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 | eval_complexity float64 0 1 |
|---|---|---|---|---|---|---|
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
#align_import linear_algebra.clifford_algebra.star from "leanprover-community/mathlib"@"4d66277cfec381260ba05c68f9ae6ce2a118031d"
variable {R : Type*} [CommRing R]
variable {M : Type*} [AddCommGroup M] [Module R M]
variable {Q : QuadraticForm R M}
namespac... | Mathlib/LinearAlgebra/CliffordAlgebra/Star.lean | 57 | 58 | theorem star_smul (r : R) (x : CliffordAlgebra Q) : star (r • x) = r • star x := by |
rw [star_def, star_def, map_smul, map_smul]
| 0.90625 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {α : Type*}
section support
v... | Mathlib/GroupTheory/Perm/Support.lean | 301 | 301 | theorem not_mem_support {x : α} : x ∉ f.support ↔ f x = x := by | simp
| 0.90625 |
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 | 371 | 372 | theorem natDegree_C_mul (a0 : a ≠ 0) : (C a * p).natDegree = p.natDegree := by |
simp only [natDegree, degree_C_mul a0]
| 0.90625 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 171 | 172 | theorem rank_finsupp' (ι : Type v) : Module.rank R (ι →₀ M) = #ι * Module.rank R M := by |
simp [rank_finsupp]
| 0.90625 |
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 36 | 36 | theorem log_im (x : ℂ) : x.log.im = x.arg := by | simp [log]
| 0.90625 |
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 83 | 83 | theorem log_ofReal_re (x : ℝ) : (log (x : ℂ)).re = Real.log x := by | simp [log_re]
| 0.90625 |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 56 | 57 | theorem nhds_zero : 𝓝 (0 : Γ₀) = ⨅ γ ≠ 0, 𝓟 (Iio γ) := by |
rw [nhds_eq_update, update_same]
| 0.90625 |
import Mathlib.Analysis.Complex.RealDeriv
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas
#align_import analysis.special_functions.exp_deriv from "leanprover-community/mathlib"@"6a5c85000ab93fe5dcfdf620676f614ba8e18c26"
noncomputable section
open Filter Asym... | Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean | 36 | 42 | theorem hasDerivAt_exp (x : ℂ) : HasDerivAt exp (exp x) x := by |
rw [hasDerivAt_iff_isLittleO_nhds_zero]
have : (1 : ℕ) < 2 := by norm_num
refine (IsBigO.of_bound ‖exp x‖ ?_).trans_isLittleO (isLittleO_pow_id this)
filter_upwards [Metric.ball_mem_nhds (0 : ℂ) zero_lt_one]
simp only [Metric.mem_ball, dist_zero_right, norm_pow]
exact fun z hz => exp_bound_sq x z hz.le
| 0.90625 |
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community/mathlib"@"9407b03373c8cd201df99d6bc5514fc2db44054f"
namespace Equiv
variable {α β : Type*} [Finite α]
noncomputable def toCompl {p q : α → Prop} (e ... | Mathlib/Logic/Equiv/Fintype.lean | 132 | 135 | theorem extendSubtype_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : p x) :
q (e.extendSubtype x) := by |
convert (e ⟨x, hx⟩).2
rw [e.extendSubtype_apply_of_mem _ hx]
| 0.90625 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 245 | 248 | theorem IsRelPrime.prod_left_iff : IsRelPrime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsRelPrime (s i) x := by |
classical
refine Finset.induction_on t (iff_of_true isRelPrime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsRelPrime.mul_left_iff, ih, Finset.forall_mem_insert]
| 0.90625 |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 149 | 151 | theorem isPiSystem_image_Iic (s : Set α) : IsPiSystem (Iic '' s) := by |
rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ -
exact ⟨a ⊓ b, inf_ind a b ha hb, Iic_inter_Iic.symm⟩
| 0.90625 |
import Mathlib.Data.PFunctor.Multivariate.W
import Mathlib.Data.QPF.Multivariate.Basic
#align_import data.qpf.multivariate.constructions.fix from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
universe u v
namespace MvQPF
open TypeVec
open MvFunctor (LiftP LiftR)
open MvFunctor
var... | Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean | 64 | 67 | theorem recF_eq {α : TypeVec n} {β : Type u} (g : F (α.append1 β) → β) (a : q.P.A)
(f' : q.P.drop.B a ⟹ α) (f : q.P.last.B a → q.P.W α) :
recF g (q.P.wMk a f' f) = g (abs ⟨a, splitFun f' (recF g ∘ f)⟩) := by |
rw [recF, MvPFunctor.wRec_eq]; rfl
| 0.90625 |
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 64 | 66 | theorem exp_log_of_neg (hx : x < 0) : exp (log x) = -x := by |
rw [exp_log_eq_abs (ne_of_lt hx)]
exact abs_of_neg hx
| 0.875 |
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.FinsuppVectorSpace
#align_import linear_algebra.tensor_product_basis from "leanprover-community/mathlib"@"f784cc6142443d9ee623a20788c282112c322081"
noncomputable section
open Set LinearMap Submodule
section CommSemiring
variable {R : T... | Mathlib/LinearAlgebra/TensorProduct/Basis.lean | 39 | 41 | theorem Basis.tensorProduct_apply (b : Basis ι R M) (c : Basis κ R N) (i : ι) (j : κ) :
Basis.tensorProduct b c (i, j) = b i ⊗ₜ c j := by |
simp [Basis.tensorProduct]
| 0.875 |
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.fin_range from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
universe u
namespace List
variable {α : Type u}
@[simp]
theorem map_coe_finRange (n : ℕ) : ((finRange n) : List (Fin n)).map (Fin.val) = ... | Mathlib/Data/List/FinRange.lean | 44 | 47 | theorem ofFn_eq_pmap {n} {f : Fin n → α} :
ofFn f = pmap (fun i hi => f ⟨i, hi⟩) (range n) fun _ => mem_range.1 := by |
rw [pmap_eq_map_attach]
exact ext_get (by simp) fun i hi1 hi2 => by simp [get_ofFn f ⟨i, hi1⟩]
| 0.875 |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.lemmas from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
open Function Nat
namespace Int
variable {a b : ℤ} {n : ℕ}
theorem natAbs_eq_iff_mul_self_eq {a b : ℤ} : a.natAbs = b.natAbs ↔ a * a = b * b := by
rw [← a... | Mathlib/Data/Int/Order/Lemmas.lean | 35 | 37 | theorem natAbs_lt_iff_mul_self_lt {a b : ℤ} : a.natAbs < b.natAbs ↔ a * a < b * b := by |
rw [← abs_lt_iff_mul_self_lt, abs_eq_natAbs, abs_eq_natAbs]
exact Int.ofNat_lt.symm
| 0.875 |
import Mathlib.Algebra.MvPolynomial.Rename
#align_import data.mv_polynomial.comap from "leanprover-community/mathlib"@"aba31c938d3243cc671be7091b28a1e0814647ee"
namespace MvPolynomial
variable {σ : Type*} {τ : Type*} {υ : Type*} {R : Type*} [CommSemiring R]
noncomputable def comap (f : MvPolynomial σ R →ₐ[R] M... | Mathlib/Algebra/MvPolynomial/Comap.lean | 48 | 50 | theorem comap_id_apply (x : σ → R) : comap (AlgHom.id R (MvPolynomial σ R)) x = x := by |
funext i
simp only [comap, AlgHom.id_apply, id, aeval_X]
| 0.875 |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.finite.card from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
noncomputable section
open scoped Classical
variable {α β γ : Type*}
def Finite.equivFin (α : Type*) [Finite α] : α ≃ Fin (Nat.card α) := by
have := (Finite.... | Mathlib/Data/Finite/Card.lean | 83 | 85 | theorem one_lt_card_iff_nontrivial [Finite α] : 1 < Nat.card α ↔ Nontrivial α := by |
haveI := Fintype.ofFinite α
simp only [Nat.card_eq_fintype_card, Fintype.one_lt_card_iff_nontrivial]
| 0.875 |
import Mathlib.MeasureTheory.Function.AEEqFun.DomAct
import Mathlib.MeasureTheory.Function.LpSpace
set_option autoImplicit true
open MeasureTheory Filter
open scoped ENNReal
namespace DomMulAct
variable {M N α E : Type*} [MeasurableSpace M] [MeasurableSpace N]
[MeasurableSpace α] [NormedAddCommGroup E] {μ : Me... | Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean | 82 | 83 | theorem smul_Lp_sub (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f - g) = c • f - c • g := by |
rintro ⟨⟨⟩, _⟩ ⟨⟨⟩, _⟩; rfl
| 0.875 |
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.NNRat.Defs
variable {ι α : Type*}
namespace NNRat
@[norm_cast]
theorem coe_list_sum (l : List ℚ≥0) : (l.sum : ℚ) = (l.map (↑)).sum :=
map_list_sum coeHom _
#align nnrat.coe_list_sum NNRat.coe_list_sum
@[norm_cast]
theorem coe_list_prod (... | Mathlib/Data/NNRat/BigOperators.lean | 52 | 55 | theorem toNNRat_prod_of_nonneg {s : Finset α} {f : α → ℚ} (hf : ∀ a ∈ s, 0 ≤ f a) :
(∏ a ∈ s, f a).toNNRat = ∏ a ∈ s, (f a).toNNRat := by |
rw [← coe_inj, coe_prod, Rat.coe_toNNRat _ (Finset.prod_nonneg hf)]
exact Finset.prod_congr rfl fun x hxs ↦ by rw [Rat.coe_toNNRat _ (hf x hxs)]
| 0.875 |
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Homeomorph
#align_import topology.algebra.group_with_zero from "leanprover-community/mathlib"@"c10e724be91096453ee3db13862b9fb9a992fef2"
open Topology Filter Function
variable {α β G₀ : Type*}
section DivConst... | Mathlib/Topology/Algebra/GroupWithZero.lean | 69 | 71 | theorem ContinuousOn.div_const (hf : ContinuousOn f s) (y : G₀) :
ContinuousOn (fun x => f x / y) s := by |
simpa only [div_eq_mul_inv] using hf.mul continuousOn_const
| 0.875 |
import Mathlib.Data.Fintype.Card
import Mathlib.Computability.Language
import Mathlib.Tactic.NormNum
#align_import computability.DFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Computability
universe u v
-- Porting note: Required as `DFA` is used in mathlib3
set_option li... | Mathlib/Computability/DFA.lean | 64 | 66 | theorem evalFrom_append_singleton (s : σ) (x : List α) (a : α) :
M.evalFrom s (x ++ [a]) = M.step (M.evalFrom s x) a := by |
simp only [evalFrom, List.foldl_append, List.foldl_cons, List.foldl_nil]
| 0.875 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : ℝ}
-- @... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 49 | 49 | theorem logb_zero : logb b 0 = 0 := by | simp [logb]
| 0.875 |
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section Inducing
variable [To... | Mathlib/Topology/Maps.lean | 132 | 134 | theorem continuous_iff (hg : Inducing g) :
Continuous f ↔ Continuous (g ∘ f) := by |
simp_rw [continuous_iff_continuousAt, hg.continuousAt_iff]
| 0.875 |
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.Dual
#align_import linear_algebra.clifford_algebra.contraction from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2... | Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 159 | 161 | theorem contractRight_mul_algebraMap (a : CliffordAlgebra Q) (r : R) :
a * algebraMap _ _ r⌊d = a⌊d * algebraMap _ _ r := by |
rw [← Algebra.commutes, contractRight_algebraMap_mul, Algebra.commutes]
| 0.875 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.Topology.Algebra.Module.Basic
open Function
structure ContinuousAffineEquiv (k P₁ P₂ : Type*) {V₁ V₂ : Type*} [Ring k]
[AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁]
[AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P... | Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean | 65 | 67 | theorem toAffineEquiv_injective : Injective (toAffineEquiv : (P₁ ≃ᵃL[k] P₂) → P₁ ≃ᵃ[k] P₂) := by |
rintro ⟨e, econt, einv_cont⟩ ⟨e', e'cont, e'inv_cont⟩ H
congr
| 0.875 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 125 | 126 | theorem Ioc_mem_nhds_iff [NoMaxOrder α] {a b x : α} : Ioc a b ∈ 𝓝 x ↔ x ∈ Ioo a b := by |
rw [← interior_Ioc, mem_interior_iff_mem_nhds]
| 0.875 |
import Mathlib.Algebra.Group.Defs
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
universe u
variable {α : Type u}
class Invertible [Mul α] [One α] (a : α) : Type u where
invOf... | Mathlib/Algebra/Group/Invertible/Defs.lean | 136 | 137 | theorem mul_invOf_mul_self_cancel [Monoid α] (a b : α) [Invertible b] : a * ⅟ b * b = a := by |
simp [mul_assoc]
| 0.875 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.pow from "leanprover-community/mathlib"@"46a64b5b4268c594af770c44d9e502afc6a515cb"
variable {R : Type*} {a b : R}
section Monoid
variable [Monoid R]
| Mathlib/Algebra/Regular/Pow.lean | 31 | 32 | theorem IsLeftRegular.pow (n : ℕ) (rla : IsLeftRegular a) : IsLeftRegular (a ^ n) := by |
simp only [IsLeftRegular, ← mul_left_iterate, rla.iterate n]
| 0.875 |
import Mathlib.Algebra.Group.Prod
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Common
#align_import group_theory.perm.basic from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
universe u v
nam... | Mathlib/GroupTheory/Perm/Basic.lean | 125 | 127 | theorem zpow_apply_comm {α : Type*} (σ : Perm α) (m n : ℤ) {x : α} :
(σ ^ m) ((σ ^ n) x) = (σ ^ n) ((σ ^ m) x) := by |
rw [← Equiv.Perm.mul_apply, ← Equiv.Perm.mul_apply, zpow_mul_comm]
| 0.875 |
import Mathlib.Algebra.Algebra.Hom
import Mathlib.RingTheory.Ideal.Quotient
#align_import algebra.ring_quot from "leanprover-community/mathlib"@"e5820f6c8fcf1b75bcd7738ae4da1c5896191f72"
universe uR uS uT uA u₄
variable {R : Type uR} [Semiring R]
variable {S : Type uS} [CommSemiring S]
variable {T : Type uT}
vari... | Mathlib/Algebra/RingQuot.lean | 62 | 64 | theorem Rel.add_right {r : R → R → Prop} ⦃a b c : R⦄ (h : Rel r b c) : Rel r (a + b) (a + c) := by |
rw [add_comm a b, add_comm a c]
exact Rel.add_left h
| 0.875 |
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 | 74 | 77 | theorem IsEquipartition.card_part_le_average_add_one (hP : P.IsEquipartition) (ht : t ∈ P.parts) :
t.card ≤ s.card / P.parts.card + 1 := by |
rw [← P.sum_card_parts]
exact Finset.EquitableOn.le_add_one hP ht
| 0.875 |
import Mathlib.Init.Algebra.Classes
import Mathlib.Init.Data.Ordering.Basic
#align_import init.data.ordering.lemmas from "leanprover-community/lean"@"4bd314f7bd5e0c9e813fc201f1279a23f13f9f1d"
universe u
namespace Ordering
@[simp]
theorem ite_eq_lt_distrib (c : Prop) [Decidable c] (a b : Ordering) :
((if c th... | Mathlib/Init/Data/Ordering/Lemmas.lean | 32 | 34 | theorem ite_eq_gt_distrib (c : Prop) [Decidable c] (a b : Ordering) :
((if c then a else b) = Ordering.gt) = if c then a = Ordering.gt else b = Ordering.gt := by |
by_cases c <;> simp [*]
| 0.875 |
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ ℵ₀
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 46 | 48 | theorem continuum_le_lift {c : Cardinal.{u}} : 𝔠 ≤ lift.{v} c ↔ 𝔠 ≤ c := by |
-- Porting note: added explicit universes
rw [← lift_continuum.{u,v}, lift_le]
| 0.875 |
import Mathlib.Data.List.Chain
#align_import data.list.destutter from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
variable {α : Type*} (l : List α) (R : α → α → Prop) [DecidableRel R] {a b : α}
namespace List
@[simp]
theorem destutter'_nil : destutter' R a [] = [a] :=
rfl
#align ... | Mathlib/Data/List/Destutter.lean | 48 | 49 | theorem destutter'_cons_pos (h : R b a) : (a :: l).destutter' R b = b :: l.destutter' R a := by |
rw [destutter', if_pos h]
| 0.875 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 167 | 169 | theorem csInf_div (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t) :
sInf (s / t) = sInf s / sSup t := by |
rw [div_eq_mul_inv, csInf_mul hs₀ hs₁ ht₀.inv ht₁.inv, csInf_inv ht₀ ht₁, div_eq_mul_inv]
| 0.875 |
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.BoundedOrder
import Mathlib.Mathport.Notation
import Mathlib.Data.Sigma.Basic
#align_import data.sigma.order from "leanprover-community/mathlib"@"1fc36cc9c8264e6e81253f88be7fb2cb6c92d76a"
namespace Sigma
variable {ι : Type*} {α : ι → Type*}
-- Porting note: I... | Mathlib/Data/Sigma/Order.lean | 89 | 96 | theorem lt_def [∀ i, LT (α i)] {a b : Σi, α i} : a < b ↔ ∃ h : a.1 = b.1, h.rec a.2 < b.2 := by |
constructor
· rintro ⟨i, a, b, h⟩
exact ⟨rfl, h⟩
· obtain ⟨i, a⟩ := a
obtain ⟨j, b⟩ := b
rintro ⟨rfl : i = j, h⟩
exact lt.fiber _ _ _ h
| 0.875 |
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.cantor_normal_form from "leanprover-community/mathlib"@"991ff3b5269848f6dd942ae8e9dd3c946035dc8b"
noncomputable section
universe u
open List
namespace Ordinal
@[elab_as_elim]
noncomputabl... | Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | 62 | 64 | theorem CNFRec_pos (b : Ordinal) {o : Ordinal} {C : Ordinal → Sort*} (ho : o ≠ 0) (H0 : C 0)
(H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :
@CNFRec b C H0 H o = H o ho (@CNFRec b C H0 H _) := by | rw [CNFRec, dif_neg ho]
| 0.875 |
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.ord_connected_component from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Interval Function OrderDual
namespace Set
variable {α : Type*} [LinearOrder α] {s t : Set α}... | Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 53 | 54 | theorem self_mem_ordConnectedComponent : x ∈ ordConnectedComponent s x ↔ x ∈ s := by |
rw [mem_ordConnectedComponent, uIcc_self, singleton_subset_iff]
| 0.875 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp
#align_import ring_theory.tensor_product from "leanprover-community/mathlib"@"88fcdc3da43943f5b01925deddaa5bf0c0e85e4e"
suppress_comp... | Mathlib/RingTheory/TensorProduct/Basic.lean | 96 | 98 | theorem baseChange_smul : (r • f).baseChange A = r • f.baseChange A := by |
ext
simp [baseChange_tmul]
| 0.875 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 121 | 122 | theorem map_nat [AddMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1]
(f : F) (n : ℕ) : f n = f 0 + n := by | simp
| 0.875 |
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Final
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
open CategoryTheory.Limits CategoryTheory.Functor Opposite
section ArbitraryUniverses
variable {C : Type u₁} [Category.{v₁}... | Mathlib/CategoryTheory/Filtered/Final.lean | 99 | 104 | theorem Functor.initial_of_exists_of_isCofiltered [IsCofilteredOrEmpty C]
(h₁ : ∀ d, ∃ c, Nonempty (F.obj c ⟶ d)) (h₂ : ∀ {d : D} {c : C} (s s' : F.obj c ⟶ d),
∃ (c' : C) (t : c' ⟶ c), F.map t ≫ s = F.map t ≫ s') : Functor.Initial F := by |
suffices ∀ d, IsCofiltered (CostructuredArrow F d) from
initial_of_isCofiltered_costructuredArrow F
exact isCofiltered_costructuredArrow_of_isCofiltered_of_exists F h₁ h₂
| 0.875 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 221 | 222 | theorem nonsing_inv_apply_not_isUnit (h : ¬IsUnit A.det) : A⁻¹ = 0 := by |
rw [inv_def, Ring.inverse_non_unit _ h, zero_smul]
| 0.875 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PowerBasis
#align_import field_theory.separable from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
... | Mathlib/FieldTheory/Separable.lean | 76 | 78 | theorem separable_X : (X : R[X]).Separable := by |
rw [separable_def, derivative_X]
exact isCoprime_one_right
| 0.875 |
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
namespace Nat
def dist (n m : ℕ) :=
n - m + (m - n)
#align nat.dist Nat.dist
-- Should be aligned to `Nat.dist.eq_def`, but that is generated on demand and isn't pr... | Mathlib/Data/Nat/Dist.lean | 45 | 46 | theorem dist_eq_sub_of_le {n m : ℕ} (h : n ≤ m) : dist n m = m - n := by |
rw [dist, tsub_eq_zero_iff_le.mpr h, zero_add]
| 0.875 |
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 | 65 | 69 | theorem oangle_add_left_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle (x + y) y = Real.arcsin (‖x‖ / ‖x + y‖) := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two h
| 0.875 |
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
open Set Filter Cardinal
universe u
variable {ι : Type u} {α β : Type u} {c : Cardinal.{u}}
class CardinalInterFilter (l : Filter α) (c : Cardinal.{... | Mathlib/Order/Filter/CardinalInter.lean | 107 | 111 | theorem eventually_cardinal_ball {S : Set ι} (hS : #S < c)
{p : α → ∀ i ∈ S, Prop} :
(∀ᶠ x in l, ∀ i hi, p x i hi) ↔ ∀ i hi, ∀ᶠ x in l, p x i hi := by |
simp only [Filter.Eventually, setOf_forall]
exact cardinal_bInter_mem hS
| 0.875 |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.List.Chain
#align_import data.bool.count from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
namespace List
@[simp]
theorem count_not_add_count (l : List Bool) (b : Bool) : count (!b) l + count b l = length l := by
-- Porting ... | Mathlib/Data/Bool/Count.lean | 33 | 34 | theorem count_add_count_not (l : List Bool) (b : Bool) : count b l + count (!b) l = length l := by |
rw [add_comm, count_not_add_count]
| 0.875 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 347 | 350 | theorem contDiffWithinAt_euclidean {n : ℕ∞} :
ContDiffWithinAt 𝕜 n f t y ↔ ∀ i, ContDiffWithinAt 𝕜 n (fun x => f x i) t y := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiffWithinAt_iff, contDiffWithinAt_pi]
rfl
| 0.875 |
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 | 97 | 100 | theorem zpow_injective (h₀ : 0 < a) (h₁ : a ≠ 1) : Injective (a ^ · : ℤ → α) := by |
rcases h₁.lt_or_lt with (H | H)
· exact (zpow_strictAnti h₀ H).injective
· exact (zpow_strictMono H).injective
| 0.875 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 619 | 620 | theorem preimage_mul_const_Ioo (a b : α) {c : α} (h : 0 < c) :
(fun x => x * c) ⁻¹' Ioo a b = Ioo (a / c) (b / c) := by | simp [← Ioi_inter_Iio, h]
| 0.875 |
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Data.Set.UnionLift
#align_import algebra.algebra.subalgebra.basic from "leanprover-community/mathlib"@"b915e9392ecb2a861e1e766f0e1df6ac481188ca"
namespace Subalgebra
open Algebra
variable {R A B : Type*} [CommSemiring R] [Semiring A] [Algebra R A] [... | Mathlib/Algebra/Algebra/Subalgebra/Directed.lean | 85 | 86 | theorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :
(iSupLift K dir f hf T hT).comp (inclusion h) = f i := by | ext; simp
| 0.875 |
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Algebra.NeZero
#align_import algebra.group_with_zero.defs from "leanprover-community/mathlib"@"2f3994e1b117b1e1da49bcfb67334f33460c3ce4"
assert_not_exists DenselyOrdered
universe u
variable {M₀ M₀' : Type*} [MulZeroOneClass M₀] [Nontrivial M₀]
instance N... | Mathlib/Algebra/GroupWithZero/NeZero.lean | 55 | 59 | theorem inv_mul_cancel (h : a ≠ 0) : a⁻¹ * a = 1 :=
calc
a⁻¹ * a = a⁻¹ * a * a⁻¹ * a⁻¹⁻¹ := by | simp [inv_ne_zero h]
_ = a⁻¹ * a⁻¹⁻¹ := by simp [h]
_ = 1 := by simp [inv_ne_zero h]
| 0.875 |
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_import number_theory.zsqrtd.gaussian_int from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open sc... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 89 | 89 | theorem to_real_re (x : ℤ[i]) : ((x.re : ℤ) : ℝ) = (x : ℂ).re := by | simp [toComplex_def]
| 0.875 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
variable ... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 80 | 83 | theorem LpAddConst_zero : LpAddConst 0 = 1 := by |
rw [LpAddConst, if_neg]
intro h
exact lt_irrefl _ h.1
| 0.875 |
import Mathlib.Algebra.Group.Support
import Mathlib.Data.Set.Pointwise.SMul
#align_import data.set.pointwise.support from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open Pointwise
open Function Set
section GroupWithZero
variable {α β γ : Type*} [GroupWithZero α] [MulAction α β]
... | Mathlib/Data/Set/Pointwise/Support.lean | 56 | 59 | theorem support_comp_inv_smul₀ [Zero γ] {c : α} (hc : c ≠ 0) (f : β → γ) :
(support fun x ↦ f (c⁻¹ • x)) = c • support f := by |
ext x
simp only [mem_smul_set_iff_inv_smul_mem₀ hc, mem_support]
| 0.875 |
import Mathlib.AlgebraicTopology.DoldKan.Projections
import Mathlib.CategoryTheory.Idempotents.FunctorCategories
import Mathlib.CategoryTheory.Idempotents.FunctorExtension
#align_import algebraic_topology.dold_kan.p_infty from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open Category... | Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 110 | 112 | theorem PInfty_idem : (PInfty : K[X] ⟶ _) ≫ PInfty = PInfty := by |
ext n
exact PInfty_f_idem n
| 0.875 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 74 | 75 | theorem apply_ne_zero_iff [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i j : V) :
¬A i j = 0 ↔ A i j = 1 := by | rw [← apply_ne_one_iff h, Classical.not_not]
| 0.875 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f... | Mathlib/Data/Finsupp/Multiset.lean | 122 | 125 | theorem toMultiset_inf [DecidableEq α] (f g : α →₀ ℕ) :
toMultiset (f ⊓ g) = toMultiset f ∩ toMultiset g := by |
ext
simp_rw [Multiset.count_inter, Finsupp.count_toMultiset, Finsupp.inf_apply, inf_eq_min]
| 0.875 |
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Set Filter Function
open TopologicalSpace Topology Filter
variable {X : Type*} {Y : Type*} {Z : Type*} {ι : Type*} {f : X → Y} {g : Y → Z}
section OpenMap
variable [Topo... | Mathlib/Topology/Maps.lean | 478 | 482 | theorem of_nonempty (h : ∀ s, IsClosed s → s.Nonempty → IsClosed (f '' s)) :
IsClosedMap f := by |
intro s hs; rcases eq_empty_or_nonempty s with h2s | h2s
· simp_rw [h2s, image_empty, isClosed_empty]
· exact h s hs h2s
| 0.875 |
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Topology.UniformSpace.Basic
#align_import topology.uniform_space.cauchy from "leanprover-community/mathlib"@"22131150f88a2d125713ffa0f4693e3355b1eb49"
universe u v
open scoped Classical
open Filter TopologicalSpace Set Uni... | Mathlib/Topology/UniformSpace/Cauchy.lean | 63 | 67 | theorem Cauchy.ultrafilter_of {l : Filter α} (h : Cauchy l) :
Cauchy (@Ultrafilter.of _ l h.1 : Filter α) := by |
haveI := h.1
have := Ultrafilter.of_le l
exact ⟨Ultrafilter.neBot _, (Filter.prod_mono this this).trans h.2⟩
| 0.875 |
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
#align_import data.typevec from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
universe u v w
@[pp_with_univ]
def TypeVec (n : ℕ) :=
Fin2 n → Type*
#align typevec TypeVec
instance {n} : Inh... | Mathlib/Data/TypeVec.lean | 60 | 62 | theorem Arrow.ext {α β : TypeVec n} (f g : α ⟹ β) :
(∀ i, f i = g i) → f = g := by |
intro h; funext i; apply h
| 0.875 |
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community/mathlib"@"9407b03373c8cd201df99d6bc5514fc2db44054f"
namespace Equiv
variable {α β : Type*} [Finite α]
noncomputable def toCompl {p q : α → Prop} (e ... | Mathlib/Logic/Equiv/Fintype.lean | 145 | 148 | theorem extendSubtype_not_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : ¬p x) :
¬q (e.extendSubtype x) := by |
convert (e.toCompl ⟨x, hx⟩).2
rw [e.extendSubtype_apply_of_not_mem _ hx]
| 0.875 |
import Mathlib.CategoryTheory.Comma.Basic
#align_import category_theory.arrow from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
namespace CategoryTheory
universe v u
-- morphism levels before object levels. See note [CategoryTheory universes].
variable {T : Type u} [Category.{v} T]
... | Mathlib/CategoryTheory/Comma/Arrow.lean | 162 | 163 | theorem hom.congr_left {f g : Arrow T} {φ₁ φ₂ : f ⟶ g} (h : φ₁ = φ₂) : φ₁.left = φ₂.left := by |
rw [h]
| 0.875 |
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 65 | 65 | theorem oangle_self_right (p₁ p₂ : P) : ∡ p₁ p₂ p₂ = 0 := by | simp [oangle]
| 0.875 |
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import analysis.special_functions.polynomials from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Finset Asymptotic... | Mathlib/Analysis/SpecialFunctions/Polynomials.lean | 84 | 88 | theorem abs_tendsto_atTop (hdeg : 0 < P.degree) :
Tendsto (fun x => abs <| eval x P) atTop atTop := by |
rcases le_total 0 P.leadingCoeff with hP | hP
· exact tendsto_abs_atTop_atTop.comp (P.tendsto_atTop_of_leadingCoeff_nonneg hdeg hP)
· exact tendsto_abs_atBot_atTop.comp (P.tendsto_atBot_of_leadingCoeff_nonpos hdeg hP)
| 0.875 |
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Fintype.Prod
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Option
#align_import group_theory.perm.option from "leanprover-community/mathlib"@"c3019c79074b0619edb4b27553a91b2e82242395"
open Equiv
@[simp]
theo... | Mathlib/GroupTheory/Perm/Option.lean | 76 | 77 | theorem Equiv.Perm.decomposeOption_symm_of_none_apply {α : Type*} [DecidableEq α] (e : Perm α)
(i : Option α) : Equiv.Perm.decomposeOption.symm (none, e) i = i.map e := by | simp
| 0.875 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.normed_space.exponential from "leanprover-community/ma... | Mathlib/Analysis/NormedSpace/Exponential.lean | 155 | 157 | theorem exp_unop [T2Space 𝔸] (x : 𝔸ᵐᵒᵖ) :
exp 𝕂 (MulOpposite.unop x) = MulOpposite.unop (exp 𝕂 x) := by |
simp_rw [exp, expSeries_sum_eq, ← MulOpposite.unop_pow, ← MulOpposite.unop_smul, tsum_unop]
| 0.875 |
import Mathlib.Algebra.Polynomial.Degree.Lemmas
open Polynomial
namespace Mathlib.Tactic.ComputeDegree
section recursion_lemmas
variable {R : Type*}
section semiring
variable [Semiring R]
theorem natDegree_C_le (a : R) : natDegree (C a) ≤ 0 := (natDegree_C a).le
theorem natDegree_natCast_le (n : ℕ) : natDeg... | Mathlib/Tactic/ComputeDegree.lean | 101 | 103 | theorem coeff_add_of_eq {n : ℕ} {a b : R} {f g : R[X]}
(h_add_left : f.coeff n = a) (h_add_right : g.coeff n = b) :
(f + g).coeff n = a + b := by | subst ‹_› ‹_›; apply coeff_add
| 0.875 |
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Nilpotent.Basic
#align_import linear_algebra.eigenspace.basic from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
universe u v... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 69 | 69 | theorem eigenspace_zero (f : End R M) : f.eigenspace 0 = LinearMap.ker f := by | simp [eigenspace]
| 0.875 |
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import algebra.order.group.with_top from "leanprover-community/mathlib"@"f178c0e25af359f6cbc72a96a243efd3b12423a3"
namespace WithTop
variable {α : Type*}
namespace LinearOrderedAddCommGroup
variable [LinearOrderedAddCommG... | Mathlib/Algebra/Order/Group/WithTop.lean | 61 | 62 | theorem top_sub {a : WithTop α} : (⊤ : WithTop α) - a = ⊤ := by |
cases a <;> rfl
| 0.875 |
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 | 164 | 164 | theorem tangentMap_id : tangentMap I I (id : M → M) = id := by | ext1 ⟨x, v⟩; simp [tangentMap]
| 0.875 |
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section Binary
variable (xs : Vector α n) (ys : Vector β n)
@[simp]
theorem mapAccumr₂_mapAccumr_left (f₁ : γ → β → σ₁ → σ₁ × ζ) (f₂ : α → σ₂ → σ₂ × γ) :
(mapAccumr₂ f₁ (mapAccumr f₂... | Mathlib/Data/Vector/MapLemmas.lean | 71 | 73 | theorem map₂_map_left (f₁ : γ → β → ζ) (f₂ : α → γ) :
map₂ f₁ (map f₂ xs) ys = map₂ (fun x y => f₁ (f₂ x) y) xs ys := by |
induction xs, ys using Vector.revInductionOn₂ <;> simp_all
| 0.875 |
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 56 | 57 | theorem rpow_def_of_pos {x : ℝ} (hx : 0 < x) (y : ℝ) : x ^ y = exp (log x * y) := by |
rw [rpow_def_of_nonneg (le_of_lt hx), if_neg (ne_of_gt hx)]
| 0.875 |
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 | 102 | 107 | theorem finsuppLeft_apply (t : (ι →₀ M) ⊗[R] N) (i : ι) :
finsuppLeft R M N ι t i = rTensor N (Finsupp.lapply i) t := by |
induction t using TensorProduct.induction_on with
| zero => simp
| tmul f n => simp only [finsuppLeft_apply_tmul_apply, rTensor_tmul, Finsupp.lapply_apply]
| add x y hx hy => simp [map_add, hx, hy]
| 0.875 |
import Mathlib.Topology.Instances.ENNReal
import Mathlib.MeasureTheory.Measure.Dirac
#align_import probability.probability_mass_function.basic from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENNReal M... | Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 107 | 108 | theorem apply_eq_zero_iff (p : PMF α) (a : α) : p a = 0 ↔ a ∉ p.support := by |
rw [mem_support_iff, Classical.not_not]
| 0.875 |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 120 | 121 | theorem tendsto_of_ne_zero {γ : Γ₀} (h : γ ≠ 0) : Tendsto f l (𝓝 γ) ↔ ∀ᶠ x in l, f x = γ := by |
rw [nhds_of_ne_zero h, tendsto_pure]
| 0.875 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe3... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 81 | 83 | theorem memℓp_zero_iff {f : ∀ i, E i} : Memℓp f 0 ↔ Set.Finite { i | f i ≠ 0 } := by |
dsimp [Memℓp]
rw [if_pos rfl]
| 0.875 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.FixedPoints.Topology
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.contracting from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classi... | Mathlib/Topology/MetricSpace/Contracting.lean | 62 | 64 | theorem one_sub_K_ne_top : (1 : ℝ≥0∞) - K ≠ ∞ := by |
norm_cast
exact ENNReal.coe_ne_top
| 0.875 |
import Mathlib.Algebra.Group.Center
#align_import group_theory.subsemigroup.centralizer from "leanprover-community/mathlib"@"cc67cd75b4e54191e13c2e8d722289a89e67e4fa"
variable {M : Type*} {S T : Set M}
namespace Set
variable (S)
@[to_additive addCentralizer " The centralizer of a subset of an additive magma. ... | Mathlib/Algebra/Group/Centralizer.lean | 102 | 105 | theorem div_mem_centralizer₀ [GroupWithZero M] (ha : a ∈ centralizer S) (hb : b ∈ centralizer S) :
a / b ∈ centralizer S := by |
rw [div_eq_mul_inv]
exact mul_mem_centralizer ha (inv_mem_centralizer₀ hb)
| 0.875 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Segment
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.between from "leanprover-community/mathlib"@"571e13cacbed7bf042fd3058c... | Mathlib/Analysis/Convex/Between.lean | 45 | 46 | theorem affineSegment_eq_segment (x y : V) : affineSegment R x y = segment R x y := by |
rw [segment_eq_image_lineMap, affineSegment]
| 0.875 |
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α β : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 28 | 28 | theorem join_singleton (l : List α) : [l].join = l := by | rw [join, join, append_nil]
| 0.875 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 113 | 116 | theorem HasDerivAt.scomp_of_eq
(hg : HasDerivAt g₁ g₁' y) (hh : HasDerivAt h h' x) (hy : y = h x) :
HasDerivAt (g₁ ∘ h) (h' • g₁') x := by |
rw [hy] at hg; exact hg.scomp x hh
| 0.875 |
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.FinsuppVectorSpace
#align_import linear_algebra.tensor_product_basis from "leanprover-community/mathlib"@"f784cc6142443d9ee623a20788c282112c322081"
noncomputable section
open Set LinearMap Submodule
section CommSemiring
variable {R : T... | Mathlib/LinearAlgebra/TensorProduct/Basis.lean | 44 | 46 | theorem Basis.tensorProduct_apply' (b : Basis ι R M) (c : Basis κ R N) (i : ι × κ) :
Basis.tensorProduct b c i = b i.1 ⊗ₜ c i.2 := by |
simp [Basis.tensorProduct]
| 0.875 |
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section General
variable {α : Type*} {g : Gen... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 38 | 38 | theorem terminatedAt_iff_s_none : g.TerminatedAt n ↔ g.s.get? n = none := by | rfl
| 0.875 |
import Mathlib.LinearAlgebra.Matrix.BilinearForm
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.LinearAlgebra.Trace
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosu... | Mathlib/RingTheory/Trace.lean | 163 | 165 | theorem trace_comp_trace [Algebra K T] [Algebra L T] [IsScalarTower K L T] [FiniteDimensional K L]
[FiniteDimensional L T] : (trace K L).comp ((trace L T).restrictScalars K) = trace K T := by |
ext; rw [LinearMap.comp_apply, LinearMap.restrictScalars_apply, trace_trace]
| 0.875 |
import Mathlib.ModelTheory.Syntax
import Mathlib.ModelTheory.Semantics
import Mathlib.Algebra.Ring.Equiv
variable {α : Type*}
namespace FirstOrder
open FirstOrder
inductive ringFunc : ℕ → Type
| add : ringFunc 2
| mul : ringFunc 2
| neg : ringFunc 1
| zero : ringFunc 0
| one : ringFunc 0
deriving D... | Mathlib/ModelTheory/Algebra/Ring/Basic.lean | 199 | 200 | theorem realize_one (v : α → R) : Term.realize v (1 : ring.Term α) = 1 := by |
simp [one_def, funMap_one, constantMap]
| 0.875 |
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.Algebra.Module.Torsion
open Submodule
variable {R : Type*} [CommRing R] (I : Ideal R)
variable {M : Type*} [AddCommGroup M] [Module R M]
namespace AdicCompletion
attribute [-simp] smul_eq_mul Algebra.id.smul_eq_mul
@[local simp]
theorem transitionMap... | Mathlib/RingTheory/AdicCompletion/Algebra.lean | 123 | 125 | theorem evalₐ_mkₐ (n : ℕ) (x : AdicCauchySequence I R) :
evalₐ I n (mkₐ I x) = Ideal.Quotient.mk (I ^ n) (x.val n) := by |
simp [mkₐ]
| 0.875 |
import Mathlib.Algebra.Group.Defs
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
universe u
variable {α : Type u}
class Invertible [Mul α] [One α] (a : α) : Type u where
invOf... | Mathlib/Algebra/Group/Invertible/Defs.lean | 141 | 142 | theorem mul_mul_invOf_self_cancel' [Monoid α] (a b : α) {_ : Invertible b} : a * b * ⅟ b = a := by |
simp [mul_assoc]
| 0.875 |
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 | 82 | 83 | theorem mem_orthogonal_singleton_iff_inner_left {u v : E} : v ∈ (𝕜 ∙ u)ᗮ ↔ ⟪v, u⟫ = 0 := by |
rw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]
| 0.875 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Contraction
variable {R M : Type*}
variable [CommRing R] [AddCommGroup M] [Module R M] {Q : QuadraticForm R M}
namespace CliffordAlgebra
variable (Q)
def invertibleιOfInvertible (m : M) [Invertible (Q m)] : Invertible (ι Q m) where
invOf := ι Q (⅟ (Q m) • m)
invO... | Mathlib/LinearAlgebra/CliffordAlgebra/Inversion.lean | 66 | 69 | theorem isUnit_of_isUnit_ι {m : M} (h : IsUnit (ι Q m)) : IsUnit (Q m) := by |
cases h.nonempty_invertible
letI := invertibleOfInvertibleι Q m
exact isUnit_of_invertible (Q m)
| 0.875 |
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι... | Mathlib/Data/Set/Lattice.lean | 67 | 68 | theorem mem_iUnion₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋃ (i) (j), s i j) ↔ ∃ i j, x ∈ s i j := by |
simp_rw [mem_iUnion]
| 0.875 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Tactic.Positivity.Core
#align_import algebra.order.field.basic from "leanprover-community/mathlib"@"8477... | Mathlib/Algebra/Order/Field/Basic.lean | 117 | 119 | theorem inv_mul_lt_iff (h : 0 < b) : b⁻¹ * a < c ↔ a < b * c := by |
rw [inv_eq_one_div, mul_comm, ← div_eq_mul_one_div]
exact div_lt_iff' h
| 0.875 |
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Pi.Basic
import Mathlib.Data.ULift
#align_import category_theory.discrete_category from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
namespace CategoryTheory
-- morphism levels before object levels. See note [Category... | Mathlib/CategoryTheory/DiscreteCategory.lean | 56 | 57 | theorem Discrete.mk_as {α : Type u₁} (X : Discrete α) : Discrete.mk X.as = X := by |
rfl
| 0.875 |
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open scoped Classical MeasureTheory NNReal ... | Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | 59 | 60 | theorem withDensityᵥ_apply (hf : Integrable f μ) {s : Set α} (hs : MeasurableSet s) :
μ.withDensityᵥ f s = ∫ x in s, f x ∂μ := by | rw [withDensityᵥ, dif_pos hf]; exact dif_pos hs
| 0.875 |
import Mathlib.Geometry.Manifold.ContMDiff.Defs
open Set Filter Function
open scoped Topology 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 H]
(I : Mode... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 252 | 253 | theorem contMDiff_one [One M'] : ContMDiff I I' n (1 : M → M') := by |
simp only [Pi.one_def, contMDiff_const]
| 0.875 |
import Mathlib.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
#align_import topology.metric_space.emetric_space from "leanprover-community/mathlib"@"c8f3055... | Mathlib/Topology/EMetricSpace/Basic.lean | 110 | 111 | theorem edist_triangle_left (x y z : α) : edist x y ≤ edist z x + edist z y := by |
rw [edist_comm z]; apply edist_triangle
| 0.875 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.Logic.Equiv.Fin
#align_import algebra.big_operators.fin from "leanprover-community/mathlib"@"cc5dd6244981976cc9da7afc4eee5682b037a013"
open Fins... | Mathlib/Algebra/BigOperators/Fin.lean | 118 | 119 | theorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by |
simp [prod_univ_succ]
| 0.875 |
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community/mathlib"@"9407b03373c8cd201df99d6bc5514fc2db44054f"
section Fintype
variable {α β : Type*} [Fintype α] [DecidableEq β] (e : Equiv.Perm α) (f : α ↪ β)
... | Mathlib/Logic/Equiv/Fintype.lean | 54 | 57 | theorem Function.Embedding.toEquivRange_eq_ofInjective :
f.toEquivRange = Equiv.ofInjective f f.injective := by |
ext
simp
| 0.875 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {α : Type*}
section support
v... | Mathlib/GroupTheory/Perm/Support.lean | 324 | 329 | theorem support_congr (h : f.support ⊆ g.support) (h' : ∀ x ∈ g.support, f x = g x) : f = g := by |
ext x
by_cases hx : x ∈ g.support
· exact h' x hx
· rw [not_mem_support.mp hx, ← not_mem_support]
exact fun H => hx (h H)
| 0.875 |
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