ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: le_of_span_le_span
File: Mathlib/LinearAlgebra/LinearIndependent.lean

Proof steps:
state_before: ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup M'
inst✝⁴ : AddCommGroup M''
inst✝³ : Module R M
inst✝² : Module R M'
inst✝¹ : Module R M''
a b : R
x y : M
inst✝ : Nontrivial R
s t u : Set M
hl : LinearIndependent (ι := { x // x ∈ u }) R Subtype.val
hsu : s ⊆ u
htu : t ⊆ u
hst : span R s ≤ span R t
⊢ s ⊆ t
state_after: ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup M'
inst✝⁴ : AddCommGroup M''
inst✝³ : Module R M
inst✝² : Module R M'
inst✝¹ : Module R M''
a b : R
x y : M
inst✝ : Nontrivial R
s t u : Set M
hl : LinearIndependent (ι := { x // x ∈ u }) R Subtype.val
hsu : s ⊆ u
htu : t ⊆ u
hst : span R s ≤ span R t
this : s ∪ t = t
⊢ s ⊆ t
tactic: have :=
    eq_of_linearIndependent_of_span_subtype (hl.mono (Set.union_subset hsu htu))
      Set.subset_union_right (Set.union_subset (Set.Subset.trans subset_span hst) subset_span)
state_before: ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup M'
inst✝⁴ : AddCommGroup M''
inst✝³ : Module R M
inst✝² : Module R M'
inst✝¹ : Module R M''
a b : R
x y : M
inst✝ : Nontrivial R
s t u : Set M
hl : LinearIndependent (ι := { x // x ∈ u }) R Subtype.val
hsu : s ⊆ u
htu : t ⊆ u
hst : span R s ≤ span R t
this : s ∪ t = t
⊢ s ⊆ t
state_after: ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup M'
inst✝⁴ : AddCommGroup M''
inst✝³ : Module R M
inst✝² : Module R M'
inst✝¹ : Module R M''
a b : R
x y : M
inst✝ : Nontrivial R
s t u : Set M
hl : LinearIndependent (ι := { x // x ∈ u }) R Subtype.val
hsu : s ⊆ u
htu : t ⊆ u
hst : span R s ≤ span R t
this : s ∪ t = t
⊢ s ⊆ s ∪ t
tactic: rw [← this]
state_before: ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup M'
inst✝⁴ : AddCommGroup M''
inst✝³ : Module R M
inst✝² : Module R M'
inst✝¹ : Module R M''
a b : R
x y : M
inst✝ : Nontrivial R
s t u : Set M
hl : LinearIndependent (ι := { x // x ∈ u }) R Subtype.val
hsu : s ⊆ u
htu : t ⊆ u
hst : span R s ≤ span R t
this : s ∪ t = t
⊢ s ⊆ s ∪ t
state_after: no goals
tactic: apply Set.subset_union_left
Theorem: Equiv.Set.sumCompl_symm_apply
File: Mathlib/Logic/Equiv/Set.lean

Proof steps:
state_before: α✝ : Sort u
β : Sort v
γ : Sort w
α : Type u_1
s : Set α
inst✝ : DecidablePred fun x => x ∈ s
x : ↑s
⊢ (Set.sumCompl s).symm ↑x = Sum.inl x
state_after: case mk
α✝ : Sort u
β : Sort v
γ : Sort w
α : Type u_1
s : Set α
inst✝ : DecidablePred fun x => x ∈ s
x : α
hx : x ∈ s
⊢ (Set.sumCompl s).symm ↑⟨x, hx⟩ = Sum.inl ⟨x, hx⟩
tactic: cases' x with x hx
state_before: case mk
α✝ : Sort u
β : Sort v
γ : Sort w
α : Type u_1
s : Set α
inst✝ : DecidablePred fun x => x ∈ s
x : α
hx : x ∈ s
⊢ (Set.sumCompl s).symm ↑⟨x, hx⟩ = Sum.inl ⟨x, hx⟩
state_after: no goals
tactic: exact Set.sumCompl_symm_apply_of_mem hx
Theorem: lift_rank_le_of_injective_injective
File: Mathlib/LinearAlgebra/Dimension/Basic.lean

Proof steps:
state_before: R : Type u
R' : Type u'
M M₁ : Type v
M' : Type v'
inst✝⁹ : Ring R
inst✝⁸ : Ring R'
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup M'
inst✝⁵ : AddCommGroup M₁
inst✝⁴ : Module R M
inst✝³ : Module R M'
inst✝² : Module R M₁
inst✝¹ : Module R' M'
inst✝ : Module R' M₁
i : R' → R
j : M →+ M'
hi : ∀ (r : R'), i r = 0 → r = 0
hj : Injective ⇑j
hc : ∀ (r : R') (m : M), j (i r • m) = r • j m
⊢ lift.{v', v} (Module.rank R M) ≤ lift.{v, v'} (Module.rank R' M')
state_after: R : Type u
R' : Type u'
M M₁ : Type v
M' : Type v'
inst✝⁹ : Ring R
inst✝⁸ : Ring R'
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup M'
inst✝⁵ : AddCommGroup M₁
inst✝⁴ : Module R M
inst✝³ : Module R M'
inst✝² : Module R M₁
inst✝¹ : Module R' M'
inst✝ : Module R' M₁
i : R' → R
j : M →+ M'
hi : ∀ (r : R'), i r = 0 → r = 0
hj : Injective ⇑j
hc : ∀ (r : R') (m : M), j (i r • m) = r • j m
⊢ ⨆ i, lift.{v', v} #↑↑i ≤ ⨆ i, lift.{v, v'} #↑↑i
tactic: simp_rw [Module.rank, lift_iSup (bddAbove_range.{v', v'} _), lift_iSup (bddAbove_range.{v, v} _)]
state_before: R : Type u
R' : Type u'
M M₁ : Type v
M' : Type v'
inst✝⁹ : Ring R
inst✝⁸ : Ring R'
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup M'
inst✝⁵ : AddCommGroup M₁
inst✝⁴ : Module R M
inst✝³ : Module R M'
inst✝² : Module R M₁
inst✝¹ : Module R' M'
inst✝ : Module R' M₁
i : R' → R
j : M →+ M'
hi : ∀ (r : R'), i r = 0 → r = 0
hj : Injective ⇑j
hc : ∀ (r : R') (m : M), j (i r • m) = r • j m
⊢ ⨆ i, lift.{v', v} #↑↑i ≤ ⨆ i, lift.{v, v'} #↑↑i
state_after: no goals
tactic: exact ciSup_mono' (bddAbove_range.{v', v} _) fun ⟨s, h⟩ ↦ ⟨⟨j '' s,
    (h.map_of_injective_injective i j hi (fun _ _ ↦ hj <| by rwa [j.map_zero]) hc).image⟩,
      lift_mk_le'.mpr ⟨(Equiv.Set.image j s hj).toEmbedding⟩⟩
Theorem: ContinuousMap.coe_sup'
File: Mathlib/Topology/ContinuousFunction/Ordered.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
ι : Type u_4
s : Finset ι
H : s.Nonempty
f : ι → C(α, β)
⊢ ⇑(s.sup' H f) = s.sup' H fun i => ⇑(f i)
state_after: case h
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
ι : Type u_4
s : Finset ι
H : s.Nonempty
f : ι → C(α, β)
x✝ : α
⊢ (s.sup' H f) x✝ = s.sup' H (fun i => ⇑(f i)) x✝
tactic: ext
state_before: case h
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : SemilatticeSup β
inst✝ : ContinuousSup β
ι : Type u_4
s : Finset ι
H : s.Nonempty
f : ι → C(α, β)
x✝ : α
⊢ (s.sup' H f) x✝ = s.sup' H (fun i => ⇑(f i)) x✝
state_after: no goals
tactic: simp [sup'_apply]
Theorem: Irrational.of_sub_rat
File: Mathlib/Data/Real/Irrational.lean

Proof steps:
state_before: q : ℚ
x y : ℝ
h : Irrational (x - ↑q)
⊢ Irrational (x + ↑(-q))
state_after: no goals
tactic: simpa only [cast_neg, sub_eq_add_neg] using h
Theorem: Ordinal.bsup_succ_le_blsub
File: Mathlib/SetTheory/Ordinal/Arithmetic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
⊢ succ (o.bsup f) ≤ o.blsub f ↔ ∃ i, ∃ (hi : i < o), f i hi = o.bsup f
state_after: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
h : succ (o.bsup f) ≤ o.blsub f
⊢ ∃ i, ∃ (hi : i < o), f i hi = o.bsup f

case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
⊢ (∃ i, ∃ (hi : i < o), f i hi = o.bsup f) → succ (o.bsup f) ≤ o.blsub f
tactic: refine ⟨fun h => ?_, ?_⟩
state_before: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
⊢ (∃ i, ∃ (hi : i < o), f i hi = o.bsup f) → succ (o.bsup f) ≤ o.blsub f
state_after: case refine_2.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
w✝¹ : Ordinal.{u}
w✝ : w✝¹ < o
hf : f w✝¹ w✝ = o.bsup f
⊢ succ (o.bsup f) ≤ o.blsub f
tactic: rintro ⟨_, _, hf⟩
state_before: case refine_2.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
w✝¹ : Ordinal.{u}
w✝ : w✝¹ < o
hf : f w✝¹ w✝ = o.bsup f
⊢ succ (o.bsup f) ≤ o.blsub f
state_after: case refine_2.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
w✝¹ : Ordinal.{u}
w✝ : w✝¹ < o
hf : f w✝¹ w✝ = o.bsup f
⊢ f w✝¹ w✝ < o.blsub f
tactic: rw [succ_le_iff, ← hf]
state_before: case refine_2.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
w✝¹ : Ordinal.{u}
w✝ : w✝¹ < o
hf : f w✝¹ w✝ = o.bsup f
⊢ f w✝¹ w✝ < o.blsub f
state_after: no goals
tactic: exact lt_blsub _ _ _
Theorem: dist_vadd_right
File: Mathlib/Analysis/Normed/Group/AddTorsor.lean

Proof steps:
state_before: α : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
inst✝² : NormedAddCommGroup W
inst✝¹ : MetricSpace Q
inst✝ : NormedAddTorsor W Q
v : V
x : P
⊢ dist x (v +ᵥ x) = ‖v‖
state_after: no goals
tactic: rw [dist_comm, dist_vadd_left]
Theorem: ZMod.valMinAbs_natCast_of_half_lt
File: Mathlib/Data/ZMod/Basic.lean

Proof steps:
state_before: m n✝ n a : ℕ
ha : n / 2 < a
ha' : a < n
⊢ (↑a).valMinAbs = ↑a - ↑n
state_after: case zero
m n a : ℕ
ha : 0 / 2 < a
ha' : a < 0
⊢ (↑a).valMinAbs = ↑a - ↑0

case succ
m n a n✝ : ℕ
ha : (n✝ + 1) / 2 < a
ha' : a < n✝ + 1
⊢ (↑a).valMinAbs = ↑a - ↑(n✝ + 1)
tactic: cases n
Theorem: Multiset.dedup_ext
File: Mathlib/Data/Multiset/Dedup.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
s t : Multiset α
⊢ s.dedup = t.dedup ↔ ∀ (a : α), a ∈ s ↔ a ∈ t
state_after: no goals
tactic: simp [Nodup.ext]
Theorem: CategoryTheory.Functor.isContinuous_comp'
File: Mathlib/CategoryTheory/Sites/Continuous.lean

Proof steps:
state_before: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
F : C ⥤ D
A : Type u
inst✝² : Category.{t, u} A
J✝ : GrothendieckTopology C
K✝ : GrothendieckTopology D
F₁ : C ⥤ D
F₂ : D ⥤ E
F₁₂ : C ⥤ E
e : F₁ ⋙ F₂ ≅ F₁₂
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology E
inst✝¹ : F₁.IsContinuous J K
inst✝ : F₂.IsContinuous K L
⊢ F₁₂.IsContinuous J L
state_after: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
F : C ⥤ D
A : Type u
inst✝² : Category.{t, u} A
J✝ : GrothendieckTopology C
K✝ : GrothendieckTopology D
F₁ : C ⥤ D
F₂ : D ⥤ E
F₁₂ : C ⥤ E
e : F₁ ⋙ F₂ ≅ F₁₂
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology E
inst✝¹ : F₁.IsContinuous J K
inst✝ : F₂.IsContinuous K L
this : (F₁ ⋙ F₂).IsContinuous J L
⊢ F₁₂.IsContinuous J L
tactic: have := Functor.isContinuous_comp F₁ F₂ J K L
state_before: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
E : Type u₃
inst✝³ : Category.{v₃, u₃} E
F : C ⥤ D
A : Type u
inst✝² : Category.{t, u} A
J✝ : GrothendieckTopology C
K✝ : GrothendieckTopology D
F₁ : C ⥤ D
F₂ : D ⥤ E
F₁₂ : C ⥤ E
e : F₁ ⋙ F₂ ≅ F₁₂
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology E
inst✝¹ : F₁.IsContinuous J K
inst✝ : F₂.IsContinuous K L
this : (F₁ ⋙ F₂).IsContinuous J L
⊢ F₁₂.IsContinuous J L
state_after: no goals
tactic: apply Functor.isContinuous_of_iso e
Theorem: List.chain'_pair
File: Mathlib/Data/List/Chain.lean

Proof steps:
state_before: α : Type u
β : Type v
R r : α → α → Prop
l l₁ l₂ : List α
a b x y : α
⊢ Chain' R [x, y] ↔ R x y
state_after: no goals
tactic: simp only [chain'_singleton, chain'_cons, and_true_iff]
Theorem: Seminorm.bddAbove_range_iff
File: Mathlib/Analysis/Seminorm.lean

Proof steps:
state_before: R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι✝ : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x : E
ι : Sort u_13
p : ι → Seminorm 𝕜 E
⊢ BddAbove (range p) ↔ ∀ (x : E), BddAbove (range fun i => (p i) x)
state_after: R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι✝ : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x : E
ι : Sort u_13
p : ι → Seminorm 𝕜 E
⊢ (∀ (a : E), BddAbove (range fun i => (DFunLike.coe ∘ p) i a)) ↔ ∀ (x : E), BddAbove (range fun i => (p i) x)
tactic: rw [Seminorm.bddAbove_iff, ← range_comp, bddAbove_range_pi]
state_before: R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι✝ : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ q : Seminorm 𝕜 E
x : E
ι : Sort u_13
p : ι → Seminorm 𝕜 E
⊢ (∀ (a : E), BddAbove (range fun i => (DFunLike.coe ∘ p) i a)) ↔ ∀ (x : E), BddAbove (range fun i => (p i) x)
state_after: no goals
tactic: rfl
Theorem: InnerProductGeometry.cos_angle_sub_of_inner_eq_zero
File: Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean

Proof steps:
state_before: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫_ℝ = 0
⊢ Real.cos (angle x (x - y)) = ‖x‖ / ‖x - y‖
state_after: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫_ℝ = 0
⊢ Real.cos (angle x (x - y)) = ‖x‖ / ‖x - y‖
tactic: rw [← neg_eq_zero, ← inner_neg_right] at h
state_before: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫_ℝ = 0
⊢ Real.cos (angle x (x - y)) = ‖x‖ / ‖x - y‖
state_after: no goals
tactic: rw [sub_eq_add_neg, cos_angle_add_of_inner_eq_zero h]
Theorem: Trivialization.symm_coordChangeL
File: Mathlib/Topology/VectorBundle/Basic.lean

Proof steps:
state_before: R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : TopologicalSpace F
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F TotalSpace.proj
x : TotalSpace F E
b✝ : B
y : E b✝
inst✝⁵ : AddCommMonoid F
inst✝⁴ : Module R F
inst✝³ : (x : B) → AddCommMonoid (E x)
inst✝² : (x : B) → Module R (E x)
e e' : Trivialization F TotalSpace.proj
inst✝¹ : Trivialization.IsLinear R e
inst✝ : Trivialization.IsLinear R e'
b : B
hb : b ∈ e'.baseSet ∩ e.baseSet
⊢ (coordChangeL R e e' b).symm = coordChangeL R e' e b
state_after: case a
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : TopologicalSpace F
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F TotalSpace.proj
x : TotalSpace F E
b✝ : B
y : E b✝
inst✝⁵ : AddCommMonoid F
inst✝⁴ : Module R F
inst✝³ : (x : B) → AddCommMonoid (E x)
inst✝² : (x : B) → Module R (E x)
e e' : Trivialization F TotalSpace.proj
inst✝¹ : Trivialization.IsLinear R e
inst✝ : Trivialization.IsLinear R e'
b : B
hb : b ∈ e'.baseSet ∩ e.baseSet
⊢ (coordChangeL R e e' b).symm.toLinearEquiv = (coordChangeL R e' e b).toLinearEquiv
tactic: apply ContinuousLinearEquiv.toLinearEquiv_injective
state_before: case a
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : TopologicalSpace F
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F TotalSpace.proj
x : TotalSpace F E
b✝ : B
y : E b✝
inst✝⁵ : AddCommMonoid F
inst✝⁴ : Module R F
inst✝³ : (x : B) → AddCommMonoid (E x)
inst✝² : (x : B) → Module R (E x)
e e' : Trivialization F TotalSpace.proj
inst✝¹ : Trivialization.IsLinear R e
inst✝ : Trivialization.IsLinear R e'
b : B
hb : b ∈ e'.baseSet ∩ e.baseSet
⊢ (coordChangeL R e e' b).symm.toLinearEquiv = (coordChangeL R e' e b).toLinearEquiv
state_after: no goals
tactic: rw [coe_coordChangeL' e' e hb, (coordChangeL R e e' b).symm_toLinearEquiv,
    coe_coordChangeL' e e' hb.symm, LinearEquiv.trans_symm, LinearEquiv.symm_symm]
Theorem: Filter.ext
File: Mathlib/Order/Filter/Basic.lean

Proof steps:
state_before: α : Type u
f g : Filter α
s t : Set α
h : ∀ (s : Set α), s ∈ f ↔ s ∈ g
⊢ f = g
state_after: no goals
tactic: simpa [filter_eq_iff, Set.ext_iff, Filter.mem_sets]
Theorem: Finset.powerset_eq_singleton_empty
File: Mathlib/Data/Finset/Powerset.lean

Proof steps:
state_before: α : Type u_1
s t : Finset α
⊢ s.powerset = {∅} ↔ s = ∅
state_after: no goals
tactic: rw [← powerset_empty, powerset_inj]
Theorem: LineDifferentiableAt.congr_of_eventuallyEq
File: Mathlib/Analysis/Calculus/LineDeriv/Basic.lean

Proof steps:
state_before: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f f₀ f₁ : E → F
f' : F
s t : Set E
x v : E
L : E →L[𝕜] F
h : LineDifferentiableAt 𝕜 f x v
hL : f₁ =ᶠ[𝓝 x] f
⊢ LineDifferentiableAt 𝕜 f₁ x v
state_after: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f f₀ f₁ : E → F
f' : F
s t : Set E
x v : E
L : E →L[𝕜] F
h : LineDifferentiableAt 𝕜 f x v
hL : f₁ =ᶠ[𝓝 x] f
⊢ (fun t => f₁ (x + t • v)) =ᶠ[𝓝 0] fun t => f (x + t • v)
tactic: apply DifferentiableAt.congr_of_eventuallyEq h
state_before: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f f₀ f₁ : E → F
f' : F
s t : Set E
x v : E
L : E →L[𝕜] F
h : LineDifferentiableAt 𝕜 f x v
hL : f₁ =ᶠ[𝓝 x] f
⊢ (fun t => f₁ (x + t • v)) =ᶠ[𝓝 0] fun t => f (x + t • v)
state_after: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F✝ : Type u_2
inst✝³ : NormedAddCommGroup F✝
inst✝² : NormedSpace 𝕜 F✝
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f f₀ f₁ : E → F✝
f' : F✝
s t : Set E
x v : E
L : E →L[𝕜] F✝
h : LineDifferentiableAt 𝕜 f x v
hL : f₁ =ᶠ[𝓝 x] f
F : 𝕜 → E := fun t => x + t • v
⊢ (fun t => f₁ (x + t • v)) =ᶠ[𝓝 0] fun t => f (x + t • v)
tactic: let F := fun (t : 𝕜) ↦ x + t • v
state_before: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F✝ : Type u_2
inst✝³ : NormedAddCommGroup F✝
inst✝² : NormedSpace 𝕜 F✝
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f f₀ f₁ : E → F✝
f' : F✝
s t : Set E
x v : E
L : E →L[𝕜] F✝
h : LineDifferentiableAt 𝕜 f x v
hL : f₁ =ᶠ[𝓝 x] f
F : 𝕜 → E := fun t => x + t • v
⊢ (fun t => f₁ (x + t • v)) =ᶠ[𝓝 0] fun t => f (x + t • v)
state_after: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F✝ : Type u_2
inst✝³ : NormedAddCommGroup F✝
inst✝² : NormedSpace 𝕜 F✝
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f f₀ f₁ : E → F✝
f' : F✝
s t : Set E
x v : E
L : E →L[𝕜] F✝
h : LineDifferentiableAt 𝕜 f x v
F : 𝕜 → E := fun t => x + t • v
hL : f₁ =ᶠ[𝓝 (F 0)] f
⊢ (fun t => f₁ (x + t • v)) =ᶠ[𝓝 0] fun t => f (x + t • v)
tactic: rw [show x = F 0 by simp [F]] at hL
state_before: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F✝ : Type u_2
inst✝³ : NormedAddCommGroup F✝
inst✝² : NormedSpace 𝕜 F✝
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f f₀ f₁ : E → F✝
f' : F✝
s t : Set E
x v : E
L : E →L[𝕜] F✝
h : LineDifferentiableAt 𝕜 f x v
F : 𝕜 → E := fun t => x + t • v
hL : f₁ =ᶠ[𝓝 (F 0)] f
⊢ (fun t => f₁ (x + t • v)) =ᶠ[𝓝 0] fun t => f (x + t • v)
state_after: no goals
tactic: exact (Continuous.continuousAt (by fun_prop)).preimage_mem_nhds hL
Theorem: Mathlib.Meta.NormNum.isNat_zpow_pos
File: Mathlib/Tactic/NormNum/Pow.lean

Proof steps:
state_before: a✝ b✝ c : ℕ
α : Type u_1
inst✝ : DivisionSemiring α
a : α
b : ℤ
nb ne : ℕ
pb : IsNat b nb
pe' : IsNat (a ^ nb) ne
⊢ IsNat (a ^ b) ne
state_after: no goals
tactic: rwa [pb.out, zpow_natCast]
Theorem: AlgebraicGeometry.PresheafedSpace.hext
File: Mathlib/Geometry/RingedSpace/PresheafedSpace.lean

Proof steps:
state_before: C : Type u_1
inst✝ : Category.{u_2, u_1} C
X Y : PresheafedSpace C
α β : X.Hom Y
w : α.base = β.base
h : HEq α.c β.c
⊢ α = β
state_after: case mk
C : Type u_1
inst✝ : Category.{u_2, u_1} C
X Y : PresheafedSpace C
β : X.Hom Y
base✝ : ↑X ⟶ ↑Y
c✝ : Y.presheaf ⟶ (Presheaf.pushforward C base✝).obj X.presheaf
w : { base := base✝, c := c✝ }.base = β.base
h : HEq { base := base✝, c := c✝ }.c β.c
⊢ { base := base✝, c := c✝ } = β
tactic: cases α
state_before: case mk
C : Type u_1
inst✝ : Category.{u_2, u_1} C
X Y : PresheafedSpace C
β : X.Hom Y
base✝ : ↑X ⟶ ↑Y
c✝ : Y.presheaf ⟶ (Presheaf.pushforward C base✝).obj X.presheaf
w : { base := base✝, c := c✝ }.base = β.base
h : HEq { base := base✝, c := c✝ }.c β.c
⊢ { base := base✝, c := c✝ } = β
state_after: case mk.mk
C : Type u_1
inst✝ : Category.{u_2, u_1} C
X Y : PresheafedSpace C
base✝¹ : ↑X ⟶ ↑Y
c✝¹ : Y.presheaf ⟶ (Presheaf.pushforward C base✝¹).obj X.presheaf
base✝ : ↑X ⟶ ↑Y
c✝ : Y.presheaf ⟶ (Presheaf.pushforward C base✝).obj X.presheaf
w : { base := base✝¹, c := c✝¹ }.base = { base := base✝, c := c✝ }.base
h : HEq { base := base✝¹, c := c✝¹ }.c { base := base✝, c := c✝ }.c
⊢ { base := base✝¹, c := c✝¹ } = { base := base✝, c := c✝ }
tactic: cases β
state_before: case mk.mk
C : Type u_1
inst✝ : Category.{u_2, u_1} C
X Y : PresheafedSpace C
base✝¹ : ↑X ⟶ ↑Y
c✝¹ : Y.presheaf ⟶ (Presheaf.pushforward C base✝¹).obj X.presheaf
base✝ : ↑X ⟶ ↑Y
c✝ : Y.presheaf ⟶ (Presheaf.pushforward C base✝).obj X.presheaf
w : { base := base✝¹, c := c✝¹ }.base = { base := base✝, c := c✝ }.base
h : HEq { base := base✝¹, c := c✝¹ }.c { base := base✝, c := c✝ }.c
⊢ { base := base✝¹, c := c✝¹ } = { base := base✝, c := c✝ }
state_after: no goals
tactic: congr
Theorem: Real.sinh_neg_iff
File: Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean

Proof steps:
state_before: x y z : ℝ
⊢ sinh x < 0 ↔ x < 0
state_after: no goals
tactic: simpa only [sinh_zero] using @sinh_lt_sinh x 0
Theorem: CategoryTheory.ShortComplex.Splitting.isoBinaryBiproduct_inv
File: Mathlib/Algebra/Homology/ShortComplex/Exact.lean

Proof steps:
state_before: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{?u.127177, u_1} C
inst✝³ : Category.{?u.127181, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
S : ShortComplex C
h : S.Splitting
inst✝ : HasBinaryBiproduct S.X₁ S.X₃
⊢ biprod.lift h.r S.g ≫ biprod.desc S.f h.s = 𝟙 S.X₂
state_after: no goals
tactic: simp [h.id]
Theorem: Polynomial.eval_natCast_map
File: Mathlib/Algebra/Polynomial/Eval.lean

Proof steps:
state_before: R : Type u
S : Type v
T : Type w
ι : Type y
a b : R
m n✝ : ℕ
inst✝¹ : Semiring R
p✝ q r : R[X]
inst✝ : Semiring S
f✝ f : R →+* S
p : R[X]
n : ℕ
⊢ eval (↑n) (map f p) = f (eval (↑n) p)
state_after: no goals
tactic: induction p using Polynomial.induction_on' with
  | h_add p q hp hq =>
    simp only [hp, hq, Polynomial.map_add, RingHom.map_add, eval_add]
  | h_monomial n r =>
    simp only [map_natCast f, eval_monomial, map_monomial, f.map_pow, f.map_mul]
Theorem: Homeomorph.subtype_apply_coe
File: Mathlib/Topology/Homeomorph.lean

Proof steps:
state_before: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Z
X' : Type u_4
Y' : Type u_5
inst✝¹ : TopologicalSpace X'
inst✝ : TopologicalSpace Y'
p : X → Prop
q : Y → Prop
h : X ≃ₜ Y
h_iff : ∀ (x : X), p x ↔ q (h x)
⊢ Continuous __spread✝⁻⁰.toFun
state_after: no goals
tactic: simpa [Equiv.coe_subtypeEquiv_eq_map] using h.continuous.subtype_map _
state_before: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Z
X' : Type u_4
Y' : Type u_5
inst✝¹ : TopologicalSpace X'
inst✝ : TopologicalSpace Y'
p : X → Prop
q : Y → Prop
h : X ≃ₜ Y
h_iff : ∀ (x : X), p x ↔ q (h x)
⊢ Continuous __spread✝⁻⁰.invFun
state_after: no goals
tactic: simpa [Equiv.coe_subtypeEquiv_eq_map] using
    h.symm.continuous.subtype_map _
Theorem: WithTop.coe_lt_coe
File: Mathlib/Order/WithBot.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : LT α
a b : α
⊢ ↑a < ↑b ↔ a < b
state_after: no goals
tactic: simp only [← toDual_lt_toDual_iff, toDual_apply_coe, WithBot.coe_lt_coe, toDual_lt_toDual]