Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k |
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import Mathlib.MeasureTheory.Integral.Lebesgue
#align_import measure_theory.measure.giry_monad from "leanprover-community/mathlib"@"56f4cd1ef396e9fd389b5d8371ee9ad91d163625"
noncomputable section
open scoped Classical
open ENNReal
open scoped Classical
open Set Filter
variable {α β : Type*}
namespace MeasureT... | Mathlib/MeasureTheory/Measure/GiryMonad.lean | 118 | 120 | theorem join_zero : (0 : Measure (Measure α)).join = 0 := by |
ext1 s hs
simp only [hs, join_apply, lintegral_zero_measure, coe_zero, Pi.zero_apply]
|
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 | 153 | 155 | theorem real_smul_negPart_nonneg (r : ℝ) (hr : 0 ≤ r) :
(r • j).negPart = r.toNNReal • j.negPart := by |
rw [real_smul_def, ← smul_negPart, if_pos hr]
|
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
@[ext]
structure YoungDiagram where
cells : Finset (ℕ × ℕ)
isLowerSet : IsLowerSet (cel... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 351 | 351 | theorem mk_mem_col_iff {μ : YoungDiagram} {i j : ℕ} : (i, j) ∈ μ.col j ↔ (i, j) ∈ μ := by | simp [col]
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Set.Finite
#align_import combinatorics.pigeonhole from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
un... | Mathlib/Combinatorics/Pigeonhole.lean | 183 | 190 | theorem exists_le_sum_fiber_of_sum_fiber_nonpos_of_nsmul_le_sum
(hf : ∀ y ∉ t, ∑ x ∈ s.filter fun x => f x = y, w x ≤ 0) (ht : t.Nonempty)
(hb : t.card • b ≤ ∑ x ∈ s, w x) : ∃ y ∈ t, b ≤ ∑ x ∈ s.filter fun x => f x = y, w x :=
exists_le_of_sum_le ht <|
calc
∑ _y ∈ t, b ≤ ∑ x ∈ s, w x := by | simpa
_ ≤ ∑ y ∈ t, ∑ x ∈ s.filter fun x => f x = y, w x :=
sum_le_sum_fiberwise_of_sum_fiber_nonpos hf
|
import Mathlib.RingTheory.Noetherian
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Module.Injective
import Mathlib.Algebra.Module.CharacterModule
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.Linear... | Mathlib/RingTheory/Flat/Basic.lean | 112 | 114 | theorem iff_lTensor_injective :
Module.Flat R M ↔ ∀ ⦃I : Ideal R⦄ (_ : I.FG), Function.Injective (lTensor M I.subtype) := by |
simpa [← comm_comp_rTensor_comp_comm_eq] using Module.Flat.iff_rTensor_injective R M
|
import Mathlib.Data.Vector.Basic
#align_import data.vector.mem from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
namespace Vector
variable {α β : Type*} {n : ℕ} (a a' : α)
@[simp]
theorem get_mem (i : Fin n) (v : Vector α n) : v.get i ∈ v.toList := by
rw [get_eq_get]
exact List.... | Mathlib/Data/Vector/Mem.lean | 31 | 35 | theorem mem_iff_get (v : Vector α n) : a ∈ v.toList ↔ ∃ i, v.get i = a := by |
simp only [List.mem_iff_get, Fin.exists_iff, Vector.get_eq_get]
exact
⟨fun ⟨i, hi, h⟩ => ⟨i, by rwa [toList_length] at hi, h⟩, fun ⟨i, hi, h⟩ =>
⟨i, by rwa [toList_length], h⟩⟩
|
import Mathlib.Data.Rat.Sqrt
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.IntervalCases
#align_import data.real.irrational from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Rat Real multiplicity
def ... | Mathlib/Data/Real/Irrational.lean | 70 | 85 | theorem irrational_nrt_of_n_not_dvd_multiplicity {x : ℝ} (n : ℕ) {m : ℤ} (hm : m ≠ 0) (p : ℕ)
[hp : Fact p.Prime] (hxr : x ^ n = m)
(hv : (multiplicity (p : ℤ) m).get (finite_int_iff.2 ⟨hp.1.ne_one, hm⟩) % n ≠ 0) :
Irrational x := by |
rcases Nat.eq_zero_or_pos n with (rfl | hnpos)
· rw [eq_comm, pow_zero, ← Int.cast_one, Int.cast_inj] at hxr
simp [hxr, multiplicity.one_right (mt isUnit_iff_dvd_one.1
(mt Int.natCast_dvd_natCast.1 hp.1.not_dvd_one)), Nat.zero_mod] at hv
refine irrational_nrt_of_notint_nrt _ _ hxr ?_ hnpos
rintro ⟨y,... |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Common
#align_import order.heyting.boundary from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
variable {α : Type*}
namespace Coheyting
variable [CoheytingAlgebra α] {a b : α}
def boundary (a : α) : α :=
a ⊓ ￢a
#align cohe... | Mathlib/Order/Heyting/Boundary.lean | 63 | 63 | theorem boundary_top : ∂ (⊤ : α) = ⊥ := by | rw [boundary, hnot_top, inf_bot_eq]
|
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {R V V' P P' : Type*}
open AffineEquiv AffineMap
namespace Affine... | Mathlib/Analysis/Convex/Side.lean | 101 | 106 | theorem WOppSide.map {s : AffineSubspace R P} {x y : P} (h : s.WOppSide x y) (f : P →ᵃ[R] P') :
(s.map f).WOppSide (f x) (f y) := by |
rcases h with ⟨p₁, hp₁, p₂, hp₂, h⟩
refine ⟨f p₁, mem_map_of_mem f hp₁, f p₂, mem_map_of_mem f hp₂, ?_⟩
simp_rw [← linearMap_vsub]
exact h.map f.linear
|
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_schmidt_ortho from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
open Finset Submodule FiniteDimensional
variable (𝕜 : Type*) {E : Type*} [RCLike �... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 58 | 60 | theorem gramSchmidt_def (f : ι → E) (n : ι) :
gramSchmidt 𝕜 f n = f n - ∑ i ∈ Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) := by |
rw [← sum_attach, attach_eq_univ, gramSchmidt]
|
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Data.ZMod.Algebra
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
import Mathlib.FieldTheory.Perfect
#align_import ring_theory.witt_vector.frobenius from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb3304bef77472"... | Mathlib/RingTheory/WittVector/Frobenius.lean | 131 | 140 | theorem map_frobeniusPoly.key₂ {n i j : ℕ} (hi : i ≤ n) (hj : j < p ^ (n - i)) :
j - v p ⟨j + 1, j.succ_pos⟩ + n = i + j + (n - i - v p ⟨j + 1, j.succ_pos⟩) := by |
generalize h : v p ⟨j + 1, j.succ_pos⟩ = m
rsuffices ⟨h₁, h₂⟩ : m ≤ n - i ∧ m ≤ j
· rw [tsub_add_eq_add_tsub h₂, add_comm i j, add_tsub_assoc_of_le (h₁.trans (Nat.sub_le n i)),
add_assoc, tsub_right_comm, add_comm i,
tsub_add_cancel_of_le (le_tsub_of_add_le_right ((le_tsub_iff_left hi).mp h₁))]
hav... |
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.NumberTheory.Liouville.Residual
import Mathlib.NumberTheory.Liouville.LiouvilleWith
import Mathlib.Analysis.PSeries
#align_import number_theory.liouville.measure from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open sc... | Mathlib/NumberTheory/Liouville/Measure.lean | 77 | 106 | theorem volume_iUnion_setOf_liouvilleWith :
volume (⋃ (p : ℝ) (_hp : 2 < p), { x : ℝ | LiouvilleWith p x }) = 0 := by |
simp only [← setOf_exists, exists_prop]
refine measure_mono_null setOf_liouvilleWith_subset_aux ?_
rw [measure_iUnion_null_iff]; intro m; rw [measure_preimage_add_right]; clear m
refine (measure_biUnion_null_iff <| to_countable _).2 fun n (hn : 1 ≤ n) => ?_
generalize hr : (2 + 1 / n : ℝ) = r
replace hr : ... |
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Algebra.Order.Group.Action
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Span
import Mathlib.RingTheory.Ideal.Basic
#align_import algebra.module.submodule.pointwise from "leanprover-community/... | Mathlib/Algebra/Module/Submodule/Pointwise.lean | 115 | 120 | theorem closure_neg (s : Set M) : span R (-s) = -span R s := by |
apply le_antisymm
· rw [span_le, coe_set_neg, ← Set.neg_subset, neg_neg]
exact subset_span
· rw [neg_le, span_le, coe_set_neg, ← Set.neg_subset]
exact subset_span
|
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.Topology.Algebra.Module.WeakDual
#align_import analysis.locally_convex.polar from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
variable {𝕜 E F : Type*}
open Topology
namespace Li... | Mathlib/Analysis/LocallyConvex/Polar.lean | 123 | 127 | theorem polar_weak_closed (s : Set E) : IsClosed[WeakBilin.instTopologicalSpace B.flip]
(B.polar s) := by |
rw [polar_eq_iInter]
refine isClosed_iInter fun x => isClosed_iInter fun _ => ?_
exact isClosed_le (WeakBilin.eval_continuous B.flip x).norm continuous_const
|
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 | 118 | 120 | theorem mulIndicator_eq_one : (mulIndicator s f = fun x => 1) ↔ Disjoint (mulSupport f) s := by |
simp only [funext_iff, mulIndicator_apply_eq_one, Set.disjoint_left, mem_mulSupport,
not_imp_not]
|
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set Function Filter
open scoped Classical NNReal Topology ENNReal
namespace MeasureTheory
open Outer... | Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 52 | 52 | theorem extend_eq_top {s : α} (h : ¬P s) : extend m s = ∞ := by | simp [extend, h]
|
import Mathlib.Order.Hom.Basic
import Mathlib.Order.BoundedOrder
#align_import order.hom.bounded from "leanprover-community/mathlib"@"f1a2caaf51ef593799107fe9a8d5e411599f3996"
open Function OrderDual
variable {F α β γ δ : Type*}
structure TopHom (α β : Type*) [Top α] [Top β] where
toFun : α → β
map_t... | Mathlib/Order/Hom/Bounded.lean | 156 | 159 | theorem map_eq_bot_iff [LE α] [OrderBot α] [PartialOrder β] [OrderBot β] [OrderIsoClass F α β]
(f : F) {a : α} : f a = ⊥ ↔ a = ⊥ := by |
letI : BotHomClass F α β := OrderIsoClass.toBotHomClass
rw [← map_bot f, (EquivLike.injective f).eq_iff]
|
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.Order.SymmDiff
#align_import measure_theory.decomposition.signed_hahn from "leanprover-community/mathlib"@"bc7d81beddb3d6c66f71449c5bc76c38cb77cf9e"
noncomputable section
open scoped Classical NNReal ENNReal MeasureTheory
variable {α β : Type*} [... | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean | 342 | 364 | theorem bddBelow_measureOfNegatives : BddBelow s.measureOfNegatives := by |
simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds]
by_contra! h
have h' : ∀ n : ℕ, ∃ y : ℝ, y ∈ s.measureOfNegatives ∧ y < -n := fun n => h (-n)
choose f hf using h'
have hf' : ∀ n : ℕ, ∃ B, MeasurableSet B ∧ s ≤[B] 0 ∧ s B < -n := by
intro n
rcases hf n with ⟨⟨B, ⟨hB₁, hBr⟩, hB₂⟩, hlt⟩
exact ⟨B, ... |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import ring_theory.algebraic from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
universe u v w
open scoped Classical
open Polynomi... | Mathlib/RingTheory/Algebraic.lean | 67 | 74 | theorem Subalgebra.isAlgebraic_iff (S : Subalgebra R A) :
S.IsAlgebraic ↔ @Algebra.IsAlgebraic R S _ _ S.algebra := by |
delta Subalgebra.IsAlgebraic
rw [Subtype.forall', Algebra.isAlgebraic_def]
refine forall_congr' fun x => exists_congr fun p => and_congr Iff.rfl ?_
have h : Function.Injective S.val := Subtype.val_injective
conv_rhs => rw [← h.eq_iff, AlgHom.map_zero]
rw [← aeval_algHom_apply, S.val_apply]
|
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Analysis.SpecialFunctions.Exp
open Filter Topology Real
namespace Polynomial
| Mathlib/Analysis/SpecialFunctions/PolynomialExp.lean | 27 | 31 | theorem tendsto_div_exp_atTop (p : ℝ[X]) : Tendsto (fun x ↦ p.eval x / exp x) atTop (𝓝 0) := by |
induction p using Polynomial.induction_on' with
| h_monomial n c => simpa [exp_neg, div_eq_mul_inv, mul_assoc]
using tendsto_const_nhds.mul (tendsto_pow_mul_exp_neg_atTop_nhds_zero n)
| h_add p q hp hq => simpa [add_div] using hp.add hq
|
import Mathlib.Topology.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
noncomputable section
open scoped Topology
open Filter Set TopologicalSpace
variable {X α : Type*} {ι : Sort*}
section BaireTheorem
variable [TopologicalSpace... | Mathlib/Topology/Baire/Lemmas.lean | 114 | 118 | theorem dense_biInter_of_Gδ {S : Set α} {f : ∀ x ∈ S, Set X} (ho : ∀ s (H : s ∈ S), IsGδ (f s H))
(hS : S.Countable) (hd : ∀ s (H : s ∈ S), Dense (f s H)) : Dense (⋂ s ∈ S, f s ‹_›) := by |
rw [biInter_eq_iInter]
haveI := hS.to_subtype
exact dense_iInter_of_Gδ (fun s => ho s s.2) fun s => hd s s.2
|
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Measure.GiryMonad
#align_import probability.kernel.basic from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open MeasureTheory
open scoped MeasureTheory ENNReal NNReal
namespace ProbabilityTheory
noncomputab... | Mathlib/Probability/Kernel/Basic.lean | 117 | 118 | theorem finset_sum_apply' (I : Finset ι) (κ : ι → kernel α β) (a : α) (s : Set β) :
(∑ i ∈ I, κ i) a s = ∑ i ∈ I, κ i a s := by | rw [finset_sum_apply, Measure.finset_sum_apply]
|
import Mathlib.Data.Multiset.Nodup
#align_import data.multiset.dedup from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {α β : Type*} [DecidableEq α]
def dedup (s : Multiset α) : Multiset α :=
Quot.liftOn s (fun l => (l.dedup : Multiset α)... | Mathlib/Data/Multiset/Dedup.lean | 112 | 113 | theorem le_dedup_self {s : Multiset α} : s ≤ dedup s ↔ Nodup s := by |
rw [le_dedup, and_iff_right le_rfl]
|
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.uniform_convergence from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
noncomputable section
open Topology Uniformity Filter S... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 138 | 139 | theorem tendstoUniformlyOn_univ : TendstoUniformlyOn F f p univ ↔ TendstoUniformly F f p := by |
simp [TendstoUniformlyOn, TendstoUniformly]
|
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
noncomputable section
open Affine
open Set
section
variable (k : Type*) {V : Type*} {P : Type*} [Ring k] [AddCommGroup V]... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 117 | 123 | theorem spanPoints_nonempty (s : Set P) : (spanPoints k s).Nonempty ↔ s.Nonempty := by |
constructor
· contrapose
rw [Set.not_nonempty_iff_eq_empty, Set.not_nonempty_iff_eq_empty]
intro h
simp [h, spanPoints]
· exact fun h => h.mono (subset_spanPoints _ _)
|
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.torus_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
variable {n : ℕ}
variable {E : Type*} [NormedAddCommGroup E]
noncomputa... | Mathlib/MeasureTheory/Integral/TorusIntegral.lean | 139 | 144 | theorem function_integrable [NormedSpace ℂ E] (hf : TorusIntegrable f c R) :
IntegrableOn (fun θ : ℝⁿ => (∏ i, R i * exp (θ i * I) * I : ℂ) • f (torusMap c R θ))
(Icc (0 : ℝⁿ) fun _ => 2 * π) volume := by |
refine (hf.norm.const_mul (∏ i, |R i|)).mono' ?_ ?_
· refine (Continuous.aestronglyMeasurable ?_).smul hf.1; continuity
simp [norm_smul, map_prod]
|
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.Data.List.Chain
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Data.Set.Pointwise.SMul
#align_import group_theor... | Mathlib/GroupTheory/CoprodI.lean | 199 | 200 | theorem of_injective (i : ι) : Function.Injective (of : M i →* _) := by |
classical exact (of_leftInverse i).injective
|
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Order.Filter.Pi
#align_import order.filter.cofinite from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Function
variable {ι α β : Type*} {l : Filter α}
namespace Filter
def cofinite : Filter α :=
comk Set.Finite finite_e... | Mathlib/Order/Filter/Cofinite.lean | 57 | 58 | theorem cofinite_eq_bot_iff : @cofinite α = ⊥ ↔ Finite α := by |
simp [← empty_mem_iff_bot, finite_univ_iff]
|
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 | 75 | 75 | theorem properDivisors.not_self_mem : ¬n ∈ properDivisors n := by | simp [properDivisors]
|
import Mathlib.CategoryTheory.Monoidal.Mon_
#align_import category_theory.monoidal.Mod_ from "leanprover-community/mathlib"@"33085c9739c41428651ac461a323fde9a2688d9b"
universe v₁ v₂ u₁ u₂
open CategoryTheory MonoidalCategory
variable (C : Type u₁) [Category.{v₁} C] [MonoidalCategory.{v₁} C]
variable {C}
struc... | Mathlib/CategoryTheory/Monoidal/Mod_.lean | 37 | 38 | theorem assoc_flip :
(A.X ◁ M.act) ≫ M.act = (α_ A.X A.X M.X).inv ≫ (A.mul ▷ M.X) ≫ M.act := by | simp
|
import Mathlib.CategoryTheory.Linear.Basic
import Mathlib.CategoryTheory.Preadditive.Biproducts
import Mathlib.LinearAlgebra.Matrix.InvariantBasisNumber
import Mathlib.Data.Set.Subsingleton
#align_import category_theory.preadditive.hom_orthogonal from "leanprover-community/mathlib"@"829895f162a1f29d0133f4b3538f4cd1fb... | Mathlib/CategoryTheory/Preadditive/HomOrthogonal.lean | 146 | 166 | theorem matrixDecomposition_comp (o : HomOrthogonal s) {α β γ : Type} [Finite α] [Fintype β]
[Finite γ] {f : α → ι} {g : β → ι} {h : γ → ι} (z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b))
(w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)) (i : ι) :
o.matrixDecomposition (z ≫ w) i = o.matrixDecomposition w i * ... |
ext ⟨c, ⟨⟩⟩ ⟨a, j_property⟩
simp only [Set.mem_preimage, Set.mem_singleton_iff] at j_property
simp only [Matrix.mul_apply, Limits.biproduct.components,
HomOrthogonal.matrixDecomposition_apply, Category.comp_id, Category.id_comp, Category.assoc,
End.mul_def, eqToHom_refl, eqToHom_trans_assoc, Finset.sum_c... |
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
#align_import linear_algebra.dfinsupp from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
variable {ι : Type*} {R : Type*} {S : Type*} {M : ι → Type*} {N : Type*}
n... | Mathlib/LinearAlgebra/DFinsupp.lean | 190 | 194 | theorem mapRange_smul (f : ∀ i, β₁ i → β₂ i) (hf : ∀ i, f i 0 = 0) (r : R)
(hf' : ∀ i x, f i (r • x) = r • f i x) (g : Π₀ i, β₁ i) :
mapRange f hf (r • g) = r • mapRange f hf g := by |
ext
simp only [mapRange_apply f, coe_smul, Pi.smul_apply, hf']
|
import Mathlib.Geometry.Manifold.MFDeriv.Basic
noncomputable section
open scoped Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {E' : Type*} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {f : E → E'}
{s : Set E} {x : E}
section MFDerivFderiv
t... | Mathlib/Geometry/Manifold/MFDeriv/FDeriv.lean | 71 | 74 | theorem mdifferentiableWithinAt_iff_differentiableWithinAt :
MDifferentiableWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') f s x ↔ DifferentiableWithinAt 𝕜 f s x := by |
simp only [mdifferentiableWithinAt_iff', mfld_simps]
exact ⟨fun H => H.2, fun H => ⟨H.continuousWithinAt, H⟩⟩
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 35 | 36 | theorem gcd_add_mul_right_right (m n k : ℕ) : gcd m (n + k * m) = gcd m n := by |
simp [gcd_rec m (n + k * m), gcd_rec m n]
|
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.IsAdjoinRoot
#align_import number_theory.kummer_dedekind from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
variable (R : Type*) {S : Type*} [CommRing R] [CommRing S] [Algebra R S]
open Ideal Polynomial DoubleQuo... | Mathlib/NumberTheory/KummerDedekind.lean | 85 | 86 | theorem conductor_eq_top_of_adjoin_eq_top (h : R<x> = ⊤) : conductor R x = ⊤ := by |
simp only [Ideal.eq_top_iff_one, mem_conductor_iff, h, mem_top, forall_const]
|
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.FreeAlgebra
#align_import algebra.star.free from "leanprover-community/mathlib"@"07c3cf2d851866ff7198219ed3fedf42e901f25c"
namespace FreeAlgebra
variable {R : Type*} [CommSemiring R] {X : Type*}
instance : StarRing (FreeAlgebra R X) where
star := MulOp... | Mathlib/Algebra/Star/Free.lean | 72 | 73 | theorem star_algebraMap (r : R) : star (algebraMap R (FreeAlgebra R X) r) = algebraMap R _ r := by |
simp [star, Star.star]
|
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {𝕜 E F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup E] [InnerProductSpace... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 116 | 123 | theorem orthogonal_eq_inter : Kᗮ = ⨅ v : K, LinearMap.ker (innerSL 𝕜 (v : E)) := by |
apply le_antisymm
· rw [le_iInf_iff]
rintro ⟨v, hv⟩ w hw
simpa using hw _ hv
· intro v hv w hw
simp only [mem_iInf] at hv
exact hv ⟨w, hw⟩
|
import Mathlib.CategoryTheory.Subobject.Limits
#align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u w
open CategoryTheory CategoryTheory.Limits
variable {ι : Type*}
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
o... | Mathlib/Algebra/Homology/ImageToKernel.lean | 136 | 141 | theorem imageToKernel_epi_comp {Z : V} (h : Z ⟶ A) [Epi h] (w) :
imageToKernel (h ≫ f) g w =
Subobject.ofLE _ _ (imageSubobject_comp_le h f) ≫
imageToKernel f g ((cancel_epi h).mp (by simpa using w : h ≫ f ≫ g = h ≫ 0)) := by |
ext
simp
|
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped NNReal Matrix
namespace Matrix
variable {R l m n α β : Type*} [Fintype l] [Fintyp... | Mathlib/Analysis/Matrix.lean | 151 | 152 | theorem nnnorm_col (v : m → α) : ‖col v‖₊ = ‖v‖₊ := by |
simp [nnnorm_def, Pi.nnnorm_def]
|
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Analysis.NormedSpace.RieszLemma
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 235 | 241 | theorem LinearMap.injective_iff_antilipschitz [FiniteDimensional 𝕜 E] (f : E →ₗ[𝕜] F) :
Injective f ↔ ∃ K > 0, AntilipschitzWith K f := by |
constructor
· rw [← LinearMap.ker_eq_bot]
exact f.exists_antilipschitzWith
· rintro ⟨K, -, H⟩
exact H.injective
|
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 151 | 152 | theorem isUnit_iff_isUnit_det : IsUnit A ↔ IsUnit A.det := by |
simp only [← nonempty_invertible_iff_isUnit, (invertibleEquivDetInvertible A).nonempty_congr]
|
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.HahnBanach.Separation
import Mathlib.LinearAlgebra.Dual
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
@[mk_iff separatingDual_def]
class SeparatingDual (R V : Type*) [Ring R] [AddCommGroup V] [TopologicalSpace V]
... | Mathlib/Analysis/NormedSpace/HahnBanach/SeparatingDual.lean | 57 | 60 | theorem exists_separating_of_ne {x y : V} (h : x ≠ y) :
∃ f : V →L[R] R, f x ≠ f y := by |
rcases exists_ne_zero (R := R) (sub_ne_zero_of_ne h) with ⟨f, hf⟩
exact ⟨f, by simpa [sub_ne_zero] using hf⟩
|
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.Logic.Equiv.Fin
#align_import algebra.big_operators.fin from "leanprover-community/mathlib"@"cc5dd6244981976cc9da7afc4eee5682b037a013"
open Fins... | Mathlib/Algebra/BigOperators/Fin.lean | 46 | 47 | theorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by |
simp [prod_eq_multiset_prod]
|
import Mathlib.Algebra.Algebra.Hom
import Mathlib.RingTheory.Ideal.Quotient
#align_import algebra.ring_quot from "leanprover-community/mathlib"@"e5820f6c8fcf1b75bcd7738ae4da1c5896191f72"
universe uR uS uT uA u₄
variable {R : Type uR} [Semiring R]
variable {S : Type uS} [CommSemiring S]
variable {T : Type uT}
vari... | Mathlib/Algebra/RingQuot.lean | 67 | 68 | theorem Rel.neg {R : Type uR} [Ring R] {r : R → R → Prop} ⦃a b : R⦄ (h : Rel r a b) :
Rel r (-a) (-b) := by | simp only [neg_eq_neg_one_mul a, neg_eq_neg_one_mul b, Rel.mul_right h]
|
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Order.LiminfLimsup
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Data.Set.La... | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 511 | 565 | theorem limsup_eq_tendsto_sum_indicator_nat_atTop (s : ℕ → Set α) :
limsup s atTop = { ω | Tendsto
(fun n ↦ ∑ k ∈ Finset.range n, (s (k + 1)).indicator (1 : α → ℕ) ω) atTop atTop } := by |
ext ω
simp only [limsup_eq_iInf_iSup_of_nat, ge_iff_le, Set.iSup_eq_iUnion, Set.iInf_eq_iInter,
Set.mem_iInter, Set.mem_iUnion, exists_prop]
constructor
· intro hω
refine tendsto_atTop_atTop_of_monotone' (fun n m hnm ↦ Finset.sum_mono_set_of_nonneg
(fun i ↦ Set.indicator_nonneg (fun _ _ ↦ zero_le... |
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 | 135 | 138 | theorem row_mulVec [Fintype n] [NonUnitalNonAssocSemiring α] (M : Matrix m n α) (v : n → α) :
Matrix.row (M *ᵥ v) = (M * Matrix.col v)ᵀ := by |
ext
rfl
|
import Mathlib.Data.Fintype.Quotient
import Mathlib.ModelTheory.Semantics
#align_import model_theory.quotients from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
namespace FirstOrder
namespace Language
variable (L : Language) {M : Type*}
open FirstOrder
open Structure
class Prest... | Mathlib/ModelTheory/Quotients.lean | 65 | 70 | theorem relMap_quotient_mk' {n : ℕ} (r : L.Relations n) (x : Fin n → M) :
(RelMap r fun i => (⟦x i⟧ : Quotient s)) ↔ @RelMap _ _ ps.toStructure _ r x := by |
change
Quotient.lift (@RelMap L M ps.toStructure n r) Prestructure.rel_equiv (Quotient.finChoice _) ↔
_
rw [Quotient.finChoice_eq, Quotient.lift_mk]
|
import Mathlib.Order.Filter.Basic
import Mathlib.Data.PFun
#align_import order.filter.partial from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
universe u v w
namespace Filter
variable {α : Type u} {β : Type v} {γ : Type w}
open Filter
def rmap (r : Rel α β) (l : Filter α) : F... | Mathlib/Order/Filter/Partial.lean | 130 | 136 | theorem rtendsto_iff_le_rcomap (r : Rel α β) (l₁ : Filter α) (l₂ : Filter β) :
RTendsto r l₁ l₂ ↔ l₁ ≤ l₂.rcomap r := by |
rw [rtendsto_def]
simp_rw [← l₂.mem_sets]
simp [Filter.le_def, rcomap, Rel.mem_image]; constructor
· exact fun h s t tl₂ => mem_of_superset (h t tl₂)
· exact fun h t tl₂ => h _ t tl₂ Set.Subset.rfl
|
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {𝕜 E F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup E] [InnerProductSpace... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 73 | 78 | theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v ∈ (𝕜 ∙ u)ᗮ ↔ ⟪u, v⟫ = 0 := by |
refine ⟨inner_right_of_mem_orthogonal (mem_span_singleton_self u), ?_⟩
intro hv w hw
rw [mem_span_singleton] at hw
obtain ⟨c, rfl⟩ := hw
simp [inner_smul_left, hv]
|
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
variable {s s₁ s₂ : Set α} {t ... | Mathlib/Data/Set/Function.lean | 360 | 363 | theorem MapsTo.coe_iterate_restrict {f : α → α} (h : MapsTo f s s) (x : s) (k : ℕ) :
h.restrict^[k] x = f^[k] x := by |
induction' k with k ih; · simp
simp only [iterate_succ', comp_apply, val_restrict_apply, ih]
|
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 | 153 | 153 | theorem sr_mul_self (i : ZMod n) : sr i * sr i = 1 := by | rw [sr_mul_sr, sub_self, one_def]
|
import Mathlib.Deprecated.Group
#align_import deprecated.ring from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
universe u v w
variable {α : Type u}
structure IsSemiringHom {α : Type u} {β : Type v} [Semiring α] [Semiring β] (f : α → β) : Prop where
map_zero : f 0 = 0
map... | Mathlib/Deprecated/Ring.lean | 58 | 63 | theorem comp (hf : IsSemiringHom f) {γ} [Semiring γ] {g : β → γ} (hg : IsSemiringHom g) :
IsSemiringHom (g ∘ f) :=
{ map_zero := by | simpa [map_zero hf] using map_zero hg
map_one := by simpa [map_one hf] using map_one hg
map_add := fun {x y} => by simp [map_add hf, map_add hg]
map_mul := fun {x y} => by simp [map_mul hf, map_mul hg] }
|
import Mathlib.Data.Finset.Pointwise
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.DFinsupp.Order
import Mathlib.Order.Interval.Finset.Basic
#align_import data.dfinsupp.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
open DFinsupp Finset
open Pointwise
vari... | Mathlib/Data/DFinsupp/Interval.lean | 125 | 132 | theorem support_rangeIcc_subset [DecidableEq ι] [∀ i, DecidableEq (α i)] :
(f.rangeIcc g).support ⊆ f.support ∪ g.support := by |
refine fun x hx => ?_
by_contra h
refine not_mem_support_iff.2 ?_ hx
rw [rangeIcc_apply, not_mem_support_iff.1 (not_mem_mono subset_union_left h),
not_mem_support_iff.1 (not_mem_mono subset_union_right h)]
exact Icc_self _
|
import Mathlib.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
#align_import topology.metric_space.emetric_space from "leanprover-community/mathlib"@"c8f3055... | Mathlib/Topology/EMetricSpace/Basic.lean | 110 | 111 | theorem edist_triangle_left (x y z : α) : edist x y ≤ edist z x + edist z y := by |
rw [edist_comm z]; apply edist_triangle
|
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 | 69 | 71 | theorem thickenedIndicatorAux_le_one (δ : ℝ) (E : Set α) (x : α) :
thickenedIndicatorAux δ E x ≤ 1 := by |
apply @tsub_le_self _ _ _ _ (1 : ℝ≥0∞)
|
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Data.Set.Function
#align_import analysis.sum_integral_comparisons from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Set MeasureTheory.MeasureSpace
variable {x₀ : ℝ} {a b : ℕ} {f : ℝ → ℝ}
theorem AntitoneOn.in... | Mathlib/Analysis/SumIntegralComparisons.lean | 126 | 147 | theorem AntitoneOn.sum_le_integral_Ico (hab : a ≤ b) (hf : AntitoneOn f (Set.Icc a b)) :
(∑ i ∈ Finset.Ico a b, f (i + 1 : ℕ)) ≤ ∫ x in a..b, f x := by |
rw [(Nat.sub_add_cancel hab).symm, Nat.cast_add]
conv =>
congr
congr
congr
rw [← zero_add a]
· skip
· skip
· skip
rw [add_comm]
rw [← Finset.sum_Ico_add, Nat.Ico_zero_eq_range]
conv =>
lhs
congr
congr
· skip
ext
rw [add_assoc, Nat.cast_add]
apply Antito... |
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 | 43 | 44 | theorem toPartENat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : toPartENat c = toNat c := by |
lift c to ℕ using h; simp
|
import Mathlib.MeasureTheory.Group.Measure
assert_not_exists NormedSpace
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {G : Type*} [MeasurableSpace G] {μ : Measure G} {g : G}
section TopologicalGroup
variable [TopologicalSpace G] [Group G] [TopologicalGroup G] [BorelSpace ... | Mathlib/MeasureTheory/Group/LIntegral.lean | 71 | 73 | theorem lintegral_eq_zero_of_isMulLeftInvariant [Regular μ] [NeZero μ] {f : G → ℝ≥0∞}
(hf : Continuous f) : ∫⁻ x, f x ∂μ = 0 ↔ f = 0 := by |
rw [lintegral_eq_zero_iff hf.measurable, hf.ae_eq_iff_eq μ continuous_zero]
|
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section Inter
@[simp]
theorem inter_nil (l : L... | Mathlib/Data/List/Lattice.lean | 183 | 184 | theorem inter_reverse {xs ys : List α} : xs.inter ys.reverse = xs.inter ys := by |
simp only [List.inter, elem_eq_mem, mem_reverse]
|
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_anti... | Mathlib/RingTheory/PowerSeries/Order.lean | 112 | 116 | theorem nat_le_order (φ : R⟦X⟧) (n : ℕ) (h : ∀ i < n, coeff R i φ = 0) : ↑n ≤ order φ := by |
by_contra H; rw [not_le] at H
have : (order φ).Dom := PartENat.dom_of_le_natCast H.le
rw [← PartENat.natCast_get this, PartENat.coe_lt_coe] at H
exact coeff_order this (h _ H)
|
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Topology.Algebra.UniformGroup
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Algebra.Algebra.Defs
import Mathlib.LinearAlgebra.Projection
import Mat... | Mathlib/Topology/Algebra/Module/Basic.lean | 172 | 175 | theorem Submodule.closure_subset_topologicalClosure_span (s : Set M) :
closure s ⊆ (span R s).topologicalClosure := by |
rw [Submodule.topologicalClosure_coe]
exact closure_mono subset_span
|
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Finset.Sum
import Mathlib.Logic.Embedding.Set
#align_import data.fintype.sum from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
universe u v
variable {α β : Type*}
open Finset
instance (α : Type u) (β : Type v) [Fintype α] [Fintyp... | Mathlib/Data/Fintype/Sum.lean | 60 | 74 | theorem image_subtype_univ_ssubset_image_univ [Fintype α] [DecidableEq β] (k : β) (b : α → β)
(hk : k ∈ Finset.image b univ) (p : β → Prop) [DecidablePred p] (hp : ¬p k) :
image (fun i : { a // p (b a) } => b ↑i) univ ⊂ image b univ := by |
constructor
· intro x hx
rcases mem_image.1 hx with ⟨y, _, hy⟩
exact hy ▸ mem_image_of_mem b (mem_univ (y : α))
· intro h
rw [mem_image] at hk
rcases hk with ⟨k', _, hk'⟩
subst hk'
have := h (mem_image_of_mem b (mem_univ k'))
rw [mem_image] at this
rcases this with ⟨j, _, hj'⟩
... |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.Logic.Equiv.Fin
#align_import algebra.big_operators.fin from "leanprover-community/mathlib"@"cc5dd6244981976cc9da7afc4eee5682b037a013"
open Fins... | Mathlib/Algebra/BigOperators/Fin.lean | 118 | 119 | theorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by |
simp [prod_univ_succ]
|
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 125 | 130 | theorem rank_submatrix_le [StrongRankCondition R] [Fintype m] (f : n → m) (e : n ≃ m)
(A : Matrix m m R) : rank (A.submatrix f e) ≤ rank A := by |
rw [rank, rank, mulVecLin_submatrix, LinearMap.range_comp, LinearMap.range_comp,
show LinearMap.funLeft R R e.symm = LinearEquiv.funCongrLeft R R e.symm from rfl,
LinearEquiv.range, Submodule.map_top]
exact Submodule.finrank_map_le _ _
|
import Mathlib.Init.Align
import Mathlib.Data.Fintype.Order
import Mathlib.Algebra.DirectLimit
import Mathlib.ModelTheory.Quotients
import Mathlib.ModelTheory.FinitelyGenerated
#align_import model_theory.direct_limit from "leanprover-community/mathlib"@"f53b23994ac4c13afa38d31195c588a1121d1860"
universe v w w' u₁... | Mathlib/ModelTheory/DirectLimit.lean | 67 | 76 | theorem coe_natLERec (m n : ℕ) (h : m ≤ n) :
(natLERec f' m n h : G' m → G' n) = Nat.leRecOn h (@fun k => f' k) := by |
obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le h
ext x
induction' k with k ih
· -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [natLERec, Nat.leRecOn_self, Embedding.refl_apply, Nat.leRecOn_self]
· -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [Nat... |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 109 | 111 | theorem compl_apply [Zero α] [One α] (i j : V) : A.compl i j = 0 ∨ A.compl i j = 1 := by |
unfold compl
split_ifs <;> simp
|
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 220 | 225 | theorem smul_eq_C_smul (x : RatFunc K) (r : K) : r • x = Polynomial.C r • x := by |
cases' x with x
-- Porting note: had to specify the induction principle manually
induction x using Localization.induction_on
rw [← ofFractionRing_smul, ← ofFractionRing_smul, Localization.smul_mk,
Localization.smul_mk, smul_eq_mul, Polynomial.smul_eq_C_mul]
|
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.GroupAction.Hom
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
variable {R S : Type*} (M : Type*) {a b : R} {s : S}
def IsSMulRegular [SMul R M] (c : R) :=
Function.Injective ((c ... | Mathlib/Algebra/Regular/SMul.lean | 202 | 205 | theorem not_zero_iff : ¬IsSMulRegular M (0 : R) ↔ Nontrivial M := by |
rw [nontrivial_iff, not_iff_comm, zero_iff_subsingleton, subsingleton_iff]
push_neg
exact Iff.rfl
|
import Mathlib.Order.RelIso.Basic
import Mathlib.Logic.Embedding.Set
import Mathlib.Logic.Equiv.Set
#align_import order.rel_iso.set from "leanprover-community/mathlib"@"ee0c179cd3c8a45aa5bffbf1b41d8dbede452865"
open Function
universe u v w
variable {α β γ δ : Type*} {r : α → α → Prop} {s : β → β → Prop} {t : γ ... | Mathlib/Order/RelIso/Set.lean | 111 | 112 | theorem RelIso.preimage_eq_image_symm (e : r ≃r s) (t : Set β) : e ⁻¹' t = e.symm '' t := by |
rw [e.symm.image_eq_preimage_symm]; rfl
|
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_anti... | Mathlib/RingTheory/PowerSeries/Order.lean | 68 | 75 | theorem order_finite_iff_ne_zero : (order φ).Dom ↔ φ ≠ 0 := by |
simp only [order]
constructor
· split_ifs with h <;> intro H
· simp only [PartENat.top_eq_none, Part.not_none_dom] at H
· exact h
· intro h
simp [h]
|
import Mathlib.LinearAlgebra.Dimension.LinearMap
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
#align_import linear_algebra.free_module.finite.matrix from "leanprover-community/mathlib"@"b1c23399f01266afe392a0d8f71f599a0dad4f7b"
universe u u' v w
variable (R : Type u) (S : Type u') (M : Type v) (N ... | Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean | 66 | 68 | theorem FiniteDimensional.rank_linearMap_self :
Module.rank S (M →ₗ[R] S) = lift.{u'} (Module.rank R M) := by |
rw [rank_linearMap, rank_self, lift_one, mul_one]
|
import Mathlib.Probability.Kernel.Disintegration.Integral
open MeasureTheory Set Filter MeasurableSpace
open scoped ENNReal MeasureTheory Topology ProbabilityTheory
namespace ProbabilityTheory
variable {α β Ω : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
[MeasurableSpace Ω] [StandardBorelSpace Ω] ... | Mathlib/Probability/Kernel/Disintegration/Unique.lean | 47 | 56 | theorem eq_condKernel_of_measure_eq_compProd' (κ : kernel α Ω) [IsSFiniteKernel κ]
(hκ : ρ = ρ.fst ⊗ₘ κ) {s : Set Ω} (hs : MeasurableSet s) :
∀ᵐ x ∂ρ.fst, κ x s = ρ.condKernel x s := by |
refine ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite
(kernel.measurable_coe κ hs) (kernel.measurable_coe ρ.condKernel hs) (fun t ht _ ↦ ?_)
conv_rhs => rw [Measure.set_lintegral_condKernel_eq_measure_prod ht hs, hκ]
simp only [Measure.compProd_apply (ht.prod hs), Set.mem_prod, ← lintegral_indicator _ ht]
... |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Function
#align_import data.set.intervals.surj_on from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
variable {α : Type*} {β : Type*} [LinearOrder α] [PartialOrder β] {f : α → β}
open Set Function
open OrderDual (toDual)... | Mathlib/Order/Interval/Set/SurjOn.lean | 26 | 32 | theorem surjOn_Ioo_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
(a b : α) : SurjOn f (Ioo a b) (Ioo (f a) (f b)) := by |
intro p hp
rcases h_surj p with ⟨x, rfl⟩
refine ⟨x, mem_Ioo.2 ?_, rfl⟩
contrapose! hp
exact fun h => h.2.not_le (h_mono <| hp <| h_mono.reflect_lt h.1)
|
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.RegularMono
#align_import category_theory.limits.shapes.kernel_pair from "leanprover-community/mathlib"@"f6bab67886fb92c3e2f539cc90a83815f69a189d"
universe v u u₂
... | Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean | 139 | 150 | theorem comp_of_mono {f₁ : X ⟶ Y} {f₂ : Y ⟶ Z} [Mono f₂] (small_k : IsKernelPair f₁ a b) :
IsKernelPair (f₁ ≫ f₂) a b :=
{ w := by | rw [small_k.w_assoc]
isLimit' := ⟨by
refine PullbackCone.isLimitAux _
(fun s => small_k.lift s.fst s.snd (by rw [← cancel_mono f₂, assoc, s.condition, assoc]))
(by simp) (by simp) ?_
intro s m hm
apply small_k.isLimit.hom_ext
apply PullbackCone.equalizer_ext small_k.cone _ _... |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Data.Nat.Factorial.BigOperators
import Mathlib.Data.ZMod.Basic
open Finset Nat
namespace ZMod
| Mathlib/Data/ZMod/Factorial.lean | 31 | 42 | theorem cast_descFactorial {n p : ℕ} (h : n ≤ p) :
(descFactorial (p - 1) n : ZMod p) = (-1) ^ n * n ! := by |
rw [descFactorial_eq_prod_range, ← prod_range_add_one_eq_factorial]
simp only [cast_prod]
nth_rw 2 [← card_range n]
rw [pow_card_mul_prod]
refine prod_congr rfl ?_
intro x hx
rw [← tsub_add_eq_tsub_tsub_swap,
Nat.cast_sub <| Nat.le_trans (Nat.add_one_le_iff.mpr (List.mem_range.mp hx)) h,
CharP.ca... |
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
import Mathlib.Order.Filter.Ker
#align_import order.filter.bases from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
set_option autoImplicit true
open Set Filter
open scoped Classical
ope... | Mathlib/Order/Filter/Bases.lean | 268 | 270 | theorem HasBasis.eq_of_same_basis (hl : l.HasBasis p s) (hl' : l'.HasBasis p s) : l = l' := by |
ext t
rw [hl.mem_iff, hl'.mem_iff]
|
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.GroupTheory.GroupAction.Ring
#align_import ring_theory.localization.basic from "leanprover-community/mathlib"@"b69c9a770ecf37eb21... | Mathlib/RingTheory/Localization/Basic.lean | 230 | 237 | theorem map_eq_zero_iff (r : R) : algebraMap R S r = 0 ↔ ∃ m : M, ↑m * r = 0 := by |
constructor
· intro h
obtain ⟨m, hm⟩ := (IsLocalization.eq_iff_exists M S).mp ((algebraMap R S).map_zero.trans h.symm)
exact ⟨m, by simpa using hm.symm⟩
· rintro ⟨m, hm⟩
rw [← (IsLocalization.map_units S m).mul_right_inj, mul_zero, ← RingHom.map_mul, hm,
RingHom.map_zero]
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 51 | 53 | theorem univ_fin2 : (univ : Finset (Fin 2)) = {0, 1} := by |
ext x
fin_cases x <;> simp
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 49 | 49 | theorem gcd_mul_left_add_right (m n k : ℕ) : gcd m (m * k + n) = gcd m n := by | simp [add_comm _ n]
|
import Mathlib.Data.Option.NAry
import Mathlib.Data.Seq.Computation
#align_import data.seq.seq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace Stream'
universe u v w
def IsSeq {α : Type u} (s : Stream' (Option α)) : Prop :=
∀ {n : ℕ}, s n = none → s (n + 1) = none
#al... | Mathlib/Data/Seq/Seq.lean | 129 | 130 | theorem not_terminates_iff {s : Seq α} : ¬s.Terminates ↔ ∀ n, (s.get? n).isSome := by |
simp only [Terminates, TerminatedAt, ← Ne.eq_def, Option.ne_none_iff_isSome, not_exists, iff_self]
|
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 208 | 209 | theorem mul_rotate' (a b c : G) : a * (b * c) = b * (c * a) := by |
simp only [mul_left_comm, mul_comm]
|
import Mathlib.Order.Disjoint
#align_import order.prop_instances from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
instance Prop.instDistribLattice : DistribLattice Prop where
sup := Or
le_sup_left := @Or.inl
le_sup_right := @Or.inr
sup_le := fun _ _ _ => Or.rec
inf := And
... | Mathlib/Order/PropInstances.lean | 72 | 80 | theorem disjoint_iff [∀ i, OrderBot (α' i)] {f g : ∀ i, α' i} :
Disjoint f g ↔ ∀ i, Disjoint (f i) (g i) := by |
classical
constructor
· intro h i x hf hg
exact (update_le_iff.mp <| h (update_le_iff.mpr ⟨hf, fun _ _ => bot_le⟩)
(update_le_iff.mpr ⟨hg, fun _ _ => bot_le⟩)).1
· intro h x hf hg i
apply h i (hf i) (hg i)
|
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.Lint.Basic
#align_import logic.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
#align_import init.ite_simp from "leanprover-communit... | Mathlib/Logic/Basic.lean | 1,131 | 1,131 | theorem exists₂_imp : (∃ x h, P x h) → b ↔ ∀ x h, P x h → b := by | simp
|
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Sup
-- can be defined with just `[Bot α]` where some lemmas hold without... | Mathlib/Data/Multiset/Lattice.lean | 84 | 85 | theorem sup_union (s₁ s₂ : Multiset α) : (s₁ ∪ s₂).sup = s₁.sup ⊔ s₂.sup := by |
rw [← sup_dedup, dedup_ext.2, sup_dedup, sup_add]; simp
|
import Mathlib.Data.Real.NNReal
import Mathlib.RingTheory.Valuation.Basic
noncomputable section
open Function Multiplicative
open scoped NNReal
variable {R : Type*} [Ring R] {Γ₀ : Type*} [LinearOrderedCommGroupWithZero Γ₀]
namespace Valuation
class RankOne (v : Valuation R Γ₀) where
hom : Γ₀ →*₀ ℝ≥0
st... | Mathlib/RingTheory/Valuation/RankOne.lean | 51 | 55 | theorem zero_of_hom_zero {x : Γ₀} (hx : hom v x = 0) : x = 0 := by |
refine (eq_of_le_of_not_lt (zero_le' (a := x)) fun h_lt ↦ ?_).symm
have hs := strictMono v h_lt
rw [_root_.map_zero, hx] at hs
exact hs.false
|
import Mathlib.Data.Bracket
import Mathlib.LinearAlgebra.Basic
#align_import algebra.lie.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u v w w₁ w₂
open Function
class LieRing (L : Type v) extends AddCommGroup L, Bracket L L where
protected add_lie : ∀ x y z ... | Mathlib/Algebra/Lie/Basic.lean | 175 | 175 | theorem sub_lie : ⁅x - y, m⁆ = ⁅x, m⁆ - ⁅y, m⁆ := by | simp [sub_eq_add_neg]
|
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.prod from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Set hiding prod_eq
open Function MeasureTheory
open Filter hiding ma... | Mathlib/MeasureTheory/Group/Prod.lean | 161 | 172 | theorem quasiMeasurePreserving_inv : QuasiMeasurePreserving (Inv.inv : G → G) μ μ := by |
refine ⟨measurable_inv, AbsolutelyContinuous.mk fun s hsm hμs => ?_⟩
rw [map_apply measurable_inv hsm, inv_preimage]
have hf : Measurable fun z : G × G => (z.2 * z.1, z.1⁻¹) :=
(measurable_snd.mul measurable_fst).prod_mk measurable_fst.inv
suffices map (fun z : G × G => (z.2 * z.1, z.1⁻¹)) (μ.prod μ) (s⁻¹ ... |
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
open scoped... | Mathlib/Analysis/Calculus/Taylor.lean | 74 | 79 | theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ +
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by |
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
|
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ring.Commute
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Order.Synonym
#align_import algebra.field.basic from "leanprover-community/mathlib"@"05101c3df9d9cfe9430edc205860c79b6d660102"
open Function ... | Mathlib/Algebra/Field/Basic.lean | 117 | 118 | theorem neg_div (a b : K) : -b / a = -(b / a) := by |
rw [neg_eq_neg_one_mul, mul_div_assoc, ← neg_eq_neg_one_mul]
|
import Mathlib.CategoryTheory.Abelian.Basic
#align_import category_theory.idempotents.basic from "leanprover-community/mathlib"@"3a061790136d13594ec10c7c90d202335ac5d854"
open CategoryTheory
open CategoryTheory.Category
open CategoryTheory.Limits
open CategoryTheory.Preadditive
open Opposite
namespace Catego... | Mathlib/CategoryTheory/Idempotents/Basic.lean | 130 | 140 | theorem split_imp_of_iso {X X' : C} (φ : X ≅ X') (p : X ⟶ X) (p' : X' ⟶ X')
(hpp' : p ≫ φ.hom = φ.hom ≫ p')
(h : ∃ (Y : C) (i : Y ⟶ X) (e : X ⟶ Y), i ≫ e = 𝟙 Y ∧ e ≫ i = p) :
∃ (Y' : C) (i' : Y' ⟶ X') (e' : X' ⟶ Y'), i' ≫ e' = 𝟙 Y' ∧ e' ≫ i' = p' := by |
rcases h with ⟨Y, i, e, ⟨h₁, h₂⟩⟩
use Y, i ≫ φ.hom, φ.inv ≫ e
constructor
· slice_lhs 2 3 => rw [φ.hom_inv_id]
rw [id_comp, h₁]
· slice_lhs 2 3 => rw [h₂]
rw [hpp', ← assoc, φ.inv_hom_id, id_comp]
|
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.projection from "leanprover-community/mathlib"@"6d584f1709bedbed9175bd9350df46599bdd7213"
noncomputable section Ring
variable {R : Type*} [Ring R] {E : Type*} [AddCommGroup E] [Module R E]
variable {F : Type*} [Ad... | Mathlib/LinearAlgebra/Projection.lean | 160 | 161 | theorem linearProjOfIsCompl_apply_left (h : IsCompl p q) (x : p) :
linearProjOfIsCompl p q h x = x := by | simp [linearProjOfIsCompl]
|
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
import Mathlib.Tactic.FieldSimp
#align_import number_theory.bernoulli from "leanprover-community/mat... | Mathlib/NumberTheory/Bernoulli.lean | 158 | 177 | theorem bernoulli'PowerSeries_mul_exp_sub_one :
bernoulli'PowerSeries A * (exp A - 1) = X * exp A := by |
ext n
-- constant coefficient is a special case
cases' n with n
· simp
rw [bernoulli'PowerSeries, coeff_mul, mul_comm X, sum_antidiagonal_succ']
suffices (∑ p ∈ antidiagonal n,
bernoulli' p.1 / p.1! * ((p.2 + 1) * p.2! : ℚ)⁻¹) = (n ! : ℚ)⁻¹ by
simpa [map_sum, Nat.factorial] using congr_arg (algeb... |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 74 | 77 | theorem HasDerivAtFilter.scomp (hg : HasDerivAtFilter g₁ g₁' (h x) L')
(hh : HasDerivAtFilter h h' x L) (hL : Tendsto h L L') :
HasDerivAtFilter (g₁ ∘ h) (h' • g₁') x L := by |
simpa using ((hg.restrictScalars 𝕜).comp x hh hL).hasDerivAtFilter
|
import Mathlib.Algebra.Order.Pointwise
import Mathlib.Analysis.NormedSpace.SphereNormEquiv
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
open Set Function Metric MeasurableSpace intervalIntegral
open s... | Mathlib/MeasureTheory/Constructions/HaarToSphere.lean | 108 | 125 | theorem measurePreserving_homeomorphUnitSphereProd :
MeasurePreserving (homeomorphUnitSphereProd E) (μ.comap (↑))
(μ.toSphere.prod (volumeIoiPow (dim E - 1))) := by |
nontriviality E
refine ⟨(homeomorphUnitSphereProd E).measurable, .symm ?_⟩
refine prod_eq_generateFrom generateFrom_measurableSet
((borel_eq_generateFrom_Iio _).symm.trans BorelSpace.measurable_eq.symm)
isPiSystem_measurableSet isPiSystem_Iio
μ.toSphere.toFiniteSpanningSetsIn (finiteSpanningSetsIn_vo... |
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Preserves.Finite
namespace CompHaus
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
universe u w
open Categor... | Mathlib/Topology/Category/CompHaus/Limits.lean | 136 | 139 | theorem pullback_snd_eq :
CompHaus.pullback.snd f g = (pullbackIsoPullback f g).hom ≫ Limits.pullback.snd := by |
dsimp [pullbackIsoPullback]
simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_π]
|
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
import Batteries.Tactic.Lint.Misc
instance {f : α → β} [DecidablePred p] : DecidablePred (p ∘ f) :=
inferInstanceAs <| DecidablePred fun x => p (f x)
@[deprecated] alias proofIrrel := proof_irrel
theorem Function.id_def : @id α = fun x => x := rfl
al... | .lake/packages/batteries/Batteries/Logic.lean | 100 | 103 | theorem eqRec_heq_iff_heq {α : Sort _} {a : α} {motive : (a' : α) → a = a' → Sort _}
(x : motive a (rfl : a = a)) {a' : α} (e : a = a') {β : Sort _} (y : β) :
HEq (@Eq.rec α a motive x a' e) y ↔ HEq x y := by |
subst e; rfl
|
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 151 | 154 | theorem dvd_gcd_iff {a : α} : a ∣ s.gcd f ↔ ∀ b ∈ s, a ∣ f b := by |
apply Iff.trans Multiset.dvd_gcd
simp only [Multiset.mem_map, and_imp, exists_imp]
exact ⟨fun k b hb ↦ k _ _ hb rfl, fun k a' b hb h ↦ h ▸ k _ hb⟩
|
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 137 | 137 | theorem prop_y (a : Solution₁ d) : d * a.y ^ 2 = a.x ^ 2 - 1 := by | rw [← a.prop]; ring
|
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Int.LeastGreatest
import Mathlib.Data.Rat.Floor
import Mathlib.Data.NNRat.Defs
#align_import algebra.order.archimedean from "leanprover-community/mathlib"@"6f413f3f7330b94c92a5a27488fdc74e6d483a78"
open Int Set
variable {α : Type*}
class Archimedean (... | Mathlib/Algebra/Order/Archimedean.lean | 64 | 81 | theorem existsUnique_zsmul_near_of_pos {a : α} (ha : 0 < a) (g : α) :
∃! k : ℤ, k • a ≤ g ∧ g < (k + 1) • a := by |
let s : Set ℤ := { n : ℤ | n • a ≤ g }
obtain ⟨k, hk : -g ≤ k • a⟩ := Archimedean.arch (-g) ha
have h_ne : s.Nonempty := ⟨-k, by simpa [s] using neg_le_neg hk⟩
obtain ⟨k, hk⟩ := Archimedean.arch g ha
have h_bdd : ∀ n ∈ s, n ≤ (k : ℤ) := by
intro n hn
apply (zsmul_le_zsmul_iff ha).mp
rw [← natCast... |
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
#align_import measure_theory.measure.finite_measure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 200 | 204 | theorem mass_zero_iff (μ : FiniteMeasure Ω) : μ.mass = 0 ↔ μ = 0 := by |
refine ⟨fun μ_mass => ?_, fun hμ => by simp only [hμ, zero_mass]⟩
apply toMeasure_injective
apply Measure.measure_univ_eq_zero.mp
rwa [← ennreal_mass, ENNReal.coe_eq_zero]
|
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