mathlib-initiative/mathlib-tactics huggingface dataset:
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
a b : (ofClass v).ValueGroup₀
hab : extensionValuation.restrict ↑⋯.choose = extensionValuation.restrict ↑⋯.choose
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
hx : v x = embedding a
hy : v y = embedding b
this : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
ha0 : embedding a ≠ embedding 0
h : x = 0
⊢ False

Tactic:
simp [h, ← hx] at ha0

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

Goal state before:
pp: case inr.succ
R : Type u_2
inst✝ : CommSemiring R
φ : R⟦X⟧
n' : ℕ
ih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)
hn : n' + 1 > 0
h₁ : ∀ (m : ℕ), φ ^ (m + 1) = φ ^ m * φ
h₂ : antidiagonal 1 = {(0, 1), (1, 0)}
⊢ (coeff 1) (φ ^ (n' + 1)) = ↑(n' + 1) * (coeff 1) φ * constantCoeff φ ^ (n' + 1 - 1)

Tactic:
h₁,

Goal state after:
case inr.succ
R : Type u_2
inst✝ : CommSemiring R
φ : R⟦X⟧
n' : ℕ
ih : n' > 0 → (coeff 1) (φ ^ n') = ↑n' * (coeff 1) φ * constantCoeff φ ^ (n' - 1)
hn : n' + 1 > 0
h₁ : ∀ (m : ℕ), φ ^ (m + 1) = φ ^ m * φ
h₂ : antidiagonal 1 = {(0, 1), (1, 0)}
⊢ (coeff 1) (φ ^ n' * φ) = ↑(n' + 1) * (coeff 1) φ * constantCoeff φ ^ (n' + 1 - 1)
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
φ ψ : MvPolynomial σ R
n : σ →₀ ℕ
⊢ (MvPowerSeries.coeff n) ↑(φ * ψ) = (MvPowerSeries.coeff n) (↑φ * ↑ψ)

Tactic:
by
    classical
    simp only [coeff_coe, MvPowerSeries.coeff_mul, coeff_mul]

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝⁴ : PartialOrder α
inst✝³ : Zero α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq α
f✝ g✝ f g x : ι →₀ α
⊢ x ∈ (f.support ∪ g.support).finsupp ⇑(f.rangeIcc g) ↔ f ≤ x ∧ x ≤ g

Tactic:
refine
        (mem_finsupp_iff_of_support_subset <| Finset.subset_of_eq <| rangeIcc_support _ _).trans ?_

Goal state after:
ι : Type u_1
α : Type u_2
inst✝⁴ : PartialOrder α
inst✝³ : Zero α
inst✝² : LocallyFiniteOrder α
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq α
f✝ g✝ f g x : ι →₀ α
⊢ (∀ (i : ι), x i ∈ (f.rangeIcc g) i) ↔ f ≤ x ∧ x ≤ g
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)
p : MvPowerSeries σ R
x✝ : σ →₀ ℕ
⊢ x✝ ∈ ((truncFinset R s) p).support → x✝ ∈ s

Tactic:
contrapose

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
s : Finset (σ →₀ ℕ)
p : MvPowerSeries σ R
x✝ : σ →₀ ℕ
⊢ x✝ ∉ s → x✝ ∉ ((truncFinset R s) p).support
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case insert
σ : Type u_1
w : σ → ℕ
R : Type u_4
ι : Type u_5
inst✝ : CommRing R
d : σ →₀ ℕ
f : MvPowerSeries σ R
g : ι → MvPowerSeries σ R
a : ι
s : Finset ι
ha : a ∉ s
ih : (∀ i ∈ s, ↑((weight w) d) < weightedOrder w (g i)) → (coeff d) (f * ∏ i ∈ s, (1 - g i)) = (coeff d) f
h : ↑((weight w) d) < weightedOrder w (g a) ∧ ∀ a ∈ s, ↑((weight w) d) < weightedOrder w (g a)
⊢ (coeff d) (f * (1 - g a) * ∏ x ∈ s, (1 - g x)) = (coeff d) f

Tactic:
mul_right_comm,

Goal state after:
case insert
σ : Type u_1
w : σ → ℕ
R : Type u_4
ι : Type u_5
inst✝ : CommRing R
d : σ →₀ ℕ
f : MvPowerSeries σ R
g : ι → MvPowerSeries σ R
a : ι
s : Finset ι
ha : a ∉ s
ih : (∀ i ∈ s, ↑((weight w) d) < weightedOrder w (g i)) → (coeff d) (f * ∏ i ∈ s, (1 - g i)) = (coeff d) f
h : ↑((weight w) d) < weightedOrder w (g a) ∧ ∀ a ∈ s, ↑((weight w) d) < weightedOrder w (g a)
⊢ (coeff d) ((f * ∏ x ∈ s, (1 - g x)) * (1 - g a)) = (coeff d) f
Module: Mathlib.RingTheory.MvPowerSeries.Trunc

Goal state before:
pp: σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
⊢ n ≠ 0 → 0 ∈ ⋯.toFinset

Tactic:
contrapose

Goal state after:
σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : Finite σ
inst✝ : CommSemiring R
⊢ 0 ∉ ⋯.toFinset → n = 0
Module: Mathlib.Topology.Algebra.TopologicallyNilpotent

Goal state before:
pp: R : Type u_1
inst✝² : TopologicalSpace R
inst✝¹ : Ring R
inst✝ : IsLinearTopology R R
a b : R
ha : IsTopologicallyNilpotent a
hb : IsTopologicallyNilpotent b
h : Commute a b
I : Ideal R
⊢ ↑I ∈ 𝓝 0 → ∃ ia, ∀ x ∈ Set.Ici ia, (a + b) ^ x ∈ ↑I

Tactic:
I_mem_nhds

Goal state after:
R : Type u_1
inst✝² : TopologicalSpace R
inst✝¹ : Ring R
inst✝ : IsLinearTopology R R
a b : R
ha : IsTopologicallyNilpotent a
hb : IsTopologicallyNilpotent b
h : Commute a b
I : Ideal R
I_mem_nhds : ↑I ∈ 𝓝 0
⊢ ∃ ia, ∀ x ∈ Set.Ici ia, (a + b) ^ x ∈ ↑I
Module: Mathlib.RingTheory.MvPowerSeries.PiTopology

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝² : TopologicalSpace R
inst✝¹ : DecidableEq σ
inst✝ : CommSemiring R
f : MvPowerSeries σ R
d n : σ →₀ ℕ
hdn : d ≤ n
⊢ (coeff d) ↑((trunc' R n) f) = (coeff d) f

Tactic:
by simp [coeff_trunc', hdn]

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹ : TopologicalSpace R
inst✝ : Semiring R
ι : Type u_3
f : ι → MvPowerSeries σ R
a : (σ →₀ ℕ) → R
h : ∀ (d : σ →₀ ℕ), HasSum (fun i ↦ (coeff d) (f i)) (a d)
⊢ ∀ (d : σ →₀ ℕ), HasSum (fun i ↦ (coeff d) (f i)) ((coeff d) a)

Tactic:
simpa using! h

Goal state after:
Module: Mathlib.Topology.Algebra.LinearTopology

Goal state before:
pp: R : Type u_1
R' : Type u_2
M : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring R'
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : Module R' M
inst✝³ : SMulCommClass R R' M
inst✝² : TopologicalSpace M
inst✝¹ : IsLinearTopology R M
inst✝ : IsLinearTopology R' M
I : Submodule R M
hI : ↑I ∈ 𝓝 0
J : Submodule R' M
hJ : ↑J ∈ 𝓝 0
J_sub_I : ↑J ⊆ ↑I
uR : Set R := univ
uR' : Set R' := univ
hRR : uR * uR ⊆ uR
hRI : uR • ↑I ⊆ ↑I
hR'J : uR' • ↑J ⊆ ↑J
hRJ : uR • ↑J ⊆ ↑I
S : Set M := ↑J ∪ uR • ↑J
S_sub_I : S ⊆ ↑I
hRS : uR • S ⊆ S
hR'S : uR' • S ⊆ S
A : AddSubgroup M := AddSubgroup.closure S
r : R
i : M
hi : i ∈ A
x y : M
x✝¹ : x ∈ AddSubgroup.closure S
x✝ : y ∈ AddSubgroup.closure S
hx : r • x ∈ A
hy : r • y ∈ A
⊢ r • (x + y) ∈ A

Tactic:
simp_rw [smul_add]; exact add_mem hx hy

Goal state after:
Module: Mathlib.Topology.Algebra.LinearTopology

Goal state before:
pp: R : Type u_1
R' : Type u_2
M : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring R'
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : Module R' M
inst✝³ : SMulCommClass R R' M
inst✝² : TopologicalSpace M
inst✝¹ : IsLinearTopology R M
inst✝ : IsLinearTopology R' M
I : Submodule R M
hI : ↑I ∈ 𝓝 0
J : Submodule R' M
hJ : ↑J ∈ 𝓝 0
J_sub_I : ↑J ⊆ ↑I
uR : Set R := univ
uR' : Set R' := univ
hRR : uR * uR ⊆ uR
hRI : uR • ↑I ⊆ ↑I
hR'J : uR' • ↑J ⊆ ↑J
hRJ : uR • ↑J ⊆ ↑I
S : Set M := ↑J ∪ uR • ↑J
S_sub_I : S ⊆ ↑I
hRS : uR • S ⊆ S
hR'S : uR' • S ⊆ S
A : AddSubgroup M := AddSubgroup.closure S
r : R
i : M
hi : i ∈ A
x : M
x✝ : x ∈ AddSubgroup.closure S
hx : r • x ∈ A
⊢ r • -x ∈ A

Tactic:
simp_rw [smul_neg]

Goal state after:
R : Type u_1
R' : Type u_2
M : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring R'
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : Module R' M
inst✝³ : SMulCommClass R R' M
inst✝² : TopologicalSpace M
inst✝¹ : IsLinearTopology R M
inst✝ : IsLinearTopology R' M
I : Submodule R M
hI : ↑I ∈ 𝓝 0
J : Submodule R' M
hJ : ↑J ∈ 𝓝 0
J_sub_I : ↑J ⊆ ↑I
uR : Set R := univ
uR' : Set R' := univ
hRR : uR * uR ⊆ uR
hRI : uR • ↑I ⊆ ↑I
hR'J : uR' • ↑J ⊆ ↑J
hRJ : uR • ↑J ⊆ ↑I
S : Set M := ↑J ∪ uR • ↑J
S_sub_I : S ⊆ ↑I
hRS : uR • S ⊆ S
hR'S : uR' • S ⊆ S
A : AddSubgroup M := AddSubgroup.closure S
r : R
i : M
hi : i ∈ A
x : M
x✝ : x ∈ AddSubgroup.closure S
hx : r • x ∈ A
⊢ -(r • x) ∈ A
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
b : ℝ≥0
hb : 1 < b
⊢ IsNonarchimedean (v.intAdicAbvDef hb)

Tactic:
intro x y
  simp only [intAdicAbvDef]
  have h_mono := (toNNReal_strictMono hb).monotone
  rw [← h_mono.map_max]
  exact h_mono <| v.intValuation.map_add x y

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

Goal state before:
pp: case neg
R : Type u_1
inst✝ : Semiring R
φ ψ : R⟦X⟧
n : ℕ
hn : ↑n < φ.order + ψ.order
i j : ℕ
hij : (i, j).1 + (i, j).2 = n
hi : φ.order ≤ ↑i
hj : ψ.order ≤ ↑j
⊢ False

Tactic:
apply ne_of_lt (lt_of_lt_of_le hn <| add_le_add hi hj)

Goal state after:
case neg
R : Type u_1
inst✝ : Semiring R
φ ψ : R⟦X⟧
n : ℕ
hn : ↑n < φ.order + ψ.order
i j : ℕ
hij : (i, j).1 + (i, j).2 = n
hi : φ.order ≤ ↑i
hj : ψ.order ≤ ↑j
⊢ ↑n = ↑i + ↑j
Module: Mathlib.Topology.Algebra.LinearTopology

Goal state before:
pp: R : Type u_1
R' : Type u_2
M : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring R'
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : Module R' M
inst✝³ : SMulCommClass R R' M
inst✝² : TopologicalSpace M
inst✝¹ : IsLinearTopology R M
inst✝ : IsLinearTopology R' M
I : Submodule R M
hI : ↑I ∈ 𝓝 0
J : Submodule R' M
hJ : ↑J ∈ 𝓝 0
J_sub_I : ↑J ⊆ ↑I
uR : Set R := univ
uR' : Set R' := univ
hRR : uR * uR ⊆ uR
hRI : uR • ↑I ⊆ ↑I
hR'J : uR' • ↑J ⊆ ↑J
hRJ : uR • ↑J ⊆ ↑I
S : Set M := ↑J ∪ uR • ↑J
S_sub_I : S ⊆ ↑I
hRS : uR • S ⊆ S
hR'S : uR' • S ⊆ S
A : AddSubgroup M := AddSubgroup.closure S
hRA : ∀ (r : R), ∀ i ∈ A, r • i ∈ A
r' : R'
i : M
hi : i ∈ A
x y : M
x✝¹ : x ∈ AddSubgroup.closure S
x✝ : y ∈ AddSubgroup.closure S
hx : r' • x ∈ A
hy : r' • y ∈ A
⊢ r' • (x + y) ∈ A

Tactic:
simp_rw [smul_add]; exact add_mem hx hy

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
b : ℝ≥0
hb : 1 < b
r : R
⊢ (v.intAdicAbv hb) r = 1 ↔ r ∉ v.asIdeal

Tactic:
contrapose

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
b : ℝ≥0
hb : 1 < b
r : R
⊢ ¬(v.intAdicAbv hb) r = 1 ↔ r ∈ v.asIdeal
Module: Mathlib.Topology.Algebra.LinearTopology

Goal state before:
pp: R : Type u_1
R' : Type u_2
M : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring R'
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : Module R' M
inst✝³ : SMulCommClass R R' M
inst✝² : TopologicalSpace M
inst✝¹ : IsLinearTopology R M
inst✝ : IsLinearTopology R' M
I : Submodule R M
hI : ↑I ∈ 𝓝 0
J : Submodule R' M
hJ : ↑J ∈ 𝓝 0
J_sub_I : ↑J ⊆ ↑I
uR : Set R := univ
uR' : Set R' := univ
hRR : uR * uR ⊆ uR
hRI : uR • ↑I ⊆ ↑I
hR'J : uR' • ↑J ⊆ ↑J
hRJ : uR • ↑J ⊆ ↑I
S : Set M := ↑J ∪ uR • ↑J
S_sub_I : S ⊆ ↑I
hRS : uR • S ⊆ S
hR'S : uR' • S ⊆ S
A : AddSubgroup M := AddSubgroup.closure S
hRA : ∀ (r : R), ∀ i ∈ A, r • i ∈ A
r' : R'
i : M
hi : i ∈ A
x : M
x✝ : x ∈ AddSubgroup.closure S
hx : r' • x ∈ A
⊢ r' • -x ∈ A

Tactic:
simp_rw [smul_neg]

Goal state after:
R : Type u_1
R' : Type u_2
M : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring R'
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : Module R' M
inst✝³ : SMulCommClass R R' M
inst✝² : TopologicalSpace M
inst✝¹ : IsLinearTopology R M
inst✝ : IsLinearTopology R' M
I : Submodule R M
hI : ↑I ∈ 𝓝 0
J : Submodule R' M
hJ : ↑J ∈ 𝓝 0
J_sub_I : ↑J ⊆ ↑I
uR : Set R := univ
uR' : Set R' := univ
hRR : uR * uR ⊆ uR
hRI : uR • ↑I ⊆ ↑I
hR'J : uR' • ↑J ⊆ ↑J
hRJ : uR • ↑J ⊆ ↑I
S : Set M := ↑J ∪ uR • ↑J
S_sub_I : S ⊆ ↑I
hRS : uR • S ⊆ S
hR'S : uR' • S ⊆ S
A : AddSubgroup M := AddSubgroup.closure S
hRA : ∀ (r : R), ∀ i ∈ A, r • i ∈ A
r' : R'
i : M
hi : i ∈ A
x : M
x✝ : x ∈ AddSubgroup.closure S
hx : r' • x ∈ A
⊢ -(r' • x) ∈ A
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: case insert
R : Type u_2
ι : Type u_3
inst✝ : CommRing R
k : ℕ
φ : R⟦X⟧
f : ι → R⟦X⟧
a : ι
s : Finset ι
ha : a ∉ s
ih : (∀ i ∈ s, ↑k < (f i).order) → (coeff k) (φ * ∏ i ∈ s, (1 - f i)) = (coeff k) φ
t : ↑k < (f a).order ∧ ∀ a ∈ s, ↑k < (f a).order
⊢ (coeff k) (φ * (1 - f a) * ∏ x ∈ s, (1 - f x)) = (coeff k) φ

Tactic:
mul_right_comm,

Goal state after:
case insert
R : Type u_2
ι : Type u_3
inst✝ : CommRing R
k : ℕ
φ : R⟦X⟧
f : ι → R⟦X⟧
a : ι
s : Finset ι
ha : a ∉ s
ih : (∀ i ∈ s, ↑k < (f i).order) → (coeff k) (φ * ∏ i ∈ s, (1 - f i)) = (coeff k) φ
t : ↑k < (f a).order ∧ ∀ a ∈ s, ↑k < (f a).order
⊢ (coeff k) ((φ * ∏ x ∈ s, (1 - f x)) * (1 - f a)) = (coeff k) φ
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
f g : R⟦X⟧
h : f ≠ 0 ∧ g ≠ 0
⊢ X ^ (f * g).order.toNat * (f * g).divXPowOrder = f * g

Tactic:
by
        simp [X_pow_order_mul_divXPowOrder]

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

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
f g : R⟦X⟧
h : f ≠ 0 ∧ g ≠ 0
⊢ f * g = X ^ f.order.toNat * f.divXPowOrder * (X ^ g.order.toNat * g.divXPowOrder)

Tactic:
by
        simp [X_pow_order_mul_divXPowOrder]

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

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : Ring R
J K : TwoSidedIdeal R
d e : σ →₀ ℕ
hK : K ≠ ⊤
h : ∀ (a : MvPowerSeries σ R), (∀ e ≤ d, (coeff e) a ∈ J) → ∀ e_1 ≤ e, (coeff e_1) a ∈ K
h' : ¬e ≤ d
x : R
a✝ : x ∈ ⊤
d' : σ →₀ ℕ
hd'_le : d' ≤ d
hd' : d' = e
⊢ x ∈ J

Tactic:
exact (h' (hd' ▸ hd'_le)).elim

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

Goal state before:
pp: R : Type u_1
inst✝¹⁷ : CommRing R
S : Type u_2
inst✝¹⁶ : CommRing S
φ : R →+* S
a : S
inst✝¹⁵ : UniformSpace R
inst✝¹⁴ : UniformSpace S
inst✝¹³ : IsUniformAddGroup R
inst✝¹² : IsTopologicalSemiring R
inst✝¹¹ : IsUniformAddGroup S
inst✝¹⁰ : T2Space S
inst✝⁹ : CompleteSpace S
inst✝⁸ : IsTopologicalRing S
inst✝⁷ : IsLinearTopology S S
hφ : Continuous ⇑φ
ha : HasEval a
T : Type u_3
inst✝⁶ : UniformSpace T
inst✝⁵ : CompleteSpace T
inst✝⁴ : T2Space T
inst✝³ : CommRing T
inst✝² : IsTopologicalRing T
inst✝¹ : IsLinearTopology T T
inst✝ : IsUniformAddGroup T
ε : S →+* T
hε : Continuous ⇑ε
⊢ Continuous ⇑(ε.comp φ)

Tactic:
simp only [RingHom.coe_comp, hε.comp hφ]

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

Goal state before:
pp: σ : Type u_1
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
a : σ → MvPowerSeries τ S
x : σ → ℕ
inst✝ : Finite τ
ha : HasSubst a
d : τ →₀ ℕ
i : σ
⊢ i ∈ {s | (coeff d) ↑((MvPowerSeries.truncTotal (x s)) (a s)) ≠ 0} → i ∈ {s | (coeff d) (a s) ≠ 0}

Tactic:
contrapose

Goal state after:
σ : Type u_1
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
a : σ → MvPowerSeries τ S
x : σ → ℕ
inst✝ : Finite τ
ha : HasSubst a
d : τ →₀ ℕ
i : σ
⊢ i ∉ {s | (coeff d) (a s) ≠ 0} → i ∉ {s | (coeff d) ↑((MvPowerSeries.truncTotal (x s)) (a s)) ≠ 0}
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
f_max : ℕ → Bool := fun n ↦ rec false (fun x x_1 ↦ true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n ↦ rec true (fun x x_1 ↦ false) n
hn : g_min 0 = true
⊢ g 0 = true

pp: case succ
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
f_max : ℕ → Bool := fun n ↦ rec false (fun x x_1 ↦ true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n ↦ rec true (fun x x_1 ↦ false) n
n✝ : ℕ
hn : g_min (n✝ + 1) = true
⊢ g (n✝ + 1) = true

Tactic:
· rw [gn]

Goal state after:
case succ
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
f_max : ℕ → Bool := fun n ↦ rec false (fun x x_1 ↦ true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n ↦ rec true (fun x x_1 ↦ false) n
n✝ : ℕ
hn : g_min (n✝ + 1) = true
⊢ g (n✝ + 1) = true
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommSemiring R
q : ℕ
f g : PowerSeries R
n : ℕ
hq : 1 < q
⊢ { toFun := Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) (f + g)) 0, map_zero' := ⋯ } n =
    ({ toFun := Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) f) 0, map_zero' := ⋯ } +
        { toFun := Function.extend (fun x ↦ q ^ x) (fun x ↦ (PowerSeries.coeff x) g) 0, map_zero' := ⋯ })
      n

Tactic:
by_cases h : ∃ a, q ^ a = n
      · obtain ⟨a, rfl⟩ := h
        simp [(Nat.pow_right_injective hq).extend_apply]
      · simp [h]

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction

Goal state before:
pp: case neg.hb
R : Type u_1
inst✝ : CommRing R
f : PowerSeries R
hf : PowerSeries.constantCoeff f = 1
n p k : ℕ
hp : Nat.Prime p
hk : 0 < k
i : ℕ
hmn : (p ^ i).Coprime n
j : ℕ
hj : p ^ j = p ^ i * n
⊢ ∃ j, p ^ j = n

Tactic:
obtain ⟨v, -, rfl⟩ := (Nat.dvd_prime_pow hp).mp (Dvd.intro_left _ hj.symm)

Goal state after:
case neg.hb
R : Type u_1
inst✝ : CommRing R
f : PowerSeries R
hf : PowerSeries.constantCoeff f = 1
p k : ℕ
hp : Nat.Prime p
hk : 0 < k
i j v : ℕ
hmn : (p ^ i).Coprime (p ^ v)
hj : p ^ j = p ^ i * p ^ v
⊢ ∃ j, p ^ j = p ^ v
Module: Mathlib.Analysis.Calculus.TangentCone.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Semiring 𝕜
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
s : Set E
x : E
inst✝ : T2Space E
hx : ¬AccPt x (𝓟 s)
y : E
hy : y ∈ tangentConeAt 𝕜 s x
⊢ y ∈ 0

Tactic:
rcases exists_fun_of_mem_tangentConeAt hy with ⟨ι, l, hl, c, d, hd₀, hds, hcd⟩

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Semiring 𝕜
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
s : Set E
x : E
inst✝ : T2Space E
hx : ¬AccPt x (𝓟 s)
y : E
hy : y ∈ tangentConeAt 𝕜 s x
ι : Type (max u_1 u_2)
l : Filter ι
hl : l.NeBot
c : ι → 𝕜
d : ι → E
hd₀ : Tendsto d l (𝓝 0)
hds : ∀ᶠ (n : ι) in l, x + d n ∈ s
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 y)
⊢ y ∈ 0
Module: Mathlib.Analysis.Calculus.FDeriv.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜 E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : ContinuousAdd E
inst✝⁶ : ContinuousSMul 𝕜 E
F : Type u_3
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜 F
inst✝³ : TopologicalSpace F
inst✝² : ContinuousAdd F
inst✝¹ : ContinuousSMul 𝕜 F
f : E → F
f' f₁' : E →L[𝕜] F
x : E
s : Set E
inst✝ : T2Space F
hf : HasFDerivWithinAt f f' s x
hg : HasFDerivWithinAt f f₁' s x
y : E
hy : y ∈ tangentConeAt 𝕜 s x
⊢ f' y = f₁' y

Tactic:
rcases exists_fun_of_mem_tangentConeAt hy with ⟨ι, l, hl, c, d, hd₀, hds, hcd⟩

Goal state after:
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜 E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : ContinuousAdd E
inst✝⁶ : ContinuousSMul 𝕜 E
F : Type u_3
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜 F
inst✝³ : TopologicalSpace F
inst✝² : ContinuousAdd F
inst✝¹ : ContinuousSMul 𝕜 F
f : E → F
f' f₁' : E →L[𝕜] F
x : E
s : Set E
inst✝ : T2Space F
hf : HasFDerivWithinAt f f' s x
hg : HasFDerivWithinAt f f₁' s x
y : E
hy : y ∈ tangentConeAt 𝕜 s x
ι : Type (max u_1 u_2)
l : Filter ι
hl : l.NeBot
c : ι → 𝕜
d : ι → E
hd₀ : Tendsto d l (𝓝 0)
hds : ∀ᶠ (n : ι) in l, x + d n ∈ s
hcd : Tendsto (fun n ↦ c n • d n) l (𝓝 y)
⊢ f' y = f₁' y
Module: Mathlib.Analysis.Calculus.FDeriv.Basic

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
h : DifferentiableWithinAt 𝕜 f univ x
⊢ HasFDerivWithinAt f (fderivWithin 𝕜 f univ x) univ x

Tactic:
exact h.hasFDerivWithinAt

Goal state after:
Module: Mathlib.Analysis.Analytic.ConvergenceRadius

Goal state before:

Tactic:
|‖p n‖ * (r : ℝ) ^ n| = ‖p n‖ * (t : ℝ) ^ n * (r / t : ℝ) ^ n := by
      simp [field, abs_mul, div_pow]
    _ ≤ C * (r / t : ℝ) ^ n := by gcongr; apply hC

Goal state after:
Module: Mathlib.Analysis.Analytic.ChangeOrigin

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
x : E
r : ℝ≥0
_h0 : 0 < r
hr : ↑‖x‖₊ + ↑r < p.radius
hr' : ↑‖x‖₊ < p.radius
⊢ ↑r ≤ (p.changeOrigin x).radius

Tactic:
apply le_radius_of_summable_nnnorm

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
x : E
r : ℝ≥0
_h0 : 0 < r
hr : ↑‖x‖₊ + ↑r < p.radius
hr' : ↑‖x‖₊ < p.radius
⊢ Summable fun n ↦ ‖p.changeOrigin x n‖₊ * r ^ n
Module: Mathlib.Analysis.Normed.Module.Multilinear.Curry

Goal state before:
pp: 𝕜 : Type u
n : ℕ
Ei : Fin n.succ → Type wEi
G : Type wG
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝² : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ContinuousMultilinearMap 𝕜 Ei G
m : (i : Fin n.succ) → Ei i
⊢ (f.curryRight (init m)) (m (last n)) = f m

Tactic:
curryRight_apply,

Goal state after:
𝕜 : Type u
n : ℕ
Ei : Fin n.succ → Type wEi
G : Type wG
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)
inst✝² : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ContinuousMultilinearMap 𝕜 Ei G
m : (i : Fin n.succ) → Ei i
⊢ f (snoc (init m) (m (last n))) = f m
Module: Mathlib.Analysis.Analytic.ChangeOrigin

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {x | AnalyticAt 𝕜 f x}

Tactic:
rw [isOpen_iff_mem_nhds]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
⊢ ∀ x ∈ {x | AnalyticAt 𝕜 f x}, {x | AnalyticAt 𝕜 f x} ∈ 𝓝 x
Module: Mathlib.Analysis.Analytic.CPolynomialDef

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
n : ℕ
r : ℝ≥0∞
finite : ∀ (m : ℕ), n ≤ m → p m = 0
pos : 0 < r
sum_eq : ∀ y ∈ Metric.eball 0 r, (∑ i ∈ Finset.range n, (p i) fun x ↦ y) = f (x + y)
y✝ : E
hy : y✝ ∈ Metric.eball 0 r
m : ℕ
hm : m ∉ Finset.range n
⊢ ((p m) fun x ↦ y✝) = 0

Tactic:
rw [Finset.mem_range, not_lt] at hm

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
n : ℕ
r : ℝ≥0∞
finite : ∀ (m : ℕ), n ≤ m → p m = 0
pos : 0 < r
sum_eq : ∀ y ∈ Metric.eball 0 r, (∑ i ∈ Finset.range n, (p i) fun x ↦ y) = f (x + y)
y✝ : E
hy : y✝ ∈ Metric.eball 0 r
m : ℕ
hm : n ≤ m
⊢ ((p m) fun x ↦ y✝) = 0
Module: Mathlib.Analysis.Analytic.CPolynomialDef

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
x : E
m : ℕ
hm : m ∉ Finset.range n
⊢ ((p m) fun x_1 ↦ x) = 0

Tactic:
rw [Finset.mem_range, not_lt] at hm

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
x : E
m : ℕ
hm : n ≤ m
⊢ ((p m) fun x_1 ↦ x) = 0