mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P Q : Fin 3 → A
⊢ (W'⁄B).addY (⇑↑f ∘ P) (⇑↑f ∘ Q) = ↑f ((W'⁄A).addY P Q)

Tactic:
← map_addY,

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P Q : Fin 3 → A
⊢ (W'⁄B).addY (⇑↑f ∘ P) (⇑↑f ∘ Q) = ((W'⁄A).map ↑f).addY (⇑↑f ∘ P) (⇑↑f ∘ Q)
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
k : ℕ
φ ψ : R⟦X⟧
h : φ * X ^ k = ψ * X ^ k
⊢ φ = ψ

Tactic:
by
  rw [PowerSeries.ext_iff] at h ⊢
  intro n
  simpa using h (n + k)

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

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
k : ℕ
φ ψ : R⟦X⟧
h : X ^ k * φ = X ^ k * ψ
⊢ φ = ψ

Tactic:
by
  rw [PowerSeries.ext_iff] at h ⊢
  intro n
  simpa using h (n + k)

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

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff m) φ = 0
m : σ →₀ ℕ
H : m - single s n + single s n = m
⊢ (coeff m) φ = (coeff m) (X s ^ n * fun m ↦ (coeff (m + single s n)) φ)

Tactic:
rw [coeff_mul, Finset.sum_eq_single (single s n, m - single s n)]

Goal state after:
case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff m) φ = 0
m : σ →₀ ℕ
H : m - single s n + single s n = m
⊢ (coeff m) φ =
    (coeff (single s n, m - single s n).1) (X s ^ n) *
      (coeff (single s n, m - single s n).2) fun m ↦ (coeff (m + single s n)) φ

case pos.h₀
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff m) φ = 0
m : σ →₀ ℕ
H : m - single s n + single s n = m
⊢ ∀ b ∈ antidiagonal m,
    b ≠ (single s n, m - single s n) → ((coeff b.1) (X s ^ n) * (coeff b.2) fun m ↦ (coeff (m + single s n)) φ) = 0

case pos.h₁
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
s : σ
n : ℕ
φ : MvPowerSeries σ R
h : ∀ (m : σ →₀ ℕ), m s < n → (coeff m) φ = 0
m : σ →₀ ℕ
H : m - single s n + single s n = m
⊢ (single s n, m - single s n) ∉ antidiagonal m →
    ((coeff (single s n, m - single s n).1) (X s ^ n) *
        (coeff (single s n, m - single s n).2) fun m ↦ (coeff (m + single s n)) φ) =
      0
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case h
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (Associates.mk v.asIdeal)
hlt : v.asIdeal ^ 2 < v.asIdeal
π : R
mem : 1 ≤ (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {π})).factors
notMem : ¬Associates.mk (v.asIdeal ^ 2) ≤ Associates.mk (Ideal.span {π})
hπ : Associates.mk (Ideal.span {π}) ≠ 0
⊢ (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {π})).factors = 1

Tactic:
Associates.mk_pow,

Goal state after:
case h
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (Associates.mk v.asIdeal)
hlt : v.asIdeal ^ 2 < v.asIdeal
π : R
mem : 1 ≤ (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {π})).factors
notMem : ¬Associates.mk v.asIdeal ^ 2 ≤ Associates.mk (Ideal.span {π})
hπ : Associates.mk (Ideal.span {π}) ≠ 0
⊢ (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {π})).factors = 1
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case h
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (Associates.mk v.asIdeal)
hlt : v.asIdeal ^ 2 < v.asIdeal
π : R
mem : 1 ≤ (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {π})).factors
notMem : ¬Associates.mk v.asIdeal ^ 2 ≤ Associates.mk (Ideal.span {π})
hπ : Associates.mk (Ideal.span {π}) ≠ 0
⊢ (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {π})).factors = 1

Tactic:
Associates.prime_pow_dvd_iff_le hπ hv,

Goal state after:
case h
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (Associates.mk v.asIdeal)
hlt : v.asIdeal ^ 2 < v.asIdeal
π : R
mem : 1 ≤ (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {π})).factors
notMem : ¬2 ≤ (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {π})).factors
hπ : Associates.mk (Ideal.span {π}) ≠ 0
⊢ (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {π})).factors = 1
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
f : MvPowerSeries σ R
m : ℕ
hf : constantCoeff f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : (range n).sum ⇑k = d ∧ k.support ⊆ range n
s : Finset ℕ := {i ∈ range n | k i = 0}
hs_def : s = {i ∈ range n | k i = 0}
hs : s ⊆ range n
i : ℕ
hi : i ∈ range n ∧ k i = 0
⊢ (coeff (k i)) f = constantCoeff f

Tactic:
rw [hi.2, coeff_zero_eq_constantCoeff]

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

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
f : MvPowerSeries σ R
m : ℕ
hf : constantCoeff f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : (range n).sum ⇑k = d ∧ k.support ⊆ range n
s : Finset ℕ := {i ∈ range n | k i = 0}
hs_def : s = {i ∈ range n | k i = 0}
hs : s ⊆ range n
hs' : ∀ i ∈ s, (coeff (k i)) f = constantCoeff f
hs'' : ∀ i ∈ s, k i = 0
this : m ≤ #s
⊢ constantCoeff f ^ #s = 0

Tactic:
obtain ⟨m', hm'⟩ := Nat.exists_eq_add_of_le this

Goal state after:
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
f : MvPowerSeries σ R
m : ℕ
hf : constantCoeff f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : (range n).sum ⇑k = d ∧ k.support ⊆ range n
s : Finset ℕ := {i ∈ range n | k i = 0}
hs_def : s = {i ∈ range n | k i = 0}
hs : s ⊆ range n
hs' : ∀ i ∈ s, (coeff (k i)) f = constantCoeff f
hs'' : ∀ i ∈ s, k i = 0
this : m ≤ #s
m' : ℕ
hm' : #s = m + m'
⊢ constantCoeff f ^ #s = 0
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
n d : R
hd : d ∈ nonZeroDivisors R
h : ∀ (v : HeightOneSpectrum R), (valuation K v) (IsLocalization.mk' K n ⟨d, hd⟩) ≤ 1
hx : IsLocalization.mk' K n ⟨d, hd⟩ * (algebraMap R K) ↑(n, ⟨d, hd⟩).2 = (algebraMap R K) (n, ⟨d, hd⟩).1
hn0 : n ≠ 0
hd0 : d ≠ 0
this : Ideal.span {d} ∣ Ideal.span {n}
⊢ IsLocalization.mk' K n ⟨d, hd⟩ ∈ (algebraMap R K).range

Tactic:
by
    obtain ⟨z, rfl⟩ := Ideal.span_singleton_le_span_singleton.1 (Ideal.le_of_dvd this)
    use z
    rw [map_mul, mul_comm, mul_eq_mul_left_iff] at hx
    exact (hx.resolve_right fun h => by simp [hd0] at h).symm

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
s : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
this✝ : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.map ↑embedding) γ)}
this : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
x✝¹ : ∃ i, {x | embedding (extension x) < embedding ↑i} ⊆ s
γ : (ofClass v).ValueGroup₀ˣ
h : {x | embedding (extension x) < embedding ↑γ} ⊆ s
x✝ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ extensionValuation.restrict x✝ < ↑((Units.map valueGroup₀_equiv_extensionValuation.toMonoidHom) γ) ↔
    embedding (extension x✝) < embedding ↑γ

Tactic:
apply this

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f : MvPowerSeries σ R
⊢ weightedOrder w f = ⊤ ↔ f = 0

Tactic:
← not_iff_not,

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f : MvPowerSeries σ R
⊢ ¬weightedOrder w f = ⊤ ↔ ¬f = 0
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
⊢ (MvPowerSeries.coeff m) ↑((monomial n) a) = (MvPowerSeries.coeff m) ((MvPowerSeries.monomial n) a)

Tactic:
rw [coeff_coe, coeff_monomial, MvPowerSeries.coeff_monomial]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n : σ →₀ ℕ
a : R
m : σ →₀ ℕ
⊢ (if n = m then a else 0) = if m = n then a else 0
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case mp.hp
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
h : Function.Surjective ⇑v
γ : Γ₀
⊢ IsClosed {a | v a = γ → ∃ a, v a = γ}

Tactic:
by_cases H : ∃ x : K, (v : K → Γ₀) x = γ

Goal state after:
case pos
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
h : Function.Surjective ⇑v
γ : Γ₀
H : ∃ x, v x = γ
⊢ IsClosed {a | v a = γ → ∃ a, v a = γ}

case neg
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
h : Function.Surjective ⇑v
γ : Γ₀
H : ¬∃ x, v x = γ
⊢ IsClosed {a | v a = γ → ∃ a, v a = γ}
Module: Mathlib.Data.Finsupp.Interval

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝ : Zero α
f✝ : ι →₀ α
i✝ : ι
a : α
f : ι →₀ α
i : ι
⊢ i ∈ f.support ↔ {f i} ≠ 0

Tactic:
← not_iff_not,

Goal state after:
ι : Type u_1
α : Type u_2
inst✝ : Zero α
f✝ : ι →₀ α
i✝ : ι
a : α
f : ι →₀ α
i : ι
⊢ i ∉ f.support ↔ ¬{f i} ≠ 0
Module: Mathlib.Data.Finsupp.Interval

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝³ : Zero α
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq ι
f✝ g✝ : ι →₀ α
i✝ : ι
a : α
f g : ι →₀ α
i : ι
⊢ i ∈ f.support ∨ i ∈ g.support ↔ Icc (f i) (g i) ≠ 0

Tactic:
← not_iff_not,

Goal state after:
ι : Type u_1
α : Type u_2
inst✝³ : Zero α
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq ι
f✝ g✝ : ι →₀ α
i✝ : ι
a : α
f g : ι →₀ α
i : ι
⊢ ¬(i ∈ f.support ∨ i ∈ g.support) ↔ ¬Icc (f i) (g i) ≠ 0
Module: Mathlib.RingTheory.MvPowerSeries.Trunc

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
s : Finset (σ →₀ ℕ)
x : σ →₀ ℕ
p : MvPowerSeries σ R
h : x ∈ s
⊢ MvPolynomial.coeff x ((truncFinset R s) p) = (coeff x) p

Tactic:
classical
  simp [truncFinset_apply, MvPolynomial.coeff_sum, h]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Trunc

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
s : Finset (σ →₀ ℕ)
x : σ →₀ ℕ
p : MvPowerSeries σ R
h : x ∉ s
⊢ MvPolynomial.coeff x ((truncFinset R s) p) = 0

Tactic:
classical
  simp [truncFinset_apply, MvPolynomial.coeff_sum, h]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case cons
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
ι : Type u_4
w : σ → ℕ
f : ι → MvPowerSeries σ R
a : ι
s : Finset ι
ha : a ∉ s
ih : ∑ i ∈ s, weightedOrder w (f i) ≤ weightedOrder w (∏ i ∈ s, f i)
⊢ weightedOrder w (f a) + ∑ x ∈ s, weightedOrder w (f x) ≤ weightedOrder w (∏ i ∈ Finset.cons a s ha, f i)

Tactic:
Finset.prod_cons ha,

Goal state after:
case cons
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
ι : Type u_4
w : σ → ℕ
f : ι → MvPowerSeries σ R
a : ι
s : Finset ι
ha : a ∉ s
ih : ∑ i ∈ s, weightedOrder w (f i) ≤ weightedOrder w (∏ i ∈ s, f i)
⊢ weightedOrder w (f a) + ∑ x ∈ s, weightedOrder w (f x) ≤ weightedOrder w (f a * ∏ x ∈ s, f x)
Module: Mathlib.RingTheory.MvPowerSeries.Trunc

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹ : DecidableEq σ
inst✝ : CommSemiring R
f g : MvPowerSeries σ R
h : f = g
⊢ ∃ᶠ (m : σ →₀ ℕ) in atTop, (trunc' R m) f = (trunc' R m) g

Tactic:
simp [h, atTop_neBot]

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
⊢ φ.order = ⊤ ↔ φ = 0

Tactic:
by
  simpa using order_finite_iff_ne_zero.not_left

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: case cons
R : Type u_2
inst✝ : CommSemiring R
ι : Type u_3
φ : ι → R⟦X⟧
a : ι
s : Finset ι
ha : a ∉ s
ih : ∑ i ∈ s, (φ i).order ≤ (∏ i ∈ s, φ i).order
⊢ (φ a).order + ∑ x ∈ s, (φ x).order ≤ (∏ i ∈ Finset.cons a s ha, φ i).order

Tactic:
Finset.prod_cons ha,

Goal state after:
case cons
R : Type u_2
inst✝ : CommSemiring R
ι : Type u_3
φ : ι → R⟦X⟧
a : ι
s : Finset ι
ha : a ∉ s
ih : ∑ i ∈ s, (φ i).order ≤ (∏ i ∈ s, φ i).order
⊢ (φ a).order + ∑ x ∈ s, (φ x).order ≤ (φ a * ∏ x ∈ s, φ x).order
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
n : ℕ
a : R
h : a ≠ 0
⊢ ((monomial n) a).order = ↑n

Tactic:
classical
  rw [order_monomial, if_neg h]

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
f : R⟦X⟧
⊢ X ^ f.order.toNat * f.divXPowOrder = f

Tactic:
by
  ext n
  rw [coeff_X_pow_mul']
  split_ifs with h
  · simp [h]
  · push Not at h
    rw [coeff_of_lt_order_toNat _ h]

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.PiTopology

Goal state before:
pp: R : Type u_1
inst✝¹ : TopologicalSpace R
inst✝ : Semiring R
ι : Type u_2
f : ι → R⟦X⟧
u : Filter ι
g : R⟦X⟧
⊢ Tendsto f u (𝓝 g) ↔ ∀ (d : ℕ), Tendsto (fun i ↦ (coeff d) (f i)) u (𝓝 ((coeff d) g))

Tactic:
by
  rw [MvPowerSeries.WithPiTopology.tendsto_iff_coeff_tendsto]
  apply (Finsupp.uniqueLinearEquiv ℕ ℕ ()).toEquiv.forall_congr
  intro d
  simp only [LinearEquiv.coe_toEquiv, Finsupp.uniqueLinearEquiv_apply, coeff]
  apply iff_of_eq
  congr
  · ext _; congr; ext; simp
  · ext; simp

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.PiTopology

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
inst✝ : TopologicalSpace R
a✝ : Nontrivial R
n m : ℕ
hm : n ≤ m
⊢ n < m + 1

Tactic:
exact Nat.lt_add_one_iff.mpr hm

Goal state after:
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case zero
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
⊢ cantorFunction c f < cantorFunction c g

Tactic:
let f_max : ℕ → Bool := fun n => Nat.rec false (fun _ _ => true) n
    have hf_max : ∀ n, f n → f_max n := by
      intro n hn
      cases n
      · rw [fn] at hn
        contradiction
      simp [f_max]
    let g_min : ℕ → Bool := fun n => Nat.rec true (fun _ _ => false) n
    have hg_min : ∀ n, g_min n → g n := by
      intro n hn
      cases n
      · rw [gn]
      simp at hn
    apply (cantorFunction_le (le_of_lt h1) h3 hf_max).trans_lt
    refine lt_of_lt_of_le ?_ (cantorFunction_le (le_of_lt h1) h3 hg_min)
    have : c / (1 - c) < 1 := by
      rw [div_lt_one, lt_sub_iff_add_lt]
      · convert! _root_.add_lt_add h2 h2
        norm_num
      rwa [sub_pos]
    convert! this
    · rw [cantorFunction_succ _ (le_of_lt h1) h3, div_eq_mul_inv, ←
        tsum_geometric_of_lt_one (le_of_lt h1) h3]
      apply zero_add
    · refine (tsum_eq_single 0 ?_).trans ?_
      · intro n hn
        cases n
        · contradiction
        simp [g_min]
      · exact cantorFunctionAux_zero _

Goal state after:
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
f g : ℕ → Bool
hfg : cantorFunction c f = cantorFunction c g
⊢ f = g

Tactic:
classical
    contrapose hfg with h
    have : ∃ n, f n ≠ g n := Function.ne_iff.mp h
    let n := Nat.find this
    have hn : ∀ k : ℕ, k < n → f k = g k := by
      intro k hk
      apply of_not_not
      exact Nat.find_min this hk
    cases fn : f n
    · apply _root_.ne_of_lt
      refine increasing_cantorFunction h1 h2 hn fn ?_
      apply Bool.eq_true_of_not_eq_false
      rw [← fn]
      apply Ne.symm
      exact Nat.find_spec this
    · apply _root_.ne_of_gt
      refine increasing_cantorFunction h1 h2 (fun k hk => (hn k hk).symm) ?_ fn
      apply Bool.eq_false_of_not_eq_true
      rw [← fn]
      apply Ne.symm
      exact Nat.find_spec this

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝² : CommRing R
τ : Type u_3
S : Type u_4
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : MvPowerSeries τ S
ha : HasSubst a
f : R⟦X⟧
e : τ →₀ ℕ
⊢ (Function.support fun d ↦ (coeff d) f • (MvPowerSeries.coeff e) (a ^ d)) =
    ⇑(Finsupp.uniqueLinearEquiv ℕ ℕ ()).toEquiv ''
      Function.support fun d ↦ (MvPowerSeries.coeff d) f • (MvPowerSeries.coeff e) (d.prod fun s e ↦ a ^ e)

Tactic:
rw [← Equiv.preimage_eq_iff_eq_image, ← Function.support_comp_eq_preimage]

Goal state after:
R : Type u_2
inst✝² : CommRing R
τ : Type u_3
S : Type u_4
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : MvPowerSeries τ S
ha : HasSubst a
f : R⟦X⟧
e : τ →₀ ℕ
⊢ Function.support
      ((fun d ↦ (coeff d) f • (MvPowerSeries.coeff e) (a ^ d)) ∘ ⇑(Finsupp.uniqueLinearEquiv ℕ ℕ ()).toEquiv) =
    Function.support fun d ↦ (MvPowerSeries.coeff d) f • (MvPowerSeries.coeff e) (d.prod fun s e ↦ a ^ e)
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_9
inst✝ : CommSemiring R
x : MvPowerSeries σ R
n : σ →₀ ℕ
h : ¬n = 0
⊢ (n.prod fun s m ↦ 0 ^ m) * x n = 0

Tactic:
convert! zero_mul _

Goal state after:
case e'_2.e'_5
σ : Type u_1
R : Type u_9
inst✝ : CommSemiring R
x : MvPowerSeries σ R
n : σ →₀ ℕ
h : ¬n = 0
⊢ (n.prod fun s m ↦ 0 ^ m) = 0
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝ : CommSemiring R
f : ι → ArithmeticFunction R
hf : ∀ (n : ℕ), ∀ᶠ (i : ι) in cofinite, (f i) n = 1 n
this✝ : UniformSpace R := ⊥
this : IsUniformInducing DFunLike.coe
⊢ ∀ (n : ℕ), ∀ᶠ (s : Finset ι) in atTop, (∏ i ∈ s, f i) n = (eulerProduct f) n

Tactic:
classical
  suffices Multipliable f from tendsto_iff.mp this.hasProd
  simp_rw [multipliable_iff_cauchySeq_finset, CauchySeq, ← this.cauchy_map_iff,
    Filter.map_map, cauchy_map_iff', Pi.uniformity, DiscreteUniformity.eq_principal_setRelId,
    tendsto_iInf, tendsto_comap_iff, tendsto_principal, Function.comp_apply, prod_atTop_atTop_eq,
    eventually_atTop_prod_self, SetRel.mem_id]
  intro n
  replace hf : ∀ k ∈ Set.Iic n, ∀ᶠ (x : ι) in cofinite, (f x) k = (1 : ArithmeticFunction R) k :=
    fun k hk ↦ hf k
  rw [← eventually_all_finite (Set.finite_Iic n), eventually_iff_exists_mem] at hf
  obtain ⟨s, hs, hs'⟩ := hf
  let t := (mem_cofinite.mp hs).toFinset
  refine ⟨t, fun u v hu hv ↦ ?_⟩
  rw [← Finset.prod_sdiff hu, ← Finset.prod_sdiff hv]
  replace hu : ∀ i ∈ u \ t, i ∈ s := by
    intro i hi
    rw [Finset.mem_sdiff, Set.Finite.mem_toFinset, Set.notMem_compl_iff] at hi
    exact hi.2
  replace hv : ∀ i ∈ v \ t, i ∈ s := by
    intro i hi
    rw [Finset.mem_sdiff, Set.Finite.mem_toFinset, Set.notMem_compl_iff] at hi
    exact hi.2
  suffices ∀ k ≤ n, (∏ x ∈ u \ t, f x) k = (∏ x ∈ v \ t, f x) k by
    rw [mul_apply, mul_apply]
    refine Finset.sum_congr rfl fun k hk ↦ ?_
    rw [this k.1 (Nat.divisor_le (Nat.fst_mem_divisors_of_mem_antidiagonal hk))]
  suffices ∀ w, (∀ i ∈ w, i ∈ s) → ∀ k ≤ n, (∏ x ∈ w, f x) k = (1 : ArithmeticFunction R) k by
    intro k hk
    rw [this (u \ t) hu k hk, this (v \ t) hv k hk]
  intro w hw
  induction w using Finset.induction_on
  case empty => simp
  case insert i w hi hw' =>
    intro k hk
    rw [← one_mul (1 : ArithmeticFunction R), Finset.prod_insert hi, mul_apply, mul_apply]
    refine Finset.sum_congr rfl fun j hj ↦ ?_
    have h1 := hs' i (hw i (Finset.mem_insert_self i w)) j.1
      ((Nat.divisor_le (Nat.fst_mem_divisors_of_mem_antidiagonal hj)).trans hk)
    have h2 := hw' (fun i hi ↦ hw i (Finset.mem_insert_of_mem hi)) j.2
      ((Nat.divisor_le (Nat.snd_mem_divisors_of_mem_antidiagonal hj)).trans hk)
    rw [h1, h2]

Goal state after:
Module: Mathlib.Analysis.Convex.EGauge

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
x : E
hc : 1 < ‖c‖
this : NontriviallyNormedField 𝕜 := { toNormedField := inst✝², non_trivial := ⋯ }
h₀ : 0 ≠ 0 ∨ ‖x‖ ≠ 0
⊢ ‖c‖ₑ ≠ 0

Tactic:
simpa using one_pos.trans hc

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Substitution

Goal state before:
pp: R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
n : ℕ
this : (coeff n) (subst (∑ i, C (P.substInvFun ↑i) * X ^ ↑i) P) = (coeff n) X
⊢ (coeff n) (subst P.substInv P) = (coeff n) (subst (∑ i, C (P.substInvFun ↑i) * X ^ ↑i) P)

Tactic:
coeff_subst',

Goal state after:
R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
n : ℕ
this : (coeff n) (subst (∑ i, C (P.substInvFun ↑i) * X ^ ↑i) P) = (coeff n) X
⊢ ∑ᶠ (d : ℕ), (coeff d) P • (coeff n) (P.substInv ^ d) = (coeff n) (subst (∑ i, C (P.substInvFun ↑i) * X ^ ↑i) P)

case hb
R : Type u_2
inst✝¹ : CommRing R
P : R⟦X⟧
hP : constantCoeff P = 0
inst✝ : Invertible ((coeff 1) P)
n : ℕ
this : (coeff n) (subst (∑ i, C (P.substInvFun ↑i) * X ^ ↑i) P) = (coeff n) X
⊢ HasSubst P.substInv
Module: Mathlib.Analysis.Calculus.TangentCone.DimOne

Goal state before:
pp: case hd₀
𝕜 : Type u_1
inst✝ : NormedDivisionRing 𝕜
s : Set 𝕜
x : 𝕜
hx : AccPt x (𝓟 s)
y : 𝕜
⊢ Tendsto (fun x_1 ↦ x_1 - x) (𝓝[≠] x) (𝓝 0)

pp: case hds
𝕜 : Type u_1
inst✝ : NormedDivisionRing 𝕜
s : Set 𝕜
x : 𝕜
hx : AccPt x (𝓟 s)
y : 𝕜
⊢ ∃ᶠ (n : 𝕜) in 𝓝[≠] x, x + (n - x) ∈ s

pp: case hcd
𝕜 : Type u_1
inst✝ : NormedDivisionRing 𝕜
s : Set 𝕜
x : 𝕜
hx : AccPt x (𝓟 s)
y : 𝕜
⊢ Tendsto (fun n ↦ (y / (n - x)) • (n - x)) (𝓝[≠] x) (𝓝 y)

Tactic:
· exact Continuous.tendsto' (by fun_prop) _ _ (by simp) |>.mono_left inf_le_left

Goal state after:
case hds
𝕜 : Type u_1
inst✝ : NormedDivisionRing 𝕜
s : Set 𝕜
x : 𝕜
hx : AccPt x (𝓟 s)
y : 𝕜
⊢ ∃ᶠ (n : 𝕜) in 𝓝[≠] x, x + (n - x) ∈ s

case hcd
𝕜 : Type u_1
inst✝ : NormedDivisionRing 𝕜
s : Set 𝕜
x : 𝕜
hx : AccPt x (𝓟 s)
y : 𝕜
⊢ Tendsto (fun n ↦ (y / (n - x)) • (n - x)) (𝓝[≠] x) (𝓝 y)
Module: Mathlib.Analysis.Calculus.FDeriv.Const

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
F : Type u_3
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : TopologicalSpace F
f : E → F
x : E
hf : Injective ⇑(fderivWithin 𝕜 f univ x)
⊢ DifferentiableWithinAt 𝕜 f univ x

Tactic:
exact differentiableWithinAt_of_fderivWithin_injective hf

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Basic

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
h : DifferentiableAt 𝕜 f x
hxs : UniqueDiffWithinAt 𝕜 s x
⊢ (fderivWithin 𝕜 f s x) 1 = (fderiv 𝕜 f x) 1

Tactic:
h.fderivWithin hxs

Goal state after:
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
h : DifferentiableAt 𝕜 f x
hxs : UniqueDiffWithinAt 𝕜 s x
⊢ (fderiv 𝕜 f x) 1 = (fderiv 𝕜 f x) 1