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

Goal state before:
pp: R : Type u
inst✝ : Semiring R
P : R[X] → Prop
p✝ : R[X]
h0✝ : 0 < p✝.degree
hC : ∀ {a : R}, a ≠ 0 → P (C a * X)
hX : ∀ {p : R[X]}, 0 < p.degree → P p → P (p * X)
hadd : ∀ {p : R[X]} {a : R}, 0 < p.degree → P p → P (p + C a)
p : R[X]
a : R
heq0 : p.coeff 0 = 0
x✝ : a ≠ 0
ih : 0 < p.degree → P p
h0 : 0 < (p + C a).degree
h : p.degree ≤ 0
⊢ 0 < (C a).degree

Tactic:
by rwa [eq_C_of_degree_le_zero h, ← C_add, heq0, zero_add] at h0

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

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
p : ℕ
f : R[X]
⊢ derivative ((expand R p) f) = (expand R p) (derivative f) * (↑p * X ^ (p - 1))

Tactic:
by
  rw [coe_expand, derivative_eval₂_C, derivative_pow, C_eq_natCast, derivative_X, mul_one]

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

Goal state before:
pp: case pos
R : Type u
inst✝¹ : CommSemiring R
S : Type v
inst✝ : CommSemiring S
p : ℕ
f : R →+* S
q : R[X]
hp : p = 0
⊢ map f ((expand R p) q) = (expand S p) (map f q)

Tactic:
simp [hp]

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

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p q : R[X]
x : R
hpq : p * q ≠ 0
hp : p ≠ 0
hq : q ≠ 0
⊢ multiplicity (X - C x) (p * q) = multiplicity (X - C x) p + rootMultiplicity x q

Tactic:
rootMultiplicity_eq_multiplicity q,

Goal state after:
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p q : R[X]
x : R
hpq : p * q ≠ 0
hp : p ≠ 0
hq : q ≠ 0
⊢ multiplicity (X - C x) (p * q) = multiplicity (X - C x) p + if q = 0 then 0 else multiplicity (X - C x) q
Module: Mathlib.Algebra.Polynomial.RingDivision

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
p : R[X]
hp : p ≠ 0
this : Decidable (∃ x, p.IsRoot x)
h : ∃ x, p.IsRoot x
x : R
hx : p.IsRoot x
hpd : 0 < p.degree
hd0 : p /ₘ (X - C x) ≠ 0
wf : (p /ₘ (X - C x)).degree < p.degree
t : Multiset R
htd : ↑t.card ≤ (p /ₘ (X - C x)).degree
htr : ∀ (a : R), count a t = rootMultiplicity a (p /ₘ (X - C x))
hdeg : (X - C x).degree ≤ p.degree
hdiv0 : p /ₘ (X - C x) ≠ 0
⊢ ↑t.card + 1 ≤ p.degree

Tactic:
rw [← degree_add_divByMonic (monic_X_sub_C x) hdeg, degree_X_sub_C, add_comm]

Goal state after:
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
p : R[X]
hp : p ≠ 0
this : Decidable (∃ x, p.IsRoot x)
h : ∃ x, p.IsRoot x
x : R
hx : p.IsRoot x
hpd : 0 < p.degree
hd0 : p /ₘ (X - C x) ≠ 0
wf : (p /ₘ (X - C x)).degree < p.degree
t : Multiset R
htd : ↑t.card ≤ (p /ₘ (X - C x)).degree
htr : ∀ (a : R), count a t = rootMultiplicity a (p /ₘ (X - C x))
hdeg : (X - C x).degree ≤ p.degree
hdiv0 : p /ₘ (X - C x) ≠ 0
⊢ 1 + ↑t.card ≤ 1 + (p /ₘ (X - C x)).degree
Module: Mathlib.Algebra.Polynomial.Expand

Goal state before:
pp: case succ
R : Type u
inst✝¹ : CommSemiring R
p : ℕ
inst✝ : ExpChar R p
f : R[X]
k : ℕ
n_ih : map (iterateFrobenius R p k) ((expand R (p ^ k)) f) = f ^ p ^ k
⊢ map (iterateFrobenius R p k) ((expand R (p ^ k)) f) ^ p =
    map (iterateFrobenius R p (k + 1)) ((expand R (p ^ (k + 1))) f)

Tactic:
← map_frobenius_expand p,

Goal state after:
case succ
R : Type u
inst✝¹ : CommSemiring R
p : ℕ
inst✝ : ExpChar R p
f : R[X]
k : ℕ
n_ih : map (iterateFrobenius R p k) ((expand R (p ^ k)) f) = f ^ p ^ k
⊢ map (frobenius R p) ((expand R p) (map (iterateFrobenius R p k) ((expand R (p ^ k)) f))) =
    map (iterateFrobenius R p (k + 1)) ((expand R (p ^ (k + 1))) f)
Module: Mathlib.Algebra.Polynomial.Div

Goal state before:
pp: R : Type u
inst✝ : Ring R
f g q r : R[X]
hg : g.Monic
h : r + g * q = f ∧ r.degree < g.degree
a✝ : Nontrivial R
⊢ r - f %ₘ g - -g * (q - f /ₘ g) = 0

Tactic:
rw [← sub_eq_zero_of_eq (h.1.trans (modByMonic_add_div f g).symm)]
        simp [mul_add, sub_eq_add_neg, add_comm, add_left_comm, add_assoc]

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

Goal state before:
pp: R : Type u
inst✝ : Ring R
p q : R[X]
hq : q.Monic
h : p %ₘ q = 0
⊢ q ∣ q * (p /ₘ q)

Tactic:
exact dvd_mul_right _ _

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

Goal state before:
pp: R : Type u
inst✝ : Ring R
p : R[X]
a : R
h0 : p ≠ 0
this : Nontrivial R
⊢ FiniteMultiplicity (X - C a) p

Tactic:
refine finiteMultiplicity_of_degree_pos_of_monic ?_ (monic_X_sub_C _) h0

Goal state after:
R : Type u
inst✝ : Ring R
p : R[X]
a : R
h0 : p ≠ 0
this : Nontrivial R
⊢ 0 < (X - C a).degree
Module: Mathlib.Algebra.Polynomial.Derivative

Goal state before:
pp: case pos
R : Type u
inst✝¹ : Semiring R
inst✝ : IsAddTorsionFree R
p : R[X]
hp : p.natDegree = 0
⊢ (derivative p).leadingCoeff = p.leadingCoeff * ↑p.natDegree

Tactic:
simp [hp]

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

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
f : R[X]
n : ℕ
c : R
m : ℕ
hm : m.succ ≠ 0
⊢ ↑n ∣ ↑n * eval c ((⇑derivative)^[m] (f ^ (n - 1) * derivative f))

Tactic:
exact dvd_mul_right _ _

Goal state after:
Module: Mathlib.Order.Filter.Cofinite

Goal state before:
pp: α : Type u_2
s t : Set α
ht : t.Finite
⊢ cofinite ⊓ 𝓟 (s \ t) = cofinite ⊓ 𝓟 s

Tactic:
rw [sdiff_eq, ← inf_principal, ← inf_assoc, inf_right_comm, ht.cofinite_inf_principal_compl]

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

Goal state before:
pp: R : Type u
inst✝ : CommRing R
k : ℕ
ind : ∀ {P : R[X]}, P.degree < ↑k → (⇑derivative)^[k] P = 0
P : R[X]
h : P.degree < ↑(k + 1)
⊢ (⇑derivative)^[k + 1] P = 0

Tactic:
by_cases P = 0
    case pos hP => simp [hP]
    case neg hP =>
      rw [Function.iterate_add_apply, Function.iterate_one]
      by_cases derivative P = 0
      case pos hP' => simp [hP']
      case neg hP' =>
        have hP'' : P.natDegree ≠ 0 := by
          contrapose hP'
          exact derivative_of_natDegree_zero hP'
        refine ind <| (natDegree_lt_iff_degree_lt hP').mp ?_
        linarith [(natDegree_lt_iff_degree_lt hP).mpr h, natDegree_derivative_lt hP'']

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

Goal state before:
pp: case pos
R : Type u
a : R
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
p : R[X]
hp : p = 0
⊢ count a p.roots = rootMultiplicity a p

Tactic:
simp [hp]

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]
hp : p ≠ 0
⊢ {x | p.IsRoot x}.Finite

Tactic:
by
  classical
  simpa only [← Finset.setOfPred_mem, Multiset.mem_toFinset, mem_roots hp]
    using p.roots.toFinset.finite_toSet

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

Goal state before:
pp: case h
σ : Type u_1
inst✝ : Finite σ
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
n : ℕ
fg : Finset ((σ →₀ ℕ) × (σ →₀ ℕ)) := ⋯
hfg : fg = Finset.antidiagonal (equivFunOnFinite.symm (Function.const σ n))
d : σ →₀ ℕ
hd : d ∈ {d | (weight w) d ≤ n}
⊢ d + (equivFunOnFinite.symm (Function.const σ n) - d) = equivFunOnFinite.symm (Function.const σ n)

Tactic:
ext x

Goal state after:
case h
σ : Type u_1
inst✝ : Finite σ
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
n : ℕ
fg : Finset ((σ →₀ ℕ) × (σ →₀ ℕ)) := Finset.antidiagonal (equivFunOnFinite.symm (Function.const σ n))
hfg : fg = Finset.antidiagonal (equivFunOnFinite.symm (Function.const σ n))
d : σ →₀ ℕ
hd : d ∈ {d | (weight w) d ≤ n}
x : σ
⊢ (d + (equivFunOnFinite.symm (Function.const σ n) - d)) x = (equivFunOnFinite.symm (Function.const σ n)) x
Module: Mathlib.RingTheory.UniqueFactorizationDomain.NormalizedFactors

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
h : normalizedFactors x = normalizedFactors y
⊢ (normalizedFactors x).prod ~ᵤ (normalizedFactors y).prod

Tactic:
by rw [h]

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

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
x : Associates α
hx : ¬x = 0
⊢ ⇑Associates.mkMonoidHom ∘ Classical.choose ⋯ = id

Tactic:
ext x

Goal state after:
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
x✝ : Associates α
hx : ¬x✝ = 0
x : Associates α
⊢ (⇑Associates.mkMonoidHom ∘ Classical.choose ⋯) x = id x
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Basic

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : IsCancelMulZero α
pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a
a b : α
ane0 : a ≠ 0
c : α
hc : ¬IsUnit c
b_eq : b = a * c
h : ¬b = 0
cne0 : c ≠ 0
⊢ (Classical.choose ⋯).prod * (Classical.choose ⋯).prod ~ᵤ a * c

Tactic:
apply Associated.mul_mul <;> apply (Classical.choose_spec (pf _ _)).2 <;> assumption

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

Goal state before:
pp: S : Type v
T : Type w
inst✝³ : CommRing T
inst✝² : CommRing S
inst✝¹ : IsDomain S
inst✝ : Algebra T S
p : T[X]
n : ℕ
⊢ (p ^ n).aroots S = n • p.aroots S

Tactic:
by
  rw [aroots_def, Polynomial.map_pow, roots_pow]

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

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
h : 0 < n
a : R
⊢ ↑(nthRootsFinset n a) = {r | r ^ n = a}

Tactic:
ext x

Goal state after:
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
h : 0 < n
a x : R
⊢ x ∈ ↑(nthRootsFinset n a) ↔ x ∈ {r | r ^ n = a}
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝ : CommRing R
p : R[X]
t : R
m : ℕ := rootMultiplicity t p
hm : m = rootMultiplicity t p
⊢ eval t (m.choose 0 • ((⇑derivative)^[m - 0] ((X - C t) ^ m) * (⇑derivative)^[0] (p /ₘ (X - C t) ^ m))) =
    m ! • eval t (p /ₘ (X - C t) ^ m)

Tactic:
m.choose_zero_right,

Goal state after:
R : Type u
inst✝ : CommRing R
p : R[X]
t : R
m : ℕ := rootMultiplicity t p
hm : m = rootMultiplicity t p
⊢ eval t (1 • ((⇑derivative)^[m - 0] ((X - C t) ^ m) * (⇑derivative)^[0] (p /ₘ (X - C t) ^ m))) =
    m ! • eval t (p /ₘ (X - C t) ^ m)
Module: Mathlib.Algebra.Polynomial.Roots

Goal state before:
pp: case inl
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R[X]
hirr : Irreducible p
hp : p.roots = 0
⊢ p.roots.card ≤ 1

Tactic:
simp [hp]

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

Goal state before:
pp: case neg.inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
p✝ q✝ : R[X]
a✝ : Nontrivial R
n : ℕ
ih : ∀ (p q : R[X]), (p * q).degree < ↑n → Associated (p * q).content (p.content * q.content)
p q : R[X]
p0 : ¬p = 0
q0 : ¬q = 0
heq : p.primPart.degree + q.primPart.degree = ↑n
⊢ Associated (p * q).content (p.content * q.content)

Tactic:
rw [p.eq_C_content_mul_primPart, q.eq_C_content_mul_primPart]

Goal state after:
case neg.inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
p✝ q✝ : R[X]
a✝ : Nontrivial R
n : ℕ
ih : ∀ (p q : R[X]), (p * q).degree < ↑n → Associated (p * q).content (p.content * q.content)
p q : R[X]
p0 : ¬p = 0
q0 : ¬q = 0
heq : p.primPart.degree + q.primPart.degree = ↑n
⊢ Associated (C p.content * p.primPart * (C q.content * q.primPart)).content
    ((C p.content * p.primPart).content * (C q.content * q.primPart).content)
Module: Mathlib.RingTheory.Polynomial.Content

Goal state before:
pp: case mpr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hq : q ≠ 0
h₁ : p.content ∣ q.content
h₂ : p.primPart ∣ q.primPart
⊢ p ∣ q

Tactic:
rw [p.eq_C_content_mul_primPart, q.eq_C_content_mul_primPart]

Goal state after:
case mpr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hq : q ≠ 0
h₁ : p.content ∣ q.content
h₂ : p.primPart ∣ q.primPart
⊢ C p.content * p.primPart ∣ C q.content * q.primPart
Module: Mathlib.Algebra.Polynomial.FieldDivision

Goal state before:
pp: R : Type u
inst✝¹ : Field R
p q : R[X]
inst✝ : DecidableEq R
hq : q ≠ 0
⊢ p ∈ normalizedFactors q ↔ Irreducible p ∧ p.Monic ∧ p ∣ q

Tactic:
by_cases hp : p = 0
  · simpa [hp] using zero_notMem_normalizedFactors _
  · rw [mem_normalizedFactors_iff' hq, normalize_eq_self_iff_monic hp]

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

Goal state before:
pp: case pos
R : Type u
S : Type v
inst✝³ : Field R
p : R[X]
inst✝² : DecidableEq R
inst✝¹ : Field S
inst✝ : DecidableEq S
f : R →+* S
hp : p = 0
⊢ map f (normalize p) = normalize (map f p)

Tactic:
simp [hp]

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

Goal state before:
pp: case neg.inr
α : Type u_1
inst✝³ : CommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
k : ℕ
habk : k ∣ p.count (a * b).factors
ha : ¬a = 0
h : p.count b.factors = 0
⊢ k ∣ p.count a.factors

Tactic:
· rw [count_mul ha hb hp, h] at habk
    exact habk

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

Goal state before:
pp: K : Type u
inst✝² : CommSemiring K
D : Type v
inst✝¹ : Semiring D
inst✝ : Algebra K D
h : IsCentral K D
x : D
⊢ x ∈ ⊥ ↔ ∃ a, x = (algebraMap K D) a

Tactic:
Algebra.mem_bot

Goal state after:
K : Type u
inst✝² : CommSemiring K
D : Type v
inst✝¹ : Semiring D
inst✝ : Algebra K D
h : IsCentral K D
x : D
⊢ x ∈ Set.range ⇑(algebraMap K D) ↔ ∃ a, x = (algebraMap K D) a
Module: Mathlib.RingTheory.Algebraic.Basic

Goal state before:
pp: R : Type u
A : Type v
inst✝³ : CommRing R
inst✝² : Ring A
inst✝¹ : Algebra R A
inst✝ : Nontrivial R
⊢ IsAlgebraic R ((algebraMap R A) 1)

Tactic:
exact isAlgebraic_algebraMap 1

Goal state after:
Module: Mathlib.Order.DirectedInverseSystem

Goal state before:
pp: ι : Type u_1
inst✝⁷ : Preorder ι
F₁ : ι → Type u_2
F₂ : ι → Type u_3
F : ι → Type u_4
X : ι → Type u_5
T₁ : ⦃i j : ι⦄ → i ≤ j → Sort u_6
f₁ : (i j : ι) → (h : i ≤ j) → T₁ h
inst✝⁶ : ⦃i j : ι⦄ → (h : i ≤ j) → FunLike (T₁ h) (F₁ i) (F₁ j)
inst✝⁵ : DirectedSystem F₁ fun x1 x2 x3 ↦ ⇑(f₁ x1 x2 x3)
T₂ : ⦃i j : ι⦄ → i ≤ j → Sort u_7
f₂ : (i j : ι) → (h : i ≤ j) → T₂ h
inst✝⁴ : ⦃i j : ι⦄ → (h : i ≤ j) → FunLike (T₂ h) (F₂ i) (F₂ j)
inst✝³ : DirectedSystem F₂ fun x1 x2 x3 ↦ ⇑(f₂ x1 x2 x3)
T : ⦃i j : ι⦄ → i ≤ j → Sort u_8
f : (i j : ι) → (h : i ≤ j) → T h
inst✝² : ⦃i j : ι⦄ → (h : i ≤ j) → FunLike (T h) (F i) (F j)
inst✝¹ : DirectedSystem F fun x1 x2 x3 ↦ ⇑(f x1 x2 x3)
inst✝ : IsDirectedOrder ι
C : Sort u_9
ih : (i : ι) → F i → C
compat : ∀ (i j : ι) (h : i ≤ j) (x : F i), ih i x = ih j ((f i j h) x)
z : DirectLimit F f
x y : (i : ι) × F i
x✝ : x ≈ y
k : ι
hxk : x.fst ≤ k
hyk : y.fst ≤ k
eq : (f x.fst k hxk) x.snd = (f y.fst k hyk) y.snd
⊢ ⋯ ▸ ih x.fst x.snd = ih y.fst y.snd

Tactic:
simp_rw [eq_rec_constant, compat _ _ hxk, compat _ _ hyk, eq]

Goal state after:
Module: Mathlib.Order.DirectedInverseSystem

Goal state before:
pp: case inr
ι : Type u_6
F : ι → Type u_7
X : ι → Type u_8
i : ι
inst✝¹ : LinearOrder ι
f : ⦃i j : ι⦄ → i ≤ j → F j → F i
equiv : (j : ↑(Iio i)) → F ↑j ≃ piLT X ↑j
nat : IsNatEquiv f equiv
equivLim : F i ≃ ↑(limit f i)
hi : IsSuccPrelimit i
inst✝ : InverseSystem f
H : ∀ (x : F i) (l : ↑(Iio i)), ↑(equivLim x) l = f ⋯ x
j k : ι
hj✝ : j ∈ Iic i
hk : k ∈ Iic i
h : k ≤ j
t : F j
hj : j < i
⊢ (piEquivLim nat equivLim hi ⟨k, hk⟩) (f h t) = piLTProj h ((piEquivLim nat equivLim hi ⟨j, hj✝⟩) t)

Tactic:
· rw [piEquivLim, piSplitLE_lt (h.trans_lt hj), piSplitLE_lt hj]; apply nat

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.RightExactness

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup N
inst✝⁵ : AddCommGroup P
inst✝⁴ : Module R M
inst✝³ : Module R N
inst✝² : Module R P
f : M →ₗ[R] N
g : N →ₗ[R] P
Q : Type u_5
inst✝¹ : AddCommGroup Q
inst✝ : Module R Q
hfg : Exact ⇑f ⇑g
hg : Surjective ⇑g
h : P → N
hgh : RightInverse h ⇑g
y : N ⊗[R] Q
⊢ (inverse_of_rightInverse Q hfg hgh) (((rTensor Q f).range.liftQ (rTensor Q g) ⋯) ((rTensor Q f).range.mkQ y)) =
    (rTensor Q f).range.mkQ y

Tactic:
simp only [Submodule.mkQ_apply, Submodule.liftQ_apply, rTensor.inverse_of_rightInverse_apply]

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

Goal state before:
pp: case inl
ι : Type u_1
G : Type u_4
inst✝² : Fintype ι
inst✝¹ : AddCommGroup G
inst✝ : Module ℚ≥0 G
f : ι → G
h✝ : IsEmpty ι
⊢ balance (balance f) = balance f

Tactic:
ext x

Goal state after:
case inl
ι : Type u_1
G : Type u_4
inst✝² : Fintype ι
inst✝¹ : AddCommGroup G
inst✝ : Module ℚ≥0 G
f : ι → G
h✝ : IsEmpty ι
x : ι
⊢ balance (balance f) x = balance f x
Module: Mathlib.Algebra.BigOperators.Balance

Goal state before:
pp: case inr
ι : Type u_1
G : Type u_4
inst✝² : Fintype ι
inst✝¹ : AddCommGroup G
inst✝ : Module ℚ≥0 G
f : ι → G
h✝ : Nonempty ι
⊢ balance (balance f) = balance f

Tactic:
ext x

Goal state after:
case inr
ι : Type u_1
G : Type u_4
inst✝² : Fintype ι
inst✝¹ : AddCommGroup G
inst✝ : Module ℚ≥0 G
f : ι → G
h✝ : Nonempty ι
x : ι
⊢ balance (balance f) x = balance f x
Module: Mathlib.Algebra.Colimit.DirectLimit

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝¹⁰ : 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✝⁹ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)
inst✝⁸ : DirectedSystem G fun x1 x2 x3 ↦ ⇑(f x1 x2 x3)
inst✝⁷ : IsDirectedOrder ι
inst✝⁶ : Semiring R
inst✝⁵ : (i : ι) → AddCommMonoid (G i)
inst✝⁴ : (i : ι) → Module R (G i)
inst✝³ : ∀ (i j : ι) (h : i ≤ j), LinearMapClass (T h) R (G i) (G j)
inst✝² : Nonempty ι
P : Type u_7
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
g₁ g₂ : DirectLimit G f →ₗ[R] P
h : ∀ (i : ι), g₁ ∘ₗ of R ι G f i = g₂ ∘ₗ of R ι G f i
⊢ g₁ = g₂

Tactic:
ext x

Goal state after:
R : Type u_1
ι : Type u_2
inst✝¹⁰ : 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✝⁹ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)
inst✝⁸ : DirectedSystem G fun x1 x2 x3 ↦ ⇑(f x1 x2 x3)
inst✝⁷ : IsDirectedOrder ι
inst✝⁶ : Semiring R
inst✝⁵ : (i : ι) → AddCommMonoid (G i)
inst✝⁴ : (i : ι) → Module R (G i)
inst✝³ : ∀ (i j : ι) (h : i ≤ j), LinearMapClass (T h) R (G i) (G j)
inst✝² : Nonempty ι
P : Type u_7
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
g₁ g₂ : DirectLimit G f →ₗ[R] P
h : ∀ (i : ι), g₁ ∘ₗ of R ι G f i = g₂ ∘ₗ of R ι G f i
x : DirectLimit G f
⊢ g₁ x = g₂ x
Module: Mathlib.Algebra.Colimit.DirectLimit

Goal state before:
pp: ι : Type u_2
inst✝⁷ : 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✝⁶ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)
inst✝⁵ : DirectedSystem G fun x1 x2 x3 ↦ ⇑(f x1 x2 x3)
inst✝⁴ : IsDirectedOrder ι
inst✝³ : (i : ι) → NonUnitalNonAssocSemiring (G i)
inst✝² : ∀ (i j : ι) (h : i ≤ j), NonUnitalRingHomClass (T h) (G i) (G j)
inst✝¹ : Nonempty ι
P : Type u_7
inst✝ : NonUnitalNonAssocSemiring P
g₁ g₂ : DirectLimit G f →ₙ+* P
h : ∀ (i : ι), g₁.comp (of G f i) = g₂.comp (of G f i)
⊢ g₁ = g₂

Tactic:
ext x

Goal state after:
ι : Type u_2
inst✝⁷ : 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✝⁶ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)
inst✝⁵ : DirectedSystem G fun x1 x2 x3 ↦ ⇑(f x1 x2 x3)
inst✝⁴ : IsDirectedOrder ι
inst✝³ : (i : ι) → NonUnitalNonAssocSemiring (G i)
inst✝² : ∀ (i j : ι) (h : i ≤ j), NonUnitalRingHomClass (T h) (G i) (G j)
inst✝¹ : Nonempty ι
P : Type u_7
inst✝ : NonUnitalNonAssocSemiring P
g₁ g₂ : DirectLimit G f →ₙ+* P
h : ∀ (i : ι), g₁.comp (of G f i) = g₂.comp (of G f i)
x : DirectLimit G f
⊢ g₁ x = g₂ x
Module: Mathlib.Algebra.Colimit.DirectLimit

Goal state before:
pp: ι : Type u_2
inst✝⁷ : 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✝⁶ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)
inst✝⁵ : DirectedSystem G fun x1 x2 x3 ↦ ⇑(f x1 x2 x3)
inst✝⁴ : IsDirectedOrder ι
inst✝³ : (i : ι) → NonAssocSemiring (G i)
inst✝² : ∀ (i j : ι) (h : i ≤ j), RingHomClass (T h) (G i) (G j)
inst✝¹ : Nonempty ι
P : Type u_7
inst✝ : NonAssocSemiring P
g₁ g₂ : DirectLimit G f →+* P
h : ∀ (i : ι), g₁.comp (of G f i) = g₂.comp (of G f i)
⊢ g₁ = g₂

Tactic:
ext x

Goal state after:
ι : Type u_2
inst✝⁷ : 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✝⁶ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)
inst✝⁵ : DirectedSystem G fun x1 x2 x3 ↦ ⇑(f x1 x2 x3)
inst✝⁴ : IsDirectedOrder ι
inst✝³ : (i : ι) → NonAssocSemiring (G i)
inst✝² : ∀ (i j : ι) (h : i ≤ j), RingHomClass (T h) (G i) (G j)
inst✝¹ : Nonempty ι
P : Type u_7
inst✝ : NonAssocSemiring P
g₁ g₂ : DirectLimit G f →+* P
h : ∀ (i : ι), g₁.comp (of G f i) = g₂.comp (of G f i)
x : DirectLimit G f
⊢ g₁ x = g₂ x
Module: Mathlib.Algebra.Colimit.DirectLimit

Goal state before:
pp: ι : Type u_2
inst✝¹⁰ : 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✝⁹ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)
inst✝⁸ : DirectedSystem G fun x1 x2 x3 ↦ ⇑(f x1 x2 x3)
inst✝⁷ : IsDirectedOrder ι
inst✝⁶ : (i : ι) → NonUnitalNonAssocSemiring (G i)
inst✝⁵ : ∀ (i j : ι) (h : i ≤ j), NonUnitalRingHomClass (T h) (G i) (G j)
inst✝⁴ : (i : ι) → StarRing (G i)
inst✝³ : ∀ (i j : ι) (h : i ≤ j), StarHomClass (T h) (G i) (G j)
inst✝² : Nonempty ι
P : Type u_7
inst✝¹ : NonUnitalNonAssocSemiring P
inst✝ : StarRing P
g₁ g₂ : DirectLimit G f →⋆ₙ+* P
h : ∀ (i : ι), g₁.comp (of G f i) = g₂.comp (of G f i)
⊢ g₁ = g₂

Tactic:
ext x

Goal state after:
ι : Type u_2
inst✝¹⁰ : 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✝⁹ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)
inst✝⁸ : DirectedSystem G fun x1 x2 x3 ↦ ⇑(f x1 x2 x3)
inst✝⁷ : IsDirectedOrder ι
inst✝⁶ : (i : ι) → NonUnitalNonAssocSemiring (G i)
inst✝⁵ : ∀ (i j : ι) (h : i ≤ j), NonUnitalRingHomClass (T h) (G i) (G j)
inst✝⁴ : (i : ι) → StarRing (G i)
inst✝³ : ∀ (i j : ι) (h : i ≤ j), StarHomClass (T h) (G i) (G j)
inst✝² : Nonempty ι
P : Type u_7
inst✝¹ : NonUnitalNonAssocSemiring P
inst✝ : StarRing P
g₁ g₂ : DirectLimit G f →⋆ₙ+* P
h : ∀ (i : ι), g₁.comp (of G f i) = g₂.comp (of G f i)
x : DirectLimit G f
⊢ g₁ x = g₂ x
Module: Mathlib.Algebra.Colimit.DirectLimit

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝¹⁰ : 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✝⁹ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)
inst✝⁸ : DirectedSystem G fun x1 x2 x3 ↦ ⇑(f x1 x2 x3)
inst✝⁷ : IsDirectedOrder ι
inst✝⁶ : CommSemiring R
inst✝⁵ : (i : ι) → Semiring (G i)
inst✝⁴ : (i : ι) → Algebra R (G i)
inst✝³ : ∀ (i j : ι) (h : i ≤ j), AlgHomClass (T h) R (G i) (G j)
inst✝² : Nonempty ι
P : Type u_7
inst✝¹ : Semiring P
inst✝ : Algebra R P
g₁ g₂ : DirectLimit G f →ₐ[R] P
h : ∀ (i : ι), g₁.comp (of G f i) = g₂.comp (of G f i)
⊢ g₁ = g₂

Tactic:
ext x

Goal state after:
R : Type u_1
ι : Type u_2
inst✝¹⁰ : 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✝⁹ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)
inst✝⁸ : DirectedSystem G fun x1 x2 x3 ↦ ⇑(f x1 x2 x3)
inst✝⁷ : IsDirectedOrder ι
inst✝⁶ : CommSemiring R
inst✝⁵ : (i : ι) → Semiring (G i)
inst✝⁴ : (i : ι) → Algebra R (G i)
inst✝³ : ∀ (i j : ι) (h : i ≤ j), AlgHomClass (T h) R (G i) (G j)
inst✝² : Nonempty ι
P : Type u_7
inst✝¹ : Semiring P
inst✝ : Algebra R P
g₁ g₂ : DirectLimit G f →ₐ[R] P
h : ∀ (i : ι), g₁.comp (of G f i) = g₂.comp (of G f i)
x : DirectLimit G f
⊢ g₁ x = g₂ x