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.blastShiftRight.twoPowShift_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/ShiftRight.lean | theorem twoPowShift_eq (aig : AIG α) (target : TwoPowShiftTarget aig w) (lhs : BitVec w)
(rhs : BitVec target.n) (assign : α → Bool)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, target.lhs.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx < target.n), ⟦aig, target.rhs.get idx hi... | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
lhs : BitVec w
assign : α → Bool
idx : Nat
hidx : idx < w
res : RefVecEntry α w
n : Nat
lvec : aig.RefVec w
rvec : aig.RefVec n
pow : Nat
rhs : BitVec { n := n, lhs := lvec, rhs := rvec, pow := pow }.n
hleft :
∀ (idx : Nat) (hidx : idx < w),
⟦... | simp only [hif1, ↓reduceIte] | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
lhs : BitVec w
assign : α → Bool
idx : Nat
hidx : idx < w
res : RefVecEntry α w
n : Nat
lvec : aig.RefVec w
rvec : aig.RefVec n
pow : Nat
rhs : BitVec { n := n, lhs := lvec, rhs := rvec, pow := pow }.n
hleft :
∀ (idx : Nat) (hidx : idx < w),
⟦... | a2117c7fa85ff087 |
Complex.hasStrictDerivAt_cpow_const | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | theorem Complex.hasStrictDerivAt_cpow_const (h : x ∈ slitPlane) :
HasStrictDerivAt (fun z : ℂ => z ^ c) (c * x ^ (c - 1)) x | x c : ℂ
h : x ∈ slitPlane
⊢ HasStrictDerivAt (fun z => z ^ c) (c * x ^ (c - 1)) x | simpa only [mul_zero, add_zero, mul_one] using
(hasStrictDerivAt_id x).cpow (hasStrictDerivAt_const x c) h | no goals | 453f16495933196c |
List.dedup_map_of_injective | Mathlib/Data/List/Dedup.lean | theorem dedup_map_of_injective [DecidableEq β] {f : α → β} (hf : Function.Injective f)
(xs : List α) :
(xs.map f).dedup = xs.dedup.map f | case neg
α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
hf : Function.Injective f
x : α
xs : List α
ih : (map f xs).dedup = map f xs.dedup
h : x ∉ xs
⊢ (f x :: map f xs).dedup = map f (x :: xs).dedup | rw [dedup_cons_of_not_mem h, dedup_cons_of_not_mem <| (mem_map_of_injective hf).not.mpr h, ih,
map_cons] | no goals | 74f40e5383066b94 |
CategoryTheory.Limits.zero_of_to_zero | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | theorem zero_of_to_zero {X : C} (f : X ⟶ 0) : f = 0 | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
X : C
f : X ⟶ 0
⊢ f = 0 | ext | no goals | efc069c4be356a65 |
Nat.pred_le_iff | Mathlib/Data/Nat/Init.lean | lemma pred_le_iff : pred m ≤ n ↔ m ≤ succ n :=
⟨le_succ_of_pred_le, by
cases m
· exact fun _ ↦ zero_le n
· exact le_of_succ_le_succ⟩
| case zero
n : ℕ
⊢ 0 ≤ n.succ → pred 0 ≤ n | exact fun _ ↦ zero_le n | no goals | d8177b02e08a865e |
Int.two_pow_two_pow_sub_pow_two_pow | Mathlib/NumberTheory/Multiplicity.lean | theorem Int.two_pow_two_pow_sub_pow_two_pow {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
emultiplicity 2 (x ^ 2 ^ n - y ^ 2 ^ n) = emultiplicity 2 (x - y) + n | x y : ℤ
n : ℕ
hxy : 4 ∣ x - y
hx : ¬2 ∣ x
⊢ emultiplicity 2 (x ^ 2 ^ n - y ^ 2 ^ n) = emultiplicity 2 (x - y) + ↑n | simp only [pow_two_pow_sub_pow_two_pow n, emultiplicity_mul Int.prime_two,
Finset.emultiplicity_prod Int.prime_two, add_comm, Nat.cast_one, Finset.sum_const,
Finset.card_range, nsmul_one, Int.two_pow_two_pow_add_two_pow_two_pow hx hxy] | no goals | a31c329cb8f6258d |
AlgebraicGeometry.PresheafedSpace.GlueData.snd_invApp_t_app' | Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean | theorem snd_invApp_t_app' (i j k : D.J) (U : Opens (pullback (D.f i j) (D.f i k)).carrier) :
∃ eq,
(π₂⁻¹ i, j, k) U ≫ (D.t k i).c.app _ ≫ (D.V (k, i)).presheaf.map (eqToHom eq) =
(D.t' k i j).c.app _ ≫ (π₁⁻¹ k, j, i) (unop _) | case w.e_unop.e_carrier.h
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
U : Opens ↑↑(pullback (D.f i j) (D.f i k))
this :
⇑(TopCat.Hom.hom (pullback.snd (D.f i j) (D.f i k)).base) =
⇑(TopCat.Hom.hom (D.t k i).base) ∘
⇑(TopCat.Hom.hom (pullback.fst (D.f k i) (D.f k j)).base) ∘ ⇑(TopCat.Hom.... | refine Function.HasLeftInverse.injective ⟨(D.t i k).base, fun x => ?_⟩ | case w.e_unop.e_carrier.h
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
U : Opens ↑↑(pullback (D.f i j) (D.f i k))
this :
⇑(TopCat.Hom.hom (pullback.snd (D.f i j) (D.f i k)).base) =
⇑(TopCat.Hom.hom (D.t k i).base) ∘
⇑(TopCat.Hom.hom (pullback.fst (D.f k i) (D.f k j)).base) ∘ ⇑(TopCat.Hom.... | e08cb0129b520e6d |
Filter.Tendsto.op_one_isBoundedUnder_le' | Mathlib/Analysis/Normed/Group/Bounded.lean | /-- A helper lemma used to prove that the (scalar or usual) product of a function that tends to one
and a bounded function tends to one. This lemma is formulated for any binary operation
`op : E → F → G` with an estimate `‖op x y‖ ≤ A * ‖x‖ * ‖y‖` for some constant A instead of
multiplication so that it can be applied ... | α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝² : SeminormedGroup E
inst✝¹ : SeminormedGroup F
inst✝ : SeminormedGroup G
f : α → E
g : α → F
l : Filter α
hf : Tendsto f l (𝓝 1)
hg : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) l (Norm.norm ∘ g)
op : E → F → G
h_op : ∃ A, ∀ (x : E) (y : F), ‖op x y‖ ≤ A * ‖x‖ * ‖y‖... | obtain ⟨A, h_op⟩ := h_op | case intro
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝² : SeminormedGroup E
inst✝¹ : SeminormedGroup F
inst✝ : SeminormedGroup G
f : α → E
g : α → F
l : Filter α
hf : Tendsto f l (𝓝 1)
hg : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) l (Norm.norm ∘ g)
op : E → F → G
A : ℝ
h_op : ∀ (x : E) (y : F), ‖op x y‖ ≤ A... | 74f45a2d5b51ad37 |
Matrix.mem_range_scalar_iff_commute_transvectionStruct | Mathlib/LinearAlgebra/Matrix/Transvection.lean | theorem _root_.Matrix.mem_range_scalar_iff_commute_transvectionStruct {M : Matrix n n R} :
M ∈ Set.range (Matrix.scalar n) ↔ ∀ t : TransvectionStruct n R, Commute t.toMatrix M | n : Type u_1
R : Type u₂
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : Fintype n
M : Matrix n n R
h : M ∈ Set.range ⇑(scalar n)
t : TransvectionStruct n R
⊢ Commute t.toMatrix M | rw [mem_range_scalar_iff_commute_stdBasisMatrix] at h | n : Type u_1
R : Type u₂
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : Fintype n
M : Matrix n n R
h : ∀ (i j : n), i ≠ j → Commute (stdBasisMatrix i j 1) M
t : TransvectionStruct n R
⊢ Commute t.toMatrix M | e29d1d2ffd73c2cb |
LieModule.iterate_toEnd_mem_lowerCentralSeries₂ | Mathlib/Algebra/Lie/Nilpotent.lean | theorem iterate_toEnd_mem_lowerCentralSeries₂ (x y : L) (m : M) (k : ℕ) :
(toEnd R L M x ∘ₗ toEnd R L M y)^[k] m ∈
lowerCentralSeries R L M (2 * k) | case succ
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x y : L
m : M
k : ℕ
ih : (⇑((toEnd R L M) x ∘ₗ (toEnd R L M) y))^[k] m ∈ lowerCentralSeries R L M (2 * k)
hk : 2 * k.su... | refine LieSubmodule.lie_mem_lie (LieSubmodule.mem_top x) ?_ | case succ
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x y : L
m : M
k : ℕ
ih : (⇑((toEnd R L M) x ∘ₗ (toEnd R L M) y))^[k] m ∈ lowerCentralSeries R L M (2 * k)
hk : 2 * k.su... | a700d4de77919e57 |
LieModule.map_lowerCentralSeries_le | Mathlib/Algebra/Lie/Nilpotent.lean | theorem map_lowerCentralSeries_le (f : M →ₗ⁅R,L⁆ M₂) :
(lowerCentralSeries R L M k).map f ≤ lowerCentralSeries R L M₂ k | R : Type u
L : Type v
M : Type w
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
k : ℕ
M₂ : Type w₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L M₂
inst✝¹ : LieModule R L M₂
inst✝ : LieModule R L M
f : M →ₗ⁅... | induction k with
| zero => simp only [lowerCentralSeries_zero, le_top]
| succ k ih =>
simp only [LieModule.lowerCentralSeries_succ, LieSubmodule.map_bracket_eq]
exact LieSubmodule.mono_lie_right ⊤ ih | no goals | 5837ece36426ed54 |
CoxeterSystem.prod_alternatingWord_eq_mul_pow | Mathlib/GroupTheory/Coxeter/Basic.lean | theorem prod_alternatingWord_eq_mul_pow (i i' : B) (m : ℕ) :
π (alternatingWord i i' m) = (if Even m then 1 else s i') * (s i * s i') ^ (m / 2) | B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
ih :
cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)
hm : Even m
h₁ : ¬Even (m + 1)
⊢ ¬2 ∣ m + 1 | rwa [← even_iff_two_dvd] | no goals | 60981bcd0df2f406 |
MeasureTheory.IsSetAlgebra.generateSetAlgebra_subset | Mathlib/MeasureTheory/SetAlgebra.lean | theorem generateSetAlgebra_subset {ℬ : Set (Set α)} (h : 𝒜 ⊆ ℬ)
(hℬ : IsSetAlgebra ℬ) : generateSetAlgebra 𝒜 ⊆ ℬ | α : Type u_1
𝒜 ℬ : Set (Set α)
h : 𝒜 ⊆ ℬ
hℬ : IsSetAlgebra ℬ
⊢ generateSetAlgebra 𝒜 ⊆ ℬ | intro s hs | α : Type u_1
𝒜 ℬ : Set (Set α)
h : 𝒜 ⊆ ℬ
hℬ : IsSetAlgebra ℬ
s : Set α
hs : s ∈ generateSetAlgebra 𝒜
⊢ s ∈ ℬ | 93ab4aef2df34ca9 |
Std.Tactic.BVDecide.Normalize.BitVec.max_ult' | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Normalize/BitVec.lean | theorem BitVec.max_ult' (a : BitVec w) : (BitVec.ult (-1#w) a) = false | w : Nat
a : BitVec w
⊢ (-1#w).ult a = false | rw [BitVec.negOne_eq_allOnes, ← Bool.not_eq_true, ← @lt_ult] | w : Nat
a : BitVec w
⊢ ¬BitVec.allOnes w < a | b59d14789a3f449f |
MeasureTheory.integral_rnDeriv_mul_log | Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean | lemma integral_rnDeriv_mul_log [SigmaFinite μ] [μ.HaveLebesgueDecomposition ν] (hμν : μ ≪ ν) :
∫ a, (μ.rnDeriv ν a).toReal * log (μ.rnDeriv ν a).toReal ∂ν = ∫ a, llr μ ν a ∂μ | α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
⊢ ∫ (a : α), (μ.rnDeriv ν a).toReal * log (μ.rnDeriv ν a).toReal ∂ν = ∫ (a : α), llr μ ν a ∂μ | simp_rw [← smul_eq_mul, integral_rnDeriv_smul hμν, llr] | no goals | de946fa119f501c8 |
Monotone.tendstoLocallyUniformly_of_forall_tendsto | Mathlib/Topology/UniformSpace/Dini.lean | /-- **Dini's theorem**: if `F n` is a monotone increasing collection of continuous functions
converging pointwise to a continuous function `f`, then `F n` converges locally uniformly to `f`. -/
lemma tendstoLocallyUniformly_of_forall_tendsto
(hF_cont : ∀ i, Continuous (F i)) (hF_mono : Monotone F) (hf : Continuous ... | ι : Type u_1
α : Type u_2
G : Type u_3
inst✝² : Preorder ι
inst✝¹ : TopologicalSpace α
inst✝ : NormedLatticeAddCommGroup G
F : ι → α → G
f : α → G
hF_cont : ∀ (i : ι), Continuous (F i)
hF_mono : Monotone F
hf : Continuous f
h_tendsto : ∀ (x : α), Tendsto (fun x_1 => F x_1 x) atTop (𝓝 (f x))
⊢ TendstoLocallyUniformly F... | refine (atTop : Filter ι).eq_or_neBot.elim (fun h ↦ ?eq_bot) (fun _ ↦ ?_) | case eq_bot
ι : Type u_1
α : Type u_2
G : Type u_3
inst✝² : Preorder ι
inst✝¹ : TopologicalSpace α
inst✝ : NormedLatticeAddCommGroup G
F : ι → α → G
f : α → G
hF_cont : ∀ (i : ι), Continuous (F i)
hF_mono : Monotone F
hf : Continuous f
h_tendsto : ∀ (x : α), Tendsto (fun x_1 => F x_1 x) atTop (𝓝 (f x))
h : atTop = ⊥
⊢... | 67a286086b03a38c |
List.chain_iff_get | Mathlib/Data/List/Chain.lean | theorem chain_iff_get {R} : ∀ {a : α} {l : List α}, Chain R a l ↔
(∀ h : 0 < length l, R a (get l ⟨0, h⟩)) ∧
∀ (i : ℕ) (h : i < l.length - 1),
R (get l ⟨i, by omega⟩) (get l ⟨i+1, by omega⟩)
| a, [] => iff_of_true (by simp) ⟨fun h => by simp at h, fun _ h => by simp at h⟩
| a, b :: t => by
rw ... | case mpr.intro.right
α : Type u
R : α → α → Prop
a b : α
t : List α
h0 : ∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)
h : ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)
⊢ (∀ (h : 0 < t.length), R b (t.get ⟨0, h⟩)) ∧ ∀ (i : ℕ) (h : i < t.length - 1), R (t.get ⟨i, ⋯⟩)... | constructor | case mpr.intro.right.left
α : Type u
R : α → α → Prop
a b : α
t : List α
h0 : ∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)
h : ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)
⊢ ∀ (h : 0 < t.length), R b (t.get ⟨0, h⟩)
case mpr.intro.right.right
α : Type u
R : α → α ... | 9aa80a65a793fb42 |
MvPowerSeries.coeff_truncFun | Mathlib/RingTheory/MvPowerSeries/Trunc.lean | theorem coeff_truncFun (m : σ →₀ ℕ) (φ : MvPowerSeries σ R) :
(truncFun n φ).coeff m = if m < n then coeff R m φ else 0 | σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n m : σ →₀ ℕ
φ : MvPowerSeries σ R
⊢ MvPolynomial.coeff m (truncFun n φ) = if m < n then (coeff R m) φ else 0 | classical
simp [truncFun, MvPolynomial.coeff_sum] | no goals | e4f1574a4807295b |
CategoryTheory.Adjunction.faithful_R_of_epi_counit_app | Mathlib/CategoryTheory/Adjunction/FullyFaithful.lean | /-- If each component of the counit is an epimorphism, then the right adjoint is faithful. -/
lemma faithful_R_of_epi_counit_app [∀ X, Epi (h.counit.app X)] : R.Faithful where
map_injective {X Y f g} hfg | C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
L : C ⥤ D
R : D ⥤ C
h : L ⊣ R
inst✝ : ∀ (X : D), Epi (h.counit.app X)
X Y : D
f g : X ⟶ Y
hfg : R.map f = R.map g
⊢ f = g | apply Epi.left_cancellation (f := h.counit.app X) | case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
L : C ⥤ D
R : D ⥤ C
h : L ⊣ R
inst✝ : ∀ (X : D), Epi (h.counit.app X)
X Y : D
f g : X ⟶ Y
hfg : R.map f = R.map g
⊢ h.counit.app X ≫ f = h.counit.app X ≫ g | 8c0966bba41f4621 |
t0Space_iff_ker_uniformity | Mathlib/Topology/UniformSpace/Separation.lean | theorem t0Space_iff_ker_uniformity : T0Space α ↔ (𝓤 α).ker = diagonal α | α : Type u
inst✝ : UniformSpace α
⊢ (∀ (a b : α), (∀ s ∈ 𝓤 α, (a, b) ∈ s) → a = b) → ∀ (x : α), ∀ s ∈ 𝓤 α, (x, x) ∈ s | exact fun _ x s hs ↦ refl_mem_uniformity hs | no goals | 653de9599b1710c3 |
KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map | Mathlib/NumberTheory/KummerDedekind.lean | theorem normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map (hI : IsMaximal I)
(hI' : I ≠ ⊥) (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) :
normalizedFactors (I.map (algebraMap R S)) =
Multiset.map
(fun f =>
((normalizedFactorsMapEquivNormaliz... | case neg
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsI... | rintro J' ⟨_, rfl⟩ | case neg.intro
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _roo... | c49652504dc8313f |
Multiset.zero_union | Mathlib/Data/Multiset/UnionInter.lean | @[simp] lemma zero_union : 0 ∪ s = s | α : Type u_1
inst✝ : DecidableEq α
s : Multiset α
⊢ 0 ∪ s = s | simp [union_def, Multiset.zero_sub] | no goals | 3737c0525f02b47d |
List.take_range | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Range.lean | theorem take_range (m n : Nat) : take m (range n) = range (min m n) | case hl
m n : Nat
⊢ (take m (range n)).length = (range (min m n)).length | simp | no goals | 6f9dd59421081959 |
Complex.hasDerivAt_logTaylor | Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean | lemma hasDerivAt_logTaylor (n : ℕ) (z : ℂ) :
HasDerivAt (logTaylor (n + 1)) (∑ j ∈ Finset.range n, (-1) ^ j * z ^ j) z | case succ
z : ℂ
n : ℕ
ih : HasDerivAt (logTaylor (n + 1)) (∑ j ∈ Finset.range n, (-1) ^ j * z ^ j) z
⊢ HasDerivAt (logTaylor (n + 1) + fun z => (-1) ^ (n + 1 + 1) * z ^ (n + 1) / ↑(n + 1))
(∑ j ∈ Finset.range (n + 1), (-1) ^ j * z ^ j) z | simp only [cpow_natCast, Nat.cast_add, Nat.cast_one, ← Nat.not_even_iff_odd,
Finset.sum_range_succ, (show (-1) ^ (n + 1 + 1) = (-1) ^ n by ring)] | case succ
z : ℂ
n : ℕ
ih : HasDerivAt (logTaylor (n + 1)) (∑ j ∈ Finset.range n, (-1) ^ j * z ^ j) z
⊢ HasDerivAt (logTaylor (n + 1) + fun z => (-1) ^ (n + 1 + 1) * z ^ (n + 1) / (↑n + 1))
(∑ j ∈ Finset.range n, (-1) ^ j * z ^ j + (-1) ^ n * z ^ n) z | 711f35464efd04b5 |
SetTheory.Game.le_birthday | Mathlib/SetTheory/Game/Birthday.lean | theorem le_birthday (x : Game) : x ≤ x.birthday.toGame | x : Game
y : PGame
hy₁ : ⟦y⟧ = x
hy₂ : y.birthday = x.birthday
⊢ y.birthday.toPGame ≤ (birthday ⟦y⟧).toPGame | rw [toPGame_le_iff, hy₁, hy₂] | no goals | e84f0f29c8990961 |
GromovHausdorff.hausdorffDist_optimal | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | theorem hausdorffDist_optimal {X : Type u} [MetricSpace X] [CompactSpace X] [Nonempty X]
{Y : Type v} [MetricSpace Y] [CompactSpace Y] [Nonempty Y] :
hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) = ghDist X Y | X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)
hp : ⟦p⟧ = toGHSpace X
hq : ⟦q⟧ = toGHSpace Y
bound : hausdorffDist ... | gcongr | no goals | e1b74305bb180259 |
Vector.eq_iff_flatten_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem eq_iff_flatten_eq {L L' : Vector (Vector α n) m} :
L = L' ↔ L.flatten = L'.flatten | α : Type u_1
n m : Nat
L : Array (Array α)
h₁ : L.size = m
h₂ : ∀ (xs : Array α), xs ∈ L → xs.size = n
L' : Array (Array α)
h₁' : L'.size = m
h₂' : ∀ (xs : Array α), xs ∈ L' → xs.size = n
h✝ : L.flatten = L'.flatten
i : Nat
h : i < (Array.map Array.size L).size
h' : i < (Array.map Array.size L').size
⊢ L'[i] ∈ L' | simp | no goals | 72d2568041cb6666 |
nodup_permsOfList | Mathlib/Data/Fintype/Perm.lean | theorem nodup_permsOfList : ∀ {l : List α}, l.Nodup → (permsOfList l).Nodup
| [], _ => by simp [permsOfList]
| a :: l, hl => by
have hl' : l.Nodup := hl.of_cons
have hln' : (permsOfList l).Nodup := nodup_permsOfList hl'
have hmeml : ∀ {f : Perm α}, f ∈ permsOfList l → f a = a := fun {f} hf =>
not_... | case refine_4
α : Type u_1
inst✝ : DecidableEq α
a : α
l : List α
hl : (a :: l).Nodup
hl' : l.Nodup
hln' : (permsOfList l).Nodup
hmeml : ∀ {f : Equiv.Perm α}, f ∈ permsOfList l → f a = a
f : Equiv.Perm α
hf₁ : f ∈ permsOfList l
hf₂ : f ∈ flatMap (fun b => List.map (fun f => Equiv.swap a b * f) (permsOfList l)) l
x : α
... | let ⟨g, hg⟩ := List.mem_map.1 hx' | case refine_4
α : Type u_1
inst✝ : DecidableEq α
a : α
l : List α
hl : (a :: l).Nodup
hl' : l.Nodup
hln' : (permsOfList l).Nodup
hmeml : ∀ {f : Equiv.Perm α}, f ∈ permsOfList l → f a = a
f : Equiv.Perm α
hf₁ : f ∈ permsOfList l
hf₂ : f ∈ flatMap (fun b => List.map (fun f => Equiv.swap a b * f) (permsOfList l)) l
x : α
... | 1438bc15e355e5ea |
Monoid.exponent_eq_iSup_orderOf' | Mathlib/GroupTheory/Exponent.lean | theorem exponent_eq_iSup_orderOf' :
exponent G = if ∃ g : G, orderOf g = 0 then 0 else ⨆ g : G, orderOf g | G : Type u
inst✝ : CommMonoid G
⊢ exponent G = if ∃ g, orderOf g = 0 then 0 else ⨆ g, orderOf g | split_ifs with h | case pos
G : Type u
inst✝ : CommMonoid G
h : ∃ g, orderOf g = 0
⊢ exponent G = 0
case neg
G : Type u
inst✝ : CommMonoid G
h : ¬∃ g, orderOf g = 0
⊢ exponent G = ⨆ g, orderOf g | 390f71f4f6a46651 |
Std.DHashMap.Raw.contains_alter_self | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean | theorem contains_alter_self [LawfulBEq α] {k : α} {f : Option (β k) → Option (β k)} (h : m.WF) :
(m.alter k f).contains k = (f (m.get? k)).isSome | α : Type u
β : α → Type v
inst✝² : BEq α
inst✝¹ : Hashable α
m : Raw α β
inst✝ : LawfulBEq α
k : α
f : Option (β k) → Option (β k)
h : m.WF
⊢ (m.alter k f).contains k = (f (m.get? k)).isSome | simp only [contains_alter h, beq_self_eq_true, reduceIte] | no goals | 055f9e26e348c7e7 |
map_wittStructureInt | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | theorem map_wittStructureInt (Φ : MvPolynomial idx ℤ) (n : ℕ) :
map (Int.castRingHom ℚ) (wittStructureInt p Φ n) =
wittStructureRat p (map (Int.castRingHom ℚ) Φ) n | case h
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH : ∀ m < n, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
⊢ ∀ (d : idx × ℕ →₀ ℕ),
(fun x => ↑x) (coeff d (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) n)).num =
coef... | intro c | case h
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℤ
n : ℕ
IH : ∀ m < n, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m
c : idx × ℕ →₀ ℕ
⊢ (fun x => ↑x) (coeff c (wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) n)).num =
coeff c (wittSt... | 4df79b6c8cd56f81 |
CategoryTheory.GrothendieckTopology.Plus.toPlus_eq_mk | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | theorem toPlus_eq_mk {X : C} {P : Cᵒᵖ ⥤ D} (x : ToType (P.obj (op X))) :
(J.toPlus P).app _ x = mk (Meq.mk ⊤ x) | C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝³ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory D FD
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cove... | simp only [ConcreteCategory.comp_apply] | C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝³ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory D FD
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cove... | 6a4d9042a1f2f255 |
MeasureTheory.intervalIntegral_tendsto_integral_Iic | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | theorem intervalIntegral_tendsto_integral_Iic (b : ℝ) (hfi : IntegrableOn f (Iic b) μ)
(ha : Tendsto a l atBot) :
Tendsto (fun i => ∫ x in a i..b, f x ∂μ) l (𝓝 <| ∫ x in Iic b, f x ∂μ) | ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.IsCountablyGenerated
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a : ι → ℝ
f : ℝ → E
b : ℝ
hfi : IntegrableOn f (Iic b) μ
ha : Tendsto a l atBot
φ : ι → Set ℝ := fun i => Ioi (a i)
hφ : AECover (μ.restrict (Iic b)) l φ
⊢ (fun i => ∫ (x : ℝ) in φ ... | filter_upwards [ha.eventually (eventually_le_atBot <| b)] with i hai | case h
ι : Type u_1
E : Type u_2
μ : Measure ℝ
l : Filter ι
inst✝² : l.IsCountablyGenerated
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a : ι → ℝ
f : ℝ → E
b : ℝ
hfi : IntegrableOn f (Iic b) μ
ha : Tendsto a l atBot
φ : ι → Set ℝ := fun i => Ioi (a i)
hφ : AECover (μ.restrict (Iic b)) l φ
i : ι
hai : a i ≤ b
... | f9439ff9429a654c |
Relation.acc_of_singleton | Mathlib/Logic/Hydra.lean | theorem acc_of_singleton [IsIrrefl α r] {s : Multiset α} (hs : ∀ a ∈ s, Acc (CutExpand r) {a}) :
Acc (CutExpand r) s | case cons
α : Type u_1
r : α → α → Prop
inst✝ : IsIrrefl α r
a : α
s : Multiset α
ihs : (∀ a ∈ s, Acc (CutExpand r) {a}) → Acc (CutExpand r) s
hs : Acc (CutExpand r) {a} ∧ ∀ x ∈ s, Acc (CutExpand r) {x}
⊢ Acc (CutExpand r) ({a} + s) | exact (hs.1.prod_gameAdd <| ihs fun a ha ↦ hs.2 a ha).of_fibration _ (cutExpand_fibration r) | no goals | 622f2e52c1100461 |
Multiset.map_lt_map | Mathlib/Data/Multiset/MapFold.lean | theorem map_lt_map {f : α → β} {s t : Multiset α} (h : s < t) : s.map f < t.map f | α : Type u_1
β : Type v
f : α → β
s t : Multiset α
h : s < t
H : map f t ≤ map f s
⊢ t.card ≤ s.card | rw [← s.card_map f, ← t.card_map f] | α : Type u_1
β : Type v
f : α → β
s t : Multiset α
h : s < t
H : map f t ≤ map f s
⊢ (map f t).card ≤ (map f s).card | 5289891e7f2deb8b |
Module.punctured_nhds_neBot | Mathlib/Topology/Algebra/Module/Basic.lean | theorem Module.punctured_nhds_neBot [Nontrivial M] [NeBot (𝓝[≠] (0 : R))] [NoZeroSMulDivisors R M]
(x : M) : NeBot (𝓝[≠] x) | case intro.refine_2
R : Type u_1
M : Type u_2
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace R
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommGroup M
inst✝⁵ : ContinuousAdd M
inst✝⁴ : Module R M
inst✝³ : ContinuousSMul R M
inst✝² : Nontrivial M
inst✝¹ : (𝓝[≠] 0).NeBot
inst✝ : NoZeroSMulDivisors R M
x y : M
hy : y ≠ 0
⊢ ∀ a ∈ ... | intro c hc | case intro.refine_2
R : Type u_1
M : Type u_2
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace R
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommGroup M
inst✝⁵ : ContinuousAdd M
inst✝⁴ : Module R M
inst✝³ : ContinuousSMul R M
inst✝² : Nontrivial M
inst✝¹ : (𝓝[≠] 0).NeBot
inst✝ : NoZeroSMulDivisors R M
x y : M
hy : y ≠ 0
c : R
hc... | 89c7f0c38835ce78 |
Metric.totallyBounded_of_finite_discretization | Mathlib/Topology/MetricSpace/Pseudo/Basic.lean | theorem totallyBounded_of_finite_discretization {s : Set α}
(H : ∀ ε > (0 : ℝ),
∃ (β : Type u) (_ : Fintype β) (F : s → β), ∀ x y, F x = F y → dist (x : α) y < ε) :
TotallyBounded s | case inr.intro.intro.intro.intro
α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
H : ∀ ε > 0, ∃ β x F, ∀ (x y : ↑s), F x = F y → dist ↑x ↑y < ε
x0 : α
hx0 : x0 ∈ s
this✝ : Inhabited ↑s
ε : ℝ
ε0 : ε > 0
β : Type u
fβ : Fintype β
F : ↑s → β
hF : ∀ (x y : ↑s), F x = F y → dist ↑x ↑y < ε
Finv : β → ↑s := Function.invFun F... | exact ⟨_, ⟨F ⟨x, xs⟩, rfl⟩, hF _ _ this.symm⟩ | no goals | 1adb2e5de1e48f63 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.insertUnitInvariant_insertUnit | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem insertUnitInvariant_insertUnit {n : Nat} (assignments0 : Array Assignment)
(assignments0_size : assignments0.size = n) (units : Array (Literal (PosFin n)))
(assignments : Array Assignment) (assignments_size : assignments.size = n)
(foundContradiction : Bool) (l : Literal (PosFin n)) :
InsertUnit... | n : Nat
assignments0 : Array Assignment
assignments0_size : assignments0.size = n
units : Array (Literal (PosFin n))
assignments : Array Assignment
assignments_size : assignments.size = n
foundContradiction : Bool
l : Literal (PosFin n)
i : Fin n
i_in_bounds : ↑i < assignments.size
l_in_bounds : l.fst.val < assignments... | intro k_eq_j | n : Nat
assignments0 : Array Assignment
assignments0_size : assignments0.size = n
units : Array (Literal (PosFin n))
assignments : Array Assignment
assignments_size : assignments.size = n
foundContradiction : Bool
l : Literal (PosFin n)
i : Fin n
i_in_bounds : ↑i < assignments.size
l_in_bounds : l.fst.val < assignments... | 4e9908103f3f6d5f |
Filter.Tendsto.if | Mathlib/Order/Filter/Tendsto.lean | theorem Tendsto.if {l₁ : Filter α} {l₂ : Filter β} {f g : α → β} {p : α → Prop}
[∀ x, Decidable (p x)] (h₀ : Tendsto f (l₁ ⊓ 𝓟 { x | p x }) l₂)
(h₁ : Tendsto g (l₁ ⊓ 𝓟 { x | ¬p x }) l₂) :
Tendsto (fun x => if p x then f x else g x) l₁ l₂ | case pos
α : Type u_1
β : Type u_2
l₁ : Filter α
l₂ : Filter β
f g : α → β
p : α → Prop
inst✝ : (x : α) → Decidable (p x)
h₀ : ∀ s ∈ l₂, {x | x ∈ {x | p x} → x ∈ f ⁻¹' s} ∈ l₁
h₁ : ∀ s ∈ l₂, {x | x ∈ {x | ¬p x} → x ∈ g ⁻¹' s} ∈ l₁
s : Set β
hs : s ∈ l₂
x : α
hp₀ : p x → x ∈ f ⁻¹' s
hp₁ : ¬p x → x ∈ g ⁻¹' s
h : p x
⊢ f ... | exacts [hp₀ h, hp₁ h] | no goals | 75dfb26837786deb |
Module.Dual.eq_of_preReflection_mapsTo' | Mathlib/LinearAlgebra/Reflection.lean | /-- This rather technical-looking lemma exists because it is exactly what is needed to establish a
uniqueness result for root data. See the doc string of `Module.Dual.eq_of_preReflection_mapsTo` for
further remarks. -/
lemma Dual.eq_of_preReflection_mapsTo' [CharZero R] [NoZeroSMulDivisors R M]
{x : M} {Φ : Set M} ... | R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : CharZero R
inst✝ : NoZeroSMulDivisors R M
x : M
Φ : Set M
hΦ₁ : Finite ↑Φ
hx : x ∈ span R Φ
f g : Dual R M
hf₁ : f x = 2
hf₂ : MapsTo (⇑(preReflection x f)) Φ Φ
hg₁ : g x = 2
hg₂ : MapsTo (⇑(preReflection x g)) Φ Φ
Φ' : S... | have hΦ'₁ : Φ'.Finite := finite_range (inclusion Submodule.subset_span) | R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : CharZero R
inst✝ : NoZeroSMulDivisors R M
x : M
Φ : Set M
hΦ₁ : Finite ↑Φ
hx : x ∈ span R Φ
f g : Dual R M
hf₁ : f x = 2
hf₂ : MapsTo (⇑(preReflection x f)) Φ Φ
hg₁ : g x = 2
hg₂ : MapsTo (⇑(preReflection x g)) Φ Φ
Φ' : S... | c008ea6743424fb0 |
Filter.tendsto_iff_ptendsto_univ | Mathlib/Order/Filter/Partial.lean | theorem tendsto_iff_ptendsto_univ (l₁ : Filter α) (l₂ : Filter β) (f : α → β) :
Tendsto f l₁ l₂ ↔ PTendsto (PFun.res f Set.univ) l₁ l₂ | α : Type u
β : Type v
l₁ : Filter α
l₂ : Filter β
f : α → β
⊢ Tendsto f l₁ l₂ ↔ Tendsto f (l₁ ⊓ 𝓟 Set.univ) l₂ | simp [principal_univ] | no goals | 6e53700d2d681faa |
CategoryTheory.IsUniversalColimit.map_reflective | Mathlib/CategoryTheory/Limits/VanKampen.lean | theorem IsUniversalColimit.map_reflective
{Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Gr.Full] [Gr.Faithful]
{F : J ⥤ D} {c : Cocone (F ⋙ Gr)}
(H : IsUniversalColimit c)
[∀ X (f : X ⟶ Gl.obj c.pt), HasPullback (Gr.map f) (adj.unit.app c.pt)]
[∀ X (f : X ⟶ Gl.obj c.pt), PreservesLimit (cospan (Gr.map ... | J : Type v'
inst✝⁶ : Category.{u', v'} J
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_2
inst✝⁴ : Category.{u_3, u_2} D
Gl : C ⥤ D
Gr : D ⥤ C
adj : Gl ⊣ Gr
inst✝³ : Gr.Full
inst✝² : Gr.Faithful
F : J ⥤ D
c : Cocone (F ⋙ Gr)
H : IsUniversalColimit c
inst✝¹ : ∀ (X : D) (f : X ⟶ Gl.obj c.pt), HasPullback (Gr.map f) (ad... | dsimp [α', c''] | J : Type v'
inst✝⁶ : Category.{u', v'} J
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_2
inst✝⁴ : Category.{u_3, u_2} D
Gl : C ⥤ D
Gr : D ⥤ C
adj : Gl ⊣ Gr
inst✝³ : Gr.Full
inst✝² : Gr.Faithful
F : J ⥤ D
c : Cocone (F ⋙ Gr)
H : IsUniversalColimit c
inst✝¹ : ∀ (X : D) (f : X ⟶ Gl.obj c.pt), HasPullback (Gr.map f) (ad... | 0ba4821bd37956f9 |
Lean.Grind.of_eq_eq_true | Mathlib/.lake/packages/lean4/src/lean/Init/Grind/Lemmas.lean | theorem of_eq_eq_true {a b : Prop} (h : (a = b) = True) : (¬a ∨ b) ∧ (¬b ∨ a) | a b : Prop
h : (a = b) = True
⊢ (¬a ∨ b) ∧ (¬b ∨ a) | by_cases a <;> by_cases b <;> simp_all | no goals | 464d659c463afb9b |
CoxeterSystem.isReflection_of_mem_rightInvSeq | Mathlib/GroupTheory/Coxeter/Inversion.lean | theorem isReflection_of_mem_rightInvSeq (ω : List B) {t : W} (ht : t ∈ ris ω) :
cs.IsReflection t | B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
t : W
ht : t ∈ cs.rightInvSeq ω
⊢ cs.IsReflection t | induction' ω with i ω ih | case nil
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
t : W
ht : t ∈ cs.rightInvSeq []
⊢ cs.IsReflection t
case cons
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
t : W
i : B
ω : List B
ih : t ∈ cs.rightInvSeq ω → cs.IsReflection t
ht : t ∈... | 257c7dfe3c310537 |
PowerSeries.hasUnitMulPowIrreducibleFactorization | Mathlib/RingTheory/PowerSeries/Inverse.lean | theorem hasUnitMulPowIrreducibleFactorization :
HasUnitMulPowIrreducibleFactorization k⟦X⟧ :=
⟨X, And.intro X_irreducible
(by
intro f hf
use f.order.lift (order_finite_iff_ne_zero.mpr hf)
use Unit_of_divided_by_X_pow_order f
simp only [Unit_of_divided_by_X_pow_order_nonzero h... | case h
k : Type u_2
inst✝ : Field k
f : k⟦X⟧
hf : f ≠ 0
⊢ X ^ f.order.lift ⋯ * divided_by_X_pow_order hf = f | exact self_eq_X_pow_order_mul_divided_by_X_pow_order hf | no goals | d531f2850aecdb0c |
WeierstrassCurve.Δ_of_isCharThreeJNeZeroNF | Mathlib/AlgebraicGeometry/EllipticCurve/NormalForms.lean | theorem Δ_of_isCharThreeJNeZeroNF : W.Δ = -64 * W.a₂ ^ 3 * W.a₆ - 432 * W.a₆ ^ 2 | R : Type u_1
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : W.IsCharThreeJNeZeroNF
⊢ W.Δ = -64 * W.a₂ ^ 3 * W.a₆ - 432 * W.a₆ ^ 2 | simp | no goals | 20d638602dd623ef |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.insertUnitInvariant_insertUnit | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem insertUnitInvariant_insertUnit {n : Nat} (assignments0 : Array Assignment)
(assignments0_size : assignments0.size = n) (units : Array (Literal (PosFin n)))
(assignments : Array Assignment) (assignments_size : assignments.size = n)
(foundContradiction : Bool) (l : Literal (PosFin n)) :
InsertUnit... | n : Nat
assignments0 : Array Assignment
assignments0_size : assignments0.size = n
units : Array (Literal (PosFin n))
assignments : Array Assignment
assignments_size : assignments.size = n
foundContradiction : Bool
l : Literal (PosFin n)
i : Fin n
i_in_bounds : ↑i < assignments.size
l_in_bounds : l.fst.val < assignments... | exact h4 ⟨k.1, k_size⟩ k_ne_j | no goals | 4e9908103f3f6d5f |
Filter.map_prod_map_const_id_principal_coprod_principal | Mathlib/Order/Filter/Prod.lean | theorem map_prod_map_const_id_principal_coprod_principal {α β ι : Type*} (a : α) (b : β) (i : ι) :
map (Prod.map (fun _ : α => b) id) ((𝓟 {a}).coprod (𝓟 {i})) =
𝓟 (({b} : Set β) ×ˢ (univ : Set ι)) | case e_s.h.mk.mp.intro.mk.intro.refl
α : Type u_6
β : Type u_7
ι : Type u_8
a : α
b : β
i : ι
a'' : α
i'' : ι
left✝ : (a'', i'') ∈ ({a}ᶜ ×ˢ {i}ᶜ)ᶜ
⊢ ((fun x => b) a'', id i'') ∈ {b} ×ˢ univ | simp | no goals | e0ef1e097c541096 |
SimpleGraph.Walk.IsTrail.even_countP_edges_iff | Mathlib/Combinatorics/SimpleGraph/Trails.lean | theorem IsTrail.even_countP_edges_iff {u v : V} {p : G.Walk u v} (ht : p.IsTrail) (x : V) :
Even (p.edges.countP fun e => x ∈ e) ↔ u ≠ v → x ≠ u ∧ x ≠ v | case pos.inr
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v x u✝ w✝ : V
huv : G.Adj u✝ x
p : G.Walk x w✝
ht : p.IsTrail ∧ s(u✝, x) ∉ p.edges
ih : Even (List.countP (fun e => decide (x ∈ e)) p.edges) ↔ x ≠ w✝ → x ≠ x ∧ x ≠ w✝
⊢ Even (List.countP (fun e => decide (x ∈ e)) p.edges + 1) ↔ ¬u✝ = w✝ → ¬x = u✝ ∧ ¬x ... | rw [Nat.even_add_one, ih, ← not_iff_not] | case pos.inr
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v x u✝ w✝ : V
huv : G.Adj u✝ x
p : G.Walk x w✝
ht : p.IsTrail ∧ s(u✝, x) ∉ p.edges
ih : Even (List.countP (fun e => decide (x ∈ e)) p.edges) ↔ x ≠ w✝ → x ≠ x ∧ x ≠ w✝
⊢ ¬¬(x ≠ w✝ → x ≠ x ∧ x ≠ w✝) ↔ ¬(¬u✝ = w✝ → ¬x = u✝ ∧ ¬x = w✝) | a61699336dc9d9f7 |
FermatLastTheoremForThree_of_FermatLastTheoremThreeGen | Mathlib/NumberTheory/FLT/Three.lean | /-- To prove `FermatLastTheoremFor 3`, it is enough to prove `FermatLastTheoremForThreeGen`. -/
lemma FermatLastTheoremForThree_of_FermatLastTheoremThreeGen
[NumberField K] [IsCyclotomicExtension {3} ℚ K] :
FermatLastTheoremForThreeGen hζ → FermatLastTheoremFor 3 | K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
H : FermatLastTheoremForThreeGen hζ
a b c : ℤ
hc : c ≠ 0
ha : ¬3 ∣ a
hb : ¬3 ∣ b
x✝ : 3 ∣ c
hcoprime : IsCoprime a b
h : a ^ 3 + b ^ 3 = c ^ 3
x : ℤ
hx : c = 3 * x
⊢ ↑b = (algebraMap ℤ (𝓞 K)) b | simp | no goals | 09422095eb0b88f1 |
WittVector.frobenius_verschiebung | Mathlib/RingTheory/WittVector/Identities.lean | theorem frobenius_verschiebung (x : 𝕎 R) : frobenius (verschiebung x) = x * p | p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
⊢ frobenius (verschiebung x) = x * ↑p | have : IsPoly p fun {R} [CommRing R] x ↦ frobenius (verschiebung x) :=
IsPoly.comp (hg := frobenius_isPoly p) (hf := verschiebung_isPoly) | p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
this : IsPoly p fun {R} [CommRing R] x => frobenius (verschiebung x)
⊢ frobenius (verschiebung x) = x * ↑p | b0d0e500717e8b33 |
mdifferentiable_prod_iff | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | theorem mdifferentiable_prod_iff (f : M → M' × N') :
MDifferentiable I (I'.prod J') f ↔
MDifferentiable I I' (Prod.fst ∘ f) ∧ MDifferentiable I J' (Prod.snd ∘ f) :=
⟨fun h => ⟨h.fst, h.snd⟩, fun h => by convert h.1.prod_mk h.2⟩
| 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : Norme... | convert h.1.prod_mk h.2 | no goals | 64b26276d5b6af7d |
SimpleGraph.IsClique.card_le_chromaticNumber | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | theorem IsClique.card_le_chromaticNumber {s : Finset V} (h : G.IsClique s) :
s.card ≤ G.chromaticNumber | case inl
V : Type u
G : SimpleGraph V
s : Finset V
h : G.IsClique ↑s
hc : G.chromaticNumber = ⊤
⊢ ↑s.card ≤ ⊤ | exact le_top | no goals | 00cfefc46dc8a026 |
HurwitzZeta.hasSum_expZeta_of_one_lt_re | Mathlib/NumberTheory/LSeries/HurwitzZeta.lean | lemma hasSum_expZeta_of_one_lt_re (a : ℝ) {s : ℂ} (hs : 1 < re s) :
HasSum (fun n : ℕ ↦ cexp (2 * π * I * a * n) / n ^ s) (expZeta a s) | case h.e'_5.h
a : ℝ
s : ℂ
hs : 1 < s.re
n : ℕ
⊢ (Complex.cos (2 * ↑π * ↑a * ↑n) + Complex.sin (2 * ↑π * ↑a * ↑n) * I) / ↑n ^ s =
Complex.cos (2 * ↑π * ↑a * ↑n) / ↑n ^ s + I * (Complex.sin (2 * ↑π * ↑a * ↑n) / ↑n ^ s) | rw [add_div, mul_div, mul_comm _ I] | no goals | cfaaf270a3748487 |
List.dropWhile_eq_drop_findIdx_not | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/TakeDrop.lean | theorem dropWhile_eq_drop_findIdx_not {xs : List α} {p : α → Bool} :
dropWhile p xs = drop (xs.findIdx (fun a => !p a)) xs | case cons
α : Type u_1
p : α → Bool
x : α
xs : List α
ih : dropWhile p xs = drop (findIdx (fun a => !p a) xs) xs
⊢ dropWhile p (x :: xs) = drop (findIdx (fun a => !p a) (x :: xs)) (x :: xs) | simp only [dropWhile_cons, ih, findIdx_cons, cond_eq_if, Bool.not_eq_eq_eq_not, Bool.not_true] | case cons
α : Type u_1
p : α → Bool
x : α
xs : List α
ih : dropWhile p xs = drop (findIdx (fun a => !p a) xs) xs
⊢ (if p x = true then drop (findIdx (fun a => !p a) xs) xs else x :: xs) =
drop (if p x = false then 0 else findIdx (fun a => !p a) xs + 1) (x :: xs) | 25293e5c152108e9 |
Batteries.RBNode.balance2_All | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/WF.lean | theorem balance2_All {l : RBNode α} {v : α} {r : RBNode α} :
(balance2 l v r).All p ↔ p v ∧ l.All p ∧ r.All p | α : Type u_1
p : α → Prop
l : RBNode α
v : α
r : RBNode α
⊢ All p (l.balance2 v r) ↔ p v ∧ All p l ∧ All p r | unfold balance2 | α : Type u_1
p : α → Prop
l : RBNode α
v : α
r : RBNode α
⊢ All p
(match l, v, r with
| a, x, node red b y (node red c z d) => node red (node black a x b) y (node black c z d)
| a, x, node red (node red b y c) z d => node red (node black a x b) y (node black c z d)
| a, x, b => node black a x b)... | 4c869e1430d76a9f |
image_le_of_liminf_slope_right_le_deriv_boundary | Mathlib/Analysis/Calculus/MeanValue.lean | theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ}
(hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
(hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
-- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x`
(bound ... | case bound
f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
x✝ : ℝ
hx : x✝ ∈ Icc a b
r : ℝ
hr : r > 0
x : ℝ
a✝¹ : x ∈ Ico ... | rw [mul_one] | case bound
f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
x✝ : ℝ
hx : x✝ ∈ Icc a b
r : ℝ
hr : r > 0
x : ℝ
a✝¹ : x ∈ Ico ... | 75498d5eb15e3649 |
Monoid.PushoutI.inf_of_range_eq_base_range | Mathlib/GroupTheory/PushoutI.lean | theorem inf_of_range_eq_base_range
(hφ : ∀ i, Injective (φ i)) {i j : ι} (hij : i ≠ j) :
(of i).range ⊓ (of j).range = (base φ).range :=
le_antisymm
(by
intro x ⟨⟨g₁, hg₁⟩, ⟨g₂, hg₂⟩⟩
by_contra hx
have hx1 : x ≠ 1 | ι : Type u_1
G : ι → Type u_2
H : Type u_3
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
hφ : ∀ (i : ι), Injective ⇑(φ i)
i j : ι
hij : i ≠ j
x : PushoutI φ
g₁ : G i
hg₁ : (of i) g₁ = x
g₂ : G j
hg₂ : (of j) g₂ = x
hx : ¬x ∈ (base φ).range
hx1 : x ≠ 1
⊢ False | have hg₁1 : g₁ ≠ 1 :=
ne_of_apply_ne (of (φ := φ) i) (by simp_all) | ι : Type u_1
G : ι → Type u_2
H : Type u_3
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
hφ : ∀ (i : ι), Injective ⇑(φ i)
i j : ι
hij : i ≠ j
x : PushoutI φ
g₁ : G i
hg₁ : (of i) g₁ = x
g₂ : G j
hg₂ : (of j) g₂ = x
hx : ¬x ∈ (base φ).range
hx1 : x ≠ 1
hg₁1 : g₁ ≠ 1
⊢ False | 7b6e5bc8fcb5d579 |
ProbabilityTheory.indepFun_iff_map_prod_eq_prod_map_map | Mathlib/Probability/Independence/Basic.lean | theorem indepFun_iff_map_prod_eq_prod_map_map {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
[IsFiniteMeasure μ] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
IndepFun f g μ ↔ μ.map (fun ω ↦ (f ω, g ω)) = (μ.map f).prod (μ.map g) | case mpr
Ω : Type u_1
β : Type u_6
β' : Type u_7
_mΩ : MeasurableSpace Ω
μ : Measure Ω
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsFiniteMeasure μ
hf : AEMeasurable f μ
hg : AEMeasurable g μ
h₀ :
∀ {s : Set β} {t : Set β'},
MeasurableSet s →
MeasurableSet t →
μ (f ... | rw [(h₀ hs ht).1, (h₀ hs ht).2, h, Measure.prod_prod] | no goals | a8deec578f15ecfc |
List.set_set_perm' | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Perm.lean | theorem set_set_perm' {as : List α} {i j : Nat} (h₁ : i < as.length) (h₂ : i + j < as.length)
(hj : 0 < j) :
(as.set i as[i + j]).set (i + j) as[i] ~ as | α : Type u_1
as : List α
i j : Nat
h₁ : i < as.length
h₂ : i + j < as.length
hj : 0 < j
this : as = take i as ++ as[i] :: drop (i + 1) (take (i + j) as) ++ as[i + j] :: drop (i + j + 1) as
⊢ ((take i as ++ as[i] :: drop (i + 1) (take (i + j) as)).set i as[i + j]).length ≤ i + j | simp | α : Type u_1
as : List α
i j : Nat
h₁ : i < as.length
h₂ : i + j < as.length
hj : 0 < j
this : as = take i as ++ as[i] :: drop (i + 1) (take (i + j) as) ++ as[i + j] :: drop (i + j + 1) as
⊢ min i as.length + (min (i + j) as.length - (i + 1) + 1) ≤ i + j | 9b51820a9079599a |
MeasureTheory.Measure.volumeIoiPow_apply_Iio | Mathlib/MeasureTheory/Constructions/HaarToSphere.lean | lemma volumeIoiPow_apply_Iio (n : ℕ) (x : Ioi (0 : ℝ)) :
volumeIoiPow n (Iio x) = ENNReal.ofReal (x.1 ^ (n + 1) / (n + 1)) | case h
n : ℕ
x : ↑(Ioi 0)
hr₀ : 0 ≤ ↑x
y : ℝ
hy : y ∈ Ioc 0 ↑x
⊢ 0 y ≤ y ^ n | exact pow_nonneg hy.1.le _ | no goals | c49724f50ecd9bbf |
Array.setIfInBounds_comm | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem setIfInBounds_comm (a b : α)
{i j : Nat} (as : Array α) (h : i ≠ j) :
(as.setIfInBounds i a).setIfInBounds j b = (as.setIfInBounds j b).setIfInBounds i a | case mk
α : Type u_1
a b : α
i j : Nat
h : i ≠ j
toList✝ : List α
⊢ ({ toList := toList✝ }.setIfInBounds i a).setIfInBounds j b =
({ toList := toList✝ }.setIfInBounds j b).setIfInBounds i a | simp [List.set_comm _ _ _ h] | no goals | e519e3d928d90261 |
Nat.psp_from_prime_psp | Mathlib/NumberTheory/FermatPsp.lean | theorem psp_from_prime_psp {b : ℕ} (b_ge_two : 2 ≤ b) {p : ℕ} (p_prime : p.Prime)
(p_gt_two : 2 < p) (not_dvd : ¬p ∣ b * (b ^ 2 - 1)) : FermatPsp (psp_from_prime b p) b | b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hi_A : 1 < A
hi_B : 1 < B
hi_AB : 1 < A * B
hi_b : 0 < b
hi_p : 1 ≤ p
hi_bsquared : 0 < b ^ 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_... | have AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) := AB_id_helper _ _ b_ge_two p_odd | b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hi_A : 1 < A
hi_B : 1 < B
hi_AB : 1 < A * B
hi_b : 0 < b
hi_p : 1 ≤ p
hi_bsquared : 0 < b ^ 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_... | 6f5ab4acf8539f9d |
CategoryTheory.braiding_leftUnitor_aux₁ | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | theorem braiding_leftUnitor_aux₁ (X : C) :
(α_ (𝟙_ C) (𝟙_ C) X).hom ≫
(𝟙_ C ◁ (β_ X (𝟙_ C)).inv) ≫ (α_ _ X _).inv ≫ ((λ_ X).hom ▷ _) =
((λ_ _).hom ▷ X) ≫ (β_ X (𝟙_ C)).inv | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X : C
⊢ (α_ (𝟙_ C) (𝟙_ C) X).hom ≫ 𝟙_ C ◁ (β_ X (𝟙_ C)).inv ≫ (α_ (𝟙_ C) X (𝟙_ C)).inv ≫ (λ_ X).hom ▷ 𝟙_ C =
(λ_ (𝟙_ C)).hom ▷ X ≫ (β_ X (𝟙_ C)).inv | monoidal | no goals | 4eff5b8031eb1d20 |
Equiv.Perm.support_extend_domain | Mathlib/GroupTheory/Perm/Support.lean | theorem support_extend_domain (f : α ≃ Subtype p) {g : Perm α} :
support (g.extendDomain f) = g.support.map f.asEmbedding | case neg
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Fintype α
β : Type u_2
inst✝² : DecidableEq β
inst✝¹ : Fintype β
p : β → Prop
inst✝ : DecidablePred p
f : α ≃ Subtype p
g : Perm α
b : β
pb : ¬p b
⊢ ∀ (x : α), ¬g x = x → ¬f.asEmbedding x = b | rintro a _ rfl | case neg
α : Type u_1
inst✝⁴ : DecidableEq α
inst✝³ : Fintype α
β : Type u_2
inst✝² : DecidableEq β
inst✝¹ : Fintype β
p : β → Prop
inst✝ : DecidablePred p
f : α ≃ Subtype p
g : Perm α
a : α
a✝ : ¬g a = a
pb : ¬p (f.asEmbedding a)
⊢ False | 18d8756ce068490b |
Std.DHashMap.Internal.List.getValueCast_alterKey_self | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem getValueCast_alterKey_self (k : α) (f : Option (β k) → Option (β k))
(l : List ((a : α) × β a)) (hl : DistinctKeys l) (hc : containsKey k (alterKey k f l)) :
haveI hc' : (f (getValueCast? k l)).isSome | α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : LawfulBEq α
k : α
f : Option (β k) → Option (β k)
l : List ((a : α) × β a)
hl : DistinctKeys l
hc : containsKey k (alterKey k f l) = true
⊢ (if h : (k == k) = true then cast ⋯ ((f (getValueCast? k l)).get ⋯) else getValueCast k l ⋯) =
(f (getValueCast? k l)).get ⋯ | simp | no goals | 076ac8519892e6be |
ProbabilityTheory.Kernel.compProdFun_iUnion | Mathlib/Probability/Kernel/Composition/CompProd.lean | theorem compProdFun_iUnion (κ : Kernel α β) (η : Kernel (α × β) γ) [IsSFiniteKernel η] (a : α)
(f : ℕ → Set (β × γ)) (hf_meas : ∀ i, MeasurableSet (f i))
(hf_disj : Pairwise (Disjoint on f)) :
compProdFun κ η a (⋃ i, f i) = ∑' i, compProdFun κ η a (f i) | case h.hn
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
a : α
f : ℕ → Set (β × γ)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf_disj : Pairwise (Disjoint on f)
h_Union : (fun b => (η (a, b)) {c | (b... | have hbcj : {(b, c)} ⊆ f j := by rw [Set.singleton_subset_iff]; exact hsj hcs | case h.hn
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
a : α
f : ℕ → Set (β × γ)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf_disj : Pairwise (Disjoint on f)
h_Union : (fun b => (η (a, b)) {c | (b... | a1a5d4db1e3c8048 |
Subgroup.commProb_quotient_le | Mathlib/GroupTheory/CommutingProbability.lean | theorem Subgroup.commProb_quotient_le [H.Normal] : commProb (G ⧸ H) ≤ commProb G * Nat.card H | case hf
G : Type u_2
inst✝² : Group G
inst✝¹ : Finite G
H : Subgroup G
inst✝ : H.Normal
⊢ Function.Surjective (ConjClasses.map (QuotientGroup.mk' H)) | exact ConjClasses.map_surjective Quotient.mk''_surjective | no goals | 3c0dfef0c93289da |
Matroid.Indep.exists_insert_of_not_isBase | Mathlib/Data/Matroid/Basic.lean | theorem Indep.exists_insert_of_not_isBase (hI : M.Indep I) (hI' : ¬M.IsBase I) (hB : M.IsBase B) :
∃ e ∈ B \ I, M.Indep (insert e I) | case neg.intro.intro
α : Type u_1
M : Matroid α
B I : Set α
hI : M.Indep I
hI' : ¬M.IsBase I
hB : M.IsBase B
B' : Set α
hB' : M.IsBase B'
hIB' : I ⊆ B'
x : α
hxB' : x ∈ B'
hx : x ∉ I
hxB : x ∉ B
e : α
he : e ∈ B \ B'
hBase : M.IsBase (insert e (B' \ {x}))
⊢ ∃ e ∈ B \ I, M.Indep (insert e I) | exact ⟨e, ⟨he.1, not_mem_subset hIB' he.2⟩,
indep_iff.2 ⟨_, hBase, insert_subset_insert (subset_diff_singleton hIB' hx)⟩⟩ | no goals | f1e599d78e3b3839 |
HomologicalComplex.homologyMap_neg | Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean | @[simp]
lemma homologyMap_neg : homologyMap (-φ) i = -homologyMap φ i | C : Type u_1
ι : Type u_2
inst✝³ : Category.{u_3, u_1} C
inst✝² : Preadditive C
c : ComplexShape ι
K L : HomologicalComplex C c
φ : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
⊢ ShortComplex.homologyMap ((shortComplexFunctor C c i).map (-φ)) =
ShortComplex.homologyMap (-(shortComplexFunctor C c i).... | rfl | no goals | fa750e3b405f0555 |
Polynomial.eval_smul | Mathlib/Algebra/Polynomial/Eval/SMul.lean | theorem eval_smul [Monoid S] [DistribMulAction S R] [IsScalarTower S R R] (s : S) (p : R[X])
(x : R) : (s • p).eval x = s • p.eval x | R : Type u
S : Type v
inst✝³ : Semiring R
inst✝² : Monoid S
inst✝¹ : DistribMulAction S R
inst✝ : IsScalarTower S R R
s : S
p : R[X]
x : R
⊢ eval x (s • p) = s • eval x p | rw [← smul_one_smul R s p, eval, eval₂_smul, RingHom.id_apply, smul_one_mul] | no goals | b23a11eb9fe61d5d |
AffineSubspace.isPreconnected_setOf_sSameSide | Mathlib/Analysis/Convex/Side.lean | theorem isPreconnected_setOf_sSameSide (s : AffineSubspace ℝ P) (x : P) :
IsPreconnected { y | s.SSameSide x y } | case neg
V : Type u_2
P : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
x : P
h : (↑s).Nonempty
hx : x ∉ s
⊢ IsPreconnected {y | s.SSameSide x y} | exact (isConnected_setOf_sSameSide hx h).isPreconnected | no goals | b6c6fe063db720d1 |
Turing.TM1.stmts₁_trans | Mathlib/Computability/PostTuringMachine.lean | theorem stmts₁_trans {q₁ q₂ : Stmt Γ Λ σ} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ | case load.inr
Γ : Type u_1
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt Γ Λ σ
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Γ → σ → σ
q : Stmt Γ Λ σ
IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q
h₁₂ : q₁ ∈ stmts₁ q
⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q) | exact Finset.mem_insert_of_mem (IH h₁₂) | no goals | e84c592f40c9658c |
Cardinal.mul_eq_self | Mathlib/SetTheory/Cardinal/Arithmetic.lean | theorem mul_eq_self {c : Cardinal} (h : ℵ₀ ≤ c) : c * c = c | c : Cardinal.{u_1}
h : ℵ₀ ≤ c
⊢ c ≤ c * c | simpa only [mul_one] using mul_le_mul_left' (one_le_aleph0.trans h) c | no goals | 2b4cf21a89b59c38 |
IsCyclotomicExtension.Rat.nrComplexPlaces_eq_totient_div_two | Mathlib/NumberTheory/Cyclotomic/Embeddings.lean | theorem nrComplexPlaces_eq_totient_div_two [h : IsCyclotomicExtension {n} ℚ K] :
haveI := IsCyclotomicExtension.numberField {n} ℚ K
nrComplexPlaces K = φ n / 2 | case neg
n : ℕ+
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
h : IsCyclotomicExtension {n} ℚ K
this : NumberField K
hn : ¬2 < n
h1 : ¬1 < ↑n
⊢ φ ↑n = 1 | convert totient_one | case h.e'_2.h.e'_1
n : ℕ+
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
h : IsCyclotomicExtension {n} ℚ K
this : NumberField K
hn : ¬2 < n
h1 : ¬1 < ↑n
⊢ ↑n = 1 | 729816392c1bf851 |
Filter.tendsto_pure | Mathlib/Order/Filter/Tendsto.lean | theorem tendsto_pure {f : α → β} {a : Filter α} {b : β} :
Tendsto f a (pure b) ↔ ∀ᶠ x in a, f x = b | α : Type u_1
β : Type u_2
f : α → β
a : Filter α
b : β
⊢ Tendsto f a (pure b) ↔ ∀ᶠ (x : α) in a, f x = b | simp only [Tendsto, le_pure_iff, mem_map', mem_singleton_iff, Filter.Eventually] | no goals | 131b36d05fa3e228 |
bdd_le_mul_tendsto_zero' | Mathlib/Topology/Algebra/Order/Field.lean | theorem bdd_le_mul_tendsto_zero' {f g : α → 𝕜} (C : 𝕜) (hf : ∀ᶠ x in l, |f x| ≤ C)
(hg : Tendsto g l (𝓝 0)) : Tendsto (fun x ↦ f x * g x) l (𝓝 0) | 𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hf : ∀ᶠ (x : α) in l, |f x| ≤ C
hg : Tendsto g l (𝓝 0)
hC : Tendsto (fun x => |C * g x|) l (𝓝 0)
⊢ Tendsto (abs ∘ fun x => f x * g x) l (𝓝 0) | apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hC | case hgf
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
C : 𝕜
hf : ∀ᶠ (x : α) in l, |f x| ≤ C
hg : Tendsto g l (𝓝 0)
hC : Tendsto (fun x => |C * g x|) l (𝓝 0)
⊢ ∀ᶠ (b : α) in l, 0 ≤ (abs ∘ fun x => f x * g x) b
case hfh
𝕜 : ... | 926bf10d1c354af6 |
CategoryTheory.Idempotents.isIdempotentComplete_iff_hasEqualizer_of_id_and_idempotent | Mathlib/CategoryTheory/Idempotents/Basic.lean | theorem isIdempotentComplete_iff_hasEqualizer_of_id_and_idempotent :
IsIdempotentComplete C ↔ ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p | case mpr
C : Type u_1
inst✝ : Category.{u_2, u_1} C
h : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p
X : C
p : X ⟶ X
hp : p ≫ p = p
this : HasEqualizer (𝟙 X) p
⊢ equalizer.ι (𝟙 X) p ≫ equalizer.lift p ⋯ = 𝟙 (equalizer (𝟙 X) p) | ext | case mpr.h
C : Type u_1
inst✝ : Category.{u_2, u_1} C
h : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p
X : C
p : X ⟶ X
hp : p ≫ p = p
this : HasEqualizer (𝟙 X) p
⊢ (equalizer.ι (𝟙 X) p ≫ equalizer.lift p ⋯) ≫ equalizer.ι (𝟙 X) p = 𝟙 (equalizer (𝟙 X) p) ≫ equalizer.ι (𝟙 X) p | b9521a8a1dfa70fd |
FractionalIdeal.finprod_heightOneSpectrum_factorization | Mathlib/RingTheory/DedekindDomain/Factorization.lean | theorem finprod_heightOneSpectrum_factorization {I : FractionalIdeal R⁰ K} (hI : I ≠ 0) {a : R}
{J : Ideal R} (haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J) :
∏ᶠ v : HeightOneSpectrum R, (v.asIdeal : FractionalIdeal R⁰ K) ^
((Associates.mk v.asIdeal).count (Associates.mk J).factors -
(As... | R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
⊢ ∏ᶠ (v : HeightOneSpectrum R),
↑v.asIdeal ^
... | have hJ := Ideal.finprod_heightOneSpectrum_factorization_coe K hJ_ne_zero | R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((As... | 61a26b1afc635e0a |
Fintype.sum_div_mul_card_choose_card | Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean | private lemma Fintype.sum_div_mul_card_choose_card :
∑ s : Finset α, (card α / ((card α - #s) * (card α).choose #s) : ℚ) =
card α * ∑ k ∈ range (card α), (↑k)⁻¹ + 1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
this :
∀ {x : ℕ},
∀ s ∈ powersetCard x univ,
↑(card α) / ((↑(card α) - ↑(#s)) * ↑((card α).choose #s)) = ↑(card α) / ((↑(card α) - ↑x) * ↑((card α).choose x))
⊢ ↑(card α) * ∑ i ∈ range (card α + 1), ↑((card α).choose i) / ((↑(card α) - ↑i) * ↑((card α).choo... | have (n) (hn : n ∈ range (card α + 1)) :
((card α).choose n / ((card α - n) * (card α).choose n) : ℚ) = (card α - n : ℚ)⁻¹ := by
rw [div_mul_cancel_right₀]
exact cast_ne_zero.2 (choose_pos <| mem_range_succ_iff.1 hn).ne' | α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
this✝ :
∀ {x : ℕ},
∀ s ∈ powersetCard x univ,
↑(card α) / ((↑(card α) - ↑(#s)) * ↑((card α).choose #s)) = ↑(card α) / ((↑(card α) - ↑x) * ↑((card α).choose x))
this : ∀ n ∈ range (card α + 1), ↑((card α).choose n) / ((↑(card α) - ↑n) * ↑((card α).choose n))... | 24303f04777eeb87 |
ContinuousMap.idealOfSet_ofIdeal_eq_closure | Mathlib/Topology/ContinuousMap/Ideals.lean | theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
idealOfSet 𝕜 (setOfIdeal I) = I.closure | case h.e'_5.h
X : Type u_1
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
g' : C(X, ℝ≥0)
hI'... | simp only [algebraMapCLM_coe, comp_apply, mul_apply, ContinuousMap.coe_coe, map_mul] | no goals | 50ad3ae1e6ad6128 |
NumberField.InfinitePlace.one_le_mult | Mathlib/NumberTheory/NumberField/Embeddings.lean | theorem one_le_mult {w : InfinitePlace K} : (1 : ℝ) ≤ mult w | K : Type u_2
inst✝ : Field K
w : InfinitePlace K
⊢ 1 ≤ w.mult | exact mult_pos | no goals | d7683f17d324e956 |
MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable | Mathlib/MeasureTheory/Integral/Layercake.lean | theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
(f_nn : 0 ≤ f) (f_mble : Measurable f)
(g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t) (g_mble : Measurable g)
(g_nn : ∀ t > 0, 0 ≤ g t) :
∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
∫⁻ t in Ioi 0, μ {a : α | ... | α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (volume.restrict (Ioi 0))] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0 ..s, g t... | apply Ioc_subset_Ioc_right | case h
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (volume.restrict (Ioi 0))] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0 .... | 6098eac6b53bede3 |
hasEigenvalue_toLin_diagonal_iff | Mathlib/LinearAlgebra/Eigenspace/Matrix.lean | /-- Eigenvalues of a diagonal linear operator are the diagonal entries. -/
lemma hasEigenvalue_toLin_diagonal_iff (d : n → R) {μ : R} [NoZeroSMulDivisors R M]
(b : Basis n R M) : HasEigenvalue (toLin b b (diagonal d)) μ ↔ ∃ i, d i = μ | case mp
R : Type u_1
n : Type u_2
M : Type u_3
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : Nontrivial R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
d : n → R
μ : R
inst✝ : NoZeroSMulDivisors R M
b : Basis n R M
this : ∀ (i : n), HasEigenvalue ((toLin b b) (diagonal d)) (d i)
hμ : ∀ (i : n), d... | have h_iSup : ⨆ μ ∈ Set.range d, eigenspace (toLin b b (diagonal d)) μ = ⊤ := by
rw [eq_top_iff, ← b.span_eq, Submodule.span_le]
rintro - ⟨i, rfl⟩
simp only [SetLike.mem_coe]
apply Submodule.mem_iSup_of_mem (d i)
apply Submodule.mem_iSup_of_mem ⟨i, rfl⟩
rw [mem_eigenspace_iff]
exact (hasEigenvector_toLin_... | case mp
R : Type u_1
n : Type u_2
M : Type u_3
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : Nontrivial R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
d : n → R
μ : R
inst✝ : NoZeroSMulDivisors R M
b : Basis n R M
this : ∀ (i : n), HasEigenvalue ((toLin b b) (diagonal d)) (d i)
hμ : ∀ (i : n), d... | 6f0451f89af97b11 |
Nat.pow_dvd_pow | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean | theorem pow_dvd_pow {m n : Nat} (a : Nat) (h : m ≤ n) : a ^ m ∣ a ^ n | case intro
m n a : Nat
h : m ≤ n
w✝ : Nat
h✝ : n = m + w✝
⊢ a ^ m ∣ a ^ n | case intro k p =>
subst p
rw [Nat.pow_add]
apply Nat.dvd_mul_right | no goals | 360da2672b6c5272 |
eventuallyEq_insert | Mathlib/Topology/Separation/Basic.lean | lemma eventuallyEq_insert [T1Space X] {s t : Set X} {x y : X} (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
(insert x s : Set X) =ᶠ[𝓝 x] (insert x t : Set X) | X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s t : Set X
x y : X
h : s =ᶠ[𝓝[{y}ᶜ] x] t
⊢ insert x s =ᶠ[𝓝 x] insert x t | simp_rw [eventuallyEq_set] at h ⊢ | X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s t : Set X
x y : X
h : ∀ᶠ (x : X) in 𝓝[{y}ᶜ] x, x ∈ s ↔ x ∈ t
⊢ ∀ᶠ (x_1 : X) in 𝓝 x, x_1 ∈ insert x s ↔ x_1 ∈ insert x t | 11cda613fdb15015 |
InnerProductGeometry.cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle | Mathlib/Geometry/Euclidean/Triangle.lean | theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) | case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
H1 :
Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖ =
Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * ... | simp (disch := field_simp_discharge) only [sub_div', div_div, mul_div_assoc',
div_mul_eq_mul_div, div_sub', neg_div', neg_sub, eq_div_iff, div_eq_iff] | case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
H1 :
Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖ =
Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * ... | 941d70c5b6a4a2d6 |
mulRothNumber_map_mul_right | Mathlib/Combinatorics/Additive/AP/Three/Defs.lean | theorem mulRothNumber_map_mul_right :
mulRothNumber (s.map <| mulRightEmbedding a) = mulRothNumber s | α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : CancelCommMonoid α
s : Finset α
a : α
⊢ mulRothNumber (map (mulRightEmbedding a) s) = mulRothNumber s | rw [← mulLeftEmbedding_eq_mulRightEmbedding, mulRothNumber_map_mul_left s a] | no goals | e5558770dc4ee284 |
Order.embedding_from_countable_to_dense | Mathlib/Order/CountableDenseLinearOrder.lean | theorem embedding_from_countable_to_dense [Countable α] [DenselyOrdered β] [Nontrivial β] :
Nonempty (α ↪o β) | case intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : LinearOrder β
inst✝² : Countable α
inst✝¹ : DenselyOrdered β
inst✝ : Nontrivial β
val✝ : Encodable α
x y : β
hxy : x < y
a : β
ha : x < a ∧ a < y
this : Nonempty ↑(Set.Ioo x y)
our_ideal : Ideal (PartialIso α ↑(Set.Ioo x y)) := ideal... | let F a := funOfIdeal a our_ideal (cofinal_meets_idealOfCofinals _ _ a) | case intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : LinearOrder β
inst✝² : Countable α
inst✝¹ : DenselyOrdered β
inst✝ : Nontrivial β
val✝ : Encodable α
x y : β
hxy : x < y
a : β
ha : x < a ∧ a < y
this : Nonempty ↑(Set.Ioo x y)
our_ideal : Ideal (PartialIso α ↑(Set.Ioo x y)) := ideal... | 39d355a3b7d10d3b |
AlgebraicGeometry.stalkMap_injective_of_isOpenMap_of_injective | Mathlib/AlgebraicGeometry/Morphisms/ClosedImmersion.lean | /-- If `f : X ⟶ Y` is open, injective, `X` is quasi-compact and `Y` is affine, then `f` is stalkwise
injective if it is injective on global sections. -/
lemma stalkMap_injective_of_isOpenMap_of_injective [CompactSpace X]
(hfopen : IsOpenMap f.base) (hfinj₁ : Function.Injective f.base)
(hfinj₂ : Function.Injecti... | X Y : Scheme
inst✝¹ : IsAffine Y
f : X ⟶ Y
inst✝ : CompactSpace ↑↑X.toPresheafedSpace
hfopen : IsOpenMap ⇑(ConcreteCategory.hom f.base)
hfinj₁ : Function.Injective ⇑(ConcreteCategory.hom f.base)
hfinj₂ : Function.Injective ⇑(ConcreteCategory.hom (Scheme.Hom.appTop f))
x : ↑↑X.toPresheafedSpace
φ : Γ(Y, ⊤) ⟶ Γ(X, ⊤) := ... | rw [← Scheme.Hom.appLE_map _ _ (homOfLE <| hwle i).op, ← Scheme.Hom.map_appLE _ le_rfl w.op] | X Y : Scheme
inst✝¹ : IsAffine Y
f : X ⟶ Y
inst✝ : CompactSpace ↑↑X.toPresheafedSpace
hfopen : IsOpenMap ⇑(ConcreteCategory.hom f.base)
hfinj₁ : Function.Injective ⇑(ConcreteCategory.hom f.base)
hfinj₂ : Function.Injective ⇑(ConcreteCategory.hom (Scheme.Hom.appTop f))
x : ↑↑X.toPresheafedSpace
φ : Γ(Y, ⊤) ⟶ Γ(X, ⊤) := ... | b31e93c621949e44 |
Tactic.NormNum.int_lcm_helper | Mathlib/Tactic/NormNum/GCD.lean | theorem int_lcm_helper {x y : ℤ} {x' y' d : ℕ}
(hx : x.natAbs = x') (hy : y.natAbs = y') (h : Nat.lcm x' y' = d) :
Int.lcm x y = d | x y : ℤ
⊢ x.lcm y = x.natAbs.lcm y.natAbs | rw [Int.lcm_def] | no goals | ca7b0ffa0f7e09e8 |
ascPochhammer_eval_eq_zero_iff | Mathlib/RingTheory/Polynomial/Pochhammer.lean | theorem ascPochhammer_eval_eq_zero_iff [IsDomain R]
(n : ℕ) (r : R) : (ascPochhammer R n).eval r = 0 ↔ ∃ k < n, k = -r | case refine_1.zero
R : Type u
inst✝¹ : Ring R
inst✝ : IsDomain R
r : R
zero' : eval r (ascPochhammer R 0) = 0
⊢ ∃ k < 0, ↑k = -r | simp only [ascPochhammer_zero, Polynomial.eval_one, one_ne_zero] at zero' | no goals | 2471204549ab0950 |
TopologicalSpace.Opens.isOpenEmbedding_obj_top | Mathlib/Topology/Category/TopCat/Opens.lean | theorem isOpenEmbedding_obj_top {X : TopCat} (U : Opens X) :
U.isOpenEmbedding.isOpenMap.functor.obj ⊤ = U | case h
X : TopCat
U : Opens ↑X
⊢ ↑(⋯.functor.obj ⊤) = ↑U | exact Set.image_univ.trans Subtype.range_coe | no goals | 697e28683ee6a884 |
div_eq_of_eq_mul' | Mathlib/Algebra/Group/Basic.lean | theorem div_eq_of_eq_mul' {a b c : G} (h : a = b * c) : a / b = c | G : Type u_3
inst✝ : CommGroup G
a b c : G
h : a = b * c
⊢ a / b = c | rw [h, div_eq_mul_inv, mul_comm, inv_mul_cancel_left] | no goals | 4bece14ce3704ff7 |
MeasureTheory.setLIntegral_mono_ae | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem setLIntegral_mono_ae {s : Set α} {f g : α → ℝ≥0∞} (hg : AEMeasurable g (μ.restrict s))
(hfg : ∀ᵐ x ∂μ, x ∈ s → f x ≤ g x) : ∫⁻ x in s, f x ∂μ ≤ ∫⁻ x in s, g x ∂μ | α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
f g : α → ℝ≥0∞
hg : AEMeasurable g (μ.restrict s)
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → f x ≤ g x
⊢ ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ | rcases exists_measurable_le_lintegral_eq (μ.restrict s) f with ⟨f', hf'm, hle, hf'⟩ | case intro.intro.intro
α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
f g : α → ℝ≥0∞
hg : AEMeasurable g (μ.restrict s)
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → f x ≤ g x
f' : α → ℝ≥0∞
hf'm : Measurable f'
hle : f' ≤ f
hf' : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, f' a ∂μ
⊢ ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s,... | 7545edfba535befe |
LinearMap.BilinForm.exists_bilinForm_self_ne_zero | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | theorem exists_bilinForm_self_ne_zero [htwo : Invertible (2 : R)] {B : BilinForm R M}
(hB₁ : B ≠ 0) (hB₂ : B.IsSymm) : ∃ x, ¬B.IsOrtho x x | R : Type u_3
M : Type u_4
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
htwo : Invertible 2
B : BilinForm R M
hB₁ : B ≠ 0
hB₂ : IsSymm B
⊢ ∃ x, ¬IsOrtho B x x | lift B to QuadraticForm R M using hB₂ with Q | case intro
R : Type u_3
M : Type u_4
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
htwo : Invertible 2
B : BilinForm R M
Q : QuadraticMap R M R
hB₁✝ hB₁ : (QuadraticMap.associatedHom ℕ) Q ≠ 0
⊢ ∃ x, ¬IsOrtho ((QuadraticMap.associatedHom ℕ) Q) x x | bb71e5ad3533db24 |
List.forIn_eq_bindList | Mathlib/.lake/packages/batteries/Batteries/Data/List/Lemmas.lean | theorem forIn_eq_bindList [Monad m] [LawfulMonad m]
(f : α → β → m (ForInStep β)) (l : List α) (init : β) :
forIn l init f = ForInStep.run <$> (ForInStep.yield init).bindList f l | case cons
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → β → m (ForInStep β)
head✝ : α
tail✝ : List α
tail_ih✝ : ∀ (init : β), forIn tail✝ init f = ForInStep.run <$> ForInStep.bindList f tail✝ (ForInStep.yield init)
init : β
⊢ (do
let x ← f head✝ init
match ... | congr | case cons.e_a
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → β → m (ForInStep β)
head✝ : α
tail✝ : List α
tail_ih✝ : ∀ (init : β), forIn tail✝ init f = ForInStep.run <$> ForInStep.bindList f tail✝ (ForInStep.yield init)
init : β
⊢ (fun x =>
match x with
| Fo... | 2ccb71a25dde806b |
Finsupp.single_le_iff | Mathlib/Data/Finsupp/Order.lean | theorem single_le_iff {i : ι} {x : α} {f : ι →₀ α} : single i x ≤ f ↔ x ≤ f i :=
(le_iff' _ _ support_single_subset).trans <| by simp
| ι : Type u_1
α : Type u_3
inst✝² : AddCommMonoid α
inst✝¹ : PartialOrder α
inst✝ : CanonicallyOrderedAdd α
i : ι
x : α
f : ι →₀ α
⊢ (∀ i_1 ∈ {i}, (single i x) i_1 ≤ f i_1) ↔ x ≤ f i | simp | no goals | 2d15b9152132b026 |
Real.negMulLog_mul | Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean | lemma negMulLog_mul (x y : ℝ) : negMulLog (x * y) = y * negMulLog x + x * negMulLog y | case neg
x y : ℝ
hx : ¬x = 0
⊢ -(x * y * log (x * y)) = y * -(x * log x) + x * -(y * log y) | by_cases hy : y = 0 | case pos
x y : ℝ
hx : ¬x = 0
hy : y = 0
⊢ -(x * y * log (x * y)) = y * -(x * log x) + x * -(y * log y)
case neg
x y : ℝ
hx : ¬x = 0
hy : ¬y = 0
⊢ -(x * y * log (x * y)) = y * -(x * log x) + x * -(y * log y) | 041c9effb14d7dc7 |
IsOpen.exists_smooth_support_eq | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | theorem IsOpen.exists_smooth_support_eq {s : Set E} (hs : IsOpen s) :
∃ f : E → ℝ, f.support = s ∧ ContDiff ℝ ∞ f ∧ Set.range f ⊆ Set.Icc 0 1 | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : IsOpen s
h's : s.Nonempty
ι : Type (max 0 u_1) := { f // support f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ∞ f ∧ range f ⊆ Icc 0 1 }
T : Set ι
T_count : T.Countable
hT : ⋃ f ∈ T, support ↑f = s
g0 : ℕ → ι
hg ... | field_simp | no goals | 415898bf3b6091fb |
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