name stringlengths 3 112 | file stringlengths 21 116 | statement stringlengths 17 8.64k | state stringlengths 7 205k | tactic stringlengths 3 4.55k | result stringlengths 7 205k | id stringlengths 16 16 |
|---|---|---|---|---|---|---|
Option.lawfulBEq_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Option/Lemmas.lean | theorem lawfulBEq_iff : LawfulBEq (Option α) ↔ LawfulBEq α | case mpr.eq_of_beq
α : Type u_1
inst✝ : BEq α
h✝ : LawfulBEq α
a b : Option α
h : (a == b) = true
⊢ a = b | simpa using h | no goals | 84ae32359a69ec10 |
ProbabilityTheory.integrable_exp_mul_abs | Mathlib/Probability/Moments/IntegrableExpMul.lean | /-- If `ω ↦ exp (t * X ω)` is integrable at `t` and `-t`, then `ω ↦ exp (t * |X ω|)` is
integrable. -/
lemma integrable_exp_mul_abs (ht_int_pos : Integrable (fun ω ↦ exp (t * X ω)) μ)
(ht_int_neg : Integrable (fun ω ↦ exp (- t * X ω)) μ) :
Integrable (fun ω ↦ exp (t * |X ω|)) μ | Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
ht_int_pos : Integrable (fun ω => rexp (t * X ω)) μ
ht_int_neg : Integrable (fun ω => rexp (-t * X ω)) μ
⊢ Integrable (fun ω => rexp (t * |X ω|)) μ | have h := integrable_exp_mul_abs_add (t := t) (μ := μ) (X := X) (v := 0) ?_ ?_ | case refine_3
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
ht_int_pos : Integrable (fun ω => rexp (t * X ω)) μ
ht_int_neg : Integrable (fun ω => rexp (-t * X ω)) μ
h : Integrable (fun ω => rexp (t * |X ω| + 0 * X ω)) μ
⊢ Integrable (fun ω => rexp (t * |X ω|)) μ
case refine_1
Ω : Type u_1
m : Measur... | 50b30d0412dd12e4 |
Multiset.revzip_powersetAux | Mathlib/Data/Multiset/Powerset.lean | theorem revzip_powersetAux {l : List α} ⦃x⦄ (h : x ∈ revzip (powersetAux l)) : x.1 + x.2 = ↑l | α : Type u_1
l : List α
x : Multiset α × Multiset α
h : x ∈ (powersetAux l).revzip
⊢ x.1 + x.2 = ↑l | rw [revzip, powersetAux_eq_map_coe, ← map_reverse, zip_map, ← revzip, List.mem_map] at h | α : Type u_1
l : List α
x : Multiset α × Multiset α
h : ∃ a ∈ l.sublists.revzip, Prod.map ofList ofList a = x
⊢ x.1 + x.2 = ↑l | 0b9e2d20137f4a32 |
Subring.exists_list_of_mem_closure | Mathlib/Algebra/Ring/Subring/Basic.lean | theorem exists_list_of_mem_closure {s : Set R} {x : R} (hx : x ∈ closure s) :
∃ L : List (List R), (∀ t ∈ L, ∀ y ∈ t, y ∈ s ∨ y = (-1 : R)) ∧ (L.map List.prod).sum = x | R : Type u
inst✝ : Ring R
s : Set R
x : R
hx : x ∈ closure s
⊢ ∃ L, (∀ t ∈ L, ∀ y ∈ t, y ∈ s ∨ y = -1) ∧ (List.map List.prod L).sum = x | rw [mem_closure_iff] at hx | R : Type u
inst✝ : Ring R
s : Set R
x : R
hx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)
⊢ ∃ L, (∀ t ∈ L, ∀ y ∈ t, y ∈ s ∨ y = -1) ∧ (List.map List.prod L).sum = x | 2fbe9fd19e92bfa4 |
isExtreme_singleton | Mathlib/Analysis/Convex/Extreme.lean | /-- x is an extreme point to A iff {x} is an extreme set of A. -/
@[simp] lemma isExtreme_singleton : IsExtreme 𝕜 A {x} ↔ x ∈ A.extremePoints 𝕜 | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : SMul 𝕜 E
A : Set E
x : E
⊢ IsExtreme 𝕜 A {x} ↔ x ∈ extremePoints 𝕜 A | refine ⟨fun hx ↦ ⟨singleton_subset_iff.1 hx.1, fun x₁ hx₁ x₂ hx₂ ↦ hx.2 hx₁ hx₂ rfl⟩, ?_⟩ | 𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : SMul 𝕜 E
A : Set E
x : E
⊢ x ∈ extremePoints 𝕜 A → IsExtreme 𝕜 A {x} | ad3c1fdb8942198d |
three_dvd_b_of_dvd_a_of_gcd_eq_one_of_case2 | Mathlib/NumberTheory/FLT/Three.lean | private lemma three_dvd_b_of_dvd_a_of_gcd_eq_one_of_case2 {a b c : ℤ} (ha : a ≠ 0)
(Hgcd : Finset.gcd {a, b, c} id = 1) (h3a : 3 ∣ a) (HF : a ^ 3 + b ^ 3 + c ^ 3 = 0)
(H : ∀ a b c : ℤ, c ≠ 0 → ¬ 3 ∣ a → ¬ 3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3) :
3 ∣ b | case pos
a b c : ℤ
ha : a ≠ 0
Hgcd : {a, b, c}.gcd id = 1
h3a : 3 ∣ a
HF : a ^ 3 + b ^ 3 + c ^ 3 = 0
H : ∀ (a b c : ℤ), c ≠ 0 → ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3
hbc : IsCoprime (-b) (-c)
h3b : ¬3 ∣ b
h3c : 3 ∣ c
⊢ False | apply h3b | case pos
a b c : ℤ
ha : a ≠ 0
Hgcd : {a, b, c}.gcd id = 1
h3a : 3 ∣ a
HF : a ^ 3 + b ^ 3 + c ^ 3 = 0
H : ∀ (a b c : ℤ), c ≠ 0 → ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3
hbc : IsCoprime (-b) (-c)
h3b : ¬3 ∣ b
h3c : 3 ∣ c
⊢ 3 ∣ b | e2b14a1dcce62c48 |
mk_mem_convexHull_prod | Mathlib/Analysis/Convex/Combination.lean | theorem mk_mem_convexHull_prod {t : Set F} {x : E} {y : F} (hx : x ∈ convexHull R s)
(hy : y ∈ convexHull R t) : (x, y) ∈ convexHull R (s ×ˢ t) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_1
E : Type u_3
F : Type u_4
inst✝⁴ : LinearOrderedField R
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module R E
inst✝ : Module R F
s : Set E
t : Set F
x : E
y : F
ι : Type
w✝¹ : Fintype ι
w : ι → R
f : ι → ... | refine ⟨ι × κ, inferInstance, fun p => w p.1 * v p.2, fun p ↦ (f p.1, g p.2),
fun p ↦ mul_nonneg (hw₀ _) (hv₀ _), h_sum, fun p ↦ ⟨hfs _, hgt _⟩, ?_⟩ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_1
E : Type u_3
F : Type u_4
inst✝⁴ : LinearOrderedField R
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module R E
inst✝ : Module R F
s : Set E
t : Set F
x : E
y : F
ι : Type
w✝¹ : Fintype ι
w : ι → R
f : ι → ... | a0f1e96c0f1383b8 |
Polynomial.Monic.irreducible_iff_irreducible_map_fraction_map | Mathlib/RingTheory/Polynomial/GaussLemma.lean | theorem Monic.irreducible_iff_irreducible_map_fraction_map [IsIntegrallyClosed R] {p : R[X]}
(h : p.Monic) : Irreducible p ↔ Irreducible (p.map <| algebraMap R K) | case intro.intro.refine_2
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : ... | by_contra h_contra | case intro.intro.refine_2
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : ... | 959796d56b359847 |
CategoryTheory.mapPair_equifibered | Mathlib/CategoryTheory/Limits/VanKampen.lean | theorem mapPair_equifibered {F F' : Discrete WalkingPair ⥤ C} (α : F ⟶ F') :
NatTrans.Equifibered α | C : Type u
inst✝ : Category.{v, u} C
F F' : Discrete WalkingPair ⥤ C
α : F ⟶ F'
⊢ 𝟙 (F.obj { as := WalkingPair.right }) ≫ α.app { as := WalkingPair.right } =
α.app { as := WalkingPair.right } ≫ 𝟙 (F'.obj { as := WalkingPair.right }) | simp only [Category.comp_id, Category.id_comp] | no goals | 75bbeab18b3afdbf |
Field.primitive_element_inf_aux_of_finite_intermediateField | Mathlib/FieldTheory/PrimitiveElement.lean | theorem primitive_element_inf_aux_of_finite_intermediateField
[Finite (IntermediateField F E)] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
⊢ ∃ γ, F⟮α, β⟯ = F⟮γ⟯ | let f : F → IntermediateField F E := fun x ↦ F⟮α + x • β⟯ | F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
α β : E
inst✝¹ : Algebra F E
inst✝ : Finite (IntermediateField F E)
f : F → IntermediateField F E := fun x => F⟮α + x • β⟯
⊢ ∃ γ, F⟮α, β⟯ = F⟮γ⟯ | 7cb38d2a2cecc9bb |
BoxIntegral.HasIntegral.of_bRiemann_eq_false_of_forall_isLittleO | Mathlib/Analysis/BoxIntegral/Basic.lean | theorem HasIntegral.of_bRiemann_eq_false_of_forall_isLittleO (hl : l.bRiemann = false)
(B : ι →ᵇᵃ[I] ℝ) (hB0 : ∀ J, 0 ≤ B J) (g : ι →ᵇᵃ[I] F) (s : Set ℝⁿ) (hs : s.Countable)
(hlH : s.Nonempty → l.bHenstock = true)
(H₁ : ∀ (c : ℝ≥0), ∀ x ∈ Box.Icc I ∩ s, ∀ ε > (0 : ℝ),
∃ δ > 0, ∀ J ≤ I, Box.Icc J ⊆ Met... | ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
hl : l.bRiemann = false
B : ι →ᵇᵃ[↑I] ℝ
hB0 : ∀ (J : Box ι), 0 ≤ B J
g : ι →ᵇᵃ[↑I] F
s... | obtain ⟨hJ, hJs⟩ := hJ | case intro
ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
hl : l.bRiemann = false
B : ι →ᵇᵃ[↑I] ℝ
hB0 : ∀ (J : Box ι), 0 ≤ B J
g : ι ... | f7fbe41cacd6cc55 |
Std.DHashMap.Internal.Raw₀.distinct_keys | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean | theorem distinct_keys [EquivBEq α] [LawfulHashable α] (h : m.1.WF) :
m.1.keys.Pairwise (fun a b => (a == b) = false) | α : Type u
β : α → Type v
m : Raw₀ α β
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.val.WF
⊢ List.Pairwise (fun a b => (a == b) = false) m.val.keys | simp_to_model using (Raw.WF.out h).distinct.distinct | no goals | d6170e75d6f0bf83 |
ComplexShape.Embedding.liftExtend_homRestrict | Mathlib/Algebra/Homology/Embedding/HomEquiv.lean | @[simp]
lemma liftExtend_homRestrict (ψ : K ⟶ L.extend e) :
e.liftExtend (e.homRestrict ψ) (e.homRestrict_hasLift ψ) = ψ | case h
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{u_4, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
L : HomologicalComplex C c
inst✝ : e.IsRelIff
ψ : K ⟶ L.extend e
i' : ι'
⊢ (e.liftExtend (e.homRestric... | by_cases hi' : ∃ i, e.f i = i' | case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{u_4, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
L : HomologicalComplex C c
inst✝ : e.IsRelIff
ψ : K ⟶ L.extend e
i' : ι'
hi' : ∃ i, e.f i = i'
⊢ (e.... | 9240214e45cb80b5 |
MeasureTheory.Measure.mkMetric_mono_smul | Mathlib/MeasureTheory/Measure/Hausdorff.lean | theorem mkMetric_mono_smul {m₁ m₂ : ℝ≥0∞ → ℝ≥0∞} {c : ℝ≥0∞} (hc : c ≠ ∞) (h0 : c ≠ 0)
(hle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂) : (mkMetric m₁ : Measure X) ≤ c • mkMetric m₂ := fun s ↦ by
rw [← OuterMeasure.coe_mkMetric, coe_smul, ← OuterMeasure.coe_mkMetric]
exact OuterMeasure.mkMetric_mono_smul hc h0 hle s
| X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
m₁ m₂ : ℝ≥0∞ → ℝ≥0∞
c : ℝ≥0∞
hc : c ≠ ⊤
h0 : c ≠ 0
hle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂
s : Set X
⊢ (OuterMeasure.mkMetric m₁) s ≤ (c • ⇑(OuterMeasure.mkMetric m₂)) s | exact OuterMeasure.mkMetric_mono_smul hc h0 hle s | no goals | 5cc82bb7072f7bff |
ContDiff.lipschitzWith_of_hasCompactSupport | Mathlib/Analysis/Calculus/ContDiff/RCLike.lean | theorem ContDiff.lipschitzWith_of_hasCompactSupport {f : E' → F'}
(hf : HasCompactSupport f) (h'f : ContDiff 𝕂 n f) (hn : 1 ≤ n) :
∃ C, LipschitzWith C f | n : WithTop ℕ∞
𝕂 : Type u_1
inst✝⁴ : RCLike 𝕂
E' : Type u_2
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕂 E'
F' : Type u_3
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace 𝕂 F'
f : E' → F'
hf : HasCompactSupport f
h'f : ContDiff 𝕂 n f
hn : 1 ≤ n
⊢ ∃ C, LipschitzWith C f | obtain ⟨C, hC⟩ := (hf.fderiv 𝕂).exists_bound_of_continuous (h'f.continuous_fderiv hn) | case intro
n : WithTop ℕ∞
𝕂 : Type u_1
inst✝⁴ : RCLike 𝕂
E' : Type u_2
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕂 E'
F' : Type u_3
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace 𝕂 F'
f : E' → F'
hf : HasCompactSupport f
h'f : ContDiff 𝕂 n f
hn : 1 ≤ n
C : ℝ
hC : ∀ (x : E'), ‖fderiv 𝕂 f x‖ ≤ C
⊢ ∃ C... | f4e30994751a1370 |
Ordinal.nfpFamily_le_fp | Mathlib/SetTheory/Ordinal/FixedPoint.lean | theorem nfpFamily_le_fp (H : ∀ i, Monotone (f i)) {a b} (ab : a ≤ b) (h : ∀ i, f i b ≤ b) :
nfpFamily f a ≤ b | case a.nil
ι : Type u_1
f : ι → Ordinal.{u} → Ordinal.{u}
H : ∀ (i : ι), Monotone (f i)
b : Ordinal.{u}
h : ∀ (i : ι), f i b ≤ b
a : Ordinal.{u}
ab : a ≤ b
⊢ List.foldr f a [] ≤ b | exact ab | no goals | ecbfdbb6d3f9b8a4 |
IsCompactOperator.continuous | Mathlib/Analysis/Normed/Operator/Compact.lean | theorem IsCompactOperator.continuous {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator f) :
Continuous f | case intro.intro
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜₁
inst✝¹¹ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝¹⁰ : RingHomIsometric σ₁₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁹ : TopologicalSpace M₁
inst✝⁸ : AddCommGroup M₁
inst✝⁷ : TopologicalSpace M₂
inst✝⁶ : AddCommGroup M₂
inst✝⁵ : Mod... | rcases (hK.totallyBounded.isVonNBounded 𝕜₂ hU).exists_pos with ⟨r, hr, hrU⟩ | case intro.intro.intro.intro
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝¹² : NontriviallyNormedField 𝕜₁
inst✝¹¹ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
inst✝¹⁰ : RingHomIsometric σ₁₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁹ : TopologicalSpace M₁
inst✝⁸ : AddCommGroup M₁
inst✝⁷ : TopologicalSpace M₂
inst✝⁶ : AddCommGroup M₂
... | d3feedeab452c721 |
CategoryTheory.ShortComplex.mono_homologyMap_of_mono_opcyclesMap' | Mathlib/Algebra/Homology/ShortComplex/Homology.lean | lemma mono_homologyMap_of_mono_opcyclesMap'
[S₁.HasHomology] [S₂.HasHomology] (h : Mono (opcyclesMap φ)) :
Mono (homologyMap φ) | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
h : Mono (opcyclesMap φ)
⊢ Mono (homologyMap φ) | have : Mono (homologyMap φ ≫ S₂.homologyι) := by
rw [homologyι_naturality φ]
apply mono_comp | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
h : Mono (opcyclesMap φ)
this : Mono (homologyMap φ ≫ S₂.homologyι)
⊢ Mono (homologyMap φ) | 5b55ce4bc8aac5ed |
ContinuousLinearMap.exists_preimage_norm_le | Mathlib/Analysis/Normed/Operator/Banach.lean | theorem exists_preimage_norm_le (surj : Surjective f) :
∃ C > 0, ∀ y, ∃ x, f x = y ∧ ‖x‖ ≤ C * ‖y‖ | case intro.intro
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝⁴ : RingHo... | choose g hg using hC | case intro.intro
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝⁴ : RingHo... | fbb58cd4816e6d13 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.mem_of_insertRupUnits | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean | theorem mem_of_insertRupUnits {n : Nat} (f : DefaultFormula n) (units : CNF.Clause (PosFin n))
(c : DefaultClause n) :
c ∈ toList (insertRupUnits f units).1 → c ∈ units.map Clause.unit ∨ c ∈ toList f | case inr.inr
n : Nat
f : DefaultFormula n
units : CNF.Clause (PosFin n)
c : DefaultClause n
hb : ∀ (l : Literal (PosFin n)), l ∈ (f.rupUnits, f.assignments, false).fst.toList → l ∈ f.rupUnits.toList ∨ l ∈ units
hl :
∀ (acc : Array (Literal (PosFin n)) × Array Assignment × Bool),
(∀ (l : Literal (PosFin n)), l ∈ a... | exact (Or.inr ∘ Or.inr ∘ Or.inr) h | no goals | d2213faecfa0fcb1 |
CategoryTheory.exists_simple_subobject | Mathlib/CategoryTheory/Noetherian.lean | theorem exists_simple_subobject {X : C} [ArtinianObject X] (h : ¬IsZero X) :
∃ Y : Subobject X, Simple (Y : C) | C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
X : C
inst✝ : ArtinianObject X
h : ¬IsZero X
this : Nontrivial (Subobject X)
⊢ ∃ Y, Simple (underlying.obj Y) | haveI := isAtomic_of_orderBot_wellFounded_lt (ArtinianObject.subobject_lt_wellFounded X) | C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
X : C
inst✝ : ArtinianObject X
h : ¬IsZero X
this✝ : Nontrivial (Subobject X)
this : IsAtomic (Subobject X)
⊢ ∃ Y, Simple (underlying.obj Y) | 7108ac09c6589114 |
MeasureTheory.Submartingale.ae_tendsto_limitProcess | Mathlib/Probability/Martingale/Convergence.lean | theorem Submartingale.ae_tendsto_limitProcess [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hbdd : ∀ n, eLpNorm (f n) 1 μ ≤ R) :
∀ᵐ ω ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (ℱ.limitProcess f μ ω)) | Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hbdd : ∀ (n : ℕ), eLpNorm (f n) 1 μ ≤ ↑R
g' : Ω → ℝ := fun ω => if h : ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) then h.choose else 0
hle : ⨆ n, ↑ℱ n ≤ m0
hg' : ∀ᵐ (ω : Ω) ∂μ.t... | have hg'm : AEStronglyMeasurable[⨆ n, ℱ n] g' (μ.trim hle) :=
(@aemeasurable_of_tendsto_metrizable_ae' _ _ (⨆ n, ℱ n) _ _ _ _ _ _ _
(fun n => ((hf.stronglyMeasurable n).measurable.mono (le_sSup ⟨n, rfl⟩ : ℱ n ≤ ⨆ n, ℱ n)
le_rfl).aemeasurable) hg').aestronglyMeasurable | Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Submartingale f ℱ μ
hbdd : ∀ (n : ℕ), eLpNorm (f n) 1 μ ≤ ↑R
g' : Ω → ℝ := fun ω => if h : ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) then h.choose else 0
hle : ⨆ n, ↑ℱ n ≤ m0
hg' : ∀ᵐ (ω : Ω) ∂μ.t... | 925ddc4c95d6c664 |
PhragmenLindelof.horizontal_strip | Mathlib/Analysis/Complex/PhragmenLindelof.lean | theorem horizontal_strip (hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo a b))
(hB : ∃ c < π / (b - a), ∃ B, f =O[comap (_root_.abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo a b)]
fun z ↦ expR (B * expR (c * |z.re|)))
(hle_a : ∀ z : ℂ, im z = a → ‖f z‖ ≤ C) (hle_b : ∀ z, im z = b → ‖f z‖ ≤ C) (hza : a ≤ im z)
(hzb : im z ≤ b)... | case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C✝ : ℝ
f : ℂ → E
z : ℂ
C : ℝ
hC₀ : 0 < C
a b : ℝ
hza : a - b < z.im
hle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C
hzb : z.im < a + b
hle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C
hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))
hab : a - b < a + b... | simp | no goals | 96530d0dcac40a71 |
LightProfinite.epi_iff_surjective | Mathlib/Topology/Category/LightProfinite/Basic.lean | theorem epi_iff_surjective {X Y : LightProfinite.{u}} (f : X ⟶ Y) :
Epi f ↔ Function.Surjective f | case mp.intro.intro.intro.intro
X Y : LightProfinite
f : X ⟶ Y
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
hf : Epi f
C : Set ((fun X => ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
U : Set ((fun X => ↑X.toTop) Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ 𝓝 y
V : Set ↑Y.toTop
hV : V ∈ {... | classical
let Z := of (ULift.{u} <| Fin 2)
let g : Y ⟶ Z := CompHausLike.ofHom _
⟨(LocallyConstant.ofIsClopen hV).map ULift.up, LocallyConstant.continuous _⟩
let h : Y ⟶ Z := CompHausLike.ofHom _ ⟨fun _ => ⟨1⟩, continuous_const⟩
have H : h = g := by
rw [← cancel_epi f]
ext x
dsimp [g, LocallyCon... | no goals | 455ee9a4bcba472a |
Nat.superFactorial_two_mul | Mathlib/Data/Nat/Factorial/SuperFactorial.lean | theorem superFactorial_two_mul : ∀ n : ℕ,
sf (2 * n) = (∏ i ∈ range n, (2 * i + 1) !) ^ 2 * 2 ^ n * n !
| 0 => rfl
| (n + 1) => by
simp only [prod_range_succ, mul_pow, mul_add, mul_one, superFactorial_succ,
superFactorial_two_mul n, factorial_succ]
ring
| n : ℕ
⊢ (2 * n + 1 + 1) * ((2 * n + 1) * (2 * n)!) *
((2 * n + 1) * (2 * n)! * ((∏ x ∈ range n, (2 * x + 1) * (2 * x)!) ^ 2 * 2 ^ n * n !)) =
(∏ x ∈ range n, (2 * x + 1) * (2 * x)!) ^ 2 * ((2 * n + 1) ^ 2 * (2 * n)! ^ 2) * 2 ^ (n + 1) * ((n + 1) * n !) | ring | no goals | 2d9182a187da3fb8 |
Algebra.FiniteType.iff_quotient_mvPolynomial'' | Mathlib/RingTheory/FiniteType.lean | theorem iff_quotient_mvPolynomial'' :
FiniteType R S ↔ ∃ (n : ℕ) (f : MvPolynomial (Fin n) R →ₐ[R] S), Surjective f | case mp.intro.intro.intro
R : Type uR
S : Type uS
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
ι : Type uS
hfintype : Fintype ι
f : MvPolynomial ι R →ₐ[R] S
hsur : Surjective ⇑f
equiv : MvPolynomial ι R ≃ₐ[R] MvPolynomial (Fin (Fintype.card ι)) R
⊢ ∃ n f, Surjective ⇑f | exact ⟨Fintype.card ι, AlgHom.comp f equiv.symm.toAlgHom, by simpa using hsur⟩ | no goals | 1f17714b519bc9b5 |
ADEInequality.admissible_of_one_lt_sumInv | Mathlib/NumberTheory/ADEInequality.lean | theorem admissible_of_one_lt_sumInv {p q r : ℕ+} (H : 1 < sumInv {p, q, r}) :
Admissible {p, q, r} | p q r : ℕ+
H : 1 < sumInv {p, q, r}
S : List ℕ+ := sort (fun x1 x2 => x1 ≤ x2) {p, q, r}
hS : List.Sorted (fun x1 x2 => x1 ≤ x2) S
hpqr : {p, q, r} = ↑S
⊢ (∃ q_1 r_1, A' q_1 r_1 = {p, q, r}) ∨
(∃ r_1, D' r_1 = {p, q, r}) ∨ E' 3 = {p, q, r} ∨ E' 4 = {p, q, r} ∨ E' 5 = {p, q, r} | rw [hpqr] | p q r : ℕ+
H : 1 < sumInv {p, q, r}
S : List ℕ+ := sort (fun x1 x2 => x1 ≤ x2) {p, q, r}
hS : List.Sorted (fun x1 x2 => x1 ≤ x2) S
hpqr : {p, q, r} = ↑S
⊢ (∃ q r, A' q r = ↑S) ∨ (∃ r, D' r = ↑S) ∨ E' 3 = ↑S ∨ E' 4 = ↑S ∨ E' 5 = ↑S | 0a2e506e5b15e061 |
Std.Sat.AIG.RefVec.zip.go_denote_mem_prefix | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/RefVecOperator/Zip.lean | theorem go_denote_mem_prefix {aig : AIG α} (curr : Nat) (hcurr : curr ≤ len)
(s : RefVec aig curr) (lhs rhs : RefVec aig len)
(f : (aig : AIG α) → BinaryInput aig → Entrypoint α) [LawfulOperator α BinaryInput f]
[chainable : LawfulZipOperator α f] (start : Nat) (hstart) :
⟦
(go aig curr s hcurr lh... | case a
α : Type
inst✝² : Hashable α
inst✝¹ : DecidableEq α
aig✝ : AIG α
len : Nat
assign : α → Bool
aig : AIG α
curr : Nat
hcurr : curr ≤ len
s : aig.RefVec curr
lhs rhs : aig.RefVec len
f : (aig : AIG α) → aig.BinaryInput → Entrypoint α
inst✝ : LawfulOperator α BinaryInput f
chainable : LawfulZipOperator α f
start : N... | apply go_le_size | no goals | 9185f603cad77490 |
Algebra.FinitePresentation.of_restrict_scalars_finitePresentation | Mathlib/RingTheory/FinitePresentation.lean | theorem of_restrict_scalars_finitePresentation [Algebra A B] [IsScalarTower R A B]
[FinitePresentation.{w₁, w₃} R B] [FiniteType R A] :
FinitePresentation.{w₂, w₃} A B | case refine_1
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : Surjective ⇑f
s :... | rintro _ (⟨x, hx, rfl⟩ | ⟨i, rfl⟩) | case refine_1.inl.intro.intro
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : S... | 765dbb818bc3202d |
AkraBazziRecurrence.rpow_p_mul_one_add_smoothingFn_ge | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | lemma rpow_p_mul_one_add_smoothingFn_ge :
∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
≤ (r i n) ^ (p a b) * (1 + ε (r i n)) | case bn_gt_one
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 + ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x => x ^ (p a b - 1)
h_main_norm :
(fun n =>... | case bn_gt_one =>
calc 1 = b i * (b i)⁻¹ := by rw [mul_inv_cancel₀ (by positivity)]
_ ≤ b i * ⌈(b i)⁻¹⌉₊ := by gcongr; exact Nat.le_ceil _
_ < b i * n := by gcongr | case le
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 + ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x => x ^ (p a b - 1)
h_main_norm :
(fun n => ‖q ↑(r... | 6b09e1ce84116b26 |
Plausible.InjectiveFunction.List.applyId_eq_self | Mathlib/Testing/Plausible/Functions.lean | theorem List.applyId_eq_self [DecidableEq α] {xs ys : List α} (x : α) :
x ∉ xs → List.applyId.{u} (xs.zip ys) x = x | α : Type u
inst✝ : DecidableEq α
xs ys : List α
x : α
h : x ∉ xs
⊢ ∀ (x_1 : α), (x, x_1) ∉ xs.zip ys | intro y hy | α : Type u
inst✝ : DecidableEq α
xs ys : List α
x : α
h : x ∉ xs
y : α
hy : (x, y) ∈ xs.zip ys
⊢ False | 1f94b1843ffbca38 |
MeasureTheory.eLpNorm_le_eLpNorm_mul_eLpNorm_of_nnnorm | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | theorem eLpNorm_le_eLpNorm_mul_eLpNorm_of_nnnorm {p q r : ℝ≥0∞}
(hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) (b : E → F → G) (c : ℝ≥0)
(h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊) [hpqr : HolderTriple p q r] :
eLpNorm (fun x => b (f x) (g x)) r μ ≤ c * eLpNorm f p μ * eLpNorm g q... | α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) ... | positivity | no goals | 73f605930a2a90d7 |
mdifferentiableAt_iff | Mathlib/Geometry/Manifold/MFDeriv/Defs.lean | theorem mdifferentiableAt_iff (f : M → M') (x : M) :
MDifferentiableAt I I' f x ↔ ContinuousAt f x ∧
DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f) (range I) ((extChartAt I x) x) | case a
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : Nor... | rfl | no goals | 3d23627734416098 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.nodup_derivedLits | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem nodup_derivedLits {n : Nat} (f : DefaultFormula n)
(f_assignments_size : f.assignments.size = n) (rupHints : Array Nat)
(f'_assignments_size : (performRupCheck f rupHints).1.assignments.size = n)
(derivedLits : CNF.Clause (PosFin n))
(derivedLits_satisfies_invariant:
DerivedLitsInvariant f... | n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
rupHints : Array Nat
f'_assignments_size : (f.performRupCheck rupHints).fst.assignments.size = n
derivedLits : CNF.Clause (PosFin n)
derivedLits_satisfies_invariant :
f.DerivedLitsInvariant f_assignments_size (f.performRupCheck rupHints).fst.ass... | rw [← j_eq_k2] at i_eq_k2 | n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
rupHints : Array Nat
f'_assignments_size : (f.performRupCheck rupHints).fst.assignments.size = n
derivedLits : CNF.Clause (PosFin n)
derivedLits_satisfies_invariant :
f.DerivedLitsInvariant f_assignments_size (f.performRupCheck rupHints).fst.ass... | b565b2925b517fa0 |
ENNReal.tendsto_finset_prod_of_ne_top | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | theorem tendsto_finset_prod_of_ne_top {ι : Type*} {f : ι → α → ℝ≥0∞} {x : Filter α} {a : ι → ℝ≥0∞}
(s : Finset ι) (h : ∀ i ∈ s, Tendsto (f i) x (𝓝 (a i))) (h' : ∀ i ∈ s, a i ≠ ∞) :
Tendsto (fun b => ∏ c ∈ s, f c b) x (𝓝 (∏ c ∈ s, a c)) | case insert.ha
α : Type u_1
ι : Type u_4
f : ι → α → ℝ≥0∞
x : Filter α
a : ι → ℝ≥0∞
a✝ : ι
s✝ : Finset ι
has : a✝ ∉ s✝
IH :
(∀ i ∈ s✝, Tendsto (f i) x (𝓝 (a i))) → (∀ i ∈ s✝, a i ≠ ⊤) → Tendsto (fun b => ∏ c ∈ s✝, f c b) x (𝓝 (∏ c ∈ s✝, a c))
h : ∀ i ∈ insert a✝ s✝, Tendsto (f i) x (𝓝 (a i))
h' : ∀ i ∈ insert a✝ s... | right | case insert.ha.h
α : Type u_1
ι : Type u_4
f : ι → α → ℝ≥0∞
x : Filter α
a : ι → ℝ≥0∞
a✝ : ι
s✝ : Finset ι
has : a✝ ∉ s✝
IH :
(∀ i ∈ s✝, Tendsto (f i) x (𝓝 (a i))) → (∀ i ∈ s✝, a i ≠ ⊤) → Tendsto (fun b => ∏ c ∈ s✝, f c b) x (𝓝 (∏ c ∈ s✝, a c))
h : ∀ i ∈ insert a✝ s✝, Tendsto (f i) x (𝓝 (a i))
h' : ∀ i ∈ insert a✝... | 42dbf6b445e9974c |
Topology.IsInducing.le_functorObj_iff | Mathlib/Topology/Category/TopCat/Opens.lean | lemma le_functorObj_iff {X Y : TopCat} {f : X ⟶ Y} (hf : IsInducing f) {U : Opens X}
{V : Opens Y} : V ≤ hf.functorObj U ↔ (Opens.map f).obj V ≤ U | case mk
X Y : TopCat
f : X ⟶ Y
hf : IsInducing ⇑(ConcreteCategory.hom f)
V : Opens ↑Y
U : Set ↑X
hU : IsOpen U
⊢ V ≤ hf.functorObj { carrier := U, is_open' := hU } ↔ (Opens.map f).obj V ≤ { carrier := U, is_open' := hU } | obtain ⟨t, ht, rfl⟩ := hf.isOpen_iff.mp hU | case mk.intro.intro
X Y : TopCat
f : X ⟶ Y
hf : IsInducing ⇑(ConcreteCategory.hom f)
V : Opens ↑Y
t : Set ↑Y
ht : IsOpen t
hU : IsOpen (⇑(ConcreteCategory.hom f) ⁻¹' t)
⊢ V ≤ hf.functorObj { carrier := ⇑(ConcreteCategory.hom f) ⁻¹' t, is_open' := hU } ↔
(Opens.map f).obj V ≤ { carrier := ⇑(ConcreteCategory.hom f) ⁻... | f5d72770c98cc6d6 |
List.perm_permutations'Aux_comm | Mathlib/Data/List/Permutation.lean | theorem perm_permutations'Aux_comm (a b : α) (l : List α) :
(permutations'Aux a l).flatMap (permutations'Aux b) ~
(permutations'Aux b l).flatMap (permutations'Aux a) | case cons.p
α : Type u_1
a b c : α
l : List α
ih : flatMap (permutations'Aux b) (permutations'Aux a l) ~ flatMap (permutations'Aux a) (permutations'Aux b l)
this :
∀ (a b : α),
flatMap (permutations'Aux b) (map (cons c) (permutations'Aux a l)) ~
map (cons b ∘ cons c) (permutations'Aux a l) ++ map (cons c) (... | refine (((this _ _).append_left _).trans ?_).trans ((this _ _).append_left _).symm | case cons.p
α : Type u_1
a b c : α
l : List α
ih : flatMap (permutations'Aux b) (permutations'Aux a l) ~ flatMap (permutations'Aux a) (permutations'Aux b l)
this :
∀ (a b : α),
flatMap (permutations'Aux b) (map (cons c) (permutations'Aux a l)) ~
map (cons b ∘ cons c) (permutations'Aux a l) ++ map (cons c) (... | a83bf3791227cbb6 |
AlgebraicGeometry.Scheme.zeroLocus_span | Mathlib/AlgebraicGeometry/Scheme.lean | lemma zeroLocus_span {U : X.Opens} (s : Set Γ(X, U)) :
X.zeroLocus (U := U) (Ideal.span s) = X.zeroLocus s | case h.refine_1
X : Scheme
U : X.Opens
s : Set ↑Γ(X, U)
x : ↑↑X.toPresheafedSpace
H : ∀ f ∈ s, x ∉ X.basicOpen f
f : ↑Γ(X, U)
⊢ x ∉ X.basicOpen 0 | simp only [Scheme.basicOpen_zero] | case h.refine_1
X : Scheme
U : X.Opens
s : Set ↑Γ(X, U)
x : ↑↑X.toPresheafedSpace
H : ∀ f ∈ s, x ∉ X.basicOpen f
f : ↑Γ(X, U)
⊢ x ∉ ⊥ | 8d238c0d3d972b9d |
RelSeries.head_smash | Mathlib/Order/RelSeries.lean | @[simp] lemma head_smash {p q : RelSeries r} (h : p.last = q.head) :
(smash p q h).head = p.head | case h.e'_3.e_a
α : Type u_1
r : Rel α α
p q : RelSeries r
h : p.last = q.head
H : p.length = 0
⊢ 0 = Fin.last p.length | aesop | no goals | 5e0aacb3d3ad0d17 |
Int.two_pow_two_pow_add_two_pow_two_pow | Mathlib/NumberTheory/Multiplicity.lean | theorem Int.two_pow_two_pow_add_two_pow_two_pow {x y : ℤ} (hx : ¬2 ∣ x) (hxy : 4 ∣ x - y) (i : ℕ) :
emultiplicity 2 (x ^ 2 ^ i + y ^ 2 ^ i) = ↑(1 : ℕ) | x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
i : ℕ
this : ∀ (x : ℤ), Odd x → x ^ 2 ^ (i + 1) % 4 = 1
⊢ ¬(1 + 1) % 4 = 0 | decide | no goals | c2c4f247c9f39fe5 |
List.mk_mem_zipIdx_iff_le_and_getElem?_sub | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Range.lean | theorem mk_mem_zipIdx_iff_le_and_getElem?_sub {k i : Nat} {x : α} {l : List α} :
(x, i) ∈ zipIdx l k ↔ k ≤ i ∧ l[i - k]? = some x | α : Type u_1
k : Nat
x : α
l : List α
m✝ : Nat
h : ¬k ≤ k + m✝
⊢ False | simp at h | no goals | 24ce2639b2cbb140 |
Monoid.CoprodI.Word.rcons_inj | Mathlib/GroupTheory/CoprodI.lean | theorem rcons_inj {i} : Function.Injective (rcons : Pair M i → Word M) | case neg.intro
ι : Type u_1
M : ι → Type u_2
inst✝¹ : (i : ι) → Monoid (M i)
inst✝ : (i : ι) → DecidableEq (M i)
i : ι
m : M i
w : Word M
h✝ : w.fstIdx ≠ some i
w' : Word M
h' : w'.fstIdx ≠ some i
hm : ¬m = 1
h : w.toList = w'.toList
he : rcons { head := m, tail := w, fstIdx_ne := h✝ } = rcons { head := m, tail := w', ... | congr | case neg.intro.e_tail
ι : Type u_1
M : ι → Type u_2
inst✝¹ : (i : ι) → Monoid (M i)
inst✝ : (i : ι) → DecidableEq (M i)
i : ι
m : M i
w : Word M
h✝ : w.fstIdx ≠ some i
w' : Word M
h' : w'.fstIdx ≠ some i
hm : ¬m = 1
h : w.toList = w'.toList
he : rcons { head := m, tail := w, fstIdx_ne := h✝ } = rcons { head := m, tail ... | 06e80c2544642218 |
Complex.det_conjAe | Mathlib/Data/Complex/Determinant.lean | theorem det_conjAe : LinearMap.det conjAe.toLinearMap = -1 | ⊢ 1 * -1 - 0 * 0 = -1 | simp | no goals | 2d12af111ab4e07f |
NoZeroSMulDivisors_of_isLocalization | Mathlib/RingTheory/Localization/Ideal.lean | lemma _root_.NoZeroSMulDivisors_of_isLocalization (Rₚ Sₚ : Type*) [CommRing Rₚ] [CommRing Sₚ]
[Algebra R Rₚ] [Algebra R Sₚ] [Algebra S Sₚ] [Algebra Rₚ Sₚ] [IsScalarTower R S Sₚ]
[IsScalarTower R Rₚ Sₚ] {M : Submonoid R} (hM : M ≤ R⁰) [IsLocalization M Rₚ]
[IsLocalization (Algebra.algebraMapSubmonoid S M) Sₚ... | case h
R : Type u_1
inst✝¹⁴ : CommRing R
S : Type u_2
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹¹ : CommRing Rₚ
inst✝¹⁰ : CommRing Sₚ
inst✝⁹ : Algebra R Rₚ
inst✝⁸ : Algebra R Sₚ
inst✝⁷ : Algebra S Sₚ
inst✝⁶ : Algebra Rₚ Sₚ
inst✝⁵ : IsScalarTower R S Sₚ
inst✝⁴ : IsScalarTower R Rₚ Sₚ
M... | simp only [IsLocalization.map_comp, ← IsScalarTower.algebraMap_eq] | no goals | d2c3122cbb91cbb5 |
MeasureTheory.eLpNorm_one_condExp_le_eLpNorm | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | theorem eLpNorm_one_condExp_le_eLpNorm (f : α → ℝ) : eLpNorm (μ[f|m]) 1 μ ≤ eLpNorm f 1 μ | α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
hm : m ≤ m0
hsig : SigmaFinite (μ.trim hm)
⊢ ∀ᵐ (x : α) ∂μ, ‖(μ[f|m]) x‖ ≤ ‖(μ[|f| |m]) x‖ | filter_upwards [condExp_mono hf hf.abs
(ae_of_all μ (fun x => le_abs_self (f x) : ∀ x, f x ≤ |f x|)),
(condExp_neg ..).symm.le.trans (condExp_mono hf.neg hf.abs
(ae_of_all μ (fun x => neg_le_abs (f x) : ∀ x, -f x ≤ |f x|)))] with x hx₁ hx₂ | case h
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
hm : m ≤ m0
hsig : SigmaFinite (μ.trim hm)
x : α
hx₁ : (μ[f|?m.9696]) x ≤ (μ[fun a => |f a| |?m.9696]) x
hx₂ : (-μ[f|?m.8598]) x ≤ (μ[fun a => |f a| |?m.8598]) x
⊢ ‖(μ[f|m]) x‖ ≤ ‖(μ[|f| |m]) x‖ | 4c3dc2a5be2f6f42 |
CategoryTheory.unit_mateEquiv | Mathlib/CategoryTheory/Adjunction/Mates.lean | theorem unit_mateEquiv (α : TwoSquare G L₁ L₂ H) (c : C) :
G.map (adj₁.unit.app c) ≫ (mateEquiv adj₁ adj₂ α).app _ =
adj₂.unit.app _ ≫ R₂.map (α.app _) | C : Type u₁
D : Type u₂
E : Type u₃
F : Type u₄
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} E
inst✝ : Category.{v₄, u₄} F
G : C ⥤ E
H : D ⥤ F
L₁ : C ⥤ D
R₁ : D ⥤ C
L₂ : E ⥤ F
R₂ : F ⥤ E
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
α : TwoSquare G L₁ L₂ H
c : C
⊢ G.map (adj₁.unit.app c) ≫ ((mat... | dsimp [mateEquiv] | C : Type u₁
D : Type u₂
E : Type u₃
F : Type u₄
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
inst✝¹ : Category.{v₃, u₃} E
inst✝ : Category.{v₄, u₄} F
G : C ⥤ E
H : D ⥤ F
L₁ : C ⥤ D
R₁ : D ⥤ C
L₂ : E ⥤ F
R₂ : F ⥤ E
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
α : TwoSquare G L₁ L₂ H
c : C
⊢ G.map (adj₁.unit.app c) ≫
... | 75158210b8e0ab27 |
Ring.multichoose_succ_succ | Mathlib/RingTheory/Binomial.lean | theorem multichoose_succ_succ (r : R) (k : ℕ) :
multichoose (r + 1) (k + 1) = multichoose r (k + 1) + multichoose (r + 1) k | R : Type u_2
inst✝³ : NonAssocSemiring R
inst✝² : Pow R ℕ
inst✝¹ : NatPowAssoc R
inst✝ : BinomialRing R
r : R
k : ℕ
⊢ (k + 1).factorial • multichoose (r + 1) (k + 1) = (k + 1).factorial • (multichoose r (k + 1) + multichoose (r + 1) k) | simp only [factorial_nsmul_multichoose_eq_ascPochhammer, smul_add] | R : Type u_2
inst✝³ : NonAssocSemiring R
inst✝² : Pow R ℕ
inst✝¹ : NatPowAssoc R
inst✝ : BinomialRing R
r : R
k : ℕ
⊢ (ascPochhammer ℕ (k + 1)).smeval (r + 1) =
(ascPochhammer ℕ (k + 1)).smeval r + (k + 1).factorial • multichoose (r + 1) k | 53a0918665dbbe2b |
Ring.smeval_ascPochhammer_neg_of_lt | Mathlib/RingTheory/Binomial.lean | theorem smeval_ascPochhammer_neg_of_lt {n k : ℕ} (h : n < k) :
smeval (ascPochhammer ℕ k) (-n : ℤ) = 0 | n k : ℕ
h : n < k
⊢ (ascPochhammer ℕ k).smeval (-↑n) = 0 | rw [show k = n + (k - n - 1) + 1 by omega, smeval_ascPochhammer_neg_add] | no goals | b22ffd58c9eb161f |
MultilinearMap.dfinsupp_ext | Mathlib/LinearAlgebra/Multilinear/DFinsupp.lean | theorem dfinsupp_ext [∀ i, DecidableEq (κ i)]
⦃f g : MultilinearMap R (fun i ↦ Π₀ j : κ i, M i j) N⦄
(h : ∀ p : Π i, κ i,
f.compLinearMap (fun i => DFinsupp.lsingle (p i)) =
g.compLinearMap (fun i => DFinsupp.lsingle (p i))) : f = g | case H
ι : Type uι
κ : ι → Type uκ
R : Type uR
M : (i : ι) → κ i → Type uM
N : Type uN
inst✝⁶ : Finite ι
inst✝⁵ : Semiring R
inst✝⁴ : (i : ι) → (k : κ i) → AddCommMonoid (M i k)
inst✝³ : AddCommMonoid N
inst✝² : (i : ι) → (k : κ i) → Module R (M i k)
inst✝¹ : Module R N
inst✝ : (i : ι) → DecidableEq (κ i)
f g : Multili... | show f (fun i ↦ x i) = g (fun i ↦ x i) | case H
ι : Type uι
κ : ι → Type uκ
R : Type uR
M : (i : ι) → κ i → Type uM
N : Type uN
inst✝⁶ : Finite ι
inst✝⁵ : Semiring R
inst✝⁴ : (i : ι) → (k : κ i) → AddCommMonoid (M i k)
inst✝³ : AddCommMonoid N
inst✝² : (i : ι) → (k : κ i) → Module R (M i k)
inst✝¹ : Module R N
inst✝ : (i : ι) → DecidableEq (κ i)
f g : Multili... | b6ddbe2bdca9e69b |
List.mem_take_iff_getElem | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/TakeDrop.lean | theorem mem_take_iff_getElem {l : List α} {a : α} :
a ∈ l.take n ↔ ∃ (i : Nat) (hm : i < min n l.length), l[i] = a | case mpr
α : Type u_1
n : Nat
l : List α
a : α
⊢ (∃ i hm, l[i] = a) → ∃ i h, (take n l)[i] = a | rintro ⟨i, hm, rfl⟩ | case mpr.intro.intro
α : Type u_1
n : Nat
l : List α
i : Nat
hm : i < min n l.length
⊢ ∃ i_1 h, (take n l)[i_1] = l[i] | ae2633f4b8825c5b |
KummerDedekind.normalizedFactorsMapEquivNormalizedFactorsMinPolyMk_symm_apply_eq_span | Mathlib/NumberTheory/KummerDedekind.lean | theorem normalizedFactorsMapEquivNormalizedFactorsMinPolyMk_symm_apply_eq_span
(hI : I.IsMaximal) {Q : R[X]}
(hQ : Q.map (Ideal.Quotient.mk I) ∈ normalizedFactors ((minpoly R x).map (Ideal.Quotient.mk I)))
(hI' : I ≠ ⊥) (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) :
((nor... | R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
Q : R[X]
hQ : Polynomial.map (Ideal.Quotient.mk I) Q ∈ normalizedFactors (Polynomial.map ... | simp | no goals | d82e13c84e2368ef |
IncidenceAlgebra.mu_toDual | Mathlib/Combinatorics/Enumerative/IncidenceAlgebra.lean | @[simp]
lemma mu_toDual (a b : α) : mu 𝕜 (toDual a) (toDual b) = mu 𝕜 b a | 𝕜 : Type u_2
α : Type u_5
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
this : DecidableRel fun x1 x2 => x1 ≤ x2 := Classical.decRel fun x1 x2 => x1 ≤ x2
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯ }
a b : ... | simp [eq_comm] | no goals | 95f4eed14862eb7a |
Matrix.twoBlockTriangular_det' | Mathlib/LinearAlgebra/Matrix/Block.lean | theorem twoBlockTriangular_det' (M : Matrix m m R) (p : m → Prop) [DecidablePred p]
(h : ∀ i, p i → ∀ j, ¬p j → M i j = 0) :
M.det = (toSquareBlockProp M p).det * (toSquareBlockProp M fun i => ¬p i).det | m : Type u_3
R : Type v
inst✝³ : CommRing R
inst✝² : DecidableEq m
inst✝¹ : Fintype m
M : Matrix m m R
p : m → Prop
inst✝ : DecidablePred p
h : ∀ (i : m), p i → ∀ (j : m), ¬p j → M i j = 0
⊢ (M.toSquareBlockProp fun i => ¬¬p i).det * (M.toSquareBlockProp fun i => ¬p i).det =
(M.toSquareBlockProp p).det * (M.toSquar... | congr 1 | case e_a
m : Type u_3
R : Type v
inst✝³ : CommRing R
inst✝² : DecidableEq m
inst✝¹ : Fintype m
M : Matrix m m R
p : m → Prop
inst✝ : DecidablePred p
h : ∀ (i : m), p i → ∀ (j : m), ¬p j → M i j = 0
⊢ (M.toSquareBlockProp fun i => ¬¬p i).det = (M.toSquareBlockProp p).det | d9f730cd9899e951 |
ENNReal.mul_lt_top_iff | Mathlib/Data/ENNReal/Order.lean | theorem mul_lt_top_iff {a b : ℝ≥0∞} : a * b < ∞ ↔ a < ∞ ∧ b < ∞ ∨ a = 0 ∨ b = 0 | case mp
a b : ℝ≥0∞
h : a * b < ⊤
⊢ a < ⊤ ∧ b < ⊤ ∨ a = 0 ∨ b = 0 | rw [← or_assoc, or_iff_not_imp_right, or_iff_not_imp_right] | case mp
a b : ℝ≥0∞
h : a * b < ⊤
⊢ ¬b = 0 → ¬a = 0 → a < ⊤ ∧ b < ⊤ | efc09a1046ed1385 |
LaurentPolynomial.comul_C_mul_T_self | Mathlib/RingTheory/Coalgebra/MonoidAlgebra.lean | theorem comul_C_mul_T_self (a : R) (n : ℤ) :
Coalgebra.comul (C a * T n) = T n ⊗ₜ[R] (C a * T n) | R : Type u_1
inst✝ : CommSemiring R
a : R
n : ℤ
⊢ CoalgebraStruct.comul (C a * T n) = T n ⊗ₜ[R] (C a * T n) | simp | no goals | ffbc6a8887a7688b |
MonotoneOn.mapsTo_Iic | Mathlib/Order/Interval/Set/Image.lean | lemma MonotoneOn.mapsTo_Iic (h : MonotoneOn f (Iic b)) : MapsTo f (Iic b) (Iic (f b)) :=
fun _ _ ↦ by aesop
| α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : Preorder α
inst✝ : Preorder β
b : α
h : MonotoneOn f (Iic b)
x✝¹ : α
x✝ : x✝¹ ∈ Iic b
⊢ f x✝¹ ∈ Iic (f b) | aesop | no goals | 20270f722a522837 |
StructureGroupoid.LocalInvariantProp.right_invariance | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | theorem right_invariance {s : Set H} {x : H} {f : H → H'} {e : PartialHomeomorph H H} (he : e ∈ G)
(hxe : x ∈ e.source) : P (f ∘ e.symm) (e.symm ⁻¹' s) (e x) ↔ P f s x | H : Type u_1
H' : Type u_3
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e : PartialHomeomorph H H
he : e ∈ G
hxe : x ∈ e.source
h : P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x... | refine hG.congr ?_ ((hG.congr_set ?_).mp this) | case refine_1
H : Type u_1
H' : Type u_3
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e : PartialHomeomorph H H
he : e ∈ G
hxe : x ∈ e.source
h : P (f ∘ ↑e.symm) (↑e.sym... | 03928e2066c2dc9b |
UV.compress_mem_compression_of_mem_compression | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | theorem compress_mem_compression_of_mem_compression (ha : a ∈ 𝓒 u v s) :
compress u v a ∈ 𝓒 u v s | case inl.intro
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
s : Finset α
u v a : α
inst✝ : DecidableEq α
left✝ : a ∈ s
ha : compress u v a ∈ s
⊢ compress u v a ∈ s ∧ compress u v a ∈ s ∨ compress u v a ∉ s ∧ ∃ b ∈ s, compress u v b = compres... | exact Or.inl ⟨ha, ha⟩ | no goals | 5fe8f7e6376119c3 |
Equiv.Perm.ofSubtype_support_disjoint | Mathlib/GroupTheory/Perm/Finite.lean | lemma ofSubtype_support_disjoint {σ : Perm α} (x : Perm (Function.fixedPoints σ)) :
_root_.Disjoint x.ofSubtype.support σ.support | α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ : Perm α
x : Perm ↑(fixedPoints ⇑σ)
⊢ ∀ a ∈ (ofSubtype x).support, ∀ b ∈ σ.support, a ≠ b | rintro a ha b hb rfl | α : Type u
inst✝¹ : DecidableEq α
inst✝ : Fintype α
σ : Perm α
x : Perm ↑(fixedPoints ⇑σ)
a : α
ha : a ∈ (ofSubtype x).support
hb : a ∈ σ.support
⊢ False | f578819eaab97f36 |
Int.testBit_lnot | Mathlib/Data/Int/Bitwise.lean | theorem testBit_lnot : ∀ n k, testBit (lnot n) k = not (testBit n k)
| (n : ℕ), k => by simp [lnot, testBit]
| -[n+1], k => by simp [lnot, testBit]
| n k : ℕ
⊢ -[n+1].lnot.testBit k = !-[n+1].testBit k | simp [lnot, testBit] | no goals | 5fa381349f24e71a |
CauSeq.sup_equiv_sup | Mathlib/Algebra/Order/CauSeq/Basic.lean | theorem sup_equiv_sup {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
a₁ ⊔ b₁ ≈ a₂ ⊔ b₂ | α : Type u_1
inst✝ : LinearOrderedField α
a₁ b₁ a₂ b₂ : CauSeq α abs
ha : a₁ ≈ a₂
hb : b₁ ≈ b₂
ε : α
ε0 : ε > 0
⊢ ∃ i, ∀ j ≥ i, |↑(a₁ ⊔ b₁ - a₂ ⊔ b₂) j| < ε | obtain ⟨ai, hai⟩ := ha ε ε0 | case intro
α : Type u_1
inst✝ : LinearOrderedField α
a₁ b₁ a₂ b₂ : CauSeq α abs
ha : a₁ ≈ a₂
hb : b₁ ≈ b₂
ε : α
ε0 : ε > 0
ai : ℕ
hai : ∀ j ≥ ai, |↑(a₁ - a₂) j| < ε
⊢ ∃ i, ∀ j ≥ i, |↑(a₁ ⊔ b₁ - a₂ ⊔ b₂) j| < ε | ba4771b7d098869c |
InfiniteGalois.fixedField_fixingSubgroup | Mathlib/FieldTheory/Galois/Infinite.lean | lemma fixedField_fixingSubgroup (L : IntermediateField k K) [IsGalois k K] :
IntermediateField.fixedField L.fixingSubgroup = L | k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
L : IntermediateField k K
inst✝ : IsGalois k K
x : K
hx : ∀ f ∈ L.fixingSubgroup, f x = x
mem : x ∈ (adjoin ↥L {x}).toIntermediateField
this : IntermediateField.fixedField ⊤ = ⊥
f : ↥(adjoin ↥L {x}).toIntermediateField ≃ₐ[↥L] ↥(adjoin ↥L {... | rcases restrictNormalHom_surjective K f with ⟨σ,hσ⟩ | case intro
k : Type u_1
K : Type u_2
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
L : IntermediateField k K
inst✝ : IsGalois k K
x : K
hx : ∀ f ∈ L.fixingSubgroup, f x = x
mem : x ∈ (adjoin ↥L {x}).toIntermediateField
this : IntermediateField.fixedField ⊤ = ⊥
f : ↥(adjoin ↥L {x}).toIntermediateField ≃ₐ[↥L] ↥(... | 0ce7ca0ea097dfc5 |
hasMFDerivAt_id | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | theorem hasMFDerivAt_id (x : M) :
HasMFDerivAt I I (@id M) x (ContinuousLinearMap.id 𝕜 (TangentSpace I x)) | 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x : M
⊢ ∀ᶠ (y : E) in 𝓝[range ↑I] ↑(extChartAt I x) x, (↑(extChar... | apply Filter.mem_of_superset (extChartAt_target_mem_nhdsWithin x) | 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x : M
⊢ (extChartAt I x).target ⊆ {x_1 | (fun y => (↑(extChartAt I... | 9740aeb7bdf02981 |
LinearMap.bot_lt_ker_of_det_eq_zero | Mathlib/LinearAlgebra/Determinant.lean | theorem bot_lt_ker_of_det_eq_zero {𝕜 : Type*} [Field 𝕜] [Module 𝕜 M] {f : M →ₗ[𝕜] M}
(hf : LinearMap.det f = 0) : ⊥ < LinearMap.ker f | M : Type u_2
inst✝² : AddCommGroup M
𝕜 : Type u_7
inst✝¹ : Field 𝕜
inst✝ : Module 𝕜 M
f : M →ₗ[𝕜] M
this : FiniteDimensional 𝕜 M
hf : IsUnit f
⊢ ¬LinearMap.det f = 0 | exact isUnit_iff_ne_zero.1 (f.isUnit_det hf) | no goals | 92fd01011447bbe2 |
AffineBasis.basisOf_vadd | Mathlib/LinearAlgebra/AffineSpace/Basis.lean | @[simp] lemma basisOf_vadd (v : V) (b : AffineBasis ι k P) : (v +ᵥ b).basisOf = b.basisOf | case h.a
ι : Type u_1
k : Type u_5
V : Type u_6
P : Type u_7
inst✝³ : AddCommGroup V
inst✝² : AffineSpace V P
inst✝¹ : Ring k
inst✝ : Module k V
v : V
b : AffineBasis ι k P
x✝ : ι
i✝ : { j // j ≠ x✝ }
⊢ ((v +ᵥ b).basisOf x✝) i✝ = (b.basisOf x✝) i✝ | simp | no goals | 1de97172839136d4 |
Algebra.FormallySmooth.iff_split_injection | Mathlib/RingTheory/Smooth/Kaehler.lean | theorem Algebra.FormallySmooth.iff_split_injection :
Algebra.FormallySmooth R S ↔ ∃ l, l ∘ₗ (kerCotangentToTensor R P S) = LinearMap.id | R P S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing P
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R P
inst✝³ : Algebra P S
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R P S
hf : Surjective ⇑(algebraMap P S)
inst✝ : FormallySmooth R P
this :
(∃ l, l ∘ₗ kerCotangentToTensor R P S = LinearMap.id) ↔
∃ g, (IsScalarTower.toA... | rw [this, ← Algebra.FormallySmooth.iff_split_surjection _ hf] | no goals | 336c618049666450 |
ENNReal.log_mul_add | Mathlib/Analysis/SpecialFunctions/Log/ENNRealLog.lean | theorem log_mul_add {x y : ℝ≥0∞} : log (x * y) = log x + log y | case inr.inl.inr.inr
y : ℝ≥0∞
y_real : 0 < y.toReal
⊢ (if y = 0 then 0 else ⊤) = ⊤ | simp only [ite_eq_right_iff, zero_ne_top, imp_false] | case inr.inl.inr.inr
y : ℝ≥0∞
y_real : 0 < y.toReal
⊢ ¬y = 0 | bf6f64107a59785f |
List.firstM_toArray | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem firstM_toArray [Alternative m] (as : List α) (f : α → m β) :
as.toArray.firstM f = as.firstM f | m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝ : Alternative m
as : List α
f : α → m β
⊢ ∀ (i : Nat), i ≤ as.length → firstM.go f as.toArray (as.length - i) = firstM f (drop (as.length - i) as) | intro i | m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝ : Alternative m
as : List α
f : α → m β
i : Nat
⊢ i ≤ as.length → firstM.go f as.toArray (as.length - i) = firstM f (drop (as.length - i) as) | 8ff826aeb0a69b7f |
ComplexShape.next_eq' | Mathlib/Algebra/Homology/ComplexShape.lean | theorem next_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.next i = j | ι : Type u_1
c : ComplexShape ι
i j : ι
h : c.Rel i j
⊢ c.next i = j | apply c.next_eq _ h | ι : Type u_1
c : ComplexShape ι
i j : ι
h : c.Rel i j
⊢ c.Rel i (c.next i) | 82dca8cb7cfa135f |
Finset.min'_image | Mathlib/Data/Finset/Max.lean | theorem min'_image [LinearOrder β] {f : α → β} (hf : Monotone f) (s : Finset α)
(h : (s.image f).Nonempty) : (s.image f).min' h = f (s.min' h.of_image) | α : Type u_2
β : Type u_3
inst✝¹ : LinearOrder α
inst✝ : LinearOrder β
f : α → β
hf : Monotone f
s : Finset α
h : (image f s).Nonempty
⊢ s.inf' ⋯ (id ∘ f) = f (s.inf' ⋯ id) | exact .symm <| comp_inf'_eq_inf'_comp _ _ fun _ _ ↦ hf.map_min | no goals | 133485904fa05aa3 |
Polynomial.IsPrimitive.irreducible_iff_irreducible_map_fraction_map | Mathlib/RingTheory/Polynomial/GaussLemma.lean | theorem IsPrimitive.irreducible_iff_irreducible_map_fraction_map {p : R[X]} (hp : p.IsPrimitive) :
Irreducible p ↔ Irreducible (p.map (algebraMap R K)) | case intro.mk.intro.mk.intro.inl
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (alg... | right | case intro.mk.intro.mk.intro.inl.h
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (a... | 8c3b311dea9aa158 |
cauchy_davenport_minOrder_mul | Mathlib/Combinatorics/Additive/CauchyDavenport.lean | /-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is... | case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
ih :
∀ (a b : Finset α),
a.Nonempty → b.Nonempty → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(#(a * b)) ∨ #a + #b ≤ #(a * b) + 1
hst : #s ≤ #t
a : ... | replace aux2 := card_mono <| mulETransformRight.fst_mul_snd_subset g (s, t) | case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
s t : Finset α
hs : s.Nonempty
ht : t.Nonempty
ih :
∀ (a b : Finset α),
a.Nonempty → b.Nonempty → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(#(a * b)) ∨ #a + #b ≤ #(a * b) + 1
hst : #s ≤ #t
a : ... | 095ce4d72a361376 |
Batteries.PairingHeapImp.Heap.noSibling_tail | Mathlib/.lake/packages/batteries/Batteries/Data/PairingHeap.lean | theorem Heap.noSibling_tail (le) (s : Heap α) : (s.tail le).NoSibling | α : Type u_1
le : α → α → Bool
s : Heap α
⊢ (tail le s).NoSibling | simp only [Heap.tail] | α : Type u_1
le : α → α → Bool
s : Heap α
⊢ ((tail? le s).getD nil).NoSibling | 55dd4cccfcf68dad |
ProbabilityTheory.IndepFun.variance_sum | Mathlib/Probability/Variance.lean | theorem IndepFun.variance_sum [IsProbabilityMeasure μ] {ι : Type*} {X : ι → Ω → ℝ}
{s : Finset ι} (hs : ∀ i ∈ s, MemLp (X i) 2 μ)
(h : Set.Pairwise ↑s fun i j => IndepFun (X i) (X j) μ) :
variance (∑ i ∈ s, X i) μ = ∑ i ∈ s, variance (X i) μ | case hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
X : ι → Ω → ℝ
k : ι
s : Finset ι
ks : k ∉ s
IH :
(∀ i ∈ s, MemLp (X i) 2 μ) →
((↑s).Pairwise fun i j => IndepFun (X i) (X j) μ) → Var[∑ i ∈ s, X i; μ] = ∑ i ∈ s, Var[X i; μ]
hs : ∀ i ∈ insert k s, MemLp (X i) 2 μ... | apply Integrable.add | case hf.hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
X : ι → Ω → ℝ
k : ι
s : Finset ι
ks : k ∉ s
IH :
(∀ i ∈ s, MemLp (X i) 2 μ) →
((↑s).Pairwise fun i j => IndepFun (X i) (X j) μ) → Var[∑ i ∈ s, X i; μ] = ∑ i ∈ s, Var[X i; μ]
hs : ∀ i ∈ insert k s, MemLp (X i) ... | 1d5529d2c6cdc1ee |
FDerivMeasurableAux.mem_A_of_differentiable | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : E} (hx : DifferentiableAt 𝕜 f x) :
∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (fderiv 𝕜 f x) r ε | case intro.intro
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
ε : ℝ
hε : 0 < ε
x : E
hx : DifferentiableAt 𝕜 f x
δ : ℝ := ε / 2 / 2
R : ℝ
R_pos : R > 0
hR : ∀ y ∈ ball... | have : r ∈ Ioc (r / 2) r := right_mem_Ioc.2 <| half_lt_self hr.1 | case intro.intro
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
ε : ℝ
hε : 0 < ε
x : E
hx : DifferentiableAt 𝕜 f x
δ : ℝ := ε / 2 / 2
R : ℝ
R_pos : R > 0
hR : ∀ y ∈ ball... | e07d58ee226eb99e |
Fin.rev_le_rev | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean | theorem rev_le_rev {i j : Fin n} : rev i ≤ rev j ↔ j ≤ i | n : Nat
i j : Fin n
⊢ i.rev ≤ j.rev ↔ j ≤ i | simp only [le_def, val_rev, Nat.sub_le_sub_iff_left (Nat.succ_le.2 j.is_lt)] | n : Nat
i j : Fin n
⊢ (↑j).succ ≤ ↑i + 1 ↔ ↑j ≤ ↑i | 40f5cf64e40f30e9 |
CategoryTheory.ShortComplex.Exact.lift' | Mathlib/Algebra/Homology/ShortComplex/Exact.lean | lemma lift' (hS : S.Exact) {A : C} (k : A ⟶ S.X₂) (hk : k ≫ S.g = 0) [Mono S.f] :
∃ (l : A ⟶ S.X₁), l ≫ S.f = k :=
⟨hS.lift k hk, by simp⟩
| C : Type u_1
inst✝³ : Category.{u_3, u_1} C
inst✝² : Preadditive C
S : ShortComplex C
inst✝¹ : Balanced C
hS : S.Exact
A : C
k : A ⟶ S.X₂
hk : k ≫ S.g = 0
inst✝ : Mono S.f
⊢ hS.lift k hk ≫ S.f = k | simp | no goals | 5bbdcd62df441fed |
BitVec.xor_allOnes | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem xor_allOnes {x : BitVec w} : x ^^^ allOnes w = ~~~ x | w : Nat
x : BitVec w
⊢ x ^^^ allOnes w = ~~~x | ext i h | case pred
w : Nat
x : BitVec w
i : Nat
h : i < w
⊢ (x ^^^ allOnes w).getLsbD i = (~~~x).getLsbD i | acdcbe648cb9eaa1 |
List.map_destutter | Mathlib/Data/List/Destutter.lean | theorem map_destutter {f : α → β} : ∀ {l : List α}, (∀ a ∈ l, ∀ b ∈ l, R a b ↔ R₂ (f a) (f b)) →
(l.destutter R).map f = (l.map f).destutter R₂
| [], hl => by simp
| [a], hl => by simp
| a :: b :: l, hl => by
have := hl a (by simp) b (by simp)
simp_rw [map_cons, destutter_cons_cons, ← this]
by_cas... | α : Type u_1
β : Type u_2
R : α → α → Prop
inst✝¹ : DecidableRel R
R₂ : β → β → Prop
inst✝ : DecidableRel R₂
f : α → β
a b : α
l : List α
hl : ∀ (a_1 : α), a_1 ∈ a :: b :: l → ∀ (b_1 : α), b_1 ∈ a :: b :: l → (R a_1 b_1 ↔ R₂ (f a_1) (f b_1))
⊢ b ∈ a :: b :: l | simp | no goals | c231877904ed9c8b |
Nat.testBit_lt_two_pow | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Bitwise/Lemmas.lean | theorem testBit_lt_two_pow {x i : Nat} (lt : x < 2^i) : x.testBit i = false | x i : Nat
lt : x < 2 ^ i
p : x.testBit i = true
⊢ False | exact Nat.not_le_of_gt lt (testBit_implies_ge p) | no goals | 0a80ec3412131aaf |
FixedDetMatrices.reduce_reduceStep | Mathlib/LinearAlgebra/Matrix/FixedDetMatrices.lean | @[simp]
lemma reduce_reduceStep {A : Δ m} (hc : (A.1 1 0) ≠ 0) :
reduce (reduceStep A) = reduce A | m : ℤ
A : FixedDetMatrix (Fin 2) ℤ m
hc : ↑A 1 0 ≠ 0
⊢ reduce A = reduce (reduceStep A) | rw [reduce, if_neg hc] | no goals | ef7300797c1d594e |
PiTensorProduct.liftAux_tprod | Mathlib/LinearAlgebra/PiTensorProduct.lean | theorem liftAux_tprod (φ : MultilinearMap R s E) (f : Π i, s i) : liftAux φ (tprod R f) = φ f | case h.e'_3
ι : Type u_1
R : Type u_4
inst✝⁴ : CommSemiring R
s : ι → Type u_7
inst✝³ : (i : ι) → AddCommMonoid (s i)
inst✝² : (i : ι) → Module R (s i)
E : Type u_9
inst✝¹ : AddCommMonoid E
inst✝ : Module R E
φ : MultilinearMap R s E
f : (i : ι) → s i
⊢ 1 • φ f = φ f | simp | no goals | 937fd2cab91e4095 |
List.modify_eraseIdx_of_lt | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Modify.lean | theorem modify_eraseIdx_of_lt (f : α → α) (i j) (l : List α) (h : j < i) :
(l.eraseIdx i).modify f j = (l.modify f j).eraseIdx i | case neg.isFalse
α : Type u_1
f : α → α
i j : Nat
l : List α
h : j < i
k : Nat
h₁ : k < (modify f j (l.eraseIdx i)).length
h₂ : k < ((modify f j l).eraseIdx i).length
h' : ¬j = k + 1
h✝ : ¬j = k
⊢ (if h' : k < i then l[k] else l[k + 1]) = if h : k < i then l[k] else l[k + 1] | split | case neg.isFalse.isTrue
α : Type u_1
f : α → α
i j : Nat
l : List α
h : j < i
k : Nat
h₁ : k < (modify f j (l.eraseIdx i)).length
h₂ : k < ((modify f j l).eraseIdx i).length
h' : ¬j = k + 1
h✝¹ : ¬j = k
h✝ : k < i
⊢ l[k] = l[k]
case neg.isFalse.isFalse
α : Type u_1
f : α → α
i j : Nat
l : List α
h : j < i
k : Nat
h₁ :... | 94a33219452b21f4 |
Nat.Linear.Poly.of_denote_eq_cancelAux | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Linear.lean | theorem Poly.of_denote_eq_cancelAux (ctx : Context) (fuel : Nat) (m₁ m₂ r₁ r₂ : Poly)
(h : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)) : denote_eq ctx (r₁.reverse ++ m₁, r₂.reverse ++ m₂) | case neg
ctx : Context
fuel : Nat
ih :
∀ (m₁ m₂ r₁ r₂ : Poly),
denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)
r₁ r₂ m₁✝ m₂✝ : Poly
k₁ : Nat
v₁ : Var
m₁ : List (Nat × Var)
k₂ : Nat
v₂ : Var
m₂ : List (Nat × Var)
hltv : ¬blt v₁ v₂ = true
h :
denote_eq ctx... | by_cases hgtv : Nat.blt v₂ v₁ <;> simp [hgtv] at h | case pos
ctx : Context
fuel : Nat
ih :
∀ (m₁ m₂ r₁ r₂ : Poly),
denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) → denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)
r₁ r₂ m₁✝ m₂✝ : Poly
k₁ : Nat
v₁ : Var
m₁ : List (Nat × Var)
k₂ : Nat
v₂ : Var
m₂ : List (Nat × Var)
hltv : ¬blt v₁ v₂ = true
hgtv : blt v₂ v₁ = ... | 77a6f41635643f9c |
Fin.castSucc_eq_zero_iff' | Mathlib/Data/Fin/Basic.lean | theorem castSucc_eq_zero_iff' [NeZero n] (a : Fin n) : castSucc a = 0 ↔ a = 0 :=
Fin.ext_iff.trans <| (Fin.ext_iff.trans <| by simp).symm
| n : ℕ
inst✝ : NeZero n
a : Fin n
⊢ ↑a = ↑0 ↔ ↑a.castSucc = ↑0 | simp | no goals | 8ac1c59215e4b145 |
measurableSet_prod_of_nonempty | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | theorem measurableSet_prod_of_nonempty {s : Set α} {t : Set β} (h : (s ×ˢ t).Nonempty) :
MeasurableSet (s ×ˢ t) ↔ MeasurableSet s ∧ MeasurableSet t | α : Type u_1
β : Type u_2
m : MeasurableSpace α
mβ : MeasurableSpace β
s : Set α
t : Set β
h : (s ×ˢ t).Nonempty
⊢ MeasurableSet (s ×ˢ t) ↔ MeasurableSet s ∧ MeasurableSet t | rcases h with ⟨⟨x, y⟩, hx, hy⟩ | case intro.mk.intro
α : Type u_1
β : Type u_2
m : MeasurableSpace α
mβ : MeasurableSpace β
s : Set α
t : Set β
x : α
y : β
hx : (x, y).1 ∈ s
hy : (x, y).2 ∈ t
⊢ MeasurableSet (s ×ˢ t) ↔ MeasurableSet s ∧ MeasurableSet t | 151f2d338c4f2c2c |
Equiv.Perm.card_support_ne_one | Mathlib/GroupTheory/Perm/Support.lean | theorem card_support_ne_one (f : Perm α) : #f.support ≠ 1 | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f : Perm α
h : f = 1
⊢ #f.support ≠ 1 | exact ne_of_eq_of_ne (card_support_eq_zero.mpr h) zero_ne_one | no goals | b9e8e2ff9e30e4de |
Nat.fast_fib_aux_eq | Mathlib/Data/Nat/Fib/Basic.lean | theorem fast_fib_aux_eq (n : ℕ) : fastFibAux n = (fib n, fib (n + 1)) | case refine_1
n : ℕ
⊢ fastFibAux 0 = (fib 0, fib (0 + 1)) | simp [fastFibAux] | no goals | 77d9424d9231d6c6 |
SemidirectProduct.lift_comp_inr | Mathlib/GroupTheory/SemidirectProduct.lean | theorem lift_comp_inr : (lift fn fg h).comp inr = fg | case h
N : Type u_1
G : Type u_2
H : Type u_3
inst✝² : Group N
inst✝¹ : Group G
inst✝ : Group H
φ : G →* MulAut N
fn : N →* H
fg : G →* H
h : ∀ (g : G), fn.comp (MulEquiv.toMonoidHom (φ g)) = (MulEquiv.toMonoidHom (MulAut.conj (fg g))).comp fn
x✝ : G
⊢ ((lift fn fg h).comp inr) x✝ = fg x✝ | simp | no goals | 8a8810150c460f22 |
Valued.continuous_valuation | Mathlib/Topology/Algebra/Valued/ValuedField.lean | theorem Valued.continuous_valuation [Valued K Γ₀] : Continuous (v : K → Γ₀) | case inr
K : Type u_1
inst✝² : DivisionRing K
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
x : K
h : x ≠ 0
v_ne : v x ≠ 0
⊢ ContinuousAt (⇑v) x | rw [ContinuousAt, WithZeroTopology.tendsto_of_ne_zero v_ne] | case inr
K : Type u_1
inst✝² : DivisionRing K
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : Valued K Γ₀
x : K
h : x ≠ 0
v_ne : v x ≠ 0
⊢ ∀ᶠ (x_1 : K) in 𝓝 x, v x_1 = v x | 40ad4a371f000982 |
StrictMonoOn.continuousWithinAt_right_of_exists_between | Mathlib/Topology/Order/MonotoneContinuity.lean | theorem StrictMonoOn.continuousWithinAt_right_of_exists_between {f : α → β} {s : Set α} {a : α}
(h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b) :
ContinuousWithinAt f (Ici a) a | α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : LinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
s : Set α
a : α
h_mono : StrictMonoOn f s
hs : s ∈ 𝓝[≥] a
hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b
ha : a ∈ Ici a
⊢ ContinuousWit... | have has : a ∈ s := mem_of_mem_nhdsWithin ha hs | α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : LinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
s : Set α
a : α
h_mono : StrictMonoOn f s
hs : s ∈ 𝓝[≥] a
hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b
ha : a ∈ Ici a
has : a ∈ s
⊢ C... | 1f1f4962019f9fb3 |
Fermat42.not_minimal | Mathlib/NumberTheory/FLT/Four.lean | theorem not_minimal {a b c : ℤ} (h : Minimal a b c) (ha2 : a % 2 = 1) (hc : 0 < c) : False | a b c : ℤ
h : Minimal a b c
ha2 : a % 2 = 1
hc : 0 < c
ht : PythagoreanTriple (a ^ 2) (b ^ 2) c
h2 : (a ^ 2).gcd (b ^ 2) = 1
ha22 : a ^ 2 % 2 = 1
m n : ℤ
ht1 : a ^ 2 = m ^ 2 - n ^ 2
ht2 : b ^ 2 = 2 * m * n
ht3 : c = m ^ 2 + n ^ 2
ht4 : m.gcd n = 1
ht5 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
ht6 : 0 ≤ m
htt : Py... | have h2 : b' ^ 2 ≤ 0 := by
rw [hs, (by ring : -d ^ 2 * m = -(d ^ 2 * m))]
exact neg_nonpos.mpr ((mul_nonneg_iff_of_pos_right h4).mpr (sq_nonneg d)) | a b c : ℤ
h : Minimal a b c
ha2 : a % 2 = 1
hc : 0 < c
ht : PythagoreanTriple (a ^ 2) (b ^ 2) c
h2✝ : (a ^ 2).gcd (b ^ 2) = 1
ha22 : a ^ 2 % 2 = 1
m n : ℤ
ht1 : a ^ 2 = m ^ 2 - n ^ 2
ht2 : b ^ 2 = 2 * m * n
ht3 : c = m ^ 2 + n ^ 2
ht4 : m.gcd n = 1
ht5 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
ht6 : 0 ≤ m
htt : P... | 58647b498e4824ae |
FirstOrder.Language.noBotOrder_of_dlo | Mathlib/ModelTheory/Order.lean | theorem noBotOrder_of_dlo [M ⊨ L.dlo] : NoBotOrder M :=
realize_noBotOrder_iff.1 (L.dlo.realize_sentence_of_mem (by
simp only [dlo, Set.union_insert, Set.union_singleton, Set.mem_insert_iff, true_or, or_true]))
| L : Language
M : Type w'
inst✝⁴ : L.IsOrdered
inst✝³ : L.Structure M
inst✝² : LE M
inst✝¹ : L.OrderedStructure M
inst✝ : M ⊨ L.dlo
⊢ L.noBotOrderSentence ∈ L.dlo | simp only [dlo, Set.union_insert, Set.union_singleton, Set.mem_insert_iff, true_or, or_true] | no goals | fa117b16514ead9b |
linearIndependent_of_top_le_span_of_card_eq_finrank | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | theorem linearIndependent_of_top_le_span_of_card_eq_finrank {ι : Type*} [Fintype ι] {b : ι → V}
(spans : ⊤ ≤ span K (Set.range b)) (card_eq : Fintype.card ι = finrank K V) :
LinearIndependent K b :=
linearIndependent_iff'.mpr fun s g dependent i i_mem_s => by
classical
by_contra gx_ne_zero
-- We'l... | K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
ι : Type u_2
inst✝ : Fintype ι
b : ι → V
spans : ⊤ ≤ span K (range b)
card_eq : Fintype.card ι = finrank K V
s : Finset ι
g : ι → K
dependent : ∑ i ∈ s, g i • b i = 0
i : ι
i_mem_s : i ∈ s
gx_ne_zero : ¬g i = 0
⊢ (b '' (Set.univ \... | rw [Set.toFinset_card, Fintype.card_ofFinset] | no goals | db71af46429798d0 |
isBigO_norm_restrict_cocompact | Mathlib/Analysis/Fourier/PoissonSummation.lean | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : f =O[cocompact ℝ] fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
(fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) =O[cocompact ℝ] (|·| ^ (-b)) | case intro.left
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
this :
∀ (x : ℝ),
‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight x))‖ ≤ ‖ContinuousMap.restrict (Icc (x - r) (x + r)) f‖
hf : (⇑f =O[atBot] fun x => |x| ^ (-b)) ∧ ⇑f =O[at... | simp_rw [norm_norm] | case intro.left
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
this :
∀ (x : ℝ),
‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight x))‖ ≤ ‖ContinuousMap.restrict (Icc (x - r) (x + r)) f‖
hf : (⇑f =O[atBot] fun x => |x| ^ (-b)) ∧ ⇑f =O[at... | 6854796ce291b958 |
heq_imp_of_eq_imp | Mathlib/Tactic/CongrExclamation.lean | theorem heq_imp_of_eq_imp {α : Sort*} {x y : α} {p : HEq x y → Prop}
(h : (he : x = y) → p (heq_of_eq he)) (he : HEq x y) : p he | case refl
α : Sort u_1
x : α
p : HEq x x → Prop
h : ∀ (he : x = x), p ⋯
⊢ p ⋯ | exact h rfl | no goals | 711d4785ea0ebfa8 |
ContMDiffSection.coe_nsmul | Mathlib/Geometry/Manifold/VectorBundle/SmoothSection.lean | theorem coe_nsmul (s : Cₛ^n⟮I; F, V⟯) (k : ℕ) : ⇑(k • s : Cₛ^n⟮I; F, V⟯) = k • ⇑s | case succ
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ :... | rfl | no goals | 5ccbbb02c50ea363 |
NumberField.InfinitePlace.card_isUnramified | Mathlib/NumberTheory/NumberField/Embeddings.lean | lemma card_isUnramified [NumberField k] [IsGalois k K] :
#{w : InfinitePlace K | w.IsUnramified k} =
#{w : InfinitePlace k | w.IsUnramifiedIn K} * finrank k K | case intro
k : Type u_1
inst✝⁵ : Field k
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra k K
inst✝² : NumberField K
inst✝¹ : NumberField k
inst✝ : IsGalois k K
this : Module.Finite k K := Finite.of_restrictScalars_finite ℚ k K
w : InfinitePlace K
hw : (fun x => x.comap (algebraMap k K)) w ∈ filter (fun w => IsUnramified... | simp only [mem_univ, forall_true_left, mem_filter, true_and] at hw | case intro
k : Type u_1
inst✝⁵ : Field k
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra k K
inst✝² : NumberField K
inst✝¹ : NumberField k
inst✝ : IsGalois k K
this : Module.Finite k K := Finite.of_restrictScalars_finite ℚ k K
w : InfinitePlace K
hw : IsUnramifiedIn K (w.comap (algebraMap k K))
⊢ #(filter (fun a => a.co... | 46af2b427709936d |
gauge_le_of_mem | Mathlib/Analysis/Convex/Gauge.lean | theorem gauge_le_of_mem (ha : 0 ≤ a) (hx : x ∈ a • s) : gauge s x ≤ a | case inr
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s : Set E
x : E
a : ℝ
ha : 0 ≤ a
hx : x ∈ a • s
ha' : 0 < a
⊢ gauge s x ≤ a | exact csInf_le gauge_set_bddBelow ⟨ha', hx⟩ | no goals | 54684b391c6fc491 |
Sum.LiftRel.isLeft_left | Mathlib/Data/Sum/Basic.lean | theorem isLeft_left (h : LiftRel r s x (inl c)) : x.isLeft | α : Type u
β : Type v
γ : Type u_1
δ : Type u_2
r : α → γ → Prop
s : β → δ → Prop
x : α ⊕ β
c : γ
h : LiftRel r s x (inl c)
⊢ x.isLeft = true | cases h | case inl
α : Type u
β : Type v
γ : Type u_1
δ : Type u_2
r : α → γ → Prop
s : β → δ → Prop
c : γ
a✝¹ : α
a✝ : r a✝¹ c
⊢ (inl a✝¹).isLeft = true | f9bcee80f5e121ef |
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