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) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTh... | Mathlib/GroupTheory/CoprodI.lean | 679 | 683 | theorem toList_ne_nil {i j} (w : NeWord M i j) : w.toList ≠ List.nil := by |
induction w
· rintro ⟨rfl⟩
· apply List.append_ne_nil_of_ne_nil_left
assumption
|
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra... | Mathlib/Algebra/BigOperators/Finprod.lean | 608 | 620 | theorem finprod_mul_distrib (hf : (mulSupport f).Finite) (hg : (mulSupport g).Finite) :
∏ᶠ i, f i * g i = (∏ᶠ i, f i) * ∏ᶠ i, g i := by |
classical
rw [finprod_eq_prod_of_mulSupport_toFinset_subset f hf Finset.subset_union_left,
finprod_eq_prod_of_mulSupport_toFinset_subset g hg Finset.subset_union_right, ←
Finset.prod_mul_distrib]
refine finprod_eq_prod_of_mulSupport_subset _ ?_
simp only [Finset.coe_union, Finite.coe_toFinset... |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 137 | 143 | theorem rpow_logb_eq_abs (hx : x ≠ 0) : b ^ logb b x = |x| := by |
apply log_injOn_pos
· simp only [Set.mem_Ioi]
apply rpow_pos_of_pos b_pos
· simp only [abs_pos, mem_Ioi, Ne, hx, not_false_iff]
rw [log_rpow b_pos, logb, log_abs]
field_simp [log_b_ne_zero b_pos b_ne_one]
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.BigOperators.Fin
im... | Mathlib/Algebra/BigOperators/Finsupp.lean | 481 | 485 | theorem equivFunOnFinite_symm_eq_sum [Fintype α] [AddCommMonoid M] (f : α → M) :
equivFunOnFinite.symm f = ∑ a, Finsupp.single a (f a) := by |
rw [← univ_sum_single (equivFunOnFinite.symm f)]
ext
simp
|
/-
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.RBMap.Alter
import Batteries.Data.List.Lemmas
/-!
# Additional lemmas for Red-black trees
-/
namespace Batteries
namespace RBNode
open RBColor... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 182 | 184 | theorem forIn_visit_eq_bindList [Monad m] [LawfulMonad m] {t : RBNode α} :
forIn.visit (m := m) f t init = (ForInStep.yield init).bindList f t.toList := by |
induction t generalizing init <;> simp [*, forIn.visit]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
#align_import measure_theory.function.conditional_expectation.condexp_L1 from "leanprover-communit... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 200 | 207 | theorem condexpIndL1_add (x y : G) :
condexpIndL1 hm μ s (x + y) = condexpIndL1 hm μ s x + condexpIndL1 hm μ s y := by |
by_cases hs : MeasurableSet s
swap; · simp_rw [condexpIndL1_of_not_measurableSet hs]; rw [zero_add]
by_cases hμs : μ s = ∞
· simp_rw [condexpIndL1_of_measure_eq_top hμs]; rw [zero_add]
· simp_rw [condexpIndL1_of_measurableSet_of_measure_ne_top hs hμs]
exact condexpIndL1Fin_add hs hμs x y
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury G. Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
impor... | Mathlib/Analysis/SpecificLimits/Normed.lean | 540 | 546 | theorem geom_series_mul_neg (x : R) (h : ‖x‖ < 1) : (∑' i : ℕ, x ^ i) * (1 - x) = 1 := by |
have := (NormedRing.summable_geometric_of_norm_lt_one x h).hasSum.mul_right (1 - x)
refine tendsto_nhds_unique this.tendsto_sum_nat ?_
have : Tendsto (fun n : ℕ ↦ 1 - x ^ n) atTop (𝓝 1) := by
simpa using tendsto_const_nhds.sub (tendsto_pow_atTop_nhds_zero_of_norm_lt_one h)
convert← this
rw [← geom_sum_m... |
/-
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 | 440 | 447 | theorem repeat_add {α : Type*} (a : Fin n → α) (m₁ m₂ : ℕ) : Fin.repeat (m₁ + m₂) a =
append (Fin.repeat m₁ a) (Fin.repeat m₂ a) ∘ cast (Nat.add_mul ..) := by |
generalize_proofs h
apply funext
rw [(Fin.rightInverse_cast h.symm).surjective.forall]
refine Fin.addCases (fun l => ?_) fun r => ?_
· simp [modNat, Nat.mod_eq_of_lt l.is_lt]
· simp [modNat, Nat.add_mod]
|
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mat... | Mathlib/NumberTheory/PythagoreanTriples.lean | 78 | 82 | theorem mul (k : ℤ) : PythagoreanTriple (k * x) (k * y) (k * z) :=
calc
k * x * (k * x) + k * y * (k * y) = k ^ 2 * (x * x + y * y) := by | ring
_ = k ^ 2 * (z * z) := by rw [h.eq]
_ = k * z * (k * z) := by ring
|
/-
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.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Sets.Opens
#align_import topology.local_homeomorph from "leanprover-community/mathlib"@"431589b... | Mathlib/Topology/PartialHomeomorph.lean | 1,084 | 1,087 | theorem prod_eq_prod_of_nonempty'
{eX eX' : PartialHomeomorph X X'} {eY eY' : PartialHomeomorph Y Y'}
(h : (eX'.prod eY').source.Nonempty) : eX.prod eY = eX'.prod eY' ↔ eX = eX' ∧ eY = eY' := by |
rw [eq_comm, prod_eq_prod_of_nonempty h, eq_comm, @eq_comm _ eY']
|
/-
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.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Mono... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,160 | 1,161 | theorem natDegree_eq_zero_iff_degree_le_zero : p.natDegree = 0 ↔ p.degree ≤ 0 := by |
rw [← nonpos_iff_eq_zero, natDegree_le_iff_degree_le, Nat.cast_zero]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.E... | Mathlib/Topology/Instances/ENNReal.lean | 603 | 605 | theorem iSup_add_iSup_le {ι ι' : Sort*} [Nonempty ι] [Nonempty ι'] {f : ι → ℝ≥0∞} {g : ι' → ℝ≥0∞}
{a : ℝ≥0∞} (h : ∀ i j, f i + g j ≤ a) : iSup f + iSup g ≤ a := by |
simp_rw [iSup_add, add_iSup]; exact iSup₂_le h
|
/-
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 | 529 | 530 | theorem ucs_eq_top_iff (k : ℕ) : N.ucs k = ⊤ ↔ LieModule.lowerCentralSeries R L M k ≤ N := by |
rw [eq_top_iff, ← lcs_le_iff]; rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264... | Mathlib/Algebra/MvPolynomial/Rename.lean | 93 | 99 | theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by |
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
#align_import category_theory.limits.shapes.reflexive from ... | Mathlib/CategoryTheory/Limits/Shapes/Reflexive.lean | 150 | 153 | theorem hasCoequalizer_of_common_section [HasReflexiveCoequalizers C] {A B : C} {f g : A ⟶ B}
(r : B ⟶ A) (rf : r ≫ f = 𝟙 _) (rg : r ≫ g = 𝟙 _) : HasCoequalizer f g := by |
letI := IsReflexivePair.mk' r rf rg
infer_instance
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky, Chris Hughes
-/
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
/-!
... | Mathlib/Data/List/Duplicate.lean | 117 | 126 | theorem duplicate_iff_sublist : x ∈+ l ↔ [x, x] <+ l := by |
induction' l with y l IH
· simp
· by_cases hx : x = y
· simp [hx, cons_sublist_cons, singleton_sublist]
· rw [duplicate_cons_iff_of_ne hx, IH]
refine ⟨sublist_cons_of_sublist y, fun h => ?_⟩
cases h
· assumption
· contradiction
|
/-
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.Circumcenter
#align_import geometry.euclidean.monge_point from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0... | Mathlib/Geometry/Euclidean/MongePoint.lean | 392 | 423 | theorem affineSpan_pair_eq_altitude_iff {n : ℕ} (s : Simplex ℝ P (n + 1)) (i : Fin (n + 2))
(p : P) :
line[ℝ, p, s.points i] = s.altitude i ↔
p ≠ s.points i ∧
p ∈ affineSpan ℝ (Set.range s.points) ∧
p -ᵥ s.points i ∈ (affineSpan ℝ (s.points '' ↑(Finset.univ.erase i))).directionᗮ := by |
rw [eq_iff_direction_eq_of_mem (mem_affineSpan ℝ (Set.mem_insert_of_mem _ (Set.mem_singleton _)))
(s.mem_altitude _),
← vsub_right_mem_direction_iff_mem (mem_affineSpan ℝ (Set.mem_range_self i)) p,
direction_affineSpan, direction_affineSpan, direction_affineSpan]
constructor
· intro h
construct... |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriente... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 212 | 216 | theorem oangle_neg_right {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
o.oangle x (-y) = o.oangle x y + π := by |
simp only [oangle, map_neg]
convert Complex.arg_neg_coe_angle _
exact o.kahler_ne_zero hx hy
|
/-
Copyright (c) 2022 Niels Voss. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Niels Voss
-/
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Order.Filter.Cofinite
#align_import number_theory.fermat_psp from "leanprover-community/mathlib"@"c0439b4877c24a117bfd... | Mathlib/NumberTheory/FermatPsp.lean | 127 | 130 | theorem fermatPsp_base_one {n : ℕ} (h₁ : 1 < n) (h₂ : ¬n.Prime) : FermatPsp n 1 := by |
refine ⟨show n ∣ 1 ^ (n - 1) - 1 from ?_, h₂, h₁⟩
exact show 0 = 1 ^ (n - 1) - 1 by
set_option tactic.skipAssignedInstances false in norm_num ▸ dvd_zero n
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.Bin... | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 207 | 210 | theorem inl_pushoutZeroZeroIso_hom (X Y : C) [HasBinaryCoproduct X Y] :
pushout.inl ≫ (pushoutZeroZeroIso X Y).hom = coprod.inl := by |
dsimp [pushoutZeroZeroIso]
simp
|
/-
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.RBMap.Alter
import Batteries.Data.List.Lemmas
/-!
# Additional lemmas for Red-black trees
-/
namespace Batteries
namespace RBNode
open RBColor... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 175 | 176 | theorem forM_eq_forM_toList [Monad m] [LawfulMonad m] {t : RBNode α} :
t.forM (m := m) f = t.toList.forM f := by | induction t <;> simp [*]
|
/-
Copyright (c) 2023 Kim Liesinger. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Liesinger
-/
import Mathlib.Algebra.Group.Defs
/-!
# Levenshtein distances
We define the Levenshtein edit distance `levenshtein C xy ys` between two `List α`,
with a customizable ... | Mathlib/Data/List/EditDistance/Defs.lean | 278 | 285 | theorem levenshtein_nil_cons (y) (ys) :
levenshtein C [] (y :: ys) = C.insert y + levenshtein C [] ys := by |
dsimp (config := { unfoldPartialApp := true }) [levenshtein, suffixLevenshtein, impl]
congr
rw [List.getLast_eq_get]
congr
rw [show (List.length _) = 1 from _]
induction ys <;> 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, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,107 | 1,110 | theorem tendsto_tan_neg_pi_div_two : Tendsto tan (𝓝[>] (-(π / 2))) atBot := by |
convert tendsto_cos_neg_pi_div_two.inv_tendsto_zero.atTop_mul_neg (by norm_num)
tendsto_sin_neg_pi_div_two using 1
simp only [Pi.inv_apply, ← div_eq_inv_mul, ← tan_eq_sin_div_cos]
|
/-
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 | 1,272 | 1,274 | theorem mul_div_mul_left_eq_div (a b c : G) : c * a / (c * b) = a / b := by |
rw [div_eq_mul_inv, mul_inv_rev, mul_comm b⁻¹ c⁻¹, mul_comm c a, mul_assoc, ← mul_assoc c,
mul_right_inv, one_mul, div_eq_mul_inv]
|
/-
Copyright (c) 2022 Alex Kontorovich and Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Mea... | Mathlib/MeasureTheory/Measure/Haar/Quotient.lean | 295 | 316 | theorem IsFundamentalDomain.QuotientMeasureEqMeasurePreimage_smulHaarMeasure {𝓕 : Set G}
(h𝓕 : IsFundamentalDomain Γ.op 𝓕 ν) (h𝓕_finite : ν 𝓕 ≠ ⊤) :
QuotientMeasureEqMeasurePreimage ν
((ν ((π ⁻¹' (K : Set (G ⧸ Γ))) ∩ 𝓕)) • haarMeasure K) := by |
set c := ν ((π ⁻¹' (K : Set (G ⧸ Γ))) ∩ 𝓕)
have c_ne_top : c ≠ ∞ := by
contrapose! h𝓕_finite
have : c ≤ ν 𝓕 := measure_mono (Set.inter_subset_right)
rw [h𝓕_finite] at this
exact top_unique this
set μ := c • haarMeasure K
have hμK : μ K = c := by simp [μ, haarMeasure_self]
haveI : SigmaFin... |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
#align_import dynamics.ergodic.ergodic from "leanprover-community/mathlib"@"809e920edfa343283cea507aedff916ea0f1bd88"
/-... | Mathlib/Dynamics/Ergodic/Ergodic.lean | 171 | 176 | theorem ae_empty_or_univ_of_image_ae_le' (hf : Ergodic f μ) (hs : MeasurableSet s)
(hs' : f '' s ≤ᵐ[μ] s) (h_fin : μ s ≠ ∞) : s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ := by |
replace hs' : s ≤ᵐ[μ] f ⁻¹' s :=
(HasSubset.Subset.eventuallyLE (subset_preimage_image f s)).trans
(hf.quasiMeasurePreserving.preimage_mono_ae hs')
exact ae_empty_or_univ_of_ae_le_preimage' hf hs hs' h_fin
|
/-
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.Data.Set.Pairwise.Basic
import Mathlib.Data.Set.Lattice
import Mathlib.Order.SuccPred.Basic
#align_import order.succ_pred.interval_succ from "leanpr... | Mathlib/Order/SuccPred/IntervalSucc.lean | 38 | 48 | theorem biUnion_Ico_Ioc_map_succ [SuccOrder α] [IsSuccArchimedean α] [LinearOrder β] {f : α → β}
(hf : Monotone f) (m n : α) : ⋃ i ∈ Ico m n, Ioc (f i) (f (succ i)) = Ioc (f m) (f n) := by |
rcases le_total n m with hnm | hmn
· rw [Ico_eq_empty_of_le hnm, Ioc_eq_empty_of_le (hf hnm), biUnion_empty]
· refine Succ.rec ?_ ?_ hmn
· simp only [Ioc_self, Ico_self, biUnion_empty]
· intro k hmk ihk
rw [← Ioc_union_Ioc_eq_Ioc (hf hmk) (hf <| le_succ _), union_comm, ← ihk]
by_cases hk : Is... |
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 119 | 140 | theorem Nat.Coprime.isPrimePow_dvd_mul {n a b : ℕ} (hab : Nat.Coprime a b) (hn : IsPrimePow n) :
n ∣ a * b ↔ n ∣ a ∨ n ∣ b := by |
rcases eq_or_ne a 0 with (rfl | ha)
· simp only [Nat.coprime_zero_left] at hab
simp [hab, Finset.filter_singleton, not_isPrimePow_one]
rcases eq_or_ne b 0 with (rfl | hb)
· simp only [Nat.coprime_zero_right] at hab
simp [hab, Finset.filter_singleton, not_isPrimePow_one]
refine
⟨?_, fun h =>
... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 385 | 386 | theorem zpow_apply_mem_support {n : ℤ} {x : α} : (f ^ n) x ∈ f.support ↔ x ∈ f.support := by |
simp only [mem_support, ne_eq, apply_zpow_apply_eq_iff]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 578 | 583 | theorem one_lt_card_support_of_ne_one {f : Perm α} (h : f ≠ 1) : 1 < f.support.card := by |
simp_rw [one_lt_card_iff, mem_support, ← not_or]
contrapose! h
ext a
specialize h (f a) a
rwa [apply_eq_iff_eq, or_self_iff, or_self_iff] at h
|
/-
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.Derivative
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.RingTheory.EuclideanDo... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 481 | 483 | theorem rootSet_C_mul_X_pow [CommRing S] [IsDomain S] [Algebra R S] {n : ℕ} (hn : n ≠ 0) {a : R}
(ha : a ≠ 0) : rootSet (C a * X ^ n) S = {0} := by |
rw [C_mul_X_pow_eq_monomial, rootSet_monomial hn ha]
|
/-
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, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib... | Mathlib/Order/SymmDiff.lean | 141 | 142 | theorem symmDiff_of_ge {a b : α} (h : b ≤ a) : a ∆ b = a \ b := by |
rw [symmDiff, sdiff_eq_bot_iff.2 h, sup_bot_eq]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Grou... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 856 | 863 | theorem isPrimitiveRoot_iff' {k : ℕ+} {ζ ξ : Rˣ} (h : IsPrimitiveRoot ζ k) :
IsPrimitiveRoot ξ k ↔ ∃ i < (k : ℕ), i.Coprime k ∧ ζ ^ i = ξ := by |
constructor
· intro hξ
obtain ⟨i, hik, rfl⟩ := h.eq_pow_of_mem_rootsOfUnity hξ.pow_eq_one
rw [h.pow_iff_coprime k.pos] at hξ
exact ⟨i, hik, hξ, rfl⟩
· rintro ⟨i, -, hi, rfl⟩; exact h.pow_of_coprime i hi
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... | Mathlib/RingTheory/Polynomial/Content.lean | 420 | 425 | theorem IsPrimitive.dvd_primPart_iff_dvd {p q : R[X]} (hp : p.IsPrimitive) (hq : q ≠ 0) :
p ∣ q.primPart ↔ p ∣ q := by |
refine ⟨fun h => h.trans (Dvd.intro_left _ q.eq_C_content_mul_primPart.symm), fun h => ?_⟩
rcases h with ⟨r, rfl⟩
apply Dvd.intro _
rw [primPart_mul hq, hp.primPart_eq]
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,775 | 1,798 | theorem Nobeling.embedding : ClosedEmbedding (Nobeling.ι S) := by |
apply Continuous.closedEmbedding
· dsimp (config := { unfoldPartialApp := true }) [ι]
refine continuous_pi ?_
intro C
rw [← IsLocallyConstant.iff_continuous]
refine ((IsLocallyConstant.tfae _).out 0 3).mpr ?_
rintro ⟨⟩
· refine IsClopen.isOpen (isClopen_compl_iff.mp ?_)
convert C.2
... |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Floris van Doorn, Gabriel Ebner, Yury Kudryashov
-/
import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Order.Interval.Finset.Nat
#align_import dat... | Mathlib/Data/Nat/Lattice.lean | 91 | 98 | theorem nonempty_of_pos_sInf {s : Set ℕ} (h : 0 < sInf s) : s.Nonempty := by |
by_contra contra
rw [Set.not_nonempty_iff_eq_empty] at contra
have h' : sInf s ≠ 0 := ne_of_gt h
apply h'
rw [Nat.sInf_eq_zero]
right
assumption
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Ta... | Mathlib/RingTheory/Valuation/Basic.lean | 421 | 438 | theorem isEquiv_of_val_le_one [LinearOrderedCommGroupWithZero Γ₀]
[LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀)
(h : ∀ {x : K}, v x ≤ 1 ↔ v' x ≤ 1) : v.IsEquiv v' := by |
intro x y
by_cases hy : y = 0; · simp [hy, zero_iff]
rw [show y = 1 * y by rw [one_mul]]
rw [← inv_mul_cancel_right₀ hy x]
iterate 2 rw [v.map_mul _ y, v'.map_mul _ y]
rw [v.map_one, v'.map_one]
constructor <;> intro H
· apply mul_le_mul_right'
replace hy := v.ne_zero_iff.mpr hy
replace H := le... |
/-
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.Data.Set.Pointwise.Interval
import Mathlib.LinearAlgebra.AffineSpace.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import ... | Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 598 | 600 | theorem apply_lineMap (f : P1 →ᵃ[k] P2) (p₀ p₁ : P1) (c : k) :
f (lineMap p₀ p₁ c) = lineMap (f p₀) (f p₁) c := by |
simp [lineMap_apply]
|
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Yuyang Zhao
-/
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.Defs
import Mathlib.Tactic.GCongr.Core
#align_import algebra.order... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 732 | 733 | theorem mul_le_of_le_one_right [PosMulMono α] (ha : 0 ≤ a) (h : b ≤ 1) : a * b ≤ a := by |
simpa only [mul_one] using mul_le_mul_of_nonneg_left h ha
|
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudriashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Analysis.Convex.Star
import Mathlib.LinearAlgebra.AffineSpace.Af... | Mathlib/Analysis/Convex/Basic.lean | 465 | 467 | theorem Convex.affinity (hs : Convex 𝕜 s) (z : E) (c : 𝕜) :
Convex 𝕜 ((fun x => z + c • x) '' s) := by |
simpa only [← image_smul, ← image_vadd, image_image] using (hs.smul c).vadd z
|
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semicon... | Mathlib/Data/Int/GCD.lean | 327 | 329 | theorem ne_zero_of_gcd {x y : ℤ} (hc : gcd x y ≠ 0) : x ≠ 0 ∨ y ≠ 0 := by |
contrapose! hc
rw [hc.left, hc.right, gcd_zero_right, natAbs_zero]
|
/-
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.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Anal... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 351 | 355 | theorem circleIntegral_def_Icc (f : ℂ → E) (c : ℂ) (R : ℝ) :
(∮ z in C(c, R), f z) = ∫ θ in Icc 0 (2 * π),
deriv (circleMap c R) θ • f (circleMap c R θ) := by |
rw [circleIntegral, intervalIntegral.integral_of_le Real.two_pi_pos.le,
Measure.restrict_congr_set Ioc_ae_eq_Icc]
|
/-
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.Order.Filter.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
im... | Mathlib/Topology/Order/Basic.lean | 292 | 294 | theorem nhdsWithin_Iic_eq' [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α}
(ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by |
simp only [nhdsWithin_Iic_eq'', biInf_inf ha, inf_principal, Ioi_inter_Iic]
|
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Ma... | Mathlib/Data/Nat/Factorization/Basic.lean | 381 | 385 | theorem ord_compl_mul (a b p : ℕ) : ord_compl[p] (a * b) = ord_compl[p] a * ord_compl[p] b := by |
if ha : a = 0 then simp [ha] else
if hb : b = 0 then simp [hb] else
simp only [ord_proj_mul p ha hb]
rw [div_mul_div_comm (ord_proj_dvd a p) (ord_proj_dvd b p)]
|
/-
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.Analysis.Filter
import Mathlib.Topology.Bases
import Mathlib.Topology.LocallyFinite
#align_import data.analysis.topology from "leanprover-communi... | Mathlib/Data/Analysis/Topology.lean | 161 | 165 | theorem ext' [T : TopologicalSpace α] {σ : Type*} {F : Ctop α σ}
(H : ∀ a s, s ∈ 𝓝 a ↔ ∃ b, a ∈ F b ∧ F b ⊆ s) : F.toTopsp = T := by |
refine TopologicalSpace.ext_nhds fun x ↦ ?_
ext s
rw [mem_nhds_toTopsp, H]
|
/-
Copyright (c) 2020 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
#align_import analysis.convex.hull from "leanprover-community/mathlib"@"92bd7b1ffeb... | Mathlib/Analysis/Convex/Hull.lean | 144 | 158 | theorem Convex.convex_remove_iff_not_mem_convexHull_remove {s : Set E} (hs : Convex 𝕜 s) (x : E) :
Convex 𝕜 (s \ {x}) ↔ x ∉ convexHull 𝕜 (s \ {x}) := by |
constructor
· rintro hsx hx
rw [hsx.convexHull_eq] at hx
exact hx.2 (mem_singleton _)
rintro hx
suffices h : s \ {x} = convexHull 𝕜 (s \ {x}) by
rw [h]
exact convex_convexHull 𝕜 _
exact
Subset.antisymm (subset_convexHull 𝕜 _) fun y hy =>
⟨convexHull_min diff_subset hs hy, by
... |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Analysis.Normed.Field.Basic
#align_import analysis.normed.mul_action from "leanprover-community/mathlib"@"bc91ed... | Mathlib/Analysis/Normed/MulAction.lean | 90 | 97 | theorem norm_smul (r : α) (x : β) : ‖r • x‖ = ‖r‖ * ‖x‖ := by |
by_cases h : r = 0
· simp [h, zero_smul α x]
· refine le_antisymm (norm_smul_le r x) ?_
calc
‖r‖ * ‖x‖ = ‖r‖ * ‖r⁻¹ • r • x‖ := by rw [inv_smul_smul₀ h]
_ ≤ ‖r‖ * (‖r⁻¹‖ * ‖r • x‖) := by gcongr; apply norm_smul_le
_ = ‖r • x‖ := by rw [norm_inv, ← mul_assoc, mul_inv_cancel (mt norm_eq_zero.... |
/-
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.MeasureTheory.OuterMeasure.Caratheodory
/-!
# Induced Outer Measure
We can extend a function defined on a subset of `Set α` to an out... | Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 290 | 297 | theorem extend_iUnion_le_tsum_nat : ∀ s : ℕ → Set α,
extend m (⋃ i, s i) ≤ ∑' i, extend m (s i) := by |
refine extend_iUnion_le_tsum_nat' MeasurableSet.iUnion ?_; intro f h
simp (config := { singlePass := true }) only [iUnion_disjointed.symm]
rw [mU (MeasurableSet.disjointed h) (disjoint_disjointed _)]
refine ENNReal.tsum_le_tsum fun i => ?_
rw [← extend_eq m, ← extend_eq m]
exact extend_mono m0 mU (Measurab... |
/-
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.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Math... | Mathlib/Data/Finset/Basic.lean | 2,059 | 2,060 | theorem erase_subset_iff_of_mem (h : a ∈ t) : s.erase a ⊆ t ↔ s ⊆ t := by |
rw [← subset_insert_iff, insert_eq_of_mem h]
|
/-
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.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "lean... | Mathlib/Analysis/Convex/Side.lean | 698 | 705 | theorem SOppSide.exists_sbtw {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y) :
∃ p ∈ s, Sbtw R x p y := by |
obtain ⟨p, hp, hw⟩ := wOppSide_iff_exists_wbtw.1 h.wOppSide
refine ⟨p, hp, hw, ?_, ?_⟩
· rintro rfl
exact h.2.1 hp
· rintro rfl
exact h.2.2 hp
|
/-
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.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLim... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 2,211 | 2,214 | theorem map_measurableEquiv_injective (e : α ≃ᵐ β) : Injective (Measure.map e) := by |
intro μ₁ μ₂ hμ
apply_fun Measure.map e.symm at hμ
simpa [map_symm_map e] using hμ
|
/-
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.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "lean... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,401 | 1,403 | theorem ContDiffAt.differentiableAt (h : ContDiffAt 𝕜 n f x) (hn : 1 ≤ n) :
DifferentiableAt 𝕜 f x := by |
simpa [hn, differentiableWithinAt_univ] using h.differentiableWithinAt
|
/-
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
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc429200506... | Mathlib/Data/Set/Image.lean | 1,322 | 1,328 | theorem Injective.compl_image_eq (hf : Injective f) (s : Set α) :
(f '' s)ᶜ = f '' sᶜ ∪ (range f)ᶜ := by |
ext y
rcases em (y ∈ range f) with (⟨x, rfl⟩ | hx)
· simp [hf.eq_iff]
· rw [mem_range, not_exists] at hx
simp [hx]
|
/-
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.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
#align_import data.nat.part_enat from "l... | Mathlib/Data/Nat/PartENat.lean | 794 | 796 | theorem withTopEquiv_symm_ofNat (n : Nat) [n.AtLeastTwo] :
withTopEquiv.symm (no_index (OfNat.ofNat n)) = OfNat.ofNat n := by |
simp
|
/-
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 | 2,642 | 2,647 | theorem pi_norm_le_iff_of_nonempty' [Nonempty ι] : ‖f‖ ≤ r ↔ ∀ b, ‖f b‖ ≤ r := by |
by_cases hr : 0 ≤ r
· exact pi_norm_le_iff_of_nonneg' hr
· exact
iff_of_false (fun h => hr <| (norm_nonneg' _).trans h) fun h =>
hr <| (norm_nonneg' _).trans <| h <| Classical.arbitrary _
|
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 87 | 88 | theorem snorm_eq_snorm' (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
snorm f p μ = snorm' f (ENNReal.toReal p) μ := by | simp [snorm, hp_ne_zero, hp_ne_top]
|
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Scott Morrison, Eric Rodriguez
-/
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count fr... | Mathlib/Data/Nat/Count.lean | 38 | 39 | theorem count_zero : count p 0 = 0 := by |
rw [count, List.range_zero, List.countP, List.countP.go]
|
/-
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, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENa... | Mathlib/RingTheory/Multiplicity.lean | 215 | 216 | theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by |
rw [Ne, eq_top_iff_not_finite, Classical.not_not]
|
/-
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.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
/-!
# Ideal... | Mathlib/Algebra/Lie/IdealOperations.lean | 119 | 121 | theorem lie_le_right : ⁅I, N⁆ ≤ N := by |
rw [lieIdeal_oper_eq_span, lieSpan_le]; rintro m ⟨x, n, hn⟩; rw [← hn]
exact N.lie_mem n.property
|
/-
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 | 880 | 880 | theorem eq_mul_inv_of_mul_eq (h : a * c = b) : a = b * c⁻¹ := by | simp [h.symm]
|
/-
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.Init.Order.LinearOrder
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Subtype
import Mathlib.Tactic.Spread
import Mathlib.Tactic.Conv... | Mathlib/Order/Basic.lean | 1,076 | 1,078 | theorem max_def_lt (x y : α) : max x y = if x < y then y else x := by |
rw [max_comm, max_def, ← ite_not]
simp only [not_le]
|
/-
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.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathli... | Mathlib/Data/ZMod/Basic.lean | 151 | 152 | theorem natCast_self' (n : ℕ) : (n + 1 : ZMod (n + 1)) = 0 := by |
rw [← Nat.cast_add_one, natCast_self (n + 1)]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Init.Data.Ordering.Lemmas
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.NormNum
#ali... | Mathlib/SetTheory/Ordinal/Notation.lean | 262 | 263 | theorem NF.zero_of_zero {e n a} (h : NF (ONote.oadd e n a)) (e0 : e = 0) : a = 0 := by |
simpa [e0, NFBelow_zero] using h.snd'
|
/-
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
-/
import Mathlib.Analysis.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.special_... | Mathlib/Analysis/SpecialFunctions/Exp.lean | 505 | 508 | theorem tendsto_exp_nhds_zero_iff {α : Type*} {l : Filter α} {f : α → ℂ} :
Tendsto (fun x => exp (f x)) l (𝓝 0) ↔ Tendsto (fun x => re (f x)) l atBot := by |
simp_rw [← comp_apply (f := exp), ← tendsto_comap_iff, comap_exp_nhds_zero, tendsto_comap_iff]
rfl
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Alex J. Best, Johan Commelin, Eric Rodriguez, Ruben Van de Velde
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.ZMod.Algebra
import Mathlib.FieldTheory.Fin... | Mathlib/FieldTheory/Finite/GaloisField.lean | 151 | 159 | theorem splits_zmod_X_pow_sub_X : Splits (RingHom.id (ZMod p)) (X ^ p - X) := by |
have hp : 1 < p := h_prime.out.one_lt
have h1 : roots (X ^ p - X : (ZMod p)[X]) = Finset.univ.val := by
convert FiniteField.roots_X_pow_card_sub_X (ZMod p)
exact (ZMod.card p).symm
have h2 := FiniteField.X_pow_card_sub_X_natDegree_eq (ZMod p) hp
-- We discharge the `p = 0` separately, to avoid typeclas... |
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Thomas Murrills
-/
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Tactic.NormNum.Basic
/-!
## `norm_num` plugin for `^`.
-/
set_option autoImplicit true
namespac... | Mathlib/Tactic/NormNum/Pow.lean | 112 | 117 | theorem intPow_negOfNat_bit1 (h1 : Nat.pow a b' = c')
(hb : nat_lit 2 * b' + nat_lit 1 = b) (hc : c' * (c' * a) = c) :
Int.pow (Int.negOfNat a) b = Int.negOfNat c := by |
rw [← hb, Int.negOfNat_eq, Int.negOfNat_eq, Int.pow_eq, pow_succ, pow_mul, neg_pow_two, ← pow_mul,
two_mul, pow_add, ← hc, ← h1]
simp [mul_assoc, mul_comm, mul_left_comm]
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 2,241 | 2,242 | theorem conjTranspose_one [DecidableEq n] [Semiring α] [StarRing α] : (1 : Matrix n n α)ᴴ = 1 := by |
simp [conjTranspose]
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Data.Countable.Basic
import Mathlib.Logic.Encodable.Basic
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.Interval.Finset.Defs
#align_import orde... | Mathlib/Order/SuccPred/LinearLocallyFinite.lean | 77 | 84 | theorem isGLB_Ioc_of_isGLB_Ioi {i j k : ι} (hij_lt : i < j) (h : IsGLB (Set.Ioi i) k) :
IsGLB (Set.Ioc i j) k := by |
simp_rw [IsGLB, IsGreatest, mem_upperBounds, mem_lowerBounds] at h ⊢
refine ⟨fun x hx ↦ h.1 x hx.1, fun x hx ↦ h.2 x ?_⟩
intro y hy
rcases le_or_lt y j with h_le | h_lt
· exact hx y ⟨hy, h_le⟩
· exact le_trans (hx j ⟨hij_lt, le_rfl⟩) h_lt.le
|
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Tactic.SplitIfs
#align_import data.nat.cast.defs from "leanprover-community/mathlib"@"a148d797a1094ab... | Mathlib/Data/Nat/Cast/Defs.lean | 144 | 146 | theorem cast_ite (P : Prop) [Decidable P] (m n : ℕ) :
((ite P m n : ℕ) : R) = ite P (m : R) (n : R) := by |
split_ifs <;> 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, 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 | 866 | 867 | theorem Ici_diff_Ioi_same : Ici a \ Ioi a = {a} := by |
rw [← Ici_diff_left, diff_diff_cancel_left (singleton_subset_iff.2 left_mem_Ici)]
|
/-
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.Basic
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import geometry.euclidean.angle.un... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | 163 | 164 | theorem angle_smul_left_of_pos (x y : V) {r : ℝ} (hr : 0 < r) : angle (r • x) y = angle x y := by |
rw [angle_comm, angle_smul_right_of_pos y x hr, angle_comm]
|
/-
Copyright (c) 2022 Cuma Kökmen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Cuma Kökmen, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.torus_... | Mathlib/MeasureTheory/Integral/TorusIntegral.lean | 84 | 85 | theorem torusMap_sub_center (c : ℂⁿ) (R : ℝⁿ) (θ : ℝⁿ) : torusMap c R θ - c = torusMap 0 R θ := by |
ext1 i; simp [torusMap]
|
/-
Copyright (c) 2022 John Nicol. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: John Nicol
-/
import Mathlib.FieldTheory.Finite.Basic
#align_import number_theory.wilson from "leanprover-community/mathlib"@"c471da714c044131b90c133701e51b877c246677"
/-!
# Wilson's the... | Mathlib/NumberTheory/Wilson.lean | 40 | 69 | theorem wilsons_lemma : ((p - 1)! : ZMod p) = -1 := by |
refine
calc
((p - 1)! : ZMod p) = ∏ x ∈ Ico 1 (succ (p - 1)), (x : ZMod p) := by
rw [← Finset.prod_Ico_id_eq_factorial, prod_natCast]
_ = ∏ x : (ZMod p)ˣ, (x : ZMod p) := ?_
_ = -1 := by
-- Porting note: `simp` is less powerful.
-- simp_rw [← Units.coeHom_apply, ← (Units... |
/-
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 | 1,045 | 1,051 | theorem setToL1_zero_left' (hT : DominatedFinMeasAdditive μ T C)
(h_zero : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0) (f : α →₁[μ] E) : setToL1 hT f = 0 := by |
suffices setToL1 hT = 0 by rw [this]; simp
refine ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT) _ _ _ _ ?_
ext1 f
rw [setToL1SCLM_zero_left' hT h_zero f, ContinuousLinearMap.zero_comp,
ContinuousLinearMap.zero_apply]
|
/-
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 | 666 | 667 | theorem contains_iff (s : String) (c : Char) : contains s c ↔ c ∈ s.1 := by |
simp [contains, any_iff]
|
/-
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.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Adjunction.Over
#align_import categ... | Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | 345 | 349 | theorem diagonalObjPullbackFstIso_inv_snd_snd {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :
(diagonalObjPullbackFstIso f g).inv ≫ pullback.snd ≫ pullback.snd =
pullback.fst ≫ pullback.snd := by |
delta diagonalObjPullbackFstIso
simp
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, E. W. Ayers
-/
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Yoneda
import Mathlib.Data.Set.L... | Mathlib/CategoryTheory/Sites/Sieves.lean | 667 | 669 | theorem functorPullback_id (R : Sieve X) : R.functorPullback (𝟭 _) = R := by |
ext
rfl
|
/-
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 | 859 | 860 | theorem norm_p_pow (n : ℕ) : ‖(p : ℚ_[p]) ^ n‖ = (p : ℝ) ^ (-n : ℤ) := by |
rw [← norm_p_zpow, zpow_natCast]
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sheaves.SheafCondition.Sites
import Mathlib.Algebra.Categor... | Mathlib/AlgebraicGeometry/Properties.lean | 149 | 157 | theorem reduce_to_affine_nbhd (P : ∀ (X : Scheme) (_ : X.carrier), Prop)
(h₁ : ∀ (R : CommRingCat) (x : PrimeSpectrum R), P (Scheme.Spec.obj <| op R) x)
(h₂ : ∀ {X Y} (f : X ⟶ Y) [IsOpenImmersion f] (x : X.carrier), P X x → P Y (f.1.base x)) :
∀ (X : Scheme) (x : X.carrier), P X x := by |
intro X x
obtain ⟨y, e⟩ := X.affineCover.Covers x
convert h₂ (X.affineCover.map (X.affineCover.f x)) y _
· rw [e]
apply h₁
|
/-
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 | 162 | 165 | theorem consCases_cons {P : (∀ i : Fin n.succ, α i) → Sort v} (h : ∀ x₀ x, P (Fin.cons x₀ x))
(x₀ : α 0) (x : ∀ i : Fin n, α i.succ) : @consCases _ _ _ h (cons x₀ x) = h x₀ x := by |
rw [consCases, cast_eq]
congr
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Eq... | Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | 418 | 421 | theorem add_assoc {X Y : C} (a b c : X ⟶ Y) : a + b + c = a + (b + c) := by |
conv_lhs =>
congr; rw [add_def]
rw [sub_add, ← add_neg, neg_sub', neg_neg]
|
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Satisfiability
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import model_theory.graph from "leanprover-community/mathlib"@"e56b8f... | Mathlib/ModelTheory/Graph.lean | 75 | 79 | theorem Theory.simpleGraph_model_iff [Language.graph.Structure V] :
V ⊨ Theory.simpleGraph ↔
(Irreflexive fun x y : V => RelMap adj ![x, y]) ∧
Symmetric fun x y : V => RelMap adj ![x, y] := by |
simp [Theory.simpleGraph]
|
/-
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.Order.Interval.Set.Basic
import Mathlib.Data.Set.NAry
import Mathlib.Order.Directed
#align_import order.bounds.basic from "leanprover... | Mathlib/Order/Bounds/Basic.lean | 982 | 984 | theorem upperBounds_insert (a : α) (s : Set α) :
upperBounds (insert a s) = Ici a ∩ upperBounds s := by |
rw [insert_eq, upperBounds_union, upperBounds_singleton]
|
/-
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.Inductions
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.RingTheory.Multiplicit... | Mathlib/Algebra/Polynomial/Div.lean | 186 | 192 | theorem zero_divByMonic (p : R[X]) : 0 /ₘ p = 0 := by |
classical
unfold divByMonic divModByMonicAux
dsimp
by_cases hp : Monic p
· rw [dif_pos hp, if_neg (mt And.right (not_not_intro rfl))]
· rw [dif_neg hp]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 975 | 985 | theorem iInf_sets_induct {f : ι → Filter α} {s : Set α} (hs : s ∈ iInf f) {p : Set α → Prop}
(uni : p univ) (ins : ∀ {i s₁ s₂}, s₁ ∈ f i → p s₂ → p (s₁ ∩ s₂)) : p s := by |
rw [mem_iInf_finite'] at hs
simp only [← Finset.inf_eq_iInf] at hs
rcases hs with ⟨is, his⟩
induction is using Finset.induction_on generalizing s with
| empty => rwa [mem_top.1 his]
| insert _ ih =>
rw [Finset.inf_insert, mem_inf_iff] at his
rcases his with ⟨s₁, hs₁, s₂, hs₂, rfl⟩
exact ins hs₁... |
/-
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.AlgebraicTopology.FundamentalGroupoid.InducedMaps
import Mathlib.Topology.Homotopy.Contractible
import Mathlib.CategoryTheory.PUnit
import Math... | Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.lean | 94 | 98 | theorem simply_connected_iff_paths_homotopic' {Y : Type*} [TopologicalSpace Y] :
SimplyConnectedSpace Y ↔
PathConnectedSpace Y ∧ ∀ {x y : Y} (p₁ p₂ : Path x y), Path.Homotopic p₁ p₂ := by |
convert simply_connected_iff_paths_homotopic (Y := Y)
simp [Path.Homotopic.Quotient, Setoid.eq_top_iff]; rfl
|
/-
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.Order.Interval.Set.Basic
import Mathlib.Data.Set.NAry
import Mathlib.Order.Directed
#align_import order.bounds.basic from "leanprover... | Mathlib/Order/Bounds/Basic.lean | 804 | 806 | theorem bddBelow_bddAbove_iff_subset_Icc : BddBelow s ∧ BddAbove s ↔ ∃ a b, s ⊆ Icc a b := by |
simp [Ici_inter_Iic.symm, subset_inter_iff, bddBelow_iff_subset_Ici,
bddAbove_iff_subset_Iic, exists_and_left, exists_and_right]
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span
#align_import algebra.algebra.tower from "leanprover-community/mathlib"@"71150516f28d9826c7... | Mathlib/Algebra/Algebra/Tower.lean | 349 | 353 | theorem map_mem_span_algebraMap_image {S T : Type*} [CommSemiring S] [Semiring T] [Algebra R S]
[Algebra R T] [Algebra S T] [IsScalarTower R S T] (x : S) (a : Set S)
(hx : x ∈ Submodule.span R a) : algebraMap S T x ∈ Submodule.span R (algebraMap S T '' a) := by |
rw [span_algebraMap_image_of_tower, mem_map]
exact ⟨x, hx, rfl⟩
|
/-
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, Apurva Nakade
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.SetTheory.Game.PGame
import Mathlib.Tactic.Abel
#align_... | Mathlib/SetTheory/Game/Basic.lean | 327 | 332 | theorem mul_moveLeft_inr {x y : PGame} {i j} :
(x * y).moveLeft (toLeftMovesMul (Sum.inr (i, j))) =
x.moveRight i * y + x * y.moveRight j - x.moveRight i * y.moveRight j := by |
cases x
cases y
rfl
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.List.Nodup
#align_import data.prod.tprod from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
/-!
# Finite products... | Mathlib/Data/Prod/TProd.lean | 99 | 103 | theorem elim_of_mem (hl : (i :: l).Nodup) (hj : j ∈ l) (v : TProd α (i :: l)) :
v.elim (mem_cons_of_mem _ hj) = TProd.elim v.2 hj := by |
apply elim_of_ne
rintro rfl
exact hl.not_mem hj
|
/-
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.GroupTheory.Finiteness
import Mathlib.RingTheory.Adjoin.Tower
import Mathlib.RingTheory.Finiteness
import Mathlib.Ri... | Mathlib/RingTheory/FiniteType.lean | 163 | 171 | theorem iff_quotient_freeAlgebra' : FiniteType R A ↔
∃ (ι : Type uA) (_ : Fintype ι) (f : FreeAlgebra R ι →ₐ[R] A), Surjective f := by |
constructor
· rw [iff_quotient_freeAlgebra]
rintro ⟨s, ⟨f, hsur⟩⟩
use { x : A // x ∈ s }, inferInstance, f
· rintro ⟨ι, ⟨hfintype, ⟨f, hsur⟩⟩⟩
letI : Fintype ι := hfintype
exact FiniteType.of_surjective (FiniteType.freeAlgebra R ι) f hsur
|
/-
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
#align_import ring_theory.polynomial.cyclotomic.roots from "le... | Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean | 56 | 59 | theorem _root_.isRoot_of_unity_iff (h : 0 < n) (R : Type*) [CommRing R] [IsDomain R] {ζ : R} :
ζ ^ n = 1 ↔ ∃ i ∈ n.divisors, (cyclotomic i R).IsRoot ζ := by |
rw [← mem_nthRoots h, nthRoots, mem_roots <| X_pow_sub_C_ne_zero h _, C_1, ←
prod_cyclotomic_eq_X_pow_sub_one h, isRoot_prod]
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.SetTheory.Cardinal.ENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
/-!
# ... | Mathlib/SetTheory/Cardinal/ToNat.lean | 168 | 170 | theorem toNat_congr {β : Type v} (e : α ≃ β) : toNat #α = toNat #β := by |
-- Porting note: Inserted universe hint below
rw [← toNat_lift, (lift_mk_eq.{_,_,v}).mpr ⟨e⟩, toNat_lift]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | Mathlib/GroupTheory/Sylow.lean | 493 | 495 | theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by |
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
|
/-
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.Calculus.InverseFunctionTheorem.FDeriv
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathl... | Mathlib/Analysis/Calculus/Implicit.lean | 293 | 297 | theorem implicitToPartialHomeomorphOfComplemented_apply_ker (hf : HasStrictFDerivAt f f' a)
(hf' : range f' = ⊤) (hker : (ker f').ClosedComplemented) (y : ker f') :
hf.implicitToPartialHomeomorphOfComplemented f f' hf' hker (y + a) = (f (y + a), y) := by |
simp only [implicitToPartialHomeomorphOfComplemented_apply, add_sub_cancel_right,
Classical.choose_spec hker]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 599 | 602 | theorem isOpenMap_prod_of_discrete_left [DiscreteTopology α] {f : α × β → γ} :
IsOpenMap f ↔ ∀ a, IsOpenMap (f ⟨a, ·⟩) := by |
simp_rw [isOpenMap_iff_nhds_le, Prod.forall, nhds_prod_eq, nhds_discrete, pure_prod, map_map]
rfl
|
/-
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.Basic
import Mathlib.RingTheory.Noetherian
#align_import algebra.lie.subalgebra from "leanprover-community/mathlib"@"6d584f1709bedbed9175bd9350d... | Mathlib/Algebra/Lie/Subalgebra.lean | 697 | 699 | theorem lieSpan_mono {t : Set L} (h : s ⊆ t) : lieSpan R L s ≤ lieSpan R L t := by |
rw [lieSpan_le]
exact Set.Subset.trans h subset_lieSpan
|
/-
Copyright (c) 2018 Kevin Buzzard, Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Patrick Massot
This file is to a certain extent based on `quotient_module.lean` by Johannes Hölzl.
-/
import Mathlib.Algebra.Group.Subgroup.Finite
import... | Mathlib/GroupTheory/QuotientGroup.lean | 602 | 607 | theorem equivQuotientZPowOfEquiv_trans :
(equivQuotientZPowOfEquiv e n).trans (equivQuotientZPowOfEquiv d n) =
equivQuotientZPowOfEquiv (e.trans d) n := by |
ext x
rw [← Quotient.out_eq' x]
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
-/
import Mathlib.MeasureTheory.OuterMeasure.OfFunction
import Mathlib.MeasureTheory.PiSystem
/-!
# The Caratheodory σ-algebra of an outer measure
Give... | Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean | 97 | 100 | theorem isCaratheodory_inter (h₁ : IsCaratheodory m s₁) (h₂ : IsCaratheodory m s₂) :
IsCaratheodory m (s₁ ∩ s₂) := by |
rw [← isCaratheodory_compl_iff, Set.compl_inter]
exact isCaratheodory_union _ (isCaratheodory_compl _ h₁) (isCaratheodory_compl _ h₂)
|
/-
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 Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#ali... | Mathlib/Algebra/Order/Group/Defs.lean | 831 | 831 | theorem inv_le_div_iff_le_mul' : a⁻¹ ≤ b / c ↔ c ≤ a * b := by | rw [inv_le_div_iff_le_mul, mul_comm]
|
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