1337xyz1337xyz/leandojo-benchmark4-v10 huggingface dataset:
Theorem: AlgebraicTopology.DoldKan.σ_comp_PInfty
File: Mathlib/AlgebraicTopology/DoldKan/Degeneracies.lean

tactic: rw [PInfty_f, σ_comp_P_eq_zero X i]
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
i : Fin (n + 1)
⊢ X.σ i ≫ PInfty.f (n + 1) = 0
state_after: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
i : Fin (n + 1)
⊢ n + 1 ≤ ↑i + (n + 1)

tactic: simp only [le_add_iff_nonneg_left, zero_le]
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
i : Fin (n + 1)
⊢ n + 1 ≤ ↑i + (n + 1)
state_after: no goals
Theorem: CategoryTheory.Idempotents.natTrans_eq
File: Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean

tactic: rw [← φ.naturality, ← assoc, ← F.map_comp]
state_before: C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{u_4, u_1} C
inst✝¹ : Category.{u_5, u_2} D
inst✝ : Category.{?u.41, u_3} E
F G : Karoubi C ⥤ D
φ : F ⟶ G
P : Karoubi C
⊢ φ.app P = F.map P.decompId_i ≫ φ.app { X := P.X, p := 𝟙 P.X, idem := ⋯ } ≫ G.map P.decompId_p
state_after: C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{u_4, u_1} C
inst✝¹ : Category.{u_5, u_2} D
inst✝ : Category.{?u.41, u_3} E
F G : Karoubi C ⥤ D
φ : F ⟶ G
P : Karoubi C
⊢ φ.app P = F.map (P.decompId_i ≫ P.decompId_p) ≫ φ.app P

tactic: conv_lhs => rw [← id_comp (φ.app P), ← F.map_id]
state_before: C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{u_4, u_1} C
inst✝¹ : Category.{u_5, u_2} D
inst✝ : Category.{?u.41, u_3} E
F G : Karoubi C ⥤ D
φ : F ⟶ G
P : Karoubi C
⊢ φ.app P = F.map (P.decompId_i ≫ P.decompId_p) ≫ φ.app P
state_after: C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{u_4, u_1} C
inst✝¹ : Category.{u_5, u_2} D
inst✝ : Category.{?u.41, u_3} E
F G : Karoubi C ⥤ D
φ : F ⟶ G
P : Karoubi C
⊢ F.map (𝟙 P) ≫ φ.app P = F.map (P.decompId_i ≫ P.decompId_p) ≫ φ.app P

tactic: congr
state_before: C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{u_4, u_1} C
inst✝¹ : Category.{u_5, u_2} D
inst✝ : Category.{?u.41, u_3} E
F G : Karoubi C ⥤ D
φ : F ⟶ G
P : Karoubi C
⊢ F.map (𝟙 P) ≫ φ.app P = F.map (P.decompId_i ≫ P.decompId_p) ≫ φ.app P
state_after: case e_a.e_a
C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{u_4, u_1} C
inst✝¹ : Category.{u_5, u_2} D
inst✝ : Category.{?u.41, u_3} E
F G : Karoubi C ⥤ D
φ : F ⟶ G
P : Karoubi C
⊢ 𝟙 P = P.decompId_i ≫ P.decompId_p

tactic: apply decompId
state_before: case e_a.e_a
C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{u_4, u_1} C
inst✝¹ : Category.{u_5, u_2} D
inst✝ : Category.{?u.41, u_3} E
F G : Karoubi C ⥤ D
φ : F ⟶ G
P : Karoubi C
⊢ 𝟙 P = P.decompId_i ≫ P.decompId_p
state_after: no goals
Theorem: FirstOrder.Language.Substructure.FG.cg
File: Mathlib/ModelTheory/FinitelyGenerated.lean

tactic: obtain ⟨s, hf, rfl⟩ := fg_def.1 h
state_before: L : Language
M : Type u_1
inst✝ : L.Structure M
N : L.Substructure M
h : N.FG
⊢ N.CG
state_after: case intro.intro
L : Language
M : Type u_1
inst✝ : L.Structure M
s : Set M
hf : s.Finite
h : ((closure L).toFun s).FG
⊢ ((closure L).toFun s).CG

tactic: exact ⟨s, hf.countable, rfl⟩
state_before: case intro.intro
L : Language
M : Type u_1
inst✝ : L.Structure M
s : Set M
hf : s.Finite
h : ((closure L).toFun s).FG
⊢ ((closure L).toFun s).CG
state_after: no goals
Theorem: Metric.Sum.dist_comm
File: Mathlib/Topology/MetricSpace/Gluing.lean

tactic: cases x <;> cases y <;> simp [Sum.dist, _root_.dist_comm, add_comm, add_left_comm, add_assoc]
state_before: X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
x y : X ⊕ Y
⊢ Sum.dist x y = Sum.dist y x
state_after: no goals
Theorem: LieSubmodule.mapsTo_pow_toEnd_sub_algebraMap
File: Mathlib/Algebra/Lie/OfAssociative.lean

tactic: rw [LinearMap.coe_pow]
state_before: R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
N : LieSubmodule R L M
x✝ : L
φ : R
k : ℕ
x : L
⊢ MapsTo ⇑(((toEnd R L M) x - (algebraMap R (Module.End R M)) φ) ^ k) ↑N ↑N
state_after: R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
N : LieSubmodule R L M
x✝ : L
φ : R
k : ℕ
x : L
⊢ MapsTo (⇑((toEnd R L M) x - (algebraMap R (Module.End R M)) φ))^[k] ↑N ↑N

tactic: exact MapsTo.iterate (fun m hm ↦ N.sub_mem (N.lie_mem hm) (N.smul_mem _ hm)) k
state_before: R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
N : LieSubmodule R L M
x✝ : L
φ : R
k : ℕ
x : L
⊢ MapsTo (⇑((toEnd R L M) x - (algebraMap R (Module.End R M)) φ))^[k] ↑N ↑N
state_after: no goals
Theorem: UpperHalfPlane.cosh_dist
File: Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean

tactic: rw [dist_eq, cosh_two_mul, cosh_sq', add_assoc, ← two_mul, sinh_arsinh, div_pow, mul_pow,
  sq_sqrt, sq (2 : ℝ), mul_assoc, ← mul_div_assoc, mul_assoc, mul_div_mul_left] <;> positivity
state_before: z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ cosh (dist z w) = 1 + dist ↑z ↑w ^ 2 / (2 * z.im * w.im)
state_after: no goals
Theorem: Submodule.le_comap_mkQ
File: Mathlib/LinearAlgebra/Quotient.lean

tactic: simpa using (comap_mono bot_le : ker p.mkQ ≤ comap p.mkQ p')
state_before: R : Type u_1
M : Type u_2
r : R
x y : M
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
p p'✝ : Submodule R M
R₂ : Type u_3
M₂ : Type u_4
inst✝² : Ring R₂
inst✝¹ : AddCommGroup M₂
inst✝ : Module R₂ M₂
τ₁₂ : R →+* R₂
p' : Submodule R (M ⧸ p)
⊢ p ≤ comap p.mkQ p'
state_after: no goals
Theorem: HomologicalComplex.homotopyCofiber.desc_f'
File: Mathlib/Algebra/Homology/HomotopyCofiber.lean

tactic: apply dif_neg hj
state_before: C : Type u_1
inst✝³ : Category.{u_3, u_1} C
inst✝² : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G K : HomologicalComplex C c
φ : F ⟶ G
inst✝¹ : HasHomotopyCofiber φ
inst✝ : DecidableRel c.Rel
α : G ⟶ K
hα : Homotopy (φ ≫ α) 0
j : ι
hj : ¬c.Rel j (c.next j)
⊢ (desc φ α hα).f j = sndX φ j ≫ α.f j
state_after: no goals
Theorem: CategoryTheory.isoClosure_le_iff
File: Mathlib/CategoryTheory/ClosedUnderIsomorphisms.lean

tactic: rw [isoClosure_eq_self]
state_before: C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
P Q : C → Prop
inst✝ : ClosedUnderIsomorphisms Q
h : P ≤ Q
⊢ isoClosure Q ≤ Q
state_after: no goals
Theorem: CategoryTheory.Limits.pullbackConeOfLeftIso_π_app_none
File: Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean

tactic: simp
state_before: C : Type u
inst✝² : Category.{v, u} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
W X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : IsIso f
⊢ (pullbackConeOfLeftIso f g).π.app none = g
state_after: no goals
Theorem: Cardinal.mk_equiv_of_lift_eq
File: Mathlib/SetTheory/Cardinal/Ordinal.lean

tactic: erw [← (lift_mk_eq'.2 ⟨.equivCongr (.refl α) (lift_mk_eq'.1 leq).some⟩).trans (lift_id'.{u,v} _),
  lift_umax.{u,v}, mk_perm_eq_two_power, lift_power, lift_natCast]
state_before: α β : Type u
β' : Type v
inst✝ : Infinite α
leq : lift.{v, u} #α = lift.{u, v} #β'
eq : #α = #β
⊢ #(α ≃ β') = 2 ^ lift.{v, u} #α
state_after: α β : Type u
β' : Type v
inst✝ : Infinite α
leq : lift.{v, u} #α = lift.{u, v} #β'
eq : #α = #β
⊢ ↑2 ^ lift.{v, u} #α = 2 ^ lift.{v, u} #α

tactic: rfl
state_before: α β : Type u
β' : Type v
inst✝ : Infinite α
leq : lift.{v, u} #α = lift.{u, v} #β'
eq : #α = #β
⊢ ↑2 ^ lift.{v, u} #α = 2 ^ lift.{v, u} #α
state_after: no goals
Theorem: Equiv.Perm.isCycle_of_prime_order''
File: Mathlib/GroupTheory/Perm/Cycle/Type.lean

tactic: rw [← one_mul (Fintype.card α), ← h2, mul_lt_mul_right (orderOf_pos σ)]
state_before: α : Type u_1
inst✝ : Fintype α
σ : Perm α
h1 : Nat.Prime (Fintype.card α)
h2 : orderOf σ = Fintype.card α
⊢ Fintype.card α < 2 * orderOf σ
state_after: α : Type u_1
inst✝ : Fintype α
σ : Perm α
h1 : Nat.Prime (Fintype.card α)
h2 : orderOf σ = Fintype.card α
⊢ 1 < 2

tactic: exact one_lt_two
state_before: α : Type u_1
inst✝ : Fintype α
σ : Perm α
h1 : Nat.Prime (Fintype.card α)
h2 : orderOf σ = Fintype.card α
⊢ 1 < 2
state_after: no goals
Theorem: Finset.image_mul_left'
File: Mathlib/Data/Finset/Pointwise.lean

tactic: simp
state_before: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁵ : DecidableEq α
inst✝⁴ : DecidableEq β
inst✝³ : Group α
inst✝² : DivisionMonoid β
inst✝¹ : FunLike F α β
inst✝ : MonoidHomClass F α β
f : F
s t : Finset α
a b : α
⊢ image (fun b => a⁻¹ * b) t = t.preimage (fun b => a * b) ⋯
state_after: no goals
Theorem: ArithmeticFunction.coe_mul
File: Mathlib/NumberTheory/LSeries/Convolution.lean

tactic: simp only [convolution, ArithmeticFunction.toArithmeticFunction_eq_self]
state_before: R : Type u_1
inst✝ : Semiring R
f g : ArithmeticFunction R
⊢ ⇑f ⍟ ⇑g = ⇑(f * g)
state_after: no goals
Theorem: Complex.continuous_cosh
File: Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean

tactic: change Continuous fun z => (exp z + exp (-z)) / 2
state_before: ⊢ Continuous cosh
state_after: ⊢ Continuous fun z => (cexp z + cexp (-z)) / 2

tactic: continuity
state_before: ⊢ Continuous fun z => (cexp z + cexp (-z)) / 2
state_after: no goals
Theorem: monovaryOn_inv
File: Mathlib/Algebra/Order/Monovary.lean

tactic: simp
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : OrderedCommGroup α
inst✝ : OrderedCommGroup β
s : Set ι
f f₁ f₂ : ι → α
g g₁ g₂ : ι → β
⊢ MonovaryOn f⁻¹ g⁻¹ s ↔ MonovaryOn f g s
state_after: no goals
Theorem: MeasurableSpace.measurableSet_injective
File: Mathlib/MeasureTheory/MeasurableSpace/Defs.lean

tactic: congr
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort u_6
s t u : Set α
MeasurableSet'✝¹ : Set α → Prop
measurableSet_empty✝¹ : MeasurableSet'✝¹ ∅
measurableSet_compl✝¹ : ∀ (s : Set α), MeasurableSet'✝¹ s → MeasurableSet'✝¹ sᶜ
measurableSet_iUnion✝¹ : ∀ (f : ℕ → Set α), (∀ (i : ℕ), MeasurableSet'✝¹ (f i)) → MeasurableSet'✝¹ (⋃ i, f i)
MeasurableSet'✝ : Set α → Prop
measurableSet_empty✝ : MeasurableSet'✝ ∅
measurableSet_compl✝ : ∀ (s : Set α), MeasurableSet'✝ s → MeasurableSet'✝ sᶜ
measurableSet_iUnion✝ : ∀ (f : ℕ → Set α), (∀ (i : ℕ), MeasurableSet'✝ (f i)) → MeasurableSet'✝ (⋃ i, f i)
x✝ : MeasurableSet = MeasurableSet
⊢ { MeasurableSet' := MeasurableSet'✝¹, measurableSet_empty := measurableSet_empty✝¹,
      measurableSet_compl := measurableSet_compl✝¹, measurableSet_iUnion := measurableSet_iUnion✝¹ } =
    { MeasurableSet' := MeasurableSet'✝, measurableSet_empty := measurableSet_empty✝,
      measurableSet_compl := measurableSet_compl✝, measurableSet_iUnion := measurableSet_iUnion✝ }
state_after: no goals
Theorem: ContDiff.locallyLipschitz
File: Mathlib/Analysis/Calculus/ContDiff/RCLike.lean

tactic: intro x
state_before: n : ℕ∞
𝕂 : Type u_1
inst✝⁴ : RCLike 𝕂
E' : Type u_2
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕂 E'
F' : Type u_3
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace 𝕂 F'
f : E' → F'
hf : ContDiff 𝕂 1 f
⊢ LocallyLipschitz f
state_after: n : ℕ∞
𝕂 : Type u_1
inst✝⁴ : RCLike 𝕂
E' : Type u_2
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕂 E'
F' : Type u_3
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace 𝕂 F'
f : E' → F'
hf : ContDiff 𝕂 1 f
x : E'
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K f t

tactic: rcases hf.contDiffAt.exists_lipschitzOnWith with ⟨K, t, ht, hf⟩
state_before: n : ℕ∞
𝕂 : Type u_1
inst✝⁴ : RCLike 𝕂
E' : Type u_2
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕂 E'
F' : Type u_3
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace 𝕂 F'
f : E' → F'
hf : ContDiff 𝕂 1 f
x : E'
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K f t
state_after: case intro.intro.intro
n : ℕ∞
𝕂 : Type u_1
inst✝⁴ : RCLike 𝕂
E' : Type u_2
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕂 E'
F' : Type u_3
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace 𝕂 F'
f : E' → F'
hf✝ : ContDiff 𝕂 1 f
x : E'
K : ℝ≥0
t : Set E'
ht : t ∈ 𝓝 ?m.48810
hf : LipschitzOnWith K f t
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K f t

tactic: use K, t
state_before: case intro.intro.intro
n : ℕ∞
𝕂 : Type u_1
inst✝⁴ : RCLike 𝕂
E' : Type u_2
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕂 E'
F' : Type u_3
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace 𝕂 F'
f : E' → F'
hf✝ : ContDiff 𝕂 1 f
x : E'
K : ℝ≥0
t : Set E'
ht : t ∈ 𝓝 ?m.48810
hf : LipschitzOnWith K f t
⊢ ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K f t
state_after: no goals
Theorem: Nat.preimage_Icc
File: Mathlib/Algebra/Order/Floor.lean

tactic: ext
state_before: F : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : LinearOrderedSemiring α
inst✝ : FloorSemiring α
a✝ : α
n : ℕ
a b : α
hb : 0 ≤ b
⊢ Nat.cast ⁻¹' Icc a b = Icc ⌈a⌉₊ ⌊b⌋₊
state_after: case h
F : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : LinearOrderedSemiring α
inst✝ : FloorSemiring α
a✝ : α
n : ℕ
a b : α
hb : 0 ≤ b
x✝ : ℕ
⊢ x✝ ∈ Nat.cast ⁻¹' Icc a b ↔ x✝ ∈ Icc ⌈a⌉₊ ⌊b⌋₊

tactic: simp [ceil_le, hb, le_floor_iff]
state_before: case h
F : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : LinearOrderedSemiring α
inst✝ : FloorSemiring α
a✝ : α
n : ℕ
a b : α
hb : 0 ≤ b
x✝ : ℕ
⊢ x✝ ∈ Nat.cast ⁻¹' Icc a b ↔ x✝ ∈ Icc ⌈a⌉₊ ⌊b⌋₊
state_after: no goals
Theorem: PartialHomeomorph.univUnitBall_apply_zero
File: Mathlib/Analysis/NormedSpace/HomeomorphBall.lean

tactic: simp [PartialHomeomorph.univUnitBall_apply]
state_before: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
⊢ ↑univUnitBall 0 = 0
state_after: no goals
Theorem: FiniteField.even_card_iff_char_two
File: Mathlib/FieldTheory/Finite/Basic.lean

tactic: rcases FiniteField.card F (ringChar F) with ⟨n, hp, h⟩
state_before: K : Type u_1
R : Type u_2
F : Type u_3
inst✝¹ : Field F
inst✝ : Fintype F
⊢ ringChar F = 2 ↔ Fintype.card F % 2 = 0
state_after: case intro.intro
K : Type u_1
R : Type u_2
F : Type u_3
inst✝¹ : Field F
inst✝ : Fintype F
n : ℕ+
hp : Nat.Prime (ringChar F)
h : Fintype.card F = ringChar F ^ ↑n
⊢ ringChar F = 2 ↔ Fintype.card F % 2 = 0

tactic: rw [h, ← Nat.even_iff, Nat.even_pow, hp.even_iff]
state_before: case intro.intro
K : Type u_1
R : Type u_2
F : Type u_3
inst✝¹ : Field F
inst✝ : Fintype F
n : ℕ+
hp : Nat.Prime (ringChar F)
h : Fintype.card F = ringChar F ^ ↑n
⊢ ringChar F = 2 ↔ Fintype.card F % 2 = 0
state_after: case intro.intro
K : Type u_1
R : Type u_2
F : Type u_3
inst✝¹ : Field F
inst✝ : Fintype F
n : ℕ+
hp : Nat.Prime (ringChar F)
h : Fintype.card F = ringChar F ^ ↑n
⊢ ringChar F = 2 ↔ ringChar F = 2 ∧ ↑n ≠ 0

tactic: simp
state_before: case intro.intro
K : Type u_1
R : Type u_2
F : Type u_3
inst✝¹ : Field F
inst✝ : Fintype F
n : ℕ+
hp : Nat.Prime (ringChar F)
h : Fintype.card F = ringChar F ^ ↑n
⊢ ringChar F = 2 ↔ ringChar F = 2 ∧ ↑n ≠ 0
state_after: no goals
Theorem: Filter.Tendsto.neg_mul_atBot
File: Mathlib/Topology/Algebra/Order/Field.lean

tactic: simpa only [mul_comm] using hg.atBot_mul_neg hC hf
state_before: 𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hC : C < 0
hf : Tendsto f l (𝓝 C)
hg : Tendsto g l atBot
⊢ Tendsto (fun x => f x * g x) l atTop
state_after: no goals
Theorem: SimpleGraph.Walk.IsPath.takeUntil
File: Mathlib/Combinatorics/SimpleGraph/Connectivity.lean

tactic: rwa [← take_spec _ h] at hc
state_before: V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
u✝ v✝ w✝ x y : V
inst✝ : DecidableEq V
u v w : V
p : G.Walk v w
hc : p.IsPath
h : u ∈ p.support
⊢ ((p.takeUntil u h).append ?m.135170).IsPath
state_after: no goals
Theorem: Function.iterate_add_minimalPeriod_eq
File: Mathlib/Dynamics/PeriodicPts.lean

tactic: rw [iterate_add_apply]
state_before: α : Type u_1
β : Type u_2
f fa : α → α
fb : β → β
x y : α
m n : ℕ
⊢ f^[n + minimalPeriod f x] x = f^[n] x
state_after: α : Type u_1
β : Type u_2
f fa : α → α
fb : β → β
x y : α
m n : ℕ
⊢ f^[n] (f^[minimalPeriod f x] x) = f^[n] x

tactic: congr
state_before: α : Type u_1
β : Type u_2
f fa : α → α
fb : β → β
x y : α
m n : ℕ
⊢ f^[n] (f^[minimalPeriod f x] x) = f^[n] x
state_after: case e_a
α : Type u_1
β : Type u_2
f fa : α → α
fb : β → β
x y : α
m n : ℕ
⊢ f^[minimalPeriod f x] x = x

tactic: exact isPeriodicPt_minimalPeriod f x
state_before: case e_a
α : Type u_1
β : Type u_2
f fa : α → α
fb : β → β
x y : α
m n : ℕ
⊢ f^[minimalPeriod f x] x = x
state_after: no goals
Theorem: Finset.insert_sdiff_insert'
File: Mathlib/Data/Finset/Basic.lean

tactic: ext
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
hab : a ≠ b
ha : a ∉ s
⊢ insert a s \ insert b s = {a}
state_after: case a
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s t u v : Finset α
a b : α
hab : a ≠ b
ha : a ∉ s
a✝ : α
⊢ a✝ ∈ insert a s \ insert b s ↔ a✝ ∈ {a}

tactic: aesop
state_before: case a
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s t u v : Finset α
a b : α
hab : a ≠ b
ha : a ∉ s
a✝ : α
⊢ a✝ ∈ insert a s \ insert b s ↔ a✝ ∈ {a}
state_after: no goals
Theorem: AddAction.orbitZMultiplesEquiv_symm_apply'
File: Mathlib/Data/ZMod/Quotient.lean

tactic: rw [AddAction.orbitZMultiplesEquiv_symm_apply, ZMod.coe_intCast]
state_before: n : ℕ
A : Type u_1
R : Type u_2
inst✝⁵ : AddGroup A
inst✝⁴ : Ring R
α✝ : Type u_3
β✝ : Type u_4
inst✝³ : Group α✝
a✝ : α✝
inst✝² : MulAction α✝ β✝
b✝ : β✝
α : Type u_5
β : Type u_6
inst✝¹ : AddGroup α
a : α
inst✝ : AddAction α β
b : β
k : ℤ
⊢ (orbitZMultiplesEquiv a b).symm ↑k = k • ⟨a, ⋯⟩ +ᵥ ⟨b, ⋯⟩
state_after: n : ℕ
A : Type u_1
R : Type u_2
inst✝⁵ : AddGroup A
inst✝⁴ : Ring R
α✝ : Type u_3
β✝ : Type u_4
inst✝³ : Group α✝
a✝ : α✝
inst✝² : MulAction α✝ β✝
b✝ : β✝
α : Type u_5
β : Type u_6
inst✝¹ : AddGroup α
a : α
inst✝ : AddAction α β
b : β
k : ℤ
⊢ (k % ↑(minimalPeriod (fun x => a +ᵥ x) b)) • ⟨a, ⋯⟩ +ᵥ ⟨b, ⋯⟩ = k • ⟨a, ⋯⟩ +ᵥ ⟨b, ⋯⟩

tactic: exact Subtype.ext (zsmul_vadd_mod_minimalPeriod a b k)
state_before: n : ℕ
A : Type u_1
R : Type u_2
inst✝⁵ : AddGroup A
inst✝⁴ : Ring R
α✝ : Type u_3
β✝ : Type u_4
inst✝³ : Group α✝
a✝ : α✝
inst✝² : MulAction α✝ β✝
b✝ : β✝
α : Type u_5
β : Type u_6
inst✝¹ : AddGroup α
a : α
inst✝ : AddAction α β
b : β
k : ℤ
⊢ (k % ↑(minimalPeriod (fun x => a +ᵥ x) b)) • ⟨a, ⋯⟩ +ᵥ ⟨b, ⋯⟩ = k • ⟨a, ⋯⟩ +ᵥ ⟨b, ⋯⟩
state_after: no goals
Theorem: IsometryEquiv.preimage_emetric_ball
File: Mathlib/Topology/MetricSpace/Isometry.lean

tactic: rw [← h.isometry.preimage_emetric_ball (h.symm x) r, h.apply_symm_apply]
state_before: ι : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝² : PseudoEMetricSpace α
inst✝¹ : PseudoEMetricSpace β
inst✝ : PseudoEMetricSpace γ
h : α ≃ᵢ β
x : β
r : ℝ≥0∞
⊢ ⇑h ⁻¹' EMetric.ball x r = EMetric.ball (h.symm x) r
state_after: no goals
Theorem: iterateFrobenius_zero_apply
File: Mathlib/Algebra/CharP/ExpChar.lean

tactic: rw [iterateFrobenius_def, pow_zero, pow_one]
state_before: R : Type u
inst✝³ : CommSemiring R
S : Type u_1
inst✝² : CommSemiring S
f : R →* S
g : R →+* S
p m n : ℕ
inst✝¹ : ExpChar R p
inst✝ : ExpChar S p
x y : R
⊢ (iterateFrobenius R p 0) x = x
state_after: no goals
Theorem: Int.cast_div_ofNat_charZero
File: Mathlib/Data/Int/CharZero.lean

tactic: rw [cast_div_charZero (Int.ofNat_dvd.mpr n_dvd), cast_natCast, cast_natCast]
state_before: α : Type u_1
β : Type u_2
k : Type u_3
inst✝¹ : DivisionRing k
inst✝ : CharZero k
m n : ℕ
n_dvd : n ∣ m
⊢ ↑(↑m / ↑n) = ↑m / ↑n
state_after: no goals
Theorem: Finset.card_pow_le'
File: Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean

tactic: simpa only [_root_.pow_zero, div_one] using card_pow_div_pow_le' hA _ _ 0
state_before: α : Type u_1
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
A B✝ C : Finset α
hA : A.Nonempty
B : Finset α
n : ℕ
⊢ ↑(B ^ n).card ≤ (↑(A / B).card / ↑A.card) ^ n * ↑A.card
state_after: no goals
Theorem: EReal.coe_ennreal_add
File: Mathlib/Data/Real/EReal.lean

tactic: cases x <;> cases y <;> rfl
state_before: x y : ℝ≥0∞
⊢ ↑(x + y) = ↑x + ↑y
state_after: no goals
Theorem: Asymptotics.isTheta_norm_right
File: Mathlib/Analysis/Asymptotics/Theta.lean

tactic: simp [IsTheta]
state_before: α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
E' : Type u_6
F' : Type u_7
G' : Type u_8
E'' : Type u_9
F'' : Type u_10
G'' : Type u_11
R : Type u_12
R' : Type u_13
𝕜 : Type u_14
𝕜' : Type u_15
inst✝¹² : Norm E
inst✝¹¹ : Norm F
inst✝¹⁰ : Norm G
inst✝⁹ : SeminormedAddCommGroup E'
inst✝⁸ : SeminormedAddCommGroup F'
inst✝⁷ : SeminormedAddCommGroup G'
inst✝⁶ : NormedAddCommGroup E''
inst✝⁵ : NormedAddCommGroup F''
inst✝⁴ : NormedAddCommGroup G''
inst✝³ : SeminormedRing R
inst✝² : SeminormedRing R'
inst✝¹ : NormedField 𝕜
inst✝ : NormedField 𝕜'
c c' c₁ c₂ : ℝ
f : α → E
g : α → F
k : α → G
f' : α → E'
g' : α → F'
k' : α → G'
f'' : α → E''
g'' : α → F''
l l' : Filter α
⊢ (f =Θ[l] fun x => ‖g' x‖) ↔ f =Θ[l] g'
state_after: no goals
Theorem: Ordinal.lt_blsub₂
File: Mathlib/SetTheory/Ordinal/Arithmetic.lean

tactic: simp only [typein_enum]
state_before: case h.e'_3
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o₁ : Ordinal.{u_4}
o₂ : Ordinal.{u_5}
op : {a : Ordinal.{u_4}} → a < o₁ → {b : Ordinal.{u_5}} → b < o₂ → Ordinal.{max (max u_4 u_5) u_6}
a : Ordinal.{u_4}
b : Ordinal.{u_5}
ha : a < o₁
hb : b < o₂
⊢ op ha hb = (fun {a} => op) ⋯ ⋯
state_after: no goals

tactic: rwa [type_lt]
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
o₁ : Ordinal.{u_4}
o₂ : Ordinal.{u_5}
op : {a : Ordinal.{u_4}} → a < o₁ → {b : Ordinal.{u_5}} → b < o₂ → Ordinal.{max (max u_4 u_5) u_6}
a : Ordinal.{u_4}
b : Ordinal.{u_5}
ha : a < o₁
hb : b < o₂
⊢ a < type fun x x_1 => x < x_1
state_after: no goals

tactic: rwa [type_lt]
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
o₁ : Ordinal.{u_4}
o₂ : Ordinal.{u_5}
op : {a : Ordinal.{u_4}} → a < o₁ → {b : Ordinal.{u_5}} → b < o₂ → Ordinal.{max (max u_4 u_5) u_6}
a : Ordinal.{u_4}
b : Ordinal.{u_5}
ha : a < o₁
hb : b < o₂
⊢ b < type fun x x_1 => x < x_1
state_after: no goals