SMul R M
inst✝¹ : IsScalarTower R M M
inst✝ : IsScalarTower R M X
r : R
x : OreLocalization S X
⊢ (r • 1) • x = r • x

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
M : Type u_3
X : Type u_4
inst✝⁶ : Monoid M
S : Submonoid M
inst✝⁵ : OreSet S
inst✝⁴ : MulAction M X
inst✝³ : SMul R X
inst✝² : SMul R M
inst✝¹ : IsScalarTower R M M
inst✝ : IsScalarTower R M X
r : R
x : OreLocalization S X
⊢ (r • 1 /ₒ 1) • x = r • x

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
R' : Type u_2
M : Type u_3
X : Type u_4
inst✝¹² : Monoid M
S : Submonoid M
inst✝¹¹ : OreSet S
inst✝¹⁰ : MulAction M X
inst✝⁹ : SMul R X
inst✝⁸ : SMul R M
inst✝⁷ : IsScalarTower R M M
inst✝⁶ : IsScalarTower R M X
inst✝⁵ : SMul R' X
inst✝⁴ : SMul R' M
inst✝³ : IsScalarTower R' M M
inst✝² : IsScalarTower R' M X
inst✝¹ : SMul R R'
inst✝ : IsScalarTower R R' M
r : R
m : R'
x : OreLocalization S X
⊢ (r • m) • x = r • m • x

Tactic:
rw [← smul_one_oreDiv_one_smul, ← smul_one_oreDiv_one_smul, ← smul_one_oreDiv_one_smul,
      ← mul_smul, mul_div_one]

Goal state after:
R : Type u_1
R' : Type u_2
M : Type u_3
X : Type u_4
inst✝¹² : Monoid M
S : Submonoid M
inst✝¹¹ : OreSet S
inst✝¹⁰ : MulAction M X
inst✝⁹ : SMul R X
inst✝⁸ : SMul R M
inst✝⁷ : IsScalarTower R M M
inst✝⁶ : IsScalarTower R M X
inst✝⁵ : SMul R' X
inst✝⁴ : SMul R' M
inst✝³ : IsScalarTower R' M M
inst✝² : IsScalarTower R' M X
inst✝¹ : SMul R R'
inst✝ : IsScalarTower R R' M
r : R
m : R'
x : OreLocalization S X
⊢ ((r • m) • 1 /ₒ 1) • x = (r • 1 * m • 1 /ₒ 1) • x
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
R' : Type u_2
M : Type u_3
X : Type u_4
inst✝¹² : Monoid M
S : Submonoid M
inst✝¹¹ : OreSet S
inst✝¹⁰ : MulAction M X
inst✝⁹ : SMul R X
inst✝⁸ : SMul R M
inst✝⁷ : IsScalarTower R M M
inst✝⁶ : IsScalarTower R M X
inst✝⁵ : SMul R' X
inst✝⁴ : SMul R' M
inst✝³ : IsScalarTower R' M M
inst✝² : IsScalarTower R' M X
inst✝¹ : SMul R R'
inst✝ : IsScalarTower R R' M
r : R
m : R'
x : OreLocalization S X
⊢ (r • m) • x = r • m • x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝³ : DecidableEq α
inst✝² : MonoidWithZero α
inst✝¹ : NoZeroDivisors α
inst✝ : Nontrivial α
x : WithTop α
n✝ n : ℕ
a : WithTop α
⊢ a ^ (n + 1) = a ^ n * a

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
R' : Type u_2
M : Type u_3
X : Type u_4
inst✝¹³ : Monoid M
S : Submonoid M
inst✝¹² : OreSet S
inst✝¹¹ : MulAction M X
inst✝¹⁰ : SMul R X
inst✝⁹ : SMul R M
inst✝⁸ : IsScalarTower R M M
inst✝⁷ : IsScalarTower R M X
inst✝⁶ : SMul R' X
inst✝⁵ : SMul R' M
inst✝⁴ : IsScalarTower R' M M
inst✝³ : IsScalarTower R' M X
inst✝² : SMul R R'
inst✝¹ : IsScalarTower R R' M
inst✝ : SMulCommClass R R' M
r : R
m : R'
x : OreLocalization S X
⊢ r • m • x = m • r • x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝³ : DecidableEq α
inst✝² : MonoidWithZero α
inst✝¹ : NoZeroDivisors α
inst✝ : Nontrivial α
x : WithTop α
n : ℕ
⊢ x ^ n = ⊤ ↔ x = ⊤ ∧ n ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝³ : DecidableEq α
inst✝² : MonoidWithZero α
inst✝¹ : NoZeroDivisors α
inst✝ : Nontrivial α
x : WithTop α
n : ℕ
⊢ x ^ n ≠ ⊤ ↔ x ≠ ⊤ ∨ n = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : MonoidWithZero α
inst✝² : NoZeroDivisors α
inst✝¹ : Nontrivial α
x : WithTop α
n : ℕ
inst✝ : Preorder α
⊢ x ^ n < ⊤ ↔ x < ⊤ ∨ n = 0

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
R' : Type u_2
M : Type u_3
X : Type u_4
inst✝¹² : Monoid M
S : Submonoid M
inst✝¹¹ : OreSet S
inst✝¹⁰ : MulAction M X
inst✝⁹ : SMul R X
inst✝⁸ : SMul R M
inst✝⁷ : IsScalarTower R M M
inst✝⁶ : IsScalarTower R M X
inst✝⁵ : SMul R' X
inst✝⁴ : SMul R' M
inst✝³ : IsScalarTower R' M M
inst✝² : IsScalarTower R' M X
inst✝¹ : SMul R R'
inst✝ : IsScalarTower R R' M
r : R
m : OreLocalization S M
x : OreLocalization S X
⊢ (r • m) • x = r • m • x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝³ : DecidableEq α
inst✝² : CommMonoidWithZero α
inst✝¹ : NoZeroDivisors α
inst✝ : Nontrivial α
a b : WithTop α
⊢ a * b = b * a

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
R' : Type u_2
M : Type u_3
X : Type u_4
inst✝¹³ : Monoid M
S : Submonoid M
inst✝¹² : OreSet S
inst✝¹¹ : MulAction M X
inst✝¹⁰ : SMul R X
inst✝⁹ : SMul R M
inst✝⁸ : IsScalarTower R M M
inst✝⁷ : IsScalarTower R M X
inst✝⁶ : SMul R' X
inst✝⁵ : SMul R' M
inst✝⁴ : IsScalarTower R' M M
inst✝³ : IsScalarTower R' M X
inst✝² : SMul R R'
inst✝¹ : IsScalarTower R R' M
inst✝ : SMulCommClass R M M
r : R
r₁ : M
s₁ : ↥S
r₂ : X
s₂ : ↥S
⊢ ↑s₁ * r • 1 = r • 1 * ↑s₁

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
R' : Type u_2
M : Type u_3
X : Type u_4
inst✝¹³ : Monoid M
S : Submonoid M
inst✝¹² : OreSet S
inst✝¹¹ : MulAction M X
inst✝¹⁰ : SMul R X
inst✝⁹ : SMul R M
inst✝⁸ : IsScalarTower R M M
inst✝⁷ : IsScalarTower R M X
inst✝⁶ : SMul R' X
inst✝⁵ : SMul R' M
inst✝⁴ : IsScalarTower R' M M
inst✝³ : IsScalarTower R' M X
inst✝² : SMul R R'
inst✝¹ : IsScalarTower R R' M
inst✝ : SMulCommClass R M M
r : R
x : OreLocalization S M
y : OreLocalization S X
⊢ r • x • y = x • r • y

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
R' : Type u_2
M : Type u_3
X : Type u_4
inst✝¹⁷ : Monoid M
S : Submonoid M
inst✝¹⁶ : OreSet S
inst✝¹⁵ : MulAction M X
inst✝¹⁴ : SMul R X
inst✝¹³ : SMul R M
inst✝¹² : IsScalarTower R M M
inst✝¹¹ : IsScalarTower R M X
inst✝¹⁰ : SMul R' X
inst✝⁹ : SMul R' M
inst✝⁸ : IsScalarTower R' M M
inst✝⁷ : IsScalarTower R' M X
inst✝⁶ : SMul R R'
inst✝⁵ : IsScalarTower R R' M
inst✝⁴ : SMul Rᵐᵒᵖ M
inst✝³ : SMul Rᵐᵒᵖ X
inst✝² : IsScalarTower Rᵐᵒᵖ M M
inst✝¹ : IsScalarTower Rᵐᵒᵖ M X
inst✝ : IsCentralScalar R M
r : R
x : OreLocalization S X
⊢ MulOpposite.op r • x = r • x

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R✝ : Type u_1
R' : Type u_2
M : Type u_3
X : Type u_4
inst✝¹⁷ : Monoid M
S : Submonoid M
inst✝¹⁶ : OreSet S
inst✝¹⁵ : MulAction M X
inst✝¹⁴ : SMul R✝ X
inst✝¹³ : SMul R✝ M
inst✝¹² : IsScalarTower R✝ M M
inst✝¹¹ : IsScalarTower R✝ M X
inst✝¹⁰ : SMul R' X
inst✝⁹ : SMul R' M
inst✝⁸ : IsScalarTower R' M M
inst✝⁷ : IsScalarTower R' M X
inst✝⁶ : SMul R✝ R'
inst✝⁵ : IsScalarTower R✝ R' M
R : Type ?u.51
inst✝⁴ : Monoid R
inst✝³ : MulAction R M
inst✝² : IsScalarTower R M M
inst✝¹ : MulAction R X
inst✝ : IsScalarTower R M X
s₁ s₂ : R
x : OreLocalization S X
⊢ (s₁ * s₂) • x = s₁ • s₂ • x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝³ : DecidableEq α
inst✝² : NonUnitalNonAssocSemiring α
inst✝¹ : PartialOrder α
inst✝ : CanonicallyOrderedAdd α
c a b : WithTop α
⊢ c * (a + b) = c * a + c * b

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R✝ : Type u_1
R' : Type u_2
M : Type u_3
X : Type u_4
inst✝¹⁷ : Monoid M
S : Submonoid M
inst✝¹⁶ : OreSet S
inst✝¹⁵ : MulAction M X
inst✝¹⁴ : SMul R✝ X
inst✝¹³ : SMul R✝ M
inst✝¹² : IsScalarTower R✝ M M
inst✝¹¹ : IsScalarTower R✝ M X
inst✝¹⁰ : SMul R' X
inst✝⁹ : SMul R' M
inst✝⁸ : IsScalarTower R' M M
inst✝⁷ : IsScalarTower R' M X
inst✝⁶ : SMul R✝ R'
inst✝⁵ : IsScalarTower R✝ R' M
R : Type ?u.51
inst✝⁴ : Monoid R
inst✝³ : MulAction R M
inst✝² : IsScalarTower R M M
inst✝¹ : MulAction R X
inst✝ : IsScalarTower R M X
x : X
s : ↥S
⊢ 1 • (x /ₒ s) = x /ₒ s

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
M : Type u_3
X : Type u_4
inst✝⁶ : Monoid M
S : Submonoid M
inst✝⁵ : OreSet S
inst✝⁴ : MulAction M X
inst✝³ : SMul R X
inst✝² : SMul R M
inst✝¹ : IsScalarTower R M M
inst✝ : IsScalarTower R M X
r : R
x : X
⊢ r • (x /ₒ 1) = r • x /ₒ 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommMonoid R
S : Submonoid R
inst✝ : OreSet S
r₁ r₂ : R
s₁ s₂ : ↥S
⊢ ↑s₂ * r₁ = r₁ * ↑s₂

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommMonoid R
S : Submonoid R
inst✝ : OreSet S
r₁ r₂ : R
s₁ s₂ : ↥S
⊢ r₁ /ₒ s₁ * (r₂ /ₒ s₂) = r₁ * r₂ /ₒ (s₁ * s₂)

Tactic:
by

Goal state after:
Module: Mathlib.Order.Iterate

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
f : α → α
x y : ℕ → α
hf : Monotone f
n : ℕ
h₀ : x 0 ≤ y 0
hx : ∀ (k : ℕ), k < n → x (k + 1) ≤ f (x k)
hy : ∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)
⊢ x n ≤ y n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝³ : DecidableEq α
inst✝² : NonUnitalNonAssocSemiring α
inst✝¹ : PartialOrder α
inst✝ : CanonicallyOrderedAdd α
a b c : WithTop α
⊢ (a + b) * c = a * c + b * c

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : CommMonoid R
S : Submonoid R
inst✝ : OreSet S
x y : OreLocalization S R
⊢ x * y = y * x

Tactic:
by

Goal state after:
Module: Mathlib.Order.Iterate

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
f : α → α
x y : ℕ → α
hf : Monotone f
n : ℕ
hn : 0 < n
h₀ : x 0 ≤ y 0
hx : ∀ (k : ℕ), k < n → x (k + 1) < f (x k)
hy : ∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)
⊢ x n < y n

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.OreLocalization.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : Monoid R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : Zero X
inst✝ : MulAction R X
⊢ 0 = 0 /ₒ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝⁵ : DecidableEq α
inst✝⁴ : CommSemiring α
inst✝³ : PartialOrder α
inst✝² : OrderBot α
inst✝¹ : CanonicallyOrderedAdd α
inst✝ : PosMulStrictMono α
b₂ : WithTop α
this : MulPosStrictMono α
a₁ b₁ : α
hb : ↑b₁ < b₂
ha : ↑a₁ < ⊤
⊢ b₂ ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Order.Iterate

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
f : α → α
x y : ℕ → α
hf : Monotone f
n : ℕ
h₀ : x 0 < y 0
hx : ∀ (k : ℕ), k < n → x (k + 1) < f (x k)
hy : ∀ (k : ℕ), k < n → f (y k) ≤ y (k + 1)
⊢ x n < y n

Tactic:
by

Goal state after:
Module: Mathlib.Order.Iterate

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
f : α → α
β : Type u_2
g : β → β
h : β → α
hf : Monotone f
H : h ∘ g ≤ f ∘ h
n : ℕ
x : β
⊢ (h ∘ g^[n]) x ≤ (f^[n] ∘ h) x

Tactic:
by

Goal state after:
Module: Mathlib.Order.Iterate

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
f : α → α
h : id ≤ f
n : ℕ
⊢ id ≤ f^[n]

Tactic:
by

Goal state after:
Module: Mathlib.Order.Iterate

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
f : α → α
h : id ≤ f
n : ℕ
x : α
⊢ f^[n] x ≤ f^[n + 1] x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝⁵ : DecidableEq α
inst✝⁴ : CommSemiring α
inst✝³ : PartialOrder α
inst✝² : OrderBot α
inst✝¹ : CanonicallyOrderedAdd α
inst✝ : PosMulStrictMono α
a₂ : WithTop α
this : MulPosStrictMono α
a₁ : α
ha : ↑a₁ < a₂
b₁ : α
ha₂ : a₂ ≠ ⊤
hb : ↑b₁ < ⊤
⊢ a₂ ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Order.Iterate

Goal state before:
pp: case hy
α : Type u_1
inst✝ : Preorder α
f g : α → α
h : Commute f g
hf : Monotone f
hg : Monotone g
x : α
hx : f x ≤ g x
n : ℕ
⊢ ∀ (k : ℕ), k < n → f (g^[k] x) ≤ g^[k + 1] x

Tactic:
simp [h.iterate_right _ _, hg.iterate _ hx]

Goal state after:
Module: Mathlib.Order.Iterate

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
f g : α → α
h : Commute f g
hf : Monotone f
hg : Monotone g
x : α
hx : f x ≤ g x
n : ℕ
⊢ f^[n] x ≤ g^[n] x

Tactic:
by

Goal state after:
Module: Mathlib.Order.Iterate

Goal state before:
pp: case hy
α : Type u_1
inst✝ : Preorder α
f g : α → α
h : Commute f g
hf : Monotone f
hg : StrictMono g
x : α
hx : f x < g x
n : ℕ
hn : 0 < n
⊢ ∀ (k : ℕ), k < n → f (g^[k] x) < g^[k + 1] x

Tactic:
simp [h.iterate_right _ _, hg.iterate _ hx]

Goal state after:
Module: Mathlib.Order.Iterate

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
f g : α → α
h : Commute f g
hf : Monotone f
hg : StrictMono g
x : α
hx : f x < g x
n : ℕ
hn : 0 < n
⊢ f^[n] x < g^[n] x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝⁵ : DecidableEq α
inst✝⁴ : CommSemiring α
inst✝³ : PartialOrder α
inst✝² : OrderBot α
inst✝¹ : CanonicallyOrderedAdd α
inst✝ : PosMulStrictMono α
a₁ a₂ b₁ b₂ : WithTop α
ha : a₁ < a₂
hb : b₁ < b₂
⊢ a₁ * b₁ < a₂ * b₂

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝⁷ : DecidableEq α
inst✝⁶ : CommSemiring α
inst✝⁵ : PartialOrder α
inst✝⁴ : OrderBot α
inst✝³ : CanonicallyOrderedAdd α
inst✝² : PosMulStrictMono α
inst✝¹ : NoZeroDivisors α
inst✝ : Nontrivial α
x✝ : 1 ≠ 0
⊢ StrictMono fun a ↦ a ^ 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝⁷ : DecidableEq α
inst✝⁶ : CommSemiring α
inst✝⁵ : PartialOrder α
inst✝⁴ : OrderBot α
inst✝³ : CanonicallyOrderedAdd α
inst✝² : PosMulStrictMono α
inst✝¹ : NoZeroDivisors α
inst✝ : Nontrivial α
n : ℕ
x✝ : n + 2 ≠ 0
x y : WithTop α
h : x < y
⊢ (fun a ↦ a ^ (n + 2)) x < (fun a ↦ a ^ (n + 2)) y

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a : WithBot α
h : a ≠ 0
⊢ a * ⊥ = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
b : WithBot α
hb : b ≠ 0
⊢ ⊥ * b = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Order.Iterate

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f g : α → α
h : Commute f g
hf : Monotone f
hg : StrictMono g
x : α
n : ℕ
hn : 0 < n
⊢ f^[n] x < g^[n] x ↔ f x < g x

Tactic:
by

Goal state after:
Module: Mathlib.Order.Iterate

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f g : α → α
h : Commute f g
hf : Monotone f
hg : StrictMono g
x : α
n : ℕ
hn : 0 < n
⊢ f^[n] x ≤ g^[n] x ↔ f x ≤ g x

Tactic:
by

Goal state after:
Module: Mathlib.Order.Iterate

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f g : α → α
h : Commute f g
hf : StrictMono f
hg : Monotone g
x : α
n : ℕ
hn : 0 < n
⊢ f^[n] x ≤ g^[n] x ↔ f x ≤ g x

Tactic:
by

Goal state after:
Module: Mathlib.Order.Iterate

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
f g : α → α
h : Commute f g
hf : Monotone f
hg : StrictMono g
x : α
n : ℕ
hn : 0 < n
⊢ f^[n] x = g^[n] x ↔ f x = g x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a b : WithBot α
⊢ a * b = if a = 0 ∨ b = 0 then 0 else map₂ (fun x1 x2 ↦ x1 * x2) a b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Iterate

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
f : α → α
x : α
hf : Monotone f
hx : x ≤ f x
n : ℕ
⊢ f^[n] x ≤ f^[n + 1] x

Tactic:
by

Goal state after:
Module: Mathlib.Order.Iterate

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
f : α → α
x : α
hf : StrictMono f
hx : x < f x
n : ℕ
⊢ f^[n] x < f^[n + 1] x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Order.Ring.WithTop

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a b : WithBot α
⊢ a * b = ⊥ ↔ a ≠ 0 ∧ b = ⊥ ∨ a = ⊥ ∧ b ≠ 0

Tactic:
by

Goal state after: