Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.Analysis.NormedSpace.Star.Basic
#align_import analysis.inner_product_space.dual from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped Classical
o... | Mathlib/Analysis/InnerProductSpace/Dual.lean | 94 | 99 | theorem ext_inner_right_basis {ι : Type*} {x y : E} (b : Basis ι 𝕜 E)
(h : ∀ i : ι, ⟪x, b i⟫ = ⟪y, b i⟫) : x = y := by |
refine ext_inner_left_basis b fun i => ?_
rw [← inner_conj_symm]
conv_rhs => rw [← inner_conj_symm]
exact congr_arg conj (h i)
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.PUnitInstances
import Mathlib.GroupTheory.Congruence.Basic
open FreeMonoid Function List Set
namespace Monoid
@[to_additive "The minimal additive congruence relation `c` on `FreeAddMonoid (M ⊕ N)`... | Mathlib/GroupTheory/Coprod/Basic.lean | 189 | 199 | theorem induction_on' {C : M ∗ N → Prop} (m : M ∗ N)
(one : C 1)
(inl_mul : ∀ m x, C x → C (inl m * x))
(inr_mul : ∀ n x, C x → C (inr n * x)) : C m := by |
rcases mk_surjective m with ⟨x, rfl⟩
induction x using FreeMonoid.recOn with
| h0 => exact one
| ih x xs ih =>
cases x with
| inl m => simpa using inl_mul m _ ih
| inr n => simpa using inr_mul n _ ih
|
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 465 | 467 | theorem toReal_smul (r : ℝ≥0) (s : ℝ≥0∞) : (r • s).toReal = r • s.toReal := by |
rw [ENNReal.smul_def, smul_eq_mul, toReal_mul, coe_toReal]
rfl
|
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
universe u v w
namespace Matrix
open Finset Matrix Polynomial
variable... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 103 | 106 | theorem charpoly_reindex (e : n ≃ m)
(M : Matrix n n R) : (reindex e e M).charpoly = M.charpoly := by |
unfold Matrix.charpoly
rw [charmatrix_reindex, Matrix.det_reindex_self]
|
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace Stream'
open Function
universe u v w
def WSeq (α) :=
Seq (Option α)
#align stream.wseq Stream'.WSeq
... | Mathlib/Data/Seq/WSeq.lean | 667 | 676 | theorem flatten_pure (s : WSeq α) : flatten (Computation.pure s) = s := by |
refine Seq.eq_of_bisim (fun s1 s2 => flatten (Computation.pure s2) = s1) ?_ rfl
intro s' s h
rw [← h]
simp only [Seq.BisimO, flatten, Seq.omap, pure_def, Seq.corec_eq, destruct_pure]
cases Seq.destruct s with
| none => simp
| some val =>
cases' val with o s'
simp
|
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 345 | 348 | theorem partiallyWellOrderedOn_insert :
PartiallyWellOrderedOn (insert a s) r ↔ PartiallyWellOrderedOn s r := by |
simp only [← singleton_union, partiallyWellOrderedOn_union,
partiallyWellOrderedOn_singleton, true_and_iff]
|
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import measure_theory.measure.measure_space from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 329 | 330 | theorem union_ae_eq_right_iff_ae_subset : (s ∪ t : Set α) =ᵐ[μ] t ↔ s ≤ᵐ[μ] t := by |
rw [union_comm, union_ae_eq_left_iff_ae_subset]
|
import Mathlib.Data.PFunctor.Univariate.M
#align_import data.qpf.univariate.basic from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
universe u
class QPF (F : Type u → Type u) [Functor F] where
P : PFunctor.{u}
abs : ∀ {α}, P α → F α
repr : ∀ {α}, F α → P α
abs_repr : ∀ {α} (... | Mathlib/Data/QPF/Univariate/Basic.lean | 279 | 294 | theorem Fix.rec_eq {α : Type _} (g : F α → α) (x : F (Fix F)) :
Fix.rec g (Fix.mk x) = g (Fix.rec g <$> x) := by |
have : recF g ∘ fixToW = Fix.rec g := by
apply funext
apply Quotient.ind
intro x
apply recF_eq_of_Wequiv
rw [fixToW]
apply Wrepr_equiv
conv =>
lhs
rw [Fix.rec, Fix.mk]
dsimp
cases' h : repr x with a f
rw [PFunctor.map_eq, recF_eq, ← PFunctor.map_eq, PFunctor.W.dest_mk, PFunc... |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 760 | 774 | theorem lintegral_indicator_le (f : α → ℝ≥0∞) (s : Set α) :
∫⁻ a, s.indicator f a ∂μ ≤ ∫⁻ a in s, f a ∂μ := by |
simp only [lintegral]
apply iSup_le (fun g ↦ (iSup_le (fun hg ↦ ?_)))
have : g ≤ f := hg.trans (indicator_le_self s f)
refine le_iSup_of_le g (le_iSup_of_le this (le_of_eq ?_))
rw [lintegral_restrict, SimpleFunc.lintegral]
congr with t
by_cases H : t = 0
· simp [H]
congr with x
simp only [mem_preim... |
import Mathlib.Algebra.Algebra.Tower
#align_import algebra.algebra.restrict_scalars from "leanprover-community/mathlib"@"c310cfdc40da4d99a10a58c33a95360ef9e6e0bf"
variable (R S M A : Type*)
@[nolint unusedArguments]
def RestrictScalars (_R _S M : Type*) : Type _ := M
#align restrict_scalars RestrictScalars
ins... | Mathlib/Algebra/Algebra/RestrictScalars.lean | 175 | 179 | theorem RestrictScalars.addEquiv_symm_map_smul_smul (r : R) (s : S) (x : M) :
(RestrictScalars.addEquiv R S M).symm ((r • s) • x) =
r • (RestrictScalars.addEquiv R S M).symm (s • x) := by |
rw [Algebra.smul_def, mul_smul]
rfl
|
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Cast.Order
#align_import data.nat.choose.bounds from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
open Nat
variable {α : Type*} [LinearOrderedSemif... | Mathlib/Data/Nat/Choose/Bounds.lean | 41 | 46 | theorem pow_le_choose (r n : ℕ) : ((n + 1 - r : ℕ) ^ r : α) / r ! ≤ n.choose r := by |
rw [div_le_iff']
· norm_cast
rw [← Nat.descFactorial_eq_factorial_mul_choose]
exact n.pow_sub_le_descFactorial r
exact mod_cast r.factorial_pos
|
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]
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 167 | 167 | theorem volume_Iic {a : ℝ} : volume (Iic a) = ∞ := by | rw [← measure_congr Iio_ae_eq_Iic]; simp
|
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped NNReal Matrix
namespace Matrix
variable {R l m n α β : Type*} [Fintype l] [Fintyp... | Mathlib/Analysis/Matrix.lean | 94 | 95 | theorem nnnorm_le_iff {r : ℝ≥0} {A : Matrix m n α} : ‖A‖₊ ≤ r ↔ ∀ i j, ‖A i j‖₊ ≤ r := by |
simp_rw [nnnorm_def, pi_nnnorm_le_iff]
|
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Order.Filter.Subsingleton
open Set
variable {α β γ δ : Type*} {l : Filter α} {f : α → β}
namespace Filter
def EventuallyConst (f : α → β) (l : Filter α) : Prop := (map f l).Subsingleton
theorem HasBasis.eventuallyConst_iff {ι : Sort*} {p : ι → Prop} {s : ι → S... | Mathlib/Order/Filter/EventuallyConst.lean | 61 | 63 | theorem eventuallyConst_pred {p : α → Prop} :
EventuallyConst p l ↔ (∀ᶠ x in l, p x) ∨ (∀ᶠ x in l, ¬p x) := by |
simp [eventuallyConst_pred', or_comm, EventuallyEq]
|
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 | 277 | 311 | theorem discr_mul_isIntegral_mem_adjoin [IsSeparable K L] [IsIntegrallyClosed R]
[IsFractionRing R K] {B : PowerBasis K L} (hint : IsIntegral R B.gen) {z : L}
(hz : IsIntegral R z) : discr K B.basis • z ∈ adjoin R ({B.gen} : Set L) := by |
have hinv : IsUnit (traceMatrix K B.basis).det := by
simpa [← discr_def] using discr_isUnit_of_basis _ B.basis
have H :
(traceMatrix K B.basis).det • (traceMatrix K B.basis) *ᵥ (B.basis.equivFun z) =
(traceMatrix K B.basis).det • fun i => trace K L (z * B.basis i) := by
congr; exact traceMatrix_o... |
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 153 | 156 | theorem cons_copy {u v w v' w'} (h : G.Adj u v) (p : G.Walk v' w') (hv : v' = v) (hw : w' = w) :
Walk.cons h (p.copy hv hw) = (Walk.cons (hv ▸ h) p).copy rfl hw := by |
subst_vars
rfl
|
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
variable {ι : Type*} [Fintype ι]
variable {M : Type*} [AddCommGroup M] (R : Type*) [Co... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 114 | 121 | theorem Matrix.Represents.mul {A A' : Matrix ι ι R} {f f' : Module.End R M} (h : A.Represents b f)
(h' : Matrix.Represents b A' f') : (A * A').Represents b (f * f') := by |
delta Matrix.Represents PiToModule.fromMatrix
rw [LinearMap.comp_apply, AlgEquiv.toLinearMap_apply, _root_.map_mul]
ext
dsimp [PiToModule.fromEnd]
rw [← h'.congr_fun, ← h.congr_fun]
rfl
|
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 641 | 652 | theorem IsPreconnected.subset_connectedComponentIn {x : α} {F : Set α} (hs : IsPreconnected s)
(hxs : x ∈ s) (hsF : s ⊆ F) : s ⊆ connectedComponentIn F x := by |
have : IsPreconnected (((↑) : F → α) ⁻¹' s) := by
refine inducing_subtype_val.isPreconnected_image.mp ?_
rwa [Subtype.image_preimage_coe, inter_eq_right.mpr hsF]
have h2xs : (⟨x, hsF hxs⟩ : F) ∈ (↑) ⁻¹' s := by
rw [mem_preimage]
exact hxs
have := this.subset_connectedComponent h2xs
rw [connecte... |
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Star.Unitary
import Mathlib.Data.Nat.ModEq
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic.Monotonicity
#align_import number_theory.pell_matiyasevic from "leanprover-community/mathlib"@"795b501869b9f... | Mathlib/NumberTheory/PellMatiyasevic.lean | 622 | 627 | theorem xn_modEq_x2n_sub {n j} (h : j ≤ 2 * n) : xn a1 (2 * n - j) + xn a1 j ≡ 0 [MOD xn a1 n] :=
(le_total j n).elim (xn_modEq_x2n_sub_lem a1) fun jn => by
have : 2 * n - j + j ≤ n + j := by |
rw [tsub_add_cancel_of_le h, two_mul]; exact Nat.add_le_add_left jn _
let t := xn_modEq_x2n_sub_lem a1 (Nat.le_of_add_le_add_right this)
rwa [tsub_tsub_cancel_of_le h, add_comm] at t
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
#align_import linear_algebra.vandermonde from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | Mathlib/LinearAlgebra/Vandermonde.lean | 165 | 168 | theorem det_vandermonde_sub {n : ℕ} (v : Fin n → R) (a : R) :
(Matrix.vandermonde fun i ↦ v i - a).det = (Matrix.vandermonde v).det := by |
rw [← det_vandermonde_add v (- a)]
simp only [← sub_eq_add_neg]
|
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import algebra.quaternion from "leanprover-community/mathlib"@"cf7a7252c19... | Mathlib/Algebra/Quaternion.lean | 767 | 767 | theorem star_mul_eq_coe : star a * a = (star a * a).re := by | ext <;> simp <;> ring
|
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.Topology.NhdsSet
import Mathlib.Algebra.Group.Defs
#align_import topology.uniform_space.basic from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Set F... | Mathlib/Topology/UniformSpace/Basic.lean | 575 | 580 | theorem uniformity_lift_le_swap {g : Set (α × α) → Filter β} {f : Filter β} (hg : Monotone g)
(h : ((𝓤 α).lift fun s => g (preimage Prod.swap s)) ≤ f) : (𝓤 α).lift g ≤ f :=
calc
(𝓤 α).lift g ≤ (Filter.map (@Prod.swap α α) <| 𝓤 α).lift g :=
lift_mono uniformity_le_symm le_rfl
_ ≤ _ := by | rw [map_lift_eq2 hg, image_swap_eq_preimage_swap]; exact h
|
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Data.Complex.Determinant
#align_import analysis.complex.operator_norm from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open ContinuousLinearMap
namespace Complex
@[simp... | Mathlib/Analysis/Complex/OperatorNorm.lean | 50 | 54 | theorem imCLM_norm : ‖imCLM‖ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
calc
1 = ‖imCLM I‖ := by | simp
_ ≤ ‖imCLM‖ := unit_le_opNorm _ _ (by simp)
|
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 | 157 | 165 | theorem degree_eq_of_le_of_coeff_ne_zero' {deg m o : WithBot ℕ} {c : R} {p : R[X]}
(h_deg_le : degree p ≤ m) (coeff_eq : coeff p (WithBot.unbot' 0 deg) = c)
(coeff_ne_zero : c ≠ 0) (deg_eq_deg : m = deg) (coeff_eq_deg : o = deg) :
degree p = deg := by |
subst coeff_eq coeff_eq_deg deg_eq_deg
rcases eq_or_ne m ⊥ with rfl|hh
· exact bot_unique h_deg_le
· obtain ⟨m, rfl⟩ := WithBot.ne_bot_iff_exists.mp hh
exact degree_eq_of_le_of_coeff_ne_zero ‹_› ‹_›
|
import Mathlib.Algebra.Field.Opposite
import Mathlib.Algebra.Group.Subgroup.ZPowers
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Order.Archimedean
import Mathlib.GroupTheory.Coset
#align_import algebra.periodic from "leanprover-community/mathlib"@"3041... | Mathlib/Algebra/Periodic.lean | 128 | 130 | theorem Periodic.const_inv_smul₀ [AddCommMonoid α] [DivisionSemiring γ] [Module γ α]
(h : Periodic f c) (a : γ) : Periodic (fun x => f (a⁻¹ • x)) (a • c) := by |
simpa only [inv_inv] using h.const_smul₀ a⁻¹
|
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.FractionalIdeal.Basic
#align_import ring_theory.fractional_ideal from "leanprover-community/mathlib"@"ed90a7d327c3a5caf65a6faf7e8a0d63c4605df7"
open IsLocalization Pointwise nonZeroDivisors
namespace FractionalIdeal
open Set Submodule
variable... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 455 | 460 | theorem mul_one_div_le_one {I : FractionalIdeal R₁⁰ K} : I * (1 / I) ≤ 1 := by |
by_cases hI : I = 0
· rw [hI, div_zero, mul_zero]
exact zero_le 1
· rw [← coe_le_coe, coe_mul, coe_div hI, coe_one]
apply Submodule.mul_one_div_le_one
|
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable s... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 420 | 446 | theorem continuous_integral_integral :
Continuous fun f : α × β →₁[μ.prod ν] E => ∫ x, ∫ y, f (x, y) ∂ν ∂μ := by |
rw [continuous_iff_continuousAt]; intro g
refine
tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_prod_left
(eventually_of_forall fun h => (L1.integrable_coeFn h).integral_prod_left) ?_
simp_rw [←
lintegral_fn_integral_sub (fun x => (‖x‖₊ : ℝ≥0∞)) (L1.integrable_coeFn _)
(L1.integrab... |
import Mathlib.Order.PropInstances
#align_import order.heyting.basic from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
open Function OrderDual
universe u
variable {ι α β : Type*}
section
variable (α β)
instance Prod.instHImp [HImp α] [HImp β] : HImp (α × β) :=
⟨fun a b => (a.1 ... | Mathlib/Order/Heyting/Basic.lean | 513 | 514 | theorem sup_sdiff_cancel' (hab : a ≤ b) (hbc : b ≤ c) : b ⊔ c \ a = c := by |
rw [sup_sdiff_eq_sup hab, sup_of_le_right hbc]
|
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
noncomputable section
ope... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 275 | 280 | theorem le_dist_coe (z w : ℍ) : w.im * (1 - Real.exp (-dist z w)) ≤ dist (z : ℂ) w :=
calc
w.im * (1 - Real.exp (-dist z w)) =
dist (z : ℂ) (w.center (dist z w)) - dist (w : ℂ) (w.center (dist z w)) := by |
rw [dist_center_dist, dist_self_center, ← Real.cosh_sub_sinh]; ring
_ ≤ dist (z : ℂ) w := sub_le_iff_le_add.2 <| dist_triangle _ _ _
|
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.Algebra.DirectSum.Algebra
suppress_compilation
open scoped TensorProduct DirectSum
variable {R ι A B : Type*}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 111 | 114 | theorem gradedComm_symm : (gradedComm R 𝒜 ℬ).symm = gradedComm R ℬ 𝒜 := by |
rw [gradedComm, gradedComm, LinearEquiv.trans_symm, LinearEquiv.symm_symm]
ext
rfl
|
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 | 674 | 675 | theorem div_le_iff_of_neg' (hc : c < 0) : b / c ≤ a ↔ c * a ≤ b := by |
rw [mul_comm, div_le_iff_of_neg hc]
|
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
noncomputable section
universe u
namespace SetTheory
open scoped PGame
namespace PGame... | Mathlib/SetTheory/Game/Nim.lean | 70 | 70 | theorem rightMoves_nim (o : Ordinal) : (nim o).RightMoves = o.out.α := by | rw [nim_def]; rfl
|
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_schmidt_ortho from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
open Finset Submodule FiniteDimensional
variable (𝕜 : Type*) {E : Type*} [RCLike �... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 68 | 72 | theorem gramSchmidt_def'' (f : ι → E) (n : ι) :
f n = gramSchmidt 𝕜 f n + ∑ i ∈ Iio n,
(⟪gramSchmidt 𝕜 f i, f n⟫ / (‖gramSchmidt 𝕜 f i‖ : 𝕜) ^ 2) • gramSchmidt 𝕜 f i := by |
convert gramSchmidt_def' 𝕜 f n
rw [orthogonalProjection_singleton, RCLike.ofReal_pow]
|
import Batteries.Data.List.Count
import Batteries.Data.Fin.Lemmas
open Nat Function
namespace List
theorem rel_of_pairwise_cons (p : (a :: l).Pairwise R) : ∀ {a'}, a' ∈ l → R a a' :=
(pairwise_cons.1 p).1 _
theorem Pairwise.of_cons (p : (a :: l).Pairwise R) : Pairwise R l :=
(pairwise_cons.1 p).2
theorem... | .lake/packages/batteries/Batteries/Data/List/Pairwise.lean | 149 | 152 | theorem pairwise_filter (p : α → Prop) [DecidablePred p] {l : List α} :
Pairwise R (filter p l) ↔ Pairwise (fun x y => p x → p y → R x y) l := by |
rw [← filterMap_eq_filter, pairwise_filterMap]
simp
|
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.Set.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.double_coset from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
-- Porting note: removed import
-- import Mathlib.Tac... | Mathlib/GroupTheory/DoubleCoset.lean | 52 | 57 | theorem doset_eq_of_mem {H K : Subgroup G} {a b : G} (hb : b ∈ doset a H K) :
doset b H K = doset a H K := by |
obtain ⟨h, hh, k, hk, rfl⟩ := mem_doset.1 hb
rw [doset, doset, ← Set.singleton_mul_singleton, ← Set.singleton_mul_singleton, mul_assoc,
mul_assoc, Subgroup.singleton_mul_subgroup hk, ← mul_assoc, ← mul_assoc,
Subgroup.subgroup_mul_singleton hh]
|
import Mathlib.Probability.Kernel.Disintegration.Integral
open MeasureTheory Set Filter MeasurableSpace
open scoped ENNReal MeasureTheory Topology ProbabilityTheory
namespace ProbabilityTheory
variable {α β Ω : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
[MeasurableSpace Ω] [StandardBorelSpace Ω] ... | Mathlib/Probability/Kernel/Disintegration/Unique.lean | 81 | 124 | theorem eq_condKernel_of_measure_eq_compProd (κ : kernel α Ω) [IsFiniteKernel κ]
(hκ : ρ = ρ.fst ⊗ₘ κ) :
∀ᵐ x ∂ρ.fst, κ x = ρ.condKernel x := by |
-- The idea is to transport the question to `ℝ` from `Ω` using `embeddingReal`
-- and then construct a measure on `α × ℝ`
let f := embeddingReal Ω
have hf := measurableEmbedding_embeddingReal Ω
set ρ' : Measure (α × ℝ) := ρ.map (Prod.map id f) with hρ'def
have hρ' : ρ'.fst = ρ.fst := by
ext s hs
rw... |
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.multivariate.basic from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
universe u v
open MvFunctor
@[pp_with_univ]
structure MvPFunctor (n : ℕ) where
A : Type u
... | Mathlib/Data/PFunctor/Multivariate/Basic.lean | 173 | 179 | theorem liftP_iff' {α : TypeVec n} (p : ∀ ⦃i⦄, α i → Prop) (a : P.A) (f : P.B a ⟹ α) :
@LiftP.{u} _ P.Obj _ α p ⟨a, f⟩ ↔ ∀ i x, p (f i x) := by |
simp only [liftP_iff, Sigma.mk.inj_iff]; constructor
· rintro ⟨_, _, ⟨⟩, _⟩
assumption
· intro
repeat' first |constructor|assumption
|
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 | 562 | 571 | theorem oangle_sub_left_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
o.oangle (x - r • o.rotation (π / 2 : ℝ) x) x = Real.arctan r := by |
by_cases hr : r = 0; · simp [hr]
have hx : x = r⁻¹ • o.rotation (π / 2 : ℝ) (-(r • o.rotation (π / 2 : ℝ) x)) := by
simp [hr, ← Real.Angle.coe_add]
rw [sub_eq_add_neg, add_comm]
nth_rw 3 [hx]
nth_rw 2 [hx]
rw [o.oangle_add_left_smul_rotation_pi_div_two, inv_inv]
simpa [hr] using h
|
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
#align_import order.chain from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open scoped Classical
open Set
variable {α β : Type*}
section Chain
variable (r : α → α → Prop)
... | Mathlib/Order/Chain.lean | 137 | 142 | theorem IsChain.exists3 (hchain : IsChain r s) [IsTrans α r] {a b c} (mem1 : a ∈ s) (mem2 : b ∈ s)
(mem3 : c ∈ s) : ∃ (z : _) (_ : z ∈ s), r a z ∧ r b z ∧ r c z := by |
rcases directedOn_iff_directed.mpr (IsChain.directed hchain) a mem1 b mem2 with ⟨z, mem4, H1, H2⟩
rcases directedOn_iff_directed.mpr (IsChain.directed hchain) z mem4 c mem3 with
⟨z', mem5, H3, H4⟩
exact ⟨z', mem5, _root_.trans H1 H3, _root_.trans H2 H3, H4⟩
|
import Mathlib.CategoryTheory.Comma.Basic
import Mathlib.CategoryTheory.PUnit
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.Logic.Small.Set
#align_import category_theory.structured_arrow from "leanprover-community/mathlib"@"8a318021995877a44630c898d... | Mathlib/CategoryTheory/Comma/StructuredArrow.lean | 244 | 246 | theorem map_id {f : StructuredArrow S T} : (map (𝟙 S)).obj f = f := by |
rw [eq_mk f]
simp
|
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Analysis.Calculus.LagrangeMultipliers
import Mathlib.LinearAlgebra.Eigenspace.Basic
#align_import analysis.inner_product_space.rayleigh from "leanprover-co... | Mathlib/Analysis/InnerProductSpace/Rayleigh.lean | 119 | 138 | theorem linearly_dependent_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : F}
(hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere (0 : F) ‖x₀‖) x₀) :
∃ a b : ℝ, (a, b) ≠ 0 ∧ a • x₀ + b • T x₀ = 0 := by |
have H : IsLocalExtrOn T.reApplyInnerSelf {x : F | ‖x‖ ^ 2 = ‖x₀‖ ^ 2} x₀ := by
convert hextr
ext x
simp [dist_eq_norm]
-- find Lagrange multipliers for the function `T.re_apply_inner_self` and the
-- hypersurface-defining function `fun x ↦ ‖x‖ ^ 2`
obtain ⟨a, b, h₁, h₂⟩ :=
IsLocalExtrOn.exists... |
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Set Filter Topology
variable {X : Type*}
def OnePoint (X : Type*) :=
Option X
#ali... | Mathlib/Topology/Compactification/OnePoint.lean | 211 | 213 | theorem isOpen_iff_of_mem' (h : ∞ ∈ s) :
IsOpen s ↔ IsCompact ((↑) ⁻¹' s : Set X)ᶜ ∧ IsOpen ((↑) ⁻¹' s : Set X) := by |
simp [isOpen_def, h]
|
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Order.Basic
import Mathlib.Order.Hom.Lattice
#align_import rin... | Mathlib/RingTheory/Filtration.lean | 234 | 238 | theorem stable_iff_exists_pow_smul_eq_of_ge :
F.Stable ↔ ∃ n₀, ∀ n ≥ n₀, F.N n = I ^ (n - n₀) • F.N n₀ := by |
refine ⟨Stable.exists_pow_smul_eq_of_ge, fun h => ⟨h.choose, fun n hn => ?_⟩⟩
rw [h.choose_spec n hn, h.choose_spec (n + 1) (by omega), smul_smul, ← pow_succ',
tsub_add_eq_add_tsub hn]
|
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace List
variable [D... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 105 | 117 | theorem cycleType_formPerm (hl : Nodup l) (hn : 2 ≤ l.length) :
cycleType l.attach.formPerm = {l.length} := by |
rw [← length_attach] at hn
rw [← nodup_attach] at hl
rw [cycleType_eq [l.attach.formPerm]]
· simp only [map, Function.comp_apply]
rw [support_formPerm_of_nodup _ hl, card_toFinset, dedup_eq_self.mpr hl]
· simp
· intro x h
simp [h, Nat.succ_le_succ_iff] at hn
· simp
· simpa using isCycle_f... |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 102 | 105 | theorem mem_support_not_mem_vars_zero {f : MvPolynomial σ R} {x : σ →₀ ℕ} (H : x ∈ f.support)
{v : σ} (h : v ∉ vars f) : x v = 0 := by |
contrapose! h
exact (mem_vars v).mpr ⟨x, H, Finsupp.mem_support_iff.mpr h⟩
|
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 377 | 394 | theorem exists_linearIsometryEquiv_eq_of_det_pos {f : V ≃ₗᵢ[ℝ] V}
(hd : 0 < LinearMap.det (f.toLinearEquiv : V →ₗ[ℝ] V)) :
∃ θ : Real.Angle, f = o.rotation θ := by |
haveI : Nontrivial V :=
FiniteDimensional.nontrivial_of_finrank_eq_succ (@Fact.out (finrank ℝ V = 2) _)
obtain ⟨x, hx⟩ : ∃ x, x ≠ (0 : V) := exists_ne (0 : V)
use o.oangle x (f x)
apply LinearIsometryEquiv.toLinearEquiv_injective
apply LinearEquiv.toLinearMap_injective
apply (o.basisRightAngleRotation ... |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.continuous_affine_map from "leanprover-community/mathlib"@"bd1fc183335ea95a9519a1630bcf901fe9326d83"
structure ContinuousAffineMap (R : T... | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | 127 | 129 | theorem mk_coe (f : P →ᴬ[R] Q) (h) : (⟨(f : P →ᵃ[R] Q), h⟩ : P →ᴬ[R] Q) = f := by |
ext
rfl
|
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 69 | 75 | theorem IsLindelof.induction_on (hs : IsLindelof s) {p : Set X → Prop}
(hmono : ∀ ⦃s t⦄, s ⊆ t → p t → p s)
(hcountable_union : ∀ (S : Set (Set X)), S.Countable → (∀ s ∈ S, p s) → p (⋃₀ S))
(hnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t) : p s := by |
let f : Filter X := ofCountableUnion p hcountable_union (fun t ht _ hsub ↦ hmono hsub ht)
have : sᶜ ∈ f := hs.compl_mem_sets_of_nhdsWithin (by simpa [f] using hnhds)
rwa [← compl_compl s]
|
import Mathlib.Topology.Sheaves.PUnit
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.Topology.Sheaves.Functors
#align_import topology.sheaves.skyscraper from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open TopologicalSpace TopCat CategoryTheory CategoryT... | Mathlib/Topology/Sheaves/Skyscraper.lean | 100 | 107 | theorem SkyscraperPresheafFunctor.map'_comp {a b c : C} (f : a ⟶ b) (g : b ⟶ c) :
SkyscraperPresheafFunctor.map' p₀ (f ≫ g) =
SkyscraperPresheafFunctor.map' p₀ f ≫ SkyscraperPresheafFunctor.map' p₀ g := by |
ext U
-- Porting note: change `simp` to `rw`
rw [NatTrans.comp_app]
simp only [SkyscraperPresheafFunctor.map'_app]
split_ifs with h <;> aesop_cat
|
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 | 142 | 156 | theorem isSymmetric_iff_inner_map_self_real (T : V →ₗ[ℂ] V) :
IsSymmetric T ↔ ∀ v : V, conj ⟪T v, v⟫_ℂ = ⟪T v, v⟫_ℂ := by |
constructor
· intro hT v
apply IsSymmetric.conj_inner_sym hT
· intro h x y
rw [← inner_conj_symm x (T y)]
rw [inner_map_polarization T x y]
simp only [starRingEnd_apply, star_div', star_sub, star_add, star_mul]
simp only [← starRingEnd_apply]
rw [h (x + y), h (x - y), h (x + Complex.I • y... |
import Batteries.Tactic.SeqFocus
namespace Ordering
@[simp] theorem swap_swap {o : Ordering} : o.swap.swap = o := by cases o <;> rfl
@[simp] theorem swap_inj {o₁ o₂ : Ordering} : o₁.swap = o₂.swap ↔ o₁ = o₂ :=
⟨fun h => by simpa using congrArg swap h, congrArg _⟩
theorem swap_then (o₁ o₂ : Ordering) : (o₁.then... | .lake/packages/batteries/Batteries/Classes/Order.lean | 26 | 27 | theorem then_eq_gt {o₁ o₂ : Ordering} : o₁.then o₂ = gt ↔ o₁ = gt ∨ o₁ = eq ∧ o₂ = gt := by |
cases o₁ <;> cases o₂ <;> decide
|
import Mathlib.Topology.Order.LeftRight
import Mathlib.Topology.Order.Monotone
#align_import topology.algebra.order.left_right_lim from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
open Set Filter
open Topology
section
variable {α β : Type*} [LinearOrder α] [TopologicalSpace β]
n... | Mathlib/Topology/Order/LeftRightLim.lean | 201 | 208 | theorem continuousWithinAt_Iio_iff_leftLim_eq :
ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x := by |
rcases eq_or_ne (𝓝[<] x) ⊥ with (h' | h')
· simp [leftLim_eq_of_eq_bot f h', ContinuousWithinAt, h']
haveI : (𝓝[Iio x] x).NeBot := neBot_iff.2 h'
refine ⟨fun h => tendsto_nhds_unique (hf.tendsto_leftLim x) h.tendsto, fun h => ?_⟩
have := hf.tendsto_leftLim x
rwa [h] at this
|
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.NumberTheory.NumberField.Discriminant
#align_import number_theory.cyclotomic.discriminant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
universe u v
open Algebra Polynomial Nat IsPrimitiveRoot PowerBasis
open s... | Mathlib/NumberTheory/Cyclotomic/Discriminant.lean | 141 | 186 | theorem discr_prime_pow [hcycl : IsCyclotomicExtension {p ^ k} K L] [hp : Fact (p : ℕ).Prime]
(hζ : IsPrimitiveRoot ζ ↑(p ^ k)) (hirr : Irreducible (cyclotomic (↑(p ^ k) : ℕ) K)) :
discr K (hζ.powerBasis K).basis =
(-1) ^ ((p ^ k : ℕ).totient / 2) * p ^ ((p : ℕ) ^ (k - 1) * ((p - 1) * k - 1)) := by |
cases' k with k k
· simp only [coe_basis, _root_.pow_zero, powerBasis_gen _ hζ, totient_one, mul_zero, mul_one,
show 1 / 2 = 0 by rfl, discr, traceMatrix]
have hζone : ζ = 1 := by simpa using hζ
rw [hζ.powerBasis_dim _, hζone, ← (algebraMap K L).map_one,
minpoly.eq_X_sub_C_of_algebraMap_inj _ (... |
import Mathlib.Computability.Encoding
import Mathlib.Logic.Small.List
import Mathlib.ModelTheory.Syntax
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import model_theory.encoding from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada"
universe u v w u' v'
namespace FirstOrder
namespace... | Mathlib/ModelTheory/Encoding.lean | 67 | 98 | theorem listDecode_encode_list (l : List (L.Term α)) :
listDecode (l.bind listEncode) = l.map Option.some := by |
suffices h : ∀ (t : L.Term α) (l : List (Sum α (Σi, L.Functions i))),
listDecode (t.listEncode ++ l) = some t::listDecode l by
induction' l with t l lih
· rfl
· rw [cons_bind, h t (l.bind listEncode), lih, List.map]
intro t
induction' t with a n f ts ih <;> intro l
· rw [listEncode, singleton... |
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.OpenPos
#align_import measure_theory.constructions.prod.basic from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb32... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 785 | 786 | theorem dirac_prod_dirac {x : α} {y : β} : (dirac x).prod (dirac y) = dirac (x, y) := by |
rw [prod_dirac, map_dirac measurable_prod_mk_right]
|
import Mathlib.Data.Multiset.Nodup
import Mathlib.Data.List.NatAntidiagonal
#align_import data.multiset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
namespace Nat
def antidiagonal (n : ℕ) : Multiset (ℕ × ℕ) :=
List.Nat.antidiagonal n
#align... | Mathlib/Data/Multiset/NatAntidiagonal.lean | 59 | 61 | theorem antidiagonal_succ {n : ℕ} :
antidiagonal (n + 1) = (0, n + 1) ::ₘ (antidiagonal n).map (Prod.map Nat.succ id) := by |
simp only [antidiagonal, List.Nat.antidiagonal_succ, map_coe, cons_coe]
|
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 213 | 218 | theorem degree_le_degree (h : coeff q (natDegree p) ≠ 0) : degree p ≤ degree q := by |
by_cases hp : p = 0
· rw [hp, degree_zero]
exact bot_le
· rw [degree_eq_natDegree hp]
exact le_degree_of_ne_zero h
|
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
noncomputable section
open Function Cardinal Set Equiv Order
open scoped Classical
open Cardinal Ordinal
universe u v w
namespace Ordinal
in... | Mathlib/SetTheory/Ordinal/Exponential.lean | 414 | 423 | theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hvb : v < b)
(hw : w < b ^ u) : log b (b ^ u * v + w) = u := by |
have hne' := (opow_mul_add_pos (zero_lt_one.trans hb).ne' u hv w).ne'
by_contra! hne
cases' lt_or_gt_of_ne hne with h h
· rw [← lt_opow_iff_log_lt hb hne'] at h
exact h.not_le ((le_mul_left _ (Ordinal.pos_iff_ne_zero.2 hv)).trans (le_add_right _ _))
· conv at h => change u < log b (b ^ u * v + w)
rw ... |
import Mathlib.Analysis.Convex.Slope
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Tactic.LinearCombination
#align_import analysis.convex.specific_functions.basic from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
open Real Set NNReal
theorem strictConvexOn_exp : St... | Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean | 67 | 94 | theorem strictConcaveOn_log_Ioi : StrictConcaveOn ℝ (Ioi 0) log := by |
apply strictConcaveOn_of_slope_strict_anti_adjacent (convex_Ioi (0 : ℝ))
intro x y z (hx : 0 < x) (hz : 0 < z) hxy hyz
have hy : 0 < y := hx.trans hxy
trans y⁻¹
· have h : 0 < z - y := by linarith
rw [div_lt_iff h]
have hyz' : 0 < z / y := by positivity
have hyz'' : z / y ≠ 1 := by
contrapo... |
import Mathlib.LinearAlgebra.Dimension.Basic
import Mathlib.SetTheory.Cardinal.ToNat
#align_import linear_algebra.finrank from "leanprover-community/mathlib"@"347636a7a80595d55bedf6e6fbd996a3c39da69a"
universe u v w
open Cardinal Submodule Module Function
variable {R : Type u} {M : Type v} {N : Type w}
variable... | Mathlib/LinearAlgebra/Dimension/Finrank.lean | 58 | 61 | theorem finrank_eq_of_rank_eq {n : ℕ} (h : Module.rank R M = ↑n) : finrank R M = n := by |
apply_fun toNat at h
rw [toNat_natCast] at h
exact mod_cast h
|
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.Derivation.Basic
#align_import data.mv_polynomial.derivation from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3"
namespace MvPolynomial
noncomputable section
variable {σ R A : Type*} [CommSemiring R] [AddCommMonoi... | Mathlib/Algebra/MvPolynomial/Derivation.lean | 96 | 114 | theorem leibniz_iff_X (D : MvPolynomial σ R →ₗ[R] A) (h₁ : D 1 = 0) :
(∀ p q, D (p * q) = p • D q + q • D p) ↔ ∀ s i, D (monomial s 1 * X i) =
(monomial s 1 : MvPolynomial σ R) • D (X i) + (X i : MvPolynomial σ R) • D (monomial s 1) := by |
refine ⟨fun H p i => H _ _, fun H => ?_⟩
have hC : ∀ r, D (C r) = 0 := by intro r; rw [C_eq_smul_one, D.map_smul, h₁, smul_zero]
have : ∀ p i, D (p * X i) = p • D (X i) + (X i : MvPolynomial σ R) • D p := by
intro p i
induction' p using MvPolynomial.induction_on' with s r p q hp hq
· rw [← mul_one r,... |
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.FieldDivision
#align_import ring_theory.polynomial.content from "leanprover-community/mathlib"@"7a030ab8eb5d99f05a891dccc49c5b5b90c947d3"
namespace Po... | Mathlib/RingTheory/Polynomial/Content.lean | 61 | 63 | theorem IsPrimitive.ne_zero [Nontrivial R] {p : R[X]} (hp : p.IsPrimitive) : p ≠ 0 := by |
rintro rfl
exact (hp 0 (dvd_zero (C 0))).ne_zero rfl
|
import Mathlib.CategoryTheory.Adjunction.Basic
open CategoryTheory
variable {C D : Type*} [Category C] [Category D]
namespace CategoryTheory.Adjunction
@[simps]
def natTransEquiv {F F' : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G') :
(G ⟶ G') ≃ (F' ⟶ F) where
toFun f := {
app := fun X ↦ F'.map... | Mathlib/CategoryTheory/Adjunction/Unique.lean | 170 | 175 | theorem leftAdjointUniq_trans_app {F F' F'' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G)
(adj3 : F'' ⊣ G) (x : C) :
(leftAdjointUniq adj1 adj2).hom.app x ≫ (leftAdjointUniq adj2 adj3).hom.app x =
(leftAdjointUniq adj1 adj3).hom.app x := by |
rw [← leftAdjointUniq_trans adj1 adj2 adj3]
rfl
|
import Mathlib.CategoryTheory.Functor.Flat
import Mathlib.CategoryTheory.Sites.Sheaf
import Mathlib.Tactic.ApplyFun
#align_import category_theory.sites.cover_preserving from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
universe w v₁ v₂ v₃ u₁ u₂ u₃
noncomputable section
open CategoryT... | Mathlib/CategoryTheory/Sites/CoverPreserving.lean | 103 | 112 | theorem Presieve.FamilyOfElements.Compatible.functorPushforward :
(x.functorPushforward G).Compatible := by |
rintro Z₁ Z₂ W g₁ g₂ f₁' f₂' H₁ H₂ eq
unfold FamilyOfElements.functorPushforward
rcases getFunctorPushforwardStructure H₁ with ⟨X₁, f₁, h₁, hf₁, rfl⟩
rcases getFunctorPushforwardStructure H₂ with ⟨X₂, f₂, h₂, hf₂, rfl⟩
suffices ℱ.val.map (g₁ ≫ h₁).op (x f₁ hf₁) = ℱ.val.map (g₂ ≫ h₂).op (x f₂ hf₂) by
simp... |
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 | 114 | 115 | theorem continuousAt_dslope_of_ne (h : b ≠ a) : ContinuousAt (dslope f a) b ↔ ContinuousAt f b := by |
simp only [← continuousWithinAt_univ, continuousWithinAt_dslope_of_ne h]
|
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.GroupTheory.GroupAction.SubMulAction
open scoped BigOperators Pointwise
namespace MulAction
section SMul
variable (G : Type*) {X : Type*} [SMul G X]
-- Change termin... | Mathlib/GroupTheory/GroupAction/Blocks.lean | 107 | 108 | theorem isBlock_singleton (a : X) : IsBlock G ({a} : Set X) := by |
simp [IsBlock.def, Classical.or_iff_not_imp_left]
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 70 | 71 | theorem taylor_monomial (i : ℕ) (k : R) : taylor r (monomial i k) = C k * (X + C r) ^ i := by |
simp [taylor_apply]
|
import Mathlib.Algebra.Module.Submodule.Lattice
import Mathlib.Algebra.Module.Submodule.LinearMap
open Function Pointwise Set
variable {R : Type*} {R₁ : Type*} {R₂ : Type*} {R₃ : Type*}
variable {M : Type*} {M₁ : Type*} {M₂ : Type*} {M₃ : Type*}
namespace Submodule
section AddCommMonoid
variable [Semiring R] [... | Mathlib/Algebra/Module/Submodule/Map.lean | 213 | 217 | theorem le_comap_pow_of_le_comap (p : Submodule R M) {f : M →ₗ[R] M} (h : p ≤ p.comap f) (k : ℕ) :
p ≤ p.comap (f ^ k) := by |
induction' k with k ih
· simp [LinearMap.one_eq_id]
· simp [LinearMap.iterate_succ, comap_comp, h.trans (comap_mono ih)]
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Card
#align_import data.multiset.fintype from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
variable {α : Type*} [DecidableEq α] {m : Multiset α}
def Multiset.ToType (m : Multiset α) : Type _ := (x : α) × Fi... | Mathlib/Data/Multiset/Fintype.lean | 194 | 202 | theorem Multiset.toEnumFinset_filter_eq (m : Multiset α) (x : α) :
(m.toEnumFinset.filter fun p ↦ x = p.1) =
(Finset.range (m.count x)).map ⟨Prod.mk x, Prod.mk.inj_left x⟩ := by |
ext ⟨y, i⟩
simp only [eq_comm, Finset.mem_filter, Multiset.mem_toEnumFinset, Finset.mem_map,
Finset.mem_range, Function.Embedding.coeFn_mk, Prod.mk.inj_iff, exists_prop,
exists_eq_right_right', and_congr_left_iff]
rintro rfl
rfl
|
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 389 | 389 | theorem smul_eq_C_mul (a : R) : a • p = C a * p := by | simp [ext_iff]
|
import Mathlib.Data.Finset.Lattice
#align_import data.finset.pairwise from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Finset
variable {α ι ι' : Type*}
instance [DecidableEq α] {r : α → α → Prop} [DecidableRel r] {s : Finset α} :
Decidable ((s : Set α).Pairwise r) :=
dec... | Mathlib/Data/Finset/Pairwise.lean | 44 | 48 | theorem PairwiseDisjoint.image_finset_of_le [DecidableEq ι] {s : Finset ι} {f : ι → α}
(hs : (s : Set ι).PairwiseDisjoint f) {g : ι → ι} (hf : ∀ a, f (g a) ≤ f a) :
(s.image g : Set ι).PairwiseDisjoint f := by |
rw [coe_image]
exact hs.image_of_le hf
|
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
open Set Filter MeasureTheory MeasurableSpace
open scoped Classical Topology NNReal ENNReal MeasureTheory
univers... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 301 | 311 | theorem measurable_of_tendsto' {ι : Type*} {f : ι → α → ℝ≥0∞} {g : α → ℝ≥0∞} (u : Filter ι)
[NeBot u] [IsCountablyGenerated u] (hf : ∀ i, Measurable (f i)) (lim : Tendsto f u (𝓝 g)) :
Measurable g := by |
rcases u.exists_seq_tendsto with ⟨x, hx⟩
rw [tendsto_pi_nhds] at lim
have : (fun y => liminf (fun n => (f (x n) y : ℝ≥0∞)) atTop) = g := by
ext1 y
exact ((lim y).comp hx).liminf_eq
rw [← this]
show Measurable fun y => liminf (fun n => (f (x n) y : ℝ≥0∞)) atTop
exact measurable_liminf fun n => hf (x... |
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
#align_import ring_theory.witt_vector.basic from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
noncomputable section
open MvPolynomial Function
variable... | Mathlib/RingTheory/WittVector/Basic.lean | 183 | 185 | theorem matrix_vecEmpty_coeff {R} (i j) :
@coeff p R (Matrix.vecEmpty i) j = (Matrix.vecEmpty i : ℕ → R) j := by |
rcases i with ⟨_ | _ | _ | _ | i_val, ⟨⟩⟩
|
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.basis from "leanprover-community/mathlib"@"6c263e4bfc2e6714de30f22178b4d0ca4d149a76"
noncomputable section
open LinearMap Matrix Set Submodule
open Matrix
section BasisToMatrix
variable {ι... | Mathlib/LinearAlgebra/Matrix/Basis.lean | 117 | 122 | theorem toMatrix_smul {R₁ S : Type*} [CommRing R₁] [Ring S] [Algebra R₁ S] [Fintype ι]
[DecidableEq ι] (x : S) (b : Basis ι R₁ S) (w : ι → S) :
(b.toMatrix (x • w)) = (Algebra.leftMulMatrix b x) * (b.toMatrix w) := by |
ext
rw [Basis.toMatrix_apply, Pi.smul_apply, smul_eq_mul, ← Algebra.leftMulMatrix_mulVec_repr]
rfl
|
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.HahnBanach.Separation
import Mathlib.LinearAlgebra.Dual
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
@[mk_iff separatingDual_def]
class SeparatingDual (R V : Type*) [Ring R] [AddCommGroup V] [TopologicalSpace V]
... | Mathlib/Analysis/NormedSpace/HahnBanach/SeparatingDual.lean | 91 | 97 | theorem dualMap_surjective_iff {W} [AddCommGroup W] [Module R W] [FiniteDimensional R W]
{f : W →ₗ[R] V} : Surjective (f.dualMap ∘ ContinuousLinearMap.toLinearMap) ↔ Injective f := by |
constructor <;> intro hf
· exact LinearMap.dualMap_surjective_iff.mp hf.of_comp
have := (separatingDual_iff_injective.mp ‹_›).comp hf
rw [← LinearMap.coe_comp] at this
exact LinearMap.flip_surjective_iff₁.mpr this
|
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 | 569 | 571 | theorem mem_rootSet {p : T[X]} {S : Type*} [CommRing S] [IsDomain S] [Algebra T S]
[NoZeroSMulDivisors T S] {a : S} : a ∈ p.rootSet S ↔ p ≠ 0 ∧ aeval a p = 0 := by |
rw [mem_rootSet', Polynomial.map_ne_zero_iff (NoZeroSMulDivisors.algebraMap_injective T S)]
|
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.basis from "leanprover-community/mathlib"@"6c263e4bfc2e6714de30f22178b4d0ca4d149a76"
noncomputable section
open LinearMap Matrix Set Submodule
open Matrix
section BasisToMatrix
variable {ι... | Mathlib/LinearAlgebra/Matrix/Basis.lean | 165 | 172 | theorem restrictScalars_toMatrix [Fintype ι] [DecidableEq ι] {S : Type*} [CommRing S] [Nontrivial S]
[Algebra R₂ S] [Module S M₂] [IsScalarTower R₂ S M₂] [NoZeroSMulDivisors R₂ S]
(b : Basis ι S M₂) (v : ι → span R₂ (Set.range b)) :
(algebraMap R₂ S).mapMatrix ((b.restrictScalars R₂).toMatrix v) =
b.t... |
ext
rw [RingHom.mapMatrix_apply, Matrix.map_apply, Basis.toMatrix_apply,
Basis.restrictScalars_repr_apply, Basis.toMatrix_apply]
|
import Mathlib.Topology.Algebra.Module.StrongTopology
import Mathlib.Topology.Algebra.Module.LocallyConvex
#align_import analysis.locally_convex.strong_topology from "leanprover-community/mathlib"@"47b12e7f2502f14001f891ca87fbae2b4acaed3f"
open Topology UniformConvergence
variable {R 𝕜₁ 𝕜₂ E F : Type*}
variab... | Mathlib/Analysis/LocallyConvex/StrongTopology.lean | 47 | 54 | theorem locallyConvexSpace (𝔖 : Set (Set E)) (h𝔖₁ : 𝔖.Nonempty)
(h𝔖₂ : DirectedOn (· ⊆ ·) 𝔖) :
LocallyConvexSpace R (UniformConvergenceCLM σ F 𝔖) := by |
apply LocallyConvexSpace.ofBasisZero _ _ _ _
(UniformConvergenceCLM.hasBasis_nhds_zero_of_basis _ _ _ h𝔖₁ h𝔖₂
(LocallyConvexSpace.convex_basis_zero R F)) _
rintro ⟨S, V⟩ ⟨_, _, hVconvex⟩ f hf g hg a b ha hb hab x hx
exact hVconvex (hf x hx) (hg x hx) ha hb hab
|
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.box_integral.box.subbox_induction from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Finset Function Filter Metric Classical Topology Filter ENNReal
noncomputable... | Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean | 101 | 103 | theorem upper_sub_lower_splitCenterBox (I : Box ι) (s : Set ι) (i : ι) :
(I.splitCenterBox s).upper i - (I.splitCenterBox s).lower i = (I.upper i - I.lower i) / 2 := by |
by_cases i ∈ s <;> field_simp [splitCenterBox] <;> field_simp [mul_two, two_mul]
|
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
theorem powerset_insert (s : Set α) (a : α)... | Mathlib/Data/Set/Image.lean | 666 | 666 | theorem exists_range_iff {p : α → Prop} : (∃ a ∈ range f, p a) ↔ ∃ i, p (f i) := by | simp
|
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Star.Unitary
import Mathlib.Data.Nat.ModEq
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic.Monotonicity
#align_import number_theory.pell_matiyasevic from "leanprover-community/mathlib"@"795b501869b9f... | Mathlib/NumberTheory/PellMatiyasevic.lean | 668 | 743 | theorem eq_of_xn_modEq_lem3 {i n} (npos : 0 < n) :
∀ {j}, i < j → j ≤ 2 * n → j ≠ n → ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2) →
xn a1 i % xn a1 n < xn a1 j % xn a1 n
| 0, ij, _, _, _ => absurd ij (Nat.not_lt_zero _)
| j + 1, ij, j2n, jnn, ntriv =>
have lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ)... |
rw [← two_mul, mul_comm,
show xn a1 n = xn a1 (n - 1 + 1) by rw [tsub_add_cancel_of_le (succ_le_of_lt npos)],
xn_succ]
exact le_trans (Nat.mul_le_mul_left _ a1) (Nat.le_add_right _ _)
have npm : (n - 1).succ = n := Nat.succ_pred_eq_of_pos npos
... |
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Pointwise
import Mathlib.GroupTheory.GroupAction.ConjAct
#align_import group_theory.subgroup.pointwise from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
open Set
open Pointwise
variable {α G A S... | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 360 | 363 | theorem conj_smul_le_of_le {P H : Subgroup G} (hP : P ≤ H) (h : H) :
MulAut.conj (h : G) • P ≤ H := by |
rintro - ⟨g, hg, rfl⟩
exact H.mul_mem (H.mul_mem h.2 (hP hg)) (H.inv_mem h.2)
|
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Basic
import Mathlib.Tactic.ApplyFun
#align_import computability.turing_machine from "leanprover-commu... | Mathlib/Computability/TuringMachine.lean | 545 | 547 | theorem Tape.move_right_left {Γ} [Inhabited Γ] (T : Tape Γ) :
(T.move Dir.right).move Dir.left = T := by |
cases T; simp [Tape.move]
|
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 223 | 225 | theorem Right.one_le_inv_iff : 1 ≤ a⁻¹ ↔ a ≤ 1 := by |
rw [← mul_le_mul_iff_right a]
simp
|
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]
|
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
section
open UniformSpace Filter Set Uniformity Topology UniformConvergence Function
variable {ι κ X X' Y Z α α' β β'... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 897 | 901 | theorem UniformEquicontinuous.closure' {A : Set Y} {u : Y → β → α}
(hA : UniformEquicontinuous (u ∘ (↑) : A → β → α)) (hu : Continuous u) :
UniformEquicontinuous (u ∘ (↑) : closure A → β → α) := by |
rw [← uniformEquicontinuousOn_univ] at hA ⊢
exact hA.closure' (Pi.continuous_restrict _ |>.comp hu)
|
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 | 128 | 129 | theorem mul_invOf_self_assoc [Monoid α] (a b : α) [Invertible a] : a * (⅟ a * b) = b := by |
rw [← mul_assoc, mul_invOf_self, one_mul]
|
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]
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 327 | 329 | theorem eval_eq_sum_range {p : R[X]} (x : R) :
p.eval x = ∑ i ∈ Finset.range (p.natDegree + 1), p.coeff i * x ^ i := by |
rw [eval_eq_sum, sum_over_range]; simp
|
import Mathlib.CategoryTheory.Sites.Spaces
import Mathlib.Topology.Sheaves.Sheaf
import Mathlib.CategoryTheory.Sites.DenseSubsite
#align_import topology.sheaves.sheaf_condition.sites from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomputable section
set_option linter.uppercaseLe... | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | 161 | 168 | theorem OpenEmbedding.compatiblePreserving (hf : OpenEmbedding f) :
CompatiblePreserving (Opens.grothendieckTopology Y) hf.isOpenMap.functor := by |
haveI : Mono f := (TopCat.mono_iff_injective f).mpr hf.inj
apply compatiblePreservingOfDownwardsClosed
intro U V i
refine ⟨(Opens.map f).obj V, eqToIso <| Opens.ext <| Set.image_preimage_eq_of_subset fun x h ↦ ?_⟩
obtain ⟨_, _, rfl⟩ := i.le h
exact ⟨_, rfl⟩
|
import Mathlib.MeasureTheory.Measure.Restrict
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter Function MeasurableSpace ENNReal
variable {α β δ ι : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace α} [MeasurableSpace β] {μ ν ν₁ ν₂: Measure α}
{s t : Set α}
section IsFinit... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 159 | 164 | theorem summable_measure_toReal [hμ : IsFiniteMeasure μ] {f : ℕ → Set α}
(hf₁ : ∀ i : ℕ, MeasurableSet (f i)) (hf₂ : Pairwise (Disjoint on f)) :
Summable fun x => (μ (f x)).toReal := by |
apply ENNReal.summable_toReal
rw [← MeasureTheory.measure_iUnion hf₂ hf₁]
exact ne_of_lt (measure_lt_top _ _)
|
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.ZMod.Basic
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Data.Fintype.BigOperators
#align_import number_theory.sum_four_squares from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
open Finset Polynomial FiniteField Equiv
the... | Mathlib/NumberTheory/SumFourSquares.lean | 34 | 42 | theorem Nat.euler_four_squares (a b c d x y z w : ℕ) :
((a : ℤ) * x - b * y - c * z - d * w).natAbs ^ 2 +
((a : ℤ) * y + b * x + c * w - d * z).natAbs ^ 2 +
((a : ℤ) * z - b * w + c * x + d * y).natAbs ^ 2 +
((a : ℤ) * w + b * z - c * y + d * x).natAbs ^ 2 =
(a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2) *... |
rw [← Int.natCast_inj]
push_cast
simp only [sq_abs, _root_.euler_four_squares]
|
import Mathlib.Data.Nat.Bits
import Mathlib.Order.Lattice
#align_import data.nat.size from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
namespace Nat
section
set_option linter.deprecated false
theorem shiftLeft_eq_mul_pow (m) : ∀ n, m <<< n = m * 2 ^ n := shiftLeft_eq _
#align nat.... | Mathlib/Data/Nat/Size.lean | 103 | 104 | theorem size_shiftLeft {m} (h : m ≠ 0) (n) : size (m <<< n) = size m + n := by |
simp only [size_shiftLeft' (shiftLeft'_ne_zero_left _ h _), ← shiftLeft'_false]
|
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 | 84 | 88 | theorem singleton_unop (x : αᵒᵖ) : ({x} : Set αᵒᵖ).unop = {unop x} := by |
ext
constructor
· apply op_injective
· apply unop_injective
|
import Mathlib.Data.Rat.Encodable
import Mathlib.Data.Real.EReal
import Mathlib.Topology.Instances.ENNReal
import Mathlib.Topology.Order.MonotoneContinuity
#align_import topology.instances.ereal from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Class... | Mathlib/Topology/Instances/EReal.lean | 86 | 90 | theorem tendsto_toReal {a : EReal} (ha : a ≠ ⊤) (h'a : a ≠ ⊥) :
Tendsto EReal.toReal (𝓝 a) (𝓝 a.toReal) := by |
lift a to ℝ using ⟨ha, h'a⟩
rw [nhds_coe, tendsto_map'_iff]
exact tendsto_id
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 659 | 686 | theorem ae_of_mem_of_ae_of_mem_inter_Ioo {μ : Measure ℝ} [NoAtoms μ] {s : Set ℝ} {p : ℝ → Prop}
(h : ∀ a b, a ∈ s → b ∈ s → a < b → ∀ᵐ x ∂μ, x ∈ s ∩ Ioo a b → p x) :
∀ᵐ x ∂μ, x ∈ s → p x := by |
/- By second-countability, we cover `s` by countably many intervals `(a, b)` (except maybe for
two endpoints, which don't matter since `μ` does not have any atom). -/
let T : s × s → Set ℝ := fun p => Ioo p.1 p.2
let u := ⋃ i : ↥s × ↥s, T i
have hfinite : (s \ u).Finite := s.finite_diff_iUnion_Ioo'
obtai... |
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 | 128 | 130 | theorem degrees_add [DecidableEq σ] (p q : MvPolynomial σ R) :
(p + q).degrees ≤ p.degrees ⊔ q.degrees := by |
simp_rw [degrees_def]; exact supDegree_add_le
|
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpace
namespace MeasureTheory
namespace Measure
variable {M : Type*} [Monoid M] [MeasurableSpace M]
@[to_additive conv "Additive convolution of measures."]
noncomputable def mconv (μ : Measure M) (ν : Measure M) :
... | Mathlib/MeasureTheory/Group/Convolution.lean | 85 | 90 | theorem mconv_comm {M : Type*} [CommMonoid M] [MeasurableSpace M] [MeasurableMul₂ M] (μ : Measure M)
(ν : Measure M) [SFinite μ] [SFinite ν] : μ ∗ ν = ν ∗ μ := by |
unfold mconv
rw [← prod_swap, map_map]
· simp [Function.comp_def, mul_comm]
all_goals { measurability }
|
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 96 | 99 | theorem rank_of_isUnit [StrongRankCondition R] [DecidableEq n] (A : Matrix n n R) (h : IsUnit A) :
A.rank = Fintype.card n := by |
obtain ⟨A, rfl⟩ := h
exact rank_unit A
|
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