1337xyz1337xyz/leandojo-benchmark4-v10 huggingface dataset:
Theorem: Finsupp.totalOn_range
File: Mathlib/LinearAlgebra/Finsupp.lean

tactic: rw [Finsupp.totalOn, LinearMap.range_eq_map, LinearMap.map_codRestrict,
  ← LinearMap.range_le_iff_comap, range_subtype, Submodule.map_top, LinearMap.range_comp,
  range_subtype]
state_before: α : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
R : Type u_5
S : Type u_6
inst✝⁹ : Semiring R
inst✝⁸ : Semiring S
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : AddCommMonoid P
inst✝² : Module R P
α' : Type u_7
M' : Type u_8
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
v : α → M
v' : α' → M'
s : Set α
⊢ LinearMap.range (Finsupp.totalOn α M R v s) = ⊤
state_after: α : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
R : Type u_5
S : Type u_6
inst✝⁹ : Semiring R
inst✝⁸ : Semiring S
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : AddCommMonoid P
inst✝² : Module R P
α' : Type u_7
M' : Type u_8
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
v : α → M
v' : α' → M'
s : Set α
⊢ span R (v '' s) ≤ map (Finsupp.total α M R v) (supported R R s)

tactic: exact (span_image_eq_map_total _ _).le
state_before: α : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
R : Type u_5
S : Type u_6
inst✝⁹ : Semiring R
inst✝⁸ : Semiring S
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : AddCommMonoid P
inst✝² : Module R P
α' : Type u_7
M' : Type u_8
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
v : α → M
v' : α' → M'
s : Set α
⊢ span R (v '' s) ≤ map (Finsupp.total α M R v) (supported R R s)
state_after: no goals
Theorem: interior_sphere'
File: Mathlib/Analysis/NormedSpace/Real.lean

tactic: rw [← frontier_closedBall' x, interior_frontier isClosed_ball]
state_before: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
r : ℝ
⊢ interior (sphere x r) = ∅
state_after: no goals
Theorem: directedOn_image
File: Mathlib/Order/Directed.lean

tactic: simp only [DirectedOn, Set.mem_image, exists_exists_and_eq_and, forall_exists_index, and_imp,
  forall_apply_eq_imp_iff₂, Order.Preimage]
state_before: α : Type u
β : Type v
ι : Sort w
r r' s✝ : α → α → Prop
s : Set β
f : β → α
⊢ DirectedOn r (f '' s) ↔ DirectedOn (f ⁻¹'o r) s
state_after: no goals
Theorem: LieSubmodule.lieSpan_eq_bot_iff
File: Mathlib/Algebra/Lie/Submodule.lean

tactic: rw [_root_.eq_bot_iff, lieSpan_le, bot_coe, subset_singleton_iff]
state_before: 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
N N' : LieSubmodule R L M
I J : LieIdeal R L
s : Set M
⊢ lieSpan R L s = ⊥ ↔ ∀ m ∈ s, m = 0
state_after: no goals
Theorem: Stream'.WSeq.tail_ofSeq
File: Mathlib/Data/Seq/WSeq.lean

tactic: simp only [tail, destruct_ofSeq, map_pure', flatten_pure]
state_before: α : Type u
β : Type v
γ : Type w
s : Seq α
⊢ (↑s).tail = ↑s.tail
state_after: α : Type u
β : Type v
γ : Type w
s : Seq α
⊢ Option.rec nil Prod.snd (Option.map (fun a => (a, ↑s.tail)) s.head) = ↑s.tail

tactic: induction' s using Seq.recOn with x s <;> simp only [ofSeq, Seq.tail_nil, Seq.head_nil,
  Option.map_none', Seq.tail_cons, Seq.head_cons, Option.map_some']
state_before: α : Type u
β : Type v
γ : Type w
s : Seq α
⊢ Option.rec nil Prod.snd (Option.map (fun a => (a, ↑s.tail)) s.head) = ↑s.tail
state_after: case h1
α : Type u
β : Type v
γ : Type w
⊢ nil = some <$> Seq.nil

tactic: rfl
state_before: case h1
α : Type u
β : Type v
γ : Type w
⊢ nil = some <$> Seq.nil
state_after: no goals
Theorem: MeasureTheory.essSup_trim
File: Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean

tactic: simp_rw [essSup]
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ : Measure α
inst✝ : NormedAddCommGroup E
hm : m ≤ m0
f : α → ℝ≥0∞
hf : Measurable f
⊢ essSup f (μ.trim hm) = essSup f μ
state_after: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ : Measure α
inst✝ : NormedAddCommGroup E
hm : m ≤ m0
f : α → ℝ≥0∞
hf : Measurable f
⊢ limsup f (ae (μ.trim hm)) = limsup f (ae μ)

tactic: exact limsup_trim hm hf
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ : Measure α
inst✝ : NormedAddCommGroup E
hm : m ≤ m0
f : α → ℝ≥0∞
hf : Measurable f
⊢ limsup f (ae (μ.trim hm)) = limsup f (ae μ)
state_after: no goals
Theorem: Rat.divInt_self'
File: Mathlib/Data/Rat/Defs.lean

tactic: simpa using divInt_mul_right (n := 1) (d := 1) hn
state_before: q : ℚ
n : ℤ
hn : n ≠ 0
⊢ n /. n = 1
state_after: no goals
Theorem: CauSeq.inf_eq_left
File: Mathlib/Algebra/Order/CauSeq/Basic.lean

tactic: simpa only [CauSeq.inf_comm] using CauSeq.inf_eq_right h
state_before: α : Type u_1
β : Type u_2
inst✝ : LinearOrderedField α
a b : CauSeq α abs
h : a ≤ b
⊢ a ⊓ b ≈ a
state_after: no goals
Theorem: Rel.card_interedges_finpartition_right
File: Mathlib/Combinatorics/SimpleGraph/Density.lean

tactic: classical
simp_rw [← P.biUnion_parts, interedges_biUnion_right, id]
rw [card_biUnion]
exact fun x hx y hy h ↦ interedges_disjoint_right r _ (P.disjoint hx hy h)
state_before: 𝕜 : Type u_1
ι : Type u_2
κ : Type u_3
α : Type u_4
β : Type u_5
inst✝² : LinearOrderedField 𝕜
r : α → β → Prop
inst✝¹ : (a : α) → DecidablePred (r a)
s✝ s₁ s₂ : Finset α
t t₁ t₂ : Finset β
a : α
b : β
δ : 𝕜
inst✝ : DecidableEq β
s : Finset α
P : Finpartition t
⊢ (interedges r s t).card = ∑ b ∈ P.parts, (interedges r s b).card
state_after: no goals

tactic: simp_rw [← P.biUnion_parts, interedges_biUnion_right, id]
state_before: 𝕜 : Type u_1
ι : Type u_2
κ : Type u_3
α : Type u_4
β : Type u_5
inst✝² : LinearOrderedField 𝕜
r : α → β → Prop
inst✝¹ : (a : α) → DecidablePred (r a)
s✝ s₁ s₂ : Finset α
t t₁ t₂ : Finset β
a : α
b : β
δ : 𝕜
inst✝ : DecidableEq β
s : Finset α
P : Finpartition t
⊢ (interedges r s t).card = ∑ b ∈ P.parts, (interedges r s b).card
state_after: 𝕜 : Type u_1
ι : Type u_2
κ : Type u_3
α : Type u_4
β : Type u_5
inst✝² : LinearOrderedField 𝕜
r : α → β → Prop
inst✝¹ : (a : α) → DecidablePred (r a)
s✝ s₁ s₂ : Finset α
t t₁ t₂ : Finset β
a : α
b : β
δ : 𝕜
inst✝ : DecidableEq β
s : Finset α
P : Finpartition t
⊢ (P.parts.biUnion fun b => interedges r s b).card = ∑ b ∈ P.parts, (interedges r s b).card

tactic: rw [card_biUnion]
state_before: 𝕜 : Type u_1
ι : Type u_2
κ : Type u_3
α : Type u_4
β : Type u_5
inst✝² : LinearOrderedField 𝕜
r : α → β → Prop
inst✝¹ : (a : α) → DecidablePred (r a)
s✝ s₁ s₂ : Finset α
t t₁ t₂ : Finset β
a : α
b : β
δ : 𝕜
inst✝ : DecidableEq β
s : Finset α
P : Finpartition t
⊢ (P.parts.biUnion fun b => interedges r s b).card = ∑ b ∈ P.parts, (interedges r s b).card
state_after: 𝕜 : Type u_1
ι : Type u_2
κ : Type u_3
α : Type u_4
β : Type u_5
inst✝² : LinearOrderedField 𝕜
r : α → β → Prop
inst✝¹ : (a : α) → DecidablePred (r a)
s✝ s₁ s₂ : Finset α
t t₁ t₂ : Finset β
a : α
b : β
δ : 𝕜
inst✝ : DecidableEq β
s : Finset α
P : Finpartition t
⊢ ∀ x ∈ P.parts, ∀ y ∈ P.parts, x ≠ y → Disjoint (interedges r s x) (interedges r s y)

tactic: exact fun x hx y hy h ↦ interedges_disjoint_right r _ (P.disjoint hx hy h)
state_before: 𝕜 : Type u_1
ι : Type u_2
κ : Type u_3
α : Type u_4
β : Type u_5
inst✝² : LinearOrderedField 𝕜
r : α → β → Prop
inst✝¹ : (a : α) → DecidablePred (r a)
s✝ s₁ s₂ : Finset α
t t₁ t₂ : Finset β
a : α
b : β
δ : 𝕜
inst✝ : DecidableEq β
s : Finset α
P : Finpartition t
⊢ ∀ x ∈ P.parts, ∀ y ∈ P.parts, x ≠ y → Disjoint (interedges r s x) (interedges r s y)
state_after: no goals
Theorem: CategoryTheory.MonoidalCategory.id_tensor_associator_naturality
File: Mathlib/CategoryTheory/Monoidal/Category.lean

tactic: rw [← tensor_id, associator_naturality]
state_before: C✝ : Type u
𝒞 : Category.{v, u} C✝
inst✝² : MonoidalCategory C✝
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : MonoidalCategory C
U V W X✝ Y✝ Z✝ X Y Z Z' : C
h : Z ⟶ Z'
⊢ (𝟙 (X ⊗ Y) ⊗ h) ≫ (α_ X Y Z').hom = (α_ X Y Z).hom ≫ (𝟙 X ⊗ 𝟙 Y ⊗ h)
state_after: no goals
Theorem: Submodule.le_self_mul_one_div
File: Mathlib/Algebra/Algebra/Operations.lean

tactic: refine (mul_one I).symm.trans_le ?_
state_before: ι : Sort uι
R : Type u
inst✝² : CommSemiring R
A : Type v
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
M N : Submodule R A
m n : A
I : Submodule R A
hI : I ≤ 1
⊢ I ≤ I * (1 / I)
state_after: ι : Sort uι
R : Type u
inst✝² : CommSemiring R
A : Type v
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
M N : Submodule R A
m n : A
I : Submodule R A
hI : I ≤ 1
⊢ I * 1 ≤ I * (1 / I)

tactic: apply mul_le_mul_right (one_le_one_div.mpr hI)
state_before: ι : Sort uι
R : Type u
inst✝² : CommSemiring R
A : Type v
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
M N : Submodule R A
m n : A
I : Submodule R A
hI : I ≤ 1
⊢ I * 1 ≤ I * (1 / I)
state_after: no goals
Theorem: Real.cosh_eq
File: Mathlib/Data/Complex/Exponential.lean

tactic: rw [cosh, exp, exp, Complex.ofReal_neg, Complex.cosh, mul_two, ← Complex.add_re, ← mul_two,
  div_mul_cancel₀ _ (two_ne_zero' ℂ), Complex.add_re]
state_before: x✝ y x : ℝ
⊢ cosh x * 2 = rexp x + rexp (-x)
state_after: no goals
Theorem: geometric_hahn_banach_closed_compact
File: Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean

tactic: simpa using ht
state_before: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : TopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
s✝ t✝ : Set E
x y : E
inst✝ : LocallyConvexSpace ℝ E
hs₁ : Convex ℝ s✝
hs₂ : IsClosed s✝
ht₁ : Convex ℝ t✝
ht₂ : IsCompact t✝
disj : Disjoint s✝ t✝
f : E →L[ℝ] ℝ
s t : ℝ
hs : ∀ a ∈ t✝, f a < s
st : s < t
ht : ∀ b ∈ s✝, t < f b
⊢ ∀ a ∈ s✝, (-f) a < -t
state_after: no goals

tactic: simpa using st
state_before: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : TopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
s✝ t✝ : Set E
x y : E
inst✝ : LocallyConvexSpace ℝ E
hs₁ : Convex ℝ s✝
hs₂ : IsClosed s✝
ht₁ : Convex ℝ t✝
ht₂ : IsCompact t✝
disj : Disjoint s✝ t✝
f : E →L[ℝ] ℝ
s t : ℝ
hs : ∀ a ∈ t✝, f a < s
st : s < t
ht : ∀ b ∈ s✝, t < f b
⊢ -t < -s
state_after: no goals

tactic: simpa using hs
state_before: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : TopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
s✝ t✝ : Set E
x y : E
inst✝ : LocallyConvexSpace ℝ E
hs₁ : Convex ℝ s✝
hs₂ : IsClosed s✝
ht₁ : Convex ℝ t✝
ht₂ : IsCompact t✝
disj : Disjoint s✝ t✝
f : E →L[ℝ] ℝ
s t : ℝ
hs : ∀ a ∈ t✝, f a < s
st : s < t
ht : ∀ b ∈ s✝, t < f b
⊢ ∀ b ∈ t✝, -s < (-f) b
state_after: no goals
Theorem: NonUnitalStarAlgebra.map_bot
File: Mathlib/Algebra/Star/NonUnitalSubalgebra.lean

tactic: simp [NonUnitalAlgebra.coe_bot, NonUnitalStarSubalgebra.coe_map]
state_before: F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : NonUnitalSemiring A
inst✝¹³ : StarRing A
inst✝¹² : Module R A
inst✝¹¹ : IsScalarTower R A A
inst✝¹⁰ : SMulCommClass R A A
inst✝⁹ : StarModule R A
inst✝⁸ : NonUnitalSemiring B
inst✝⁷ : StarRing B
inst✝⁶ : Module R B
inst✝⁵ : IsScalarTower R B B
inst✝⁴ : SMulCommClass R B B
inst✝³ : StarModule R B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F R A B
inst✝ : NonUnitalStarAlgHomClass F R A B
f : F
⊢ ↑(map f ⊥) = ↑⊥
state_after: no goals
Theorem: Nat.Icc_succ_left
File: Mathlib/Order/Interval/Finset/Nat.lean

tactic: ext x
state_before: a b c : ℕ
⊢ Icc a.succ b = Ioc a b
state_after: case a
a b c x : ℕ
⊢ x ∈ Icc a.succ b ↔ x ∈ Ioc a b

tactic: rw [mem_Icc, mem_Ioc, succ_le_iff]
state_before: case a
a b c x : ℕ
⊢ x ∈ Icc a.succ b ↔ x ∈ Ioc a b
state_after: no goals
Theorem: MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded'
File: Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean

tactic: simpa only [ENNReal.ofReal_coe_nnreal] using hbounded
state_before: α : Type u_1
ι : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
l : Filter ι
inst✝² : NormedAddCommGroup E
inst✝¹ : l.NeBot
inst✝ : l.IsCountablyGenerated
φ : ι → Set α
hφ : AECover μ l φ
f : α → E
I : ℝ≥0
hfm : AEStronglyMeasurable f μ
hbounded : ∀ᶠ (i : ι) in l, ∫⁻ (x : α) in φ i, ↑‖f x‖₊ ∂μ ≤ ↑I
⊢ ∀ᶠ (i : ι) in l, ∫⁻ (x : α) in φ i, ↑‖f x‖₊ ∂μ ≤ ENNReal.ofReal ↑I
state_after: no goals
Theorem: Equiv.Perm.pow_eq_on_of_mem_support
File: Mathlib/GroupTheory/Perm/Support.lean

tactic: induction' k with k hk
state_before: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
k : ℕ
⊢ ∀ x ∈ f.support ∩ g.support, (f ^ k) x = (g ^ k) x
state_after: case zero
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
⊢ ∀ x ∈ f.support ∩ g.support, (f ^ 0) x = (g ^ 0) x

case succ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
k : ℕ
hk : ∀ x ∈ f.support ∩ g.support, (f ^ k) x = (g ^ k) x
⊢ ∀ x ∈ f.support ∩ g.support, (f ^ (k + 1)) x = (g ^ (k + 1)) x

tactic: simp
state_before: case zero
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
⊢ ∀ x ∈ f.support ∩ g.support, (f ^ 0) x = (g ^ 0) x
state_after: no goals

tactic: intro x hx
state_before: case succ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
k : ℕ
hk : ∀ x ∈ f.support ∩ g.support, (f ^ k) x = (g ^ k) x
⊢ ∀ x ∈ f.support ∩ g.support, (f ^ (k + 1)) x = (g ^ (k + 1)) x
state_after: case succ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
k : ℕ
hk : ∀ x ∈ f.support ∩ g.support, (f ^ k) x = (g ^ k) x
x : α
hx : x ∈ f.support ∩ g.support
⊢ (f ^ (k + 1)) x = (g ^ (k + 1)) x

tactic: rw [pow_succ, mul_apply, pow_succ, mul_apply, h _ hx, hk]
state_before: case succ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
k : ℕ
hk : ∀ x ∈ f.support ∩ g.support, (f ^ k) x = (g ^ k) x
x : α
hx : x ∈ f.support ∩ g.support
⊢ (f ^ (k + 1)) x = (g ^ (k + 1)) x
state_after: case succ.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
k : ℕ
hk : ∀ x ∈ f.support ∩ g.support, (f ^ k) x = (g ^ k) x
x : α
hx : x ∈ f.support ∩ g.support
⊢ g x ∈ f.support ∩ g.support

tactic: rwa [mem_inter, apply_mem_support, ← h _ hx, apply_mem_support, ← mem_inter]
state_before: case succ.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
k : ℕ
hk : ∀ x ∈ f.support ∩ g.support, (f ^ k) x = (g ^ k) x
x : α
hx : x ∈ f.support ∩ g.support
⊢ g x ∈ f.support ∩ g.support
state_after: no goals
Theorem: Hyperreal.isSt_symm_real
File: Mathlib/Data/Real/Hyperreal.lean

tactic: rw [isSt_real_iff_eq, isSt_real_iff_eq, eq_comm]
state_before: r s : ℝ
⊢ (↑r).IsSt s ↔ (↑s).IsSt r
state_after: no goals
Theorem: MeasureTheory.Measure.ext_iff_of_iUnion_eq_univ
File: Mathlib/MeasureTheory/Measure/Restrict.lean

tactic: rw [← restrict_iUnion_congr, hs, restrict_univ, restrict_univ]
state_before: R : Type u_1
α : Type u_2
β : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
inst✝ : Countable ι
s : ι → Set α
hs : ⋃ i, s i = univ
⊢ μ = ν ↔ ∀ (i : ι), μ.restrict (s i) = ν.restrict (s i)
state_after: no goals
Theorem: Real.tendsto_exp_nhds_zero_nhds_one
File: Mathlib/Analysis/SpecialFunctions/Exp.lean

tactic: convert continuous_exp.tendsto 0
state_before: α : Type u_1
x y z : ℝ
l : Filter α
⊢ Tendsto rexp (𝓝 0) (𝓝 1)
state_after: case h.e'_5.h.e'_3
α : Type u_1
x y z : ℝ
l : Filter α
⊢ 1 = rexp 0

tactic: simp
state_before: case h.e'_5.h.e'_3
α : Type u_1
x y z : ℝ
l : Filter α
⊢ 1 = rexp 0
state_after: no goals
Theorem: HasFTaylorSeriesUpTo.eq_iteratedFDeriv
File: Mathlib/Analysis/Calculus/ContDiff/Defs.lean

tactic: rw [← iteratedFDerivWithin_univ]
state_before: 𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type uX
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x✝ x₀ : E
c : F
m✝ n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
h : HasFTaylorSeriesUpTo n f p
m : ℕ
hmn : ↑m ≤ n
x : E
⊢ p x m = iteratedFDeriv 𝕜 m f x
state_after: 𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type uX
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x✝ x₀ : E
c : F
m✝ n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
h : HasFTaylorSeriesUpTo n f p
m : ℕ
hmn : ↑m ≤ n
x : E
⊢ p x m = iteratedFDerivWithin 𝕜 m f univ x

tactic: rw [← hasFTaylorSeriesUpToOn_univ_iff] at h
state_before: 𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type uX
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x✝ x₀ : E
c : F
m✝ n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
h : HasFTaylorSeriesUpTo n f p
m : ℕ
hmn : ↑m ≤ n
x : E
⊢ p x m = iteratedFDerivWithin 𝕜 m f univ x
state_after: 𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type uX
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x✝ x₀ : E
c : F
m✝ n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
h : HasFTaylorSeriesUpToOn n f p univ
m : ℕ
hmn : ↑m ≤ n
x : E
⊢ p x m = iteratedFDerivWithin 𝕜 m f univ x

tactic: exact h.eq_iteratedFDerivWithin_of_uniqueDiffOn hmn uniqueDiffOn_univ (mem_univ _)
state_before: 𝕜 : Type u
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type uX
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x✝ x₀ : E
c : F
m✝ n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
h : HasFTaylorSeriesUpToOn n f p univ
m : ℕ
hmn : ↑m ≤ n
x : E
⊢ p x m = iteratedFDerivWithin 𝕜 m f univ x
state_after: no goals
Theorem: TypeVec.subtypeVal_nil
File: Mathlib/Data/TypeVec.lean

tactic: rintro ⟨⟩
state_before: n : ℕ
α : TypeVec.{u} 0
ps : α ⟹ repeat 0 Prop
⊢ ∀ (x : Fin2 0), subtypeVal ps x = nilFun x
state_after: no goals
Theorem: IntervalIntegrable.intervalIntegrable_norm_iff
File: Mathlib/MeasureTheory/Integral/IntervalIntegral.lean

tactic: simp_rw [intervalIntegrable_iff, IntegrableOn]
state_before: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝ : NormedAddCommGroup E
f✝ : ℝ → E
a✝ b✝ c d : ℝ
μ✝ ν : Measure ℝ
f : ℝ → E
μ : Measure ℝ
a b : ℝ
hf : AEStronglyMeasurable f (μ.restrict (Ι a b))
⊢ IntervalIntegrable (fun t => ‖f t‖) μ a b ↔ IntervalIntegrable f μ a b
state_after: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝ : NormedAddCommGroup E
f✝ : ℝ → E
a✝ b✝ c d : ℝ
μ✝ ν : Measure ℝ
f : ℝ → E
μ : Measure ℝ
a b : ℝ
hf : AEStronglyMeasurable f (μ.restrict (Ι a b))
⊢ Integrable (fun t => ‖f t‖) (μ.restrict (Ι a b)) ↔ Integrable f (μ.restrict (Ι a b))

tactic: exact integrable_norm_iff hf
state_before: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝ : NormedAddCommGroup E
f✝ : ℝ → E
a✝ b✝ c d : ℝ
μ✝ ν : Measure ℝ
f : ℝ → E
μ : Measure ℝ
a b : ℝ
hf : AEStronglyMeasurable f (μ.restrict (Ι a b))
⊢ Integrable (fun t => ‖f t‖) (μ.restrict (Ι a b)) ↔ Integrable f (μ.restrict (Ι a b))
state_after: no goals
Theorem: Set.ordConnectedComponent_inter
File: Mathlib/Order/Interval/Set/OrdConnectedComponent.lean

tactic: simp [ordConnectedComponent, setOf_and]
state_before: α : Type u_1
inst✝ : LinearOrder α
s✝ t✝ : Set α
x✝ y z : α
s t : Set α
x : α
⊢ (s ∩ t).ordConnectedComponent x = s.ordConnectedComponent x ∩ t.ordConnectedComponent x
state_after: no goals
Theorem: tendsto_nat_ceil_div_atTop
File: Mathlib/Analysis/SpecificLimits/Basic.lean

tactic: simpa using tendsto_nat_ceil_mul_div_atTop (zero_le_one' R)
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
R : Type u_4
inst✝³ : TopologicalSpace R
inst✝² : LinearOrderedField R
inst✝¹ : OrderTopology R
inst✝ : FloorRing R
⊢ Tendsto (fun x => ↑⌈x⌉₊ / x) atTop (𝓝 1)
state_after: no goals
Theorem: Equiv.toHomeomorph_symm
File: Mathlib/Topology/Homeomorph.lean

tactic: convert (he _).symm
state_before: X : Type u_1
Y : Type u_2
Z✝ : Type u_3
Z : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
e : X ≃ Y
he : ∀ (s : Set Y), IsOpen (⇑e ⁻¹' s) ↔ IsOpen s
s : Set X
⊢ IsOpen (⇑e.symm ⁻¹' s) ↔ IsOpen s
state_after: case h.e'_2.h.e'_3
X : Type u_1
Y : Type u_2
Z✝ : Type u_3
Z : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
e : X ≃ Y
he : ∀ (s : Set Y), IsOpen (⇑e ⁻¹' s) ↔ IsOpen s
s : Set X
⊢ s = ⇑e ⁻¹' (⇑e.symm ⁻¹' s)

tactic: simp
state_before: case h.e'_2.h.e'_3
X : Type u_1
Y : Type u_2
Z✝ : Type u_3
Z : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
e : X ≃ Y
he : ∀ (s : Set Y), IsOpen (⇑e ⁻¹' s) ↔ IsOpen s
s : Set X
⊢ s = ⇑e ⁻¹' (⇑e.symm ⁻¹' s)
state_after: no goals
Theorem: IsCyclotomicExtension.Rat.Three.eta_sq_add_eta_add_one
File: Mathlib/NumberTheory/Cyclotomic/Three.lean

tactic: rw [eta_sq]
state_before: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
⊢ ↑η ^ 2 + ↑η + 1 = 0
state_after: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
⊢ -↑η - 1 + ↑η + 1 = 0

tactic: ring
state_before: K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
u : (𝓞 K)ˣ
x : 𝓞 K
⊢ -↑η - 1 + ↑η + 1 = 0
state_after: no goals
Theorem: Function.LeftInverse.preimage_preimage
File: Mathlib/Data/Set/Image.lean

tactic: rw [← preimage_comp, h.comp_eq_id, preimage_id]
state_before: α : Type u_1
β : Type u_2
ι : Sort u_3
f : α → β
g : β → α
h : LeftInverse g f
s : Set α
⊢ f ⁻¹' (g ⁻¹' s) = s
state_after: no goals
Theorem: Matroid.Base.base_of_basis_superset
File: Mathlib/Data/Matroid/Basic.lean

tactic: by_contra h
state_before: α : Type u_1
M : Matroid α
B X I : Set α
hB : M.Base B
hBX : B ⊆ X
hIX : M.Basis I X
⊢ M.Base I
state_after: α : Type u_1
M : Matroid α
B X I : Set α
hB : M.Base B
hBX : B ⊆ X
hIX : M.Basis I X
h : ¬M.Base I
⊢ False

tactic: obtain ⟨e,heBI,he⟩ := hIX.indep.exists_insert_of_not_base h hB
state_before: α : Type u_1
M : Matroid α
B X I : Set α
hB : M.Base B
hBX : B ⊆ X
hIX : M.Basis I X
h : ¬M.Base I
⊢ False
state_after: case intro.intro
α : Type u_1
M : Matroid α
B X I : Set α
hB : M.Base B
hBX : B ⊆ X
hIX : M.Basis I X
h : ¬M.Base I
e : α
heBI : e ∈ B \ I
he : M.Indep (insert e I)
⊢ False

tactic: exact heBI.2 (hIX.mem_of_insert_indep (hBX heBI.1) he)
state_before: case intro.intro
α : Type u_1
M : Matroid α
B X I : Set α
hB : M.Base B
hBX : B ⊆ X
hIX : M.Basis I X
h : ¬M.Base I
e : α
heBI : e ∈ B \ I
he : M.Indep (insert e I)
⊢ False
state_after: no goals