mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Polynomial.Derivative

Goal state before:
pp: R : Type u
inst✝ : CommRing R
S : Finset R
k : ℕ
ind : (⇑derivative)^[k] (∏ a ∈ S, (X - C a)) = ↑k ! * ∑ T ∈ powersetCard (#S - k) S, ∏ a ∈ T, (X - C a)
hk : k + 1 ≤ #S
⊢ (∑ x ∈ powersetCard (#S - (k + 1)) S ×ˢ S with
      match x with
      | (T, i) => i ∉ T,
      match x with
      | (T, i) => ∏ a ∈ T, (X - C a)) =
    ∑ p ∈ ?r, ∏ a ∈ p.1, (X - C a)

pp: case r
R : Type u
inst✝ : CommRing R
S : Finset R
k : ℕ
ind : (⇑derivative)^[k] (∏ a ∈ S, (X - C a)) = ↑k ! * ∑ T ∈ powersetCard (#S - k) S, ∏ a ∈ T, (X - C a)
hk : k + 1 ≤ #S
⊢ Finset (Finset R × R)

pp: case h
R : Type u
inst✝ : CommRing R
S : Finset R
k : ℕ
ind : (⇑derivative)^[k] (∏ a ∈ S, (X - C a)) = ↑k ! * ∑ T ∈ powersetCard (#S - k) S, ∏ a ∈ T, (X - C a)
hk : k + 1 ≤ #S
⊢ ∀ (p : Finset R × R), p ∈ ?r ↔ p.1 ∈ powersetCard (#S - (k + 1)) S ∧ p.2 ∈ S \ p.1

Tactic:
]

Goal state after:
case h
R : Type u
inst✝ : CommRing R
S : Finset R
k : ℕ
ind : (⇑derivative)^[k] (∏ a ∈ S, (X - C a)) = ↑k ! * ∑ T ∈ powersetCard (#S - k) S, ∏ a ∈ T, (X - C a)
hk : k + 1 ≤ #S
⊢ ∀ (p : Finset R × R),
    p ∈
        {x ∈ powersetCard (#S - (k + 1)) S ×ˢ S |
          match x with
          | (T, i) => i ∉ T} ↔
      p.1 ∈ powersetCard (#S - (k + 1)) S ∧ p.2 ∈ S \ p.1
Module: Mathlib.Algebra.Polynomial.Derivative

Goal state before:
pp: R : Type u
inst✝ : CommRing R
S : Finset R
k : ℕ
ind : (⇑derivative)^[k] (∏ a ∈ S, (X - C a)) = ↑k ! * ∑ T ∈ powersetCard (#S - k) S, ∏ a ∈ T, (X - C a)
hk : k + 1 ≤ #S
⊢ ∑ T ∈ powersetCard (#S - (k + 1)) S, ∑ i ∈ S \ T, ∏ a ∈ T, (X - C a) =
    ∑ i ∈ powersetCard (#S - (k + 1)) S, (↑k + 1) * ∏ a ∈ i, (X - C a)

Tactic:
]

Goal state after:
R : Type u
inst✝ : CommRing R
S : Finset R
k : ℕ
ind : (⇑derivative)^[k] (∏ a ∈ S, (X - C a)) = ↑k ! * ∑ T ∈ powersetCard (#S - k) S, ∏ a ∈ T, (X - C a)
hk : k + 1 ≤ #S
⊢ ∑ T ∈ powersetCard (#S - (k + 1)) S, ∑ i ∈ S \ T, ∏ a ∈ T, (X - C a) =
    ∑ i ∈ powersetCard (#S - (k + 1)) S, (↑k + 1) * ∏ a ∈ i, (X - C a)
Module: Mathlib.Algebra.Polynomial.Derivative

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
inst✝¹ : NoZeroDivisors R
n : ℕ
chn : ↑n ≠ 0
a : R[X]
inst✝ : Nontrivial R
⊢ derivative (a ^ n) = 0 ↔ derivative a = 0

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommSemiring R
inst✝¹ : NoZeroDivisors R
n : ℕ
chn : ↑n ≠ 0
a : R[X]
inst✝ : Nontrivial R
⊢ derivative (a ^ n) = 0 ↔ derivative a = 0
Module: Mathlib.Algebra.BigOperators.Associated

Goal state before:
pp: case cons.inl.refine_2
M₀ : Type u_3
inst✝¹ : CommMonoidWithZero M₀
inst✝ : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀),
    x * y ∈ closure {r | IsUnit r ∨ Prime r} →
      (∀ y ∈ s, IsUnit y ∨ Prime y) → s.prod = x * y → x ∈ closure {r | IsUnit r ∨ Prime r}
x y : M₀
hxy : x * y ∈ closure {r | IsUnit r ∨ Prime r}
hprod : c * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₁ : IsUnit c
k : M₀
hk : c * k = 1
⊢ c * s.prod = x * y ∨ k = 0

Tactic:
]

Goal state after:
case cons.inl.refine_2
M₀ : Type u_3
inst✝¹ : CommMonoidWithZero M₀
inst✝ : IsCancelMulZero M₀
c : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀),
    x * y ∈ closure {r | IsUnit r ∨ Prime r} →
      (∀ y ∈ s, IsUnit y ∨ Prime y) → s.prod = x * y → x ∈ closure {r | IsUnit r ∨ Prime r}
x y : M₀
hxy : x * y ∈ closure {r | IsUnit r ∨ Prime r}
hprod : c * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₁ : IsUnit c
k : M₀
hk : c * k = 1
⊢ c * s.prod = x * y ∨ k = 0
Module: Mathlib.Algebra.BigOperators.Associated

Goal state before:
pp: case cons.inr.inl
M₀ : Type u_3
inst✝¹ : CommMonoidWithZero M₀
inst✝ : IsCancelMulZero M₀
c✝ : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀),
    x * y ∈ closure {r | IsUnit r ∨ Prime r} →
      (∀ y ∈ s, IsUnit y ∨ Prime y) → s.prod = x * y → x ∈ closure {r | IsUnit r ∨ Prime r}
x y : M₀
hxy : x * y ∈ closure {r | IsUnit r ∨ Prime r}
hprod : c✝ * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₂ : Prime c✝
c : M₀
hc : x = c✝ * c
⊢ c✝ * c ∈ closure {r | IsUnit r ∨ Prime r}

Tactic:
]

Goal state after:
case cons.inr.inl
M₀ : Type u_3
inst✝¹ : CommMonoidWithZero M₀
inst✝ : IsCancelMulZero M₀
c✝ : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀),
    x * y ∈ closure {r | IsUnit r ∨ Prime r} →
      (∀ y ∈ s, IsUnit y ∨ Prime y) → s.prod = x * y → x ∈ closure {r | IsUnit r ∨ Prime r}
x y : M₀
hxy : x * y ∈ closure {r | IsUnit r ∨ Prime r}
hprod : c✝ * s.prod = x * y
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₂ : Prime c✝
c : M₀
hc : x = c✝ * c
⊢ c✝ * c ∈ closure {r | IsUnit r ∨ Prime r}
Module: Mathlib.Algebra.BigOperators.Associated

Goal state before:
pp: case cons.inr.inl
M₀ : Type u_3
inst✝¹ : CommMonoidWithZero M₀
inst✝ : IsCancelMulZero M₀
c✝ : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀),
    x * y ∈ closure {r | IsUnit r ∨ Prime r} →
      (∀ y ∈ s, IsUnit y ∨ Prime y) → s.prod = x * y → x ∈ closure {r | IsUnit r ∨ Prime r}
x y : M₀
hxy : x * y ∈ closure {r | IsUnit r ∨ Prime r}
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₂ : Prime c✝
c : M₀
hprod : c✝ * s.prod = c✝ * (c * y)
hc : x = c✝ * c
⊢ c✝ * c ∈ closure {r | IsUnit r ∨ Prime r}

Tactic:
]

Goal state after:
case cons.inr.inl
M₀ : Type u_3
inst✝¹ : CommMonoidWithZero M₀
inst✝ : IsCancelMulZero M₀
c✝ : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀),
    x * y ∈ closure {r | IsUnit r ∨ Prime r} →
      (∀ y ∈ s, IsUnit y ∨ Prime y) → s.prod = x * y → x ∈ closure {r | IsUnit r ∨ Prime r}
x y : M₀
hxy : x * y ∈ closure {r | IsUnit r ∨ Prime r}
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₂ : Prime c✝
c : M₀
hprod : c✝ * s.prod = c✝ * (c * y)
hc : x = c✝ * c
⊢ c✝ * c ∈ closure {r | IsUnit r ∨ Prime r}
Module: Mathlib.Algebra.BigOperators.Associated

Goal state before:
pp: case cons.inr.inl.refine_2
M₀ : Type u_3
inst✝¹ : CommMonoidWithZero M₀
inst✝ : IsCancelMulZero M₀
c✝ : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀),
    x * y ∈ closure {r | IsUnit r ∨ Prime r} →
      (∀ y ∈ s, IsUnit y ∨ Prime y) → s.prod = x * y → x ∈ closure {r | IsUnit r ∨ Prime r}
x y : M₀
hxy : x * y ∈ closure {r | IsUnit r ∨ Prime r}
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₂ : Prime c✝
c : M₀
hprod : c✝ * s.prod = c✝ * (c * y)
hc : x = c✝ * c
⊢ s.prod ∈ closure {r | IsUnit r ∨ Prime r}

Tactic:
]

Goal state after:
case cons.inr.inl.refine_2
M₀ : Type u_3
inst✝¹ : CommMonoidWithZero M₀
inst✝ : IsCancelMulZero M₀
c✝ : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀),
    x * y ∈ closure {r | IsUnit r ∨ Prime r} →
      (∀ y ∈ s, IsUnit y ∨ Prime y) → s.prod = x * y → x ∈ closure {r | IsUnit r ∨ Prime r}
x y : M₀
hxy : x * y ∈ closure {r | IsUnit r ∨ Prime r}
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₂ : Prime c✝
c : M₀
hprod : c✝ * s.prod = c✝ * (c * y)
hc : x = c✝ * c
⊢ s.prod ∈ closure {r | IsUnit r ∨ Prime r}
Module: Mathlib.Algebra.BigOperators.Associated

Goal state before:
pp: case cons.inr.inr
M₀ : Type u_3
inst✝¹ : CommMonoidWithZero M₀
inst✝ : IsCancelMulZero M₀
c✝ : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀),
    x * y ∈ closure {r | IsUnit r ∨ Prime r} →
      (∀ y ∈ s, IsUnit y ∨ Prime y) → s.prod = x * y → x ∈ closure {r | IsUnit r ∨ Prime r}
x y : M₀
hxy : x * y ∈ closure {r | IsUnit r ∨ Prime r}
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₂ : Prime c✝
c : M₀
hprod : c✝ * s.prod = c✝ * (x * c)
hc : y = c✝ * c
⊢ x ∈ closure {r | IsUnit r ∨ Prime r}

Tactic:
]

Goal state after:
case cons.inr.inr
M₀ : Type u_3
inst✝¹ : CommMonoidWithZero M₀
inst✝ : IsCancelMulZero M₀
c✝ : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀),
    x * y ∈ closure {r | IsUnit r ∨ Prime r} →
      (∀ y ∈ s, IsUnit y ∨ Prime y) → s.prod = x * y → x ∈ closure {r | IsUnit r ∨ Prime r}
x y : M₀
hxy : x * y ∈ closure {r | IsUnit r ∨ Prime r}
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₂ : Prime c✝
c : M₀
hprod : c✝ * s.prod = c✝ * (x * c)
hc : y = c✝ * c
⊢ x ∈ closure {r | IsUnit r ∨ Prime r}
Module: Mathlib.Algebra.BigOperators.Associated

Goal state before:
pp: case cons.inr.inr
M₀ : Type u_3
inst✝¹ : CommMonoidWithZero M₀
inst✝ : IsCancelMulZero M₀
c✝ : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀),
    x * y ∈ closure {r | IsUnit r ∨ Prime r} →
      (∀ y ∈ s, IsUnit y ∨ Prime y) → s.prod = x * y → x ∈ closure {r | IsUnit r ∨ Prime r}
x y : M₀
hxy : x * y ∈ closure {r | IsUnit r ∨ Prime r}
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₂ : Prime c✝
c : M₀
hprod : c✝ * s.prod = c✝ * (x * c)
hc : y = c✝ * c
⊢ s.prod ∈ closure {r | IsUnit r ∨ Prime r}

Tactic:
]

Goal state after:
case cons.inr.inr
M₀ : Type u_3
inst✝¹ : CommMonoidWithZero M₀
inst✝ : IsCancelMulZero M₀
c✝ : M₀
s : Multiset M₀
hind :
  ∀ (x y : M₀),
    x * y ∈ closure {r | IsUnit r ∨ Prime r} →
      (∀ y ∈ s, IsUnit y ∨ Prime y) → s.prod = x * y → x ∈ closure {r | IsUnit r ∨ Prime r}
x y : M₀
hxy : x * y ∈ closure {r | IsUnit r ∨ Prime r}
hs : ∀ a ∈ s, IsUnit a ∨ Prime a
ha₂ : Prime c✝
c : M₀
hprod : c✝ * s.prod = c✝ * (x * c)
hc : y = c✝ * c
⊢ s.prod ∈ closure {r | IsUnit r ∨ Prime r}
Module: Mathlib.Algebra.BigOperators.Associated

Goal state before:
pp: case cons
M₀ : Type u_3
inst✝² : CommMonoidWithZero M₀
inst✝¹ : IsCancelMulZero M₀
inst✝ : (a : M₀) → DecidablePred (Associated a)
a : M₀
s : Multiset M₀
induct : ∀ (n : M₀), (∀ a ∈ s, Prime a) → (∀ a ∈ s, a ∣ n) → (∀ (a : M₀), countP (Associated a) s ≤ 1) → s.prod ∣ n
n : M₀
h : ∀ a_1 ∈ a ::ₘ s, Prime a_1
div : ∀ a_1 ∈ a ::ₘ s, a_1 ∣ n
uniq : ∀ (a_1 : M₀), countP (Associated a_1) (a ::ₘ s) ≤ 1
⊢ a * s.prod ∣ n

Tactic:
]

Goal state after:
case cons
M₀ : Type u_3
inst✝² : CommMonoidWithZero M₀
inst✝¹ : IsCancelMulZero M₀
inst✝ : (a : M₀) → DecidablePred (Associated a)
a : M₀
s : Multiset M₀
induct : ∀ (n : M₀), (∀ a ∈ s, Prime a) → (∀ a ∈ s, a ∣ n) → (∀ (a : M₀), countP (Associated a) s ≤ 1) → s.prod ∣ n
n : M₀
h : ∀ a_1 ∈ a ::ₘ s, Prime a_1
div : ∀ a_1 ∈ a ::ₘ s, a_1 ∣ n
uniq : ∀ (a_1 : M₀), countP (Associated a_1) (a ::ₘ s) ≤ 1
⊢ a * s.prod ∣ n
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
⊢ ∀ (a : α), Irreducible a ↔ Prime a

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
⊢ ∀ (a : α), Irreducible a ↔ Prime a
Module: Mathlib.Data.Finsupp.Weight

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝¹ : Semiring R
inst✝ : DecidableEq σ
s : σ
f : σ →₀ R
⊢ f s = f s

Tactic:
]

Goal state after:
Module: Mathlib.Data.Finsupp.Weight

Goal state before:
pp: σ : Type u_1
M : Type u_2
R : Type u_3
inst✝³ : Semiring R
w : σ → M
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Fintype σ
f : σ →₀ R
⊢ ∑ i, f i • w i = ∑ i, f i • w i

Tactic:
]

Goal state after:
Module: Mathlib.Data.Finsupp.Weight

Goal state before:
pp: σ : Type u_1
M : Type u_2
R : Type u_3
inst✝⁴ : Semiring R
w : σ → M
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
hw : ∀ (i : σ), w i ≠ 0
n : R
s : σ
h : ¬w s = 0 → n = 0
⊢ n = 0

Tactic:
]

Goal state after:
σ : Type u_1
M : Type u_2
R : Type u_3
inst✝⁴ : Semiring R
w : σ → M
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
hw : ∀ (i : σ), w i ≠ 0
n : R
s : σ
h : ¬w s = 0 → n = 0
⊢ n = 0
Module: Mathlib.Algebra.BigOperators.Associated

Goal state before:
pp: case h
M₀ : Type u_3
inst✝² : CommMonoidWithZero M₀
inst✝¹ : IsCancelMulZero M₀
inst✝ : (a : M₀) → DecidablePred (Associated a)
a : M₀
s : Multiset M₀
induct : ∀ (n : M₀), (∀ a ∈ s, Prime a) → (∀ a ∈ s, a ∣ n) → (∀ (a : M₀), countP (Associated a) s ≤ 1) → s.prod ∣ n
h : ∀ a_1 ∈ a ::ₘ s, Prime a_1
uniq : ∀ (a_1 : M₀), countP (Associated a_1) (a ::ₘ s) ≤ 1
k : M₀
div : ∀ a_1 ∈ a ::ₘ s, a_1 ∣ a * k
b : M₀
b_in_s : b ∈ s
b_div_n : b ∣ a * k
a_prime : Prime a
b_prime : Prime b
b_div_a : b ∣ a
assoc : b ~ᵤ a
this : ¬∃ a_1 ∈ s, a ~ᵤ a_1
⊢ False

Tactic:
]

Goal state after:
case h
M₀ : Type u_3
inst✝² : CommMonoidWithZero M₀
inst✝¹ : IsCancelMulZero M₀
inst✝ : (a : M₀) → DecidablePred (Associated a)
a : M₀
s : Multiset M₀
induct : ∀ (n : M₀), (∀ a ∈ s, Prime a) → (∀ a ∈ s, a ∣ n) → (∀ (a : M₀), countP (Associated a) s ≤ 1) → s.prod ∣ n
h : ∀ a_1 ∈ a ::ₘ s, Prime a_1
uniq : ∀ (a_1 : M₀), countP (Associated a_1) (a ::ₘ s) ≤ 1
k : M₀
div : ∀ a_1 ∈ a ::ₘ s, a_1 ∣ a * k
b : M₀
b_in_s : b ∈ s
b_div_n : b ∣ a * k
a_prime : Prime a
b_prime : Prime b
b_div_a : b ∣ a
assoc : b ~ᵤ a
this : ¬∃ a_1 ∈ s, a ~ᵤ a_1
⊢ False
Module: Mathlib.Data.Finsupp.Weight

Goal state before:
pp: σ : Type u_1
M : Type u_2
R : Type u_3
inst✝⁴ : Semiring R
w : σ → M
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Nontrivial R
inst✝ : NonTorsionWeight R w
s : σ
h : 1 • w s = 0
⊢ False

Tactic:
]

Goal state after:
σ : Type u_1
M : Type u_2
R : Type u_3
inst✝⁴ : Semiring R
w : σ → M
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Nontrivial R
inst✝ : NonTorsionWeight R w
s : σ
h : 1 • w s = 0
⊢ False
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic

Goal state before:
pp: case cons
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : IsCancelMulZero α
p : α
f : Multiset α
ih : ∀ {g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : (p ::ₘ f).prod ~ᵤ g.prod
b : α
hbg : b ∈ g
hb : p ~ᵤ b
this : DecidableEq α
⊢ Multiset.Rel Associated (p ::ₘ f) (b ::ₘ g.erase b)

Tactic:
]

Goal state after:
case cons
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : IsCancelMulZero α
p : α
f : Multiset α
ih : ∀ {g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : (p ::ₘ f).prod ~ᵤ g.prod
b : α
hbg : b ∈ g
hb : p ~ᵤ b
this : DecidableEq α
⊢ Multiset.Rel Associated (p ::ₘ f) (b ::ₘ g.erase b)
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : IsCancelMulZero α
p : α
f : Multiset α
ih : ∀ {g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : (p ::ₘ f).prod ~ᵤ g.prod
b : α
hbg : b ∈ g
hb : p ~ᵤ b
this : DecidableEq α
q : α
hq : q ∈ f
⊢ q ∈ p ::ₘ f

Tactic:
by simp [hq]

Goal state after:
Module: Mathlib.Data.Finsupp.Weight

Goal state before:
pp: σ : Type u_1
M : Type u_2
w : σ → M
inst✝ : AddCommMonoid M
f : σ →₀ ℕ
i : σ
hi : f i ≠ 0
⊢ ((f - single i 1).sum fun i c ↦ c • w i) + w i = ((f - single i 1).sum fun i c ↦ c • w i) + w i

pp: σ : Type u_1
M : Type u_2
w : σ → M
inst✝ : AddCommMonoid M
f : σ →₀ ℕ
i : σ
hi : f i ≠ 0
⊢ 0 • w i = 0

pp: case h_zero
σ : Type u_1
M : Type u_2
w : σ → M
inst✝ : AddCommMonoid M
f : σ →₀ ℕ
i : σ
hi : f i ≠ 0
⊢ ∀ (a : σ), 0 • w a = 0

pp: case h_add
σ : Type u_1
M : Type u_2
w : σ → M
inst✝ : AddCommMonoid M
f : σ →₀ ℕ
i : σ
hi : f i ≠ 0
⊢ ∀ (a : σ) (b₁ b₂ : ℕ), (b₁ + b₂) • w a = b₁ • w a + b₂ • w a

Tactic:
]

Goal state after:
σ : Type u_1
M : Type u_2
w : σ → M
inst✝ : AddCommMonoid M
f : σ →₀ ℕ
i : σ
hi : f i ≠ 0
⊢ 0 • w i = 0

case h_zero
σ : Type u_1
M : Type u_2
w : σ → M
inst✝ : AddCommMonoid M
f : σ →₀ ℕ
i : σ
hi : f i ≠ 0
⊢ ∀ (a : σ), 0 • w a = 0

case h_add
σ : Type u_1
M : Type u_2
w : σ → M
inst✝ : AddCommMonoid M
f : σ →₀ ℕ
i : σ
hi : f i ≠ 0
⊢ ∀ (a : σ) (b₁ b₂ : ℕ), (b₁ + b₂) • w a = b₁ • w a + b₂ • w a
Module: Mathlib.Algebra.BigOperators.Associated

Goal state before:
pp: ι : Type u_1
M : Type u_2
inst✝ : CommMonoid M
p : Finset ι
f : ι → M
⊢ Associates.mk (∏ x ∈ p, f x) = Associates.mk (∏ i ∈ p, f i)

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.BigOperators.Associated

Goal state before:
pp: M : Type u_2
inst✝ : CommMonoid M
p q : Multiset M
⊢ Multiset.Rel Associated p q ↔ Multiset.Rel (fun a b ↦ Associates.mk a = Associates.mk b) p q

Tactic:
]

Goal state after:
M : Type u_2
inst✝ : CommMonoid M
p q : Multiset M
⊢ Multiset.Rel Associated p q ↔ Multiset.Rel (fun a b ↦ Associates.mk a = Associates.mk b) p q
Module: Mathlib.Data.Finsupp.Weight

Goal state before:
pp: case pos
σ : Type u_1
w : σ → ℕ
s : σ
hs : w s ≠ 0
f : σ →₀ ℕ
h : s ∈ f.support
⊢ f s ≤ f s • w s + ∑ x ∈ f.support \ {s}, f x • w x

Tactic:
]

Goal state after:
case pos
σ : Type u_1
w : σ → ℕ
s : σ
hs : w s ≠ 0
f : σ →₀ ℕ
h : s ∈ f.support
⊢ f s ≤ f s • w s + ∑ x ∈ f.support \ {s}, f x • w x
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : IsCancelMulZero α
p : α
f : Multiset α
ih : ∀ {g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : (p ::ₘ f).prod ~ᵤ g.prod
b : α
hbg : b ∈ g
hb : p ~ᵤ b
this : DecidableEq α
⊢ (p ::ₘ f).prod ~ᵤ g.prod

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : IsCancelMulZero α
p : α
f : Multiset α
ih : ∀ {g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g
g : Multiset α
hf : ∀ x ∈ p ::ₘ f, Prime x
hg : ∀ x ∈ g, Prime x
hfg : (p ::ₘ f).prod ~ᵤ g.prod
b : α
hbg : b ∈ g
hb : p ~ᵤ b
this : DecidableEq α
⊢ (p ::ₘ f).prod ~ᵤ g.prod
Module: Mathlib.Algebra.BigOperators.Associated

Goal state before:
pp: M : Type u_2
inst✝ : CommMonoid M
p q : Multiset (Associates M)
h : p ≤ q
this✝ : DecidableEq (Associates M)
this : p.prod ≤ q.prod
⊢ p.prod ≤ q.prod

Tactic:
]

Goal state after:
M : Type u_2
inst✝ : CommMonoid M
p q : Multiset (Associates M)
h : p ≤ q
this✝ : DecidableEq (Associates M)
this : p.prod ≤ q.prod
⊢ p.prod ≤ q.prod
Module: Mathlib.Data.Finsupp.Weight

Goal state before:
pp: case neg
σ : Type u_1
w : σ → ℕ
s : σ
hs : w s ≠ 0
f : σ →₀ ℕ
h : f s = 0
⊢ 0 ≤ ∑ x ∈ f.support, f x • w x

Tactic:
]

Goal state after:
case neg
σ : Type u_1
w : σ → ℕ
s : σ
hs : w s ≠ 0
f : σ →₀ ℕ
h : f s = 0
⊢ 0 ≤ ∑ x ∈ f.support, f x • w x
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hp0 : p ≠ 0
⊢ ↑(Classical.choose ⋯).card ≤ p.degree

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hp0 : p ≠ 0
⊢ ↑(Classical.choose ⋯).card ≤ p.degree
Module: Mathlib.Data.Finsupp.Weight

Goal state before:
pp: σ : Type u_1
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : PartialOrder M
inst✝ : IsOrderedAddMonoid M
w : σ → M
hw : ∀ (s : σ), 0 ≤ w s
s : σ
f : σ →₀ ℕ
hs : s ∈ f.support
⊢ f s • w s ≤ ∑ x ∈ f.support, f x • w x

Tactic:
]

Goal state after:
σ : Type u_1
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : PartialOrder M
inst✝ : IsOrderedAddMonoid M
w : σ → M
hw : ∀ (s : σ), 0 ≤ w s
s : σ
f : σ →₀ ℕ
hs : s ∈ f.support
⊢ f s • w s ≤ ∑ x ∈ f.support, f x • w x
Module: Mathlib.Data.Finsupp.Weight

Goal state before:
pp: σ : Type u_1
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : PartialOrder M
inst✝ : IsOrderedAddMonoid M
w : σ → M
hw : ∀ (s : σ), 0 ≤ w s
s : σ
f : σ →₀ ℕ
hs : s ∈ f.support
⊢ f s • w s ≤ f s • w s + ∑ x ∈ f.support \ {s}, f x • w x

Tactic:
]

Goal state after:
σ : Type u_1
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : PartialOrder M
inst✝ : IsOrderedAddMonoid M
w : σ → M
hw : ∀ (s : σ), 0 ≤ w s
s : σ
f : σ →₀ ℕ
hs : s ∈ f.support
⊢ f s • w s ≤ f s • w s + ∑ x ∈ f.support \ {s}, f x • w x
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
a : R
hp0 : 0 < p.degree
⊢ (p + C (-a)).degree = p.degree

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
a : R
hp0 : 0 < p.degree
⊢ (p + C (-a)).degree = p.degree
Module: Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
⊢ (Multiset.map normalize (Classical.choose ⋯)).prod ~ᵤ a

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
⊢ (Multiset.map normalize (Classical.choose ⋯)).prod ~ᵤ a
Module: Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
⊢ (Multiset.map (Associates.mk ∘ normalize) (Classical.choose ⋯)).prod =
    (Multiset.map Associates.mk (Classical.choose ⋯)).prod

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
⊢ (Multiset.map (Associates.mk ∘ normalize) (Classical.choose ⋯)).prod =
    (Multiset.map Associates.mk (Classical.choose ⋯)).prod
Module: Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors

Goal state before:
pp: case e_f
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
x✝ : α
⊢ Associates.mk x✝ = Associates.mk x✝

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors

Goal state before:
pp: α : Type u_2
inst✝² : CommMonoidWithZero α
inst✝¹ : StrongNormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
⊢ normalize (factors a).prod = normalize (factors a).prod

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: case neg
R : Type u
a : R
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
p : R[X]
hp : ¬p = 0
⊢ count a (Classical.choose ⋯) = rootMultiplicity a p

Tactic:
]

Goal state after:
case neg
R : Type u
a : R
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
p : R[X]
hp : ¬p = 0
⊢ count a (Classical.choose ⋯) = rootMultiplicity a p
Module: Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
⊢ ∀ x ∈ Multiset.map normalize (if h : a = 0 then 0 else Classical.choose ⋯), Prime x

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
⊢ ∀ x ∈ Multiset.map normalize (if h : a = 0 then 0 else Classical.choose ⋯), Prime x
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: R : Type u
a : R
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
⊢ p ≠ 0 ∧ p.IsRoot a ↔ p ≠ 0 ∧ p.IsRoot a

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors

Goal state before:
pp: case neg
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : ¬a = 0
y : α
hy : y ∈ Classical.choose ⋯
hx : normalize y ∈ Multiset.map (fun x ↦ normalize x) (Classical.choose ⋯)
⊢ Prime y

Tactic:
]

Goal state after:
case neg
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : ¬a = 0
y : α
hy : y ∈ Classical.choose ⋯
hx : normalize y ∈ Multiset.map (fun x ↦ normalize x) (Classical.choose ⋯)
⊢ Prime y
Module: Mathlib.Data.Finsupp.Weight

Goal state before:
pp: case mp
σ : Type u_1
M : Type u_4
inst✝⁴ : AddCommMonoid M
inst✝³ : PartialOrder M
inst✝² : IsOrderedAddMonoid M
inst✝¹ : CanonicallyOrderedAdd M
w : σ → M
inst✝ : NonTorsionWeight ℕ w
f : σ →₀ ℕ
h : (weight w) f = 0
s : σ
hs : ¬f s = 0
⊢ w s ≤ (weight w) f

Tactic:
]

Goal state after:
case mp
σ : Type u_1
M : Type u_4
inst✝⁴ : AddCommMonoid M
inst✝³ : PartialOrder M
inst✝² : IsOrderedAddMonoid M
inst✝¹ : CanonicallyOrderedAdd M
w : σ → M
inst✝ : NonTorsionWeight ℕ w
f : σ →₀ ℕ
h : (weight w) f = 0
s : σ
hs : ¬f s = 0
⊢ w s ≤ (weight w) f
Module: Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
⊢ ∀ x ∈ Multiset.map normalize (if h : a = 0 then 0 else Classical.choose ⋯), normalize x = x

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
⊢ ∀ x ∈ Multiset.map normalize (if h : a = 0 then 0 else Classical.choose ⋯), normalize x = x