mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case h
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x z : R
hxz : x ∣ z
r s p g : R
hxy : ¬g = 0
hyz : g * s ∣ z
hr : x = gcd x (g * s) * r
hp : z * g = x * (g * s) * p
⊢ x * (g * s) * p = x * s * p * g

Tactic:
]

Goal state after:
case h
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x z : R
hxz : x ∣ z
r s p g : R
hxy : ¬g = 0
hyz : g * s ∣ z
hr : x = gcd x (g * s) * r
hp : z * g = x * (g * s) * p
⊢ x * (g * s) * p = x * s * p * g
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case h
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x z : R
hxz : x ∣ z
r s p g : R
hxy : ¬g = 0
hyz : g * s ∣ z
hr : x = gcd x (g * s) * r
hp : z * g = x * (g * s) * p
⊢ x * g * s * p = x * s * p * g

Tactic:
]

Goal state after:
case h
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x z : R
hxz : x ∣ z
r s p g : R
hxy : ¬g = 0
hyz : g * s ∣ z
hr : x = gcd x (g * s) * r
hp : z * g = x * (g * s) * p
⊢ x * g * s * p = x * s * p * g
Module: Mathlib.RingTheory.Ideal.Maps

Goal state before:
pp: R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Semiring S
inst✝ : FunLike F R S
rc : RingHomClass F R S
f : F
⊢ Function.Injective ⇑f ↔ ∀ (x : R), x ∈ ⇑f ⁻¹' {0} ↔ x ∈ ↑⊥

Tactic:
]

Goal state after:
R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Semiring S
inst✝ : FunLike F R S
rc : RingHomClass F R S
f : F
⊢ Function.Injective ⇑f ↔ ∀ (x : R), x ∈ ⇑f ⁻¹' {0} ↔ x ∈ ↑⊥
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
I : Ideal R
inst✝ : I.IsTwoSided
ι : Type u_4
s : Set ι
f : ι → M
x : M
⊢ x ∈ I • span R (f '' s) ↔ x ∈ I • span R (f '' s)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Maps

Goal state before:
pp: R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Semiring S
inst✝ : FunLike F R S
rc : RingHomClass F R S
f : F
⊢ ker f = ⊥ ↔ ker f = ⊥

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Maps

Goal state before:
pp: R : Type u
S : Type v
T : Type w
inst✝² : Ring R
inst✝¹ : Semiring S
inst✝ : Semiring T
g : T →+* R
f : R →+* S
hf : Function.Injective ⇑f
⊢ Ideal.comap g ⊥ = Ideal.comap g ⊥

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case neg
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y z : R
hxz : x ∣ z
hyz : y ∣ z
hxy : ¬gcd x y = 0
r : R
hr : x = gcd x y * r
s : R
hs : y = gcd x y * s
⊢ x * y ∣ z * (x * gcdA x y) + z * (y * gcdB x y)

Tactic:
]

Goal state after:
case neg
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y z : R
hxz : x ∣ z
hyz : y ∣ z
hxy : ¬gcd x y = 0
r : R
hr : x = gcd x y * r
s : R
hs : y = gcd x y * s
⊢ x * y ∣ z * (x * gcdA x y) + z * (y * gcdB x y)
Module: Mathlib.RingTheory.Ideal.Maps

Goal state before:
pp: R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : FunLike F R S
rc : RingHomClass F R S
f : F
x y : R
⊢ f x = f y ↔ f x = f y

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case neg.h₁
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y z : R
hxz : x ∣ z
hyz : y ∣ z
hxy : ¬gcd x y = 0
r : R
hr : x = gcd x y * r
s : R
hs : y = gcd x y * s
⊢ x * y ∣ x * (z * gcdA x y)

Tactic:
]

Goal state after:
case neg.h₁
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y z : R
hxz : x ∣ z
hyz : y ∣ z
hxy : ¬gcd x y = 0
r : R
hr : x = gcd x y * r
s : R
hs : y = gcd x y * s
⊢ x * y ∣ x * (z * gcdA x y)
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u
inst✝ : Semiring R
⊢ ⊤ = ⊤

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Maps

Goal state before:
pp: R : Type u_1
M : Type u_2
M' : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
f : M →ₗ[R] M'
hf : Function.Injective ⇑f
x : R
h : ∀ (m : M'), x • m = 0
⊢ ∀ (m : M), x • m = 0

Tactic:
]

Goal state after:
R : Type u_1
M : Type u_2
M' : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
f : M →ₗ[R] M'
hf : Function.Injective ⇑f
x : R
h : ∀ (m : M'), x • m = 0
⊢ ∀ (m : M), x • m = 0
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u
inst✝ : Semiring R
I J : Ideal R
⊢ (∃ y ∈ I, ∃ z ∈ J, y + z = 1) ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Maps

Goal state before:
pp: R : Type u_1
M : Type u_2
M' : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
f : M →ₗ[R] M'
hf : Function.Injective ⇑f
x : R
h : ∀ (m : M'), x • m = 0
m : M
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case h₂
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y z : R
hxz : x ∣ z
hyz : y ∣ z
hxy : ¬gcd x y = 0
r : R
hr : x = gcd x y * r
s : R
hs : y = gcd x y * s
⊢ y * x ∣ y * (z * gcdB x y)

Tactic:
]

Goal state after:
case h₂
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y z : R
hxz : x ∣ z
hyz : y ∣ z
hxy : ¬gcd x y = 0
r : R
hr : x = gcd x y * r
s : R
hs : y = gcd x y * s
⊢ y * x ∣ y * (z * gcdB x y)
Module: Mathlib.RingTheory.Ideal.Maps

Goal state before:
pp: R : Type u_1
M : Type u_2
M' : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
f : M →ₗ[R] M'
hf : Function.Surjective ⇑f
x : R
h : ∀ (m : M), x • m = 0
⊢ ∀ (m : M'), x • m = 0

Tactic:
]

Goal state after:
R : Type u_1
M : Type u_2
M' : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
f : M →ₗ[R] M'
hf : Function.Surjective ⇑f
x : R
h : ∀ (m : M), x • m = 0
⊢ ∀ (m : M'), x • m = 0
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: case zero
R : Type u
inst✝ : Semiring R
I : Ideal R
n : ℕ
h : 0 ≤ n
⊢ I ^ n ≤ ⊤

Tactic:
]

Goal state after:
case zero
R : Type u
inst✝ : Semiring R
I : Ideal R
n : ℕ
h : 0 ≤ n
⊢ I ^ n ≤ ⊤
Module: Mathlib.RingTheory.Ideal.Maps

Goal state before:
pp: R : Type u_1
M : Type u_2
M' : Type u_3
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
f : M →ₗ[R] M'
hf : Function.Surjective ⇑f
x : R
h : ∀ (m : M), x • m = 0
m : M
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Maps

Goal state before:
pp: R : Type u_4
M : Type u_5
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
⊢ annihilator R M ≤ ⊥ ↔ FaithfulSMul R M

Tactic:
]

Goal state after:
R : Type u_4
M : Type u_5
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
⊢ annihilator R M ≤ ⊥ ↔ FaithfulSMul R M
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x : R
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Maps

Goal state before:
pp: case refine_1
R : Type u_4
M : Type u_5
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
H : annihilator R M ≤ ⊥
r s : R
H' : ∀ (a : M), r • a = s • a
⊢ r - s = 0

Tactic:
]

Goal state after:
case refine_1
R : Type u_4
M : Type u_5
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
H : annihilator R M ≤ ⊥
r s : R
H' : ∀ (a : M), r • a = s • a
⊢ r - s = 0
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: case succ
R : Type u
inst✝ : Semiring R
I : Ideal R
m n : ℕ
h : m + 1 ≤ m + 1 + n
⊢ I ^ n * I ^ (m + 1) ≤ I ^ (m + 1)

Tactic:
]

Goal state after:
case succ
R : Type u
inst✝ : Semiring R
I : Ideal R
m n : ℕ
h : m + 1 ≤ m + 1 + n
⊢ I ^ n * I ^ (m + 1) ≤ I ^ (m + 1)
Module: Mathlib.RingTheory.Ideal.Maps

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
h : annihilator R M = ⊤
m m' : M
⊢ 1 • m = 1 • m'

Tactic:
]

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
h : annihilator R M = ⊤
m m' : M
⊢ 1 • m = 1 • m'
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case mp
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y : R
hxy : x = 0 ∨ y / gcd x y = 0
⊢ x = 0 ∨ y = 0

Tactic:
]

Goal state after:
case mp
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y : R
hxy : x = 0 ∨ y / gcd x y = 0
⊢ x = 0 ∨ y = 0
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: case zero
R : Type u
inst✝ : Semiring R
I J : Ideal R
e : I ≤ J
⊢ 1 ≤ 1

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: case succ
R : Type u
inst✝ : Semiring R
I J : Ideal R
e : I ≤ J
n✝ : ℕ
hn : I ^ n✝ ≤ J ^ n✝
⊢ I ^ n✝ * I ≤ J ^ n✝ * J

Tactic:
]

Goal state after:
case succ
R : Type u
inst✝ : Semiring R
I J : Ideal R
e : I ≤ J
n✝ : ℕ
hn : I ^ n✝ ≤ J ^ n✝
⊢ I ^ n✝ * I ≤ J ^ n✝ * J
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case pos
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y : R
hxy : x = 0 ∨ y / gcd x y = 0
hy : y / gcd x y = 0
hgxy : x = 0 ∧ y = 0
⊢ y = 0

Tactic:
]

Goal state after:
case pos
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y : R
hxy : x = 0 ∨ y / gcd x y = 0
hy : y / gcd x y = 0
hgxy : x = 0 ∧ y = 0
⊢ y = 0
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u
inst✝¹ : Semiring R
I J K L : Ideal R
inst✝ : J.IsTwoSided
a✝ b : R
ha : a✝ ∈ I * J
i : R
hi : i ∈ I
j : R
hj : j ∈ J
⊢ i * (j * b) ∈ I * J

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : Semiring R
I J K L : Ideal R
inst✝ : J.IsTwoSided
a✝ b : R
ha : a✝ ∈ I * J
i : R
hi : i ∈ I
j : R
hj : j ∈ J
⊢ i * (j * b) ∈ I * J
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case neg
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x r s g : R
hr : x = g * r
hgxy : ¬g = 0
hxy : x = 0 ∨ g * s / gcd x (g * s) = 0
hy : s = 0
⊢ g * s = 0

Tactic:
]

Goal state after:
case neg
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x r s g : R
hr : x = g * r
hgxy : ¬g = 0
hxy : x = 0 ∨ g * s / gcd x (g * s) = 0
hy : s = 0
⊢ g * s = 0
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u
inst✝¹ : Semiring R
I J K L : Ideal R
inst✝ : J.IsTwoSided
a✝ b : R
ha : a✝ ∈ I * J
x y : R
hx : x * b ∈ I * J
hy : y * b ∈ I * J
⊢ x * b + y * b ∈ I * J

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : Semiring R
I J K L : Ideal R
inst✝ : J.IsTwoSided
a✝ b : R
ha : a✝ ∈ I * J
x y : R
hx : x * b ∈ I * J
hy : y * b ∈ I * J
⊢ x * b + y * b ∈ I * J
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case neg
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x r s g : R
hr : x = g * r
hgxy : ¬g = 0
hxy : x = 0 ∨ g * s / gcd x (g * s) = 0
hy : s = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case neg
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y : R
hxy : x = 0 ∨ y / gcd x y = 0
hy : y / gcd x y = 0
hgxy : ¬gcd x y = 0
⊢ y = 0

Tactic:
rcases gcd_dvd x y with ⟨⟨r, hr⟩, ⟨s, hs⟩⟩
      generalize gcd x y = g at hr hs hy hgxy ⊢
      subst hs
      rw [mul_div_cancel_left₀ _ hgxy] at hy
      rw [hy, mul_zero]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case mpr.inl
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y : R
hx : x = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case mpr.inr
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y : R
hy : y = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y : R
⊢ gcd x y * (x * y / gcd x y) = x * y

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y : R
⊢ gcd x y * (x * y / gcd x y) = x * y
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u
inst✝¹ : Semiring R
I J K L : Ideal R
inst✝ : I.IsTwoSided
m n : ℕ
⊢ ⊤.IsTwoSided

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : Semiring R
I J K L : Ideal R
inst✝ : I.IsTwoSided
m n : ℕ
⊢ ⊤.IsTwoSided
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case pos
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y : R
h : gcd x y = 0
⊢ 0 = x * y

Tactic:
]

Goal state after:
case pos
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y : R
h : gcd x y = 0
⊢ 0 = x * y
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u
inst✝¹ : Semiring R
I J K L : Ideal R
inst✝ : I.IsTwoSided
m n x✝¹ : ℕ
x✝ : (I ^ x✝¹).IsTwoSided
⊢ IsTwoSided (I ^ x✝¹ * I)

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : Semiring R
I J K L : Ideal R
inst✝ : I.IsTwoSided
m n x✝¹ : ℕ
x✝ : (I ^ x✝¹).IsTwoSided
⊢ IsTwoSided (I ^ x✝¹ * I)
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case pos
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y : R
h : x = 0 ∧ y = 0
⊢ 0 = x * y

Tactic:
]

Goal state after:
case pos
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y : R
h : x = 0 ∧ y = 0
⊢ 0 = x * y
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: case inl
R : Type u
inst✝¹ : Semiring R
I : Ideal R
inst✝ : I.IsTwoSided
m : ℕ
⊢ I ^ m = I ^ m

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case pos
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y : R
h : x = 0 ∧ y = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u
inst✝¹ : Semiring R
I : Ideal R
inst✝ : I.IsTwoSided
n : ℕ
⊢ I * I ^ n = I * I ^ n

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case neg
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
y r s g : R
h : ¬g = 0
hs : y = gcd (g * r) y * s
⊢ g * (r * y) = g * (r * y)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u
inst✝¹ : Semiring R
r s : R
inst✝ : (span {r}).IsTwoSided
⊢ span {r * s} = span {r * s}

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case neg
R : Type u
inst✝ : EuclideanDomain R
a b c : R
ha : a ≠ 0
hcb : c ∣ b
hc : ¬c = 0
⊢ a * b = a * b

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u
inst✝² : Semiring R
inst✝¹ : NoZeroDivisors R
I J : Ideal R
inst✝ : I.IsTwoSided
hI : I ≠ ⊥
hJ : J ≠ ⊥
⊢ ¬(I = ⊥ ∨ J = ⊥)

Tactic:
]

Goal state after:
R : Type u
inst✝² : Semiring R
inst✝¹ : NoZeroDivisors R
I J : Ideal R
inst✝ : I.IsTwoSided
hI : I ≠ ⊥
hJ : J ≠ ⊥
⊢ ¬(I = ⊥ ∨ J = ⊥)
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u
inst✝¹ : Semiring R
I J K : Ideal R
inst✝ : I.IsTwoSided
h : 1 ∈ I ⊔ J
i : R
hi : i ∈ I ⊔ K
⊢ i ∈ I ⊔ J * K

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : Semiring R
I J K : Ideal R
inst✝ : I.IsTwoSided
h : 1 ∈ I ⊔ J
i : R
hi : i ∈ I ⊔ K
⊢ i ∈ I ⊔ J * K
Module: Mathlib.RingTheory.Ideal.Maps

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
r : R
⊢ r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = 0

Tactic:
simp_rw [annihilator, Module.mem_annihilator, Subtype.forall, Subtype.ext_iff]; rfl

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case inr.inr
R : Type u
inst✝ : EuclideanDomain R
c d : R
hc0 : c ≠ 0
hd0 : d ≠ 0
k1 k2 : R
⊢ k1 * k2 = k1 * k2

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Maps

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x✝¹ : R
x✝ : x✝¹ ∈ ⊤
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : y ≠ 0
h2 : y ∣ x
⊢ x / y + z = (x + y * z) / y

Tactic:
]

Goal state after:
R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : y ≠ 0
h2 : y ∣ x
⊢ x / y + z = (x + y * z) / y
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u
inst✝¹ : Semiring R
I J K : Ideal R
inst✝ : I.IsTwoSided
h : ∃ y ∈ I, ∃ z ∈ J, y + z = 1
i : R
hi : ∃ y ∈ I, ∃ z ∈ K, y + z = i
⊢ ∃ y ∈ I, ∃ z ∈ J * K, y + z = i

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : Semiring R
I J K : Ideal R
inst✝ : I.IsTwoSided
h : ∃ y ∈ I, ∃ z ∈ J, y + z = 1
i : R
hi : ∃ y ∈ I, ∃ z ∈ K, y + z = i
⊢ ∃ y ∈ I, ∃ z ∈ J * K, y + z = i
Module: Mathlib.RingTheory.Ideal.Maps

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
⊢ N = ⊥ ↔ N = ⊥

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : y ≠ 0
h2 : y ∣ x
⊢ x + y * z = x + y * z

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : y ≠ 0
h2 : y ∣ x
⊢ (x + y * z) / y = x / y + z

Tactic:
]

Goal state after:
R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : y ≠ 0
h2 : y ∣ x
⊢ (x + y * z) / y = x / y + z
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : y ≠ 0
h2 : y ∣ x
⊢ x / y - z = (x - y * z) / y

Tactic:
]

Goal state after:
R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : y ≠ 0
h2 : y ∣ x
⊢ x / y - z = (x - y * z) / y
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : y ≠ 0
h2 : y ∣ x
⊢ x - y * z = x - y * z

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : y ≠ 0
h2 : y ∣ x
⊢ (x - y * z) / y = x / y - z

Tactic:
]

Goal state after:
R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : y ≠ 0
h2 : y ∣ x
⊢ (x - y * z) / y = x / y - z
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u
inst✝¹ : Semiring R
I J K : Ideal R
inst✝ : I.IsTwoSided
i1 : R
hi1 : i1 ∈ I
j : R
hj : j ∈ J
h : i1 + j = 1
i' : R
hi' : i' ∈ I
k : R
hk : k ∈ K
⊢ i' + k = i' + k

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : z ≠ 0
h2 : z ∣ y
⊢ x + y / z = (z * x + y) / z

Tactic:
]

Goal state after:
R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : z ≠ 0
h2 : z ∣ y
⊢ x + y / z = (z * x + y) / z
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : z ≠ 0
h2 : z ∣ y
⊢ z * x + y = z * x + y

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : z ≠ 0
h2 : z ∣ y
⊢ (z * x + y) / z = x + y / z

Tactic:
]

Goal state after:
R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : z ≠ 0
h2 : z ∣ y
⊢ (z * x + y) / z = x + y / z
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : z ≠ 0
h2 : z ∣ y
⊢ x - y / z = (z * x - y) / z

Tactic:
]

Goal state after:
R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : z ≠ 0
h2 : z ∣ y
⊢ x - y / z = (z * x - y) / z
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : z ≠ 0
h2 : z ∣ y
⊢ z * x - y = z * x - y

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : z ≠ 0
h2 : z ∣ y
⊢ (z * x - y) / z = x - y / z

Tactic:
]

Goal state after:
R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : z ≠ 0
h2 : z ∣ y
⊢ (z * x - y) / z = x - y / z
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case inr
R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : y ∣ x
h2 : y * z ∣ x
hz : z ≠ 0
⊢ x / y = x / y

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Maps

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N P : Submodule R M
⊢ ⨅ i, (↑i).annihilator = ⨅ x, (↑x).annihilator

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case inr
R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : y ∣ x
h2 : y * z ∣ x
hy : y ≠ 0
⊢ x / y / z = x / (y * z)

Tactic:
]

Goal state after:
case inr
R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : y ∣ x
h2 : y * z ∣ x
hy : y ≠ 0
⊢ x / y / z = x / (y * z)
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝ : EuclideanDomain R
x y z t : R
h1 : y ≠ 0
h2 : t ≠ 0
h3 : y ∣ x
h4 : t ∣ z
⊢ t * x + y * z = t * x + y * z

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u
inst✝¹ : Semiring R
I J K : Ideal R
inst✝ : J.IsTwoSided
h : 1 ∈ I ⊔ K
i : R
hi : i ∈ I ⊔ J
⊢ i ∈ I ⊔ J * K

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : Semiring R
I J K : Ideal R
inst✝ : J.IsTwoSided
h : 1 ∈ I ⊔ K
i : R
hi : i ∈ I ⊔ J
⊢ i ∈ I ⊔ J * K
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝ : EuclideanDomain R
x y z t : R
h1 : y ≠ 0
h2 : t ≠ 0
h3 : y ∣ x
h4 : t ∣ z
⊢ t * x - y * z = t * x - y * z

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : y ≠ 0
a : R
ha : x = y * a
⊢ a = z ↔ y * a = y * z

Tactic:
]

Goal state after:
R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : y ≠ 0
a : R
ha : x = y * a
⊢ a = z ↔ y * a = y * z
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u
inst✝¹ : Semiring R
I J K : Ideal R
inst✝ : J.IsTwoSided
h : ∃ y ∈ I, ∃ z ∈ K, y + z = 1
i : R
hi : ∃ y ∈ I, ∃ z ∈ J, y + z = i
⊢ ∃ y ∈ I, ∃ z ∈ J * K, y + z = i

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : Semiring R
I J K : Ideal R
inst✝ : J.IsTwoSided
h : ∃ y ∈ I, ∃ z ∈ K, y + z = 1
i : R
hi : ∃ y ∈ I, ∃ z ∈ J, y + z = i
⊢ ∃ y ∈ I, ∃ z ∈ J * K, y + z = i
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝ : EuclideanDomain R
x y z : R
h1 : z ≠ 0
h2 : z ∣ y
⊢ z * x = y ↔ z * x = y

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: R : Type u
inst✝ : EuclideanDomain R
x y z t : R
h1 : y ≠ 0
h2 : t ≠ 0
h3 : y ∣ x
h4 : t ∣ z
⊢ t * x = y * z ↔ t * x = y * z

pp: R : Type u
inst✝ : EuclideanDomain R
x y z t : R
h1 : y ≠ 0
h2 : t ≠ 0
h3 : y ∣ x
h4 : t ∣ z
⊢ t ∣ y * z

Tactic:
]

Goal state after:
R : Type u
inst✝ : EuclideanDomain R
x y z t : R
h1 : y ≠ 0
h2 : t ≠ 0
h3 : y ∣ x
h4 : t ∣ z
⊢ t ∣ y * z
Module: Mathlib.RingTheory.Ideal.Maps

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
r : R
⊢ (∀ n ∈ span R s, r • n = 0) ↔ ∀ (n : ↑s), r • ↑n = 0

Tactic:
]

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set M
r : R
⊢ (∀ n ∈ span R s, r • n = 0) ↔ ∀ (n : ↑s), r • ↑n = 0
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case h
R : Type u
inst✝ : EuclideanDomain R
x y z t : R
h1 : y ≠ 0
h2 : t ≠ 0
h3 : y ∣ x
a : R
ha : z = t * a
⊢ t * (a * y) = t * (a * y)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u
inst✝¹ : Semiring R
I J K : Ideal R
inst✝ : J.IsTwoSided
i1 : R
hi1 : i1 ∈ I
k : R
hk : k ∈ K
h : i1 + k = 1
i' : R
hi' : i' ∈ I
j : R
hj : j ∈ J
⊢ i' + j = i' + j

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u
inst✝¹ : Semiring R
I J K : Ideal R
inst✝ : J.IsTwoSided
h : J ⊔ I = ⊤
⊢ I * K ⊔ J = K ⊔ J

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : Semiring R
I J K : Ideal R
inst✝ : J.IsTwoSided
h : J ⊔ I = ⊤
⊢ I * K ⊔ J = K ⊔ J