Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sรฉbastien Gouรซzel, Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Subspace
import Mathlib.LinearAlgebra.SesquilinearForm
/-!
# Orthogonal complements of su... | simp only [iInf_coe]
/-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
instance instOrthogonalCompleteSpace [CompleteSpace E] : CompleteSpace Kแฎ :=
K.isClosed_orthogonal.completeSpace_coe
variable (๐ E)
| Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 116 | 123 |
/-
Copyright (c) 2019 Sรฉbastien Gouรซzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sรฉbastien Gouรซzel
-/
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,411 | 1,427 | |
/-
Copyright (c) 2023 Rรฉmy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rรฉmy Degenne
-/
import Mathlib.Probability.Independence.Kernel
import Mathlib.Probability.Kernel.Condexp
/-!
# Conditional Independence
We define conditional independence of sets/ฯ-alg... | simp only [iCondIndepFun_iff_iCondIndep]
rw [iCondIndep_iff]
exact fun i โฆ (hf i).comap_le
lemma condIndepFun_iff_condIndep {ฮฒ ฮณ : Type*} [mฮฒ : MeasurableSpace ฮฒ]
[mฮณ : MeasurableSpace ฮณ] (f : ฮฉ โ ฮฒ) (g : ฮฉ โ ฮณ) (ฮผ : Measure ฮฉ) [IsFiniteMeasure ฮผ] :
CondIndepFun m' hm' f g ฮผ
โ CondIndep m' (Measura... | Mathlib/Probability/Independence/Conditional.lean | 327 | 335 |
/-
Copyright (c) 2023 Yaรซl Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Triangle.Basic
/-!
# Construct a tripartite graph from its triangles
This file contains the constru... | /-- This lemma reorders the elements of a triangle in the tripartite graph. It turns a triangle
`{x, y, z}` into a triangle `{a, b, c}` where `a : ฮฑ `, `b : ฮฒ`, `c : ฮณ`. -/
lemma graph_triple โฆx y zโฆ :
(graph t).Adj x y โ (graph t).Adj x z โ (graph t).Adj y z โ โ a b c,
({inโ a, inโ b, inโ c} : Finset (ฮฑ โ ฮฒ โ ฮณ... | Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean | 143 | 151 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | This one instead uses a `NeZero n` typeclass hypothesis.
-/
@[simp]
protected theorem zero_le' [NeZero n] (a : Fin n) : 0 โค a :=
Nat.zero_le a.val
| Mathlib/Data/Fin/Basic.lean | 205 | 210 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hรถlzl, Yaรซl Dillies
-/
import Mathlib.Algebra.Group.PUnit
import Mathlib.Algebra.Group.ULift
import Mathlib.Analysis.Normed.Group.Basic
/-!
# Product of norme... | refine Pi.apply_single (fun i (x : G i) โฆ โxโโ) ?_ i y b
simp
simp [Pi.nnnorm_def, H, Pi.single_apply, Finset.sup_ite, Finset.filter_eq']
| Mathlib/Analysis/Normed/Group/Constructions.lean | 402 | 404 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.RingTheory.GradedAlgebra.Basic
/-!
# Results about the grading structure of the clifford algebra
The mai... |
theorem evenOdd_mul_le (i j : ZMod 2) : evenOdd Q i * evenOdd Q j โค evenOdd Q (i + j) := by
simp_rw [evenOdd, Submodule.iSup_eq_span, Submodule.span_mul_span]
apply Submodule.span_mono
simp_rw [Set.iUnion_mul, Set.mul_iUnion, Set.iUnion_subset_iff, Set.mul_subset_iff]
rintro โจxi, rflโฉ โจyi, rflโฉ x hx y hy
ref... | Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean | 49 | 55 |
/-
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
-/
import Mathlib.Data.Subtype
import Mathlib.Order.Defs.LinearOrder
import Mathlib.Order.Notation
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.... | Mathlib/Order/Basic.lean | 1,553 | 1,554 | |
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Data.Complex.Basic
import Mathlib.MeasureTheory.Integral.CircleIntegral
/-!
# Circle integral transform
In this file we define the circle integral tr... | let V : โ โ โ โ โ := fun ฮธ w => circleTransformDeriv R z w (fun _ => 1) ฮธ
obtain โจX, -, HX2โฉ := (isCompact_sphere z R).exists_isMaxOn
(NormedSpace.sphere_nonempty.2 hR.le) hf.norm
refine โจโV b aโ * โf Xโ, ฮต', hฮต', H.trans (ball_subset_ball hr.le), fun y v hv โฆ ?_โฉ
obtain โจy1, hy1, hfunโฉ :=
Periodic.exis... | Mathlib/MeasureTheory/Integral/CircleTransform.lean | 133 | 152 |
/-
Copyright (c) 2023 Rรฉmy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rรฉmy Degenne, Peter Pfaffelhuber
-/
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Set.Accumulate
import Mathlib.Data.Set.Pairwise.Lattice
import Mathlib.MeasureTheory.PiSystem
/-!... |
lemma subset_of_diffUnion_disjointOfDiffUnion (hC : IsSetSemiring C) (hs : s โ C) (hI : โI โ C)
(t : Set ฮฑ) (ht : t โ (hC.disjointOfDiffUnion hs hI : Set (Set ฮฑ))) :
t โ s \ โโ I := by
revert t ht
rw [โ sUnion_subset_iff, hC.diff_sUnion_eq_sUnion_disjointOfDiffUnion hs hI]
lemma subset_of_mem_disjointOfDi... | Mathlib/MeasureTheory/SetSemiring.lean | 254 | 261 |
/-
Copyright (c) 2016 Johannes Hรถlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hรถlzl, Mario Carneiro
-/
import Mathlib.Data.Set.Defs
import Mathlib.Logic.Basic
import Mathlib.Logic.ExistsUnique
import Mathlib.Logic.Nonempty
import Mathlib.Logic.Nontrivia... |
theorem Function.LeftInverse.eq_rec_on_eq {ฮณ : ฮฒ โ Sort v} {f : ฮฑ โ ฮฒ} {g : ฮฒ โ ฮฑ}
(h : Function.LeftInverse g f) (C : โ a : ฮฑ, ฮณ (f a)) (a : ฮฑ) :
-- TODO: mathlib3 uses `(congr_arg f (h a)).recOn (C (g (f a)))` for LHS
@Eq.recOn ฮฒ (f (g (f a))) (fun x _ โฆ ฮณ x) (f a) (congr_arg f (h a)) (C (g (f a))) = C a... | Mathlib/Logic/Function/Basic.lean | 1,007 | 1,016 |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | instance distribLattice : DistribLattice G.Subgraph :=
{ show DistribLattice G.Subgraph from
verts_spanningCoe_injective.distribLattice _
(fun _ _ => rfl) fun _ _ => rfl with
le := fun x y => x.verts โ y.verts โง โ โฆv w : Vโฆ, x.Adj v w โ y.Adj v w }
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 437 | 441 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.Connected.Basic
/-!
# Locally connected topological spaces
A topological space is **locally connected** if each neighborhood filter admi... |
protected theorem IsOpen.connectedComponentIn [LocallyConnectedSpace ฮฑ] {F : Set ฮฑ} {x : ฮฑ}
(hF : IsOpen F) : IsOpen (connectedComponentIn F x) := by
rw [isOpen_iff_mem_nhds]
intro y hy
rw [connectedComponentIn_eq hy]
| Mathlib/Topology/Connected/LocallyConnected.lean | 70 | 75 |
/-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Junyan Xu, Jack McKoen
-/
import Mathlib.RingTheory.Valuation.ValuationRing
import Mathlib.RingTheory.Localization.AsSubring
import Mathlib.Algebra.Ring.Subring.Pointwise
impor... | right_inv a := by ext; rfl
map_mul' a b := by ext; rfl
@[simp]
theorem coe_unitGroupMulEquiv_apply (a : A.unitGroup) :
((A.unitGroupMulEquiv a : A) : K) = ((a : Kหฃ) : K) := rfl
@[simp]
| Mathlib/RingTheory/Valuation/ValuationSubring.lean | 436 | 443 |
/-
Copyright (c) 2020 Frรฉdรฉric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frรฉdรฉric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... |
/-- The real part in an `RCLike` field, as a continuous linear map. -/
noncomputable def reCLM : K โL[โ] โ :=
reLm.mkContinuous 1 fun x => by
rw [one_mul]
exact abs_re_le_norm x
@[simp, rclike_simps, norm_cast]
theorem reCLM_coe : ((reCLM : K โL[โ] โ) : K โโ[โ] โ) = reLm :=
| Mathlib/Analysis/RCLike/Basic.lean | 956 | 964 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Analytic.Inverse
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.Deriv... | h.hasDerivAt.deriv
/-- If a function is analytic on a set `s` in a complete space, so is its derivative. -/
protected theorem AnalyticOnNhd.deriv [CompleteSpace F] (h : AnalyticOnNhd ๐ f s) :
AnalyticOnNhd ๐ (deriv f) s :=
| Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 426 | 430 |
/-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... | ext : 1 <;> assumption
/- We prove that the `SubNegMonoid`s are equal because they are one
field away from `Sub` and `Neg`, enabling use of `injection`. -/
have hโ : (instโ.toAddCommGroup).toAddGroup.toSubNegMonoid
= (instโ.toAddCommGroup).toAddGroup.toSubNegMonoid :=
congrArg (@AddGroup.toSub... | Mathlib/Algebra/Ring/Ext.lean | 317 | 330 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaรซl Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... | theorem le_symmDiff_sup_left (a b : ฮฑ) : b โค (a โ b) โ a :=
symmDiff_comm a b โธ le_symmDiff_sup_right ..
end GeneralizedCoheytingAlgebra
section GeneralizedHeytingAlgebra
variable [GeneralizedHeytingAlgebra ฮฑ] (a b c : ฮฑ)
| Mathlib/Order/SymmDiff.lean | 189 | 196 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Transfer
/-!
# The Schur-Zassenhaus Theorem
In this file we prove the Schur-Zassenhaus theorem.
## Main results
- `Subgroup.exists_ri... | have h6 : Nat.Coprime (Nat.card (N.comap K.subtype)) (N.comap K.subtype).index := by
rw [h4]
exact h1.coprime_dvd_left (card_comap_dvd_of_injective N K.subtype Subtype.coe_injective)
obtain โจH, hHโฉ := h2 K h5 h6
replace hH : Nat.card (H.map K.subtype) = N.index := by
rw [โ relindex_bot_left, โ relinde... | Mathlib/GroupTheory/SchurZassenhaus.lean | 167 | 188 |
/-
Copyright (c) 2022 Antoine Labelle, Rรฉmi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rรฉmi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
/-!
# Rewriting arrows and paths along vertex... | Mathlib/Combinatorics/Quiver/Cast.lean | 136 | 139 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.FieldTheory.SplittingField.IsSplittingField
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.RingTheory.Algebra... | ยท exact (Classical.choose_spec H).1
ยท exact irreducible_X
/-- See note [fact non-instances]. -/
theorem fact_irreducible_factor (f : K[X]) : Fact (Irreducible (factor f)) :=
| Mathlib/FieldTheory/SplittingField/Construction.lean | 55 | 59 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Data.Nat.EvenOddRec
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
/-!
# Elliptic divisibility sequences
... |
/-- The canonical example of a normalised EDS `W : โค โ R`, with initial values
`W(0) = 0`, `W(1) = 1`, `W(2) = b`, `W(3) = c`, and `W(4) = d * b`.
This is defined in terms of `preNormEDS` whose even terms differ by a factor of `b`. -/
def normEDS (n : โค) : R :=
preNormEDS (b ^ 4) c d n * if Even n then b else 1
@[... | Mathlib/NumberTheory/EllipticDivisibilitySequence.lean | 291 | 300 |
/-
Copyright (c) 2017 Johannes Hรถlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hรถlzl
-/
import Mathlib.Logic.Encodable.Lattice
import Mathlib.Order.Filter.AtTopBot.Finset
import Mathlib.Topology.Algebra.InfiniteSum.Group
/-!
# Infinite sums and product... |
end Int
section pnat
@[to_additive]
theorem pnat_multipliable_iff_multipliable_succ {ฮฑ : Type*} [TopologicalSpace ฮฑ] [CommMonoid ฮฑ]
| Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 537 | 543 |
/-
Copyright (c) 2017 Johannes Hรถlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hรถlzl, Mario Carneiro, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 546 | 547 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sรถnne, Sรฉbastien Gouรซzel,
Rรฉmy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | lemma inv_natCast_cpow_ofReal_pos {n : โ} (hn : n โ 0) (x : โ) :
0 < ((n : โ) ^ (x : โ))โปยน := by
refine RCLike.inv_pos_of_pos ?_
rw [show (n : โ) ^ (x : โ) = (n : โ) ^ (x : โ) from rfl, โ ofReal_cpow n.cast_nonneg']
positivity
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 1,025 | 1,029 |
/-
Copyright (c) 2020 Rรฉmy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rรฉmy Degenne, Sรฉbastien Gouรซzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | eLpNorm' f q ฮผ = eLpNorm' g q ฮผ :=
eLpNorm'_congr_nnnorm_ae <| hfg.mono fun _x hx => NNReal.eq hx
theorem eLpNorm'_congr_ae {f g : ฮฑ โ ฮต} (hfg : f =แต[ฮผ] g) : eLpNorm' f q ฮผ = eLpNorm' g q ฮผ :=
eLpNorm'_congr_enorm_ae (hfg.fun_comp _)
theorem eLpNormEssSup_congr_ae {f g : ฮฑ โ ฮต} (hfg : f =แต[ฮผ] g) :
eLpNorm... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 347 | 356 |
/-
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... |
protected theorem mvPolynomial (ฮน : Type*) [Finite ฮน] : FiniteType R (MvPolynomial ฮน R) := by
cases nonempty_fintype ฮน
classical
exact
โจโจFinset.univ.image MvPolynomial.X, by
| Mathlib/RingTheory/FiniteType.lean | 86 | 91 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mat... | theorem _root_.Sbtw.angleโโโ_eq_pi {pโ pโ pโ : P} (h : Sbtw โ pโ pโ pโ) : โ pโ pโ pโ = ฯ := by
rw [โ h.angleโโโ_eq_pi, angle_comm]
/-- The angle between three points is ฯ if and only if the second point is strictly between the
other two. -/
theorem angle_eq_pi_iff_sbtw {pโ pโ pโ : P} : โ pโ pโ pโ = ฯ โ Sbtw โ pโ pโ ... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 268 | 275 |
/-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara
-/
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
/-!
# L... |
section LogDeriv
/-- The logarithmic derivative of a sequence of functions converging locally uniformly to a
function is the logarithmic derivative of the limit function. -/
theorem logDeriv_tendsto {ฮน : Type*} {p : Filter ฮน} (f : ฮน โ โ โ โ) (g : โ โ โ)
{s : Set โ} (hs : IsOpen s) (x : s) (hF : TendstoLocallyUnif... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 191 | 200 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.MvPowerSer... | theorem mul_X_pow_injective {k : โ} : Function.Injective (ยท * X ^ k : RโฆXโง โ RโฆXโง) :=
fun _ _ โฆ mul_X_pow_cancel
| Mathlib/RingTheory/PowerSeries/Basic.lean | 431 | 433 |
/-
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.Data.Fin.Tuple.Basic
/-!
# Lists from functions
Theorems and lemmas for dealing with `List.ofFn`, which converts a function on `Fin n` to a list
of l... | (โ l : List ฮฑ, P l) โ โ (n) (f : Fin n โ ฮฑ), P (List.ofFn f) :=
equivSigmaTuple.symm.surjective.forall.trans Sigma.forall
/-- `Fin.sigma_eq_iff_eq_comp_cast` may be useful to work with the RHS of this expression. -/
theorem ofFn_inj' {m n : โ} {f : Fin m โ ฮฑ} {g : Fin n โ ฮฑ} :
ofFn f = ofFn g โ (โจm, fโฉ : ฮฃn,... | Mathlib/Data/List/OfFn.lean | 201 | 206 |
/-
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
-/
import Mathlib.Data.List.Forall2
/-!
# zip & unzip
This file provides results about `List.zipWith`, `List.zip` and `List.unzip` (definitions are in
core ... | (init, tail) โ zip l.inits l.tails โ init ++ tail = l := by
induction' l with hd tl ih generalizing init tail <;> simp_rw [tails, inits, zip_cons_cons]
ยท simp
ยท constructor <;> rw [mem_cons, zip_map_left, mem_map, Prod.exists]
ยท rintro (โจrfl, rflโฉ | โจ_, _, h, rfl, rflโฉ)
ยท simp
| Mathlib/Data/List/Zip.lean | 125 | 130 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
/-!
# Tuples of types, and their categorica... |
theorem repeatEq_iff_eq {ฮฑ : TypeVec n} {i x y} :
ofRepeat (repeatEq ฮฑ i (prod.mk _ x y)) โ x = y := by
induction i with
| Mathlib/Data/TypeVec.lean | 491 | 494 |
/-
Copyright (c) 2023 Shogo Saito. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shogo Saito. Adapted for mathlib by Hunter Monroe
-/
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.ChineseRemainder
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.Nat.Pai... | private def supOfSeq (a : Fin m โ โ) : โ := max m (Finset.sup .univ a) + 1
private def coprimes (a : Fin m โ โ) : Fin m โ โ := fun i => (i + 1) * (supOfSeq a)! + 1
lemma coprimes_lt (a : Fin m โ โ) (i) : a i < coprimes a i := by
have hโ : a i < supOfSeq a :=
Nat.lt_add_one_iff.mpr (le_max_of_le_right <| Finset.... | Mathlib/Logic/Godel/GodelBetaFunction.lean | 63 | 69 |
/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Algebra.Order.Ring.Canonical
/-!
# Distance function on โ
This file defines a simple dista... | Mathlib/Data/Nat/Dist.lean | 107 | 110 | |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.Algebra.Ring.Subring.Units
import Mathlib.LinearAlgebra.LinearIndepende... | Mathlib/LinearAlgebra/Ray.lean | 717 | 721 | |
/-
Copyright (c) 2023 Yaรซl Dillies, Vladimir Ivanov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies, Vladimir Ivanov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.... | refine le_antisymm ?_ le_truncatedSup
simp_rw [truncatedSup_of_mem (subset_lowerClosure ha), sup'_le_iff, mem_filter]
rintro b โจhb, habโฉ
exact (hs.eq ha hb hab).ge
| Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean | 148 | 151 |
/-
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, Johannes Hรถlzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... |
theorem Ici_iSupโ (f : โ i, ฮบ i โ ฮฑ) : Ici (โจ (i) (j), f i j) = โ (i) (j), Ici (f i j) := by
simp_rw [Ici_iSup]
theorem Iic_iInfโ (f : โ i, ฮบ i โ ฮฑ) : Iic (โจ
(i) (j), f i j) = โ (i) (j), Iic (f i j) := by
simp_rw [Iic_iInf]
theorem Ici_sSup (s : Set ฮฑ) : Ici (sSup s) = โ a โ s, Ici a := by rw [sSup_eq_iSup, Ici_... | Mathlib/Data/Set/Lattice.lean | 1,268 | 1,277 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Order.Antidiag.Finsupp
import Mathlib.Data.Finsupp.Weight
import Mathlib.Tactic.Linarith
import Mathlib.LinearAlgebra.Pi
import Mat... |
instance : Mul (MvPowerSeries ฯ R) :=
letI := Classical.decEq ฯ
โจfun ฯ ฯ n => โ p โ antidiagonal n, coeff R p.1 ฯ * coeff R p.2 ฯโฉ
theorem coeff_mul [DecidableEq ฯ] :
coeff R n (ฯ * ฯ) = โ p โ antidiagonal n, coeff R p.1 ฯ * coeff R p.2 ฯ := by
refine Finset.sum_congr ?_ fun _ _ => rfl
rw [Subsingleton.el... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 213 | 222 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 631 | 636 | |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
/-!
# Infimum separation
This file defines the extended infimum separation of a set. This is approximately dual to the
diame... | Mathlib/Topology/MetricSpace/Infsep.lean | 537 | 539 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Continuous
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.To... | noncomputable instance : NormedDivisionRing โ where
dist_eq _ _ := rfl
| Mathlib/Analysis/Quaternion.lean | 83 | 84 |
/-
Copyright (c) 2023 Mohanad ahmed. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mohanad Ahmed
-/
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.SemiringInverse
/-! # Block Matrices from Rows and Columns
This file provides the basic definiti... | @[deprecated (since := "2024-12-11")] alias vecMul_fromColumns := vecMul_fromCols
@[simp]
lemma sumElim_vecMul_fromRows [Fintype mโ] [Fintype mโ] (Bโ : Matrix mโ n R) (Bโ : Matrix mโ n R)
(vโ : mโ โ R) (vโ : mโ โ R) :
| Mathlib/Data/Matrix/ColumnRowPartitioned.lean | 213 | 217 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Analysis.Normed.Ring.Lemmas
/-!
# The integers as normed ring
This file contains basic facts about the integers as normed ring.
Recall that `โnโ` de... | simp only [Units.coe_neg_one, Units.val_one, nnnorm_neg, nnnorm_one]
theorem norm_coe_units (e : โคหฃ) : โ(e : โค)โ = 1 := by
| Mathlib/Analysis/NormedSpace/Int.lean | 24 | 26 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
/-!
# Weights and roots of Lie modules and Lie algebras with respect to Cartan subalge... | @[simp]
theorem rootSpace_zero_eq (H : LieSubalgebra R L) [H.IsCartanSubalgebra] [IsNoetherian R L] :
rootSpace H 0 = H.toLieSubmodule := by
rw [โ LieSubmodule.toSubmodule_inj, โ coe_zeroRootSubalgebra,
| Mathlib/Algebra/Lie/Weights/Cartan.lean | 239 | 242 |
/-
Copyright (c) 2022 Marรญa Inรฉs de Frutos-Fernรกndez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, Marรญa Inรฉs de Frutos-Fernรกndez
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Algeb... | simp only [weight, LinearMap.toAddMonoidHom_coe, Finsupp.linearCombination_apply, Finsupp.sum,
sum_eq_zero_iff, Finsupp.mem_support_iff, Ne, smul_eq_zero, not_forall, not_or,
and_self_left, exists_prop]
simp only [DFunLike.ext_iff, Finsupp.coe_zero, Pi.zero_apply, not_forall] at hd
obtain โจi, hi... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 512 | 519 |
/-
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.Analysis.Convex.Between
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.Analysis.Normed.Affine.AddTorsor
import Mathlib.Analysis.Normed.Aff... | dist pโ p < max (dist pโ p) (dist pโ p) := by
have hpโpโ : pโ -แตฅ p โ pโ -แตฅ p := by simpa using h.left_ne_right
rw [Sbtw, โ wbtw_vsub_const_iff p, Wbtw, affineSegment_eq_segment, โ insert_endpoints_openSegment,
Set.mem_insert_iff, Set.mem_insert_iff] at h
rcases h with โจh | h | h, hpโpโ, hpโpโโฉ
ยท rw [vsu... | Mathlib/Analysis/Convex/StrictConvexBetween.lean | 29 | 43 |
/-
Copyright (c) 2021 Frรฉdรฉric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frรฉdรฉric Dupuis
-/
import Mathlib.Algebra.Group.Subgroup.Defs
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
import Mathlib.Algebra.Star.Rat
/-!
# Self-adjoint, sk... | refine โจfun hrx โฆ ?_, .smul hrโฉ
lift r to Rหฃ using hu
| Mathlib/Algebra/Star/SelfAdjoint.lean | 293 | 294 |
/-
Copyright (c) 2019 Johannes Hรถlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hรถlzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Nonneg.Field
import Mathlib.Data.Rat.Cast.Defs
import Mathlib.Tactic.Positivity.Basic
/-!
# Some exiled lemmas about cas... | โ (Int.cast_commute _ _).mul_pow, โ div_eq_mul_inv]
@[simp]
theorem cast_inv_nat (n : โ) : ((nโปยน : โ) : ฮฑ) = (n : ฮฑ)โปยน := by
rcases n with - | n
| Mathlib/Data/Rat/Cast/Lemmas.lean | 28 | 32 |
/-
Copyright (c) 2020 Sรฉbastien Gouรซzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sรฉbastien Gouรซzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin.Basic
imp... | apply iSup_mono'
intro h
simp only [le_refl, le_sup_iff, exists_prop, and_true]
intro n
cases eq_zero_or_pos r with
| inl hr =>
rw [hr]
cases n <;> simp
| Mathlib/Analysis/Analytic/Basic.lean | 361 | 368 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.P... | -- instance {R : Type u} {ฯ : Type v} [CommRing R] [IsDomain R] :
-- IsDomain (MvPolynomial ฯ R)[X] := inferInstance
theorem map_mvPolynomial_eq_evalโ {S : Type*} [CommSemiring S] [Finite ฯ]
(ฯ : MvPolynomial ฯ R โ+* S) (p : MvPolynomial ฯ R) :
ฯ p = MvPolynomial.evalโ (ฯ.comp MvPolynomial.C) (fun s => ฯ (... | Mathlib/RingTheory/Polynomial/Basic.lean | 1,061 | 1,068 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic
import Mathlib.Tactic.ComputeDegree
/-!
# Division polynomials of Weierst... |
lemma natDegree_preฮจ'_pos {n : โ} (hn : 2 < n) (h : (n : R) โ 0) : 0 < (W.preฮจ' n).natDegree := by
simp only [W.natDegree_preฮจ' h, Nat.div_pos_iff, zero_lt_two, true_and]
split_ifs <;>
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean | 250 | 253 |
/-
Copyright (c) 2020 Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jalex Stark, Kim Morrison, Eric Wieser, Oliver Nash, Wen Yang
-/
import Mathlib.Data.Matrix.Basic
/-!
# Matrices with a single non-zero element.
This file provides `Matrix.stdBasisMatri... | map_smul' _ _:= smul_stdBasisMatrix _ _ _ _ |>.symm
section ext
| Mathlib/Data/Matrix/Basis.lean | 133 | 135 |
/-
Copyright (c) 2024 Joรซl Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joรซl Riou
-/
import Mathlib.Algebra.Category.ModuleCat.ChangeOfRings
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.RingTheory.Kaehler.Basic
/-!
# The differentials of a morphis... | @[simp]
lemma d_mul (b b' : B) : D.d (b * b') = b โข D.d b' + b' โข D.d b := by simp [d]
| Mathlib/Algebra/Category/ModuleCat/Differentials/Basic.lean | 72 | 73 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Sh... |
@[simp, reassoc]
theorem t_snd (i j : ๐ฐ.J) : t ๐ฐ f g i j โซ pullback.snd _ _ =
pullback.fst _ _ โซ pullback.fst _ _ := by
| Mathlib/AlgebraicGeometry/Pullbacks.lean | 71 | 74 |
/-
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 Hรถlzl, Mario Carneiro, Sรฉbastien Gouรซzel
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Order.Interval.Finset.Na... | Mathlib/Topology/EMetricSpace/Basic.lean | 724 | 726 | |
/-
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... | Mathlib/Data/Vector3.lean | 280 | 291 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sรฉbastien Gouรซzel
-/
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
impo... | rw [deriv_pow', interior_Ici]
exact fun x (hx : 0 < x) y _ hxy => mul_lt_mul_of_pos_left
(pow_lt_pow_leftโ hxy hx.le <| Nat.sub_ne_zero_of_lt hn) (by positivity)
/-- `x^n`, `n : โ` is strictly convex on the whole real line whenever `n โ 0` is even. -/
| Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 40 | 44 |
/-
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.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... | (bโ bโ : P) : s.weightedVSubOfPoint p bโ w +แตฅ bโ = s.weightedVSubOfPoint p bโ w +แตฅ bโ := by
rw [weightedVSubOfPoint_apply, weightedVSubOfPoint_apply, โ @vsub_eq_zero_iff_eq V,
vadd_vsub_assoc, vsub_vadd_eq_vsub_sub, โ add_sub_assoc, add_comm, add_sub_assoc, โ
sum_sub_distrib]
conv_lhs =>
congr
ยท... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 123 | 135 |
/-
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.Data.Fin.Tuple.Basic
/-!
# Lists from functions
Theorems and lemmas for dealing with `List.ofFn`, which converts a function on `Fin n` to a list
of l... | simp_rw [ofFn_add, Fin.append_left, Fin.append_right]
/-- This breaks a list of `m*n` items into `m` groups each containing `n` elements. -/
theorem ofFn_mul {m n} (f : Fin (m * n) โ ฮฑ) :
List.ofFn f = List.flatten (List.ofFn fun i : Fin m => List.ofFn fun j : Fin n => f โจi * n + j,
calc
โi * n + j < (... | Mathlib/Data/List/OfFn.lean | 70 | 77 |
/-
Copyright (c) 2020 Sรฉbastien Gouรซzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sรฉbastien Gouรซzel
-/
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.LinearAlgebra.Multilinear.Curry
/-!
# Currying and uncurrying continuous multilinear maps
... | (continuousMultilinearCurryFin1 ๐ G G').symm f v = f (v 0) :=
| Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean | 464 | 464 |
/-
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.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 1,336 | 1,337 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hรถlzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,246 | 1,247 | |
/-
Copyright (c) 2021 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.Constructions
import Mathlib.Topology.Order.OrderClosed
/-!
# Topological lattices
In this file we define mixin classes `ContinuousI... | Tendsto (s.inf' hne f) l (๐ (s.inf' hne g)) :=
finset_sup'_nhds (L := Lแตแต) hne hs
lemma finset_inf'_nhds_apply [SemilatticeInf L] [ContinuousInf L]
(hne : s.Nonempty) (hs : โ i โ s, Tendsto (f i) l (๐ (g i))) :
Tendsto (fun a โฆ s.inf' hne (f ยท a)) l (๐ (s.inf' hne g)) :=
finset_sup'_nhds_apply (L :=... | Mathlib/Topology/Order/Lattice.lean | 136 | 144 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathli... | (fun i =>
not_imp_comm.mp fun hi =>
Finset.mem_erase.mpr
โจmt (fun h : i = k => show c i = 0 from h.symm โธ hk) hi, Finset.mem_univ iโฉ)
k (Finset.not_mem_erase k Finset.univ) A_eq
theorem det_eq_of_forall_row_eq_smul_add_pred_aux {n : โ} (k : Fin (n + 1)) :
โ (c : Fin n โ R) (_hc : โ ... | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 540 | 575 |
/-
Copyright (c) 2017 Johannes Hรถlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hรถlzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 1,382 | 1,383 | |
/-
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 ... | theorem inv_whiskerLeft (f : a โถ b) {g h : b โถ c} (ฮท : g โถ h) [IsIso ฮท] :
inv (f โ ฮท) = f โ inv ฮท := by
| Mathlib/CategoryTheory/Bicategory/Basic.lean | 201 | 202 |
/-
Copyright (c) 2017 Johannes Hรถlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hรถlzl, Yury Kudryashov, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | ยท rw [borel_eq_generateFrom_of_subbasis (@OrderTopology.topology_eq_generate_intervals ฮฑ _ _ _)]
letI : MeasurableSpace ฮฑ := MeasurableSpace.generateFrom (range Iio)
have H : โ a : ฮฑ, MeasurableSet (Iio a) := fun a => GenerateMeasurable.basic _ โจ_, rflโฉ
refine generateFrom_le ?_
rintro _ โจa, rfl | rfl... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 54 | 74 |
/-
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.Geometry.Euclidean.Altitude
import Mathlib.Geometry.Euclidean.Circumcenter
/-!
# Monge point and orthocenter
This file defines the orthocenter of a trian... | rcases ho with โจt, hto, hstโฉ
rw [hst] at hps
rcases exists_dist_eq_circumradius_of_subset_insert_orthocenter hto hps hpi with โจc, _, hcโฉ
exact Cospherical.affineIndependent โจc, t.circumradius, hcโฉ Set.Subset.rfl hpi
| Mathlib/Geometry/Euclidean/MongePoint.lean | 562 | 566 |
/-
Copyright (c) 2017 Johannes Hรถlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hรถlzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 2,234 | 2,236 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
i... | classical
by_contra h
exact w (isIso_of_mono_of_nonzero h)
theorem id_nonzero (X : C) [Simple.{v} X] : ๐ X โ 0 :=
| Mathlib/CategoryTheory/Simple.lean | 96 | 100 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.FiniteDimensional
import Mathlib.Da... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 586 | 589 | |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Affine
import Mathlib.LinearAlgebra.FreeModule.Norm
import Mathlib.RingTheory.ClassGroup
import Mat... |
variable {W} in
lemma degree_norm_ne_one [IsDomain R] (x : W.CoordinateRing) :
(Algebra.norm R[X] x).degree โ 1 := by
rcases exists_smul_basis_eq x with โจp, q, rflโฉ
rw [degree_norm_smul_basis]
rcases p.degree with (_ | _ | _ | _) <;> cases q.degree
any_goals rintro (_ | _)
| Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean | 462 | 469 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hรถlzl, Yuyang Zhao
-/
import Mathlib.Algebra.Group.Units.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra... | @[to_additive (attr := simp) add_pos_iff]
theorem one_lt_mul_iff : 1 < a * b โ 1 < a โจ 1 < b := by
simp only [one_lt_iff_ne_one, Ne, mul_eq_one, not_and_or]
end PartialOrder
end CommMonoid
namespace NeZero
theorem pos {M} [AddZeroClass M] [PartialOrder M] [CanonicallyOrderedAdd M]
(a : M) [NeZero a] : 0 < a :... | Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean | 273 | 285 |
/-
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.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
/-!
# Chain homotopies
We define chain... | (succ (n + 1) (mkInductiveAuxโ (n + 1))).2โฉ
section
/-- An auxiliary construction for `mkInductive`.
| Mathlib/Algebra/Homology/Homotopy.lean | 477 | 481 |
/-
Copyright (c) 2022 Yaรซl Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Regularity.Bound
import Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise
import Mathlib.Comb... | theorem card_eq_of_mem_parts_chunk (hs : s โ (chunk hP G ฮต hU).parts) :
#s = m โจ #s = m + 1 := by
unfold chunk at hs
split_ifs at hs <;> exact card_eq_of_mem_parts_equitabilise hs
theorem m_le_card_of_mem_chunk_parts (hs : s โ (chunk hP G ฮต hU).parts) : m โค #s :=
| Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean | 180 | 185 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Order.Filter.Tendsto
import Mathlib.Data.PFun
/-!
# `Tendsto` for relations and partial functions
This file generalizes `Filter` definitions from funct... | def pmap (f : ฮฑ โ. ฮฒ) (l : Filter ฮฑ) : Filter ฮฒ :=
Filter.rmap f.graph' l
@[simp]
theorem mem_pmap (f : ฮฑ โ. ฮฒ) (l : Filter ฮฑ) (s : Set ฮฒ) : s โ l.pmap f โ f.core s โ l :=
| Mathlib/Order/Filter/Partial.lean | 188 | 192 |
/-
Copyright (c) 2023 Yaรซl Dillies, Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies, Christopher Hoskin
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Or... |
@[simp] lemma supClosure_empty : supClosure (โ
: Set ฮฑ) = โ
:= by simp
@[simp] lemma supClosure_singleton : supClosure {a} = {a} := by simp
@[simp] lemma supClosure_univ : supClosure (Set.univ : Set ฮฑ) = Set.univ := by simp
@[simp] lemma upperBounds_supClosure (s : Set ฮฑ) : upperBounds (supClosure s) = upperBounds s ... | Mathlib/Order/SupClosed.lean | 307 | 316 |
/-
Copyright (c) 2020 Sรฉbastien Gouรซzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sรฉbastien Gouรซzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ContMDiff.Defs
/-!
## Basic properties of `C^n` functions between manifolds
In this file, we show that stan... | @[deprecated (since := "2024-11-20")] alias smoothOn_id := contMDiffOn_id
theorem contMDiffAt_id : ContMDiffAt I I n (id : M โ M) x :=
| Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 167 | 169 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Basic
import Mathlib.RingTheory.RootsOfUnity.Minpoly
/-!
# Roots of cyclotomic polynomials.
We gather results abou... | exact (IsPrimitive.irreducible_iff_irreducible_map_fraction_map (cyclotomic.isPrimitive n โค)).1
(cyclotomic.irreducible hpos)
/-- If `n โ m`, then `(cyclotomic n โ)` and `(cyclotomic m โ)` are coprime. -/
| Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean | 192 | 195 |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
/-!
# Preserving (co)kernels
Constructions to relate the ... | rw [โ cancel_mono (kernel.ฮน _)]
simp [PreservesKernel.iso]
| Mathlib/CategoryTheory/Limits/Preserves/Shapes/Kernels.lean | 139 | 141 |
/-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Kevin Klinge
-/
import Mathlib.Algebra.Group.Submonoid.Defs
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.GroupTheor... | Mathlib/RingTheory/OreLocalization/OreSet.lean | 152 | 159 | |
/-
Copyright (c) 2020 Nicolรฒ Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolรฒ Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.... | ยท intro x y ex ey; rw [map_add, ex, ey, zero_add]
theorem Derivation.liftKaehlerDifferential_apply (D : Derivation R S M) (x) :
D.liftKaehlerDifferential ((KaehlerDifferential.ideal R S).toCotangent x) =
D.tensorProductTo x := rfl
theorem Derivation.liftKaehlerDifferential_comp (D : Derivation R S M) :
... | Mathlib/RingTheory/Kaehler/Basic.lean | 267 | 274 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
/-!
# Permutations from a list
A list `l : List ฮฑ` ... | intro H
rw [nodup_iff_injective_get, Function.Injective] at h
specialize h H
| Mathlib/GroupTheory/Perm/List.lean | 196 | 198 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Winston Yin
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Topology.Algebra.Order.Floor
import Mathlib.Topology.MetricSpace.Contracting
/-!
# P... | โจv, tMin, tMax, tโ, xโ, C, โจR, hpl.hRโฉ, L, { hpl with htโ := tโ.property }โฉ
variable {v : E โ E} (tโ : โ) {xโ : E}
/-- A time-independent, continuously differentiable ODE satisfies the hypotheses of the
Picard-Lindelรถf theorem. -/
theorem exists_isPicardLindelof_const_of_contDiffAt (hv : ContDiffAt โ 1 v xโ) :
... | Mathlib/Analysis/ODE/PicardLindelof.lean | 358 | 366 |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
/-!
# Infimum separation
This file defines the extended infimum separation of a set. This is approximately dual to the
diame... | ยท exact Or.inl (iInf_le_of_le _ (iInfโ_le hz hyz))
ยท rw [edist_comm]
exact Or.inl (iInf_le_of_le _ (iInfโ_le hy hyz.symm))
ยท exact Or.inr (einfsep_le_edist_of_mem hy hz hyz)
theorem einfsep_triple (hxy : x โ y) (hyz : y โ z) (hxz : x โ z) :
einfsep ({x, y, z} : Set ฮฑ) = edist x y โ edist x z โ edist y z ... | Mathlib/Topology/MetricSpace/Infsep.lean | 187 | 194 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hรถlzl
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Group/Defs.lean | 944 | 945 | |
/-
Copyright (c) 2023 Frรฉdรฉric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frรฉdรฉric Dupuis
-/
import Mathlib.Analysis.Normed.Group.Uniform
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Completeness of normed groups
This file includes a completeness c... | lemma Metric.exists_subseq_summable_dist_of_cauchySeq (u : โ โ ฮฑ) (hu : CauchySeq u) :
โ f : โ โ โ, StrictMono f โง Summable fun i => dist (u (f (i+1))) (u (f i)) := by
obtain โจf, hfโ, hfโโฉ := Metric.exists_subseq_bounded_of_cauchySeq u hu
(fun n => (1 / (2 : โ))^n) (fun n => by positivity)
refine โจf, hfโ, ?... | Mathlib/Analysis/Normed/Group/Completeness.lean | 39 | 45 |
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Basic Translation Lemmas Between Functions Defined for Continued... | end GenContFract
| Mathlib/Algebra/ContinuedFractions/Translations.lean | 155 | 159 |
/-
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.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Tactic.AdaptationNote
/-!
# Inversion in an affine sp... | theorem inversion_inversion (c : P) {R : โ} (hR : R โ 0) (x : P) :
inversion c R (inversion c R x) = x := by
| Mathlib/Geometry/Euclidean/Inversion/Basic.lean | 107 | 108 |
/-
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
-/
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Measure theory in the product of groups
In this file we show propertie... | end LeftInvariant
section RightInvariant
@[to_additive measurePreserving_prod_add_right]
theorem measurePreserving_prod_mul_right [IsMulRightInvariant ฮฝ] :
MeasurePreserving (fun z : G ร G => (z.1, z.2 * z.1)) (ฮผ.prod ฮฝ) (ฮผ.prod ฮฝ) :=
MeasurePreserving.skew_product (g := fun x y => y * x) (MeasurePreserving.id ... | Mathlib/MeasureTheory/Group/Prod.lean | 339 | 348 |
/-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.DedekindDomain.Dvr
import Mathlib.RingTheory.DedekindDomain.Ideal
/-!
# Criteria under which a Dedekind domain is a PID
This file contains some ... | rw [Finset.mem_filter, Finset.mem_union, Finset.mem_singleton, Set.mem_setOf,
Multiset.mem_toFinset]
exact
and_iff_right_of_imp fun hP =>
or_iff_not_imp_left.mpr (IsLocalization.OverPrime.mem_normalizedFactors_of_isPrime S p hp0 hP)
| Mathlib/RingTheory/DedekindDomain/PID.lean | 248 | 262 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Topology.Order.MonotoneContinuity
import M... | /-- If `f x - x` is an integer number `m` for some point `x`, then `ฯ f = m`.
On the circle this means that a map with a fixed point has rotation number zero. -/
theorem translationNumber_of_eq_add_int {x : โ} {m : โค} (h : f x = x + m) : ฯ f = m :=
le_antisymm (translationNumber_le_of_le_add_int f <| le_of_eq h)
... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 725 | 733 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Nat.Prime.Factorial
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.Analysis.Normed.Ring.Lemmas
/-!
# Lemmas of Gauss and Eisenstein
... | calc
x * q / p โค p / 2 * q / p := by have := le_of_lt_succ (mem_Ico.mp hx).2; gcongr
_ โค _ := Nat.div_mul_div_le_div _ _ _
_ = _ := by
rw [โ card_sigma]
exact card_nbij' (fun a โฆ โจa.1, a.2โฉ) (fun a โฆ โจa.1, a.2โฉ)
(by simp +contextual only [mem_filter, mem... | Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean | 169 | 188 |
/-
Copyright (c) 2023 Joรซl Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joรซl Riou
-/
import Mathlib.Algebra.Homology.Additive
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.Algebra.Homology.ShortComplex.Preadditive
import Mathlib.Tactic.Linar... | section
variable [K.HasHomology i]
@[reassoc]
lemma comp_liftCycles {A' A : C} (k : A โถ K.X i) (j : ฮน) (hj : c.next i = j)
(hk : k โซ K.d i j = 0) (ฮฑ : A' โถ A) :
ฮฑ โซ K.liftCycles k j hj hk = K.liftCycles (ฮฑ โซ k) j hj (by rw [assoc, hk, comp_zero]) := by
simp only [โ cancel_mono (K.iCycles i), assoc, liftCycle... | Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean | 161 | 171 |
/-
Copyright (c) 2017 Johannes Hรถlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hรถlzl, Mario Carneiro, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Group.Submonoid.Basic
import M... | exact โจK, K_mem_l, K_compactโฉ
have l_le_coprod : l โค (๐ 0).coprod (๐ 0) := by
rw [l_def, โ coprod_cocompact]
| Mathlib/Topology/Algebra/Monoid.lean | 458 | 460 |
/-
Copyright (c) 2023 Yaรซl Dillies, Vladimir Ivanov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaรซl Dillies, Vladimir Ivanov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.... | rw [coe_sups, upperClosure_sups, UpperSet.mem_sup_iff]
variable [DecidableLE ฮฑ] [BoundedOrder ฮฑ]
lemma truncatedSup_infs (hs : a โ lowerClosure s) (ht : a โ lowerClosure t) :
truncatedSup (s โผ t) a = truncatedSup s a โ truncatedSup t a := by
| Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean | 255 | 260 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Ralf Stephan, Neil Strickland, Ruben Van de Velde
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Positive.Ring
import Mathlib.... | constructor <;> intro h
ยท rcases h with โจ_, rflโฉ
apply dvd_mul_right
ยท rcases h with โจa, hโฉ
obtain โจn, rflโฉ := Nat.exists_eq_succ_of_ne_zero (n := a) <| by
rintro rfl
simp only [mul_zero, ne_zero] at h
use โจn.succ, n.succ_posโฉ
| Mathlib/Data/PNat/Basic.lean | 327 | 334 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Products
/-!
# Pullbacks and pushouts in the category of topological spaces
-... | ext x
constructor
ยท rintro โจy, rflโฉ
| Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 130 | 132 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Order.Antidiag.Finsupp
import Mathlib.Data.Finsupp.Weight
import Mathlib.Tactic.Linarith
import Mathlib.LinearAlgebra.Pi
import Mat... |
theorem algebraMap_apply {r : R} :
algebraMap R (MvPowerSeries ฯ A) r = C ฯ A (algebraMap R A r) := by
change (MvPowerSeries.map ฯ (algebraMap R A)).comp (C ฯ R) r = _
| Mathlib/RingTheory/MvPowerSeries/Basic.lean | 744 | 747 |
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