mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Group.SeparationQuotient

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝¹ : SeminormedAddCommGroup M
inst✝ : SeminormedAddCommGroup N
f : NormedAddGroupHom M N
hf : ∀ (x : M), ‖x‖ = 0 → f x = 0
x : SeparationQuotient M
⊢ ‖(liftNormedAddGroupHom f hf) x‖ ≤ ‖f‖ * ‖x‖

Tactic:
by
  obtain ⟨x, rfl⟩ := surjective_mk x
  exact le_opNorm f x

Goal state after:
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u
inst✝⁴ : NontriviallyNormedField K
hu : IsUltrametricDist K
L : Type u_2
inst✝³ : NormedField L
inst✝² : NormedAlgebra K L
inst✝¹ : Algebra.IsAlgebraic K L
inst✝ : CompleteSpace K
x : L
⊢ (toMulAlgebraNorm K L).toAlgebraNorm x = (spectralAlgNorm K L) x

Tactic:
spectralNorm_unique (f := (toMulAlgebraNorm K L).toAlgebraNorm)
      (MulRingNorm.isPowMul (toMulAlgebraNorm K L).toMulRingNorm)

Goal state after:
K : Type u
inst✝⁴ : NontriviallyNormedField K
hu : IsUltrametricDist K
L : Type u_2
inst✝³ : NormedField L
inst✝² : NormedAlgebra K L
inst✝¹ : Algebra.IsAlgebraic K L
inst✝ : CompleteSpace K
x : L
⊢ (spectralAlgNorm K L) x = (spectralAlgNorm K L) x
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: R : Type u_1
K : Type u
inst✝¹¹ : NontriviallyNormedField K
L✝ : Type v
inst✝¹⁰ : Field L✝
inst✝⁹ : Algebra K L✝
inst✝⁸ : Algebra.IsAlgebraic K L✝
hu : IsUltrametricDist K
inst✝⁷ : CompleteSpace K
E : Type u_2
L : Type u_3
inst✝⁶ : Field L
inst✝⁵ : Algebra K L
inst✝⁴ : Algebra.IsAlgebraic K L
inst✝³ : NormedField E
inst✝² : NormedAlgebra K E
inst✝¹ : Algebra E L
inst✝ : IsScalarTower K E L
x✝⁴ : NormedField L := normedField K L
x✝³ : NormedAlgebra K L := normedAlgebra K L
x✝² : Algebra.IsAlgebraic K E := Algebra.IsAlgebraic.tower_bot K E L
x✝¹ : E
x✝ : L
⊢ ‖(algebraMap E L) x✝¹‖ = ‖x✝¹‖ ∨ x✝ = 0

Tactic:
simp only [NormedAlgebra.norm_eq_spectralNorm K]

Goal state after:
R : Type u_1
K : Type u
inst✝¹¹ : NontriviallyNormedField K
L✝ : Type v
inst✝¹⁰ : Field L✝
inst✝⁹ : Algebra K L✝
inst✝⁸ : Algebra.IsAlgebraic K L✝
hu : IsUltrametricDist K
inst✝⁷ : CompleteSpace K
E : Type u_2
L : Type u_3
inst✝⁶ : Field L
inst✝⁵ : Algebra K L
inst✝⁴ : Algebra.IsAlgebraic K L
inst✝³ : NormedField E
inst✝² : NormedAlgebra K E
inst✝¹ : Algebra E L
inst✝ : IsScalarTower K E L
x✝⁴ : NormedField L := normedField K L
x✝³ : NormedAlgebra K L := normedAlgebra K L
x✝² : Algebra.IsAlgebraic K E := Algebra.IsAlgebraic.tower_bot K E L
x✝¹ : E
x✝ : L
⊢ spectralNorm K L ((algebraMap E L) x✝¹) = spectralNorm K E x✝¹ ∨ x✝ = 0
Module: Mathlib.Analysis.Normed.Unbundled.SpectralNorm

Goal state before:
pp: K : Type u
inst✝⁸ : NontriviallyNormedField K
L : Type v
inst✝⁷ : Field L
inst✝⁶ : Algebra K L
hu : IsUltrametricDist K
inst✝⁵ : CompleteSpace K
x : L
E : Type u_2
inst✝⁴ : Field E
inst✝³ : Algebra K E
inst✝² : Algebra L E
inst✝¹ : IsScalarTower K L E
inst✝ : Algebra.IsAlgebraic K E
a : E
ha : a ∈ ((mapAlg K E) (minpoly K x)).roots
⊢ (spectralMulAlgNorm K E) a = (spectralMulAlgNorm K E) ((algebraMap L E) x)

Tactic:
by
  simp only [spectralMulAlgNorm_def, spectralNorm]
  have : (aeval a) (minpoly K ((algebraMap L E) x)) = 0 := by
    simp only [mem_roots', IsRoot.def] at ha
    rw [← ha.2, mapAlg_eq_map, minpoly.algebraMap_eq (algebraMap L E).injective, aeval_def,
      eval_map]
  rw [← minpoly.eq_of_root (Algebra.IsAlgebraic.isAlgebraic ((algebraMap L E) x)) this]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Summable

Goal state before:
pp: ι : Type u_1
R : Type u_2
inst✝¹ : NormedCommRing R
inst✝ : NormOneClass R
f : ι → R
hf : Summable fun i ↦ ‖f i‖
i : ι
⊢ ‖‖1 + f i‖ - 1‖ ≤ ‖f i‖

Tactic:
simpa using abs_norm_sub_norm_le (1 + f i) 1

Goal state after:
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q : { P // IsLprojection X P }
⊢ ↑P = ↑P * ↑(P ⊔ Q)

Tactic:
coe_sup,

Goal state after:
X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q : { P // IsLprojection X P }
⊢ ↑P = ↑P * (↑P + ↑Q - ↑P * ↑Q)
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q : { P // IsLprojection X P }
⊢ ↑Q = ↑Q * ↑(P ⊔ Q)

Tactic:
coe_sup,

Goal state after:
X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q : { P // IsLprojection X P }
⊢ ↑Q = ↑Q * (↑P + ↑Q - ↑P * ↑Q)
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ ↑P = ↑P * ↑R → ↑Q = ↑Q * ↑R → ↑(P ⊔ Q) = ↑((P ⊔ Q) ⊓ R)

Tactic:
coe_sup,

Goal state after:
X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ ↑P = ↑P * ↑R → ↑Q = ↑Q * ↑R → ↑P + ↑Q - ↑P * ↑Q = ↑((P ⊔ Q) ⊓ R)
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ ↑P = ↑P * ↑R → ↑Q = ↑Q * ↑R → ↑P + ↑Q - ↑P * ↑Q = ↑(P ⊔ Q) * ↑R

Tactic:
coe_sup,

Goal state after:
X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ ↑P = ↑P * ↑R → ↑Q = ↑Q * ↑R → ↑P + ↑Q - ↑P * ↑Q = (↑P + ↑Q - ↑P * ↑Q) * ↑R
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ ↑(P ⊔ Q) * ↑(P ⊔ R) = ↑P + ↑Q * ↑R * ↑Pᶜ

Tactic:
coe_sup,

Goal state after:
X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ (↑P + ↑Q - ↑P * ↑Q) * ↑(P ⊔ R) = ↑P + ↑Q * ↑R * ↑Pᶜ
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ (↑P + ↑Q - ↑P * ↑Q) * ↑(P ⊔ R) = ↑P + ↑Q * ↑R * ↑Pᶜ

Tactic:
coe_sup,

Goal state after:
X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
⊢ (↑P + ↑Q - ↑P * ↑Q) * (↑P + ↑R - ↑P * ↑R) = ↑P + ↑Q * ↑R * ↑Pᶜ
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
e₁ : ↑((P ⊔ Q) ⊓ (P ⊔ R)) = ↑P + ↑Q * ↑R * ↑Pᶜ
⊢ ↑(P ⊔ Q) * ↑(P ⊔ R) * ↑(P ⊔ Q ⊓ R) = ↑P + ↑Q * ↑R * ↑Pᶜ

Tactic:
coe_sup,

Goal state after:
X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
e₁ : ↑((P ⊔ Q) ⊓ (P ⊔ R)) = ↑P + ↑Q * ↑R * ↑Pᶜ
⊢ (↑P + ↑Q - ↑P * ↑Q) * ↑(P ⊔ R) * ↑(P ⊔ Q ⊓ R) = ↑P + ↑Q * ↑R * ↑Pᶜ
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
e₁ : ↑((P ⊔ Q) ⊓ (P ⊔ R)) = ↑P + ↑Q * ↑R * ↑Pᶜ
⊢ (↑P + ↑Q - ↑P * ↑Q) * ↑(P ⊔ R) * ↑(P ⊔ Q ⊓ R) = ↑P + ↑Q * ↑R * ↑Pᶜ

Tactic:
coe_sup,

Goal state after:
X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
e₁ : ↑((P ⊔ Q) ⊓ (P ⊔ R)) = ↑P + ↑Q * ↑R * ↑Pᶜ
⊢ (↑P + ↑Q - ↑P * ↑Q) * (↑P + ↑R - ↑P * ↑R) * ↑(P ⊔ Q ⊓ R) = ↑P + ↑Q * ↑R * ↑Pᶜ
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
e₁ : ↑((P ⊔ Q) ⊓ (P ⊔ R)) = ↑P + ↑Q * ↑R * ↑Pᶜ
⊢ (↑P + ↑Q - ↑P * ↑Q) * (↑P + ↑R - ↑P * ↑R) * ↑(P ⊔ Q ⊓ R) = ↑P + ↑Q * ↑R * ↑Pᶜ

Tactic:
coe_sup,

Goal state after:
X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P Q R : { P // IsLprojection X P }
e₁ : ↑((P ⊔ Q) ⊓ (P ⊔ R)) = ↑P + ↑Q * ↑R * ↑Pᶜ
⊢ (↑P + ↑Q - ↑P * ↑Q) * (↑P + ↑R - ↑P * ↑R) * (↑P + ↑(Q ⊓ R) - ↑P * ↑(Q ⊓ R)) = ↑P + ↑Q * ↑R * ↑Pᶜ
Module: Mathlib.Analysis.Normed.Module.MStructure

Goal state before:
pp: X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P : { P // IsLprojection X P }
⊢ 1 = ↑(P ⊔ Pᶜ)

Tactic:
coe_sup,

Goal state after:
X : Type u_1
inst✝³ : NormedAddCommGroup X
M : Type u_2
inst✝² : Ring M
inst✝¹ : Module M X
inst✝ : FaithfulSMul M X
P : { P // IsLprojection X P }
⊢ 1 = ↑P + ↑Pᶜ - ↑P * ↑Pᶜ
Module: Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm

Goal state before:
pp: ι : Type u_1
inst✝⁵ : Fintype ι
𝕜 : Type u_2
E : ι → Type u_3
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type u_4
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : (⨂[𝕜] (i : ι), E i) →L[𝕜] F
⊢ ‖(liftEquiv 𝕜 E F).symm f‖ ≤ ‖f‖

Tactic:
by
      rw [liftEquiv_symm_apply]
      exact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Perturbation.StrictByFinite

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : CompleteSpace 𝕜
E : Type u_2
F : Type u_3
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
A : Submodule 𝕜 E
A_closed : IsClosed[inst✝⁵] ↑A
A_cofg : A.CoFG
h_ker : Disjoint (↑u).ker A
range_u : (↑u).range = ⊤
range_u_restr : IsClosed[inst✝²] ↑(↑(u.domRestrict A)).range
S : Submodule 𝕜 E
ker_le_S : (↑u).ker ≤ S
this✝ : FiniteDimensional 𝕜 ↥S
S_compl_A : IsTopCompl S A
uS_compl_uA : IsTopCompl (map (↑u) S) (map (↑u) A)
this : T2Space ↥(map (↑u) S)
uₛ : ↥S →L[𝕜] ↥(map (↑u) S) := u.restrict ⋯
⊢ IsStrictMap ⇑u ↔ IsStrictMap ⇑(u.domRestrict A)

Tactic:
set uₐ : A →L[𝕜] map u.toLinearMap A := u.restrict (fun _ ↦ mem_map_of_mem)

Goal state after:
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : CompleteSpace 𝕜
E : Type u_2
F : Type u_3
inst✝⁹ : AddCommGroup E
inst✝⁸ : Module 𝕜 E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul 𝕜 F
u : E →L[𝕜] F
A : Submodule 𝕜 E
A_closed : IsClosed[inst✝⁵] ↑A
A_cofg : A.CoFG
h_ker : Disjoint (↑u).ker A
range_u : (↑u).range = ⊤
range_u_restr : IsClosed[inst✝²] ↑(↑(u.domRestrict A)).range
S : Submodule 𝕜 E
ker_le_S : (↑u).ker ≤ S
this✝ : FiniteDimensional 𝕜 ↥S
S_compl_A : IsTopCompl S A
uS_compl_uA : IsTopCompl (map (↑u) S) (map (↑u) A)
this : T2Space ↥(map (↑u) S)
uₛ : ↥S →L[𝕜] ↥(map (↑u) S) := u.restrict ⋯
uₐ : ↥A →L[𝕜] ↥(map (↑u) A) := u.restrict ⋯
⊢ IsStrictMap ⇑u ↔ IsStrictMap ⇑(u.domRestrict A)
Module: Mathlib.Analysis.Normed.Lp.lpHolder

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
E : ι → Type u_3
F : ι → Type u_4
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → NormedAddCommGroup (F i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (F i)
p q : ℝ≥0∞
H : Type u_6
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : NormedSpace 𝕜 H
inst✝³ : CompleteSpace H
inst✝² : Fact (1 ≤ p)
inst✝¹ : Fact (1 ≤ q)
inst✝ : p.HolderConjugate q
B : (i : ι) → E i →L[𝕜] F i →L[𝕜] H
K : ℝ≥0
hBK : ∀ (i : ι), ‖B i‖ ≤ ↑K
⊢ ‖ContinuousLinearMap.postcomp (↥(lp F q)) (tsumCLM 𝕜 ι H)‖ * ‖holderL 1 B hBK‖ ≤ 1 * ↑K

Tactic:
gcongr
    · exact ContinuousLinearMap.norm_postcomp_le _ |>.trans norm_tsumCLM_le
    · exact norm_holderL_le 1 B hBK

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Int

Goal state before:
pp: e : ℤˣ
⊢ ↑1 = 1

Tactic:
NNReal.coe_one

Goal state after:
e : ℤˣ
⊢ 1 = 1
Module: Mathlib.Analysis.Normed.Order.UpperLower

Goal state before:
pp: ι : Type u_2
inst✝ : Finite ι
s : Set (ι → ℝ)
x y : ι → ℝ
hs : IsLowerSet s
hx : x ∈ closure s
h : ∀ (i : ι), y i < x i
val✝ : Fintype ι
ε : ℝ
hε : 0 < ε
hxy : ∀ (i : ι), y i + ε < x i
z : ι → ℝ
hz : z ∈ s
hxz : ∀ (b : ι), dist (x b) (z b) < ε
i : ι
⊢ |x i - z i| ≤ ε

Tactic:
rw [← Real.norm_eq_abs, ← dist_eq_norm]

Goal state after:
ι : Type u_2
inst✝ : Finite ι
s : Set (ι → ℝ)
x y : ι → ℝ
hs : IsLowerSet s
hx : x ∈ closure s
h : ∀ (i : ι), y i < x i
val✝ : Fintype ι
ε : ℝ
hε : 0 < ε
hxy : ∀ (i : ι), y i + ε < x i
z : ι → ℝ
hz : z ∈ s
hxz : ∀ (b : ι), dist (x b) (z b) < ε
i : ι
⊢ dist (x i) (z i) ≤ ε
Module: Mathlib.Analysis.ODE.Transform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
s : Set ℝ
dt : ℝ
⊢ IsIntegralCurveOn (γ ∘ fun x ↦ x + dt) (v ∘ fun x ↦ x + dt) (-dt +ᵥ s) ↔ IsIntegralCurveOn γ v s

Tactic:
refine ⟨fun hγ ↦ ?_, fun hγ ↦ hγ.comp_add _⟩

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
γ : ℝ → E
v : ℝ → E → E
s : Set ℝ
dt : ℝ
hγ : IsIntegralCurveOn (γ ∘ fun x ↦ x + dt) (v ∘ fun x ↦ x + dt) (-dt +ᵥ s)
⊢ IsIntegralCurveOn γ v s
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ : E
a r L : ℝ≥0
inst✝ : CompleteSpace E
⊢ CompleteSpace (FunSpace t₀ x₀ r L)

Tactic:
rw [completeSpace_iff_isComplete_range isUniformInducing_toContinuousMap]
  apply IsClosed.isComplete
  rw [range_toContinuousMap, ofPred_and]
  apply isClosed_setOfPred_lipschitzWith L |>.preimage continuous_coeFun |>.inter
  simp_rw [mem_closedBall_iff_norm]
  exact isClosed_le (by fun_prop) (by fun_prop)

Goal state after:
Module: Mathlib.Analysis.ODE.ExistUnique

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
v : ℝ → E → E
s : ℝ → Set E
K : ℝ≥0
f g : ℝ → E
a b t₀ : ℝ
hv : ∀ t ∈ Ioo a b, LipschitzOnWith K (v t) (s t)
ht : t₀ ∈ Ioo a b
hf : ∀ t ∈ Ioo a b, HasDerivAt f (v t (f t)) t ∧ f t ∈ s t
hg : ∀ t ∈ Ioo a b, HasDerivAt g (v t (g t)) t ∧ g t ∈ s t
heq : f t₀ = g t₀
⊢ EqOn f g (Ioo a b)

Tactic:
by
  intro t' ht'
  rcases lt_or_ge t' t₀ with (h | h)
  · have hss : Icc t' t₀ ⊆ Ioo a b :=
      fun _ ht'' ↦ ⟨lt_of_lt_of_le ht'.1 ht''.1, lt_of_le_of_lt ht''.2 ht.2⟩
    exact ODE_solution_unique_of_mem_Icc_left
      (fun t'' ht'' ↦ hv t'' ((Ioc_subset_Icc_self.trans hss) ht''))
      (HasDerivAt.continuousOn fun _ ht'' ↦ (hf _ <| hss ht'').1)
      (fun _ ht'' ↦ (hf _ <| hss <| Ioc_subset_Icc_self ht'').1.hasDerivWithinAt)
      (fun _ ht'' ↦ (hf _ <| hss <| Ioc_subset_Icc_self ht'').2)
      (HasDerivAt.continuousOn fun _ ht'' ↦ (hg _ <| hss ht'').1)
      (fun _ ht'' ↦ (hg _ <| hss <| Ioc_subset_Icc_self ht'').1.hasDerivWithinAt)
      (fun _ ht'' ↦ (hg _ <| hss <| Ioc_subset_Icc_self ht'').2) heq
      ⟨le_rfl, le_of_lt h⟩
  · have hss : Icc t₀ t' ⊆ Ioo a b :=
      fun _ ht'' ↦ ⟨lt_of_lt_of_le ht.1 ht''.1, lt_of_le_of_lt ht''.2 ht'.2⟩
    exact ODE_solution_unique_of_mem_Icc_right
      (fun t'' ht'' ↦ hv t'' ((Ico_subset_Icc_self.trans hss) ht''))
      (HasDerivAt.continuousOn fun _ ht'' ↦ (hf _ <| hss ht'').1)
      (fun _ ht'' ↦ (hf _ <| hss <| Ico_subset_Icc_self ht'').1.hasDerivWithinAt)
      (fun _ ht'' ↦ (hf _ <| hss <| Ico_subset_Icc_self ht'').2)
      (HasDerivAt.continuousOn fun _ ht'' ↦ (hg _ <| hss ht'').1)
      (fun _ ht'' ↦ (hg _ <| hss <| Ico_subset_Icc_self ht'').1.hasDerivWithinAt)
      (fun _ ht'' ↦ (hg _ <| hss <| Ico_subset_Icc_self ht'').2) heq
      ⟨h, le_rfl⟩

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ x : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
hx : x ∈ closedBall x₀ ↑r
α β : FunSpace t₀ x₀ r L
n : ℕ
⊢ dist ((next hf hx)^[n] α) ((next hf hx)^[n] β) ≤ (↑K * max (tmax - ↑t₀) (↑t₀ - tmin)) ^ n / ↑n ! * dist α β

Tactic:
rw [← MetricSpace.isometry_induced FunSpace.toContinuousMap FunSpace.toContinuousMap.injective
    |>.dist_eq, ContinuousMap.dist_le]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ x : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
hx : x ∈ closedBall x₀ ↑r
α β : FunSpace t₀ x₀ r L
n : ℕ
⊢ ∀ (x_1 : ↑(Icc tmin tmax)),
    dist ((toContinuousMap ((next hf hx)^[n] α)) x_1) ((toContinuousMap ((next hf hx)^[n] β)) x_1) ≤
      (↑K * max (tmax - ↑t₀) (↑t₀ - tmin)) ^ n / ↑n ! * dist α β

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ x : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
hx : x ∈ closedBall x₀ ↑r
α β : FunSpace t₀ x₀ r L
n : ℕ
⊢ 0 ≤ (↑K * max (tmax - ↑t₀) (↑t₀ - tmin)) ^ n / ↑n ! * dist α β
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hP : ¬P = 0
hQ : Q = 0
⊢ (fun x ↦ eval x P / eval x Q) ~[atTop] fun x ↦ P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)

Tactic:
simp [hQ, IsEquivalent.refl]

Goal state after:
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ x : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
hx : x ∈ closedBall x₀ ↑r
α : FunSpace t₀ x₀ r L
n : ℕ
⊢ dist α ((next hf hx)^[n] α) ≤
    (∑ i ∈ Finset.range n, (↑K * max (tmax - ↑t₀) (↑t₀ - tmin)) ^ i / ↑i !) * dist α (next hf hx α)

Tactic:
nth_rw 1 [← iterate_zero_apply (next hf hx) α]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ x : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
hx : x ∈ closedBall x₀ ↑r
α : FunSpace t₀ x₀ r L
n : ℕ
⊢ dist ((next hf hx)^[0] α) ((next hf hx)^[n] α) ≤
    (∑ i ∈ Finset.range n, (↑K * max (tmax - ↑t₀) (↑t₀ - tmin)) ^ i / ↑i !) * dist α (next hf hx α)
Module: Mathlib.Analysis.ODE.PicardLindelof

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ x : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
hx : x ∈ closedBall x₀ ↑r
α : FunSpace t₀ x₀ r L
m : ℕ
C : ℝ≥0
hm : LipschitzWith C (next hf hx)^[m]
n : ℕ
⊢ dist α ((next hf hx)^[m]^[n] α) ≤
    ((∑ i ∈ Finset.range m, (↑K * max (tmax - ↑t₀) (↑t₀ - tmin)) ^ i / ↑i !) * ∑ i ∈ Finset.range n, ↑C ^ i) *
      dist α (next hf hx α)

Tactic:
nth_rw 1 [← iterate_zero_apply (next hf hx) α]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E → E
tmin tmax : ℝ
t₀ : ↑(Icc tmin tmax)
x₀ x : E
a r L K : ℝ≥0
hf : IsPicardLindelof f t₀ x₀ a r L K
hx : x ∈ closedBall x₀ ↑r
α : FunSpace t₀ x₀ r L
m : ℕ
C : ℝ≥0
hm : LipschitzWith C (next hf hx)^[m]
n : ℕ
⊢ dist ((next hf hx)^[0] α) ((next hf hx)^[m]^[n] α) ≤
    ((∑ i ∈ Finset.range m, (↑K * max (tmax - ↑t₀) (↑t₀ - tmin)) ^ i / ↑i !) * ∑ i ∈ Finset.range n, ↑C ^ i) *
      dist α (next hf hx α)
Module: Mathlib.Analysis.Polynomial.Fourier

Goal state before:
pp: p : ℂ[X]
this :
  ∑ b ∈ Finset.map { toFun := Nat.cast, inj' := ⋯ } p.support, ‖if 0 ≤ b then p.coeff b.natAbs else 0‖ ^ 2 =
    ∫ (t : AddCircle (2 * π)), ‖↑↑((ContinuousMap.toLp 2 haarAddCircle ℂ) (toAddCircle p)) t‖ ^ 2 ∂haarAddCircle
⊢ ∑ i ∈ p.support, ‖p.coeff i‖ ^ 2 = Real.circleAverage (fun θ ↦ ‖eval θ p‖ ^ 2) 0 1

Tactic:
simp only [Finset.sum_map, Function.Embedding.coeFn_mk, Nat.cast_nonneg,
    ↓reduceIte, Int.natAbs_natCast, ContinuousMap.coe_toLp] at this

Goal state after:
p : ℂ[X]
this :
  ∑ x ∈ p.support, ‖p.coeff x‖ ^ 2 =
    ∫ (t : AddCircle (2 * π)), ‖↑(ContinuousMap.toAEEqFun haarAddCircle (toAddCircle p)) t‖ ^ 2 ∂haarAddCircle
⊢ ∑ i ∈ p.support, ‖p.coeff i‖ ^ 2 = Real.circleAverage (fun θ ↦ ‖eval θ p‖ ^ 2) 0 1
Module: Mathlib.Analysis.Polynomial.Basic

Goal state before:
pp: case pos
𝕜 : Type u_1
inst✝³ : NormedField 𝕜
inst✝² : LinearOrder 𝕜
inst✝¹ : IsStrictOrderedRing 𝕜
P Q : 𝕜[X]
inst✝ : OrderTopology 𝕜
hP : ¬P = 0
hQ : Q = 0
⊢ (fun x ↦ eval x P / eval x Q) ~[atBot] fun x ↦ P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)

Tactic:
simp [hQ, IsEquivalent.refl]

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

Goal state before:
pp: R : Type u_1
σ : Type u_2
v : R → ℝ
c : σ → ℝ
inst✝¹ : Ring R
inst✝ : DecidableEq σ
f g : MvPowerSeries σ R
vMulEq : ∀ (a b : R), v (a * b) = v a * v b
vna : IsNonarchimedean v
vNeg : ∀ (a : R), v a = v (-a)
hbfg : HasGaussNorm v c (f * g)
i₀ j₀ : σ →₀ ℕ
hi₀ : (v ((coeff i₀) f) * i₀.prod fun x1 x2 ↦ c x1 ^ x2) = gaussNorm v c f
hj₀ : (v ((coeff j₀) g) * j₀.prod fun x1 x2 ↦ c x1 ^ x2) = gaussNorm v c g
hdom' :
  ∀ p ∈ Finset.antidiagonal (i₀ + j₀),
    p ≠ (i₀, j₀) → v ((coeff p.1) f * (coeff p.2) g) < v ((coeff i₀) f) * v ((coeff j₀) g)
⊢ ((i₀ + j₀).prod fun x1 x2 ↦ c x1 ^ x2) = (i₀.prod fun x1 x2 ↦ c x1 ^ x2) * j₀.prod fun x1 x2 ↦ c x1 ^ x2

Tactic:
simp [Finsupp.prod_add_index', pow_add]

Goal state after:
Module: Mathlib.Analysis.Real.Pi.Bounds

Goal state before:
pp: n : ℕ
this : π < (√(2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)
⊢ (√(2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2) =
    2 ^ (n + 1) * √(2 - sqrtTwoAddSeries 0 n) + 1 / 4 ^ n

Tactic:
rw [add_mul]

Goal state after:
n : ℕ
this : π < (√(2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)
⊢ √(2 - sqrtTwoAddSeries 0 n) / 2 * 2 ^ (n + 2) + 1 / (2 ^ n) ^ 3 / 4 * 2 ^ (n + 2) =
    2 ^ (n + 1) * √(2 - sqrtTwoAddSeries 0 n) + 1 / 4 ^ n
Module: Mathlib.Analysis.Real.Pi.Bounds

Goal state before:
pp: n : ℕ
a : ℝ
h : 2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ (↑0 / ↑1).sqrtTwoAddSeries n
h₂ : 1 / 4 ^ n ≤ a
⊢ 2 ^ (n + 1) * √(2 - sqrtTwoAddSeries 0 n) ≤ a - 1 / 4 ^ n

Tactic:
← le_div_iff₀',

Goal state after:
n : ℕ
a : ℝ
h : 2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ (↑0 / ↑1).sqrtTwoAddSeries n
h₂ : 1 / 4 ^ n ≤ a
⊢ √(2 - sqrtTwoAddSeries 0 n) ≤ (a - 1 / 4 ^ n) / 2 ^ (n + 1)

n : ℕ
a : ℝ
h : 2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ (↑0 / ↑1).sqrtTwoAddSeries n
h₂ : 1 / 4 ^ n ≤ a
⊢ 0 < 2 ^ (n + 1)
Module: Mathlib.Analysis.Polynomial.MahlerMeasure

Goal state before:
pp: case neg
A : Type u_2
inst✝ : NormedRing A
p : A[X]
v : A →+* ℂ
hv : Isometry ⇑v
hp : ¬p.leadingCoeff = 0
hv_ne : v p.leadingCoeff ≠ 0
hv_norm : ‖v p.leadingCoeff‖ = ‖p.leadingCoeff‖
⊢ ‖p.leadingCoeff‖ ≤ p.mapMahlerMeasure v

Tactic:
grw [← hv_norm, ← leadingCoeff_map_of_leadingCoeff_ne_zero v hv_ne,
      leadingCoeff_le_mahlerMeasure, mapMahlerMeasure]

Goal state after:
Module: Mathlib.NumberTheory.Real.Irrational

Goal state before:
pp: x : ℝ
h : Irrational x
m : ℤ
⊢ x ≠ ↑m

Tactic:
by
  rw [← Rat.cast_intCast]
  exact h.ne_rat _

Goal state after:
Module: Mathlib.NumberTheory.Real.Irrational

Goal state before:
pp: x : ℝ
h : Irrational (1 / x)
⊢ Irrational (↑1 / x)

Tactic:
cast_one

Goal state after:
x : ℝ
h : Irrational (1 / x)
⊢ Irrational (1 / x)
Module: Mathlib.Analysis.Real.Pi.Wallis

Goal state before:
pp: case succ
n : ℕ
IH : W n = 2 ^ (4 * n) * ↑n ! ^ 4 / (↑(2 * n)! ^ 2 * (2 * ↑n + 1))
⊢ W (n + 1) = 2 ^ (4 * (n + 1)) * ↑(n + 1)! ^ 4 / (↑(2 * (n + 1))! ^ 2 * (2 * ↑(n + 1) + 1))

Tactic:
| succ n IH =>

Goal state after: