ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: RingEquiv.zeroRingProd_apply
File: Mathlib/Algebra/Ring/Prod.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
S : Type u_5
S' : Type u_6
T : Type u_7
T' : Type u_8
inst✝⁴ : NonAssocSemiring R
inst✝³ : NonAssocSemiring S
inst✝² : NonAssocSemiring R'
inst✝¹ : NonAssocSemiring S'
inst✝ : Subsingleton S
⊢ ∀ (x y : R),
    { toFun := fun x => (0, x), invFun := Prod.snd, left_inv := ⋯, right_inv := ⋯ }.toFun (x + y) =
      { toFun := fun x => (0, x), invFun := Prod.snd, left_inv := ⋯, right_inv := ⋯ }.toFun x +
        { toFun := fun x => (0, x), invFun := Prod.snd, left_inv := ⋯, right_inv := ⋯ }.toFun y
state_after: no goals
tactic: simp
state_before: α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
S : Type u_5
S' : Type u_6
T : Type u_7
T' : Type u_8
inst✝⁴ : NonAssocSemiring R
inst✝³ : NonAssocSemiring S
inst✝² : NonAssocSemiring R'
inst✝¹ : NonAssocSemiring S'
inst✝ : Subsingleton S
⊢ ∀ (x y : R),
    { toFun := fun x => (0, x), invFun := Prod.snd, left_inv := ⋯, right_inv := ⋯ }.toFun (x * y) =
      { toFun := fun x => (0, x), invFun := Prod.snd, left_inv := ⋯, right_inv := ⋯ }.toFun x *
        { toFun := fun x => (0, x), invFun := Prod.snd, left_inv := ⋯, right_inv := ⋯ }.toFun y
state_after: no goals
tactic: simp
state_before: α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
S : Type u_5
S' : Type u_6
T : Type u_7
T' : Type u_8
inst✝⁴ : NonAssocSemiring R
inst✝³ : NonAssocSemiring S
inst✝² : NonAssocSemiring R'
inst✝¹ : NonAssocSemiring S'
inst✝ : Subsingleton S
x : S × R
⊢ (fun x => (0, x)) x.2 = x
state_after: case mk
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
S : Type u_5
S' : Type u_6
T : Type u_7
T' : Type u_8
inst✝⁴ : NonAssocSemiring R
inst✝³ : NonAssocSemiring S
inst✝² : NonAssocSemiring R'
inst✝¹ : NonAssocSemiring S'
inst✝ : Subsingleton S
fst✝ : S
snd✝ : R
⊢ (fun x => (0, x)) (fst✝, snd✝).2 = (fst✝, snd✝)
tactic: cases x
state_before: case mk
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
S : Type u_5
S' : Type u_6
T : Type u_7
T' : Type u_8
inst✝⁴ : NonAssocSemiring R
inst✝³ : NonAssocSemiring S
inst✝² : NonAssocSemiring R'
inst✝¹ : NonAssocSemiring S'
inst✝ : Subsingleton S
fst✝ : S
snd✝ : R
⊢ (fun x => (0, x)) (fst✝, snd✝).2 = (fst✝, snd✝)
state_after: no goals
tactic: simp [eq_iff_true_of_subsingleton]
Theorem: left_lt_add_div_two
File: Mathlib/Algebra/Order/Field/Basic.lean

Proof steps:
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : LinearOrderedSemifield α
a b c d e : α
m n : ℤ
⊢ a < (a + b) / 2 ↔ a < b
state_after: no goals
tactic: simp [lt_div_iff, mul_two]
Theorem: SimpleGraph.TripartiteFromTriangles.farFromTriangleFree
File: Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
𝕜 : Type u_4
inst✝⁷ : LinearOrderedField 𝕜
t : Finset (α × β × γ)
a a' : α
b b' : β
c c' : γ
x✝ : α × β × γ
ε✝ : 𝕜
inst✝⁶ : DecidableEq α
inst✝⁵ : DecidableEq β
inst✝⁴ : DecidableEq γ
inst✝³ : Fintype α
inst✝² : Fintype β
inst✝¹ : Fintype γ
inst✝ : ExplicitDisjoint t
ε : 𝕜
ht : ε * ↑((Fintype.card α + Fintype.card β + Fintype.card γ) ^ 2) ≤ ↑t.card
x : α × β × γ
hx : x ∈ t
⊢ x ∈ ((graph t).cliqueFinset 3).preimage ⇑toTriangle ⋯
state_after: no goals
tactic: simpa using toTriangle_is3Clique hx
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
𝕜 : Type u_4
inst✝⁷ : LinearOrderedField 𝕜
t : Finset (α × β × γ)
a a' : α
b b' : β
c c' : γ
x : α × β × γ
ε✝ : 𝕜
inst✝⁶ : DecidableEq α
inst✝⁵ : DecidableEq β
inst✝⁴ : DecidableEq γ
inst✝³ : Fintype α
inst✝² : Fintype β
inst✝¹ : Fintype γ
inst✝ : ExplicitDisjoint t
ε : 𝕜
ht : ε * ↑((Fintype.card α + Fintype.card β + Fintype.card γ) ^ 2) ≤ ↑t.card
⊢ ε * ↑(Fintype.card (α ⊕ β ⊕ γ) ^ 2) ≤ ↑(map toTriangle t).card
state_after: no goals
tactic: simpa [add_assoc] using ht
Theorem: Inducing.comp
File: Mathlib/Topology/Maps/Basic.lean

Proof steps:
state_before: X : Type u_1
Y : Type u_2
Z : Type u_3
ι : Type u_4
f : X → Y
g : Y → Z
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
hg : Inducing g
hf : Inducing f
⊢ inst✝² = TopologicalSpace.induced (g ∘ f) inst✝
state_after: no goals
tactic: rw [hf.induced, hg.induced, induced_compose]
Theorem: CategoryTheory.ShortComplex.RightHomologyData.map_opcyclesMap'
File: Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean

Proof steps:
state_before: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_4, u_1} C
inst✝⁶ : Category.{u_3, u_2} D
inst✝⁵ : HasZeroMorphisms C
inst✝⁴ : HasZeroMorphisms D
S : ShortComplex C
h₁✝ : S.LeftHomologyData
h₂✝ : S.RightHomologyData
F✝ : C ⥤ D
inst✝³ : F✝.PreservesZeroMorphisms
hl : S.LeftHomologyData
hr : S.RightHomologyData
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : hr₁.IsPreservedBy F
inst✝ : hr₂.IsPreservedBy F
⊢ F.map (opcyclesMap' φ hr₁ hr₂) = opcyclesMap' (F.mapShortComplex.map φ) (hr₁.map F) (hr₂.map F)
state_after: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_4, u_1} C
inst✝⁶ : Category.{u_3, u_2} D
inst✝⁵ : HasZeroMorphisms C
inst✝⁴ : HasZeroMorphisms D
S : ShortComplex C
h₁✝ : S.LeftHomologyData
h₂✝ : S.RightHomologyData
F✝ : C ⥤ D
inst✝³ : F✝.PreservesZeroMorphisms
hl : S.LeftHomologyData
hr : S.RightHomologyData
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : hr₁.IsPreservedBy F
inst✝ : hr₂.IsPreservedBy F
γ : RightHomologyMapData φ hr₁ hr₂
⊢ F.map (opcyclesMap' φ hr₁ hr₂) = opcyclesMap' (F.mapShortComplex.map φ) (hr₁.map F) (hr₂.map F)
tactic: have γ : ShortComplex.RightHomologyMapData φ hr₁ hr₂ := default
state_before: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_4, u_1} C
inst✝⁶ : Category.{u_3, u_2} D
inst✝⁵ : HasZeroMorphisms C
inst✝⁴ : HasZeroMorphisms D
S : ShortComplex C
h₁✝ : S.LeftHomologyData
h₂✝ : S.RightHomologyData
F✝ : C ⥤ D
inst✝³ : F✝.PreservesZeroMorphisms
hl : S.LeftHomologyData
hr : S.RightHomologyData
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : hr₁.IsPreservedBy F
inst✝ : hr₂.IsPreservedBy F
γ : RightHomologyMapData φ hr₁ hr₂
⊢ F.map (opcyclesMap' φ hr₁ hr₂) = opcyclesMap' (F.mapShortComplex.map φ) (hr₁.map F) (hr₂.map F)
state_after: no goals
tactic: rw [γ.opcyclesMap'_eq, (γ.map F).opcyclesMap'_eq,  ShortComplex.RightHomologyMapData.map_φQ]
Theorem: schnirelmannDensity_eq_one_iff
File: Mathlib/Combinatorics/Schnirelmann.lean

Proof steps:
state_before: A : Set ℕ
inst✝ : DecidablePred fun x => x ∈ A
⊢ schnirelmannDensity A = 1 ↔ {0}ᶜ ⊆ A
state_after: A : Set ℕ
inst✝ : DecidablePred fun x => x ∈ A
⊢ 1 ≤ schnirelmannDensity A ↔ {0}ᶜ ⊆ A
tactic: rw [le_antisymm_iff, and_iff_right schnirelmannDensity_le_one]
state_before: A : Set ℕ
inst✝ : DecidablePred fun x => x ∈ A
⊢ 1 ≤ schnirelmannDensity A ↔ {0}ᶜ ⊆ A
state_after: case mp
A : Set ℕ
inst✝ : DecidablePred fun x => x ∈ A
⊢ 1 ≤ schnirelmannDensity A → {0}ᶜ ⊆ A

case mpr
A : Set ℕ
inst✝ : DecidablePred fun x => x ∈ A
⊢ {0}ᶜ ⊆ A → 1 ≤ schnirelmannDensity A
tactic: constructor
Theorem: Polynomial.degree_quadratic_le
File: Mathlib/Algebra/Polynomial/Degree/Definitions.lean

Proof steps:
state_before: R : Type u
S : Type v
a b c d : R
n m : ℕ
inst✝ : Semiring R
p✝ p q : R[X]
ι : Type u_1
⊢ (C a * X ^ 2 + C b * X + C c).degree ≤ 2
state_after: no goals
tactic: simpa only [add_assoc] using
    degree_add_le_of_degree_le (degree_C_mul_X_pow_le 2 a)
      (le_trans degree_linear_le <| WithBot.coe_le_coe.mpr one_le_two)
Theorem: CategoryTheory.epi_iff_isIso_inl
File: Mathlib/CategoryTheory/Limits/EpiMono.lean

Proof steps:
state_before: C : Type u_1
inst✝ : Category.{u_2, u_1} C
X Y : C
f : X ⟶ Y
c : PushoutCocone f f
hc : IsColimit c
⊢ Epi f ↔ IsIso c.inl
state_after: C : Type u_1
inst✝ : Category.{u_2, u_1} C
X Y : C
f : X ⟶ Y
c : PushoutCocone f f
hc : IsColimit c
⊢ c.inl = c.inr ↔ IsIso c.inl
tactic: rw [epi_iff_inl_eq_inr hc]
state_before: C : Type u_1
inst✝ : Category.{u_2, u_1} C
X Y : C
f : X ⟶ Y
c : PushoutCocone f f
hc : IsColimit c
⊢ c.inl = c.inr ↔ IsIso c.inl
state_after: case mp
C : Type u_1
inst✝ : Category.{u_2, u_1} C
X Y : C
f : X ⟶ Y
c : PushoutCocone f f
hc : IsColimit c
⊢ c.inl = c.inr → IsIso c.inl

case mpr
C : Type u_1
inst✝ : Category.{u_2, u_1} C
X Y : C
f : X ⟶ Y
c : PushoutCocone f f
hc : IsColimit c
⊢ IsIso c.inl → c.inl = c.inr
tactic: constructor
Theorem: equicontinuousWithinAt_iInf_rng
File: Mathlib/Topology/UniformSpace/Equicontinuity.lean

Proof steps:
state_before: ι : Type u_1
κ : Type u_2
X : Type u_3
X' : Type u_4
Y : Type u_5
Z : Type u_6
α : Type u_7
α' : Type u_8
β : Type u_9
β' : Type u_10
γ : Type u_11
𝓕 : Type u_12
tX : TopologicalSpace X
tY : TopologicalSpace Y
tZ : TopologicalSpace Z
uα : UniformSpace α
uβ : UniformSpace β
uγ : UniformSpace γ
u : κ → UniformSpace α'
F : ι → X → α'
S : Set X
x₀ : X
⊢ EquicontinuousWithinAt F S x₀ ↔ ∀ (k : κ), EquicontinuousWithinAt F S x₀
state_after: ι : Type u_1
κ : Type u_2
X : Type u_3
X' : Type u_4
Y : Type u_5
Z : Type u_6
α : Type u_7
α' : Type u_8
β : Type u_9
β' : Type u_10
γ : Type u_11
𝓕 : Type u_12
tX : TopologicalSpace X
tY : TopologicalSpace Y
tZ : TopologicalSpace Z
uα : UniformSpace α
uβ : UniformSpace β
uγ : UniformSpace γ
u : κ → UniformSpace α'
F : ι → X → α'
S : Set X
x₀ : X
⊢ ContinuousWithinAt (⇑UniformFun.ofFun ∘ swap F) S x₀ ↔ ∀ (k : κ), ContinuousWithinAt (⇑UniformFun.ofFun ∘ swap F) S x₀
tactic: simp only [equicontinuousWithinAt_iff_continuousWithinAt (uα := _), topologicalSpace]
state_before: ι : Type u_1
κ : Type u_2
X : Type u_3
X' : Type u_4
Y : Type u_5
Z : Type u_6
α : Type u_7
α' : Type u_8
β : Type u_9
β' : Type u_10
γ : Type u_11
𝓕 : Type u_12
tX : TopologicalSpace X
tY : TopologicalSpace Y
tZ : TopologicalSpace Z
uα : UniformSpace α
uβ : UniformSpace β
uγ : UniformSpace γ
u : κ → UniformSpace α'
F : ι → X → α'
S : Set X
x₀ : X
⊢ ContinuousWithinAt (⇑UniformFun.ofFun ∘ swap F) S x₀ ↔ ∀ (k : κ), ContinuousWithinAt (⇑UniformFun.ofFun ∘ swap F) S x₀
state_after: ι : Type u_1
κ : Type u_2
X : Type u_3
X' : Type u_4
Y : Type u_5
Z : Type u_6
α : Type u_7
α' : Type u_8
β : Type u_9
β' : Type u_10
γ : Type u_11
𝓕 : Type u_12
tX : TopologicalSpace X
tY : TopologicalSpace Y
tZ : TopologicalSpace Z
uα : UniformSpace α
uβ : UniformSpace β
uγ : UniformSpace γ
u : κ → UniformSpace α'
F : ι → X → α'
S : Set X
x₀ : X
⊢ Tendsto (⇑UniformFun.ofFun ∘ swap F) (𝓝[S] x₀) (𝓝 ((⇑UniformFun.ofFun ∘ swap F) x₀)) ↔
    ∀ (k : κ), Tendsto (⇑UniformFun.ofFun ∘ swap F) (𝓝[S] x₀) (𝓝 ((⇑UniformFun.ofFun ∘ swap F) x₀))
tactic: unfold ContinuousWithinAt
state_before: ι : Type u_1
κ : Type u_2
X : Type u_3
X' : Type u_4
Y : Type u_5
Z : Type u_6
α : Type u_7
α' : Type u_8
β : Type u_9
β' : Type u_10
γ : Type u_11
𝓕 : Type u_12
tX : TopologicalSpace X
tY : TopologicalSpace Y
tZ : TopologicalSpace Z
uα : UniformSpace α
uβ : UniformSpace β
uγ : UniformSpace γ
u : κ → UniformSpace α'
F : ι → X → α'
S : Set X
x₀ : X
⊢ Tendsto (⇑UniformFun.ofFun ∘ swap F) (𝓝[S] x₀) (𝓝 ((⇑UniformFun.ofFun ∘ swap F) x₀)) ↔
    ∀ (k : κ), Tendsto (⇑UniformFun.ofFun ∘ swap F) (𝓝[S] x₀) (𝓝 ((⇑UniformFun.ofFun ∘ swap F) x₀))
state_after: no goals
tactic: rw [UniformFun.iInf_eq, toTopologicalSpace_iInf, nhds_iInf, tendsto_iInf]
Theorem: UpperSet.coe_iSup
File: Mathlib/Order/UpperLower/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
κ : ι → Sort u_5
inst✝ : LE α
S : Set (UpperSet α)
s t : UpperSet α
a : α
f : ι → UpperSet α
⊢ ↑(⨆ i, f i) = ⋂ i, ↑(f i)
state_after: no goals
tactic: simp [iSup]
Theorem: AddSubgroup.prod_eq_bot_iff
File: Mathlib/Algebra/Group/Subgroup/Basic.lean

Proof steps:
state_before: G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝⁵ : Group G
inst✝⁴ : Group G'
inst✝³ : Group G''
A : Type u_4
inst✝² : AddGroup A
H✝ K✝ : Subgroup G
k : Set G
N : Type u_5
inst✝¹ : Group N
P : Type u_6
inst✝ : Group P
H : Subgroup G
K : Subgroup N
⊢ H.prod K = ⊥ ↔ H = ⊥ ∧ K = ⊥
state_after: no goals
tactic: simpa only [← Subgroup.toSubmonoid_eq] using Submonoid.prod_eq_bot_iff
Theorem: AlgebraicGeometry.is_localization_basicOpen_of_qcqs
File: Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean

Proof steps:
state_before: X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : X.Opens
hU : IsCompact U.carrier
hU' : IsQuasiSeparated U.carrier
f : ↑Γ(X, U)
⊢ IsLocalization.Away f ↑Γ(X, X.basicOpen f)
state_after: case map_units'
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : X.Opens
hU : IsCompact U.carrier
hU' : IsQuasiSeparated U.carrier
f : ↑Γ(X, U)
⊢ ∀ (y : ↥(Submonoid.powers f)), IsUnit ((algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑y)

case surj'
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : X.Opens
hU : IsCompact U.carrier
hU' : IsQuasiSeparated U.carrier
f : ↑Γ(X, U)
⊢ ∀ (z : ↑Γ(X, X.basicOpen f)),
    ∃ x, z * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑x.2 = (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) x.1

case exists_of_eq
X✝ Y : Scheme
f✝ : X✝ ⟶ Y
X : Scheme
U : X.Opens
hU : IsCompact U.carrier
hU' : IsQuasiSeparated U.carrier
f : ↑Γ(X, U)
⊢ ∀ {x y : ↑Γ(X, U)},
    (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) x = (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) y → ∃ c, ↑c * x = ↑c * y
tactic: constructor
Theorem: Real.rpow_lt_rpow_left_iff_of_base_lt_one
File: Mathlib/Analysis/SpecialFunctions/Pow/Real.lean

Proof steps:
state_before: x y z : ℝ
n : ℕ
hx0 : 0 < x
hx1 : x < 1
⊢ x ^ y < x ^ z ↔ z < y
state_after: no goals
tactic: rw [lt_iff_not_le, rpow_le_rpow_left_iff_of_base_lt_one hx0 hx1, lt_iff_not_le]
Theorem: Finset.ne_insert_of_not_mem
File: Mathlib/Data/Finset/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s✝ t✝ u v : Finset α
a✝ b : α
s t : Finset α
a : α
h : a ∉ s
⊢ s ≠ insert a t
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s✝ t✝ u v : Finset α
a✝ b : α
s t : Finset α
a : α
h : s = insert a t
⊢ a ∈ s
tactic: contrapose! h
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s✝ t✝ u v : Finset α
a✝ b : α
s t : Finset α
a : α
h : s = insert a t
⊢ a ∈ s
state_after: no goals
tactic: simp [h]
Theorem: Hyperreal.infiniteNeg_mul_of_infiniteNeg_not_infinitesimal_pos
File: Mathlib/Data/Real/Hyperreal.lean

Proof steps:
state_before: x y : ℝ*
⊢ x.InfiniteNeg → ¬y.Infinitesimal → 0 < y → (x * y).InfiniteNeg
state_after: x y : ℝ*
⊢ (-x).InfinitePos → ¬y.Infinitesimal → 0 < y → (-x * y).InfinitePos
tactic: rw [← infinitePos_neg, ← infinitePos_neg, neg_mul_eq_neg_mul]
state_before: x y : ℝ*
⊢ (-x).InfinitePos → ¬y.Infinitesimal → 0 < y → (-x * y).InfinitePos
state_after: no goals
tactic: exact infinitePos_mul_of_infinitePos_not_infinitesimal_pos
Theorem: exists_seq_tendsto_sSup
File: Mathlib/Topology/Order/IsLUB.lean

Proof steps:
state_before: α✝ : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁹ : TopologicalSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : LinearOrder α✝
inst✝⁶ : LinearOrder β
inst✝⁵ : OrderTopology α✝
inst✝⁴ : OrderTopology β
α : Type u_4
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : FirstCountableTopology α
S : Set α
hS : S.Nonempty
hS' : BddAbove S
⊢ ∃ u, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ (n : ℕ), u n ∈ S
state_after: case intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁹ : TopologicalSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : LinearOrder α✝
inst✝⁶ : LinearOrder β
inst✝⁵ : OrderTopology α✝
inst✝⁴ : OrderTopology β
α : Type u_4
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : FirstCountableTopology α
S : Set α
hS : S.Nonempty
hS' : BddAbove S
u : ℕ → α
hu : Monotone u ∧ (∀ (n : ℕ), u n ≤ sSup S) ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ (n : ℕ), u n ∈ S
⊢ ∃ u, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ (n : ℕ), u n ∈ S
tactic: rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩
state_before: case intro
α✝ : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁹ : TopologicalSpace α✝
inst✝⁸ : TopologicalSpace β
inst✝⁷ : LinearOrder α✝
inst✝⁶ : LinearOrder β
inst✝⁵ : OrderTopology α✝
inst✝⁴ : OrderTopology β
α : Type u_4
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : FirstCountableTopology α
S : Set α
hS : S.Nonempty
hS' : BddAbove S
u : ℕ → α
hu : Monotone u ∧ (∀ (n : ℕ), u n ≤ sSup S) ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ (n : ℕ), u n ∈ S
⊢ ∃ u, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ (n : ℕ), u n ∈ S
state_after: no goals
tactic: exact ⟨u, hu.1, hu.2.2⟩
Theorem: Set.image_neg_Iic
File: Mathlib/Data/Set/Pointwise/Interval.lean

Proof steps:
state_before: α : Type u_1
inst✝ : OrderedAddCommGroup α
a b c : α
⊢ Neg.neg '' Iic a = Ici (-a)
state_after: no goals
tactic: simp
Theorem: FiniteDimensional.right
File: Mathlib/FieldTheory/Tower.lean

Proof steps:
state_before: F : Type u
K : Type v
A : Type w
inst✝⁶ : DivisionRing F
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup A
inst✝³ : Module F K
inst✝² : Module K A
inst✝¹ : Module F A
inst✝ : IsScalarTower F K A
hf : FiniteDimensional F A
b : Finset A
hb : span F ↑b = ⊤
⊢ restrictScalars F (span K ↑b) = restrictScalars F ⊤
state_after: F : Type u
K : Type v
A : Type w
inst✝⁶ : DivisionRing F
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup A
inst✝³ : Module F K
inst✝² : Module K A
inst✝¹ : Module F A
inst✝ : IsScalarTower F K A
hf : FiniteDimensional F A
b : Finset A
hb : span F ↑b = ⊤
⊢ ↑b ⊆ ↑(restrictScalars F (span K ↑b))
tactic: rw [Submodule.restrictScalars_top, eq_top_iff, ← hb, Submodule.span_le]
state_before: F : Type u
K : Type v
A : Type w
inst✝⁶ : DivisionRing F
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup A
inst✝³ : Module F K
inst✝² : Module K A
inst✝¹ : Module F A
inst✝ : IsScalarTower F K A
hf : FiniteDimensional F A
b : Finset A
hb : span F ↑b = ⊤
⊢ ↑b ⊆ ↑(restrictScalars F (span K ↑b))
state_after: no goals
tactic: exact Submodule.subset_span
Theorem: zpow_add_one₀
File: Mathlib/Algebra/GroupWithZero/Basic.lean

Proof steps:
state_before: α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝ : GroupWithZero G₀
a b c : G₀
ha : a ≠ 0
n : ℕ
⊢ a ^ (↑n + 1) = a ^ ↑n * a
state_after: no goals
tactic: simp only [← Int.ofNat_succ, zpow_natCast, pow_succ]
state_before: α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝ : GroupWithZero G₀
a b c : G₀
ha : a ≠ 0
⊢ a ^ (Int.negSucc 0 + 1) = a ^ Int.negSucc 0 * a
state_after: no goals
tactic: erw [zpow_zero, zpow_negSucc, pow_one, inv_mul_cancel ha]
state_before: α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝ : GroupWithZero G₀
a b c : G₀
ha : a ≠ 0
n : ℕ
⊢ a ^ (Int.negSucc (n + 1) + 1) = a ^ Int.negSucc (n + 1) * a
state_after: no goals
tactic: rw [Int.negSucc_eq, zpow_neg, Int.neg_add, Int.neg_add_cancel_right, zpow_neg, ← Int.ofNat_succ,
      zpow_natCast, zpow_natCast, pow_succ' _ (n + 1), mul_inv_rev, mul_assoc, inv_mul_cancel ha,
      mul_one]
Theorem: LinearMap.ker_le_comap
File: Mathlib/Algebra/Module/Submodule/Ker.lean

Proof steps:
state_before: R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
K : Type u_5
M : Type u_6
M₁ : Type u_7
M₂ : Type u_8
M₃ : Type u_9
V : Type u_10
V₂ : Type u_11
inst✝¹² : Semiring R
inst✝¹¹ : Semiring R₂
inst✝¹⁰ : Semiring R₃
inst✝⁹ : AddCommMonoid M
inst✝⁸ : AddCommMonoid M₂
inst✝⁷ : AddCommMonoid M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝⁶ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝⁵ : Module R M
inst✝⁴ : Module R₂ M₂
inst✝³ : Module R₃ M₃
σ₂₁ : R₂ →+* R
τ₁₂ : R →+* R₂
τ₂₃ : R₂ →+* R₃
τ₁₃ : R →+* R₃
inst✝² : RingHomCompTriple τ₁₂ τ₂₃ τ₁₃
F : Type u_12
inst✝¹ : FunLike F M M₂
inst✝ : SemilinearMapClass F τ₁₂ M M₂
p : Submodule R₂ M₂
f : M →ₛₗ[τ₁₂] M₂
x : M
hx : x ∈ ker f
⊢ x ∈ comap f p
state_after: no goals
tactic: simp [mem_ker.mp hx]
Theorem: Tropical.succ_nsmul
File: Mathlib/Algebra/Tropical/Basic.lean

Proof steps:
state_before: R✝ : Type u
inst✝² : LinearOrderedAddCommMonoidWithTop R✝
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : OrderTop R
x : Tropical R
n : ℕ
⊢ (n + 1) • x = x
state_after: case zero
R✝ : Type u
inst✝² : LinearOrderedAddCommMonoidWithTop R✝
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : OrderTop R
x : Tropical R
⊢ (0 + 1) • x = x

case succ
R✝ : Type u
inst✝² : LinearOrderedAddCommMonoidWithTop R✝
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : OrderTop R
x : Tropical R
n : ℕ
IH : (n + 1) • x = x
⊢ (n + 1 + 1) • x = x
tactic: induction' n with n IH
Theorem: SimpleGraph.TripartiteFromTriangles.Graph.in₀₂_iff'
File: Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
𝕜 : Type u_4
inst✝ : LinearOrderedField 𝕜
t : Finset (α × β × γ)
a a' : α
b b' : β
c c' : γ
x : α × β × γ
ε : 𝕜
⊢ (graph t).Adj (in₀ a) (in₂ c) → ∃ x ∈ t, x.1 = a ∧ x.2.2 = c
state_after: case in₀₂
α : Type u_1
β : Type u_2
γ : Type u_3
𝕜 : Type u_4
inst✝ : LinearOrderedField 𝕜
t : Finset (α × β × γ)
a a' : α
b b' : β
c c' : γ
x : α × β × γ
ε : 𝕜
b✝ : β
a✝ : (a, b✝, c) ∈ t
⊢ ∃ x ∈ t, x.1 = a ∧ x.2.2 = c
tactic: rintro ⟨⟩
state_before: case in₀₂
α : Type u_1
β : Type u_2
γ : Type u_3
𝕜 : Type u_4
inst✝ : LinearOrderedField 𝕜
t : Finset (α × β × γ)
a a' : α
b b' : β
c c' : γ
x : α × β × γ
ε : 𝕜
b✝ : β
a✝ : (a, b✝, c) ∈ t
⊢ ∃ x ∈ t, x.1 = a ∧ x.2.2 = c
state_after: no goals
tactic: exact ⟨_, ‹_›, by simp⟩
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
𝕜 : Type u_4
inst✝ : LinearOrderedField 𝕜
t : Finset (α × β × γ)
a a' : α
b b' : β
c c' : γ
x : α × β × γ
ε : 𝕜
⊢ (∃ x ∈ t, x.1 = a ∧ x.2.2 = c) → (graph t).Adj (in₀ a) (in₂ c)
state_after: case intro.mk.mk.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
𝕜 : Type u_4
inst✝ : LinearOrderedField 𝕜
t : Finset (α × β × γ)
a' : α
b✝ b' : β
c' : γ
x : α × β × γ
ε : 𝕜
a : α
b : β
c : γ
h : (a, b, c) ∈ t
⊢ (graph t).Adj (in₀ (a, b, c).1) (in₂ (a, b, c).2.2)
tactic: rintro ⟨⟨a, b, c⟩, h, rfl, rfl⟩
state_before: case intro.mk.mk.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
𝕜 : Type u_4
inst✝ : LinearOrderedField 𝕜
t : Finset (α × β × γ)
a' : α
b✝ b' : β
c' : γ
x : α × β × γ
ε : 𝕜
a : α
b : β
c : γ
h : (a, b, c) ∈ t
⊢ (graph t).Adj (in₀ (a, b, c).1) (in₂ (a, b, c).2.2)
state_after: case intro.mk.mk.intro.intro.a
α : Type u_1
β : Type u_2
γ : Type u_3
𝕜 : Type u_4
inst✝ : LinearOrderedField 𝕜
t : Finset (α × β × γ)
a' : α
b✝ b' : β
c' : γ
x : α × β × γ
ε : 𝕜
a : α
b : β
c : γ
h : (a, b, c) ∈ t
⊢ ((a, b, c).1, ?intro.mk.mk.intro.intro.b, (a, b, c).2.2) ∈ t

case intro.mk.mk.intro.intro.b
α : Type u_1
β : Type u_2
γ : Type u_3
𝕜 : Type u_4
inst✝ : LinearOrderedField 𝕜
t : Finset (α × β × γ)
a' : α
b✝ b' : β
c' : γ
x : α × β × γ
ε : 𝕜
a : α
b : β
c : γ
h : (a, b, c) ∈ t
⊢ β
tactic: constructor
state_before: case intro.mk.mk.intro.intro.a
α : Type u_1
β : Type u_2
γ : Type u_3
𝕜 : Type u_4
inst✝ : LinearOrderedField 𝕜
t : Finset (α × β × γ)
a' : α
b✝ b' : β
c' : γ
x : α × β × γ
ε : 𝕜
a : α
b : β
c : γ
h : (a, b, c) ∈ t
⊢ ((a, b, c).1, ?intro.mk.mk.intro.intro.b, (a, b, c).2.2) ∈ t

case intro.mk.mk.intro.intro.b
α : Type u_1
β : Type u_2
γ : Type u_3
𝕜 : Type u_4
inst✝ : LinearOrderedField 𝕜
t : Finset (α × β × γ)
a' : α
b✝ b' : β
c' : γ
x : α × β × γ
ε : 𝕜
a : α
b : β
c : γ
h : (a, b, c) ∈ t
⊢ β
state_after: no goals
tactic: assumption
Theorem: List.enumFrom_cons'
File: Mathlib/Data/List/Enum.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
n : ℕ
x : α
xs : List α
⊢ enumFrom n (x :: xs) = (n, x) :: map (Prod.map Nat.succ id) (enumFrom n xs)
state_after: no goals
tactic: rw [enumFrom_cons, Nat.add_comm, ← map_fst_add_enumFrom_eq_enumFrom]
Theorem: Submonoid.LocalizationMap.eq'
File: Mathlib/GroupTheory/MonoidLocalization/Basic.lean

Proof steps:
state_before: M : Type u_1
inst✝² : CommMonoid M
S : Submonoid M
N : Type u_2
inst✝¹ : CommMonoid N
P : Type u_3
inst✝ : CommMonoid P
f : S.LocalizationMap N
a₁ b₁ : M
a₂ b₂ : ↥S
⊢ f.mk' a₁ a₂ = f.mk' b₁ b₂ ↔ (Localization.r S) (a₁, a₂) (b₁, b₂)
state_after: no goals
tactic: rw [f.eq, Localization.r_iff_exists]