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 |
|---|---|---|---|---|---|---|
Std.Tactic.BVDecide.BVExpr.bitblast.denote_blastUmod | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Umod.lean | theorem denote_blastUmod (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... | case hright
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 := ... | intro idx hidx | case hright
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 := ... | 3032a72cd8852392 |
List.Nodup.cyclicPermutations | Mathlib/Data/List/Rotate.lean | theorem Nodup.cyclicPermutations {l : List α} (hn : Nodup l) :
Nodup (cyclicPermutations l) | case inr.mk.mk
α : Type u
l : List α
hn : l.Nodup
hl : l ≠ []
i : ℕ
hi : i < l.cyclicPermutations.length
j : ℕ
hj : j < l.cyclicPermutations.length
h : l.cyclicPermutations.get ⟨i, hi⟩ = l.cyclicPermutations.get ⟨j, hj⟩
⊢ ⟨i, hi⟩ = ⟨j, hj⟩ | simp only [length_cyclicPermutations_of_ne_nil l hl] at hi hj | case inr.mk.mk
α : Type u
l : List α
hn : l.Nodup
hl : l ≠ []
i : ℕ
hi✝ : i < l.cyclicPermutations.length
j : ℕ
hj✝ : j < l.cyclicPermutations.length
h : l.cyclicPermutations.get ⟨i, hi✝⟩ = l.cyclicPermutations.get ⟨j, hj✝⟩
hi : i < l.length
hj : j < l.length
⊢ ⟨i, hi✝⟩ = ⟨j, hj✝⟩ | 2e9cb67be0f29a76 |
Subgroup.mulSingle_mem_pi | Mathlib/Algebra/Group/Subgroup/Basic.lean | theorem mulSingle_mem_pi [DecidableEq η] {I : Set η} {H : ∀ i, Subgroup (f i)} (i : η) (x : f i) :
Pi.mulSingle i x ∈ pi I H ↔ i ∈ I → x ∈ H i | case mp
η : Type u_7
f : η → Type u_8
inst✝¹ : (i : η) → Group (f i)
inst✝ : DecidableEq η
I : Set η
H : (i : η) → Subgroup (f i)
i : η
x : f i
⊢ Pi.mulSingle i x ∈ pi I H → i ∈ I → x ∈ H i | intro h hi | case mp
η : Type u_7
f : η → Type u_8
inst✝¹ : (i : η) → Group (f i)
inst✝ : DecidableEq η
I : Set η
H : (i : η) → Subgroup (f i)
i : η
x : f i
h : Pi.mulSingle i x ∈ pi I H
hi : i ∈ I
⊢ x ∈ H i | 1c8d5b24fe240cda |
DirectSum.linearEquivFunOnFintype_lof | Mathlib/Algebra/DirectSum/Module.lean | theorem linearEquivFunOnFintype_lof [Fintype ι] (i : ι) (m : M i) :
(linearEquivFunOnFintype R ι M) (lof R ι M i m) = Pi.single i m | case h
R : Type u
inst✝⁴ : Semiring R
ι : Type v
M : ι → Type w
inst✝³ : (i : ι) → AddCommMonoid (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
i : ι
m : M i
a : ι
⊢ (linearEquivFunOnFintype R ι M) ((lof R ι M i) m) a = Pi.single i m a | change (DFinsupp.equivFunOnFintype (lof R ι M i m)) a = _ | case h
R : Type u
inst✝⁴ : Semiring R
ι : Type v
M : ι → Type w
inst✝³ : (i : ι) → AddCommMonoid (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
i : ι
m : M i
a : ι
⊢ DFinsupp.equivFunOnFintype ((lof R ι M i) m) a = Pi.single i m a | 7f56508017376b53 |
CategoryTheory.classifier_isSheaf | Mathlib/CategoryTheory/Sites/Closed.lean | theorem classifier_isSheaf : Presieve.IsSheaf J₁ (Functor.closedSieves J₁) | case refine_1.mk.mk.a.h
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows
M : Sieve (Opposite.unop (Opposite.op X))
hM : J₁.IsClosed M
N : Sieve (Opposite.unop (Opposite.op X))
hN : J₁.IsClosed N
hM₂ : x.IsAm... | have q : ∀ ⦃Z : C⦄ (g : Z ⟶ X) (_ : S g), M.pullback g = N.pullback g :=
fun Z g hg => congr_arg Subtype.val ((hM₂ g hg).trans (hN₂ g hg).symm) | case refine_1.mk.mk.a.h
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows
M : Sieve (Opposite.unop (Opposite.op X))
hM : J₁.IsClosed M
N : Sieve (Opposite.unop (Opposite.op X))
hN : J₁.IsClosed N
hM₂ : x.IsAm... | 902347477e841054 |
Associates.dvd_of_mem_factors | Mathlib/RingTheory/UniqueFactorizationDomain/FactorSet.lean | theorem dvd_of_mem_factors {a p : Associates α} (hm : p ∈ factors a) :
p ∣ a | α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a p : Associates α
hm : p ∈ a.factors
⊢ p ∣ a | rcases eq_or_ne a 0 with rfl | ha0 | case inl
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
p : Associates α
hm : p ∈ factors 0
⊢ p ∣ 0
case inr
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
a p : Associates α
hm : p ∈ a.factors
ha0 : a ≠ 0
⊢ p ∣ a | 5299e19f6694fb2c |
FirstOrder.Language.dlo_age | Mathlib/ModelTheory/Order.lean | lemma dlo_age [Language.order.Structure M] [Mdlo : M ⊨ Language.order.dlo] [Nonempty M] :
Language.order.age M = {M : CategoryTheory.Bundled.{w'} Language.order.Structure |
Finite M ∧ M ⊨ Language.order.linearOrderTheory} | case h
M : Type w'
inst✝¹ : Language.order.Structure M
Mdlo : M ⊨ Language.order.dlo
inst✝ : Nonempty M
N : CategoryTheory.Bundled Language.order.Structure
⊢ N ∈ {N | FG Language.order ↑N ∧ Nonempty (↑N ↪[Language.order] M)} ↔
N ∈ {M | Finite ↑M ∧ ↑M ⊨ Language.order.linearOrderTheory} | refine ⟨fun ⟨hF, h⟩ => ⟨hF.finite, Theory.IsUniversal.models_of_embedding h.some⟩,
fun ⟨hF, h⟩ => ⟨FG.of_finite, ?_⟩⟩ | case h
M : Type w'
inst✝¹ : Language.order.Structure M
Mdlo : M ⊨ Language.order.dlo
inst✝ : Nonempty M
N : CategoryTheory.Bundled Language.order.Structure
x✝ : N ∈ {M | Finite ↑M ∧ ↑M ⊨ Language.order.linearOrderTheory}
hF : Finite ↑N
h : ↑N ⊨ Language.order.linearOrderTheory
⊢ Nonempty (↑N ↪[Language.order] M) | 833eff06b402bf5b |
upperSemicontinuousWithinAt_univ_iff | Mathlib/Topology/Semicontinuous.lean | theorem upperSemicontinuousWithinAt_univ_iff :
UpperSemicontinuousWithinAt f univ x ↔ UpperSemicontinuousAt f x | α : Type u_1
inst✝¹ : TopologicalSpace α
β : Type u_2
inst✝ : Preorder β
f : α → β
x : α
⊢ UpperSemicontinuousWithinAt f univ x ↔ UpperSemicontinuousAt f x | simp [UpperSemicontinuousWithinAt, UpperSemicontinuousAt, nhdsWithin_univ] | no goals | ccaa880ab8e0e06e |
not_countable_complex | Mathlib/Data/Complex/Cardinality.lean | theorem not_countable_complex : ¬(Set.univ : Set ℂ).Countable | ⊢ ℵ₀ < 𝔠 | apply cantor | no goals | b54c9fa69209a03d |
LieSubmodule.lowerCentralSeries_tensor_eq_baseChange | Mathlib/Algebra/Lie/Nilpotent.lean | @[simp]
lemma LieSubmodule.lowerCentralSeries_tensor_eq_baseChange (k : ℕ) :
lowerCentralSeries A (A ⊗[R] L) (A ⊗[R] M) k =
(lowerCentralSeries R L M k).baseChange A | R : Type u_1
A : 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 : CommRing A
inst✝ : Algebra R A
k : ℕ
⊢ lowerCentralSeries A (A ⊗[R] L) (A ⊗[R] M) k = baseChange A (... | induction k with
| zero => simp
| succ k ih => simp only [lowerCentralSeries_succ, ih, ← baseChange_top, lie_baseChange] | no goals | e15d4e487430dd8d |
EuclideanDomain.dvd_div_of_mul_dvd | Mathlib/Algebra/EuclideanDomain/Basic.lean | theorem dvd_div_of_mul_dvd {a b c : R} (h : a * b ∣ c) : b ∣ c / a | case inl
R : Type u
inst✝ : EuclideanDomain R
b c : R
h : 0 * b ∣ c
⊢ b ∣ c / 0 | simp only [div_zero, dvd_zero] | no goals | aadbf8668220469f |
Ordinal.isLimit_add_iff | Mathlib/SetTheory/Ordinal/Arithmetic.lean | theorem isLimit_add_iff {a b} : IsLimit (a + b) ↔ IsLimit b ∨ b = 0 ∧ IsLimit a | case neg.h
a b : Ordinal.{u_4}
h : (a + b).IsLimit
h' : ¬b = 0
⊢ a < a + b | suffices a + 0 < a + b by simpa only [add_zero] using this | case neg.h
a b : Ordinal.{u_4}
h : (a + b).IsLimit
h' : ¬b = 0
⊢ a + 0 < a + b | da5dc0b68c8ce554 |
IsDiscreteValuationRing.aux_pid_of_ufd_of_unique_irreducible | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | theorem aux_pid_of_ufd_of_unique_irreducible (R : Type u) [CommRing R] [IsDomain R]
[UniqueFactorizationMonoid R] (h₁ : ∃ p : R, Irreducible p)
(h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q) :
IsPrincipalIdealRing R | case h.a
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₁ : ∃ p, Irreducible p
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
I : Ideal R
I0 : ¬I = ⊥
x : R
hxI : x ∈ I
hx0 : x ≠ 0
p : R
left✝ : Irreducible p
H : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x
ex :... | intro r hr | case h.a
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₁ : ∃ p, Irreducible p
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
I : Ideal R
I0 : ¬I = ⊥
x : R
hxI : x ∈ I
hx0 : x ≠ 0
p : R
left✝ : Irreducible p
H : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x
ex :... | 378c4d5b6b1ec199 |
blimsup_cthickening_mul_ae_eq | Mathlib/MeasureTheory/Covering/LiminfLimsup.lean | theorem blimsup_cthickening_mul_ae_eq (p : ℕ → Prop) (s : ℕ → Set α) {M : ℝ} (hM : 0 < M)
(r : ℕ → ℝ) (hr : Tendsto r atTop (𝓝 0)) :
(blimsup (fun i => cthickening (M * r i) (s i)) atTop p : Set α) =ᵐ[μ]
(blimsup (fun i => cthickening (r i) (s i)) atTop p : Set α) | case pos
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
∀ (p : ℕ → ... | simp [r', hi] | no goals | 8749d72cf710712d |
CategoryTheory.Presieve.isSheaf_coverage | Mathlib/CategoryTheory/Sites/Coverage.lean | theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type*) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) | C : Type u_2
inst✝ : Category.{u_3, u_2} C
K : Coverage C
P : Cᵒᵖ ⥤ Type u_1
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : K.Saturate X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → K.Saturate Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ K.covering X, IsSeparatedFor P R ∧ ∀ (x : FamilyOfElements P R)... | rw [← P.map_comp, ← P.map_comp, ← op_comp, ← op_comp, hz, hz] | C : Type u_2
inst✝ : Category.{u_3, u_2} C
K : Coverage C
P : Cᵒᵖ ⥤ Type u_1
X✝ : C
S✝ : Sieve X✝
X : C
R S : Sieve X
a✝¹ : K.Saturate X R
a✝ : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, R.arrows f → K.Saturate Y (Sieve.pullback f S)
Y : C
f : Y ⟶ X
H :
∀ {X : C},
∀ R ∈ K.covering X, IsSeparatedFor P R ∧ ∀ (x : FamilyOfElements P R)... | b0ee5f44dbe2a8c4 |
WeierstrassCurve.Affine.addPolynomial_slope | Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | lemma addPolynomial_slope {x₁ x₂ y₁ y₂ : F} (h₁ : W.Equation x₁ y₁) (h₂ : W.Equation x₂ y₂)
(hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)) : W.addPolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂) =
-((X - C x₁) * (X - C x₂) * (X - C (W.addX x₁ x₂ <| W.slope x₁ x₂ y₁ y₂))) | case pos.c
F : Type u
inst✝ : Field F
W : Affine F
x₁ y₁ : F
h₁ h₂ : y₁ ^ 2 + W.a₁ * x₁ * y₁ + W.a₃ * y₁ = x₁ ^ 3 + W.a₂ * x₁ ^ 2 + W.a₄ * x₁ + W.a₆
hxy : ¬(x₁ = x₁ ∧ y₁ = W.negY x₁ y₁)
hy : y₁ - (-y₁ - W.a₁ * x₁ - W.a₃) ≠ 0
⊢ { a := 1,
b :=
-((3 * x₁ ^ 2 + 2 * W.a₂ * x₁ + W.a₄ - W.a₁ * y₁) / (y₁ - W.... | field_simp [hy] | case pos.c
F : Type u
inst✝ : Field F
W : Affine F
x₁ y₁ : F
h₁ h₂ : y₁ ^ 2 + W.a₁ * x₁ * y₁ + W.a₃ * y₁ = x₁ ^ 3 + W.a₂ * x₁ ^ 2 + W.a₄ * x₁ + W.a₆
hxy : ¬(x₁ = x₁ ∧ y₁ = W.negY x₁ y₁)
hy : y₁ - (-y₁ - W.a₁ * x₁ - W.a₃) ≠ 0
⊢ 2 * x₁ * (3 * x₁ ^ 2 + 2 * W.a₂ * x₁ + W.a₄ - W.a₁ * y₁) ^ 2 * (y₁ - (-y₁ - W.a₁ * x₁ - W.a₃)... | 5805a29f228063e8 |
CategoryTheory.oneTruncation₂_toNerve₂Mk' | Mathlib/AlgebraicTopology/SimplicialSet/NerveAdjunction.lean | theorem oneTruncation₂_toNerve₂Mk' : oneTruncation₂.map (toNerve₂.mk' F hyp) = F | case refine_3.edge.w
C : Type u
inst✝ : SmallCategory C
X✝ : SSet.Truncated 2
F : oneTruncation₂.obj X✝ ⟶ oneTruncation₂.obj (nerveFunctor₂.obj (Cat.of C))
hyp :
∀ (φ : X✝.obj (op { obj := [2], property := ⋯ })),
(F ≫ (OneTruncation₂.ofNerve₂.natIso.app (Cat.of C)).hom).map (ev02₂ φ) =
(F ≫ (OneTruncation₂.... | obtain ⟨g, rfl, rfl⟩ := g | case refine_3.edge.w.mk
C : Type u
inst✝ : SmallCategory C
X : SSet.Truncated 2
F : oneTruncation₂.obj X ⟶ oneTruncation₂.obj (nerveFunctor₂.obj (Cat.of C))
hyp :
∀ (φ : X.obj (op { obj := [2], property := ⋯ })),
(F ≫ (OneTruncation₂.ofNerve₂.natIso.app (Cat.of C)).hom).map (ev02₂ φ) =
(F ≫ (OneTruncation₂.... | 71748558c57225f2 |
MeasurableSet.everywherePosSubset | Mathlib/MeasureTheory/Measure/EverywherePos.lean | protected lemma _root_.MeasurableSet.everywherePosSubset [OpensMeasurableSpace α]
(hs : MeasurableSet s) :
MeasurableSet (μ.everywherePosSubset s) | case intro.intro
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : OpensMeasurableSpace α
hs : MeasurableSet s
u : Set α
u_open : IsOpen u
hu : μ.everywherePosSubset s = s \ u
⊢ MeasurableSet (μ.everywherePosSubset s) | rw [hu] | case intro.intro
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : OpensMeasurableSpace α
hs : MeasurableSet s
u : Set α
u_open : IsOpen u
hu : μ.everywherePosSubset s = s \ u
⊢ MeasurableSet (s \ u) | eb10019edf998a4d |
IsOpen.analyticOn_iff_analyticOnNhd | Mathlib/Analysis/Analytic/Within.lean | /-- On open sets, `AnalyticOnNhd` and `AnalyticOn` coincide -/
lemma IsOpen.analyticOn_iff_analyticOnNhd {f : E → F} {s : Set E} (hs : IsOpen s) :
AnalyticOn 𝕜 f s ↔ AnalyticOnNhd 𝕜 f s | case a
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
hs : IsOpen s
hf : AnalyticOn 𝕜 f s
x : E
m : x ∈ s
r : ℝ
r0 : r > 0
rs : Metric.ball x r ⊆ s
p : FormalM... | apply rs | case a.a
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
hs : IsOpen s
hf : AnalyticOn 𝕜 f s
x : E
m : x ∈ s
r : ℝ
r0 : r > 0
rs : Metric.ball x r ⊆ s
p : Forma... | 1b818455a3407b83 |
List.forIn'_loop_congr | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Monadic.lean | theorem forIn'_loop_congr [Monad m] {as bs : List α}
{f : (a' : α) → a' ∈ as → β → m (ForInStep β)}
{g : (a' : α) → a' ∈ bs → β → m (ForInStep β)}
{b : β} (ha : ∃ ys, ys ++ xs = as) (hb : ∃ ys, ys ++ xs = bs)
(h : ∀ a m m' b, f a m b = g a m' b) : forIn'.loop as f xs b ha = forIn'.loop bs g xs b hb | case nil
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝ : Monad m
as bs : List α
f : (a' : α) → a' ∈ as → β → m (ForInStep β)
g : (a' : α) → a' ∈ bs → β → m (ForInStep β)
h : ∀ (a : α) (m_1 : a ∈ as) (m' : a ∈ bs) (b : β), f a m_1 b = g a m' b
b : β
ha : ∃ ys, ys ++ [] = as
hb : ∃ ys, ys ++ [] = bs
⊢ forIn'.lo... | simp [forIn'.loop] | no goals | 91c1263f341f7e7e |
UV.compression_self | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | theorem compression_self (u : α) (s : Finset α) : 𝓒 u u s = s | case h.e'_2.h.e'_4
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
inst✝ : DecidableEq α
u : α
s : Finset α
⊢ filter (fun a => a ∉ s) (image (compress u u) s) = ∅ | refine eq_empty_of_forall_not_mem fun a ha ↦ ?_ | case h.e'_2.h.e'_4
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
inst✝ : DecidableEq α
u : α
s : Finset α
a : α
ha : a ∈ filter (fun a => a ∉ s) (image (compress u u) s)
⊢ False | de0a413978276aa7 |
Submodule.spanRank_toENat_eq_iInf_finset_card | Mathlib/Algebra/Module/SpanRank.lean | lemma spanRank_toENat_eq_iInf_finset_card (p : Submodule R M) :
p.spanRank.toENat =
⨅ (s : {s : Set M // s.Finite ∧ span R s = p}), (s.2.1.toFinset.card : ℕ∞) | R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
⊢ ⨅ s, ⨅ (_ : span R s = p), s.encard = ⨅ s, ↑⋯.toFinset.card | rcases (eq_or_ne (⨅ (s : Set M) (_ : span R s = p), s.encard) ⊤) with (h1 | h2) | case inl
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h1 : ⨅ s, ⨅ (_ : span R s = p), s.encard = ⊤
⊢ ⨅ s, ⨅ (_ : span R s = p), s.encard = ⨅ s, ↑⋯.toFinset.card
case inr
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
... | f74636938a98e2b3 |
CategoryTheory.PreGaloisCategory.autMap_comp | Mathlib/CategoryTheory/Galois/GaloisObjects.lean | @[simp]
lemma autMap_comp {X Y Z : C} [IsConnected X] [IsGalois Y] [IsGalois Z] (f : X ⟶ Y)
(g : Y ⟶ Z) : autMap (f ≫ g) = autMap g ∘ autMap f | C : Type u₁
inst✝⁴ : Category.{u₂, u₁} C
inst✝³ : GaloisCategory C
X Y Z : C
inst✝² : IsConnected X
inst✝¹ : IsGalois Y
inst✝ : IsGalois Z
f : X ⟶ Y
g : Y ⟶ Z
⊢ autMap (f ≫ g) = autMap g ∘ autMap f | refine funext fun σ ↦ autMap_unique _ σ _ ?_ | C : Type u₁
inst✝⁴ : Category.{u₂, u₁} C
inst✝³ : GaloisCategory C
X Y Z : C
inst✝² : IsConnected X
inst✝¹ : IsGalois Y
inst✝ : IsGalois Z
f : X ⟶ Y
g : Y ⟶ Z
σ : Aut X
⊢ (f ≫ g) ≫ ((autMap g ∘ autMap f) σ).hom = σ.hom ≫ f ≫ g | ef647b00a365a2e8 |
Padic.norm_le_pow_iff_norm_lt_pow_add_one | Mathlib/NumberTheory/Padics/PadicNumbers.lean | theorem norm_le_pow_iff_norm_lt_pow_add_one (x : ℚ_[p]) (n : ℤ) :
‖x‖ ≤ (p : ℝ) ^ n ↔ ‖x‖ < (p : ℝ) ^ (n + 1) | case pos
p : ℕ
hp : Fact (Nat.Prime p)
x : ℚ_[p]
n : ℤ
aux : ∀ (n : ℤ), 0 < ↑p ^ n
hx0 : x = 0
⊢ ‖x‖ ≤ ↑p ^ n ↔ ‖x‖ < ↑p ^ (n + 1) | simp [hx0, norm_zero, aux, le_of_lt (aux _)] | no goals | a9f8ab08c755e528 |
SimpleGraph.turanGraph_cliqueFree | Mathlib/Combinatorics/SimpleGraph/Turan.lean | theorem turanGraph_cliqueFree (hr : 0 < r) : (turanGraph n r).CliqueFree (r + 1) | case intro.mk
n r : ℕ
hr : 0 < r
f : Fin (r + 1) ↪ Fin n
ha : ∀ {a b : Fin (r + 1)}, (turanGraph n r).Adj (f a) (f b) ↔ ⊤.Adj a b
⊢ False | simp only [turanGraph, top_adj] at ha | case intro.mk
n r : ℕ
hr : 0 < r
f : Fin (r + 1) ↪ Fin n
ha : ∀ {a b : Fin (r + 1)}, ↑(f a) % r ≠ ↑(f b) % r ↔ a ≠ b
⊢ False | f1f16760de3c27cc |
hasConstantSpeedOnWith_iff_variationOnFromTo_eq | Mathlib/Analysis/ConstantSpeed.lean | theorem hasConstantSpeedOnWith_iff_variationOnFromTo_eq :
HasConstantSpeedOnWith f s l ↔ LocallyBoundedVariationOn f s ∧
∀ ⦃x⦄ (_ : x ∈ s) ⦃y⦄ (_ : y ∈ s), variationOnFromTo f s x y = l * (y - x) | case mpr
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
⊢ (LocallyBoundedVariationOn f s ∧ ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → variationOnFromTo f s x y = ↑l * (y - x)) →
∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x)) | rintro h x xs y ys xy | case mpr
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : ℝ → E
s : Set ℝ
l : ℝ≥0
h : LocallyBoundedVariationOn f s ∧ ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → variationOnFromTo f s x y = ↑l * (y - x)
x : ℝ
xs : x ∈ s
y : ℝ
ys : y ∈ s
xy : x ≤ y
⊢ eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x)) | 0a4250968eca8999 |
AkraBazziRecurrence.isLittleO_deriv_one_sub_smoothingFn | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | lemma isLittleO_deriv_one_sub_smoothingFn :
deriv (fun x => 1 - ε x) =o[atTop] fun (x : ℝ) => x⁻¹ := calc
deriv (fun x => 1 - ε x) =ᶠ[atTop] fun z => -(deriv ε z) | ⊢ (fun z => -deriv ε z) =o[atTop] fun x => x⁻¹ | rw [isLittleO_neg_left] | ⊢ deriv ε =o[atTop] fun x => x⁻¹ | fe232c00b4bddc81 |
isSquare_iff_exists_sq | Mathlib/Algebra/Group/Even.lean | @[to_additive even_iff_exists_two_nsmul]
lemma isSquare_iff_exists_sq (a : α) : IsSquare a ↔ ∃ r, a = r ^ 2 | α : Type u_2
inst✝ : Monoid α
a : α
⊢ IsSquare a ↔ ∃ r, a = r ^ 2 | simp [IsSquare, pow_two] | no goals | 3f2fe248829aa57f |
cardinalMk_eq_cardinalMk_field_pow_rank | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | theorem cardinalMk_eq_cardinalMk_field_pow_rank (K V : Type u) [Ring K] [StrongRankCondition K]
[AddCommGroup V] [Module K V] [Module.Free K V] [Module.Finite K V] :
#V = #K ^ Module.rank K V | K V : Type u
inst✝⁵ : Ring K
inst✝⁴ : StrongRankCondition K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : Free K V
inst✝ : Module.Finite K V
⊢ #V = #K ^ Module.rank K V | simpa using lift_cardinalMk_eq_lift_cardinalMk_field_pow_lift_rank K V | no goals | 4dc3238ebd0a171b |
Finset.pow_right_strictMono | Mathlib/Geometry/Group/Growth/LinearLowerBound.lean | @[to_additive]
lemma pow_right_strictMono (hX₁ : 1 ∈ X) (hXclosure : (closure (X : Set G) : Set G).Infinite) :
StrictMono fun n ↦ X ^ n | case inr
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
hX₁ : 1 ∈ X
hXclosure : (↑(closure ↑X)).Infinite
hX : X.Nontrivial
⊢ StrictMono fun n => X ^ n | have h n : (X ^ (n - 1) : Set G) ≠ closure (X : Set G) :=
fun h ↦ by simp [← h, ← coe_pow] at hXclosure | case inr
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
hX₁ : 1 ∈ X
hXclosure : (↑(closure ↑X)).Infinite
hX : X.Nontrivial
h : ∀ (n : ℕ), ↑X ^ (n - 1) ≠ ↑(closure ↑X)
⊢ StrictMono fun n => X ^ n | 41f61d6f7eb2df9f |
FirstOrder.Language.equiv_between_cg | Mathlib/ModelTheory/PartialEquiv.lean | theorem equiv_between_cg (M_cg : Structure.CG L M) (N_cg : Structure.CG L N)
(g : L.FGEquiv M N)
(ext_dom : L.IsExtensionPair M N)
(ext_cod : L.IsExtensionPair N M) :
∃ f : M ≃[L] N, g ≤ f.toEmbedding.toPartialEquiv | L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
ext_dom : L.IsExtensionPair M N
ext_cod : L.IsExtensionPair N M
X : Set M
X_count : X.Countable
X_gen : (closure L).toFun X = ⊤
Y : Set N
Y_count : Y.Countable
Y_gen : (closure L).toFun Y = ⊤
x✝⁴ : Countable ↑X
x✝³ : Enco... | intro y hy | L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
ext_dom : L.IsExtensionPair M N
ext_cod : L.IsExtensionPair N M
X : Set M
X_count : X.Countable
X_gen : (closure L).toFun X = ⊤
Y : Set N
Y_count : Y.Countable
Y_gen : (closure L).toFun Y = ⊤
x✝⁴ : Countable ↑X
x✝³ : Enco... | d1b972cd720b6e06 |
Complex.tan_arctan | Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean | theorem tan_arctan {z : ℂ} (h₁ : z ≠ I) (h₂ : z ≠ -I) : tan (arctan z) = z | z : ℂ
h₁ : z ≠ I
h₂ : z ≠ -I
⊢ (cexp (-z.arctan * I) - cexp (z.arctan * I)) * I / 2 / ((cexp (z.arctan * I) + cexp (-z.arctan * I)) / 2) = z | rw [div_div_eq_mul_div, div_mul_cancel₀ _ two_ne_zero, ← div_mul_eq_mul_div,
← mul_div_mul_right _ _ (exp_ne_zero (arctan z * I)), sub_mul, add_mul,
← exp_add, neg_mul, neg_add_cancel, exp_zero, ← exp_add, ← two_mul] | z : ℂ
h₁ : z ≠ I
h₂ : z ≠ -I
⊢ (1 - cexp (2 * (z.arctan * I))) / (cexp (2 * (z.arctan * I)) + 1) * I = z | b2b39565dc385b44 |
EuclideanGeometry.dist_div_cos_oangle_right_of_oangle_eq_pi_div_two | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | theorem dist_div_cos_oangle_right_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
(h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : dist p₃ p₂ / Real.Angle.cos (∡ p₂ p₃ p₁) = dist p₁ p₃ | V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ : P
h : ∡ p₁ p₂ p₃ = ↑(π / 2)
⊢ dist p₃ p₂ / (∡ p₂ p₃ p₁).cos = dist p₁ p₃ | have hs : (∡ p₂ p₃ p₁).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two] | V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ : P
h : ∡ p₁ p₂ p₃ = ↑(π / 2)
hs : (∡ p₂ p₃ p₁).sign = 1
⊢ dist p₃ p₂ / (∡ p₂ p₃ p₁).cos = dist p₁ p₃ | e82a55585bbba2df |
MeasureTheory.MemLp.exists_boundedContinuous_eLpNorm_sub_le | Mathlib/MeasureTheory/Function/ContinuousMapDense.lean | theorem MemLp.exists_boundedContinuous_eLpNorm_sub_le [μ.WeaklyRegular] (hp : p ≠ ∞) {f : α → E}
(hf : MemLp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
∃ g : α →ᵇ E, eLpNorm (f - (g : α → E)) p μ ≤ ε ∧ MemLp g p μ | α : Type u_1
inst✝⁶ : TopologicalSpace α
inst✝⁵ : NormalSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
inst✝¹ : NormedSpace ℝ E
inst✝ : μ.WeaklyRegular
hp : p ≠ ⊤
f : α → E
hf : MemLp f p μ
ε✝ : ℝ≥0∞
hε✝ : ε✝ ≠ 0
c : E
t : Set α
ht : Measurable... | rcases exists_Lp_half E μ p hε with ⟨δ, δpos, hδ⟩ | case intro.intro
α : Type u_1
inst✝⁶ : TopologicalSpace α
inst✝⁵ : NormalSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
inst✝¹ : NormedSpace ℝ E
inst✝ : μ.WeaklyRegular
hp : p ≠ ⊤
f : α → E
hf : MemLp f p μ
ε✝ : ℝ≥0∞
hε✝ : ε✝ ≠ 0
c : E
t : Set ... | 7e6196982c59c5fe |
exists_seq_forall_proj_of_forall_finite | Mathlib/Order/KonigLemma.lean | theorem exists_seq_forall_proj_of_forall_finite {α : ℕ → Type*} [Finite (α 0)] [∀ i, Nonempty (α i)]
(π : {i j : ℕ} → (hij : i ≤ j) → α j → α i)
(π_refl : ∀ ⦃i⦄ (a : α i), π rfl.le a = a)
(π_trans : ∀ ⦃i j k⦄ (hij : i ≤ j) (hjk : j ≤ k) a, π hij (π hjk a) = π (hij.trans hjk) a)
(hfin : ∀ i a, {b : α (i+... | α : ℕ → Type u_1
inst✝¹ : Finite (α 0)
inst✝ : ∀ (i : ℕ), Nonempty (α i)
π : {i j : ℕ} → i ≤ j → α j → α i
π_refl : ∀ ⦃i : ℕ⦄ (a : α i), π ⋯ a = a
π_trans : ∀ ⦃i j k : ℕ⦄ (hij : i ≤ j) (hjk : j ≤ k) (a : α k), π hij (π hjk a) = π ⋯ a
hfin : ∀ (i : ℕ) (a : α i), {b | π ⋯ b = a}.Finite
αs : Type u_1 := (i : ℕ) × α i
x✝¹ ... | rw [← π_trans h.1 h'.1 c.2, h'.2, h.2] | no goals | 271fcb2e96bcf388 |
Finset.image₂_left_comm | Mathlib/Data/Finset/NAry.lean | theorem image₂_left_comm {γ : Type*} {u : Finset γ} {f : α → δ → ε} {g : β → γ → δ}
{f' : α → γ → δ'} {g' : β → δ' → ε} (h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
image₂ f s (image₂ g t u) = image₂ g' t (image₂ f' s u) :=
coe_injective <| by
push_cast
exact image2_left_comm h_left_comm
| α : Type u_1
β : Type u_3
δ : Type u_7
δ' : Type u_8
ε : Type u_9
inst✝² : DecidableEq δ'
inst✝¹ : DecidableEq ε
s : Finset α
t : Finset β
inst✝ : DecidableEq δ
γ : Type u_14
u : Finset γ
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
⊢ i... | exact image2_left_comm h_left_comm | no goals | e951e1d0d2442dd3 |
Real.strictConcaveOn_rpow | Mathlib/Analysis/Convex/SpecificFunctions/Pow.lean | lemma strictConcaveOn_rpow {p : ℝ} (hp₀ : 0 < p) (hp₁ : p < 1) :
StrictConcaveOn ℝ (Set.Ici 0) fun x : ℝ ↦ x ^ p | p : ℝ
hp₀ : 0 < p
hp₁ : p < 1
x : ℝ
hx : x ∈ Ici 0
y : ℝ
hy : y ∈ Ici 0
hxy : x ≠ y
a b : ℝ
ha : 0 < a
hb : 0 < b
hab : a + b = 1
⊢ a • (fun x => x ^ p) x + b • (fun x => x ^ p) y < (fun x => x ^ p) (a • x + b • y) | let x' : ℝ≥0 := ⟨x, hx⟩ | p : ℝ
hp₀ : 0 < p
hp₁ : p < 1
x : ℝ
hx : x ∈ Ici 0
y : ℝ
hy : y ∈ Ici 0
hxy : x ≠ y
a b : ℝ
ha : 0 < a
hb : 0 < b
hab : a + b = 1
x' : ℝ≥0 := ⟨x, hx⟩
⊢ a • (fun x => x ^ p) x + b • (fun x => x ^ p) y < (fun x => x ^ p) (a • x + b • y) | e355702cbbe8ec4b |
List.getElem_tails | Mathlib/Data/List/Infix.lean | theorem getElem_tails (l : List α) (n : Nat) (h : n < (tails l).length) :
(tails l)[n] = l.drop n | case nil
α : Type u_1
n : ℕ
h : n < [].tails.length
⊢ [].tails[n] = drop n [] | simp | no goals | d9f4a04274408191 |
WithSeminorms.toLocallyConvexSpace | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | theorem WithSeminorms.toLocallyConvexSpace {p : SeminormFamily 𝕜 E ι} (hp : WithSeminorms p) :
LocallyConvexSpace ℝ E | case hbasis
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝⁷ : Nonempty ι
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedSpace ℝ 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Module ℝ E
inst✝¹ : IsScalarTower ℝ 𝕜 E
inst✝ : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
this : IsTopologicalAddGroup ... | rw [hp.1, AddGroupFilterBasis.nhds_eq _, AddGroupFilterBasis.N_zero] | case hbasis
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝⁷ : Nonempty ι
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedSpace ℝ 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Module ℝ E
inst✝¹ : IsScalarTower ℝ 𝕜 E
inst✝ : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
this : IsTopologicalAddGroup ... | 60979115a7f92b1c |
SimpleGraph.ConnectedComponent.mem_supp_congr_adj | Mathlib/Combinatorics/SimpleGraph/Path.lean | lemma mem_supp_congr_adj {v w : V} (c : G.ConnectedComponent) (hadj : G.Adj v w) :
v ∈ c.supp ↔ w ∈ c.supp | V : Type u
G : SimpleGraph V
v w : V
c : G.ConnectedComponent
hadj : G.Adj v w
⊢ G.connectedComponentMk v = c ↔ G.connectedComponentMk w = c | constructor <;> intro h <;> simp only [← h] <;> apply connectedComponentMk_eq_of_adj | case mp.a
V : Type u
G : SimpleGraph V
v w : V
c : G.ConnectedComponent
hadj : G.Adj v w
h : G.connectedComponentMk v = c
⊢ G.Adj w v
case mpr.a
V : Type u
G : SimpleGraph V
v w : V
c : G.ConnectedComponent
hadj : G.Adj v w
h : G.connectedComponentMk w = c
⊢ G.Adj v w | 02aeef40deecd781 |
MeasureTheory.integral_comp_mul_left_Ioi | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | theorem integral_comp_mul_left_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 < b) :
(∫ x in Ioi a, g (b * x)) = b⁻¹ • ∫ x in Ioi (b * a), g x | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : ℝ → E
a b : ℝ
hb : 0 < b
this : ∀ (c : ℝ), MeasurableSet (Ioi c)
⊢ ∫ (x : ℝ), (Ioi a).indicator (fun x => g (b * x)) x ∂volume = ∫ (x : ℝ), (Ioi (b * a)).indicator g (b * x) | congr | case e_f
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : ℝ → E
a b : ℝ
hb : 0 < b
this : ∀ (c : ℝ), MeasurableSet (Ioi c)
⊢ (fun x => (Ioi a).indicator (fun x => g (b * x)) x) = fun x => (Ioi (b * a)).indicator g (b * x) | 321e4575ea9df033 |
CategoryTheory.Limits.coequalizer.π_colimMap_desc | Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | theorem coequalizer.π_colimMap_desc {X' Y' Z : C} (f' g' : X' ⟶ Y') [HasCoequalizer f' g']
(p : X ⟶ X') (q : Y ⟶ Y') (wf : f ≫ q = p ≫ f') (wg : g ≫ q = p ≫ g') (h : Y' ⟶ Z)
(wh : f' ≫ h = g' ≫ h) :
coequalizer.π f g ≫ colimMap (parallelPairHom f g f' g' p q wf wg) ≫ coequalizer.desc h wh =
q ≫ h | C : Type u
inst✝² : Category.{v, u} C
X Y : C
f g : X ⟶ Y
inst✝¹ : HasCoequalizer f g
X' Y' Z : C
f' g' : X' ⟶ Y'
inst✝ : HasCoequalizer f' g'
p : X ⟶ X'
q : Y ⟶ Y'
wf : f ≫ q = p ≫ f'
wg : g ≫ q = p ≫ g'
h : Y' ⟶ Z
wh : f' ≫ h = g' ≫ h
⊢ π f g ≫ colimMap (parallelPairHom f g f' g' p q wf wg) ≫ desc h wh = q ≫ h | rw [ι_colimMap_assoc, parallelPairHom_app_one, coequalizer.π_desc] | no goals | 219f49726852a3a1 |
InnerProductGeometry.angle_neg_neg | Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | theorem angle_neg_neg (x y : V) : angle (-x) (-y) = angle x y | V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ angle (-x) (-y) = angle x y | unfold angle | V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ arccos (inner (-x) (-y) / (‖-x‖ * ‖-y‖)) = arccos (inner x y / (‖x‖ * ‖y‖)) | b9daa6990569b326 |
integral_Ioi_rpow_of_lt | Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | theorem integral_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 < c) :
∫ t : ℝ in Ioi c, t ^ a = -c ^ (a + 1) / (a + 1) | a : ℝ
ha : a < -1
c : ℝ
hc : 0 < c
⊢ ∫ (t : ℝ) in Ioi c, t ^ a = -c ^ (a + 1) / (a + 1) | have hd : ∀ x ∈ Ici c, HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x := by
intro x hx
convert (hasDerivAt_rpow_const (p := a + 1) (Or.inl (hc.trans_le hx).ne')).div_const _ using 1
field_simp [show a + 1 ≠ 0 from ne_of_lt (by linarith), mul_comm] | a : ℝ
ha : a < -1
c : ℝ
hc : 0 < c
hd : ∀ x ∈ Ici c, HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x
⊢ ∫ (t : ℝ) in Ioi c, t ^ a = -c ^ (a + 1) / (a + 1) | 63db95eb21c4b87c |
HurwitzZeta.evenKernel_functional_equation | Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | lemma evenKernel_functional_equation (a : UnitAddCircle) (x : ℝ) :
evenKernel a x = 1 / x ^ (1 / 2 : ℝ) * cosKernel a (1 / x) | case inr.H
x : ℝ
hx : 0 < x
a : ℝ
h1 : I * ↑(1 / x) = -1 / (I * ↑x)
hx' : I * ↑x ≠ 0
⊢ cexp (-↑π * ↑a ^ 2 * ↑x) *
(1 / (-I * (I * ↑x)) ^ (1 / 2) * cexp (-↑π * I * (↑a * I * ↑x) ^ 2 / (I * ↑x)) *
jacobiTheta₂ (↑a * I * ↑x / (I * ↑x)) (-1 / (I * ↑x))) =
↑(1 / x ^ (1 / 2)) * jacobiTheta₂ (↑a) (I * ↑(1 / ... | have h2 : a * I * x / (I * x) = a := by
rw [div_eq_iff hx']
ring | case inr.H
x : ℝ
hx : 0 < x
a : ℝ
h1 : I * ↑(1 / x) = -1 / (I * ↑x)
hx' : I * ↑x ≠ 0
h2 : ↑a * I * ↑x / (I * ↑x) = ↑a
⊢ cexp (-↑π * ↑a ^ 2 * ↑x) *
(1 / (-I * (I * ↑x)) ^ (1 / 2) * cexp (-↑π * I * (↑a * I * ↑x) ^ 2 / (I * ↑x)) *
jacobiTheta₂ (↑a * I * ↑x / (I * ↑x)) (-1 / (I * ↑x))) =
↑(1 / x ^ (1 / 2)... | ad24ce6b479b05e9 |
Ideal.exists_radical_pow_le_of_fg | Mathlib/RingTheory/Finiteness/Ideal.lean | theorem exists_radical_pow_le_of_fg {R : Type*} [CommSemiring R] (I : Ideal R) (h : I.radical.FG) :
∃ n : ℕ, I.radical ^ n ≤ I | case refine_2.intro
R : Type u_3
inst✝ : CommSemiring R
I : Ideal R
h : I.radical.FG
J K : Submodule R R
hJ : J ≤ I.radical → ∃ n, J ^ n ≤ I
hK : K ≤ I.radical → ∃ n, K ^ n ≤ I
hJK : J ⊔ K ≤ I.radical
n : ℕ
hn : J ^ n ≤ I
⊢ ∃ n, (J ⊔ K) ^ n ≤ I | obtain ⟨m, hm⟩ := hK fun x hx => hJK <| Ideal.mem_sup_right hx | case refine_2.intro.intro
R : Type u_3
inst✝ : CommSemiring R
I : Ideal R
h : I.radical.FG
J K : Submodule R R
hJ : J ≤ I.radical → ∃ n, J ^ n ≤ I
hK : K ≤ I.radical → ∃ n, K ^ n ≤ I
hJK : J ⊔ K ≤ I.radical
n : ℕ
hn : J ^ n ≤ I
m : ℕ
hm : K ^ m ≤ I
⊢ ∃ n, (J ⊔ K) ^ n ≤ I | 28279910d767e50e |
EuclideanGeometry.exists_of_range_subset_orthocentricSystem | Mathlib/Geometry/Euclidean/MongePoint.lean | theorem exists_of_range_subset_orthocentricSystem {t : Triangle ℝ P}
(ho : t.orthocenter ∉ Set.range t.points) {p : Fin 3 → P}
(hps : Set.range p ⊆ insert t.orthocenter (Set.range t.points)) (hpi : Function.Injective p) :
(∃ i₁ i₂ i₃ j₂ j₃ : Fin 3,
i₁ ≠ i₂ ∧ i₁ ≠ i₃ ∧ i₂ ≠ i₃ ∧ (∀ i : Fin 3, i = i₁ ∨ ... | V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
ho : t.orthocenter ∉ Set.range t.points
p : Fin 3 → P
hpi : Function.Injective p
i₁ : Fin 3
h₁ : p i₁ = t.orthocenter
i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ :... | exact hps | no goals | 090685f526132b06 |
CategoryTheory.Equivalence.unit_inverse_comp | Mathlib/CategoryTheory/Equivalence.lean | theorem unit_inverse_comp (e : C ≌ D) (Y : D) :
e.unit.app (e.inverse.obj Y) ≫ e.inverse.map (e.counit.app Y) = 𝟙 (e.inverse.obj Y) | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
Y : D
⊢ e.unit.app ((𝟭 C).obj (e.inverse.obj Y)) ≫
(e.functor ⋙ e.inverse).map (e.unit.1 (e.inverse.obj Y)) ≫
(((𝟙 (e.inverse.obj (e.functor.obj (e.inverse.toPrefunctor.1 (e.functor.obj (e.inverse.obj Y))))) ≫
... | erw [id_comp] | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
Y : D
⊢ e.unit.app ((𝟭 C).obj (e.inverse.obj Y)) ≫
(e.functor ⋙ e.inverse).map (e.unit.1 (e.inverse.obj Y)) ≫
(((e.inverse.mapIso (e.counitIso.app (e.functor.obj (e.inverse.obj (e.functor.obj (e.inverse.obj Y))))))... | 559e5ef37eba7881 |
CategoryTheory.Limits.BinaryBiconeMorphism.ext | Mathlib/CategoryTheory/Limits/Shapes/BinaryBiproducts.lean | theorem BinaryBiconeMorphism.ext {P Q : C} {c c' : BinaryBicone P Q}
(f g : c ⟶ c') (w : f.hom = g.hom) : f = g | C : Type uC
inst✝¹ : Category.{uC', uC} C
inst✝ : HasZeroMorphisms C
P Q : C
c c' : BinaryBicone P Q
f g : c ⟶ c'
w : f.hom = g.hom
⊢ f = g | cases f | case mk
C : Type uC
inst✝¹ : Category.{uC', uC} C
inst✝ : HasZeroMorphisms C
P Q : C
c c' : BinaryBicone P Q
g : c ⟶ c'
hom✝ : c.pt ⟶ c'.pt
wfst✝ : hom✝ ≫ c'.fst = c.fst
wsnd✝ : hom✝ ≫ c'.snd = c.snd
winl✝ : c.inl ≫ hom✝ = c'.inl
winr✝ : c.inr ≫ hom✝ = c'.inr
w : { hom := hom✝, wfst := wfst✝, wsnd := wsnd✝, winl := win... | 356d919e305d0a6c |
convexIndependent_set_iff_inter_convexHull_subset | Mathlib/Analysis/Convex/Independent.lean | theorem convexIndependent_set_iff_inter_convexHull_subset {s : Set E} :
ConvexIndependent 𝕜 ((↑) : s → E) ↔ ∀ t, t ⊆ s → s ∩ convexHull 𝕜 t ⊆ t | case mpr
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
⊢ (∀ t ⊆ s, s ∩ (convexHull 𝕜) t ⊆ t) → ConvexIndependent 𝕜 Subtype.val | intro hc t x h | case mpr
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
hc : ∀ t ⊆ s, s ∩ (convexHull 𝕜) t ⊆ t
t : Set { x // x ∈ s }
x : { x // x ∈ s }
h : ↑x ∈ (convexHull 𝕜) (Subtype.val '' t)
⊢ x ∈ t | f772aaa05649ce6c |
List.Ico.nodup | Mathlib/Data/List/Intervals.lean | theorem nodup (n m : ℕ) : Nodup (Ico n m) | n m : ℕ
⊢ (range' n (m - n)).Nodup | simp [nodup_range'] | no goals | d71af4ed1530723e |
RCLike.wInner_neg_left | Mathlib/Analysis/RCLike/Inner.lean | @[simp] lemma wInner_neg_left (w : ι → ℝ) (f g : ∀ i, E i) : ⟪-f, g⟫_[𝕜, w] = -⟪f, g⟫_[𝕜, w] | ι : Type u_1
𝕜 : Type u_3
E : ι → Type u_4
inst✝³ : Fintype ι
inst✝² : RCLike 𝕜
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (E i)
w : ι → ℝ
f g : (i : ι) → E i
⊢ wInner w (-f) g = -wInner w f g | simp [wInner] | no goals | e9c4cb49671717f8 |
Ordnode.Valid'.merge_aux₁ | Mathlib/Data/Ordmap/Ordset.lean | theorem Valid'.merge_aux₁ {o₁ o₂ ls ll lx lr rs rl rx rr t}
(hl : Valid' o₁ (@Ordnode.node α ls ll lx lr) o₂) (hr : Valid' o₁ (.node rs rl rx rr) o₂)
(h : delta * ls < rs) (v : Valid' o₁ t rx) (e : size t = ls + size rl) :
Valid' o₁ (.balanceL t rx rr) o₂ ∧ size (.balanceL t rx rr) = ls + rs | case refine_1
α : Type u_1
inst✝ : Preorder α
o₁ : WithBot α
o₂ : WithTop α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
rs : ℕ
rl : Ordnode α
rx : α
rr t : Ordnode α
hl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂
hr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂
h : 3 * (ll.size + lr.size + 1) < rl.size + rr.size + 1
v : Vali... | rw [e] | case refine_1
α : Type u_1
inst✝ : Preorder α
o₁ : WithBot α
o₂ : WithTop α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
rs : ℕ
rl : Ordnode α
rx : α
rr t : Ordnode α
hl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂
hr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂
h : 3 * (ll.size + lr.size + 1) < rl.size + rr.size + 1
v : Vali... | dde6393e47c80664 |
AlgebraicGeometry.isIso_of_comp_eq_sigmaSpec | Mathlib/AlgebraicGeometry/PointsPi.lean | /-- If `V` is a locally closed subscheme of `Spec (Π Rᵢ)` containing `∐ Spec Rᵢ`, then
`V = Spec (Π Rᵢ)`. -/
lemma isIso_of_comp_eq_sigmaSpec {V : Scheme}
(f : (∐ fun i ↦ Spec (R i)) ⟶ V) (g : V ⟶ Spec (.of (Π i, R i)))
[IsImmersion g] [CompactSpace V]
(hU' : f ≫ g = sigmaSpec R) : IsIso g | ι : Type u
R : ι → CommRingCat
V : Scheme
f : (∐ fun i => Spec (R i)) ⟶ V
inst✝¹ : CompactSpace ↑↑V.toPresheafedSpace
I : Ideal ↑(CommRingCat.of ((i : ι) → ↑(R i)))
e : V ≅ Spec (CommRingCat.of (↑(CommRingCat.of ((i : ι) → ↑(R i))) ⧸ I))
inst✝ : IsImmersion (e.hom ≫ Spec.map (CommRingCat.ofHom (Ideal.Quotient.mk I)))
h... | rwa [Category.assoc] | no goals | 8616e9c378ddcd3e |
SimpleGraph.three_le_egirth | Mathlib/Combinatorics/SimpleGraph/Girth.lean | lemma three_le_egirth : 3 ≤ G.egirth | case neg
α : Type u_1
G : SimpleGraph α
h : ∃ a w, w.IsCycle ∧ G.egirth = ↑w.length
⊢ 3 ≤ G.egirth | have ⟨_, _, _⟩ := h | case neg
α : Type u_1
G : SimpleGraph α
h : ∃ a w, w.IsCycle ∧ G.egirth = ↑w.length
w✝¹ : α
w✝ : G.Walk w✝¹ w✝¹
h✝ : w✝.IsCycle ∧ G.egirth = ↑w✝.length
⊢ 3 ≤ G.egirth | ea32850d40d27a7e |
PadicInt.isUnit_den | Mathlib/NumberTheory/Padics/RingHoms.lean | theorem isUnit_den (r : ℚ) (h : ‖(r : ℚ_[p])‖ ≤ 1) : IsUnit (r.den : ℤ_[p]) | p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
h : ‖↑r‖ ≤ 1
norm_denom_lt : ‖↑r.den‖ < 1
⊢ False | have hr : ‖(r * r.den : ℚ_[p])‖ = ‖(r.num : ℚ_[p])‖ := by
congr
rw_mod_cast [@Rat.mul_den_eq_num r] | p : ℕ
hp_prime : Fact (Nat.Prime p)
r : ℚ
h : ‖↑r‖ ≤ 1
norm_denom_lt : ‖↑r.den‖ < 1
hr : ‖↑r * ↑r.den‖ = ‖↑r.num‖
⊢ False | 639c7872f389f578 |
ENNReal.sub_iInf | Mathlib/Data/ENNReal/Real.lean | theorem sub_iInf : (a - ⨅ i, f i) = ⨆ i, a - f i | ι : Sort u_1
f : ι → ℝ≥0∞
a : ℝ≥0∞
⊢ a - ⨅ i, f i = ⨆ i, a - f i | refine eq_of_forall_ge_iff fun c => ?_ | ι : Sort u_1
f : ι → ℝ≥0∞
a c : ℝ≥0∞
⊢ a - ⨅ i, f i ≤ c ↔ ⨆ i, a - f i ≤ c | 3474e7e67084f94c |
Ideal.disjoint_nonZeroDivisors_of_mem_minimalPrimes | Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean | /-- minimal primes are contained in zero divisors. -/
lemma Ideal.disjoint_nonZeroDivisors_of_mem_minimalPrimes {p : Ideal R} (hp : p ∈ minimalPrimes R) :
Disjoint (p : Set R) (nonZeroDivisors R) | R : Type u_1
inst✝ : CommSemiring R
p : Ideal R
hp : p ∈ minimalPrimes R
⊢ Disjoint ↑p ↑(nonZeroDivisors R) | rw [← Set.subset_compl_iff_disjoint_right, Set.subset_def] | R : Type u_1
inst✝ : CommSemiring R
p : Ideal R
hp : p ∈ minimalPrimes R
⊢ ∀ x ∈ ↑p, x ∈ (↑(nonZeroDivisors R))ᶜ | 9300e742a04a71be |
Finsupp.sigma_sum | Mathlib/Data/Finsupp/Basic.lean | theorem sigma_sum [AddCommMonoid N] (f : (Σi : ι, αs i) → M → N) :
l.sum f = ∑ i ∈ splitSupport l, (split l i).sum fun (a : αs i) b => f ⟨i, a⟩ b | ι : Type u_4
M : Type u_5
N : Type u_7
αs : ι → Type u_13
inst✝¹ : Zero M
l : (i : ι) × αs i →₀ M
inst✝ : AddCommMonoid N
f : (i : ι) × αs i → M → N
⊢ l.sum f = ∑ i ∈ l.splitSupport, (l.split i).sum fun a b => f ⟨i, a⟩ b | simp only [sum, sigma_support, sum_sigma, split_apply] | no goals | f5fb3ed0e52a01dc |
CategoryTheory.Equivalence.counitInv_app_functor | Mathlib/CategoryTheory/Equivalence.lean | theorem counitInv_app_functor (e : C ≌ D) (X : C) :
e.counitInv.app (e.functor.obj X) = e.functor.map (e.unit.app X) | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
X : C
⊢ e.functor.map (e.unit.app X) = e.counitInv.app (e.functor.obj X) | erw [← Iso.comp_hom_eq_id (e.counitIso.app _), functor_unit_comp] | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
X : C
⊢ 𝟙 (e.functor.obj X) = 𝟙 ((𝟭 D).obj (e.functor.toPrefunctor.1 ((𝟭 C).obj X))) | 751abebe57bf6dbb |
Polynomial.exists_root_of_degree_eq_one | Mathlib/Algebra/Polynomial/FieldDivision.lean | theorem exists_root_of_degree_eq_one (h : degree p = 1) : ∃ x, IsRoot p x :=
⟨-((p.coeff 1)⁻¹ * p.coeff 0), by
rw [← mem_roots (by simp [← zero_le_degree_iff, h])]
simp [roots_degree_eq_one h]⟩
| R : Type u
inst✝ : Field R
p : R[X]
h : p.degree = 1
⊢ p.IsRoot (-((p.coeff 1)⁻¹ * p.coeff 0)) | rw [← mem_roots (by simp [← zero_le_degree_iff, h])] | R : Type u
inst✝ : Field R
p : R[X]
h : p.degree = 1
⊢ -((p.coeff 1)⁻¹ * p.coeff 0) ∈ p.roots | 48d63a9d139e135c |
Equiv.Perm.IsCycle.isCycle_pow_pos_of_lt_prime_order | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | theorem IsCycle.isCycle_pow_pos_of_lt_prime_order [Finite β] {f : Perm β} (hf : IsCycle f)
(hf' : (orderOf f).Prime) (n : ℕ) (hn : 0 < n) (hn' : n < orderOf f) : IsCycle (f ^ n) | β : Type u_3
inst✝ : Finite β
f : Perm β
hf : f.IsCycle
hf' : Nat.Prime (orderOf f)
n : ℕ
hn : 0 < n
hn' : n < orderOf f
val✝ : Fintype β
⊢ ¬orderOf f ∣ n | exact Nat.not_dvd_of_pos_of_lt hn hn' | no goals | a913e4a3e336e84b |
Rat.one_mul | Mathlib/.lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean | theorem one_mul (a : Rat) : 1 * a = a | a : Rat
⊢ 1 * a = a | simp [mul_def, normalize_self] | no goals | 5c4efd5fe8500fc7 |
Matrix.mem_subfield_of_mul_eq_one_of_mem_subfield_right | Mathlib/LinearAlgebra/Matrix/BaseChange.lean | lemma mem_subfield_of_mul_eq_one_of_mem_subfield_right
(h_mem : ∀ i j, A i j ∈ K) (i : n) (j : m) :
B i j ∈ K | m : Type u_1
n : Type u_2
L : Type u_3
inst✝³ : Finite m
inst✝² : Fintype n
inst✝¹ : DecidableEq m
inst✝ : Field L
e : m ≃ n
K : Subfield L
A : Matrix m n L
B : Matrix n m L
hAB : A * B = 1
h_mem : ∀ (i : m) (j : n), A i j ∈ K
i : n
j : m
val✝ : Fintype m
A' : Matrix m m ↥K := of fun i j => ⟨A.submatrix id (⇑e) i j, ⋯⟩... | replace hB := congr_arg (fun C ↦ C.map K.subtype) hB | m : Type u_1
n : Type u_2
L : Type u_3
inst✝³ : Finite m
inst✝² : Fintype n
inst✝¹ : DecidableEq m
inst✝ : Field L
e : m ≃ n
K : Subfield L
A : Matrix m n L
B : Matrix n m L
hAB : A * B = 1
h_mem : ∀ (i : m) (j : n), A i j ∈ K
i : n
j : m
val✝ : Fintype m
A' : Matrix m m ↥K := of fun i j => ⟨A.submatrix id (⇑e) i j, ⋯⟩... | 8d2ecd23f1ae3281 |
QuasispectrumRestricts.isometric_cfc | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Isometric.lean | theorem isometric_cfc (f : C(S, R)) (halg : Isometry (algebraMap R S)) (h0 : p 0)
(h : ∀ a, p a ↔ q a ∧ QuasispectrumRestricts a f) :
NonUnitalIsometricContinuousFunctionalCalculus R A p where
toNonUnitalContinuousFunctionalCalculus := QuasispectrumRestricts.cfc f
halg.isUniformEmbedding h0 h
isometric ... | case intro.refine_2
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar ... | refine ContinuousMap.dist_le dist_nonneg |>.mpr fun x ↦ ?_ | case intro.refine_2
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar ... | d83e17800d426a00 |
ModelWithCorners.boundary_prod | Mathlib/Geometry/Manifold/IsManifold/InteriorBoundary.lean | /-- The boundary of `M × N` is `∂M × N ∪ (M × ∂N)`. -/
lemma boundary_prod :
(I.prod J).boundary (M × N) = Set.prod univ (J.boundary N) ∪ Set.prod (I.boundary M) univ | 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 : NormedSpac... | rw [compl_prod_eq_union] | no goals | b2e07c83f52d0bc6 |
isRegular_mul_and_mul_iff | Mathlib/Algebra/Regular/Basic.lean | theorem isRegular_mul_and_mul_iff :
IsRegular (a * b) ∧ IsRegular (b * a) ↔ IsRegular a ∧ IsRegular b | case refine_1
R : Type u_1
inst✝ : Semigroup R
a b : R
⊢ IsRegular (a * b) ∧ IsRegular (b * a) → IsRegular a ∧ IsRegular b | rintro ⟨ab, ba⟩ | case refine_1.intro
R : Type u_1
inst✝ : Semigroup R
a b : R
ab : IsRegular (a * b)
ba : IsRegular (b * a)
⊢ IsRegular a ∧ IsRegular b | 720eed5536016320 |
Cardinal.isRegular_aleph'_succ | Mathlib/SetTheory/Cardinal/Cofinality.lean | theorem isRegular_aleph'_succ {o : Ordinal} (h : ω ≤ o) : IsRegular (aleph' (succ o)) | o : Ordinal.{u_1}
h : ω ≤ o
⊢ (aleph' (succ o)).IsRegular | rw [aleph'_succ] | o : Ordinal.{u_1}
h : ω ≤ o
⊢ (succ (aleph' o)).IsRegular | 0a3ee084f1b26a92 |
MeasureTheory.locallyIntegrableOn_of_locallyIntegrable_restrict | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | theorem locallyIntegrableOn_of_locallyIntegrable_restrict [OpensMeasurableSpace X]
(hf : LocallyIntegrable f (μ.restrict s)) : LocallyIntegrableOn f s μ | X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hf : LocallyIntegrable f (μ.restrict s)
⊢ LocallyIntegrableOn f s μ | intro x _ | X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : OpensMeasurableSpace X
hf : LocallyIntegrable f (μ.restrict s)
x : X
a✝ : x ∈ s
⊢ IntegrableAtFilter f (𝓝[s] x) μ | 0d90e38c24dc4abb |
SimpleGraph.ConnectedComponent.eq_of_common_vertex | Mathlib/Combinatorics/SimpleGraph/Path.lean | lemma eq_of_common_vertex {v : V} {c c' : ConnectedComponent G} (hc : v ∈ c.supp)
(hc' : v ∈ c'.supp) : c = c' | V : Type u
G : SimpleGraph V
v : V
c c' : G.ConnectedComponent
hc : v ∈ c.supp
hc' : v ∈ c'.supp
⊢ c = c' | simp only [mem_supp_iff] at * | V : Type u
G : SimpleGraph V
v : V
c c' : G.ConnectedComponent
hc : G.connectedComponentMk v = c
hc' : G.connectedComponentMk v = c'
⊢ c = c' | 474fe99f85af1b4b |
Nat.coprime_of_probablePrime | Mathlib/NumberTheory/FermatPsp.lean | theorem coprime_of_probablePrime {n b : ℕ} (h : ProbablePrime n b) (h₁ : 1 ≤ n) (h₂ : 1 ≤ b) :
Nat.Coprime n b | case pos.H.intro.intro
k : ℕ
hk : Prime k
m : ℕ
h₁ : 1 ≤ k * m
h₃ : 2 ≤ k * m
j : ℕ
h₂ : 1 ≤ k * j
h : k ∣ (k * j) ^ (k * m - 1) - 1
⊢ k * m - 1 ≠ 0 | omega | no goals | 55d29b10e4363e3c |
Real.Angle.cos_eq_iff_coe_eq_or_eq_neg | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | theorem cos_eq_iff_coe_eq_or_eq_neg {θ ψ : ℝ} :
cos θ = cos ψ ↔ (θ : Angle) = ψ ∨ (θ : Angle) = -ψ | case mpr.inr.intro
θ ψ : ℝ
k : ℤ
H : θ - -ψ = 2 * π * ↑k
⊢ cos θ = cos ψ | rw [← sub_eq_zero, cos_sub_cos, ← sub_neg_eq_add, H, mul_assoc 2 π k,
mul_div_cancel_left₀ _ (two_ne_zero' ℝ), mul_comm π _, sin_int_mul_pi, mul_zero,
zero_mul] | no goals | 231d1921c276f456 |
Matroid.Indep.augment | Mathlib/Data/Matroid/Restrict.lean | theorem Indep.augment (hI : M.Indep I) (hJ : M.Indep J) (hIJ : I.encard < J.encard) :
∃ e ∈ J \ I, M.Indep (insert e I) | α : Type u_1
M : Matroid α
I J : Set α
hI : M.Indep I
hJ : M.Indep J
hIJ : I.encard < J.encard
he : ∀ e ∈ J \ I, ¬M.Indep (insert e I)
hb : M.IsBasis I (I ∪ J)
⊢ False | obtain ⟨J', hJ', hJJ'⟩ := hJ.subset_isBasis_of_subset I.subset_union_right | case intro.intro
α : Type u_1
M : Matroid α
I J : Set α
hI : M.Indep I
hJ : M.Indep J
hIJ : I.encard < J.encard
he : ∀ e ∈ J \ I, ¬M.Indep (insert e I)
hb : M.IsBasis I (I ∪ J)
J' : Set α
hJ' : M.IsBasis J' (I ∪ J)
hJJ' : J ⊆ J'
⊢ False | c69faa8b545b0044 |
RatFunc.map_apply_div | Mathlib/FieldTheory/RatFunc/Basic.lean | theorem map_apply_div {R F : Type*} [CommRing R] [IsDomain R]
[FunLike F K[X] R[X]] [MonoidWithZeroHomClass F K[X] R[X]]
(φ : F) (hφ : K[X]⁰ ≤ R[X]⁰.comap φ) (p q : K[X]) :
map φ hφ (algebraMap _ _ p / algebraMap _ _ q) =
algebraMap _ _ (φ p) / algebraMap _ _ (φ q) | K : Type u
inst✝⁵ : CommRing K
inst✝⁴ : IsDomain K
R : Type u_1
F : Type u_2
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : FunLike F K[X] R[X]
inst✝ : MonoidWithZeroHomClass F K[X] R[X]
φ : F
hφ : K[X]⁰ ≤ Submonoid.comap φ R[X]⁰
p q : K[X]
⊢ (map φ hφ) ((algebraMap K[X] (RatFunc K)) p / (algebraMap K[X] (RatFunc K))... | rcases eq_or_ne q 0 with (rfl | hq) | case inl
K : Type u
inst✝⁵ : CommRing K
inst✝⁴ : IsDomain K
R : Type u_1
F : Type u_2
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : FunLike F K[X] R[X]
inst✝ : MonoidWithZeroHomClass F K[X] R[X]
φ : F
hφ : K[X]⁰ ≤ Submonoid.comap φ R[X]⁰
p : K[X]
⊢ (map φ hφ) ((algebraMap K[X] (RatFunc K)) p / (algebraMap K[X] (RatF... | beba2d03782a6c42 |
right_div_gcd_ne_zero | Mathlib/RingTheory/EuclideanDomain.lean | theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 | case intro.intro
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p q : R
hq : q ≠ 0
r : R
hr : q = GCDMonoid.gcd p q * r
pq0 : GCDMonoid.gcd p q ≠ 0
r0 : r ≠ 0
⊢ q / GCDMonoid.gcd p q ≠ 0 | nth_rw 1 [hr] | case intro.intro
R : Type u_1
inst✝¹ : EuclideanDomain R
inst✝ : GCDMonoid R
p q : R
hq : q ≠ 0
r : R
hr : q = GCDMonoid.gcd p q * r
pq0 : GCDMonoid.gcd p q ≠ 0
r0 : r ≠ 0
⊢ GCDMonoid.gcd p q * r / GCDMonoid.gcd p q ≠ 0 | 2b644cd2124d946c |
UpperHalfPlane.ofComplex_apply_of_im_pos | Mathlib/Analysis/Complex/UpperHalfPlane/Topology.lean | lemma ofComplex_apply_of_im_pos {z : ℂ} (hz : 0 < z.im) :
ofComplex z = ⟨z, hz⟩ | z : ℂ
hz : 0 < z.im
⊢ ↑ofComplex z = ⟨z, hz⟩ | simpa only [coe_mk_subtype] using ofComplex_apply ⟨z, hz⟩ | no goals | 4db9d917fcad6c8d |
OrderIso.tendsto_atTop_iff | Mathlib/Order/Filter/AtTopBot/Map.lean | theorem tendsto_atTop_iff {l : Filter γ} {f : γ → α} (e : α ≃o β) :
Tendsto (fun x => e (f x)) l atTop ↔ Tendsto f l atTop | α : Type u_3
β : Type u_4
γ : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
l : Filter γ
f : γ → α
e : α ≃o β
⊢ Tendsto (fun x => e (f x)) l atTop ↔ Tendsto f l atTop | rw [← e.comap_atTop, tendsto_comap_iff, Function.comp_def] | no goals | debf757d8c057f9a |
Perfect.exists_nat_bool_injection | Mathlib/Topology/MetricSpace/Perfect.lean | theorem Perfect.exists_nat_bool_injection
(hC : Perfect C) (hnonempty : C.Nonempty) [CompleteSpace α] :
∃ f : (ℕ → Bool) → α, range f ⊆ C ∧ Continuous f ∧ Injective f | case false.true
α : Type u_1
inst✝¹ : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst✝ : CompleteSpace α
u : ℕ → ℝ≥0∞
upos' : ∀ (n : ℕ), u n ∈ Ioo 0 1
hu : Tendsto u atTop (nhds 0)
upos : ∀ (n : ℕ), 0 < u n
P : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }
C0 C1 : {C : Set α} → Perfect C → C.No... | exact hdisj _ _ _ | no goals | ef9f9f33b845189a |
Polynomial.flt_catalan_aux | Mathlib/NumberTheory/FLT/Polynomial.lean | theorem Polynomial.flt_catalan_aux
{p q r : ℕ} {a b c : k[X]} {u v w : k}
(heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0)
(hp : 0 < p) (hq : 0 < q) (hr : 0 < r)
(hineq : q * r + r * p + p * q ≤ p * q * r)
(chp : (p : k) ≠ 0) (chq : (q : k) ≠ 0) (chr : (r : k) ≠ 0)
(ha : a ≠ 0) (hb : b ≠ 0) (... | k : Type u_1
inst✝ : Field k
p q r : ℕ
u v w : k
hp : 0 < p
hq : 0 < q
hr : 0 < r
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
d : ℕ
a b c : k[X]
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
eq... | omega | no goals | ce7003012aac2b7e |
IsDenseInducing.extend_Z_bilin_key | Mathlib/Topology/Algebra/UniformGroup/Defs.lean | theorem extend_Z_bilin_key (x₀ : α) (y₀ : γ) : ∃ U ∈ comap e (𝓝 x₀), ∃ V ∈ comap f (𝓝 y₀),
∀ x ∈ U, ∀ x' ∈ U, ∀ (y) (_ : y ∈ V) (y') (_ : y' ∈ V),
(fun p : β × δ => φ p.1 p.2) (x', y') - (fun p : β × δ => φ p.1 p.2) (x, y) ∈ W' | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹² : TopologicalSpace α
inst✝¹¹ : AddCommGroup α
inst✝¹⁰ : IsTopologicalAddGroup α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : AddCommGroup β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : A... | exists U₁ ∩ U₂, inter_mem U₁_nhd U₂_nhd, V₁ ∩ V₂, inter_mem V₁_nhd V₂_nhd | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹² : TopologicalSpace α
inst✝¹¹ : AddCommGroup α
inst✝¹⁰ : IsTopologicalAddGroup α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : AddCommGroup β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : A... | 7e13666b5fc1d57e |
Nat.sum_range_choose_halfway | Mathlib/Data/Nat/Choose/Sum.lean | theorem sum_range_choose_halfway (m : ℕ) : (∑ i ∈ range (m + 1), (2 * m + 1).choose i) = 4 ^ m :=
have : (∑ i ∈ range (m + 1), (2 * m + 1).choose (2 * m + 1 - i)) =
∑ i ∈ range (m + 1), (2 * m + 1).choose i :=
sum_congr rfl fun i hi ↦ choose_symm <| by linarith [mem_range.1 hi]
mul_right_injective₀ two_ne... | m : ℕ
this : ∑ i ∈ range (m + 1), (2 * m + 1).choose (2 * m + 1 - i) = ∑ i ∈ range (m + 1), (2 * m + 1).choose i
⊢ 2 ^ (2 * m + 1) = 2 * 4 ^ m | rw [pow_succ, pow_mul, mul_comm] | m : ℕ
this : ∑ i ∈ range (m + 1), (2 * m + 1).choose (2 * m + 1 - i) = ∑ i ∈ range (m + 1), (2 * m + 1).choose i
⊢ 2 * (2 ^ 2) ^ m = 2 * 4 ^ m | e86ec317b0f3446d |
Submonoid.card_le_one_iff_eq_bot | Mathlib/Algebra/Group/Submonoid/Membership.lean | theorem card_le_one_iff_eq_bot : card S ≤ 1 ↔ S = ⊥ :=
⟨fun h =>
(eq_bot_iff_forall _).2 fun x hx => by
simpa [Subtype.ext_iff] using card_le_one_iff.1 h ⟨x, hx⟩ 1,
fun h => by simp [h]⟩
| M : Type u_1
inst✝¹ : Monoid M
S : Submonoid M
inst✝ : Fintype ↥S
h : S = ⊥
⊢ card ↥S ≤ 1 | simp [h] | no goals | 14d4abd5463560fe |
Algebra.transcendental_iff_not_isAlgebraic | Mathlib/RingTheory/Algebraic/Defs.lean | theorem Algebra.transcendental_iff_not_isAlgebraic :
Algebra.Transcendental R A ↔ ¬ Algebra.IsAlgebraic R A | R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
⊢ Algebra.Transcendental R A ↔ ¬Algebra.IsAlgebraic R A | simp [isAlgebraic_def, transcendental_def, Transcendental] | no goals | 7fd698d13ad8de4c |
MeasureTheory.MemLp.integrable_norm_rpow' | Mathlib/MeasureTheory/Function/L1Space/Integrable.lean | theorem MemLp.integrable_norm_rpow' [IsFiniteMeasure μ] {f : α → β} {p : ℝ≥0∞} (hf : MemLp f p μ) :
Integrable (fun x : α => ‖f x‖ ^ p.toReal) μ | case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
inst✝ : IsFiniteMeasure μ
f : α → β
p : ℝ≥0∞
hf : MemLp f p μ
h_zero : ¬p = 0
h_top : p = ⊤
⊢ Integrable (fun x => ‖f x‖ ^ p.toReal) μ | simp [h_top, integrable_const] | no goals | 6a1793d4ed5a42ea |
AlgebraicClosure.Monics.map_eq_prod | Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean | theorem Monics.map_eq_prod {f : Monics k} :
f.1.map (algebraMap k (AlgebraicClosure k)) =
∏ i, map (Ideal.Quotient.mk <| maxIdeal k) (X - C (MvPolynomial.X ⟨f, i⟩)) | k : Type u
inst✝ : Field k
f : Monics k
⊢ map (algebraMap k (AlgebraicClosure k)) ↑f =
∏ i : Fin (↑f).natDegree, map (Ideal.Quotient.mk (maxIdeal k)) (X - C (MvPolynomial.X ⟨f, i⟩)) | ext | case a
k : Type u
inst✝ : Field k
f : Monics k
n✝ : ℕ
⊢ (map (algebraMap k (AlgebraicClosure k)) ↑f).coeff n✝ =
(∏ i : Fin (↑f).natDegree, map (Ideal.Quotient.mk (maxIdeal k)) (X - C (MvPolynomial.X ⟨f, i⟩))).coeff n✝ | 4c41eea2302d83ca |
maximalIdeal_isPrincipal_of_isDedekindDomain | Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean | theorem maximalIdeal_isPrincipal_of_isDedekindDomain [IsLocalRing R] [IsDomain R]
[IsDedekindDomain R] : (maximalIdeal R).IsPrincipal | case pos.refine_1
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : (Ideal.span {a}).radical = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.... | exact (IsFractionRing.to_map_eq_zero_iff (K := K)).not.mpr ha₂ | no goals | e801db6f819d1797 |
CategoryTheory.Pretriangulated.Opposite.distinguished_cocone_triangle | Mathlib/CategoryTheory/Triangulated/Opposite/Pretriangulated.lean | lemma distinguished_cocone_triangle {X Y : Cᵒᵖ} (f : X ⟶ Y) :
∃ (Z : Cᵒᵖ) (g : Y ⟶ Z) (h : Z ⟶ X⟦(1 : ℤ)⟧),
Triangle.mk f g h ∈ distinguishedTriangles C | C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X Y : Cᵒᵖ
f : X ⟶ Y
Z : C
g : Z ⟶ Opposite.unop Y
h : Opposite.unop X ⟶ (shiftFunctor C 1).obj Z
H : Triangle.mk g f.unop h ∈ Pretri... | simp | no goals | 272fd5c34afae8cc |
Finset.card_eq_succ_iff_cons | Mathlib/Data/Finset/Card.lean | theorem card_eq_succ_iff_cons :
#s = n + 1 ↔ ∃ a t, ∃ (h : a ∉ t), cons a t h = s ∧ #t = n :=
⟨cons_induction_on s (by simp) fun a s _ _ _ => ⟨a, s, by simp_all⟩,
fun ⟨a, t, _, hs, _⟩ => by simpa [← hs]⟩
| α : Type u_1
s✝ : Finset α
n : ℕ
a : α
s : Finset α
x✝² : a ∉ s
x✝¹ : #s = n + 1 → ∃ a t, ∃ (h : a ∉ t), cons a t h = s ∧ #t = n
x✝ : #(cons a s x✝²) = n + 1
⊢ ∃ (h : a ∉ s), cons a s h = cons a s x✝² ∧ #s = n | simp_all | no goals | 96502733de50c25a |
Basis.mapCoeffs_apply | Mathlib/LinearAlgebra/Basis/Defs.lean | theorem mapCoeffs_apply (i : ι) : b.mapCoeffs f h i = b i :=
apply_eq_iff.mpr <| by
-- Porting note: in Lean 3, these were automatically inferred from the definition of
-- `mapCoeffs`.
letI : Module R' R := Module.compHom R (↑f.symm : R' →+* R)
haveI : IsScalarTower R' R M :=
{ smul_assoc := fun x... | ι : Type u_1
R : Type u_3
M : Type u_6
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
b : Basis ι R M
R' : Type u_10
inst✝¹ : Semiring R'
inst✝ : Module R' M
f : R ≃+* R'
h : ∀ (c : R) (x : M), f c • x = c • x
i : ι
⊢ (b.mapCoeffs f h).repr (b i) = single i 1 | letI : Module R' R := Module.compHom R (↑f.symm : R' →+* R) | ι : Type u_1
R : Type u_3
M : Type u_6
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
b : Basis ι R M
R' : Type u_10
inst✝¹ : Semiring R'
inst✝ : Module R' M
f : R ≃+* R'
h : ∀ (c : R) (x : M), f c • x = c • x
i : ι
this : Module R' R := Module.compHom R ↑f.symm
⊢ (b.mapCoeffs f h).repr (b i) = single... | 1c7b16a64e9a7022 |
Std.DHashMap.Raw.Const.getD_of_isEmpty | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean | theorem getD_of_isEmpty [EquivBEq α] [LawfulHashable α] (h : m.WF) {a : α} {fallback : β} :
m.isEmpty = true → getD m a fallback = fallback | 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
a : α
fallback : β
⊢ m.isEmpty = true → getD m a fallback = fallback | simp_to_raw using Raw₀.Const.getD_of_isEmpty ⟨m, _⟩ | no goals | 2f134cb384174e34 |
intervalIntegral.integral_interval_add_interval_comm | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | theorem integral_interval_add_interval_comm (hab : IntervalIntegrable f μ a b)
(hcd : IntervalIntegrable f μ c d) (hac : IntervalIntegrable f μ a c) :
((∫ x in a..b, f x ∂μ) + ∫ x in c..d, f x ∂μ) =
(∫ x in a..d, f x ∂μ) + ∫ x in c..b, f x ∂μ | E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c d : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hcd : IntervalIntegrable f μ c d
hac : IntervalIntegrable f μ a c
⊢ ∫ (x : ℝ) in a..b, f x ∂μ + ∫ (x : ℝ) in c..d, f x ∂μ = ∫ (x : ℝ) in a..d, f x ∂μ + ∫ (x : ℝ) in c..b, f x ∂μ | rw [← integral_add_adjacent_intervals hac hcd, add_assoc, add_left_comm,
integral_add_adjacent_intervals hac (hac.symm.trans hab), add_comm] | no goals | 14b22fd0694e14de |
EMetric.edist_le_infEdist_add_ediam | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | theorem edist_le_infEdist_add_ediam (hy : y ∈ s) : edist x y ≤ infEdist x s + diam s | α : Type u
inst✝ : PseudoEMetricSpace α
x y : α
s : Set α
hy : y ∈ s
⊢ edist x y ≤ infEdist x s + diam s | simp_rw [infEdist, ENNReal.iInf_add] | α : Type u
inst✝ : PseudoEMetricSpace α
x y : α
s : Set α
hy : y ∈ s
⊢ edist x y ≤ ⨅ i ∈ s, edist x i + diam s | 02e32c68f53769e8 |
Orientation.inner_mul_inner_add_areaForm_mul_areaForm' | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | theorem inner_mul_inner_add_areaForm_mul_areaForm' (a x : E) :
⟪a, x⟫ • innerₛₗ ℝ a + ω a x • ω a = ‖a‖ ^ 2 • innerₛₗ ℝ x | case neg.«_@».Mathlib.Analysis.InnerProductSpace.TwoDim._hyg.6483.«0»
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
a x : E
ha : ¬a = 0
⊢ inner a x * ‖a‖ ^ 2 = ‖a‖ ^ 2 * inner a x | ring | no goals | 4675dd0b22e71b62 |
univLE_iff_exists_embedding | Mathlib/SetTheory/Cardinal/UnivLE.lean | theorem univLE_iff_exists_embedding : UnivLE.{u, v} ↔ Nonempty (Ordinal.{u} ↪ Ordinal.{v}) | ⊢ UnivLE.{u, v} ↔ Nonempty (Ordinal.{u} ↪ Ordinal.{v}) | rw [univLE_iff_cardinal_le] | ⊢ univ.{u, v + 1} ≤ univ.{v, u + 1} ↔ Nonempty (Ordinal.{u} ↪ Ordinal.{v}) | 04156f59f87b829f |
Subgroup.leftCoset_cover_filter_FiniteIndex_aux | Mathlib/GroupTheory/CosetCover.lean | theorem leftCoset_cover_filter_FiniteIndex_aux
[DecidablePred (FiniteIndex : Subgroup G → Prop)] :
(⋃ k ∈ s.filter (fun i => (H i).FiniteIndex), g k • (H k : Set G) = Set.univ) ∧
(1 ≤ ∑ i ∈ s, ((H i).index : ℚ)⁻¹) ∧
(∑ i ∈ s, ((H i).index : ℚ)⁻¹ = 1 → Set.PairwiseDisjoint
(s.filter (fun i =>... | case refine_2
G : Type u_1
inst✝¹ : Group G
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
inst✝ : DecidablePred FiniteIndex
D : Subgroup G := ⨅ k ∈ Finset.filter (fun i => (H i).FiniteIndex) s, H k
hD : D.FiniteIndex
hD_le : ∀ {i : ι}, i ∈ s → (H i).FiniteIndex → D ≤ ... | rw [hdensity] | case refine_2
G : Type u_1
inst✝¹ : Group G
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
inst✝ : DecidablePred FiniteIndex
D : Subgroup G := ⨅ k ∈ Finset.filter (fun i => (H i).FiniteIndex) s, H k
hD : D.FiniteIndex
hD_le : ∀ {i : ι}, i ∈ s → (H i).FiniteIndex → D ≤ ... | 6c7a447596ed19f6 |
List.map_mk_sublist_sym2 | Mathlib/Data/List/Sym.lean | theorem map_mk_sublist_sym2 (x : α) (xs : List α) (h : x ∈ xs) :
map (fun y ↦ s(x, y)) xs <+ xs.sym2 | case cons
α : Type u_1
x x' : α
xs : List α
ih : x ∈ xs → map (fun y => s(x, y)) xs <+ xs.sym2
h : x ∈ x' :: xs
⊢ s(x, x') :: map (fun y => s(x, y)) xs <+ s(x', x') :: (map (fun y => s(x', y)) xs ++ xs.sym2) | cases h with
| head =>
exact (sublist_append_left _ _).cons₂ _
| tail _ h =>
refine .cons _ ?_
rw [← singleton_append]
refine .append ?_ (ih h)
rw [singleton_sublist, mem_map]
exact ⟨_, h, Sym2.eq_swap⟩ | no goals | c6231b97dc942a4b |
ZLattice.covolume.tendsto_card_div_pow'' | Mathlib/Algebra/Module/ZLattice/Covolume.lean | theorem tendsto_card_div_pow'' [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
{s : Set E} (hs₁ : IsBounded s) (hs₂ : MeasurableSet s)
(hs₃ : volume (frontier ((b.ofZLatticeBasis ℝ).equivFun '' s)) = 0):
Tendsto (fun n : ℕ ↦ (Nat.card (s ∩ (n : ℝ)⁻¹ • L : Set E) : ℝ) / n ^ card ι)
atTop (𝓝... | case h.e_a.e_a
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁵ : DiscreteTopology ↥L
inst✝⁴ : IsZLattice ℝ L
ι : Type u_2
inst✝³ : Fintype ι
b : Basis ι ℤ ↥L
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set E
hs₁ : Bornology.IsBounded s
... | refine Nat.card_congr <| ((b.ofZLatticeBasis ℝ).equivFun.toEquiv.subtypeEquiv fun x ↦ ?_).symm | case h.e_a.e_a
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁵ : DiscreteTopology ↥L
inst✝⁴ : IsZLattice ℝ L
ι : Type u_2
inst✝³ : Fintype ι
b : Basis ι ℤ ↥L
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set E
hs₁ : Bornology.IsBounded s
... | 730c1ed62d95b3cb |
map_prod | Mathlib/Algebra/BigOperators/Group/Finset/Defs.lean | theorem map_prod [CommMonoid β] [CommMonoid γ] {G : Type*} [FunLike G β γ] [MonoidHomClass G β γ]
(g : G) (f : α → β) (s : Finset α) : g (∏ x ∈ s, f x) = ∏ x ∈ s, g (f x) | α : Type u_3
β : Type u_4
γ : Type u_5
inst✝³ : CommMonoid β
inst✝² : CommMonoid γ
G : Type u_6
inst✝¹ : FunLike G β γ
inst✝ : MonoidHomClass G β γ
g : G
f : α → β
s : Finset α
⊢ g (∏ x ∈ s, f x) = ∏ x ∈ s, g (f x) | simp only [Finset.prod_eq_multiset_prod, map_multiset_prod, Multiset.map_map] | α : Type u_3
β : Type u_4
γ : Type u_5
inst✝³ : CommMonoid β
inst✝² : CommMonoid γ
G : Type u_6
inst✝¹ : FunLike G β γ
inst✝ : MonoidHomClass G β γ
g : G
f : α → β
s : Finset α
⊢ (Multiset.map (⇑g ∘ f) s.val).prod = (Multiset.map (fun x => g (f x)) s.val).prod | 6bed3386895df0d8 |
RelSeries.smash_succ_castAdd | Mathlib/Order/RelSeries.lean | lemma smash_succ_castAdd {p q : RelSeries r} (h : p.last = q.head)
(i : Fin p.length) : p.smash q h (i.castAdd q.length).succ = p i.succ | case h.e'_3.e_a
α : Type u_1
r : Rel α α
p q : RelSeries r
h : p.last = q.head
i : Fin p.length
H : ¬↑i + 1 < p.length
⊢ i.succ = Fin.last p.length | ext | case h.e'_3.e_a.h
α : Type u_1
r : Rel α α
p q : RelSeries r
h : p.last = q.head
i : Fin p.length
H : ¬↑i + 1 < p.length
⊢ ↑i.succ = ↑(Fin.last p.length) | fd7866c3f8830725 |
cube_of_castHom_ne_zero | Mathlib/NumberTheory/FLT/Three.lean | private lemma cube_of_castHom_ne_zero {n : ZMod 9} :
castHom (show 3 ∣ 9 by norm_num) (ZMod 3) n ≠ 0 → n ^ 3 = 1 ∨ n ^ 3 = 8 | n : ZMod 9
⊢ (castHom ⋯ (ZMod 3)) n ≠ 0 → n ^ 3 = 1 ∨ n ^ 3 = 8 | revert n | ⊢ ∀ {n : ZMod 9}, (castHom ⋯ (ZMod 3)) n ≠ 0 → n ^ 3 = 1 ∨ n ^ 3 = 8 | 09117387c84ad9bd |
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