ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Analysis/SpecialFunctions/Exp.lean
α : Type u_1
x y z : ℝ
l : Filter α
f g : α → ℝ
\u22a2 ((fun x => rexp (f x)) =o[l] fun x => rexp (g x)) ↔ Tendsto (fun x => g x - f x) l atTop
-- Mathlib/Data/PNat/Prime.lean
n : ℕ+
h : n < 2
\u22a2 n = 1
-- Mathlib/Data/PNat/Prime.lean
n : ℕ+
h : n < 2
\u22a2 n ≤ 1
-- Mathlib/Data/PNat/Prime.lean
n : ℕ+
h : n < 2
\u22a2 1 ≤ n
-- Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
Ω : Type u_1
inst✝⁵ : MeasurableSpace Ω
R : Type u_2
inst✝⁴ : SMul R ℝ≥0
inst✝³ : SMul R ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0 ℝ≥0
c : R
μ : FiniteMeasure Ω
s : Set Ω
\u22a2 (c • μ) s = c • μ s
-- Mathlib/SetTheory/Ordinal/Arithmetic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
a b : Ordinal.{u_4}
l : a.IsLimit
b0 : 0 < b
\u22a2 (a * b).IsLimit
-- Mathlib/SetTheory/Ordinal/Arithmetic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
a : Ordinal.{u_4}
l : a.IsLimit
b0 : 0 < 0
\u22a2 (a * 0).IsLimit
-- Mathlib/SetTheory/Ordinal/Arithmetic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
a : Ordinal.{u_4}
l : a.IsLimit
b : Ordinal.{u_4}
b0 : 0 < succ b
\u22a2 (a * succ b).IsLimit
-- Mathlib/SetTheory/Ordinal/Arithmetic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
a b : Ordinal.{u_4}
l : a.IsLimit
b0 : 0 < b
lb : b.IsLimit
\u22a2 (a * b).IsLimit
-- Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
ζ : R
n : ℕ
h : IsPrimitiveRoot ζ n
\u22a2 (cyclotomic' n R).degree = ↑n.totient
-- Mathlib/CategoryTheory/CofilteredSystem.lean
J : Type u
inst✝¹ : Category.{u_1, u} J
F : J ⥤ Type v
i j k : J
s : Set (F.obj i)
inst✝ : IsCofilteredOrEmpty J
\u22a2 F.IsMittagLeffler ↔ ∀ (j : J), ∃ i f, ∀ ⦃k : J⦄ (g : k ⟶ i), range (F.map f) ⊆ range (F.map (g ≫ f))
-- Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : Ring 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y : E
a b : 𝕜
f : E →ᵃ[𝕜] F
h : a + b = 1
\u22a2 f (a • x + b • y) = a • f x + b • f y
-- Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : Ring 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y : E
a b : 𝕜
f : E →ᵃ[𝕜] F
h : a + b = 1
\u22a2 f (b • (y -ᵥ x) + x) = b • (f y -ᵥ f x) + f x
-- Mathlib/Algebra/MonoidAlgebra/Basic.lean
k : Type u₁
G : Type u₂
H : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Semiring k
inst✝¹ : Algebra R k
inst✝ : AddMonoid G
r : R
\u22a2 (↑↑__src✝).toFun ((algebraMap R k) r) = (algebraMap R k[G]) r
-- Mathlib/Algebra/MonoidAlgebra/Basic.lean
k : Type u₁
G : Type u₂
H : Type u_1
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : Semiring k
inst✝¹ : Algebra R k
inst✝ : AddMonoid G
r : R
x✝ : G
\u22a2 ((↑↑__src✝).toFun ((algebraMap R k) r)) x✝ = ((algebraMap R k[G]) r) x✝
-- Mathlib/Probability/Distributions/Uniform.lean
α : Type u_1
β : Type u_2
γ : Type u_3
s : Multiset α
hs : s ≠ 0
\u22a2 ∀ (x : α), x ∈ (ofMultiset s hs).support ↔ x ∈ ↑s.toFinset
-- Mathlib/CategoryTheory/Bicategory/Functor/Lax.lean
B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
F : LaxFunctor B C
a b c d : B
f : a ⟶ b
g : b ⟶ c
h : c ⟶ d
\u22a2 F.map f ◁ F.mapComp g h ≫ F.mapComp f (g ≫ h) = (α_ (F.map f) (F.map g) (F.map h)).inv ≫ F.mapComp f g ▷ F.map h ≫ F.mapComp (f ≫ g) h ≫ F.map₂ (α_ f g h).hom
-- Mathlib/MeasureTheory/Function/UnifTight.lean
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ✝ : Measure α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p μ
hf_tendsto : Tendsto (fun n => eLpNorm (f n) p μ) atTop (𝓝 0)
ε : ℝ≥0
hε : 0 < ε
\u22a2 ∃ s, μ s ≠ ⊤ ∧ ∀ (i : ℕ), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε
-- Mathlib/MeasureTheory/Function/UnifTight.lean
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ✝ : Measure α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p μ
hf_tendsto : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n) p μ ≤ ε
ε : ℝ≥0
hε : 0 < ε
\u22a2 ∃ s, μ s ≠ ⊤ ∧ ∀ (i : ℕ), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε
-- Mathlib/MeasureTheory/Function/UnifTight.lean
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ✝ : Measure α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p μ
hf_tendsto : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n) p μ ≤ ε
ε : ℝ≥0
hε : 0 < ε
N : ℕ
hNε : ∀ n ≥ N, eLpNorm (f n) p μ ≤ ↑ε
\u22a2 ∃ s, μ s ≠ ⊤ ∧ ∀ (i : ℕ), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε
-- Mathlib/MeasureTheory/Function/UnifTight.lean
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ✝ : Measure α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p μ
hf_tendsto : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n) p μ ≤ ε
ε : ℝ≥0
hε : 0 < ε
N : ℕ
hNε : ∀ n ≥ N, eLpNorm (f n) p μ ≤ ↑ε
F : Fin N → α → β := fun n => f ↑n
hF : ∀ (n : Fin N), Memℒp (F n) p μ
\u22a2 ∃ s, μ s ≠ ⊤ ∧ ∀ (i : ℕ), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε
-- Mathlib/MeasureTheory/Function/UnifTight.lean
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ✝ : Measure α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp' : p ≠ ⊤
hf : ∀ (n : ℕ), Memℒp (f n) p μ
hf_tendsto : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n) p μ ≤ ε
ε : ℝ≥0
hε : 0 < ε
N : ℕ
hNε : ∀ n ≥ N, eLpNorm (f n) p μ ≤ ↑ε
F : Fin N → α → β := fun n => f ↑n
hF : ∀ (n : Fin N), Memℒp (F n) p μ
s : Set α
hμs : μ s ≠ ⊤
hFε : ∀ (i : Fin N), eLpNorm (sᶜ.indicator (F i)) p μ ≤ ↑ε
\u22a2 ∃ s, μ s ≠ ⊤ ∧ ∀ (i : ℕ), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε
-- Mathlib/Algebra/Lie/Weights/Cartan.lean
R : Type u_1
L : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra R L
H : LieSubalgebra R L
inst✝⁴ : IsNilpotent R ↥H
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
h : zeroRootSubalgebra R L H = H
\u22a2 (zeroRootSubalgebra R L H).normalizer = zeroRootSubalgebra R L H
-- Mathlib/Topology/Connected/Clopen.lean
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s✝ t u v s : Set α
hs : IsClosed s
\u22a2 IsPreconnected s ↔ ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
-- Mathlib/Topology/Connected/Clopen.lean
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s✝ t u✝ v✝ s : Set α
hs : IsClosed s
H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
u v : Set α
hu : IsClosed u
hv : IsClosed v
hss : s ⊆ u ∪ v
huv : Disjoint u v
\u22a2 s ⊆ u ∨ s ⊆ v
-- Mathlib/Topology/Connected/Clopen.lean
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s✝ t u✝ v✝ s : Set α
hs : IsClosed s
H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set α
hu : IsClosed u
hv : IsClosed v
hss : s ⊆ u ∪ v
huv : s ∩ (u ∩ v) = ∅
\u22a2 s ⊆ u ∨ s ⊆ v
-- Mathlib/Topology/Connected/Clopen.lean
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s✝ t u✝ v✝ s : Set α
hs : IsClosed s
H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set α
hu : IsClosed u
hv : IsClosed v
hss : s ⊆ u ∪ v
huv : s ∩ (u ∩ v) = ∅
H1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∩ s ∨ s ⊆ v ∩ s
\u22a2 s ⊆ u ∨ s ⊆ v
-- Mathlib/Topology/Connected/Clopen.lean
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s✝ t u✝ v✝ s : Set α
hs : IsClosed s
H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set α
hu : IsClosed u
hv : IsClosed v
hss : s ⊆ u ∪ v
huv : s ∩ (u ∩ v) = ∅
H1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∧ s ⊆ s ∨ s ⊆ v ∧ s ⊆ s
\u22a2 s ⊆ u ∨ s ⊆ v
-- Mathlib/Topology/Connected/Clopen.lean
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s✝ t u✝ v✝ s : Set α
hs : IsClosed s
H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set α
hu : IsClosed u
hv : IsClosed v
hss : s ⊆ u ∪ v
huv : s ∩ (u ∩ v) = ∅
H1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v
\u22a2 s ⊆ u ∨ s ⊆ v
-- Mathlib/Topology/Connected/Clopen.lean
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s✝ t u✝ v✝ s : Set α
hs : IsClosed s
H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set α
hu : IsClosed u
hv : IsClosed v
hss : s ⊆ u ∪ v
huv : s ∩ (u ∩ v) = ∅
H1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v
\u22a2 s ⊆ u ∩ s ∪ v ∩ s
-- Mathlib/Topology/Connected/Clopen.lean
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s✝ t u✝ v✝ s : Set α
hs : IsClosed s
H : ∀ (u v : Set α), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set α
hu : IsClosed u
hv : IsClosed v
hss : s ⊆ u ∪ v
huv : s ∩ (u ∩ v) = ∅
H1 : IsClosed (u ∩ s) → IsClosed (v ∩ s) → s ⊆ u ∩ s ∪ v ∩ s → Disjoint (u ∩ s) (v ∩ s) → s ⊆ u ∨ s ⊆ v
\u22a2 Disjoint (u ∩ s) (v ∩ s)
-- Mathlib/LinearAlgebra/LinearIndependent.lean
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid M'
inst✝³ : AddCommMonoid M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
f : M →ₗ[R] M'
hf_inj : ker f = ⊥
h : LinearIndependent R v
\u22a2 Disjoint (span R (Set.range v)) (ker f)
-- Mathlib/RingTheory/Discriminant.lean
A : Type u
B : Type v
C : Type z
ι : Type w
inst✝¹⁶ : DecidableEq ι
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
inst✝¹² : CommRing C
inst✝¹¹ : Algebra A C
ι' : Type u_1
inst✝¹⁰ : Fintype ι'
inst✝⁹ : Fintype ι
inst✝⁸ : DecidableEq ι'
K : Type u
L : Type v
E : Type z
inst✝⁷ : Field K
inst✝⁶ : Field L
inst✝⁵ : Field E
inst✝⁴ : Algebra K L
inst✝³ : Algebra K E
inst✝² : Module.Finite K L
inst✝¹ : IsAlgClosed E
b : ι → L
pb : PowerBasis K L
e : Fin pb.dim ≃ (L →ₐ[K] E)
inst✝ : Algebra.IsSeparable K L
\u22a2 (algebraMap K E) (discr K ⇑pb.basis) = ∏ i : Fin pb.dim, ∏ j ∈ Ioi i, ((e j) pb.gen - (e i) pb.gen) ^ 2
-- Mathlib/RingTheory/Discriminant.lean
A : Type u
B : Type v
C : Type z
ι : Type w
inst✝¹⁶ : DecidableEq ι
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
inst✝¹² : CommRing C
inst✝¹¹ : Algebra A C
ι' : Type u_1
inst✝¹⁰ : Fintype ι'
inst✝⁹ : Fintype ι
inst✝⁸ : DecidableEq ι'
K : Type u
L : Type v
E : Type z
inst✝⁷ : Field K
inst✝⁶ : Field L
inst✝⁵ : Field E
inst✝⁴ : Algebra K L
inst✝³ : Algebra K E
inst✝² : Module.Finite K L
inst✝¹ : IsAlgClosed E
b : ι → L
pb : PowerBasis K L
e : Fin pb.dim ≃ (L →ₐ[K] E)
inst✝ : Algebra.IsSeparable K L
\u22a2 ∏ x : Fin pb.dim, (∏ j ∈ Ioi x, ((e j) pb.gen - (e x) pb.gen)) ^ 2 = ∏ i : Fin pb.dim, ∏ j ∈ Ioi i, ((e j) pb.gen - (e i) pb.gen) ^ 2
-- Mathlib/RingTheory/Discriminant.lean
A : Type u
B : Type v
C : Type z
ι : Type w
inst✝¹⁶ : DecidableEq ι
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Algebra A B
inst✝¹² : CommRing C
inst✝¹¹ : Algebra A C
ι' : Type u_1
inst✝¹⁰ : Fintype ι'
inst✝⁹ : Fintype ι
inst✝⁸ : DecidableEq ι'
K : Type u
L : Type v
E : Type z
inst✝⁷ : Field K
inst✝⁶ : Field L
inst✝⁵ : Field E
inst✝⁴ : Algebra K L
inst✝³ : Algebra K E
inst✝² : Module.Finite K L
inst✝¹ : IsAlgClosed E
b : ι → L
pb : PowerBasis K L
e : Fin pb.dim ≃ (L →ₐ[K] E)
inst✝ : Algebra.IsSeparable K L
i : Fin pb.dim
\u22a2 (∏ j ∈ Ioi i, ((e j) pb.gen - (e i) pb.gen)) ^ 2 = ∏ j ∈ Ioi i, ((e j) pb.gen - (e i) pb.gen) ^ 2
-- Mathlib/Data/Nat/Factorization/Basic.lean
a b✝ m n✝ p✝ n p b : ℕ
hb : n ≤ p ^ b
\u22a2 n.factorization p ≤ b
-- Mathlib/Order/Partition/Equipartition.lean
α : Type u_1
inst✝ : DecidableEq α
s t : Finset α
P : Finpartition s
hP : P.IsEquipartition
\u22a2 (filter (fun p => p.card = s.card / P.parts.card + 1) P.parts).card = s.card % P.parts.card
-- Mathlib/Order/Partition/Equipartition.lean
α : Type u_1
inst✝ : DecidableEq α
s t : Finset α
P : Finpartition s
hP : P.IsEquipartition
z : ∑ i ∈ P.parts, i.card = s.card
\u22a2 (filter (fun p => p.card = s.card / P.parts.card + 1) P.parts).card = s.card % P.parts.card
-- Mathlib/Order/Partition/Equipartition.lean
α : Type u_1
inst✝ : DecidableEq α
s t : Finset α
P : Finpartition s
hP : P.IsEquipartition
z : (filter (fun x => x.card = s.card / P.parts.card + 1) P.parts).card + P.parts.card * (s.card / P.parts.card) = s.card
\u22a2 (filter (fun p => p.card = s.card / P.parts.card + 1) P.parts).card = s.card % P.parts.card
-- Mathlib/Data/PNat/Factors.lean
m n : ℕ+
\u22a2 (m.gcd n).factorMultiset = m.factorMultiset ⊓ n.factorMultiset
-- Mathlib/Data/PNat/Factors.lean
m n : ℕ+
\u22a2 (m.gcd n).factorMultiset ≤ m.factorMultiset ⊓ n.factorMultiset
-- Mathlib/Data/PNat/Factors.lean
m n : ℕ+
\u22a2 m.factorMultiset ⊓ n.factorMultiset ≤ (m.gcd n).factorMultiset
-- Mathlib/Data/Nat/Prime/Basic.lean
n a c : ℕ
\u22a2 ∃ n0, ∀ n ≥ n0, a * c ^ n < (n - 1)!
-- Mathlib/Data/Nat/Prime/Basic.lean
n a c n0 : ℕ
hn : n0 > 1
h : ∀ n ≥ n0, (a * c) ^ n < (n - 1)!
\u22a2 ∃ n0, ∀ n ≥ n0, a * c ^ n < (n - 1)!
-- Mathlib/Data/Nat/Prime/Basic.lean
n✝ a c n0 : ℕ
hn✝ : n0 > 1
h : ∀ n ≥ n0, (a * c) ^ n < (n - 1)!
n : ℕ
hn : n ≥ n0
\u22a2 a * c ^ n ≤ (a * c) ^ n
-- Mathlib/Data/Nat/Prime/Basic.lean
n✝ a c n0 : ℕ
hn✝ : n0 > 1
h : ∀ n ≥ n0, (a * c) ^ n < (n - 1)!
n : ℕ
hn : n ≥ n0
\u22a2 a * c ^ n ≤ a ^ n * c ^ n
-- Mathlib/Data/Nat/Prime/Basic.lean
n✝ a c n0 : ℕ
hn✝ : n0 > 1
h : ∀ n ≥ n0, (a * c) ^ n < (n - 1)!
n : ℕ
hn : n ≥ n0
\u22a2 n ≠ 0
-- Mathlib/Computability/PartrecCode.lean
x✝ : Unit
p : ℕ
\u22a2 Nat.Partrec.Code.G (x✝, List.map (fun n => let a := ofNat (ℕ × Code) n; List.map (evaln a.1 a.2) (List.range a.1)) (List.range p)).2 = some (let a := ofNat (ℕ × Code) p; List.map (evaln a.1 a.2) (List.range a.1))
-- Mathlib/Computability/PartrecCode.lean
x✝ : Unit
p : ℕ
\u22a2 List.map (fun n => Nat.rec Option.none (fun n_1 n_ih => rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2) (fun cf cg x x => do let x ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range p)) ((unpair p).1, cf) n let y ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range p)) ((unpair p).1, cg) n some (Nat.pair x y)) (fun cf cg x x => do let x ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range p)) ((unpair p).1, cg) n Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range p)) ((unpair p).1, cf) x) (fun cf cg x x => Nat.rec (Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range p)) ((unpair p).1, cf) (unpair n).1) (fun n_2 n_ih => do let i ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range p)) (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 n_2) Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range p)) ((unpair p).1, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i))) (unpair n).2) (fun cf x => do let x ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range p)) ((unpair p).1, cf) n Nat.rec (some (unpair n).2) (fun n_2 n_ih => Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range p)) (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) x) (ofNat Code (unpair p).2)) (unpair p).1) (List.range (unpair p).1) = List.map (evaln (unpair p).1 (ofNat Code (unpair p).2)) (List.range (unpair p).1)
-- Mathlib/Computability/PartrecCode.lean
x✝ : Unit
p n : ℕ
\u22a2 n ∈ List.range (unpair p).1 → Nat.rec Option.none (fun n_1 n_ih => rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2) (fun cf cg x x => do let x ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range p)) ((unpair p).1, cf) n let y ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range p)) ((unpair p).1, cg) n some (Nat.pair x y)) (fun cf cg x x => do let x ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range p)) ((unpair p).1, cg) n Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range p)) ((unpair p).1, cf) x) (fun cf cg x x => Nat.rec (Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range p)) ((unpair p).1, cf) (unpair n).1) (fun n_2 n_ih => do let i ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range p)) (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 n_2) Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range p)) ((unpair p).1, cg) (Nat.pair (unpair n).1 (Nat.pair n_2 i))) (unpair n).2) (fun cf x => do let x ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range p)) ((unpair p).1, cf) n Nat.rec (some (unpair n).2) (fun n_2 n_ih => Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range p)) (n_1, ofNat Code (unpair p).2) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) x) (ofNat Code (unpair p).2)) (unpair p).1 = evaln (unpair p).1 (ofNat Code (unpair p).2) n
-- Mathlib/Computability/PartrecCode.lean
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
c : Code
k' : ℕ
nk : n ∈ List.range (k' + 1)
k : ℕ := k' + 1
\u22a2 rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2) (fun cf cg x x => do let x ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cf) n let y ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cg) n some (Nat.pair x y)) (fun cf cg x x => do let x ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cg) n Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cf) x) (fun cf cg x x => Nat.rec (Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cf) (unpair n).1) (fun n_1 n_ih => do let i ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k', c) (Nat.pair (unpair n).1 n_1) Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i))) (unpair n).2) (fun cf x => do let x ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cf) n Nat.rec (some (unpair n).2) (fun n_1 n_ih => Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k', c) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) x) c = evaln (k' + 1) c n
-- Mathlib/Computability/PartrecCode.lean
x✝ : Unit
p n : ℕ
this : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))
c : Code
k' : ℕ
k : ℕ := k' + 1
nk : n ≤ k'
\u22a2 rec (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2) (fun cf cg x x => do let x ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cf) n let y ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cg) n some (Nat.pair x y)) (fun cf cg x x => do let x ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cg) n Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cf) x) (fun cf cg x x => Nat.rec (Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cf) (unpair n).1) (fun n_1 n_ih => do let i ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k', c) (Nat.pair (unpair n).1 n_1) Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cg) (Nat.pair (unpair n).1 (Nat.pair n_1 i))) (unpair n).2) (fun cf x => do let x ← Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k' + 1, cf) n Nat.rec (some (unpair n).2) (fun n_1 n_ih => Nat.Partrec.Code.lup (List.map (fun n => List.map (evaln (unpair n).1 (ofNat Code (unpair n).2)) (List.range (unpair n).1)) (List.range (Nat.pair (k' + 1) (encode c)))) (k', c) (Nat.pair (unpair n).1 ((unpair n).2 + 1))) x) c = evaln (k' + 1) c n