1337xyz1337xyz/leandojo-benchmark4-v10 huggingface dataset:
Theorem: ContinuousWithinAt.nhdsWithin_extChartAt_symm_preimage_inter_range
File: Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean

tactic: rw [← (extChartAt I x).image_source_inter_eq', ← map_extChartAt_nhdsWithin_eq_image,
  ← map_extChartAt_nhdsWithin, nhdsWithin_inter_of_mem']
state_before: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
E' : Type u_5
M' : Type u_6
H' : Type u_7
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : TopologicalSpace H
inst✝⁶ : TopologicalSpace M
f✝ f' : PartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : TopologicalSpace H'
inst✝² : TopologicalSpace M'
I' : ModelWithCorners 𝕜 E' H'
s t : Set M
inst✝¹ : ChartedSpace H M
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
hc : ContinuousWithinAt f s x
⊢ 𝓝[↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I] ↑(extChartAt I x) x =
    𝓝[(extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source)] ↑(extChartAt I x) x
state_after: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
E' : Type u_5
M' : Type u_6
H' : Type u_7
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : TopologicalSpace H
inst✝⁶ : TopologicalSpace M
f✝ f' : PartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : TopologicalSpace H'
inst✝² : TopologicalSpace M'
I' : ModelWithCorners 𝕜 E' H'
s t : Set M
inst✝¹ : ChartedSpace H M
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
hc : ContinuousWithinAt f s x
⊢ f ⁻¹' (extChartAt I' (f x)).source ∈ 𝓝[s] x

tactic: exact hc (extChartAt_source_mem_nhds _ _)
state_before: 𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
E' : Type u_5
M' : Type u_6
H' : Type u_7
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : TopologicalSpace H
inst✝⁶ : TopologicalSpace M
f✝ f' : PartialHomeomorph M H
I : ModelWithCorners 𝕜 E H
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : TopologicalSpace H'
inst✝² : TopologicalSpace M'
I' : ModelWithCorners 𝕜 E' H'
s t : Set M
inst✝¹ : ChartedSpace H M
inst✝ : ChartedSpace H' M'
f : M → M'
x : M
hc : ContinuousWithinAt f s x
⊢ f ⁻¹' (extChartAt I' (f x)).source ∈ 𝓝[s] x
state_after: no goals
Theorem: ModularForm.add_slash
File: Mathlib/NumberTheory/ModularForms/SlashActions.lean

tactic: ext1
state_before: Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
⊢ (f + g) ∣[k]A = f ∣[k]A + g ∣[k]A
state_after: case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x✝ : ℍ
⊢ ((f + g) ∣[k]A) x✝ = (f ∣[k]A + g ∣[k]A) x✝

tactic: simp only [slash, Pi.add_apply, denom, zpow_neg]
state_before: case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x✝ : ℍ
⊢ ((f + g) ∣[k]A) x✝ = (f ∣[k]A + g ∣[k]A) x✝
state_after: case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x✝ : ℍ
⊢ (f (A • x✝) + g (A • x✝)) * ↑(↑↑A).det ^ (k - 1) * ((↑(↑↑A 1 0) * ↑x✝ + ↑(↑↑A 1 1)) ^ k)⁻¹ =
    f (A • x✝) * ↑(↑↑A).det ^ (k - 1) * ((↑(↑↑A 1 0) * ↑x✝ + ↑(↑↑A 1 1)) ^ k)⁻¹ +
      g (A • x✝) * ↑(↑↑A).det ^ (k - 1) * ((↑(↑↑A 1 0) * ↑x✝ + ↑(↑↑A 1 1)) ^ k)⁻¹

tactic: ring
state_before: case h
Γ : Subgroup SL(2, ℤ)
k✝ : ℤ
f✝ : ℍ → ℂ
k : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x✝ : ℍ
⊢ (f (A • x✝) + g (A • x✝)) * ↑(↑↑A).det ^ (k - 1) * ((↑(↑↑A 1 0) * ↑x✝ + ↑(↑↑A 1 1)) ^ k)⁻¹ =
    f (A • x✝) * ↑(↑↑A).det ^ (k - 1) * ((↑(↑↑A 1 0) * ↑x✝ + ↑(↑↑A 1 1)) ^ k)⁻¹ +
      g (A • x✝) * ↑(↑↑A).det ^ (k - 1) * ((↑(↑↑A 1 0) * ↑x✝ + ↑(↑↑A 1 1)) ^ k)⁻¹
state_after: no goals
Theorem: FiniteDimensional.finrank_le_finrank_of_rank_le_rank
File: Mathlib/LinearAlgebra/Dimension/Finrank.lean

tactic: simpa only [toNat_lift] using toNat_le_toNat h (lift_lt_aleph0.mpr h')
state_before: R : Type u
M : Type v
N : Type w
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
h : lift.{w, v} (Module.rank R M) ≤ lift.{v, w} (Module.rank R N)
h' : Module.rank R N < ℵ₀
⊢ finrank R M ≤ finrank R N
state_after: no goals
Theorem: Set.pairwise_union
File: Mathlib/Data/Set/Pairwise/Basic.lean

tactic: simp only [Set.Pairwise, mem_union, or_imp, forall_and]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
r p q : α → α → Prop
f g : ι → α
s t u : Set α
a b : α
⊢ (s ∪ t).Pairwise r ↔ s.Pairwise r ∧ t.Pairwise r ∧ ∀ a ∈ s, ∀ b ∈ t, a ≠ b → r a b ∧ r b a
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
r p q : α → α → Prop
f g : ι → α
s t u : Set α
a b : α
⊢ (((∀ x ∈ s, ∀ x_2 ∈ s, x ≠ x_2 → r x x_2) ∧ ∀ x ∈ s, ∀ x_2 ∈ t, x ≠ x_2 → r x x_2) ∧
      (∀ x ∈ t, ∀ x_2 ∈ s, x ≠ x_2 → r x x_2) ∧ ∀ x ∈ t, ∀ x_2 ∈ t, x ≠ x_2 → r x x_2) ↔
    (∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → x ≠ y → r x y) ∧
      (∀ ⦃x : α⦄, x ∈ t → ∀ ⦃y : α⦄, y ∈ t → x ≠ y → r x y) ∧
        (∀ x ∈ s, ∀ x_2 ∈ t, x ≠ x_2 → r x x_2) ∧ ∀ x ∈ s, ∀ x_2 ∈ t, x ≠ x_2 → r x_2 x

tactic: exact
  ⟨fun H => ⟨H.1.1, H.2.2, H.1.2, fun x hx y hy hne => H.2.1 y hy x hx hne.symm⟩,
   fun H => ⟨⟨H.1, H.2.2.1⟩, fun x hx y hy hne => H.2.2.2 y hy x hx hne.symm, H.2.1⟩⟩
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
r p q : α → α → Prop
f g : ι → α
s t u : Set α
a b : α
⊢ (((∀ x ∈ s, ∀ x_2 ∈ s, x ≠ x_2 → r x x_2) ∧ ∀ x ∈ s, ∀ x_2 ∈ t, x ≠ x_2 → r x x_2) ∧
      (∀ x ∈ t, ∀ x_2 ∈ s, x ≠ x_2 → r x x_2) ∧ ∀ x ∈ t, ∀ x_2 ∈ t, x ≠ x_2 → r x x_2) ↔
    (∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → x ≠ y → r x y) ∧
      (∀ ⦃x : α⦄, x ∈ t → ∀ ⦃y : α⦄, y ∈ t → x ≠ y → r x y) ∧
        (∀ x ∈ s, ∀ x_2 ∈ t, x ≠ x_2 → r x x_2) ∧ ∀ x ∈ s, ∀ x_2 ∈ t, x ≠ x_2 → r x_2 x
state_after: no goals
Theorem: LieSubalgebra.mem_normalizer_iff'
File: Mathlib/Algebra/Lie/Normalizer.lean

tactic: rw [Subtype.forall']
state_before: R : Type u_1
L : Type u_2
M : Type u_3
M' : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : AddCommGroup M'
inst✝² : Module R M'
inst✝¹ : LieRingModule L M'
inst✝ : LieModule R L M'
H : LieSubalgebra R L
x : L
⊢ x ∈ H.normalizer ↔ ∀ y ∈ H, ⁅y, x⁆ ∈ H
state_after: R : Type u_1
L : Type u_2
M : Type u_3
M' : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : AddCommGroup M'
inst✝² : Module R M'
inst✝¹ : LieRingModule L M'
inst✝ : LieModule R L M'
H : LieSubalgebra R L
x : L
⊢ x ∈ H.normalizer ↔ ∀ (x_1 : ↥H), ⁅↑x_1, x⁆ ∈ H

tactic: rfl
state_before: R : Type u_1
L : Type u_2
M : Type u_3
M' : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : AddCommGroup M'
inst✝² : Module R M'
inst✝¹ : LieRingModule L M'
inst✝ : LieModule R L M'
H : LieSubalgebra R L
x : L
⊢ x ∈ H.normalizer ↔ ∀ (x_1 : ↥H), ⁅↑x_1, x⁆ ∈ H
state_after: no goals
Theorem: Algebra.FiniteType.of_surjective
File: Mathlib/RingTheory/FiniteType.lean

tactic: convert hRA.1.map f
state_before: R : Type uR
S : Type uS
A : Type uA
B : Type uB
M : Type uM
N : Type uN
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Semiring A
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
hRA : FiniteType R A
f : A →ₐ[R] B
hf : Surjective ⇑f
⊢ ⊤.FG
state_after: case h.e'_6
R : Type uR
S : Type uS
A : Type uA
B : Type uB
M : Type uM
N : Type uN
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Semiring A
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
hRA : FiniteType R A
f : A →ₐ[R] B
hf : Surjective ⇑f
⊢ ⊤ = Subalgebra.map f ⊤

tactic: simpa only [map_top f, @eq_comm _ ⊤, eq_top_iff, AlgHom.mem_range] using hf
state_before: case h.e'_6
R : Type uR
S : Type uS
A : Type uA
B : Type uB
M : Type uM
N : Type uN
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Semiring A
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
hRA : FiniteType R A
f : A →ₐ[R] B
hf : Surjective ⇑f
⊢ ⊤ = Subalgebra.map f ⊤
state_after: no goals
Theorem: CompleteLattice.WellFounded.finite_ne_bot_of_independent
File: Mathlib/Order/CompactlyGenerated/Basic.lean

tactic: refine Finite.of_finite_image (Finite.subset ?_ (image_subset_range t _)) ht.injOn
state_before: ι✝ : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι✝ → α
hwf : WellFounded fun x x_1 => x > x_1
ι : Type u_3
t : ι → α
ht : Independent t
⊢ {i | t i ≠ ⊥}.Finite
state_after: ι✝ : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι✝ → α
hwf : WellFounded fun x x_1 => x > x_1
ι : Type u_3
t : ι → α
ht : Independent t
⊢ (range t).Finite

tactic: exact WellFounded.finite_of_setIndependent hwf ht.setIndependent_range
state_before: ι✝ : Sort u_1
α : Type u_2
inst✝ : CompleteLattice α
f : ι✝ → α
hwf : WellFounded fun x x_1 => x > x_1
ι : Type u_3
t : ι → α
ht : Independent t
⊢ (range t).Finite
state_after: no goals
Theorem: NonarchAddGroupSeminorm.toZeroHom_eq_coe
File: Mathlib/Analysis/Normed/Group/Seminorm.lean

tactic: rfl
state_before: ι : Type u_1
R : Type u_2
R' : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
inst✝³ : FunLike F E ℝ
inst✝² : AddGroup E
inst✝¹ : AddGroup F
inst✝ : AddGroup G
p q : NonarchAddGroupSeminorm E
⊢ ⇑p.toZeroHom = ⇑p
state_after: no goals
Theorem: ContinuousLinearEquiv.ext₁
File: Mathlib/Topology/Algebra/Module/Basic.lean

tactic: rw [← smul_eq_mul, map_smul, h, map_smul]
state_before: R₁ : Type u_1
R₂ : Type u_2
R₃ : Type u_3
inst✝²⁵ : Semiring R₁
inst✝²⁴ : Semiring R₂
inst✝²³ : Semiring R₃
σ₁₂ : R₁ →+* R₂
σ₂₁ : R₂ →+* R₁
inst✝²² : RingHomInvPair σ₁₂ σ₂₁
inst✝²¹ : RingHomInvPair σ₂₁ σ₁₂
σ₂₃ : R₂ →+* R₃
σ₃₂ : R₃ →+* R₂
inst✝²⁰ : RingHomInvPair σ₂₃ σ₃₂
inst✝¹⁹ : RingHomInvPair σ₃₂ σ₂₃
σ₁₃ : R₁ →+* R₃
σ₃₁ : R₃ →+* R₁
inst✝¹⁸ : RingHomInvPair σ₁₃ σ₃₁
inst✝¹⁷ : RingHomInvPair σ₃₁ σ₁₃
inst✝¹⁶ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁵ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁
M₁ : Type u_4
inst✝¹⁴ : TopologicalSpace M₁
inst✝¹³ : AddCommMonoid M₁
M'₁ : Type u_5
inst✝¹² : TopologicalSpace M'₁
inst✝¹¹ : AddCommMonoid M'₁
M₂ : Type u_6
inst✝¹⁰ : TopologicalSpace M₂
inst✝⁹ : AddCommMonoid M₂
M₃ : Type u_7
inst✝⁸ : TopologicalSpace M₃
inst✝⁷ : AddCommMonoid M₃
M₄ : Type u_8
inst✝⁶ : TopologicalSpace M₄
inst✝⁵ : AddCommMonoid M₄
inst✝⁴ : Module R₁ M₁
inst✝³ : Module R₁ M'₁
inst✝² : Module R₂ M₂
inst✝¹ : Module R₃ M₃
inst✝ : TopologicalSpace R₁
f g : R₁ ≃L[R₁] M₁
h : f 1 = g 1
x : R₁
⊢ f (x * 1) = g (x * 1)
state_after: no goals
Theorem: Int.natAbs_eq_iff_mul_self_eq
File: Mathlib/Data/Int/Order/Lemmas.lean

tactic: rw [← abs_eq_iff_mul_self_eq, abs_eq_natAbs, abs_eq_natAbs]
state_before: a✝ b✝ : ℤ
n : ℕ
a b : ℤ
⊢ a.natAbs = b.natAbs ↔ a * a = b * b
state_after: a✝ b✝ : ℤ
n : ℕ
a b : ℤ
⊢ a.natAbs = b.natAbs ↔ ↑a.natAbs = ↑b.natAbs

tactic: exact Int.natCast_inj.symm
state_before: a✝ b✝ : ℤ
n : ℕ
a b : ℤ
⊢ a.natAbs = b.natAbs ↔ ↑a.natAbs = ↑b.natAbs
state_after: no goals
Theorem: cfc_le_iff
File: Mathlib/Topology/ContinuousFunction/FunctionalCalculus.lean

tactic: rw [cfc_apply f a, cfc_apply g a, cfcHom_le_iff (show p a from ha), ContinuousMap.le_def]
state_before: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁴ : OrderedCommRing R
inst✝¹³ : StarRing R
inst✝¹² : MetricSpace R
inst✝¹¹ : TopologicalRing R
inst✝¹⁰ : ContinuousStar R
inst✝⁹ : ∀ (α : Type ?u.783396) [inst : TopologicalSpace α], StarOrderedRing C(α, R)
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Ring A
inst✝⁶ : StarRing A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : Algebra R A
inst✝² : StarModule R A
inst✝¹ : ContinuousFunctionalCalculus R p
inst✝ : NonnegSpectrumClass R A
f g : R → R
a : A
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝
⊢ cfc f a ≤ cfc g a ↔ ∀ x ∈ spectrum R a, f x ≤ g x
state_after: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁴ : OrderedCommRing R
inst✝¹³ : StarRing R
inst✝¹² : MetricSpace R
inst✝¹¹ : TopologicalRing R
inst✝¹⁰ : ContinuousStar R
inst✝⁹ : ∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Ring A
inst✝⁶ : StarRing A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : Algebra R A
inst✝² : StarModule R A
inst✝¹ : ContinuousFunctionalCalculus R p
inst✝ : NonnegSpectrumClass R A
f g : R → R
a : A
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝
⊢ (∀ (a_1 : ↑(spectrum R a)),
      { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } a_1 ≤
        { toFun := (spectrum R a).restrict g, continuous_toFun := ⋯ } a_1) ↔
    ∀ x ∈ spectrum R a, f x ≤ g x

tactic: simp
state_before: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁴ : OrderedCommRing R
inst✝¹³ : StarRing R
inst✝¹² : MetricSpace R
inst✝¹¹ : TopologicalRing R
inst✝¹⁰ : ContinuousStar R
inst✝⁹ : ∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Ring A
inst✝⁶ : StarRing A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : Algebra R A
inst✝² : StarModule R A
inst✝¹ : ContinuousFunctionalCalculus R p
inst✝ : NonnegSpectrumClass R A
f g : R → R
a : A
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝
⊢ (∀ (a_1 : ↑(spectrum R a)),
      { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } a_1 ≤
        { toFun := (spectrum R a).restrict g, continuous_toFun := ⋯ } a_1) ↔
    ∀ x ∈ spectrum R a, f x ≤ g x
state_after: no goals
Theorem: hasFDerivWithinAt_inter'
File: Mathlib/Analysis/Calculus/FDeriv/Basic.lean

tactic: simp [HasFDerivWithinAt, nhdsWithin_restrict'' s h]
state_before: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
h : t ∈ 𝓝[s] x
⊢ HasFDerivWithinAt f f' (s ∩ t) x ↔ HasFDerivWithinAt f f' s x
state_after: no goals
Theorem: FractionalIdeal.count_coe
File: Mathlib/RingTheory/DedekindDomain/Factorization.lean

tactic: rw [count_well_defined K (J := J) (a := 1), Ideal.span_singleton_one, sub_eq_self,
  Nat.cast_eq_zero, ← Ideal.one_eq_top, Associates.mk_one, Associates.factors_one,
  Associates.count_zero v.associates_irreducible]
state_before: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
J : Ideal R
hJ : J ≠ 0
⊢ count K v ↑J = ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors)
state_after: case hI
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
J : Ideal R
hJ : J ≠ 0
⊢ ↑J ≠ 0

case h_aJ
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
J : Ideal R
hJ : J ≠ 0
⊢ ↑J = spanSingleton R⁰ ((algebraMap R K) 1)⁻¹ * ↑J

tactic: simpa only [ne_eq, coeIdeal_eq_zero]
state_before: case hI
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
J : Ideal R
hJ : J ≠ 0
⊢ ↑J ≠ 0
state_after: no goals

tactic: simp only [_root_.map_one, inv_one, spanSingleton_one, one_mul]
state_before: case h_aJ
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
J : Ideal R
hJ : J ≠ 0
⊢ ↑J = spanSingleton R⁰ ((algebraMap R K) 1)⁻¹ * ↑J
state_after: no goals
Theorem: MvPolynomial.optionEquivRight_X_some
File: Mathlib/Algebra/MvPolynomial/Equiv.lean

tactic: simp [optionEquivRight_apply, aeval_X]
state_before: R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
s : σ →₀ ℕ
inst✝ : CommSemiring R
x : S₁
⊢ (optionEquivRight R S₁) (X (Option.some x)) = X x
state_after: no goals
Theorem: MonoidAlgebra.lift_apply
File: Mathlib/Algebra/MonoidAlgebra/Basic.lean

tactic: simp only [lift_apply', Algebra.smul_def]
state_before: k : Type u₁
G : Type u₂
H : Type u_1
R : Type u_2
inst✝⁶ : CommSemiring k
inst✝⁵ : Monoid G
inst✝⁴ : Monoid H
A : Type u₃
inst✝³ : Semiring A
inst✝² : Algebra k A
B : Type u_3
inst✝¹ : Semiring B
inst✝ : Algebra k B
F : G →* A
f : MonoidAlgebra k G
⊢ ((lift k G A) F) f = sum f fun a b => b • F a
state_after: no goals
Theorem: jacobiSym.quadratic_reciprocity_one_mod_four
File: Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean

tactic: rw [quadratic_reciprocity (odd_iff.mpr (odd_of_mod_four_eq_one ha)) hb, pow_mul,
  neg_one_pow_div_two_of_one_mod_four ha, one_pow, one_mul]
state_before: a b : ℕ
ha : a % 4 = 1
hb : Odd b
⊢ J(↑a | b) = J(↑b | a)
state_after: no goals
Theorem: CategoryTheory.Presheaf.isLocallySurjective_of_le
File: Mathlib/CategoryTheory/Sites/LocallySurjective.lean

tactic: apply hJK
state_before: C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝¹ : Category.{v', u'} A
inst✝ : ConcreteCategory A
K : GrothendieckTopology C
hJK : J ≤ K
F G : Cᵒᵖ ⥤ A
f : F ⟶ G
h : IsLocallySurjective J f
U✝ : C
s : (forget A).obj (G.obj { unop := U✝ })
⊢ imageSieve f s ∈ K.sieves U✝
state_after: case a
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝¹ : Category.{v', u'} A
inst✝ : ConcreteCategory A
K : GrothendieckTopology C
hJK : J ≤ K
F G : Cᵒᵖ ⥤ A
f : F ⟶ G
h : IsLocallySurjective J f
U✝ : C
s : (forget A).obj (G.obj { unop := U✝ })
⊢ imageSieve f s ∈ J.sieves U✝

tactic: exact h.1 _
state_before: case a
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝¹ : Category.{v', u'} A
inst✝ : ConcreteCategory A
K : GrothendieckTopology C
hJK : J ≤ K
F G : Cᵒᵖ ⥤ A
f : F ⟶ G
h : IsLocallySurjective J f
U✝ : C
s : (forget A).obj (G.obj { unop := U✝ })
⊢ imageSieve f s ∈ J.sieves U✝
state_after: no goals
Theorem: Complex.logTaylor_zero
File: Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean

tactic: funext
state_before: ⊢ logTaylor 0 = fun x => 0
state_after: case h
x✝ : ℂ
⊢ logTaylor 0 x✝ = 0

tactic: simp only [logTaylor, Finset.range_zero, Nat.odd_iff_not_even, Int.cast_pow, Int.cast_neg,
  Int.cast_one, Finset.sum_empty]
state_before: case h
x✝ : ℂ
⊢ logTaylor 0 x✝ = 0
state_after: no goals
Theorem: ENNReal.iInf_mul_right'
File: Mathlib/Topology/Instances/ENNReal.lean

tactic: simpa only [mul_comm a] using iInf_mul_left' h h0
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
a✝ b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
ι : Sort u_4
f : ι → ℝ≥0∞
a : ℝ≥0∞
h : a = ⊤ → ⨅ i, f i = 0 → ∃ i, f i = 0
h0 : a = 0 → Nonempty ι
⊢ ⨅ i, f i * a = (⨅ i, f i) * a
state_after: no goals
Theorem: BumpCovering.toPOUFun_zero_of_zero
File: Mathlib/Topology/PartitionOfUnity.lean

tactic: rw [toPOUFun, h, zero_mul]
state_before: ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
f : BumpCovering ι X s
i : ι
x : X
h : (f i) x = 0
⊢ f.toPOUFun i x = 0
state_after: no goals
Theorem: Equiv.extendSubtype_apply_of_not_mem
File: Mathlib/Logic/Equiv/Fintype.lean

tactic: dsimp only [extendSubtype]
state_before: α : Type u_1
β : Type u_2
inst✝² : Finite α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
e : { x // p x } ≃ { x // q x }
x : α
hx : ¬p x
⊢ e.extendSubtype x = ↑(e.toCompl ⟨x, hx⟩)
state_after: α : Type u_1
β : Type u_2
inst✝² : Finite α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
e : { x // p x } ≃ { x // q x }
x : α
hx : ¬p x
⊢ (e.subtypeCongr e.toCompl) x = ↑(e.toCompl ⟨x, hx⟩)

tactic: simp only [subtypeCongr, Equiv.trans_apply, Equiv.sumCongr_apply]
state_before: α : Type u_1
β : Type u_2
inst✝² : Finite α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
e : { x // p x } ≃ { x // q x }
x : α
hx : ¬p x
⊢ (e.subtypeCongr e.toCompl) x = ↑(e.toCompl ⟨x, hx⟩)
state_after: α : Type u_1
β : Type u_2
inst✝² : Finite α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
e : { x // p x } ≃ { x // q x }
x : α
hx : ¬p x
⊢ (sumCompl q) (Sum.map (⇑e) (⇑e.toCompl) ((sumCompl p).symm x)) = ↑(e.toCompl ⟨x, hx⟩)

tactic: rw [sumCompl_apply_symm_of_neg _ _ hx, Sum.map_inr, sumCompl_apply_inr]
state_before: α : Type u_1
β : Type u_2
inst✝² : Finite α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
e : { x // p x } ≃ { x // q x }
x : α
hx : ¬p x
⊢ (sumCompl q) (Sum.map (⇑e) (⇑e.toCompl) ((sumCompl p).symm x)) = ↑(e.toCompl ⟨x, hx⟩)
state_after: no goals
Theorem: IsLocalization.smul_mk'_self
File: Mathlib/RingTheory/Localization/Basic.lean

tactic: rw [smul_mk', mk'_mul_cancel_left]
state_before: R : Type u_1
inst✝⁴ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝³ : CommSemiring S
inst✝² : Algebra R S
P : Type u_3
inst✝¹ : CommSemiring P
inst✝ : IsLocalization M S
m : ↥M
r : R
⊢ ↑m • mk' S r m = (algebraMap R S) r
state_after: no goals
Theorem: Set.image_pair
File: Mathlib/Data/Set/Image.lean

tactic: simp only [image_insert_eq, image_singleton]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : α → β
s t : Set α
f : α → β
a b : α
⊢ f '' {a, b} = {f a, f b}
state_after: no goals
Theorem: MeasureTheory.SimpleFunc.bind_const
File: Mathlib/MeasureTheory/Function/SimpleFunc.lean

tactic: ext
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : MeasurableSpace α
f : α →ₛ β
⊢ f.bind (const α) = f
state_after: case H
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : MeasurableSpace α
f : α →ₛ β
a✝ : α
⊢ ↑(f.bind (const α)) a✝ = ↑f a✝

tactic: simp
state_before: case H
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : MeasurableSpace α
f : α →ₛ β
a✝ : α
⊢ ↑(f.bind (const α)) a✝ = ↑f a✝
state_after: no goals
Theorem: ne_zero_of_irreducible_X_pow_sub_C
File: Mathlib/FieldTheory/KummerExtension.lean

tactic: rintro rfl
state_before: K : Type u
inst✝ : Field K
n : ℕ
a : K
H : Irreducible (X ^ n - C a)
⊢ n ≠ 0
state_after: K : Type u
inst✝ : Field K
a : K
H : Irreducible (X ^ 0 - C a)
⊢ False

tactic: rw [pow_zero, ← C.map_one, ← map_sub] at H
state_before: K : Type u
inst✝ : Field K
a : K
H : Irreducible (X ^ 0 - C a)
⊢ False
state_after: K : Type u
inst✝ : Field K
a : K
H : Irreducible (C (1 - a))
⊢ False

tactic: exact not_irreducible_C _ H
state_before: K : Type u
inst✝ : Field K
a : K
H : Irreducible (C (1 - a))
⊢ False
state_after: no goals