mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Order.Filter.Cofinite

Goal state before:
pp: case refine_2
ι : Type u_1
α : ι → Type u_4
β : ι → Type u_5
f : (i : ι) → α i → β i
hf : ∀ᶠ (i : ι) in cofinite, Surjective (f i)
l : (i : ι) → Filter (α i)
I : Set ι
s : (i : ι) → Set (α i)
hI : I.Finite
hs : ∀ i ∈ I, s i ∈ l i
⊢ ∀ᶠ (i : ι) in cofinite, f i '' (I, s).1.piecewise (fun i ↦ id ((I, s).2 i)) (fun x ↦ univ) i = univ

Tactic:
· filter_upwards [hf, hI.compl_mem_cofinite] with i hsurj (hiI : i ∉ I)
    simp [hiI, hsurj.range_eq]

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

Goal state before:
pp: M₀ : Type u_3
inst✝² : CommMonoidWithZero M₀
inst✝¹ : IsCancelMulZero M₀
inst✝ : Subsingleton M₀ˣ
s : Finset M₀
n : M₀
h : ∀ a ∈ s, Prime a
div : ∀ a ∈ s, a ∣ n
⊢ ∏ p ∈ s, p ∣ n

Tactic:
exact
      Multiset.prod_primes_dvd n (by simpa only [Multiset.map_id', Finset.mem_def] using h)
        (by simpa only [Multiset.map_id', Finset.mem_def] using div)
        (by
          simp only [Multiset.map_id', associated_eq_eq, Multiset.countP_eq_card_filter,
            ← s.val.count_eq_card_filter_eq, ← Multiset.nodup_iff_count_le_one, s.nodup])

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

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
a b : R
ha : IsUnit a
f : R → R := fun x ↦ a * x + b
hf : Function.Bijective f
x : R
⊢ rootMultiplicity x (p.comp (C a * X + C b)) = rootMultiplicity (f x) p

Tactic:
rootMultiplicity_comp_C_mul_X_add_C p a b x ha

Goal state after:
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
a b : R
ha : IsUnit a
f : R → R := fun x ↦ a * x + b
hf : Function.Bijective f
x : R
⊢ rootMultiplicity (a * x + b) p = rootMultiplicity (f x) p
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic

Goal state before:
pp: case h
α : Type u_1
inst✝ : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
a✝ : s.prod ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ p, a ~ᵤ p ∧ s = {p}
u : αˣ
hu : (p ::ₘ s).prod * ↑u = a
hs : ∀ b ∈ p ::ₘ s, Prime b
⊢ a ~ᵤ p ∧ p ::ₘ s = {p}

Tactic:
have hs0 : s = 0 := by
      by_contra hs0
      obtain ⟨q, hq⟩ := Multiset.exists_mem_of_ne_zero hs0
      apply (hs q (by simp [hq])).2.1
      refine (ha.isUnit_or_isUnit (?_ : _ = p * ↑u * (s.erase q).prod * _)).resolve_left ?_
      · rw [mul_right_comm _ _ q, mul_assoc, ← Multiset.prod_cons, Multiset.cons_erase hq, ← hu,
          mul_comm, mul_comm p _, mul_assoc]
        simp
      apply mt isUnit_of_mul_isUnit_left (mt isUnit_of_mul_isUnit_left _)
      apply (hs p (Multiset.mem_cons_self _ _)).2.1

Goal state after:
case h
α : Type u_1
inst✝ : CommMonoidWithZero α
a : α
f : Multiset α
ha : Irreducible a
pfa : (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
this : DecidableEq α
p : α
s : Multiset α
a✝ : s.prod ~ᵤ a → (∀ b ∈ s, Prime b) → ∃ p, a ~ᵤ p ∧ s = {p}
u : αˣ
hu : (p ::ₘ s).prod * ↑u = a
hs : ∀ b ∈ p ::ₘ s, Prime b
hs0 : s = 0
⊢ a ~ᵤ p ∧ p ::ₘ s = {p}
Module: Mathlib.Data.Finsupp.Weight

Goal state before:
pp: case zero
σ : Type u_5
s : Set ℕ
hn : 0 ≠ 0
⊢ ⇑degree ⁻¹' (0 • s) = 0 • ⇑degree ⁻¹' s

Tactic:
contradiction

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

Goal state before:
pp: R : Type u
a : R
inst✝¹ : CommRing R
inst✝ : IsDomain R
ha : a ≠ 0
n : ℕ
⊢ (C a * X ^ n).roots = n • {0}

Tactic:
roots_C_mul _ ha,

Goal state after:
R : Type u
a : R
inst✝¹ : CommRing R
inst✝ : IsDomain R
ha : a ≠ 0
n : ℕ
⊢ (X ^ n).roots = n • {0}
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: case a
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
s : Multiset R
⊢ 0 ∉ Multiset.map (fun a ↦ X - C a) s

Tactic:
· rw [Multiset.mem_map]
    rintro ⟨a, -, h⟩
    exact X_sub_C_ne_zero a h

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

Goal state before:
pp: case mpr
α : Type u_1
inst✝³ : CommMonoidWithZero α
inst✝² : NormalizationMonoid α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Subsingleton αˣ
p x : α
hx : x ≠ 0
hprime : Prime p
hdvd : p ∣ x
q : α
hqmem : q ∈ normalizedFactors x
hqeq : p = q
⊢ p ∈ normalizedFactors x

Tactic:
exact hqeq ▸ hqmem

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Content

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
r : R
⊢ (C r).content = normalize r

Tactic:
content

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
r : R
⊢ (C r).support.gcd (C r).coeff = normalize r
Module: Mathlib.RingTheory.Polynomial.Content

Goal state before:
pp: case neg
R : Type u_2
inst✝¹ : CommRing R
inst✝ : StrongNormalizedGCDMonoid R
r : R
p : R[X]
h0 : ¬r = 0
⊢ (C r * p).support.gcd (C r * p).coeff = p.support.gcd fun x ↦ r * p.coeff x

Tactic:
refine congr (congr rfl ?_) ?_

Goal state after:
case neg.refine_1
R : Type u_2
inst✝¹ : CommRing R
inst✝ : StrongNormalizedGCDMonoid R
r : R
p : R[X]
h0 : ¬r = 0
⊢ (C r * p).support = p.support

case neg.refine_2
R : Type u_2
inst✝¹ : CommRing R
inst✝ : StrongNormalizedGCDMonoid R
r : R
p : R[X]
h0 : ¬r = 0
⊢ (C r * p).coeff = fun x ↦ r * p.coeff x
Module: Mathlib.RingTheory.Polynomial.Content

Goal state before:
pp: case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
r : R
p : R[X]
h0 : ¬r = 0
⊢ (C r * p).content = p.support.gcd fun x ↦ r * p.coeff x

Tactic:
content

Goal state after:
case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
r : R
p : R[X]
h0 : ¬r = 0
⊢ (C r * p).support.gcd (C r * p).coeff = p.support.gcd fun x ↦ r * p.coeff x
Module: Mathlib.RingTheory.Polynomial.Content

Goal state before:
pp: case neg
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
r : R
p : R[X]
h0 : ¬r = 0
⊢ (C r * p).support.gcd (C r * p).coeff = p.support.gcd fun x ↦ r * p.coeff x

Tactic:
refine congr (congr rfl ?_) ?_

Goal state after:
case neg.refine_1
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
r : R
p : R[X]
h0 : ¬r = 0
⊢ (C r * p).support = p.support

case neg.refine_2
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
r : R
p : R[X]
h0 : ¬r = 0
⊢ (C r * p).coeff = fun x ↦ r * p.coeff x
Module: Mathlib.RingTheory.Polynomial.Content

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
r : R
k : ℕ
⊢ normalize (C r * X ^ k).content = normalize r

Tactic:
normalize_eq_normalize_iff_associated

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
r : R
k : ℕ
⊢ Associated (C r * X ^ k).content r
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic

Goal state before:
pp: case refine_1
R : Type u_2
inst✝¹ : CommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a : R
a✝ : 0 ≠ 0
b✝ : R
⊢ ∃ a' b' c', IsRelPrime a' b' ∧ c' * a' = 0 ∧ c' * b' = b✝

Tactic:
contradiction

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

Goal state before:
pp: R : Type u
inst✝ : CommRing R
p : R[X]
t : R
hpt : p.IsRoot t
h✝ : ¬p = 0
g : R[X]
hndvd✝ : ¬X - C t ∣ g
m : ℕ := rootMultiplicity t p
hnzd : ↑m ∈ nonZeroDivisors R
hp : p = (X - C t) ^ m * g
hm : m - 1 + 1 = m
hndvd : ¬(X - C t) ^ m ∣ derivative p
h : derivative p = 0
⊢ (X - C t) ^ m ∣ derivative p

Tactic:
by rw [h]; exact dvd_zero _

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

Goal state before:
pp: case neg.inl
R : Type u_2
inst✝¹ : CommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
a p : R
p_prime : Prime p
a' b' c' : R
coprime : IsRelPrime a' b'
a_ne_zero : c' * a' ≠ 0
ih_a : c' * a' ≠ 0 → ∀ (b : R), ∃ a'_1 b' c'_1, IsRelPrime a'_1 b' ∧ c'_1 * a'_1 = c' * a' ∧ c'_1 * b' = b
pa_ne_zero : p * (c' * a') ≠ 0
h : ¬p ∣ c' * b'
q : R
q_dvd_pa' : q ∣ p * a'
q_dvd_b' : q ∣ b'
p_dvd_q : p ∣ q
this : p ∣ c' * b'
⊢ IsUnit q

Tactic:
contradiction

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

Goal state before:
pp: case mp
α : Type u_1
M : Type u_2
inst✝² : CommMonoidWithZero M
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : Fintype Mˣ
y : M
hy : y ≠ 0
this✝² : Nontrivial M
this✝¹ : StrongNormalizationMonoid M
this✝ : DecidableEq M
this : DecidableEq (Associates M)
s : Multiset M × Mˣ
hs : s.1 ≤ normalizedFactors y
⊢ ↑s.2 * s.1.prod ∣ y

Tactic:
rw [(unit_associated_one.mul_right s.fst.prod).dvd_iff_dvd_left, one_mul,
      ← (prod_normalizedFactors hy).dvd_iff_dvd_right]

Goal state after:
case mp
α : Type u_1
M : Type u_2
inst✝² : CommMonoidWithZero M
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : Fintype Mˣ
y : M
hy : y ≠ 0
this✝² : Nontrivial M
this✝¹ : StrongNormalizationMonoid M
this✝ : DecidableEq M
this : DecidableEq (Associates M)
s : Multiset M × Mˣ
hs : s.1 ≤ normalizedFactors y
⊢ s.1.prod ∣ (normalizedFactors y).prod
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : Field R
p : R[X]
this : p.degree ≤ 0
hc : p.coeff 0 = 0
h : degree 0 = 0
⊢ False

Tactic:
contradiction

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

Goal state before:
pp: K : Type u_1
inst✝ : Ring K
f : K[X]
a : K
⊢ f %ₘ (X - C a) = f - (X - C a) * (f /ₘ (X - C a))

Tactic:
modByMonic_eq_sub_mul_div

Goal state after:
K : Type u_1
inst✝ : Ring K
f : K[X]
a : K
⊢ f - (X - C a) * (f /ₘ (X - C a)) = f - (X - C a) * (f /ₘ (X - C a))
Module: Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet

Goal state before:
pp: case inl
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
a✝ : Nontrivial α
hk0 : ¬k = 0
p : Associates α
hp : Irreducible p
h : a * b = 0
h✝ : a = 0
⊢ False

Tactic:
contradiction

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

Goal state before:
pp: case inr
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
a✝ : Nontrivial α
hk0 : ¬k = 0
p : Associates α
hp : Irreducible p
h : a * b = 0
h✝ : b = 0
⊢ False

Tactic:
contradiction

Goal state after:
Module: Mathlib.RingTheory.Algebraic.Basic

Goal state before:
pp: R : Type u
S : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Nontrivial R
M : Submonoid R
inst✝ : IsLocalization M S
x : S
⊢ IsAlgebraic R x

Tactic:
obtain rfl | hx := eq_or_ne x 0

Goal state after:
case inl
R : Type u
S : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Nontrivial R
M : Submonoid R
inst✝ : IsLocalization M S
⊢ IsAlgebraic R 0

case inr
R : Type u
S : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Nontrivial R
M : Submonoid R
inst✝ : IsLocalization M S
x : S
hx : x ≠ 0
⊢ IsAlgebraic R x
Module: Mathlib.RingTheory.Algebraic.Basic

Goal state before:
pp: R : Type u
S : Type u_1
A : Type v
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Ring A
inst✝⁴ : Algebra R A
inst✝³ : Algebra R S
inst✝² : Algebra S A
inst✝¹ : IsScalarTower R S A
inst✝ : Nontrivial R
⊢ ∀ (x : ↥⊥), IsAlgebraic R x

Tactic:
rintro ⟨_, x, rfl⟩; exact isAlgebraic_algebraMap _

Goal state after:
Module: Mathlib.RingTheory.Algebraic.Basic

Goal state before:
pp: case refine_1
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
A : Subalgebra K L
x : ↥A
p : K[X]
h : 0 ≠ 0
⊢ (aeval x) 0 = 0 → (↑x)⁻¹ ∈ A

Tactic:
contradiction

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

Goal state before:
pp: R : Type u_1
inst✝⁶ : Semiring R
ι : Type u_2
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommMonoid (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
inst✝² : DecidableEq ι
inst✝¹ : Nonempty ι
inst✝ : IsDirectedOrder ι
z✝ : DirectLimit G f
ind : ι
z : DirectSum ι G
p q : DirectSum ι fun i ↦ G i
x✝¹ : ∃ i x, (of R ι G f i) x = Quotient.mk'' p
x✝ : ∃ i x, (of R ι G f i) x = Quotient.mk'' q
i : ι
x : G i
ihx : (of R ι G f i) x = Quotient.mk'' p
j : ι
y : G j
ihy : (of R ι G f j) y = Quotient.mk'' q
k : ι
hik : i ≤ k
hjk : j ≤ k
⊢ (of R ι G f i) x + (of R ι G f j) y = Quotient.mk'' (p + q)

Tactic:
ihx,

Goal state after:
R : Type u_1
inst✝⁶ : Semiring R
ι : Type u_2
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : (i : ι) → AddCommMonoid (G i)
inst✝³ : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
inst✝² : DecidableEq ι
inst✝¹ : Nonempty ι
inst✝ : IsDirectedOrder ι
z✝ : DirectLimit G f
ind : ι
z : DirectSum ι G
p q : DirectSum ι fun i ↦ G i
x✝¹ : ∃ i x, (of R ι G f i) x = Quotient.mk'' p
x✝ : ∃ i x, (of R ι G f i) x = Quotient.mk'' q
i : ι
x : G i
ihx : (of R ι G f i) x = Quotient.mk'' p
j : ι
y : G j
ihy : (of R ι G f j) y = Quotient.mk'' q
k : ι
hik : i ≤ k
hjk : j ≤ k
⊢ Quotient.mk'' p + (of R ι G f j) y = Quotient.mk'' (p + q)
Module: Mathlib.RingTheory.Finiteness.Small

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Small.{u, u_1} R
P : Submodule R M
hP : P.FG
⊢ Small.{u, u_2} ↥P

Tactic:
rw [fg_iff_exists_fin_generating_family] at hP

Goal state after:
R : Type u_1
M : Type u_2
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Small.{u, u_1} R
P : Submodule R M
hP : ∃ n s, span R (Set.range s) = P
⊢ Small.{u, u_2} ↥P
Module: Mathlib.Algebra.Central.TensorProduct

Goal state before:
pp: K : Type u_1
B : Type u_2
C : Type u_3
inst✝⁴ : CommSemiring K
inst✝³ : Semiring B
inst✝² : Semiring C
inst✝¹ : Algebra K B
inst✝ : Algebra K C
c : C
hc0 : ∀ (b : C), b * c = c * b
bc : B ⊗[K] C
⊢ bc * includeRight c = includeRight c * bc

Tactic:
induction bc using TensorProduct.induction_on with
  | zero => simp
  | tmul b c' => simp [hc0]
  | add _ _ _ _ => simp_all [add_mul, mul_add]

Goal state after:
Module: Mathlib.Algebra.Colimit.DirectLimit

Goal state before:
pp: ι : Type u_2
inst✝⁶ : Preorder ι
G : ι → Type u_3
T : ⦃i j : ι⦄ → i ≤ j → Type u_6
f : (x x_1 : ι) → (h : x ≤ x_1) → T h
inst✝⁵ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)
inst✝⁴ : DirectedSystem G fun x1 x2 x3 ↦ ⇑(f x1 x2 x3)
inst✝³ : IsDirectedOrder ι
inst✝² : Nonempty ι
inst✝¹ : (i : ι) → One (G i)
inst✝ : ∀ (i j : ι) (h : i ≤ j), OneHomClass (T h) (G i) (G j)
⊢ map₀ f 1 = 1

Tactic:
rw [map₀, Pi.one_apply, one_def]

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

Goal state before:
pp: ι : Type u_1
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : Module ℚ≥0 M
s : Finset ι
f g : ι → M
t : Finset ι
hst : s = t
h : ∀ i ∈ t, f i = g i
⊢ 𝔼 i ∈ s, f i = 𝔼 i ∈ t, g i

Tactic:
by rw [expect, expect, sum_congr hst h, hst]

Goal state after:
Module: Mathlib.Algebra.BigOperators.Group.Finset.Interval

Goal state before:
pp: case succ
α : Type u_1
inst✝ : CommGroup α
f : ℤ → α
b : ℕ
ihb : ∏ n ∈ Ico (-↑b) ↑b, f n / f (n + 1) = f (-↑b) / f ↑b
⊢ ∏ n ∈ Ico (-↑(b + 1)) ↑(b + 1), f n / f (n + 1) = f (-↑(b + 1)) / f ↑(b + 1)

Tactic:
simp only [Nat.cast_add_one, Ico_succ_succ]

Goal state after:
case succ
α : Type u_1
inst✝ : CommGroup α
f : ℤ → α
b : ℕ
ihb : ∏ n ∈ Ico (-↑b) ↑b, f n / f (n + 1) = f (-↑b) / f ↑b
⊢ ∏ x ∈ Ico (-↑b) ↑b ∪ {-(↑b + 1), ↑b}, f x / f (x + 1) = f (-(↑b + 1)) / f (↑b + 1)
Module: Mathlib.Data.List.Sym

Goal state before:
pp: case swap
α : Type u_1
xs✝ ys : List α
x y : α
xs : List α
⊢ s(y, y) :: s(x, y) :: (List.map (fun y_1 ↦ s(y, y_1)) xs ++ s(x, x) :: (List.map (fun y ↦ s(x, y)) xs ++ xs.sym2)) ~
    s(x, x) :: s(x, y) :: (List.map (fun y ↦ s(x, y)) xs ++ s(y, y) :: (List.map (fun y_1 ↦ s(y, y_1)) xs ++ xs.sym2))

Tactic:
refine Perm.trans (Perm.swap ..) (Perm.trans (Perm.cons _ ?_) (Perm.swap ..))

Goal state after:
case swap
α : Type u_1
xs✝ ys : List α
x y : α
xs : List α
⊢ s(y, y) :: (List.map (fun y_1 ↦ s(y, y_1)) xs ++ s(x, x) :: (List.map (fun y ↦ s(x, y)) xs ++ xs.sym2)) ~
    s(x, x) :: (List.map (fun y ↦ s(x, y)) xs ++ s(y, y) :: (List.map (fun y_1 ↦ s(y, y_1)) xs ++ xs.sym2))
Module: Mathlib.Data.List.Sym

Goal state before:
pp: α : Type u_1
β : Type u_3
f : α → β
n✝ : ℕ
xs : List α
n : ℕ
⊢ List.sym (n + 1) (map f []) = map (Sym.map f) (List.sym (n + 1) [])

Tactic:
by simp [List.sym]

Goal state after:
Module: Mathlib.Data.List.Sym

Goal state before:
pp: α : Type u_1
n : ℕ
xs✝ ys✝ : List α
h : xs✝ <+ ys✝
ys xs : List α
x✝ : xs <+ ys
⊢ List.sym 0 xs <+ List.sym 0 ys

Tactic:
by simp [List.sym]

Goal state after:
Module: Mathlib.Data.List.Sym

Goal state before:
pp: α : Type u_1
n : ℕ
xs x✝ : List α
h : x✝.Nodup
⊢ (List.sym 0 x✝).Nodup

Tactic:
by simp [List.sym]

Goal state after:
Module: Mathlib.Data.List.Sym

Goal state before:
pp: α : Type u_1
n✝ : ℕ
xs : List α
n : ℕ
h : [].Nodup
⊢ (List.sym (n + 1) []).Nodup

Tactic:
by simp [List.sym]

Goal state after:
Module: Mathlib.Data.List.Sym

Goal state before:
pp: α : Type u_1
n✝ : ℕ
xs : List α
n : ℕ
⊢ (List.sym (n + 1) []).length = [].length.multichoose (n + 1)

Tactic:
by simp [List.sym]

Goal state after:
Module: Mathlib.RingTheory.IsTensorProduct

Goal state before:
pp: R : Type u_1
M : Type v₁
N : Type v₂
S : Type v₃
inst✝¹⁹ : AddCommMonoid M
inst✝¹⁸ : AddCommMonoid N
inst✝¹⁷ : CommSemiring R
inst✝¹⁶ : CommSemiring S
inst✝¹⁵ : Algebra R S
inst✝¹⁴ : Module R M
inst✝¹³ : Module R N
inst✝¹² : Module S N
inst✝¹¹ : IsScalarTower R S N
T : Type u_4
O : Type u_5
inst✝¹⁰ : CommSemiring T
inst✝⁹ : Algebra R T
inst✝⁸ : Algebra S T
inst✝⁷ : IsScalarTower R S T
inst✝⁶ : AddCommMonoid O
inst✝⁵ : Module R O
inst✝⁴ : Module S O
inst✝³ : Module T O
inst✝² : IsScalarTower S T O
inst✝¹ : IsScalarTower R S O
inst✝ : IsScalarTower R T O
f : M →ₗ[R] N
hf : IsBaseChange S f
h : N →ₗ[S] O
hc : IsBaseChange T (↑R h ∘ₗ f)
⊢ ∀ (Q : Type (max v₂ u_5 u_4)) [inst : AddCommMonoid Q] [inst_1 : Module S Q] [inst_2 : Module T Q]
    [inst_3 : IsScalarTower S T Q] (g : N →ₗ[S] Q), ∃! g', ↑S g' ∘ₗ h = g

Tactic:
intro Q _ _ _ _ r

Goal state after:
R : Type u_1
M : Type v₁
N : Type v₂
S : Type v₃
inst✝²³ : AddCommMonoid M
inst✝²² : AddCommMonoid N
inst✝²¹ : CommSemiring R
inst✝²⁰ : CommSemiring S
inst✝¹⁹ : Algebra R S
inst✝¹⁸ : Module R M
inst✝¹⁷ : Module R N
inst✝¹⁶ : Module S N
inst✝¹⁵ : IsScalarTower R S N
T : Type u_4
O : Type u_5
inst✝¹⁴ : CommSemiring T
inst✝¹³ : Algebra R T
inst✝¹² : Algebra S T
inst✝¹¹ : IsScalarTower R S T
inst✝¹⁰ : AddCommMonoid O
inst✝⁹ : Module R O
inst✝⁸ : Module S O
inst✝⁷ : Module T O
inst✝⁶ : IsScalarTower S T O
inst✝⁵ : IsScalarTower R S O
inst✝⁴ : IsScalarTower R T O
f : M →ₗ[R] N
hf : IsBaseChange S f
h : N →ₗ[S] O
hc : IsBaseChange T (↑R h ∘ₗ f)
Q : Type (max v₂ u_5 u_4)
inst✝³ : AddCommMonoid Q
inst✝² : Module S Q
inst✝¹ : Module T Q
inst✝ : IsScalarTower S T Q
r : N →ₗ[S] Q
⊢ ∃! g', ↑S g' ∘ₗ h = r
Module: Mathlib.RingTheory.IsTensorProduct

Goal state before:
pp: case refine_2
R : Type u_1
M : Type v₁
N : Type v₂
S : Type v₃
inst✝²³ : AddCommMonoid M
inst✝²² : AddCommMonoid N
inst✝²¹ : CommSemiring R
inst✝²⁰ : CommSemiring S
inst✝¹⁹ : Algebra R S
inst✝¹⁸ : Module R M
inst✝¹⁷ : Module R N
inst✝¹⁶ : Module S N
inst✝¹⁵ : IsScalarTower R S N
T : Type u_4
O : Type u_5
inst✝¹⁴ : CommSemiring T
inst✝¹³ : Algebra R T
inst✝¹² : Algebra S T
inst✝¹¹ : IsScalarTower R S T
inst✝¹⁰ : AddCommMonoid O
inst✝⁹ : Module R O
inst✝⁸ : Module S O
inst✝⁷ : Module T O
inst✝⁶ : IsScalarTower S T O
inst✝⁵ : IsScalarTower R S O
inst✝⁴ : IsScalarTower R T O
f : M →ₗ[R] N
hf : IsBaseChange S f
h : N →ₗ[S] O
hc : IsBaseChange T (↑R h ∘ₗ f)
Q : Type (max v₂ u_5 u_4)
inst✝³ : AddCommMonoid Q
inst✝² : Module S Q
inst✝¹ : Module T Q
inst✝ : IsScalarTower S T Q
r : N →ₗ[S] Q
this✝¹ : Module R Q := Module.restrictScalars R S Q
this✝ : IsScalarTower R S Q
this : IsScalarTower R T Q
r' : M →ₗ[R] Q := ↑R r ∘ₗ f
q : O →ₗ[T] Q := hc.lift r'
q' : O →ₗ[T] Q
hq' : ↑R (↑S q' ∘ₗ h) = ↑R r
⊢ ↑R r ∘ₗ f = ↑R q ∘ₗ ↑R h ∘ₗ f

Tactic:
hc.lift_comp

Goal state after:
case refine_2
R : Type u_1
M : Type v₁
N : Type v₂
S : Type v₃
inst✝²³ : AddCommMonoid M
inst✝²² : AddCommMonoid N
inst✝²¹ : CommSemiring R
inst✝²⁰ : CommSemiring S
inst✝¹⁹ : Algebra R S
inst✝¹⁸ : Module R M
inst✝¹⁷ : Module R N
inst✝¹⁶ : Module S N
inst✝¹⁵ : IsScalarTower R S N
T : Type u_4
O : Type u_5
inst✝¹⁴ : CommSemiring T
inst✝¹³ : Algebra R T
inst✝¹² : Algebra S T
inst✝¹¹ : IsScalarTower R S T
inst✝¹⁰ : AddCommMonoid O
inst✝⁹ : Module R O
inst✝⁸ : Module S O
inst✝⁷ : Module T O
inst✝⁶ : IsScalarTower S T O
inst✝⁵ : IsScalarTower R S O
inst✝⁴ : IsScalarTower R T O
f : M →ₗ[R] N
hf : IsBaseChange S f
h : N →ₗ[S] O
hc : IsBaseChange T (↑R h ∘ₗ f)
Q : Type (max v₂ u_5 u_4)
inst✝³ : AddCommMonoid Q
inst✝² : Module S Q
inst✝¹ : Module T Q
inst✝ : IsScalarTower S T Q
r : N →ₗ[S] Q
this✝¹ : Module R Q := Module.restrictScalars R S Q
this✝ : IsScalarTower R S Q
this : IsScalarTower R T Q
r' : M →ₗ[R] Q := ↑R r ∘ₗ f
q : O →ₗ[T] Q := hc.lift r'
q' : O →ₗ[T] Q
hq' : ↑R (↑S q' ∘ₗ h) = ↑R r
⊢ ↑R r ∘ₗ f = r'
Module: Mathlib.RingTheory.IsTensorProduct

Goal state before:
pp: R : Type u_1
S : Type v₃
inst✝²⁷ : CommSemiring R
inst✝²⁶ : CommSemiring S
inst✝²⁵ : Algebra R S
T : Type u_4
inst✝²⁴ : CommSemiring T
inst✝²³ : Algebra R T
inst✝²² : Algebra S T
inst✝²¹ : IsScalarTower R S T
R' : Type u_6
S' : Type u_7
inst✝²⁰ : CommSemiring R'
inst✝¹⁹ : CommSemiring S'
inst✝¹⁸ : Algebra R R'
inst✝¹⁷ : Algebra S S'
inst✝¹⁶ : Algebra R' S'
inst✝¹⁵ : Algebra R S'
inst✝¹⁴ : IsScalarTower R R' S'
inst✝¹³ : IsScalarTower R S S'
T' : Type u_8
inst✝¹² : CommSemiring T'
inst✝¹¹ : Algebra R T'
inst✝¹⁰ : Algebra S' T'
inst✝⁹ : Algebra S T'
inst✝⁸ : Algebra T T'
inst✝⁷ : Algebra R' T'
inst✝⁶ : IsScalarTower R T T'
inst✝⁵ : IsScalarTower S T T'
inst✝⁴ : IsScalarTower S S' T'
inst✝³ : IsScalarTower R R' T'
inst✝² : IsScalarTower R S' T'
inst✝¹ : IsScalarTower R' S' T'
inst✝ : IsPushout R S R' S'
f : R' →ₗ[R] S' := (toAlgHom R R' S').toLinearMap
x : R
⊢ (algebraMap S' T') ((algebraMap S S') ((algebraMap R S) x)) = (algebraMap S T') ((algebraMap R S) x)

Tactic:
← algebraMap_apply S S' T'

Goal state after:
R : Type u_1
S : Type v₃
inst✝²⁷ : CommSemiring R
inst✝²⁶ : CommSemiring S
inst✝²⁵ : Algebra R S
T : Type u_4
inst✝²⁴ : CommSemiring T
inst✝²³ : Algebra R T
inst✝²² : Algebra S T
inst✝²¹ : IsScalarTower R S T
R' : Type u_6
S' : Type u_7
inst✝²⁰ : CommSemiring R'
inst✝¹⁹ : CommSemiring S'
inst✝¹⁸ : Algebra R R'
inst✝¹⁷ : Algebra S S'
inst✝¹⁶ : Algebra R' S'
inst✝¹⁵ : Algebra R S'
inst✝¹⁴ : IsScalarTower R R' S'
inst✝¹³ : IsScalarTower R S S'
T' : Type u_8
inst✝¹² : CommSemiring T'
inst✝¹¹ : Algebra R T'
inst✝¹⁰ : Algebra S' T'
inst✝⁹ : Algebra S T'
inst✝⁸ : Algebra T T'
inst✝⁷ : Algebra R' T'
inst✝⁶ : IsScalarTower R T T'
inst✝⁵ : IsScalarTower S T T'
inst✝⁴ : IsScalarTower S S' T'
inst✝³ : IsScalarTower R R' T'
inst✝² : IsScalarTower R S' T'
inst✝¹ : IsScalarTower R' S' T'
inst✝ : IsPushout R S R' S'
f : R' →ₗ[R] S' := (toAlgHom R R' S').toLinearMap
x : R
⊢ (algebraMap S T') ((algebraMap R S) x) = (algebraMap S T') ((algebraMap R S) x)
Module: Mathlib.CategoryTheory.ObjectProperty.Basic

Goal state before:
pp: C : Type u
inst✝ : CategoryStruct.{v, u} C
ι : Type u'
X : ι → C
Y : C
⊢ ofObj X Y ↔ ∃ i, X i = Y

Tactic:
by
  constructor
  · rintro ⟨i⟩
    exact ⟨i, rfl⟩
  · rintro ⟨i, rfl⟩
    exact ⟨i⟩

Goal state after:
Module: Mathlib.Combinatorics.Quiver.Symmetric

Goal state before:
pp: V : Type u_2
inst✝ : Quiver V
h : HasInvolutiveReverse V
a b : V
f : a ⟶ b
g : b ⟶ a
⊢ f = reverse g ↔ reverse f = g

Tactic:
rw [← reverse_inj, reverse_reverse]

Goal state after:
Module: Mathlib.Combinatorics.Quiver.Path

Goal state before:
pp: case refine_1
V : Type u_1
inst✝ : Quiver V
a b : V
p : Path a b
h : p.length ≠ 0
⊢ ∃ c p' e, p = p'.cons e

Tactic:
cases p with
    | nil => simp at h
    | cons p' e => exact ⟨_, p', e, rfl⟩

Goal state after:
Module: Mathlib.CategoryTheory.Functor.ReflectsIso.Basic

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
D : Type u_2
inst✝³ : Category.{v_2, u_2} D
E : Type u_3
inst✝² : Category.{v_3, u_3} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : F.ReflectsIsomorphisms
inst✝ : G.ReflectsIsomorphisms
A✝ B✝ : C
f : A✝ ⟶ B✝
hf : IsIso (G.map (F.map f))
⊢ IsIso f

Tactic:
have := isIso_of_reflects_iso (F.map f) G

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
D : Type u_2
inst✝³ : Category.{v_2, u_2} D
E : Type u_3
inst✝² : Category.{v_3, u_3} E
F : C ⥤ D
G : D ⥤ E
inst✝¹ : F.ReflectsIsomorphisms
inst✝ : G.ReflectsIsomorphisms
A✝ B✝ : C
f : A✝ ⟶ B✝
hf : IsIso (G.map (F.map f))
this : IsIso (F.map f)
⊢ IsIso f