mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: case cons
R : Type u
I : Type v
inst✝ : CommSemiring R
x : R
s : I → R
t✝ : Finset I
b : I
t : Finset I
hbt : b ∉ t
ih : (∀ i ∈ t, IsCoprime (s i) x) → IsCoprime (∏ i ∈ t, s i) x
h : ∀ i ∈ Finset.cons b t hbt, IsCoprime (s i) x
⊢ IsCoprime (s b * ∏ x ∈ t, s x) x

Tactic:
]

Goal state after:
case cons
R : Type u
I : Type v
inst✝ : CommSemiring R
x : R
s : I → R
t✝ : Finset I
b : I
t : Finset I
hbt : b ∉ t
ih : (∀ i ∈ t, IsCoprime (s i) x) → IsCoprime (∏ i ∈ t, s i) x
h : ∀ i ∈ Finset.cons b t hbt, IsCoprime (s i) x
⊢ IsCoprime (s b * ∏ x ∈ t, s x) x
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: case cons
R : Type u
I : Type v
inst✝ : CommSemiring R
x : R
s : I → R
t✝ : Finset I
b : I
t : Finset I
hbt : b ∉ t
ih : (∀ i ∈ t, IsCoprime (s i) x) → IsCoprime (∏ i ∈ t, s i) x
h : IsCoprime (s b) x ∧ ∀ x_1 ∈ t, IsCoprime (s x_1) x
⊢ IsCoprime (s b * ∏ x ∈ t, s x) x

Tactic:
]

Goal state after:
case cons
R : Type u
I : Type v
inst✝ : CommSemiring R
x : R
s : I → R
t✝ : Finset I
b : I
t : Finset I
hbt : b ∉ t
ih : (∀ i ∈ t, IsCoprime (s i) x) → IsCoprime (∏ i ∈ t, s i) x
h : IsCoprime (s b) x ∧ ∀ x_1 ∈ t, IsCoprime (s x_1) x
⊢ IsCoprime (s b * ∏ x ∈ t, s x) x
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: R : Type u
I : Type v
inst✝ : CommSemiring R
x : R
s : I → R
t✝ : Finset I
b : I
t : Finset I
hbt : b ∉ t
ih : IsCoprime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsCoprime (s i) x
⊢ (IsCoprime (s b) x ∧ ∀ i ∈ t, IsCoprime (s i) x) ↔ IsCoprime (s b) x ∧ ∀ x_1 ∈ t, IsCoprime (s x_1) x

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: case insert
R : Type u
I : Type v
inst✝ : CommSemiring R
z : R
s : I → R
t : Finset I
a : I
r : Finset I
har : a ∉ r
ih : (↑r).Pairwise (IsCoprime on s) → (∀ i ∈ r, s i ∣ z) → ∏ x ∈ r, s x ∣ z
Hs : (↑(insert a r)).Pairwise (IsCoprime on s)
Hs1 : ∀ i ∈ insert a r, s i ∣ z
⊢ s a * ∏ x ∈ r, s x ∣ z

Tactic:
]

Goal state after:
case insert
R : Type u
I : Type v
inst✝ : CommSemiring R
z : R
s : I → R
t : Finset I
a : I
r : Finset I
har : a ∉ r
ih : (↑r).Pairwise (IsCoprime on s) → (∀ i ∈ r, s i ∣ z) → ∏ x ∈ r, s x ∣ z
Hs : (↑(insert a r)).Pairwise (IsCoprime on s)
Hs1 : ∀ i ∈ insert a r, s i ∣ z
⊢ s a * ∏ x ∈ r, s x ∣ z
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: case cons
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
⊢ (∃ μ, ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1) ↔
    Pairwise (IsCoprime on fun a ↦ s ↑a) ∧ ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)

Tactic:
]

Goal state after:
case cons
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
⊢ (∃ μ, ∑ i ∈ cons a t hat, μ i * ∏ j ∈ cons a t hat \ {i}, s j = 1) ↔
    Pairwise (IsCoprime on fun a ↦ s ↑a) ∧ ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
x : I
hx : x ∈ t
⊢ a ∈ insert a t ∧ ¬a = x

Tactic:
]

Goal state after:
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
x : I
hx : x ∈ t
⊢ a ∈ insert a t ∧ ¬a = x
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: case cons.mp
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
⊢ Pairwise (IsCoprime on fun a ↦ s ↑a) ∧ ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)

Tactic:
]

Goal state after:
case cons.mp
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
⊢ Pairwise (IsCoprime on fun a ↦ s ↑a) ∧ ∀ b ∈ t, IsCoprime (s a) (s b) ∧ IsCoprime (s b) (s a)
Module: Mathlib.Tactic.Order.ToInt

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
n : ℕ
val : Fin n → α
li : List α := List.ofFn val
sli : List α := li.mergeSort fun a b ↦ decide (a ≤ b)
i : Fin n
⊢ val i ∈ li

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : LinearOrder α
n : ℕ
val : Fin n → α
li : List α := List.ofFn val
sli : List α := li.mergeSort fun a b ↦ decide (a ≤ b)
i : Fin n
⊢ val i ∈ li
Module: Mathlib.Tactic.Order.ToInt

Goal state before:
pp: case neg
α : Type u_1
inst✝ : LinearOrder α
n : ℕ
val : Fin n → α
li : List α := List.ofFn val
sli : List α := li.mergeSort fun a b ↦ decide (a ≤ b)
this : ∀ (i : Fin n), ∃ j, sli[j] = val i
i j : Fin n
pf✝ : ∀ (j : Fin sli.length), ↑j < sli.length
hi : ∃ j, sli[↑j] = val i
hj : ∃ j_1, sli[↑j_1] = val j
h_eq : ¬sli[↑hi.choose] = sli[↑hj.choose]
⊢ val i ≤ val j ↔ ↑hi.choose ≤ ↑hj.choose

Tactic:
]

Goal state after:
case neg
α : Type u_1
inst✝ : LinearOrder α
n : ℕ
val : Fin n → α
li : List α := List.ofFn val
sli : List α := li.mergeSort fun a b ↦ decide (a ≤ b)
this : ∀ (i : Fin n), ∃ j, sli[j] = val i
i j : Fin n
pf✝ : ∀ (j : Fin sli.length), ↑j < sli.length
hi : ∃ j, sli[↑j] = val i
hj : ∃ j_1, sli[↑j_1] = val j
h_eq : ¬sli[↑hi.choose] = sli[↑hj.choose]
⊢ val i ≤ val j ↔ ↑hi.choose ≤ ↑hj.choose
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: case cons.mp.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
⊢ ∑ i ∈ t, (Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) + μ * fun x ↦ s a) i * ∏ j ∈ t \ {i}, s j = 1

Tactic:
]

Goal state after:
case cons.mp.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
⊢ ∑ i ∈ t, (Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) + μ * fun x ↦ s a) i * ∏ j ∈ t \ {i}, s j = 1
Module: Mathlib.Tactic.Order.ToInt

Goal state before:
pp: case neg
α : Type u_1
inst✝ : LinearOrder α
n : ℕ
val : Fin n → α
li : List α := List.ofFn val
sli : List α := li.mergeSort fun a b ↦ decide (a ≤ b)
this✝ : ∀ (i : Fin n), ∃ j, sli[j] = val i
i j : Fin n
pf✝ : ∀ (j : Fin sli.length), ↑j < sli.length
hi : ∃ j, sli[↑j] = val i
hj : ∃ j_1, sli[↑j_1] = val j
h_eq : ¬sli[↑hi.choose] = sli[↑hj.choose]
this :
  ∀ (i j : Fin (li.mergeSort fun a b ↦ decide (a ≤ b)).length),
    i < j →
      decide ((li.mergeSort fun a b ↦ decide (a ≤ b)).get i ≤ (li.mergeSort fun a b ↦ decide (a ≤ b)).get j) = true
⊢ sli[↑hi.choose] ≤ sli[↑hj.choose] ↔ ↑hi.choose ≤ ↑hj.choose

Tactic:
]

Goal state after:
case neg
α : Type u_1
inst✝ : LinearOrder α
n : ℕ
val : Fin n → α
li : List α := List.ofFn val
sli : List α := li.mergeSort fun a b ↦ decide (a ≤ b)
this✝ : ∀ (i : Fin n), ∃ j, sli[j] = val i
i j : Fin n
pf✝ : ∀ (j : Fin sli.length), ↑j < sli.length
hi : ∃ j, sli[↑j] = val i
hj : ∃ j_1, sli[↑j_1] = val j
h_eq : ¬sli[↑hi.choose] = sli[↑hj.choose]
this :
  ∀ (i j : Fin (li.mergeSort fun a b ↦ decide (a ≤ b)).length),
    i < j →
      decide ((li.mergeSort fun a b ↦ decide (a ≤ b)).get i ≤ (li.mergeSort fun a b ↦ decide (a ≤ b)).get j) = true
⊢ sli[↑hi.choose] ≤ sli[↑hj.choose] ↔ ↑hi.choose ≤ ↑hj.choose
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: case cons.mp.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
⊢ t.sum ?m.248 =
    ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j)

pp: case cons.mp.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
⊢ ∀ x ∈ t, (Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) + μ * fun x ↦ s a) x * ∏ j ∈ t \ {x}, s j = ?m.248 x

pp: R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
⊢ I → R

Tactic:
]

Goal state after:
case cons.mp.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
⊢ ∀ x ∈ t,
    (Pi.single (Exists.choose h) (μ a * s (Exists.choose h)) + μ * fun x ↦ s a) x * ∏ j ∈ t \ {x}, s j =
      Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j
Module: Mathlib.Algebra.Algebra.Operations

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
x : A
r : R
⊢ (fun r ↦ r • 1) r = x ↔ r • 1 = x

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Algebra.Operations

Goal state before:
pp: R : Type u
inst✝² : Semiring R
A : Type v
inst✝¹ : Semiring A
inst✝ : Module R A
n : ℕ
⊢ ↑n = ↑n

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Algebra.Operations

Goal state before:
pp: R : Type u
inst✝² : Semiring R
A : Type v
inst✝¹ : Semiring A
inst✝ : Module R A
x✝ : A
⊢ x✝ ∈ (R ∙ 1).toSubMulAction ↔ ∃ r, r • 1 = x✝

Tactic:
]

Goal state after:
R : Type u
inst✝² : Semiring R
A : Type v
inst✝¹ : Semiring A
inst✝ : Module R A
x✝ : A
⊢ x✝ ∈ (R ∙ 1).toSubMulAction ↔ ∃ r, r • 1 = x✝
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: case pos
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx✝ : x ∈ t
hx : x = Exists.choose h
⊢ μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x =
    μ a * s (Exists.choose h) * ∏ j ∈ t \ {Exists.choose h}, s j

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Algebra.Operations

Goal state before:
pp: R : Type u
inst✝⁶ : Semiring R
A : Type v
inst✝⁵ : Semiring A
inst✝⁴ : Module R A
M : Type u_1
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
A' : Submodule R A
M' : Submodule R M
r : R
m✝ : M
hm✝ : m✝ ∈ (A'.toAddSubmonoid • M'.toAddSubmonoid).carrier
a : A
ha : a ∈ A'.toAddSubmonoid
m : M
hm : m ∈ M'.toAddSubmonoid
⊢ (r • a) • m ∈ (A'.toAddSubmonoid • M'.toAddSubmonoid).carrier

Tactic:
]

Goal state after:
R : Type u
inst✝⁶ : Semiring R
A : Type v
inst✝⁵ : Semiring A
inst✝⁴ : Module R A
M : Type u_1
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
A' : Submodule R A
M' : Submodule R M
r : R
m✝ : M
hm✝ : m✝ ∈ (A'.toAddSubmonoid • M'.toAddSubmonoid).carrier
a : A
ha : a ∈ A'.toAddSubmonoid
m : M
hm : m ∈ M'.toAddSubmonoid
⊢ (r • a) • m ∈ (A'.toAddSubmonoid • M'.toAddSubmonoid).carrier
Module: Mathlib.Algebra.Algebra.Operations

Goal state before:
pp: R : Type u
inst✝⁶ : Semiring R
A : Type v
inst✝⁵ : Semiring A
inst✝⁴ : Module R A
M : Type u_1
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
A' : Submodule R A
M' : Submodule R M
r : R
m : M
hm : m ∈ (A'.toAddSubmonoid • M'.toAddSubmonoid).carrier
m₁ m₂ : M
h₁ : r • m₁ ∈ (A'.toAddSubmonoid • M'.toAddSubmonoid).carrier
h₂ : r • m₂ ∈ (A'.toAddSubmonoid • M'.toAddSubmonoid).carrier
⊢ r • m₁ + r • m₂ ∈ (A'.toAddSubmonoid • M'.toAddSubmonoid).carrier

Tactic:
]

Goal state after:
R : Type u
inst✝⁶ : Semiring R
A : Type v
inst✝⁵ : Semiring A
inst✝⁴ : Module R A
M : Type u_1
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
A' : Submodule R A
M' : Submodule R M
r : R
m : M
hm : m ∈ (A'.toAddSubmonoid • M'.toAddSubmonoid).carrier
m₁ m₂ : M
h₁ : r • m₁ ∈ (A'.toAddSubmonoid • M'.toAddSubmonoid).carrier
h₂ : r • m₂ ∈ (A'.toAddSubmonoid • M'.toAddSubmonoid).carrier
⊢ r • m₁ + r • m₂ ∈ (A'.toAddSubmonoid • M'.toAddSubmonoid).carrier
Module: Mathlib.Algebra.Algebra.Operations

Goal state before:
pp: R : Type u
inst✝⁶ : Semiring R
A : Type v
inst✝⁵ : Semiring A
inst✝⁴ : Module R A
M : Type u_1
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
N : Submodule R M
n✝ : M
a✝ : n✝ ∈ N
⊢ 0 ∈ ⊥

Tactic:
]

Goal state after:
R : Type u
inst✝⁶ : Semiring R
A : Type v
inst✝⁵ : Semiring A
inst✝⁴ : Module R A
M : Type u_1
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
N : Submodule R M
n✝ : M
a✝ : n✝ ∈ N
⊢ 0 ∈ ⊥
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: case neg
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx✝ : x ∈ t
hx : ¬x = Exists.choose h
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Algebra.Operations

Goal state before:
pp: R : Type u
inst✝⁶ : Semiring R
A : Type v
inst✝⁵ : Semiring A
inst✝⁴ : Module R A
M : Type u_1
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
I J : Submodule R A
N : Submodule R M
p : M
hp : p ∈ N
m : A
hm : m ∈ I
n : A
hn : n ∈ J
hmn : m + n ∈ I ⊔ J
⊢ m • p + n • p ∈ I • N ⊔ J • N

Tactic:
]

Goal state after:
R : Type u
inst✝⁶ : Semiring R
A : Type v
inst✝⁵ : Semiring A
inst✝⁴ : Module R A
M : Type u_1
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
I J : Submodule R A
N : Submodule R M
p : M
hp : p ∈ N
m : A
hm : m ∈ I
n : A
hn : n ∈ J
hmn : m + n ∈ I ⊔ J
⊢ m • p + n • p ∈ I • N ⊔ J • N
Module: Mathlib.Algebra.Algebra.Operations

Goal state before:
pp: case refine_1
R : Type u
inst✝⁶ : Semiring R
A : Type v
inst✝⁵ : Semiring A
inst✝⁴ : Module R A
M : Type u_1
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
N : Submodule R M
m : M
hm : m ∈ N
r : R
⊢ r • m ∈ N

Tactic:
]

Goal state after:
case refine_1
R : Type u
inst✝⁶ : Semiring R
A : Type v
inst✝⁵ : Semiring A
inst✝⁴ : Module R A
M : Type u_1
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
N : Submodule R M
m : M
hm : m ∈ N
r : R
⊢ r • m ∈ N
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: case e'_2.e'_6
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ✝ : μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
hμ :
  ∑ x ∈ t,
      (Pi.single (Exists.choose h) (μ a * s (Exists.choose h) * ∏ x ∈ t \ {Exists.choose h}, s x) x +
        μ x * ∏ j ∈ insert a t \ {x}, s j) =
    1
x : I
hx : x ∈ t
⊢ s a * ∏ x ∈ t \ {x}, s x = (fun x ↦ s a) x * ∏ j ∈ t \ {x}, s j

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Algebra.Operations

Goal state before:
pp: case refine_2
R : Type u
inst✝⁶ : Semiring R
A : Type v
inst✝⁵ : Semiring A
inst✝⁴ : Module R A
M : Type u_1
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
N : Submodule R M
m : M
hm : m ∈ N
⊢ 1 • m ∈ 1 • N

Tactic:
]

Goal state after:
case refine_2
R : Type u
inst✝⁶ : Semiring R
A : Type v
inst✝⁵ : Semiring A
inst✝⁴ : Module R A
M : Type u_1
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Module A M
inst✝ : IsScalarTower R A M
N : Submodule R M
m : M
hm : m ∈ N
⊢ 1 • m ∈ 1 • N
Module: Mathlib.Algebra.Algebra.Operations

Goal state before:
pp: R : Type u
inst✝³ : Semiring R
A : Type v
inst✝² : Semiring A
inst✝¹ : Module R A
inst✝ : IsScalarTower R A A
M : Submodule R A
⊢ M = M

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Algebra.Operations

Goal state before:
pp: R : Type u
inst✝³ : Semiring R
A : Type v
inst✝² : Semiring A
inst✝¹ : Module R A
inst✝ : IsScalarTower R A A
M : Submodule R A
n : ℕ
h : n ≠ 0
⊢ M * M ^ n = M * M ^ n

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Algebra.Operations

Goal state before:
pp: R : Type u
inst✝³ : Semiring R
A : Type v
inst✝² : Semiring A
inst✝¹ : Module R A
inst✝ : IsScalarTower R A A
n : ℕ
x✝ : n + 2 ≠ 0
⊢ ⊥ = ⊥

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Algebra.Operations

Goal state before:
pp: case succ
R : Type u
inst✝³ : Semiring R
A : Type v
inst✝² : Semiring A
inst✝¹ : Module R A
inst✝ : IsScalarTower R A A
M : Submodule R A
n : ℕ
ih : n ≠ 0 → (M ^ n).toAddSubmonoid = M.toAddSubmonoid ^ n
h : n + 1 ≠ 0
⊢ (M ^ n).toAddSubmonoid * M.toAddSubmonoid = M.toAddSubmonoid ^ n * M.toAddSubmonoid

Tactic:
]

Goal state after:
case succ
R : Type u
inst✝³ : Semiring R
A : Type v
inst✝² : Semiring A
inst✝¹ : Module R A
inst✝ : IsScalarTower R A A
M : Submodule R A
n : ℕ
ih : n ≠ 0 → (M ^ n).toAddSubmonoid = M.toAddSubmonoid ^ n
h : n + 1 ≠ 0
⊢ (M ^ n).toAddSubmonoid * M.toAddSubmonoid = M.toAddSubmonoid ^ n * M.toAddSubmonoid
Module: Mathlib.Algebra.Algebra.Operations

Goal state before:
pp: case succ.zero
R : Type u
inst✝³ : Semiring R
A : Type v
inst✝² : Semiring A
inst✝¹ : Module R A
inst✝ : IsScalarTower R A A
M : Submodule R A
ih : 0 ≠ 0 → (M ^ 0).toAddSubmonoid = M.toAddSubmonoid ^ 0
h : 0 + 1 ≠ 0
⊢ M.toAddSubmonoid = M.toAddSubmonoid

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Algebra.Operations

Goal state before:
pp: case succ.succ
R : Type u
inst✝³ : Semiring R
A : Type v
inst✝² : Semiring A
inst✝¹ : Module R A
inst✝ : IsScalarTower R A A
M : Submodule R A
n : ℕ
ih : n + 1 ≠ 0 → (M ^ (n + 1)).toAddSubmonoid = M.toAddSubmonoid ^ (n + 1)
h : n + 1 + 1 ≠ 0
⊢ M.toAddSubmonoid ^ (n + 1) * M.toAddSubmonoid = M.toAddSubmonoid ^ (n + 1) * M.toAddSubmonoid

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Algebra.Operations

Goal state before:
pp: case inl
R : Type u
inst✝³ : Semiring R
A : Type v
inst✝² : Semiring A
inst✝¹ : Module R A
inst✝ : IsScalarTower R A A
M : Submodule R A
⊢ 1 ≤ toAddSubmonoid 1

Tactic:
]

Goal state after:
case inl
R : Type u
inst✝³ : Semiring R
A : Type v
inst✝² : Semiring A
inst✝¹ : Module R A
inst✝ : IsScalarTower R A A
M : Submodule R A
⊢ 1 ≤ toAddSubmonoid 1
Module: Mathlib.RingTheory.Coprime.Lemmas

Goal state before:
pp: case cons.mp.refine_2.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
⊢ μ a * ∏ x ∈ t, s x + t.sum ?m.978 = μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j

pp: case cons.mp.refine_2.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
⊢ ∀ x ∈ t, (μ x * ∏ j ∈ t \ {x}, s j) * s a = ?m.978 x

pp: R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
⊢ I → R

Tactic:
]

Goal state after:
case cons.mp.refine_2.refine_1
R : Type u
I : Type v
inst✝¹ : CommSemiring R
s : I → R
t✝ : Finset I
inst✝ : DecidableEq I
a : I
t : Finset I
hat : a ∉ t
h : t.Nonempty
ih : (∃ μ, ∑ i ∈ t, μ i * ∏ j ∈ t \ {i}, s j = 1) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)
mem : ∀ x ∈ t, a ∈ insert a t \ {x}
μ : I → R
hμ : μ a * ∏ j ∈ t, s j + ∑ x ∈ t, μ x * ∏ j ∈ insert a t \ {x}, s j = 1
b : I
hb : b ∈ t
⊢ ∀ x ∈ t, (μ x * ∏ j ∈ t \ {x}, s j) * s a = μ x * ∏ j ∈ insert a t \ {x}, s j