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 |
|---|---|---|---|---|---|---|
Module.End.injOn_genEigenspace | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | lemma injOn_genEigenspace [NoZeroSMulDivisors R M] (f : End R M) (k : ℕ∞) :
InjOn (f.genEigenspace · k) {μ | f.genEigenspace μ k ≠ ⊥} | R : Type v
M : Type w
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
f : End R M
k : ℕ∞
⊢ InjOn (fun x => (f.genEigenspace x) k) {μ | (f.genEigenspace μ) k ≠ ⊥} | rintro μ₁ _ μ₂ hμ₂ hμ₁₂ | R : Type v
M : Type w
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
f : End R M
k : ℕ∞
μ₁ : R
a✝ : μ₁ ∈ {μ | (f.genEigenspace μ) k ≠ ⊥}
μ₂ : R
hμ₂ : μ₂ ∈ {μ | (f.genEigenspace μ) k ≠ ⊥}
hμ₁₂ : (fun x => (f.genEigenspace x) k) μ₁ = (fun x => (f.genEigenspace x) k) μ₂
⊢ μ₁... | a0c686e86c7fd170 |
Set.unbounded_lt_iff | Mathlib/Order/Bounded.lean | theorem unbounded_lt_iff [LinearOrder α] : Unbounded (· < ·) s ↔ ∀ a, ∃ b ∈ s, a ≤ b | α : Type u_1
s : Set α
inst✝ : LinearOrder α
⊢ Unbounded (fun x1 x2 => x1 < x2) s ↔ ∀ (a : α), ∃ b ∈ s, a ≤ b | simp only [Unbounded, not_lt] | no goals | b1b0ca051c937751 |
Std.DHashMap.Raw.Const.size_le_size_insertMany_list | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean | theorem size_le_size_insertMany_list [EquivBEq α] [LawfulHashable α] (h : m.WF)
{l : List (α × β)} :
m.size ≤ (insertMany m l).size | α : Type u
inst✝³ : BEq α
inst✝² : Hashable α
β : Type v
m : Raw α fun x => β
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.WF
l : List (α × β)
⊢ m.size ≤ (insertMany m l).size | simp_to_raw using Raw₀.Const.size_le_size_insertMany_list ⟨m, _⟩ | no goals | a2a816818387ca7a |
Std.DHashMap.Internal.List.getValueCast?_insertList_of_mem | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem getValueCast?_insertList_of_mem [BEq α] [LawfulBEq α]
{l toInsert : List ((a : α) × β a)}
{k k' : α} (k_beq : k == k') {v : β k}
(distinct_l : DistinctKeys l)
(distinct_toInsert : toInsert.Pairwise (fun a b => (a.1 == b.1) = false))
(mem : ⟨k, v⟩ ∈ toInsert) :
getValueCast? k' (insertLis... | α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : LawfulBEq α
l toInsert : List ((a : α) × β a)
k k' : α
k_beq : (k == k') = true
v : β k
distinct_l : DistinctKeys l
distinct_toInsert : DistinctKeys toInsert
mem : ⟨k, v⟩ ∈ toInsert
this : getEntry? k' (insertList l toInsert) = getEntry? k' toInsert
⊢ (Std.DHashMap.Inter... | rw [getEntry?_of_mem distinct_toInsert k_beq mem] at this | α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : LawfulBEq α
l toInsert : List ((a : α) × β a)
k k' : α
k_beq : (k == k') = true
v : β k
distinct_l : DistinctKeys l
distinct_toInsert : DistinctKeys toInsert
mem : ⟨k, v⟩ ∈ toInsert
this : getEntry? k' (insertList l toInsert) = some ⟨k, v⟩
⊢ (Std.DHashMap.Internal.List.O... | 82c18001be4198ba |
rootsOfUnity.integer_power_of_ringEquiv | Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean | theorem rootsOfUnity.integer_power_of_ringEquiv (g : L ≃+* L) :
∃ m : ℤ, ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ m : Lˣ) | case intro
L : Type u
inst✝² : CommRing L
inst✝¹ : IsDomain L
n : ℕ
inst✝ : NeZero n
g : L ≃+* L
m : ℤ
hm : ∀ (g_1 : ↥(rootsOfUnity n L)), ((↑g).restrictRootsOfUnity n).toMonoidHom g_1 = g_1 ^ m
⊢ ∃ m, ∀ (t : ↥(rootsOfUnity n L)), g ↑↑t = ↑(↑t ^ m) | exact ⟨m, fun t ↦ Units.ext_iff.1 <| SetCoe.ext_iff.2 <| hm t⟩ | no goals | 6b741b3c54da15ad |
Std.DHashMap.Internal.List.Const.modifyKey_eq_alterKey | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem modifyKey_eq_alterKey (k : α) (f : β → β) (l : List ((_ : α) × β)) :
modifyKey k f l = alterKey k (·.map f) l | α : Type u
β : Type v
inst✝ : BEq α
k : α
f : β → β
l : List ((_ : α) × β)
⊢ modifyKey k f l = alterKey k (fun x => Option.map f x) l | rw [modifyKey, alterKey, Option.map.eq_def] | α : Type u
β : Type v
inst✝ : BEq α
k : α
f : β → β
l : List ((_ : α) × β)
⊢ (match getValue? k l with
| none => l
| some v => replaceEntry k (f v) l) =
match
match getValue? k l with
| some x => some (f x)
| none => none with
| none => eraseKey k l
| some v => insertEntry k v l | 4c12c679cb341031 |
MeasureTheory.Measure.prod_prod | Mathlib/MeasureTheory/Measure/Prod.lean | theorem prod_prod (s : Set α) (t : Set β) : μ.prod ν (s ×ˢ t) = μ s * ν t | case a
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
s : Set α
t : Set β
S : Set α := toMeasurable μ s
T : Set β := toMeasurable ν t
hSTm : MeasurableSet (S ×ˢ T)
⊢ (μ.prod ν) (s ×ˢ t) ≤ μ s * ν t | calc
μ.prod ν (s ×ˢ t) ≤ μ.prod ν (S ×ˢ T) := by gcongr <;> apply subset_toMeasurable
_ = μ S * ν T := by
rw [prod_apply hSTm]
simp_rw [S, mk_preimage_prod_right_eq_if, measure_if,
lintegral_indicator (measurableSet_toMeasurable _ _), lintegral_const,
restrict_apply_univ, mul_comm]
_ = μ s * ν... | no goals | 3a4325941a637749 |
CategoryTheory.Limits.coprod.pentagon | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | theorem coprod.pentagon (W X Y Z : C) :
coprod.map (coprod.associator W X Y).hom (𝟙 Z) ≫
(coprod.associator W (X ⨿ Y) Z).hom ≫ coprod.map (𝟙 W) (coprod.associator X Y Z).hom =
(coprod.associator (W ⨿ X) Y Z).hom ≫ (coprod.associator W X (Y ⨿ Z)).hom | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasBinaryCoproducts C
W X Y Z : C
⊢ map (associator W X Y).hom (𝟙 Z) ≫ (associator W (X ⨿ Y) Z).hom ≫ map (𝟙 W) (associator X Y Z).hom =
(associator (W ⨿ X) Y Z).hom ≫ (associator W X (Y ⨿ Z)).hom | simp | no goals | 025eb35b3fee2ad8 |
CategoryTheory.GrothendieckTopology.coversTop_iff_of_isTerminal | Mathlib/CategoryTheory/Sites/CoversTop.lean | lemma coversTop_iff_of_isTerminal (X : C) (hX : IsTerminal X)
{I : Type*} (Y : I → C) :
J.CoversTop Y ↔ Sieve.ofObjects Y X ∈ J X | case mpr
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X : C
hX : IsTerminal X
I : Type u_1
Y : I → C
h : Sieve.ofObjects Y X ∈ J X
W : C
⊢ Sieve.ofObjects Y W ∈ J W | apply J.superset_covering _ (J.pullback_stable (hX.from W) h) | C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X : C
hX : IsTerminal X
I : Type u_1
Y : I → C
h : Sieve.ofObjects Y X ∈ J X
W : C
⊢ Sieve.pullback (hX.from W) (Sieve.ofObjects Y X) ≤ Sieve.ofObjects Y W | 2ecd555335ab9063 |
Finsupp.indicator_eq_sum_attach_single | Mathlib/Algebra/BigOperators/Finsupp.lean | lemma indicator_eq_sum_attach_single [AddCommMonoid M] {s : Finset α} (f : ∀ a ∈ s, M) :
indicator s f = ∑ x ∈ s.attach, single ↑x (f x x.2) | α : Type u_1
M : Type u_8
inst✝ : AddCommMonoid M
s : Finset α
f : (a : α) → a ∈ s → M
x✝¹ : { x // x ∈ s }
x✝ : x✝¹ ∈ s.attach
⊢ single (↑x✝¹) ((indicator s f) ↑x✝¹) = single (↑x✝¹) (f ↑x✝¹ ⋯) | rw [indicator_of_mem] | no goals | 4a42177640682c3f |
tendsto_nhdsWithin_iff_subtype | Mathlib/Topology/ContinuousOn.lean | theorem tendsto_nhdsWithin_iff_subtype {s : Set α} {a : α} (h : a ∈ s) (f : α → β) (l : Filter β) :
Tendsto f (𝓝[s] a) l ↔ Tendsto (s.restrict f) (𝓝 ⟨a, h⟩) l | α : Type u_1
β : Type u_2
inst✝ : TopologicalSpace α
s : Set α
a : α
h : a ∈ s
f : α → β
l : Filter β
⊢ Tendsto (f ∘ Subtype.val) (𝓝 ⟨a, h⟩) l ↔ Tendsto (s.restrict f) (𝓝 ⟨a, h⟩) l | rfl | no goals | c34ed15966db3f6f |
LSeries.term_convolution' | Mathlib/NumberTheory/LSeries/Convolution.lean | /-- We give an expression of the `LSeries.term` of the convolution of two functions
in terms of an a priori infinite sum over all pairs `(k, m)` with `k * m = n`
(the set we sum over is infinite when `n = 0`). This is the version needed for the
proof that `L (f ⍟ g) = L f * L g`. -/
lemma term_convolution' (f g : ℕ → ℂ... | f g : ℕ → ℂ
s : ℂ
hS : (fun p => p.1 * p.2) ⁻¹' {0} = {0} ×ˢ univ ∪ univ ×ˢ {0}
⊢ ∀ (p : ↑((fun p => p.1 * p.2) ⁻¹' {0})), term f s (↑p).1 * term g s (↑p).2 = 0 | rintro ⟨⟨_, _⟩, hp⟩ | case mk.mk
f g : ℕ → ℂ
s : ℂ
hS : (fun p => p.1 * p.2) ⁻¹' {0} = {0} ×ˢ univ ∪ univ ×ˢ {0}
fst✝ snd✝ : ℕ
hp : (fst✝, snd✝) ∈ (fun p => p.1 * p.2) ⁻¹' {0}
⊢ term f s (↑⟨(fst✝, snd✝), hp⟩).1 * term g s (↑⟨(fst✝, snd✝), hp⟩).2 = 0 | 0320871b3b1f5c99 |
MeasureTheory.eLpNormEssSup_indicator_const_eq | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | lemma eLpNormEssSup_indicator_const_eq (s : Set α) (c : G) (hμs : μ s ≠ 0) :
eLpNormEssSup (s.indicator fun _ : α => c) μ = ‖c‖ₑ | α : Type u_1
G : Type u_5
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup G
s : Set α
c : G
hμs : μ s ≠ 0
h : eLpNormEssSup (s.indicator fun x => c) μ < ‖c‖ₑ
h' : μ {a | ‖c‖ₑ ≤ ‖s.indicator (fun x => c) a‖ₑ} = 0
x : α
hx_mem : x ∈ s
⊢ x ∈ {a | ‖c‖ₑ ≤ ‖s.indicator (fun x => c) a‖ₑ} | rw [Set.mem_setOf_eq, Set.indicator_of_mem hx_mem, enorm_eq_nnnorm] | no goals | 4bfba53b43399385 |
DFinsupp.card_Ioc | Mathlib/Data/DFinsupp/Interval.lean | lemma card_Ioc : #(Ioc f g) = (∏ i ∈ f.support ∪ g.support, #(Icc (f i) (g i))) - 1 | ι : Type u_1
α : ι → Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : (i : ι) → DecidableEq (α i)
inst✝² : (i : ι) → PartialOrder (α i)
inst✝¹ : (i : ι) → Zero (α i)
inst✝ : (i : ι) → LocallyFiniteOrder (α i)
f g : Π₀ (i : ι), α i
⊢ #(Ioc f g) = ∏ i ∈ f.support ∪ g.support, #(Icc (f i) (g i)) - 1 | rw [card_Ioc_eq_card_Icc_sub_one, card_Icc] | no goals | 1f338799efd87f48 |
Lean.Omega.IntList.dot_of_left_zero | Mathlib/.lake/packages/lean4/src/lean/Init/Omega/IntList.lean | theorem dot_of_left_zero (w : ∀ x, x ∈ xs → x = 0) : dot xs ys = 0 | case cons.cons
x : Int
xs : List Int
ih : ∀ {ys : IntList}, (∀ (x : Int), x ∈ xs → x = 0) → dot xs ys = 0
w : ∀ (x_1 : Int), x_1 ∈ x :: xs → x_1 = 0
y : Int
ys : List Int
⊢ dot (x :: xs) (y :: ys) = 0 | rw [dot_cons₂, w x (by simp), ih] | case cons.cons
x : Int
xs : List Int
ih : ∀ {ys : IntList}, (∀ (x : Int), x ∈ xs → x = 0) → dot xs ys = 0
w : ∀ (x_1 : Int), x_1 ∈ x :: xs → x_1 = 0
y : Int
ys : List Int
⊢ 0 * y + 0 = 0
case cons.cons
x : Int
xs : List Int
ih : ∀ {ys : IntList}, (∀ (x : Int), x ∈ xs → x = 0) → dot xs ys = 0
w : ∀ (x_1 : Int), x_1 ∈ x... | fa16c8cd41b04385 |
IsUnifLocDoublingMeasure.closedBall_mem_vitaliFamily_of_dist_le_mul | Mathlib/MeasureTheory/Covering/DensityTheorem.lean | theorem closedBall_mem_vitaliFamily_of_dist_le_mul {K : ℝ} {x y : α} {r : ℝ} (h : dist x y ≤ K * r)
(rpos : 0 < r) : closedBall y r ∈ (vitaliFamily μ K).setsAt x | case pos.h.inl.left
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ ((4 * K + 3... | apply closedBall_subset_closedBall' | case pos.h.inl.left.h
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ ((4 * K +... | 60c120fcf237fc0d |
ProbabilityTheory.measure_condCDF_univ | Mathlib/Probability/Kernel/Disintegration/CondCDF.lean | theorem measure_condCDF_univ (ρ : Measure (α × ℝ)) (a : α) : (condCDF ρ a).measure univ = 1 | α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
a : α
⊢ (condCDF ρ a).measure univ = 1 | rw [← ENNReal.ofReal_one, ← sub_zero (1 : ℝ)] | α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
a : α
⊢ (condCDF ρ a).measure univ = ENNReal.ofReal (1 - 0) | 1b16a5159393d6ca |
LinearMap.range_eq_map | Mathlib/Algebra/Module/Submodule/Range.lean | theorem range_eq_map [RingHomSurjective τ₁₂] (f : F) : range f = map f ⊤ | R : Type u_1
R₂ : Type u_2
M : Type u_5
M₂ : Type u_6
inst✝⁸ : Semiring R
inst✝⁷ : Semiring R₂
inst✝⁶ : AddCommMonoid M
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
τ₁₂ : R →+* R₂
F : Type u_10
inst✝² : FunLike F M M₂
inst✝¹ : SemilinearMapClass F τ₁₂ M M₂
inst✝ : RingHomSurjective τ₁₂
f : F
⊢ ra... | ext | case h
R : Type u_1
R₂ : Type u_2
M : Type u_5
M₂ : Type u_6
inst✝⁸ : Semiring R
inst✝⁷ : Semiring R₂
inst✝⁶ : AddCommMonoid M
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
τ₁₂ : R →+* R₂
F : Type u_10
inst✝² : FunLike F M M₂
inst✝¹ : SemilinearMapClass F τ₁₂ M M₂
inst✝ : RingHomSurjective τ₁₂
f :... | 57c153fc58aa9513 |
ContinuousLinearMap.iteratedFDerivWithin_comp_left | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | theorem ContinuousLinearMap.iteratedFDerivWithin_comp_left {f : E → F} (g : F →L[𝕜] G)
(hf : ContDiffWithinAt 𝕜 n f s x) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {i : ℕ} (hi : i ≤ n) :
iteratedFDerivWithin 𝕜 i (g ∘ f) s x =
g.compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f s x) | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
x : E
n : WithTop ℕ∞
f : E → F
g : F →L[𝕜] G
hf : ContDi... | simp | no goals | 832a54e17c3023f1 |
differentiableWithinAt_localInvariantProp | Mathlib/Geometry/Manifold/MFDeriv/Defs.lean | theorem differentiableWithinAt_localInvariantProp :
(contDiffGroupoid 1 I).LocalInvariantProp (contDiffGroupoid 1 I')
(DifferentiableWithinAtProp I I') :=
{ is_local | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorn... | exact u_open.mem_nhds xu | no goals | 3d1405d37db22f10 |
BoxIntegral.norm_integral_le_of_norm_le | Mathlib/Analysis/BoxIntegral/Basic.lean | theorem norm_integral_le_of_norm_le {g : ℝⁿ → ℝ} (hle : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ g x)
(μ : Measure ℝⁿ) [IsLocallyFiniteMeasure μ] (hg : Integrable I l g μ.toBoxAdditive.toSMul) :
‖(integral I l f μ.toBoxAdditive.toSMul : E)‖ ≤ integral I l g μ.toBoxAdditive.toSMul | case neg
ι : Type u
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
I : Box ι
inst✝¹ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
g : (ι → ℝ) → ℝ
hle : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ g x
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hg : Integrable I l g μ.toBoxAdditive.toSMul
hfi : ¬Integrable ... | rw [integral, dif_neg hfi, norm_zero] | case neg
ι : Type u
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
I : Box ι
inst✝¹ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
g : (ι → ℝ) → ℝ
hle : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ g x
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hg : Integrable I l g μ.toBoxAdditive.toSMul
hfi : ¬Integrable ... | fabc40b1fa159acd |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.sat_of_insertRat | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RatAddSound.lean | theorem sat_of_insertRat {n : Nat} (f : DefaultFormula n)
(hf : f.ratUnits = #[] ∧ AssignmentsInvariant f) (c : DefaultClause n) (p : PosFin n → Bool)
(pf : p ⊨ f) :
(insertRatUnits f (negate c)).2 = true → p ⊨ c | case intro.inr.inl.intro.intro.intro.intro.intro.intro
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).2.snd = true
i : PosFin n
hboth : (List.fold... | simp only [negate, Literal.negate, List.mem_map, Prod.mk.injEq, Prod.exists, Bool.exists_bool,
Bool.not_false, Bool.not_true, hf.1, Array.toList_toArray, List.find?, List.not_mem_nil, or_false]
at ib_in_insertUnit_fold | case intro.inr.inl.intro.intro.intro.intro.intro.intro
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).2.snd = true
i : PosFin n
hboth : (List.fold... | c0b439908684a182 |
discrim_eq_zero_iff | Mathlib/Algebra/QuadraticDiscriminant.lean | theorem discrim_eq_zero_iff (ha : a ≠ 0) :
discrim a b c = 0 ↔ (∃! x, a * (x * x) + b * x + c = 0) | K : Type u_1
inst✝¹ : Field K
inst✝ : NeZero 2
a b c : K
ha : a ≠ 0
hd : discrim a b c = 0
⊢ ∃! x, a * (x * x) + b * x + c = 0 | simp_rw [quadratic_eq_zero_iff_of_discrim_eq_zero ha hd, existsUnique_eq] | no goals | b032d9789f00b01a |
imaginaryPart_smul | Mathlib/Data/Complex/Module.lean | theorem imaginaryPart_smul (z : ℂ) (a : A) : ℑ (z • a) = z.re • ℑ a + z.im • ℜ a | A : Type u_1
inst✝³ : AddCommGroup A
inst✝² : Module ℂ A
inst✝¹ : StarAddMonoid A
inst✝ : StarModule ℂ A
z : ℂ
a : A
⊢ ?m.175621 | congrm (ℑ ($((re_add_im z).symm) • a)) | no goals | 72cc35a2c5277b8f |
Matrix.sum_cramer_apply | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | theorem sum_cramer_apply {β} (s : Finset β) (f : n → β → α) (i : n) :
(∑ x ∈ s, cramer A (fun j => f j x) i) = cramer A (fun j : n => ∑ x ∈ s, f j x) i :=
calc
(∑ x ∈ s, cramer A (fun j => f j x) i) = (∑ x ∈ s, cramer A fun j => f j x) i :=
(Finset.sum_apply i s _).symm
_ = cramer A (fun j : n => ∑ ... | n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
β : Type u_1
s : Finset β
f : n → β → α
i : n
⊢ (∑ x ∈ s, A.cramer fun j => f j x) i = A.cramer (fun j => ∑ x ∈ s, f j x) i | rw [sum_cramer, cramer_apply, cramer_apply] | n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
β : Type u_1
s : Finset β
f : n → β → α
i : n
⊢ (A.updateCol i (∑ x ∈ s, fun j => f j x)).det = (A.updateCol i fun j => ∑ x ∈ s, f j x).det | a0c8c2461d8700d4 |
Rat.fract_inv_num_lt_num_of_pos | Mathlib/Data/Rat/Floor.lean | theorem fract_inv_num_lt_num_of_pos {q : ℚ} (q_pos : 0 < q) : (fract q⁻¹).num < q.num | q : ℚ
q_pos : 0 < q
q_num_pos : 0 < q.num
q_num_abs_eq_q_num : ↑q.num.natAbs = q.num
q_inv : ℚ := ↑q.den / ↑q.num
q_inv_def : q_inv = ↑q.den / ↑q.num
q_inv_eq : q⁻¹ = q_inv
this✝ : q⁻¹.num < (⌊q⁻¹⌋ + 1) * ↑q⁻¹.den
this : q⁻¹.num < ⌊q⁻¹⌋ * ↑q⁻¹.den + ↑q⁻¹.den
⊢ q⁻¹.num - ⌊q⁻¹⌋ * ↑q⁻¹.den < ↑q⁻¹.den | rwa [← sub_lt_iff_lt_add'] at this | no goals | d2870a8c7e295972 |
List.Nodup.isCycleOn_formPerm | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | theorem Nodup.isCycleOn_formPerm (h : l.Nodup) :
l.formPerm.IsCycleOn { a | a ∈ l } | α : Type u_2
inst✝ : DecidableEq α
l : List α
h : l.Nodup
a : α
ha : idxOf a l < l.length
b : α
hb : idxOf b l < l.length
⊢ l.formPerm.SameCycle a b | rw [← List.getElem_idxOf ha, ← List.getElem_idxOf hb] | α : Type u_2
inst✝ : DecidableEq α
l : List α
h : l.Nodup
a : α
ha : idxOf a l < l.length
b : α
hb : idxOf b l < l.length
⊢ l.formPerm.SameCycle l[idxOf a l] l[idxOf b l] | d4fd1a414e146e95 |
Matrix.isAddUnit_detp_smul_mul_adjp | Mathlib/LinearAlgebra/Matrix/SemiringInverse.lean | theorem isAddUnit_detp_smul_mul_adjp (hAB : A * B = 1) :
IsAddUnit (detp 1 A • (B * adjp (-1) B) + detp (-1) A • (B * adjp 1 B)) | case intro.intro.hr
n : Type u_1
R : Type u_3
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommSemiring R
A B : Matrix n n R
hAB : A * B = 1
s t : ℤˣ
i j k : n
hk : k ∈ univ
σ : Perm n
hσ : sign σ = t ∧ σ j = k
τ : Perm n
hτ : sign τ = s
h : ¬σ * τ = 1
l : n
hl1 : l ≠ (σ * τ) l
hl2 : l ∈ {τ⁻¹ j}ᶜ
h0 : ∀ (k : n), k... | rw [← prod_equiv τ h0 fun _ _ ↦ rfl, ← prod_mul_distrib, ← mul_prod_erase _ _ hl2, ← smul_eq_mul] | case intro.intro.hr
n : Type u_1
R : Type u_3
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommSemiring R
A B : Matrix n n R
hAB : A * B = 1
s t : ℤˣ
i j k : n
hk : k ∈ univ
σ : Perm n
hσ : sign σ = t ∧ σ j = k
τ : Perm n
hτ : sign τ = s
h : ¬σ * τ = 1
l : n
hl1 : l ≠ (σ * τ) l
hl2 : l ∈ {τ⁻¹ j}ᶜ
h0 : ∀ (k : n), k... | cfc0571f345e1ba6 |
subsingleton_of_disjoint_isClopen | Mathlib/Topology/Connected/Clopen.lean | /-- In a preconnected space, any disjoint family of non-empty clopen subsets has at most one
element. -/
lemma subsingleton_of_disjoint_isClopen
(h_clopen : ∀ i, IsClopen (s i)) :
Subsingleton ι | α : Type u
ι : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : PreconnectedSpace α
s : ι → Set α
h_disj : Pairwise (Disjoint on s)
h_clopen : ∀ (i : ι), IsClopen (s i)
h_nonempty : ∀ (i : ι), s i ≠ ∅
⊢ ¬Nontrivial ι | by_contra contra | α : Type u
ι : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : PreconnectedSpace α
s : ι → Set α
h_disj : Pairwise (Disjoint on s)
h_clopen : ∀ (i : ι), IsClopen (s i)
h_nonempty : ∀ (i : ι), s i ≠ ∅
contra : Nontrivial ι
⊢ False | 6e05e6622254f214 |
Filter.HasBasis.exists_antitone_subbasis | Mathlib/Order/Filter/CountablyGenerated.lean | theorem HasBasis.exists_antitone_subbasis {f : Filter α} [h : f.IsCountablyGenerated]
{p : ι' → Prop} {s : ι' → Set α} (hs : f.HasBasis p s) :
∃ x : ℕ → ι', (∀ i, p (x i)) ∧ f.HasAntitoneBasis fun i => s (x i) | case intro
α : Type u_1
ι' : Sort u_5
f : Filter α
h : f.IsCountablyGenerated
p : ι' → Prop
s : ι' → Set α
hs : f.HasBasis p s
x' : ℕ → Set α
hx' : f = ⨅ i, 𝓟 (x' i)
this : ∀ (i : ℕ), x' i ∈ f
x : ℕ → { i // p i } := fun n => Nat.recOn n (hs.index (x' 0) ⋯) fun n xn => hs.index (x' (n + 1) ∩ s ↑xn) ⋯
x_anti : Antitone... | refine ⟨fun i => (x i).1, fun i => (x i).2, ?_⟩ | case intro
α : Type u_1
ι' : Sort u_5
f : Filter α
h : f.IsCountablyGenerated
p : ι' → Prop
s : ι' → Set α
hs : f.HasBasis p s
x' : ℕ → Set α
hx' : f = ⨅ i, 𝓟 (x' i)
this : ∀ (i : ℕ), x' i ∈ f
x : ℕ → { i // p i } := fun n => Nat.recOn n (hs.index (x' 0) ⋯) fun n xn => hs.index (x' (n + 1) ∩ s ↑xn) ⋯
x_anti : Antitone... | 25d2caa8c01b800f |
Finset.sup_div₀ | Mathlib/Algebra/Order/GroupWithZero/Finset.lean | lemma sup_div₀ [LinearOrderedCommGroupWithZero G₀] [OrderBot G₀] {a : G₀} (ha : 0 < a)
(s : Finset ι) (f : ι → G₀) : s.sup f / a = s.sup fun i ↦ f i / a | case inr
ι : Type u_1
G₀ : Type u_3
inst✝¹ : LinearOrderedCommGroupWithZero G₀
inst✝ : OrderBot G₀
a : G₀
ha : 0 < a
s : Finset ι
f : ι → G₀
hs : s.Nonempty
⊢ s.sup f / a = s.sup fun i => f i / a | rw [← Finset.sup'_eq_sup hs, ← Finset.sup'_eq_sup hs, sup'_div₀ (ha := ha)] | no goals | f14336314318b796 |
MvQPF.liftP_iff | Mathlib/Data/QPF/Multivariate/Basic.lean | theorem liftP_iff {α : TypeVec n} (p : ∀ ⦃i⦄, α i → Prop) (x : F α) :
LiftP p x ↔ ∃ a f, x = abs ⟨a, f⟩ ∧ ∀ i j, p (f i j) | case mp.intro.mk
n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
α : TypeVec.{u} n
p : ⦃i : Fin2 n⦄ → α i → Prop
x : F α
y : F fun i => Subtype p
hy : (fun i => Subtype.val) <$$> y = x
a : (P F).A
f : (P F).B a ⟹ fun i => Subtype p
h : repr y = ⟨a, f⟩
⊢ ∃ a f, x = abs ⟨a, f⟩ ∧ ∀ (i : Fin2 n) (j : (P F).B a i), p (f i j) | use a, fun i j => (f i j).val | case h
n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
α : TypeVec.{u} n
p : ⦃i : Fin2 n⦄ → α i → Prop
x : F α
y : F fun i => Subtype p
hy : (fun i => Subtype.val) <$$> y = x
a : (P F).A
f : (P F).B a ⟹ fun i => Subtype p
h : repr y = ⟨a, f⟩
⊢ x = abs ⟨a, fun i j => ↑(f i j)⟩ ∧ ∀ (i : Fin2 n) (j : (P F).B a i), p ((fun ... | 63f9b99313f568d9 |
finrank_top | Mathlib/LinearAlgebra/Dimension/Finrank.lean | theorem finrank_top : finrank R (⊤ : Submodule R M) = finrank R M | R : Type u
M : Type v
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ toNat (Module.rank R ↥⊤) = toNat (Module.rank R M) | simp [rank_top] | no goals | 26d886d5066f9570 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.restoreAssignments_performRupCheck | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem restoreAssignments_performRupCheck {n : Nat} (f : DefaultFormula n) (f_assignments_size : f.assignments.size = n)
(rupHints : Array Nat) :
restoreAssignments (performRupCheck f rupHints).1.assignments (performRupCheck f rupHints).2.1
=
f.assignments | case intro.h₂.inl.intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
rupHints : Array Nat
f' : DefaultFormula n := (f.performRupCheck rupHints).fst
f'_def : f' = (f.performRupCheck rupHints).fst
f'_assignments_size : f'.assignments.size = n
derivedLits : CNF.Clause (PosFin n)
derivedLits_sat... | simp only [← derivedLits_arr_def] | case intro.h₂.inl.intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
rupHints : Array Nat
f' : DefaultFormula n := (f.performRupCheck rupHints).fst
f'_def : f' = (f.performRupCheck rupHints).fst
f'_assignments_size : f'.assignments.size = n
derivedLits : CNF.Clause (PosFin n)
derivedLits_sat... | 39cd436ee84b1d45 |
one_lt_mabs | Mathlib/Algebra/Order/Group/Unbundled/Abs.lean | @[to_additive (attr := simp) abs_pos] lemma one_lt_mabs : 1 < |a|ₘ ↔ a ≠ 1 | case inr.inr
α : Type u_1
inst✝² : Group α
inst✝¹ : LinearOrder α
inst✝ : MulLeftMono α
a : α
ha : 1 < a
⊢ 1 < mabs a ↔ a ≠ 1 | simp [mabs_of_one_lt ha, ha, ha.ne'] | no goals | 246ff238fd37060c |
IsPredArchimedean.pred_findAtom | Mathlib/Order/SuccPred/Tree.lean | @[simp]
lemma pred_findAtom (r : α) : Order.pred (findAtom r) = ⊥ | α : Type u_1
inst✝⁴ : PartialOrder α
inst✝³ : PredOrder α
inst✝² : IsPredArchimedean α
inst✝¹ : OrderBot α
inst✝ : DecidableEq α
r : α
⊢ Order.pred (Order.pred^[Nat.find ⋯ - 1] r) = ⊥ | generalize h : Nat.find (bot_le (a := r)).exists_pred_iterate = n | α : Type u_1
inst✝⁴ : PartialOrder α
inst✝³ : PredOrder α
inst✝² : IsPredArchimedean α
inst✝¹ : OrderBot α
inst✝ : DecidableEq α
r : α
n : ℕ
h : Nat.find ⋯ = n
⊢ Order.pred (Order.pred^[n - 1] r) = ⊥ | 292f95ee3094926c |
PythagoreanTriple.even_odd_of_coprime | Mathlib/NumberTheory/PythagoreanTriples.lean | theorem even_odd_of_coprime (hc : Int.gcd x y = 1) :
x % 2 = 0 ∧ y % 2 = 1 ∨ x % 2 = 1 ∧ y % 2 = 0 | case inr.inr.intro.intro.intro
z x0 y0 : ℤ
hx : (x0 * 2 + 1) % 2 = 1
hy : (y0 * 2 + 1) % 2 = 1
h : PythagoreanTriple (x0 * 2 + 1) (y0 * 2 + 1) z
hc : (x0 * 2 + 1).gcd (y0 * 2 + 1) = 1
⊢ 2 % 4 % 4 = 2 | decide | no goals | 48f64ea05953c676 |
Finsupp.antidiagonal_single | Mathlib/Data/Finsupp/Antidiagonal.lean | theorem antidiagonal_single (a : α) (n : ℕ) :
antidiagonal (single a n) = (antidiagonal n).map
(Function.Embedding.prodMap ⟨_, single_injective a⟩ ⟨_, single_injective a⟩) | case h.mk
α : Type u
inst✝ : DecidableEq α
a : α
n : ℕ
x y : α →₀ ℕ
⊢ x + y = single a n ↔ ∃ a_1 b, a_1 + b = n ∧ single a a_1 = x ∧ single a b = y | constructor | case h.mk.mp
α : Type u
inst✝ : DecidableEq α
a : α
n : ℕ
x y : α →₀ ℕ
⊢ x + y = single a n → ∃ a_2 b, a_2 + b = n ∧ single a a_2 = x ∧ single a b = y
case h.mk.mpr
α : Type u
inst✝ : DecidableEq α
a : α
n : ℕ
x y : α →₀ ℕ
⊢ (∃ a_1 b, a_1 + b = n ∧ single a a_1 = x ∧ single a b = y) → x + y = single a n | 7873ab140ae48427 |
RingHom.finitePresentation_isStableUnderBaseChange | Mathlib/RingTheory/RingHom/FinitePresentation.lean | theorem finitePresentation_isStableUnderBaseChange :
IsStableUnderBaseChange @FinitePresentation | case h₂
⊢ ∀ ⦃R S T : Type u_1⦄ [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]
[inst_4 : Algebra R T],
(algebraMap R T).FinitePresentation → Algebra.TensorProduct.includeLeftRingHom.FinitePresentation | introv h | case h₂
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : (algebraMap R T).FinitePresentation
⊢ Algebra.TensorProduct.includeLeftRingHom.FinitePresentation | ecddf4004478cb5e |
SimpleGraph.Subgraph.Connected.adj_union | Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean | lemma Connected.adj_union {H K : G.Subgraph}
(Hconn : H.Connected) (Kconn : K.Connected) {u v : V} (uH : u ∈ H.verts) (vK : v ∈ K.verts)
(huv : G.Adj u v) :
((⊤ : G.Subgraph).induce {u, v} ⊔ H ⊔ K).Connected | case refine_1
V : Type u
G : SimpleGraph V
H K : G.Subgraph
Hconn : H.Connected
Kconn : K.Connected
u v : V
uH : u ∈ H.verts
vK : v ∈ K.verts
huv : G.Adj u v
⊢ (⊤.induce {u, v} ⊓ H).verts.Nonempty | exact ⟨u, by simp [uH]⟩ | no goals | 92d126dba17e5301 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.sat_of_insertRat | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RatAddSound.lean | theorem sat_of_insertRat {n : Nat} (f : DefaultFormula n)
(hf : f.ratUnits = #[] ∧ AssignmentsInvariant f) (c : DefaultClause n) (p : PosFin n → Bool)
(pf : p ⊨ f) :
(insertRatUnits f (negate c)).2 = true → p ⊨ c | case intro.inr.inl.intro.intro.intro.intro.intro.intro.intro.inr.intro.intro.h
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).2.snd = true
i : Pos... | rw [i'_eq_i] at i_true_in_c | case intro.inr.inl.intro.intro.intro.intro.intro.intro.intro.inr.intro.intro.h
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).2.snd = true
i : Pos... | c0b439908684a182 |
IsUltrametricDist.ball_eq_of_mem | Mathlib/Topology/MetricSpace/Ultra/Basic.lean | lemma ball_eq_of_mem {x y : X} {r : ℝ} (h : y ∈ ball x r) : ball x r = ball y r | case h
X : Type u_1
inst✝¹ : PseudoMetricSpace X
inst✝ : IsUltrametricDist X
x y : X
r : ℝ
h : y ∈ ball x r
a : X
⊢ a ∈ ball x r ↔ a ∈ ball y r | simp_rw [mem_ball] at h ⊢ | case h
X : Type u_1
inst✝¹ : PseudoMetricSpace X
inst✝ : IsUltrametricDist X
x y : X
r : ℝ
a : X
h : dist y x < r
⊢ dist a x < r ↔ dist a y < r | e1447400c896ac2c |
AffineSubspace.setOf_sOppSide_eq_image2 | Mathlib/Analysis/Convex/Side.lean | theorem setOf_sOppSide_eq_image2 {s : AffineSubspace R P} {x p : P} (hx : x ∉ s) (hp : p ∈ s) :
{ y | s.SOppSide x y } = Set.image2 (fun (t : R) q => t • (x -ᵥ p) +ᵥ q) (Set.Iio 0) s | case h.mp.intro.intro.intro.intro.inr.inr.intro.intro.intro.intro
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x p : P
hx : x ∉ s
hp : p ∈ s
y : P
hy : y ∉ s
p₂ : P
hp₂ : p₂ ∈ s
r₁ r₂ : R
hr₁ : 0 < r₁
hr₂ : ... | rw [div_eq_inv_mul, ← smul_smul, neg_smul, h, smul_neg, smul_smul,
inv_mul_cancel₀ hr₂.ne.symm, one_smul, neg_vsub_eq_vsub_rev, vsub_vadd] | no goals | 839d674f40d68f94 |
Complex.tendsto_tsum_powerSeries_nhdsWithin_stolzSet | Mathlib/Analysis/Complex/AbelLimit.lean | theorem tendsto_tsum_powerSeries_nhdsWithin_stolzSet
(h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) {M : ℝ} :
Tendsto (fun z ↦ ∑' n, f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l) | f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖... | calc
_ = ‖l - g z‖ := by rw [norm_sub_rev]
_ = ‖l - g z - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i +
(1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ := by rw [sub_add_cancel _]
_ ≤ ‖l - g z - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ +
‖(1 - z) * ∑ i ∈ r... | no goals | e1f8d437480849f9 |
mul_div_assoc | Mathlib/Algebra/Group/Defs.lean | theorem mul_div_assoc (a b c : G) : a * b / c = a * (b / c) | G : Type u_1
inst✝ : DivInvMonoid G
a b c : G
⊢ a * b / c = a * (b / c) | rw [div_eq_mul_inv, div_eq_mul_inv, mul_assoc _ _ _] | no goals | a1c0e0e7ec74d8be |
IncidenceAlgebra.mu_prod_mu | Mathlib/Combinatorics/Enumerative/IncidenceAlgebra.lean | /-- The Möbius function on a product order. Based on lemma 2.1.13 of Incidence Algebras by Spiegel
and O'Donnell. -/
@[simp]
lemma mu_prod_mu : (mu 𝕜).prod (mu 𝕜) = (mu 𝕜 : IncidenceAlgebra 𝕜 (α × β)) | 𝕜 : Type u_2
α : Type u_5
β : Type u_6
inst✝⁸ : Ring 𝕜
inst✝⁷ : PartialOrder α
inst✝⁶ : PartialOrder β
inst✝⁵ : LocallyFiniteOrder α
inst✝⁴ : LocallyFiniteOrder β
inst✝³ : DecidableEq α
inst✝² : DecidableEq β
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
inst✝ : DecidableRel fun x1 x2 => x1 ≤ x2
x✝⁵ x✝⁴ x✝³ : α
x✝² x✝¹ ... | simp | no goals | 75516926b8530a97 |
unitInterval.eq_one_or_eq_zero_of_le_mul | Mathlib/Topology/UnitInterval.lean | lemma eq_one_or_eq_zero_of_le_mul {i j : I} (h : i ≤ j * i) : i = 0 ∨ j = 1 | i j : ↑I
h : 0 < ↑i ∧ ↑j < 1
⊢ ↑j * ↑i < ↑i | simpa using mul_lt_mul_of_pos_right h.right h.left | no goals | 93bd4eed0e6ac1e2 |
DistLat.inv_hom_apply | Mathlib/Order/Category/DistLat.lean | lemma inv_hom_apply {X Y : DistLat} (e : X ≅ Y) (x : X) : e.inv (e.hom x) = x | X Y : DistLat
e : X ≅ Y
x : ↑X
⊢ (ConcreteCategory.hom e.inv) ((ConcreteCategory.hom e.hom) x) = x | simp | no goals | ccb6d78e0847694f |
Filter.IsCobounded.frequently_ge | Mathlib/Order/LiminfLimsup.lean | /-- For nontrivial filters in linear orders, coboundedness for `≤` implies frequent boundedness
from below. -/
lemma IsCobounded.frequently_ge [LinearOrder α] [NeBot f] (cobdd : IsCobounded (· ≤ ·) f) :
∃ l, ∃ᶠ x in f, l ≤ x | case intro.inl
α : Type u_1
f : Filter α
inst✝¹ : LinearOrder α
inst✝ : f.NeBot
t : α
ht : ∀ (a : α), (∀ᶠ (x : α) in f, (fun x1 x2 => x1 ≤ x2) x a) → (fun x1 x2 => x1 ≤ x2) t a
tbot : IsBot t
⊢ ∃ l, ∃ᶠ (x : α) in f, l ≤ x | exact ⟨t, .of_forall fun r ↦ tbot r⟩ | no goals | ecc5bf07ddfe8868 |
MvPolynomial.support_esymm'' | Mathlib/RingTheory/MvPolynomial/Symmetric/Defs.lean | theorem support_esymm'' [DecidableEq σ] [Nontrivial R] (n : ℕ) :
(esymm σ R n).support =
(powersetCard n (univ : Finset σ)).biUnion fun t =>
(Finsupp.single (∑ i ∈ t, Finsupp.single i 1) (1 : R)).support | σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
n : ℕ
s t : Finset σ
hst : s ≠ t
h : ∑ i ∈ t, Finsupp.single i 1 = ∑ i ∈ s, Finsupp.single i 1
this : (t.biUnion fun i => (Finsupp.single i 1).support) = s.biUnion fun i => (Finsupp.single i 1).support
hsing... | have ht := biUnion_congr (of_eq_true (eq_self t)) (hsingle t) | σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : Fintype σ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
n : ℕ
s t : Finset σ
hst : s ≠ t
h : ∑ i ∈ t, Finsupp.single i 1 = ∑ i ∈ s, Finsupp.single i 1
this : (t.biUnion fun i => (Finsupp.single i 1).support) = s.biUnion fun i => (Finsupp.single i 1).support
hsing... | 91d19fd65d77a5f5 |
BitVec.not_neg | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem not_neg (x : BitVec w) : ~~~(-x) = x + -1#w | case neg
w : Nat
x : BitVec (w + 1)
hx : ¬x.toNat = 0
⊢ (2 ^ (w + 1) - x.toNat) % 2 ^ (w + 1) = 2 ^ (w + 1) - 1 - (x.toNat + (2 ^ (w + 1) - 1 % 2 ^ (w + 1))) % 2 ^ (w + 1) | rw [show (_ - 1 % _) = _ by rw [Nat.mod_eq_of_lt (by omega)],
show _ + (_ - 1) = (x.toNat - 1) + 2^(w + 1) by omega,
Nat.add_mod_right,
show (x.toNat - 1) % _ = _ by rw [Nat.mod_eq_of_lt (by omega)],
show (_ - x.toNat) % _ = _ by rw [Nat.mod_eq_of_lt (by omega)]] | case neg
w : Nat
x : BitVec (w + 1)
hx : ¬x.toNat = 0
⊢ 2 ^ (w + 1) - x.toNat = 2 ^ (w + 1) - 1 - (x.toNat - 1) | d3765816d380f089 |
PerfectClosure.mk_eq_iff | Mathlib/FieldTheory/PerfectClosure.lean | theorem mk_eq_iff (x y : ℕ × K) :
mk K p x = mk K p y ↔ ∃ z, (frobenius K p)^[y.1 + z] x.2 = (frobenius K p)^[x.1 + z] y.2 | case mp
K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x y : ℕ × K
⊢ mk K p x = mk K p y → ∃ z, (⇑(frobenius K p))^[y.1 + z] x.2 = (⇑(frobenius K p))^[x.1 + z] y.2 | intro H | case mp
K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x y : ℕ × K
H : mk K p x = mk K p y
⊢ ∃ z, (⇑(frobenius K p))^[y.1 + z] x.2 = (⇑(frobenius K p))^[x.1 + z] y.2 | 50fcb88ed45b5508 |
alternatingGroup.nontrivial_of_three_le_card | Mathlib/GroupTheory/SpecificGroups/Alternating.lean | theorem nontrivial_of_three_le_card (h3 : 3 ≤ card α) : Nontrivial (alternatingGroup α) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h3 : 3 ≤ card α
⊢ 1 < 2 | decide | no goals | ab07403d8f3d56b2 |
Language.map_kstar | Mathlib/Computability/Language.lean | theorem map_kstar (f : α → β) (l : Language α) : map f l∗ = (map f l)∗ | α : Type u_1
β : Type u_2
f : α → β
l : Language α
⊢ (map f) (⨆ i, l ^ i) = ⨆ i, (map f) l ^ i | simp_rw [← map_pow] | α : Type u_1
β : Type u_2
f : α → β
l : Language α
⊢ (map f) (⨆ i, l ^ i) = ⨆ i, (map f) (l ^ i) | 03c14b98312e4bfa |
Std.Tactic.BVDecide.BVExpr.bitblast.go_decl_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Impl/Expr.lean | theorem bitblast.go_decl_eq (aig : AIG BVBit) (expr : BVExpr w) :
∀ (idx : Nat) (h1) (h2), (go aig expr).val.aig.decls[idx]'h2 = aig.decls[idx]'h1 | case append.h2
w idx l✝ r✝ : Nat
lhs : BVExpr l✝
rhs : BVExpr r✝
lih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig lhs).val.aig.decls.size),
(go aig lhs).val.aig.decls[idx] = aig.decls[idx]
rih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig rhs).val.aig.decls.size)... | apply Nat.le_trans <;> assumption | no goals | f92988d5a1595b39 |
CochainComplex.HomComplex.Cochain.comp_assoc | Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | /-- The associativity of the composition of cochains. -/
lemma comp_assoc {n₁ n₂ n₃ n₁₂ n₂₃ n₁₂₃ : ℤ}
(z₁ : Cochain F G n₁) (z₂ : Cochain G K n₂) (z₃ : Cochain K L n₃)
(h₁₂ : n₁ + n₂ = n₁₂) (h₂₃ : n₂ + n₃ = n₂₃) (h₁₂₃ : n₁ + n₂ + n₃ = n₁₂₃) :
(z₁.comp z₂ h₁₂).comp z₃ (show n₁₂ + n₃ = n₁₂₃ by rw [← h₁₂, h₁₂₃... | case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K L : CochainComplex C ℤ
n₁ n₂ n₃ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
z₃ : Cochain K L n₃
p q : ℤ
hpq : p + (n₁ + n₂ + n₃) = q
⊢ ((z₁.comp z₂ ⋯).comp z₃ ⋯).v p q hpq = (z₁.comp (z₂.comp z₃ ⋯) ⋯).v p q hpq | rw [comp_v _ _ rfl p (p + n₁ + n₂) q (add_assoc _ _ _).symm (by omega),
comp_v z₁ z₂ rfl p (p + n₁) (p + n₁ + n₂) (by omega) (by omega),
comp_v z₁ (z₂.comp z₃ rfl) (add_assoc n₁ n₂ n₃).symm p (p + n₁) q (by omega) (by omega),
comp_v z₂ z₃ rfl (p + n₁) (p + n₁ + n₂) q (by omega) (by omega), assoc] | no goals | 57cd4e02b56f42c2 |
Finsupp.linearCombination_onFinset | Mathlib/LinearAlgebra/Finsupp/LinearCombination.lean | theorem linearCombination_onFinset {s : Finset α} {f : α → R} (g : α → M)
(hf : ∀ a, f a ≠ 0 → a ∈ s) :
linearCombination R g (Finsupp.onFinset s f hf) = Finset.sum s fun x : α => f x • g x | α : Type u_1
M : Type u_2
R : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Finset α
f : α → R
g : α → M
hf : ∀ (a : α), f a ≠ 0 → a ∈ s
x : α
a✝ : x ∈ s
h : f x • g x ≠ 0
⊢ f x ≠ 0 | contrapose! h | α : Type u_1
M : Type u_2
R : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Finset α
f : α → R
g : α → M
hf : ∀ (a : α), f a ≠ 0 → a ∈ s
x : α
a✝ : x ∈ s
h : f x = 0
⊢ f x • g x = 0 | 072dad5e571c8914 |
CategoryTheory.isCoseparating_iff_mono | Mathlib/CategoryTheory/Generator/Basic.lean | theorem isCoseparating_iff_mono (𝒢 : Set C)
[∀ A : C, HasProduct fun f : ΣG : 𝒢, A ⟶ (G : C) => (f.1 : C)] :
IsCoseparating 𝒢 ↔ ∀ A : C, Mono (Pi.lift (@Sigma.snd 𝒢 fun G => A ⟶ (G : C))) | case refine_2
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
𝒢 : Set C
inst✝ : ∀ (A : C), HasProduct fun f => ↑f.fst
h : ∀ (A : C), Mono (Pi.lift Sigma.snd)
X Y : C
f g : X ⟶ Y
hh : ∀ G ∈ 𝒢, ∀ (h : Y ⟶ G), f ≫ h = g ≫ h
⊢ f = g | haveI := h Y | case refine_2
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
𝒢 : Set C
inst✝ : ∀ (A : C), HasProduct fun f => ↑f.fst
h : ∀ (A : C), Mono (Pi.lift Sigma.snd)
X Y : C
f g : X ⟶ Y
hh : ∀ G ∈ 𝒢, ∀ (h : Y ⟶ G), f ≫ h = g ≫ h
this : Mono (Pi.lift Sigma.snd)
⊢ f = g | ae856e477809f8db |
Std.Tactic.BVDecide.BVExpr.bitblast.denote_blastUdiv | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Udiv.lean | theorem denote_blastUdiv (aig : AIG α) (lhs rhs : BitVec w) (assign : α → Bool)
(input : BinaryRefVec aig w)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, input.lhs.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, input.rhs.get idx hidx, assign⟧ = rhs.getLsbD i... | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
lhs rhs : BitVec w
assign : α → Bool
input : aig.BinaryRefVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := input.lhs.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := ... | rw [Normalize.BitVec.zero_lt_iff_zero_neq] | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
lhs rhs : BitVec w
assign : α → Bool
input : aig.BinaryRefVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := input.lhs.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := ... | 1e06923b53c4a62f |
WithBot.succ_mono | Mathlib/Order/SuccPred/WithBot.lean | lemma succ_mono : Monotone (succ : WithBot α → α)
| ⊥, _, _ => by simp
| (a : α), ⊥, hab => by simp at hab
| (a : α), (b : α), hab => Order.succ_le_succ (by simpa using hab)
| α : Type u_1
inst✝² : Preorder α
inst✝¹ : OrderBot α
inst✝ : SuccOrder α
x✝¹ : WithBot α
x✝ : ⊥ ≤ x✝¹
⊢ ⊥.succ ≤ x✝¹.succ | simp | no goals | ef72efd54259d7ea |
MeasureTheory.Measure.addHaar_image_homothety | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | theorem addHaar_image_homothety (x : E) (r : ℝ) (s : Set E) :
μ (AffineMap.homothety x r '' s) = ENNReal.ofReal (abs (r ^ finrank ℝ E)) * μ s :=
calc
μ (AffineMap.homothety x r '' s) = μ ((fun y => y + x) '' (r • (fun y => y + -x) '' s)) | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
x : E
r : ℝ
s : Set E
⊢ μ (⇑(AffineMap.homothety x r) '' s) = μ ((fun x_1 => r • (x_1 - x) + x) '' s) | rfl | no goals | b87771ad3884117d |
IntermediateField.Lifts.nonempty_algHom_of_exist_lifts_finset | Mathlib/FieldTheory/Extension.lean | theorem nonempty_algHom_of_exist_lifts_finset [alg : Algebra.IsAlgebraic F E]
(h : ∀ S : Finset E, ∃ σ : Lifts F E K, (S : Set E) ⊆ σ.carrier) :
Nonempty (E →ₐ[F] K) | case intro.intro
F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Field K
inst✝¹ : Algebra F E
inst✝ : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ σ, ↑S ⊆ ↑σ.carrier
this✝ : ⊥.IsExtendible
ϕ : Lifts F E K
hϕ : Maximal (fun x => x ∈ {ϕ | ϕ.IsExtendible}) ϕ
this : ϕ.... | let _ : Algebra ϕ.carrier K := ϕ.emb.toAlgebra | case intro.intro
F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Field K
inst✝¹ : Algebra F E
inst✝ : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ σ, ↑S ⊆ ↑σ.carrier
this✝ : ⊥.IsExtendible
ϕ : Lifts F E K
hϕ : Maximal (fun x => x ∈ {ϕ | ϕ.IsExtendible}) ϕ
this : ϕ.... | 18d2833b17837530 |
ProbabilityTheory.Kernel.IndepSets.indep_aux | Mathlib/Probability/Independence/Kernel.lean | theorem IndepSets.indep_aux {m₂ m : MeasurableSpace Ω}
{κ : Kernel α Ω} {μ : Measure α} [IsZeroOrMarkovKernel κ] {p1 p2 : Set (Set Ω)} (h2 : m₂ ≤ m)
(hp2 : IsPiSystem p2) (hpm2 : m₂ = generateFrom p2) (hyp : IndepSets p1 p2 κ μ) {t1 t2 : Set Ω}
(ht1 : t1 ∈ p1) (ht1m : MeasurableSet[m] t1) (ht2m : Measurable... | case inr.basic
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
h : IsMarkov... | exact hyp t1 u ht1 hu | no goals | 7296e6275e034385 |
SimpleGraph.Preconnected.boxProd | Mathlib/Combinatorics/SimpleGraph/Prod.lean | theorem Preconnected.boxProd (hG : G.Preconnected) (hH : H.Preconnected) :
(G □ H).Preconnected | case intro.intro
α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
hG : G.Preconnected
hH : H.Preconnected
x y : α × β
w₁ : G.Walk x.1 y.1
w₂ : H.Walk x.2 y.2
⊢ (G □ H).Reachable x y | exact ⟨(w₁.boxProdLeft _ _).append (w₂.boxProdRight _ _)⟩ | no goals | 9563a2a9ce79f351 |
exists_msmooth_support_eq_eq_one_iff | Mathlib/Geometry/Manifold/PartitionOfUnity.lean | theorem exists_msmooth_support_eq_eq_one_iff
{s t : Set M} (hs : IsOpen s) (ht : IsClosed t) (h : t ⊆ s) :
∃ f : M → ℝ, ContMDiff I 𝓘(ℝ) ∞ f ∧ range f ⊆ Icc 0 1 ∧ support f = s
∧ (∀ x, x ∈ t ↔ f x = 1) | E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
s t : Set M
hs :... | apply lt_of_le_of_ne (g_pos x) (Ne.symm ?_) | E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
s t : Set M
hs :... | 2508fb22dbe6e145 |
MulAction.IsBlock.subsingleton_of_card_lt | Mathlib/GroupTheory/GroupAction/Blocks.lean | theorem subsingleton_of_card_lt [Finite X] (hB : IsBlock G B)
(hB' : Nat.card X < 2 * Set.ncard (orbit G B)) :
B.Subsingleton | G : Type u_1
inst✝³ : Group G
X : Type u_2
inst✝² : MulAction G X
inst✝¹ : IsPretransitive G X
B : Set X
inst✝ : Finite X
hB : IsBlock G B
hB' : Nat.card X < 2 * (orbit G B).ncard
this : B.ncard < 2
⊢ B.Subsingleton | rw [Nat.lt_succ_iff, Set.ncard_le_one_iff_eq] at this | G : Type u_1
inst✝³ : Group G
X : Type u_2
inst✝² : MulAction G X
inst✝¹ : IsPretransitive G X
B : Set X
inst✝ : Finite X
hB : IsBlock G B
hB' : Nat.card X < 2 * (orbit G B).ncard
this : B = ∅ ∨ ∃ a, B = {a}
⊢ B.Subsingleton | 97bb74a7b2c4a12d |
Primrec.of_equiv_symm | Mathlib/Computability/Primrec.lean | theorem of_equiv_symm {β} {e : β ≃ α} :
haveI := Primcodable.ofEquiv α e
Primrec e.symm :=
letI := Primcodable.ofEquiv α e
encode_iff.1 (show Primrec fun a => encode (e (e.symm a)) by simp [Primrec.encode])
| α : Type u_1
inst✝ : Primcodable α
β : Type u_4
e : β ≃ α
this : Primcodable β := Primcodable.ofEquiv α e
⊢ Primrec fun a => encode (e (e.symm a)) | simp [Primrec.encode] | no goals | 596b50cd3bdff543 |
Polynomial.coprime_of_root_cyclotomic | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | theorem coprime_of_root_cyclotomic {n : ℕ} (hpos : 0 < n) {p : ℕ} [hprime : Fact p.Prime] {a : ℕ}
(hroot : IsRoot (cyclotomic n (ZMod p)) (Nat.castRingHom (ZMod p) a)) : a.Coprime p | case a
n : ℕ
hpos : 0 < n
p : ℕ
hprime : Fact (Nat.Prime p)
a : ℕ
hroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)
h : p ∣ a
⊢ False | replace h := (ZMod.natCast_zmod_eq_zero_iff_dvd a p).2 h | case a
n : ℕ
hpos : 0 < n
p : ℕ
hprime : Fact (Nat.Prime p)
a : ℕ
hroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)
h : ↑a = 0
⊢ False | 0720e362f4eb9006 |
Nat.modEq_digits_sum | Mathlib/Data/Nat/Digits.lean | theorem modEq_digits_sum (b b' : ℕ) (h : b' % b = 1) (n : ℕ) : n ≡ (digits b' n).sum [MOD b] | case h.e'_3.h.e'_3
b b' : ℕ
h : b' % b = 1
n : ℕ
⊢ 1 = b' % b | exact h.symm | no goals | 854602fdaff1f255 |
Lean.Data.AC.Context.evalList_sort_congr | Mathlib/.lake/packages/lean4/src/lean/Init/Data/AC.lean | theorem Context.evalList_sort_congr
(ctx : Context α)
(h : Commutative ctx.op)
(h₂ : evalList α ctx a = evalList α ctx b)
(h₃ : a ≠ [])
(h₄ : b ≠ [])
: evalList α ctx (sort.loop a c) = evalList α ctx (sort.loop b c) | case cons.h₂.cons.nil
α : Sort u_1
ctx : Context α
h : Commutative ctx.op
c : Nat
tail✝ : List Nat
ih :
∀ {a b : List Nat},
evalList α ctx a = evalList α ctx b →
a ≠ [] → b ≠ [] → evalList α ctx (sort.loop a tail✝) = evalList α ctx (sort.loop b tail✝)
a : Nat
as : List Nat
h₃ : a :: as ≠ []
h₄ : [] ≠ []
h₂ ... | apply absurd h₄ | case cons.h₂.cons.nil
α : Sort u_1
ctx : Context α
h : Commutative ctx.op
c : Nat
tail✝ : List Nat
ih :
∀ {a b : List Nat},
evalList α ctx a = evalList α ctx b →
a ≠ [] → b ≠ [] → evalList α ctx (sort.loop a tail✝) = evalList α ctx (sort.loop b tail✝)
a : Nat
as : List Nat
h₃ : a :: as ≠ []
h₄ : [] ≠ []
h₂ ... | 21bdef0510a8391b |
ContinuousAffineMap.norm_comp_le | Mathlib/Analysis/Normed/Affine/ContinuousAffineMap.lean | theorem norm_comp_le (g : W₂ →ᴬ[𝕜] V) : ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f 0‖ | 𝕜 : Type u_1
V : Type u_3
W : Type u_4
W₂ : Type u_5
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedAddCommGroup W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →ᴬ[𝕜] W
g : W₂ →ᴬ[𝕜] V
⊢ ‖f.comp g‖ ≤ ‖f‖ * ‖g‖ + ‖f ... | rw [norm_def, max_le_iff] | 𝕜 : Type u_1
V : Type u_3
W : Type u_4
W₂ : Type u_5
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedAddCommGroup W₂
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedSpace 𝕜 V
inst✝¹ : NormedSpace 𝕜 W
inst✝ : NormedSpace 𝕜 W₂
f : V →ᴬ[𝕜] W
g : W₂ →ᴬ[𝕜] V
⊢ ‖(f.comp g) 0‖ ≤ ‖f‖ * ‖g‖ +... | aa55b874b2229b96 |
Zsqrtd.le_arch | Mathlib/NumberTheory/Zsqrtd/Basic.lean | theorem le_arch (a : ℤ√d) : ∃ n : ℕ, a ≤ n | case intro.intro.zero
d : ℕ
a : ℤ√↑d
x : ℕ
h : a ≤ { re := ↑x, im := ↑0 }
⊢ { re := 0, im := 0 }.Nonneg | trivial | no goals | 8d26cc96bf8abedc |
Std.Tactic.BVDecide.BVExpr.bitblast.blastMul.go_denote_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Mul.lean | theorem go_denote_eq {w : Nat} (aig : AIG BVBit) (curr : Nat) (hcurr : curr + 1 ≤ w)
(acc : AIG.RefVec aig w) (lhs rhs : AIG.RefVec aig w) (lexpr rexpr : BitVec w) (assign : Assignment)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, lhs.get idx hidx, assign.toAIGAssignment⟧ = lexpr.getLsbD idx)
(hright : ∀ ... | case isTrue.isFalse.hleft
w : Nat
aig : AIG BVBit
curr : Nat
hcurr : curr + 1 ≤ w
acc lhs rhs : aig.RefVec w
lexpr rexpr : BitVec w
assign : Assignment
hleft :
∀ (idx : Nat) (hidx : idx < w), ⟦assign.toAIGAssignment, { aig := aig, ref := lhs.get idx hidx }⟧ = lexpr.getLsbD idx
hright :
∀ (idx : Nat) (hidx : idx < w... | simp only [RefVec.get_cast, Ref.cast_eq] | case isTrue.isFalse.hleft
w : Nat
aig : AIG BVBit
curr : Nat
hcurr : curr + 1 ≤ w
acc lhs rhs : aig.RefVec w
lexpr rexpr : BitVec w
assign : Assignment
hleft :
∀ (idx : Nat) (hidx : idx < w), ⟦assign.toAIGAssignment, { aig := aig, ref := lhs.get idx hidx }⟧ = lexpr.getLsbD idx
hright :
∀ (idx : Nat) (hidx : idx < w... | 45666f07d8b1c2e0 |
hasBasis_nhdsSet_Iic_Iio | Mathlib/Topology/Order/NhdsSet.lean | theorem hasBasis_nhdsSet_Iic_Iio (a : α) [h : Nonempty (Ioi a)] :
HasBasis (𝓝ˢ (Iic a)) (a < ·) Iio | α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
a : α
h : Nonempty ↑(Ioi a)
⊢ (𝓝ˢ (Iic a)).HasBasis (fun x => a < x) Iio | refine ⟨fun s ↦ ⟨fun hs ↦ ?_, fun ⟨b, hab, hb⟩ ↦ mem_of_superset (Iio_mem_nhdsSet_Iic hab) hb⟩⟩ | α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
a : α
h : Nonempty ↑(Ioi a)
s : Set α
hs : s ∈ 𝓝ˢ (Iic a)
⊢ ∃ i, a < i ∧ Iio i ⊆ s | 7165672eb2730c40 |
Module.DirectLimit.exists_eq_of_of_eq | Mathlib/Algebra/Colimit/Module.lean | theorem exists_eq_of_of_eq {i x y} (h : of R ι G f i x = of R ι G f i y) :
∃ j hij, f i j hij x = f i j hij y | R : Type u_1
inst✝⁶ : Semiring R
ι : Type u_2
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : DecidableEq ι
inst✝³ : (i : ι) → AddCommMonoid (G i)
inst✝² : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
inst✝¹ : DirectedSystem G fun x1 x2 x3 => ⇑(f x1 x2 x3)
inst✝ : IsDirected ι fun x1 x2 => x1 ≤ x2
i : ι... | have := Nonempty.intro i | R : Type u_1
inst✝⁶ : Semiring R
ι : Type u_2
inst✝⁵ : Preorder ι
G : ι → Type u_3
inst✝⁴ : DecidableEq ι
inst✝³ : (i : ι) → AddCommMonoid (G i)
inst✝² : (i : ι) → Module R (G i)
f : (i j : ι) → i ≤ j → G i →ₗ[R] G j
inst✝¹ : DirectedSystem G fun x1 x2 x3 => ⇑(f x1 x2 x3)
inst✝ : IsDirected ι fun x1 x2 => x1 ≤ x2
i : ι... | db7d351d31db8aaa |
Quotient.inductionOn₃ | Mathlib/.lake/packages/lean4/src/lean/Init/Core.lean | theorem inductionOn₃
{s₃ : Setoid φ}
{motive : Quotient s₁ → Quotient s₂ → Quotient s₃ → Prop}
(q₁ : Quotient s₁)
(q₂ : Quotient s₂)
(q₃ : Quotient s₃)
(h : (a : α) → (b : β) → (c : φ) → motive (Quotient.mk s₁ a) (Quotient.mk s₂ b) (Quotient.mk s₃ c))
: motive q₁ q₂ q₃ | α : Sort uA
β : Sort uB
φ : Sort uC
s₁ : Setoid α
s₂ : Setoid β
s₃ : Setoid φ
motive : Quotient s₁ → Quotient s₂ → Quotient s₃ → Prop
q₁ : Quotient s₁
q₂ : Quotient s₂
q₃ : Quotient s₃
h : ∀ (a : α) (b : β) (c : φ), motive (Quotient.mk s₁ a) (Quotient.mk s₂ b) (Quotient.mk s₃ c)
⊢ motive q₁ q₂ q₃ | induction q₁ using Quotient.ind | case a
α : Sort uA
β : Sort uB
φ : Sort uC
s₁ : Setoid α
s₂ : Setoid β
s₃ : Setoid φ
motive : Quotient s₁ → Quotient s₂ → Quotient s₃ → Prop
q₂ : Quotient s₂
q₃ : Quotient s₃
h : ∀ (a : α) (b : β) (c : φ), motive (Quotient.mk s₁ a) (Quotient.mk s₂ b) (Quotient.mk s₃ c)
a✝ : α
⊢ motive (Quotient.mk s₁ a✝) q₂ q₃ | b768315a7eea770d |
Lean.Order.List.monotone_filterM | Mathlib/.lake/packages/lean4/src/lean/Init/Internal/Order/Lemmas.lean | theorem monotone_filterM
{m : Type → Type v} [Monad m] [∀ α, PartialOrder (m α)] [MonoBind m] {α : Type}
(f : γ → α → m Bool) (xs : List α) (hmono : monotone f) :
monotone (fun x => xs.filterM (f x)) | case hmono₁
γ : Type w
inst✝³ : PartialOrder γ
m : Type → Type v
inst✝² : Monad m
inst✝¹ : (α : Type) → PartialOrder (m α)
inst✝ : MonoBind m
α : Type
f : γ → α → m Bool
xs : List α
hmono : monotone f
⊢ monotone fun x => List.filterAuxM (f x) xs [] | exact monotone_filterAuxM f xs [] hmono | no goals | 03c2246583d08433 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastArithShiftRight.go_denote_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/ShiftRight.lean | theorem go_denote_eq (aig : AIG α) (distance : AIG.RefVec aig n) (curr : Nat)
(hcurr : curr ≤ n - 1) (acc : AIG.RefVec aig w)
(lhs : BitVec w) (rhs : BitVec n) (assign : α → Bool)
(hacc : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, acc.get idx hidx, assign⟧ = (BitVec.sshiftRightRec lhs rhs curr).getLsbD idx)
... | case isTrue.hacc
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
n w : Nat
aig : AIG α
distance : aig.RefVec n
curr : Nat
hcurr : curr ≤ n - 1
acc : aig.RefVec w
lhs : BitVec w
rhs : BitVec n
assign : α → Bool
hacc :
∀ (idx : Nat) (hidx : idx < w),
⟦assign, { aig := aig, ref := acc.get idx hidx }⟧ = (lhs.sshif... | rw [twoPowShift_eq (lhs := BitVec.sshiftRightRec lhs rhs curr)] | case isTrue.hacc.hleft
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
n w : Nat
aig : AIG α
distance : aig.RefVec n
curr : Nat
hcurr : curr ≤ n - 1
acc : aig.RefVec w
lhs : BitVec w
rhs : BitVec n
assign : α → Bool
hacc :
∀ (idx : Nat) (hidx : idx < w),
⟦assign, { aig := aig, ref := acc.get idx hidx }⟧ = (lhs... | a62c971c970b9dfe |
MeasureTheory.AEEqFun.Integrable.add | Mathlib/MeasureTheory/Function/L1Space/AEEqFun.lean | theorem Integrable.add {f g : α →ₘ[μ] β} : Integrable f → Integrable g → Integrable (f + g) | α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : α →ₘ[μ] β
⊢ f.Integrable → g.Integrable → (f + g).Integrable | refine induction_on₂ f g fun f hf g hg hfi hgi => ?_ | α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f✝ g✝ : α →ₘ[μ] β
f : α → β
hf : AEStronglyMeasurable f μ
g : α → β
hg : AEStronglyMeasurable g μ
hfi : (mk f hf).Integrable
hgi : (mk g hg).Integrable
⊢ (mk f hf + mk g hg).Integrable | 48646018570a5bc9 |
exists_Ioo_extr_on_Icc | Mathlib/Topology/Order/Rolle.lean | theorem exists_Ioo_extr_on_Icc (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) :
∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c | X : Type u_1
Y : Type u_2
inst✝⁶ : ConditionallyCompleteLinearOrder X
inst✝⁵ : DenselyOrdered X
inst✝⁴ : TopologicalSpace X
inst✝³ : OrderTopology X
inst✝² : LinearOrder Y
inst✝¹ : TopologicalSpace Y
inst✝ : OrderTopology Y
f : X → Y
a b : X
hab : a < b
hfc : ContinuousOn f (Icc a b)
hfI : f a = f b
ne : (Icc a b).None... | rw [h] | no goals | d612389e0169ee4b |
MvPolynomial.constantCoeff_rename | Mathlib/Algebra/MvPolynomial/Rename.lean | theorem constantCoeff_rename {τ : Type*} (f : σ → τ) (φ : MvPolynomial σ R) :
constantCoeff (rename f φ) = constantCoeff φ | case h_add
σ : Type u_1
R : Type u_4
inst✝ : CommSemiring R
τ : Type u_6
f : σ → τ
φ : MvPolynomial σ R
⊢ ∀ (p q : MvPolynomial σ R),
constantCoeff ((rename f) p) = constantCoeff p →
constantCoeff ((rename f) q) = constantCoeff q → constantCoeff ((rename f) (p + q)) = constantCoeff (p + q) | intro p q hp hq | case h_add
σ : Type u_1
R : Type u_4
inst✝ : CommSemiring R
τ : Type u_6
f : σ → τ
φ p q : MvPolynomial σ R
hp : constantCoeff ((rename f) p) = constantCoeff p
hq : constantCoeff ((rename f) q) = constantCoeff q
⊢ constantCoeff ((rename f) (p + q)) = constantCoeff (p + q) | 99e6581b3c86be32 |
LinearMap.lTensor_sub | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | theorem lTensor_sub (f g : N →ₗ[R] P) : (f - g).lTensor M = f.lTensor M - g.lTensor M | R : Type u_1
inst✝⁶ : CommSemiring R
M : Type u_2
N : Type u_3
P : Type u_4
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup N
inst✝³ : AddCommGroup P
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R P
f g : N →ₗ[R] P
⊢ (lTensorHom M) (f - g) = (lTensorHom M) f - (lTensorHom M) g | exact (lTensorHom (R := R) (N := N) (P := P) M).map_sub f g | no goals | 731dd0d94b8c6625 |
Uniform.tendsto_nhds_right | Mathlib/Topology/UniformSpace/Basic.lean | theorem tendsto_nhds_right {f : Filter β} {u : β → α} {a : α} :
Tendsto u f (𝓝 a) ↔ Tendsto (fun x => (a, u x)) f (𝓤 α) | α : Type ua
β : Type ub
inst✝ : UniformSpace α
f : Filter β
u : β → α
a : α
⊢ Tendsto (Prod.mk a ∘ u) f (𝓤 α) ↔ Tendsto (fun x => (a, u x)) f (𝓤 α) | rfl | no goals | e444ec12a4c4ae35 |
Subalgebra.eq_bot_of_rank_le_one | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | theorem eq_bot_of_rank_le_one (h : Module.rank F S ≤ 1) [Module.Free F S] : S = ⊥ | case intro.mk
F : Type u_1
E : Type u_2
inst✝⁴ : CommRing F
inst✝³ : StrongRankCondition F
inst✝² : Ring E
inst✝¹ : Algebra F E
S : Subalgebra F E
h : Module.rank F ↥S ≤ 1
inst✝ : Free F ↥S
a✝ : Nontrivial E
κ : Type u_2
b : Basis κ F ↥S
⊢ S = ⊥ | by_cases h1 : Module.rank F S = 1 | case pos
F : Type u_1
E : Type u_2
inst✝⁴ : CommRing F
inst✝³ : StrongRankCondition F
inst✝² : Ring E
inst✝¹ : Algebra F E
S : Subalgebra F E
h : Module.rank F ↥S ≤ 1
inst✝ : Free F ↥S
a✝ : Nontrivial E
κ : Type u_2
b : Basis κ F ↥S
h1 : Module.rank F ↥S = 1
⊢ S = ⊥
case neg
F : Type u_1
E : Type u_2
inst✝⁴ : CommRing... | 9ac2057737ac9eb2 |
BoundedContinuousFunction.lipschitz_comp | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | theorem lipschitz_comp {G : β → γ} {C : ℝ≥0} (H : LipschitzWith C G) :
LipschitzWith C (comp G H : (α →ᵇ β) → α →ᵇ γ) :=
LipschitzWith.of_dist_le_mul fun f g =>
(dist_le (mul_nonneg C.2 dist_nonneg)).2 fun x =>
calc
dist (G (f x)) (G (g x)) ≤ C * dist (f x) (g x) := H.dist_le_mul _ _
_ ≤... | case h
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
G : β → γ
C : ℝ≥0
H : LipschitzWith C G
f g : α →ᵇ β
x : α
⊢ dist (f x) (g x) ≤ dist f g | apply dist_coe_le_dist | no goals | 8b4dbb01a1e3a73a |
Vector.eq_iff_flatten_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem eq_iff_flatten_eq {L L' : Vector (Vector α n) m} :
L = L' ↔ L.flatten = L'.flatten | case of.of.mp
α : Type u_1
n m : Nat
L : Array (Array α)
h₁ : L.size = m
h₂ : ∀ (xs : Array α), xs ∈ L → xs.size = n
L' : Array (Array α)
h₁' : L'.size = m
h₂' : ∀ (xs : Array α), xs ∈ L' → xs.size = n
⊢ Array.map (fun x => { toArray := x.val, size_toArray := ⋯ }) L.attach =
Array.map (fun x => { toArray := x.val... | intro h | case of.of.mp
α : Type u_1
n m : Nat
L : Array (Array α)
h₁ : L.size = m
h₂ : ∀ (xs : Array α), xs ∈ L → xs.size = n
L' : Array (Array α)
h₁' : L'.size = m
h₂' : ∀ (xs : Array α), xs ∈ L' → xs.size = n
h :
Array.map (fun x => { toArray := x.val, size_toArray := ⋯ }) L.attach =
Array.map (fun x => { toArray := x.v... | 72d2568041cb6666 |
real_inner_div_norm_mul_norm_eq_neg_one_iff | Mathlib/Analysis/InnerProductSpace/Basic.lean | theorem real_inner_div_norm_mul_norm_eq_neg_one_iff (x y : F) :
⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = -1 ↔ x ≠ 0 ∧ ∃ r : ℝ, r < 0 ∧ y = r • x | F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ (x ≠ 0 ∧ ∃ x_1, 0 < -x_1 ∧ -y = -x_1 • x) ↔ x ≠ 0 ∧ ∃ r < 0, y = r • x | refine Iff.rfl.and (exists_congr fun r => ?_) | F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
r : ℝ
⊢ 0 < -r ∧ -y = -r • x ↔ r < 0 ∧ y = r • x | f7464562b660032b |
LinearMap.BilinForm.span_singleton_sup_orthogonal_eq_top | Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean | theorem span_singleton_sup_orthogonal_eq_top {B : BilinForm K V} {x : V} (hx : ¬B.IsOrtho x x) :
(K ∙ x) ⊔ B.orthogonal (K ∙ x) = ⊤ | V : Type u_5
K : Type u_6
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
B : BilinForm K V
x : V
hx : ¬B.IsOrtho x x
⊢ Submodule.span K {x} ⊔ ker (B.toLinHomAux₁ x) = ⊤ | exact LinearMap.span_singleton_sup_ker_eq_top _ hx | no goals | fa0f717d19df8262 |
ProbabilityTheory.Kernel.deterministic_apply' | Mathlib/Probability/Kernel/Basic.lean | theorem deterministic_apply' {f : α → β} (hf : Measurable f) (a : α) {s : Set β}
(hs : MeasurableSet s) : deterministic f hf a s = s.indicator (fun _ => 1) (f a) | α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : Measurable f
a : α
s : Set β
hs : MeasurableSet s
⊢ ({ toFun := fun a => Measure.dirac (f a), measurable' := ⋯ } a) s = s.indicator (fun x => 1) (f a) | change Measure.dirac (f a) s = s.indicator 1 (f a) | α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
f : α → β
hf : Measurable f
a : α
s : Set β
hs : MeasurableSet s
⊢ (Measure.dirac (f a)) s = s.indicator 1 (f a) | 0e51885f87331fa2 |
LocallyFinite.countable_univ | Mathlib/Topology/Compactness/SigmaCompact.lean | theorem LocallyFinite.countable_univ {f : ι → Set X} (hf : LocallyFinite f)
(hne : ∀ i, (f i).Nonempty) : (univ : Set ι).Countable | case intro
X : Type u_1
ι : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : SigmaCompactSpace X
f : ι → Set X
hf : LocallyFinite f
hne : ∀ (i : ι), (f i).Nonempty
this : ∀ (n : ℕ), {i | (f i ∩ compactCovering X n).Nonempty}.Finite
i : ι
x✝ : i ∈ univ
x : X
hx : x ∈ f i
⊢ i ∈ ⋃ i, {i_1 | (f i_1 ∩ compactCovering X i).Nonem... | rcases iUnion_eq_univ_iff.1 (iUnion_compactCovering X) x with ⟨n, hn⟩ | case intro.intro
X : Type u_1
ι : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : SigmaCompactSpace X
f : ι → Set X
hf : LocallyFinite f
hne : ∀ (i : ι), (f i).Nonempty
this : ∀ (n : ℕ), {i | (f i ∩ compactCovering X n).Nonempty}.Finite
i : ι
x✝ : i ∈ univ
x : X
hx : x ∈ f i
n : ℕ
hn : x ∈ compactCovering X n
⊢ i ∈ ⋃ i, {... | 7464f3928fba8b2f |
MeasureTheory.tendsto_condExp_unique | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | theorem tendsto_condExp_unique (fs gs : ℕ → α → E) (f g : α → E)
(hfs_int : ∀ n, Integrable (fs n) μ) (hgs_int : ∀ n, Integrable (gs n) μ)
(hfs : ∀ᵐ x ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x)))
(hgs : ∀ᵐ x ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))) (bound_fs : α → ℝ)
(h_int_bound_fs : Integrab... | case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedLatticeAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
fs gs : ℕ → α → E
f g : α → E
hfs_int : ∀ (n : ℕ), Integrable (fs n) μ
hgs_int : ∀ (n : ℕ), Integrable (gs n) μ
hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) a... | rfl | no goals | a9df86dd1e158e77 |
LinearMap.polyCharpolyAux_baseChange | Mathlib/Algebra/Module/LinearMap/Polynomial.lean | lemma polyCharpolyAux_baseChange (A : Type*) [CommRing A] [Algebra R A] :
polyCharpolyAux (tensorProduct _ _ _ _ ∘ₗ φ.baseChange A) (basis A b) (basis A bₘ) =
(polyCharpolyAux φ b bₘ).map (MvPolynomial.map (algebraMap R A)) | case h₀
R : Type u_1
L : Type u_2
M : Type u_3
ι : Type u_5
ιM : Type u_7
inst✝¹⁰ : CommRing R
inst✝⁹ : AddCommGroup L
inst✝⁸ : Module R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
φ : L →ₗ[R] Module.End R M
inst✝⁵ : Fintype ι
inst✝⁴ : Fintype ιM
inst✝³ : DecidableEq ι
inst✝² : DecidableEq ιM
b : Basis ι R L
bₘ : Bas... | rw [this, if_neg H, map_zero] | no goals | 43f09ebe5fe966ae |
BitVec.getMsbD_or | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem getMsbD_or {x y : BitVec w} : (x ||| y).getMsbD i = (x.getMsbD i || y.getMsbD i) | w i : Nat
x y : BitVec w
⊢ (x ||| y).getMsbD i = (x.getMsbD i || y.getMsbD i) | simp only [getMsbD] | w i : Nat
x y : BitVec w
⊢ (decide (i < w) && (x ||| y).getLsbD (w - 1 - i)) =
(decide (i < w) && x.getLsbD (w - 1 - i) || decide (i < w) && y.getLsbD (w - 1 - i)) | b7b212b8c727554b |
String.extract.go₂_append_left | Mathlib/.lake/packages/batteries/Batteries/Data/String/Lemmas.lean | theorem extract.go₂_append_left : ∀ (s t : List Char) (i e : Nat),
e = utf8Len s + i → go₂ (s ++ t) ⟨i⟩ ⟨e⟩ = s
| [], t, i, _, rfl => by cases t <;> simp [go₂]
| c :: cs, t, i, _, rfl => by
simp only [List.cons_append, utf8Len_cons, go₂, Pos.ext_iff, ne_add_utf8Size_add_self, ↓reduceIte,
Pos.addChar_eq, List.... | c : Char
cs t : List Char
i : Nat
⊢ go₂ (cs ++ t) { byteIdx := i + c.utf8Size } { byteIdx := utf8Len cs + c.utf8Size + i } = cs | apply go₂_append_left | case a
c : Char
cs t : List Char
i : Nat
⊢ utf8Len cs + c.utf8Size + i = utf8Len cs + (i + c.utf8Size) | f179773ad64697cb |
Metric.Sigma.dist_same | Mathlib/Topology/MetricSpace/Gluing.lean | theorem dist_same (i : ι) (x y : E i) : dist (Sigma.mk i x) ⟨i, y⟩ = dist x y | ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x y : E i
⊢ dist ⟨i, x⟩ ⟨i, y⟩ = dist x y | simp [Dist.dist, Sigma.dist] | no goals | 27687575e71f2783 |
mul_eq_of_eq_mul_inv₀ | Mathlib/Algebra/GroupWithZero/Units/Basic.lean | /-- A variant of `eq_mul_inv_iff_mul_eq₀` that moves the nonzero hypothesis to another variable. -/
lemma mul_eq_of_eq_mul_inv₀ (ha : a ≠ 0) (h : a = c * b⁻¹) : a * b = c | G₀ : Type u_3
inst✝ : GroupWithZero G₀
a c : G₀
ha : a ≠ 0
h : a = c * 0⁻¹
⊢ False | simp [ha] at h | no goals | a1b58083849146a3 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.limplies_of_assignmentsInvariant | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean | theorem limplies_of_assignmentsInvariant {n : Nat} (f : DefaultFormula n)
(f_AssignmentsInvariant : AssignmentsInvariant f) :
Limplies (PosFin n) f f.assignments | case neg
n : Nat
f : DefaultFormula n
p : PosFin n → Bool
pf : p ⊨ f
hsize : f.assignments.size = n
i : PosFin n
f_AssignmentsInvariant :
hasAssignment (decide (p i = false)) f.assignments[i.val] = true → Limplies (PosFin n) f (i, decide (p i = false))
h✝ : ¬hasAssignment (decide (p i = false)) f.assignments[i.val] =... | next h => simp_all [getElem!, i.2.2, decidableGetElem?] | no goals | 324db374b482bd66 |
LieModule.disjoint_genWeightSpaceOf | Mathlib/Algebra/Lie/Weights/Basic.lean | lemma disjoint_genWeightSpaceOf [NoZeroSMulDivisors R M] {x : L} {φ₁ φ₂ : R} (h : φ₁ ≠ φ₂) :
Disjoint (genWeightSpaceOf M φ₁ x) (genWeightSpaceOf M φ₂ x) | R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : LieRing.IsNilpotent L
inst✝ : NoZeroSMulDivisors R M
x : L
φ₁ φ₂ : R
h : φ₁ ≠ φ₂
⊢ Disjoint ↑(genWeightSpaceOf M ... | dsimp [genWeightSpaceOf] | R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : LieRing.IsNilpotent L
inst✝ : NoZeroSMulDivisors R M
x : L
φ₁ φ₂ : R
h : φ₁ ≠ φ₂
⊢ Disjoint (((toEnd R L M) x).ma... | f2856c44a97b7b31 |
CategoryTheory.NatTrans.epi_iff_epi_app' | Mathlib/CategoryTheory/Adjunction/Evaluation.lean | theorem NatTrans.epi_iff_epi_app' {F G : C ⥤ D} (η : F ⟶ G) : Epi η ↔ ∀ c, Epi (η.app c) | C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : ∀ (a b : C), HasProductsOfShape (a ⟶ b) D
F G : C ⥤ D
η : F ⟶ G
⊢ Epi η ↔ ∀ (c : C), Epi (η.app c) | constructor | case mp
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : ∀ (a b : C), HasProductsOfShape (a ⟶ b) D
F G : C ⥤ D
η : F ⟶ G
⊢ Epi η → ∀ (c : C), Epi (η.app c)
case mpr
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : ∀ (a b : C), HasProduct... | 7c954a3aed68ec0c |
exists_pow_lt_of_lt_one | Mathlib/Algebra/Order/Archimedean/Basic.lean | theorem exists_pow_lt_of_lt_one (hx : 0 < x) (hy : y < 1) : ∃ n : ℕ, y ^ n < x | case neg.intro
α : Type u_1
inst✝² : LinearOrderedSemifield α
inst✝¹ : Archimedean α
x y : α
inst✝ : ExistsAddOfLE α
hx : 0 < x
hy : y < 1
y_pos : 0 < y
q : ℕ
hq : x⁻¹ < y⁻¹ ^ q
⊢ ∃ n, y ^ n < x | exact ⟨q, by rwa [inv_pow, inv_lt_inv₀ hx (pow_pos y_pos _)] at hq⟩ | no goals | 21e7fde79a3a92b2 |
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