ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: WeierstrassCurve.Affine.baseChange_slope
File: Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean

Proof steps:
state_before: R✝ : Type u
inst✝²³ : CommRing R✝
W✝ : Affine R✝
R : Type r
inst✝²² : CommRing R
W : Affine R
S : Type s
inst✝²¹ : CommRing S
inst✝²⁰ : Algebra R S
A : Type u
inst✝¹⁹ : CommRing A
inst✝¹⁸ : Algebra R A
inst✝¹⁷ : Algebra S A
inst✝¹⁶ : IsScalarTower R S A
B : Type v
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Algebra R B
inst✝¹³ : Algebra S B
inst✝¹² : IsScalarTower R S B
f✝ : A →ₐ[S] B
F : Type u
inst✝¹¹ : Field F
inst✝¹⁰ : Algebra R F
inst✝⁹ : Algebra S F
inst✝⁸ : IsScalarTower R S F
K : Type v
inst✝⁷ : Field K
inst✝⁶ : Algebra R K
inst✝⁵ : Algebra S K
inst✝⁴ : IsScalarTower R S K
f : F →ₐ[S] K
L : Type w
inst✝³ : Field L
inst✝² : Algebra R L
inst✝¹ : Algebra S L
inst✝ : IsScalarTower R S L
g : K →ₐ[S] L
x₁ x₂ y₁ y₂ : F
⊢ (baseChange W K).toAffine.slope (f x₁) (f x₂) (f y₁) (f y₂) = f ((baseChange W F).toAffine.slope x₁ x₂ y₁ y₂)
state_after: R✝ : Type u
inst✝²³ : CommRing R✝
W✝ : Affine R✝
R : Type r
inst✝²² : CommRing R
W : Affine R
S : Type s
inst✝²¹ : CommRing S
inst✝²⁰ : Algebra R S
A : Type u
inst✝¹⁹ : CommRing A
inst✝¹⁸ : Algebra R A
inst✝¹⁷ : Algebra S A
inst✝¹⁶ : IsScalarTower R S A
B : Type v
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Algebra R B
inst✝¹³ : Algebra S B
inst✝¹² : IsScalarTower R S B
f✝ : A →ₐ[S] B
F : Type u
inst✝¹¹ : Field F
inst✝¹⁰ : Algebra R F
inst✝⁹ : Algebra S F
inst✝⁸ : IsScalarTower R S F
K : Type v
inst✝⁷ : Field K
inst✝⁶ : Algebra R K
inst✝⁵ : Algebra S K
inst✝⁴ : IsScalarTower R S K
f : F →ₐ[S] K
L : Type w
inst✝³ : Field L
inst✝² : Algebra R L
inst✝¹ : Algebra S L
inst✝ : IsScalarTower R S L
g : K →ₐ[S] L
x₁ x₂ y₁ y₂ : F
⊢ (baseChange W K).toAffine.slope (f x₁) (f x₂) (f y₁) (f y₂) =
    (baseChange W K).toAffine.slope (f.toRingHom x₁) (f.toRingHom x₂) (f.toRingHom y₁) (f.toRingHom y₂)
tactic: erw [← map_slope, map_baseChange]
state_before: R✝ : Type u
inst✝²³ : CommRing R✝
W✝ : Affine R✝
R : Type r
inst✝²² : CommRing R
W : Affine R
S : Type s
inst✝²¹ : CommRing S
inst✝²⁰ : Algebra R S
A : Type u
inst✝¹⁹ : CommRing A
inst✝¹⁸ : Algebra R A
inst✝¹⁷ : Algebra S A
inst✝¹⁶ : IsScalarTower R S A
B : Type v
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Algebra R B
inst✝¹³ : Algebra S B
inst✝¹² : IsScalarTower R S B
f✝ : A →ₐ[S] B
F : Type u
inst✝¹¹ : Field F
inst✝¹⁰ : Algebra R F
inst✝⁹ : Algebra S F
inst✝⁸ : IsScalarTower R S F
K : Type v
inst✝⁷ : Field K
inst✝⁶ : Algebra R K
inst✝⁵ : Algebra S K
inst✝⁴ : IsScalarTower R S K
f : F →ₐ[S] K
L : Type w
inst✝³ : Field L
inst✝² : Algebra R L
inst✝¹ : Algebra S L
inst✝ : IsScalarTower R S L
g : K →ₐ[S] L
x₁ x₂ y₁ y₂ : F
⊢ (baseChange W K).toAffine.slope (f x₁) (f x₂) (f y₁) (f y₂) =
    (baseChange W K).toAffine.slope (f.toRingHom x₁) (f.toRingHom x₂) (f.toRingHom y₁) (f.toRingHom y₂)
state_after: no goals
tactic: rfl
Theorem: GenContFract.conts_recurrenceAux
File: Mathlib/Algebra/ContinuedFractions/ContinuantsRecurrence.lean

Proof steps:
state_before: K : Type u_1
g : GenContFract K
n : ℕ
inst✝ : DivisionRing K
gp ppred pred : Pair K
nth_s_eq : g.s.get? n = some gp
nth_contsAux_eq : g.contsAux n = ppred
succ_nth_contsAux_eq : g.contsAux (n + 1) = pred
⊢ g.conts (n + 1) = { a := gp.b * pred.a + gp.a * ppred.a, b := gp.b * pred.b + gp.a * ppred.b }
state_after: no goals
tactic: simp [nth_cont_eq_succ_nth_contAux,
    contsAux_recurrence nth_s_eq nth_contsAux_eq succ_nth_contsAux_eq]
Theorem: Asymptotics.superpolynomialDecay_mul_const_iff
File: Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
l : Filter α
k f g g' : α → β
inst✝² : TopologicalSpace β
inst✝¹ : Field β
inst✝ : ContinuousMul β
c : β
hc0 : c ≠ 0
h : SuperpolynomialDecay l k fun n => f n * c
x : α
⊢ f x * c * c⁻¹ = f x
state_after: no goals
tactic: simp [mul_assoc, mul_inv_cancel hc0]
Theorem: Ordering.swap_inj
File: .lake/packages/batteries/Batteries/Classes/Order.lean

Proof steps:
state_before: o₁ o₂ : Ordering
h : o₁.swap = o₂.swap
⊢ o₁ = o₂
state_after: no goals
tactic: simpa using congrArg swap h
Theorem: Matrix.comp_equiv_symm_dotProduct
File: Mathlib/Data/Matrix/Basic.lean

Proof steps:
state_before: l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
m' : o → Type u_5
n' : o → Type u_6
R : Type u_7
S : Type u_8
α : Type v
β : Type w
γ : Type u_9
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : NonUnitalNonAssocSemiring α
u v w : m → α
x y : n → α
e : m ≃ n
x✝¹ : m
x✝ : x✝¹ ∈ Finset.univ
⊢ (u ∘ ⇑e.symm) (e x✝¹) * x (e x✝¹) = u x✝¹ * (x ∘ ⇑e) x✝¹
state_after: no goals
tactic: simp only [Function.comp, Equiv.symm_apply_apply]
Theorem: image_range_orderOf
File: Mathlib/GroupTheory/OrderOfElement.lean

Proof steps:
state_before: G : Type u_1
H : Type u_2
A : Type u_3
α : Type u_4
β : Type u_5
inst✝² : Group G
inst✝¹ : Fintype G
x : G
n : ℕ
inst✝ : DecidableEq G
⊢ Finset.image (fun i => x ^ i) (Finset.range (orderOf x)) = (↑(zpowers x)).toFinset
state_after: case a
G : Type u_1
H : Type u_2
A : Type u_3
α : Type u_4
β : Type u_5
inst✝² : Group G
inst✝¹ : Fintype G
x✝ : G
n : ℕ
inst✝ : DecidableEq G
x : G
⊢ x ∈ Finset.image (fun i => x✝ ^ i) (Finset.range (orderOf x✝)) ↔ x ∈ (↑(zpowers x✝)).toFinset
tactic: ext x
state_before: case a
G : Type u_1
H : Type u_2
A : Type u_3
α : Type u_4
β : Type u_5
inst✝² : Group G
inst✝¹ : Fintype G
x✝ : G
n : ℕ
inst✝ : DecidableEq G
x : G
⊢ x ∈ Finset.image (fun i => x✝ ^ i) (Finset.range (orderOf x✝)) ↔ x ∈ (↑(zpowers x✝)).toFinset
state_after: no goals
tactic: rw [Set.mem_toFinset, SetLike.mem_coe, mem_zpowers_iff_mem_range_orderOf]
Theorem: Complex.angle_mul_left
File: Mathlib/Analysis/Complex/Angle.lean

Proof steps:
state_before: a x✝ y✝ : ℂ
ha : a ≠ 0
x y : ℂ
⊢ angle (a * x) (a * y) = angle x y
state_after: no goals
tactic: obtain rfl | hx := eq_or_ne x 0 <;> obtain rfl | hy := eq_or_ne y 0 <;>
    simp [angle_eq_abs_arg, mul_div_mul_left, *]
Theorem: Associated.refl
File: Mathlib/Algebra/Associated/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : Monoid α
x : α
⊢ x * ↑1 = x
state_after: no goals
tactic: simp
Theorem: Set.ordConnected_iff_uIcc_subset_right
File: Mathlib/Order/Interval/Set/OrdConnected.lean

Proof steps:
state_before: α : Type u_1
inst✝ : LinearOrder α
s : Set α
x : α
hx : x ∈ s
⊢ s.OrdConnected ↔ ∀ ⦃y : α⦄, y ∈ s → [[y, x]] ⊆ s
state_after: no goals
tactic: simp_rw [ordConnected_iff_uIcc_subset_left hx, uIcc_comm]
Theorem: ContinuousLinearEquiv.image_symm_eq_preimage
File: Mathlib/Topology/Algebra/Module/Basic.lean

Proof steps:
state_before: R₁ : Type u_1
R₂ : Type u_2
R₃ : Type u_3
inst✝²⁴ : Semiring R₁
inst✝²³ : Semiring R₂
inst✝²² : Semiring R₃
σ₁₂ : R₁ →+* R₂
σ₂₁ : R₂ →+* R₁
inst✝²¹ : RingHomInvPair σ₁₂ σ₂₁
inst✝²⁰ : RingHomInvPair σ₂₁ σ₁₂
σ₂₃ : R₂ →+* R₃
σ₃₂ : R₃ →+* R₂
inst✝¹⁹ : RingHomInvPair σ₂₃ σ₃₂
inst✝¹⁸ : RingHomInvPair σ₃₂ σ₂₃
σ₁₃ : R₁ →+* R₃
σ₃₁ : R₃ →+* R₁
inst✝¹⁷ : RingHomInvPair σ₁₃ σ₃₁
inst✝¹⁶ : RingHomInvPair σ₃₁ σ₁₃
inst✝¹⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁴ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁
M₁ : Type u_4
inst✝¹³ : TopologicalSpace M₁
inst✝¹² : AddCommMonoid M₁
M'₁ : Type u_5
inst✝¹¹ : TopologicalSpace M'₁
inst✝¹⁰ : AddCommMonoid M'₁
M₂ : Type u_6
inst✝⁹ : TopologicalSpace M₂
inst✝⁸ : AddCommMonoid M₂
M₃ : Type u_7
inst✝⁷ : TopologicalSpace M₃
inst✝⁶ : AddCommMonoid M₃
M₄ : Type u_8
inst✝⁵ : TopologicalSpace M₄
inst✝⁴ : AddCommMonoid M₄
inst✝³ : Module R₁ M₁
inst✝² : Module R₁ M'₁
inst✝¹ : Module R₂ M₂
inst✝ : Module R₃ M₃
e : M₁ ≃SL[σ₁₂] M₂
s : Set M₂
⊢ ⇑e.symm '' s = ⇑e ⁻¹' s
state_after: no goals
tactic: rw [e.symm.image_eq_preimage, e.symm_symm]
Theorem: isPrimePow_iff_factorization_eq_single
File: Mathlib/Data/Nat/Factorization/PrimePow.lean

Proof steps:
state_before: R : Type u_1
inst✝ : CommMonoidWithZero R
n✝ p : R
k n : ℕ
⊢ IsPrimePow n ↔ ∃ p k, 0 < k ∧ n.factorization = Finsupp.single p k
state_after: R : Type u_1
inst✝ : CommMonoidWithZero R
n✝ p : R
k n : ℕ
⊢ (∃ p k, Nat.Prime p ∧ 0 < k ∧ p ^ k = n) ↔ ∃ p k, 0 < k ∧ n.factorization = Finsupp.single p k
tactic: rw [isPrimePow_nat_iff]
state_before: R : Type u_1
inst✝ : CommMonoidWithZero R
n✝ p : R
k n : ℕ
⊢ (∃ p k, Nat.Prime p ∧ 0 < k ∧ p ^ k = n) ↔ ∃ p k, 0 < k ∧ n.factorization = Finsupp.single p k
state_after: R : Type u_1
inst✝ : CommMonoidWithZero R
n✝ p✝ : R
k✝ n p k : ℕ
⊢ Nat.Prime p ∧ 0 < k ∧ p ^ k = n ↔ 0 < k ∧ n.factorization = Finsupp.single p k
tactic: refine exists₂_congr fun p k => ?_
state_before: R : Type u_1
inst✝ : CommMonoidWithZero R
n✝ p✝ : R
k✝ n p k : ℕ
⊢ Nat.Prime p ∧ 0 < k ∧ p ^ k = n ↔ 0 < k ∧ n.factorization = Finsupp.single p k
state_after: case mp
R : Type u_1
inst✝ : CommMonoidWithZero R
n✝ p✝ : R
k✝ n p k : ℕ
⊢ Nat.Prime p ∧ 0 < k ∧ p ^ k = n → 0 < k ∧ n.factorization = Finsupp.single p k

case mpr
R : Type u_1
inst✝ : CommMonoidWithZero R
n✝ p✝ : R
k✝ n p k : ℕ
⊢ 0 < k ∧ n.factorization = Finsupp.single p k → Nat.Prime p ∧ 0 < k ∧ p ^ k = n
tactic: constructor
Theorem: Order.succ_lt_succ_iff
File: Mathlib/Order/SuccPred/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝² : Preorder α
inst✝¹ : SuccOrder α
a b : α
inst✝ : NoMaxOrder α
⊢ succ a < succ b ↔ a < b
state_after: no goals
tactic: simp
Theorem: Set.neg_singleton
File: Mathlib/Data/Set/Pointwise/Basic.lean

Proof steps:
state_before: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : InvolutiveInv α
s t : Set α
a✝ a : α
⊢ {a}⁻¹ = {a⁻¹}
state_after: no goals
tactic: rw [← image_inv, image_singleton]
Theorem: LinearEquiv.dualMap_refl
File: Mathlib/LinearAlgebra/Dual.lean

Proof steps:
state_before: R : Type u
inst✝⁴ : CommSemiring R
M₁ : Type v
M₂ : Type v'
inst✝³ : AddCommMonoid M₁
inst✝² : Module R M₁
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R M₂
⊢ (refl R M₁).dualMap = refl R (Dual R M₁)
state_after: case h.h
R : Type u
inst✝⁴ : CommSemiring R
M₁ : Type v
M₂ : Type v'
inst✝³ : AddCommMonoid M₁
inst✝² : Module R M₁
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R M₂
x✝¹ : Dual R M₁
x✝ : M₁
⊢ ((refl R M₁).dualMap x✝¹) x✝ = ((refl R (Dual R M₁)) x✝¹) x✝
tactic: ext
state_before: case h.h
R : Type u
inst✝⁴ : CommSemiring R
M₁ : Type v
M₂ : Type v'
inst✝³ : AddCommMonoid M₁
inst✝² : Module R M₁
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R M₂
x✝¹ : Dual R M₁
x✝ : M₁
⊢ ((refl R M₁).dualMap x✝¹) x✝ = ((refl R (Dual R M₁)) x✝¹) x✝
state_after: no goals
tactic: rfl
Theorem: TwoSidedIdeal.iSup_ringCon
File: Mathlib/RingTheory/TwoSidedIdeal/Lattice.lean

Proof steps:
state_before: R : Type u_1
inst✝ : NonUnitalNonAssocRing R
ι : Type u_2
I : ι → TwoSidedIdeal R
⊢ (⨆ i, I i).ringCon = ⨆ i, (I i).ringCon
state_after: R : Type u_1
inst✝ : NonUnitalNonAssocRing R
ι : Type u_2
I : ι → TwoSidedIdeal R
⊢ sSup (ringCon '' Set.range fun i => I i) = sSup (Set.range fun i => (I i).ringCon)
tactic: simp only [iSup, sSup_ringCon]
state_before: R : Type u_1
inst✝ : NonUnitalNonAssocRing R
ι : Type u_2
I : ι → TwoSidedIdeal R
⊢ sSup (ringCon '' Set.range fun i => I i) = sSup (Set.range fun i => (I i).ringCon)
state_after: case e_a
R : Type u_1
inst✝ : NonUnitalNonAssocRing R
ι : Type u_2
I : ι → TwoSidedIdeal R
⊢ (ringCon '' Set.range fun i => I i) = Set.range fun i => (I i).ringCon
tactic: congr
state_before: case e_a
R : Type u_1
inst✝ : NonUnitalNonAssocRing R
ι : Type u_2
I : ι → TwoSidedIdeal R
⊢ (ringCon '' Set.range fun i => I i) = Set.range fun i => (I i).ringCon
state_after: case e_a.h
R : Type u_1
inst✝ : NonUnitalNonAssocRing R
ι : Type u_2
I : ι → TwoSidedIdeal R
x✝ : RingCon R
⊢ (x✝ ∈ ringCon '' Set.range fun i => I i) ↔ x✝ ∈ Set.range fun i => (I i).ringCon
tactic: ext
state_before: case e_a.h
R : Type u_1
inst✝ : NonUnitalNonAssocRing R
ι : Type u_2
I : ι → TwoSidedIdeal R
x✝ : RingCon R
⊢ (x✝ ∈ ringCon '' Set.range fun i => I i) ↔ x✝ ∈ Set.range fun i => (I i).ringCon
state_after: no goals
tactic: simp
Theorem: Polynomial.roots_prod_X_sub_C
File: Mathlib/Algebra/Polynomial/Roots.lean

Proof steps:
state_before: R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommRing R
inst✝ : IsDomain R
p q : R[X]
s : Finset R
⊢ (∏ a ∈ s, (X - C a)).roots = s.val
state_after: R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommRing R
inst✝ : IsDomain R
p q : R[X]
s : Finset R
⊢ (s.val.bind fun i => (X - C i).roots) = s.val

R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommRing R
inst✝ : IsDomain R
p q : R[X]
s : Finset R
⊢ ∏ a ∈ s, (X - C a) ≠ 0
tactic: apply (roots_prod (fun a => X - C a) s ?_).trans
Theorem: ContinuousMultilinearMap.hasFDerivAt
File: Mathlib/Analysis/Calculus/FDeriv/Analytic.lean

Proof steps:
state_before: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E✝ : Type u
inst✝⁷ : NormedAddCommGroup E✝
inst✝⁶ : NormedSpace 𝕜 E✝
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
n : ℕ∞
x : (i : ι) → E i
inst✝ : DecidableEq ι
⊢ HasFDerivAt (⇑f) (f.linearDeriv x) x
state_after: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E✝ : Type u
inst✝⁷ : NormedAddCommGroup E✝
inst✝⁶ : NormedSpace 𝕜 E✝
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
n : ℕ∞
x : (i : ι) → E i
inst✝ : DecidableEq ι
⊢ HasFDerivAt (⇑f) ((continuousMultilinearCurryFin1 𝕜 ((i : ι) → E i) F) (f.toFormalMultilinearSeries.changeOrigin x 1))
    x
tactic: rw [← changeOrigin_toFormalMultilinearSeries]
state_before: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E✝ : Type u
inst✝⁷ : NormedAddCommGroup E✝
inst✝⁶ : NormedSpace 𝕜 E✝
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
n : ℕ∞
x : (i : ι) → E i
inst✝ : DecidableEq ι
⊢ HasFDerivAt (⇑f) ((continuousMultilinearCurryFin1 𝕜 ((i : ι) → E i) F) (f.toFormalMultilinearSeries.changeOrigin x 1))
    x
state_after: case h.e'_11
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E✝ : Type u
inst✝⁷ : NormedAddCommGroup E✝
inst✝⁶ : NormedSpace 𝕜 E✝
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
n : ℕ∞
x : (i : ι) → E i
inst✝ : DecidableEq ι
⊢ x = 0 + x
tactic: convert f.hasFiniteFPowerSeriesOnBall.hasFDerivAt (y := x) ENNReal.coe_lt_top
state_before: case h.e'_11
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E✝ : Type u
inst✝⁷ : NormedAddCommGroup E✝
inst✝⁶ : NormedSpace 𝕜 E✝
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
n : ℕ∞
x : (i : ι) → E i
inst✝ : DecidableEq ι
⊢ x = 0 + x
state_after: no goals
tactic: rw [zero_add]
Theorem: IntermediateField.map_comap_eq_self
File: Mathlib/FieldTheory/Adjoin.lean

Proof steps:
state_before: K : Type u_1
L : Type u_2
L' : Type u_3
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Field L'
inst✝¹ : Algebra K L
inst✝ : Algebra K L'
f : L →ₐ[K] L'
S : IntermediateField K L'
h : S ≤ f.fieldRange
⊢ map f (comap f S) = S
state_after: no goals
tactic: simpa only [inf_of_le_left h] using map_comap_eq f S
Theorem: MeasureTheory.integral_const
File: Mathlib/MeasureTheory/Integral/Bochner.lean

Proof steps:
state_before: α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : CompleteSpace F
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
c : E
⊢ ∫ (x : α), c ∂μ = (μ univ).toReal • c
state_after: case inl
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : CompleteSpace F
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
c : E
hμ : μ univ < ⊤
⊢ ∫ (x : α), c ∂μ = (μ univ).toReal • c

case inr
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : CompleteSpace F
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
c : E
hμ : μ univ = ⊤
⊢ ∫ (x : α), c ∂μ = (μ univ).toReal • c
tactic: cases' (@le_top _ _ _ (μ univ)).lt_or_eq with hμ hμ
Theorem: contDiffAt_zero
File: Mathlib/Analysis/Calculus/ContDiff/Defs.lean

Proof steps:
state_before: 𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type uX
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x x₀ : E
c : F
m n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
⊢ ContDiffAt 𝕜 0 f x ↔ ∃ u ∈ 𝓝 x, ContinuousOn f u
state_after: 𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type uX
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x x₀ : E
c : F
m n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
⊢ ContDiffWithinAt 𝕜 0 f univ x ↔ ∃ u ∈ 𝓝 x, ContinuousOn f u
tactic: rw [← contDiffWithinAt_univ]
state_before: 𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type uX
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x x₀ : E
c : F
m n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
⊢ ContDiffWithinAt 𝕜 0 f univ x ↔ ∃ u ∈ 𝓝 x, ContinuousOn f u
state_after: no goals
tactic: simp [contDiffWithinAt_zero, nhdsWithin_univ]
Theorem: Set.preimage_subset_preimage_iff
File: Mathlib/Data/Set/Image.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t✝ s t : Set α
f : β → α
hs : s ⊆ range f
⊢ f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t
state_after: case mp
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t✝ s t : Set α
f : β → α
hs : s ⊆ range f
⊢ f ⁻¹' s ⊆ f ⁻¹' t → s ⊆ t

case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t✝ s t : Set α
f : β → α
hs : s ⊆ range f
⊢ s ⊆ t → f ⁻¹' s ⊆ f ⁻¹' t
tactic: constructor
state_before: case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t✝ s t : Set α
f : β → α
hs : s ⊆ range f
⊢ s ⊆ t → f ⁻¹' s ⊆ f ⁻¹' t
state_after: case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t✝ s t : Set α
f : β → α
hs : s ⊆ range f
h : s ⊆ t
x : β
⊢ x ∈ f ⁻¹' s → x ∈ f ⁻¹' t
tactic: intro h x
state_before: case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s✝ t✝ s t : Set α
f : β → α
hs : s ⊆ range f
h : s ⊆ t
x : β
⊢ x ∈ f ⁻¹' s → x ∈ f ⁻¹' t
state_after: no goals
tactic: apply h
Theorem: Subgroup.op_closure
File: Mathlib/Algebra/Group/Subgroup/MulOpposite.lean

Proof steps:
state_before: ι : Sort u_1
G : Type u_2
inst✝ : Group G
s : Set G
⊢ (closure s).op = closure (MulOpposite.unop ⁻¹' s)
state_after: ι : Sort u_1
G : Type u_2
inst✝ : Group G
s : Set G
⊢ sInf {a | s ⊆ MulOpposite.op ⁻¹' ↑a} = sInf {K | MulOpposite.unop ⁻¹' s ⊆ ↑K}
tactic: simp_rw [closure, op_sInf, Set.preimage_setOf_eq, Subgroup.unop_coe]
state_before: ι : Sort u_1
G : Type u_2
inst✝ : Group G
s : Set G
⊢ sInf {a | s ⊆ MulOpposite.op ⁻¹' ↑a} = sInf {K | MulOpposite.unop ⁻¹' s ⊆ ↑K}
state_after: case e_a.h
ι : Sort u_1
G : Type u_2
inst✝ : Group G
s : Set G
a : Subgroup Gᵐᵒᵖ
⊢ a ∈ {a | s ⊆ MulOpposite.op ⁻¹' ↑a} ↔ a ∈ {K | MulOpposite.unop ⁻¹' s ⊆ ↑K}
tactic: congr with a
state_before: case e_a.h
ι : Sort u_1
G : Type u_2
inst✝ : Group G
s : Set G
a : Subgroup Gᵐᵒᵖ
⊢ a ∈ {a | s ⊆ MulOpposite.op ⁻¹' ↑a} ↔ a ∈ {K | MulOpposite.unop ⁻¹' s ⊆ ↑K}
state_after: no goals
tactic: exact MulOpposite.unop_surjective.forall
Theorem: minpoly.algEquiv_apply
File: Mathlib/FieldTheory/Adjoin.lean

Proof steps:
state_before: K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
x y : L
hx : IsAlgebraic K x
h_mp : minpoly K x = minpoly K y
⊢ (algEquiv hx h_mp) (AdjoinSimple.gen K x) = AdjoinSimple.gen K y
state_after: K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
x y : L
hx : IsAlgebraic K x
h_mp : minpoly K x = minpoly K y
hy : IsAlgebraic K y
⊢ (algEquiv hx h_mp) (AdjoinSimple.gen K x) = AdjoinSimple.gen K y
tactic: have hy : IsAlgebraic K y := ⟨minpoly K x, ne_zero hx.isIntegral, (h_mp ▸ aeval _ _)⟩
state_before: K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
x y : L
hx : IsAlgebraic K x
h_mp : minpoly K x = minpoly K y
hy : IsAlgebraic K y
⊢ (algEquiv hx h_mp) (AdjoinSimple.gen K x) = AdjoinSimple.gen K y
state_after: no goals
tactic: rw [algEquiv, trans_apply, ← adjoinRootEquivAdjoin_apply_root K hx.isIntegral,
    symm_apply_apply, trans_apply, AdjoinRoot.algEquivOfEq_apply_root,
    adjoinRootEquivAdjoin_apply_root K hy.isIntegral]
Theorem: min_def_lt
File: Mathlib/Order/Basic.lean

Proof steps:
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
π : ι → Type u_4
inst✝ : LinearOrder α
p : α → Prop
x✝ y✝ x y : α
⊢ min x y = if x < y then x else y
state_after: ι : Type u_1
α : Type u_2
β : Type u_3
π : ι → Type u_4
inst✝ : LinearOrder α
p : α → Prop
x✝ y✝ x y : α
⊢ (if ¬y ≤ x then x else y) = if x < y then x else y
tactic: rw [min_comm, min_def, ← ite_not]
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
π : ι → Type u_4
inst✝ : LinearOrder α
p : α → Prop
x✝ y✝ x y : α
⊢ (if ¬y ≤ x then x else y) = if x < y then x else y
state_after: no goals
tactic: simp only [not_le]