ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: AlgEquiv.arrowCongr_trans
File: Mathlib/Algebra/Algebra/Equiv.lean

Proof steps:
state_before: R : Type uR
A₁ : Type uA₁
A₂ : Type uA₂
A₃ : Type uA₃
A₁' : Type uA₁'
A₂' : Type uA₂'
A₃' : Type uA₃'
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring A₁
inst✝¹⁰ : Semiring A₂
inst✝⁹ : Semiring A₃
inst✝⁸ : Semiring A₁'
inst✝⁷ : Semiring A₂'
inst✝⁶ : Semiring A₃'
inst✝⁵ : Algebra R A₁
inst✝⁴ : Algebra R A₂
inst✝³ : Algebra R A₃
inst✝² : Algebra R A₁'
inst✝¹ : Algebra R A₂'
inst✝ : Algebra R A₃'
e e₁ : A₁ ≃ₐ[R] A₂
e₁' : A₁' ≃ₐ[R] A₂'
e₂ : A₂ ≃ₐ[R] A₃
e₂' : A₂' ≃ₐ[R] A₃'
⊢ (e₁.trans e₂).arrowCongr (e₁'.trans e₂') = (e₁.arrowCongr e₁').trans (e₂.arrowCongr e₂')
state_after: case H.H
R : Type uR
A₁ : Type uA₁
A₂ : Type uA₂
A₃ : Type uA₃
A₁' : Type uA₁'
A₂' : Type uA₂'
A₃' : Type uA₃'
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring A₁
inst✝¹⁰ : Semiring A₂
inst✝⁹ : Semiring A₃
inst✝⁸ : Semiring A₁'
inst✝⁷ : Semiring A₂'
inst✝⁶ : Semiring A₃'
inst✝⁵ : Algebra R A₁
inst✝⁴ : Algebra R A₂
inst✝³ : Algebra R A₃
inst✝² : Algebra R A₁'
inst✝¹ : Algebra R A₂'
inst✝ : Algebra R A₃'
e e₁ : A₁ ≃ₐ[R] A₂
e₁' : A₁' ≃ₐ[R] A₂'
e₂ : A₂ ≃ₐ[R] A₃
e₂' : A₂' ≃ₐ[R] A₃'
x✝¹ : A₁ →ₐ[R] A₁'
x✝ : A₃
⊢ (((e₁.trans e₂).arrowCongr (e₁'.trans e₂')) x✝¹) x✝ = (((e₁.arrowCongr e₁').trans (e₂.arrowCongr e₂')) x✝¹) x✝
tactic: ext
state_before: case H.H
R : Type uR
A₁ : Type uA₁
A₂ : Type uA₂
A₃ : Type uA₃
A₁' : Type uA₁'
A₂' : Type uA₂'
A₃' : Type uA₃'
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring A₁
inst✝¹⁰ : Semiring A₂
inst✝⁹ : Semiring A₃
inst✝⁸ : Semiring A₁'
inst✝⁷ : Semiring A₂'
inst✝⁶ : Semiring A₃'
inst✝⁵ : Algebra R A₁
inst✝⁴ : Algebra R A₂
inst✝³ : Algebra R A₃
inst✝² : Algebra R A₁'
inst✝¹ : Algebra R A₂'
inst✝ : Algebra R A₃'
e e₁ : A₁ ≃ₐ[R] A₂
e₁' : A₁' ≃ₐ[R] A₂'
e₂ : A₂ ≃ₐ[R] A₃
e₂' : A₂' ≃ₐ[R] A₃'
x✝¹ : A₁ →ₐ[R] A₁'
x✝ : A₃
⊢ (((e₁.trans e₂).arrowCongr (e₁'.trans e₂')) x✝¹) x✝ = (((e₁.arrowCongr e₁').trans (e₂.arrowCongr e₂')) x✝¹) x✝
state_after: no goals
tactic: rfl
Theorem: Encodable.mem_decode₂'
File: Mathlib/Logic/Encodable/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝ : Encodable α
n : ℕ
a : α
⊢ a ∈ decode₂ α n ↔ a ∈ decode n ∧ encode a = n
state_after: no goals
tactic: simpa [decode₂, bind_eq_some] using
    ⟨fun ⟨_, h₁, rfl, h₂⟩ => ⟨h₁, h₂⟩, fun ⟨h₁, h₂⟩ => ⟨_, h₁, rfl, h₂⟩⟩
Theorem: LinearMap.BilinForm.dualSubmodule_flip_dualSubmodule_of_basis
File: Mathlib/LinearAlgebra/BilinearForm/DualLattice.lean

Proof steps:
state_before: R : Type u_4
S : Type u_2
M : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : Field S
inst✝⁵ : AddCommGroup M
inst✝⁴ : Algebra R S
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
B : BilinForm S M
ι : Type u_1
inst✝ : Finite ι
hB : B.Nondegenerate
b : Basis ι S M
⊢ B.flip.dualSubmodule (B.dualSubmodule (Submodule.span R (Set.range ⇑b))) = Submodule.span R (Set.range ⇑b)
state_after: no goals
tactic: classical
  letI := FiniteDimensional.of_fintype_basis b
  rw [dualSubmodule_span_of_basis B hB, dualSubmodule_span_of_basis _ hB.flip,
    dualBasis_flip_dualBasis B hB]
Theorem: InnerProductGeometry.cos_angle_mul_norm_mul_norm
File: Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean

Proof steps:
state_before: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x✝ y✝ x y : V
⊢ cos (angle x y) * (‖x‖ * ‖y‖) = ⟪x, y⟫_ℝ
state_after: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x✝ y✝ x y : V
⊢ ‖x‖ * ‖y‖ = 0 → ⟪x, y⟫_ℝ = 0
tactic: rw [cos_angle, div_mul_cancel_of_imp]
state_before: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x✝ y✝ x y : V
⊢ ‖x‖ * ‖y‖ = 0 → ⟪x, y⟫_ℝ = 0
state_after: no goals
tactic: simp (config := { contextual := true }) [or_imp]
Theorem: CategoryTheory.Adjunction.homEquiv_symm_id
File: Mathlib/CategoryTheory/Adjunction/Basic.lean

Proof steps:
state_before: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
X' X✝ : C
Y Y' X : D
⊢ (adj.homEquiv (G.obj X) X).symm (𝟙 (G.obj X)) = adj.counit.app X
state_after: no goals
tactic: simp
Theorem: Cardinal.ord_aleph0
File: Mathlib/SetTheory/Ordinal/Arithmetic.lean

Proof steps:
state_before: o : Ordinal.{u}
h : o < ω
⊢ o < ℵ₀.ord
state_after: case intro.intro
o : Ordinal.{0}
h' : o < type fun x x_1 => x < x_1
h : Ordinal.lift.{u, 0} o < ω
⊢ Ordinal.lift.{u, 0} o < ℵ₀.ord
tactic: rcases Ordinal.lt_lift_iff.1 h with ⟨o, rfl, h'⟩
state_before: case intro.intro
o : Ordinal.{0}
h' : o < type fun x x_1 => x < x_1
h : Ordinal.lift.{u, 0} o < ω
⊢ (typein (fun x x_1 => x < x_1) (enum (fun x x_1 => x < x_1) o h')).card < ℵ₀
state_after: no goals
tactic: exact lt_aleph0_iff_fintype.2 ⟨Set.fintypeLTNat _⟩
Theorem: StarSubalgebra.toSubalgebra_injective
File: Mathlib/Algebra/Star/Subalgebra.lean

Proof steps:
state_before: F : Type u_1
R : Type u_2
A : Type u_3
B : Type u_4
C : Type u_5
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : Semiring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra R A
inst✝⁸ : StarModule R A
inst✝⁷ : Semiring B
inst✝⁶ : StarRing B
inst✝⁵ : Algebra R B
inst✝⁴ : StarModule R B
inst✝³ : Semiring C
inst✝² : StarRing C
inst✝¹ : Algebra R C
inst✝ : StarModule R C
S T : StarSubalgebra R A
h : S.toSubalgebra = T.toSubalgebra
x : A
⊢ x ∈ S ↔ x ∈ T
state_after: no goals
tactic: rw [← mem_toSubalgebra, ← mem_toSubalgebra, h]
Theorem: SymOptionSuccEquiv.decode_encode
File: Mathlib/Data/Sym/Basic.lean

Proof steps:
state_before: α : Type u_1
n : ℕ
inst✝ : DecidableEq α
s : Sym (Option α) n.succ
⊢ decode (encode s) = s
state_after: case pos
α : Type u_1
n : ℕ
inst✝ : DecidableEq α
s : Sym (Option α) n.succ
h : none ∈ s
⊢ decode (encode s) = s

case neg
α : Type u_1
n : ℕ
inst✝ : DecidableEq α
s : Sym (Option α) n.succ
h : none ∉ s
⊢ decode (encode s) = s
tactic: by_cases h : none ∈ s
Theorem: Real.mk_mul
File: Mathlib/Data/Real/Basic.lean

Proof steps:
state_before: x y : ℝ
f g : CauSeq ℚ abs
⊢ mk (f * g) = mk f * mk g
state_after: no goals
tactic: simp [mk, ← ofCauchy_mul]
Theorem: Complex.tan_zero
File: Mathlib/Data/Complex/Exponential.lean

Proof steps:
state_before: x y : ℂ
⊢ tan 0 = 0
state_after: no goals
tactic: simp [tan]
Theorem: AddSubgroup.addOrderOf_dvd_natCard
File: Mathlib/GroupTheory/OrderOfElement.lean

Proof steps:
state_before: G : Type u_1
H : Type u_2
A : Type u_3
α : Type u_4
β : Type u_5
inst✝¹ : Group G
inst✝ : Fintype G
x : G
n : ℕ
s : Subgroup G
hx : x ∈ s
⊢ orderOf x ∣ Nat.card ↥s
state_after: no goals
tactic: simpa using orderOf_dvd_natCard (⟨x, hx⟩ : s)
Theorem: inf_sdiff_right_comm
File: Mathlib/Order/BooleanAlgebra.lean

Proof steps:
state_before: α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x \ z ⊓ y = (x ⊓ y) \ z
state_after: no goals
tactic: rw [inf_comm x, inf_comm, inf_sdiff_assoc]
Theorem: AddMonoidHom.toNatLinearMap_injective
File: Mathlib/Algebra/Module/LinearMap/Defs.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
S : Type u_6
S₃ : Type u_7
T : Type u_8
M : Type u_9
M₁ : Type u_10
M₂ : Type u_11
M₃ : Type u_12
N₁ : Type u_13
N₂ : Type u_14
N₃ : Type u_15
ι : Type u_16
inst✝¹ : AddCommMonoid M
inst✝ : AddCommMonoid M₂
⊢ Injective toNatLinearMap
state_after: R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
k : Type u_5
S : Type u_6
S₃ : Type u_7
T : Type u_8
M : Type u_9
M₁ : Type u_10
M₂ : Type u_11
M₃ : Type u_12
N₁ : Type u_13
N₂ : Type u_14
N₃ : Type u_15
ι : Type u_16
inst✝¹ : AddCommMonoid M
inst✝ : AddCommMonoid M₂
f g : M →+ M₂
h : f.toNatLinearMap = g.toNatLinearMap
⊢ f = g
tactic: intro f g h
state_before: R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
k : Type u_5
S : Type u_6
S₃ : Type u_7
T : Type u_8
M : Type u_9
M₁ : Type u_10
M₂ : Type u_11
M₃ : Type u_12
N₁ : Type u_13
N₂ : Type u_14
N₃ : Type u_15
ι : Type u_16
inst✝¹ : AddCommMonoid M
inst✝ : AddCommMonoid M₂
f g : M →+ M₂
h : f.toNatLinearMap = g.toNatLinearMap
⊢ f = g
state_after: case h
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
k : Type u_5
S : Type u_6
S₃ : Type u_7
T : Type u_8
M : Type u_9
M₁ : Type u_10
M₂ : Type u_11
M₃ : Type u_12
N₁ : Type u_13
N₂ : Type u_14
N₃ : Type u_15
ι : Type u_16
inst✝¹ : AddCommMonoid M
inst✝ : AddCommMonoid M₂
f g : M →+ M₂
h : f.toNatLinearMap = g.toNatLinearMap
x : M
⊢ f x = g x
tactic: ext x
state_before: case h
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
k : Type u_5
S : Type u_6
S₃ : Type u_7
T : Type u_8
M : Type u_9
M₁ : Type u_10
M₂ : Type u_11
M₃ : Type u_12
N₁ : Type u_13
N₂ : Type u_14
N₃ : Type u_15
ι : Type u_16
inst✝¹ : AddCommMonoid M
inst✝ : AddCommMonoid M₂
f g : M →+ M₂
h : f.toNatLinearMap = g.toNatLinearMap
x : M
⊢ f x = g x
state_after: no goals
tactic: exact LinearMap.congr_fun h x
Theorem: CategoryTheory.obj_μ_zero_app
File: Mathlib/CategoryTheory/Monoidal/End.lean

Proof steps:
state_before: C : Type u
inst✝² : Category.{v, u} C
M : Type u_1
inst✝¹ : Category.{u_2, u_1} M
inst✝ : MonoidalCategory M
F : MonoidalFunctor M (C ⥤ C)
m₁ m₂ : M
X : C
⊢ (F.μ (𝟙_ M) m₂).app ((F.obj m₁).obj X) ≫
      (F.μ m₁ (𝟙_ M ⊗ m₂)).app X ≫ (F.map (α_ m₁ (𝟙_ M) m₂).inv).app X ≫ (F.μIso (m₁ ⊗ 𝟙_ M) m₂).inv.app X =
    (F.μ (𝟙_ M) m₂).app ((F.obj m₁).obj X) ≫
      (F.map (λ_ m₂).hom).app ((F.obj m₁).obj X) ≫ (F.obj m₂).map ((F.map (ρ_ m₁).inv).app X)
state_after: C : Type u
inst✝² : Category.{v, u} C
M : Type u_1
inst✝¹ : Category.{u_2, u_1} M
inst✝ : MonoidalCategory M
F : MonoidalFunctor M (C ⥤ C)
m₁ m₂ : M
X : C
⊢ (F.μ (𝟙_ M) m₂).app ((F.obj m₁).obj X) ≫
      (F.μ m₁ (𝟙_ M ⊗ m₂)).app X ≫ (F.map (α_ m₁ (𝟙_ M) m₂).inv).app X ≫ (F.μIso (m₁ ⊗ 𝟙_ M) m₂).inv.app X =
    (F.obj m₂).map (F.εIso.inv.app ((F.obj m₁).obj X) ≫ (F.map (ρ_ m₁).inv).app X)
tactic: rw [← obj_ε_inv_app_assoc, ← Functor.map_comp]
state_before: C : Type u
inst✝² : Category.{v, u} C
M : Type u_1
inst✝¹ : Category.{u_2, u_1} M
inst✝ : MonoidalCategory M
F : MonoidalFunctor M (C ⥤ C)
m₁ m₂ : M
X : C
⊢ (F.μ (𝟙_ M) m₂).app ((F.obj m₁).obj X) ≫
      (F.μ m₁ (𝟙_ M ⊗ m₂)).app X ≫ (F.map (α_ m₁ (𝟙_ M) m₂).inv).app X ≫ (F.μIso (m₁ ⊗ 𝟙_ M) m₂).inv.app X =
    (F.obj m₂).map (F.εIso.inv.app ((F.obj m₁).obj X) ≫ (F.map (ρ_ m₁).inv).app X)
state_after: no goals
tactic: simp
Theorem: compl_sup_eq_top
File: Mathlib/Order/BooleanAlgebra.lean

Proof steps:
state_before: α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α
⊢ xᶜ ⊔ x = ⊤
state_after: no goals
tactic: rw [sup_comm, sup_compl_eq_top]
Theorem: Units.continuousOn_inv₀_spectrum
File: Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean

Proof steps:
state_before: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Semifield R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : TopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : HasContinuousInv₀ R
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
f : R → R
a✝ : A
inst✝ : UniqueContinuousFunctionalCalculus R A
a : Aˣ
⊢ spectrum R ↑a ⊆ {0}ᶜ
state_after: no goals
tactic: simpa only [Set.subset_compl_singleton_iff] using spectrum.zero_not_mem R a.isUnit
Theorem: le_smul_iff_one_le_left
File: Mathlib/Algebra/Order/Module/Defs.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
a a₁ a₂ : α
b b₁ b₂ : β
inst✝⁷ : Monoid α
inst✝⁶ : Zero α
inst✝⁵ : Zero β
inst✝⁴ : MulAction α β
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : SMulPosMono α β
inst✝ : SMulPosReflectLE α β
hb : 0 < b
⊢ b ≤ a • b ↔ 1 • b ≤ a • b
state_after: no goals
tactic: rw [one_smul]
Theorem: RCLike.natCast_re
File: Mathlib/Analysis/RCLike/Basic.lean

Proof steps:
state_before: K : Type u_1
E : Type u_2
inst✝ : RCLike K
z : K
n : ℕ
⊢ re ↑n = ↑n
state_after: no goals
tactic: rw [← ofReal_natCast, ofReal_re]
Theorem: Multiset.map_hcongr
File: Mathlib/Data/Multiset/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type v
γ : Type u_2
β' : Type v
m : Multiset α
f : α → β
f' : α → β'
h : β = β'
hf : ∀ a ∈ m, HEq (f a) (f' a)
⊢ HEq (map f m) (map f' m)
state_after: α : Type u_1
β : Type v
γ : Type u_2
m : Multiset α
f f' : α → β
hf : ∀ a ∈ m, HEq (f a) (f' a)
⊢ HEq (map f m) (map f' m)
tactic: subst h
state_before: α : Type u_1
β : Type v
γ : Type u_2
m : Multiset α
f f' : α → β
hf : ∀ a ∈ m, HEq (f a) (f' a)
⊢ HEq (map f m) (map f' m)
state_after: α : Type u_1
β : Type v
γ : Type u_2
m : Multiset α
f f' : α → β
hf : ∀ a ∈ m, f a = f' a
⊢ HEq (map f m) (map f' m)
tactic: simp at hf
state_before: α : Type u_1
β : Type v
γ : Type u_2
m : Multiset α
f f' : α → β
hf : ∀ a ∈ m, f a = f' a
⊢ HEq (map f m) (map f' m)
state_after: no goals
tactic: simp [map_congr rfl hf]
Theorem: PowerSeries.exp_pow_eq_rescale_exp
File: Mathlib/RingTheory/PowerSeries/WellKnown.lean

Proof steps:
state_before: A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
k : ℕ
⊢ exp A ^ k = (rescale ↑k) (exp A)
state_after: case zero
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
⊢ exp A ^ 0 = (rescale ↑0) (exp A)

case succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
k : ℕ
h : exp A ^ k = (rescale ↑k) (exp A)
⊢ exp A ^ (k + 1) = (rescale ↑(k + 1)) (exp A)
tactic: induction' k with k h
Theorem: Finset.nonempty_Ioo
File: Mathlib/Order/Interval/Finset/Basic.lean

Proof steps:
state_before: ι : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : LocallyFiniteOrder α
a a₁ a₂ b b₁ b₂ c x : α
inst✝ : DenselyOrdered α
⊢ (Ioo a b).Nonempty ↔ a < b
state_after: no goals
tactic: rw [← coe_nonempty, coe_Ioo, Set.nonempty_Ioo]
Theorem: measurableSet_integrable
File: Mathlib/MeasureTheory/Constructions/Prod/Integral.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
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α → β → E
hf : StronglyMeasurable (uncurry f)
⊢ MeasurableSet {x | Integrable (f x) ν}
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α → β → E
hf : StronglyMeasurable (uncurry f)
⊢ MeasurableSet {x | HasFiniteIntegral (f x) ν}
tactic: simp_rw [Integrable, hf.of_uncurry_left.aestronglyMeasurable, true_and_iff]
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α → β → E
hf : StronglyMeasurable (uncurry f)
⊢ MeasurableSet {x | HasFiniteIntegral (f x) ν}
state_after: no goals
tactic: exact measurableSet_lt (Measurable.lintegral_prod_right hf.ennnorm) measurable_const
Theorem: LipschitzWith.dist_iterate_succ_le_geometric
File: Mathlib/Topology/MetricSpace/Lipschitz.lean

Proof steps:
state_before: α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝² : PseudoMetricSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
K : ℝ≥0
f✝ : α → β
x✝ y : α
r : ℝ
f : α → α
hf : LipschitzWith K f
x : α
n : ℕ
⊢ dist (f^[n] x) (f^[n + 1] x) ≤ dist x (f x) * ↑K ^ n
state_after: α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝² : PseudoMetricSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
K : ℝ≥0
f✝ : α → β
x✝ y : α
r : ℝ
f : α → α
hf : LipschitzWith K f
x : α
n : ℕ
⊢ dist (f^[n] x) ((f^[n] ∘ f) x) ≤ ↑K ^ n * dist x (f x)
tactic: rw [iterate_succ, mul_comm]
state_before: α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝² : PseudoMetricSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
K : ℝ≥0
f✝ : α → β
x✝ y : α
r : ℝ
f : α → α
hf : LipschitzWith K f
x : α
n : ℕ
⊢ dist (f^[n] x) ((f^[n] ∘ f) x) ≤ ↑K ^ n * dist x (f x)
state_after: no goals
tactic: simpa only [NNReal.coe_pow] using (hf.iterate n).dist_le_mul x (f x)
Theorem: edist_mul_mul_le
File: Mathlib/Analysis/Normed/Group/Uniform.lean

Proof steps:
state_before: 𝓕 : Type u_1
α : Type u_2
E : Type u_3
F : Type u_4
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁✝ a₂✝ b b₁✝ b₂✝ : E
r r₁ r₂ : ℝ
a₁ a₂ b₁ b₂ : E
⊢ edist (a₁ * a₂) (b₁ * b₂) ≤ edist a₁ b₁ + edist a₂ b₂
state_after: 𝓕 : Type u_1
α : Type u_2
E : Type u_3
F : Type u_4
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁✝ a₂✝ b b₁✝ b₂✝ : E
r r₁ r₂ : ℝ
a₁ a₂ b₁ b₂ : E
⊢ ↑(nndist (a₁ * a₂) (b₁ * b₂)) ≤ ↑(nndist a₁ b₁) + ↑(nndist a₂ b₂)
tactic: simp only [edist_nndist]
state_before: 𝓕 : Type u_1
α : Type u_2
E : Type u_3
F : Type u_4
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁✝ a₂✝ b b₁✝ b₂✝ : E
r r₁ r₂ : ℝ
a₁ a₂ b₁ b₂ : E
⊢ ↑(nndist (a₁ * a₂) (b₁ * b₂)) ≤ ↑(nndist a₁ b₁) + ↑(nndist a₂ b₂)
state_after: 𝓕 : Type u_1
α : Type u_2
E : Type u_3
F : Type u_4
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁✝ a₂✝ b b₁✝ b₂✝ : E
r r₁ r₂ : ℝ
a₁ a₂ b₁ b₂ : E
⊢ nndist (a₁ * a₂) (b₁ * b₂) ≤ nndist a₁ b₁ + nndist a₂ b₂
tactic: norm_cast
state_before: 𝓕 : Type u_1
α : Type u_2
E : Type u_3
F : Type u_4
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁✝ a₂✝ b b₁✝ b₂✝ : E
r r₁ r₂ : ℝ
a₁ a₂ b₁ b₂ : E
⊢ nndist (a₁ * a₂) (b₁ * b₂) ≤ nndist a₁ b₁ + nndist a₂ b₂
state_after: no goals
tactic: apply nndist_mul_mul_le
Theorem: CoxeterSystem.IsReflection.length_mul_right_ne
File: Mathlib/GroupTheory/Coxeter/Inversion.lean

Proof steps:
state_before: B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
t : W
ht : cs.IsReflection t
w : W
⊢ cs.length (t * w) ≠ cs.length w
state_after: B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
t : W
ht : cs.IsReflection t
w : W
⊢ cs.lengthParity (t * w) ≠ cs.lengthParity w
tactic: suffices cs.lengthParity (t * w) ≠ cs.lengthParity w by
    contrapose! this
    simp only [lengthParity_eq_ofAdd_length, this]
state_before: B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
t : W
ht : cs.IsReflection t
w : W
⊢ cs.lengthParity (t * w) ≠ cs.lengthParity w
state_after: case intro.intro
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w✝ w : W
i : B
⊢ cs.lengthParity (w * cs.simple i * w⁻¹ * w✝) ≠ cs.lengthParity w✝
tactic: rcases ht with ⟨w, i, rfl⟩
state_before: case intro.intro
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w✝ w : W
i : B
⊢ cs.lengthParity (w * cs.simple i * w⁻¹ * w✝) ≠ cs.lengthParity w✝
state_after: no goals
tactic: simp [lengthParity_simple]
Theorem: eq_of_prime_pow_eq
File: Mathlib/Algebra/Associated/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R : Type u_5
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : Unique Rˣ
p₁ p₂ : R
k₁ k₂ : ℕ
hp₁ : Prime p₁
hp₂ : Prime p₂
hk₁ : 0 < k₁
h : p₁ ^ k₁ = p₂ ^ k₂
⊢ p₁ = p₂
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R : Type u_5
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : Unique Rˣ
p₁ p₂ : R
k₁ k₂ : ℕ
hp₁ : Prime p₁
hp₂ : Prime p₂
hk₁ : 0 < k₁
h : p₁ ^ k₁ ~ᵤ p₂ ^ k₂
⊢ p₁ ~ᵤ p₂
tactic: rw [← associated_iff_eq] at h ⊢
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
R : Type u_5
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : Unique Rˣ
p₁ p₂ : R
k₁ k₂ : ℕ
hp₁ : Prime p₁
hp₂ : Prime p₂
hk₁ : 0 < k₁
h : p₁ ^ k₁ ~ᵤ p₂ ^ k₂
⊢ p₁ ~ᵤ p₂
state_after: no goals
tactic: apply h.of_pow_associated_of_prime hp₁ hp₂ hk₁
Theorem: EMetric.diam_le_iff
File: Mathlib/Topology/EMetricSpace/Basic.lean

Proof steps:
state_before: α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x y z : α
ε ε₁ ε₂ : ℝ≥0∞
s t : Set α
d : ℝ≥0∞
⊢ diam s ≤ d ↔ ∀ x ∈ s, ∀ y ∈ s, edist x y ≤ d
state_after: no goals
tactic: simp only [diam, iSup_le_iff]
Theorem: CategoryTheory.IsPullback.of_right'
File: Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean

Proof steps:
state_before: C : Type u₁
inst✝ : Category.{v₁, u₁} C
P X Y Z : C
fst : P ⟶ X
snd : P ⟶ Y
f : X ⟶ Z
g : Y ⟶ Z
X₁₁ X₁₂ X₁₃ X₂₁ X₂₂ X₂₃ : C
h₁₂ : X₁₂ ⟶ X₁₃
h₂₁ : X₂₁ ⟶ X₂₂
h₂₂ : X₂₂ ⟶ X₂₃
h₁₃ : X₁₁ ⟶ X₁₃
v₁₁ : X₁₁ ⟶ X₂₁
v₁₂ : X₁₂ ⟶ X₂₂
v₁₃ : X₁₃ ⟶ X₂₃
s : IsPullback h₁₃ v₁₁ v₁₃ (h₂₁ ≫ h₂₂)
t : IsPullback h₁₂ v₁₂ v₁₃ h₂₂
⊢ h₁₃ ≫ v₁₃ = (v₁₁ ≫ h₂₁) ≫ h₂₂
state_after: no goals
tactic: rw [s.w, Category.assoc]
Theorem: CategoryTheory.ShortComplex.ShortExact.singleTriangleIso_hom_hom₃
File: Mathlib/Algebra/Homology/DerivedCategory/SingleTriangle.lean

Proof steps:
state_before: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S : ShortComplex C
hS : S.ShortExact
⊢ hS.singleTriangle ≅ triangleOfSES ⋯
state_after: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S : ShortComplex C
hS : S.ShortExact
e : (SingleFunctors.evaluation C (DerivedCategory C) 0).obj (singleFunctors C) ≅
  (SingleFunctors.evaluation C (DerivedCategory C) 0).obj ((CochainComplex.singleFunctors C).postcomp Q) :=
  (SingleFunctors.evaluation C (DerivedCategory C) 0).mapIso (singleFunctorsPostcompQIso C)
⊢ hS.singleTriangle ≅ triangleOfSES ⋯
tactic: let e := (SingleFunctors.evaluation _ _ 0).mapIso (singleFunctorsPostcompQIso C)
state_before: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S : ShortComplex C
hS : S.ShortExact
e : (SingleFunctors.evaluation C (DerivedCategory C) 0).obj (singleFunctors C) ≅
  (SingleFunctors.evaluation C (DerivedCategory C) 0).obj ((CochainComplex.singleFunctors C).postcomp Q) :=
  (SingleFunctors.evaluation C (DerivedCategory C) 0).mapIso (singleFunctorsPostcompQIso C)
⊢ hS.singleTriangle ≅ triangleOfSES ⋯
state_after: case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S : ShortComplex C
hS : S.ShortExact
e : (SingleFunctors.evaluation C (DerivedCategory C) 0).obj (singleFunctors C) ≅
  (SingleFunctors.evaluation C (DerivedCategory C) 0).obj ((CochainComplex.singleFunctors C).postcomp Q) :=
  (SingleFunctors.evaluation C (DerivedCategory C) 0).mapIso (singleFunctorsPostcompQIso C)
⊢ hS.singleTriangle.mor₁ ≫ (e.app S.X₂).hom = (e.app S.X₁).hom ≫ (triangleOfSES ⋯).mor₁

case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S : ShortComplex C
hS : S.ShortExact
e : (SingleFunctors.evaluation C (DerivedCategory C) 0).obj (singleFunctors C) ≅
  (SingleFunctors.evaluation C (DerivedCategory C) 0).obj ((CochainComplex.singleFunctors C).postcomp Q) :=
  (SingleFunctors.evaluation C (DerivedCategory C) 0).mapIso (singleFunctorsPostcompQIso C)
⊢ hS.singleTriangle.mor₂ ≫ (e.app S.X₃).hom = (e.app S.X₂).hom ≫ (triangleOfSES ⋯).mor₂

case refine_3
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S : ShortComplex C
hS : S.ShortExact
e : (SingleFunctors.evaluation C (DerivedCategory C) 0).obj (singleFunctors C) ≅
  (SingleFunctors.evaluation C (DerivedCategory C) 0).obj ((CochainComplex.singleFunctors C).postcomp Q) :=
  (SingleFunctors.evaluation C (DerivedCategory C) 0).mapIso (singleFunctorsPostcompQIso C)
⊢ hS.singleTriangle.mor₃ ≫ (shiftFunctor (DerivedCategory C) 1).map (e.app S.X₁).hom =
    (e.app S.X₃).hom ≫ (triangleOfSES ⋯).mor₃
tactic: refine Triangle.isoMk _ _ (e.app S.X₁) (e.app S.X₂) (e.app S.X₃) ?_ ?_ ?_
Theorem: Associated.mul_left
File: Mathlib/Algebra/Associated/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : Monoid α
a b c : α
h : b ~ᵤ c
⊢ a * b ~ᵤ a * c
state_after: case intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : Monoid α
a b : α
d : αˣ
⊢ a * b ~ᵤ a * (b * ↑d)
tactic: obtain ⟨d, rfl⟩ := h
state_before: case intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : Monoid α
a b : α
d : αˣ
⊢ a * b ~ᵤ a * (b * ↑d)
state_after: no goals
tactic: exact ⟨d, mul_assoc _ _ _⟩