Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Data.Finite.Card
#align_import group_theory.subgroup.finite from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
variable {G : Type*} [Group G]
variable {A : Type*} [AddGroup A]
... | Mathlib/Algebra/Group/Subgroup/Finite.lean | 127 | 137 | theorem eq_top_of_card_eq [Finite H] (h : Nat.card H = Nat.card G) :
H = ⊤ := by |
have : Nonempty H := ⟨1, one_mem H⟩
have h' : Nat.card H ≠ 0 := Nat.card_pos.ne'
have : Finite G := (Nat.finite_of_card_ne_zero (h ▸ h'))
have : Fintype G := Fintype.ofFinite G
have : Fintype H := Fintype.ofFinite H
rw [Nat.card_eq_fintype_card, Nat.card_eq_fintype_card] at h
rw [SetLike.ext'_iff, coe_to... |
import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
#align list.length_enum_from List.enumFrom_length
#align list.length_enum List.enum_length
@[simp]
theorem get?_enumFrom :
∀ n (l : List α) m, get? (enumFrom n l) m = (get? l m).map fun a => (n + m, a)
| n, [], m => rfl
| n, a :: l, 0 =... | Mathlib/Data/List/Enum.lean | 169 | 170 | theorem enumFrom_eq_nil {n : ℕ} {l : List α} : List.enumFrom n l = [] ↔ l = [] := by |
cases l <;> simp
|
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {α : Type v} {β : Type w}
namespace Matrix
def col (w : m → α) : Matrix m Unit α :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 200 | 204 | theorem updateRow_apply [DecidableEq m] {i' : m} :
updateRow M i b i' j = if i' = i then b j else M i' j := by |
by_cases h : i' = i
· rw [h, updateRow_self, if_pos rfl]
· rw [updateRow_ne h, if_neg h]
|
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
inductive DihedralGroup (n : ℕ) : Type
| r : ZMod n → DihedralGroup n
| sr : ZMod n → DihedralGroup n
derivin... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 146 | 149 | theorem r_one_pow_n : r (1 : ZMod n) ^ n = 1 := by |
rw [r_one_pow, one_def]
congr 1
exact ZMod.natCast_self _
|
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 581 | 581 | theorem sub_zero (a : Ordinal) : a - 0 = a := by | simpa only [zero_add] using add_sub_cancel 0 a
|
import Mathlib.Init.Core
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
open Polynomial Algebra FiniteD... | Mathlib/NumberTheory/Cyclotomic/Basic.lean | 132 | 150 | theorem trans (C : Type w) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C]
[hS : IsCyclotomicExtension S A B] [hT : IsCyclotomicExtension T B C]
(h : Function.Injective (algebraMap B C)) : IsCyclotomicExtension (S ∪ T) A C := by |
refine ⟨fun hn => ?_, fun x => ?_⟩
· cases' hn with hn hn
· obtain ⟨b, hb⟩ := ((isCyclotomicExtension_iff _ _ _).1 hS).1 hn
refine ⟨algebraMap B C b, ?_⟩
exact hb.map_of_injective h
· exact ((isCyclotomicExtension_iff _ _ _).1 hT).1 hn
· refine adjoin_induction (((isCyclotomicExtension_iff T ... |
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topology.LocallyFinite
open Set Filter Topology TopologicalSpace Classical Function
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [Topolog... | Mathlib/Topology/Compactness/Compact.lean | 57 | 64 | theorem IsCompact.compl_mem_sets_of_nhdsWithin (hs : IsCompact s) {f : Filter X}
(hf : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, tᶜ ∈ f) : sᶜ ∈ f := by |
refine hs.compl_mem_sets fun x hx => ?_
rcases hf x hx with ⟨t, ht, hst⟩
replace ht := mem_inf_principal.1 ht
apply mem_inf_of_inter ht hst
rintro x ⟨h₁, h₂⟩ hs
exact h₂ (h₁ hs)
|
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Parity
#align_import algebra.group_power.order from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
-- We should need only a minimal development of sets in order to get her... | Mathlib/Algebra/Order/Ring/Basic.lean | 32 | 32 | theorem map_neg (x : R) : f (-x) = f x := by | rw [← neg_one_mul, map_mul, map_neg_one, one_mul]
|
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Basic
#align_import algebra.euclidean_domain.basic from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d0... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 203 | 204 | theorem xgcdAux_val (x y : R) : xgcdAux x 1 0 y 0 1 = (gcd x y, xgcd x y) := by |
rw [xgcd, ← xgcdAux_fst x y 1 0 0 1]
|
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.Tactic.IntervalCases
#align_import group_theory.p_gr... | Mathlib/GroupTheory/PGroup.lean | 54 | 65 | theorem iff_card [Fact p.Prime] [Fintype G] : IsPGroup p G ↔ ∃ n : ℕ, card G = p ^ n := by |
have hG : card G ≠ 0 := card_ne_zero
refine ⟨fun h => ?_, fun ⟨n, hn⟩ => of_card hn⟩
suffices ∀ q ∈ Nat.factors (card G), q = p by
use (card G).factors.length
rw [← List.prod_replicate, ← List.eq_replicate_of_mem this, Nat.prod_factors hG]
intro q hq
obtain ⟨hq1, hq2⟩ := (Nat.mem_factors hG).mp hq
... |
import Mathlib.Logic.Equiv.Option
import Mathlib.Order.RelIso.Basic
import Mathlib.Order.Disjoint
import Mathlib.Order.WithBot
import Mathlib.Tactic.Monotonicity.Attr
import Mathlib.Util.AssertExists
#align_import order.hom.basic from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
open ... | Mathlib/Order/Hom/Basic.lean | 945 | 947 | theorem refl_trans (e : α ≃o β) : (refl α).trans e = e := by |
ext x
rfl
|
import Mathlib.CategoryTheory.Closed.Cartesian
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
#align_import category_theory.closed.functor from "leanprover-community/mathlib"@"cea27692b3fdeb328a2ddba6aabf181754543184"
noncomputable secti... | Mathlib/CategoryTheory/Closed/Functor.lean | 100 | 103 | theorem uncurry_expComparison (A B : C) :
CartesianClosed.uncurry ((expComparison F A).app B) =
inv (prodComparison F _ _) ≫ F.map ((exp.ev _).app _) := by |
rw [uncurry_eq, expComparison_ev]
|
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
noncomputable section
open scoped Classical
open ... | Mathlib/Geometry/Euclidean/Basic.lean | 442 | 450 | theorem orthogonalProjection_vadd_eq_self {s : AffineSubspace ℝ P} [Nonempty s]
[HasOrthogonalProjection s.direction] {p : P} (hp : p ∈ s) {v : V} (hv : v ∈ s.directionᗮ) :
orthogonalProjection s (v +ᵥ p) = ⟨p, hp⟩ := by |
have h := vsub_orthogonalProjection_mem_direction_orthogonal s (v +ᵥ p)
rw [vadd_vsub_assoc, Submodule.add_mem_iff_right _ hv] at h
refine (eq_of_vsub_eq_zero ?_).symm
ext
refine Submodule.disjoint_def.1 s.direction.orthogonal_disjoint _ ?_ h
exact (_ : s.direction).2
|
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
noncomputable section
open scoped Classical
open Topology Filter
open TopologicalSpace Set Metric Filter Function
attribute [local simp... | Mathlib/Topology/MetricSpace/PiNat.lean | 247 | 251 | theorem cylinder_eq_res (x : ℕ → α) (n : ℕ) :
cylinder x n = { y | res y n = res x n } := by |
ext y
dsimp [cylinder]
rw [res_eq_res]
|
import Mathlib.Topology.Gluing
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits
#align_import algebraic_geometry.presheafed_space.gluing from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
set_option linter.uppercaseLean... | Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean | 456 | 488 | theorem π_ιInvApp_π (i j : D.J) (U : Opens (D.U i).carrier) :
D.diagramOverOpenπ U i ≫ D.ιInvAppπEqMap U ≫ D.ιInvApp U ≫ D.diagramOverOpenπ U j =
D.diagramOverOpenπ U j := by |
-- Porting note: originally, the proof of monotonicity was left a blank and proved in the end
-- but Lean 4 doesn't like this any more, so the proof is restructured
rw [← @cancel_mono (f := (componentwiseDiagram 𝖣.diagram.multispan _).map
(Quiver.Hom.op (WalkingMultispan.Hom.snd (i, j))) ≫ 𝟙 _) _ _ (by
... |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 80 | 81 | theorem preimage_coe_Ico_top : (some : α → WithTop α) ⁻¹' Ico a ⊤ = Ici a := by |
simp [← Ici_inter_Iio]
|
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.theta from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter
open Topology
namespace Asymptotics
set_option linter.uppercaseLean3 false -- is_Theta
v... | Mathlib/Analysis/Asymptotics/Theta.lean | 214 | 216 | theorem IsTheta.tendsto_zero_iff (h : f'' =Θ[l] g'') :
Tendsto f'' l (𝓝 0) ↔ Tendsto g'' l (𝓝 0) := by |
simp only [← isLittleO_one_iff ℝ, h.isLittleO_congr_left]
|
import Mathlib.CategoryTheory.Limits.Creates
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.IsConnected
#align_import category_theory.limits.constructions.over.connected from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
universe v u
-- morphism levels before o... | Mathlib/CategoryTheory/Limits/Constructions/Over/Connected.lean | 60 | 62 | theorem raised_cone_lowers_to_original [IsConnected J] {B : C} {F : J ⥤ Over B}
(c : Cone (F ⋙ forget B)) :
(forget B).mapCone (raiseCone c) = c := by | aesop_cat
|
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Topology.MetricSpace.ThickenedIndicator
open MeasureTheory Topology Metric Filter Set ENNReal NNReal
open scoped Topology ENNReal NNReal BoundedContinuousFunction
section auxiliary
namespace MeasureTheory
variable {Ω : Type*} [TopologicalSpace Ω] [Mea... | Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean | 110 | 119 | theorem tendsto_lintegral_thickenedIndicator_of_isClosed {Ω : Type*} [MeasurableSpace Ω]
[PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ] {F : Set Ω}
(F_closed : IsClosed F) {δs : ℕ → ℝ} (δs_pos : ∀ n, 0 < δs n)
(δs_lim : Tendsto δs atTop (𝓝 0)) :
Tendsto (fun n ↦ lin... |
apply measure_of_cont_bdd_of_tendsto_indicator μ F_closed.measurableSet
(fun n ↦ thickenedIndicator (δs_pos n) F) fun n ω ↦ thickenedIndicator_le_one (δs_pos n) F ω
have key := thickenedIndicator_tendsto_indicator_closure δs_pos δs_lim F
rwa [F_closed.closure_eq] at key
|
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 127 | 130 | theorem IsPerfectMatching.even_card {M : Subgraph G} [Fintype V] (h : M.IsPerfectMatching) :
Even (Fintype.card V) := by |
classical
simpa only [h.2.card_verts] using IsMatching.even_card h.1
|
import Mathlib.Algebra.Module.Equiv
import Mathlib.Algebra.Module.Hom
import Mathlib.Algebra.Module.Prod
import Mathlib.Algebra.Module.Submodule.Range
import Mathlib.Data.Set.Finite
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Tactic.Abel
#align_import linear_algebra.basic from "leanprover-c... | Mathlib/LinearAlgebra/Basic.lean | 138 | 138 | theorem ofEq_rfl : ofEq p p rfl = LinearEquiv.refl R p := by | ext; rfl
|
import Mathlib.Algebra.ContinuedFractions.Computation.Basic
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.computation.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
open Generali... | Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | 163 | 165 | theorem of_h_eq_intFractPair_seq1_fst_b : (of v).h = (IntFractPair.seq1 v).fst.b := by |
cases aux_seq_eq : IntFractPair.seq1 v
simp [of, aux_seq_eq]
|
import Mathlib.Order.Filter.Prod
#align_import order.filter.n_ary from "leanprover-community/mathlib"@"78f647f8517f021d839a7553d5dc97e79b508dea"
open Function Set
open Filter
namespace Filter
variable {α α' β β' γ γ' δ δ' ε ε' : Type*} {m : α → β → γ} {f f₁ f₂ : Filter α}
{g g₁ g₂ : Filter β} {h h₁ h₂ : Filt... | Mathlib/Order/Filter/NAry.lean | 160 | 160 | theorem map₂_right [NeBot f] : map₂ (fun _ y => y) f g = g := by | rw [map₂_swap, map₂_left]
|
import Mathlib.Data.Fintype.Card
import Mathlib.Computability.Language
import Mathlib.Tactic.NormNum
#align_import computability.DFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Computability
universe u v
-- Porting note: Required as `DFA` is used in mathlib3
set_option li... | Mathlib/Computability/DFA.lean | 98 | 98 | theorem mem_accepts (x : List α) : x ∈ M.accepts ↔ M.evalFrom M.start x ∈ M.accept := by | rfl
|
import Mathlib.GroupTheory.GroupAction.BigOperators
import Mathlib.Logic.Equiv.Fin
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Module.Prod
import Mathlib.Algebra.Module.Submodule.Ker
#align_import linear_algebra.pi from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
un... | Mathlib/LinearAlgebra/Pi.lean | 249 | 253 | theorem update_apply (f : (i : ι) → M₂ →ₗ[R] φ i) (c : M₂) (i j : ι) (b : M₂ →ₗ[R] φ i) :
(update f i b j) c = update (fun i => f i c) i (b c) j := by |
by_cases h : j = i
· rw [h, update_same, update_same]
· rw [update_noteq h, update_noteq h]
|
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 197 | 198 | theorem mul_ite {α} [Mul α] (P : Prop) [Decidable P] (a b c : α) :
(a * if P then b else c) = if P then a * b else a * c := by | split_ifs <;> rfl
|
import Mathlib.ModelTheory.Syntax
import Mathlib.ModelTheory.Semantics
import Mathlib.Algebra.Ring.Equiv
variable {α : Type*}
namespace FirstOrder
open FirstOrder
inductive ringFunc : ℕ → Type
| add : ringFunc 2
| mul : ringFunc 2
| neg : ringFunc 1
| zero : ringFunc 0
| one : ringFunc 0
deriving D... | Mathlib/ModelTheory/Algebra/Ring/Basic.lean | 190 | 192 | theorem realize_neg (x : ring.Term α) (v : α → R) :
Term.realize v (-x) = -Term.realize v x := by |
simp [neg_def, funMap_neg]
|
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.Disjoint
#align_import data.real.basic from "leanprover-community/mathlib"@"cb42593171ba005beaaf4549fcfe0dece9ada4c9"
open scoped Classical
open Pointwise CauSeq
namespace Real
... | Mathlib/Data/Real/Archimedean.lean | 356 | 360 | theorem not_bddAbove_coe : ¬ (BddAbove <| range (fun (x : ℚ) ↦ (x : ℝ))) := by |
dsimp only [BddAbove, upperBounds]
rw [Set.not_nonempty_iff_eq_empty]
ext
simpa using exists_rat_gt _
|
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import linear_algebra.trace from "leanprover-community/mathlib"@"4cf7ca0e69e048b006674cf4499e5c7d296a89e0"
noncomputable section
universe u v w
namespace LinearMap
open Matrix
open FiniteDimensional
open Tensor... | Mathlib/LinearAlgebra/Trace.lean | 294 | 307 | theorem trace_conj' (f : M →ₗ[R] M) (e : M ≃ₗ[R] N) : trace R N (e.conj f) = trace R M f := by |
classical
by_cases hM : ∃ s : Finset M, Nonempty (Basis s R M)
· obtain ⟨s, ⟨b⟩⟩ := hM
haveI := Module.Finite.of_basis b
haveI := (Module.free_def R M).mpr ⟨_, ⟨b⟩⟩
haveI := Module.Finite.of_basis (b.map e)
haveI := (Module.free_def R N).mpr ⟨_, ⟨(b.map e).reindex (e.toEquiv.image _)⟩⟩
rw [e.... |
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import number_theory.von_mangoldt from "leanprover-community/mathlib"@"c946d6097a6925ad16d7ec55677bbc977f9846de"
namespace ArithmeticFunction
open Finset Nat
open scoped Arit... | Mathlib/NumberTheory/VonMangoldt.lean | 131 | 131 | theorem zeta_mul_vonMangoldt : (ζ : ArithmeticFunction ℝ) * Λ = log := by | rw [mul_comm]; simp
|
import Mathlib.FieldTheory.Perfect
#align_import field_theory.perfect_closure from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v
open Function
section
variable (K : Type u) [CommRing K] (p : ℕ) [Fact p.Prime] [CharP K p]
@[mk_iff]
inductive PerfectClosure.R : ℕ × K → ℕ... | Mathlib/FieldTheory/PerfectClosure.lean | 354 | 362 | theorem mk_pow (x : ℕ × K) (n : ℕ) : mk K p x ^ n = mk K p (x.1, x.2 ^ n) := by |
induction n with
| zero =>
rw [pow_zero, pow_zero, one_def, mk_eq_iff]
exact ⟨0, by simp_rw [← coe_iterateFrobenius, map_one]⟩
| succ n ih =>
rw [pow_succ, pow_succ, ih, mk_mul_mk, mk_eq_iff]
exact ⟨0, by simp_rw [iterate_frobenius, add_zero, mul_pow, ← pow_mul,
← pow_add, mul_assoc, ← pow_... |
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι... | Mathlib/Data/Set/Lattice.lean | 917 | 920 | theorem biInter_inter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t := by |
haveI : Nonempty s := hs.to_subtype
simp [biInter_eq_iInter, ← iInter_inter]
|
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Topology Manifold
open Set Bundle
section DerivativesProperties
variable
{𝕜 : Type*} [NontriviallyNormedFiel... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 765 | 770 | theorem tangentMap_comp_at (p : TangentBundle I M) (hg : MDifferentiableAt I' I'' g (f p.1))
(hf : MDifferentiableAt I I' f p.1) :
tangentMap I I'' (g ∘ f) p = tangentMap I' I'' g (tangentMap I I' f p) := by |
simp only [tangentMap, mfld_simps]
rw [mfderiv_comp p.1 hg hf]
rfl
|
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Matrix
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Topology.UniformSpace.Matrix
#align_import analysis.normed_space.matrix_exponential from "l... | Mathlib/Analysis/NormedSpace/MatrixExponential.lean | 80 | 81 | theorem exp_diagonal (v : m → 𝔸) : exp 𝕂 (diagonal v) = diagonal (exp 𝕂 v) := by |
simp_rw [exp_eq_tsum, diagonal_pow, ← diagonal_smul, ← diagonal_tsum]
|
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RCLike
open scoped ComplexConjugate
variable {𝕜 E F G : Type... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 168 | 171 | theorem eq_adjoint_iff (A : E →L[𝕜] F) (B : F →L[𝕜] E) : A = B† ↔ ∀ x y, ⟪A x, y⟫ = ⟪x, B y⟫ := by |
refine ⟨fun h x y => by rw [h, adjoint_inner_left], fun h => ?_⟩
ext x
exact ext_inner_right 𝕜 fun y => by simp only [adjoint_inner_left, h x y]
|
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
noncomputable section
open Polynomial
namespace P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 133 | 143 | theorem coeff_hermite_of_odd_add {n k : ℕ} (hnk : Odd (n + k)) : coeff (hermite n) k = 0 := by |
induction' n with n ih generalizing k
· rw [zero_add k] at hnk
exact coeff_hermite_of_lt hnk.pos
· cases' k with k
· rw [Nat.succ_add_eq_add_succ] at hnk
rw [coeff_hermite_succ_zero, ih hnk, neg_zero]
· rw [coeff_hermite_succ_succ, ih, ih, mul_zero, sub_zero]
· rwa [Nat.succ_add_eq_add_su... |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 92 | 93 | theorem derivative_C_mul_X (a : R) : derivative (C a * X) = C a := by |
simp [C_mul_X_eq_monomial, derivative_monomial, Nat.cast_one, mul_one]
|
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
#align_import analysis.normed_space.multilinear from "leanprover-community/mathlib"@"f40476639bac089693a489c9e354ebd75dc0f886"
suppress_compilation
noncomputable section
open NNReal Finset Metric ContinuousMultilinearMap Fin Function
universe u v v' wE wE... | Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean | 290 | 293 | theorem ContinuousMultilinearMap.uncurry_curryRight (f : ContinuousMultilinearMap 𝕜 Ei G) :
f.curryRight.uncurryRight = f := by |
ext m
rw [uncurryRight_apply, curryRight_apply, snoc_init_self]
|
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.Diagonal
import Mathlib.CategoryTheory.MorphismProperty.Composition
universe v u
namespace CategoryTheory
open Limits
namespace MorphismProperty
variable {C : Type u} [Category.{v} C]
def StableUnderBaseChange (P : ... | Mathlib/CategoryTheory/MorphismProperty/Limits.lean | 83 | 92 | theorem StableUnderBaseChange.baseChange_map [HasPullbacks C] {P : MorphismProperty C}
(hP : StableUnderBaseChange P) {S S' : C} (f : S' ⟶ S) {X Y : Over S} (g : X ⟶ Y)
(H : P g.left) : P ((Over.baseChange f).map g).left := by |
let e :=
pullbackRightPullbackFstIso Y.hom f g.left ≪≫
pullback.congrHom (g.w.trans (Category.comp_id _)) rfl
have : e.inv ≫ pullback.snd = ((Over.baseChange f).map g).left := by
ext <;> dsimp [e] <;> simp
rw [← this, hP.respectsIso.cancel_left_isIso]
exact hP.snd _ _ H
|
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Set Filter Topology
variable {X : Type*}
def OnePoint (X : Type*) :=
Option X
#ali... | Mathlib/Topology/Compactification/OnePoint.lean | 357 | 359 | theorem tendsto_nhds_infty' {α : Type*} {f : OnePoint X → α} {l : Filter α} :
Tendsto f (𝓝 ∞) l ↔ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ (↑)) (coclosedCompact X) l := by |
simp [nhds_infty_eq, and_comm]
|
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.function.simple_func_dense_lp from "leanprover-community/mathlib"@"5a2df4cd59cb31e97a516d4603a14bed5c2f9425"
noncomputable section
set_option linter.uppercaseLean3 false
open Set Func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 77 | 82 | theorem norm_approxOn_y₀_le [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f) {s : Set E}
{y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] (x : β) (n : ℕ) :
‖approxOn f hf s y₀ h₀ n x - y₀‖ ≤ ‖f x - y₀‖ + ‖f x - y₀‖ := by |
have := edist_approxOn_y0_le hf h₀ x n
repeat rw [edist_comm y₀, edist_eq_coe_nnnorm_sub] at this
exact mod_cast this
|
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Order.Circular
import Mathlib.Data.List.TFAE
import Mathlib.Data.Set.Lattice
#align_import a... | Mathlib/Algebra/Order/ToIntervalMod.lean | 453 | 454 | theorem toIcoMod_sub_zsmul (a b : α) (m : ℤ) : toIcoMod hp a (b - m • p) = toIcoMod hp a b := by |
rw [sub_eq_add_neg, ← neg_smul, toIcoMod_add_zsmul]
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 395 | 396 | theorem map_fract_sub_fract_eq (x : ℝ) : f (fract x) - fract x = f x - x := by |
rw [Int.fract, f.map_sub_int, sub_sub_sub_cancel_right]
|
import Mathlib.Data.Nat.SuccPred
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Algebra.Order.Ring.WithTop
#align_import data.enat.basic from "leanprover-community/mathlib"@"ceb887ddf3344dab425292e497fa2af91498437c"
def ENat : Type :=
WithTop ℕ
deriving Zero,
--... | Mathlib/Data/ENat/Basic.lean | 256 | 261 | theorem nat_induction {P : ℕ∞ → Prop} (a : ℕ∞) (h0 : P 0) (hsuc : ∀ n : ℕ, P n → P n.succ)
(htop : (∀ n : ℕ, P n) → P ⊤) : P a := by |
have A : ∀ n : ℕ, P n := fun n => Nat.recOn n h0 hsuc
cases a
· exact htop A
· exact A _
|
import Mathlib.Algebra.Category.ModuleCat.EpiMono
import Mathlib.CategoryTheory.ConcreteCategory.Elementwise
#align_import algebra.category.Module.kernels from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
set_option linter.uppercaseLean3 false
open CategoryTheory CategoryTheory.Limits... | Mathlib/Algebra/Category/ModuleCat/Kernels.lean | 137 | 140 | theorem cokernel_π_ext {M N : ModuleCat.{u} R} (f : M ⟶ N) {x y : N} (m : M) (w : x = y + f m) :
cokernel.π f x = cokernel.π f y := by |
subst w
simpa only [map_add, add_right_eq_self] using cokernel.condition_apply f m
|
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f"
noncomputable section
open scoped Classical
open Topology TopologicalSpace Set Filter Function
universe u v
variable {X : Type u} {Y :... | Mathlib/Topology/Constructions.lean | 480 | 487 | theorem continuous_inf_dom_right₂ {X Y Z} {f : X → Y → Z} {ta1 ta2 : TopologicalSpace X}
{tb1 tb2 : TopologicalSpace Y} {tc1 : TopologicalSpace Z}
(h : by haveI := ta2; haveI := tb2; exact Continuous fun p : X × Y => f p.1 p.2) : by
haveI := ta1 ⊓ ta2; haveI := tb1 ⊓ tb2; exact Continuous fun p : X × Y => f... |
have ha := @continuous_inf_dom_right _ _ id ta1 ta2 ta2 (@continuous_id _ (id _))
have hb := @continuous_inf_dom_right _ _ id tb1 tb2 tb2 (@continuous_id _ (id _))
have h_continuous_id := @Continuous.prod_map _ _ _ _ ta2 tb2 (ta1 ⊓ ta2) (tb1 ⊓ tb2) _ _ ha hb
exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ h h... |
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.Group
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.cyclic from "leanprover-community/mathli... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 105 | 107 | theorem Nontrivial.of_not_isCyclic (nc : ¬IsCyclic α) : Nontrivial α := by |
contrapose! nc
exact @isCyclic_of_subsingleton _ _ (not_nontrivial_iff_subsingleton.mp nc)
|
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Topology.MetricSpace.IsometricSMul
import Mathlib.Topology.Sequences
#align_import analysis.normed.group.basic from "leanprover-community/mat... | Mathlib/Analysis/Normed/Group/Basic.lean | 590 | 590 | theorem norm_one' : ‖(1 : E)‖ = 0 := by | rw [← dist_one_right, dist_self]
|
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open ... | Mathlib/Algebra/Homology/Homotopy.lean | 321 | 329 | theorem comp_nullHomotopicMap' (f : C ⟶ D) (hom : ∀ i j, c.Rel j i → (D.X i ⟶ E.X j)) :
f ≫ nullHomotopicMap' hom = nullHomotopicMap' fun i j hij => f.f i ≫ hom i j hij := by |
ext n
erw [comp_nullHomotopicMap]
congr
ext i j
split_ifs
· rfl
· rw [comp_zero]
|
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Finite.Set
#align_import combinatorics.simple_graph.ends.defs from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
universe u
variable {V : Type u} (G : SimpleGraph V... | Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean | 71 | 75 | theorem componentComplMk_eq_of_adj (G : SimpleGraph V) {v w : V} (vK : v ∉ K) (wK : w ∉ K)
(a : G.Adj v w) : G.componentComplMk vK = G.componentComplMk wK := by |
rw [ConnectedComponent.eq]
apply Adj.reachable
exact a
|
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
d... | Mathlib/NumberTheory/Divisors.lean | 382 | 386 | theorem divisors_prime_pow {p : ℕ} (pp : p.Prime) (k : ℕ) :
divisors (p ^ k) = (Finset.range (k + 1)).map ⟨(p ^ ·), Nat.pow_right_injective pp.two_le⟩ := by |
ext a
rw [mem_divisors_prime_pow pp]
simp [Nat.lt_succ, eq_comm]
|
import Mathlib.Topology.Instances.Real
import Mathlib.Order.Filter.Archimedean
#align_import analysis.subadditive from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Set Filter Topology
def Subadditive (u : ℕ → ℝ) : Prop :=
∀ m n, u (m + n) ≤ u m + u n
#al... | Mathlib/Analysis/Subadditive.lean | 62 | 81 | theorem eventually_div_lt_of_div_lt {L : ℝ} {n : ℕ} (hn : n ≠ 0) (hL : u n / n < L) :
∀ᶠ p in atTop, u p / p < L := by |
/- It suffices to prove the statement for each arithmetic progression `(n * · + r)`. -/
refine .atTop_of_arithmetic hn fun r _ => ?_
/- `(k * u n + u r) / (k * n + r)` tends to `u n / n < L`, hence
`(k * u n + u r) / (k * n + r) < L` for sufficiently large `k`. -/
have A : Tendsto (fun x : ℝ => (u n + u r / ... |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 196 | 205 | theorem mellin_convergent_iff_norm [NormedSpace ℂ E] {f : ℝ → E} {T : Set ℝ} (hT : T ⊆ Ioi 0)
(hT' : MeasurableSet T) (hfc : AEStronglyMeasurable f <| volume.restrict <| Ioi 0) {s : ℂ} :
IntegrableOn (fun t : ℝ => (t : ℂ) ^ (s - 1) • f t) T ↔
IntegrableOn (fun t : ℝ => t ^ (s.re - 1) * ‖f t‖) T := by |
have : AEStronglyMeasurable (fun t : ℝ => (t : ℂ) ^ (s - 1) • f t) (volume.restrict T) := by
refine ((ContinuousAt.continuousOn ?_).aestronglyMeasurable hT').smul (hfc.mono_set hT)
exact fun t ht => continuousAt_ofReal_cpow_const _ _ (Or.inr <| ne_of_gt (hT ht))
rw [IntegrableOn, ← integrable_norm_iff this... |
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.Data.Set.Finite
import Mathlib.Data.Set.Subsingleton
import Mathlib.Topology.Category.TopCat.Limits.Konig
import Mathlib.Tactic.AdaptationNote
#align_import category_theory.cofiltered_system from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e... | Mathlib/CategoryTheory/CofilteredSystem.lean | 68 | 76 | theorem nonempty_sections_of_finite_cofiltered_system.init {J : Type u} [SmallCategory J]
[IsCofilteredOrEmpty J] (F : J ⥤ Type u) [hf : ∀ j, Finite (F.obj j)]
[hne : ∀ j, Nonempty (F.obj j)] : F.sections.Nonempty := by |
let F' : J ⥤ TopCat := F ⋙ TopCat.discrete
haveI : ∀ j, DiscreteTopology (F'.obj j) := fun _ => ⟨rfl⟩
haveI : ∀ j, Finite (F'.obj j) := hf
haveI : ∀ j, Nonempty (F'.obj j) := hne
obtain ⟨⟨u, hu⟩⟩ := TopCat.nonempty_limitCone_of_compact_t2_cofiltered_system.{u} F'
exact ⟨u, hu⟩
|
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community/mathlib"@"2f3994e1b117b1e1da49bcfb67334f33460c3ce4"
open Nat
namespace Nat
def choose : ℕ → ℕ → ℕ
| _, 0 => 1
| 0, _ + 1 => 0
| n + 1, k + 1 => choose n k + choose n ... | Mathlib/Data/Nat/Choose/Basic.lean | 385 | 387 | theorem multichoose_succ_succ (n k : ℕ) :
multichoose (n + 1) (k + 1) = multichoose n (k + 1) + multichoose (n + 1) k := by |
simp [multichoose]
|
import Mathlib.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Analysis.Normed.Group.ZeroAtInfty
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Ana... | Mathlib/Analysis/Distribution/SchwartzSpace.lean | 613 | 629 | theorem _root_.Function.HasTemperateGrowth.norm_iteratedFDeriv_le_uniform_aux {f : E → F}
(hf_temperate : f.HasTemperateGrowth) (n : ℕ) :
∃ (k : ℕ) (C : ℝ), 0 ≤ C ∧ ∀ N ≤ n, ∀ x : E, ‖iteratedFDeriv ℝ N f x‖ ≤ C * (1 + ‖x‖) ^ k := by |
choose k C f using hf_temperate.2
use (Finset.range (n + 1)).sup k
let C' := max (0 : ℝ) ((Finset.range (n + 1)).sup' (by simp) C)
have hC' : 0 ≤ C' := by simp only [C', le_refl, Finset.le_sup'_iff, true_or_iff, le_max_iff]
use C', hC'
intro N hN x
rw [← Finset.mem_range_succ_iff] at hN
refine le_trans... |
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Set.Function
import Mathlib.Logic.Pairwise
#align_import algebra.group.pi from "leanprover-community/mathlib"@"e4bc74cbaf429d706cb9140902f7ca6c431e75a4"
assert_not_exists AddMonoidWithOne
assert_not_exists Mono... | Mathlib/Algebra/Group/Pi/Lemmas.lean | 335 | 344 | theorem Pi.mulSingle_commute [∀ i, MulOneClass <| f i] :
Pairwise fun i j => ∀ (x : f i) (y : f j), Commute (mulSingle i x) (mulSingle j y) := by |
intro i j hij x y; ext k
by_cases h1 : i = k;
· subst h1
simp [hij]
by_cases h2 : j = k;
· subst h2
simp [hij]
simp [h1, h2]
|
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Support
#align_import algebra.indicator_function from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
assert_not_exists MonoidWithZero
open Function
variable {α β ι M N : Type*}
namespace Set
section One
variable [On... | Mathlib/Algebra/Group/Indicator.lean | 97 | 98 | theorem mulIndicator_eq_self : s.mulIndicator f = f ↔ mulSupport f ⊆ s := by |
simp only [funext_iff, subset_def, mem_mulSupport, mulIndicator_apply_eq_self, not_imp_comm]
|
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.polynomial.quotient from "leanprover-community/mathlib"@"4f840b8d28320b20c87db17b3a6eef3d325fca87"
set_option linter.uppercaseLean3 false
open Polynomial
... | Mathlib/RingTheory/Polynomial/Quotient.lean | 150 | 154 | theorem polynomialQuotientEquivQuotientPolynomial_symm_mk (I : Ideal R) (f : R[X]) :
I.polynomialQuotientEquivQuotientPolynomial.symm (Quotient.mk _ f) = f.map (Quotient.mk I) := by |
rw [polynomialQuotientEquivQuotientPolynomial, RingEquiv.symm_mk, RingEquiv.coe_mk,
Equiv.coe_fn_mk, Quotient.lift_mk, coe_eval₂RingHom, eval₂_eq_eval_map, ← Polynomial.map_map,
← eval₂_eq_eval_map, Polynomial.eval₂_C_X]
|
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, S... | Mathlib/Order/BooleanAlgebra.lean | 284 | 299 | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z :=
sdiff_unique
(calc
y ⊓ (x ⊔ z) ⊔ y \ x ⊓ y \ z = (y ⊓ (x ⊔ z) ⊔ y \ x) ⊓ (y ⊓ (x ⊔ z) ⊔ y \ z) := by |
rw [sup_inf_left]
_ = (y ⊓ x ⊔ y ⊓ z ⊔ y \ x) ⊓ (y ⊓ x ⊔ y ⊓ z ⊔ y \ z) := by rw [@inf_sup_left _ _ y]
_ = (y ⊓ z ⊔ (y ⊓ x ⊔ y \ x)) ⊓ (y ⊓ x ⊔ (y ⊓ z ⊔ y \ z)) := by ac_rfl
_ = (y ⊓ z ⊔ y) ⊓ (y ⊓ x ⊔ y) := by rw [sup_inf_sdiff, sup_inf_sdiff]
_ = (y ⊔ y ⊓ z) ⊓ (y ⊔ y ⊓ x) := by ac_rf... |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
#align_import linear_algebra.free_module.pid from "leanprover-community/mathlib"@"d87199d51218d36a0a42c66c82d147b5a7ff87b3"
universe u v
section Ring
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/FreeModule/PID.lean | 72 | 81 | theorem eq_bot_of_generator_maximal_submoduleImage_eq_zero {N O : Submodule R M} (b : Basis ι R O)
(hNO : N ≤ O) {ϕ : O →ₗ[R] R} (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
[(ϕ.submoduleImage N).IsPrincipal] (hgen : generator (ϕ.submoduleImage N) = 0) : N = ⊥ := by |
rw [Submodule.eq_bot_iff]
intro x hx
refine (mk_eq_zero _ _).mp (show (⟨x, hNO hx⟩ : O) = 0 from b.ext_elem fun i ↦ ?_)
rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ
rw [LinearEquiv.map_zero, Finsupp.zero_apply]
refine (Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| hϕ (Finsupp.lapply i ∘ₗ ↑b.repr)... |
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Finiteness
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import group_theory.index from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace Subgroup
open Ca... | Mathlib/GroupTheory/Index.lean | 134 | 135 | theorem inf_relindex_right : (H ⊓ K).relindex K = H.relindex K := by |
rw [relindex, relindex, inf_subgroupOf_right]
|
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
va... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 163 | 165 | theorem real_smul_negPart_neg (r : ℝ) (hr : r < 0) :
(r • j).negPart = (-r).toNNReal • j.posPart := by |
rw [real_smul_def, ← smul_posPart, if_neg (not_le.2 hr), neg_negPart]
|
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.SupIndep
#align_import group_theory.noncomm_pi_coprod from "leanprover-community/mathlib"@"6f9f36364eae3f42368b04858fd66d6d9ae730d8"
... | Mathlib/GroupTheory/NoncommPiCoprod.lean | 159 | 170 | theorem noncommPiCoprod_mrange :
MonoidHom.mrange (noncommPiCoprod ϕ hcomm) = ⨆ i : ι, MonoidHom.mrange (ϕ i) := by |
letI := Classical.decEq ι
apply le_antisymm
· rintro x ⟨f, rfl⟩
refine Submonoid.noncommProd_mem _ _ _ (fun _ _ _ _ h => hcomm h _ _) (fun i _ => ?_)
apply Submonoid.mem_sSup_of_mem
· use i
simp
· refine iSup_le ?_
rintro i x ⟨y, rfl⟩
exact ⟨Pi.mulSingle i y, noncommPiCoprod_mulSingle _... |
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Shapes.Diagonal
#align_import algebraic_geometry.pullbacks from "leanprover-community/mathlib"@"7316286ff2942aa14e540add9058c6b0aa1c8070"
set_opt... | Mathlib/AlgebraicGeometry/Pullbacks.lean | 272 | 277 | theorem gluedLiftPullbackMap_fst (i j : 𝒰.J) :
gluedLiftPullbackMap 𝒰 f g s i j ≫ pullback.fst =
pullback.fst ≫
(pullbackSymmetry _ _).hom ≫
pullback.map _ _ _ _ (𝟙 _) s.snd f (Category.id_comp _).symm s.condition := by |
simp [gluedLiftPullbackMap]
|
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variable [TopologicalSpace α]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 1,026 | 1,029 | theorem Filter.EventuallyEq.congr_continuousWithinAt {f g : α → β} {s : Set α} {x : α}
(h : f =ᶠ[𝓝[s] x] g) (hx : f x = g x) :
ContinuousWithinAt f s x ↔ ContinuousWithinAt g s x := by |
rw [ContinuousWithinAt, hx, tendsto_congr' h, ContinuousWithinAt]
|
import Mathlib.Algebra.Jordan.Basic
import Mathlib.Algebra.Module.Defs
#align_import algebra.symmetrized from "leanprover-community/mathlib"@"933547832736be61a5de6576e22db351c6c2fbfd"
open Function
def SymAlg (α : Type*) : Type _ :=
α
#align sym_alg SymAlg
postfix:max "ˢʸᵐ" => SymAlg
namespace SymAlg
varia... | Mathlib/Algebra/Symmetrized.lean | 325 | 326 | theorem unsym_mul_self [Semiring α] [Invertible (2 : α)] (a : αˢʸᵐ) :
unsym (a * a) = unsym a * unsym a := by | rw [mul_def, unsym_sym, ← two_mul, invOf_mul_self_assoc]
|
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Support
#align_import algebra.indicator_function from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
assert_not_exists MonoidWithZero
open Function
variable {α β ι M N : Type*}
namespace Set
section One
variable [On... | Mathlib/Algebra/Group/Indicator.lean | 286 | 290 | theorem mulIndicator_one_preimage (s : Set M) :
t.mulIndicator 1 ⁻¹' s ∈ ({Set.univ, ∅} : Set (Set α)) := by |
classical
rw [mulIndicator_one', preimage_one]
split_ifs <;> simp
|
import Mathlib.Data.List.Range
import Mathlib.Data.List.Perm
#align_import data.list.sigma from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a71d317643cf7fb5fbb"
universe u v
namespace List
variable {α : Type u} {β : α → Type v} {l l₁ l₂ : List (Sigma β)}
def keys : List (Sigma β) → List α :=
map ... | Mathlib/Data/List/Sigma.lean | 311 | 315 | theorem lookupAll_length_le_one (a : α) {l : List (Sigma β)} (h : l.NodupKeys) :
length (lookupAll a l) ≤ 1 := by |
have := Nodup.sublist ((lookupAll_sublist a l).map _) h
rw [map_map] at this
rwa [← nodup_replicate, ← map_const]
|
import Mathlib.Data.List.Forall2
#align_import data.list.zip from "leanprover-community/mathlib"@"134625f523e737f650a6ea7f0c82a6177e45e622"
-- Make sure we don't import algebra
assert_not_exists Monoid
universe u
open Nat
namespace List
variable {α : Type u} {β γ δ ε : Type*}
#align list.zip_with_cons_cons Li... | Mathlib/Data/List/Zip.lean | 272 | 278 | theorem get?_zip_with_eq_some (f : α → β → γ) (l₁ : List α) (l₂ : List β) (z : γ) (i : ℕ) :
(zipWith f l₁ l₂).get? i = some z ↔
∃ x y, l₁.get? i = some x ∧ l₂.get? i = some y ∧ f x y = z := by |
induction l₁ generalizing l₂ i
· simp [zipWith]
· cases l₂ <;> simp only [zipWith, get?, exists_false, and_false_iff, false_and_iff]
cases i <;> simp [*]
|
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ι : Type*}
namespace CharTwo
section CommSemiring
variable [CommSemiring R] [CharP R 2]
theorem add_sq (x y... | Mathlib/Algebra/CharP/Two.lean | 115 | 116 | theorem sum_mul_self (s : Finset ι) (f : ι → R) :
((∑ i ∈ s, f i) * ∑ i ∈ s, f i) = ∑ i ∈ s, f i * f i := by | simp_rw [← pow_two, sum_sq]
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 443 | 445 | theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by |
rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)]
simp [neg_div, Real.arccos_neg]
|
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.MeasureTheory.Group.FundamentalDomain
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.RingTheory.Localization.Module
#align_import algebra.module.zlattice from "leanprover-community/mathlib"@"a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3"
n... | Mathlib/Algebra/Module/Zlattice/Basic.lean | 74 | 79 | theorem fundamentalDomain_reindex {ι' : Type*} (e : ι ≃ ι') :
fundamentalDomain (b.reindex e) = fundamentalDomain b := by |
ext
simp_rw [mem_fundamentalDomain, Basis.repr_reindex_apply]
rw [Equiv.forall_congr' e]
simp_rw [implies_true]
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Card
#align_import data.multiset.fintype from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
variable {α : Type*} [DecidableEq α] {m : Multiset α}
def Multiset.ToType (m : Multiset α) : Type _ := (x : α) × Fi... | Mathlib/Data/Multiset/Fintype.lean | 242 | 244 | theorem Multiset.card_coe (m : Multiset α) : Fintype.card m = Multiset.card m := by |
rw [Fintype.card_congr m.coeEquiv]
simp only [Fintype.card_coe, card_toEnumFinset]
|
import Mathlib.Topology.CompactOpen
noncomputable section
open scoped Topology
open Filter
variable {X ι : Type*} {Y : ι → Type*} [TopologicalSpace X] [∀ i, TopologicalSpace (Y i)]
namespace ContinuousMap
| Mathlib/Topology/ContinuousFunction/Sigma.lean | 44 | 54 | theorem embedding_sigmaMk_comp [Nonempty X] :
Embedding (fun g : Σ i, C(X, Y i) ↦ (sigmaMk g.1).comp g.2) where
toInducing := inducing_sigma.2
⟨fun i ↦ (sigmaMk i).inducing_comp embedding_sigmaMk.toInducing, fun i ↦
let ⟨x⟩ := ‹Nonempty X›
⟨_, (isOpen_sigma_fst_preimage {i}).preimage (continuous_e... |
· rintro ⟨i, g⟩ ⟨i', g'⟩ h
obtain ⟨rfl, hg⟩ : i = i' ∧ HEq (⇑g) (⇑g') :=
Function.eq_of_sigmaMk_comp <| congr_arg DFunLike.coe h
simpa using hg
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localization.Basic
#align_import data.polynomial.laurent from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Polynomial Func... | Mathlib/Algebra/Polynomial/Laurent.lean | 209 | 212 | theorem single_eq_C_mul_T (r : R) (n : ℤ) :
(Finsupp.single n r : R[T;T⁻¹]) = (C r * T n : R[T;T⁻¹]) := by |
-- Porting note: was `convert single_mul_single.symm`
simp [C, T, single_mul_single]
|
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
namespace List
open Nat
@[simp] theorem mem_toArray {a : α} {l : List α} : a ∈ l.toArray ↔ a ∈ l := by
simp [Array.mem_def]
@[simp]
theorem drop_one : ∀ l : List α, drop 1 l =... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 1,462 | 1,469 | theorem findIdxs_cons :
(x :: xs : List α).findIdxs p =
bif p x then 0 :: (xs.findIdxs p).map (· + 1) else (xs.findIdxs p).map (· + 1) := by |
dsimp [findIdxs]
rw [cond_eq_if]
split <;>
· simp only [Nat.zero_add, foldrIdx_start, Nat.add_zero, cons.injEq, true_and]
apply findIdxs_cons_aux
|
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 473 | 475 | theorem toNonUnitalRing_injective :
Function.Injective (@toNonUnitalRing R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
|
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Segment
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.between from "leanprover-community/mathlib"@"571e13cacbed7bf042fd3058c... | Mathlib/Analysis/Convex/Between.lean | 45 | 46 | theorem affineSegment_eq_segment (x y : V) : affineSegment R x y = segment R x y := by |
rw [segment_eq_image_lineMap, affineSegment]
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
#align_import analysis.special_functions.trigonometric.inverse from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter
open S... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 346 | 346 | theorem arcsin_eq_pi_div_two_sub_arccos (x : ℝ) : arcsin x = π / 2 - arccos x := by | simp [arccos]
|
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 783 | 786 | theorem Function.Surjective.connectedSpace [ConnectedSpace α] [TopologicalSpace β]
{f : α → β} (hf : Surjective f) (hf' : Continuous f) : ConnectedSpace β := by |
rw [connectedSpace_iff_univ, ← hf.range_eq]
exact isConnected_range hf'
|
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Contraction
import Mathlib.RingTheory.TensorProduct.Basic
#align_import representation_... | Mathlib/RepresentationTheory/Basic.lean | 248 | 252 | theorem smul_ofModule_asModule (r : MonoidAlgebra k G) (m : (ofModule M).asModule) :
(RestrictScalars.addEquiv k _ _) ((ofModule M).asModuleEquiv (r • m)) =
r • (RestrictScalars.addEquiv k _ _) ((ofModule M).asModuleEquiv (G := G) m) := by |
dsimp
simp only [AddEquiv.apply_symm_apply, ofModule_asAlgebraHom_apply_apply]
|
import Mathlib.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.Order.Basic
#align_import topology.fiber_bundle.trivialization from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open TopologicalSpace Filter Set Bundle Function
... | Mathlib/Topology/FiberBundle/Trivialization.lean | 118 | 118 | theorem mem_source : x ∈ e.source ↔ proj x ∈ e.baseSet := by | rw [e.source_eq, mem_preimage]
|
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import analysis.ODE.gronwall from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Metric Set Asymptotics Fil... | Mathlib/Analysis/ODE/Gronwall.lean | 156 | 176 | theorem dist_le_of_approx_trajectories_ODE_of_mem
(hf : ContinuousOn f (Icc a b))
(hf' : ∀ t ∈ Ico a b, HasDerivWithinAt f (f' t) (Ici t) t)
(f_bound : ∀ t ∈ Ico a b, dist (f' t) (v t (f t)) ≤ εf)
(hfs : ∀ t ∈ Ico a b, f t ∈ s t)
(hg : ContinuousOn g (Icc a b))
(hg' : ∀ t ∈ Ico a b, HasDerivWith... |
simp only [dist_eq_norm] at ha ⊢
have h_deriv : ∀ t ∈ Ico a b, HasDerivWithinAt (fun t => f t - g t) (f' t - g' t) (Ici t) t :=
fun t ht => (hf' t ht).sub (hg' t ht)
apply norm_le_gronwallBound_of_norm_deriv_right_le (hf.sub hg) h_deriv ha
intro t ht
have := dist_triangle4_right (f' t) (g' t) (v t (f t))... |
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic
namespace Ring
open Mathlib.Meta Qq NormNum Lean.Meta AtomM
open Lean (MetaM Expr mkRawNatLit)
def instCommSemiringNat : CommSe... | Mathlib/Tactic/Ring/Basic.lean | 668 | 670 | theorem pow_bit1 (_ : (a : R) ^ k = b) (_ : b * b = c) (_ : c * a = d) :
a ^ (Nat.add (Nat.mul (nat_lit 2) k) (nat_lit 1)) = d := by |
subst_vars; simp [Nat.succ_mul, pow_add]
|
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 89 | 91 | theorem thickenedIndicatorAux_one_of_mem_closure (δ : ℝ) (E : Set α) {x : α}
(x_mem : x ∈ closure E) : thickenedIndicatorAux δ E x = 1 := by |
rw [← thickenedIndicatorAux_closure_eq, thickenedIndicatorAux_one δ (closure E) x_mem]
|
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Prod
import Mathlib.Tactic.Common
variable {ι G₁ G₂ : Type*} {G : ι → Type*} [Semigroup G₁] [Semigroup G₂] [∀ i, Semigroup (G i)]
| Mathlib/Algebra/Divisibility/Prod.lean | 16 | 20 | theorem prod_dvd_iff {x y : G₁ × G₂} :
x ∣ y ↔ x.1 ∣ y.1 ∧ x.2 ∣ y.2 := by |
cases x; cases y
simp only [dvd_def, Prod.exists, Prod.mk_mul_mk, Prod.mk.injEq,
exists_and_left, exists_and_right, and_self, true_and]
|
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
#align_import category_theory.limits.constructions.epi_mono from "leanprover-community/mathlib"@"f7baecbb54bd0f24f228576f97b1752fc3c9b318"
... | Mathlib/CategoryTheory/Limits/Constructions/EpiMono.lean | 32 | 36 | theorem preserves_mono_of_preservesLimit {X Y : C} (f : X ⟶ Y) [PreservesLimit (cospan f f) F]
[Mono f] : Mono (F.map f) := by |
have := isLimitPullbackConeMapOfIsLimit F _ (PullbackCone.isLimitMkIdId f)
simp_rw [F.map_id] at this
apply PullbackCone.mono_of_isLimitMkIdId _ this
|
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable s... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 527 | 532 | theorem setIntegral_prod_mul {L : Type*} [RCLike L] (f : α → L) (g : β → L) (s : Set α)
(t : Set β) :
∫ z in s ×ˢ t, f z.1 * g z.2 ∂μ.prod ν = (∫ x in s, f x ∂μ) * ∫ y in t, g y ∂ν := by |
-- Porting note: added
rw [← Measure.prod_restrict s t]
apply integral_prod_mul
|
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.NormedSpace.Ray
#align_import analysis.convex.strict_convex_space from "leanprover-... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 152 | 158 | theorem combo_mem_ball_of_ne (hx : x ∈ closedBall z r) (hy : y ∈ closedBall z r) (hne : x ≠ y)
(ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) : a • x + b • y ∈ ball z r := by |
rcases eq_or_ne r 0 with (rfl | hr)
· rw [closedBall_zero, mem_singleton_iff] at hx hy
exact (hne (hx.trans hy.symm)).elim
· simp only [← interior_closedBall _ hr] at hx hy ⊢
exact strictConvex_closedBall ℝ z r hx hy hne ha hb hab
|
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.Data.Nat.PartENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
universe u v
open Function
variable {α : Type u}
namespace Cardinal
noncomputable def toPartENat : Cardinal →+o PartEN... | Mathlib/SetTheory/Cardinal/PartENat.lean | 112 | 113 | theorem mk_toPartENat_eq_coe_card [Fintype α] : toPartENat #α = Fintype.card α := by |
simp
|
import Mathlib.Probability.Kernel.MeasurableIntegral
#align_import probability.kernel.composition from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7cc45d0b"
open MeasureTheory
open scoped ENNReal
namespace ProbabilityTheory
namespace kernel
variable {α β ι : Type*} {mα : MeasurableSpace α}... | Mathlib/Probability/Kernel/Composition.lean | 151 | 154 | theorem compProdFun_eq_tsum (κ : kernel α β) [IsSFiniteKernel κ] (η : kernel (α × β) γ)
[IsSFiniteKernel η] (a : α) (hs : MeasurableSet s) :
compProdFun κ η a s = ∑' (n) (m), compProdFun (seq κ n) (seq η m) a s := by |
simp_rw [compProdFun_tsum_left κ η a s, compProdFun_tsum_right _ η a hs]
|
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.NumberTheory.Divisors
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Tactic.Zify
#align_import rin... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 187 | 190 | theorem mem_rootsOfUnity_iff_mem_nthRoots {ζ : Rˣ} :
ζ ∈ rootsOfUnity k R ↔ (ζ : R) ∈ nthRoots k (1 : R) := by |
simp only [mem_rootsOfUnity, mem_nthRoots k.pos, Units.ext_iff, Units.val_one,
Units.val_pow_eq_pow_val]
|
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
universe u v
va... | Mathlib/FieldTheory/RatFunc/Defs.lean | 195 | 199 | theorem mk_eq_mk {p q p' q' : K[X]} (hq : q ≠ 0) (hq' : q' ≠ 0) :
RatFunc.mk p q = RatFunc.mk p' q' ↔ p * q' = p' * q := by |
rw [mk_def_of_ne _ hq, mk_def_of_ne _ hq', ofFractionRing_injective.eq_iff,
IsLocalization.mk'_eq_iff_eq', -- Porting note: removed `[anonymous], [anonymous]`
(IsFractionRing.injective K[X] (FractionRing K[X])).eq_iff]
|
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import analysis.ODE.gronwall from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Metric Set Asymptotics Fil... | Mathlib/Analysis/ODE/Gronwall.lean | 73 | 76 | theorem hasDerivAt_gronwallBound_shift (δ K ε x a : ℝ) :
HasDerivAt (fun y => gronwallBound δ K ε (y - a)) (K * gronwallBound δ K ε (x - a) + ε) x := by |
convert (hasDerivAt_gronwallBound δ K ε _).comp x ((hasDerivAt_id x).sub_const a) using 1
rw [id, mul_one]
|
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Basic
import Mathlib.Tactic.ApplyFun
#align_import computability.turing_machine from "leanprover-commu... | Mathlib/Computability/TuringMachine.lean | 630 | 632 | theorem Tape.mk'_nth_nat {Γ} [Inhabited Γ] (L R : ListBlank Γ) (n : ℕ) :
(Tape.mk' L R).nth n = R.nth n := by |
rw [← Tape.right₀_nth, Tape.mk'_right₀]
|
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
section
open UniformSpace Filter Set Uniformity Topology UniformConvergence Function
variable {ι κ X X' Y Z α α' β β'... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 741 | 745 | theorem UniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u : α → β}
(hu : UniformInducing u) : EquicontinuousAt F x₀ ↔ EquicontinuousAt ((u ∘ ·) ∘ F) x₀ := by |
have := (UniformFun.postcomp_uniformInducing (α := ι) hu).inducing
rw [equicontinuousAt_iff_continuousAt, equicontinuousAt_iff_continuousAt, this.continuousAt_iff]
rfl
|
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open AffineMap
variable {k E PE : Type*} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE]
def ... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 47 | 48 | theorem slope_same (f : k → PE) (a : k) : (slope f a a : E) = 0 := by |
rw [slope, sub_self, inv_zero, zero_smul]
|
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 537 | 538 | theorem coeff_mul_X_sub_C {p : R[X]} {r : R} {a : ℕ} :
coeff (p * (X - C r)) (a + 1) = coeff p a - coeff p (a + 1) * r := by | simp [mul_sub]
|
import Mathlib.Order.Interval.Set.OrdConnectedComponent
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.t5 from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Filter Set Function OrderDual Topology Interval
variable {X : Type*} [LinearOrder X] [Topological... | Mathlib/Topology/Order/T5.lean | 66 | 71 | theorem compl_section_ordSeparatingSet_mem_nhdsWithin_Iic (hd : Disjoint s (closure t))
(ha : a ∈ s) : (ordConnectedSection <| ordSeparatingSet s t)ᶜ ∈ 𝓝[≤] a := by |
have hd' : Disjoint (ofDual ⁻¹' s) (closure <| ofDual ⁻¹' t) := hd
have ha' : toDual a ∈ ofDual ⁻¹' s := ha
simpa only [dual_ordSeparatingSet, dual_ordConnectedSection] using
compl_section_ordSeparatingSet_mem_nhdsWithin_Ici hd' ha'
|
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