Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#alig... | Mathlib/Algebra/Polynomial/Roots.lean | 248 | 253 | theorem roots_pow (p : R[X]) (n : ℕ) : (p ^ n).roots = n • p.roots := by |
induction' n with n ihn
· rw [pow_zero, roots_one, zero_smul, empty_eq_zero]
· rcases eq_or_ne p 0 with (rfl | hp)
· rw [zero_pow n.succ_ne_zero, roots_zero, smul_zero]
· rw [pow_succ, roots_mul (mul_ne_zero (pow_ne_zero _ hp) hp), ihn, add_smul, one_smul]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
impor... | Mathlib/Dynamics/PeriodicPts.lean | 539 | 543 | theorem nodup_periodicOrbit : (periodicOrbit f x).Nodup := by |
rw [periodicOrbit, Cycle.nodup_coe_iff, List.nodup_map_iff_inj_on (List.nodup_range _)]
intro m hm n hn hmn
rw [List.mem_range] at hm hn
rwa [iterate_eq_iterate_iff_of_lt_minimalPeriod hm hn] at hmn
|
/-
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.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Ma... | Mathlib/Computability/TuringMachine.lean | 1,324 | 1,325 | theorem stmts₁_self {q : Stmt₁} : q ∈ stmts₁ q := by |
cases q <;> simp only [stmts₁, Finset.mem_insert_self, Finset.mem_singleton_self]
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padic... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 121 | 135 | theorem norm_zero_iff (f : PadicSeq p) : f.norm = 0 ↔ f ≈ 0 := by |
constructor
· intro h
by_contra hf
unfold norm at h
split_ifs at h
· contradiction
apply hf
intro ε hε
exists stationaryPoint hf
intro j hj
have heq := stationaryPoint_spec hf le_rfl hj
simpa [h, heq]
· intro h
simp [norm, h]
|
/-
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.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTh... | Mathlib/RingTheory/Localization/Basic.lean | 720 | 722 | theorem ringEquivOfRingEquiv_eq {j : R ≃+* P} (H : M.map j.toMonoidHom = T) (x) :
ringEquivOfRingEquiv S Q j H ((algebraMap R S) x) = algebraMap P Q (j x) := by |
simp
|
/-
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.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_im... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 323 | 330 | theorem image_coe_ball (z : ℍ) (r : ℝ) :
((↑) : ℍ → ℂ) '' ball (α := ℍ) z r = ball ↑(z.center r) (z.im * Real.sinh r) := by |
ext w; constructor
· rintro ⟨w, hw, rfl⟩
exact dist_lt_iff_dist_coe_center_lt.1 hw
· intro hw
lift w to ℍ using im_pos_of_dist_center_le (ball_subset_closedBall hw)
exact mem_image_of_mem _ (dist_lt_iff_dist_coe_center_lt.2 hw)
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 765 | 771 | theorem setToL1S_neg {T : Set α → E →L[ℝ] F} (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0)
(h_add : FinMeasAdditive μ T) (f : α →₁ₛ[μ] E) : setToL1S T (-f) = -setToL1S T f := by |
simp_rw [setToL1S]
have : simpleFunc.toSimpleFunc (-f) =ᵐ[μ] ⇑(-simpleFunc.toSimpleFunc f) :=
neg_toSimpleFunc f
rw [SimpleFunc.setToSimpleFunc_congr T h_zero h_add (SimpleFunc.integrable _) this]
exact SimpleFunc.setToSimpleFunc_neg T h_add (SimpleFunc.integrable f)
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl
-/
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryPro... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 1,903 | 1,904 | theorem pushoutCoconeOfRightIso_ι_app_none : (pushoutCoconeOfRightIso f g).ι.app none = f := by |
simp
|
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedS... | Mathlib/Analysis/RCLike/Basic.lean | 484 | 485 | theorem normSq_conj (z : K) : normSq (conj z) = normSq z := by |
simp only [normSq_apply, neg_mul, mul_neg, neg_neg, rclike_simps]
|
/-
Copyright (c) 2021 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
#align_import data.real.pi.wallis from "leanprover-community/mathlib"@"980755c33b9168bc82f774f665eaa27878140fac"
/-... | Mathlib/Data/Real/Pi/Wallis.lean | 85 | 88 | theorem W_le (k : ℕ) : W k ≤ π / 2 := by |
rw [← div_le_one pi_div_two_pos, div_eq_inv_mul]
rw [W_eq_integral_sin_pow_div_integral_sin_pow, div_le_one (integral_sin_pow_pos _)]
apply integral_sin_pow_succ_le
|
/-
Copyright (c) 2023 Bulhwi Cha. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bulhwi Cha, Mario Carneiro
-/
import Batteries.Data.Char
import Batteries.Data.List.Lemmas
import Batteries.Data.String.Basic
import Batteries.Tactic.Lint.Misc
import Batteries.Tactic.SeqF... | .lake/packages/batteries/Batteries/Data/String/Lemmas.lean | 279 | 280 | theorem find_of_valid (p s) : find s p = ⟨utf8Len (s.1.takeWhile (!p ·))⟩ := by |
simpa using findAux_of_valid p [] s.1 []
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.calculus.... | Mathlib/Analysis/Calculus/TangentCone.lean | 367 | 379 | theorem uniqueDiffWithinAt_convex {s : Set G} (conv : Convex ℝ s) (hs : (interior s).Nonempty)
{x : G} (hx : x ∈ closure s) : UniqueDiffWithinAt ℝ s x := by |
rcases hs with ⟨y, hy⟩
suffices y - x ∈ interior (tangentConeAt ℝ s x) by
refine ⟨Dense.of_closure ?_, hx⟩
simp [(Submodule.span ℝ (tangentConeAt ℝ s x)).eq_top_of_nonempty_interior'
⟨y - x, interior_mono Submodule.subset_span this⟩]
rw [mem_interior_iff_mem_nhds]
replace hy : interior s ∈ 𝓝 y... |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Int.ModEq
import Mat... | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 113 | 114 | theorem cycleOf_apply_apply_self (f : Perm α) (x : α) : cycleOf f x (f x) = f (f x) := by |
convert cycleOf_apply_apply_pow_self f x 1 using 1
|
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.F... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 214 | 219 | theorem isCycle_of_prime_order {σ : Perm α} (h1 : (orderOf σ).Prime)
(h2 : σ.support.card < 2 * orderOf σ) : σ.IsCycle := by |
obtain ⟨n, hn⟩ := cycleType_prime_order h1
rw [← σ.sum_cycleType, hn, Multiset.sum_replicate, nsmul_eq_mul, Nat.cast_id,
mul_lt_mul_right (orderOf_pos σ), Nat.succ_lt_succ_iff, Nat.lt_succ_iff, Nat.le_zero] at h2
rw [← card_cycleType_eq_one, hn, card_replicate, h2]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Math... | Mathlib/Analysis/SpecificLimits/Basic.lean | 286 | 287 | theorem hasSum_geometric_two : HasSum (fun n : ℕ ↦ ((1 : ℝ) / 2) ^ n) 2 := by |
convert hasSum_geometric_of_lt_one _ _ <;> norm_num
|
/-
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 Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 1,426 | 1,432 | theorem multiplicative_of_isTotal (p : α → Prop) (hswap : ∀ {a b}, p a → p b → f a b * f b a = 1)
(hmul : ∀ {a b c}, r a b → r b c → p a → p b → p c → f a c = f a b * f b c) {a b c : α}
(pa : p a) (pb : p b) (pc : p c) : f a c = f a b * f b c := by |
apply multiplicative_of_symmetric_of_isTotal (fun a b => p a ∧ p b) r f fun _ _ => And.symm
· simp_rw [and_imp]; exact @hswap
· exact fun rab rbc pab _pbc pac => hmul rab rbc pab.1 pab.2 pac.2
exacts [⟨pa, pb⟩, ⟨pb, pc⟩, ⟨pa, pc⟩]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Basic
#align_import data.multiset.range from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
/-! # `Mu... | Mathlib/Data/Multiset/Range.lean | 73 | 75 | theorem range_add_eq_union (a b : ℕ) : range (a + b) = range a ∪ (range b).map (a + ·) := by |
rw [range_add, add_eq_union_iff_disjoint]
apply range_disjoint_map_add
|
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanpro... | Mathlib/Data/Sym/Sym2.lean | 275 | 276 | theorem map_map {g : β → γ} {f : α → β} (x : Sym2 α) : map g (map f x) = map (g ∘ f) x := by |
revert x; apply Sym2.ind; aesop
|
/-
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, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 368 | 374 | theorem span_smul_eq_of_isUnit (s : Set M) (r : R) (hr : IsUnit r) : span R (r • s) = span R s := by |
apply le_antisymm
· apply span_smul_le
· convert span_smul_le (r • s) ((hr.unit⁻¹ : _) : R)
rw [smul_smul]
erw [hr.unit.inv_val]
rw [one_smul]
|
/-
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.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
/-!
# Dropping or taki... | Mathlib/Data/List/DropRight.lean | 210 | 211 | theorem rtakeWhile_concat_neg (x : α) (h : ¬p x) : rtakeWhile p (l ++ [x]) = [] := by |
rw [rtakeWhile_concat, if_neg h]
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Order.EuclideanAbsoluteValue
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Polynomial.Field... | Mathlib/Algebra/Polynomial/Degree/CardPowDegree.lean | 79 | 83 | theorem cardPowDegree_apply [DecidableEq Fq] (p : Fq[X]) :
cardPowDegree p = if p = 0 then 0 else (Fintype.card Fq : ℤ) ^ natDegree p := by |
rw [cardPowDegree]
dsimp
convert rfl
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 1,372 | 1,379 | theorem indexOf_inj [DecidableEq α] {l : List α} {x y : α} (hx : x ∈ l) (hy : y ∈ l) :
indexOf x l = indexOf y l ↔ x = y :=
⟨fun h => by
have x_eq_y :
get l ⟨indexOf x l, indexOf_lt_length.2 hx⟩ =
get l ⟨indexOf y l, indexOf_lt_length.2 hy⟩ := by |
simp only [h]
simp only [indexOf_get] at x_eq_y; exact x_eq_y, fun h => by subst h; rfl⟩
|
/-
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.GroupTheory.Abelianization
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schreier from "leanpro... | Mathlib/GroupTheory/Schreier.lean | 188 | 200 | theorem card_commutator_le_of_finite_commutatorSet [Finite (commutatorSet G)] :
Nat.card (_root_.commutator G) ≤ cardCommutatorBound (Nat.card (commutatorSet G)) := by |
have h1 := index_center_le_pow (closureCommutatorRepresentatives G)
have h2 := card_commutator_dvd_index_center_pow (closureCommutatorRepresentatives G)
rw [card_commutatorSet_closureCommutatorRepresentatives] at h1 h2
rw [card_commutator_closureCommutatorRepresentatives] at h2
replace h1 :=
h1.trans
... |
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.Algebra.Algebra.Subalgebra.Directed
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
imp... | Mathlib/FieldTheory/Adjoin.lean | 686 | 689 | theorem sup_toSubalgebra_of_isAlgebraic_left [Algebra.IsAlgebraic K E1] :
(E1 ⊔ E2).toSubalgebra = E1.toSubalgebra ⊔ E2.toSubalgebra := by |
have := sup_toSubalgebra_of_isAlgebraic_right E2 E1
rwa [sup_comm (a := E1), sup_comm (a := E1.toSubalgebra)]
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.... | Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | 189 | 204 | theorem is_ideal_adic_pow {J : Ideal R} (h : IsAdic J) {n : ℕ} (hn : 0 < n) : IsAdic (J ^ n) := by |
rw [isAdic_iff] at h ⊢
constructor
· intro m
rw [← pow_mul]
apply h.left
· intro V hV
cases' h.right V hV with m hm
use m
refine Set.Subset.trans ?_ hm
cases n
· exfalso
exact Nat.not_succ_le_zero 0 hn
rw [← pow_mul, Nat.succ_mul]
apply Ideal.pow_le_pow_right
apply... |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib... | Mathlib/Data/Fin/Tuple/Basic.lean | 1,044 | 1,045 | theorem find_eq_none_iff {n : ℕ} {p : Fin n → Prop} [DecidablePred p] :
find p = none ↔ ∀ i, ¬p i := by | rw [← not_exists, ← isSome_find_iff]; cases find p <;> simp
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Group.Prod
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Logic... | Mathlib/GroupTheory/Perm/Basic.lean | 566 | 567 | theorem mul_swap_mul_self (i j : α) (σ : Perm α) : σ * Equiv.swap i j * Equiv.swap i j = σ := by |
rw [mul_assoc, swap_mul_self, mul_one]
|
/-
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.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import ... | Mathlib/Data/List/Chain.lean | 449 | 472 | theorem Chain'.cons_of_le [LinearOrder α] {a : α} {as m : List α}
(ha : List.Chain' (· > ·) (a :: as)) (hm : List.Chain' (· > ·) m) (hmas : m ≤ as) :
List.Chain' (· > ·) (a :: m) := by |
cases m with
| nil => simp only [List.chain'_singleton]
| cons b bs =>
apply hm.cons
cases as with
| nil =>
simp only [le_iff_lt_or_eq, or_false] at hmas
exact (List.Lex.not_nil_right (·<·) _ hmas).elim
| cons a' as =>
rw [List.chain'_cons] at ha
refine gt_of_gt_of_ge ha.1... |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Scott Morrison
-/
import Mathlib.Data.Opposite
import Mathlib.Tactic.Cases
#align_import combinatorics.quiver.basic from "leanprover-community/mathlib"@"56adee5b5eef9e734d8227... | Mathlib/Combinatorics/Quiver/Basic.lean | 138 | 140 | theorem congr_map {U V : Type*} [Quiver U] [Quiver V] (F : U ⥤q V) {X Y : U} {f g : X ⟶ Y}
(h : f = g) : F.map f = F.map g := by |
rw [h]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.pol... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 187 | 198 | theorem U_neg_sub_one (n : ℤ) : U R (-n - 1) = -U R (n - 1) := by |
induction n using Polynomial.Chebyshev.induct with
| zero => simp
| one => simp
| add_two n ih1 ih2 =>
have h₁ := U_add_one R n
have h₂ := U_sub_two R (-n - 1)
linear_combination (norm := ring_nf) 2 * (X:R[X]) * ih1 - ih2 + h₁ + h₂
| neg_add_one n ih1 ih2 =>
have h₁ := U_eq R n
have h₂ :=... |
/-
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.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mat... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 1,054 | 1,059 | theorem isLocalRingHom_iff_comap_closedPoint {S : Type v} [CommSemiring S] [LocalRing S]
(f : R →+* S) : IsLocalRingHom f ↔ PrimeSpectrum.comap f (closedPoint S) = closedPoint R := by |
-- Porting note: inline `this` does **not** work
have := (local_hom_TFAE f).out 0 4
rw [this, PrimeSpectrum.ext_iff]
rfl
|
/-
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.InnerProductSpace.Projection
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
#align_... | Mathlib/Geometry/Euclidean/Basic.lean | 197 | 205 | theorem eq_of_dist_eq_of_dist_eq_of_finrank_eq_two [FiniteDimensional ℝ V] (hd : finrank ℝ V = 2)
{c₁ c₂ p₁ p₂ p : P} {r₁ r₂ : ℝ} (hc : c₁ ≠ c₂) (hp : p₁ ≠ p₂) (hp₁c₁ : dist p₁ c₁ = r₁)
(hp₂c₁ : dist p₂ c₁ = r₁) (hpc₁ : dist p c₁ = r₁) (hp₁c₂ : dist p₁ c₂ = r₂)
(hp₂c₂ : dist p₂ c₂ = r₂) (hpc₂ : dist p c₂ = ... |
rw [direction_top, finrank_top]
exact hd
eq_of_dist_eq_of_dist_eq_of_mem_of_finrank_eq_two hd' (mem_top ℝ V _) (mem_top ℝ V _)
(mem_top ℝ V _) (mem_top ℝ V _) (mem_top ℝ V _) hc hp hp₁c₁ hp₂c₁ hpc₁ hp₁c₂ hp₂c₂ hpc₂
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSe... | Mathlib/RingTheory/PowerSeries/WellKnown.lean | 218 | 220 | theorem map_cos : map f (cos A) = cos A' := by |
ext
simp [cos, apply_ite f]
|
/-
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.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
/-!
... | Mathlib/Data/ENNReal/Real.lean | 375 | 377 | theorem ofReal_inv_of_pos {x : ℝ} (hx : 0 < x) : ENNReal.ofReal x⁻¹ = (ENNReal.ofReal x)⁻¹ := by |
rw [ENNReal.ofReal, ENNReal.ofReal, ← @coe_inv (Real.toNNReal x) (by simp [hx]), coe_inj,
← Real.toNNReal_inv]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Geometry.RingedSpace.PresheafedSpace
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.Topology.Sheaves.Stalks
#align_import algebraic_geometr... | Mathlib/Geometry/RingedSpace/Stalks.lean | 108 | 121 | theorem restrictStalkIso_inv_eq_ofRestrict {U : TopCat} (X : PresheafedSpace.{_, _, v} C)
{f : U ⟶ (X : TopCat.{v})} (h : OpenEmbedding f) (x : U) :
(X.restrictStalkIso h x).inv = stalkMap (X.ofRestrict h) x := by |
-- We can't use `ext` here due to https://github.com/leanprover/std4/pull/159
refine colimit.hom_ext fun V => ?_
induction V with | h V => ?_
let i : (h.isOpenMap.functorNhds x).obj ((OpenNhds.map f x).obj V) ⟶ V :=
homOfLE (Set.image_preimage_subset f _)
erw [Iso.comp_inv_eq, colimit.ι_map_assoc, colimi... |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Subsemigroup.Basic
#align_import group_theory.subsemigroup.membership from "leanprover-community/mathlib"@"6cb77a8eaff0ddd100e87b1591c6d3a... | Mathlib/Algebra/Group/Subsemigroup/Membership.lean | 109 | 112 | theorem mem_sSup_of_mem {S : Set (Subsemigroup M)} {s : Subsemigroup M} (hs : s ∈ S) :
∀ {x : M}, x ∈ s → x ∈ sSup S := by |
have : s ≤ sSup S := le_sSup hs
tauto
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Rat.Basic
import Batteries.Tactic.SeqFocus
/-! # Additional lemmas about the Rational Numbers -/
namespace Rat
theorem ext : {p q : Rat} → p.... | .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean | 58 | 60 | theorem normalize_mul_right {a : Nat} (d0 : d ≠ 0) (a0 : a ≠ 0) :
normalize (n * a) (d * a) (Nat.mul_ne_zero d0 a0) = normalize n d d0 := by |
rw [← normalize_mul_left (d0 := d0) a0]; congr 1 <;> [apply Int.mul_comm; apply Nat.mul_comm]
|
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.compression.uv from "leanprover-community/mathlib"@"6f8ab7de1c4b78a68a... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 185 | 190 | theorem compress_mem_compression (ha : a ∈ s) : compress u v a ∈ 𝓒 u v s := by |
rw [mem_compression]
by_cases h : compress u v a ∈ s
· rw [compress_idem]
exact Or.inl ⟨h, h⟩
· exact Or.inr ⟨h, a, ha, rfl⟩
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.locally_convex.continuous_of_bounded from "leanprover-commun... | Mathlib/Analysis/LocallyConvex/ContinuousOfBounded.lean | 96 | 166 | theorem LinearMap.continuousAt_zero_of_locally_bounded (f : E →ₛₗ[σ] F)
(hf : ∀ s, IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (f '' s)) : ContinuousAt f 0 := by |
-- Assume that f is not continuous at 0
by_contra h
-- We use a decreasing balanced basis for 0 : E and a balanced basis for 0 : F
-- and reformulate non-continuity in terms of these bases
rcases (nhds_basis_balanced 𝕜 E).exists_antitone_subbasis with ⟨b, bE1, bE⟩
simp only [_root_.id] at bE
have bE' : ... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathli... | Mathlib/Data/Nat/Digits.lean | 773 | 780 | theorem eleven_dvd_of_palindrome (p : (digits 10 n).Palindrome) (h : Even (digits 10 n).length) :
11 ∣ n := by |
let dig := (digits 10 n).map fun n : ℕ => (n : ℤ)
replace h : Even dig.length := by rwa [List.length_map]
refine eleven_dvd_iff.2 ⟨0, (?_ : dig.alternatingSum = 0)⟩
have := dig.alternatingSum_reverse
rw [(p.map _).reverse_eq, _root_.pow_succ', h.neg_one_pow, mul_one, neg_one_zsmul] at this
exact eq_zero_of... |
/-
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.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlg... | Mathlib/Algebra/Lie/Nilpotent.lean | 155 | 163 | theorem trivial_iff_lower_central_eq_bot : IsTrivial L M ↔ lowerCentralSeries R L M 1 = ⊥ := by |
constructor <;> intro h
· erw [eq_bot_iff, LieSubmodule.lieSpan_le]; rintro m ⟨x, n, hn⟩; rw [← hn, h.trivial]; simp
· rw [LieSubmodule.eq_bot_iff] at h; apply IsTrivial.mk; intro x m; apply h
apply LieSubmodule.subset_lieSpan
-- Porting note: was `use x, m; rfl`
simp only [LieSubmodule.top_coe, Subt... |
/-
Copyright (c) 2021 Gabriel Moise. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Moise, Yaël Dillies, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 85 | 89 | theorem incMatrix_apply_mul_incMatrix_apply_of_not_adj (hab : a ≠ b) (h : ¬G.Adj a b) :
G.incMatrix R a e * G.incMatrix R b e = 0 := by |
rw [incMatrix_apply_mul_incMatrix_apply, Set.indicator_of_not_mem]
rw [G.incidenceSet_inter_incidenceSet_of_not_adj h hab]
exact Set.not_mem_empty e
|
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.Sigma
#align_import data.list.alist from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a71d317643cf7fb5fbb"
/-!
# Associ... | Mathlib/Data/List/AList.lean | 388 | 399 | theorem insertRec_insert {C : AList β → Sort*} (H0 : C ∅)
(IH : ∀ (a : α) (b : β a) (l : AList β), a ∉ l → C l → C (l.insert a b)) {c : Sigma β}
{l : AList β} (h : c.1 ∉ l) :
@insertRec α β _ C H0 IH (l.insert c.1 c.2) = IH c.1 c.2 l h (@insertRec α β _ C H0 IH l) := by |
cases' l with l hl
suffices HEq (@insertRec α β _ C H0 IH ⟨c :: l, nodupKeys_cons.2 ⟨h, hl⟩⟩)
(IH c.1 c.2 ⟨l, hl⟩ h (@insertRec α β _ C H0 IH ⟨l, hl⟩)) by
cases c
apply eq_of_heq
convert this <;> rw [insert_of_neg h]
rw [insertRec]
apply cast_heq
|
/-
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.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Ring.Action.Subobjects
import Mathlib.Algebra.Ring.Equiv... | Mathlib/Algebra/Ring/Subsemiring/Basic.lean | 848 | 856 | theorem closure_addSubmonoid_closure {s : Set R} :
closure ↑(AddSubmonoid.closure s) = closure s := by |
ext x
refine ⟨fun hx => ?_, fun hx => closure_mono AddSubmonoid.subset_closure hx⟩
rintro - ⟨H, rfl⟩
rintro - ⟨J, rfl⟩
refine (AddSubmonoid.mem_closure.mp (mem_closure_iff.mp hx)) H.toAddSubmonoid fun y hy => ?_
refine (Submonoid.mem_closure.mp hy) H.toSubmonoid fun z hz => ?_
exact (AddSubmonoid.mem_clo... |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import ring_theory.graded_algebra.homogeneous_localizati... | Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean | 543 | 546 | theorem den_smul_val (f : HomogeneousLocalization 𝒜 x) :
f.den • f.val = algebraMap _ _ f.num := by |
rw [eq_num_div_den, Localization.mk_eq_mk', IsLocalization.smul_mk']
exact IsLocalization.mk'_mul_cancel_left _ ⟨_, _⟩
|
/-
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.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0... | Mathlib/Data/Int/Log.lean | 223 | 224 | theorem log_inv (b : ℕ) (r : R) : log b r⁻¹ = -clog b r := by |
rw [← inv_inv r, clog_inv, neg_neg, inv_inv]
|
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.Order.Filter.AtTopBot
#align_import order.filter.indicator_function from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b8... | Mathlib/Order/Filter/IndicatorFunction.lean | 89 | 94 | theorem mulIndicator_biUnion_finset_eventuallyEq {ι} [One β] (s : ι → Set α) (f : α → β) (a : α) :
(fun n : Finset ι => mulIndicator (⋃ i ∈ n, s i) f a) =ᶠ[atTop]
fun _ ↦ mulIndicator (iUnion s) f a := by |
rw [iUnion_eq_iUnion_finset s]
apply Monotone.mulIndicator_eventuallyEq_iUnion
exact fun _ _ ↦ biUnion_subset_biUnion_left
|
/-
Copyright (c) 2023 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.FieldTheory.Fixed
import Mathlib.RepresentationTheory.GroupCohomology.LowDegree
import Mathlib.LinearAlgebra.LinearIndependent
/-!
# Hilbert's T... | Mathlib/RepresentationTheory/GroupCohomology/Hilbert90.lean | 80 | 96 | theorem isMulOneCoboundary_of_isMulOneCocycle_of_aut_to_units
(f : (L ≃ₐ[K] L) → Lˣ) (hf : IsMulOneCocycle f) :
IsMulOneCoboundary f := by |
/- Let `z : L` be such that `∑ f(h) * h(z) ≠ 0`, for `h ∈ Aut_K(L)` -/
obtain ⟨z, hz⟩ : ∃ z, aux f z ≠ 0 :=
not_forall.1 (fun H => aux_ne_zero f <| funext <| fun x => H x)
have : aux f z = ∑ h, f h * h z := by simp [aux, Finsupp.total, Finsupp.sum_fintype]
/- Let `β = (∑ f(h) * h(z))⁻¹.` -/
use (Units.mk0 (a... |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalcu... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 1,220 | 1,222 | theorem integrableOn_Ioi_comp_mul_right_iff (f : ℝ → E) (c : ℝ) {a : ℝ} (ha : 0 < a) :
IntegrableOn (fun x => f (x * a)) (Ioi c) ↔ IntegrableOn f (Ioi <| c * a) := by |
simpa only [mul_comm, mul_zero] using integrableOn_Ioi_comp_mul_left_iff f c ha
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
#align_import data.int.bitwise from "leanprover-community/mathlib"@"0743cc... | Mathlib/Data/Int/Bitwise.lean | 228 | 231 | theorem bit_val (b n) : bit b n = 2 * n + cond b 1 0 := by |
cases b
· apply (bit0_val n).trans (add_zero _).symm
· apply bit1_val
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Heather Macbeth
-/
import Mathlib.Topology.FiberBundle.Trivialization
import Mathlib.Topology.Order.LeftRightNhds
#align_import topology.fiber_... | Mathlib/Topology/FiberBundle/Basic.lean | 709 | 711 | theorem mem_localTrivAt_baseSet (b : B) : b ∈ (Z.localTrivAt b).baseSet := by |
rw [localTrivAt, ← baseSet_at]
exact Z.mem_baseSet_at b
|
/-
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
-/
import Mathlib.Topology.Constructions
import Mathlib.Topology.ContinuousOn
#align_import topology.bases from "leanprover-community/mathlib"@"bcfa7268... | Mathlib/Topology/Bases.lean | 417 | 418 | theorem separableSpace_iff_countable [DiscreteTopology α] : SeparableSpace α ↔ Countable α := by |
simp [separableSpace_iff, countable_univ_iff]
|
/-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Init.Data.List.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.Con... | Mathlib/Data/Nat/Bits.lean | 592 | 595 | theorem bits_append_bit (n : ℕ) (b : Bool) (hn : n = 0 → b = true) :
(bit b n).bits = b :: n.bits := by |
rw [Nat.bits, binaryRec_eq']
simpa
|
/-
Copyright (c) 2021 Alex J. Best. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Yaël Dillies
-/
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#a... | Mathlib/Algebra/Order/Pointwise.lean | 183 | 194 | theorem smul_Ioo : r • Ioo a b = Ioo (r • a) (r • b) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Ioo]
constructor
· rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩
constructor
· exact (mul_lt_mul_left hr).mpr a_h_left_left
· exact (mul_lt_mul_left hr).mpr a_h_left_right
· rintro ⟨a_left, a_right⟩
use x / r
refine ⟨⟨(lt_div_iff' hr).mpr... |
/-
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.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPower... | Mathlib/RingTheory/PowerSeries/Basic.lean | 545 | 547 | theorem map_C (r : R) : map f (C _ r) = C _ (f r) := by |
ext
simp [coeff_C, apply_ite f]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Jireh Loreaux
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Data.Fintype.Order
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.Analysis.Norm... | Mathlib/Analysis/NormedSpace/PiLp.lean | 247 | 249 | theorem dist_eq_iSup (f g : PiLp ∞ α) : dist f g = ⨆ i, dist (f i) (g i) := by |
dsimp [dist]
exact if_neg ENNReal.top_ne_zero
|
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
/-!
# Reflections, inversions, and inversion sequences
Throughout this file, `B` is a type and... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 88 | 93 | theorem length_mul_left_ne (w : W) : ℓ (w * t) ≠ ℓ w := by |
suffices cs.lengthParity (w * t) ≠ cs.lengthParity w by
contrapose! this
simp only [lengthParity_eq_ofAdd_length, this]
rcases ht with ⟨w, i, rfl⟩
simp [lengthParity_simple]
|
/-
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
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib... | Mathlib/Deprecated/Submonoid.lean | 370 | 380 | theorem exists_list_of_mem_closure {s : Set M} {a : M} (h : a ∈ Closure s) :
∃ l : List M, (∀ x ∈ l, x ∈ s) ∧ l.prod = a := by |
induction h with
| @basic a ha => exists [a]; simp [ha]
| one => exists []; simp
| mul _ _ ha hb =>
rcases ha with ⟨la, ha, eqa⟩
rcases hb with ⟨lb, hb, eqb⟩
exists la ++ lb
simp only [List.mem_append, or_imp, List.prod_append, eqa.symm, eqb.symm, and_true]
exact fun a => ⟨ha a, hb a⟩
|
/-
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.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
/-!
# Compl... | Mathlib/GroupTheory/Complement.lean | 541 | 544 | theorem toEquiv_apply {f : G ⧸ H → G} (hf : ∀ q, (f q : G ⧸ H) = q) (q : G ⧸ H) :
(toEquiv (range_mem_leftTransversals hf) q : G) = f q := by |
refine (Subtype.ext_iff.mp ?_).trans (Subtype.coe_mk (f q) ⟨q, rfl⟩)
exact (toEquiv (range_mem_leftTransversals hf)).apply_eq_iff_eq_symm_apply.mpr (hf q).symm
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "lean... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 160 | 166 | theorem χ₈_int_eq_if_mod_eight (n : ℤ) :
χ₈ n = if n % 2 = 0 then 0 else if n % 8 = 1 ∨ n % 8 = 7 then 1 else -1 := by |
have help :
∀ m : ℤ, 0 ≤ m → m < 8 → χ₈ m = if m % 2 = 0 then 0 else if m = 1 ∨ m = 7 then 1 else -1 := by
decide
rw [← Int.emod_emod_of_dvd n (by decide : (2 : ℤ) ∣ 8), ← ZMod.intCast_mod n 8]
exact help (n % 8) (Int.emod_nonneg n (by norm_num)) (Int.emod_lt n (by norm_num))
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison, Adam Topaz
-/
import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCat... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 501 | 504 | theorem δ_comp_δ_self {n} {i : Fin (n + 2)} :
X.δ i ≫ X.δ (Fin.castSucc i) = X.δ i ≫ X.δ i.succ := by |
dsimp [δ]
simp only [← X.map_comp, SimplexCategory.δ_comp_δ_self]
|
/-
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.ExteriorAlgebra.Basic
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import linear_algebra.exterior_algebra.grading from "leanprover-com... | Mathlib/LinearAlgebra/ExteriorAlgebra/Grading.lean | 64 | 80 | theorem GradedAlgebra.liftι_eq (i : ℕ) (x : ⋀[R]^i M) :
GradedAlgebra.liftι R M x = DirectSum.of (fun i => ⋀[R]^i M) i x := by |
cases' x with x hx
dsimp only [Subtype.coe_mk, DirectSum.lof_eq_of]
-- Porting note: original statement was
-- refine Submodule.pow_induction_on_left' _ (fun r => ?_) (fun x y i hx hy ihx ihy => ?_)
-- (fun m hm i x hx ih => ?_) hx
-- but it created invalid goals
induction hx using Submodule.pow_indu... |
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer, Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.Geometry.RingedSpace.SheafedSpace
import Mathlib.Topology.Sheaves.Stalks
import Ma... | Mathlib/Geometry/RingedSpace/Basic.lean | 58 | 79 | theorem isUnit_res_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U)) (x : U)
(h : IsUnit (X.presheaf.germ x f)) :
∃ (V : Opens X) (i : V ⟶ U) (_ : x.1 ∈ V), IsUnit (X.presheaf.map i.op f) := by |
obtain ⟨g', heq⟩ := h.exists_right_inv
obtain ⟨V, hxV, g, rfl⟩ := X.presheaf.germ_exist x.1 g'
let W := U ⊓ V
have hxW : x.1 ∈ W := ⟨x.2, hxV⟩
-- Porting note: `erw` can't write into `HEq`, so this is replaced with another `HEq` in the
-- desired form
replace heq : (X.presheaf.germ ⟨x.val, hxW⟩) ((X.pres... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"7... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 134 | 138 | theorem natDegree_mul_C_eq_of_mul_ne_zero (h : p.leadingCoeff * a ≠ 0) :
(p * C a).natDegree = p.natDegree := by |
refine eq_natDegree_of_le_mem_support (natDegree_mul_C_le p a) ?_
refine mem_support_iff.mpr ?_
rwa [coeff_mul_C]
|
/-
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.Convex.StrictConvexBetween
import Mathlib.Geometry.Euclidean.Basic
#align_import geometry.euclidean.sphere.basic from "leanprover-community/mathl... | Mathlib/Geometry/Euclidean/Sphere/Basic.lean | 188 | 190 | theorem cospherical_singleton (p : P) : Cospherical ({p} : Set P) := by |
use p
simp
|
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.Algebra.Module.Submodule.Ker
/-!
# Iterate maps and comaps of submodules
Some preliminary work for establishing the strong rank condition for noetherian rings.
Giv... | Mathlib/Algebra/Module/Submodule/IterateMapComap.lean | 65 | 79 | theorem iterateMapComap_eq_succ (K : Submodule R N)
(m : ℕ) (heq : f.iterateMapComap i m K = f.iterateMapComap i (m + 1) K)
(hf : Surjective f) (hi : Injective i) (n : ℕ) :
f.iterateMapComap i n K = f.iterateMapComap i (n + 1) K := by |
induction n with
| zero =>
contrapose! heq
induction m with
| zero => exact heq
| succ m ih =>
rw [iterateMapComap, iterateMapComap, iterate_succ', iterate_succ']
exact fun H ↦ ih (map_injective_of_injective hi (comap_injective_of_surjective hf H))
| succ n ih =>
rw [iterateMapCom... |
/-
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, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,723 | 1,726 | theorem sup_typein_limit {o : Ordinal} (ho : ∀ a, a < o → succ a < o) :
sup.{u, u} (typein ((· < ·) : o.out.α → o.out.α → Prop)) = o := by |
-- Porting note: `rwa` → `rw` & `assumption`
rw [(sup_eq_lsub_iff_succ.{u, u} (typein (· < ·))).2] <;> rw [lsub_typein o]; assumption
|
/-
Copyright (c) 2023 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Sébastien Gouëzel, Jireh Loreaux
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.NormedSpace.WithLp
/-!
# `L^p` distance on products of two metric space... | Mathlib/Analysis/NormedSpace/ProdLp.lean | 616 | 619 | theorem prod_nnnorm_eq_add (hp : p ≠ ∞) (f : WithLp p (α × β)) :
‖f‖₊ = (‖f.fst‖₊ ^ p.toReal + ‖f.snd‖₊ ^ p.toReal) ^ (1 / p.toReal) := by |
ext
simp [prod_norm_eq_add (p.toReal_pos_iff_ne_top.mpr hp)]
|
/-
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.Algebra.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathli... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 996 | 1,011 | theorem norm_eq_zero {d : ℤ} (h_nonsquare : ∀ n : ℤ, d ≠ n * n) (a : ℤ√d) : norm a = 0 ↔ a = 0 := by |
refine ⟨fun ha => (Zsqrtd.ext_iff _ _).mpr ?_, fun h => by rw [h, norm_zero]⟩
dsimp only [norm] at ha
rw [sub_eq_zero] at ha
by_cases h : 0 ≤ d
· obtain ⟨d', rfl⟩ := Int.eq_ofNat_of_zero_le h
haveI : Nonsquare d' := ⟨fun n h => h_nonsquare n <| mod_cast h⟩
exact divides_sq_eq_zero_z ha
· push_neg a... |
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Measurable
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calc... | Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 1,588 | 1,595 | theorem integral_comp_mul_deriv''' {a b : ℝ} {f f' : ℝ → ℝ} {g : ℝ → ℝ}
(hf : ContinuousOn f [[a, b]])
(hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x)
(hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))) (hg1 : IntegrableOn g (f '' [[a, b]]))
(hg2 : IntegrableOn (fun x =... |
have hg2' : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]] := by simpa [mul_comm] using hg2
simpa [mul_comm] using integral_comp_smul_deriv''' hf hff' hg_cont hg1 hg2'
|
/-
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.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlg... | Mathlib/Algebra/Lie/Nilpotent.lean | 255 | 261 | theorem exists_forall_pow_toEnd_eq_zero [hM : IsNilpotent R L M] :
∃ k : ℕ, ∀ x : L, toEnd R L M x ^ k = 0 := by |
obtain ⟨k, hM⟩ := hM
use k
intro x; ext m
rw [LinearMap.pow_apply, LinearMap.zero_apply, ← @LieSubmodule.mem_bot R L M, ← hM]
exact iterate_toEnd_mem_lowerCentralSeries R L M x m k
|
/-
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.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat... | Mathlib/Data/Num/Lemmas.lean | 1,532 | 1,532 | theorem of_to_int (n : ZNum) : ((n : ℤ) : ZNum) = n := by | rw [← ofInt'_eq, of_to_int']
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Module.LinearMap.Basic
#align_import algebra.algebra.basic from "leanprover-community/... | Mathlib/Algebra/Algebra/Defs.lean | 300 | 309 | theorem algebra_ext {R : Type*} [CommSemiring R] {A : Type*} [Semiring A] (P Q : Algebra R A)
(h : ∀ r : R, (haveI := P; algebraMap R A r) = haveI := Q; algebraMap R A r) :
P = Q := by |
replace h : P.toRingHom = Q.toRingHom := DFunLike.ext _ _ h
have h' : (haveI := P; (· • ·) : R → A → A) = (haveI := Q; (· • ·) : R → A → A) := by
funext r a
rw [P.smul_def', Q.smul_def', h]
rcases P with @⟨⟨P⟩⟩
rcases Q with @⟨⟨Q⟩⟩
congr
|
/-
Copyright (c) 2019 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Topology.UniformSpace.Equiv
#align_import topology.uniform_space.abstract_completion from "leanp... | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | 158 | 161 | theorem extend_unique (hf : UniformContinuous f) {g : hatα → β} (hg : UniformContinuous g)
(h : ∀ a : α, f a = g (ι a)) : pkg.extend f = g := by |
apply pkg.funext pkg.continuous_extend hg.continuous
simpa only [pkg.extend_coe hf] using h
|
/-
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.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
/-!
# Dropping or taki... | Mathlib/Data/List/DropRight.lean | 166 | 174 | theorem rdropWhile_eq_self_iff : rdropWhile p l = l ↔ ∀ hl : l ≠ [], ¬p (l.getLast hl) := by |
simp only [rdropWhile, reverse_eq_iff, dropWhile_eq_self_iff, getLast_eq_get]
refine ⟨fun h hl => ?_, fun h hl => ?_⟩
· rw [← length_pos, ← length_reverse] at hl
have := h hl
rwa [get_reverse'] at this
· rw [length_reverse, length_pos] at hl
have := h hl
rwa [get_reverse']
|
/-
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.Topology.Order.Basic
import Mathlib.Data.Set.Pointwise.Basic
/-!
# Neighborhoods to the left and to the right on an ... | Mathlib/Topology/Order/LeftRightNhds.lean | 288 | 293 | theorem mem_nhdsWithin_Iic_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α}
{s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s :=
calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl
_ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s :=
mem_nhdsWithin_Ici_i... | simp only [dual_Icc]; rfl
|
/-
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, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Map
#align_import linear_algebra.basic fr... | Mathlib/Algebra/Module/Submodule/Ker.lean | 293 | 294 | theorem ker_comp_of_ker_eq_bot (f : M →ₛₗ[τ₁₂] M₂) {g : M₂ →ₛₗ[τ₂₃] M₃} (hg : ker g = ⊥) :
ker (g.comp f : M →ₛₗ[τ₁₃] M₃) = ker f := by | rw [ker_comp, hg, Submodule.comap_bot]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.... | Mathlib/LinearAlgebra/Determinant.lean | 256 | 267 | theorem det_smul {𝕜 : Type*} [Field 𝕜] {M : Type*} [AddCommGroup M] [Module 𝕜 M] (c : 𝕜)
(f : M →ₗ[𝕜] M) :
LinearMap.det (c • f) = c ^ FiniteDimensional.finrank 𝕜 M * LinearMap.det f := by |
by_cases H : ∃ s : Finset M, Nonempty (Basis s 𝕜 M)
· have : FiniteDimensional 𝕜 M := by
rcases H with ⟨s, ⟨hs⟩⟩
exact FiniteDimensional.of_fintype_basis hs
simp only [← det_toMatrix (FiniteDimensional.finBasis 𝕜 M), LinearEquiv.map_smul,
Fintype.card_fin, Matrix.det_smul]
· classical
... |
/-
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
-/
import Mathlib.Topology.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd575... | Mathlib/Topology/Order.lean | 347 | 349 | theorem discreteTopology_iff_singleton_mem_nhds [TopologicalSpace α] :
DiscreteTopology α ↔ ∀ x : α, {x} ∈ 𝓝 x := by |
simp only [← singletons_open_iff_discrete, isOpen_iff_mem_nhds, mem_singleton_iff, forall_eq]
|
/-
Copyright (c) 2022 Yuyang Zhao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuyang Zhao
-/
import Mathlib.Algebra.Order.Floor
import Mathlib.Data.Nat.Prime
/-!
# Existence of a sufficiently large prime for which `a * c ^ p / (p - 1)! < 1`
This is a technical r... | Mathlib/Algebra/Order/Floor/Prime.lean | 36 | 40 | theorem exists_prime_mul_pow_div_factorial_lt_one [LinearOrderedField K] [FloorRing K]
(n : ℕ) (a c : K) :
∃ p > n, p.Prime ∧ a * c ^ p / (p - 1)! < 1 := by |
simp_rw [div_lt_one (α := K) (Nat.cast_pos.mpr (Nat.factorial_pos _))]
exact exists_prime_mul_pow_lt_factorial ..
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanpro... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 501 | 502 | theorem totalVariation_zero : (0 : SignedMeasure α).totalVariation = 0 := by |
simp [totalVariation, toJordanDecomposition_zero]
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Top... | Mathlib/Analysis/Normed/Group/Basic.lean | 883 | 885 | theorem lipschitzOnWith_iff_norm_div_le {f : E → F} {C : ℝ≥0} :
LipschitzOnWith C f s ↔ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ‖f x / f y‖ ≤ C * ‖x / y‖ := by |
simp only [lipschitzOnWith_iff_dist_le_mul, dist_eq_norm_div]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
#align_import data.pnat.prime from "leanprover-community/mathlib"@"09597669f0242... | Mathlib/Data/PNat/Prime.lean | 296 | 299 | theorem Coprime.factor_eq_gcd_right_right {a b m n : ℕ+} (cop : m.Coprime n) (am : a ∣ m)
(bn : b ∣ n) : a = m.gcd (b * a) := by |
rw [gcd_comm]
apply Coprime.factor_eq_gcd_right cop am bn
|
/-
Copyright (c) 2020 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Basic
/-!
# Properties of `List.reduceOption`
In this file we prove basic lemmas about `List.reduceOption`.
-/
namespace List
variable ... | Mathlib/Data/List/ReduceOption.lean | 59 | 61 | theorem reduceOption_length_le (l : List (Option α)) : l.reduceOption.length ≤ l.length := by |
rw [length_eq_reduceOption_length_add_filter_none]
apply Nat.le_add_right
|
/-
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.LinearAlgebra.Ray
import Mathlib.LinearAlgebra.Determinant
#align_import linear_algebra.orientation from "leanprover-community/mathlib"@"0c1d80f5a86b36c1d... | Mathlib/LinearAlgebra/Orientation.lean | 346 | 348 | theorem abs_det_adjustToOrientation [Nonempty ι] (e : Basis ι R M)
(x : Orientation R M ι) (v : ι → M) : |(e.adjustToOrientation x).det v| = |e.det v| := by |
cases' e.det_adjustToOrientation x with h h <;> simp [h]
|
/-
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.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.RingTheory.Loca... | Mathlib/AlgebraicGeometry/AffineScheme.lean | 237 | 250 | theorem isBasis_basicOpen (X : Scheme) [IsAffine X] :
Opens.IsBasis (Set.range (X.basicOpen : X.presheaf.obj (op ⊤) → Opens X)) := by |
delta Opens.IsBasis
convert PrimeSpectrum.isBasis_basic_opens.inducing
(TopCat.homeoOfIso (Scheme.forgetToTop.mapIso X.isoSpec)).inducing using 1
ext
simp only [Set.mem_image, exists_exists_eq_and]
constructor
· rintro ⟨_, ⟨x, rfl⟩, rfl⟩
refine ⟨_, ⟨_, ⟨x, rfl⟩, rfl⟩, ?_⟩
exact congr_arg Opens.... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingle... | Mathlib/Combinatorics/Enumerative/Composition.lean | 1,017 | 1,029 | theorem CompositionAsSet.toComposition_boundaries (c : CompositionAsSet n) :
c.toComposition.boundaries = c.boundaries := by |
ext j
simp only [c.mem_boundaries_iff_exists_blocks_sum_take_eq, Composition.boundaries, Finset.mem_map]
constructor
· rintro ⟨i, _, hi⟩
refine ⟨i.1, ?_, ?_⟩
· simpa [c.card_boundaries_eq_succ_length] using i.2
· simp [Composition.boundary, Composition.sizeUpTo, ← hi]
· rintro ⟨i, i_lt, hi⟩
r... |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheory.Products.Basic
#align_import cat... | Mathlib/CategoryTheory/Monoidal/Category.lean | 297 | 299 | theorem whiskerLeft_hom_inv (X : C) {Y Z : C} (f : Y ≅ Z) :
X ◁ f.hom ≫ X ◁ f.inv = 𝟙 (X ⊗ Y) := by |
rw [← whiskerLeft_comp, hom_inv_id, whiskerLeft_id]
|
/-
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
#align_import algebra.lie.skew_adjoint from "leanp... | Mathlib/Algebra/Lie/SkewAdjoint.lean | 103 | 112 | theorem Matrix.isSkewAdjoint_bracket {A B : Matrix n n R} (hA : A ∈ skewAdjointMatricesSubmodule J)
(hB : B ∈ skewAdjointMatricesSubmodule J) : ⁅A, B⁆ ∈ skewAdjointMatricesSubmodule J := by |
simp only [mem_skewAdjointMatricesSubmodule] at *
change ⁅A, B⁆ᵀ * J = J * (-⁅A, B⁆)
change Aᵀ * J = J * (-A) at hA
change Bᵀ * J = J * (-B) at hB
rw [Matrix.lie_transpose, LieRing.of_associative_ring_bracket,
LieRing.of_associative_ring_bracket, sub_mul, mul_assoc, mul_assoc, hA, hB, ← mul_assoc,
← ... |
/-
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, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_imp... | Mathlib/Order/Interval/Set/Basic.lean | 696 | 697 | theorem Icc_eq_empty_iff : Icc a b = ∅ ↔ ¬a ≤ b := by |
rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Icc]
|
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Scott Morrison
-/
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.List.InsertNth
import Mathlib.Logic.Relation
import Mathlib... | Mathlib/SetTheory/Game/PGame.lean | 647 | 649 | theorem zero_le_lf {x : PGame} : 0 ≤ x ↔ ∀ j, 0 ⧏ x.moveRight j := by |
rw [le_iff_forall_lf]
simp
|
/-
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 Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 169 | 171 | theorem rpow_nonneg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : 0 ≤ x ^ y := by |
rw [rpow_def_of_nonneg hx]; split_ifs <;>
simp only [zero_le_one, le_refl, le_of_lt (exp_pos _)]
|
/-
Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Hom.End
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Algebra.SMulWithZero
imp... | Mathlib/Algebra/Module/Defs.lean | 259 | 259 | theorem neg_smul_neg : -r • -x = r • x := by | rw [neg_smul, smul_neg, neg_neg]
|
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 912 | 914 | theorem exists_ne_one_of_nontrivial (H : Subgroup G) [Nontrivial H] :
∃ x ∈ H, x ≠ 1 := by |
rwa [← Subgroup.nontrivial_iff_exists_ne_one]
|
/-
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.Order.RelClasses
import Mathlib.Order.Interval.Set.Basic
#align_import order.bounded from "leanprover-community/mathlib"@"aba5... | Mathlib/Order/Bounded.lean | 301 | 306 | theorem bounded_inter_not (H : ∀ a b, ∃ m, ∀ c, r c a ∨ r c b → r c m) (a : α) :
Bounded r (s ∩ { b | ¬r b a }) ↔ Bounded r s := by |
refine ⟨?_, Bounded.mono inter_subset_left⟩
rintro ⟨b, hb⟩
cases' H a b with m hm
exact ⟨m, fun c hc => hm c (or_iff_not_imp_left.2 fun hca => hb c ⟨hc, hca⟩)⟩
|
/-
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.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
/-!
... | Mathlib/Data/ENNReal/Real.lean | 581 | 582 | theorem toReal_iSup (hf : ∀ i, f i ≠ ∞) : (iSup f).toReal = ⨆ i, (f i).toReal := by |
simp only [ENNReal.toReal, toNNReal_iSup hf, NNReal.coe_iSup]
|
/-
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.Order.Field.Power
import Mathlib.Data.Int.LeastGreatest
import Mathlib.Data.Rat.Floor
import Mathlib.Data.NNRat.Defs
#align_import algebra.ord... | Mathlib/Algebra/Order/Archimedean.lean | 334 | 335 | theorem exists_pos_rat_lt {x : α} (x0 : 0 < x) : ∃ q : ℚ, 0 < q ∧ (q : α) < x := by |
simpa only [Rat.cast_pos] using exists_rat_btwn x0
|
/-
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, Floris van Doorn
-/
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,774 | 1,776 | theorem _root_.Set.countable_infinite_iff_nonempty_denumerable {α : Type*} {s : Set α} :
s.Countable ∧ s.Infinite ↔ Nonempty (Denumerable s) := by |
rw [nonempty_denumerable_iff, ← Set.infinite_coe_iff, countable_coe_iff]
|
/-
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 Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 2,021 | 2,024 | theorem directedOn_sUnion {r} {S : Set (Set α)} (hd : DirectedOn (· ⊆ ·) S)
(h : ∀ x ∈ S, DirectedOn r x) : DirectedOn r (⋃₀ S) := by |
rw [sUnion_eq_iUnion]
exact directedOn_iUnion (directedOn_iff_directed.mp hd) (fun i ↦ h i.1 i.2)
|
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.BernoulliPolynomials
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathl... | Mathlib/NumberTheory/ZetaValues.lean | 179 | 192 | theorem summable_bernoulli_fourier {k : ℕ} (hk : 2 ≤ k) :
Summable (fun n => -k ! / (2 * π * I * n) ^ k : ℤ → ℂ) := by |
have :
∀ n : ℤ, -(k ! : ℂ) / (2 * π * I * n) ^ k = -k ! / (2 * π * I) ^ k * (1 / (n : ℂ) ^ k) := by
intro n; rw [mul_one_div, div_div, ← mul_pow]
simp_rw [this]
refine Summable.mul_left _ <| .of_norm ?_
have : (fun x : ℤ => ‖1 / (x : ℂ) ^ k‖) = fun x : ℤ => |1 / (x : ℝ) ^ k| := by
ext1 x
rw [... |
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