ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: ENNReal.sub_iInf
File: Mathlib/Data/ENNReal/Real.lean

Proof steps:
state_before: ι : Sort u_1
f g : ι → ℝ≥0∞
a b c d : ℝ≥0∞
r p q : ℝ≥0
⊢ a - ⨅ i, f i = ⨆ i, a - f i
state_after: ι : Sort u_1
f g : ι → ℝ≥0∞
a b c✝ d : ℝ≥0∞
r p q : ℝ≥0
c : ℝ≥0∞
⊢ a - ⨅ i, f i ≤ c ↔ ⨆ i, a - f i ≤ c
tactic: refine eq_of_forall_ge_iff fun c => ?_
state_before: ι : Sort u_1
f g : ι → ℝ≥0∞
a b c✝ d : ℝ≥0∞
r p q : ℝ≥0
c : ℝ≥0∞
⊢ a - ⨅ i, f i ≤ c ↔ ⨆ i, a - f i ≤ c
state_after: ι : Sort u_1
f g : ι → ℝ≥0∞
a b c✝ d : ℝ≥0∞
r p q : ℝ≥0
c : ℝ≥0∞
⊢ a ≤ ⨅ i, f i + c ↔ ⨆ i, a - f i ≤ c
tactic: rw [tsub_le_iff_right, add_comm, iInf_add]
state_before: ι : Sort u_1
f g : ι → ℝ≥0∞
a b c✝ d : ℝ≥0∞
r p q : ℝ≥0
c : ℝ≥0∞
⊢ a ≤ ⨅ i, f i + c ↔ ⨆ i, a - f i ≤ c
state_after: no goals
tactic: simp [tsub_le_iff_right, sub_eq_add_neg, add_comm]
Theorem: AddSubgroup.closure_closure_coe_preimage
File: Mathlib/Algebra/Group/Subgroup/Basic.lean

Proof steps:
state_before: G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : Group G''
A : Type u_4
inst✝ : AddGroup A
H K : Subgroup G
k✝ k : Set G
x✝¹ : { x // x ∈ ↑(closure k) }
x : G
hx : x ∈ ↑(closure k)
x✝ : ⟨x, hx⟩ ∈ ⊤
⊢ ⟨x, hx⟩ ∈ closure (Subtype.val ⁻¹' k)
state_after: case refine_1
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : Group G''
A : Type u_4
inst✝ : AddGroup A
H K : Subgroup G
k✝ k : Set G
x✝¹ : { x // x ∈ ↑(closure k) }
x : G
hx : x ∈ ↑(closure k)
x✝ : ⟨x, hx⟩ ∈ ⊤
g : G
hg : g ∈ k
⊢ ⟨g, ⋯⟩ ∈ closure (Subtype.val ⁻¹' k)

case refine_2
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : Group G''
A : Type u_4
inst✝ : AddGroup A
H K : Subgroup G
k✝ k : Set G
x✝¹ : { x // x ∈ ↑(closure k) }
x : G
hx : x ∈ ↑(closure k)
x✝ : ⟨x, hx⟩ ∈ ⊤
⊢ ⟨1, ⋯⟩ ∈ closure (Subtype.val ⁻¹' k)

case refine_3
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : Group G''
A : Type u_4
inst✝ : AddGroup A
H K : Subgroup G
k✝ k : Set G
x✝¹ : { x // x ∈ ↑(closure k) }
x : G
hx : x ∈ ↑(closure k)
x✝ : ⟨x, hx⟩ ∈ ⊤
g₁ : G
g₂ : g₁ ∈ closure k
hg₁ : G
hg₂ : hg₁ ∈ closure k
⊢ ⟨g₁, g₂⟩ ∈ closure (Subtype.val ⁻¹' k) →
    ⟨hg₁, hg₂⟩ ∈ closure (Subtype.val ⁻¹' k) → ⟨g₁ * hg₁, ⋯⟩ ∈ closure (Subtype.val ⁻¹' k)

case refine_4
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝³ : Group G
inst✝² : Group G'
inst✝¹ : Group G''
A : Type u_4
inst✝ : AddGroup A
H K : Subgroup G
k✝ k : Set G
x✝¹ : { x // x ∈ ↑(closure k) }
x : G
hx : x ∈ ↑(closure k)
x✝ : ⟨x, hx⟩ ∈ ⊤
g : G
hg : g ∈ closure k
⊢ ⟨g, hg⟩ ∈ closure (Subtype.val ⁻¹' k) → ⟨g⁻¹, ⋯⟩ ∈ closure (Subtype.val ⁻¹' k)
tactic: refine closure_induction' (fun g hg => ?_) ?_ (fun g₁ g₂ hg₁ hg₂ => ?_) (fun g hg => ?_) hx
Theorem: Units.dvd_mul_right
File: Mathlib/Algebra/Divisibility/Units.lean

Proof steps:
state_before: α : Type u_1
inst✝ : Monoid α
a b : α
u : αˣ
x✝ : a ∣ b * ↑u
c : α
Eq : b * ↑u = a * c
⊢ b = a * (c * ↑u⁻¹)
state_after: no goals
tactic: rw [← mul_assoc, ← Eq, Units.mul_inv_cancel_right]
Theorem: Polynomial.reverse_C_add
File: Mathlib/Algebra/Polynomial/Reverse.lean

Proof steps:
state_before: R : Type u_1
inst✝ : Semiring R
f p : R[X]
t : R
⊢ (C t + p).reverse = C t * X ^ p.natDegree + p.reverse
state_after: no goals
tactic: rw [add_comm, reverse_add_C, add_comm]
Theorem: Submodule.singleton_smul
File: Mathlib/Algebra/Algebra/Operations.lean

Proof steps:
state_before: ι : Sort uι
R : Type u
inst✝² : CommSemiring R
A : Type v
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
M✝ N : Submodule R A
m n a : A
M : Submodule R A
⊢ Set.up {a} • M = map (LinearMap.mulLeft R a) M
state_after: ι : Sort uι
R : Type u
inst✝² : CommSemiring R
A : Type v
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
M✝ N : Submodule R A
m n a : A
M : Submodule R A
⊢ Set.up {a} • span R ↑M = map (LinearMap.mulLeft R a) M
tactic: conv_lhs => rw [← span_eq M]
state_before: ι : Sort uι
R : Type u
inst✝² : CommSemiring R
A : Type v
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
M✝ N : Submodule R A
m n a : A
M : Submodule R A
⊢ Set.up {a} • span R ↑M = map (LinearMap.mulLeft R a) M
state_after: ι : Sort uι
R : Type u
inst✝² : CommSemiring R
A : Type v
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
M✝ N : Submodule R A
m n a : A
M : Submodule R A
⊢ span R ((fun x => a * x) '' ↑M) = map (LinearMap.mulLeft R a) M
tactic: rw [smul_def, SetSemiring.down_up, span_mul_span, singleton_mul]
state_before: ι : Sort uι
R : Type u
inst✝² : CommSemiring R
A : Type v
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
M✝ N : Submodule R A
m n a : A
M : Submodule R A
⊢ span R ((fun x => a * x) '' ↑M) = map (LinearMap.mulLeft R a) M
state_after: no goals
tactic: exact (map (LinearMap.mulLeft R a) M).span_eq
Theorem: Nat.sq_sub_sq
File: Mathlib/Data/Nat/Defs.lean

Proof steps:
state_before: a✝ b✝ c d m n k : ℕ
p q : ℕ → Prop
a b : ℕ
⊢ a ^ 2 - b ^ 2 = (a + b) * (a - b)
state_after: no goals
tactic: simpa [Nat.pow_succ] using Nat.mul_self_sub_mul_self_eq a b
Theorem: Group.covariant_swap_iff_contravariant_swap
File: Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean

Proof steps:
state_before: M : Type u_1
N : Type u_2
μ : M → N → N
r : N → N → Prop
inst✝¹ : CovariantClass M N μ r
inst✝ : Group N
⊢ Covariant N N (swap fun x x_1 => x * x_1) r ↔ Contravariant N N (swap fun x x_1 => x * x_1) r
state_after: case refine_1
M : Type u_1
N : Type u_2
μ : M → N → N
r : N → N → Prop
inst✝¹ : CovariantClass M N μ r
inst✝ : Group N
h : Covariant N N (swap fun x x_1 => x * x_1) r
a b c : N
bc : r (swap (fun x x_1 => x * x_1) a b) (swap (fun x x_1 => x * x_1) a c)
⊢ r b c

case refine_2
M : Type u_1
N : Type u_2
μ : M → N → N
r : N → N → Prop
inst✝¹ : CovariantClass M N μ r
inst✝ : Group N
h : Contravariant N N (swap fun x x_1 => x * x_1) r
a b c : N
bc : r b c
⊢ r (swap (fun x x_1 => x * x_1) a b) (swap (fun x x_1 => x * x_1) a c)
tactic: refine ⟨fun h a b c bc ↦ ?_, fun h a b c bc ↦ ?_⟩
Theorem: Rel.edgeDensity_comm
File: Mathlib/Combinatorics/SimpleGraph/Density.lean

Proof steps:
state_before: 𝕜 : Type u_1
ι : Type u_2
κ : Type u_3
α : Type u_4
β : Type u_5
r : α → α → Prop
inst✝ : DecidableRel r
s✝ s₁ s₂ t✝ t₁ t₂ : Finset α
a b : α
hr✝ hr : Symmetric r
s t : Finset α
⊢ edgeDensity r s t = edgeDensity r t s
state_after: no goals
tactic: rw [edgeDensity, mul_comm, card_interedges_comm hr, edgeDensity]
Theorem: List.mem_keys_kunion
File: Mathlib/Data/List/Sigma.lean

Proof steps:
state_before: α : Type u
β : α → Type v
l l₁✝ l₂✝ : List (Sigma β)
inst✝ : DecidableEq α
a : α
l₁ l₂ : List (Sigma β)
⊢ a ∈ (l₁.kunion l₂).keys ↔ a ∈ l₁.keys ∨ a ∈ l₂.keys
state_after: no goals
tactic: induction l₁ generalizing l₂ with
  | nil => simp
  | cons s l₁ ih => by_cases h : a = s.1 <;> [simp [h]; simp [h, ih]]
Theorem: SimpleGraph.Walk.support_reverse
File: Mathlib/Combinatorics/SimpleGraph/Walk.lean

Proof steps:
state_before: V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
u v : V
p : G.Walk u v
⊢ p.reverse.support = p.support.reverse
state_after: no goals
tactic: induction p <;> simp [support_append, *]
Theorem: Finset.image₂_singleton_right
File: Mathlib/Data/Finset/NAry.lean

Proof steps:
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
γ' : Type u_6
δ : Type u_7
δ' : Type u_8
ε : Type u_9
ε' : Type u_10
ζ : Type u_11
ζ' : Type u_12
ν : Type u_13
inst✝⁷ : DecidableEq α'
inst✝⁶ : DecidableEq β'
inst✝⁵ : DecidableEq γ
inst✝⁴ : DecidableEq γ'
inst✝³ : DecidableEq δ
inst✝² : DecidableEq δ'
inst✝¹ : DecidableEq ε
inst✝ : DecidableEq ε'
f f' : α → β → γ
g g' : α → β → γ → δ
s s' : Finset α
t t' : Finset β
u u' : Finset γ
a a' : α
b b' : β
c x : γ
⊢ x ∈ image₂ f s {b} ↔ x ∈ image (fun a => f a b) s
state_after: no goals
tactic: simp
Theorem: HasFDerivAt.comp_hasDerivAt_of_eq
File: Mathlib/Analysis/Calculus/Deriv/Comp.lean

Proof steps:
state_before: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x : 𝕜
s t : Set 𝕜
L L₁ L₂ : Filter 𝕜
l : F → E
l' : F →L[𝕜] E
y : F
hl : HasFDerivAt l l' y
hf : HasDerivAt f f' x
hy : y = f x
⊢ HasDerivAt (l ∘ f) (l' f') x
state_after: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x : 𝕜
s t : Set 𝕜
L L₁ L₂ : Filter 𝕜
l : F → E
l' : F →L[𝕜] E
y : F
hl : HasFDerivAt l l' (f x)
hf : HasDerivAt f f' x
hy : y = f x
⊢ HasDerivAt (l ∘ f) (l' f') x
tactic: rw [hy] at hl
state_before: 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x : 𝕜
s t : Set 𝕜
L L₁ L₂ : Filter 𝕜
l : F → E
l' : F →L[𝕜] E
y : F
hl : HasFDerivAt l l' (f x)
hf : HasDerivAt f f' x
hy : y = f x
⊢ HasDerivAt (l ∘ f) (l' f') x
state_after: no goals
tactic: exact hl.comp_hasDerivAt x hf
Theorem: Polynomial.eq_zero_of_eq_zero
File: Mathlib/Algebra/Polynomial/Basic.lean

Proof steps:
state_before: R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
p✝ q : R[X]
h : 0 = 1
p : R[X]
⊢ p = 0
state_after: no goals
tactic: rw [← one_smul R p, ← h, zero_smul]
Theorem: Set.integer_carrier
File: Mathlib/RingTheory/DedekindDomain/SInteger.lean

Proof steps:
state_before: R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
S : Set (HeightOneSpectrum R)
K : Type v
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
x✝ : K
⊢ x✝ ∈ {x | ∀ v ∉ S, v.valuation x ≤ 1} ↔ x✝ ∈ ↑(⨅ v, ⨅ (_ : v ∉ S), v.valuation.valuationSubring.toSubring)
state_after: no goals
tactic: simp [SetLike.mem_coe, Subring.mem_iInf]
Theorem: frobenius_comp_frobeniusEquiv_symm
File: Mathlib/FieldTheory/Perfect.lean

Proof steps:
state_before: R : Type u_1
p m n : ℕ
inst✝² : CommSemiring R
inst✝¹ : ExpChar R p
inst✝ : PerfectRing R p
⊢ (frobenius R p).comp ↑(frobeniusEquiv R p).symm = RingHom.id R
state_after: case a
R : Type u_1
p m n : ℕ
inst✝² : CommSemiring R
inst✝¹ : ExpChar R p
inst✝ : PerfectRing R p
x✝ : R
⊢ ((frobenius R p).comp ↑(frobeniusEquiv R p).symm) x✝ = (RingHom.id R) x✝
tactic: ext
state_before: case a
R : Type u_1
p m n : ℕ
inst✝² : CommSemiring R
inst✝¹ : ExpChar R p
inst✝ : PerfectRing R p
x✝ : R
⊢ ((frobenius R p).comp ↑(frobeniusEquiv R p).symm) x✝ = (RingHom.id R) x✝
state_after: no goals
tactic: simp
Theorem: UniformCauchySeqOnFilter.mono_right
File: Mathlib/Topology/UniformSpace/UniformConvergence.lean

Proof steps:
state_before: α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
s s' : Set α
x : α
p : Filter ι
p' : Filter α
g : ι → α
p'' : Filter α
hf : UniformCauchySeqOnFilter F p p'
hp : p'' ≤ p'
u : Set (β × β)
hu : u ∈ 𝓤 β
this : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p'', (F x.1.1 x.2, F x.1.2 x.2) ∈ u
⊢ ∀ (x : (ι × ι) × α), (F x.1.1 x.2, F x.1.2 x.2) ∈ u → (F x.1.1 x.2, F x.1.2 x.2) ∈ u
state_after: no goals
tactic: simp
Theorem: Finsupp.neLocus_sub_right
File: Mathlib/Data/Finsupp/NeLocus.lean

Proof steps:
state_before: α : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
inst✝² : DecidableEq α
inst✝¹ : DecidableEq N
inst✝ : AddGroup N
f f₁ f₂ g g₁ g₂ : α →₀ N
⊢ (f₁ - g).neLocus (f₂ - g) = f₁.neLocus f₂
state_after: no goals
tactic: simpa only [sub_eq_add_neg] using neLocus_add_right _ _ _
Theorem: CategoryTheory.ShortComplex.opcyclesMap'_comp
File: Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean

Proof steps:
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ₁ : S₁ ⟶ S₂
φ₂ : S₂ ⟶ S₃
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
h₃ : S₃.RightHomologyData
⊢ opcyclesMap' (φ₁ ≫ φ₂) h₁ h₃ = opcyclesMap' φ₁ h₁ h₂ ≫ opcyclesMap' φ₂ h₂ h₃
state_after: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ₁ : S₁ ⟶ S₂
φ₂ : S₂ ⟶ S₃
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
h₃ : S₃.RightHomologyData
γ₁ : RightHomologyMapData φ₁ h₁ h₂ := rightHomologyMapData φ₁ h₁ h₂
⊢ opcyclesMap' (φ₁ ≫ φ₂) h₁ h₃ = opcyclesMap' φ₁ h₁ h₂ ≫ opcyclesMap' φ₂ h₂ h₃
tactic: let γ₁ := rightHomologyMapData φ₁ h₁ h₂
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ₁ : S₁ ⟶ S₂
φ₂ : S₂ ⟶ S₃
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
h₃ : S₃.RightHomologyData
γ₁ : RightHomologyMapData φ₁ h₁ h₂ := rightHomologyMapData φ₁ h₁ h₂
⊢ opcyclesMap' (φ₁ ≫ φ₂) h₁ h₃ = opcyclesMap' φ₁ h₁ h₂ ≫ opcyclesMap' φ₂ h₂ h₃
state_after: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ₁ : S₁ ⟶ S₂
φ₂ : S₂ ⟶ S₃
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
h₃ : S₃.RightHomologyData
γ₁ : RightHomologyMapData φ₁ h₁ h₂ := rightHomologyMapData φ₁ h₁ h₂
γ₂ : RightHomologyMapData φ₂ h₂ h₃ := rightHomologyMapData φ₂ h₂ h₃
⊢ opcyclesMap' (φ₁ ≫ φ₂) h₁ h₃ = opcyclesMap' φ₁ h₁ h₂ ≫ opcyclesMap' φ₂ h₂ h₃
tactic: let γ₂ := rightHomologyMapData φ₂ h₂ h₃
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ₁ : S₁ ⟶ S₂
φ₂ : S₂ ⟶ S₃
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
h₃ : S₃.RightHomologyData
γ₁ : RightHomologyMapData φ₁ h₁ h₂ := rightHomologyMapData φ₁ h₁ h₂
γ₂ : RightHomologyMapData φ₂ h₂ h₃ := rightHomologyMapData φ₂ h₂ h₃
⊢ opcyclesMap' (φ₁ ≫ φ₂) h₁ h₃ = opcyclesMap' φ₁ h₁ h₂ ≫ opcyclesMap' φ₂ h₂ h₃
state_after: no goals
tactic: rw [γ₁.opcyclesMap'_eq, γ₂.opcyclesMap'_eq, (γ₁.comp γ₂).opcyclesMap'_eq,
    RightHomologyMapData.comp_φQ]
Theorem: Ordinal.IsNormal.le_set'
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
f : Ordinal.{u_4} → Ordinal.{u_5}
o : Ordinal.{u_5}
H : IsNormal f
p : Set α
p0 : p.Nonempty
g : α → Ordinal.{u_4}
b : Ordinal.{u_4}
H₂ : ∀ (o : Ordinal.{u_4}), b ≤ o ↔ ∀ a ∈ p, g a ≤ o
⊢ f b ≤ o ↔ ∀ a ∈ p, f (g a) ≤ o
state_after: no goals
tactic: simpa [H₂] using H.le_set (g '' p) (p0.image g) b
Theorem: Equiv.extendSubtype_apply_of_not_mem
File: Mathlib/Logic/Equiv/Fintype.lean

Proof steps:
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: 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.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: 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
⊢ (sumCompl q) (Sum.map (⇑e) (⇑e.toCompl) ((sumCompl p).symm x)) = ↑(e.toCompl ⟨x, hx⟩)
state_after: no goals
tactic: rw [sumCompl_apply_symm_of_neg _ _ hx, Sum.map_inr, sumCompl_apply_inr]
Theorem: ProbabilityTheory.IsMeasurableRatCDF.measure_stieltjesFunction_univ
File: Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : IsMeasurableRatCDF f
a : α
⊢ (hf.stieltjesFunction a).measure univ = 1
state_after: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : IsMeasurableRatCDF f
a : α
⊢ (hf.stieltjesFunction a).measure univ = ENNReal.ofReal (1 - 0)
tactic: rw [← ENNReal.ofReal_one, ← sub_zero (1 : ℝ)]
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
f : α → ℚ → ℝ
hf : IsMeasurableRatCDF f
a : α
⊢ (hf.stieltjesFunction a).measure univ = ENNReal.ofReal (1 - 0)
state_after: no goals
tactic: exact StieltjesFunction.measure_univ _ (tendsto_stieltjesFunction_atBot hf a)
    (tendsto_stieltjesFunction_atTop hf a)
Theorem: Rat.cast_eq_zero
File: Mathlib/Data/Rat/Cast/CharZero.lean

Proof steps:
state_before: F : Type u_1
ι : Type u_2
α : Type u_3
β : Type u_4
inst✝¹ : DivisionRing α
inst✝ : CharZero α
n : ℚ
⊢ ↑n = 0 ↔ n = 0
state_after: no goals
tactic: rw [← cast_zero, cast_inj]
Theorem: Multiset.Nat.map_swap_antidiagonal
File: Mathlib/Data/Multiset/NatAntidiagonal.lean

Proof steps:
state_before: n : ℕ
⊢ map Prod.swap (antidiagonal n) = antidiagonal n
state_after: no goals
tactic: rw [antidiagonal, map_coe, List.Nat.map_swap_antidiagonal, coe_reverse]
Theorem: CategoryTheory.finrank_hom_simple_simple_eq_zero_of_not_iso
File: Mathlib/CategoryTheory/Preadditive/Schur.lean

Proof steps:
state_before: C : Type u_1
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Preadditive C
𝕜 : Type u_2
inst✝⁴ : DivisionRing 𝕜
inst✝³ : HasKernels C
inst✝² : Linear 𝕜 C
X Y : C
inst✝¹ : Simple X
inst✝ : Simple Y
h : (X ≅ Y) → False
f : X ⟶ Y
⊢ f = 0
state_after: C : Type u_1
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Preadditive C
𝕜 : Type u_2
inst✝⁴ : DivisionRing 𝕜
inst✝³ : HasKernels C
inst✝² : Linear 𝕜 C
X Y : C
inst✝¹ : Simple X
inst✝ : Simple Y
h : (X ≅ Y) → False
f : X ⟶ Y
p : ¬IsIso f ↔ ¬f ≠ 0
⊢ f = 0
tactic: have p := not_congr (isIso_iff_nonzero f)
state_before: C : Type u_1
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Preadditive C
𝕜 : Type u_2
inst✝⁴ : DivisionRing 𝕜
inst✝³ : HasKernels C
inst✝² : Linear 𝕜 C
X Y : C
inst✝¹ : Simple X
inst✝ : Simple Y
h : (X ≅ Y) → False
f : X ⟶ Y
p : ¬IsIso f ↔ ¬f ≠ 0
⊢ f = 0
state_after: C : Type u_1
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Preadditive C
𝕜 : Type u_2
inst✝⁴ : DivisionRing 𝕜
inst✝³ : HasKernels C
inst✝² : Linear 𝕜 C
X Y : C
inst✝¹ : Simple X
inst✝ : Simple Y
h : (X ≅ Y) → False
f : X ⟶ Y
p : ¬IsIso f ↔ f = 0
⊢ f = 0
tactic: simp only [Classical.not_not, Ne] at p
state_before: C : Type u_1
inst✝⁶ : Category.{u_3, u_1} C
inst✝⁵ : Preadditive C
𝕜 : Type u_2
inst✝⁴ : DivisionRing 𝕜
inst✝³ : HasKernels C
inst✝² : Linear 𝕜 C
X Y : C
inst✝¹ : Simple X
inst✝ : Simple Y
h : (X ≅ Y) → False
f : X ⟶ Y
p : ¬IsIso f ↔ f = 0
⊢ f = 0
state_after: no goals
tactic: exact p.mp fun _ => h (asIso f)
Theorem: Real.volume_eq_stieltjes_id
File: Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean

Proof steps:
state_before: ι : Type u_1
inst✝ : Fintype ι
⊢ volume = StieltjesFunction.id.measure
state_after: ι : Type u_1
inst✝ : Fintype ι
this : StieltjesFunction.id.measure.IsAddLeftInvariant
⊢ volume = StieltjesFunction.id.measure
tactic: haveI : IsAddLeftInvariant StieltjesFunction.id.measure :=
    ⟨fun a =>
      Eq.symm <|
        Real.measure_ext_Ioo_rat fun p q => by
          simp only [Measure.map_apply (measurable_const_add a) measurableSet_Ioo,
            sub_sub_sub_cancel_right, StieltjesFunction.measure_Ioo, StieltjesFunction.id_leftLim,
            StieltjesFunction.id_apply, id, preimage_const_add_Ioo]⟩
state_before: ι : Type u_1
inst✝ : Fintype ι
this : StieltjesFunction.id.measure.IsAddLeftInvariant
⊢ volume = StieltjesFunction.id.measure
state_after: ι : Type u_1
inst✝ : Fintype ι
this : StieltjesFunction.id.measure.IsAddLeftInvariant
A : StieltjesFunction.id.measure ↑(stdOrthonormalBasis ℝ ℝ).toBasis.parallelepiped = 1
⊢ volume = StieltjesFunction.id.measure
tactic: have A : StieltjesFunction.id.measure (stdOrthonormalBasis ℝ ℝ).toBasis.parallelepiped = 1 := by
    change StieltjesFunction.id.measure (parallelepiped (stdOrthonormalBasis ℝ ℝ)) = 1
    rcases parallelepiped_orthonormalBasis_one_dim (stdOrthonormalBasis ℝ ℝ) with (H | H) <;>
      simp only [H, StieltjesFunction.measure_Icc, StieltjesFunction.id_apply, id, tsub_zero,
        StieltjesFunction.id_leftLim, sub_neg_eq_add, zero_add, ENNReal.ofReal_one]
state_before: ι : Type u_1
inst✝ : Fintype ι
this : StieltjesFunction.id.measure.IsAddLeftInvariant
A : StieltjesFunction.id.measure ↑(stdOrthonormalBasis ℝ ℝ).toBasis.parallelepiped = 1
⊢ volume = StieltjesFunction.id.measure
state_after: ι : Type u_1
inst✝ : Fintype ι
this : StieltjesFunction.id.measure.IsAddLeftInvariant
A : StieltjesFunction.id.measure ↑(stdOrthonormalBasis ℝ ℝ).toBasis.parallelepiped = 1
⊢ volume = 1 • addHaarMeasure (stdOrthonormalBasis ℝ ℝ).toBasis.parallelepiped
tactic: conv_rhs =>
    rw [addHaarMeasure_unique StieltjesFunction.id.measure
        (stdOrthonormalBasis ℝ ℝ).toBasis.parallelepiped, A]
state_before: ι : Type u_1
inst✝ : Fintype ι
this : StieltjesFunction.id.measure.IsAddLeftInvariant
A : StieltjesFunction.id.measure ↑(stdOrthonormalBasis ℝ ℝ).toBasis.parallelepiped = 1
⊢ volume = 1 • addHaarMeasure (stdOrthonormalBasis ℝ ℝ).toBasis.parallelepiped
state_after: no goals
tactic: simp only [volume, Basis.addHaar, one_smul]
Theorem: SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj
File: Mathlib/Combinatorics/SimpleGraph/Subgraph.lean

Proof steps:
state_before: ι : Sort u_1
V : Type u
W : Type v
G : SimpleGraph V
G' : SimpleGraph W
u v : V
h : G.Adj u v
⊢ G.singletonSubgraph v ≤ G.subgraphOfAdj h
state_after: no goals
tactic: simp
Theorem: PadicInt.cast_toZModPow
File: Mathlib/NumberTheory/Padics/RingHoms.lean

Proof steps:
state_before: p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
x✝ : ℤ_[p]
m n : ℕ
h : m ≤ n
x : ℤ_[p]
⊢ ((toZModPow n) x).cast = (toZModPow m) x
state_after: p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
x✝ : ℤ_[p]
m n : ℕ
h : m ≤ n
x : ℤ_[p]
⊢ ((toZModPow n) x).cast = ((ZMod.castHom ⋯ (ZMod (p ^ m))).comp (toZModPow n)) x
tactic: rw [← zmod_cast_comp_toZModPow _ _ h]
state_before: p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
x✝ : ℤ_[p]
m n : ℕ
h : m ≤ n
x : ℤ_[p]
⊢ ((toZModPow n) x).cast = ((ZMod.castHom ⋯ (ZMod (p ^ m))).comp (toZModPow n)) x
state_after: no goals
tactic: rfl
Theorem: Ordnode.size_balanceR
File: Mathlib/Data/Ordmap/Ordset.lean

Proof steps:
state_before: α : Type u_1
l : Ordnode α
x : α
r : Ordnode α
hl : l.Balanced
hr : r.Balanced
sl : l.Sized
sr : r.Sized
H : (∃ l', Raised l.size l' ∧ BalancedSz l' r.size) ∨ ∃ r', Raised r' r.size ∧ BalancedSz l.size r'
⊢ (l.balanceR x r).size = l.size + r.size + 1
state_after: no goals
tactic: rw [balanceR_eq_balance' hl hr sl sr H, size_balance' sl sr]
Theorem: GenContFract.dens_stable_of_terminated
File: Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean

Proof steps:
state_before: K : Type u_1
g : GenContFract K
n m : ℕ
inst✝ : DivisionRing K
n_le_m : n ≤ m
terminatedAt_n : g.TerminatedAt n
⊢ g.dens m = g.dens n
state_after: no goals
tactic: simp only [den_eq_conts_b, conts_stable_of_terminated n_le_m terminatedAt_n]
Theorem: Submonoid.LocalizationMap.lift_mk'_spec
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
g : M →* P
hg : ∀ (y : ↥S), IsUnit (g ↑y)
x : M
v : P
y : ↥S
⊢ (f.lift hg) (f.mk' x y) = v ↔ g x = g ↑y * v
state_after: 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
g : M →* P
hg : ∀ (y : ↥S), IsUnit (g ↑y)
x : M
v : P
y : ↥S
⊢ g x * ↑((IsUnit.liftRight (g.restrict S) hg) y)⁻¹ = v ↔ g x = g ↑y * v
tactic: rw [f.lift_mk' hg]
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
g : M →* P
hg : ∀ (y : ↥S), IsUnit (g ↑y)
x : M
v : P
y : ↥S
⊢ g x * ↑((IsUnit.liftRight (g.restrict S) hg) y)⁻¹ = v ↔ g x = g ↑y * v
state_after: no goals
tactic: exact mul_inv_left hg _ _ _
Theorem: NNReal.tsum_mul_right
File: Mathlib/Topology/Instances/NNReal.lean

Proof steps:
state_before: α : Type u_1
f : α → ℝ≥0
a : ℝ≥0
⊢ ↑(∑' (x : α), f x * a) = ↑((∑' (x : α), f x) * a)
state_after: no goals
tactic: simp only [coe_tsum, NNReal.coe_mul, tsum_mul_right]