Loogle!

Result

Found 434 declarations mentioning AbsoluteValue. Of these, only the first 200 are shown.

AbsoluteValue 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
(R : Type u_4) (S : Type u_5) [Semiring R] [Semiring S] [PartialOrder S] : Type (max u_4 u_5)
AbsoluteValue.IsNontrivial 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} [Semiring R] {S : Type u_5} [Semiring S] [PartialOrder S] (v : AbsoluteValue R S) : Prop
AbsoluteValue.Simps.apply 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (f : AbsoluteValue R S) : R → S
AbsoluteValue.funLike 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] : FunLike (AbsoluteValue R S) R S
IsAbsoluteValue.toAbsoluteValue 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{S : Type u_4} [Semiring S] [PartialOrder S] {R : Type u_5} [Semiring R] (abv : R → S) [IsAbsoluteValue abv] : AbsoluteValue R S
AbsoluteValue.toMulHom 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (self : AbsoluteValue R S) : R →ₙ* S
AbsoluteValue.toMonoidWithZeroHom 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) [IsDomain S] [Nontrivial R] : R →*₀ S
AbsoluteValue.mulHomClass 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] : MulHomClass (AbsoluteValue R S) R S
AbsoluteValue.nonnegHomClass 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] : NonnegHomClass (AbsoluteValue R S) R S
AbsoluteValue.zeroHomClass 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] : ZeroHomClass (AbsoluteValue R S) R S
AbsoluteValue.monoidWithZeroHomClass 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] [IsDomain S] [Nontrivial R] : MonoidWithZeroHomClass (AbsoluteValue R S) R S
AbsoluteValue.toMonoidHom 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) [IsDomain S] [Nontrivial R] : R →* S
AbsoluteValue.abs 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{S : Type u_4} [Ring S] [LinearOrder S] [IsStrictOrderedRing S] : AbsoluteValue S S
AbsoluteValue.isAbsoluteValue 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{S : Type u_4} [Semiring S] [PartialOrder S] {R : Type u_5} [Semiring R] (abv : AbsoluteValue R S) : IsAbsoluteValue ⇑abv
AbsoluteValue.instInhabited 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{S : Type u_4} [Ring S] [LinearOrder S] [IsStrictOrderedRing S] : Inhabited (AbsoluteValue S S)
AbsoluteValue.subadditiveHomClass 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] : SubadditiveHomClass (AbsoluteValue R S) R S
IsAbsoluteValue.toAbsoluteValue_apply 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{S : Type u_4} [Semiring S] [PartialOrder S] {R : Type u_5} [Semiring R] (abv : R → S) [IsAbsoluteValue abv] (a✝ : R) : (IsAbsoluteValue.toAbsoluteValue abv) a✝ = abv a✝
AbsoluteValue.trivial 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} [Semiring R] [DecidablePred fun x => x = 0] [NoZeroDivisors R] {S : Type u_5} [Semiring S] [PartialOrder S] [IsOrderedRing S] [Nontrivial S] : AbsoluteValue R S
AbsoluteValue.nonneg 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) (x : R) : 0 ≤ abv x
AbsoluteValue.map_zero 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) : abv 0 = 0
AbsoluteValue.nonneg' 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (self : AbsoluteValue R S) (x : R) : 0 ≤ self.toFun x
AbsoluteValue.ne_zero 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) {x : R} : x ≠ 0 → abv x ≠ 0
AbsoluteValue.eq_zero 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) {x : R} : abv x = 0 ↔ x = 0
AbsoluteValue.ne_zero_iff 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) {x : R} : abv x ≠ 0 ↔ x ≠ 0
AbsoluteValue.eq_zero' 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (self : AbsoluteValue R S) (x : R) : self.toFun x = 0 ↔ x = 0
AbsoluteValue.pos 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) {x : R} : x ≠ 0 → 0 < abv x
AbsoluteValue.nonpos_iff 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) {x : R} : abv x ≤ 0 ↔ x = 0
AbsoluteValue.pos_iff 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) {x : R} : 0 < abv x ↔ x ≠ 0
AbsoluteValue.map_one 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) [IsDomain S] [Nontrivial R] : abv 1 = 1
AbsoluteValue.ext 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] ⦃f g : AbsoluteValue R S⦄ : (∀ (x : R), f x = g x) → f = g
AbsoluteValue.ext_iff 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] {f g : AbsoluteValue R S} : f = g ↔ ∀ (x : R), f x = g x
AbsoluteValue.apply_nat_le_self 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) [IsDomain S] [IsOrderedRing S] (n : ℕ) : abv ↑n ≤ ↑n
AbsoluteValue.coe_toMonoidWithZeroHom 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) [IsDomain S] [Nontrivial R] : ⇑abv.toMonoidWithZeroHom = ⇑abv
AbsoluteValue.not_isNontrivial_apply 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} [Semiring R] {S : Type u_5} [Semiring S] [PartialOrder S] {v : AbsoluteValue R S} (hv : ¬v.IsNontrivial) {x : R} (hx : x ≠ 0) : v x = 1
AbsoluteValue.coe_toMulHom 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) : ⇑abv.toMulHom = ⇑abv
AbsoluteValue.not_isNontrivial_iff 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} [Semiring R] {S : Type u_5} [Semiring S] [PartialOrder S] (v : AbsoluteValue R S) : ¬v.IsNontrivial ↔ ∀ (x : R), x ≠ 0 → v x = 1
AbsoluteValue.map_sub_eq_zero_iff 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Ring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) (a b : R) : abv (a - b) = 0 ↔ a = b
AbsoluteValue.isNontrivial_iff_ne_trivial 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} [Semiring R] {S : Type u_5} [Semiring S] [PartialOrder S] [IsOrderedRing S] [DecidablePred fun x => x = 0] [NoZeroDivisors R] [Nontrivial S] (v : AbsoluteValue R S) : v.IsNontrivial ↔ v ≠ AbsoluteValue.trivial
AbsoluteValue.addGroupSeminormClass 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_2} {S : Type u_3} [CommRing S] [PartialOrder S] [IsOrderedRing S] [Ring R] [NoZeroDivisors S] : AddGroupSeminormClass (AbsoluteValue R S) R S
AbsoluteValue.abs_apply 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{S : Type u_4} [Ring S] [LinearOrder S] [IsStrictOrderedRing S] (a : S) : AbsoluteValue.abs a = |a|
AbsoluteValue.instMulRingNormClassOfNontrivialOfIsDomain 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_2} {S : Type u_3} [CommRing S] [PartialOrder S] [IsOrderedRing S] [Ring R] [NoZeroDivisors S] [Nontrivial R] [IsDomain S] : MulRingNormClass (AbsoluteValue R S) R S
AbsoluteValue.coe_toMonoidHom 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) [IsDomain S] [Nontrivial R] : ⇑abv.toMonoidHom = ⇑abv
AbsoluteValue.map_pow 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) [IsDomain S] [Nontrivial R] (a : R) (n : ℕ) : abv (a ^ n) = abv a ^ n
AbsoluteValue.map_mul 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) (x y : R) : abv (x * y) = abv x * abv y
AbsoluteValue.listSum_le 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) [AddLeftMono S] (l : List R) : abv l.sum ≤ (List.map (⇑abv) l).sum
AbsoluteValue.map_one_of_isLeftRegular 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) (h : IsLeftRegular (abv 1)) : abv 1 = 1
AbsoluteValue.trivial_apply 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} [Semiring R] [DecidablePred fun x => x = 0] [NoZeroDivisors R] {S : Type u_5} [Semiring S] [PartialOrder S] [IsOrderedRing S] [Nontrivial S] {x : R} (hx : x ≠ 0) : AbsoluteValue.trivial x = 1
AbsoluteValue.add_le 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) (x y : R) : abv (x + y) ≤ abv x + abv y
AbsoluteValue.add_le' 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (self : AbsoluteValue R S) (x y : R) : self.toFun (x + y) ≤ self.toFun x + self.toFun y
AbsoluteValue.map_neg 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_2} {S : Type u_3} [CommRing S] [PartialOrder S] [IsOrderedRing S] [Ring R] (abv : AbsoluteValue R S) [NoZeroDivisors S] (a : R) : abv (-a) = abv a
AbsoluteValue.apply_natAbs_eq 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_2} {S : Type u_3} [CommRing S] [PartialOrder S] [IsOrderedRing S] [Ring R] (abv : AbsoluteValue R S) [NoZeroDivisors S] (x : ℤ) : abv ↑x.natAbs = abv ↑x
AbsoluteValue.le_sub 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Ring R] [Ring S] [PartialOrder S] [IsOrderedRing S] (abv : AbsoluteValue R S) (a b : R) : abv a - abv b ≤ abv (a - b)
AbsoluteValue.map_sub 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_2} {S : Type u_3} [CommRing S] [PartialOrder S] [IsOrderedRing S] [Ring R] (abv : AbsoluteValue R S) [NoZeroDivisors S] (a b : R) : abv (a - b) = abv (b - a)
AbsoluteValue.sub_le 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Ring R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) (a b c : R) : abv (a - c) ≤ abv (a - b) + abv (b - c)
AbsoluteValue.le_add 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_2} {S : Type u_3} [CommRing S] [PartialOrder S] [IsOrderedRing S] [Ring R] (abv : AbsoluteValue R S) [NoZeroDivisors S] (a b : R) : abv a - abv b ≤ abv (a + b)
AbsoluteValue.sub_le_add 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_2} {S : Type u_3} [CommRing S] [PartialOrder S] [IsOrderedRing S] [Ring R] (abv : AbsoluteValue R S) [NoZeroDivisors S] (a b : R) : abv (a - b) ≤ abv a + abv b
AbsoluteValue.mk 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (toMulHom : R →ₙ* S) (nonneg' : ∀ (x : R), 0 ≤ toMulHom.toFun x) (eq_zero' : ∀ (x : R), toMulHom.toFun x = 0 ↔ x = 0) (add_le' : ∀ (x y : R), toMulHom.toFun (x + y) ≤ toMulHom.toFun x + toMulHom.toFun y) : AbsoluteValue R S
AbsoluteValue.IsNontrivial.exists_abv_gt_one 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_2} {S : Type u_3} [Field R] [Semifield S] [LinearOrder S] [IsStrictOrderedRing S] [ExistsAddOfLE S] {v : AbsoluteValue R S} (h : v.IsNontrivial) : ∃ x, 1 < v x
AbsoluteValue.IsNontrivial.exists_abv_lt_one 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_2} {S : Type u_3} [Field R] [Semifield S] [LinearOrder S] [IsStrictOrderedRing S] [ExistsAddOfLE S] {v : AbsoluteValue R S} (h : v.IsNontrivial) : ∃ x, x ≠ 0 ∧ v x < 1
AbsoluteValue.eq_on_nat_iff_eq_on_int 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_2} {S : Type u_3} [CommRing S] [PartialOrder S] [IsOrderedRing S] [Ring R] [NoZeroDivisors S] {f g : AbsoluteValue R S} : (∀ (n : ℕ), f ↑n = g ↑n) ↔ ∀ (n : ℤ), f ↑n = g ↑n
AbsoluteValue.coe_mk 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [PartialOrder S] (f : R →ₙ* S) {h₁ : ∀ (x : R), 0 ≤ f.toFun x} {h₂ : ∀ (x : R), f.toFun x = 0 ↔ x = 0} {h₃ : ∀ (x y : R), f.toFun (x + y) ≤ f.toFun x + f.toFun y} : ⇑{ toMulHom := f, nonneg' := h₁, eq_zero' := h₂, add_le' := h₃ } = ⇑f
AbsoluteValue.abs_abv_sub_le_abv_sub 📋 Mathlib.Algebra.Order.AbsoluteValue.Basic
{R : Type u_4} {S : Type u_5} [Ring R] [CommRing S] [LinearOrder S] [IsStrictOrderedRing S] (abv : AbsoluteValue R S) (a b : R) : |abv a - abv b| ≤ abv (a - b)
AbsoluteValue.sum_le 📋 Mathlib.Algebra.Order.BigOperators.Ring.Finset
{ι : Type u_1} {R : Type u_2} {S : Type u_3} [Semiring R] [Semiring S] [PartialOrder S] [IsOrderedRing S] (abv : AbsoluteValue R S) (s : Finset ι) (f : ι → R) : abv (∑ i ∈ s, f i) ≤ ∑ i ∈ s, abv (f i)
AbsoluteValue.map_prod 📋 Mathlib.Algebra.Order.BigOperators.Ring.Finset
{ι : Type u_1} {R : Type u_2} {S : Type u_3} [CommSemiring R] [Nontrivial R] [CommRing S] [LinearOrder S] [IsStrictOrderedRing S] (abv : AbsoluteValue R S) (f : ι → R) (s : Finset ι) : abv (∏ i ∈ s, f i) = ∏ i ∈ s, abv (f i)
AbsoluteValue.comp 📋 Mathlib.Topology.Algebra.Ring.Basic
{R : Type u_1} {S : Type u_2} {T : Type u_3} [Semiring T] [Semiring R] [Semiring S] [PartialOrder S] (v : AbsoluteValue R S) {f : T →+* R} (hf : Function.Injective ⇑f) : AbsoluteValue T S
AbsoluteValue.comp_apply 📋 Mathlib.Topology.Algebra.Ring.Basic
{R : Type u_1} {S : Type u_2} {T : Type u_3} [Semiring T] [Semiring R] [Semiring S] [PartialOrder S] (v : AbsoluteValue R S) {f : T →+* R} (hf : Function.Injective ⇑f) (x : T) : (v.comp hf) x = v (f x)
AbsoluteValue.toNormedRing 📋 Mathlib.Analysis.Normed.Ring.Basic
{R : Type u_5} [Ring R] (v : AbsoluteValue R ℝ) : NormedRing R
AbsoluteValue.toNormedField 📋 Mathlib.Analysis.Normed.Field.Basic
{K : Type u_3} [Field K] (v : AbsoluteValue K ℝ) : NormedField K
AbsoluteValue.tendsto_div_one_add_pow_nhds_one 📋 Mathlib.Analysis.SpecificLimits.Normed
{R : Type u_2} {S : Type u_3} [Field R] [Field S] [LinearOrder S] [TopologicalSpace S] [IsStrictOrderedRing S] [Archimedean S] [_i : OrderTopology S] {v : AbsoluteValue R S} {a : R} (ha : v a < 1) : Filter.Tendsto (fun n => v (1 / (1 + a ^ n))) Filter.atTop (nhds 1)
AbsoluteValue.tendsto_div_one_add_pow_nhds_zero 📋 Mathlib.Analysis.SpecificLimits.Normed
{R : Type u_2} {S : Type u_3} [Field R] [Field S] [LinearOrder S] [TopologicalSpace S] [IsStrictOrderedRing S] [Archimedean S] [_i : OrderTopology S] {v : AbsoluteValue R S} {a : R} (ha : 1 < v a) : Filter.Tendsto (fun n => v (1 / (1 + a ^ n))) Filter.atTop (nhds 0)
AbsoluteValue.IsEuclidean 📋 Mathlib.Algebra.Order.AbsoluteValue.Euclidean
{R : Type u_1} {S : Type u_2} [EuclideanDomain R] [Semiring S] [PartialOrder S] (abv : AbsoluteValue R S) : Prop
AbsoluteValue.IsEuclidean.map_lt_map_iff 📋 Mathlib.Algebra.Order.AbsoluteValue.Euclidean
{R : Type u_1} {S : Type u_2} [EuclideanDomain R] [Semiring S] [PartialOrder S] {abv : AbsoluteValue R S} {x y : R} (h : abv.IsEuclidean) : abv x < abv y ↔ EuclideanDomain.r x y
AbsoluteValue.IsEuclidean.map_lt_map_iff' 📋 Mathlib.Algebra.Order.AbsoluteValue.Euclidean
{R : Type u_1} {S : Type u_2} [EuclideanDomain R] [Semiring S] [PartialOrder S] {abv : AbsoluteValue R S} (self : abv.IsEuclidean) {x y : R} : abv x < abv y ↔ EuclideanDomain.r x y
AbsoluteValue.IsEuclidean.mk 📋 Mathlib.Algebra.Order.AbsoluteValue.Euclidean
{R : Type u_1} {S : Type u_2} [EuclideanDomain R] [Semiring S] [PartialOrder S] {abv : AbsoluteValue R S} (map_lt_map_iff' : ∀ {x y : R}, abv x < abv y ↔ EuclideanDomain.r x y) : abv.IsEuclidean
AbsoluteValue.IsEuclidean.sub_mod_lt 📋 Mathlib.Algebra.Order.AbsoluteValue.Euclidean
{R : Type u_1} {S : Type u_2} [EuclideanDomain R] [Semiring S] [PartialOrder S] {abv : AbsoluteValue R S} (h : abv.IsEuclidean) (a : R) {b : R} (hb : b ≠ 0) : abv (a % b) < abv b
Polynomial.cardPowDegree 📋 Mathlib.Algebra.Polynomial.Degree.CardPowDegree
{Fq : Type u_1} [Field Fq] [Fintype Fq] : AbsoluteValue (Polynomial Fq) ℤ
Polynomial.cardPowDegree_zero 📋 Mathlib.Algebra.Polynomial.Degree.CardPowDegree
{Fq : Type u_1} [Field Fq] [Fintype Fq] : Polynomial.cardPowDegree 0 = 0
Polynomial.cardPowDegree_nonzero 📋 Mathlib.Algebra.Polynomial.Degree.CardPowDegree
{Fq : Type u_1} [Field Fq] [Fintype Fq] (p : Polynomial Fq) (hp : p ≠ 0) : Polynomial.cardPowDegree p = ↑(Fintype.card Fq) ^ p.natDegree
Polynomial.cardPowDegree_apply 📋 Mathlib.Algebra.Polynomial.Degree.CardPowDegree
{Fq : Type u_1} [Field Fq] [Fintype Fq] [DecidableEq Fq] (p : Polynomial Fq) : Polynomial.cardPowDegree p = if p = 0 then 0 else ↑(Fintype.card Fq) ^ p.natDegree
IsDedekindDomain.HeightOneSpectrum.intAdicAbv 📋 Mathlib.RingTheory.DedekindDomain.AdicValuation
{R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) : AbsoluteValue R ℝ
IsDedekindDomain.HeightOneSpectrum.adicAbv 📋 Mathlib.RingTheory.DedekindDomain.AdicValuation
{R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) : AbsoluteValue K ℝ
IsDedekindDomain.HeightOneSpectrum.intAdicAbv_le_one 📋 Mathlib.RingTheory.DedekindDomain.AdicValuation
{R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (r : R) : (v.intAdicAbv hb) r ≤ 1
IsDedekindDomain.HeightOneSpectrum.isNonarchimedean_intAdicAbv 📋 Mathlib.RingTheory.DedekindDomain.AdicValuation
{R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) : IsNonarchimedean ⇑(v.intAdicAbv hb)
IsDedekindDomain.HeightOneSpectrum.isNonarchimedean_adicAbv 📋 Mathlib.RingTheory.DedekindDomain.AdicValuation
{R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) : IsNonarchimedean ⇑(v.adicAbv hb)
IsDedekindDomain.HeightOneSpectrum.intAdicAbv_eq_one_iff 📋 Mathlib.RingTheory.DedekindDomain.AdicValuation
{R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (r : R) : (v.intAdicAbv hb) r = 1 ↔ r ∉ v.asIdeal
IsDedekindDomain.HeightOneSpectrum.intAdicAbv_lt_one_iff 📋 Mathlib.RingTheory.DedekindDomain.AdicValuation
{R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (r : R) : (v.intAdicAbv hb) r < 1 ↔ r ∈ v.asIdeal
IsDedekindDomain.HeightOneSpectrum.adicAbv_coe_le_one 📋 Mathlib.RingTheory.DedekindDomain.AdicValuation
{R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (r : R) : (v.adicAbv hb) ((algebraMap R K) r) ≤ 1
IsDedekindDomain.HeightOneSpectrum.adicAbv_of_algebraMap 📋 Mathlib.RingTheory.DedekindDomain.AdicValuation
{R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (r : R) : (v.adicAbv hb) ((algebraMap R K) r) = (v.intAdicAbv hb) r
IsDedekindDomain.HeightOneSpectrum.adicAbv_coe_eq_one_iff 📋 Mathlib.RingTheory.DedekindDomain.AdicValuation
{R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (r : R) : (v.adicAbv hb) ((algebraMap R K) r) = 1 ↔ r ∉ v.asIdeal
IsDedekindDomain.HeightOneSpectrum.adicAbv_coe_lt_one_iff 📋 Mathlib.RingTheory.DedekindDomain.AdicValuation
{R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (r : R) : (v.adicAbv hb) ((algebraMap R K) r) < 1 ↔ r ∈ v.asIdeal
IsDedekindDomain.HeightOneSpectrum.adicAbv_of_mk' 📋 Mathlib.RingTheory.DedekindDomain.AdicValuation
{R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) {b : NNReal} (hb : 1 < b) (r : R) {s : ↥(nonZeroDivisors R)} : (v.adicAbv hb) (IsLocalization.mk' K r s) = (v.intAdicAbv hb) r / (v.intAdicAbv hb) ↑s
NumberField.place 📋 Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings
{K : Type u_1} [Field K] {A : Type u_2} [NormedDivisionRing A] (φ : K →+* A) : AbsoluteValue K ℝ
NumberField.ComplexEmbedding.place_conjugate 📋 Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings
{K : Type u_1} [Field K] (φ : K →+* ℂ) : NumberField.place (NumberField.ComplexEmbedding.conjugate φ) = NumberField.place φ
NumberField.place_apply 📋 Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings
{K : Type u_1} [Field K] {A : Type u_2} [NormedDivisionRing A] (φ : K →+* A) (x : K) : (NumberField.place φ) x = ‖φ x‖
NumberField.IsFinitePlace 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
{K : Type u_1} [Field K] [NumberField K] (w : AbsoluteValue K ℝ) : Prop
NumberField.HeightOneSpectrum.adicAbv 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] : AbsoluteValue K ℝ
NumberField.RingOfIntegers.HeightOneSpectrum.adicAbv 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] : AbsoluteValue K ℝ
NumberField.HeightOneSpectrum.NumberField.RingOfIntegers.HeightOneSpectrum.adicAbv 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] : AbsoluteValue K ℝ
NumberField.HeightOneSpectrum.isNonarchimedean_adicAbv 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] : IsNonarchimedean ⇑(NumberField.HeightOneSpectrum.adicAbv K v)
NumberField.RingOfIntegers.HeightOneSpectrum.isNonarchimedean_adicAbv 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] : IsNonarchimedean ⇑(NumberField.HeightOneSpectrum.adicAbv K v)
NumberField.HeightOneSpectrum.NumberField.RingOfIntegers.HeightOneSpectrum.isNonarchimedean_adicAbv 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] : IsNonarchimedean ⇑(NumberField.HeightOneSpectrum.adicAbv K v)
NumberField.HeightOneSpectrum.adicAbv_intCast_le_one 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] (n : ℤ) : (NumberField.HeightOneSpectrum.adicAbv K v) ↑n ≤ 1
NumberField.RingOfIntegers.HeightOneSpectrum.adicAbv_intCast_le_one 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] (n : ℤ) : (NumberField.HeightOneSpectrum.adicAbv K v) ↑n ≤ 1
NumberField.HeightOneSpectrum.NumberField.RingOfIntegers.HeightOneSpectrum.adicAbv_intCast_le_one 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] (n : ℤ) : (NumberField.HeightOneSpectrum.adicAbv K v) ↑n ≤ 1
NumberField.HeightOneSpectrum.adicAbv_natCast_le_one 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] (n : ℕ) : (NumberField.HeightOneSpectrum.adicAbv K v) ↑n ≤ 1
NumberField.RingOfIntegers.HeightOneSpectrum.adicAbv_natCast_le_one 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] (n : ℕ) : (NumberField.HeightOneSpectrum.adicAbv K v) ↑n ≤ 1
NumberField.HeightOneSpectrum.NumberField.RingOfIntegers.HeightOneSpectrum.adicAbv_natCast_le_one 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] (n : ℕ) : (NumberField.HeightOneSpectrum.adicAbv K v) ↑n ≤ 1
NumberField.FinitePlace.isFinitePlace 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
{K : Type u_1} [Field K] [NumberField K] (v : NumberField.FinitePlace K) : NumberField.IsFinitePlace ↑v
NumberField.HeightOneSpectrum.adicAbv_add_le_max 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] (x y : K) : (NumberField.HeightOneSpectrum.adicAbv K v) (x + y) ≤ max ((NumberField.HeightOneSpectrum.adicAbv K v) x) ((NumberField.HeightOneSpectrum.adicAbv K v) y)
NumberField.RingOfIntegers.HeightOneSpectrum.adicAbv_add_le_max 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] (x y : K) : (NumberField.HeightOneSpectrum.adicAbv K v) (x + y) ≤ max ((NumberField.HeightOneSpectrum.adicAbv K v) x) ((NumberField.HeightOneSpectrum.adicAbv K v) y)
NumberField.HeightOneSpectrum.NumberField.RingOfIntegers.HeightOneSpectrum.adicAbv_add_le_max 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] (x y : K) : (NumberField.HeightOneSpectrum.adicAbv K v) (x + y) ≤ max ((NumberField.HeightOneSpectrum.adicAbv K v) x) ((NumberField.HeightOneSpectrum.adicAbv K v) y)
NumberField.isFinitePlace_iff 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
{K : Type u_1} [Field K] [NumberField K] (v : AbsoluteValue K ℝ) : NumberField.IsFinitePlace v ↔ ∃ w, ↑w = v
NumberField.FinitePlace.coe_apply 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
{K : Type u_1} [Field K] [NumberField K] (v : NumberField.FinitePlace K) (x : K) : v x = ↑v x
NumberField.FinitePlace.norm_embedding 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
{K : Type u_1} [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] (x : K) : ‖(NumberField.FinitePlace.embedding v) x‖ = (NumberField.HeightOneSpectrum.adicAbv K v) x
NumberField.HeightOneSpectrum.adicAbv_def 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] {x : K} : (NumberField.HeightOneSpectrum.adicAbv K v) x = ↑((WithZeroMulInt.toNNReal ⋯) ((IsDedekindDomain.HeightOneSpectrum.valuation K v) x))
NumberField.RingOfIntegers.HeightOneSpectrum.adicAbv_def 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] {x : K} : (NumberField.HeightOneSpectrum.adicAbv K v) x = ↑((WithZeroMulInt.toNNReal ⋯) ((IsDedekindDomain.HeightOneSpectrum.valuation K v) x))
NumberField.HeightOneSpectrum.NumberField.RingOfIntegers.HeightOneSpectrum.adicAbv_def 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] {x : K} : (NumberField.HeightOneSpectrum.adicAbv K v) x = ↑((WithZeroMulInt.toNNReal ⋯) ((IsDedekindDomain.HeightOneSpectrum.valuation K v) x))
NumberField.toNNReal_valued_eq_adicAbv 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] (x : WithVal (IsDedekindDomain.HeightOneSpectrum.valuation K v)) : ↑((WithZeroMulInt.toNNReal ⋯) (Valued.v x)) = (NumberField.HeightOneSpectrum.adicAbv K v) ((WithVal.equiv (IsDedekindDomain.HeightOneSpectrum.valuation K v)) x)
NumberField.HeightOneSpectrum.toNNReal_valued_eq_adicAbv 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] (x : WithVal (IsDedekindDomain.HeightOneSpectrum.valuation K v)) : ↑((WithZeroMulInt.toNNReal ⋯) (Valued.v x)) = (NumberField.HeightOneSpectrum.adicAbv K v) ((WithVal.equiv (IsDedekindDomain.HeightOneSpectrum.valuation K v)) x)
NumberField.HeightOneSpectrum.NumberField.toNNReal_valued_eq_adicAbv 📋 Mathlib.NumberTheory.NumberField.Completion.FinitePlace
(K : Type u_1) [Field K] {R : Type u_3} [CommRing R] [Algebra R K] [IsDedekindDomain R] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) [Ring.HasFiniteQuotients R] [Infinite R] (x : WithVal (IsDedekindDomain.HeightOneSpectrum.valuation K v)) : ↑((WithZeroMulInt.toNNReal ⋯) (Valued.v x)) = (NumberField.HeightOneSpectrum.adicAbv K v) ((WithVal.equiv (IsDedekindDomain.HeightOneSpectrum.valuation K v)) x)
padicNormE 📋 Mathlib.NumberTheory.Padics.PadicNumbers
{p : ℕ} [hp : Fact (Nat.Prime p)] : AbsoluteValue ℚ_[p] ℚ
Padic.limSeq 📋 Mathlib.NumberTheory.Padics.PadicNumbers
{p : ℕ} [Fact (Nat.Prime p)] (f : CauSeq ℚ_[p] ⇑padicNormE) : ℕ → ℚ
Padic.padicNormE.is_norm 📋 Mathlib.NumberTheory.Padics.PadicNumbers
{p : ℕ} [hp : Fact (Nat.Prime p)] (q : ℚ_[p]) : ↑(padicNormE q) = ‖q‖
padicNormE.eq_padic_norm' 📋 Mathlib.NumberTheory.Padics.PadicNumbers
{p : ℕ} [Fact (Nat.Prime p)] (q : ℚ) : padicNormE ↑q = padicNorm p q
Padic.exi_rat_seq_conv_cauchy 📋 Mathlib.NumberTheory.Padics.PadicNumbers
{p : ℕ} [Fact (Nat.Prime p)] (f : CauSeq ℚ_[p] ⇑padicNormE) : IsCauSeq (padicNorm p) (Padic.limSeq f)
padicNormE.image' 📋 Mathlib.NumberTheory.Padics.PadicNumbers
{p : ℕ} [Fact (Nat.Prime p)] {q : ℚ_[p]} : q ≠ 0 → ∃ n, padicNormE q = ↑p ^ (-n)
Padic.rat_dense' 📋 Mathlib.NumberTheory.Padics.PadicNumbers
{p : ℕ} [Fact (Nat.Prime p)] (q : ℚ_[p]) {ε : ℚ} (hε : 0 < ε) : ∃ r, padicNormE (q - ↑r) < ε
padicNormE.defn 📋 Mathlib.NumberTheory.Padics.PadicNumbers
{p : ℕ} [Fact (Nat.Prime p)] (f : PadicSeq p) {ε : ℚ} (hε : 0 < ε) : ∃ N, ∀ i ≥ N, padicNormE (Padic.mk f - ↑(↑f i)) < ε
padicNormE.nonarchimedean' 📋 Mathlib.NumberTheory.Padics.PadicNumbers
{p : ℕ} [Fact (Nat.Prime p)] (q r : ℚ_[p]) : padicNormE (q + r) ≤ max (padicNormE q) (padicNormE r)
Padic.complete' 📋 Mathlib.NumberTheory.Padics.PadicNumbers
{p : ℕ} [Fact (Nat.Prime p)] (f : CauSeq ℚ_[p] ⇑padicNormE) : ∃ q, ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (q - ↑f i) < ε
Padic.complete'' 📋 Mathlib.NumberTheory.Padics.PadicNumbers
{p : ℕ} [Fact (Nat.Prime p)] (f : CauSeq ℚ_[p] ⇑padicNormE) : ∃ q, ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑f i - q) < ε
Padic.exi_rat_seq_conv 📋 Mathlib.NumberTheory.Padics.PadicNumbers
{p : ℕ} [Fact (Nat.Prime p)] (f : CauSeq ℚ_[p] ⇑padicNormE) {ε : ℚ} (hε : 0 < ε) : ∃ N, ∀ i ≥ N, padicNormE (↑f i - ↑(Padic.limSeq f i)) < ε
padicNormE.add_eq_max_of_ne' 📋 Mathlib.NumberTheory.Padics.PadicNumbers
{p : ℕ} [Fact (Nat.Prime p)] {q r : ℚ_[p]} : padicNormE q ≠ padicNormE r → padicNormE (q + r) = max (padicNormE q) (padicNormE r)
WithAbs 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) : Type u_1
WithAbs.normedRing 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} [Ring R] (v : AbsoluteValue R ℝ) : NormedRing (WithAbs v)
WithAbs.instInhabited 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) : Inhabited (WithAbs v)
WithAbs.instSemiring 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) : Semiring (WithAbs v)
WithAbs.ofAbs 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] {v : AbsoluteValue R S} (self : WithAbs v) : R
WithAbs.toAbs 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) (ofAbs : R) : WithAbs v
WithAbs.instNontrivial 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) [Nontrivial R] : Nontrivial (WithAbs v)
WithAbs.instUnique 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) [Unique R] : Unique (WithAbs v)
WithAbs.instCommSemiring 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [CommSemiring R] (v : AbsoluteValue R S) : CommSemiring (WithAbs v)
WithAbs.instRing 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Ring R] (v : AbsoluteValue R S) : Ring (WithAbs v)
WithAbs.instSMul 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] {R : Type u_3} {T : Type u_4} [Semiring R] (v : AbsoluteValue R S) [SMul R T] : SMul (WithAbs v) T
WithAbs.instSMul_1 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] {R : Type u_3} {T : Type u_4} [Semiring R] (v : AbsoluteValue R S) [SMul T R] : SMul T (WithAbs v)
WithAbs.instCommRing 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [CommRing R] (v : AbsoluteValue R S) : CommRing (WithAbs v)
WithAbs.ofAbs_bijective 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) : Function.Bijective WithAbs.ofAbs
WithAbs.ofAbs_injective 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) : Function.Injective WithAbs.ofAbs
WithAbs.ofAbs_surjective 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) : Function.Surjective WithAbs.ofAbs
WithAbs.toAbs_bijective 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) : Function.Bijective (WithAbs.toAbs v)
WithAbs.toAbs_injective 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) : Function.Injective (WithAbs.toAbs v)
WithAbs.toAbs_surjective 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) : Function.Surjective (WithAbs.toAbs v)
WithAbs.ofAbs_toAbs 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) (x : R) : (WithAbs.toAbs v x).ofAbs = x
AbsoluteValue.under 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{L : Type u_3} (K : Type u_4) {S : Type u_5} [CommSemiring K] [Semiring L] [Algebra K L] [FaithfulSMul K L] [PartialOrder S] [Semiring S] (w : AbsoluteValue L S) : AbsoluteValue K S
WithAbs.instAlgebra 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] {R : Type u_3} {T : Type u_4} [CommSemiring R] [Semiring T] [Algebra R T] (v : AbsoluteValue T S) : Algebra R (WithAbs v)
WithAbs.instAlgebra_right 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] {R : Type u_3} {T : Type u_4} [CommSemiring R] [Semiring T] [Algebra R T] (v : AbsoluteValue T S) : Algebra R (WithAbs v)
WithAbs.instModule_left 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] {R : Type u_3} {T : Type u_4} [Semiring R] (v : AbsoluteValue R S) [AddCommMonoid T] [Module R T] : Module (WithAbs v) T
WithAbs.moduleLeft 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] {R : Type u_3} {T : Type u_4} [Semiring R] (v : AbsoluteValue R S) [AddCommMonoid T] [Module R T] : Module (WithAbs v) T
AbsoluteValue.LiesOver 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{L : Type u_3} {K : Type u_4} {S : Type u_5} [CommSemiring K] [Semiring L] [Algebra K L] [FaithfulSMul K L] [PartialOrder S] [Semiring S] (w : AbsoluteValue L S) (v : AbsoluteValue K S) : Prop
WithAbs.instFaithfulSMul 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] {R : Type u_3} {T : Type u_4} [Semiring R] (v : AbsoluteValue R S) [SMul R T] [FaithfulSMul R T] : FaithfulSMul (WithAbs v) T
WithAbs.instFaithfulSMul_1 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] {R : Type u_3} {T : Type u_4} [Semiring R] (v : AbsoluteValue R S) [SMul T R] [FaithfulSMul T R] : FaithfulSMul T (WithAbs v)
WithAbs.algebraLeft 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] {R : Type u_3} (T : Type u_4) [CommSemiring R] [Semiring T] [Algebra R T] (v : AbsoluteValue R S) : Algebra (WithAbs v) T
WithAbs.instAlgebra_left 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] {R : Type u_3} (T : Type u_4) [CommSemiring R] [Semiring T] [Algebra R T] (v : AbsoluteValue R S) : Algebra (WithAbs v) T
WithAbs.toAbs_ofAbs 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) (x : WithAbs v) : WithAbs.toAbs v x.ofAbs = x
WithAbs.instModule 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] {R : Type u_3} {T : Type u_4} [Semiring R] (v : AbsoluteValue R S) [Semiring T] [Module T R] : Module T (WithAbs v)
WithAbs.instModule_right 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] {R : Type u_3} {T : Type u_4} [Semiring R] (v : AbsoluteValue R S) [Semiring T] [Module T R] : Module T (WithAbs v)
WithAbs.algEquiv 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] (R : Type u_3) {T : Type u_4} [CommSemiring R] [Semiring T] [Algebra R T] (v : AbsoluteValue T S) : WithAbs v ≃ₐ[R] T
AbsoluteValue.instLiesOverUnder 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{L : Type u_3} {K : Type u_4} {S : Type u_5} [CommSemiring K] [Semiring L] [Algebra K L] [FaithfulSMul K L] [PartialOrder S] [Semiring S] (w : AbsoluteValue L S) : w.LiesOver (AbsoluteValue.under K w)
WithAbs.instIsScalarTower 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] {R : Type u_3} {T : Type u_4} [Semiring R] (v : AbsoluteValue R S) {P : Type u_5} [SMul T P] [SMul R T] [SMul R P] [IsScalarTower R T P] : IsScalarTower (WithAbs v) T P
WithAbs.instIsScalarTower_1 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] {R : Type u_3} {T : Type u_4} [Semiring R] (v : AbsoluteValue R S) {P : Type u_5} [SMul P R] [SMul T R] [SMul P T] [IsScalarTower P T R] : IsScalarTower P T (WithAbs v)
WithAbs.instIsScalarTower_2 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] {R : Type u_3} {T : Type u_4} [Semiring R] (v : AbsoluteValue R S) {P : Type u_5} [SMul P R] [SMul P T] [SMul R T] [IsScalarTower P R T] : IsScalarTower P (WithAbs v) T
WithAbs.norm_eq_abv' 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} [Ring R] (v : AbsoluteValue R ℝ) (x : R) : ‖WithAbs.toAbs v x‖ = v x
WithAbs.norm_toAbs_eq 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} [Ring R] (v : AbsoluteValue R ℝ) (x : R) : ‖WithAbs.toAbs v x‖ = v x
AbsoluteValue.LiesOver.comp_eq 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{L : Type u_3} {K : Type u_4} {S : Type u_5} {inst✝ : CommSemiring K} {inst✝¹ : Semiring L} {inst✝² : Algebra K L} {inst✝³ : FaithfulSMul K L} {inst✝⁴ : PartialOrder S} {inst✝⁵ : Semiring S} (w : AbsoluteValue L S) (v : AbsoluteValue K S) [self : w.LiesOver v] : AbsoluteValue.under K w = v
AbsoluteValue.LiesOver.mk 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{L : Type u_3} {K : Type u_4} {S : Type u_5} [CommSemiring K] [Semiring L] [Algebra K L] [FaithfulSMul K L] [PartialOrder S] [Semiring S] {w : AbsoluteValue L S} {v : AbsoluteValue K S} (under_eq : AbsoluteValue.under K w = v) : w.LiesOver v
AbsoluteValue.LiesOver.under_eq 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{L : Type u_3} {K : Type u_4} {S : Type u_5} {inst✝ : CommSemiring K} {inst✝¹ : Semiring L} {inst✝² : Algebra K L} {inst✝³ : FaithfulSMul K L} {inst✝⁴ : PartialOrder S} {inst✝⁵ : Semiring S} (w : AbsoluteValue L S) (v : AbsoluteValue K S) [self : w.LiesOver v] : AbsoluteValue.under K w = v
WithAbs.ofAbs_zero 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) : WithAbs.ofAbs 0 = 0
AbsoluteValue.under_def 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{L : Type u_3} (K : Type u_4) {S : Type u_5} [CommSemiring K] [Semiring L] [Algebra K L] [FaithfulSMul K L] [PartialOrder S] [Semiring S] (w : AbsoluteValue L S) : AbsoluteValue.under K w = w.comp ⋯
WithAbs.smul_left_def 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] {R : Type u_3} {T : Type u_4} [Semiring R] (v : AbsoluteValue R S) [SMul R T] (x : WithAbs v) (t : T) : x • t = x.ofAbs • t
WithAbs.map 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) {T : Type u_3} [Semiring T] (w : AbsoluteValue T S) (f : R →+* T) : WithAbs v →+* WithAbs w
WithAbs.norm_eq_abv 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} [Ring R] (v : AbsoluteValue R ℝ) (x : WithAbs v) : ‖x‖ = v x.ofAbs
WithAbs.norm_eq_apply_ofAbs 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} [Ring R] (v : AbsoluteValue R ℝ) (x : WithAbs v) : ‖x‖ = v x.ofAbs
WithAbs.equiv 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) : WithAbs v ≃+* R
WithAbs.toAbs_zero 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) : WithAbs.toAbs v 0 = 0
WithAbs.linearEquiv 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] (R : Type u_3) {T : Type u_4} [Semiring R] [Semiring T] [Module R T] (v : AbsoluteValue T S) : WithAbs v ≃ₗ[R] T
WithAbs.toAbs_eq_zero 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) {x : R} : WithAbs.toAbs v x = 0 ↔ x = 0
WithAbs.ofAbs_eq_zero 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) {x : WithAbs v} : x.ofAbs = 0 ↔ x = 0
WithAbs.ofAbs_one 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) : WithAbs.ofAbs 1 = 1
WithAbs.smul_right_def 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] {R : Type u_3} {T : Type u_4} [Semiring R] (v : AbsoluteValue R S) [SMul T R] (t : T) (x : WithAbs v) : t • x = WithAbs.toAbs v (t • x.ofAbs)
WithAbs.map_id 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) : WithAbs.map v v (RingHom.id R) = RingHom.id (WithAbs v)
WithAbs.toAbs_one 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) : WithAbs.toAbs v 1 = 1
WithAbs.ofAbs_pow 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) (x : WithAbs v) (n : ℕ) : (x ^ n).ofAbs = x.ofAbs ^ n
WithAbs.toAbs_pow 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) (x : R) (n : ℕ) : WithAbs.toAbs v (x ^ n) = WithAbs.toAbs v x ^ n
WithAbs.toAbs_add 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) (x y : R) : WithAbs.toAbs v (x + y) = WithAbs.toAbs v x + WithAbs.toAbs v y
WithAbs.toAbs_mul 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) (x y : R) : WithAbs.toAbs v (x * y) = WithAbs.toAbs v x * WithAbs.toAbs v y
WithAbs.ofAbs_add 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) (x y : WithAbs v) : (x + y).ofAbs = x.ofAbs + y.ofAbs
WithAbs.ofAbs_mul 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v : AbsoluteValue R S) (x y : WithAbs v) : (x * y).ofAbs = x.ofAbs * y.ofAbs
WithAbs.algEquiv_apply 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{S : Type u_2} [Semiring S] [PartialOrder S] {R : Type u_3} {T : Type u_4} [CommSemiring R] [Semiring T] [Algebra R T] (v : AbsoluteValue T S) (x : WithAbs v) : (WithAbs.algEquiv R v) x = x.ofAbs
WithAbs.equivWithAbs 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Semiring R] (v w : AbsoluteValue R S) : WithAbs v ≃+* WithAbs w
WithAbs.ofAbs_neg 📋 Mathlib.Analysis.Normed.Ring.WithAbs
{R : Type u_1} {S : Type u_2} [Semiring S] [PartialOrder S] [Ring R] (v : AbsoluteValue R S) (x : WithAbs v) : (-x).ofAbs = -x.ofAbs

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

By constant:
🔍 Real.sin
finds all lemmas whose statement somehow mentions the sine function.
By lemma name substring:
🔍 "differ"
finds all lemmas that have "differ" somewhere in their lemma name.
By subexpression:
🔍 _ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

The pattern can also be non-linear, as in
🔍 Real.sqrt ?a * Real.sqrt ?a

If the pattern has parameters, they are matched in any order. Both of these will find List.map:
🔍 (?a -> ?b) -> List ?a -> List ?b
🔍 List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍 |- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍 |- _ < _ → tsum _ < tsum _
will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.
You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c