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/-
Copyright (c) 2022 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.Topology.ContinuousMap.Algebra
import Mathlib.Data.Set.... | /-- under suitable mild assumptions on `𝕜`, the character space is a closed set in
`WeakDual 𝕜 A`. -/
protected theorem isClosed [Nontrivial 𝕜] [T2Space 𝕜] [ContinuousMul 𝕜] :
IsClosed (characterSpace 𝕜 A) := by
rw [eq_set_map_one_map_mul, Set.setOf_and]
refine IsClosed.inter (isClosed_eq (eval_continuous... | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | 166 | 176 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.List.TakeDrop
import Mathlib.Data.List.Induction
/-!
# Prefixes, suffixes, infixes
This file proves properties about
* `List.isPrefix`: `l₁` is ... | Mathlib/Data/List/Infix.lean | 458 | 458 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.CategoryTheory.ComposableArrows
/-!
# Exact sequences
A sequence of `n` composable arrows `S : ComposableArr... | -- Without `decide := true`, simp gets stuck at `hi : autoParam False _auto✝`
zero i hi := by simp +decide at hi
lemma isComplex₁ (S : ComposableArrows C 1) : S.IsComplex where
zero i hi := by omega
| Mathlib/Algebra/Homology/ExactSequence.lean | 102 | 106 |
/-
Copyright (c) 2020 Bhavik Mehta, Edward Ayers, Thomas Read. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Edward Ayers, Thomas Read
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.Cate... | Mathlib/CategoryTheory/Closed/Cartesian.lean | 380 | 384 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 909 | 912 | |
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathl... | -- `simp` can prove this using `genEigenspace_zero`
lemma mem_genEigenspace_zero {f : End R M} {μ : R} {x : M} :
x ∈ f.genEigenspace μ 0 ↔ x = 0 := by
rw [← Nat.cast_zero, mem_genEigenspace_nat, pow_zero, LinearMap.mem_ker, Module.End.one_apply]
| Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 123 | 127 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Module.Convex
import Mathlib.Analysis.Normed.Module.Ray
import Mathlib.Analysis.NormedSpace.Pointwise
/-!
# Strictly conv... | div_eq_inv_mul, div_lt_one hxy] at this
theorem lt_norm_sub_of_not_sameRay (h : ¬SameRay ℝ x y) : ‖x‖ - ‖y‖ < ‖x - y‖ := by
nth_rw 1 [← sub_add_cancel x y] at h ⊢
exact sub_lt_iff_lt_add.2 (norm_add_lt_of_not_sameRay fun H' => h <| H'.add_left SameRay.rfl)
theorem abs_lt_norm_sub_of_not_sameRay (h : ¬SameRay ... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 178 | 189 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Util.Notation3
import Mathlib.Tactic.TypeStar
/-!
# Alternate definition of `Vector` in terms of `Fin2`
This file provid... | | 0, f => f []
| succ k, f => ∀ x : α, VectorAll k fun v => f (x :: v)
theorem exists_vector_zero (f : Vector3 α 0 → Prop) : Exists f ↔ f [] :=
⟨fun ⟨v, fv⟩ => by rw [← eq_nil v]; exact fv, fun f0 => ⟨[], f0⟩⟩
theorem exists_vector_succ (f : Vector3 α (succ n) → Prop) : Exists f ↔ ∃ x v, f (x :: v) :=
⟨fun ⟨v... | Mathlib/Data/Vector3.lean | 201 | 215 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Algebra.Ring.Regular
import Mathlib.Data.Int.Order.Units
import Mathlib.RingTheory.Coprime.Basic
/-!
# Unit... | · exact Or.inr (Or.inr (Or.inr p.mirror_neg))
/-- A unit trinomial is irreducible if it is coprime with its mirror -/
theorem irreducible_of_isCoprime (hp : p.IsUnitTrinomial) (h : IsCoprime p p.mirror) :
Irreducible p :=
irreducible_of_coprime hp fun _ => h.isUnit_of_dvd'
end IsUnitTrinomial
end Polynomia... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 311 | 343 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Int.Order.Basic
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Compare
impor... | (hf hb).not_isMax ha
| Mathlib/Order/Monotone/Basic.lean | 267 | 268 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly meas... | (hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f ⊓ g) μ := by
refine
⟨fun n => hf.approx n ⊓ hg.approx n, fun n => ?_, fun x =>
(hf.tendsto_approx x).inf_nhds (hg.tendsto_approx x)⟩
refine (measure_mono (support_inf _ _)).trans_lt ?_
exact measure_union_lt_top_iff.mpr ⟨hf.fin_support_appr... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 1,125 | 1,130 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,094 | 1,098 | |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Right-angled triangles
This file proves ba... | rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ←
inner_neg_left, neg_vsub_eq_vsub_rev] at h
rw [← @vsub_ne_zero V, eq_comm, ← @vsub_eq_zero_iff_eq V, or_comm] at h0
rw [angle, ← vsub_add_vsub_cancel p₁ p₂ p₃, add_comm]
exact angle_add_pos_of_inner_eq_zero h h0
| Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 360 | 365 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Fin.Tuple.Basic
/-!
# Matrix and vector notation
This file defines notation for vectors and matrices. Given `a b c d : α`,
the notation a... | let_expr Matrix.vecCons α en x xs' ei := ← Meta.whnfR e | return .continue
let some i := ei.int? | return .continue
let (xs, etailn, tail) ← matchVecConsPrefix en xs'
| Mathlib/Data/Fin/VecNotation.lean | 157 | 159 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Eric Wieser
-/
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.Tactic.AdaptationNote
import Mathlib.LinearAlgebra.Multilinear.Curry
/-!
# Tenso... | private theorem map_add_smul_aux [DecidableEq ι] (i : ι) (x : Π i, s i) (u : s i →ₗ[R] t i) :
(fun j ↦ update f i u j (x j)) = update (fun j ↦ (f j) (x j)) i (u (x i)) := by
ext j
| Mathlib/LinearAlgebra/PiTensorProduct.lean | 547 | 549 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Reduced
import Mathlib.RingTheory.IntegralDomain
-- TODO: remove Mathlib.Algebra.CharP.Reduced and move the last two lemmas to Lemmas
/-... | simp_all only [mem_rootsOfUnity, pow_mul, one_pow]
| Mathlib/RingTheory/RootsOfUnity/Basic.lean | 93 | 94 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | any_goals rw [Nat.bitwise_zero_right, ← Bool.cond_eq_ite, this, ← bit_to_nat, pb1]
all_goals
| Mathlib/Data/Num/Lemmas.lean | 792 | 793 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Primrec
import Mathlib.Data.Nat.PSub
import Mathlib.Data.PFun
/-!
# The partial recursive functions
The partial recursive functions are... | @[deprecated (since := "2025-02-21")] alias sum_inr := Computable.sumInr
theorem list_cons : Computable₂ (@List.cons α) :=
| Mathlib/Computability/Partrec.lean | 313 | 315 |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.GroupTheory.GroupAction.Ring
/-!
# Relating unital and non-unital subs... | Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean | 356 | 364 | |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.EffectiveEpi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
import Mathlib.Tactic.ApplyFun
/-!
# Effective ... | g₁ ≫ Sigma.desc e = g₂ ≫ Sigma.desc e := by
apply_fun ((Sigma.desc fun a ↦ pullback.fst g₁ (Sigma.ι X a)) ≫ ·) using
(fun a b ↦ (cancel_epi _).mp)
ext a
simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_pt, Cofan.mk_ι_app]
rw [← Category.assoc, pullback.condition]
simp only [Category.ass... | Mathlib/CategoryTheory/EffectiveEpi/Coproduct.lean | 61 | 93 |
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Kim Morrison, Eric Rodriguez
-/
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import M... |
theorem surjective_count_of_infinite_setOf (h : {n | p n}.Infinite) :
Function.Surjective (Nat.count p) :=
fun n => ⟨nth p n, count_nth_of_infinite h n⟩
theorem count_nth_succ {n : ℕ} (hn : ∀ hf : (setOf p).Finite, n < #hf.toFinset) :
| Mathlib/Data/Nat/Nth.lean | 376 | 381 |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Lemmas
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.BigOperators.Group... |
/-- decomposing `x : ι → R` as a sum along the canonical basis -/
theorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [NonAssocSemiring R]
(x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by
| Mathlib/Algebra/BigOperators/Pi.lean | 61 | 64 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Bryan Gin-ge Chen, Patrick Massot, Wen Yang, Johan Commelin
-/
import Mathlib.Data.Set.Finite.Range
import Mathlib.Order.Partition.Finpartition
/-!
# Equivalenc... | protected def Partition.orderIso : Setoid α ≃o { C : Set (Set α) // IsPartition C } where
toFun r := ⟨r.classes, empty_not_mem_classes, classes_eqv_classes⟩
invFun C := mkClasses C.1 C.2.2
left_inv := mkClasses_classes
right_inv C := by rw [Subtype.ext_iff_val, ← classes_mkClasses C.1 C.2]
map_rel_iff' {r s} ... | Mathlib/Data/Setoid/Partition.lean | 255 | 262 |
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.UnorderedInterva... | Mathlib/Data/Complex/Basic.lean | 937 | 937 | |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
/-!
# Martingales
A family of functions `f : ι → Ω → E` is a martingale wit... | refine EventuallyLE.trans ?_ (condExp_add (hf.integrable j) (hg.integrable j) _).symm.le
filter_upwards [hf.2.1 i j hij, hg.2.1 i j hij]
intros
refine add_le_add ?_ ?_ <;> assumption
theorem add_martingale [Preorder E] [AddLeftMono E] (hf : Submartingale f ℱ μ)
| Mathlib/Probability/Martingale/Basic.lean | 204 | 209 |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Computability.AkraBazzi.GrowsPolynomially
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
/-!
# Divid... | calc 0 < (n : ℝ) ^ p a b * (1 + 0) := by aesop (add safe Real.rpow_pos_of_pos)
_ ≤ asympBound g a b n := by
simp only [asympBound_def']
gcongr n^p a b * (1 + ?_)
have := R.g_nonneg
aesop (add safe Real.rpow_nonneg,
... | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 603 | 678 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Category.Grp.Preadditive
import Mathlib.Tactic.Linarith
import Mathlib.Cat... | simp only [ofHom, ofHoms_v_comp_d]
@[simp]
lemma d_comp_ofHom_v (φ : F ⟶ G) (p' p q : ℤ) (hpq : p + 0 = q) :
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 175 | 178 |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Polynomial.Taylor
import Mathlib.FieldTheory.RatFunc.AsPolynomial
/-!
# Laurent expansions of rational functions
## Main declarations
* `R... | intro x hx
have : x = taylor (r - r) x := by simp
rwa [this, sub_eq_add_neg, ← taylor_taylor, ← taylor_mul,
LinearMap.map_eq_zero_iff _ (taylor_injective _), mul_right_mem_nonZeroDivisors_eq_zero_iff hp,
LinearMap.map_eq_zero_iff _ (taylor_injective _)] at hx
/-- The Laurent expansion of rational functio... | Mathlib/FieldTheory/Laurent.lean | 39 | 45 |
/-
Copyright (c) 2023 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Combinatorics.SimpleGraph.Subgraph
/-!
# Local graph operations
... | Sym2.isDiag_iff_proj_eq]
lemma sup_edge_of_adj (h : G.Adj s t) : G ⊔ edge s t = G := by
| Mathlib/Combinatorics/SimpleGraph/Operations.lean | 177 | 179 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... | simp_rw [← Ioi_inter_Iio, Ioi_pred_eq_insert, insert_inter_of_mem (mem_Iio.2 h)]
| Mathlib/Order/SuccPred/Basic.lean | 890 | 891 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 1,374 | 1,380 | |
/-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Norm.Defs
import Mathlib.RingTheory.Trace.Defs
/-!
# Relation between Norms and Traces
-/
| lemma Algebra.norm_one_add_smul {A B} [CommRing A] [CommRing B] [Algebra A B]
[Module.Free A B] [Module.Finite A B] (a : A) (x : B) :
∃ r : A, Algebra.norm A (1 + a • x) = 1 + Algebra.trace A B x * a + r * a ^ 2 := by
classical
let ι := Module.Free.ChooseBasisIndex A B
let b : Basis ι A B := Module.Free.c... | Mathlib/RingTheory/NormTrace.lean | 14 | 24 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.Order.Interval.Finset.Basic
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tact... | simp only [mem_Ico]
rintro i ⟨_, im⟩ j ⟨_, jm⟩ Hij
rw [← tsub_tsub_cancel_of_le (this _ im), Hij, tsub_tsub_cancel_of_le (this _ jm)]
· have : n + 1 - k ≤ n + 1 - m := by
rw [tsub_le_tsub_iff_left h]
exact hkm
| Mathlib/Algebra/BigOperators/Intervals.lean | 211 | 216 |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.Normal.... | classical
have hne := (multiplicity_pos_of_dvd hf).ne'
refine ⟨_, n, y, hne, H, ?_⟩
| Mathlib/FieldTheory/SeparableDegree.lean | 596 | 598 |
/-
Copyright (c) 2024 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.Analysis.Calculus.Deriv.Pi
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.Analysis.InnerProductSpace.NormP... |
The grid-lines operation `GridLines.T` on a nonnegative function on a finitary product type is
less than or equal to the grid-lines operation of its partial integral in one co-ordinate
(the latter intuitively considered as a function on a space "one dimension down"). -/
theorem T_insert_le_T_lmarginal_singleton [∀ i, ... | Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean | 105 | 206 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.CategoryTheory.Shift.Quotient
import Mathlib.CategoryTheory.Linear.LinearF... | ((CategoryTheory.shiftFunctorAdd' (CochainComplex C ℤ) a b ab h).hom.app K).f n =
(K.XIsoOfEq (by dsimp; rw [← h, add_assoc, add_comm a])).hom := by
subst h
| Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean | 149 | 151 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Hom.BoundedLattice
/-!
# Heyting algebra morphisms
A Heyting homomorphism between two Heyting algebras is a bounded lattice homomorphism that prese... | DFunLike.ext' h
| Mathlib/Order/Heyting/Hom.lean | 379 | 380 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | Filter.Tendsto (fun t => ((U t).orthogonalProjection x : E)) atTop (𝓝 x) := by
have : (⨆ i, U i).topologicalClosure.HasOrthogonalProjection := by
rw [top_unique hU']
| Mathlib/Analysis/InnerProductSpace/Projection.lean | 897 | 899 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Monad
import Mathlib.LinearAlgebra.Charpoly.ToMatrix
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.Li... | MvPolynomial.comp_eval₂Hom, charpoly.univ_map_eval₂Hom]
congr
ext
rw [of_apply, Function.curry_apply, toMvPolynomial_eval_eq_apply, LinearEquiv.symm_apply_apply]
rfl
open LinearMap in
lemma polyCharpolyAux_eval_eq_toMatrix_charpoly_coeff (x : L) (i : ℕ) :
| Mathlib/Algebra/Module/LinearMap/Polynomial.lean | 280 | 287 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Basic
/-!
# Intervals as multisets
This file defines intervals as multisets.
## Main declarations
In a `LocallyFiniteOrder`,
* `M... | Mathlib/Order/Interval/Multiset.lean | 369 | 370 | |
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Thomas Murrills
-/
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Tactic.NormNum.Basic
/-!
## `norm_num` plugin for `^`.
-/
assert_not_exists RelIso
namespace Ma... | (h2 : IsNatPowT (Nat.pow a b = c) a b' c') : IsNatPowT p a b' c' :=
⟨h2.run' ∘ h1.run'⟩
| Mathlib/Tactic/NormNum/Pow.lean | 43 | 44 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Yury Kudryashov
-/
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Logic.Function.Iterate
import Mathlib.Logic.Relation
/-!
# R... | ∀ {a : α} {l : List α}, Chain R a l ↔ l = [] ∨ Forall₂ R (a :: dropLast l) l
| a, [] => by simp
| a, b :: l => by
by_cases h : l = [] <;>
| Mathlib/Data/List/Chain.lean | 62 | 65 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Filter.Prod
/-!
# N-ary maps of filter
This file defines the binary and ternary maps of filters. This is mostly useful to define pointwise
operatio... | (map₂_swap _ _ _).trans <| by simp_rw [h_comm]
theorem map₂_left_comm {m : α → δ → ε} {n : β → γ → δ} {m' : α → γ → δ'} {n' : β → δ' → ε}
| Mathlib/Order/Filter/NAry.lean | 177 | 179 |
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Data.Nat.GCD.Bas... | end Nat
/-! ### Divisibility over ℤ -/
namespace Int
theorem gcd_def (i j : ℤ) : gcd i j = Nat.gcd i.natAbs j.natAbs := rfl
| Mathlib/Data/Int/GCD.lean | 146 | 154 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.LinearAlgebra.FreeM... | ⟨fun ⟨h⟩ => h.toLinearEquiv.finrank_eq, fun h =>
FiniteDimensional.nonempty_continuousLinearEquiv_of_finrank_eq h⟩
/-- A continuous linear equivalence between two finite-dimensional topological vector spaces over a
| Mathlib/Topology/Algebra/Module/FiniteDimension.lean | 402 | 405 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Principal ordinals
We define principal or indecomposable ordinals, and we prove the standa... | apply not_lt_of_le
suffices e : ω ^ log ω o * n + o = o by
simpa only [e] using le_add_right (ω ^ log ω o * ↑n) o
induction' n with n IH
· simp [Nat.cast_zero, mul_zero, zero_add]
· simp only [Nat.cast_succ, mul_add_one, add_assoc, this, IH]
theorem principal_add_opow_of_principal_add {a} (ha... | Mathlib/SetTheory/Ordinal/Principal.lean | 209 | 230 |
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Order.RelClasses
/-!
# Euclidean domains
This file introduces Euclidean domains and provides the extende... | satisfying `gcd x y = x * a + y * b`. -/
def xgcd (x y : R) : R × R :=
(xgcdAux x 1 0 y 0 1).2
| Mathlib/Algebra/EuclideanDomain/Defs.lean | 209 | 211 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Action.Faithful
import Mathlib.Algebra.Grou... | Mathlib/Algebra/Group/Submonoid/Operations.lean | 1,140 | 1,142 | |
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
/-!
# Bicategories
In this file we define typeclass for bicategories.
A bicategory `B` consists of
* objects `a : B`,
* 1-morphisms ... | @[reassoc (attr := simp)]
theorem pentagon_hom_hom_inv_inv_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
(α_ f (g ≫ h) i).hom ≫ f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv =
(α_ f g h).inv ▷ i ≫ (α_ (f ≫ g) h i).hom :=
eq_of_inv_eq_inv (by simp)
| Mathlib/CategoryTheory/Bicategory/Basic.lean | 259 | 263 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | QuotientAddGroup.equivIcoMod hp a 0 = ⟨toIcoMod hp a 0, toIcoMod_mem_Ico hp a _⟩ :=
rfl
| Mathlib/Algebra/Order/ToIntervalMod.lean | 685 | 686 |
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearG... | let f : (Fin 2 → ℝ) →ₗ[ℝ] ℂ := π₀.smulRight (z : ℂ) + π₁.smulRight 1
have f_def : ⇑f = fun p : Fin 2 → ℝ => (p 0 : ℂ) * ↑z + p 1 := by
ext1
dsimp only [π₀, π₁, f, LinearMap.coe_proj, real_smul, LinearMap.coe_smulRight,
LinearMap.add_apply]
rw [mul_one]
have :
(fun p : Fin 2 → ℤ => normSq ((p... | Mathlib/NumberTheory/Modular.lean | 117 | 161 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.... | use k
intro x; ext m
rw [Module.End.pow_apply, LinearMap.zero_apply, ← @LieSubmodule.mem_bot R L M, ← hM]
exact iterate_toEnd_mem_lowerCentralSeries R L M x m k
theorem isNilpotent_toEnd_of_isNilpotent [IsNilpotent L M] (x : L) :
_root_.IsNilpotent (toEnd R L M x) := by
change ∃ k, toEnd R L M x ^ k = 0
... | Mathlib/Algebra/Lie/Nilpotent.lean | 326 | 337 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Da... | open Finset
theorem sub_one_mul_sum_div_pow_eq_sub_sum_digits {p : ℕ}
(L : List ℕ) {h_nonempty} (h_ne_zero : L.getLast h_nonempty ≠ 0) (h_lt : ∀ l ∈ L, l < p) :
(p - 1) * ∑ i ∈ range L.length, (ofDigits p L) / p ^ i.succ = (ofDigits p L) - L.sum := by
obtain h | rfl | h : 1 < p ∨ 1 = p ∨ p < 1 := trichotomou... | Mathlib/Data/Nat/Digits.lean | 550 | 559 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic
deprecated_module (since := "2025-04-13")
| Mathlib/MeasureTheory/Integral/Periodic.lean | 319 | 329 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.GroupTheory.GroupAction.Hom
import ... |
variable {M}
variable {P : Type*} [CommSemiring P] [MulSemiringAction M P]
variable {Q : Type*} [CommSemiring Q] [MulSemiringAction M Q]
| Mathlib/Algebra/Polynomial/GroupRingAction.lean | 108 | 112 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... |
/-- The uniform neighborhoods of all points of a dense set cover the whole space. -/
| Mathlib/Topology/UniformSpace/Basic.lean | 237 | 238 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Order.Antidiag.Prod
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Multiset.NatAntidiagonal
/-!
# Antidiagonals in ℕ × ℕ as finset... | ⟨Nat.succ, Nat.succ_injective⟩)) <|
by simp)
(by simp) := by
simp_rw [antidiagonal_succ (n + 1), antidiagonal_succ', Finset.map_cons, map_map]
rfl
theorem antidiagonal.fst_lt {n : ℕ} {kl : ℕ × ℕ} (hlk : kl ∈ antidiagonal n) : kl.1 < n + 1 :=
Nat.lt_succ_of_le <| antidiagonal.fst... | Mathlib/Data/Finset/NatAntidiagonal.lean | 89 | 99 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Max
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Multiset.Sort
import Mathlib.Order.RelIso.Set
/-!
# Construct a sorted list f... | s.max' (card_pos.mp (h.symm ▸ hz)) := by
simp [orderEmbOfFin_apply, max'_eq_sorted_last, h]
/-- `orderEmbOfFin {a} h` sends any argument to `a`. -/
@[simp]
theorem orderEmbOfFin_singleton (a : α) (i : Fin 1) :
orderEmbOfFin {a} (card_singleton a) i = a := by
rw [Subsingleton.elim i ⟨0, Nat.zero_lt_one⟩, ... | Mathlib/Data/Finset/Sort.lean | 219 | 226 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Peter Nelson
-/
import Mathlib.Order.Antichain
/-!
# Minimality and Maximality
This file proves basic facts about minimality and maximality
of an element with respect to ... |
theorem Minimal.and_left (h : Minimal P x) (hQ : Q x) : Minimal (fun x ↦ (Q x ∧ P x)) x :=
h.mono (fun _ ↦ And.right) ⟨hQ, h.prop⟩
| Mathlib/Order/Minimal.lean | 114 | 117 |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.DirectSum.LinearMap
import Mathlib.Algebra.Lie.Weights.Cartan
import Mathlib.Data.Int.Interval
import Mathlib.LinearAlgebra.Trace
import Mathlib.Ring... | rcases n with (n | n)
| Mathlib/Algebra/Lie/Weights/Chain.lean | 318 | 318 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... | wellFounded_iff_wellFounded_subrel.mpr (·.induction_on fun ⟨_, _, wo⟩ ↦
RelHomClass.wellFounded (enum _) wo.wf)
| Mathlib/SetTheory/Ordinal/Basic.lean | 542 | 544 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.GroupWithZero.N... | Mathlib/RingTheory/Localization/Basic.lean | 1,365 | 1,367 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | rw [ENNReal.ofReal, ENNReal.ofReal, coe_lt_coe, Real.toNNReal_lt_toNNReal_iff h]
| Mathlib/Data/ENNReal/Real.lean | 159 | 160 |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.LinearAlgebra.Alternating.Basic
/-!
# Exterior Algebras
We construct the e... | · rintro ⟨rfl, rfl⟩
rw [LinearMap.map_zero, RingHom.map_zero]
@[simp]
theorem ι_ne_one [Nontrivial R] (x : M) : ι R x ≠ 1 := by
| Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean | 220 | 224 |
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algeb... | (e : ι → ℕ), DirectSum.IsInternal fun i => torsionBy R M <| p i ^ e i := by
classical
refine ⟨_, ?_, _, _, ?_, _, Submodule.isInternal_prime_power_torsion_of_pid hM⟩
· exact Finset.fintypeCoeSort _
· rintro ⟨p, hp⟩
have hP := prime_of_factor p (Multiset.mem_toFinset.mp hp)
haveI := Ideal.isPrime... | Mathlib/Algebra/Module/PID.lean | 89 | 98 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.PreservesHomology
import Mathlib.Algebra.Homology.ShortComplex.Abelian
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import... |
lemma exact_of_f_is_kernel (hS : IsLimit (KernelFork.ofι S.f S.zero))
[S.HasHomology] : S.Exact := by
rw [exact_iff_epi_toCycles]
have : IsSplitEpi S.toCycles :=
| Mathlib/Algebra/Homology/ShortComplex/Exact.lean | 404 | 408 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Lie.TensorProduct
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.RingTheory.TensorPro... | lemma baseChange_top : (⊤ : LieSubmodule R L M).baseChange A = ⊤ := by
simp only [baseChange, top_toSubmodule, Submodule.baseChange_top,
Submodule.bot_toAddSubmonoid]
rfl
| Mathlib/Algebra/Lie/BaseChange.lean | 206 | 210 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.InfiniteSum.Defs
imp... | calc
∏' x, f x = ∏' x, (ite (x = b) (f x) 1 * update f b 1 x) :=
tprod_congr fun n ↦ by split_ifs with h <;> simp [update_apply, h]
| Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 589 | 591 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
/-!
# Preadditive monoidal categories
A monoidal category is `M... | { preserves := fun {b} i => ⟨isBilimitOfTotal _ (by
dsimp
simp_rw [← id_tensorHom]
| Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 118 | 120 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | finite_of_encard_le_coe h.le
| Mathlib/Data/Set/Card.lean | 152 | 153 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Int.Notation
import Mathlib.Data.Nat.BinaryRec
import Mathlib.L... | Mathlib/Algebra/Group/Defs.lean | 1,262 | 1,263 | |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.LinearAlgebra.BilinearForm.Hom
import Mathlib.LinearAlgebra.Dual.Lemmas
/-!
# Bilinear form
This file defines various properties of ... | B.compLeft_injective b <| ext fun v w => by rw [compLeft_apply, compLeft_apply, hψ₁, hψ₂]
| Mathlib/LinearAlgebra/BilinearForm/Properties.lean | 220 | 221 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Ordering.Lemmas
import Mathlib.Data.PNat.Basic
import Mathlib.SetTheory.Ordinal.Principal
import Ma... | | Ordering.gt, (h : repr a > repr b)=> (ne_of_gt h e).elim
| Ordering.eq, h => h,
congr_arg _⟩
| Mathlib/SetTheory/Ordinal/Notation.lean | 320 | 322 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.AddGroupWithTop
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.ENat.Defs
import... | Order.succ_eq_add_one m
theorem add_one_le_iff (hm : m ≠ ⊤) : m + 1 ≤ n ↔ m < n :=
Order.add_one_le_iff_of_not_isMax (not_isMax_iff_ne_top.mpr hm)
theorem one_le_iff_ne_zero : 1 ≤ n ↔ n ≠ 0 :=
| Mathlib/Data/ENat/Basic.lean | 273 | 278 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | theorem sphere_eq_empty_of_neg (hε : ε < 0) : sphere x ε = ∅ :=
Set.eq_empty_iff_forall_not_mem.mpr fun _y hy => (nonneg_of_mem_sphere hy).not_lt hε
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 415 | 416 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.Ess... | @[simp]
theorem limit.pre_pre [h : HasLimit (D ⋙ E ⋙ F)] : haveI : HasLimit ((D ⋙ E) ⋙ F) := h
limit.pre F E ≫ limit.pre (E ⋙ F) D = limit.pre F (D ⋙ E) := by
haveI : HasLimit ((D ⋙ E) ⋙ F) := h
ext j; erw [assoc, limit.pre_π, limit.pre_π, limit.pre_π]; rfl
variable {E F}
| Mathlib/CategoryTheory/Limits/HasLimits.lean | 374 | 380 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.MonoidAlgebra.Defs
/-!
# Division of `AddMonoidAlgebra` by monomials
This file is most important for when `G = ℕ` (polynomials) or `G = σ →₀ ℕ` (mu... | ext
rw [AddMonoidAlgebra.divOf_apply, of'_apply, mul_single_apply_aux, mul_one]
intro c hc
rw [add_comm]
exact add_right_inj _
| Mathlib/Algebra/MonoidAlgebra/Division.lean | 105 | 109 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Antoine Chambert-Loir
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.IndexNormal
import Mathlib.GroupTheory.Perm.F... | theorem alternatingGroup_eq_sign_ker : alternatingGroup α = sign.ker :=
rfl
namespace Equiv.Perm
| Mathlib/GroupTheory/SpecificGroups/Alternating.lean | 77 | 80 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 823 | 824 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Order.Filter.Cofinite
/-!
# Computational realization of filters (experimental)
This file provides infrastructure to compute with filters.
## Main d... | ⟨f.sets,
{ f := Subtype.val
| Mathlib/Data/Analysis/Filter.lean | 112 | 113 |
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Matrix
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.Tactic.NoncommRing
/-!
# Lie algebras of skew-adjoint endomorphisms o... | rw [mem_skewAdjointSubmodule] at *
have hfg : IsAdjointPair B B (f * g) (g * f) := by rw [← neg_mul_neg g f]; exact hg.comp hf
have hgf : IsAdjointPair B B (g * f) (f * g) := by rw [← neg_mul_neg f g]; exact hf.comp hg
change IsAdjointPair B B (f * g - g * f) (-(f * g - g * f)); rw [neg_sub]
exact hfg.sub hgf... | Mathlib/Algebra/Lie/SkewAdjoint.lean | 46 | 53 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | · use 1; simpa using hr.trans_lt hε
· simpa [lt_div_iff₀ hr] using exists_norm_lt 𝕜 (div_pos hε hr)
rw [← set_smul_mem_nhds_zero_iff (norm_pos_iff.1 hk₀), smul_closedBall_zero hk₀] at hp
exact mem_of_superset hp <| p.closedBall_mono hk.le
| Mathlib/Analysis/Seminorm.lean | 1,061 | 1,065 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer
import Mathlib.CategoryTheory.Limits.Shapes.SplitEqualizer
import Mathlib.CategoryTheory.Limits.Preserves.Bas... | /-- Any functor preserves coequalizers of split pairs. -/
instance (priority := 1) preservesSplitCoequalizers (f g : X ⟶ Y) [HasSplitCoequalizer f g] :
PreservesColimit (parallelPair f g) G := by
apply
preservesColimit_of_preserves_colimit_cocone
(HasSplitCoequalizer.isSplitCoequalizer f g).isCoequalize... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Equalizers.lean | 224 | 231 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Johan Commelin, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Equivalence
import Mathlib.CategoryTheory.Yoneda
/-!
# Adjunctions between functors
`F ⊣ G` represents the dat... |
theorem homEquiv_naturality_right (f : F.obj X ⟶ Y) (g : Y ⟶ Y') :
(adj.homEquiv X Y') (f ≫ g) = (adj.homEquiv X Y) f ≫ G.map g := by
simp
| Mathlib/CategoryTheory/Adjunction/Basic.lean | 214 | 217 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Right-angled triangles
This file proves ba... | rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ←
inner_neg_left, neg_vsub_eq_vsub_rev] at h
rw [angle, dist_eq_norm_vsub' V p₃ p₂, dist_eq_norm_vsub V p₁ p₃, ← vsub_add_vsub_cancel p₁ p₂ p₃,
add_comm, angle_add_eq_arccos_of_inner_eq_zero h]
| Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 334 | 338 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.SimplicialObject.Split
import Mathlib.AlgebraicTopology.DoldKan.Degeneracies
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
/-!
# Split si... |
variable [Preadditive C]
theorem decomposition_id (Δ : SimplexCategoryᵒᵖ) :
| Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean | 53 | 56 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | (H : ∃ (j : Subtype p), BddBelow (range (fun (i : s j) ↦ f i))) :
liminf f v = ⨆ (j : Subtype p), ⨅ (i : s (liminf_reparam f s p j)), f i := by
| Mathlib/Order/LiminfLimsup.lean | 975 | 976 |
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.Field.ZMod
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.LocalRing.ResidueField.... | obtain ⟨e, he⟩ := Ideal.mem_span_singleton.mp (PadicInt.appr_spec n x)
rw [sub_eq_iff_eq_add] at he
obtain ⟨N, hN⟩ := padicNormE.defn s (ε := p ^ (-n : ℤ))
(by simp only [zpow_neg, zpow_natCast, inv_pos]; exact_mod_cast Nat.pos_of_neZero _)
replace hN := hN (N + n) (Nat.le_add_right N n)
rw [← hs, he, ← R... | Mathlib/NumberTheory/Padics/RingHoms.lean | 688 | 698 |
/-
Copyright (c) 2020 Alexander Bentkamp, Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Sébastien Gouëzel, Eric Wieser
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Data.... |
lemma Complex.coe_realPart (z : ℂ) : (ℜ z : ℂ) = z.re := calc
(ℜ z : ℂ) = (↑(ℜ (↑z.re + ↑z.im * I))) := by congrm (ℜ $((re_add_im z).symm))
| Mathlib/Data/Complex/Module.lean | 486 | 488 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.MeasureTheory... | /-! ### Measure of balls -/
theorem addHaar_ball_center {E : Type*} [NormedAddCommGroup E] [MeasurableSpace E] [BorelSpace E]
(μ : Measure E) [IsAddHaarMeasure μ] (x : E) (r : ℝ) : μ (ball x r) = μ (ball (0 : E) r) := by
have : ball (0 : E) r = (x + ·) ⁻¹' ball x r := by simp [preimage_add_ball]
rw [this, meas... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 411 | 420 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... | f = (ρ_ X).inv ≫ f ▷ 𝟙_ C ≫ (ρ_ Y).hom := by
simp
| Mathlib/CategoryTheory/Monoidal/Category.lean | 486 | 487 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Algebra.Ring.Pointwise.Set
import Mathlib.Order.Filter.AtTopBot.CompleteLattice
import Mathlib.Order.Filter.AtTopBot.G... | constructor
· rintro ⟨u, au, as⟩
rcases exists_between au with ⟨v, hv⟩
exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩
· rintro ⟨u, au, as⟩
exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩
@[deprecated (since := "2024-12-22")]
| Mathlib/Topology/Order/LeftRightNhds.lean | 131 | 138 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Measure.Typeclasses.SFinite
/-!
# Restriction of a measure to a sub-σ-algebra
## Main definitions
* `MeasureTheory.Measure.trim`: restric... |
theorem ae_of_ae_trim (hm : m ≤ m0) {μ : Measure α} {P : α → Prop} (h : ∀ᵐ x ∂μ.trim hm, P x) :
∀ᵐ x ∂μ, P x :=
| Mathlib/MeasureTheory/Measure/Trim.lean | 69 | 71 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,148 | 1,173 | |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Heather Macbeth, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Analysis.Normed.Group.Uniform
import Mathlib.Topo... | calc
‖∑ k ∈ Ico N n, f k‖ ≤ ∑ k ∈ _, ‖f k‖ := norm_sum_le _ _
_ ≤ ∑ k ∈ _, g k := sum_le_sum fun x _ => hf x
_ ≤ ‖∑ k ∈ _, g k‖ := le_abs_self _
_ < ε := hg
theorem cauchySeq_finset_of_summable_norm {f : ι → E} (hf : Summable fun a => ‖f a‖) :
CauchySeq fun s : Finset ι => ∑ a ∈ s, f a :=
cauch... | Mathlib/Analysis/Normed/Group/InfiniteSum.lean | 78 | 89 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Group.Semiconj.Units
/-!
# Lemmas about semiconjugate elements in a `GroupWithZero`.
-/
ass... |
@[simp]
| Mathlib/Algebra/GroupWithZero/Semiconj.lean | 29 | 30 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathli... | Mathlib/Data/Finset/Image.lean | 807 | 807 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.FreeAlgebra
import Mathlib.RingTheory.Adjoin.Polynomial
import Mathlib.RingTheory.Adjoin.Tower
import Mathlib.RingTheory.Ideal.Quotient.Operati... | exact ⟨r • P, map_smul _ _ _⟩
variable (R M)
/-- If an additive monoid `M` is finitely generated then `R[M]` is of finite
type. -/
instance finiteType_of_fg [CommRing R] [h : AddMonoid.FG M] :
FiniteType R R[M] := by
obtain ⟨S, hS⟩ := h.fg_top
exact (FiniteType.freeAlgebra R (S : Set M)).of_surjective
... | Mathlib/RingTheory/FiniteType.lean | 451 | 470 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Topology.Instances.ENNReal.Lemmas
import Mathlib.MeasureTheory.Measure.Dirac
/-!
# Probability mass functions
This file is about probabil... | Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 368 | 371 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | Mathlib/Data/Fintype/Basic.lean | 507 | 510 |
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