ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
this✝ : Inhabited I
m : Products I →₀ ℤ
hmmem : m ∈ Finsupp.supported ℤ ℤ {m | m < (↑l).Tail}
hmsum : (m.sum fun i a => a • Products.eval (C' C ho) i) = (Linear_CC' C hsC ho) (Products.eval C ↑l)
this : (Linear_CC' C hsC ho) (Products.eval C ↑l) = (Linear_CC' C hsC ho) (m.sum fun i a => a • (List.map (e C) (term I ho :: ↑i)).prod)
hse : LinearMap.range (ShortComplex.mk (ModuleCat.ofHom (πs C o)) (ModuleCat.ofHom (Linear_CC' C hsC ho)) ⋯).f = LinearMap.ker (ShortComplex.mk (ModuleCat.ofHom (πs C o)) (ModuleCat.ofHom (Linear_CC' C hsC ho)) ⋯).g
\u22a2 False
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
this✝ : Inhabited I
m : Products I →₀ ℤ
hmmem : m ∈ Finsupp.supported ℤ ℤ {m | m < (↑l).Tail}
hmsum : (m.sum fun i a => a • Products.eval (C' C ho) i) = (Linear_CC' C hsC ho) (Products.eval C ↑l)
this : (Linear_CC' C hsC ho) (Products.eval C ↑l) = (Linear_CC' C hsC ho) (m.sum fun i a => a • (List.map (e C) (term I ho :: ↑i)).prod)
hse : LinearMap.range (πs C o) = LinearMap.ker (Linear_CC' C hsC ho)
\u22a2 False
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
this✝ : Inhabited I
m : Products I →₀ ℤ
hmmem : m ∈ Finsupp.supported ℤ ℤ {m | m < (↑l).Tail}
hmsum : (m.sum fun i a => a • Products.eval (C' C ho) i) = (Linear_CC' C hsC ho) (Products.eval C ↑l)
this : (Products.eval C ↑l - m.sum fun i a => a • (List.map (e C) (term I ho :: ↑i)).prod) ∈ LinearMap.range (πs C o)
hse : LinearMap.range (πs C o) = LinearMap.ker (Linear_CC' C hsC ho)
\u22a2 False
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
this : Inhabited I
m : Products I →₀ ℤ
hmmem : m ∈ Finsupp.supported ℤ ℤ {m | m < (↑l).Tail}
hmsum : (m.sum fun i a => a • Products.eval (C' C ho) i) = (Linear_CC' C hsC ho) (Products.eval C ↑l)
hse : LinearMap.range (πs C o) = LinearMap.ker (Linear_CC' C hsC ho)
n : LocallyConstant ↑(π C fun x => ord I x < o) ℤ
hn : (πs C o) n = Products.eval C ↑l - m.sum fun i a => a • (List.map (e C) (term I ho :: ↑i)).prod
\u22a2 False
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
this : Inhabited I
m : Products I →₀ ℤ
hmmem : m ∈ Finsupp.supported ℤ ℤ {m | m < (↑l).Tail}
hmsum : (m.sum fun i a => a • Products.eval (C' C ho) i) = (Linear_CC' C hsC ho) (Products.eval C ↑l)
hse : LinearMap.range (πs C o) = LinearMap.ker (Linear_CC' C hsC ho)
n : LocallyConstant ↑(π C fun x => ord I x < o) ℤ
hn : ((πs C o) n + m.sum fun i a => a • (List.map (e C) (term I ho :: ↑i)).prod) = Products.eval C ↑l
\u22a2 False
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
this : Inhabited I
m : Products I →₀ ℤ
hmmem : m ∈ Finsupp.supported ℤ ℤ {m | m < (↑l).Tail}
hmsum : (m.sum fun i a => a • Products.eval (C' C ho) i) = (Linear_CC' C hsC ho) (Products.eval C ↑l)
hse : LinearMap.range (πs C o) = LinearMap.ker (Linear_CC' C hsC ho)
n : LocallyConstant ↑(π C fun x => ord I x < o) ℤ
hn : ((πs C o) n + m.sum fun i a => a • (List.map (e C) (term I ho :: ↑i)).prod) = Products.eval C ↑l
hn' : n ∈ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
\u22a2 False
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
this : Inhabited I
m : Products I →₀ ℤ
hmmem : m ∈ Finsupp.supported ℤ ℤ {m | m < (↑l).Tail}
hmsum : (m.sum fun i a => a • Products.eval (C' C ho) i) = (Linear_CC' C hsC ho) (Products.eval C ↑l)
hse : LinearMap.range (πs C o) = LinearMap.ker (Linear_CC' C hsC ho)
n : LocallyConstant ↑(π C fun x => ord I x < o) ℤ
hn : ((πs C o) n + m.sum fun i a => a • (List.map (e C) (term I ho :: ↑i)).prod) = Products.eval C ↑l
hn' : ∃ c, (c.sum fun i a => a • eval (π C fun x => ord I x < o) i) = n
\u22a2 False
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
this : Inhabited I
m : Products I →₀ ℤ
hmmem : m ∈ Finsupp.supported ℤ ℤ {m | m < (↑l).Tail}
hmsum : (m.sum fun i a => a • Products.eval (C' C ho) i) = (Linear_CC' C hsC ho) (Products.eval C ↑l)
hse : LinearMap.range (πs C o) = LinearMap.ker (Linear_CC' C hsC ho)
n : LocallyConstant ↑(π C fun x => ord I x < o) ℤ
hn : ((πs C o) n + m.sum fun i a => a • (List.map (e C) (term I ho :: ↑i)).prod) = Products.eval C ↑l
w : { l // Products.isGood (π C fun x => ord I x < o) l } →₀ ℤ
hc : (w.sum fun i a => a • eval (π C fun x => ord I x < o) i) = n
\u22a2 False
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
this : Inhabited I
m : Products I →₀ ℤ
hmmem : m ∈ Finsupp.supported ℤ ℤ {m | m < (↑l).Tail}
hmsum : (m.sum fun i a => a • Products.eval (C' C ho) i) = (Linear_CC' C hsC ho) (Products.eval C ↑l)
hse : LinearMap.range (πs C o) = LinearMap.ker (Linear_CC' C hsC ho)
n : LocallyConstant ↑(π C fun x => ord I x < o) ℤ
w : { l // Products.isGood (π C fun x => ord I x < o) l } →₀ ℤ
hn : ((w.sum fun a b => (πs C o) (b • eval (π C fun x => ord I x < o) a)) + m.sum fun i a => a • (List.map (e C) (term I ho :: ↑i)).prod) = Products.eval C ↑l
hc : (w.sum fun i a => a • eval (π C fun x => ord I x < o) i) = n
\u22a2 False
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
this : Inhabited I
m : Products I →₀ ℤ
hmmem : m ∈ Finsupp.supported ℤ ℤ {m | m < (↑l).Tail}
hmsum : (m.sum fun i a => a • Products.eval (C' C ho) i) = (Linear_CC' C hsC ho) (Products.eval C ↑l)
hse : LinearMap.range (πs C o) = LinearMap.ker (Linear_CC' C hsC ho)
n : LocallyConstant ↑(π C fun x => ord I x < o) ℤ
w : { l // Products.isGood (π C fun x => ord I x < o) l } →₀ ℤ
hn : ((w.sum fun a b => (πs C o) (b • eval (π C fun x => ord I x < o) a)) + m.sum fun i a => a • (List.map (e C) (term I ho :: ↑i)).prod) = Products.eval C ↑l
hc : (w.sum fun i a => a • eval (π C fun x => ord I x < o) i) = n
\u22a2 Products.eval C ↑l ∈ Submodule.span ℤ (Products.eval C '' {m | m < ↑l})
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
this : Inhabited I
m : Products I →₀ ℤ
hmmem : m ∈ Finsupp.supported ℤ ℤ {m | m < (↑l).Tail}
hmsum : (m.sum fun i a => a • Products.eval (C' C ho) i) = (Linear_CC' C hsC ho) (Products.eval C ↑l)
hse : LinearMap.range (πs C o) = LinearMap.ker (Linear_CC' C hsC ho)
n : LocallyConstant ↑(π C fun x => ord I x < o) ℤ
w : { l // Products.isGood (π C fun x => ord I x < o) l } →₀ ℤ
hn : ((w.sum fun a b => (πs C o) (b • eval (π C fun x => ord I x < o) a)) + m.sum fun i a => a • (List.map (e C) (term I ho :: ↑i)).prod) = Products.eval C ↑l
hc : (w.sum fun i a => a • eval (π C fun x => ord I x < o) i) = n
\u22a2 ((w.sum fun a b => (πs C o) (b • eval (π C fun x => ord I x < o) a)) + m.sum fun i a => a • (List.map (e C) (term I ho :: ↑i)).prod) ∈ Submodule.span ℤ (Products.eval C '' {m | m < ↑l})
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
this : Inhabited I
m : Products I →₀ ℤ
hmmem : m ∈ Finsupp.supported ℤ ℤ {m | m < (↑l).Tail}
hmsum : (m.sum fun i a => a • Products.eval (C' C ho) i) = (Linear_CC' C hsC ho) (Products.eval C ↑l)
hse : LinearMap.range (πs C o) = LinearMap.ker (Linear_CC' C hsC ho)
n : LocallyConstant ↑(π C fun x => ord I x < o) ℤ
w : { l // Products.isGood (π C fun x => ord I x < o) l } →₀ ℤ
hn : ((w.sum fun a b => (πs C o) (b • eval (π C fun x => ord I x < o) a)) + m.sum fun i a => a • (List.map (e C) (term I ho :: ↑i)).prod) = Products.eval C ↑l
hc : (w.sum fun i a => a • eval (π C fun x => ord I x < o) i) = n
\u22a2 (w.sum fun a b => (πs C o) (b • eval (π C fun x => ord I x < o) a)) ∈ Submodule.span ℤ (Products.eval C '' {m | m < ↑l})
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
l : ↑(MaxProducts C ho)
h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))
this : Inhabited I
m : Products I →₀ ℤ
hmmem : m ∈ Finsupp.supported ℤ ℤ {m | m < (↑l).Tail}
hmsum : (m.sum fun i a => a • Products.eval (C' C ho) i) = (Linear_CC' C hsC ho) (Products.eval C ↑l)
hse : LinearMap.range (πs C o) = LinearMap.ker (Linear_CC' C hsC ho)
n : LocallyConstant ↑(π C fun x => ord I x < o) ℤ
w : { l // Products.isGood (π C fun x => ord I x < o) l } →₀ ℤ
hn : ((w.sum fun a b => (πs C o) (b • eval (π C fun x => ord I x < o) a)) + m.sum fun i a => a • (List.map (e C) (term I ho :: ↑i)).prod) = Products.eval C ↑l
hc : (w.sum fun i a => a • eval (π C fun x => ord I x < o) i) = n
\u22a2 (m.sum fun i a => a • (List.map (e C) (term I ho :: ↑i)).prod) ∈ Submodule.span ℤ (Products.eval C '' {m | m < ↑l})
-- Mathlib/Probability/Kernel/Disintegration/Density.lean
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
h0 : ¬(ν a) (countablePartitionSet n x) = 0
\u22a2 ((κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x)).toReal ≤ ((κ' a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x)).toReal
-- Mathlib/Probability/Kernel/Disintegration/Density.lean
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
h0 : ¬(ν a) (countablePartitionSet n x) = 0
h_le : (κ' a) (countablePartitionSet n x ×ˢ s) ≤ (ν a) (countablePartitionSet n x)
\u22a2 ((κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x)).toReal ≤ ((κ' a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x)).toReal
-- Mathlib/Probability/Kernel/Disintegration/Density.lean
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
h0 : ¬(ν a) (countablePartitionSet n x) = 0
h_le : (κ' a) (countablePartitionSet n x ×ˢ s) ≤ (ν a) (countablePartitionSet n x)
\u22a2 (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ≤ (κ' a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x)
-- Mathlib/Probability/Kernel/Disintegration/Density.lean
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
h0 : ¬(ν a) (countablePartitionSet n x) = 0
h_le : (κ' a) (countablePartitionSet n x ×ˢ s) ≤ (ν a) (countablePartitionSet n x)
\u22a2 (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ≠ ⊤
-- Mathlib/Probability/Kernel/Disintegration/Density.lean
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
h0 : ¬(ν a) (countablePartitionSet n x) = 0
h_le : (κ' a) (countablePartitionSet n x ×ˢ s) ≤ (ν a) (countablePartitionSet n x)
\u22a2 (κ' a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ≠ ⊤
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean
x : ℂ
\u22a2 sin x = 1 ↔ ∃ k, ↑π / 2 + ↑k * (2 * ↑π) = x
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean
x : ℂ
\u22a2 (∃ k, ↑k * (2 * ↑π) = x - ↑π / 2) ↔ ∃ k, ↑π / 2 + ↑k * (2 * ↑π) = x
-- Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean
a : ℂ
r : ℝ
ha : 0 < a.re
hr : 0 < r
\u22a2 ∫ (t : ℝ) in Ioi 0, ↑t ^ (a - 1) * cexp (-(↑r * ↑t)) = (1 / ↑r) ^ a * Gamma a
-- Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean
a : ℂ
r : ℝ
ha : 0 < a.re
hr : 0 < r
aux : (1 / ↑r) ^ a = 1 / ↑r * (1 / ↑r) ^ (a - 1)
\u22a2 ∫ (t : ℝ) in Ioi 0, ↑t ^ (a - 1) * cexp (-(↑r * ↑t)) = (1 / ↑r) ^ a * Gamma a
-- Mathlib/Data/Fin/Tuple/Basic.lean
m n : ℕ
α✝ : Fin (n + 1) → Type u
x : α✝ 0
q : (i : Fin (n + 1)) → α✝ i
p : (i : Fin n) → α✝ i.succ
i : Fin n
y : α✝ i.succ
z : α✝ 0
α : Type u_1
a : Fin n → α
h : 1 * n = n
\u22a2 repeat 1 a = a ∘ cast h
-- Mathlib/Data/Fin/Tuple/Basic.lean
m n : ℕ
α✝ : Fin (n + 1) → Type u
x : α✝ 0
q : (i : Fin (n + 1)) → α✝ i
p : (i : Fin n) → α✝ i.succ
i : Fin n
y : α✝ i.succ
z : α✝ 0
α : Type u_1
a : Fin n → α
h : 1 * n = n
\u22a2 ∀ (x : Fin (1 * n)), repeat 1 a x = (a ∘ cast h) x
-- Mathlib/Data/Fin/Tuple/Basic.lean
m n : ℕ
α✝ : Fin (n + 1) → Type u
x : α✝ 0
q : (i : Fin (n + 1)) → α✝ i
p : (i : Fin n) → α✝ i.succ
i : Fin n
y : α✝ i.succ
z : α✝ 0
α : Type u_1
a : Fin n → α
h : 1 * n = n
\u22a2 ∀ (x : Fin n), repeat 1 a (cast ⋯ x) = (a ∘ cast h) (cast ⋯ x)
-- Mathlib/Data/Fin/Tuple/Basic.lean
m n : ℕ
α✝ : Fin (n + 1) → Type u
x : α✝ 0
q : (i : Fin (n + 1)) → α✝ i
p : (i : Fin n) → α✝ i.succ
i✝ : Fin n
y : α✝ i✝.succ
z : α✝ 0
α : Type u_1
a : Fin n → α
h : 1 * n = n
i : Fin n
\u22a2 repeat 1 a (cast ⋯ i) = (a ∘ cast h) (cast ⋯ i)
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
n : ℕ
hn : 0 < n
s : ℂ
\u22a2 ‖↑n ^ s‖ = ↑n ^ s.re
-- Mathlib/Algebra/Module/Defs.lean
α : Type u_1
R : Type u_2
k : Type u_3
S : Type u_4
M : Type u_5
M₂ : Type u_6
M₃ : Type u_7
ι : Type u_8
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
r s : R
x y : M
\u22a2 2 • x = x + x
-- Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f✝ : ℝ → E
g' g φ : ℝ → ℝ
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hcont : ContinuousOn g [[a, b]]
hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x
hpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x
\u22a2 IntervalIntegrable g' volume a b
-- Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f✝ : ℝ → E
g' g φ : ℝ → ℝ
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hcont : ContinuousOn g [[a, b]]
hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x
hpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x
hab : a ≤ b
\u22a2 IntervalIntegrable g' volume a b
-- Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f✝ : ℝ → E
g' g φ : ℝ → ℝ
a b : ℝ
inst✝ : CompleteSpace E
f f' : ℝ → E
hcont : ContinuousOn g [[a, b]]
hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x
hpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x
hab : b ≤ a
\u22a2 IntervalIntegrable g' volume a b
-- Mathlib/Data/ENNReal/Basic.lean
α : Type u_1
a b c d : ℝ≥0∞
r p q : ℝ≥0
ι : Sort u_2
f : ι → ℝ≥0
s : Set ℝ≥0
\u22a2 ↑r ∈ upperBounds (ofNNReal '' s) ↔ r ∈ upperBounds s
-- Mathlib/Data/Nat/Nth.lean
p : ℕ → Prop
inst✝ : DecidablePred p
n : ℕ
hn : ∀ (hf : (setOf p).Finite), n < hf.toFinset.card
\u22a2 count p (nth p n + 1) = n + 1
-- Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
x y : ℝ
hy : 0 < y
\u22a2 x < log y ↔ rexp x < y
-- Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
R : Type u
S : Type v
inst✝ : CommSemiring R
\u22a2 zeroLocus ∘ SetLike.coe '' minimalPrimes R = irreducibleComponents (PrimeSpectrum R)
-- Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
R : Type u
S : Type v
inst✝ : CommSemiring R
\u22a2 Set.EqOn ((zeroLocus ∘ SetLike.coe) ∘ vanishingIdeal) id (irreducibleComponents (PrimeSpectrum R))
-- Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
R : Type u
S : Type v
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : s ∈ irreducibleComponents (PrimeSpectrum R)
\u22a2 ((zeroLocus ∘ SetLike.coe) ∘ vanishingIdeal) s = id s
-- Mathlib/Data/Nat/Choose/Sum.lean
R : Type u_1
n : ℕ
\u22a2 ∑ m ∈ range (n + 1), (-1) ^ m * ↑(n.choose m) = if n = 0 then 1 else 0
-- Mathlib/Data/Seq/WSeq.lean
α : Type u
β : Type v
γ : Type w
s✝ t✝ : WSeq α
h✝ : ∀ (n : ℕ), s✝.get? n ~ t✝.get? n
s t : WSeq α
h : (fun s t => ∀ (n : ℕ), s.get? n ~ t.get? n) s t
\u22a2 Computation.LiftRel (LiftRelO (fun x x_1 => x = x_1) fun s t => ∀ (n : ℕ), s.get? n ~ t.get? n) s.destruct t.destruct
-- Mathlib/Data/Seq/WSeq.lean
α : Type u
β : Type v
γ : Type w
s✝ t✝ : WSeq α
h✝ : ∀ (n : ℕ), s✝.get? n ~ t✝.get? n
s t : WSeq α
h : (fun s t => ∀ (n : ℕ), s.get? n ~ t.get? n) s t
\u22a2 s.destruct.Terminates ↔ t.destruct.Terminates
-- Mathlib/Data/Seq/WSeq.lean
α : Type u
β : Type v
γ : Type w
s✝ t✝ : WSeq α
h✝ : ∀ (n : ℕ), s✝.get? n ~ t✝.get? n
s t : WSeq α
h : (fun s t => ∀ (n : ℕ), s.get? n ~ t.get? n) s t
\u22a2 ∀ {a b : Option (α × WSeq α)}, a ∈ s.destruct → b ∈ t.destruct → LiftRelO (fun x x_1 => x = x_1) (fun s t => ∀ (n : ℕ), s.get? n ~ t.get? n) a b
-- 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 : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R
g : R →+* S
x : 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)
n : ℕ
\u22a2 ((WittVector.map g) (f x)).coeff n = (f ((WittVector.map 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 : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R
g : R →+* S
x : 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)
n : ℕ
\u22a2 (eval₂Hom (g.comp (algebraMap ℤ R)) fun i => g (x.coeff i)) (φ n) = (aeval ((WittVector.map g) x).coeff) (φ 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 : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R
g : R →+* S
x : 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)
n : ℕ
\u22a2 (fun i => g (x.coeff i)) = ((WittVector.map g) x).coeff
-- 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 : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R
g : R →+* S
x : 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)
n x✝ : ℕ
\u22a2 g (x.coeff x✝) = ((WittVector.map g) x).coeff x✝
-- Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean
N : ℕ
x✝ : SL(2, ℤ)
\u22a2 x✝ ∈ Gamma 1 ↔ x✝ ∈ ⊤
-- Mathlib/Topology/MetricSpace/HausdorffDistance.lean
ι : Sort u_1
α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
s t u : Set α
x y✝ : α
Φ : α → β
y : α
hy : y ∈ s
\u22a2 infEdist x s ≠ ⊤
-- Mathlib/Data/Bool/AllAny.lean
α : Type u_1
p : α → Prop
inst✝ : DecidablePred p
l : List α
a : α
\u22a2 (l.all fun a => decide (p a)) = true ↔ ∀ (a : α), a ∈ l → p a
-- Mathlib/Logic/Equiv/Fin.lean
m n k : ℕ
h₁ : m ≤ k
h₂ : k < m + n
\u22a2 finAddFlip ⟨k, h₂⟩ = ⟨k - m, ⋯⟩
-- Mathlib/Logic/Equiv/Fin.lean
m n k : ℕ
h₁ : m ≤ k
h₂ : k < m + n
\u22a2 k = ↑(Fin.natAdd m ⟨k - m, ⋯⟩)
-- Mathlib/Algebra/Module/Submodule/Bilinear.lean
ι : Sort uι
R : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : AddCommMonoid P
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R P
f : M →ₗ[R] N →ₗ[R] P
q : Submodule R N
m : M
hm : m ∈ ⊥
n : N
hn : n ∈ q
\u22a2 (f m) n ∈ ⊥
-- Mathlib/Algebra/Module/Submodule/Bilinear.lean
ι : Sort uι
R : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : AddCommMonoid P
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R P
f : M →ₗ[R] N →ₗ[R] P
q : Submodule R N
m : M
hm : m = 0
n : N
hn : n ∈ q
\u22a2 (f m) n = 0
-- Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean
α : Type u
g : Ordinal.{u} → α
h_inj : InjOn g (Iio (succ #α).ord)
\u22a2 False
-- Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean
α : Type u
g : Ordinal.{u} → α
h_inj : InjOn g (Iio (succ #α).ord)
h : lift.{u, u + 1} #↑(Iio (succ #α).ord) ≤ lift.{u + 1, u} #α
\u22a2 False
-- Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean
α : Type u
g : Ordinal.{u} → α
h_inj : InjOn g (Iio (succ #α).ord)
h : lift.{u, u + 1} #↑(Iio (succ #α).ord) ≤ lift.{u + 1, u} #α
mk_initialSeg_subtype : #↑(Iio (succ #α).ord) = lift.{u + 1, u} (succ #α)
\u22a2 False
-- Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean
α : Type u
g : Ordinal.{u} → α
h_inj : InjOn g (Iio (succ #α).ord)
h : succ #α ≤ #α
mk_initialSeg_subtype : #↑(Iio (succ #α).ord) = lift.{u + 1, u} (succ #α)
\u22a2 False
-- Mathlib/Topology/LocallyClosed.lean
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
f : X → Y
\u22a2 closure s ∩ coborder s = s