ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/LinearAlgebra/Multilinear/Basic.lean
R : Type uR
S : Type uS
ι : Type uι
n : ℕ
M : Fin n.succ → Type v
M₁ : ι → Type v₁
M₂ : Type v₂
M₃ : Type v₃
M' : Type v'
inst✝⁸ : Semiring R
inst✝⁷ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁶ : (i : ι) → Module R (M₁ i)
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M₂
inst✝³ : Semiring S
inst✝² : Module S M₂
inst✝¹ : SMulCommClass R S M₂
ι' : Type u_1
σ : ι ≃ ι'
f : MultilinearMap R (fun i => M₁ (σ.symm i)) M₂
inst✝ : DecidableEq ι
m : (i : ι) → M₁ i
i : ι
c : R
this : DecidableEq ι' := σ.symm.decidableEq
x : M₁ (σ.symm (σ i))

⊢ (⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ)) (update m (σ.symm (σ i)) (c • x)) = c • (⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ)) (update m (σ.symm (σ i)) x)
-- Mathlib/LinearAlgebra/Multilinear/Basic.lean
R : Type uR
S : Type uS
ι : Type uι
n : ℕ
M : Fin n.succ → Type v
M₁ : ι → Type v₁
M₂ : Type v₂
M₃ : Type v₃
M' : Type v'
inst✝⁷ : Semiring R
inst✝⁶ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁵ : (i : ι) → Module R (M₁ i)
inst✝⁴ : AddCommMonoid M₂
inst✝³ : Module R M₂
inst✝² : Semiring S
inst✝¹ : Module S M₂
inst✝ : SMulCommClass R S M₂
ι' : Type u_1
σ : ι ≃ ι'
f₁ f₂ : MultilinearMap R M₁ M₂
x✝ : (i : ι') → M₁ (σ.symm i)

⊢ ((fun f => { toFun := ⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm, map_add' := ⋯, map_smul' := ⋯ }) (f₁ + f₂)) x✝ = ((fun f => { toFun := ⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm, map_add' := ⋯, map_smul' := ⋯ }) f₁ + (fun f => { toFun := ⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm, map_add' := ⋯, map_smul' := ⋯ }) f₂) x✝
-- Mathlib/LinearAlgebra/Multilinear/Basic.lean
R : Type uR
S : Type uS
ι : Type uι
n : ℕ
M : Fin n.succ → Type v
M₁ : ι → Type v₁
M₂ : Type v₂
M₃ : Type v₃
M' : Type v'
inst✝⁷ : Semiring R
inst✝⁶ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁵ : (i : ι) → Module R (M₁ i)
inst✝⁴ : AddCommMonoid M₂
inst✝³ : Module R M₂
inst✝² : Semiring S
inst✝¹ : Module S M₂
inst✝ : SMulCommClass R S M₂
ι' : Type u_1
σ : ι ≃ ι'
c : S
f : MultilinearMap R M₁ M₂
x✝ : (i : ι') → M₁ (σ.symm i)

⊢ ({ toFun := fun f => { toFun := ⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm, map_add' := ⋯, map_smul' := ⋯ }, map_add' := ⋯ }.toFun (c • f)) x✝ = ((RingHom.id S) c • { toFun := fun f => { toFun := ⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm, map_add' := ⋯, map_smul' := ⋯ }, map_add' := ⋯ }.toFun f) x✝
-- Mathlib/LinearAlgebra/Multilinear/Basic.lean
R : Type uR
S : Type uS
ι : Type uι
n : ℕ
M : Fin n.succ → Type v
M₁ : ι → Type v₁
M₂ : Type v₂
M₃ : Type v₃
M' : Type v'
inst✝⁷ : Semiring R
inst✝⁶ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁵ : (i : ι) → Module R (M₁ i)
inst✝⁴ : AddCommMonoid M₂
inst✝³ : Module R M₂
inst✝² : Semiring S
inst✝¹ : Module S M₂
inst✝ : SMulCommClass R S M₂
ι' : Type u_1
σ : ι ≃ ι'
f : MultilinearMap R M₁ M₂
x✝ : (i : ι) → M₁ i

⊢ ((fun f => { toFun := ⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ), map_add' := ⋯, map_smul' := ⋯ }) ({ toFun := fun f => { toFun := ⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm, map_add' := ⋯, map_smul' := ⋯ }, map_add' := ⋯, map_smul' := ⋯ }.toFun f)) x✝ = f x✝
-- Mathlib/LinearAlgebra/Multilinear/Basic.lean
R : Type uR
S : Type uS
ι : Type uι
n : ℕ
M : Fin n.succ → Type v
M₁ : ι → Type v₁
M₂ : Type v₂
M₃ : Type v₃
M' : Type v'
inst✝⁷ : Semiring R
inst✝⁶ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁵ : (i : ι) → Module R (M₁ i)
inst✝⁴ : AddCommMonoid M₂
inst✝³ : Module R M₂
inst✝² : Semiring S
inst✝¹ : Module S M₂
inst✝ : SMulCommClass R S M₂
ι' : Type u_1
σ : ι ≃ ι'
f : MultilinearMap R (fun i => M₁ (σ.symm i)) M₂
x✝ : (i : ι') → M₁ (σ.symm i)

⊢ ({ toFun := fun f => { toFun := ⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm, map_add' := ⋯, map_smul' := ⋯ }, map_add' := ⋯, map_smul' := ⋯ }.toFun ((fun f => { toFun := ⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ), map_add' := ⋯, map_smul' := ⋯ }) f)) x✝ = f x✝
-- Mathlib/Algebra/BigOperators/Finprod.lean
α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M : Type u_5
N : Type u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f : α → M
s : Finset α
h : mulSupport f ⊆ ↑s

⊢ ∏ᶠ (i : α), f i = ∏ i ∈ s, f i
-- Mathlib/Algebra/BigOperators/Finprod.lean
α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M : Type u_5
N : Type u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f : α → M
s : Finset α
h : mulSupport f ⊆ ↑s
A : mulSupport (f ∘ PLift.down) = ⇑Equiv.plift.symm '' mulSupport f

⊢ ∏ᶠ (i : α), f i = ∏ i ∈ s, f i
-- Mathlib/Algebra/BigOperators/Finprod.lean
α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M : Type u_5
N : Type u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f : α → M
s : Finset α
h : mulSupport f ⊆ ↑s
A : mulSupport (f ∘ PLift.down) = ⇑Equiv.plift.symm '' mulSupport f
this : mulSupport (f ∘ PLift.down) ⊆ ↑(Finset.map Equiv.plift.symm.toEmbedding s)

⊢ ∏ᶠ (i : α), f i = ∏ i ∈ s, f i
-- Mathlib/Algebra/BigOperators/Finprod.lean
α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M : Type u_5
N : Type u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f : α → M
s : Finset α
h : mulSupport f ⊆ ↑s
A : mulSupport (f ∘ PLift.down) = ⇑Equiv.plift.symm '' mulSupport f
this : mulSupport (f ∘ PLift.down) ⊆ ↑(Finset.map Equiv.plift.symm.toEmbedding s)

⊢ ∏ i ∈ Finset.map Equiv.plift.symm.toEmbedding s, f i.down = ∏ i ∈ s, f i
-- Mathlib/Algebra/BigOperators/Finprod.lean
α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M : Type u_5
N : Type u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f : α → M
s : Finset α
h : mulSupport f ⊆ ↑s
A : mulSupport (f ∘ PLift.down) = ⇑Equiv.plift.symm '' mulSupport f
this : mulSupport (f ∘ PLift.down) ⊆ ↑(Finset.map Equiv.plift.symm.toEmbedding s)

⊢ ∏ x ∈ s, f (Equiv.plift.symm x).down = ∏ i ∈ s, f i
-- Mathlib/Algebra/Module/LinearMap/Polynomial.lean
m : Type u_1
n : Type u_2
o : Type u_3
R : Type u_4
S : Type u_5
inst✝³ : Fintype n
inst✝² : Fintype o
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
M : Matrix m n R
i : m

⊢ constantCoeff (M.toMvPolynomial i) = 0
-- Mathlib/Data/Set/Pointwise/Interval.lean
α : Type u_1
inst✝ : OrderedAddCommGroup α
a b c : α

⊢ -Ioc a b = Ico (-b) (-a)
-- Mathlib/Probability/Process/Stopping.lean
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹ : Preorder ι
f : Filtration ι m
τ : Ω → ι
inst✝ : PredOrder ι
hτ : IsStoppingTime f τ
i : ι
hi_min : ¬IsMin i

⊢ MeasurableSet {ω | τ ω < i}
-- Mathlib/Probability/Process/Stopping.lean
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹ : Preorder ι
f : Filtration ι m
τ : Ω → ι
inst✝ : PredOrder ι
hτ : IsStoppingTime f τ
i : ι
hi_min : ¬IsMin i
this : {ω | τ ω < i} = τ ⁻¹' Set.Iic (pred i)

⊢ MeasurableSet {ω | τ ω < i}
-- Mathlib/Probability/Process/Stopping.lean
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹ : Preorder ι
f : Filtration ι m
τ : Ω → ι
inst✝ : PredOrder ι
hτ : IsStoppingTime f τ
i : ι
hi_min : ¬IsMin i
this : {ω | τ ω < i} = τ ⁻¹' Set.Iic (pred i)

⊢ MeasurableSet (τ ⁻¹' Set.Iic (pred i))
-- Mathlib/LinearAlgebra/AffineSpace/Slope.lean
k : Type u_1
E : Type u_2
PE : Type u_3
inst✝³ : Field k
inst✝² : AddCommGroup E
inst✝¹ : Module k E
inst✝ : AddTorsor E PE
f : k → E
c : PE
a b : k

⊢ slope (fun x => f x +ᵥ c) a b = slope f a b
-- Mathlib/Data/Setoid/Basic.lean
α : Type u_1
β : Type u_2
r : α → α → Prop
s : Setoid α
h : ∀ (x y : α), r x y → s.Rel x y

⊢ EqvGen.Setoid r ≤ s
-- Mathlib/Data/Setoid/Basic.lean
α : Type u_1
β : Type u_2
r : α → α → Prop
s : Setoid α
h : ∀ (x y : α), r x y → s.Rel x y

⊢ sInf {s | ∀ ⦃x y : α⦄, r x y → s.Rel x y} ≤ s
-- Mathlib/Data/List/Sigma.lean
α : Type u
β : α → Type v
l✝ l₁✝ l₂✝ : List (Sigma β)
inst✝ : DecidableEq α
a₁ a₂ : α
l : List (Sigma β)
h : ¬a₁ = a₂
ha₂ : a₂ ∈ l.keys
b₁ : β a₁
l₁ l₂ : List (Sigma β)
a₁_nin_l₁ : a₁ ∉ l₁.keys
x✝ : a₂ ∈ (l₁ ++ ⟨a₁, b₁⟩ :: l₂).keys
ha₁ : a₁ ∈ (l₁ ++ ⟨a₁, b₁⟩ :: l₂).keys
h' : a₂ ∈ l₁.keys

⊢ kerase a₂ (l₁ ++ l₂) = kerase a₁ (kerase a₂ (l₁ ++ ⟨a₁, b₁⟩ :: l₂))
-- Mathlib/Data/List/Sigma.lean
α : Type u
β : α → Type v
l✝ l₁✝ l₂✝ : List (Sigma β)
inst✝ : DecidableEq α
a₁ a₂ : α
l : List (Sigma β)
h : ¬a₁ = a₂
ha₂ : a₂ ∈ l.keys
b₁ : β a₁
l₁ l₂ : List (Sigma β)
a₁_nin_l₁ : a₁ ∉ l₁.keys
x✝ : a₂ ∈ (l₁ ++ ⟨a₁, b₁⟩ :: l₂).keys
ha₁ : a₁ ∈ (l₁ ++ ⟨a₁, b₁⟩ :: l₂).keys
h' : a₂ ∉ l₁.keys

⊢ kerase a₂ (l₁ ++ l₂) = kerase a₁ (kerase a₂ (l₁ ++ ⟨a₁, b₁⟩ :: l₂))
-- Mathlib/Data/List/Sigma.lean
α : Type u
β : α → Type v
l✝ l₁ l₂ : List (Sigma β)
inst✝ : DecidableEq α
a₁ a₂ : α
l : List (Sigma β)
h : ¬a₁ = a₂
ha₁ : a₁ ∈ l.keys
ha₂ : a₂ ∉ l.keys

⊢ kerase a₂ (kerase a₁ l) = kerase a₁ (kerase a₂ l)
-- Mathlib/Data/List/Sigma.lean
α : Type u
β : α → Type v
l✝ l₁ l₂ : List (Sigma β)
inst✝ : DecidableEq α
a₁ a₂ : α
l : List (Sigma β)
h : ¬a₁ = a₂
ha₁ : a₁ ∉ l.keys

⊢ kerase a₂ (kerase a₁ l) = kerase a₁ (kerase a₂ l)
-- Mathlib/Data/Fin/Tuple/Reflection.lean
m n : ℕ
α : Type u_1
β : Type u_2
γ : Type u_3
P : (Fin 0 → α) → Prop

⊢ Exists P ↔ ∃ x, P x
-- Mathlib/Data/Fin/Tuple/Reflection.lean
m n : ℕ
α : Type u_1
β : Type u_2
γ : Type u_3
P : (Fin 0 → α) → Prop

⊢ P Fin.elim0 ↔ P finZeroElim
-- Mathlib/Data/Fin/Tuple/Reflection.lean
m n✝ : ℕ
α : Type u_1
β : Type u_2
γ : Type u_3
n : ℕ
P : (Fin n.succ → α) → Prop

⊢ Exists P ↔ ∃ x, P x
-- Mathlib/Deprecated/Group.lean
α : Type u
β : Type v
inst✝² : MulOneClass α
inst✝¹ : MulOneClass β
f : α → β
hf : IsMonoidHom f
γ : Type u_1
inst✝ : MulOneClass γ
g : β → γ
hg : IsMonoidHom g

⊢ g (f 1) = 1
-- .lake/packages/batteries/Batteries/Data/List/EraseIdx.lean
α : Type u
β : Type v
x : α
l : List α
k : Nat

⊢ x ∈ l.eraseIdx k ↔ ∃ i, i ≠ k ∧ l[i]? = some x
-- .lake/packages/batteries/Batteries/Data/List/EraseIdx.lean
α : Type u
β : Type v
x : α
l : List α
k : Nat

⊢ (∃ i, i ≠ k ∧ ∃ x_1, l[i]? = some x) ↔ ∃ i, i ≠ k ∧ l[i]? = some x
-- .lake/packages/batteries/Batteries/Data/List/EraseIdx.lean
α : Type u
β : Type v
x : α
l : List α
k i : Nat

⊢ (∃ x_1, l[i]? = some x) ↔ l[i]? = some x
-- Mathlib/Topology/MetricSpace/PiNat.lean
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : s.Nonempty

⊢ ∃ f, (∀ x ∈ s, f x = x) ∧ range f = s ∧ LipschitzWith 1 f
-- Mathlib/Topology/MetricSpace/PiNat.lean
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some

⊢ ∃ f, (∀ x ∈ s, f x = x) ∧ range f = s ∧ LipschitzWith 1 f
-- Mathlib/RingTheory/WittVector/IsPoly.lean
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R
h : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), (ghostComponent n) (f x) = (ghostComponent n) (g x)
φ : ℕ → MvPolynomial ℕ ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => (aeval x.coeff) (φ n)
ψ : ℕ → MvPolynomial ℕ ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => (aeval x.coeff) (ψ n)
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ : 𝕎 R✝
n : ℕ

⊢ (f x✝).coeff n = (g x✝).coeff n
-- Mathlib/RingTheory/WittVector/IsPoly.lean
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R
h : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), (ghostComponent n) (f x) = (ghostComponent n) (g x)
φ : ℕ → MvPolynomial ℕ ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => (aeval x.coeff) (φ n)
ψ : ℕ → MvPolynomial ℕ ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => (aeval x.coeff) (ψ n)
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ : 𝕎 R✝
n : ℕ

⊢ ∀ (n : ℕ), (bind₁ φ) (wittPolynomial p ℤ n) = (bind₁ ψ) (wittPolynomial p ℤ n)
-- Mathlib/RingTheory/WittVector/IsPoly.lean
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R
h : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), (ghostComponent n) (f x) = (ghostComponent n) (g x)
φ : ℕ → MvPolynomial ℕ ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => (aeval x.coeff) (φ n)
ψ : ℕ → MvPolynomial ℕ ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => (aeval x.coeff) (ψ n)
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ : 𝕎 R✝
n k : ℕ

⊢ (bind₁ φ) (wittPolynomial p ℤ k) = (bind₁ ψ) (wittPolynomial p ℤ k)
-- Mathlib/RingTheory/WittVector/IsPoly.lean
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R
h : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), (ghostComponent n) (f x) = (ghostComponent n) (g x)
φ : ℕ → MvPolynomial ℕ ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => (aeval x.coeff) (φ n)
ψ : ℕ → MvPolynomial ℕ ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => (aeval x.coeff) (ψ n)
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ : 𝕎 R✝
n k : ℕ

⊢ ∀ (x : ℕ → ℤ), (MvPolynomial.eval x) ((bind₁ φ) (wittPolynomial p ℤ k)) = (MvPolynomial.eval x) ((bind₁ ψ) (wittPolynomial p ℤ k))
-- Mathlib/RingTheory/WittVector/IsPoly.lean
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R
h : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), (ghostComponent n) (f x) = (ghostComponent n) (g x)
φ : ℕ → MvPolynomial ℕ ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => (aeval x.coeff) (φ n)
ψ : ℕ → MvPolynomial ℕ ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => (aeval x.coeff) (ψ n)
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ : 𝕎 R✝
n k : ℕ
x : ℕ → ℤ

⊢ (MvPolynomial.eval x) ((bind₁ φ) (wittPolynomial p ℤ k)) = (MvPolynomial.eval x) ((bind₁ ψ) (wittPolynomial p ℤ k))
-- Mathlib/RingTheory/WittVector/IsPoly.lean
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R
h : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), (ghostComponent n) (f x) = (ghostComponent n) (g x)
φ : ℕ → MvPolynomial ℕ ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => (aeval x.coeff) (φ n)
ψ : ℕ → MvPolynomial ℕ ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => (aeval x.coeff) (ψ n)
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ : 𝕎 R✝
n k : ℕ
x : ℕ → ℤ

⊢ (eval₂Hom ((MvPolynomial.eval x).comp C) fun i => (MvPolynomial.eval x) (φ i)) (wittPolynomial p ℤ k) = (eval₂Hom ((MvPolynomial.eval x).comp C) fun i => (MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k)
-- Mathlib/RingTheory/WittVector/IsPoly.lean
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R
φ : ℕ → MvPolynomial ℕ ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => (aeval x.coeff) (φ n)
ψ : ℕ → MvPolynomial ℕ ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => (aeval x.coeff) (ψ n)
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ : 𝕎 R✝
n k : ℕ
x : ℕ → ℤ
h : (ghostComponent k) (f (mk p fun i => { down := x i })) = (ghostComponent k) (g (mk p fun i => { down := x i }))

⊢ (eval₂Hom ((MvPolynomial.eval x).comp C) fun i => (MvPolynomial.eval x) (φ i)) (wittPolynomial p ℤ k) = (eval₂Hom ((MvPolynomial.eval x).comp C) fun i => (MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k)
-- Mathlib/RingTheory/WittVector/IsPoly.lean
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R
φ : ℕ → MvPolynomial ℕ ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => (aeval x.coeff) (φ n)
ψ : ℕ → MvPolynomial ℕ ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => (aeval x.coeff) (ψ n)
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ : 𝕎 R✝
n k : ℕ
x : ℕ → ℤ
h : (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ)) (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) = (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ)) (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)

⊢ (eval₂Hom ((MvPolynomial.eval x).comp C) fun i => (MvPolynomial.eval x) (φ i)) (wittPolynomial p ℤ k) = (eval₂Hom ((MvPolynomial.eval x).comp C) fun i => (MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k)
-- Mathlib/RingTheory/WittVector/IsPoly.lean
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R
φ : ℕ → MvPolynomial ℕ ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => (aeval x.coeff) (φ n)
ψ : ℕ → MvPolynomial ℕ ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => (aeval x.coeff) (ψ n)
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ : 𝕎 R✝
n k : ℕ
x : ℕ → ℤ
h : (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ)) (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) = (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ)) (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)

⊢ ULift.ringEquiv.symm ((eval₂Hom ((MvPolynomial.eval x).comp C) fun i => (MvPolynomial.eval x) (φ i)) (wittPolynomial p ℤ k)) = ULift.ringEquiv.symm ((eval₂Hom ((MvPolynomial.eval x).comp C) fun i => (MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k))
-- Mathlib/RingTheory/WittVector/IsPoly.lean
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R
φ : ℕ → MvPolynomial ℕ ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => (aeval x.coeff) (φ n)
ψ : ℕ → MvPolynomial ℕ ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => (aeval x.coeff) (ψ n)
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ : 𝕎 R✝
n k : ℕ
x : ℕ → ℤ
h : (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ)) (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) = (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ)) (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)

⊢ (eval₂Hom ((↑ULift.ringEquiv.symm).comp ((MvPolynomial.eval x).comp C)) fun i => ↑ULift.ringEquiv.symm ((MvPolynomial.eval x) (φ i))) (wittPolynomial p ℤ k) = (eval₂Hom ((↑ULift.ringEquiv.symm).comp ((MvPolynomial.eval x).comp C)) fun i => ↑ULift.ringEquiv.symm ((MvPolynomial.eval x) (ψ i))) (wittPolynomial p ℤ k)
-- Mathlib/RingTheory/WittVector/IsPoly.lean
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R
φ : ℕ → MvPolynomial ℕ ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => (aeval x.coeff) (φ n)
ψ : ℕ → MvPolynomial ℕ ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => (aeval x.coeff) (ψ n)
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ : 𝕎 R✝
n k : ℕ
x : ℕ → ℤ
h : (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ)) (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) = (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ)) (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)

⊢ (eval₂Hom ((↑ULift.ringEquiv.symm).comp ((MvPolynomial.eval x).comp C)) fun i => ↑ULift.ringEquiv.symm ((MvPolynomial.eval x) (φ i))) (wittPolynomial p ℤ k) = (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ)) (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)
-- Mathlib/RingTheory/WittVector/IsPoly.lean
p : ℕ
R S : Type u
σ : Type u_1
idx : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R
φ : ℕ → MvPolynomial ℕ ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n => (aeval x.coeff) (φ n)
ψ : ℕ → MvPolynomial ℕ ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n => (aeval x.coeff) (ψ n)
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ : 𝕎 R✝
n k : ℕ
x : ℕ → ℤ
h : (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ)) (f (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k) = (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ)) (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)

⊢ (eval₂Hom ((↑ULift.ringEquiv.symm).comp ((MvPolynomial.eval x).comp C)) fun i => ↑ULift.ringEquiv.symm ((MvPolynomial.eval x) (ψ i))) (wittPolynomial p ℤ k) = (eval₂Hom (algebraMap ℤ (ULift.{u, 0} ℤ)) (g (mk p fun i => { down := x i })).coeff) (wittPolynomial p ℤ k)
-- Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫_ℝ = 0
h0 : x = 0 ∨ y ≠ 0

⊢ ‖y‖ / Real.tan (angle x (x - y)) = ‖x‖
-- Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫_ℝ = 0
h0 : x = 0 ∨ y ≠ 0

⊢ ‖y‖ / Real.tan (angle x (x - y)) = ‖x‖
-- Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫_ℝ = 0
h0 : x = 0 ∨ -y ≠ 0

⊢ ‖y‖ / Real.tan (angle x (x - y)) = ‖x‖
-- Mathlib/Geometry/Manifold/MFDeriv/Basic.lean
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
f f₀ f₁ : M → M'
x : M
s t : Set M
g : M' → M''
u : Set M'
Is : SmoothManifoldWithCorners I M
I's : SmoothManifoldWithCorners I' M'
I''s : SmoothManifoldWithCorners I'' M''
f' f₀' f₁' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)
g' : TangentSpace I' (f x) →L[𝕜] TangentSpace I'' (g (f x))
U : UniqueMDiffWithinAt I s x
h : HasMFDerivWithinAt I I' f s x f'
h₁ : HasMFDerivWithinAt I I' f s x f₁'

⊢ f' = f₁'
-- Mathlib/Algebra/Field/Basic.lean
α : Type u_1
β : Type u_2
K : Type u_3
inst✝ : DivisionSemiring α
a b c d : α
ha : a ≠ 0
hb : b ≠ 0

⊢ 1 / a * (a + b) * (1 / b) = 1 / a + 1 / b
-- Mathlib/Combinatorics/Additive/Corner/Roth.lean
G : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : Fintype G
A✝ B : Finset (G × G)
a b c d x y : G
n : ℕ
ε✝ ε : ℝ
hε : 0 < ε
hG : cornersTheoremBound ε ≤ Fintype.card G
A : Finset G
hAε : ε * ↑(Fintype.card G) ≤ ↑A.card
hA : ThreeAPFree ↑A

⊢ False
-- Mathlib/Analysis/Normed/Ring/Seminorm.lean
F : Type u_1
R✝ : Type u_2
S : Type u_3
x y : R✝
r : ℝ
inst✝⁴ : NonAssocRing R✝
inst✝³ : DecidableEq R✝
inst✝² : NoZeroDivisors R✝
inst✝¹ : Nontrivial R✝
R : Type u_4
inst✝ : Ring R
f g : MulRingNorm R
c : ℝ
hcpos : 0 < c
h : (fun x => f x ^ c) = ⇑g

⊢ g.equiv f
-- Mathlib/Analysis/Normed/Ring/Seminorm.lean
F : Type u_1
R✝ : Type u_2
S : Type u_3
x y : R✝
r : ℝ
inst✝⁴ : NonAssocRing R✝
inst✝³ : DecidableEq R✝
inst✝² : NoZeroDivisors R✝
inst✝¹ : Nontrivial R✝
R : Type u_4
inst✝ : Ring R
f g : MulRingNorm R
c : ℝ
hcpos : 0 < c
h : (fun x => f x ^ c) = ⇑g

⊢ 0 < 1 / c ∧ (fun x => g x ^ (1 / c)) = ⇑f
-- Mathlib/Analysis/Normed/Ring/Seminorm.lean
F : Type u_1
R✝ : Type u_2
S : Type u_3
x✝ y : R✝
r : ℝ
inst✝⁴ : NonAssocRing R✝
inst✝³ : DecidableEq R✝
inst✝² : NoZeroDivisors R✝
inst✝¹ : Nontrivial R✝
R : Type u_4
inst✝ : Ring R
f g : MulRingNorm R
c : ℝ
hcpos : 0 < c
h : (fun x => f x ^ c) = ⇑g
x : R

⊢ g x ^ (1 / c) = f x
-- Mathlib/Data/List/Rotate.lean
α : Type u
l : List α
n m : ℕ
hml : m < l.length

⊢ (l.rotate n).get? m = l.get? ((m + n) % l.length)
-- Mathlib/Data/List/Rotate.lean
α : Type u
l : List α
n m : ℕ
hml : m < l.length

⊢ some l[(m + n) % l.length] = l[(m + n) % l.length]?