Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 |
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import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.dslope from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open scoped Classical Topology Filter
open Function Set Filter
variable {𝕜 E : Type*} [NontriviallyNormed... | Mathlib/Analysis/Calculus/Dslope.lean | 46 | 52 | theorem ContinuousLinearMap.dslope_comp {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
(f : E →L[𝕜] F) (g : 𝕜 → E) (a b : 𝕜) (H : a = b → DifferentiableAt 𝕜 g a) :
dslope (f ∘ g) a b = f (dslope g a b) := by |
rcases eq_or_ne b a with (rfl | hne)
· simp only [dslope_same]
exact (f.hasFDerivAt.comp_hasDerivAt b (H rfl).hasDerivAt).deriv
· simpa only [dslope_of_ne _ hne] using f.toLinearMap.slope_comp g a b
| 4 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.normed_space.add_torsor from "leanprover-community/mathlib"@... | Mathlib/Analysis/NormedSpace/AddTorsor.lean | 36 | 41 | theorem AffineSubspace.isClosed_direction_iff (s : AffineSubspace 𝕜 Q) :
IsClosed (s.direction : Set W) ↔ IsClosed (s : Set Q) := by |
rcases s.eq_bot_or_nonempty with (rfl | ⟨x, hx⟩); · simp [isClosed_singleton]
rw [← (IsometryEquiv.vaddConst x).toHomeomorph.symm.isClosed_image,
AffineSubspace.coe_direction_eq_vsub_set_right hx]
rfl
| 4 |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Lang... | Mathlib/ModelTheory/Definability.lean | 125 | 130 | theorem definable_finset_inf {ι : Type*} {f : ι → Set (α → M)} (hf : ∀ i, A.Definable L (f i))
(s : Finset ι) : A.Definable L (s.inf f) := by |
classical
refine Finset.induction definable_univ (fun i s _ h => ?_) s
rw [Finset.inf_insert]
exact (hf i).inter h
| 4 |
import Mathlib.Data.Finset.Sum
import Mathlib.Data.Sum.Order
import Mathlib.Order.Interval.Finset.Defs
#align_import data.sum.interval from "leanprover-community/mathlib"@"48a058d7e39a80ed56858505719a0b2197900999"
open Function Sum
namespace Finset
variable {α₁ α₂ β₁ β₂ γ₁ γ₂ : Type*}
section SumLift₂
variabl... | Mathlib/Data/Sum/Interval.lean | 68 | 73 | theorem inr_mem_sumLift₂ {c₂ : γ₂} :
inr c₂ ∈ sumLift₂ f g a b ↔ ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ c₂ ∈ g a₂ b₂ := by |
rw [mem_sumLift₂, or_iff_right]
· simp only [inr.injEq, exists_and_left, exists_eq_left']
rintro ⟨_, _, c₂, _, _, h, _⟩
exact inr_ne_inl h
| 4 |
import Mathlib.Probability.Kernel.Composition
#align_import probability.kernel.invariance from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7cc45d0b"
open MeasureTheory
open scoped MeasureTheory ENNReal ProbabilityTheory
namespace ProbabilityTheory
variable {α β γ : Type*} {mα : MeasurableSp... | Mathlib/Probability/Kernel/Invariance.lean | 87 | 92 | theorem Invariant.comp [IsSFiniteKernel κ] (hκ : Invariant κ μ) (hη : Invariant η μ) :
Invariant (κ ∘ₖ η) μ := by |
cases' isEmpty_or_nonempty α with _ hα
· exact Subsingleton.elim _ _
· simp_rw [Invariant, ← comp_const_apply_eq_bind (κ ∘ₖ η) μ hα.some, comp_assoc, hη.comp_const,
hκ.comp_const, const_apply]
| 4 |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
namespace WithBot
instance : WellFoundedRelation (WithBot ℕ) where
rel := (· < ·)
wf := IsWellFounde... | Mathlib/Data/Nat/WithBot.lean | 81 | 85 | theorem add_one_le_of_lt {n m : WithBot ℕ} (h : n < m) : n + 1 ≤ m := by |
cases n
· exact bot_le
cases m
exacts [(not_lt_bot h).elim, WithBot.coe_le_coe.2 (WithBot.coe_lt_coe.1 h)]
| 4 |
import Batteries.Data.Sum.Basic
import Batteries.Logic
open Function
namespace Sum
@[simp] protected theorem «forall» {p : α ⊕ β → Prop} :
(∀ x, p x) ↔ (∀ a, p (inl a)) ∧ ∀ b, p (inr b) :=
⟨fun h => ⟨fun _ => h _, fun _ => h _⟩, fun ⟨h₁, h₂⟩ => Sum.rec h₁ h₂⟩
@[simp] protected theorem «exists» {p : α ⊕ β ... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 33 | 38 | theorem forall_sum {γ : α ⊕ β → Sort _} (p : (∀ ab, γ ab) → Prop) :
(∀ fab, p fab) ↔ (∀ fa fb, p (Sum.rec fa fb)) := by |
refine ⟨fun h fa fb => h _, fun h fab => ?_⟩
have h1 : fab = Sum.rec (fun a => fab (Sum.inl a)) (fun b => fab (Sum.inr b)) := by
ext ab; cases ab <;> rfl
rw [h1]; exact h _ _
| 4 |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_space.symmetric from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open RCLike
open ComplexConjugate
variable ... | Mathlib/Analysis/InnerProductSpace/Symmetric.lean | 115 | 120 | theorem IsSymmetric.coe_reApplyInnerSelf_apply {T : E →L[𝕜] E} (hT : IsSymmetric (T : E →ₗ[𝕜] E))
(x : E) : (T.reApplyInnerSelf x : 𝕜) = ⟪T x, x⟫ := by |
rsuffices ⟨r, hr⟩ : ∃ r : ℝ, ⟪T x, x⟫ = r
· simp [hr, T.reApplyInnerSelf_apply]
rw [← conj_eq_iff_real]
exact hT.conj_inner_sym x x
| 4 |
import Mathlib.Data.Vector.Basic
set_option autoImplicit true
namespace Vector
def snoc : Vector α n → α → Vector α (n+1) :=
fun xs x => append xs (x ::ᵥ Vector.nil)
section Simp
variable (xs : Vector α n)
@[simp]
theorem snoc_cons : (x ::ᵥ xs).snoc y = x ::ᵥ (xs.snoc y) :=
rfl
@[simp]
theorem snoc_nil... | Mathlib/Data/Vector/Snoc.lean | 48 | 52 | theorem reverse_snoc : reverse (xs.snoc x) = x ::ᵥ (reverse xs) := by |
cases xs
simp only [reverse, snoc, cons, toList_mk]
congr
simp [toList, Vector.append, Append.append]
| 4 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
#align_import analysis.special_functions.trigonometric.arctan from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Real
open Set Filter
open scoped Topology Real
theorem tan_add {x y : ℝ}
... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 56 | 60 | theorem continuousOn_tan : ContinuousOn tan {x | cos x ≠ 0} := by |
suffices ContinuousOn (fun x => sin x / cos x) {x | cos x ≠ 0} by
have h_eq : (fun x => sin x / cos x) = tan := by ext1 x; rw [tan_eq_sin_div_cos]
rwa [h_eq] at this
exact continuousOn_sin.div continuousOn_cos fun x => id
| 4 |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 121 | 126 | theorem exists_finite_submodule_left_of_finite (s : Set (M ⊗[R] N)) (hs : s.Finite) :
∃ M' : Submodule R M, Module.Finite R M' ∧ s ⊆ LinearMap.range (M'.subtype.rTensor N) := by |
obtain ⟨M', _, hfin, _, h⟩ := exists_finite_submodule_of_finite s hs
refine ⟨M', hfin, ?_⟩
rw [mapIncl, ← LinearMap.rTensor_comp_lTensor] at h
exact h.trans (LinearMap.range_comp_le_range _ _)
| 4 |
import Mathlib.RingTheory.RingHomProperties
import Mathlib.RingTheory.IntegralClosure
#align_import ring_theory.ring_hom.integral from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
namespace RingHom
open scoped TensorProduct
open TensorProduct Algebra.TensorProduct
theorem isIntegra... | Mathlib/RingTheory/RingHom/Integral.lean | 28 | 32 | theorem isIntegral_respectsIso : RespectsIso fun f => f.IsIntegral := by |
apply isIntegral_stableUnderComposition.respectsIso
introv x
rw [← e.apply_symm_apply x]
apply RingHom.isIntegralElem_map
| 4 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 274 | 279 | theorem normAtPlace_smul (w : InfinitePlace K) (x : E K) (c : ℝ) :
normAtPlace w (c • x) = |c| * normAtPlace w x := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]
split_ifs
· rw [Prod.smul_fst, Pi.smul_apply, norm_smul, Real.norm_eq_abs]
· rw [Prod.smul_snd, Pi.smul_apply, norm_smul, Real.norm_eq_abs, Complex.norm_eq_abs]
| 4 |
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algebra.Ring.Int
#align_import algebra.order.field.power from "leanprover-community/mathlib"@"acb3d204d4ee883eb686f45d486a2a6811a01329"
... | Mathlib/Algebra/Order/Field/Power.lean | 155 | 159 | theorem Odd.zpow_neg_iff (hn : Odd n) : a ^ n < 0 ↔ a < 0 := by |
refine ⟨lt_imp_lt_of_le_imp_le (zpow_nonneg · _), fun ha ↦ ?_⟩
obtain ⟨k, rfl⟩ := hn
rw [zpow_add_one₀ ha.ne]
exact mul_neg_of_pos_of_neg (Even.zpow_pos (even_two_mul _) ha.ne) ha
| 4 |
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Finset
namespace Nat
variable (p : ℕ → Prop)
section Count
variable [DecidablePred p]
def count (n : ℕ) : ℕ :=
(List.range n).... | Mathlib/Data/Nat/Count.lean | 133 | 137 | theorem count_injective {m n : ℕ} (hm : p m) (hn : p n) (heq : count p m = count p n) : m = n := by |
by_contra! h : m ≠ n
wlog hmn : m < n
· exact this hn hm heq.symm h.symm (h.lt_or_lt.resolve_left hmn)
· simpa [heq] using count_strict_mono hm hmn
| 4 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.monoid from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {M : Type*} ... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 58 | 62 | theorem Ioc_add_bij : BijOn (· + d) (Ioc a b) (Ioc (a + d) (b + d)) := by |
rw [← Ioi_inter_Iic, ← Ioi_inter_Iic]
exact
(Ioi_add_bij a d).inter_mapsTo (fun x hx => add_le_add_right hx _) fun x hx =>
le_of_add_le_add_right hx.2
| 4 |
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 "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Set
variable {F α β γ : Type*}
variable [Monoid α] {s t : Set α}... | 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]
| 4 |
import Mathlib.Data.ZMod.Quotient
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.Peel
#align_import group_... | Mathlib/GroupTheory/Exponent.lean | 151 | 155 | theorem pow_exponent_eq_one (g : G) : g ^ exponent G = 1 := by |
by_cases h : ExponentExists G
· simp_rw [exponent, dif_pos h]
exact (Nat.find_spec h).2 g
· simp_rw [exponent, dif_neg h, pow_zero]
| 4 |
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.Weierstrass
#align_import topology.continuous_function.stone_weierstrass fro... | 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
| 4 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e600... | 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]
| 4 |
import Mathlib.CategoryTheory.Sites.Sheaf
#align_import category_theory.sites.plus from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory.GrothendieckTopology
open CategoryTheory
open CategoryTheory.Limits
open Opposite
universe w v u
variable {C : Type u} [Ca... | 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]
| 4 |
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.function.ae_measurable_sequence from "leanprover-community/mathlib"@"d003c55042c3cd08aefd1ae9a42ef89441cdaaf3"
open MeasureTheory
open scoped Classical
variable {ι : Sort*} {α β γ... | 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
| 4 |
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
#align_import field_theory.minpoly.is_integrally_closed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Classical Polynomial
open Polynomial Set... | 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 _
| 4 |
import Mathlib.Geometry.Manifold.ContMDiff.Basic
open Set ChartedSpace SmoothManifoldWithCorners
open scoped Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace... | 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
| 4 |
import Mathlib.Data.List.Count
import Mathlib.Data.List.Dedup
import Mathlib.Data.List.InsertNth
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Permutation
import Mathlib.Data.Nat.Factorial.Basic
#align_import data.list.perm from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
... | Mathlib/Data/List/Perm.lean | 142 | 146 | theorem perm_comp_perm : (Perm ∘r Perm : List α → List α → Prop) = Perm := by |
funext a c; apply propext
constructor
· exact fun ⟨b, hab, hba⟩ => Perm.trans hab hba
· exact fun h => ⟨a, Perm.refl a, h⟩
| 4 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.Module.Defs
import Mathlib.Tactic.Abel
namespace Finset
variable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] (f : ℕ → R) (g : ℕ → M) {m n : ℕ}
-- The partial sum of `g`, starting from zero
local notation "G " n:80 => ∑ i ∈ range n, g i
... | Mathlib/Algebra/BigOperators/Module.lean | 63 | 69 | theorem sum_range_by_parts :
∑ i ∈ range n, f i • g i =
f (n - 1) • G n - ∑ i ∈ range (n - 1), (f (i + 1) - f i) • G (i + 1) := by |
by_cases hn : n = 0
· simp [hn]
· rw [range_eq_Ico, sum_Ico_by_parts f g (Nat.pos_of_ne_zero hn), sum_range_zero, smul_zero,
sub_zero, range_eq_Ico]
| 4 |
import Mathlib.LinearAlgebra.Alternating.Basic
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import linear_algebra.alternating from "leanprover-community/mathlib"@"0c1d80f5a86b36c1db32e021e8d19ae7809d5b79"
suppress_compilation
open TensorProduct
vari... | Mathlib/LinearAlgebra/Alternating/DomCoprod.lean | 212 | 222 | theorem MultilinearMap.domCoprod_alternization_coe [DecidableEq ιa] [DecidableEq ιb]
(a : MultilinearMap R' (fun _ : ιa => Mᵢ) N₁) (b : MultilinearMap R' (fun _ : ιb => Mᵢ) N₂) :
MultilinearMap.domCoprod (MultilinearMap.alternatization a)
(MultilinearMap.alternatization b) =
∑ σa : Perm ιa, ∑ σb : P... |
simp_rw [← MultilinearMap.domCoprod'_apply, MultilinearMap.alternatization_coe]
simp_rw [TensorProduct.sum_tmul, TensorProduct.tmul_sum, _root_.map_sum,
← TensorProduct.smul_tmul', TensorProduct.tmul_smul]
rfl
| 4 |
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.ordinal.principal from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
universe u v w
noncomputable section
open Order
namespace Ordinal
-- Porting note: commented out, doesn't seem necessary
--local infixr:0 "^" => ... | Mathlib/SetTheory/Ordinal/Principal.lean | 62 | 66 | theorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0 := by |
refine ⟨fun h => ?_, fun h a b ha hb => ?_⟩
· rw [← lt_one_iff_zero]
exact h zero_lt_one zero_lt_one
· rwa [lt_one_iff_zero, ha, hb] at *
| 4 |
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ι : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 426 | 431 | theorem irreducible_mul_leadingCoeff_inv {p : K[X]} :
Irreducible (p * C (leadingCoeff p)⁻¹) ↔ Irreducible p := by |
by_cases hp0 : p = 0
· simp [hp0]
exact irreducible_mul_isUnit
(isUnit_C.mpr (IsUnit.mk0 _ (inv_ne_zero (leadingCoeff_ne_zero.mpr hp0))))
| 4 |
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.RingTheory.Prime
import Mathlib.RingTheory.Polynomial.Content
import Mathlib.RingTheory.Ideal.Quotient
#align_import ring_theory.eisenstein_criterion from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
open Polynomial Ideal.Quotient
v... | Mathlib/RingTheory/EisensteinCriterion.lean | 52 | 61 | theorem le_natDegree_of_map_eq_mul_X_pow {n : ℕ} {P : Ideal R} (hP : P.IsPrime) {q : R[X]}
{c : Polynomial (R ⧸ P)} (hq : map (mk P) q = c * X ^ n) (hc0 : c.degree = 0) :
n ≤ q.natDegree :=
Nat.cast_le.1
(calc
↑n = degree (q.map (mk P)) := by |
rw [hq, degree_mul, hc0, zero_add, degree_pow, degree_X, nsmul_one]
_ ≤ degree q := degree_map_le _ _
_ ≤ natDegree q := degree_le_natDegree
)
| 4 |
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.PEmpty
#align_import category_theory.monoidal.of_chosen_finite_products.basic from "leanprover-community/mathlib"@"95a87616d63b3cb49d3fe678d416fbe9c4217bf4"
universe v u
names... | Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean | 242 | 246 | theorem tensor_id (X₁ X₂ : C) : tensorHom ℬ (𝟙 X₁) (𝟙 X₂) = 𝟙 (tensorObj ℬ X₁ X₂) := by |
apply IsLimit.hom_ext (ℬ _ _).isLimit;
rintro ⟨⟨⟩⟩ <;>
· dsimp [tensorHom]
simp
| 4 |
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import ring_theory.graded_algebra.homogeneous_ideal from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
open SetLike Direc... | Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean | 102 | 107 | theorem HomogeneousIdeal.ext' {I J : HomogeneousIdeal 𝒜} (h : ∀ i, ∀ x ∈ 𝒜 i, x ∈ I ↔ x ∈ J) :
I = J := by |
ext
rw [I.isHomogeneous.mem_iff, J.isHomogeneous.mem_iff]
apply forall_congr'
exact fun i ↦ h i _ (decompose 𝒜 _ i).2
| 4 |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 137 | 142 | theorem finsuppRight_apply (t : M ⊗[R] (ι →₀ N)) (i : ι) :
finsuppRight R M N ι t i = lTensor M (Finsupp.lapply i) t := by |
induction t using TensorProduct.induction_on with
| zero => simp
| tmul m f => simp [finsuppRight_apply_tmul_apply]
| add x y hx hy => simp [map_add, hx, hy]
| 4 |
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin
import Mathlib.Order.Partition.Finpartition
#align_import order.partition.equipartition from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
open Finset Fintype
namespace Finpartition
variable {α : Type*} [DecidableEq α] ... | Mathlib/Order/Partition/Equipartition.lean | 61 | 66 | theorem IsEquipartition.card_part_eq_average_iff (hP : P.IsEquipartition) (ht : t ∈ P.parts) :
t.card = s.card / P.parts.card ↔ t.card ≠ s.card / P.parts.card + 1 := by |
have a := hP.card_parts_eq_average ht
have b : ¬(t.card = s.card / P.parts.card ∧ t.card = s.card / P.parts.card + 1) := by
by_contra h; exact absurd (h.1 ▸ h.2) (lt_add_one _).ne
tauto
| 4 |
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.PEquiv
#align_import data.matrix.pequiv from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
namespace PEquiv
open Matrix
universe u v
variable {k l m n : Type*}
variable {α : Type v}
open Matrix
def toMatrix [DecidableEq n] [Zer... | Mathlib/Data/Matrix/PEquiv.lean | 62 | 67 | theorem mul_matrix_apply [Fintype m] [DecidableEq m] [Semiring α] (f : l ≃. m) (M : Matrix m n α)
(i j) : (f.toMatrix * M :) i j = Option.casesOn (f i) 0 fun fi => M fi j := by |
dsimp [toMatrix, Matrix.mul_apply]
cases' h : f i with fi
· simp [h]
· rw [Finset.sum_eq_single fi] <;> simp (config := { contextual := true }) [h, eq_comm]
| 4 |
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace α] [MeasurableSpace β] {s : Set α}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure α :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 115 | 119 | theorem empty_of_count_eq_zero' (s_mble : MeasurableSet s) (hsc : count s = 0) : s = ∅ := by |
have hs : s.Finite := by
rw [← count_apply_lt_top' s_mble, hsc]
exact WithTop.zero_lt_top
simpa [count_apply_finite' hs s_mble] using hsc
| 4 |
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.ordinal.principal from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
universe u v w
noncomputable section
open Order
namespace Ordinal
-- Porting note: commented out, doesn't seem necessary
--local infixr:0 "^" => ... | Mathlib/SetTheory/Ordinal/Principal.lean | 69 | 74 | theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)
(ho : Principal op o) (n : ℕ) : (op a)^[n] a < o := by |
induction' n with n hn
· rwa [Function.iterate_zero]
· rw [Function.iterate_succ']
exact ho hao hn
| 4 |
import Mathlib.FieldTheory.SeparableDegree
import Mathlib.FieldTheory.IsSepClosed
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField Field
noncomputable section
universe u v w
variable (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E]
variable (K : Type w) [Field K] [... | Mathlib/FieldTheory/SeparableClosure.lean | 115 | 121 | theorem separableClosure.map_eq_of_separableClosure_eq_bot [Algebra E K] [IsScalarTower F E K]
(h : separableClosure E K = ⊥) :
(separableClosure F E).map (IsScalarTower.toAlgHom F E K) = separableClosure F K := by |
refine le_antisymm (map_le_of_algHom _) (fun x hx ↦ ?_)
obtain ⟨y, rfl⟩ := mem_bot.1 <| h ▸ mem_separableClosure_iff.2
(mem_separableClosure_iff.1 hx |>.map_minpoly E)
exact ⟨y, (map_mem_separableClosure_iff <| IsScalarTower.toAlgHom F E K).mp hx, rfl⟩
| 4 |
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Category.ModuleCat.Abelian
import Mathlib.Algebra.Category.ModuleCat.Subobject
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import algebra.homology.Module from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225... | Mathlib/Algebra/Homology/ModuleCat.lean | 72 | 79 | theorem cycles'Map_toCycles' (f : C ⟶ D) {i : ι} (x : LinearMap.ker (C.dFrom i)) :
(cycles'Map f i) (toCycles' x) = toCycles' ⟨f.f i x.1, by
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
rw [LinearMap.mem_ker]; erw [Hom.comm_from_apply, x.2, map_zero]⟩ := by |
ext
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [cycles'Map_arrow_apply, toKernelSubobject_arrow, toKernelSubobject_arrow]
rfl
| 4 |
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.ProdSigma
#align_import model_theory.semantics from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
universe u v w u' v'
namespace FirstOrder
namespace Language
variable {L : Language.{u, v}} {... | Mathlib/ModelTheory/Semantics.lean | 117 | 122 | theorem realize_functions_apply₂ {f : L.Functions 2} {t₁ t₂ : L.Term α} {v : α → M} :
(f.apply₂ t₁ t₂).realize v = funMap f ![t₁.realize v, t₂.realize v] := by |
rw [Functions.apply₂, Term.realize]
refine congr rfl (funext (Fin.cases ?_ ?_))
· simp only [Matrix.cons_val_zero]
· simp only [Matrix.cons_val_succ, Matrix.cons_val_fin_one, forall_const]
| 4 |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.Topology.LocalAtTarget
#align_import algebraic_geometry.morphisms.universally_closed from "leanprover-community/mathlib"@"a8ae1b3f7979249a0af6bc7cf20c1f6bf656ca73"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite TopologicalS... | Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean | 72 | 76 | theorem topologically_isClosedMap_respectsIso : RespectsIso (topologically @IsClosedMap) := by |
apply MorphismProperty.respectsIso_of_isStableUnderComposition
intro _ _ f hf
have : IsIso f := hf
exact (TopCat.homeoOfIso (Scheme.forgetToTop.mapIso (asIso f))).isClosedMap
| 4 |
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.StrongTopology
#align_import analysis.normed_space.compact_operator from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Function Set Filter Bornology Metric Pointwise Topology
def IsCompactOperat... | Mathlib/Analysis/NormedSpace/CompactOperator.lean | 252 | 257 | theorem IsCompactOperator.comp_clm [AddCommMonoid M₂] [Module R₂ M₂] {f : M₂ → M₃}
(hf : IsCompactOperator f) (g : M₁ →SL[σ₁₂] M₂) : IsCompactOperator (f ∘ g) := by |
have := g.continuous.tendsto 0
rw [map_zero] at this
rcases hf with ⟨K, hK, hKf⟩
exact ⟨K, hK, this hKf⟩
| 4 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Topology.Algebra.Order.Group
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.floor from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Filter Function Int Set Topology
variable {α β γ : Type*} [LinearOrdere... | Mathlib/Topology/Algebra/Order/Floor.lean | 88 | 93 | theorem tendsto_floor_left_pure_ceil_sub_one (x : α) :
Tendsto (floor : α → ℤ) (𝓝[<] x) (pure (⌈x⌉ - 1)) :=
have h₁ : ↑(⌈x⌉ - 1) < x := by | rw [cast_sub, cast_one, sub_lt_iff_lt_add]; exact ceil_lt_add_one _
have h₂ : x ≤ ↑(⌈x⌉ - 1) + 1 := by rw [cast_sub, cast_one, sub_add_cancel]; exact le_ceil _
tendsto_pure.2 <| mem_of_superset (Ico_mem_nhdsWithin_Iio' h₁) fun _y hy =>
floor_eq_on_Ico _ _ ⟨hy.1, hy.2.trans_le h₂⟩
| 4 |
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.Perm
import Mathlib.Data.List.Range
#align_import data.list.sublists from "leanprover-community/mathlib"@"ccad6d5093bd2f5c6ca621fc74674cce51355af6"
universe u v w
variable {α : Type u} {β : Type v} {γ : Type w}
open Nat
namespace List
@[simp]
theo... | Mathlib/Data/List/Sublists.lean | 61 | 66 | theorem sublists'_eq_sublists'Aux (l : List α) :
sublists' l = l.foldr (fun a r => sublists'Aux a r r) [[]] := by |
simp only [sublists', sublists'Aux_eq_array_foldl]
rw [← List.foldr_hom Array.toList]
· rfl
· intros _ _; congr <;> simp
| 4 |
import Batteries.Classes.Order
namespace Batteries.PairingHeapImp
inductive Heap (α : Type u) where
| nil : Heap α
| node (a : α) (child sibling : Heap α) : Heap α
deriving Repr
def Heap.size : Heap α → Nat
| .nil => 0
| .node _ c s => c.size + 1 + s.size
def Heap.singleton (a : α) : Heap α := .... | .lake/packages/batteries/Batteries/Data/PairingHeap.lean | 113 | 117 | theorem Heap.noSibling_tail (le) (s : Heap α) : (s.tail le).NoSibling := by |
simp only [Heap.tail]
match eq : s.tail? le with
| none => cases s with cases eq | nil => constructor
| some tl => exact Heap.noSibling_tail? eq
| 4 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.Topology.Order.LeftRightLim
#align_import measure_theory.measure.stieltjes from "leanprover-community/mathlib"@"20d5763051978e9bc6428578ed070445df6a18b3"
noncomputable section
open scoped Classical
open Set Filter Function ENNReal NNReal T... | Mathlib/MeasureTheory/Measure/Stieltjes.lean | 138 | 142 | theorem countable_leftLim_ne (f : StieltjesFunction) : Set.Countable { x | leftLim f x ≠ f x } := by |
refine Countable.mono ?_ f.mono.countable_not_continuousAt
intro x hx h'x
apply hx
exact tendsto_nhds_unique (f.mono.tendsto_leftLim x) (h'x.tendsto.mono_left nhdsWithin_le_nhds)
| 4 |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.ldl from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
variable {𝕜 : Type*} [RCLike 𝕜]
variable {n : Type*} [LinearOrder n] [IsWellOrder n (· < ·)... | Mathlib/LinearAlgebra/Matrix/LDL.lean | 93 | 97 | theorem LDL.lowerInv_triangular {i j : n} (hij : i < j) : LDL.lowerInv hS i j = 0 := by |
rw [←
@gramSchmidt_triangular 𝕜 (n → 𝕜) _ (_ : _) (InnerProductSpace.ofMatrix hS.transpose) n _ _ _
i j hij (Pi.basisFun 𝕜 n),
Pi.basisFun_repr, LDL.lowerInv]
| 4 |
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.Polynomial.SeparableDegree
open scoped Classical Polynomial
open FiniteDimensional Polynomial Interm... | Mathlib/FieldTheory/SeparableDegree.lean | 168 | 172 | theorem finSepDegree_self : finSepDegree F F = 1 := by |
have : Cardinal.mk (Emb F F) = 1 := le_antisymm
(Cardinal.le_one_iff_subsingleton.2 AlgHom.subsingleton)
(Cardinal.one_le_iff_ne_zero.2 <| Cardinal.mk_ne_zero _)
rw [finSepDegree, Nat.card, this, Cardinal.one_toNat]
| 4 |
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.measure.haar.inner_product_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open FiniteDimensional MeasureTheory MeasureTheory.Measure Set
var... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 61 | 66 | theorem OrthonormalBasis.volume_parallelepiped (b : OrthonormalBasis ι ℝ F) :
volume (parallelepiped b) = 1 := by |
haveI : Fact (finrank ℝ F = finrank ℝ F) := ⟨rfl⟩
let o := (stdOrthonormalBasis ℝ F).toBasis.orientation
rw [← o.measure_eq_volume]
exact o.measure_orthonormalBasis b
| 4 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.TypeTags
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Algebra.Ring.Nat
#align_import data.nat.cast.basic from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
assert_not_e... | Mathlib/Data/Nat/Cast/Basic.lean | 159 | 164 | theorem ext_nat'' [MonoidWithZeroHomClass F ℕ A] (f g : F) (h_pos : ∀ {n : ℕ}, 0 < n → f n = g n) :
f = g := by |
apply DFunLike.ext
rintro (_ | n)
· simp [map_zero f, map_zero g]
· exact h_pos n.succ_pos
| 4 |
import Mathlib.Probability.Process.Filtration
import Mathlib.Topology.Instances.Discrete
#align_import probability.process.adapted from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Order TopologicalSpace
open scoped Classical MeasureTheory NNReal ENNReal Topology
namespa... | Mathlib/Probability/Process/Adapted.lean | 99 | 105 | theorem Filtration.adapted_natural [MetrizableSpace β] [mβ : MeasurableSpace β] [BorelSpace β]
{u : ι → Ω → β} (hum : ∀ i, StronglyMeasurable[m] (u i)) :
Adapted (Filtration.natural u hum) u := by |
intro i
refine StronglyMeasurable.mono ?_ (le_iSup₂_of_le i (le_refl i) le_rfl)
rw [stronglyMeasurable_iff_measurable_separable]
exact ⟨measurable_iff_comap_le.2 le_rfl, (hum i).isSeparable_range⟩
| 4 |
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.Pointwise
#align_import algebra.monoid_algebra.support from "leanprover-community/mathlib"@"16749fc4661828cba18cd0f4e3c5eb66a8e80598"
open scoped Pointwise
universe u₁ u₂ u₃
namespace MonoidAlgebra
open Finset Finsupp
variable {k : Type u₁} ... | Mathlib/Algebra/MonoidAlgebra/Support.lean | 25 | 30 | theorem support_mul [Mul G] [DecidableEq G] (a b : MonoidAlgebra k G) :
(a * b).support ⊆ a.support * b.support := by |
rw [MonoidAlgebra.mul_def]
exact support_sum.trans <| biUnion_subset.2 fun _x hx ↦
support_sum.trans <| biUnion_subset.2 fun _y hy ↦
support_single_subset.trans <| singleton_subset_iff.2 <| mem_image₂_of_mem hx hy
| 4 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 109 | 113 | theorem areaForm_apply_self (x : E) : ω x x = 0 := by |
rw [areaForm_to_volumeForm]
refine o.volumeForm.map_eq_zero_of_eq ![x, x] ?_ (?_ : (0 : Fin 2) ≠ 1)
· simp
· norm_num
| 4 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.monoid from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {M : Type*} ... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 51 | 55 | theorem Ioo_add_bij : BijOn (· + d) (Ioo a b) (Ioo (a + d) (b + d)) := by |
rw [← Ioi_inter_Iio, ← Ioi_inter_Iio]
exact
(Ioi_add_bij a d).inter_mapsTo (fun x hx => add_lt_add_right hx _) fun x hx =>
lt_of_add_lt_add_right hx.2
| 4 |
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Periodic
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic.Monotonicity
#align_import data.nat.totient from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8"
open Finset
namespace Nat
... | Mathlib/Data/Nat/Totient.lean | 117 | 126 | theorem _root_.ZMod.card_units_eq_totient (n : ℕ) [NeZero n] [Fintype (ZMod n)ˣ] :
Fintype.card (ZMod n)ˣ = φ n :=
calc
Fintype.card (ZMod n)ˣ = Fintype.card { x : ZMod n // x.val.Coprime n } :=
Fintype.card_congr ZMod.unitsEquivCoprime
_ = φ n := by |
obtain ⟨m, rfl⟩ : ∃ m, n = m + 1 := exists_eq_succ_of_ne_zero NeZero.out
simp only [totient, Finset.card_eq_sum_ones, Fintype.card_subtype, Finset.sum_filter, ←
Fin.sum_univ_eq_sum_range, @Nat.coprime_comm (m + 1)]
rfl
| 4 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 629 | 634 | theorem oangle_left_eq_arctan_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
∡ p₃ p₁ p₂ = Real.arctan (dist p₃ p₂ / dist p₁ p₂) := by |
have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm,
angle_eq_arctan_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(left_ne_of_oangle_eq_pi_div_two h)]
| 4 |
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
#align_import measure_theory.function.simple_func_dense from "leanprover-community/mathlib"@"7317149f12f55affbc900fc873d0d422485122b9"
open Set Function Filter TopologicalSpace ENNReal EMetric Finset
... | Mathlib/MeasureTheory/Function/SimpleFuncDense.lean | 116 | 121 | theorem tendsto_nearestPt {e : ℕ → α} {x : α} (hx : x ∈ closure (range e)) :
Tendsto (fun N => nearestPt e N x) atTop (𝓝 x) := by |
refine (atTop_basis.tendsto_iff nhds_basis_eball).2 fun ε hε => ?_
rcases EMetric.mem_closure_iff.1 hx ε hε with ⟨_, ⟨N, rfl⟩, hN⟩
rw [edist_comm] at hN
exact ⟨N, trivial, fun n hn => (edist_nearestPt_le e x hn).trans_lt hN⟩
| 4 |
import Mathlib.Topology.UniformSpace.AbsoluteValue
import Mathlib.Topology.Instances.Real
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.UniformSpace.Completion
#align_import topology.uniform_space.compare_reals from "leanprover-community/mathlib"@"e1a7bdeb4fd826b7e71d130d34988f0a2d26a177"
open Set... | Mathlib/Topology/UniformSpace/CompareReals.lean | 60 | 65 | theorem Rat.uniformSpace_eq :
(AbsoluteValue.abs : AbsoluteValue ℚ ℚ).uniformSpace = PseudoMetricSpace.toUniformSpace := by |
ext s
rw [(AbsoluteValue.hasBasis_uniformity _).mem_iff, Metric.uniformity_basis_dist_rat.mem_iff]
simp only [Rat.dist_eq, AbsoluteValue.abs_apply, ← Rat.cast_sub, ← Rat.cast_abs, Rat.cast_lt,
abs_sub_comm]
| 4 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Topology.MetricSpace.CauSeqFilter
#align_import analysis.special_functions.exponential from "leanprover-community/mathlib"@"e1a18cad9cd462973d760af7de36b05776b8811c"
open Filter RCLike ContinuousMultili... | Mathlib/Analysis/SpecialFunctions/Exponential.lean | 220 | 224 | theorem Complex.exp_eq_exp_ℂ : Complex.exp = NormedSpace.exp ℂ := by |
refine funext fun x => ?_
rw [Complex.exp, exp_eq_tsum_div]
have : CauSeq.IsComplete ℂ norm := Complex.instIsComplete
exact tendsto_nhds_unique x.exp'.tendsto_limit (expSeries_div_summable ℝ x).hasSum.tendsto_sum_nat
| 4 |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {𝕜 E F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup E] [InnerProductSpace... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 127 | 131 | theorem isClosed_orthogonal : IsClosed (Kᗮ : Set E) := by |
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL 𝕜 (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
| 4 |
import Batteries.Data.Fin.Basic
namespace Fin
attribute [norm_cast] val_last
protected theorem le_antisymm_iff {x y : Fin n} : x = y ↔ x ≤ y ∧ y ≤ x :=
Fin.ext_iff.trans Nat.le_antisymm_iff
protected theorem le_antisymm {x y : Fin n} (h1 : x ≤ y) (h2 : y ≤ x) : x = y :=
Fin.le_antisymm_iff.2 ⟨h1, h2⟩
@[simp... | .lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean | 80 | 85 | theorem foldl_succ_last (f : α → Fin (n+1) → α) (x) :
foldl (n+1) f x = f (foldl n (f · ·.castSucc) x) (last n) := by |
rw [foldl_succ]
induction n generalizing x with
| zero => simp [foldl_succ, Fin.last]
| succ n ih => rw [foldl_succ, ih (f · ·.succ), foldl_succ]; simp [succ_castSucc]
| 4 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Tactic.IntervalCases
#align_import geometry.euclidean.triangle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped Classica... | Mathlib/Geometry/Euclidean/Triangle.lean | 62 | 67 | theorem norm_sub_sq_eq_norm_sq_add_norm_sq_sub_two_mul_norm_mul_norm_mul_cos_angle (x y : V) :
‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - 2 * ‖x‖ * ‖y‖ * Real.cos (angle x y) := by |
rw [show 2 * ‖x‖ * ‖y‖ * Real.cos (angle x y) = 2 * (Real.cos (angle x y) * (‖x‖ * ‖y‖)) by ring,
cos_angle_mul_norm_mul_norm, ← real_inner_self_eq_norm_mul_norm, ←
real_inner_self_eq_norm_mul_norm, ← real_inner_self_eq_norm_mul_norm, real_inner_sub_sub_self,
sub_add_eq_add_sub]
| 4 |
import Mathlib.Algebra.Group.Center
import Mathlib.Data.Int.Cast.Lemmas
#align_import group_theory.subsemigroup.center from "leanprover-community/mathlib"@"1ac8d4304efba9d03fa720d06516fac845aa5353"
variable {M : Type*}
namespace Set
variable (M)
@[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) :... | Mathlib/Algebra/Ring/Center.lean | 81 | 86 | theorem neg_mem_center [NonUnitalNonAssocRing M] {a : M} (ha : a ∈ Set.center M) :
-a ∈ Set.center M where
comm _ := by | rw [← neg_mul_comm, ← ha.comm, neg_mul_comm]
left_assoc _ _ := by rw [neg_mul, ha.left_assoc, neg_mul, neg_mul]
mid_assoc _ _ := by rw [← neg_mul_comm, ha.mid_assoc, neg_mul_comm, neg_mul]
right_assoc _ _ := by rw [mul_neg, ha.right_assoc, mul_neg, mul_neg]
| 4 |
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Ty... | Mathlib/Algebra/Polynomial/Roots.lean | 69 | 73 | theorem card_roots (hp0 : p ≠ 0) : (Multiset.card (roots p) : WithBot ℕ) ≤ degree p := by |
classical
unfold roots
rw [dif_neg hp0]
exact (Classical.choose_spec (exists_multiset_roots hp0)).1
| 4 |
import Mathlib.Data.Set.Basic
open Function
universe u v
namespace Set
section Nontrivial
variable {α : Type u} {a : α} {s t : Set α}
protected def Nontrivial (s : Set α) : Prop :=
∃ x ∈ s, ∃ y ∈ s, x ≠ y
#align set.nontrivial Set.Nontrivial
theorem nontrivial_of_mem_mem_ne {x y} (hx : x ∈ s) (hy : y ∈... | Mathlib/Data/Set/Subsingleton.lean | 194 | 198 | theorem Nontrivial.exists_ne (hs : s.Nontrivial) (z) : ∃ x ∈ s, x ≠ z := by |
by_contra! H
rcases hs with ⟨x, hx, y, hy, hxy⟩
rw [H x hx, H y hy] at hxy
exact hxy rfl
| 4 |
import Mathlib.RingTheory.RootsOfUnity.Basic
universe u
variable {L : Type u} [CommRing L] [IsDomain L]
variable (n : ℕ+)
theorem rootsOfUnity.integer_power_of_ringEquiv (g : L ≃+* L) :
∃ m : ℤ, ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ m : Lˣ) := by
obtain ⟨m, hm⟩ := MonoidHom.map_cyclic ((g : L ≃* L).re... | Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean | 120 | 125 | theorem toFun_unique (g : L ≃+* L) (c : ZMod (Fintype.card (rootsOfUnity n L)))
(hc : ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ c.val : Lˣ)) : c = χ₀ n g := by |
apply IsCyclic.ext rfl (fun ζ ↦ ?_)
specialize hc ζ
suffices ((ζ ^ c.val : Lˣ) : L) = (ζ ^ (χ₀ n g).val : Lˣ) by exact_mod_cast this
rw [← toFun_spec g ζ, hc]
| 4 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.arsinh from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Function Filter Set
open scoped Topology
name... | Mathlib/Analysis/SpecialFunctions/Arsinh.lean | 69 | 73 | theorem arsinh_neg (x : ℝ) : arsinh (-x) = -arsinh x := by |
rw [← exp_eq_exp, exp_arsinh, exp_neg, exp_arsinh]
apply eq_inv_of_mul_eq_one_left
rw [neg_sq, neg_add_eq_sub, add_comm x, mul_comm, ← sq_sub_sq, sq_sqrt, add_sub_cancel_right]
exact add_nonneg zero_le_one (sq_nonneg _)
| 4 |
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.Separation
import Mathlib.Topology.Support
#align_import topology.uniform_space.compact from "leanprover-community/mathlib"@"735b22f8f9ff9792cf4212d7cb051c4c994bc685"
open scoped Cla... | Mathlib/Topology/UniformSpace/Compact.lean | 69 | 75 | theorem unique_uniformity_of_compact [t : TopologicalSpace γ] [CompactSpace γ]
{u u' : UniformSpace γ} (h : u.toTopologicalSpace = t) (h' : u'.toTopologicalSpace = t) :
u = u' := by |
refine UniformSpace.ext ?_
have : @CompactSpace γ u.toTopologicalSpace := by rwa [h]
have : @CompactSpace γ u'.toTopologicalSpace := by rwa [h']
rw [@compactSpace_uniformity _ u, compactSpace_uniformity, h, h']
| 4 |
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Topology.Order.MonotoneConvergence
#align_import analysis.box_integral.box.basic from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Fu... | Mathlib/Analysis/BoxIntegral/Box/Basic.lean | 181 | 185 | theorem injective_coe : Injective ((↑) : Box ι → Set (ι → ℝ)) := by |
rintro ⟨l₁, u₁, h₁⟩ ⟨l₂, u₂, h₂⟩ h
simp only [Subset.antisymm_iff, coe_subset_coe, le_iff_bounds] at h
congr
exacts [le_antisymm h.2.1 h.1.1, le_antisymm h.1.2 h.2.2]
| 4 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Topology.Algebra.Module.FiniteDimension
variable {𝕜 E A B : Type*} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E]
[CommSemiring A] {z : E} {... | Mathlib/Analysis/Analytic/Polynomial.lean | 47 | 52 | theorem AnalyticAt.aeval_mvPolynomial (hf : ∀ i, AnalyticAt 𝕜 (f · i) z) (p : MvPolynomial σ A) :
AnalyticAt 𝕜 (fun x ↦ aeval (f x) p) z := by |
apply p.induction_on (fun k ↦ ?_) (fun p q hp hq ↦ ?_) fun p i hp ↦ ?_ -- `refine` doesn't work
· simp_rw [aeval_C]; apply analyticAt_const
· simp_rw [map_add]; exact hp.add hq
· simp_rw [map_mul, aeval_X]; exact hp.mul (hf i)
| 4 |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
open List
def Cycle (α : Type*) : Type _ :=
Quotient (IsRotated.setoid α)
#align cycle Cycle
namespace Cycle
variable {α : Type*}
--... | Mathlib/Data/List/Cycle.lean | 605 | 610 | theorem Subsingleton.congr {s : Cycle α} (h : Subsingleton s) :
∀ ⦃x⦄ (_hx : x ∈ s) ⦃y⦄ (_hy : y ∈ s), x = y := by |
induction' s using Quot.inductionOn with l
simp only [length_subsingleton_iff, length_coe, mk_eq_coe, le_iff_lt_or_eq, Nat.lt_add_one_iff,
length_eq_zero, length_eq_one, Nat.not_lt_zero, false_or_iff] at h
rcases h with (rfl | ⟨z, rfl⟩) <;> simp
| 4 |
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Conj
#align_import category_theory.adjunction.mates from "leanprover-community/mathlib"@"cea27692b3fdeb328a2ddba6aabf181754543184"
universe v₁ v₂ v₃ v₄ u₁ u₂ u₃ u₄
namespace CategoryTheory
open Category
variable {C : Type u₁} {D : Typ... | Mathlib/CategoryTheory/Adjunction/Mates.lean | 118 | 124 | theorem unit_transferNatTrans (f : G ⋙ L₂ ⟶ L₁ ⋙ H) (X : C) :
G.map (adj₁.unit.app X) ≫ (transferNatTrans adj₁ adj₂ f).app _ =
adj₂.unit.app _ ≫ R₂.map (f.app _) := by |
dsimp [transferNatTrans]
rw [← adj₂.unit_naturality_assoc, ← R₂.map_comp, ← Functor.comp_map G L₂, f.naturality_assoc,
Functor.comp_map, ← H.map_comp]
dsimp; simp
| 4 |
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.pi from "leanprover-community/mathlib"@"e4bc74cbaf429d706cb9140902f7ca6c431e75a4"
-- Porting note: Added, since dot nota... | Mathlib/Order/Interval/Set/Pi.lean | 112 | 118 | theorem disjoint_pi_univ_Ioc_update_left_right {x y : ∀ i, α i} {i₀ : ι} {m : α i₀} :
Disjoint (pi univ fun i ↦ Ioc (x i) (update y i₀ m i))
(pi univ fun i ↦ Ioc (update x i₀ m i) (y i)) := by |
rw [disjoint_left]
rintro z h₁ h₂
refine (h₁ i₀ (mem_univ _)).2.not_lt ?_
simpa only [Function.update_same] using (h₂ i₀ (mem_univ _)).1
| 4 |
import Batteries.Data.List.Lemmas
namespace List
universe u v
variable {α : Type u} {β : Type v}
@[simp] theorem eraseIdx_zero (l : List α) : eraseIdx l 0 = tail l := by cases l <;> rfl
theorem eraseIdx_eq_take_drop_succ :
∀ (l : List α) (i : Nat), l.eraseIdx i = l.take i ++ l.drop (i + 1)
| nil, _ => by s... | .lake/packages/batteries/Batteries/Data/List/EraseIdx.lean | 76 | 80 | theorem mem_eraseIdx_iff_get? {x : α} {l} {k} : x ∈ eraseIdx l k ↔ ∃ i ≠ k, l.get? i = x := by |
simp only [mem_eraseIdx_iff_get, Fin.exists_iff, exists_and_left, get_eq_iff, exists_prop]
refine exists_congr fun i => and_congr_right' <| and_iff_right_of_imp fun h => ?_
obtain ⟨h, -⟩ := get?_eq_some.1 h
exact h
| 4 |
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Defs
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
variable {α : Type*}
open Opposite
namespace Set
protected def op (s : Set α) : Set αᵒᵖ :=
unop ⁻¹' s
#align set.op Set.op
protected def u... | Mathlib/Data/Set/Opposite.lean | 92 | 96 | theorem singleton_op_unop (x : α) : ({op x} : Set αᵒᵖ).unop = {x} := by |
ext
constructor
· apply op_injective
· apply unop_injective
| 4 |
import Mathlib.Geometry.Manifold.MFDeriv.FDeriv
noncomputable section
open scoped Manifold
open Bundle Set Topology
section SpecificFunctions
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H)... | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 291 | 297 | theorem tangentMapWithin_prod_fst {s : Set (M × M')} {p : TangentBundle (I.prod I') (M × M')}
(hs : UniqueMDiffWithinAt (I.prod I') s p.proj) :
tangentMapWithin (I.prod I') I Prod.fst s p = ⟨p.proj.1, p.2.1⟩ := by |
simp only [tangentMapWithin]
rw [mfderivWithin_fst]
· rcases p with ⟨⟩; rfl
· exact hs
| 4 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section SigmaLift
variable {α β γ : ι → Type*} [DecidableEq ι]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 204 | 208 | theorem sigmaLift_eq_empty : sigmaLift f a b = ∅ ↔ ∀ h : a.1 = b.1, f (h ▸ a.2) b.2 = ∅ := by |
simp_rw [sigmaLift]
split_ifs with h
· simp [h, forall_prop_of_true h]
· simp [h, forall_prop_of_false h]
| 4 |
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.NormNum.Basic
#align_import data.nat.prime_norm_num from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
open Nat Qq Lean Meta
namespace Mathlib.Meta.NormNum
theorem not_prime_mul_of_ble (a b n : ℕ) (h : a * b = n) (h₁ : a.ble 1 = fals... | Mathlib/Tactic/NormNum/Prime.lean | 90 | 95 | theorem minFacHelper_3 {n k k' : ℕ} (e : k + 2 = k') (nk : (n % k).beq 0 = false)
(h : MinFacHelper n k) : MinFacHelper n k' := by |
refine minFacHelper_1 e h λ h2 ↦ ?_
have nk := Nat.ne_of_beq_eq_false nk
rw [← Nat.dvd_iff_mod_eq_zero, ← h2] at nk
exact nk <| minFac_dvd n
| 4 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Analysis.NormedSpace.Span
suppress_compilation
open Bornology
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `ₗ` subscript variables are for special... | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 99 | 107 | theorem opNorm_zero_iff [RingHomIsometric σ₁₂] : ‖f‖ = 0 ↔ f = 0 :=
Iff.intro
(fun hn => ContinuousLinearMap.ext fun x => norm_le_zero_iff.1
(calc
_ ≤ ‖f‖ * ‖x‖ := le_opNorm _ _
_ = _ := by | rw [hn, zero_mul]))
(by
rintro rfl
exact opNorm_zero)
| 4 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 110 | 114 | theorem natDegree_opRingEquiv (p : R[X]ᵐᵒᵖ) : (opRingEquiv R p).natDegree = (unop p).natDegree := by |
by_cases p0 : p = 0
· simp only [p0, _root_.map_zero, natDegree_zero, unop_zero]
· simp only [p0, natDegree_eq_support_max', Ne, AddEquivClass.map_eq_zero_iff, not_false_iff,
support_opRingEquiv, unop_eq_zero_iff]
| 4 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Function Set Filter
open scoped Topology Filter
variable... | Mathlib/Analysis/Calculus/BumpFunction/Basic.lean | 172 | 176 | theorem support_eq : Function.support f = Metric.ball c f.rOut := by |
simp only [toFun, support_comp_eq_preimage, ContDiffBumpBase.support _ _ f.one_lt_rOut_div_rIn]
ext x
simp only [mem_ball_iff_norm, sub_zero, norm_smul, mem_preimage, Real.norm_eq_abs, abs_inv,
abs_of_pos f.rIn_pos, ← div_eq_inv_mul, div_lt_div_right f.rIn_pos]
| 4 |
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.function.simple_func_dense_lp from "leanprover-community/mathlib"@"5a2df4cd59cb31e97a516d4603a14bed5c2f9425"
noncomputable section
set_option linter.uppercaseLean3 false
open Set Func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 68 | 74 | theorem nnnorm_approxOn_le [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f) {s : Set E}
{y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] (x : β) (n : ℕ) :
‖approxOn f hf s y₀ h₀ n x - f x‖₊ ≤ ‖f x - y₀‖₊ := by |
have := edist_approxOn_le hf h₀ x n
rw [edist_comm y₀] at this
simp only [edist_nndist, nndist_eq_nnnorm] at this
exact mod_cast this
| 4 |
import Mathlib.ModelTheory.Satisfiability
#align_import model_theory.types from "leanprover-community/mathlib"@"98bd247d933fb581ff37244a5998bd33d81dd46d"
set_option linter.uppercaseLean3 false
universe u v w w'
open Cardinal Set
open scoped Classical
open Cardinal FirstOrder
namespace FirstOrder
namespace La... | Mathlib/ModelTheory/Types.lean | 147 | 151 | theorem nonempty_iff : Nonempty (T.CompleteType α) ↔ T.IsSatisfiable := by |
rw [← isSatisfiable_onTheory_iff (lhomWithConstants_injective L α)]
rw [nonempty_iff_univ_nonempty, nonempty_iff_ne_empty, Ne, not_iff_comm,
← union_empty ((L.lhomWithConstants α).onTheory T), ← setOf_subset_eq_empty_iff]
simp
| 4 |
import Mathlib.Data.Fintype.Quotient
import Mathlib.ModelTheory.Semantics
#align_import model_theory.quotients from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
namespace FirstOrder
namespace Language
variable (L : Language) {M : Type*}
open FirstOrder
open Structure
class Prest... | Mathlib/ModelTheory/Quotients.lean | 57 | 62 | theorem funMap_quotient_mk' {n : ℕ} (f : L.Functions n) (x : Fin n → M) :
(funMap f fun i => (⟦x i⟧ : Quotient s)) = ⟦@funMap _ _ ps.toStructure _ f x⟧ := by |
change
Quotient.map (@funMap L M ps.toStructure n f) Prestructure.fun_equiv (Quotient.finChoice _) =
_
rw [Quotient.finChoice_eq, Quotient.map_mk]
| 4 |
import Mathlib.Data.Nat.Prime
#align_import data.int.nat_prime from "leanprover-community/mathlib"@"422e70f7ce183d2900c586a8cda8381e788a0c62"
open Nat
namespace Int
theorem not_prime_of_int_mul {a b : ℤ} {c : ℕ} (ha : a.natAbs ≠ 1) (hb : b.natAbs ≠ 1)
(hc : a * b = (c : ℤ)) : ¬Nat.Prime c :=
not_prime_mul... | Mathlib/Data/Int/NatPrime.lean | 24 | 33 | theorem succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul {p : ℕ} (p_prime : Nat.Prime p) {m n : ℤ}
{k l : ℕ} (hpm : ↑(p ^ k) ∣ m) (hpn : ↑(p ^ l) ∣ n) (hpmn : ↑(p ^ (k + l + 1)) ∣ m * n) :
↑(p ^ (k + 1)) ∣ m ∨ ↑(p ^ (l + 1)) ∣ n :=
have hpm' : p ^ k ∣ m.natAbs := Int.natCast_dvd_natCast.1 <| Int.dvd_natAbs.2 hpm
ha... |
rw [← Int.natAbs_mul]; apply Int.natCast_dvd_natCast.1 <| Int.dvd_natAbs.2 hpmn
let hsd := Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn'
hsd.elim (fun hsd1 => Or.inl (by apply Int.dvd_natAbs.1; apply Int.natCast_dvd_natCast.2 hsd1))
fun hsd2 => Or.inr (by apply Int.dvd_natAbs.1; appl... | 4 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Order.SupIndep
import Mathlib.Order.Atoms
#align_import order.partition.finpartition from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Finset Function
variable {α : Type*}
@[ext]
structure Finpartition [Lattice α]... | Mathlib/Order/Partition/Finpartition.lean | 178 | 182 | theorem ne_bot {b : α} (hb : b ∈ P.parts) : b ≠ ⊥ := by |
intro h
refine P.not_bot_mem (?_)
rw [h] at hb
exact hb
| 4 |
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
namespace List
variable {α : Type*}
section Sym
protected def sym : (n : ℕ) → List α → List (Sym α n)
| 0, _ => [.nil]
| _, [] => []
| n + 1, x :: xs => ((x :: xs).sym n |>.map fun p => x ::ₛ p) ++ xs.sym (n + 1)
variable {xs ys : List α} ... | Mathlib/Data/List/Sym.lean | 165 | 169 | theorem sym_one_eq : xs.sym 1 = xs.map (· ::ₛ .nil) := by |
induction xs with
| nil => simp only [List.sym, Nat.succ_eq_add_one, Nat.reduceAdd, map_nil]
| cons x xs ih =>
rw [map_cons, ← ih, List.sym, List.sym, map_singleton, singleton_append]
| 4 |
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.Convex.Deriv
#align_import analysis.convex.specific_functions.deriv from "leanprover-communi... | Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 40 | 44 | theorem strictConvexOn_pow {n : ℕ} (hn : 2 ≤ n) : StrictConvexOn ℝ (Ici 0) fun x : ℝ => x ^ n := by |
apply StrictMonoOn.strictConvexOn_of_deriv (convex_Ici _) (continuousOn_pow _)
rw [deriv_pow', interior_Ici]
exact fun x (hx : 0 < x) y _ hxy => mul_lt_mul_of_pos_left
(pow_lt_pow_left hxy hx.le <| Nat.sub_ne_zero_of_lt hn) (by positivity)
| 4 |
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
suppress_compilation
open Set
variable {R L : Type*} [CommRing R] [LieRing L] [LieAlgebra R L]
(H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R H]
{M : Type*} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L ... | Mathlib/Algebra/Lie/Weights/Cartan.lean | 135 | 141 | theorem mapsTo_toEnd_weightSpace_add_of_mem_rootSpace (α χ : H → R)
{x : L} (hx : x ∈ rootSpace H α) :
MapsTo (toEnd R L M x) (weightSpace M χ) (weightSpace M (α + χ)) := by |
intro m hm
let x' : rootSpace H α := ⟨x, hx⟩
let m' : weightSpace M χ := ⟨m, hm⟩
exact (rootSpaceWeightSpaceProduct R L H M α χ (α + χ) rfl (x' ⊗ₜ m')).property
| 4 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
#align_import linear_algebra.matrix.mv_polynomial from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
set_option linter.uppercaseLean3 false
varia... | Mathlib/LinearAlgebra/Matrix/MvPolynomial.lean | 75 | 80 | theorem det_mvPolynomialX_ne_zero [DecidableEq m] [Fintype m] [CommRing R] [Nontrivial R] :
det (mvPolynomialX m m R) ≠ 0 := by |
intro h_det
have := congr_arg Matrix.det (mvPolynomialX_mapMatrix_eval (1 : Matrix m m R))
rw [det_one, ← RingHom.map_det, h_det, RingHom.map_zero] at this
exact zero_ne_one this
| 4 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 482 | 491 | theorem schur_complement_eq₁₁ [Fintype m] [DecidableEq m] [Fintype n] {A : Matrix m m 𝕜}
(B : Matrix m n 𝕜) (D : Matrix n n 𝕜) (x : m → 𝕜) (y : n → 𝕜) [Invertible A]
(hA : A.IsHermitian) :
(star (x ⊕ᵥ y)) ᵥ* (fromBlocks A B Bᴴ D) ⬝ᵥ (x ⊕ᵥ y) =
(star (x + (A⁻¹ * B) *ᵥ y)) ᵥ* A ⬝ᵥ (x + (A⁻¹ * B) *ᵥ... |
simp [Function.star_sum_elim, fromBlocks_mulVec, vecMul_fromBlocks, add_vecMul,
dotProduct_mulVec, vecMul_sub, Matrix.mul_assoc, vecMul_mulVec, hA.eq,
conjTranspose_nonsing_inv, star_mulVec]
abel
| 4 |
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Trace
import Mathlib.RingTheory.Norm
#align_import ring_theory.discriminant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
universe u v w z
open scoped Matrix
open Matrix FiniteDimensional Fintype Polynomial Fin... | Mathlib/RingTheory/Discriminant.lean | 161 | 167 | theorem discr_powerBasis_eq_prod (e : Fin pb.dim ≃ (L →ₐ[K] E)) [IsSeparable K L] :
algebraMap K E (discr K pb.basis) =
∏ i : Fin pb.dim, ∏ j ∈ Ioi i, (e j pb.gen - e i pb.gen) ^ 2 := by |
rw [discr_eq_det_embeddingsMatrixReindex_pow_two K E pb.basis e,
embeddingsMatrixReindex_eq_vandermonde, det_transpose, det_vandermonde, ← prod_pow]
congr; ext i
rw [← prod_pow]
| 4 |
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 122 | 127 | theorem withDensity_smul (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) :
μ.withDensity (r • f) = r • μ.withDensity f := by |
refine Measure.ext fun s hs => ?_
rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,
smul_eq_mul, ← lintegral_const_mul r hf]
simp only [Pi.smul_apply, smul_eq_mul]
| 4 |
import Mathlib.Mathport.Rename
set_option autoImplicit true
namespace Thunk
#align thunk.mk Thunk.mk
-- Porting note: Added `Thunk.ext` to get `ext` tactic to work.
@[ext]
| Mathlib/Lean/Thunk.lean | 20 | 24 | theorem ext {α : Type u} {a b : Thunk α} (eq : a.get = b.get) : a = b := by |
have ⟨_⟩ := a
have ⟨_⟩ := b
congr
exact funext fun _ ↦ eq
| 4 |
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Set Fin Filter Function
open scoped NNReal Topology
section Real
variab... | Mathlib/Analysis/Calculus/ContDiff/RCLike.lean | 43 | 49 | theorem ContDiffAt.hasStrictFDerivAt' {f : E' → F'} {f' : E' →L[𝕂] F'} {x : E'}
(hf : ContDiffAt 𝕂 n f x) (hf' : HasFDerivAt f f' x) (hn : 1 ≤ n) :
HasStrictFDerivAt f f' x := by |
rcases hf 1 hn with ⟨u, H, p, hp⟩
simp only [nhdsWithin_univ, mem_univ, insert_eq_of_mem] at H
have := hp.hasStrictFDerivAt le_rfl H
rwa [hf'.unique this.hasFDerivAt]
| 4 |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp Ad... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 88 | 92 | theorem degrees_monomial (s : σ →₀ ℕ) (a : R) : degrees (monomial s a) ≤ toMultiset s := by |
classical
refine (supDegree_single s a).trans_le ?_
split_ifs
exacts [bot_le, le_rfl]
| 4 |
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.ConcreteCategory
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryT... | Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean | 227 | 234 | theorem widePullback_ext {B : C} {ι : Type w} {X : ι → C} (f : ∀ j : ι, X j ⟶ B)
[HasWidePullback B X f] [PreservesLimit (wideCospan B X f) (forget C)]
(x y : ↑(widePullback B X f)) (h₀ : base f x = base f y) (h : ∀ j, π f j x = π f j y) :
x = y := by |
apply Concrete.limit_ext
rintro (_ | j)
· exact h₀
· apply h
| 4 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ι : Sort u} {α : Type v} {β : Type w}
open Set
open OrderDual (toDual)
namespace Set
section UnionIxx
variable [LinearOrder α] {s ... | Mathlib/Order/Interval/Set/Disjoint.lean | 201 | 205 | theorem IsGLB.biUnion_Ioi_eq (h : IsGLB s a) : ⋃ x ∈ s, Ioi x = Ioi a := by |
refine (iUnion₂_subset fun x hx => ?_).antisymm fun x hx => ?_
· exact Ioi_subset_Ioi (h.1 hx)
· rcases h.exists_between hx with ⟨y, hys, _, hyx⟩
exact mem_biUnion hys hyx
| 4 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Ring.Regular
import Mathlib.Tactic.Common
#align_import algebra.gcd_monoid.basic from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
variable {α : Type*}
-- Porting note: mathlib3 had a `@[protect_proj]` here, but adding `protect... | Mathlib/Algebra/GCDMonoid/Basic.lean | 162 | 166 | theorem normUnit_mul_normUnit (a : α) : normUnit (a * normUnit a) = 1 := by |
nontriviality α using Subsingleton.elim a 0
obtain rfl | h := eq_or_ne a 0
· rw [normUnit_zero, zero_mul, normUnit_zero]
· rw [normUnit_mul h (Units.ne_zero _), normUnit_coe_units, mul_inv_eq_one]
| 4 |
import Batteries.Data.List.Lemmas
import Batteries.Tactic.Classical
import Mathlib.Tactic.TypeStar
import Mathlib.Mathport.Rename
#align_import data.list.tfae from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
namespace List
def TFAE (l : List Prop) : Prop :=
∀ x ∈ l, ∀ y ∈ l, x ↔ ... | Mathlib/Data/List/TFAE.lean | 91 | 96 | theorem forall_tfae {α : Type*} (l : List (α → Prop)) (H : ∀ a : α, (l.map (fun p ↦ p a)).TFAE) :
(l.map (fun p ↦ ∀ a, p a)).TFAE := by |
simp only [TFAE, List.forall_mem_map_iff]
intros p₁ hp₁ p₂ hp₂
exact forall_congr' fun a ↦ H a (p₁ a) (mem_map_of_mem (fun p ↦ p a) hp₁)
(p₂ a) (mem_map_of_mem (fun p ↦ p a) hp₂)
| 4 |
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