Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.path_connected from "leanprover... | Mathlib/Topology/Connected/PathConnected.lean | 581 | 586 | theorem trans_pi_eq_pi_trans (γ₀ : ∀ i, Path (as i) (bs i)) (γ₁ : ∀ i, Path (bs i) (cs i)) :
(Path.pi γ₀).trans (Path.pi γ₁) = Path.pi fun i => (γ₀ i).trans (γ₁ i) := by |
ext t i
unfold Path.trans
simp only [Path.coe_mk_mk, Function.comp_apply, pi_coe]
split_ifs <;> rfl
|
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216... | Mathlib/Data/Bool/Basic.lean | 109 | 109 | theorem and_elim_left : ∀ {a b : Bool}, a && b → a := by | decide
|
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Algebra.Polynomial.Splits
import... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 336 | 345 | theorem degree_cyclotomic (n : ℕ) (R : Type*) [Ring R] [Nontrivial R] :
(cyclotomic n R).degree = Nat.totient n := by |
rw [← map_cyclotomic_int]
rw [degree_map_eq_of_leadingCoeff_ne_zero (Int.castRingHom R) _]
· cases' n with k
· simp only [cyclotomic, degree_one, dif_pos, Nat.totient_zero, CharP.cast_eq_zero]
rw [← degree_cyclotomic' (Complex.isPrimitiveRoot_exp k.succ (Nat.succ_ne_zero k))]
exact (int_cyclotomic_sp... |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.ca... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 327 | 330 | theorem hasFDerivAt_iff_isLittleO_nhds_zero :
HasFDerivAt f f' x ↔ (fun h : E => f (x + h) - f x - f' h) =o[𝓝 0] fun h => h := by |
rw [HasFDerivAt, hasFDerivAtFilter_iff_isLittleO, ← map_add_left_nhds_zero x, isLittleO_map]
simp [(· ∘ ·)]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Data.Set.Lattice
#... | Mathlib/Order/Interval/Set/Pi.lean | 168 | 169 | theorem image_update_Ioc_left (f : ∀ i, α i) (i : ι) (a : α i) :
update f i '' Ioc a (f i) = Ioc (update f i a) f := by | simpa using image_update_Ioc f i a (f i)
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 309 | 316 | theorem toDual_total_left (f : ι →₀ R) (i : ι) :
b.toDual (Finsupp.total ι M R b f) (b i) = f i := by |
rw [Finsupp.total_apply, Finsupp.sum, _root_.map_sum, LinearMap.sum_apply]
simp_rw [LinearMap.map_smul, LinearMap.smul_apply, toDual_apply, smul_eq_mul, mul_boole,
Finset.sum_ite_eq']
split_ifs with h
· rfl
· rw [Finsupp.not_mem_support_iff.mp h]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "... | Mathlib/Data/Set/Function.lean | 1,603 | 1,606 | theorem piecewise_le_piecewise {δ : α → Type*} [∀ i, Preorder (δ i)] {s : Set α}
[∀ j, Decidable (j ∈ s)] {f₁ f₂ g₁ g₂ : ∀ i, δ i} (h₁ : ∀ i ∈ s, f₁ i ≤ g₁ i)
(h₂ : ∀ i ∉ s, f₂ i ≤ g₂ i) : s.piecewise f₁ f₂ ≤ s.piecewise g₁ g₂ := by |
apply piecewise_le <;> intros <;> simp [*]
|
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localizati... | Mathlib/Algebra/Polynomial/Laurent.lean | 373 | 380 | theorem leftInverse_trunc_toLaurent :
Function.LeftInverse (trunc : R[T;T⁻¹] → R[X]) Polynomial.toLaurent := by |
refine fun f => f.induction_on' ?_ ?_
· intro f g hf hg
simp only [hf, hg, _root_.map_add]
· intro n r
simp only [Polynomial.toLaurent_C_mul_T, trunc_C_mul_T, Int.natCast_nonneg, Int.toNat_natCast,
if_true]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 821 | 823 | theorem Orthonormal.inner_finsupp_eq_sum_left {v : ι → E} (hv : Orthonormal 𝕜 v) (l₁ l₂ : ι →₀ 𝕜) :
⟪Finsupp.total ι E 𝕜 v l₁, Finsupp.total ι E 𝕜 v l₂⟫ = l₁.sum fun i y => conj y * l₂ i := by |
simp only [l₁.total_apply _, Finsupp.sum_inner, hv.inner_right_finsupp, smul_eq_mul]
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 506 | 509 | theorem Gamma_mul_Gamma_one_sub (s : ℝ) : Gamma s * Gamma (1 - s) = π / sin (π * s) := by |
simp_rw [← Complex.ofReal_inj, Complex.ofReal_div, Complex.ofReal_sin, Complex.ofReal_mul, ←
Complex.Gamma_ofReal, Complex.ofReal_sub, Complex.ofReal_one]
exact Complex.Gamma_mul_Gamma_one_sub s
|
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
/-!
# Special values of Hurwitz and Riemann zeta functions
This file gives t... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 76 | 97 | theorem sinZeta_two_mul_nat_add_one (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
sinZeta x (2 * k + 1) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k + 1) / 2 / (2 * k + 1)! *
((Polynomial.bernoulli (2 * k + 1)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by |
rw [← (hasSum_nat_sinZeta x (?_ : 1 < re (2 * k + 1))).tsum_eq]
refine Eq.trans ?_ <| (congr_arg ofReal' (hasSum_one_div_nat_pow_mul_sin hk hx).tsum_eq).trans ?_
· rw [ofReal_tsum]
refine tsum_congr fun n ↦ ?_
rw [mul_comm (1 / _), mul_one_div, ofReal_div, mul_assoc (2 * π), mul_comm x n, ← mul_assoc]
... |
/-
Copyright (c) 2023 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Filtered.Final
/-!
# Finally small categories
A category given by `(J : Type u) [Category.{v} J]` is `w`-finally small if there exists a... | Mathlib/CategoryTheory/Limits/FinallySmall.lean | 175 | 179 | theorem initiallySmall_iff_exists_small_weakly_initial_set [IsCofilteredOrEmpty J] :
InitiallySmall.{v} J ↔ ∃ (s : Set J) (_ : Small.{v} s), ∀ i, ∃ j ∈ s, Nonempty (j ⟶ i) := by |
refine ⟨fun _ => InitiallySmall.exists_small_weakly_initial_set _, fun h => ?_⟩
rcases h with ⟨s, hs, hs'⟩
exact initiallySmall_of_small_weakly_initial_set s hs'
|
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot, Eric Wieser
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Unique
import Mathlib.Tacti... | Mathlib/Algebra/Group/Pi/Basic.lean | 604 | 606 | theorem elim_div_div [Div γ] : Sum.elim (a / a') (b / b') = Sum.elim a b / Sum.elim a' b' := by |
ext x
cases x <;> rfl
|
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpace
/-!
# The multiplicative and additive convolution of measures
In... | Mathlib/MeasureTheory/Group/Convolution.lean | 77 | 81 | theorem add_mconv [MeasurableMul₂ M] (μ : Measure M) (ν : Measure M) (ρ : Measure M) [SFinite μ]
[SFinite ν] [SFinite ρ] : (μ + ν) ∗ ρ = μ ∗ ρ + ν ∗ ρ := by |
unfold mconv
rw [add_prod, map_add]
measurability
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "lean... | Mathlib/Analysis/Convex/Side.lean | 766 | 782 | theorem setOf_wSameSide_eq_image2 {s : AffineSubspace R P} {x p : P} (hx : x ∉ s) (hp : p ∈ s) :
{ y | s.WSameSide x y } = Set.image2 (fun (t : R) q => t • (x -ᵥ p) +ᵥ q) (Set.Ici 0) s := by |
ext y
simp_rw [Set.mem_setOf, Set.mem_image2, Set.mem_Ici]
constructor
· rw [wSameSide_iff_exists_left hp, or_iff_right hx]
rintro ⟨p₂, hp₂, h | h | ⟨r₁, r₂, hr₁, hr₂, h⟩⟩
· rw [vsub_eq_zero_iff_eq] at h
exact False.elim (hx (h.symm ▸ hp))
· rw [vsub_eq_zero_iff_eq] at h
refine ⟨0, le_r... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 445 | 445 | theorem inv_eq_one_div (x : G) : x⁻¹ = 1 / x := by | rw [div_eq_mul_inv, one_mul]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Set
import Mathlib.Data.Nat.Set
import Mathlib.Data.Set.Prod
import Mathlib.Data.ULift
import Mathlib.Order.Bounds.Basic
import Mathlib.Order... | Mathlib/Order/CompleteLattice.lean | 1,220 | 1,221 | theorem inf_iInf [Nonempty ι] {f : ι → α} {a : α} : (a ⊓ ⨅ x, f x) = ⨅ x, a ⊓ f x := by |
rw [iInf_inf_eq, iInf_const]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Order.Lattice
import Mathlib.Order.ULift
import Mathlib.Tactic.PushNeg
#align_import order.bounded_order from "leanprover-community/mathlib"@"70d50ecf... | Mathlib/Order/BoundedOrder.lean | 752 | 754 | theorem coe_eq_top_iff [OrderTop α] [OrderTop (Subtype p)] (htop : p ⊤) {x : { x // p x }} :
(x : α) = ⊤ ↔ x = ⊤ := by |
rw [← coe_top htop, ext_iff]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Order.Basic
#align_import analysis.convex.strict from "leanprove... | Mathlib/Analysis/Convex/Strict.lean | 333 | 341 | theorem StrictConvex.eq_of_openSegment_subset_frontier [Nontrivial 𝕜] [DenselyOrdered 𝕜]
(hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s) (h : openSegment 𝕜 x y ⊆ frontier s) :
x = y := by |
obtain ⟨a, ha₀, ha₁⟩ := DenselyOrdered.dense (0 : 𝕜) 1 zero_lt_one
classical
by_contra hxy
exact
(h ⟨a, 1 - a, ha₀, sub_pos_of_lt ha₁, add_sub_cancel _ _, rfl⟩).2
(hs hx hy hxy ha₀ (sub_pos_of_lt ha₁) <| add_sub_cancel _ _)
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Finset.Prod
#align_import data.fintype.prod from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df08... | Mathlib/Data/Fintype/Prod.lean | 31 | 34 | theorem toFinset_prod (s : Set α) (t : Set β) [Fintype s] [Fintype t] [Fintype (s ×ˢ t)] :
(s ×ˢ t).toFinset = s.toFinset ×ˢ t.toFinset := by |
ext
simp
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 922 | 924 | theorem leftpad_length (n : Nat) (a : α) (l : List α) :
(leftpad n a l).length = max n l.length := by |
simp only [leftpad, length_append, length_replicate, Nat.sub_add_eq_max]
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.GammaCompN
import Mathlib.AlgebraicTopology.DoldKan.NReflectsIso
#align_import algebraic_topology.dold_kan.n_comp_gamma from "leanprov... | Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean | 216 | 226 | theorem compatibility_Γ₂N₁_Γ₂N₂_natTrans (X : SimplicialObject C) :
Γ₂N₁.natTrans.app X =
(Γ₂N₂ToKaroubiIso.app X).inv ≫
Γ₂N₂.natTrans.app ((toKaroubi (SimplicialObject C)).obj X) := by |
rw [Γ₂N₂.natTrans_app_f_app]
dsimp only [Karoubi.decompId_i_toKaroubi, Karoubi.decompId_p_toKaroubi, Functor.comp_map,
NatTrans.comp_app]
rw [N₂.map_id, Γ₂.map_id, Iso.app_inv]
dsimp only [toKaroubi]
erw [id_comp]
rw [comp_id, Iso.inv_hom_id_app_assoc]
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_... | Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean | 148 | 151 | theorem decomp (f : V →ᴬ[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by |
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Option.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sigma.Basic
import Mathlib... | Mathlib/Logic/Equiv/Basic.lean | 789 | 792 | theorem prodCongr_refl_right (e : β₁ ≃ β₂) :
prodCongr e (Equiv.refl α₁) = prodCongrLeft fun _ => e := by |
ext ⟨a, b⟩ : 1
simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
i... | Mathlib/LinearAlgebra/Basis.lean | 981 | 983 | theorem Basis.constr_apply_fintype [Fintype ι] (b : Basis ι R M) (f : ι → M') (x : M) :
(constr (M' := M') b S f : M → M') x = ∑ i, b.equivFun x i • f i := by |
simp [b.constr_apply, b.equivFun_apply, Finsupp.sum_fintype]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 288 | 289 | theorem toIocDiv_sub_zsmul (a b : α) (m : ℤ) : toIocDiv hp a (b - m • p) = toIocDiv hp a b - m := by |
rw [sub_eq_add_neg, ← neg_smul, toIocDiv_add_zsmul, sub_eq_add_neg]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
#align_import measure_theory.function.conditional_expectation.condexp_L1 from "leanprover-communit... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 220 | 227 | theorem condexpIndL1_smul' [NormedSpace ℝ F] [SMulCommClass ℝ 𝕜 F] (c : 𝕜) (x : F) :
condexpIndL1 hm μ s (c • x) = c • condexpIndL1 hm μ s x := by |
by_cases hs : MeasurableSet s
swap; · simp_rw [condexpIndL1_of_not_measurableSet hs]; rw [smul_zero]
by_cases hμs : μ s = ∞
· simp_rw [condexpIndL1_of_measure_eq_top hμs]; rw [smul_zero]
· simp_rw [condexpIndL1_of_measurableSet_of_measure_ne_top hs hμs]
exact condexpIndL1Fin_smul' hs hμs c x
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 756 | 761 | theorem getLast?_append {l₁ l₂ : List α} {x : α} (h : x ∈ l₂.getLast?) :
x ∈ (l₁ ++ l₂).getLast? := by |
cases l₂
· contradiction
· rw [List.getLast?_append_cons]
exact h
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Order.SuccPred.Basic
#align_import order.succ_pred.relation from "leanprover-community/mathlib"@"9aba7801eeecebb61f58a5763c2b6dd1b47dc6ef"
/-!
# ... | Mathlib/Order/SuccPred/Relation.lean | 84 | 87 | theorem transGen_of_succ_of_reflexive (r : α → α → Prop) {n m : α} (hr : Reflexive r)
(h1 : ∀ i ∈ Ico n m, r i (succ i)) (h2 : ∀ i ∈ Ico m n, r (succ i) i) : TransGen r n m := by |
rcases eq_or_ne m n with (rfl | hmn); · exact TransGen.single (hr m)
exact transGen_of_succ_of_ne r h1 h2 hmn.symm
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Multiset.Sort
import Mathlib.Data.PNat.Basic
import Mathlib.Data.PN... | Mathlib/NumberTheory/ADEInequality.lean | 175 | 195 | theorem lt_three {p q r : ℕ+} (hpq : p ≤ q) (hqr : q ≤ r) (H : 1 < sumInv {p, q, r}) : p < 3 := by |
have h3 : (0 : ℚ) < 3 := by norm_num
contrapose! H
rw [sumInv_pqr]
have h3q := H.trans hpq
have h3r := h3q.trans hqr
have hp: (p : ℚ)⁻¹ ≤ 3⁻¹ := by
rw [inv_le_inv _ h3]
· assumption_mod_cast
· norm_num
have hq: (q : ℚ)⁻¹ ≤ 3⁻¹ := by
rw [inv_le_inv _ h3]
· assumption_mod_cast
· nor... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analys... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 616 | 617 | theorem mul_rpow_of_ne_top {x y : ℝ≥0∞} (hx : x ≠ ⊤) (hy : y ≠ ⊤) (z : ℝ) :
(x * y) ^ z = x ^ z * y ^ z := by | simp [*, mul_rpow_eq_ite]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import rin... | Mathlib/RingTheory/Algebraic.lean | 439 | 447 | theorem inv_eq_of_root_of_coeff_zero_ne_zero {x : L} {p : K[X]} (aeval_eq : aeval x p = 0)
(coeff_zero_ne : p.coeff 0 ≠ 0) : x⁻¹ = -(aeval x (divX p) / algebraMap _ _ (p.coeff 0)) := by |
convert inv_eq_of_aeval_divX_ne_zero (p := p) (L := L)
(mt (fun h => (algebraMap K L).injective ?_) coeff_zero_ne) using 1
· rw [aeval_eq, zero_sub, div_neg]
rw [RingHom.map_zero]
convert aeval_eq
conv_rhs => rw [← divX_mul_X_add p]
rw [AlgHom.map_add, AlgHom.map_mul, h, zero_mul, zero_add, aeval_C]
|
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd0... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 500 | 502 | theorem rowLen_ofRowLens {w : List ℕ} {hw : w.Sorted (· ≥ ·)} (i : Fin w.length) :
(ofRowLens w hw).rowLen i = w.get i := by |
simp [rowLen, Nat.find_eq_iff, mem_ofRowLens]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Ma... | Mathlib/Analysis/Analytic/Basic.lean | 537 | 540 | theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
HasFPowerSeriesAt (f + g) (pf + pg) x := by |
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Antoine Chambert-Loir
-/
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Group.Hom.CompType... | Mathlib/GroupTheory/GroupAction/Hom.lean | 479 | 481 | theorem toAddMonoidHom_injective {f g : A →ₑ+[φ] B} (h : (f : A →+ B) = (g : A →+ B)) : f = g := by |
ext a
exact DFunLike.congr_fun h a
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
... | Mathlib/LinearAlgebra/Prod.lean | 487 | 489 | theorem prod_eq_sup_map (p : Submodule R M) (q : Submodule R M₂) :
p.prod q = p.map (LinearMap.inl R M M₂) ⊔ q.map (LinearMap.inr R M M₂) := by |
rw [← map_coprod_prod, coprod_inl_inr, map_id]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.A... | Mathlib/RingTheory/AdjoinRoot.lean | 600 | 602 | theorem minpoly_powerBasis_gen_of_monic (hf : f.Monic) (hf' : f ≠ 0 := hf.ne_zero) :
minpoly K (powerBasis hf').gen = f := by |
rw [minpoly_powerBasis_gen hf', hf.leadingCoeff, inv_one, C.map_one, mul_one]
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Topology.EMetricSpace.Paracompact
import Mathlib.Topology.Instances.ENNReal
import Mathlib.Analysis.Convex.PartitionOfUnity
#align_import topology.m... | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | 42 | 61 | theorem eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
(hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
∀ᶠ p : ℝ≥0∞ × X in 𝓝 0 ×ˢ 𝓝 x, ∀ i, p.2 ∈ K i → closedBall p.2 p.1 ⊆ U i := by |
suffices ∀ i, x ∈ K i → ∀ᶠ p : ℝ≥0∞ × X in 𝓝 0 ×ˢ 𝓝 x, closedBall p.2 p.1 ⊆ U i by
apply mp_mem ((eventually_all_finite (hfin.point_finite x)).2 this)
(mp_mem (@tendsto_snd ℝ≥0∞ _ (𝓝 0) _ _ (hfin.iInter_compl_mem_nhds hK x)) _)
apply univ_mem'
rintro ⟨r, y⟩ hxy hyU i hi
simp only [mem_iInter... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
#align_import linear_algebra.dfinsup... | Mathlib/LinearAlgebra/DFinsupp.lean | 516 | 519 | theorem Independent.dfinsupp_sumAddHom_injective {p : ι → AddSubgroup N} (h : Independent p) :
Function.Injective (sumAddHom fun i => (p i).subtype) := by |
rw [← independent_map_orderIso_iff (AddSubgroup.toIntSubmodule : AddSubgroup N ≃o _)] at h
exact h.dfinsupp_lsum_injective
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory... | Mathlib/CategoryTheory/Limits/HasLimits.lean | 463 | 469 | theorem limit.post_post {E : Type u''} [Category.{v''} E] (H : D ⥤ E) [h : HasLimit ((F ⋙ G) ⋙ H)] :
-- H G (limit F) ⟶ H (limit (F ⋙ G)) ⟶ limit ((F ⋙ G) ⋙ H) equals
-- H G (limit F) ⟶ limit (F ⋙ (G ⋙ H))
haveI : HasLimit (F ⋙ G ⋙ H) := h
H.map (limit.post F G) ≫ limit.post (F ⋙ G) H = limit.post F (G ... |
haveI : HasLimit (F ⋙ G ⋙ H) := h
ext; erw [assoc, limit.post_π, ← H.map_comp, limit.post_π, limit.post_π]; rfl
|
/-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Closed.Monoidal
import Mathlib.... | Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean | 503 | 506 | theorem coevaluation_comp_leftAdjointMate {X Y : C} [HasLeftDual X] [HasLeftDual Y] (f : X ⟶ Y) :
η_ (ᘁY) Y ≫ (ᘁf) ▷ Y = η_ (ᘁX) X ≫ (ᘁX) ◁ f := by |
apply_fun (tensorRightHomEquiv _ (ᘁY) Y _).symm
simp
|
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm
#align_import n... | Mathlib/NumberTheory/Cyclotomic/Rat.lean | 300 | 325 | theorem zeta_sub_one_prime_of_two_pow [IsCyclotomicExtension {(2 : ℕ+) ^ (k + 1)} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑((2 : ℕ+) ^ (k + 1))) :
Prime (hζ.toInteger - 1) := by |
letI := IsCyclotomicExtension.numberField {(2 : ℕ+) ^ (k + 1)} ℚ K
refine Ideal.prime_of_irreducible_absNorm_span (fun h ↦ ?_) ?_
· apply hζ.pow_ne_one_of_pos_of_lt zero_lt_one (one_lt_pow (by decide) (by simp))
rw [sub_eq_zero] at h
simpa using congrArg (algebraMap _ K) h
rw [Nat.irreducible_iff_prime... |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalcu... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 1,021 | 1,027 | theorem _root_.HasCompactSupport.integral_Iic_deriv_eq (hf : ContDiff ℝ 1 f)
(h2f : HasCompactSupport f) (b : ℝ) : ∫ x in Iic b, deriv f x = f b := by |
have := fun x (_ : x ∈ Iio b) ↦ hf.differentiable le_rfl x |>.hasDerivAt
rw [integral_Iic_of_hasDerivAt_of_tendsto hf.continuous.continuousWithinAt this, sub_zero]
· refine hf.continuous_deriv le_rfl |>.integrable_of_hasCompactSupport h2f.deriv |>.integrableOn
rw [hasCompactSupport_iff_eventuallyEq, Filter.coc... |
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl
-/
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryPro... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 1,524 | 1,528 | theorem pushoutComparison_map_desc (f : X ⟶ Y) (g : X ⟶ Z) [HasPushout f g]
[HasPushout (G.map f) (G.map g)] {W : C} {h : Y ⟶ W} {k : Z ⟶ W} (w : f ≫ h = g ≫ k) :
pushoutComparison G f g ≫ G.map (pushout.desc _ _ w) =
pushout.desc (G.map h) (G.map k) (by simp only [← G.map_comp, w]) := by |
ext <;> simp [← G.map_comp]
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathl... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 769 | 777 | theorem restrict_smul {v : VectorMeasure α M} {i : Set α} (c : R) :
(c • v).restrict i = c • v.restrict i := by |
by_cases hi : MeasurableSet i
· ext j hj
simp [restrict_apply _ hi hj]
· simp only [restrict_not_measurable _ hi]
-- `smul_zero` does not work since we do not require `ContinuousAdd`
ext j
simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGrou... | Mathlib/Topology/Algebra/Group/Basic.lean | 1,778 | 1,787 | theorem compact_covered_by_mul_left_translates {K V : Set G} (hK : IsCompact K)
(hV : (interior V).Nonempty) : ∃ t : Finset G, K ⊆ ⋃ g ∈ t, (g * ·) ⁻¹' V := by |
obtain ⟨t, ht⟩ : ∃ t : Finset G, K ⊆ ⋃ x ∈ t, interior ((x * ·) ⁻¹' V) := by
refine
hK.elim_finite_subcover (fun x => interior <| (x * ·) ⁻¹' V) (fun x => isOpen_interior) ?_
cases' hV with g₀ hg₀
refine fun g _ => mem_iUnion.2 ⟨g₀ * g⁻¹, ?_⟩
refine preimage_interior_subset_interior_preimage (c... |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Sets.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
/-!
# Properties ... | Mathlib/Topology/LocalAtTarget.lean | 66 | 72 | theorem IsClosedMap.restrictPreimage (H : IsClosedMap f) (s : Set β) :
IsClosedMap (s.restrictPreimage f) := by |
intro t
suffices ∀ u, IsClosed u → Subtype.val ⁻¹' u = t →
∃ v, IsClosed v ∧ Subtype.val ⁻¹' v = s.restrictPreimage f '' t by
simpa [isClosed_induced_iff]
exact fun u hu e => ⟨f '' u, H u hu, by simp [← e, image_restrictPreimage]⟩
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.LocalAtTarget
import Mathlib.AlgebraicGeometry.Morphisms.Basic
#align_import algebraic_geometry.morphisms.open_immersion from "leanprover-community... | Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.lean | 53 | 74 | theorem isOpenImmersion_is_local_at_target : PropertyIsLocalAtTarget @IsOpenImmersion := by |
constructor
· exact isOpenImmersion_respectsIso
· intros; infer_instance
· intro X Y f 𝒰 H
rw [isOpenImmersion_iff_stalk]
constructor
· apply (openEmbedding_iff_openEmbedding_of_iSup_eq_top 𝒰.iSup_opensRange f.1.base.2).mpr
intro i
have := ((isOpenImmersion_respectsIso.arrow_iso_iff
... |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.Order.LocalExtr
import Mathlib.Topology.Order.IntermediateValue
import Mathlib.Topology.Support
import Mathlib.Topology.Order... | Mathlib/Topology/Algebra/Order/Compact.lean | 265 | 268 | theorem IsCompact.exists_isMinOn [ClosedIicTopology α] {s : Set β} (hs : IsCompact s)
(ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, IsMinOn f s x := by |
rcases (hs.image_of_continuousOn hf).exists_isLeast (ne_s.image f) with ⟨_, ⟨x, hxs, rfl⟩, hx⟩
exact ⟨x, hxs, forall_mem_image.1 hx⟩
|
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Ba... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 457 | 459 | theorem det_eq_sum_mul_adjugate_col (A : Matrix n n α) (j : n) :
det A = ∑ i : n, A i j * adjugate A j i := by |
simpa only [det_transpose, ← adjugate_transpose] using det_eq_sum_mul_adjugate_row Aᵀ j
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 816 | 834 | theorem inj_on_of_surj_on_of_ncard_le {t : Set β} (f : ∀ a ∈ s, β) (hf : ∀ a ha, f a ha ∈ t)
(hsurj : ∀ b ∈ t, ∃ a ha, f a ha = b) (hst : s.ncard ≤ t.ncard) ⦃a₁⦄ (ha₁ : a₁ ∈ s) ⦃a₂⦄
(ha₂ : a₂ ∈ s) (ha₁a₂ : f a₁ ha₁ = f a₂ ha₂) (hs : s.Finite := by | toFinite_tac) :
a₁ = a₂ := by
classical
set f' : s → t := fun x ↦ ⟨f x.1 x.2, hf _ _⟩
have hsurj : f'.Surjective := by
rintro ⟨y, hy⟩
obtain ⟨a, ha, rfl⟩ := hsurj y hy
simp only [Subtype.mk.injEq, Subtype.exists]
exact ⟨_, ha, rfl⟩
haveI := hs.fintype
haveI := Fintype.ofSurjective _ hsurj... |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 2,425 | 2,426 | theorem closure_le_normalClosure {s : Set G} : closure s ≤ normalClosure s := by |
simp only [subset_normalClosure, closure_le]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Ring.Regular
import Mathlib.Tactic.Common
#align_import algebra.gcd_monoid.basic from "leanp... | Mathlib/Algebra/GCDMonoid/Basic.lean | 889 | 890 | theorem lcm_eq_right_iff [NormalizedGCDMonoid α] (a b : α) (h : normalize b = b) :
lcm a b = b ↔ a ∣ b := by | simpa only [lcm_comm b a] using lcm_eq_left_iff b a h
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f6856... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 386 | 393 | theorem ContMDiffWithinAt.mdifferentiableWithinAt (hf : ContMDiffWithinAt I I' n f s x)
(hn : 1 ≤ n) : MDifferentiableWithinAt I I' f s x := by |
suffices h : MDifferentiableWithinAt I I' f (s ∩ f ⁻¹' (extChartAt I' (f x)).source) x by
rwa [mdifferentiableWithinAt_inter'] at h
apply hf.1.preimage_mem_nhdsWithin
exact extChartAt_source_mem_nhds I' (f x)
rw [mdifferentiableWithinAt_iff]
exact ⟨hf.1.mono inter_subset_left, (hf.2.differentiableWit... |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalcu... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 988 | 1,003 | theorem integral_Iic_of_hasDerivAt_of_tendsto (hcont : ContinuousWithinAt f (Iic a) a)
(hderiv : ∀ x ∈ Iio a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a))
(hf : Tendsto f atBot (𝓝 m)) : ∫ x in Iic a, f' x = f a - m := by |
have hcont : ContinuousOn f (Iic a) := by
intro x hx
rcases hx.out.eq_or_lt with rfl|hx
· exact hcont
· exact (hderiv x hx).continuousAt.continuousWithinAt
refine tendsto_nhds_unique (intervalIntegral_tendsto_integral_Iic a f'int tendsto_id) ?_
apply Tendsto.congr' _ (hf.const_sub _)
filter_upw... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"... | Mathlib/Topology/Constructions.lean | 1,768 | 1,771 | theorem IsClosed.trans (ht : IsClosed t) (hs : IsClosed s) : IsClosed (t : Set X) := by |
rcases isClosed_induced_iff.mp ht with ⟨s', hs', rfl⟩
rw [Subtype.image_preimage_coe]
exact hs.inter hs'
|
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subg... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,086 | 1,100 | theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) :
G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by |
ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists,
not_and, and_congr_right_iff]
intro
constructor
· intro hs
refine Sym2.ind ?_
rintro ⟨v', hv'⟩ ⟨w', hw'⟩
simp only [Sym2.map_pair_eq, Sym2.eq]
contrapose!
rintro (_ | _) <;... |
/-
Copyright (c) 2020 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Ring.Subsemiring.Basic
#align_import ring_theory.subring.basic from "leanprover-community/math... | Mathlib/Algebra/Ring/Subring/Basic.lean | 681 | 682 | theorem coe_iInf {ι : Sort*} {S : ι → Subring R} : (↑(⨅ i, S i) : Set R) = ⋂ i, S i := by |
simp only [iInf, coe_sInf, Set.biInter_range]
|
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.Real.Basic
import Mathlib.Data.ENNReal.Real
import Mathlib.Data.Sign
#align_import data.real.ereal from "leanprover-community/mathlib"@"2196ab363eb... | Mathlib/Data/Real/EReal.lean | 970 | 971 | theorem neg_lt_iff_neg_lt {a b : EReal} : -a < b ↔ -b < a := by |
rw [← neg_lt_neg_iff, neg_neg]
|
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.Algebra.Ring.Subri... | Mathlib/LinearAlgebra/Ray.lean | 102 | 111 | theorem trans (hxy : SameRay R x y) (hyz : SameRay R y z) (hy : y = 0 → x = 0 ∨ z = 0) :
SameRay R x z := by |
rcases eq_or_ne x 0 with (rfl | hx); · exact zero_left z
rcases eq_or_ne z 0 with (rfl | hz); · exact zero_right x
rcases eq_or_ne y 0 with (rfl | hy);
· exact (hy rfl).elim (fun h => (hx h).elim) fun h => (hz h).elim
rcases hxy.exists_pos hx hy with ⟨r₁, r₂, hr₁, hr₂, h₁⟩
rcases hyz.exists_pos hy hz with ... |
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Algebra.Field.Opposite
import Mathlib.Algebra.Group.Subgroup.ZPowers
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Rin... | Mathlib/Algebra/Periodic.lean | 572 | 574 | theorem Antiperiodic.mul_const' [DivisionSemiring α] [Neg β] (h : Antiperiodic f c) {a : α}
(ha : a ≠ 0) : Antiperiodic (fun x => f (x * a)) (c / a) := by |
simpa only [div_eq_mul_inv] using h.mul_const ha
|
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Tactic.Ring
import Mathlib.Data.PNat.Prime
#align_import data.pnat.xgcd from "leanprover-community/mathlib"@"6afc9b06856ad973f6a2619e3e8a0a8d537a58f... | Mathlib/Data/PNat/Xgcd.lean | 443 | 445 | theorem gcdA'_coe : (gcdA' a b : ℕ) = gcdW a b + gcdX a b := by |
dsimp [gcdA', gcdX, gcdW, XgcdType.w]
rw [add_right_comm]
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.RBMap.WF
/-!
# Path operations; `modify` and `alter`
This develops the necessary theorems to construct the `modify` and `alter` functions on `... | .lake/packages/batteries/Batteries/Data/RBMap/Alter.lean | 34 | 38 | theorem zoom_fill' (cut : α → Ordering) (t : RBNode α) (path : Path α) :
fill' (zoom cut t path) = path.fill t := by |
induction t generalizing path with
| nil => rfl
| node _ _ _ _ iha ihb => unfold zoom; split <;> [apply iha; apply ihb; rfl]
|
/-
Copyright (c) 2022 Floris van Doorn, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.To... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 522 | 528 | theorem Trivialization.contMDiffOn_iff {f : M → TotalSpace F E} {s : Set M}
(he : MapsTo f s e.source) :
ContMDiffOn IM (IB.prod 𝓘(𝕜, F)) n f s ↔
ContMDiffOn IM IB n (fun x => (f x).proj) s ∧
ContMDiffOn IM 𝓘(𝕜, F) n (fun x ↦ (e (f x)).2) s := by |
simp only [ContMDiffOn, ← forall_and]
exact forall₂_congr fun x hx ↦ e.contMDiffWithinAt_iff IB (he hx)
|
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.spec... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 92 | 94 | theorem inv_logb_div_base {a b : ℝ} (h₁ : a ≠ 0) (h₂ : b ≠ 0) (c : ℝ) :
(logb (a / b) c)⁻¹ = (logb a c)⁻¹ - (logb b c)⁻¹ := by |
simp_rw [inv_logb]; exact logb_div h₁ h₂
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Adam Topaz, Johan Commelin, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.C... | Mathlib/CategoryTheory/Abelian/Exact.lean | 162 | 164 | theorem exact_cokernel : Exact f (cokernel.π f) := by |
rw [exact_iff]
aesop_cat
|
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# Lindelöf sets and Lindelöf spaces
## Mai... | Mathlib/Topology/Compactness/Lindelof.lean | 418 | 433 | theorem Tendsto.isLindelof_insert_range_of_coLindelof {f : X → Y} {y}
(hf : Tendsto f (coLindelof X) (𝓝 y)) (hfc : Continuous f) :
IsLindelof (insert y (range f)) := by |
intro l hne _ hle
by_cases hy : ClusterPt y l
· exact ⟨y, Or.inl rfl, hy⟩
simp only [clusterPt_iff, not_forall, ← not_disjoint_iff_nonempty_inter, not_not] at hy
rcases hy with ⟨s, hsy, t, htl, hd⟩
rcases mem_coLindelof.1 (hf hsy) with ⟨K, hKc, hKs⟩
have : f '' K ∈ l := by
filter_upwards [htl, le_pri... |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Prod
#align_import analysis.calculus.fderiv.bilinear from "leanprover-community/mathlib"@"e... | Mathlib/Analysis/Calculus/FDeriv/Bilinear.lean | 51 | 74 | theorem IsBoundedBilinearMap.hasStrictFDerivAt (h : IsBoundedBilinearMap 𝕜 b) (p : E × F) :
HasStrictFDerivAt b (h.deriv p) p := by |
simp only [HasStrictFDerivAt]
simp only [← map_add_left_nhds_zero (p, p), isLittleO_map]
set T := (E × F) × E × F
calc
_ = fun x ↦ h.deriv (x.1 - x.2) (x.2.1, x.1.2) := by
ext ⟨⟨x₁, y₁⟩, ⟨x₂, y₂⟩⟩
rcases p with ⟨x, y⟩
simp only [map_sub, deriv_apply, Function.comp_apply, Prod.mk_add_mk, h... |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, François Dupuis
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Filter.Extr
import Mathlib.Tactic.GCongr
#align_import analysis.convex.function fr... | Mathlib/Analysis/Convex/Function.lean | 233 | 243 | theorem ConvexOn.convex_le (hf : ConvexOn 𝕜 s f) (r : β) : Convex 𝕜 ({ x ∈ s | f x ≤ r }) :=
fun x hx y hy a b ha hb hab =>
⟨hf.1 hx.1 hy.1 ha hb hab,
calc
f (a • x + b • y) ≤ a • f x + b • f y := hf.2 hx.1 hy.1 ha hb hab
_ ≤ a • r + b • r := by |
gcongr
· exact hx.2
· exact hy.2
_ = r := Convex.combo_self hab r
⟩
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.List.Count
import Mathlib.Data.List.Dedup
import Mathlib.Data.List.InsertNth
import Mathlib.Data.List.Lat... | Mathlib/Data/List/Perm.lean | 416 | 421 | theorem Perm.bagInter_left (l : List α) {t₁ t₂ : List α} (p : t₁ ~ t₂) :
l.bagInter t₁ = l.bagInter t₂ := by |
induction' l with a l IH generalizing t₁ t₂ p; · simp
by_cases h : a ∈ t₁
· simp [h, p.subset h, IH (p.erase _)]
· simp [h, mt p.mem_iff.2 h, IH p]
|
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## HNN Extensions of Groups
This file defines the HNN extension of a group `G`, `HNN... | Mathlib/GroupTheory/HNNExtension.lean | 395 | 410 | theorem unitsSMul_cancels_iff (u : ℤˣ) (w : NormalWord d) :
Cancels (-u) (unitsSMul φ u w) ↔ ¬ Cancels u w := by |
by_cases h : Cancels u w
· simp only [unitsSMul, h, dite_true, not_true_eq_false, iff_false]
induction w using consRecOn with
| ofGroup => simp [Cancels, unitsSMulWithCancel]
| cons g u' w h1 h2 _ =>
intro hc
apply not_cancels_of_cons_hyp _ _ h2
simp only [Cancels, cons_head, cons_toL... |
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Category.GroupCat.Basic
import Mathlib.CategoryTheory.ConcreteCategory.ReflectsIso
import Mathlib.Algebra.Ring... | Mathlib/Algebra/Category/Ring/Basic.lean | 674 | 677 | theorem commRingIsoToRingEquiv_symm_toRingHom {X Y : CommRingCat} (i : X ≅ Y) :
i.commRingCatIsoToRingEquiv.symm.toRingHom = i.inv := by |
ext
rfl
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 332 | 334 | theorem eval_eq_sum_range' {p : R[X]} {n : ℕ} (hn : p.natDegree < n) (x : R) :
p.eval x = ∑ i ∈ Finset.range n, p.coeff i * x ^ i := by |
rw [eval_eq_sum, p.sum_over_range' _ _ hn]; simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Tactic.ApplyFun
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.Separation
#align_import topology.... | Mathlib/Topology/UniformSpace/Separation.lean | 186 | 191 | theorem Filter.Tendsto.inseparable_iff_uniformity {l : Filter β} [NeBot l] {f g : β → α} {a b : α}
(ha : Tendsto f l (𝓝 a)) (hb : Tendsto g l (𝓝 b)) :
Inseparable a b ↔ Tendsto (fun x ↦ (f x, g x)) l (𝓤 α) := by |
refine ⟨fun h ↦ (ha.prod_mk_nhds hb).mono_right h.nhds_le_uniformity, fun h ↦ ?_⟩
rw [inseparable_iff_clusterPt_uniformity]
exact (ClusterPt.of_le_nhds (ha.prod_mk_nhds hb)).mono h
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
import Mathlib.Order.WellFounded
#align_import order.succ_pred.basic from "l... | Mathlib/Order/SuccPred/Basic.lean | 667 | 669 | theorem pred_lt_pred_iff_of_not_isMin (ha : ¬IsMin a) (hb : ¬IsMin b) :
pred a < pred b ↔ a < b := by |
rw [pred_lt_iff_of_not_isMin ha, le_pred_iff_of_not_isMin hb]
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Thin
#align_import category_theory.limits.shapes.wide_pullbacks from "lean... | Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean | 387 | 393 | theorem hom_ext (g1 g2 : X ⟶ widePullback _ _ arrows) : (∀ j : J,
g1 ≫ π arrows j = g2 ≫ π arrows j) → g1 ≫ base arrows = g2 ≫ base arrows → g1 = g2 := by |
intro h1 h2
apply limit.hom_ext
rintro (_ | _)
· apply h2
· apply h1
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Topology.UniformSpace.Com... | Mathlib/Topology/Algebra/UniformGroup.lean | 827 | 860 | theorem extend_Z_bilin : Continuous (extend (de.prod df) (fun p : β × δ => φ p.1 p.2)) := by |
refine continuous_extend_of_cauchy _ ?_
rintro ⟨x₀, y₀⟩
constructor
· apply NeBot.map
apply comap_neBot
intro U h
rcases mem_closure_iff_nhds.1 ((de.prod df).dense (x₀, y₀)) U h with ⟨x, x_in, ⟨z, z_x⟩⟩
exists z
aesop
· suffices map (fun p : (β × δ) × β × δ => (fun p : β × δ => φ p.1 p.2)... |
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Init.Core
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#ali... | Mathlib/NumberTheory/Cyclotomic/Basic.lean | 100 | 103 | theorem iff_singleton :
IsCyclotomicExtension {n} A B ↔
(∃ r : B, IsPrimitiveRoot r n) ∧ ∀ x, x ∈ adjoin A {b : B | b ^ (n : ℕ) = 1} := by |
simp [isCyclotomicExtension_iff]
|
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck, David Loeffler
-/
import Mathlib.Algebra.Module.Submodule.Basic
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Al... | Mathlib/Order/Filter/ZeroAndBoundedAtFilter.lean | 51 | 53 | theorem ZeroAtFilter.smul [TopologicalSpace β] [Zero 𝕜] [Zero β]
[SMulWithZero 𝕜 β] [ContinuousConstSMul 𝕜 β] {l : Filter α} {f : α → β} (c : 𝕜)
(hf : ZeroAtFilter l f) : ZeroAtFilter l (c • f) := by | simpa using hf.const_smul c
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.List.Basic
#align_import data.bool.all_any from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
/-!
# Boolean quantifi... | Mathlib/Data/Bool/AllAny.lean | 33 | 34 | theorem all_iff_forall_prop : (all l fun a => p a) ↔ ∀ a ∈ l, p a := by |
simp only [all_iff_forall, decide_eq_true_iff]
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 2,518 | 2,522 | theorem pmap_congr {p q : α → Prop} {f : ∀ a, p a → β} {g : ∀ a, q a → β} (l : List α) {H₁ H₂}
(h : ∀ a ∈ l, ∀ (h₁ h₂), f a h₁ = g a h₂) : pmap f l H₁ = pmap g l H₂ := by |
induction' l with _ _ ih
· rfl
· rw [pmap, pmap, h _ (mem_cons_self _ _), ih fun a ha => h a (mem_cons_of_mem _ ha)]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.... | Mathlib/LinearAlgebra/Multilinear/Basic.lean | 1,258 | 1,265 | theorem mkPiRing_apply_one_eq_self [Fintype ι] (f : MultilinearMap R (fun _ : ι => R) M₂) :
MultilinearMap.mkPiRing R ι (f fun _ => 1) = f := by |
ext m
have : m = fun i => m i • (1 : R) := by
ext j
simp
conv_rhs => rw [this, f.map_smul_univ]
rfl
|
/-
Copyright (c) 2020 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Tree.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum.C... | Mathlib/Tactic/CancelDenoms/Core.lean | 55 | 56 | theorem add_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :
n * (e1 + e2) = t1 + t2 := by | simp [left_distrib, *]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_imp... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 101 | 101 | theorem toComplex_im (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ).im = y := by | simp [toComplex_def]
|
/-
Copyright (c) 2023 Bulhwi Cha. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bulhwi Cha, Mario Carneiro
-/
import Batteries.Data.Char
import Batteries.Data.List.Lemmas
import Batteries.Data.String.Basic
import Batteries.Tactic.Lint.Misc
import Batteries.Tactic.SeqF... | .lake/packages/batteries/Batteries/Data/String/Lemmas.lean | 628 | 629 | theorem foldl_eq (f : α → Char → α) (s a) : foldl f a s = s.1.foldl f a := by |
simpa using foldlAux_of_valid f [] s.1 [] a
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury G. Kudryashov, Scott Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.BigOperators.Finsupp
impor... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 496 | 507 | theorem mapDomain_mul {α : Type*} {β : Type*} {α₂ : Type*} [Semiring β] [Mul α] [Mul α₂]
{F : Type*} [FunLike F α α₂] [MulHomClass F α α₂] (f : F) (x y : MonoidAlgebra β α) :
mapDomain f (x * y) = mapDomain f x * mapDomain f y := by |
simp_rw [mul_def, mapDomain_sum, mapDomain_single, map_mul]
rw [Finsupp.sum_mapDomain_index]
· congr
ext a b
rw [Finsupp.sum_mapDomain_index]
· simp
· simp [mul_add]
· simp
· simp [add_mul]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 1,203 | 1,219 | theorem affineSpan_induction' {s : Set P} {p : ∀ x, x ∈ affineSpan k s → Prop}
(mem : ∀ (y) (hys : y ∈ s), p y (subset_affineSpan k _ hys))
(smul_vsub_vadd :
∀ (c : k) (u hu v hv w hw),
p u hu →
p v hv → p w hw → p (c • (u -ᵥ v) +ᵥ w) (AffineSubspace.smul_vsub_vadd_mem _ _ hu hv hw))
... |
refine Exists.elim ?_ fun (hx : x ∈ affineSpan k s) (hc : p x hx) => hc
-- Porting note: Lean couldn't infer the motive
refine affineSpan_induction (p := fun y => ∃ z, p y z) h ?_ ?_
· exact fun y hy => ⟨subset_affineSpan _ _ hy, mem y hy⟩
· exact fun c u v w hu hv hw =>
Exists.elim hu fun hu' hu =>
... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
import Mathlib.Data.Option.Basic
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad... | Mathlib/Computability/PartrecCode.lean | 957 | 1,016 | theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a.2 :=
have :
Primrec₂ fun (_ : Unit) (n : ℕ) =>
let a := ofNat (ℕ × Code) n
(List.range a.1).map (evaln a.1 a.2) :=
Primrec.nat_strong_rec _ (hG.comp Primrec.snd).to₂ fun _ p => by
simp only [G, prod_ofNat_val, ofNat_n... |
simp
rw [this]
generalize p.unpair.1 = k
generalize ofNat Code p.unpair.2 = c
intro nk
cases' k with k'
· simp [evaln]
let k := k' + 1
simp only [show k'.succ = k from rfl]
simp? [Nat.lt_succ_iff] at nk says simp only [List.mem_range, Nat.lt_succ_iff] at nk... |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrabl... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 235 | 255 | theorem tendsto_setIntegral_of_monotone {ι : Type*} [Countable ι] [SemilatticeSup ι]
{s : ι → Set X} (hsm : ∀ i, MeasurableSet (s i)) (h_mono : Monotone s)
(hfi : IntegrableOn f (⋃ n, s n) μ) :
Tendsto (fun i => ∫ x in s i, f x ∂μ) atTop (𝓝 (∫ x in ⋃ n, s n, f x ∂μ)) := by |
have hfi' : ∫⁻ x in ⋃ n, s n, ‖f x‖₊ ∂μ < ∞ := hfi.2
set S := ⋃ i, s i
have hSm : MeasurableSet S := MeasurableSet.iUnion hsm
have hsub : ∀ {i}, s i ⊆ S := @(subset_iUnion s)
rw [← withDensity_apply _ hSm] at hfi'
set ν := μ.withDensity fun x => ‖f x‖₊ with hν
refine Metric.nhds_basis_closedBall.tendsto_... |
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Scott Morrison, Eric Rodriguez
-/
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count fr... | Mathlib/Data/Nat/Count.lean | 102 | 103 | theorem count_lt_count_succ_iff {n : ℕ} : count p n < count p (n + 1) ↔ p n := by |
by_cases h : p n <;> simp [count_succ, h]
|
/-
Copyright (c) 2022 Yuyang Zhao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuyang Zhao
-/
import Batteries.Classes.Order
namespace Batteries.PairingHeapImp
/--
A `Heap` is the nodes of the pairing heap.
Each node have two pointers: `child` going to the first c... | .lake/packages/batteries/Batteries/Data/PairingHeap.lean | 256 | 260 | theorem Heap.WF.tail? (hwf : (s : Heap α).WF le) : s.tail? le = some tl →
tl.WF le := by |
simp only [Heap.tail?]; intro eq
match eq₂ : s.deleteMin le, eq with
| some (a, tl), rfl => exact hwf.deleteMin eq₂
|
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semicon... | Mathlib/Data/Int/GCD.lean | 445 | 446 | theorem lcm_mul_left {m n k : ℤ} : (m * n).lcm (m * k) = natAbs m * n.lcm k := by |
simp_rw [Int.lcm, natAbs_mul, Nat.lcm_mul_left]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Init.Data.Fin.Basic
#align_import data.list.nodup from "leanprover-... | Mathlib/Data/List/Nodup.lean | 191 | 195 | theorem count_eq_of_nodup [DecidableEq α] {a : α} {l : List α} (d : Nodup l) :
count a l = if a ∈ l then 1 else 0 := by |
split_ifs with h
· exact count_eq_one_of_mem d h
· exact count_eq_zero_of_not_mem h
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Logic.Function.Conjugate
#align_import logic.function.iterate from "leanprover-community/mathlib"@"792a2a264169d64986541c6f8f7e3bbb6acb6295"
/-!
# ... | Mathlib/Logic/Function/Iterate.lean | 156 | 161 | theorem comp_iterate (h : Commute f g) (n : ℕ) : (f ∘ g)^[n] = f^[n] ∘ g^[n] := by |
induction n with
| zero => rfl
| succ n ihn =>
funext x
simp only [ihn, (h.iterate_right n).eq, iterate_succ, comp_apply]
|
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
/-!
# The length function, reduced words, and descents
Throughout this file, `B` is a type and `... | Mathlib/GroupTheory/Coxeter/Length.lean | 152 | 159 | theorem length_eq_one_iff {w : W} : ℓ w = 1 ↔ ∃ i : B, w = s i := by |
constructor
· intro h
rcases cs.exists_reduced_word w with ⟨ω, hω, rfl⟩
rcases List.length_eq_one.mp (hω.trans h) with ⟨i, rfl⟩
exact ⟨i, cs.wordProd_singleton i⟩
· rintro ⟨i, rfl⟩
exact cs.length_simple i
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Algebra.Order.Floor
import Mathlib.Topology.Algebra.Order.Group
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.floor fro... | Mathlib/Topology/Algebra/Order/Floor.lean | 101 | 105 | theorem tendsto_ceil_right_pure_floor_add_one (x : α) :
Tendsto (ceil : α → ℤ) (𝓝[>] x) (pure (⌊x⌋ + 1)) :=
have : ↑(⌊x⌋ + 1) - 1 ≤ x := by | rw [cast_add, cast_one, add_sub_cancel_right]; exact floor_le _
tendsto_pure.2 <| mem_of_superset (Ioc_mem_nhdsWithin_Ioi' <| lt_succ_floor _) fun _y hy =>
ceil_eq_on_Ioc _ _ ⟨this.trans_lt hy.1, hy.2⟩
|
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Anatole Dedecker
-/
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.Algebra.Equicontinuity
import Mathlib.Topology.MetricSpace.Equicontinuity
import Mathlib.Topology... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 361 | 365 | theorem WithSeminorms.separating_iff_T1 (hp : WithSeminorms p) :
(∀ x, x ≠ 0 → ∃ i, p i x ≠ 0) ↔ T1Space E := by |
refine ⟨WithSeminorms.T1_of_separating hp, ?_⟩
intro
exact WithSeminorms.separating_of_T1 hp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | Mathlib/Order/Lattice.lean | 261 | 262 | theorem sup_sup_distrib_right (a b c : α) : a ⊔ b ⊔ c = a ⊔ c ⊔ (b ⊔ c) := by |
rw [sup_sup_sup_comm, sup_idem]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Logic.Equiv.Nat
import Mathlib.Data.PNat.Basic
import Mathlib.Order.Directed
import Mathlib.Data.Countable.Defs
import Mathli... | Mathlib/Logic/Encodable/Basic.lean | 326 | 335 | theorem decode_ge_two (n) (h : 2 ≤ n) : (decode n : Option Bool) = none := by |
suffices decodeSum n = none by
change (decodeSum n).bind _ = none
rw [this]
rfl
have : 1 ≤ n / 2 := by
rw [Nat.le_div_iff_mul_le]
exacts [h, by decide]
cases' exists_eq_succ_of_ne_zero (_root_.ne_of_gt this) with m e
simp only [decodeSum, boddDiv2_eq, div2_val]; cases bodd n <;> simp [e]
|
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheo... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 208 | 211 | theorem exp_ne_zero_of_hasGenEigenvalue {f : End R M} {μ : R} {k : ℕ}
(h : f.HasGenEigenvalue μ k) : k ≠ 0 := by |
rintro rfl
exact h LinearMap.ker_id
|
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