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 |
|---|---|---|---|---|---|---|
Filter.liminf_sdiff | Mathlib/Order/LiminfLimsup.lean | theorem liminf_sdiff [NeBot f] (a : α) : liminf u f \ a = liminf (fun b => u b \ a) f | α : Type u_1
β : Type u_2
inst✝¹ : CompleteBooleanAlgebra α
f : Filter β
u : β → α
inst✝ : f.NeBot
a : α
⊢ liminf u f \ a = liminf (fun b => u b \ a) f | simp only [sdiff_eq, inf_comm _ aᶜ, inf_liminf] | no goals | 3142b483c8bd7f08 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastUdiv.blastDivSubtractShift_decl_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Impl/Operations/Udiv.lean | theorem blastDivSubtractShift_decl_eq (aig : AIG α) (falseRef trueRef : AIG.Ref aig)
(n d : AIG.RefVec aig w) (wn wr : Nat) (q r : AIG.RefVec aig w) :
∀ (idx : Nat) (h1) (h2),
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).aig.decls[idx]'h2 = aig.decls[idx]'h1 | case h2.h.h.h.h
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
falseRef trueRef : aig.Ref
n d : aig.RefVec w
wn wr : Nat
q r : aig.RefVec w
res : BlastDivSubtractShiftOutput aig w
hres :
{
aig :=
(AIG.RefVec.ite
(AIG.RefVec.ite
(BVPred.mkUlt
... | assumption | no goals | 76230fc57d8c20e2 |
Submodule.goursatFst_prod_goursatSnd_le | Mathlib/LinearAlgebra/Goursat.lean | lemma goursatFst_prod_goursatSnd_le : L.goursatFst.prod L.goursatSnd ≤ L | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
L : Submodule R (M × N)
⊢ L.goursatFst.prod L.goursatSnd ≤ L | simpa only [← toAddSubgroup_le, goursatFst_toAddSubgroup, goursatSnd_toAddSubgroup]
using L.toAddSubgroup.goursatFst_prod_goursatSnd_le | no goals | 006e5b2502a4ed1e |
MeasureTheory.hasFDerivAt_convolution_right_with_param | Mathlib/Analysis/Convolution.lean | theorem hasFDerivAt_convolution_right_with_param {g : P → G → E'} {s : Set P} {k : Set G}
(hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 1 (↿g) (s ×ˢ univ)) (q₀ : P × G)
(hq₀ : q₀.1 ∈ s) :
HasFDerivAt (fun q : P × G => (f ⋆... | 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : Measura... | refine (hasFDerivAt_zero_of_eventually_const 0 ?_).fderiv | 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : Measura... | 681ec723c7fd0b70 |
MeasureTheory.Measure.integral_isMulLeftInvariant_isMulRightInvariant_combo | Mathlib/MeasureTheory/Measure/Haar/Unique.lean | /-- In a group with a left invariant measure `μ` and a right invariant measure `ν`, one can express
integrals with respect to `μ` as integrals with respect to `ν` up to a constant scaling factor
(given in the statement as `∫ x, g x ∂μ` where `g` is a fixed reference function) and an
explicit density `y ↦ 1/∫ z, g (z⁻¹ ... | case h'f
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeas... | let L := tsupport g | case h'f
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeas... | 6b9d060f3308d3d5 |
mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt | Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean | theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
(hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
(hz : p • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
z ∈ adjoin R ({B.gen} : Set L) | case neg.hi.convert_2.intro
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClose... | rw [Algebra.smul_def, mul_assoc, ← mul_sub, _root_.map_mul, algebraMap_apply R K L, map_pow,
Algebra.norm_algebraMap, _root_.map_mul, algebraMap_apply R K L, Algebra.norm_algebraMap,
finrank B, ← hr, PowerBasis.norm_gen_eq_coeff_zero_minpoly,
minpoly.isIntegrallyClosed_eq_field_fractions' K hBint, coeff_map,
sh... | case neg.hi.convert_2.intro
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClose... | fa4f10e8cbdbb4c7 |
CategoryTheory.mem_essImage_of_unit_isSplitMono | Mathlib/CategoryTheory/Adjunction/Reflective.lean | theorem mem_essImage_of_unit_isSplitMono [Reflective i] {A : C}
[IsSplitMono ((reflectorAdjunction i).unit.app A)] : i.essImage A | C : Type u₁
D : Type u₂
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
i : D ⥤ C
inst✝¹ : Reflective i
A : C
inst✝ : IsSplitMono ((reflectorAdjunction i).unit.app A)
η : 𝟭 C ⟶ reflector i ⋙ i := (reflectorAdjunction i).unit
this✝¹ : IsIso (η.app (i.obj ((reflector i).obj A)))
this✝ : Epi (η.app A)
this : Is... | exact (reflectorAdjunction i).mem_essImage_of_unit_isIso A | no goals | 78eecab7fff04ed5 |
HNNExtension.NormalWord.unitsSMul_cancels_iff | Mathlib/GroupTheory/HNNExtension.lean | theorem unitsSMul_cancels_iff (u : ℤˣ) (w : NormalWord d) :
Cancels (-u) (unitsSMul φ u w) ↔ ¬ Cancels u w | case pos.cons
G : Type u_1
inst✝ : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
d : TransversalPair G A B
u : ℤˣ
g : G
u' : ℤˣ
w : NormalWord d
h1 : w.head ∈ d.set u'
h2 : ∀ u'_1 ∈ Option.map Prod.fst w.toList.head?, w.head ∈ toSubgroup A B u' → u' = u'_1
a✝ : ∀ (h : Cancels u w), ¬Cancels (-u) (unitsSMulWithCancel φ u w ⋯)
h... | cases h.2 | case pos.cons.refl
G : Type u_1
inst✝ : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
d : TransversalPair G A B
u : ℤˣ
g : G
w : NormalWord d
a✝ : ∀ (h : Cancels u w), ¬Cancels (-u) (unitsSMulWithCancel φ u w ⋯)
h1 : w.head ∈ d.set (-u)
h2 : ∀ u' ∈ Option.map Prod.fst w.toList.head?, w.head ∈ toSubgroup A B (-u) → -u = u'
h✝ :... | b3a9a1f389e0c3f6 |
AlgebraicGeometry.genericPoint_eq_of_isOpenImmersion | Mathlib/AlgebraicGeometry/FunctionField.lean | theorem genericPoint_eq_of_isOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [H : IsOpenImmersion f]
[hX : IrreducibleSpace X] [IrreducibleSpace Y] :
f.base (genericPoint X) = genericPoint Y | case h.e'_4
X Y : Scheme
f : X ⟶ Y
H : IsOpenImmersion f
hX : IrreducibleSpace ↑↑X.toPresheafedSpace
inst✝ : IrreducibleSpace ↑↑Y.toPresheafedSpace
⊢ Set.univ = closure (⇑(ConcreteCategory.hom f.base) '' Set.univ) | symm | case h.e'_4
X Y : Scheme
f : X ⟶ Y
H : IsOpenImmersion f
hX : IrreducibleSpace ↑↑X.toPresheafedSpace
inst✝ : IrreducibleSpace ↑↑Y.toPresheafedSpace
⊢ closure (⇑(ConcreteCategory.hom f.base) '' Set.univ) = Set.univ | 1e9165b2d519e678 |
List.Perm.pmap | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Perm.lean | theorem Perm.pmap {p : α → Prop} (f : ∀ a, p a → β) {l₁ l₂ : List α} (p : l₁ ~ l₂) {H₁ H₂} :
pmap f l₁ H₁ ~ pmap f l₂ H₂ | α : Type u_1
β : Type u_2
p✝ : α → Prop
f : (a : α) → p✝ a → β
l₁ l₂ : List α
p : l₁ ~ l₂
H₁ : ∀ (a : α), a ∈ l₁ → p✝ a
H₂ : ∀ (a : α), a ∈ l₂ → p✝ a
⊢ List.pmap f l₁ H₁ ~ List.pmap f l₂ H₂ | induction p with
| nil => simp
| cons x _p IH => simp [IH, Perm.cons]
| swap x y => simp [swap]
| trans _p₁ p₂ IH₁ IH₂ => exact IH₁.trans (IH₂ (H₁ := fun a m => H₂ a (p₂.subset m))) | no goals | b6857377dabdd618 |
ContinuousLinearMap.opNorm_le_of_unit_norm | Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean | theorem opNorm_le_of_unit_norm [NormedSpace ℝ E] [NormedSpace ℝ F] {f : E →L[ℝ] F} {C : ℝ}
(hC : 0 ≤ C) (hf : ∀ x, ‖x‖ = 1 → ‖f x‖ ≤ C) : ‖f‖ ≤ C | E : Type u_4
F : Type u_5
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
f : E →L[ℝ] F
C : ℝ
hC : 0 ≤ C
hf : ∀ (x : E), ‖x‖ = 1 → ‖f x‖ ≤ C
⊢ ‖f‖ ≤ C | refine opNorm_le_bound' f hC fun x hx => ?_ | E : Type u_4
F : Type u_5
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
f : E →L[ℝ] F
C : ℝ
hC : 0 ≤ C
hf : ∀ (x : E), ‖x‖ = 1 → ‖f x‖ ≤ C
x : E
hx : ‖x‖ ≠ 0
⊢ ‖f x‖ ≤ C * ‖x‖ | 7859156de0e60578 |
FirstOrder.Language.Substructure.cg_iff_empty_or_exists_nat_generating_family | Mathlib/ModelTheory/FinitelyGenerated.lean | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N | L : Language
M : Type u_1
inst✝ : L.Structure M
N : L.Substructure M
⊢ (∃ S, S.Countable ∧ (closure L).toFun S = N) ↔ ↑N = ∅ ∨ ∃ s, (closure L).toFun (range s) = N | constructor | case mp
L : Language
M : Type u_1
inst✝ : L.Structure M
N : L.Substructure M
⊢ (∃ S, S.Countable ∧ (closure L).toFun S = N) → ↑N = ∅ ∨ ∃ s, (closure L).toFun (range s) = N
case mpr
L : Language
M : Type u_1
inst✝ : L.Structure M
N : L.Substructure M
⊢ (↑N = ∅ ∨ ∃ s, (closure L).toFun (range s) = N) → ∃ S, S.Countable ... | 9cbb3eaa8b379543 |
HurwitzZeta.hurwitzZetaEven_one_sub | Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | /-- If `s` is not in `-ℕ`, and either `a ≠ 0` or `s ≠ 1`, then
`hurwitzZetaEven a (1 - s)` is an explicit multiple of `cosZeta s`. -/
lemma hurwitzZetaEven_one_sub (a : UnitAddCircle) {s : ℂ}
(hs : ∀ (n : ℕ), s ≠ -n) (hs' : a ≠ 0 ∨ s ≠ 1) :
hurwitzZetaEven a (1 - s) = 2 * (2 * π) ^ (-s) * Gamma s * cos (π * s /... | a : UnitAddCircle
s : ℂ
hs : ∀ (n : ℕ), s ≠ -↑n
hs' : a ≠ 0 ∨ s ≠ 1
⊢ a ≠ 0 ∨ 1 - s ≠ 0 | simpa [sub_eq_zero, eq_comm (a := s)] using hs' | no goals | ba1c42484b38c301 |
SSet.horn.hom_ext | Mathlib/AlgebraicTopology/SimplicialSet/Horn.lean | /-- Two morphisms from a horn are equal if they are equal on all suitable faces. -/
protected
lemma hom_ext {n : ℕ} {i : Fin (n+2)} {S : SSet} (σ₁ σ₂ : Λ[n+1, i] ⟶ S)
(h : ∀ (j) (h : j ≠ i), σ₁.app _ (face i j h) = σ₂.app _ (face i j h)) :
σ₁ = σ₂ | n : ℕ
i : Fin (n + 2)
S : SSet
σ₁ σ₂ : Λ[n + 1, i] ⟶ S
h : ∀ (j : Fin (n + 2)) (h : j ≠ i), σ₁.app (op ⦋n⦌) (face i j h) = σ₂.app (op ⦋n⦌) (face i j h)
m : ℕ
f : Λ[n + 1, i] _⦋m⦌
f' : unop (op ⦋m⦌) ⟶ ⦋n + 1⦌
hf : (stdSimplex.objEquiv ⦋n + 1⦌ (op ⦋m⦌)).symm f' = ↑f
⊢ ∃ j, ¬j = i ∧ ∀ (k : Fin ((unop (op ⦋m⦌)).len + 1)), ... | obtain ⟨f, hf'⟩ := f | case mk
n : ℕ
i : Fin (n + 2)
S : SSet
σ₁ σ₂ : Λ[n + 1, i] ⟶ S
h : ∀ (j : Fin (n + 2)) (h : j ≠ i), σ₁.app (op ⦋n⦌) (face i j h) = σ₂.app (op ⦋n⦌) (face i j h)
m : ℕ
f' : unop (op ⦋m⦌) ⟶ ⦋n + 1⦌
f : Δ[n + 1] _⦋m⦌
hf' : Set.range ⇑(asOrderHom f) ∪ {i} ≠ Set.univ
hf : (stdSimplex.objEquiv ⦋n + 1⦌ (op ⦋m⦌)).symm f' = ↑⟨f,... | 81aed7a2f2a51d1a |
Sylow.mapSurjective_surjective | Mathlib/GroupTheory/Sylow.lean | theorem mapSurjective_surjective (p : ℕ) [Fact p.Prime] :
Function.Surjective (Sylow.mapSurjective hf : Sylow p G → Sylow p G') | G : Type u_1
inst✝³ : Group G
inst✝² : Finite G
G' : Type u_2
inst✝¹ : Group G'
f : G →* G'
hf : Function.Surjective ⇑f
p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ Function.Surjective (mapSurjective hf) | have : Finite G' := Finite.of_surjective f hf | G : Type u_1
inst✝³ : Group G
inst✝² : Finite G
G' : Type u_2
inst✝¹ : Group G'
f : G →* G'
hf : Function.Surjective ⇑f
p : ℕ
inst✝ : Fact (Nat.Prime p)
this : Finite G'
⊢ Function.Surjective (mapSurjective hf) | 0fc3ef39797b02b0 |
VectorFourier.norm_fourierPowSMulRight_iteratedFDeriv_fourierIntegral_le | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | theorem norm_fourierPowSMulRight_iteratedFDeriv_fourierIntegral_le [FiniteDimensional ℝ V]
{μ : Measure V} [Measure.IsAddHaarMeasure μ] {K N : ℕ∞} (hf : ContDiff ℝ N f)
(h'f : ∀ (k n : ℕ), k ≤ K → n ≤ N → Integrable (fun v ↦ ‖v‖^k * ‖iteratedFDeriv ℝ n f v‖) μ)
{k n : ℕ} (hk : k ≤ K) (hn : n ≤ N) {w : W} :
... | E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : NormedSpace ℝ V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : FiniteDimensional ℝ V
μ : ... | intro i hi j hj | E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : NormedSpace ℝ V
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : FiniteDimensional ℝ V
μ : ... | 8318f0ae43be69ad |
UpperHalfPlane.contMDiffAt_ofComplex | Mathlib/Analysis/Complex/UpperHalfPlane/Manifold.lean | lemma contMDiffAt_ofComplex {n : WithTop ℕ∞} {z : ℂ} (hz : 0 < z.im) :
ContMDiffAt 𝓘(ℂ) 𝓘(ℂ) n ofComplex z | case left
n : WithTop ℕ∞
z : ℂ
hz : 0 < z.im
⊢ Tendsto (Subtype.val ∘ ↑ofComplex) (nhds z) (nhds ↑(↑ofComplex z)) | refine Tendsto.congr' (eventuallyEq_coe_comp_ofComplex hz).symm ?_ | case left
n : WithTop ℕ∞
z : ℂ
hz : 0 < z.im
⊢ Tendsto id (nhds z) (nhds ↑(↑ofComplex z)) | 99c8225fb9438f5f |
subsingleton_floorSemiring | Mathlib/Algebra/Order/Floor.lean | theorem subsingleton_floorSemiring {α} [LinearOrderedSemiring α] :
Subsingleton (FloorSemiring α) | α : Type u_4
inst✝ : LinearOrderedSemiring α
H₁ H₂ : FloorSemiring α
this✝ : FloorSemiring.ceil = FloorSemiring.ceil
this : FloorSemiring.floor = FloorSemiring.floor
⊢ H₁ = H₂ | cases H₁ | case mk
α : Type u_4
inst✝ : LinearOrderedSemiring α
H₂ : FloorSemiring α
floor✝ ceil✝ : α → ℕ
floor_of_neg✝ : ∀ {a : α}, a < 0 → floor✝ a = 0
gc_floor✝ : ∀ {a : α} {n : ℕ}, 0 ≤ a → (n ≤ floor✝ a ↔ ↑n ≤ a)
gc_ceil✝ : GaloisConnection ceil✝ Nat.cast
this✝ : FloorSemiring.ceil = FloorSemiring.ceil
this : FloorSemiring.fl... | 6d1650047452147a |
EisensteinSeries.r1_aux_bound | Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean | /-- For `c, d ∈ ℝ` with `1 ≤ d ^ 2`, we have `r1 z ≤ |c * z + d| ^ 2`. -/
lemma r1_aux_bound (c : ℝ) {d : ℝ} (hd : 1 ≤ d ^ 2) :
r1 z ≤ (c * z.re + d) ^ 2 + (c * z.im) ^ 2 | z : ℍ
c d : ℝ
hd : 1 ≤ d ^ 2
⊢ r1 z ≤ (c * z.re + d) ^ 2 + (c * z.im) ^ 2 | have H1 : (c * z.re + d) ^ 2 + (c * z.im) ^ 2 =
c ^ 2 * (z.re ^ 2 + z.im ^ 2) + d * 2 * c * z.re + d ^ 2 := by ring | z : ℍ
c d : ℝ
hd : 1 ≤ d ^ 2
H1 : (c * z.re + d) ^ 2 + (c * z.im) ^ 2 = c ^ 2 * (z.re ^ 2 + z.im ^ 2) + d * 2 * c * z.re + d ^ 2
⊢ r1 z ≤ (c * z.re + d) ^ 2 + (c * z.im) ^ 2 | 9235d86a26f3640b |
Turing.ToPartrec.code_is_ok | Mathlib/Computability/TMConfig.lean | theorem code_is_ok (c) : Code.Ok c | case cons.e_a.h
f fs : Code
IHf : f.Ok
IHfs : fs.Ok
k : Cont
v✝ v : List ℕ
⊢ eval step ?m.141342 = do
let x ← fs.eval v✝
eval step (Cfg.ret k (v.headI :: x))
case cons.e_a.h
f fs : Code
IHf : f.Ok
IHfs : fs.Ok
k : Cont
v✝ v : List ℕ
⊢ Reaches step (Cfg.ret (Cont.cons₁ fs v✝ k) v) ?m.141342
f fs : Code
IHf : f... | swap | case cons.e_a.h
f fs : Code
IHf : f.Ok
IHfs : fs.Ok
k : Cont
v✝ v : List ℕ
⊢ Reaches step (Cfg.ret (Cont.cons₁ fs v✝ k) v) ?m.141342
case cons.e_a.h
f fs : Code
IHf : f.Ok
IHfs : fs.Ok
k : Cont
v✝ v : List ℕ
⊢ eval step ?m.141342 = do
let x ← fs.eval v✝
eval step (Cfg.ret k (v.headI :: x))
f fs : Code
IHf : f... | 1263cc60b426aeed |
IsExposed.eq_inter_halfSpace | Mathlib/Analysis/Convex/Exposed.lean | theorem eq_inter_halfSpace [Nontrivial 𝕜] {A B : Set E} (hAB : IsExposed 𝕜 A B) :
∃ l : E →L[𝕜] 𝕜, ∃ a, B = { x ∈ A | a ≤ l x } | case inl
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : OrderedRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : TopologicalSpace E
inst✝¹ : Module 𝕜 E
inst✝ : Nontrivial 𝕜
A : Set E
hAB : IsExposed 𝕜 A ∅
⊢ ∅ = {x | x ∈ A ∧ 1 ≤ 0 x} | rw [eq_comm, eq_empty_iff_forall_not_mem] | case inl
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : OrderedRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : TopologicalSpace E
inst✝¹ : Module 𝕜 E
inst✝ : Nontrivial 𝕜
A : Set E
hAB : IsExposed 𝕜 A ∅
⊢ ∀ (x : E), x ∉ {x | x ∈ A ∧ 1 ≤ 0 x} | 8f4e1e362f92e274 |
Algebra.Extension.CotangentSpace.map_comp | Mathlib/RingTheory/Kaehler/CotangentComplex.lean | lemma map_comp (f : Hom P P') (g : Hom P' P'') :
CotangentSpace.map (g.comp f) =
(CotangentSpace.map g).restrictScalars S ∘ₗ CotangentSpace.map f | case h.tmul
R : Type u
S : Type v
inst✝²² : CommRing R
inst✝²¹ : CommRing S
inst✝²⁰ : Algebra R S
P : Extension R S
R' : Type u'
S' : Type v'
inst✝¹⁹ : CommRing R'
inst✝¹⁸ : CommRing S'
inst✝¹⁷ : Algebra R' S'
P' : Extension R' S'
inst✝¹⁶ : Algebra R R'
inst✝¹⁵ : Algebra S S'
inst✝¹⁴ : Algebra R S'
inst✝¹³ : IsScalarTo... | obtain ⟨y, rfl⟩ := KaehlerDifferential.tensorProductTo_surjective _ _ y | case h.tmul.intro
R : Type u
S : Type v
inst✝²² : CommRing R
inst✝²¹ : CommRing S
inst✝²⁰ : Algebra R S
P : Extension R S
R' : Type u'
S' : Type v'
inst✝¹⁹ : CommRing R'
inst✝¹⁸ : CommRing S'
inst✝¹⁷ : Algebra R' S'
P' : Extension R' S'
inst✝¹⁶ : Algebra R R'
inst✝¹⁵ : Algebra S S'
inst✝¹⁴ : Algebra R S'
inst✝¹³ : IsSc... | 36b1f3e0f0214876 |
Ideal.isPrime_ideal_prod_top | Mathlib/RingTheory/Ideal/Prod.lean | theorem isPrime_ideal_prod_top {I : Ideal R} [h : I.IsPrime] : (prod I (⊤ : Ideal S)).IsPrime | case mem_or_mem'.mk.mk.intro
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
I : Ideal R
h : I.IsPrime
r₁ : R
s₁ : S
r₂ : R
s₂ : S
h₁ : (r₁, s₁) * (r₂, s₂) ∈ ↑(comap (RingHom.fst R S) I)
right✝ : (r₁, s₁) * (r₂, s₂) ∈ ↑(comap (RingHom.snd R S) ⊤)
⊢ (r₁, s₁) ∈ I.prod ⊤ ∨ (r₂, s₂) ∈ I.prod ⊤ | rcases h.mem_or_mem h₁ with h | h | case mem_or_mem'.mk.mk.intro.inl
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
I : Ideal R
h✝ : I.IsPrime
r₁ : R
s₁ : S
r₂ : R
s₂ : S
h₁ : (r₁, s₁) * (r₂, s₂) ∈ ↑(comap (RingHom.fst R S) I)
right✝ : (r₁, s₁) * (r₂, s₂) ∈ ↑(comap (RingHom.snd R S) ⊤)
h : (r₁, s₁).1 ∈ I
⊢ (r₁, s₁) ∈ I.prod ⊤ ∨ (r₂, s₂) ∈ I... | e99515d75b39b235 |
CategoryTheory.NonPreadditiveAbelian.σ_comp | Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | theorem σ_comp {X Y : C} (f : X ⟶ Y) : σ ≫ f = Limits.prod.map f f ≫ σ | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
f : X ⟶ Y
g : (CokernelCofork.ofπ σ ⋯).pt ⟶ Y
hg : Cofork.π (CokernelCofork.ofπ σ ⋯) ≫ g = prod.map f f ≫ σ
⊢ prod.lift (𝟙 X) 0 ≫ σ ≫ g = g | rw [← Category.assoc, lift_σ, Category.id_comp] | no goals | b7d15c6a05369a0f |
CategoryTheory.IsCofilteredOrEmpty.of_exists_of_isCofiltered_of_fullyFaithful | Mathlib/CategoryTheory/Filtered/Final.lean | theorem IsCofilteredOrEmpty.of_exists_of_isCofiltered_of_fullyFaithful [IsCofilteredOrEmpty D]
[F.Full] [F.Faithful] (h : ∀ d, ∃ c, Nonempty (F.obj c ⟶ d)) : IsCofilteredOrEmpty C | case intro.intro
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝² : IsCofilteredOrEmpty D
inst✝¹ : F.Full
inst✝ : F.Faithful
h : ∀ (d : D), ∃ c, Nonempty (F.obj c ⟶ d)
d : Dᵒᵖ
c : C
f : F.obj c ⟶ unop d
⊢ ∃ c, Nonempty (d ⟶ F.op.obj c) | exact ⟨op c, ⟨f.op⟩⟩ | no goals | a95d8844f0cff275 |
Finset.ofColex_ne_ofColex | Mathlib/Combinatorics/Colex.lean | lemma ofColex_ne_ofColex {s t : Colex α} : ofColex s ≠ ofColex t ↔ s ≠ t | α : Type u_1
s t : Colex α
⊢ s.ofColex ≠ t.ofColex ↔ s ≠ t | simp | no goals | 4a3884f9aa9180f8 |
Nat.mod_def | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Div/Basic.lean | theorem mod_def (m k : Nat) : m % k = m - k * (m / k) | m k : Nat
⊢ m % k = m - k * (m / k) | rw [Nat.sub_eq_of_eq_add] | m k : Nat
⊢ m = m % k + k * (m / k) | f10f5dd1b2641e77 |
PrimeSpectrum.existsUnique_idempotent_basicOpen_eq_of_isClopen | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | @[stacks 00EE]
lemma existsUnique_idempotent_basicOpen_eq_of_isClopen {s : Set (PrimeSpectrum R)}
(hs : IsClopen s) : ∃! e : R, IsIdempotentElem e ∧ s = basicOpen e | case refine_1.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : ... | refine ⟨x, ?_, subset_antisymm ?_ ?_⟩ | case refine_1.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤... | f72a236e53847d1c |
BitVec.DivModState.umod_eq_of_lawful | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean | theorem DivModState.umod_eq_of_lawful {qr : DivModState w}
(h : DivModState.Lawful {n, d} qr)
(h_final : qr.wn = 0) :
n % d = qr.r | w : Nat
n d : BitVec w
qr : DivModState w
h : Lawful { n := n, d := d } qr
h_final✝ : qr.wn = 0
h_final : True
hdiv : n.toNat >>> 0 = d.toNat * qr.q.toNat + qr.r.toNat
⊢ d.toNat * BitVec.toNat ?m.63092 + qr.r.toNat = n.toNat
w : Nat
n d : BitVec w
qr : DivModState w
h : Lawful { n := n, d := d } qr
h_final : qr.wn = 0... | exact hdiv.symm | no goals | 0b594396734385e1 |
Multiset.sub_add_eq_sub_sub | Mathlib/Data/Multiset/AddSub.lean | protected lemma sub_add_eq_sub_sub : s - (t + u) = s - t - u | case a
α : Type u_1
inst✝ : DecidableEq α
s t u : Multiset α
a✝ : α
⊢ count a✝ (s - (t + u)) = count a✝ (s - t - u) | simp [Nat.sub_add_eq] | no goals | f6dd2d36c336bc75 |
MulAction.movedBy_mem_fixedBy_of_commute | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | theorem movedBy_mem_fixedBy_of_commute {g h : G} (comm : Commute g h) :
(fixedBy α g)ᶜ ∈ fixedBy (Set α) h | α : Type u_1
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
g h : G
comm : Commute g h
⊢ (fixedBy α g)ᶜ ∈ fixedBy (Set α) h | rw [mem_fixedBy, Set.smul_set_compl, fixedBy_mem_fixedBy_of_commute comm] | no goals | 94f70af15c35631c |
Ideal.Quotient.index_eq_zero | Mathlib/Algebra/CharP/Quotient.lean | theorem Ideal.Quotient.index_eq_zero {R : Type*} [CommRing R] (I : Ideal R) :
(↑I.toAddSubgroup.index : R ⧸ I) = 0 | R : Type u_1
inst✝ : CommRing R
I : Ideal R
⊢ ↑(if x : Finite (R ⧸ Submodule.toAddSubgroup I) then Fintype.card (R ⧸ Submodule.toAddSubgroup I) else 0) = 0 | split_ifs with hq | case pos
R : Type u_1
inst✝ : CommRing R
I : Ideal R
hq : Finite (R ⧸ Submodule.toAddSubgroup I)
⊢ ↑(Fintype.card (R ⧸ Submodule.toAddSubgroup I)) = 0
case neg
R : Type u_1
inst✝ : CommRing R
I : Ideal R
hq : ¬Finite (R ⧸ Submodule.toAddSubgroup I)
⊢ ↑0 = 0 | 64a79a192da7e338 |
Dioph.ex_dioph | Mathlib/NumberTheory/Dioph.lean | theorem ex_dioph {S : Set (α ⊕ β → ℕ)} : Dioph S → Dioph {v | ∃ x, v ⊗ x ∈ S}
| ⟨γ, p, pe⟩ =>
⟨β ⊕ γ, p.map ((inl ⊗ inr ∘ inl) ⊗ inr ∘ inr), fun v =>
⟨fun ⟨x, hx⟩ =>
let ⟨t, ht⟩ := (pe _).1 hx
⟨x ⊗ t, by
simp; rw [show (v ⊗ x ⊗ t) ∘ ((inl ⊗ inr ∘ inl) ⊗ inr ∘ inr) = (v ⊗ x) ⊗ t fro... | case inr
α β : Type u
S : Set (α ⊕ β → ℕ)
γ : Type u
p : Poly ((α ⊕ β) ⊕ γ)
pe : ∀ (v : α ⊕ β → ℕ), S v ↔ ∃ t, p (v ⊗ t) = 0
v : α → ℕ
x✝ : {v | ∃ x, v ⊗ x ∈ S} v
x : β → ℕ
hx : v ⊗ x ∈ S
t : γ → ℕ
ht : p ((v ⊗ x) ⊗ t) = 0
b : γ
⊢ ((v ⊗ x ⊗ t) ∘ ((inl ⊗ inr ∘ inl) ⊗ inr ∘ inr)) (inr b) = ((v ⊗ x) ⊗ t) (inr b) | rfl | no goals | c388e394bd185bab |
t2Space_of_properSMul_of_t2Group | Mathlib/Topology/Algebra/ProperAction/Basic.lean | theorem t2Space_of_properSMul_of_t2Group [h_proper : ProperSMul G X] [T2Space G] : T2Space X | case refine_2
G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace X
h_proper : ProperSMul G X
inst✝ : T2Space G
f : X → G × X := fun x => (1, x)
this : range f = {1} ×ˢ univ
⊢ IsClosed (range f) | rw [this] | case refine_2
G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace X
h_proper : ProperSMul G X
inst✝ : T2Space G
f : X → G × X := fun x => (1, x)
this : range f = {1} ×ˢ univ
⊢ IsClosed ({1} ×ˢ univ) | e99ce87ed540aaac |
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 shiftRight
w idx m✝ n✝ : Nat
lhs : BVExpr m✝
rhs : BVExpr n✝
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... | have := (bitblast.go aig lhs).property | case shiftRight
w idx m✝ n✝ : Nat
lhs : BVExpr m✝
rhs : BVExpr n✝
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... | f92988d5a1595b39 |
ProbabilityTheory.iteratedDeriv_two_cgf | Mathlib/Probability/Moments/MGFAnalytic.lean | lemma iteratedDeriv_two_cgf (h : v ∈ interior (integrableExpSet X μ)) :
iteratedDeriv 2 (cgf X μ) v
= μ[fun ω ↦ (X ω)^2 * exp (v * X ω)] / mgf X μ v - deriv (cgf X μ) v ^ 2 | case e_a
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : ¬μ = 0
h_mem : ∀ᶠ (y : ℝ) in 𝓝 v, y ∈ interior (integrableExpSet X μ)
h_d_cgf : deriv (cgf X μ) =ᶠ[𝓝 v] fun u => (∫ (x : Ω), (fun ω => X ω * rexp (u * X ω)) x ∂μ) / mgf X μ u
h_d_mgf : deriv (mgf X μ... | exact (mgf_pos' hμ (interior_subset (s := integrableExpSet X μ) h)).ne' | no goals | 24002d4e2edb89d3 |
MeasureTheory.upcrossings_lt_top_iff | Mathlib/Probability/Martingale/Upcrossing.lean | theorem upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossingsBefore a b f N ω ≤ k | Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
this : upcrossings a b f ω < ⊤ ↔ ∃ k, upcrossings a b f ω ≤ ↑k
⊢ (∃ k, ∀ (i : ℕ), ↑(upcrossingsBefore a b f i ω) ≤ ↑k) ↔ ∃ k, ∀ (N : ℕ), upcrossingsBefore a b f N ω ≤ k | constructor <;> rintro ⟨k, hk⟩ | case mp.intro
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
this : upcrossings a b f ω < ⊤ ↔ ∃ k, upcrossings a b f ω ≤ ↑k
k : ℝ≥0
hk : ∀ (i : ℕ), ↑(upcrossingsBefore a b f i ω) ≤ ↑k
⊢ ∃ k, ∀ (N : ℕ), upcrossingsBefore a b f N ω ≤ k
case mpr.intro
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
ω : Ω
this : upcrossings a b f ω < ⊤ ↔ ∃ ... | ad04e390c118cf1b |
LiouvilleWith.add_rat | Mathlib/NumberTheory/Transcendental/Liouville/LiouvilleWith.lean | theorem add_rat (h : LiouvilleWith p x) (r : ℚ) : LiouvilleWith p (x + r) | case intro.intro.intro.intro.intro
p x : ℝ
h : LiouvilleWith p x
r : ℚ
C : ℝ
_hC₀ : 0 < C
hC : ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p
n : ℕ
hn : 1 ≤ n
m : ℤ
hne : x ≠ ↑m / ↑n
hlt : |x - ↑m / ↑n| < C / ↑n ^ p
⊢ ∃ m, x + ↑r ≠ ↑m / ↑(r.den • id n) ∧ |x + ↑r - ↑m / ↑(r.den • id n)| < ↑r.... | have : (↑(r.den * m + r.num * n : ℤ) / ↑(r.den • id n) : ℝ) = m / n + r := by
rw [Algebra.id.smul_eq_mul, id]
nth_rewrite 4 [← Rat.num_div_den r]
push_cast
rw [add_div, mul_div_mul_left _ _ (by positivity), mul_div_mul_right _ _ (by positivity)] | case intro.intro.intro.intro.intro
p x : ℝ
h : LiouvilleWith p x
r : ℚ
C : ℝ
_hC₀ : 0 < C
hC : ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p
n : ℕ
hn : 1 ≤ n
m : ℤ
hne : x ≠ ↑m / ↑n
hlt : |x - ↑m / ↑n| < C / ↑n ^ p
this : ↑(↑r.den * m + r.num * ↑n) / ↑(r.den • id n) = ↑m / ↑n + ↑r
⊢ ∃ m, x ... | 7652465b354d5b90 |
NNRat.addSubmonoid_closure_range_pow | Mathlib/Data/Rat/Star.lean | @[simp] lemma addSubmonoid_closure_range_pow {n : ℕ} (hn₀ : n ≠ 0) :
closure (range fun x : ℚ≥0 ↦ x ^ n) = ⊤ | n : ℕ
hn₀ : n ≠ 0
x : ℚ≥0
⊢ x = (x.num * x.den ^ (n - 1)) • (↑x.den)⁻¹ ^ n | rw [nsmul_eq_mul] | n : ℕ
hn₀ : n ≠ 0
x : ℚ≥0
⊢ x = ↑(x.num * x.den ^ (n - 1)) * (↑x.den)⁻¹ ^ n | 7fc9ce8a3377d839 |
Bimod.TensorBimod.left_assoc' | Mathlib/CategoryTheory/Monoidal/Bimod.lean | theorem left_assoc' :
(R.mul ▷ _) ≫ actLeft P Q = (α_ R.X R.X _).hom ≫ (R.X ◁ actLeft P Q) ≫ actLeft P Q | C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (α_ (R.X ⊗ R.X) P.X Q.X).inv ≫
((α_ R.X R.X P.X).hom ▷ Q.X ≫ (R.X ◁ P.actLeft) ▷ Q.X ≫ P.act... | monoidal | no goals | bad1b93937126f5d |
ENNReal.lintegral_mul_le_one_of_lintegral_rpow_eq_one | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | theorem lintegral_mul_le_one_of_lintegral_rpow_eq_one {p q : ℝ} (hpq : p.IsConjExponent q)
{f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf_norm : ∫⁻ a, f a ^ p ∂μ = 1)
(hg_norm : ∫⁻ a, g a ^ q ∂μ = 1) : (∫⁻ a, (f * g) a ∂μ) ≤ 1 | case hf
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.IsConjExponent q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hf_norm : ∫⁻ (a : α), f a ^ p ∂μ = 1
hg_norm : ∫⁻ (a : α), g a ^ q ∂μ = 1
⊢ AEMeasurable (fun a => f a ^ p * (ENNReal.ofReal p)⁻¹) μ | exact (hf.pow_const _).mul_const _ | no goals | bea204bbfdd8c8d8 |
Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin | Mathlib/RingTheory/Adjoin/Basic.lean | theorem pow_smul_mem_of_smul_subset_of_mem_adjoin [CommSemiring B] [Algebra R B] [Algebra A B]
[IsScalarTower R A B] (r : A) (s : Set B) (B' : Subalgebra R B) (hs : r • s ⊆ B') {x : B}
(hx : x ∈ adjoin R s) (hr : algebraMap A B r ∈ B') : ∃ n₀ : ℕ, ∀ n ≥ n₀, r ^ n • x ∈ B' | R : Type uR
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : CommSemiring B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
r : A
s : Set B
B' : Subalgebra R B
hs : r • s ⊆ ↑B'
hr : (algebraMap A B) r ∈ B'
l : ↑↑(Submonoid.closure s) →₀ R
n₁ : ↥... | apply Submonoid.closure_mono hs (n₂ a) | no goals | c0ccc6b7c63e5bab |
Array.flatMap_mkArray | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem flatMap_mkArray {β} (f : α → Array β) : (mkArray n a).flatMap f = (mkArray n (f a)).flatten | α : Type u_1
n : Nat
a : α
β : Type u_2
f : α → Array β
⊢ flatMap f (mkArray n a) = (mkArray n (f a)).flatten | rw [← toList_inj] | α : Type u_1
n : Nat
a : α
β : Type u_2
f : α → Array β
⊢ (flatMap f (mkArray n a)).toList = (mkArray n (f a)).flatten.toList | 3ca0b1f8e526e2cc |
List.Chain.rel | Mathlib/Data/List/Chain.lean | theorem Chain.rel [IsTrans α R] (hl : l.Chain R a) (hb : b ∈ l) : R a b | α : Type u
R : α → α → Prop
l : List α
a b : α
inst✝ : IsTrans α R
hl : Pairwise R (a :: l)
hb : b ∈ l
⊢ R a b | exact rel_of_pairwise_cons hl hb | no goals | 72301b273c3b93a5 |
Matrix.Pivot.exists_list_transvec_mul_mul_list_transvec_eq_diagonal_induction | Mathlib/LinearAlgebra/Matrix/Transvection.lean | theorem exists_list_transvec_mul_mul_list_transvec_eq_diagonal_induction
(IH :
∀ M : Matrix (Fin r) (Fin r) 𝕜,
∃ (L₀ L₀' : List (TransvectionStruct (Fin r) 𝕜)) (D₀ : Fin r → 𝕜),
(L₀.map toMatrix).prod * M * (L₀'.map toMatrix).prod = diagonal D₀)
(M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Uni... | 𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
IH :
∀ (M : Matrix (Fin r) (Fin r) 𝕜),
∃ L₀ L₀' D₀, (List.map toMatrix L₀).prod * M * (List.map toMatrix L₀').prod = diagonal D₀
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
L₁ L₁' : List (TransvectionStruct (Fin r ⊕ Unit) 𝕜)
hM : ((List.map toMatrix L₁).prod * M * (List.map ... | simpa [M', c, Matrix.mul_assoc] | no goals | 1db945989babbcf8 |
Polynomial.coeff_divByMonic_X_sub_C | Mathlib/Algebra/Polynomial/Div.lean | theorem coeff_divByMonic_X_sub_C (p : R[X]) (a : R) (n : ℕ) :
(p /ₘ (X - C a)).coeff n = ∑ i ∈ Icc (n + 1) p.natDegree, a ^ (i - (n + 1)) * p.coeff i | case inr
R : Type u
inst✝ : Ring R
p : R[X]
a : R
n : ℕ
this :
∀ (n : ℕ), p.natDegree ≤ n → (p /ₘ (X - C a)).coeff n = ∑ i ∈ Icc (n + 1) p.natDegree, a ^ (i - (n + 1)) * p.coeff i
h : ¬p.natDegree ≤ n
⊢ (p /ₘ (X - C a)).coeff n = ∑ i ∈ Icc (n + 1) p.natDegree, a ^ (i - (n + 1)) * p.coeff i | refine Nat.decreasingInduction' (fun n hn _ ih ↦ ?_) (le_of_not_le h) ?_ | case inr.refine_1
R : Type u
inst✝ : Ring R
p : R[X]
a : R
n✝ : ℕ
this :
∀ (n : ℕ), p.natDegree ≤ n → (p /ₘ (X - C a)).coeff n = ∑ i ∈ Icc (n + 1) p.natDegree, a ^ (i - (n + 1)) * p.coeff i
h : ¬p.natDegree ≤ n✝
n : ℕ
hn : n < p.natDegree
x✝ : n✝ ≤ n
ih : (p /ₘ (X - C a)).coeff (n + 1) = ∑ i ∈ Icc (n + 1 + 1) p.natDe... | b5156308e39e2482 |
Ordnode.Valid'.map_aux | Mathlib/Data/Ordmap/Ordset.lean | theorem Valid'.map_aux {β} [Preorder β] {f : α → β} (f_strict_mono : StrictMono f) {t a₁ a₂}
(h : Valid' a₁ t a₂) :
Valid' (Option.map f a₁) (map f t) (Option.map f a₂) ∧ (map f t).size = t.size | case node.intro.intro.sz.left
α : Type u_1
inst✝¹ : Preorder α
β : Type u_2
inst✝ : Preorder β
f : α → β
f_strict_mono : StrictMono f
size✝ : ℕ
l✝ : Ordnode α
x✝ : α
r✝ : Ordnode α
a₁ : WithBot α
a₂ : WithTop α
h : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂
t_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ... | rw [t_l_size, t_r_size] | case node.intro.intro.sz.left
α : Type u_1
inst✝¹ : Preorder α
β : Type u_2
inst✝ : Preorder β
f : α → β
f_strict_mono : StrictMono f
size✝ : ℕ
l✝ : Ordnode α
x✝ : α
r✝ : Ordnode α
a₁ : WithBot α
a₂ : WithTop α
h : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂
t_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option.map f ... | ff36f5e3d5092df5 |
PiTensorProduct.lifts_add | Mathlib/LinearAlgebra/PiTensorProduct.lean | /-- If elements `p,q` of `FreeAddMonoid (R × Π i, s i)` lift elements `x,y` of `⨂[R] i, s i`
respectively, then `p + q` lifts `x + y`.
-/
lemma lifts_add {x y : ⨂[R] i, s i} {p q : FreeAddMonoid (R × Π i, s i)}
(hp : p ∈ lifts x) (hq : q ∈ lifts y) : p + q ∈ lifts (x + y) | ι : Type u_1
R : Type u_4
inst✝² : CommSemiring R
s : ι → Type u_7
inst✝¹ : (i : ι) → AddCommMonoid (s i)
inst✝ : (i : ι) → Module R (s i)
x y : ⨂[R] (i : ι), s i
p q : FreeAddMonoid (R × ((i : ι) → s i))
hp : p ∈ x.lifts
hq : q ∈ y.lifts
⊢ ↑p + ↑q = x + y | rw [hp, hq] | no goals | d4629aec80161898 |
FreeAlgebra.adjoin_range_ι | Mathlib/Algebra/FreeAlgebra.lean | theorem adjoin_range_ι : Algebra.adjoin R (Set.range (ι R : X → FreeAlgebra R X)) = ⊤ | case h_add
R : Type u_1
inst✝ : CommSemiring R
X : Type u_2
S : Subalgebra R (FreeAlgebra R X) := Algebra.adjoin R (Set.range (ι R))
x y : FreeAlgebra R X
hx : x ∈ S
hy : y ∈ S
⊢ x + y ∈ S | exact S.add_mem hx hy | no goals | cec84737beae3660 |
AffineSubspace.direction_mk' | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace/Defs.lean | theorem direction_mk' (p : P) (direction : Submodule k V) :
(mk' p direction).direction = direction | case h.mp.intro.intro.intro.intro.intro.intro.intro.intro
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : P
direction : Submodule k V
v : V
p₁ : P
v₁ : V
hv₁ : v₁ ∈ direction
hp₁ : p₁ = v₁ +ᵥ p
p₂ : P
hv : v = p₁ -ᵥ p₂
v₂ : V
hv₂ : v₂ ∈ dire... | rw [hv, hp₁, hp₂, vadd_vsub_vadd_cancel_right] | case h.mp.intro.intro.intro.intro.intro.intro.intro.intro
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : P
direction : Submodule k V
v : V
p₁ : P
v₁ : V
hv₁ : v₁ ∈ direction
hp₁ : p₁ = v₁ +ᵥ p
p₂ : P
hv : v = p₁ -ᵥ p₂
v₂ : V
hv₂ : v₂ ∈ dire... | 4871a469895a4a3c |
zpow_right_anti₀ | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | lemma zpow_right_anti₀ (ha₀ : 0 < a) (ha₁ : a ≤ 1) : Antitone fun n : ℤ ↦ a ^ n | G₀ : Type u_2
inst✝⁴ : GroupWithZero G₀
inst✝³ : PartialOrder G₀
inst✝² : ZeroLEOneClass G₀
inst✝¹ : PosMulReflectLT G₀
a : G₀
inst✝ : PosMulMono G₀
ha₀ : 0 < a
ha₁ : a ≤ 1
n : ℤ
⊢ a ^ n * a ≤ a ^ n | exact mul_le_of_le_one_right (zpow_nonneg ha₀.le _) ha₁ | no goals | 0b4c6dc9e11e8a54 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.confirmRupHint_preserves_invariant_helper | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem confirmRupHint_preserves_invariant_helper {n : Nat} (f : DefaultFormula n)
(f_assignments_size : f.assignments.size = n)
(acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool) (hsize : acc.1.size = n)
(l : Literal (PosFin n)) (ih : DerivedLitsInvariant f f_assignments_size acc.1 hsize acc.2.... | case isTrue
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.... | all_goals
simp +decide [getElem!, l_eq_i, i_in_bounds, h1, decidableGetElem?] at h | no goals | 02ca1706b397b254 |
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
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : maximalIdeal R = ⊥
⊢ Submodule.IsPrincipal ⊥ | infer_instance | no goals | e801db6f819d1797 |
SimpleGraph.triangle_split_helper | Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean | /-- A subset of the triangles constructed in a weird way to make them easy to count. -/
private lemma triangle_split_helper [DecidableEq α] :
(s \ (badVertices G ε s t ∪ badVertices G ε s u)).biUnion
(fun x ↦ (G.interedges {y ∈ t | G.Adj x y} {y ∈ u | G.Adj x y}).image (x, ·)) ⊆
(s ×ˢ t ×ˢ u).filter (fu... | case mk.mk
α : Type u_1
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
s t u : Finset α
inst✝ : DecidableEq α
x : α
hx : x ∈ s
y z : α
hy : y ∈ t
xy : G.Adj x y
hz : z ∈ u
xz : G.Adj x z
yz : G.Adj y z
⊢ x ∈ s ∧ y ∈ t ∧ z ∈ u ∧ G.Adj x y ∧ G.Adj x z ∧ G.Adj y z | exact ⟨hx, hy, hz, xy, xz, yz⟩ | no goals | a4679bcedf9587ce |
ProfiniteGrp.denseRange_toLimit | Mathlib/Topology/Algebra/Category/ProfiniteGrp/Limits.lean | theorem denseRange_toLimit (P : ProfiniteGrp.{u}) : DenseRange (toLimit P) | case h
P : ProfiniteGrp.{u}
U : Set ↑(limit (P.toFiniteQuotientFunctor ⋙ forget₂ FiniteGrp.{u} ProfiniteGrp.{u})).toProfinite.toTop
s :
Set
((j : OpenNormalSubgroup ↑P.toProfinite.toTop) →
↑((P.toFiniteQuotientFunctor ⋙ forget₂ FiniteGrp.{u} ProfiniteGrp.{u}).obj j).toProfinite.toTop)
hsO : IsOpen s
hsv : S... | apply hJ2 | case h.a
P : ProfiniteGrp.{u}
U : Set ↑(limit (P.toFiniteQuotientFunctor ⋙ forget₂ FiniteGrp.{u} ProfiniteGrp.{u})).toProfinite.toTop
s :
Set
((j : OpenNormalSubgroup ↑P.toProfinite.toTop) →
↑((P.toFiniteQuotientFunctor ⋙ forget₂ FiniteGrp.{u} ProfiniteGrp.{u}).obj j).toProfinite.toTop)
hsO : IsOpen s
hsv :... | 8969731dd4e83c84 |
ProbabilityTheory.Kernel.compProd_eq_sum_compProd_right | Mathlib/Probability/Kernel/Composition/CompProd.lean | theorem compProd_eq_sum_compProd_right (κ : Kernel α β) (η : Kernel (α × β) γ)
[IsSFiniteKernel η] : κ ⊗ₖ η = Kernel.sum fun n => κ ⊗ₖ seq η n | case pos
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
hκ : IsSFiniteKernel κ
⊢ κ ⊗ₖ η = Kernel.sum fun n => κ ⊗ₖ η.seq n
case neg
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ... | swap | case neg
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
hκ : ¬IsSFiniteKernel κ
⊢ κ ⊗ₖ η = Kernel.sum fun n => κ ⊗ₖ η.seq n
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
m... | 95e47d4c49db6f31 |
Array.mapIdx_set | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/MapIdx.lean | theorem mapIdx_set {l : Array α} {i : Nat} {h : i < l.size} {a : α} :
(l.set i a).mapIdx f = (l.mapIdx f).set i (f i a) (by simpa) | case mk
α : Type u_1
α✝ : Type u_2
f : Nat → α → α✝
i : Nat
a : α
l : List α
h : i < { toList := l }.size
⊢ mapIdx f ({ toList := l }.set i a h) = (mapIdx f { toList := l }).set i (f i a) ⋯ | simp [List.mapIdx_set] | no goals | 6b94a1f6965c8a48 |
hasFPowerSeriesAt_iff | Mathlib/Analysis/Analytic/Basic.lean | theorem hasFPowerSeriesAt_iff :
HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) | case intro.intro.refine_1.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
h : ∀ ⦃y : 𝕜⦄, dist y 0 < r → HasSum (fun n => y ^ n • p.coeff n) (f (z₀ + y))
z : 𝕜
z_... | simp only [ENNReal.coe_pos] | case intro.intro.refine_1.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
r : ℝ
r_pos : r > 0
h : ∀ ⦃y : 𝕜⦄, dist y 0 < r → HasSum (fun n => y ^ n • p.coeff n) (f (z₀ + y))
z : 𝕜
z_... | 72d24d0576ad9ebc |
Finset.subset_set_biUnion_of_mem | Mathlib/Order/CompleteLattice/Finset.lean | theorem subset_set_biUnion_of_mem {s : Finset α} {f : α → Set β} {x : α} (h : x ∈ s) :
f x ⊆ ⋃ y ∈ s, f y :=
show f x ≤ ⨆ y ∈ s, f y from le_iSup_of_le x <| by simp only [h, iSup_pos, le_refl]
| α : Type u_2
β : Type u_3
s : Finset α
f : α → Set β
x : α
h : x ∈ s
⊢ f x ≤ ⨆ (_ : x ∈ s), f x | simp only [h, iSup_pos, le_refl] | no goals | 1b7e0d4a1b062cd4 |
IsCoprime.add_mul_left_right | Mathlib/RingTheory/Coprime/Basic.lean | theorem add_mul_left_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (y + x * z) | R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime x (y + x * z) | rw [isCoprime_comm] | R : Type u
inst✝ : CommRing R
x y : R
h : IsCoprime x y
z : R
⊢ IsCoprime (y + x * z) x | 8e203d872b104c31 |
CochainComplex.HomComplex.Cochain.rightUnshift_smul | Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean | @[simp]
lemma rightUnshift_smul {n' a : ℤ} (γ : Cochain K (L⟦a⟧) n') (n : ℤ) (hn : n' + a = n) (x : R) :
(x • γ).rightUnshift n hn = x • γ.rightUnshift n hn | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L : CochainComplex C ℤ
n' a : ℤ
γ : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n'
n : ℤ
hn : n' + a = n
x : R
⊢ (x • γ).rightUnshift n hn = x • γ.rightUnshift n hn | change (rightShiftLinearEquiv R K L n a n' hn).symm (x • γ) = _ | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L : CochainComplex C ℤ
n' a : ℤ
γ : Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n'
n : ℤ
hn : n' + a = n
x : R
⊢ (rightShiftLinearEquiv R K L n a n' hn).symm (x • γ) = x • γ.rightUns... | d2ca77d274bb6190 |
Real.tendsto_toNNReal_atTop_iff | Mathlib/Topology/Instances/NNReal/Lemmas.lean | theorem _root_.Real.tendsto_toNNReal_atTop_iff {l : Filter α} {f : α → ℝ} :
Tendsto (fun x ↦ (f x).toNNReal) l atTop ↔ Tendsto f l atTop | α : Type u_1
l : Filter α
f : α → ℝ
⊢ Tendsto (fun x => (f x).toNNReal) l atTop ↔ Tendsto f l atTop | rw [← Real.comap_toNNReal_atTop, tendsto_comap_iff, Function.comp_def] | no goals | b7641e2f4d27383b |
Ideal.natAbs_det_basis_change | Mathlib/RingTheory/Ideal/Norm/AbsNorm.lean | theorem natAbs_det_basis_change {ι : Type*} [Fintype ι] [DecidableEq ι] (b : Basis ι ℤ S)
(I : Ideal S) (bI : Basis ι ℤ I) : (b.det ((↑) ∘ bI)).natAbs = Ideal.absNorm I | S : Type u_1
inst✝⁶ : CommRing S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ S
I : Ideal S
bI : Basis ι ℤ ↥I
⊢ (b.det (Subtype.val ∘ ⇑bI)).natAbs = absNorm I | let e := b.equiv bI (Equiv.refl _) | S : Type u_1
inst✝⁶ : CommRing S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ S
I : Ideal S
bI : Basis ι ℤ ↥I
e : S ≃ₗ[ℤ] ↥I := b.equiv bI (Equiv.refl ι)
⊢ (b.det (Subtype.val ∘ ⇑bI)).natAbs = ab... | 0271342b99129e4f |
IsCoprime.prod_left_iff | Mathlib/RingTheory/Coprime/Lemmas.lean | theorem IsCoprime.prod_left_iff : IsCoprime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsCoprime (s i) x | R : Type u
I : Type v
inst✝ : CommSemiring R
x : R
s : I → R
t : Finset I
⊢ IsCoprime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsCoprime (s i) x | refine Finset.induction_on t (iff_of_true isCoprime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_ | R : Type u
I : Type v
inst✝ : CommSemiring R
x : R
s : I → R
t✝ : Finset I
b : I
t : Finset I
hbt : b ∉ t
ih : IsCoprime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsCoprime (s i) x
⊢ IsCoprime (∏ i ∈ insert b t, s i) x ↔ ∀ i ∈ insert b t, IsCoprime (s i) x | 9d3d4bcf53392746 |
Vector.map_eq_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem map_eq_iff {f : α → β} {l : Vector α n} {l' : Vector β n} :
map f l = l' ↔ ∀ i (h : i < n), l'[i] = f l[i] | case mk.mk.mp
α : Type u_1
β : Type u_2
f : α → β
l : Array α
l' : Array β
h' : l'.size = l.size
⊢ (∀ (i : Nat), l'[i]? = Option.map f l[i]?) → ∀ (i : Nat) (h : i < l.size), l'[i] = f l[i] | intro w i h | case mk.mk.mp
α : Type u_1
β : Type u_2
f : α → β
l : Array α
l' : Array β
h' : l'.size = l.size
w : ∀ (i : Nat), l'[i]? = Option.map f l[i]?
i : Nat
h : i < l.size
⊢ l'[i] = f l[i] | 9e0ae22d1d915b68 |
Batteries.RBNode.Ordered.unique | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | theorem Ordered.unique [@TransCmp α cmp] (ht : Ordered cmp t)
(hx : x ∈ t) (hy : y ∈ t) (e : cmp x y = .eq) : x = y | case node.inr.inr.inr.inl
α : Type u_1
cmp : α → α → Ordering
x y : α
inst✝ : TransCmp cmp
e : cmp x y = Ordering.eq
c✝ : RBColor
l : RBNode α
v✝ : α
r : RBNode α
ihl : Ordered cmp l → x ∈ l → y ∈ l → x = y
ihr : Ordered cmp r → x ∈ r → y ∈ r → x = y
ht : Ordered cmp (node c✝ l v✝ r)
lx : All (fun x => cmpLT cmp x v✝) ... | cases e.symm.trans <| OrientedCmp.cmp_eq_gt.2
((All_def.1 lx _ hy).trans (All_def.1 xr _ hx)).1 | no goals | 77450b4176612ab8 |
contractLeft_assoc_coevaluation' | Mathlib/LinearAlgebra/Coevaluation.lean | theorem contractLeft_assoc_coevaluation' :
(contractLeft K V).lTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).toLinearMap ∘ₗ (coevaluation K V).rTensor V =
(TensorProduct.rid K _).symm.toLinearMap ∘ₗ (TensorProduct.lid K _).toLinearMap | case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (LinearMap.lTensor V (contractLeft K V))
((TensorPro... | rw [lid_tmul, one_smul, rid_symm_apply] | case H.h
K : Type u
inst✝³ : Field K
V : Type v
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)
j : ↑(Basis.ofVectorSpaceIndex K V)
⊢ (LinearMap.lTensor V (contractLeft K V))
((TensorPro... | fa558da7fb2ef0c7 |
FermatLastTheoremForThreeGen.Solution.lambda_sq_div_u₅_mul | Mathlib/NumberTheory/FLT/Three.lean | private lemma lambda_sq_div_u₅_mul : λ ^ 2 ∣ S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 | K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
⊢ 3 * (S.multiplicity - 1) = 2 + (3 * S.multiplicity - 5) | have := S.two_le_multiplicity | K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
this : 2 ≤ S.multiplicity
⊢ 3 * (S.multiplicity - 1) = 2 + (3 * S.multiplicity - 5) | 49199d47aec8e219 |
LinearMap.IsSymmetric.iSup_eigenspace_inf_eigenspace_of_commute | Mathlib/Analysis/InnerProductSpace/JointEigenspace.lean | theorem iSup_eigenspace_inf_eigenspace_of_commute (hB : B.IsSymmetric) (hAB : Commute A B) :
(⨆ γ, eigenspace A α ⊓ eigenspace B γ) = eigenspace A α | case e_p
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
α : 𝕜
A B : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
hB : B.IsSymmetric
hAB : Commute A B
⊢ ⨆ i, (genEigenspace (B.restrict ⋯) i) 1 = ⊤ | simpa only [genEigenspace_eq_eigenspace, Submodule.orthogonal_eq_bot_iff]
using orthogonalComplement_iSup_eigenspaces_eq_bot <|
hB.restrict_invariant <| mapsTo_genEigenspace_of_comm hAB α 1 | no goals | 602f1bc013c8706e |
norm_mk_lt' | Mathlib/Analysis/Normed/Group/Quotient.lean | theorem norm_mk_lt' (S : AddSubgroup M) (m : M) {ε : ℝ} (hε : 0 < ε) :
∃ s ∈ S, ‖m + s‖ < ‖mk' S m‖ + ε | case intro.intro
M : Type u_1
inst✝ : SeminormedAddCommGroup M
S : AddSubgroup M
m : M
ε : ℝ
hε : 0 < ε
n : M
hn : -m + n ∈ S
hn' : ‖n‖ < ‖(mk' S) m‖ + ε
⊢ ∃ s ∈ S, ‖m + s‖ < ‖(mk' S) m‖ + ε | use -m + n, hn | case right
M : Type u_1
inst✝ : SeminormedAddCommGroup M
S : AddSubgroup M
m : M
ε : ℝ
hε : 0 < ε
n : M
hn : -m + n ∈ S
hn' : ‖n‖ < ‖(mk' S) m‖ + ε
⊢ ‖m + (-m + n)‖ < ‖(mk' S) m‖ + ε | 7e118629953e7ad0 |
PresentedMonoid.ext | Mathlib/Algebra/PresentedMonoid/Basic.lean | theorem ext {M : Type*} [Monoid M] (rels : FreeMonoid α → FreeMonoid α → Prop)
{φ ψ : PresentedMonoid rels →* M} (hx : ∀ (x : α), φ (.of rels x) = ψ (.of rels x)) :
φ = ψ | α : Type u_2
M : Type u_3
inst✝ : Monoid M
rels : FreeMonoid α → FreeMonoid α → Prop
φ ψ : PresentedMonoid rels →* M
hx : ∀ (x : α), φ (of rels x) = ψ (of rels x)
⊢ EqOn (⇑φ) (⇑ψ) (range (of rels)) | apply eqOn_range.mpr | α : Type u_2
M : Type u_3
inst✝ : Monoid M
rels : FreeMonoid α → FreeMonoid α → Prop
φ ψ : PresentedMonoid rels →* M
hx : ∀ (x : α), φ (of rels x) = ψ (of rels x)
⊢ ⇑φ ∘ of rels = ⇑ψ ∘ of rels | 4e13ad6c6062fd98 |
EuclideanGeometry.angle_const_sub | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | theorem angle_const_sub (v : V) (v₁ v₂ v₃ : V) : ∠ (v - v₁) (v - v₂) (v - v₃) = ∠ v₁ v₂ v₃ | V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
v v₁ v₂ v₃ : V
⊢ ∠ (v - v₁) (v - v₂) (v - v₃) = ∠ v₁ v₂ v₃ | simpa only [vsub_eq_sub] using angle_const_vsub v v₁ v₂ v₃ | no goals | fb1bdb894af22abd |
Nat.mul_pow | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean | theorem mul_pow (a b n : Nat) : (a * b) ^ n = a ^ n * b ^ n | a b n : Nat
⊢ (a * b) ^ n = a ^ n * b ^ n | induction n with
| zero => rw [Nat.pow_zero, Nat.pow_zero, Nat.pow_zero, Nat.mul_one]
| succ _ ih => rw [Nat.pow_succ, Nat.pow_succ, Nat.pow_succ, Nat.mul_mul_mul_comm, ih] | no goals | a47bdc1fc08287a8 |
MulAction.zpow_smul_mod_minimalPeriod | Mathlib/Dynamics/PeriodicPts/Defs.lean | theorem zpow_smul_mod_minimalPeriod (n : ℤ) :
a ^ (n % (minimalPeriod (a • ·) b : ℤ)) • b = a ^ n • b | α : Type v
G : Type u
inst✝¹ : Group G
inst✝ : MulAction G α
a : G
b : α
n : ℤ
⊢ a ^ (n % ↑(minimalPeriod (fun x => a • x) b)) • b = a ^ n • b | rw [← period_eq_minimalPeriod, zpow_mod_period_smul] | no goals | 4712540e5a9d4a97 |
comap_map_eq_map_of_isLocalization_algebraMapSubmonoid | Mathlib/RingTheory/Trace/Quotient.lean | lemma comap_map_eq_map_of_isLocalization_algebraMapSubmonoid :
(Ideal.map (algebraMap R Sₚ) p).comap (algebraMap S Sₚ) = pS | case intro
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
p : Ideal R
inst✝⁵ : p.IsMaximal
Sₚ : Type u_4
inst✝⁴ : CommRing Sₚ
inst✝³ : Algebra S Sₚ
inst✝² : Algebra R Sₚ
inst✝¹ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝ : IsScalarTower R S Sₚ
x : S
hx ... | refine ⟨β, β * α - 1, ?_, ?_⟩ | case intro.refine_1
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
p : Ideal R
inst✝⁵ : p.IsMaximal
Sₚ : Type u_4
inst✝⁴ : CommRing Sₚ
inst✝³ : Algebra S Sₚ
inst✝² : Algebra R Sₚ
inst✝¹ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝ : IsScalarTower R S Sₚ
... | d6f88c93445b65e1 |
Prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd | Mathlib/Algebra/Prime/Lemmas.lean | theorem Prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd [CancelCommMonoidWithZero M] {p a b : M}
{n : ℕ} (hp : Prime p) (hpow : p ^ n.succ ∣ a ^ n.succ * b ^ n) (hb : ¬p ^ 2 ∣ b) : p ∣ a | M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p a : M
n : ℕ
hp : Prime p
x : M
hb : ¬p ^ 2 ∣ p * x
hbdiv : p ∣ (p * x) ^ n
y : M
hy : a ^ n.succ * (p * x) ^ n = p ^ n.succ * y
⊢ p ^ n * (a ^ n.succ * x ^ n) = p ^ n * (p * y) | rw [← mul_assoc _ p, ← pow_succ, ← hy, mul_pow, ← mul_assoc (a ^ n.succ), mul_comm _ (p ^ n),
mul_assoc] | no goals | a203319e403717ea |
Tree.treesOfNumNodesEq_succ | Mathlib/Combinatorics/Enumerative/Catalan.lean | theorem treesOfNumNodesEq_succ (n : ℕ) :
treesOfNumNodesEq (n + 1) =
(antidiagonal n).biUnion fun ij =>
pairwiseNode (treesOfNumNodesEq ij.1) (treesOfNumNodesEq ij.2) | n : ℕ
⊢ ((antidiagonal n).attach.biUnion fun ijh => pairwiseNode (treesOfNumNodesEq (↑ijh).1) (treesOfNumNodesEq (↑ijh).2)) =
(antidiagonal n).biUnion fun ij => pairwiseNode (treesOfNumNodesEq ij.1) (treesOfNumNodesEq ij.2) | ext | case h
n : ℕ
a✝ : Tree Unit
⊢ (a✝ ∈
(antidiagonal n).attach.biUnion fun ijh =>
pairwiseNode (treesOfNumNodesEq (↑ijh).1) (treesOfNumNodesEq (↑ijh).2)) ↔
a✝ ∈ (antidiagonal n).biUnion fun ij => pairwiseNode (treesOfNumNodesEq ij.1) (treesOfNumNodesEq ij.2) | 8ad380b3ed743c12 |
CategoryTheory.IsVanKampenColimit.whiskerEquivalence | Mathlib/CategoryTheory/Limits/VanKampen.lean | theorem IsVanKampenColimit.whiskerEquivalence {K : Type*} [Category K] (e : J ≌ K)
{F : K ⥤ C} {c : Cocone F} (hc : IsVanKampenColimit c) :
IsVanKampenColimit (c.whisker e.functor) | case h.e'_2.a.mpr
J : Type v'
inst✝² : Category.{u', v'} J
C : Type u
inst✝¹ : Category.{v, u} C
K : Type u_3
inst✝ : Category.{u_4, u_3} K
e : J ≌ K
F : K ⥤ C
c : Cocone F
hc : IsVanKampenColimit c
F' : J ⥤ C
c' : Cocone F'
α : F' ⟶ e.functor ⋙ F
f : c'.pt ⟶ (Cocone.whisker e.functor c).pt
e' : α ≫ (Cocone.whisker e.f... | exact IsPullback.of_vert_isIso ⟨by simp⟩ | no goals | 04b6c653df528b06 |
AlgebraicTopology.DoldKan.PInfty_on_Γ₀_splitting_summand_eq_self | Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean | theorem PInfty_on_Γ₀_splitting_summand_eq_self (K : ChainComplex C ℕ) {n : ℕ} :
((Γ₀.splitting K).cofan _).inj (Splitting.IndexSet.id (op ⦋n⦌)) ≫
(PInfty : K[Γ₀.obj K] ⟶ _).f n =
((Γ₀.splitting K).cofan _).inj (Splitting.IndexSet.id (op ⦋n⦌)) | case succ
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
K : ChainComplex C ℕ
n : ℕ
⊢ ((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)) ≫ (P (n + 1)).f (n + 1) =
((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1... | exact (HigherFacesVanish.on_Γ₀_summand_id K n).comp_P_eq_self | no goals | fdf1ba0fb1743d80 |
Order.exists_series_of_le_height | Mathlib/Order/KrullDimension.lean | /-- There exist a series ending in a element for any length up to the element’s height. -/
lemma exists_series_of_le_height (a : α) {n : ℕ} (h : n ≤ height a) :
∃ p : LTSeries α, p.last = a ∧ p.length = n | α : Type u_1
inst✝ : Preorder α
n m : ℕ
p : LTSeries α
hne : Nonempty { p_1 // RelSeries.last p_1 = RelSeries.last p }
ha : ∀ (p_1 : LTSeries α), RelSeries.last p_1 = RelSeries.last p → p_1.length ≠ n
hp : p.length = m
hnm : n < m
⊢ (RelSeries.drop p ⟨m - n, ⋯⟩).last = RelSeries.last p | simp | no goals | 0e15cb2199a1ee03 |
hasFTaylorSeriesUpToOn_succ_iff_left | Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean | theorem hasFTaylorSeriesUpToOn_succ_iff_left {n : ℕ} :
HasFTaylorSeriesUpToOn (n + 1) f p s ↔
HasFTaylorSeriesUpToOn n f p s ∧
(∀ x ∈ s, HasFDerivWithinAt (fun y => p y n) (p x n.succ).curryLeft s x) ∧
ContinuousOn (fun x => p x (n + 1)) s | case neg
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
p : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
h :
HasFTaylorSeriesUpToOn (↑n) f p s ∧
(∀ x ∈ s, HasFDer... | exact h.2.2 | no goals | 5ea2ef581950d21b |
ascPochhammer_eval_neg_eq_descPochhammer | Mathlib/RingTheory/Polynomial/Pochhammer.lean | theorem ascPochhammer_eval_neg_eq_descPochhammer (r : R) : ∀ (k : ℕ),
(ascPochhammer R k).eval (-r) = (-1)^k * (descPochhammer R k).eval r
| 0 => by
rw [ascPochhammer_zero, descPochhammer_zero]
simp only [eval_one, pow_zero, mul_one]
| (k+1) => by
rw [ascPochhammer_succ_right, mul_add, eval_add, eva... | R : Type u
inst✝ : Ring R
r : R
⊢ eval (-r) (ascPochhammer R 0) = (-1) ^ 0 * eval r (descPochhammer R 0) | rw [ascPochhammer_zero, descPochhammer_zero] | R : Type u
inst✝ : Ring R
r : R
⊢ eval (-r) 1 = (-1) ^ 0 * eval r 1 | 91e878a9eb30688b |
MonovaryOn.sum_smul_comp_perm_le_sum_smul | Mathlib/Algebra/Order/Rearrangement.lean | theorem MonovaryOn.sum_smul_comp_perm_le_sum_smul (hfg : MonovaryOn f g s)
(hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i | case pos.inr
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝⁴ : LinearOrderedSemiring α
inst✝³ : ExistsAddOfLE α
inst✝² : LinearOrderedCancelAddCommMonoid β
inst✝¹ : Module α β
inst✝ : PosSMulMono α β
s✝ : Finset ι
f : ι → α
g : ι → β
a : ι
s : Finset ι
has : a ∉ s
hamax : ∀ x ∈ s, toLex (g x, f x) ≤ toLex (g a, f a)
hind... | exact mem_of_mem_insert_of_ne (hσ fun h ↦ hax <| h.symm.trans h₁) hax | no goals | 4fd7a15670de10aa |
MeasureTheory.lintegral_smul_measure | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem lintegral_smul_measure (c : ℝ≥0∞) (f : α → ℝ≥0∞) : ∫⁻ a, f a ∂c • μ = c * ∫⁻ a, f a ∂μ | α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
f : α → ℝ≥0∞
⊢ ∫⁻ (a : α), f a ∂c • μ = c * ∫⁻ (a : α), f a ∂μ | simp only [lintegral, iSup_subtype', SimpleFunc.lintegral_smul, ENNReal.mul_iSup, smul_eq_mul] | no goals | c16ad3405766c04f |
Order.le_succ_iff_eq_or_le | Mathlib/Order/SuccPred/Basic.lean | theorem le_succ_iff_eq_or_le : a ≤ succ b ↔ a = succ b ∨ a ≤ b | case pos
α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : SuccOrder α
a b : α
hb : IsMax b
⊢ a ≤ succ b ↔ a = succ b ∨ a ≤ b | rw [hb.succ_eq, or_iff_right_of_imp le_of_eq] | no goals | ca762971dc330b2f |
HasFPowerSeriesWithinOnBall.fderivWithin_of_mem_of_analyticOn | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | theorem HasFPowerSeriesWithinOnBall.fderivWithin_of_mem_of_analyticOn
(hr : HasFPowerSeriesWithinOnBall f p s x r)
(h : AnalyticOn 𝕜 f s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
HasFPowerSeriesWithinOnBall (fderivWithin 𝕜 f s) p.derivSeries s x r | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
hr : HasFPowerSeriesWithinOnBall f p s x r
h : AnalyticOn 𝕜 f s
hs... | simpa [edist_zero_eq_enorm] using h'y | no goals | d39a108ffb050c55 |
List.zipIdxLE_trans | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sort/Lemmas.lean | theorem zipIdxLE_trans (trans : ∀ a b c, le a b → le b c → le a c)
(a b c : α × Nat) : zipIdxLE le a b → zipIdxLE le b c → zipIdxLE le a c | case isTrue.isTrue.isTrue
α : Type u_1
le : α → α → Bool
trans : ∀ (a b c : α), le a b = true → le b c = true → le a c = true
a b c : α × Nat
ab₂ : le a.fst b.fst = true
ba₂ : le b.fst a.fst = true
bc₂ : le b.fst c.fst = true
ab₁ : a.snd ≤ b.snd
h : le c.fst b.fst = false ∨ b.snd ≤ c.snd
⊢ le c.fst a.fst = false ∨ a.sn... | rcases h with (cd₂ | bc₁) | case isTrue.isTrue.isTrue.inl
α : Type u_1
le : α → α → Bool
trans : ∀ (a b c : α), le a b = true → le b c = true → le a c = true
a b c : α × Nat
ab₂ : le a.fst b.fst = true
ba₂ : le b.fst a.fst = true
bc₂ : le b.fst c.fst = true
ab₁ : a.snd ≤ b.snd
cd₂ : le c.fst b.fst = false
⊢ le c.fst a.fst = false ∨ a.snd ≤ c.snd
... | 4b08ab1e530cfafb |
GroupWithZero.isOpen_singleton_zero | Mathlib/Topology/Algebra/Monoid.lean | lemma GroupWithZero.isOpen_singleton_zero [GroupWithZero M] [TopologicalSpace M]
[ContinuousMul M] [CompactSpace M] [T1Space M] :
IsOpen {(0 : M)} | case intro.intro.intro
M : Type u_3
inst✝⁴ : GroupWithZero M
inst✝³ : TopologicalSpace M
inst✝² : ContinuousMul M
inst✝¹ : CompactSpace M
inst✝ : T1Space M
U : Set M
hU : IsOpen U
h0U : 0 ∈ U
h1U : 1 ∉ U
⊢ IsOpen {0} | obtain ⟨W, hW, hW'⟩ := exists_mem_nhds_zero_mul_subset isCompact_univ (hU.mem_nhds h0U) | case intro.intro.intro.intro.intro
M : Type u_3
inst✝⁴ : GroupWithZero M
inst✝³ : TopologicalSpace M
inst✝² : ContinuousMul M
inst✝¹ : CompactSpace M
inst✝ : T1Space M
U : Set M
hU : IsOpen U
h0U : 0 ∈ U
h1U : 1 ∉ U
W : Set M
hW : W ∈ 𝓝 0
hW' : univ * W ⊆ U
⊢ IsOpen {0} | dc8e2a6913e2be45 |
Polynomial.sub_one_pow_totient_lt_cyclotomic_eval | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | theorem sub_one_pow_totient_lt_cyclotomic_eval {n : ℕ} {q : ℝ} (hn' : 2 ≤ n) (hq' : 1 < q) :
(q - 1) ^ totient n < (cyclotomic n ℝ).eval q | case convert_6
n : ℕ
q : ℝ
hn' : 2 ≤ n
hq' : 1 < q
hn : 0 < n
hq : 0 < q
hfor : ∀ ζ' ∈ primitiveRoots n ℂ, q - 1 ≤ ‖↑q - ζ'‖
ζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)
hζ : IsPrimitiveRoot ζ n
hex : ∃ ζ' ∈ primitiveRoots n ℂ, q - 1 < ‖↑q - ζ'‖
this : ¬eval (↑q) (cyclotomic n ℂ) = 0
⊢ ∃ i ∈ (primitiveRoots n ℂ... | simp only [Subtype.coe_mk, Finset.mem_attach, exists_true_left, Subtype.exists, ←
NNReal.coe_lt_coe, ← Units.val_lt_val, Units.val_mk0 _, coe_nnnorm] | case convert_6
n : ℕ
q : ℝ
hn' : 2 ≤ n
hq' : 1 < q
hn : 0 < n
hq : 0 < q
hfor : ∀ ζ' ∈ primitiveRoots n ℂ, q - 1 ≤ ‖↑q - ζ'‖
ζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)
hζ : IsPrimitiveRoot ζ n
hex : ∃ ζ' ∈ primitiveRoots n ℂ, q - 1 < ‖↑q - ζ'‖
this : ¬eval (↑q) (cyclotomic n ℂ) = 0
⊢ ∃ a, ∃ (_ : a ∈ primitive... | 0ef7adcad44e6a5e |
Finset.small_pos_neg_neg_mul | Mathlib/Combinatorics/Additive/SmallTripling.lean | @[to_additive]
private lemma small_pos_neg_neg_mul (hA : #(A ^ 3) ≤ K * #A) : #(A * A⁻¹ * A⁻¹) ≤ K ^ 2 * #A | G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
⊢ ↑(#(A * A⁻¹ * A⁻¹)) ≤ K ^ 2 * ↑(#A) | simpa using small_neg_pos_pos_mul (A := A⁻¹) (by simpa) | no goals | 5a530b69a1256eb1 |
PrincipalSeg.top_rel_top | Mathlib/Order/InitialSeg.lean | theorem top_rel_top {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} [IsWellOrder γ t]
(f : r ≺i s) (g : s ≺i t) (h : r ≺i t) : t h.top g.top | α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
inst✝ : IsWellOrder γ t
f : r ≺i s
g : s ≺i t
h : r ≺i t
⊢ t h.top g.top | rw [Subsingleton.elim h (f.trans g)] | α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
inst✝ : IsWellOrder γ t
f : r ≺i s
g : s ≺i t
h : r ≺i t
⊢ t (f.trans g).top g.top | e7eeb80e8731e3bf |
hasSum_mellin_pi_mul_sq' | Mathlib/NumberTheory/LSeries/MellinEqDirichlet.lean | /-- Tailored version for odd Jacobi theta functions. -/
lemma hasSum_mellin_pi_mul_sq' {a : ι → ℂ} {r : ι → ℝ} {F : ℝ → ℂ} {s : ℂ} (hs : 0 < s.re)
(hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * r i * rexp (-π * r i ^ 2 * t)) (F t))
(h_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re) :
HasSum (fun i ↦ Gammaℝ (s + 1) *... | case h.e'_5.h
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i => if r i = 0 then 0 else a i * ↑(r i) * ↑(rexp (-π * r i ^ 2 * t))) (F t)
h_sum : Summable fun i => ‖a i‖ / |r i| ^ s.re
hs₁ : s ≠ 0
hs₂ : 0 < (s + 1).re
hs₃ : s + 1 ≠ 0
this :
∀ (i : ι) (... | rcases eq_or_ne (r i) 0 with h | h | case h.e'_5.h.inl
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
r : ι → ℝ
F : ℝ → ℂ
s : ℂ
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i => if r i = 0 then 0 else a i * ↑(r i) * ↑(rexp (-π * r i ^ 2 * t))) (F t)
h_sum : Summable fun i => ‖a i‖ / |r i| ^ s.re
hs₁ : s ≠ 0
hs₂ : 0 < (s + 1).re
hs₃ : s + 1 ≠ 0
this :
∀ (i : ... | 22936c8a231e7fbd |
Subgroup.IsComplement.equiv_fst_eq_self_iff_mem | Mathlib/GroupTheory/Complement.lean | theorem equiv_fst_eq_self_iff_mem {g : G} (h1 : 1 ∈ T) :
((hST.equiv g).fst : G) = g ↔ g ∈ S | case mpr
G : Type u_1
inst✝ : Group G
S T : Set G
hST : IsComplement S T
g : G
h1 : 1 ∈ T
h : g ∈ S
⊢ ↑(hST.equiv g).1 = g | rw [hST.equiv_fst_eq_self_of_mem_of_one_mem h1 h] | no goals | becd34388e09e225 |
IsCyclotomicExtension.Rat.Three.lambda_dvd_or_dvd_sub_one_or_dvd_add_one | Mathlib/NumberTheory/Cyclotomic/Three.lean | /-- Let `(x : 𝓞 K)`. Then we have that `λ` divides one amongst `x`, `x - 1` and `x + 1`. -/
lemma lambda_dvd_or_dvd_sub_one_or_dvd_add_one [NumberField K] [IsCyclotomicExtension {3} ℚ K] :
λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1 | K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
x : 𝓞 K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1 | let _ := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ}) | K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
x : 𝓞 K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
this : Finite (𝓞 K ⧸ Ideal.span {λ})
x✝ : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})
⊢ λ ∣ x ∨ λ ∣ x - 1 ∨ λ ∣ x + 1 | 40b2b58fa758600e |
SimpleGraph.Walk.IsPath.snd_of_toSubgraph_adj | Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean | lemma snd_of_toSubgraph_adj {u v v'} {p : G.Walk u v} (hp : p.IsPath)
(hadj : p.toSubgraph.Adj u v') : p.snd = v' | V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
hp : p.IsPath
i : ℕ
hl1 : p.getVert i = u
hadj : p.toSubgraph.Adj u (p.getVert (i + 1))
hi :
(p.getVert i = u ∧ p.getVert (i + 1) = p.getVert (i + 1) ∨ p.getVert i = p.getVert (i + 1) ∧ p.getVert (i + 1) = u) ∧
i < p.length
⊢ 0 ∈ {i | i ≤ p.length} | rw [Set.mem_setOf] | V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
hp : p.IsPath
i : ℕ
hl1 : p.getVert i = u
hadj : p.toSubgraph.Adj u (p.getVert (i + 1))
hi :
(p.getVert i = u ∧ p.getVert (i + 1) = p.getVert (i + 1) ∨ p.getVert i = p.getVert (i + 1) ∧ p.getVert (i + 1) = u) ∧
i < p.length
⊢ 0 ≤ p.length | 6a4c9bd87e294aeb |
Set.univ_div_univ | Mathlib/Algebra/Group/Pointwise/Set/Basic.lean | @[to_additive (attr := simp)]
lemma univ_div_univ : (univ / univ : Set α) = univ | α : Type u_2
inst✝ : DivisionMonoid α
⊢ univ / univ = univ | simp [div_eq_mul_inv] | no goals | ea0f1a3b1c0a164a |
Affine.Simplex.sum_pointsWithCircumcenter | Mathlib/Geometry/Euclidean/Circumcenter.lean | theorem sum_pointsWithCircumcenter {α : Type*} [AddCommMonoid α] {n : ℕ}
(f : PointsWithCircumcenterIndex n → α) :
∑ i, f i = (∑ i : Fin (n + 1), f (pointIndex i)) + f circumcenterIndex | α : Type u_3
inst✝ : AddCommMonoid α
n : ℕ
f : PointsWithCircumcenterIndex n → α
h : univ = insert circumcenterIndex (Finset.map (pointIndexEmbedding n) univ)
⊢ circumcenterIndex ∉ Finset.map (pointIndexEmbedding n) univ | simp_rw [Finset.mem_map, not_exists] | α : Type u_3
inst✝ : AddCommMonoid α
n : ℕ
f : PointsWithCircumcenterIndex n → α
h : univ = insert circumcenterIndex (Finset.map (pointIndexEmbedding n) univ)
⊢ ∀ (x : Fin (n + 1)), ¬(x ∈ univ ∧ (pointIndexEmbedding n) x = circumcenterIndex) | 248891fce28689d6 |
ZMod.castHom_bijective | Mathlib/Data/ZMod/Basic.lean | theorem castHom_bijective [Fintype R] (h : Fintype.card R = n) :
Function.Bijective (ZMod.castHom (dvd_refl n) R) | n : ℕ
R : Type u_1
inst✝² : Ring R
inst✝¹ : CharP R n
inst✝ : Fintype R
h : Fintype.card R = n
hn : n = 0
⊢ False | rw [hn] at h | n : ℕ
R : Type u_1
inst✝² : Ring R
inst✝¹ : CharP R n
inst✝ : Fintype R
h : Fintype.card R = 0
hn : n = 0
⊢ False | f5948c31783484d7 |
IsIntegralCurve.periodic_xor_injective | Mathlib/Geometry/Manifold/IntegralCurve/ExistUnique.lean | /-- A global integral curve is injective xor periodic with positive period. -/
lemma IsIntegralCurve.periodic_xor_injective [BoundarylessManifold I M]
(hγ : IsIntegralCurve γ v)
(hv : ContMDiff I I.tangent 1 (fun x ↦ (⟨x, v x⟩ : TangentBundle I M))) :
Xor' (∃ T > 0, Periodic γ T) (Injective γ) | case neg
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I 1 M
γ : ℝ → M
v : (x : M) → TangentSpace I x
inst✝¹ : T2Space M
inst✝ : BoundarylessMa... | exact hγ.periodic_of_eq hv heq | no goals | 51957a500664b235 |
four_mul_le_sq_add | Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean | /-- Binary, squared, and division-free **arithmetic mean-geometric mean inequality**
(aka AM-GM inequality) for linearly ordered commutative semirings. -/
lemma four_mul_le_sq_add [ExistsAddOfLE α] [MulPosStrictMono α]
[AddLeftReflectLE α] [AddLeftMono α]
(a b : α) : 4 * a * b ≤ (a + b) ^ 2 | α : Type u
inst✝⁵ : CommSemiring α
inst✝⁴ : LinearOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : MulPosStrictMono α
inst✝¹ : AddLeftReflectLE α
inst✝ : AddLeftMono α
a b : α
⊢ a ^ 2 + b ^ 2 + 2 * a * b = a ^ 2 + 2 * a * b + b ^ 2 | rw [add_right_comm] | no goals | 38ca5601c78dc32a |
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