tasksource/leandojo huggingface dataset:
Theorem: Polynomial.coeff_mul_X_zero
File: Mathlib/Data/Polynomial/Coeff.lean

tactic: simp
state_before: R : Type u
S : Type v
a b : R
n m : ℕ
inst✝ : Semiring R
p✝ q r p : R[X]
⊢ coeff (p * X) 0 = 0
Theorem: PosNum.one_sub'
File: Mathlib/Data/Num/Lemmas.lean

tactic: cases a <;> rfl
state_before: a : PosNum
⊢ sub' 1 a = toZNumNeg (pred' a)
Theorem: inv_pow_lt_inv_pow_of_lt
File: Mathlib/Algebra/Order/Field/Basic.lean

tactic: convert one_div_pow_lt_one_div_pow_of_lt a1 mn using 1 <;> simp
state_before: ι : Type ?u.128272
α : Type u_1
β : Type ?u.128278
inst✝ : LinearOrderedSemifield α
a b c d e : α
m✝ n✝ : ℤ
a1 : 1 < a
m n : ℕ
mn : m < n
⊢ (a ^ n)⁻¹ < (a ^ m)⁻¹
Theorem: mem_nhdsWithin_iff_eventuallyEq
File: Mathlib/Topology/ContinuousOn.lean

tactic: simp_rw [mem_nhdsWithin_iff_eventually, eventuallyEq_set, mem_inter_iff, iff_self_and]
state_before: α : Type u_1
β : Type ?u.6991
γ : Type ?u.6994
δ : Type ?u.6997
inst✝ : TopologicalSpace α
s t : Set α
x : α
⊢ t ∈ 𝓝[s] x ↔ s =ᶠ[𝓝 x] s ∩ t
Theorem: Int.bodd_mul
File: Mathlib/Data/Int/Bitwise.lean

tactic: cases' m with m m <;> cases' n with n n <;>
simp only [ofNat_eq_coe, ofNat_mul_negSucc, negSucc_mul_ofNat, ofNat_mul_ofNat,
           negSucc_mul_negSucc] <;>
simp only [negSucc_coe, bodd_neg, bodd_coe, ←Nat.bodd_mul]
state_before: m n : ℤ
⊢ bodd (m * n) = (bodd m && bodd n)
Theorem: Infinite.exists_subset_card_eq
File: Mathlib/Data/Fintype/Card.lean

tactic: rw [card_map, card_range]
state_before: α✝ : Type ?u.80738
β : Type ?u.80741
γ : Type ?u.80744
α : Type u_1
inst✝ : Infinite α
n : ℕ
⊢ card (map (natEmbedding α) (range n)) = n
Theorem: MeasureTheory.VectorMeasure.restrict_le_restrict_union
File: Mathlib/MeasureTheory/Measure/VectorMeasure.lean

tactic: rw [Set.union_eq_iUnion]
state_before: α : Type u_1
β : Type ?u.578132
m : MeasurableSpace α
M : Type u_2
inst✝² : TopologicalSpace M
inst✝¹ : OrderedAddCommMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
i j : Set α
hi₁ : MeasurableSet i
hi₂ : restrict v i ≤ restrict w i
hj₁ : MeasurableSet j
hj₂ : restrict v j ≤ restrict w j
⊢ restrict v (i ∪ j) ≤ restrict w (i ∪ j)

tactic: refine' restrict_le_restrict_countable_iUnion v w _ _
state_before: α : Type u_1
β : Type ?u.578132
m : MeasurableSpace α
M : Type u_2
inst✝² : TopologicalSpace M
inst✝¹ : OrderedAddCommMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
i j : Set α
hi₁ : MeasurableSet i
hi₂ : restrict v i ≤ restrict w i
hj₁ : MeasurableSet j
hj₂ : restrict v j ≤ restrict w j
⊢ restrict v (⋃ (b : Bool), bif b then i else j) ≤ restrict w (⋃ (b : Bool), bif b then i else j)

tactic: measurability
state_before: case refine'_1
α : Type u_1
β : Type ?u.578132
m : MeasurableSpace α
M : Type u_2
inst✝² : TopologicalSpace M
inst✝¹ : OrderedAddCommMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
i j : Set α
hi₁ : MeasurableSet i
hi₂ : restrict v i ≤ restrict w i
hj₁ : MeasurableSet j
hj₂ : restrict v j ≤ restrict w j
⊢ ∀ (b : Bool), MeasurableSet (bif b then i else j)

tactic: rintro (_ | _) <;> simpa
state_before: case refine'_2
α : Type u_1
β : Type ?u.578132
m : MeasurableSpace α
M : Type u_2
inst✝² : TopologicalSpace M
inst✝¹ : OrderedAddCommMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
i j : Set α
hi₁ : MeasurableSet i
hi₂ : restrict v i ≤ restrict w i
hj₁ : MeasurableSet j
hj₂ : restrict v j ≤ restrict w j
⊢ ∀ (b : Bool), restrict v (bif b then i else j) ≤ restrict w (bif b then i else j)
Theorem: SimplexCategory.hom_zero_zero
File: Mathlib/AlgebraicTopology/SimplexCategory.lean

tactic: ext : 3
state_before: f : [0] ⟶ [0]
⊢ f = 𝟙 [0]

tactic: apply @Subsingleton.elim (Fin 1)
state_before: case a.h.h
f : [0] ⟶ [0]
x✝ : Fin (len [0] + 1)
⊢ ↑(Hom.toOrderHom f) x✝ = ↑(Hom.toOrderHom (𝟙 [0])) x✝
Theorem: Computable.pair
File: Mathlib/Computability/Partrec.lean

tactic: cases decode (α := α) n <;> simp [Seq.seq]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
σ : Type ?u.40884
inst✝³ : Primcodable α
inst✝² : Primcodable β
inst✝¹ : Primcodable γ
inst✝ : Primcodable σ
f : α → β
g : α → γ
hf : Computable f
hg : Computable g
n : ℕ
⊢ (Seq.seq (Nat.pair <$> Part.bind ↑(decode n) fun a => map encode (↑f a)) fun x =>
      Part.bind ↑(decode n) fun a => map encode (↑g a)) =
    Part.bind ↑(decode n) fun a => map encode ((↑fun a => (f a, g a)) a)
Theorem: Con.sInf_toSetoid
File: Mathlib/GroupTheory/Congruence.lean

tactic: rw [← hr]
state_before: M : Type u_1
N : Type ?u.19770
P : Type ?u.19773
inst✝² : Mul M
inst✝¹ : Mul N
inst✝ : Mul P
c✝ : Con M
S : Set (Con M)
x y : M
h : Rel (sInf S).toSetoid x y
r : Setoid M
x✝ : r ∈ toSetoid '' S
c : Con M
hS : c ∈ S
hr : c.toSetoid = r
⊢ Rel r x y

tactic: exact h c hS
state_before: M : Type u_1
N : Type ?u.19770
P : Type ?u.19773
inst✝² : Mul M
inst✝¹ : Mul N
inst✝ : Mul P
c✝ : Con M
S : Set (Con M)
x y : M
h : Rel (sInf S).toSetoid x y
r : Setoid M
x✝ : r ∈ toSetoid '' S
c : Con M
hS : c ∈ S
hr : c.toSetoid = r
⊢ Rel c.toSetoid x y
Theorem: Set.pow_subset_pow_of_one_mem
File: Mathlib/Data/Set/Pointwise/Basic.lean

tactic: induction' n, hn using Nat.le_induction with _ _ ih
state_before: F : Type ?u.59752
α : Type u_1
β : Type ?u.59758
γ : Type ?u.59761
inst✝ : Monoid α
s t : Set α
a : α
m n : ℕ
hs : 1 ∈ s
hn : m ≤ n
⊢ s ^ m ⊆ s ^ n

tactic: exact Subset.rfl
state_before: case base
F : Type ?u.59752
α : Type u_1
β : Type ?u.59758
γ : Type ?u.59761
inst✝ : Monoid α
s t : Set α
a : α
m n : ℕ
hs : 1 ∈ s
⊢ s ^ m ⊆ s ^ m

tactic: dsimp only
state_before: case succ
F : Type ?u.59752
α : Type u_1
β : Type ?u.59758
γ : Type ?u.59761
inst✝ : Monoid α
s t : Set α
a : α
m n : ℕ
hs : 1 ∈ s
n✝ : ℕ
hn✝ : m ≤ n✝
ih : s ^ m ⊆ s ^ n✝
⊢ s ^ m ⊆ s ^ (n✝ + 1)

tactic: rw [pow_succ]
state_before: case succ
F : Type ?u.59752
α : Type u_1
β : Type ?u.59758
γ : Type ?u.59761
inst✝ : Monoid α
s t : Set α
a : α
m n : ℕ
hs : 1 ∈ s
n✝ : ℕ
hn✝ : m ≤ n✝
ih : s ^ m ⊆ s ^ n✝
⊢ s ^ m ⊆ s ^ (n✝ + 1)

tactic: exact ih.trans (subset_mul_right _ hs)
state_before: case succ
F : Type ?u.59752
α : Type u_1
β : Type ?u.59758
γ : Type ?u.59761
inst✝ : Monoid α
s t : Set α
a : α
m n : ℕ
hs : 1 ∈ s
n✝ : ℕ
hn✝ : m ≤ n✝
ih : s ^ m ⊆ s ^ n✝
⊢ s ^ m ⊆ s * s ^ n✝
Theorem: BoxIntegral.integralSum_add
File: Mathlib/Analysis/BoxIntegral/Basic.lean

tactic: simp only [integralSum, Pi.add_apply, (vol _).map_add, Finset.sum_add_distrib]
state_before: ι : Type u
E : Type v
F : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
I J : Box ι
π✝ : TaggedPrepartition I
f g : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
π : TaggedPrepartition I
⊢ integralSum (f + g) vol π = integralSum f vol π + integralSum g vol π
Theorem: Set.mulIndicator_const_preimage
File: Mathlib/Algebra/IndicatorFunction.lean

tactic: classical
  rw [mulIndicator_const_preimage_eq_union]
  split_ifs <;> simp
state_before: α : Type u_1
β : Type ?u.29472
ι : Type ?u.29475
M : Type u_2
N : Type ?u.29481
inst✝¹ : One M
inst✝ : One N
s✝ t : Set α
f g : α → M
a✝ : α
U : Set α
s : Set M
a : M
⊢ (mulIndicator U fun x => a) ⁻¹' s ∈ {univ, U, Uᶜ, ∅}

tactic: rw [mulIndicator_const_preimage_eq_union]
state_before: α : Type u_1
β : Type ?u.29472
ι : Type ?u.29475
M : Type u_2
N : Type ?u.29481
inst✝¹ : One M
inst✝ : One N
s✝ t : Set α
f g : α → M
a✝ : α
U : Set α
s : Set M
a : M
⊢ (mulIndicator U fun x => a) ⁻¹' s ∈ {univ, U, Uᶜ, ∅}

tactic: split_ifs <;> simp
state_before: α : Type u_1
β : Type ?u.29472
ι : Type ?u.29475
M : Type u_2
N : Type ?u.29481
inst✝¹ : One M
inst✝ : One N
s✝ t : Set α
f g : α → M
a✝ : α
U : Set α
s : Set M
a : M
⊢ ((if a ∈ s then U else ∅) ∪ if 1 ∈ s then Uᶜ else ∅) ∈ {univ, U, Uᶜ, ∅}
Theorem: algebraMap_monotone
File: Mathlib/Algebra/Order/Algebra.lean

tactic: rw [Algebra.algebraMap_eq_smul_one, Algebra.algebraMap_eq_smul_one, ← sub_nonneg, ← sub_smul]
state_before: R : Type u_1
A : Type u_2
a✝ b✝ : A
r : R
inst✝³ : OrderedCommRing R
inst✝² : OrderedRing A
inst✝¹ : Algebra R A
inst✝ : OrderedSMul R A
a b : R
h : a ≤ b
⊢ ↑(algebraMap R A) a ≤ ↑(algebraMap R A) b

tactic: trans (b - a) • (0 : A)
state_before: R : Type u_1
A : Type u_2
a✝ b✝ : A
r : R
inst✝³ : OrderedCommRing R
inst✝² : OrderedRing A
inst✝¹ : Algebra R A
inst✝ : OrderedSMul R A
a b : R
h : a ≤ b
⊢ 0 ≤ (b - a) • 1

tactic: simp
state_before: R : Type u_1
A : Type u_2
a✝ b✝ : A
r : R
inst✝³ : OrderedCommRing R
inst✝² : OrderedRing A
inst✝¹ : Algebra R A
inst✝ : OrderedSMul R A
a b : R
h : a ≤ b
⊢ 0 ≤ (b - a) • 0

tactic: exact smul_le_smul_of_nonneg zero_le_one (sub_nonneg.mpr h)
state_before: R : Type u_1
A : Type u_2
a✝ b✝ : A
r : R
inst✝³ : OrderedCommRing R
inst✝² : OrderedRing A
inst✝¹ : Algebra R A
inst✝ : OrderedSMul R A
a b : R
h : a ≤ b
⊢ (b - a) • 0 ≤ (b - a) • 1
Theorem: FiberPrebundle.isOpen_target_of_mem_pretrivializationAtlas_inter
File: Mathlib/Topology/FiberBundle/Basic.lean

tactic: letI := a.totalSpaceTopology
state_before: ι : Type ?u.58328
B : Type u_1
F : Type u_2
X : Type ?u.58337
inst✝³ : TopologicalSpace X
E : B → Type u_3
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
inst✝ : (x : B) → TopologicalSpace (E x)
a : FiberPrebundle F E
e✝ e e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.pretrivializationAtlas
⊢ IsOpen (e'.target ∩ ↑(LocalEquiv.symm e'.toLocalEquiv) ⁻¹' e.source)

tactic: obtain ⟨u, hu1, hu2⟩ := continuousOn_iff'.mp (a.continuous_symm_of_mem_pretrivializationAtlas he')
  e.source (a.isOpen_source e)
state_before: ι : Type ?u.58328
B : Type u_1
F : Type u_2
X : Type ?u.58337
inst✝³ : TopologicalSpace X
E : B → Type u_3
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
inst✝ : (x : B) → TopologicalSpace (E x)
a : FiberPrebundle F E
e✝ e e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.pretrivializationAtlas
this : TopologicalSpace (TotalSpace E) := totalSpaceTopology a
⊢ IsOpen (e'.target ∩ ↑(LocalEquiv.symm e'.toLocalEquiv) ⁻¹' e.source)

tactic: rw [inter_comm, hu2]
state_before: case intro.intro
ι : Type ?u.58328
B : Type u_1
F : Type u_2
X : Type ?u.58337
inst✝³ : TopologicalSpace X
E : B → Type u_3
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
inst✝ : (x : B) → TopologicalSpace (E x)
a : FiberPrebundle F E
e✝ e e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.pretrivializationAtlas
this : TopologicalSpace (TotalSpace E) := totalSpaceTopology a
u : Set (B × F)
hu1 : IsOpen u
hu2 : ↑(LocalEquiv.symm e'.toLocalEquiv) ⁻¹' e.source ∩ e'.target = u ∩ e'.target
⊢ IsOpen (e'.target ∩ ↑(LocalEquiv.symm e'.toLocalEquiv) ⁻¹' e.source)

tactic: exact hu1.inter e'.open_target
state_before: case intro.intro
ι : Type ?u.58328
B : Type u_1
F : Type u_2
X : Type ?u.58337
inst✝³ : TopologicalSpace X
E : B → Type u_3
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
inst✝ : (x : B) → TopologicalSpace (E x)
a : FiberPrebundle F E
e✝ e e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.pretrivializationAtlas
this : TopologicalSpace (TotalSpace E) := totalSpaceTopology a
u : Set (B × F)
hu1 : IsOpen u
hu2 : ↑(LocalEquiv.symm e'.toLocalEquiv) ⁻¹' e.source ∩ e'.target = u ∩ e'.target
⊢ IsOpen (u ∩ e'.target)
Theorem: tsum_of_nnnorm_bounded
File: Mathlib/Analysis/Normed/Group/InfiniteSum.lean

tactic: simp only [← NNReal.coe_le_coe, ← NNReal.hasSum_coe, coe_nnnorm] at *
state_before: ι : Type u_1
α : Type ?u.20330
E : Type u_2
F : Type ?u.20336
inst✝¹ : SeminormedAddCommGroup E
inst✝ : SeminormedAddCommGroup F
f : ι → E
g : ι → ℝ≥0
a : ℝ≥0
hg : HasSum g a
h : ∀ (i : ι), ‖f i‖₊ ≤ g i
⊢ ‖∑' (i : ι), f i‖₊ ≤ a

tactic: exact tsum_of_norm_bounded hg h
state_before: ι : Type u_1
α : Type ?u.20330
E : Type u_2
F : Type ?u.20336
inst✝¹ : SeminormedAddCommGroup E
inst✝ : SeminormedAddCommGroup F
f : ι → E
g : ι → ℝ≥0
a : ℝ≥0
hg : HasSum (fun a => ↑(g a)) ↑a
h : ∀ (i : ι), ‖f i‖ ≤ ↑(g i)
⊢ ‖∑' (i : ι), f i‖ ≤ ↑a
Theorem: Std.RBNode.find?_eq_zoom
File: Std/Data/RBMap/Alter.lean

tactic: unfold find? zoom
state_before: α : Type u_1
cut : α → Ordering
c✝ : RBColor
l✝ : RBNode α
v✝ : α
r✝ : RBNode α
x✝ : optParam (Path α) Path.root
⊢ find? cut (node c✝ l✝ v✝ r✝) = root? (zoom cut (node c✝ l✝ v✝ r✝) x✝).fst

tactic: split <;> [apply find?_eq_zoom; apply find?_eq_zoom; rfl]
state_before: α : Type u_1
cut : α → Ordering
c✝ : RBColor
l✝ : RBNode α
v✝ : α
r✝ : RBNode α
x✝ : optParam (Path α) Path.root
⊢ (match cut v✝ with
    | Ordering.lt => find? cut l✝
    | Ordering.gt => find? cut r✝
    | Ordering.eq => some v✝) =
    root?
      (match cut v✝ with
        | Ordering.lt => zoom cut l✝ (Path.left c✝ x✝ v✝ r✝)
        | Ordering.gt => zoom cut r✝ (Path.right c✝ l✝ v✝ x✝)
        | Ordering.eq => (node c✝ l✝ v✝ r✝, x✝)).fst
Theorem: List.destutter'_eq_self_iff
File: Mathlib/Data/List/Destutter.lean

tactic: suffices Chain' R (a::l) by
  assumption
state_before: α : Type u_1
l : List α
R : α → α → Prop
inst✝ : DecidableRel R
a✝ b a : α
h : destutter' R a l = a :: l
⊢ Chain R a l

tactic: rw [← h]
state_before: α : Type u_1
l : List α
R : α → α → Prop
inst✝ : DecidableRel R
a✝ b a : α
h : destutter' R a l = a :: l
⊢ Chain' R (a :: l)

tactic: exact l.destutter'_is_chain' R a
state_before: α : Type u_1
l : List α
R : α → α → Prop
inst✝ : DecidableRel R
a✝ b a : α
h : destutter' R a l = a :: l
⊢ Chain' R (destutter' R a l)

tactic: assumption
state_before: α : Type u_1
l : List α
R : α → α → Prop
inst✝ : DecidableRel R
a✝ b a : α
h : destutter' R a l = a :: l
this : Chain' R (a :: l)
⊢ Chain R a l
Theorem: Polynomial.divByMonic_one
File: Mathlib/Data/Polynomial/Div.lean

tactic: conv_rhs => rw [← modByMonic_add_div p monic_one]; simp
state_before: R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : CommRing R
p✝ q p : R[X]
⊢ p /ₘ 1 = p
Theorem: Subgroup.relindex_self
File: Mathlib/GroupTheory/Index.lean

tactic: rw [relindex, subgroupOf_self, index_top]
state_before: G : Type u_1
inst✝ : Group G
H K L : Subgroup G
⊢ relindex H H = 1
Theorem: MultilinearMap.uncurry_curryLeft
File: Mathlib/LinearAlgebra/Multilinear/Basic.lean

tactic: ext m
state_before: R : Type u
ι : Type u'
n : ℕ
M : Fin (Nat.succ n) → Type v
M₁ : ι → Type v₁
M₂ : Type v₂
M₃ : Type v₃
M' : Type v'
inst✝⁶ : CommSemiring R
inst✝⁵ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)
inst✝⁴ : AddCommMonoid M'
inst✝³ : AddCommMonoid M₂
inst✝² : (i : Fin (Nat.succ n)) → Module R (M i)
inst✝¹ : Module R M'
inst✝ : Module R M₂
f : MultilinearMap R M M₂
⊢ LinearMap.uncurryLeft (curryLeft f) = f

tactic: simp
state_before: case H
R : Type u
ι : Type u'
n : ℕ
M : Fin (Nat.succ n) → Type v
M₁ : ι → Type v₁
M₂ : Type v₂
M₃ : Type v₃
M' : Type v'
inst✝⁶ : CommSemiring R
inst✝⁵ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)
inst✝⁴ : AddCommMonoid M'
inst✝³ : AddCommMonoid M₂
inst✝² : (i : Fin (Nat.succ n)) → Module R (M i)
inst✝¹ : Module R M'
inst✝ : Module R M₂
f : MultilinearMap R M M₂
m : (i : Fin (Nat.succ n)) → M i
⊢ ↑(LinearMap.uncurryLeft (curryLeft f)) m = ↑f m
Theorem: MeasureTheory.FiniteMeasure.continuous_testAgainstNN_eval
File: Mathlib/MeasureTheory/Measure/FiniteMeasure.lean

tactic: show Continuous ((fun φ : WeakDual ℝ≥0 (Ω →ᵇ ℝ≥0) => φ f) ∘ toWeakDualBCNN)
state_before: Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type ?u.105418
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
f : Ω →ᵇ ℝ≥0
⊢ Continuous fun μ => testAgainstNN μ f

tactic: refine Continuous.comp ?_ (toWeakDualBCNN_continuous (Ω := Ω))
state_before: Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type ?u.105418
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
f : Ω →ᵇ ℝ≥0
⊢ Continuous ((fun φ => ↑φ f) ∘ toWeakDualBCNN)

tactic: exact @WeakBilin.eval_continuous _ _ _ _ _ _ ContinuousLinearMap.module _ _ _ _
state_before: Ω : Type u_1
inst✝⁶ : MeasurableSpace Ω
R : Type ?u.105418
inst✝⁵ : SMul R ℝ≥0
inst✝⁴ : SMul R ℝ≥0∞
inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
f : Ω →ᵇ ℝ≥0
⊢ Continuous fun φ => ↑φ f
Theorem: DihedralGroup.sr_mul_self
File: Mathlib/GroupTheory/SpecificGroups/Dihedral.lean

tactic: rw [sr_mul_sr, sub_self, one_def]
state_before: n : ℕ
i : ZMod n
⊢ sr i * sr i = 1
Theorem: SemidirectProduct.lift_inl
File: Mathlib/GroupTheory/SemidirectProduct.lean

tactic: simp [lift]
state_before: N : Type u_2
G : Type u_3
H : Type u_1
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
f₁ : N →* H
f₂ : G →* H
h :
  ∀ (g : G),
    MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑f₂ g))) f₁
n : N
⊢ ↑(lift f₁ f₂ h) (↑inl n) = ↑f₁ n
Theorem: YoungDiagram.mem_transpose
File: Mathlib/Combinatorics/Young/YoungDiagram.lean

tactic: simp [transpose]
state_before: μ : YoungDiagram
c : ℕ × ℕ
⊢ c ∈ transpose μ ↔ Prod.swap c ∈ μ
Theorem: linearIndependent_fin2
File: Mathlib/LinearAlgebra/LinearIndependent.lean

tactic: rw [linearIndependent_fin_succ, linearIndependent_unique_iff, range_unique, mem_span_singleton,
  not_exists, show Fin.tail f default = f 1 by rw [← Fin.succ_zero_eq_one]; rfl]
state_before: ι : Type u'
ι' : Type ?u.1252914
R : Type ?u.1252917
K : Type u_1
M : Type ?u.1252923
M' : Type ?u.1252926
M'' : Type ?u.1252929
V : Type u
V' : Type ?u.1252934
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : AddCommGroup V'
inst✝¹ : Module K V
inst✝ : Module K V'
v : ι → V
s t : Set V
x y z : V
f : Fin 2 → V
⊢ LinearIndependent K f ↔ f 1 ≠ 0 ∧ ∀ (a : K), a • f 1 ≠ f 0

tactic: rw [← Fin.succ_zero_eq_one]
state_before: ι : Type u'
ι' : Type ?u.1252914
R : Type ?u.1252917
K : Type u_1
M : Type ?u.1252923
M' : Type ?u.1252926
M'' : Type ?u.1252929
V : Type u
V' : Type ?u.1252934
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : AddCommGroup V'
inst✝¹ : Module K V
inst✝ : Module K V'
v : ι → V
s t : Set V
x y z : V
f : Fin 2 → V
⊢ Fin.tail f default = f 1

tactic: rfl
state_before: ι : Type u'
ι' : Type ?u.1252914
R : Type ?u.1252917
K : Type u_1
M : Type ?u.1252923
M' : Type ?u.1252926
M'' : Type ?u.1252929
V : Type u
V' : Type ?u.1252934
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : AddCommGroup V'
inst✝¹ : Module K V
inst✝ : Module K V'
v : ι → V
s t : Set V
x y z : V
f : Fin 2 → V
⊢ Fin.tail f default = f (Fin.succ 0)
Theorem: summable_div_const_iff
File: Mathlib/Topology/Algebra/InfiniteSum/Ring.lean

tactic: simpa only [div_eq_mul_inv] using summable_mul_right_iff (inv_ne_zero h)
state_before: ι : Type u_2
κ : Type ?u.15580
R : Type ?u.15583
α : Type u_1
inst✝² : DivisionSemiring α
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSemiring α
f g : ι → α
a a₁ a₂ : α
h : a ≠ 0
⊢ (Summable fun i => f i / a) ↔ Summable f
Theorem: eventually_homothety_image_subset_of_finite_subset_interior
File: Mathlib/Analysis/NormedSpace/AddTorsor.lean

tactic: suffices ∀ y ∈ t, ∀ᶠ δ in 𝓝 (1 : 𝕜), homothety x δ y ∈ s by
  simp_rw [Set.image_subset_iff]
  exact (Filter.eventually_all_finite ht).mpr this
state_before: α : Type ?u.95109
V : Type ?u.95112
P : Type ?u.95115
W : Type u_3
Q : Type u_1
inst✝⁸ : SeminormedAddCommGroup V
inst✝⁷ : PseudoMetricSpace P
inst✝⁶ : NormedAddTorsor V P
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : MetricSpace Q
inst✝³ : NormedAddTorsor W Q
𝕜 : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : NormedSpace 𝕜 W
x : Q
s t : Set Q
ht : Set.Finite t
h : t ⊆ interior s
⊢ ∀ᶠ (δ : 𝕜) in 𝓝 1, ↑(homothety x δ) '' t ⊆ s

tactic: intro y hy
state_before: α : Type ?u.95109
V : Type ?u.95112
P : Type ?u.95115
W : Type u_3
Q : Type u_1
inst✝⁸ : SeminormedAddCommGroup V
inst✝⁷ : PseudoMetricSpace P
inst✝⁶ : NormedAddTorsor V P
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : MetricSpace Q
inst✝³ : NormedAddTorsor W Q
𝕜 : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : NormedSpace 𝕜 W
x : Q
s t : Set Q
ht : Set.Finite t
h : t ⊆ interior s
⊢ ∀ (y : Q), y ∈ t → ∀ᶠ (δ : 𝕜) in 𝓝 1, ↑(homothety x δ) y ∈ s

tactic: exact eventually_homothety_mem_of_mem_interior 𝕜 x (h hy)
state_before: α : Type ?u.95109
V : Type ?u.95112
P : Type ?u.95115
W : Type u_3
Q : Type u_1
inst✝⁸ : SeminormedAddCommGroup V
inst✝⁷ : PseudoMetricSpace P
inst✝⁶ : NormedAddTorsor V P
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : MetricSpace Q
inst✝³ : NormedAddTorsor W Q
𝕜 : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : NormedSpace 𝕜 W
x : Q
s t : Set Q
ht : Set.Finite t
h : t ⊆ interior s
y : Q
hy : y ∈ t
⊢ ∀ᶠ (δ : 𝕜) in 𝓝 1, ↑(homothety x δ) y ∈ s

tactic: simp_rw [Set.image_subset_iff]
state_before: α : Type ?u.95109
V : Type ?u.95112
P : Type ?u.95115
W : Type u_3
Q : Type u_1
inst✝⁸ : SeminormedAddCommGroup V
inst✝⁷ : PseudoMetricSpace P
inst✝⁶ : NormedAddTorsor V P
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : MetricSpace Q
inst✝³ : NormedAddTorsor W Q
𝕜 : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : NormedSpace 𝕜 W
x : Q
s t : Set Q
ht : Set.Finite t
h : t ⊆ interior s
this : ∀ (y : Q), y ∈ t → ∀ᶠ (δ : 𝕜) in 𝓝 1, ↑(homothety x δ) y ∈ s
⊢ ∀ᶠ (δ : 𝕜) in 𝓝 1, ↑(homothety x δ) '' t ⊆ s

tactic: exact (Filter.eventually_all_finite ht).mpr this
state_before: α : Type ?u.95109
V : Type ?u.95112
P : Type ?u.95115
W : Type u_3
Q : Type u_1
inst✝⁸ : SeminormedAddCommGroup V
inst✝⁷ : PseudoMetricSpace P
inst✝⁶ : NormedAddTorsor V P
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : MetricSpace Q
inst✝³ : NormedAddTorsor W Q
𝕜 : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : NormedSpace 𝕜 W
x : Q
s t : Set Q
ht : Set.Finite t
h : t ⊆ interior s
this : ∀ (y : Q), y ∈ t → ∀ᶠ (δ : 𝕜) in 𝓝 1, ↑(homothety x δ) y ∈ s
⊢ ∀ᶠ (δ : 𝕜) in 𝓝 1, t ⊆ (fun a => ↑(homothety x δ) a) ⁻¹' s