Context stringlengths 285 6.98k | file_name stringlengths 21 79 | start int64 14 184 | end int64 18 184 | theorem stringlengths 25 1.34k | proof stringlengths 5 3.43k |
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/-
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.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
/-!
# Linear equivalences of tensor products as isometrie... | Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean | 79 | 85 | theorem tmul_comp_tensorComm (Q₁ : QuadraticForm R M₁) (Q₂ : QuadraticForm R M₂) :
(Q₂.tmul Q₁).comp (TensorProduct.comm R M₁ M₂) = Q₁.tmul Q₂ := by |
refine (QuadraticForm.associated_rightInverse R).injective ?_
ext m₁ m₂ m₁' m₂'
dsimp [-associated_apply]
simp only [associated_tmul, QuadraticForm.associated_comp]
exact mul_comm _ _
|
/-
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.BigOperators.Group.List
import Mathlib.Data.List.OfFn
import Mathlib.Data.Set.Pointwise.Basic
#align_import data.set.pointwise.list_of_fn from "lean... | Mathlib/Data/Set/Pointwise/ListOfFn.lean | 26 | 31 | theorem mem_prod_list_ofFn {a : α} {s : Fin n → Set α} :
a ∈ (List.ofFn s).prod ↔ ∃ f : ∀ i : Fin n, s i, (List.ofFn fun i ↦ (f i : α)).prod = a := by |
induction' n with n ih generalizing a
· simp_rw [List.ofFn_zero, List.prod_nil, Fin.exists_fin_zero_pi, eq_comm, Set.mem_one]
· simp_rw [List.ofFn_succ, List.prod_cons, Fin.exists_fin_succ_pi, Fin.cons_zero, Fin.cons_succ,
mem_mul, @ih, exists_exists_eq_and, SetCoe.exists, exists_prop]
|
/-
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.GroupTheory.QuotientGroup
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.class_group from "leanprover-community/mathlib"@"565eb991... | Mathlib/RingTheory/ClassGroup.lean | 61 | 63 | theorem coe_toPrincipalIdeal (x : Kˣ) :
(toPrincipalIdeal R K x : FractionalIdeal R⁰ K) = spanSingleton _ (x : K) := by |
simp only [toPrincipalIdeal]; rfl
|
/-
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.Multiset.Bind
#align_import data.multiset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
/-!
# The fold ope... | Mathlib/Data/Multiset/Fold.lean | 108 | 110 | theorem fold_union_inter [DecidableEq α] (s₁ s₂ : Multiset α) (b₁ b₂ : α) :
((s₁ ∪ s₂).fold op b₁ * (s₁ ∩ s₂).fold op b₂) = s₁.fold op b₁ * s₂.fold op b₂ := by |
rw [← fold_add op, union_add_inter, fold_add op]
|
/-
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.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
#align_import measure_theory.m... | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 102 | 104 | theorem index_empty {V : Set G} : index ∅ V = 0 := by |
simp only [index, Nat.sInf_eq_zero]; left; use ∅
simp only [Finset.card_empty, empty_subset, mem_setOf_eq, eq_self_iff_true, and_self_iff]
|
/-
Copyright (c) 2022 Mantas Bakšys. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mantas Bakšys
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Data.Prod.L... | Mathlib/Algebra/Order/Rearrangement.lean | 62 | 108 | theorem MonovaryOn.sum_smul_comp_perm_le_sum_smul (hfg : MonovaryOn f g s)
(hσ : { x | σ x ≠ x } ⊆ s) : (∑ i ∈ s, f i • g (σ i)) ≤ ∑ i ∈ s, f i • g i := by |
classical
revert hσ σ hfg
-- Porting note: Specify `p` to get around `∀ {σ}` in the current goal.
apply Finset.induction_on_max_value (fun i ↦ toLex (g i, f i))
(p := fun t ↦ ∀ {σ : Perm ι}, MonovaryOn f g t → { x | σ x ≠ x } ⊆ t →
(∑ i ∈ t, f i • g (σ i)) ≤ ∑ i ∈ t, f i • g i) s
· simp... |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Simon Hudon, Kenny Lau
-/
import Mathlib.Data.Multiset.Bind
import Mathlib.Control.Traversable.Lemmas
import Mathlib.Control.Traversable.Instances
#align_... | Mathlib/Data/Multiset/Functor.lean | 102 | 105 | theorem id_traverse {α : Type*} (x : Multiset α) : traverse (pure : α → Id α) x = x := by |
refine Quotient.inductionOn x ?_
intro
simp [traverse, Coe.coe]
|
/-
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.CharP.Invertible
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
#align_import linear_algebra.affine_space.midpoint_zero from "leanprover-... | Mathlib/LinearAlgebra/AffineSpace/MidpointZero.lean | 45 | 47 | theorem homothety_one_half {k : Type*} {V P : Type*} [Field k] [CharZero k] [AddCommGroup V]
[Module k V] [AddTorsor V P] (a b : P) : homothety a (1 / 2 : k) b = midpoint k a b := by |
rw [one_div, homothety_inv_two]
|
/-
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.Order.Filter.Basic
import Mathlib.Data.Set.Countable
#align_import order.filter.countable_Inter from "leanprover-community/mathlib"@"b9e46fe10... | Mathlib/Order/Filter/CountableInter.lean | 65 | 68 | theorem eventually_countable_forall [Countable ι] {p : α → ι → Prop} :
(∀ᶠ x in l, ∀ i, p x i) ↔ ∀ i, ∀ᶠ x in l, p x i := by |
simpa only [Filter.Eventually, setOf_forall] using
@countable_iInter_mem _ _ l _ _ fun i => { x | p x i }
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam, Yury Kudryashov
-/
import Mathlib.Algebra.MvPolynomial.Derivation
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.pderiv from "leanprover... | Mathlib/Algebra/MvPolynomial/PDeriv.lean | 96 | 96 | theorem pderiv_X_self (i : σ) : pderiv i (X i : MvPolynomial σ R) = 1 := by | classical simp
|
/-
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.Finsupp
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.RingTheory.GradedAlgebra.Basic
#... | Mathlib/Algebra/MonoidAlgebra/Grading.lean | 67 | 69 | theorem mem_grade_iff (m : M) (a : R[M]) : a ∈ grade R m ↔ a.support ⊆ {m} := by |
rw [← Finset.coe_subset, Finset.coe_singleton]
rfl
|
/-
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.RingTheory.DiscreteValuationRing.Basic
import Mathlib.RingTheory.MvPowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.Basic
import M... | Mathlib/RingTheory/PowerSeries/Inverse.lean | 100 | 102 | theorem constantCoeff_invOfUnit (φ : R⟦X⟧) (u : Rˣ) :
constantCoeff R (invOfUnit φ u) = ↑u⁻¹ := by |
rw [← coeff_zero_eq_constantCoeff_apply, coeff_invOfUnit, if_pos rfl]
|
/-
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 | 82 | 82 | theorem cons_zero : cons x p 0 = x := by | simp [cons]
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Adjunction.Unique
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Sites.Sheaf
import Mathlib.Ca... | Mathlib/CategoryTheory/Sites/Sheafification.lean | 96 | 97 | theorem sheafifyMap_id (P : Cᵒᵖ ⥤ D) : sheafifyMap J (𝟙 P) = 𝟙 (sheafify J P) := by |
simp [sheafifyMap, sheafify]
|
/-
Copyright (c) 2021 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Adam Topaz, Johan Commelin
-/
import Mathlib.Algebra.Homology.Additive
import Mathlib.AlgebraicTopology.MooreComplex
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Catego... | Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean | 132 | 135 | theorem obj_d_eq (X : SimplicialObject C) (n : ℕ) :
(AlternatingFaceMapComplex.obj X).d (n + 1) n
= ∑ i : Fin (n + 2), (-1 : ℤ) ^ (i : ℕ) • X.δ i := by |
apply ChainComplex.of_d
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Complex.Circle
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Meas... | Mathlib/Analysis/Fourier/FourierTransform.lean | 84 | 92 | theorem fourierIntegral_smul_const (e : AddChar 𝕜 𝕊) (μ : Measure V)
(L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E) (r : ℂ) :
fourierIntegral e μ L (r • f) = r • fourierIntegral e μ L f := by |
ext1 w
-- Porting note: was
-- simp only [Pi.smul_apply, fourierIntegral, smul_comm _ r, integral_smul]
simp only [Pi.smul_apply, fourierIntegral, ← integral_smul]
congr 1 with v
rw [smul_comm]
|
/-
Copyright (c) 2021 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Analysis.InnerProductSpace.Projection
#align_import analysis.convex.cone.dual from "leanprover-commu... | Mathlib/Analysis/Convex/Cone/InnerDual.lean | 125 | 127 | theorem innerDualCone_eq_iInter_innerDualCone_singleton :
(s.innerDualCone : Set H) = ⋂ i : s, (({↑i} : Set H).innerDualCone : Set H) := by |
rw [← ConvexCone.coe_iInf, ← innerDualCone_iUnion, iUnion_of_singleton_coe]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.Po... | Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | 73 | 78 | theorem le_prod_of_submultiplicative_on_pred (f : M → N) (p : M → Prop) (h_one : f 1 = 1)
(h_mul : ∀ x y, p x → p y → f (x * y) ≤ f x * f y) (hp_mul : ∀ x y, p x → p y → p (x * y))
(g : ι → M) {s : Finset ι} (hs : ∀ i ∈ s, p (g i)) : f (∏ i ∈ s, g i) ≤ ∏ i ∈ s, f (g i) := by |
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
· simp [h_one]
· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Heather Macbeth
-/
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Math... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 116 | 121 | theorem abs_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A) :
|(f : C(X, ℝ))| ∈ A.topologicalClosure := by |
let f' := attachBound (f : C(X, ℝ))
let abs : C(Set.Icc (-‖f‖) ‖f‖, ℝ) := { toFun := fun x : Set.Icc (-‖f‖) ‖f‖ => |(x : ℝ)| }
change abs.comp f' ∈ A.topologicalClosure
apply comp_attachBound_mem_closure
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Homology.ImageToKernel
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.GradedObject
#align_import algebra.hom... | Mathlib/Algebra/Homology/Homology.lean | 97 | 100 | theorem boundaries_eq_bot [HasZeroObject V] {j} (h : ¬c.Rel (c.prev j) j) : C.boundaries j = ⊥ := by |
rw [eq_bot_iff]
refine imageSubobject_le _ 0 ?_
rw [C.dTo_eq_zero h, zero_comp]
|
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.Bas... | Mathlib/Algebra/Regular/Basic.lean | 91 | 94 | theorem IsRightRegular.left_of_commute {a : R}
(ca : ∀ b, Commute a b) (h : IsRightRegular a) : IsLeftRegular a := by |
simp_rw [@Commute.symm_iff R _ a] at ca
exact fun x y xy => h <| (ca x).trans <| xy.trans <| (ca y).symm
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | 92 | 113 | theorem integral_abs_condexp_le (f : α → ℝ) : ∫ x, |(μ[f|m]) x| ∂μ ≤ ∫ x, |f x| ∂μ := by |
by_cases hm : m ≤ m0
swap
· simp_rw [condexp_of_not_le hm, Pi.zero_apply, abs_zero, integral_zero]
positivity
by_cases hfint : Integrable f μ
swap
· simp only [condexp_undef hfint, Pi.zero_apply, abs_zero, integral_const, Algebra.id.smul_eq_mul,
mul_zero]
positivity
rw [integral_eq_lintegra... |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathli... | Mathlib/FieldTheory/Separable.lean | 52 | 54 | theorem not_separable_zero [Nontrivial R] : ¬Separable (0 : R[X]) := by |
rintro ⟨x, y, h⟩
simp only [derivative_zero, mul_zero, add_zero, zero_ne_one] at h
|
/-
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 | 92 | 93 | theorem incMatrix_of_not_mem_incidenceSet (h : e ∉ G.incidenceSet a) : G.incMatrix R a e = 0 := by |
rw [incMatrix_apply, Set.indicator_of_not_mem h]
|
/-
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.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.... | Mathlib/Data/Nat/GCD/Basic.lean | 89 | 90 | theorem gcd_self_add_right (m n : ℕ) : gcd m (m + n) = gcd m n := by |
rw [add_comm, gcd_add_self_right]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
/-!
# Finite int... | Mathlib/Order/Interval/Finset/Fin.lean | 84 | 85 | theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b := by |
simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]
|
/-
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.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducib... | Mathlib/Topology/Connected/Basic.lean | 116 | 120 | theorem isPreconnected_of_forall_pair {s : Set α}
(H : ∀ x ∈ s, ∀ y ∈ s, ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t) :
IsPreconnected s := by |
rcases eq_empty_or_nonempty s with (rfl | ⟨x, hx⟩)
exacts [isPreconnected_empty, isPreconnected_of_forall x fun y => H x hx y]
|
/-
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 | 52 | 55 | theorem Duplicate.mem_cons_self (h : x ∈+ x :: l) : x ∈ l := by |
cases' h with _ h _ _ h
· exact h
· exact h.mem
|
/-
Copyright (c) 2020 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.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.Topology.Sheaves.SheafCondition.PairwiseInt... | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | 86 | 94 | theorem w : res F U ≫ leftRes F U = res F U ≫ rightRes F U := by |
dsimp [res, leftRes, rightRes]
-- Porting note: `ext` can't see `limit.hom_ext` applies here:
-- See https://github.com/leanprover-community/mathlib4/issues/5229
refine limit.hom_ext (fun _ => ?_)
simp only [limit.lift_π, limit.lift_π_assoc, Fan.mk_π_app, Category.assoc]
rw [← F.map_comp]
rw [← F.map_com... |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.... | Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 63 | 68 | theorem Finsupp.toFreeAbelianGroup_comp_toFinsupp :
toFreeAbelianGroup.comp toFinsupp = AddMonoidHom.id (FreeAbelianGroup X) := by |
ext
rw [toFreeAbelianGroup, toFinsupp, AddMonoidHom.comp_apply, lift.of,
liftAddHom_apply_single, AddMonoidHom.flip_apply, smulAddHom_apply, one_smul,
AddMonoidHom.id_apply]
|
/-
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.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62... | Mathlib/Algebra/Group/Support.lean | 88 | 90 | theorem mulSupport_update_of_ne_one [DecidableEq α] (f : α → M) (x : α) {y : M} (hy : y ≠ 1) :
mulSupport (update f x y) = insert x (mulSupport f) := by |
ext a; rcases eq_or_ne a x with rfl | hne <;> simp [*]
|
/-
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.Pullbacks
import Mathlib.AlgebraicGeometry.AffineScheme
#align_import algebraic_geometry.limits from "leanprover-community/mathlib"@"70fd9... | Mathlib/AlgebraicGeometry/Limits.lean | 133 | 139 | theorem bot_isAffineOpen (X : Scheme) : IsAffineOpen (⊥ : Opens X.carrier) := by |
convert rangeIsAffineOpenOfOpenImmersion (initial.to X)
ext
-- Porting note: added this `erw` to turn LHS to `False`
erw [Set.mem_empty_iff_false]
rw [false_iff_iff]
exact fun x => isEmptyElim (show (⊥_ Scheme).carrier from x.choose)
|
/-
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.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d449... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 119 | 120 | theorem midpoint_vsub_right (p₁ p₂ : P) : midpoint R p₁ p₂ -ᵥ p₂ = (⅟ 2 : R) • (p₁ -ᵥ p₂) := by |
rw [midpoint_comm, midpoint_vsub_left]
|
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Sheaf
#align_import category_theory.sites.plus from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# The... | Mathlib/CategoryTheory/Sites/Plus.lean | 81 | 86 | theorem diagramNatTrans_zero [Preadditive D] (X : C) (P Q : Cᵒᵖ ⥤ D) :
J.diagramNatTrans (0 : P ⟶ Q) X = 0 := by |
ext : 2
refine Multiequalizer.hom_ext _ _ _ (fun i => ?_)
dsimp
rw [zero_comp, Multiequalizer.lift_ι, comp_zero]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Eric Wieser
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.RingTheory.TensorProduct.Basic
#align_import ring_theory.matrix_algebra from "leanprover-community/mathl... | Mathlib/RingTheory/MatrixAlgebra.lean | 89 | 89 | theorem invFun_zero : invFun R A n 0 = 0 := by | simp [invFun]
|
/-
Copyright (c) 2023 Sidharth Hariharan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Sidharth Hariharan
-/
import Mathlib.Algebra.Polynomial.Div
import Mathlib.Logic.Function.Basic
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Ta... | Mathlib/Algebra/Polynomial/PartialFractions.lean | 60 | 79 | theorem div_eq_quo_add_rem_div_add_rem_div (f : R[X]) {g₁ g₂ : R[X]} (hg₁ : g₁.Monic)
(hg₂ : g₂.Monic) (hcoprime : IsCoprime g₁ g₂) :
∃ q r₁ r₂ : R[X],
r₁.degree < g₁.degree ∧
r₂.degree < g₂.degree ∧ (f : K) / (↑g₁ * ↑g₂) = ↑q + ↑r₁ / ↑g₁ + ↑r₂ / ↑g₂ := by |
rcases hcoprime with ⟨c, d, hcd⟩
refine
⟨f * d /ₘ g₁ + f * c /ₘ g₂, f * d %ₘ g₁, f * c %ₘ g₂, degree_modByMonic_lt _ hg₁,
degree_modByMonic_lt _ hg₂, ?_⟩
have hg₁' : (↑g₁ : K) ≠ 0 := by
norm_cast
exact hg₁.ne_zero
have hg₂' : (↑g₂ : K) ≠ 0 := by
norm_cast
exact hg₂.ne_zero
have hfc ... |
/-
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.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib... | Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 130 | 134 | theorem contractLeft_ι_mul (a : M) (b : CliffordAlgebra Q) :
d⌋(ι Q a * b) = d a • b - ι Q a * (d⌋b) := by |
-- Porting note: Lean cannot figure out anymore the third argument
refine foldr'_ι_mul _ _ ?_ _ _ _
exact fun m x fx ↦ contractLeftAux_contractLeftAux Q d m x fx
|
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
/-!
# Row and column matrices
This file provides results about row and column matrices
## Main definitions
* `Matrix.row r... | Mathlib/Data/Matrix/RowCol.lean | 117 | 120 | theorem row_vecMul [Fintype m] [NonUnitalNonAssocSemiring α] (M : Matrix m n α) (v : m → α) :
Matrix.row (v ᵥ* M) = Matrix.row v * M := by |
ext
rfl
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Data.Int.LeastGreatest
#align_import data.int.conditionally_complete_order from "leanprove... | Mathlib/Data/Int/ConditionallyCompleteOrder.lean | 99 | 101 | theorem csInf_mem {s : Set ℤ} (h1 : s.Nonempty) (h2 : BddBelow s) : sInf s ∈ s := by |
convert (leastOfBdd _ (Classical.choose_spec h2) h1).2.1
exact dif_pos ⟨h1, h2⟩
|
/-
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.Topology.Homotopy.Path
import Mathlib.Topology.Homotopy.Equiv
#align_import topology.homotopy.contractible from "leanprover-community/mathlib"... | Mathlib/Topology/Homotopy/Contractible.lean | 113 | 117 | theorem hequiv [ContractibleSpace X] [ContractibleSpace Y] :
Nonempty (X ≃ₕ Y) := by |
rcases ContractibleSpace.hequiv_unit' (X := X) with ⟨h⟩
rcases ContractibleSpace.hequiv_unit' (X := Y) with ⟨h'⟩
exact ⟨h.trans h'.symm⟩
|
/-
Copyright (c) 2019 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.Fixed... | Mathlib/Topology/MetricSpace/Contracting.lean | 68 | 76 | theorem edist_inequality (hf : ContractingWith K f) {x y} (h : edist x y ≠ ∞) :
edist x y ≤ (edist x (f x) + edist y (f y)) / (1 - K) :=
suffices edist x y ≤ edist x (f x) + edist y (f y) + K * edist x y by
rwa [ENNReal.le_div_iff_mul_le (Or.inl hf.one_sub_K_ne_zero) (Or.inl one_sub_K_ne_top),
mul_comm,... | rw [edist_comm y, add_right_comm]
_ ≤ edist x (f x) + edist y (f y) + K * edist x y := add_le_add le_rfl (hf.2 _ _)
|
/-
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.Order.Filter.Lift
import Mathlib.Topology.Separation
import Mathlib.Order.Interval.Set.Monotone
#align_import topology.filter from "leanprover-commu... | Mathlib/Topology/Filter.lean | 55 | 56 | theorem isOpen_setOf_mem {s : Set α} : IsOpen { l : Filter α | s ∈ l } := by |
simpa only [Iic_principal] using isOpen_Iic_principal
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
#align_import topology.metric_space.hausdorff_dimension from "leanp... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 133 | 135 | theorem le_dimH_of_hausdorffMeasure_eq_top {s : Set X} {d : ℝ≥0} (h : μH[d] s = ∞) :
↑d ≤ dimH s := by |
rw [dimH_def]; exact le_iSup₂ (α := ℝ≥0∞) d h
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Quotient
import Mathlib.Combinatorics.Quiver.Path
#align_import category_theory.path_category fro... | Mathlib/CategoryTheory/PathCategory.lean | 103 | 119 | theorem lift_unique {C} [Category C] (φ : V ⥤q C) (Φ : Paths V ⥤ C)
(hΦ : of ⋙q Φ.toPrefunctor = φ) : Φ = lift φ := by |
subst_vars
fapply Functor.ext
· rintro X
rfl
· rintro X Y f
dsimp [lift]
induction' f with _ _ p f' ih
· simp only [Category.comp_id]
apply Functor.map_id
· simp only [Category.comp_id, Category.id_comp] at ih ⊢
-- Porting note: Had to do substitute `p.cons f'` and `f'.toPath` b... |
/-
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.Basis
import Mathlib.Algebra.FreeAlgebra
import Mathlib.LinearAlgebra.FinsuppVectorSpace
import Mathlib.LinearAlgebra.FreeModule.StrongRankCond... | Mathlib/LinearAlgebra/FreeAlgebra.lean | 44 | 47 | theorem rank_eq [CommRing R] [Nontrivial R] :
Module.rank R (FreeAlgebra R X) = Cardinal.lift.{u} (Cardinal.mk (List X)) := by |
rw [← (Basis.mk_eq_rank'.{_,_,_,u} (basisFreeMonoid R X)).trans (Cardinal.lift_id _),
Cardinal.lift_umax'.{v,u}, FreeMonoid]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5... | Mathlib/Topology/MetricSpace/PiNat.lean | 80 | 85 | theorem apply_eq_of_lt_firstDiff {x y : ∀ n, E n} {n : ℕ} (hn : n < firstDiff x y) : x n = y n := by |
rw [firstDiff_def] at hn
split_ifs at hn with h
· convert Nat.find_min (ne_iff.1 h) hn
simp
· exact (not_lt_zero' hn).elim
|
/-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Group... | Mathlib/GroupTheory/GroupAction/Blocks.lean | 85 | 87 | theorem IsBlock.def {B : Set X} :
IsBlock G B ↔ ∀ g g' : G, g • B = g' • B ∨ Disjoint (g • B) (g' • B) := by |
apply Set.pairwiseDisjoint_range_iff
|
/-
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 | 93 | 104 | theorem dist_affineCombination {ι : Type*} {s : Finset ι} {w₁ w₂ : ι → ℝ} (p : ι → P)
(h₁ : ∑ i ∈ s, w₁ i = 1) (h₂ : ∑ i ∈ s, w₂ i = 1) : by
have a₁ := s.affineCombination ℝ p w₁
have a₂ := s.affineCombination ℝ p w₂
exact dist a₁ a₂ * dist a₁ a₂ = (-∑ i₁ ∈ s, ∑ i₂ ∈ s,
(w₁ - w₂) i₁ * (w₁ ... |
dsimp only
rw [dist_eq_norm_vsub V (s.affineCombination ℝ p w₁) (s.affineCombination ℝ p w₂), ←
@inner_self_eq_norm_mul_norm ℝ, Finset.affineCombination_vsub]
have h : (∑ i ∈ s, (w₁ - w₂) i) = 0 := by
simp_rw [Pi.sub_apply, Finset.sum_sub_distrib, h₁, h₂, sub_self]
exact inner_weightedVSub p h p h
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Truncated
import Mathlib.RingTheory.WittVector.Identities
import Mathlib.NumberTheory.Padics.RingHoms
#align_im... | Mathlib/RingTheory/WittVector/Compare.lean | 107 | 112 | theorem commutes_symm' {m : ℕ} (hm : n ≤ m) (x : TruncatedWittVector p m (ZMod p)) :
(zmodEquivTrunc p n).symm (truncate hm x) =
ZMod.castHom (pow_dvd_pow p hm) _ ((zmodEquivTrunc p m).symm x) := by |
apply (zmodEquivTrunc p n).injective
rw [← commutes' _ _ hm]
simp
|
/-
Copyright (c) 2021 Roberto Alvarez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Roberto Alvarez
-/
import Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Logic.Equiv.TransferInstance
import Ma... | Mathlib/Topology/Homotopy/HomotopyGroup.lean | 74 | 78 | theorem insertAt_boundary (i : N) {t₀ : I} {t}
(H : (t₀ = 0 ∨ t₀ = 1) ∨ t ∈ boundary { j // j ≠ i }) : insertAt i ⟨t₀, t⟩ ∈ boundary N := by |
obtain H | ⟨j, H⟩ := H
· use i; rwa [funSplitAt_symm_apply, dif_pos rfl]
· use j; rwa [funSplitAt_symm_apply, dif_neg j.prop, Subtype.coe_eta]
|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | Mathlib/MeasureTheory/Measure/Portmanteau.lean | 105 | 123 | theorem le_measure_compl_liminf_of_limsup_measure_le {ι : Type*} {L : Filter ι} {μ : Measure Ω}
{μs : ι → Measure Ω} [IsProbabilityMeasure μ] [∀ i, IsProbabilityMeasure (μs i)] {E : Set Ω}
(E_mble : MeasurableSet E) (h : (L.limsup fun i => μs i E) ≤ μ E) :
μ Eᶜ ≤ L.liminf fun i => μs i Eᶜ := by |
rcases L.eq_or_neBot with rfl | hne
· simp only [liminf_bot, le_top]
have meas_Ec : μ Eᶜ = 1 - μ E := by
simpa only [measure_univ] using measure_compl E_mble (measure_lt_top μ E).ne
have meas_i_Ec : ∀ i, μs i Eᶜ = 1 - μs i E := by
intro i
simpa only [measure_univ] using measure_compl E_mble (measur... |
/-
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 | 76 | 82 | theorem maybeNormalize_eq_normalize {num : Int} {den g : Nat} (den_nz reduced)
(hn : ↑g ∣ num) (hd : g ∣ den) :
maybeNormalize num den g den_nz reduced = normalize num den (mt (by simp [·]) den_nz) := by |
simp only [maybeNormalize_eq, mk_eq_normalize, Int.div_eq_ediv_of_dvd hn]
have : g ≠ 0 := mt (by simp [·]) den_nz
rw [← normalize_mul_right _ this, Int.ediv_mul_cancel hn]
congr 1; exact Nat.div_mul_cancel hd
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.split from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf973... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 65 | 70 | theorem coe_splitLower : (splitLower I i x : Set (ι → ℝ)) = ↑I ∩ { y | y i ≤ x } := by |
rw [splitLower, coe_mk']
ext y
simp only [mem_univ_pi, mem_Ioc, mem_inter_iff, mem_coe, mem_setOf_eq, forall_and, ← Pi.le_def,
le_update_iff, le_min_iff, and_assoc, and_forall_ne (p := fun j => y j ≤ upper I j) i, mem_def]
rw [and_comm (a := y i ≤ x)]
|
/-
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.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Ideal
import Mathlib.... | Mathlib/RingTheory/Localization/Submodule.lean | 48 | 49 | theorem coeSubmodule_bot : coeSubmodule S (⊥ : Ideal R) = ⊥ := by |
rw [coeSubmodule, Submodule.map_bot]
|
/-
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.Algebra.Module.LinearMap.Basic
import Mathlib.RingTheory.HahnSeries.Basic
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a... | Mathlib/RingTheory/HahnSeries/Addition.lean | 166 | 168 | theorem sub_coeff' {x y : HahnSeries Γ R} : (x - y).coeff = x.coeff - y.coeff := by |
ext
simp [sub_eq_add_neg]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kevin Buzzard
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.P... | Mathlib/NumberTheory/Bernoulli.lean | 104 | 106 | theorem bernoulli'_zero : bernoulli' 0 = 1 := by |
rw [bernoulli'_def]
norm_num
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Tactic.AdaptationNote
#align_import geometry.eu... | Mathlib/Geometry/Euclidean/Inversion/Basic.lean | 69 | 69 | theorem inversion_self (c : P) (R : ℝ) : inversion c R c = c := by | simp [inversion]
|
/-
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.MeasurableSpace.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.function.ae_measurable_sequence from... | Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean | 81 | 86 | theorem fun_prop_of_mem_aeSeqSet (hf : ∀ i, AEMeasurable (f i) μ) {x : α} (hx : x ∈ aeSeqSet hf p) :
p x fun n => f n x := by |
have h_eq : (fun n => f n x) = fun n => aeSeq hf p n x :=
funext fun n => (aeSeq_eq_fun_of_mem_aeSeqSet hf hx n).symm
rw [h_eq]
exact prop_of_mem_aeSeqSet hf hx
|
/-
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, Johannes Hölzl
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.RelIso.Basic
#align_import order.ord_continuous from "leanprover... | Mathlib/Order/OrdContinuous.lean | 98 | 99 | theorem le_iff (hf : LeftOrdContinuous f) (h : Injective f) {x y} : f x ≤ f y ↔ x ≤ y := by |
simp only [← sup_eq_right, ← hf.map_sup, h.eq_iff]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
impo... | Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 48 | 54 | theorem Even.strictConvexOn_pow {n : ℕ} (hn : Even n) (h : n ≠ 0) :
StrictConvexOn ℝ Set.univ fun x : ℝ => x ^ n := by |
apply StrictMono.strictConvexOn_univ_of_deriv (continuous_pow n)
rw [deriv_pow']
replace h := Nat.pos_of_ne_zero h
exact StrictMono.const_mul (Odd.strictMono_pow <| Nat.Even.sub_odd h hn <| Nat.odd_iff.2 rfl)
(Nat.cast_pos.2 h)
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Scott Morrison, Ainsley Pahljina
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathli... | Mathlib/NumberTheory/LucasLehmer.lean | 93 | 95 | theorem succ_mersenne (k : ℕ) : mersenne k + 1 = 2 ^ k := by |
rw [mersenne, tsub_add_cancel_of_le]
exact one_le_pow_of_one_le (by norm_num) k
|
/-
Copyright (c) 2020 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Tree.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum.C... | Mathlib/Tactic/CancelDenoms/Core.lean | 45 | 47 | theorem div_subst {α} [Field α] {n1 n2 k e1 e2 t1 : α}
(h1 : n1 * e1 = t1) (h2 : n2 / e2 = 1) (h3 : n1 * n2 = k) : k * (e1 / e2) = t1 := by |
rw [← h3, mul_assoc, mul_div_left_comm, h2, ← mul_assoc, h1, mul_comm, one_mul]
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
/-!
# Real sign fu... | Mathlib/Data/Real/Sign.lean | 39 | 39 | theorem sign_of_pos {r : ℝ} (hr : 0 < r) : sign r = 1 := by | rw [sign, if_pos hr, if_neg hr.not_lt]
|
/-
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 | 99 | 100 | theorem T_eq (n : ℤ) : T R n = 2 * X * T R (n - 1) - T R (n - 2) := by |
linear_combination (norm := ring_nf) T_add_two R (n - 2)
|
/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
... | Mathlib/Data/Nat/Dist.lean | 103 | 104 | theorem dist_mul_left (k n m : ℕ) : dist (k * n) (k * m) = k * dist n m := by |
rw [mul_comm k n, mul_comm k m, dist_mul_right, mul_comm]
|
/-
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.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_e... | Mathlib/FieldTheory/PrimitiveElement.lean | 86 | 96 | theorem primitive_element_inf_aux_exists_c (f g : F[X]) :
∃ c : F, ∀ α' ∈ (f.map ϕ).roots, ∀ β' ∈ (g.map ϕ).roots, -(α' - α) / (β' - β) ≠ ϕ c := by |
let sf := (f.map ϕ).roots
let sg := (g.map ϕ).roots
let s := (sf.bind fun α' => sg.map fun β' => -(α' - α) / (β' - β)).toFinset
let s' := s.preimage ϕ fun x _ y _ h => ϕ.injective h
obtain ⟨c, hc⟩ := Infinite.exists_not_mem_finset s'
simp_rw [s', s, Finset.mem_preimage, Multiset.mem_toFinset, Multiset.mem_... |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Floris van Doorn, Mario Carneiro
-/
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
import Batteries.Tactic.Lint.Misc
instance {f :... | .lake/packages/batteries/Batteries/Logic.lean | 42 | 43 | theorem congrArg₂ (f : α → β → γ) {x x' : α} {y y' : β}
(hx : x = x') (hy : y = y') : f x y = f x' y' := by | subst hx hy; rfl
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Quotient
import Mathlib.Combinatorics.Quiver.Path
#align_import category_theory.path_category fro... | Mathlib/CategoryTheory/PathCategory.lean | 87 | 90 | theorem lift_toPath {C} [Category C] (φ : V ⥤q C) {X Y : V} (f : X ⟶ Y) :
(lift φ).map f.toPath = φ.map f := by |
dsimp [Quiver.Hom.toPath, lift]
simp
|
/-
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.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Adjugate
#align_import linear_algebra.matrix.non... | Mathlib/LinearAlgebra/Matrix/Nondegenerate.lean | 50 | 63 | theorem nondegenerate_of_det_ne_zero [DecidableEq m] {M : Matrix m m A} (hM : M.det ≠ 0) :
Nondegenerate M := by |
intro v hv
ext i
specialize hv (M.cramer (Pi.single i 1))
refine (mul_eq_zero.mp ?_).resolve_right hM
convert hv
simp only [mulVec_cramer M (Pi.single i 1), dotProduct, Pi.smul_apply, smul_eq_mul]
rw [Finset.sum_eq_single i, Pi.single_eq_same, mul_one]
· intro j _ hj
simp [hj]
· intros
have :... |
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Martin Zinkevich, Rémy Degenne
-/
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system fro... | Mathlib/MeasureTheory/PiSystem.lean | 79 | 82 | theorem IsPiSystem.singleton {α} (S : Set α) : IsPiSystem ({S} : Set (Set α)) := by |
intro s h_s t h_t _
rw [Set.mem_singleton_iff.1 h_s, Set.mem_singleton_iff.1 h_t, Set.inter_self,
Set.mem_singleton_iff]
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.MeasurableSpace.Defs
/-!
# σ-algebra of sets invariant under a self-map
In this file we define `MeasurableSpace.invariants (f : α → α)... | Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean | 58 | 60 | theorem measurable_invariants_dom {f : α → α} {g : α → β} :
Measurable[invariants f] g ↔ Measurable g ∧ ∀ s, MeasurableSet s → (g ∘ f) ⁻¹' s = g ⁻¹' s := by |
simp only [Measurable, ← forall_and]; rfl
|
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Eric Rodriguez
-/
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Localization.NormTrace
#align_import number_theory.number_field.norm from "... | Mathlib/NumberTheory/NumberField/Norm.lean | 104 | 106 | theorem norm_norm [IsSeparable K L] [Algebra F L] [IsSeparable F L] [FiniteDimensional F L]
[IsScalarTower K F L] (x : 𝓞 L) : norm K (norm F x) = norm K x := by |
rw [RingOfIntegers.ext_iff, coe_norm, coe_norm, coe_norm, Algebra.norm_norm]
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.multivariate.basic from "leanprover-... | Mathlib/Data/PFunctor/Multivariate/Basic.lean | 142 | 144 | theorem comp.get_map (f : α ⟹ β) (x : comp P Q α) :
comp.get (f <$$> x) = (fun i (x : Q i α) => f <$$> x) <$$> comp.get x := by |
rfl
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Heather Macbeth
-/
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Math... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 69 | 77 | theorem polynomial_comp_attachBound (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g : ℝ[X]) :
(g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (f : C(X, ℝ)).attachBound =
Polynomial.aeval f g := by |
ext
simp only [ContinuousMap.coe_comp, Function.comp_apply, ContinuousMap.attachBound_apply_coe,
Polynomial.toContinuousMapOn_apply, Polynomial.aeval_subalgebra_coe,
Polynomial.aeval_continuousMap_apply, Polynomial.toContinuousMap_apply]
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2... |
/-
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 | 71 | 73 | theorem dist_left_midpoint_eq_dist_right_midpoint (p1 p2 : P) :
dist p1 (midpoint ℝ p1 p2) = dist p2 (midpoint ℝ p1 p2) := by |
rw [dist_left_midpoint (𝕜 := ℝ) p1 p2, dist_right_midpoint (𝕜 := ℝ) p1 p2]
|
/-
Copyright (c) 2021 Martin Dvorak. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Dvorak, Kyle Miller, Eric Wieser
-/
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Math... | Mathlib/LinearAlgebra/CrossProduct.lean | 119 | 122 | theorem triple_product_permutation (u v w : Fin 3 → R) : u ⬝ᵥ v ×₃ w = v ⬝ᵥ w ×₃ u := by |
simp_rw [cross_apply, vec3_dotProduct]
norm_num
ring
|
/-
Copyright (c) 2019 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Paul Lezeau, Junyan Xu
-/
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
#align_import... | Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 50 | 55 | theorem isIntegrallyClosed_eq_field_fractions [IsDomain S] {s : S} (hs : IsIntegral R s) :
minpoly K (algebraMap S L s) = (minpoly R s).map (algebraMap R K) := by |
refine (eq_of_irreducible_of_monic ?_ ?_ ?_).symm
· exact ((monic hs).irreducible_iff_irreducible_map_fraction_map).1 (irreducible hs)
· rw [aeval_map_algebraMap, aeval_algebraMap_apply, aeval, map_zero]
· exact (monic hs).map _
|
/-
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.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "lea... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 47 | 51 | theorem snormEssSup_add_le {f g : α → E} :
snormEssSup (f + g) μ ≤ snormEssSup f μ + snormEssSup g μ := by |
refine le_trans (essSup_mono_ae (eventually_of_forall fun x => ?_)) (ENNReal.essSup_add_le _ _)
simp_rw [Pi.add_apply, ← ENNReal.coe_add, ENNReal.coe_le_coe]
exact nnnorm_add_le _ _
|
/-
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, Mario Carneiro
-/
/-!
# Definitions and properties of `coprime`
-/
namespace Nat
/-!
### `coprime`
See also `nat.coprime_of_dvd` and `nat.coprime_o... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 49 | 51 | theorem Coprime.gcd_mul_left_cancel_right (n : Nat)
(H : Coprime k m) : gcd m (k * n) = gcd m n := by |
rw [gcd_comm m n, gcd_comm m (k * n), H.gcd_mul_left_cancel n]
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.NatTrans
import Mathlib.CategoryTheory.Iso
#align_import category_theory.functor.categor... | Mathlib/CategoryTheory/Functor/Category.lean | 121 | 122 | theorem hcomp_id_app {H : D ⥤ E} (α : F ⟶ G) (X : C) : (α ◫ 𝟙 H).app X = H.map (α.app X) := by |
simp
|
/-
Copyright (c) 2015 Leonardo de Moura. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.List.Basic
import Mathlib.Data.Sigma.Basic
#align_import data.list.prod_sigma from "leanprover-community/mathlib"@"dd71334d... | Mathlib/Data/List/ProdSigma.lean | 45 | 48 | theorem mem_product {l₁ : List α} {l₂ : List β} {a : α} {b : β} :
(a, b) ∈ l₁ ×ˢ l₂ ↔ a ∈ l₁ ∧ b ∈ l₂ := by |
simp_all [SProd.sprod, product, mem_bind, mem_map, Prod.ext_iff, exists_prop, and_left_comm,
exists_and_left, exists_eq_left, exists_eq_right]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mat... | Mathlib/CategoryTheory/Monoidal/Mon_.lean | 75 | 76 | theorem one_mul_hom {Z : C} (f : Z ⟶ M.X) : (M.one ⊗ f) ≫ M.mul = (λ_ Z).hom ≫ f := by |
rw [tensorHom_def'_assoc, M.one_mul, leftUnitor_naturality]
|
/-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Bicategory.Basic
#align_import category_theory.bicategory.strict from "leanprover-community/mathlib"@"d... | Mathlib/CategoryTheory/Bicategory/Strict.lean | 78 | 81 | theorem whiskerLeft_eqToHom {a b c : B} (f : a ⟶ b) {g h : b ⟶ c} (η : g = h) :
f ◁ eqToHom η = eqToHom (congr_arg₂ (· ≫ ·) rfl η) := by |
cases η
simp only [whiskerLeft_id, eqToHom_refl]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Eric Wieser
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Pointwise
import Mathlib.Data.Real.Archimedean
#align_import data.real.p... | Mathlib/Data/Real/Pointwise.lean | 75 | 84 | theorem Real.sInf_smul_of_nonpos (ha : a ≤ 0) (s : Set ℝ) : sInf (a • s) = a • sSup s := by |
obtain rfl | hs := s.eq_empty_or_nonempty
· rw [smul_set_empty, Real.sInf_empty, Real.sSup_empty, smul_zero]
obtain rfl | ha' := ha.eq_or_lt
· rw [zero_smul_set hs, zero_smul]
exact csInf_singleton 0
by_cases h : BddAbove s
· exact ((OrderIso.smulRightDual ℝ ha').map_csSup' hs h).symm
· rw [Real.sInf... |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathli... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 130 | 133 | theorem card_edgeFinset_le_card_choose_two : G.edgeFinset.card ≤ (Fintype.card V).choose 2 := by |
classical
rw [← card_edgeFinset_top_eq_card_choose_two]
exact card_le_card (edgeFinset_mono le_top)
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ContMDiff.Basic
/-!
## Smoothness of charts and local structomorphisms
We show that the model with corners,... | Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean | 105 | 110 | theorem contMDiffOn_extend_symm (he : e ∈ maximalAtlas I M) :
ContMDiffOn 𝓘(𝕜, E) I n (e.extend I).symm (I '' e.target) := by |
refine (contMDiffOn_symm_of_mem_maximalAtlas he).comp
(contMDiffOn_model_symm.mono <| image_subset_range _ _) ?_
simp_rw [image_subset_iff, PartialEquiv.restr_coe_symm, I.toPartialEquiv_coe_symm,
preimage_preimage, I.left_inv, preimage_id']; rfl
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.AEEqFun.DomAct
import Mathlib.MeasureTheory.Function.LpSpace
/-!
# Action of `Mᵈᵐᵃ` on `Lᵖ` spaces
In this file we define ac... | Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean | 78 | 79 | theorem smul_Lp_neg (c : Mᵈᵐᵃ) (f : Lp E p μ) : c • (-f) = -(c • f) := by |
rcases f with ⟨⟨_⟩, _⟩; rfl
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Topology.PartialHomeomorph... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean | 101 | 105 | theorem approximatesLinearOn_iff_lipschitzOnWith {f : E → F} {f' : E →L[𝕜] F} {s : Set E}
{c : ℝ≥0} : ApproximatesLinearOn f f' s c ↔ LipschitzOnWith c (f - ⇑f') s := by |
have : ∀ x y, f x - f y - f' (x - y) = (f - f') x - (f - f') y := fun x y ↦ by
simp only [map_sub, Pi.sub_apply]; abel
simp only [this, lipschitzOnWith_iff_norm_sub_le, ApproximatesLinearOn]
|
/-
Copyright (c) 2023 Mantas Bakšys, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mantas Bakšys, Yaël Dillies
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Rearrangement
i... | Mathlib/Algebra/Order/Chebyshev.lean | 57 | 64 | theorem MonovaryOn.sum_smul_sum_le_card_smul_sum (hfg : MonovaryOn f g s) :
((∑ i ∈ s, f i) • ∑ i ∈ s, g i) ≤ s.card • ∑ i ∈ s, f i • g i := by |
classical
obtain ⟨σ, hσ, hs⟩ := s.countable_toSet.exists_cycleOn
rw [← card_range s.card, sum_smul_sum_eq_sum_perm hσ]
exact
sum_le_card_nsmul _ _ _ fun n _ =>
hfg.sum_smul_comp_perm_le_sum_smul fun x hx => hs fun h => hx <| IsFixedPt.perm_pow h _
|
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.LinearAlgebra.Alternating.Basic
#align_import linear_algebra.exterior_algeb... | Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean | 97 | 98 | theorem comp_ι_sq_zero (g : ExteriorAlgebra R M →ₐ[R] A) (m : M) : g (ι R m) * g (ι R m) = 0 := by |
rw [← AlgHom.map_mul, ι_sq_zero, AlgHom.map_zero]
|
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Hom.Defs
#align_import algebra.group.ext from "leanprover-community/mathlib"@"e574b1a4e891376b0ef974b926da39e05da... | Mathlib/Algebra/Group/Ext.lean | 56 | 59 | theorem CommMonoid.toMonoid_injective {M : Type u} :
Function.Injective (@CommMonoid.toMonoid M) := by |
rintro ⟨⟩ ⟨⟩ h
congr
|
/-
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.Finset.Lattice
import Mathlib.Data.Multiset.Powerset
#align_import data.finset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4... | Mathlib/Data/Finset/Powerset.lean | 99 | 113 | theorem powerset_insert [DecidableEq α] (s : Finset α) (a : α) :
powerset (insert a s) = s.powerset ∪ s.powerset.image (insert a) := by |
ext t
simp only [exists_prop, mem_powerset, mem_image, mem_union, subset_insert_iff]
by_cases h : a ∈ t
· constructor
· exact fun H => Or.inr ⟨_, H, insert_erase h⟩
· intro H
cases' H with H H
· exact Subset.trans (erase_subset a t) H
· rcases H with ⟨u, hu⟩
rw [← hu.2]
... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum
#align_import number_theory.legendre_sy... | Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean | 121 | 133 | theorem quadratic_reciprocity (hp : p ≠ 2) (hq : q ≠ 2) (hpq : p ≠ q) :
legendreSym q p * legendreSym p q = (-1) ^ (p / 2 * (q / 2)) := by |
have hp₁ := (Prime.eq_two_or_odd <| @Fact.out p.Prime _).resolve_left hp
have hq₁ := (Prime.eq_two_or_odd <| @Fact.out q.Prime _).resolve_left hq
have hq₂ : ringChar (ZMod q) ≠ 2 := (ringChar_zmod_n q).substr hq
have h :=
quadraticChar_odd_prime ((ringChar_zmod_n p).substr hp) hq ((ringChar_zmod_n p).subst... |
/-
Copyright (c) 2023 Luke Mantle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Mantle
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.h... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 92 | 99 | theorem coeff_hermite_of_lt {n k : ℕ} (hnk : n < k) : coeff (hermite n) k = 0 := by |
obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_lt hnk
clear hnk
induction' n with n ih generalizing k
· apply coeff_C
· have : n + k + 1 + 2 = n + (k + 2) + 1 := by ring
rw [coeff_hermite_succ_succ, add_right_comm, this, ih k, ih (k + 2),
mul_zero, sub_zero]
|
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Analysis.SpecialFunctions.Complex.LogBounds
/-!
# Complex arctangent
This file defines the complex arctangent `Complex.arctan` as
$$\arctan z = -\frac i2 \lo... | Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean | 26 | 46 | theorem tan_arctan {z : ℂ} (h₁ : z ≠ I) (h₂ : z ≠ -I) : tan (arctan z) = z := by |
unfold tan sin cos
rw [div_div_eq_mul_div, div_mul_cancel₀ _ two_ne_zero, ← div_mul_eq_mul_div,
-- multiply top and bottom by `exp (arctan z * I)`
← mul_div_mul_right _ _ (exp_ne_zero (arctan z * I)), sub_mul, add_mul,
← exp_add, neg_mul, add_left_neg, exp_zero, ← exp_add, ← two_mul]
have z₁ : 1 + z ... |
/-
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.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Set.Pointwise.Basic
#align_import data.set.pointwise.big_operators from "leanprover-community/mathlib"... | Mathlib/Data/Set/Pointwise/BigOperators.lean | 58 | 79 | theorem mem_finset_prod (t : Finset ι) (f : ι → Set α) (a : α) :
(a ∈ ∏ i ∈ t, f i) ↔ ∃ (g : ι → α) (_ : ∀ {i}, i ∈ t → g i ∈ f i), ∏ i ∈ t, g i = a := by |
classical
induction' t using Finset.induction_on with i is hi ih generalizing a
· simp_rw [Finset.prod_empty, Set.mem_one]
exact ⟨fun h ↦ ⟨fun _ ↦ a, fun hi ↦ False.elim (Finset.not_mem_empty _ hi), h.symm⟩,
fun ⟨_, _, hf⟩ ↦ hf.symm⟩
rw [Finset.prod_insert hi, Set.mem_mul]
simp_rw [Fins... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.RingTheory.MatrixAlgebra
#align_impor... | Mathlib/RingTheory/PolynomialAlgebra.lean | 80 | 82 | theorem toFunLinear_mul_tmul_mul_aux_1 (p : R[X]) (k : ℕ) (h : Decidable ¬p.coeff k = 0) (a : A) :
ite (¬coeff p k = 0) (a * (algebraMap R A) (coeff p k)) 0 =
a * (algebraMap R A) (coeff p k) := by | classical split_ifs <;> simp [*]
|
/-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Bicategory.Basic
#align_import category_theory.bicategory.strict from "leanprover-community/mathlib"@"d... | Mathlib/CategoryTheory/Bicategory/Strict.lean | 85 | 88 | theorem eqToHom_whiskerRight {a b c : B} {f g : a ⟶ b} (η : f = g) (h : b ⟶ c) :
eqToHom η ▷ h = eqToHom (congr_arg₂ (· ≫ ·) η rfl) := by |
cases η
simp only [id_whiskerRight, eqToHom_refl]
|
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
/-!
# Row and column matrices
This file provides results about row and column matrices
## Main definitions
* `Matrix.row r... | Mathlib/Data/Matrix/RowCol.lean | 67 | 69 | theorem col_smul [SMul R α] (x : R) (v : m → α) : col (x • v) = x • col v := by |
ext
rfl
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
Some proofs and docs came from `algebra/commute` (c) Neil Strickland
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Init.Logic
import Mathlib.Tactic.Cases
#alig... | Mathlib/Algebra/Group/Semiconj/Defs.lean | 97 | 97 | theorem one_right (a : M) : SemiconjBy a 1 1 := by | rw [SemiconjBy, mul_one, one_mul]
|
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