ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/NumberTheory/LSeries/MellinEqDirichlet.lean
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
h_sum : Summable fun i => ‖a i‖ / |r i| ^ s.re
hs' : 0 < (s / 2).re
hF : ∀ t ∈ Ioi 0, HasSum (fun i => if r i ^ 2 = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)

⊢ Summable fun i => ‖a i‖ / (r i ^ 2) ^ (s / 2).re
-- Mathlib/LinearAlgebra/QuadraticForm/Prod.lean
ι : Type u_1
R : Type u_2
M₁ : Type u_3
M₂ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
P : Type u_7
Mᵢ : ι → Type u_8
Nᵢ : ι → Type u_9
inst✝⁷ : CommSemiring R
inst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)
inst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)
inst✝⁴ : AddCommMonoid P
inst✝³ : (i : ι) → Module R (Mᵢ i)
inst✝² : (i : ι) → Module R (Nᵢ i)
inst✝¹ : Module R P
inst✝ : Fintype ι
Q : (i : ι) → QuadraticMap R (Mᵢ i) P
Q' : (i : ι) → QuadraticMap R (Nᵢ i) P
e : (i : ι) → (Q i).IsometryEquiv (Q' i)
x : (i : ι) → Mᵢ i

⊢ (QuadraticMap.pi Q') ((↑(LinearEquiv.piCongrRight fun i => (e i).toLinearEquiv)).toFun x) = (QuadraticMap.pi Q) x
-- Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p p' : P
hp : p ∈ s
hp' : dist p' s.center ≤ s.radius
h : ¬p' = p

⊢ Wbtw ℝ p p' (s.secondInter p (p' -ᵥ p))
-- Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p p' : P
hp : p ∈ s
hp' : dist p' s.center ≤ s.radius
h : ¬p' = p

⊢ p ≠ s.secondInter p (p' -ᵥ p)
-- Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p p' : P
hp : p ∈ s
hp' : dist p' s.center ≤ s.radius
h : ¬p' = p
he : p = s.secondInter p (p' -ᵥ p)

⊢ False
-- Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p p' : P
hp : p ∈ s
hp' : dist p' s.center ≤ s.radius
h : ¬p' = p
he : 0 = ⟪p -ᵥ p', p -ᵥ s.center⟫_ℝ

⊢ False
-- Mathlib/Algebra/BigOperators/Group/Multiset.lean
F : Type u_1
ι : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
s : Multiset ℤ
n : ℤ

⊢ s.sum % n = (map (fun x => x % n) s).sum % n
-- Mathlib/Topology/Support.lean
X : Type u_1
α : Type u_2
α' : Type u_3
β : Type u_4
γ : Type u_5
δ : Type u_6
M : Type u_7
E : Type u_8
R : Type u_9
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace α'
inst✝² : One β
inst✝¹ : One γ
inst✝ : One δ
g : β → γ
f : α → β
f₂ : α → γ
m : β → γ → δ
x : α

⊢ x ∉ mulTSupport f ↔ f =ᶠ[𝓝 x] 1
-- Mathlib/Algebra/Field/Subfield.lean
K : Type u
L : Type v
M : Type w
inst✝² : DivisionRing K
inst✝¹ : DivisionRing L
inst✝ : DivisionRing M
S : Set (Subfield K)
Sne : S.Nonempty
hS : DirectedOn (fun x x_1 => x ≤ x_1) S
x : K

⊢ x ∈ ↑(sSup S) ↔ x ∈ ⋃ s ∈ S, ↑s
-- Mathlib/Data/TypeVec.lean
n : ℕ
α : TypeVec.{u_1} (n + 1)
β : TypeVec.{u_2} (n + 1)
f g : α ⟹ β
h₀ : dropFun f = dropFun g
h₁ : lastFun f = lastFun g

⊢ f = g
-- Mathlib/Data/TypeVec.lean
n : ℕ
α : TypeVec.{u_1} (n + 1)
β : TypeVec.{u_2} (n + 1)
f g : α ⟹ β
h₀ : dropFun f = dropFun g
h₁ : lastFun f = lastFun g
x : Fin2 (n + 1)

⊢ f x = g x
-- Mathlib/NumberTheory/Cyclotomic/Basic.lean
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
h : ⊥ = ⊤

⊢ IsCyclotomicExtension {1} A B
-- Mathlib/NumberTheory/Cyclotomic/Basic.lean
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
h : ⊥ = ⊤

⊢ {1} = ∅ ∪ {1}
-- Mathlib/Computability/TuringMachine.lean
K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
M : Λ → Stmt₂
q : Stmt₂
v : σ
T : ListBlank ((i : K) → Option (Γ i))
k : K
S : (k : K) → List (Γ k)
hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse
o : StAct k
IH : ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))}, (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) → ∃ b, TrCfg (TM2.stepAux q v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b

⊢ ∃ b, TrCfg (TM2.stepAux (stRun o q) v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal (stRun o q)) v (Tape.mk' ∅ (addBottom T))) b
-- Mathlib/Computability/TuringMachine.lean
K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
M : Λ → Stmt₂
q : Stmt₂
v : σ
T : ListBlank ((i : K) → Option (Γ i))
k : K
S : (k : K) → List (Γ k)
hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse
o : StAct k
IH : ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))}, (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) → ∃ b, TrCfg (TM2.stepAux q v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b

⊢ ∃ b, TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b
-- Mathlib/Computability/TuringMachine.lean
K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
M : Λ → Stmt₂
q : Stmt₂
v : σ
T : ListBlank ((i : K) → Option (Γ i))
k : K
S : (k : K) → List (Γ k)
hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse
o : StAct k
IH : ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))}, (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) → ∃ b, TrCfg (TM2.stepAux q v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b
hgo : Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }

⊢ ∃ b, TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b
-- Mathlib/Computability/TuringMachine.lean
K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
M : Λ → Stmt₂
q : Stmt₂
v : σ
T : ListBlank ((i : K) → Option (Γ i))
k : K
S : (k : K) → List (Γ k)
hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse
o : StAct k
IH : ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))}, (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) → ∃ b, TrCfg (TM2.stepAux q v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b
hgo : Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }
T' : ListBlank ((k : K) → Option (Γ k))
hT' : ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite ?m.379536 (S k) o) k_1)).reverse
hrun : TM1.stepAux (trStAct ?m.379535 o) ?m.379536 ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) = TM1.stepAux ?m.379535 (stVar ?m.379536 (S k) o) ((Tape.move Dir.right)^[(update S k (stWrite ?m.379536 (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))

⊢ ∃ b, TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b
-- Mathlib/Computability/TuringMachine.lean
K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
M : Λ → Stmt₂
q : Stmt₂
v : σ
T : ListBlank ((i : K) → Option (Γ i))
k : K
S : (k : K) → List (Γ k)
hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse
o : StAct k
IH : ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))}, (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) → ∃ b, TrCfg (TM2.stepAux q v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b
hgo : Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }
T' : ListBlank ((k : K) → Option (Γ k))
hT' : ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite ?m.379536 (S k) o) k_1)).reverse
hrun : TM1.stepAux (trStAct ?m.379535 o) ?m.379536 ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) = TM1.stepAux ?m.379535 (stVar ?m.379536 (S k) o) ((Tape.move Dir.right)^[(update S k (stWrite ?m.379536 (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))
this : Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } (TM1.stepAux (tr M (go k o q)) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))))

⊢ ∃ b, TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b
-- Mathlib/Computability/TuringMachine.lean
K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
M : Λ → Stmt₂
q : Stmt₂
v : σ
T : ListBlank ((i : K) → Option (Γ i))
k : K
S : (k : K) → List (Γ k)
hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse
o : StAct k
IH : ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))}, (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) → ∃ b, TrCfg (TM2.stepAux q v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b
hgo : Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }
T' : ListBlank ((k : K) → Option (Γ k))
hT' : ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite v (S k) o) k_1)).reverse
hrun : TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) = TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o) ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))
this : Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } (match match none with | some val => false | none => true with | true => TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o) ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T'))) | false => TM1.stepAux (goto fun x x => go k o q) v (Tape.move Dir.right ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)))))

⊢ ∃ b, TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b
-- Mathlib/Computability/TuringMachine.lean
K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
M : Λ → Stmt₂
q : Stmt₂
v : σ
T : ListBlank ((i : K) → Option (Γ i))
k : K
S : (k : K) → List (Γ k)
hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse
o : StAct k
IH : ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))}, (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) → ∃ b, TrCfg (TM2.stepAux q v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b
hgo : Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }
T' : ListBlank ((k : K) → Option (Γ k))
hT' : ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite v (S k) o) k_1)).reverse
hrun : TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) = TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o) ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))
this : Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } (match match none with | some val => false | none => true with | true => TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o) ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T'))) | false => TM1.stepAux (goto fun x x => go k o q) v (Tape.move Dir.right ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)))))
c : TM1.Cfg Γ' Λ' σ
gc : TrCfg (TM2.stepAux q ?m.383145 (update S k (stWrite v (S k) o))) c
rc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) ?m.383145 (Tape.mk' ∅ (addBottom T'))) c

⊢ ∃ b, TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x => go k o q) v (Tape.mk' ∅ (addBottom T))) b
-- Mathlib/Computability/TuringMachine.lean
K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
M : Λ → Stmt₂
q : Stmt₂
v : σ
T : ListBlank ((i : K) → Option (Γ i))
k : K
S : (k : K) → List (Γ k)
hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse
o : StAct k
IH : ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))}, (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) → ∃ b, TrCfg (TM2.stepAux q v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b
hgo : Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }
T' : ListBlank ((k : K) → Option (Γ k))
hT' : ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite v (S k) o) k_1)).reverse
hrun : TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) = TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o) ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))
this : Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } (match match none with | some val => false | none => true with | true => TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o) ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T'))) | false => TM1.stepAux (goto fun x x => go k o q) v (Tape.move Dir.right ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)))))
c : TM1.Cfg Γ' Λ' σ
gc : TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) c
rc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T'))) c

⊢ ReflTransGen (fun a b => b ∈ TM1.step (tr M) a) (TM1.stepAux (tr M (ret q)) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T'))) c
-- Mathlib/Computability/TuringMachine.lean
K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
M : Λ → Stmt₂
q : Stmt₂
v : σ
T : ListBlank ((i : K) → Option (Γ i))
k : K
S : (k : K) → List (Γ k)
hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse
o : StAct k
IH : ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))}, (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) → ∃ b, TrCfg (TM2.stepAux q v S) b ∧ Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b
hgo : Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }
T' : ListBlank ((k : K) → Option (Γ k))
hT' : ∀ (k_1 : K), ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite v (S k) o) k_1)).reverse
hrun : TM1.stepAux (trStAct (goto fun x x => ret q) o) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) = TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o) ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))
this : Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) } (match match none with | some val => false | none => true with | true => TM1.stepAux (goto fun x x => ret q) (stVar v (S k) o) ((Tape.move Dir.right)^[(update S k (stWrite v (S k) o) k).length] (Tape.mk' ∅ (addBottom T'))) | false => TM1.stepAux (goto fun x x => go k o q) v (Tape.move Dir.right ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)))))
c : TM1.Cfg Γ' Λ' σ
gc : TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) c
rc : Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T'))) c

⊢ ReflTransGen (fun a b => b ∈ TM1.step (tr M) a) (bif true then TM1.stepAux (trNormal q) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T')) else TM1.stepAux (move Dir.left (goto fun x x => ret q)) (stVar v (S k) o) (Tape.mk' ∅ (addBottom T'))) c
-- Mathlib/InformationTheory/Hamming.lean
α : Type u_1
ι : Type u_2
β : ι → Type u_3
inst✝² : Fintype ι
inst✝¹ : (i : ι) → DecidableEq (β i)
γ : ι → Type u_4
inst✝ : (i : ι) → DecidableEq (γ i)
x y z : (i : ι) → β i

⊢ hammingDist x y ≤ hammingDist x z + hammingDist y z
-- Mathlib/InformationTheory/Hamming.lean
α : Type u_1
ι : Type u_2
β : ι → Type u_3
inst✝² : Fintype ι
inst✝¹ : (i : ι) → DecidableEq (β i)
γ : ι → Type u_4
inst✝ : (i : ι) → DecidableEq (γ i)
x y z : (i : ι) → β i

⊢ hammingDist x y ≤ hammingDist x z + hammingDist z y
-- Mathlib/LinearAlgebra/Eigenspace/Basic.lean
K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End R M
μ : R
v : M
hv : f.HasEigenvector μ v
n : ℕ

⊢ (f ^ n) v = μ ^ n • v
-- Mathlib/Algebra/Group/Units.lean
α : Type u
M : Type u_1
N : Type u_2
inst✝ : Monoid M
a✝ b c a : M
h : IsUnit a
y : M

⊢ (fun x => x * a) (y * ↑h.unit⁻¹) = y
-- Mathlib/NumberTheory/PythagoreanTriples.lean
x y z : ℤ
h : PythagoreanTriple x y z
h0 : ¬x.gcd y = 0

⊢ ↑(x.gcd y) ∣ z
-- Mathlib/NumberTheory/PythagoreanTriples.lean
z : ℤ
k : ℕ
x0 y0 : ℤ
left✝ : 0 < k
h2 : x0.gcd y0 = 1
h : PythagoreanTriple (x0 * ↑k) (y0 * ↑k) z
h0 : ¬(x0 * ↑k).gcd (y0 * ↑k) = 0

⊢ ↑((x0 * ↑k).gcd (y0 * ↑k)) ∣ z
-- Mathlib/NumberTheory/PythagoreanTriples.lean
z : ℤ
k : ℕ
x0 y0 : ℤ
left✝ : 0 < k
h2 : x0.gcd y0 = 1
h : PythagoreanTriple (x0 * ↑k) (y0 * ↑k) z
h0 : ¬(x0 * ↑k).gcd (y0 * ↑k) = 0

⊢ ↑k ∣ z
-- Mathlib/NumberTheory/PythagoreanTriples.lean
z : ℤ
k : ℕ
x0 y0 : ℤ
left✝ : 0 < k
h2 : x0.gcd y0 = 1
h : PythagoreanTriple (x0 * ↑k) (y0 * ↑k) z
h0 : ¬(x0 * ↑k).gcd (y0 * ↑k) = 0

⊢ ↑k ^ 2 ∣ x0 * ↑k * (x0 * ↑k) + y0 * ↑k * (y0 * ↑k)
-- Mathlib/NumberTheory/PythagoreanTriples.lean
z : ℤ
k : ℕ
x0 y0 : ℤ
left✝ : 0 < k
h2 : x0.gcd y0 = 1
h : PythagoreanTriple (x0 * ↑k) (y0 * ↑k) z
h0 : ¬(x0 * ↑k).gcd (y0 * ↑k) = 0

⊢ ↑k ^ 2 ∣ ↑k ^ 2 * (x0 * x0 + y0 * y0)
-- Mathlib/Logic/Equiv/Basic.lean
α : Sort u_2
inst✝ : DecidableEq α
β : Sort u_1
f : α ≃ β
a : α
b : β

⊢ (f.setValue a b) a = b
-- Mathlib/LinearAlgebra/AffineSpace/Combination.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
ι : Type u_4
s✝ : Finset ι
ι₂ : Type u_5
s₂ : Finset ι₂
s : Finset ι
p : ι → V
w : ι → k
hw : ∑ i ∈ s, w i = 1

⊢ (affineCombination k s p) w = ∑ i ∈ s, w i • p i
-- Mathlib/SetTheory/Cardinal/Basic.lean
α✝ β : Type u
c : Cardinal.{?u.203972}
α : Type u
ι : Type v
A : ι → Set α
s : Set ι

⊢ lift.{v, u} #↑(⋃ x ∈ s, A x) ≤ lift.{u, v} #↑s * ⨆ x, lift.{v, u} #↑(A ↑x)
-- Mathlib/SetTheory/Cardinal/Basic.lean
α✝ β : Type u
c : Cardinal.{?u.203972}
α : Type u
ι : Type v
A : ι → Set α
s : Set ι

⊢ lift.{v, u} #↑(⋃ x, A ↑x) ≤ lift.{u, v} #↑s * ⨆ x, lift.{v, u} #↑(A ↑x)
-- Mathlib/Analysis/Calculus/Deriv/Basic.lean
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f✝ f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x✝ : 𝕜
s t : Set 𝕜
L L₁ L₂ : Filter 𝕜
f : 𝕜 → 𝕜
x y : 𝕜

⊢ (fderiv 𝕜 f x) y = deriv f x * y
-- Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean
R : Type u
S : Type v
a b : R
n✝ m : ℕ
inst✝ : Semiring R
p q r : R[X]
n : ℕ

⊢ ↑n ≤ (X ^ n).trailingDegree
-- Mathlib/Algebra/Lie/OfAssociative.lean
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x y : L
z : M
n : ℕ

⊢ (φ x ^ n) ⁅y, z⁆ = ∑ ij ∈ antidiagonal n, n.choose ij.1 • ⁅((ad R L) x ^ ij.1) y, (φ x ^ ij.2) z⁆
-- Mathlib/Data/Set/Image.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
p : Set β → Prop

⊢ (∃ s ⊆ range f, p s) ↔ ∃ s, p (f '' s)
-- Mathlib/Data/Set/Image.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
f✝ : ι → α
s t : Set α
f : α → β
p : Set β → Prop

⊢ (∃ s ⊆ range f, p s) ↔ ∃ a ∈ 𝒫 range f, p a
-- Mathlib/Algebra/Lie/Classical.lean
n : Type u_1
p : Type u_2
q : Type u_3
l : Type u_4
R : Type u₂
inst✝⁵ : DecidableEq n
inst✝⁴ : DecidableEq p
inst✝³ : DecidableEq q
inst✝² : DecidableEq l
inst✝¹ : CommRing R
inst✝ : Fintype n
A : Matrix n n R

⊢ A ∈ so n R ↔ Aᵀ = -A
-- Mathlib/Algebra/Lie/Classical.lean
n : Type u_1
p : Type u_2
q : Type u_3
l : Type u_4
R : Type u₂
inst✝⁵ : DecidableEq n
inst✝⁴ : DecidableEq p
inst✝³ : DecidableEq q
inst✝² : DecidableEq l
inst✝¹ : CommRing R
inst✝ : Fintype n
A : Matrix n n R

⊢ IsSkewAdjoint 1 A ↔ Aᵀ = -A
-- Mathlib/CategoryTheory/Functor/KanExtension/Pointwise.lean
C : Type u_1
D : Type u_2
H : Type u_3
inst✝² : Category.{?u.51683, u_1} C
inst✝¹ : Category.{?u.51687, u_2} D
inst✝ : Category.{?u.51691, u_3} H
L : C ⥤ D
F : C ⥤ H
E✝ E'✝ : L.RightExtension F
Y : D
E E' : L.RightExtension F
φ : E ⟶ E'
G : StructuredArrow Y L

⊢ φ.left.app Y ≫ E'.left.map G.hom ≫ E'.hom.app G.right = E.left.map G.hom ≫ E.hom.app G.right
-- Mathlib/CategoryTheory/Functor/KanExtension/Pointwise.lean
C : Type u_1
D : Type u_2
H : Type u_3
inst✝² : Category.{?u.51683, u_1} C
inst✝¹ : Category.{?u.51687, u_2} D
inst✝ : Category.{?u.51691, u_3} H
L : C ⥤ D
F : C ⥤ H
E✝ E'✝ : L.RightExtension F
Y : D
E E' : L.RightExtension F
φ : E ⟶ E'
G : StructuredArrow Y L

⊢ φ.left.app Y ≫ E'.left.map G.hom ≫ E'.hom.app G.right = E.left.map G.hom ≫ (((whiskeringLeft C D H).obj L).map φ.left ≫ E'.hom).app G.right
-- Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
P X Y Z : C
fst : P ⟶ X
snd : P ⟶ Y
f : X ⟶ Z
g : Y ⟶ Z
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
b : BinaryBicone X Y
h : b.IsBilimit

⊢ IsPullback b.inr 0 b.fst 0
-- Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
P X Y Z : C
fst : P ⟶ X
snd : P ⟶ Y
f : X ⟶ Z
g : Y ⟶ Z
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
b : BinaryBicone X Y
h : b.IsBilimit

⊢ IsPullback 0 b.inr 0 b.fst