Context stringlengths 295 65.3k | file_name stringlengths 21 74 | start int64 14 1.41k | end int64 20 1.41k | theorem stringlengths 27 1.42k | proof stringlengths 0 4.57k |
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/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | Mathlib/Data/ENNReal/Real.lean | 442 | 444 | theorem add_iInf {a : ℝ≥0∞} : a + iInf f = ⨅ b, a + f b := by | rw [add_comm, iInf_add]; simp [add_comm] |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Nat.Totient
import Mathlib.Data.ZMod.Aut
import Mathlib.Data.ZMod.QuotientGroup
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Sub... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 414 | 433 | theorem IsCyclic.image_range_card (ha : ∀ x : α, x ∈ zpowers a) :
Finset.image (fun i => a ^ i) (range (Nat.card α)) = univ := by | rw [← orderOf_eq_card_of_forall_mem_zpowers ha, IsCyclic.image_range_orderOf ha]
@[to_additive]
lemma IsCyclic.ext [Finite G] [IsCyclic G] {d : ℕ} {a b : ZMod d}
(hGcard : Nat.card G = d) (h : ∀ t : G, t ^ a.val = t ^ b.val) : a = b := by
have : NeZero (Nat.card G) := ⟨Nat.card_pos.ne'⟩
obtain ⟨g, hg⟩ := IsCyc... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | Mathlib/Data/ENNReal/Real.lean | 246 | 248 | theorem ofReal_le_iff_le_toReal {a : ℝ} {b : ℝ≥0∞} (hb : b ≠ ∞) :
ENNReal.ofReal a ≤ b ↔ a ≤ ENNReal.toReal b := by | lift b to ℝ≥0 using hb |
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Data.Nat.Factors
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
impo... | Mathlib/NumberTheory/FLT/Four.lean | 298 | 302 | theorem fermatLastTheoremFour : FermatLastTheoremFor 4 := by | rw [fermatLastTheoremFor_iff_int]
intro a b c ha hb _ heq
apply @not_fermat_42 _ _ (c ^ 2) ha hb
rw [heq]; ring |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | Mathlib/Algebra/Order/Field/Basic.lean | 272 | 275 | theorem le_iff_forall_one_lt_le_mul₀ {α : Type*}
[Semifield α] [LinearOrder α] [IsStrictOrderedRing α]
{a b : α} (hb : 0 ≤ b) : a ≤ b ↔ ∀ ε, 1 < ε → a ≤ b * ε := by | refine ⟨fun h _ hε ↦ h.trans <| le_mul_of_one_le_right hb hε.le, fun h ↦ ?_⟩ |
/-
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 Batteries.Tactic.Lint.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.ZeroLEOne... | Mathlib/Tactic/Linarith/Lemmas.lean | 42 | 44 | theorem le_of_le_of_eq {a b : α} (ha : a ≤ 0) (hb : b = 0) : a + b ≤ 0 := by | simp [*] |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Floris van Doorn, Gabriel Ebner, Yury Kudryashov
-/
import Mathlib.Data.Set.Accumulate
import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Order.Int... | Mathlib/Data/Nat/Lattice.lean | 86 | 88 | theorem nonempty_of_pos_sInf {s : Set ℕ} (h : 0 < sInf s) : s.Nonempty := by | by_contra contra
rw [Set.not_nonempty_iff_eq_empty] at contra |
/-
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.Analysis.SpecialFunctions.ExpDeriv
/-!
# Grönwall's inequality
The main technical result of this file is the Grönwall-like inequality
`norm_le_gron... | Mathlib/Analysis/ODE/Gronwall.lean | 73 | 76 | theorem gronwallBound_ε0 (δ K x : ℝ) : gronwallBound δ K 0 x = δ * exp (K * x) := by | by_cases hK : K = 0
· simp only [gronwallBound_K0, hK, zero_mul, exp_zero, add_zero, mul_one]
· simp only [gronwallBound_of_K_ne_0 hK, zero_div, zero_mul, add_zero] |
/-
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.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp.VectorSpace
import Mathlib.LinearAlge... | Mathlib/LinearAlgebra/TensorProduct/Basis.lean | 44 | 46 | theorem Basis.tensorProduct_apply' (b : Basis ι S M) (c : Basis κ R N) (i : ι × κ) :
Basis.tensorProduct b c i = b i.1 ⊗ₜ c i.2 := by | simp [Basis.tensorProduct] |
/-
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
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | Mathlib/Data/Set/Card.lean | 916 | 919 | theorem ncard_eq_ncard_iff_ncard_diff_eq_ncard_diff (hs : s.Finite := by | toFinite_tac)
(ht : t.Finite := by toFinite_tac) : s.ncard = t.ncard ↔ (s \ t).ncard = (t \ s).ncard := by
rw [← ncard_inter_add_ncard_diff_eq_ncard s t hs, ← ncard_inter_add_ncard_diff_eq_ncard t s ht,
inter_comm, add_right_inj] |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Defs
/-!
# Binomial coefficients
This file defines binomial coeffic... | Mathlib/Data/Nat/Choose/Basic.lean | 197 | 200 | theorem choose_symm_add {a b : ℕ} : choose (a + b) a = choose (a + b) b :=
choose_symm_of_eq_add rfl
theorem choose_symm_half (m : ℕ) : choose (2 * m + 1) (m + 1) = choose (2 * m + 1) m := by | |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Ring.Parity
import Mathlib.Combinatorics.SimpleGraph.Path
/-!
# Trails and Eulerian trails
This module contains additional theory about trails, in... | Mathlib/Combinatorics/SimpleGraph/Trails.lean | 119 | 125 | theorem IsEulerian.edgesFinset_eq [Fintype G.edgeSet] {u v : V} {p : G.Walk u v}
(h : p.IsEulerian) : h.isTrail.edgesFinset = G.edgeFinset := by | ext e
simp [h.mem_edges_iff]
theorem IsEulerian.even_degree_iff {x u v : V} {p : G.Walk u v} (ht : p.IsEulerian) [Fintype V]
[DecidableRel G.Adj] : Even (G.degree x) ↔ u ≠ v → x ≠ u ∧ x ≠ v := by |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.Normal.... | Mathlib/FieldTheory/SeparableDegree.lean | 807 | 809 | theorem IsSeparable.of_algebra_isSeparable_of_isSeparable [Algebra E K] [IsScalarTower F E K]
[Algebra.IsSeparable F E] {x : K} (hsep : IsSeparable E x) : IsSeparable F x := by | set f := minpoly E x with hf |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.Algebra.Equiv.TransferInstance
import Mathlib.RingTheory.Finiteness.Cardinality
/-!
# Orzech property of rings
In this file we define the following property of ring... | Mathlib/RingTheory/OrzechProperty.lean | 69 | 82 | theorem injective_of_surjective_of_injective
{N : Type w} [AddCommMonoid N] [Module R N]
(i f : N →ₗ[R] M) (hi : Injective i) (hf : Surjective f) : Injective f := by | obtain ⟨n, g, hg⟩ := Module.Finite.exists_fin' R M
haveI := small_of_surjective hg
letI := Equiv.addCommMonoid (equivShrink M).symm
letI := Equiv.module R (equivShrink M).symm
let j : Shrink.{u} M ≃ₗ[R] M := Equiv.linearEquiv R (equivShrink M).symm
haveI := Module.Finite.equiv j.symm
let i' := j.symm.toLine... |
/-
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.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | Mathlib/Data/Matroid/Basic.lean | 969 | 972 | theorem exists_isBasis_union_inter_isBasis (M : Matroid α) (X Y : Set α)
(hX : X ⊆ M.E := by | aesop_mat) (hY : Y ⊆ M.E := by aesop_mat) :
∃ I, M.IsBasis I (X ∪ Y) ∧ M.IsBasis (I ∩ Y) Y :=
let ⟨J, hJ⟩ := M.exists_isBasis Y |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Data.Finset.Max
import Mathlib.Data.Sy... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 351 | 354 | theorem degree_le_maxDegree [DecidableRel G.Adj] (v : V) : G.degree v ≤ G.maxDegree := by | obtain ⟨t, ht : _ = _⟩ := Finset.max_of_mem (mem_image_of_mem (fun v => G.degree v) (mem_univ v))
have := Finset.le_max_of_eq (mem_image_of_mem _ (mem_univ v)) ht
rwa [maxDegree, ht] |
/-
Copyright (c) 2022 Jon Eugster. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jon Eugster
-/
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient.Basic
import Mathlib.Tactic.FieldSimp
/-!
# Equal and mixed characteristic
In commutative ... | Mathlib/Algebra/CharP/MixedCharZero.lean | 201 | 206 | theorem pnatCast_eq_natCast [Fact (∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I))] (n : ℕ+) :
((n : Rˣ) : R) = ↑n := by | change ((PNat.isUnit_natCast (R := R) n).unit : R) = ↑n
simp only [IsUnit.unit_spec]
/-- Equal characteristic implies `ℚ`-algebra. -/ |
/-
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.Init
import Mathlib.Data.Int.Init
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 569 | 570 | theorem div_div : a / b / c = a / (b * c) := by | simp |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.Dynamics.Minimal
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.MeasureTheory.Group.Meas... | Mathlib/MeasureTheory/Group/Action.lean | 308 | 321 | theorem measure_isOpen_pos_of_smulInvariant_of_ne_zero (hμ : μ ≠ 0) (hU : IsOpen U)
(hne : U.Nonempty) : 0 < μ U :=
let ⟨_K, hK, hμK⟩ := Regular.exists_isCompact_not_null.mpr hμ
measure_isOpen_pos_of_smulInvariant_of_compact_ne_zero G hK hμK hU hne
@[to_additive]
theorem measure_pos_iff_nonempty_of_smulInvaria... | |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.UniqueFactorizationDomai... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 235 | 250 | theorem finprod_heightOneSpectrum_factorization_principal_fraction {n : R} (hn : n ≠ 0) (d : ↥R⁰) :
∏ᶠ v : HeightOneSpectrum R, (v.asIdeal : FractionalIdeal R⁰ K) ^
((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {n} : Ideal R)).factors -
(Associates.mk v.asIdeal).count (Associates.mk ((Id... | have hd_ne_zero : (algebraMap R K) (d : R) ≠ 0 :=
map_ne_zero_of_mem_nonZeroDivisors _ (IsFractionRing.injective R K) d.property
have h0 : spanSingleton R⁰ (mk' K n d) ≠ 0 := by
rw [spanSingleton_ne_zero_iff, IsFractionRing.mk'_eq_div, ne_eq, div_eq_zero_iff, not_or]
exact ⟨(map_ne_zero_iff (algebraMap R ... |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Nonneg.Field
import Mathlib.Data.Rat.Cast.Defs
import Mathlib.Tactic.Positivity.Basic
/-!
# Some exiled lemmas about cas... | Mathlib/Data/Rat/Cast/Lemmas.lean | 44 | 51 | theorem cast_nnratCast {K} [DivisionRing K] (q : ℚ≥0) :
((q : ℚ) : K) = (q : K) := by | rw [Rat.cast_def, NNRat.cast_def, NNRat.cast_def]
have hn := @num_div_eq_of_coprime q.num q.den ?hdp q.coprime_num_den
on_goal 1 => have hd := @den_div_eq_of_coprime q.num q.den ?hdp q.coprime_num_den
case hdp => simpa only [Int.ofNat_pos] using q.den_pos
simp only [Int.cast_natCast, Nat.cast_inj] at hn hd
rw... |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Kevin Buzzard, Jujian Zhang, Fangming Li
-/
import Mathlib.Algebra.DirectSum.Algebra
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Algebra.DirectSum.Internal
... | Mathlib/RingTheory/GradedAlgebra/Basic.lean | 340 | 343 | theorem coe_decompose_mul_of_right_mem (n) [Decidable (i ≤ n)] (b_mem : b ∈ 𝒜 i) :
(decompose 𝒜 (a * b) n : A) = if i ≤ n then decompose 𝒜 a (n - i) * b else 0 := by | lift b to 𝒜 i using b_mem
rw [decompose_mul, decompose_coe, coe_mul_of_apply] |
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Tactic.IntervalCases
/-!
# Cubics and discriminants
This file defines cubic polynomials ... | Mathlib/Algebra/CubicDiscriminant.lean | 220 | 222 | theorem monic_of_c_eq_one (ha : P.a = 0) (hb : P.b = 0) (hc : P.c = 1) : P.toPoly.Monic := by | nontriviality R
rw [Monic, leadingCoeff_of_c_ne_zero ha hb (hc ▸ one_ne_zero), hc] |
/-
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.Complex
import Qq
/-! # P... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 647 | 656 | theorem rpow_lt_rpow_left_iff_of_base_lt_one (hx0 : 0 < x) (hx1 : x < 1) :
x ^ y < x ^ z ↔ z < y := by | rw [lt_iff_not_le, rpow_le_rpow_left_iff_of_base_lt_one hx0 hx1, lt_iff_not_le]
theorem rpow_lt_one {x z : ℝ} (hx1 : 0 ≤ x) (hx2 : x < 1) (hz : 0 < z) : x ^ z < 1 := by
rw [← one_rpow z]
exact rpow_lt_rpow hx1 hx2 hz
theorem rpow_le_one {x z : ℝ} (hx1 : 0 ≤ x) (hx2 : x ≤ 1) (hz : 0 ≤ z) : x ^ z ≤ 1 := by
rw [← ... |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
/-!
# Permutations from a list
A list `l : List α` ... | Mathlib/GroupTheory/Perm/List.lean | 100 | 103 | theorem formPerm_apply_of_not_mem (h : x ∉ l) : formPerm l x = x :=
not_imp_comm.1 mem_of_formPerm_apply_ne h
theorem formPerm_apply_mem_of_mem (h : x ∈ l) : formPerm l x ∈ l := by | |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebra.Group.TypeTags.Finite
import Mathlib.Combinatorics.Enumerative... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 204 | 211 | theorem cycleType_prime_order {σ : Perm α} (hσ : (orderOf σ).Prime) :
∃ n : ℕ, σ.cycleType = Multiset.replicate (n + 1) (orderOf σ) := by | refine ⟨Multiset.card σ.cycleType - 1, eq_replicate.2 ⟨?_, fun n hn ↦ ?_⟩⟩
· rw [tsub_add_cancel_of_le]
rw [Nat.succ_le_iff, card_cycleType_pos, Ne, ← orderOf_eq_one_iff]
exact hσ.ne_one
· exact (hσ.eq_one_or_self_of_dvd n (dvd_of_mem_cycleType hn)).resolve_left
(one_lt_of_mem_cycleType hn).ne' |
/-
Copyright (c) 2023 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck, Ruben Van de Velde
-/
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
impor... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 64 | 66 | theorem iteratedDerivWithin_const_smul (c : R) (hf : ContDiffWithinAt 𝕜 n f s x) :
iteratedDerivWithin n (c • f) s x = c • iteratedDerivWithin n f s x := by | simp_rw [iteratedDerivWithin] |
/-
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.Ring.Associated
import Mathlib.Algebra.Ring.Regular
/-!
# Monoids with normalization functions, `gcd`, and `lcm`
This file de... | Mathlib/Algebra/GCDMonoid/Basic.lean | 468 | 479 | theorem dvd_mul_gcd_iff_dvd_mul [GCDMonoid α] {m n k : α} : k ∣ m * gcd k n ↔ k ∣ m * n :=
⟨fun h => h.trans (mul_dvd_mul dvd_rfl (gcd_dvd_right k n)), dvd_mul_gcd_of_dvd_mul⟩
/-- Represent a divisor of `m * n` as a product of a divisor of `m` and a divisor of `n`.
Note: In general, this representation is highly no... | cases h
by_cases h0 : gcd k m = 0 |
/-
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.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... | Mathlib/Order/SuccPred/Basic.lean | 452 | 455 | theorem Ioo_succ_right_eq_insert_of_not_isMax (h₁ : a < b) (h₂ : ¬IsMax b) :
Ioo a (succ b) = insert b (Ioo a b) := by | simp_rw [← Iio_inter_Ioi, Iio_succ_eq_insert_of_not_isMax h₂, insert_inter_of_mem (mem_Ioi.2 h₁)] |
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Analysis.SpecialFunctions.Complex.LogBounds
/-!
# Complex arctangent
This file defines the complex arctangent `Complex.arctan` as
$$\arctan z = -\frac i2 \lo... | Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean | 64 | 77 | theorem arctan_tan {z : ℂ} (h₀ : z ≠ π / 2) (h₁ : -(π / 2) < z.re) (h₂ : z.re ≤ π / 2) :
arctan (tan z) = z := by | have h := cos_ne_zero_of_arctan_bounds h₀ h₁ h₂
unfold arctan tan
-- multiply top and bottom by `cos z`
rw [← mul_div_mul_right (1 + _) _ h, add_mul, sub_mul, one_mul, ← mul_rotate, mul_div_cancel₀ _ h]
conv_lhs =>
enter [2, 1, 2]
rw [sub_eq_add_neg, ← neg_mul, ← sin_neg, ← cos_neg]
rw [← exp_mul_I, ←... |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Algebra.Group.Fin.Tuple
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Alge... | Mathlib/Data/Matrix/Notation.lean | 190 | 193 | theorem cons_dotProduct_cons (x : α) (v : Fin n → α) (y : α) (w : Fin n → α) :
dotProduct (vecCons x v) (vecCons y w) = x * y + dotProduct v w := by | simp
end DotProduct |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Nat.ModEq
/-!
# Congruences modulo an integer
This file defines the equivalence relation `a ≡ b [ZMOD n]` on the integers, similarly to how
`Data.N... | Mathlib/Data/Int/ModEq.lean | 77 | 78 | theorem modEq_iff_dvd : a ≡ b [ZMOD n] ↔ n ∣ b - a := by | rw [ModEq, eq_comm] |
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SetFamily.Shadow
/-!
# UV-compressions
This file defines UV-compression. It is an operation on a set family that reduces its ... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 194 | 200 | theorem card_compression (u v : α) (s : Finset α) : #(𝓒 u v s) = #s := by | rw [compression, card_union_of_disjoint compress_disjoint, filter_image,
card_image_of_injOn compress_injOn, ← card_union_of_disjoint (disjoint_filter_filter_neg s _ _),
filter_union_filter_neg_eq]
theorem le_of_mem_compression_of_not_mem (h : a ∈ 𝓒 u v s) (ha : a ∉ s) : u ≤ a := by
rw [mem_compression] at ... |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Data.Finset.Fin
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Interval.Set.Fin
/-!
# Finite intervals in `Fin n`
This fi... | Mathlib/Order/Interval/Finset/Fin.lean | 241 | 242 | theorem finsetImage_castLE_Iic (h : n ≤ m) :
(Iic a).image (castLE h) = Iic (castLE h a) := by | simp [← coe_inj] |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
/-! # Measurability of the line derivative
We prove in `me... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 33 | 38 | theorem measurableSet_lineDifferentiableAt (hf : Continuous f) :
MeasurableSet {x : E | LineDifferentiableAt 𝕜 f x v} := by | borelize 𝕜
let g : E → 𝕜 → F := fun x t ↦ f (x + t • v)
have hg : Continuous g.uncurry := by fun_prop
exact measurable_prodMk_right (measurableSet_of_differentiableAt_with_param 𝕜 hg) |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
/-!
# Equicontinuity of a family of functions
Let `X` be a topological space and `α` a `UniformS... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 633 | 638 | theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X}
{p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p₁ s₁)
(hα : (𝓤 α).HasBasis p₂ s₂) :
EquicontinuousAt F x₀ ↔
∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x ∈ s₁ k₁, ∀ i, (F i x₀, ... | rw [equicontinuousAt_iff_continuousAt, ContinuousAt, |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | Mathlib/Topology/Order/DenselyOrdered.lean | 350 | 352 | theorem tendsto_Iio_atTop {f : β → Iio a} :
Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by | rw [← comap_coe_Iio_nhdsLT, tendsto_comap_iff]; rfl |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Set.Prod
/-!
# N-ary images of sets
This file defines `Set.image2`, the binary image of sets.
This is mostly useful to define pointwise oper... | Mathlib/Data/Set/NAry.lean | 325 | 329 | theorem image2_inter_union_subset {f : α → α → β} {s t : Set α} (hf : ∀ a b, f a b = f b a) :
image2 f (s ∩ t) (s ∪ t) ⊆ image2 f s t := by | rw [inter_comm]
exact image2_inter_union_subset_union.trans (union_subset (image2_comm hf).subset Subset.rfl) |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.ExistsOfLE
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Unbundled.Basic
impor... | Mathlib/Algebra/Order/Sub/Basic.lean | 25 | 28 | theorem tsub_add_cancel_iff_le : b - a + a = b ↔ a ≤ b := by | rw [add_comm]
exact add_tsub_cancel_iff_le |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | Mathlib/MeasureTheory/Function/Jacobian.lean | 751 | 757 | theorem aemeasurable_toNNReal_abs_det_fderivWithin (hs : MeasurableSet s)
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) :
AEMeasurable (fun x => |(f' x).det|.toNNReal) (μ.restrict s) := by | apply measurable_real_toNNReal.comp_aemeasurable
refine continuous_abs.measurable.comp_aemeasurable ?_
refine ContinuousLinearMap.continuous_det.measurable.comp_aemeasurable ?_
exact aemeasurable_fderivWithin μ hs hf' |
/-
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, Heather Macbeth
-/
import Mathlib.Geometry.Manifold.VectorBundle.Basic
/-! # Tangent bundles
This file defines the tangent bundle as a `C^n` vector bundle.
Let `... | Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean | 54 | 63 | theorem contDiffOn_fderiv_coord_change [IsManifold I (n + 1) M]
(i j : atlas H M) :
ContDiffOn 𝕜 n (fderivWithin 𝕜 (j.1.extend I ∘ (i.1.extend I).symm) (range I))
((i.1.extend I).symm ≫ j.1.extend I).source := by | have h : ((i.1.extend I).symm ≫ j.1.extend I).source ⊆ range I := by
rw [i.1.extend_coord_change_source]; apply image_subset_range
intro x hx
refine (ContDiffWithinAt.fderivWithin_right ?_ I.uniqueDiffOn le_rfl
<| h hx).mono h
refine (PartialHomeomorph.contDiffOn_extend_coord_change (subset_maximalAtlas j... |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Support
import Mathlib.Data.Set.SymmDiff
/-!
# Indicator function
- `Set.indicator (s : Set α) (f ... | Mathlib/Algebra/Group/Indicator.lean | 348 | 351 | theorem mulIndicator_mul_eq_left {f g : α → M} (h : Disjoint (mulSupport f) (mulSupport g)) :
(mulSupport f).mulIndicator (f * g) = f := by | refine (mulIndicator_congr fun x hx => ?_).trans mulIndicator_mulSupport
have : g x = 1 := nmem_mulSupport.1 (disjoint_left.1 h hx) |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.CauSeq.Basic
import Mathlib.Algebra.Ring.Action.Rat
import Mathlib.Tactic.FastInstance
/-!
# Cauchy completion
This fi... | Mathlib/Algebra/Order/CauSeq/Completion.lean | 364 | 368 | theorem lim_inv {f : CauSeq β abv} (hf : ¬LimZero f) : lim (inv f hf) = (lim f)⁻¹ :=
have hl : lim f ≠ 0 := by | rwa [← lim_eq_zero_iff] at hf
lim_eq_of_equiv_const <|
show LimZero (inv f hf - const abv (lim f)⁻¹) from
have h₁ : ∀ (g f : CauSeq β abv) (hf : ¬LimZero f), LimZero (g - f * inv f hf * g) := |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Kim Morrison, Apurva Nakade, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.SetTheory.PGame.Algebra
import Mathl... | Mathlib/SetTheory/Game/Basic.lean | 112 | 114 | theorem not_lf : ∀ {x y : Game}, ¬Game.LF x y ↔ y ≤ x := by | rintro ⟨x⟩ ⟨y⟩
exact PGame.not_lf |
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Dynamics.PeriodicPts.Defs
import Mathlib.GroupTheory.G... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 96 | 98 | theorem fixedBy_mul (m₁ m₂ : M) : fixedBy α m₁ ∩ fixedBy α m₂ ⊆ fixedBy α (m₁ * m₂) := by | intro a ⟨h₁, h₂⟩
rw [mem_fixedBy, mul_smul, h₂, h₁] |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Filter.Prod
/-!
# N-ary maps of filter
This file defines the binary and ternary maps of filters. This is mostly useful to define pointwise
operatio... | Mathlib/Order/Filter/NAry.lean | 91 | 91 | theorem map₂_neBot_iff : (map₂ m f g).NeBot ↔ f.NeBot ∧ g.NeBot := by | simp [neBot_iff, not_or] |
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Order.Partition.Equipartition
/-!
# Turán's theorem
In this file we prove Turán's theorem, the first importan... | Mathlib/Combinatorics/SimpleGraph/Turan.lean | 74 | 82 | theorem turanGraph_eq_top : turanGraph n r = ⊤ ↔ r = 0 ∨ n ≤ r := by | simp_rw [SimpleGraph.ext_iff, funext_iff, turanGraph, top_adj, eq_iff_iff, not_iff_not]
refine ⟨fun h ↦ ?_, ?_⟩
· contrapose! h
use ⟨0, (Nat.pos_of_ne_zero h.1).trans h.2⟩, ⟨r, h.2⟩
simp [h.1.symm]
· rintro (rfl | h) a b
· simp [Fin.val_inj]
· rw [Nat.mod_eq_of_lt (a.2.trans_le h), Nat.mod_eq_of_l... |
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov, Kevin H. Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Tendsto
/-!
# Product and coproduct filters
In this file we define `F... | Mathlib/Order/Filter/Prod.lean | 452 | 453 | theorem bot_coprod_bot : (⊥ : Filter α).coprod (⊥ : Filter β) = ⊥ := by | simp |
/-
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.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... | Mathlib/Analysis/Convex/Between.lean | 498 | 501 | theorem Wbtw.sameRay_vsub {x y z : P} (h : Wbtw R x y z) : SameRay R (y -ᵥ x) (z -ᵥ y) := by | rcases h with ⟨t, ⟨ht0, ht1⟩, rfl⟩
simp_rw [lineMap_apply]
rcases ht0.lt_or_eq with (ht0' | rfl); swap; · simp |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 166 | 177 | theorem image_of_equiv {ν : Measure β} (h : IsFundamentalDomain G s μ) (f : α ≃ β)
(hf : QuasiMeasurePreserving f.symm ν μ) (e : H ≃ G)
(hef : ∀ g, Semiconj f (e g • ·) (g • ·)) : IsFundamentalDomain H (f '' s) ν := by | rw [f.image_eq_preimage]
refine h.preimage_of_equiv hf e.symm.bijective fun g x => ?_
rcases f.surjective x with ⟨x, rfl⟩
rw [← hef _ _, f.symm_apply_apply, f.symm_apply_apply, e.apply_symm_apply]
@[to_additive]
theorem pairwise_aedisjoint_of_ac {ν} (h : IsFundamentalDomain G s μ) (hν : ν ≪ μ) :
Pairwise fun... |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
/-!
# Changing the index type of a matrix
This file concerns the map... | Mathlib/LinearAlgebra/Matrix/Reindex.lean | 66 | 70 | theorem reindexLinearEquiv_comp (e₁ : m ≃ m') (e₂ : n ≃ n') (e₁' : m' ≃ m'') (e₂' : n' ≃ n'') :
reindexLinearEquiv R A e₁' e₂' ∘ reindexLinearEquiv R A e₁ e₂ =
reindexLinearEquiv R A (e₁.trans e₁') (e₂.trans e₂') := by | rw [← reindexLinearEquiv_trans]
rfl |
/-
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, Kim Morrison, Alex Keizer
-/
import Mathlib.Data.List.OfFn
import Batteries.Data.List.Perm
import Mathlib.Data.List.Nodup
/-!
# Lists of elements of `Fin n`... | Mathlib/Data/List/FinRange.lean | 25 | 27 | theorem finRange_eq_pmap_range (n : ℕ) : finRange n = (range n).pmap Fin.mk (by simp) := by | apply List.ext_getElem <;> simp [finRange] |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Ring.Basic
/-!
# Infini... | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | 81 | 82 | theorem hasSum_div_const_iff (h : a₂ ≠ 0) : HasSum (fun i ↦ f i / a₂) (a₁ / a₂) ↔ HasSum f a₁ := by | simpa only [div_eq_mul_inv] using hasSum_mul_right_iff (inv_ne_zero h) |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | Mathlib/Data/Set/Prod.lean | 293 | 295 | theorem image_prodMk_subset_prod_right (ha : a ∈ s) : Prod.mk a '' t ⊆ s ×ˢ t := by | rintro _ ⟨b, hb, rfl⟩
exact ⟨ha, hb⟩ |
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Data.Complex.Basic
import Mathlib.MeasureTheory.Integral.CircleIntegral
/-!
# Circle integral transform
In this file we define the circle integral tr... | Mathlib/MeasureTheory/Integral/CircleTransform.lean | 98 | 106 | theorem continuousOn_norm_circleTransformBoundingFunction {R r : ℝ} (hr : r < R) (z : ℂ) :
ContinuousOn ((‖·‖) ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ) := by | have : ContinuousOn (circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ) := by
apply_rules [ContinuousOn.smul, continuousOn_const]
· simp only [deriv_circleMap]
apply_rules [ContinuousOn.mul, (continuous_circleMap 0 R).comp_continuousOn continuousOn_snd,
continuousOn_const]
· simpa o... |
/-
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, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | Mathlib/Algebra/Polynomial/Basic.lean | 822 | 826 | theorem monomial_left_inj {a : R} (ha : a ≠ 0) {i j : ℕ} :
monomial i a = monomial j a ↔ i = j := by | simp only [← ofFinsupp_single, ofFinsupp.injEq, Finsupp.single_left_inj ha]
theorem binomial_eq_binomial {k l m n : ℕ} {u v : R} (hu : u ≠ 0) (hv : v ≠ 0) : |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Rat
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.PNat.Defs... | Mathlib/Data/Rat/Lemmas.lean | 247 | 252 | theorem inv_natCast_den_of_pos {a : ℕ} (ha0 : 0 < a) : (a : ℚ)⁻¹.den = a := by | rw [← Int.ofNat_inj, ← Int.cast_natCast a, inv_intCast_den_of_pos]
rwa [Int.natCast_pos]
@[simp]
theorem inv_intCast_num (a : ℤ) : (a : ℚ)⁻¹.num = Int.sign a := by |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Ideal.Operations
/-!
# Maps on modules and ideals
Main definitions include `Ideal.map`, `Ideal.comap`, `RingHom.ker`, `Module.annihilator`
and `Subm... | Mathlib/RingTheory/Ideal/Maps.lean | 691 | 702 | theorem ker_coe_equiv (f : R ≃+* S) : ker (f : R →+* S) = ⊥ := by | ext; simp
theorem ker_coe_toRingHom : ker (f : R →+* S) = ker f := rfl
@[simp]
theorem ker_equiv {F' : Type*} [EquivLike F' R S] [RingEquivClass F' R S] (f : F') :
ker f = ⊥ := by
ext; simp
lemma ker_equiv_comp (f : R →+* S) (e : S ≃+* T) :
ker (e.toRingHom.comp f) = RingHom.ker f := by |
/-
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
-/
import Mathlib.MeasureTheory.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 451 | 455 | theorem restrict_biInf {ι} {I : Set ι} (hI : I.Nonempty) (s : Set α) (m : ι → OuterMeasure α) :
restrict s (⨅ i ∈ I, m i) = ⨅ i ∈ I, restrict s (m i) := by | haveI := hI.to_subtype
rw [← iInf_subtype'', ← iInf_subtype'']
exact restrict_iInf _ _ |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Eric Wieser
-/
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Matrices as a normed space
In this file we provide the fo... | Mathlib/Analysis/Matrix.lean | 88 | 89 | 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] |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | Mathlib/NumberTheory/ArithmeticFunction.lean | 577 | 580 | theorem map_div_of_coprime [GroupWithZero R] {f : ArithmeticFunction R}
(hf : IsMultiplicative f) {l d : ℕ} (hdl : d ∣ l) (hl : (l / d).Coprime d) (hd : f d ≠ 0) :
f (l / d) = f l / f d := by | apply (div_eq_of_eq_mul hd ..).symm |
/-
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.Group.Hom.CompTypeclasses
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Notation.Prod
import Mathlib.Algebra.Ring.Act... | Mathlib/GroupTheory/GroupAction/Hom.lean | 560 | 561 | theorem id_apply (x : A) : DistribMulActionHom.id M x = x := by | rfl |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Da... | Mathlib/Data/Nat/Digits.lean | 267 | 287 | theorem digits_eq_nil_iff_eq_zero {b n : ℕ} : digits b n = [] ↔ n = 0 := by | constructor
· intro h
have : ofDigits b (digits b n) = ofDigits b [] := by rw [h]
convert this
rw [ofDigits_digits]
· rintro rfl
simp
theorem digits_ne_nil_iff_ne_zero {b n : ℕ} : digits b n ≠ [] ↔ n ≠ 0 :=
not_congr digits_eq_nil_iff_eq_zero
theorem digits_eq_cons_digits_div {b n : ℕ} (h : 1 < ... |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... | Mathlib/Order/Heyting/Basic.lean | 336 | 336 | theorem Codisjoint.himp_inf_cancel_left (h : Codisjoint a b) : b ⇨ a ⊓ b = a := by | |
/-
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
-/
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Topology.UniformSpac... | Mathlib/Topology/UniformSpace/Cauchy.lean | 316 | 324 | theorem isComplete_iff_clusterPt {s : Set α} :
IsComplete s ↔ ∀ l, Cauchy l → l ≤ 𝓟 s → ∃ x ∈ s, ClusterPt x l :=
forall₃_congr fun _ hl _ => exists_congr fun _ => and_congr_right fun _ =>
le_nhds_iff_adhp_of_cauchy hl
theorem isComplete_iff_ultrafilter {s : Set α} :
IsComplete s ↔ ∀ l : Ultrafilter α, ... | refine ⟨fun h l => h l, fun H => isComplete_iff_clusterPt.2 fun l hl hls => ?_⟩
haveI := hl.1 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Ideal.Operations
/-!
# Maps on modules and ideals
Main definitions include `Ideal.map`, `Ideal.comap`, `RingHom.ker`, `Module.annihilator`
and `Subm... | Mathlib/RingTheory/Ideal/Maps.lean | 715 | 716 | theorem ker_eq_bot_iff_eq_zero : ker f = ⊥ ↔ ∀ x, f x = 0 → x = 0 := by | rw [← injective_iff_map_eq_zero f, injective_iff_ker_eq_bot] |
/-
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.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.Meas... | Mathlib/Analysis/Convolution.lean | 956 | 999 | theorem _root_.HasCompactSupport.hasFDerivAt_convolution_right (hcg : HasCompactSupport g)
(hf : LocallyIntegrable f μ) (hg : ContDiff 𝕜 1 g) (x₀ : G) :
HasFDerivAt (f ⋆[L, μ] g) ((f ⋆[L.precompR G, μ] fderiv 𝕜 g) x₀) x₀ := by | rcases hcg.eq_zero_or_finiteDimensional 𝕜 hg.continuous with (rfl | fin_dim)
· have : fderiv 𝕜 (0 : G → E') = 0 := fderiv_const (0 : E')
simp only [this, convolution_zero, Pi.zero_apply]
exact hasFDerivAt_const (0 : F) x₀
have : ProperSpace G := FiniteDimensional.proper_rclike 𝕜 G
set L' := L.precompR ... |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Dynamics.BirkhoffSum.Average
/-!
# Birkhoff average in a normed space
In this file we prove some lemmas about ... | Mathlib/Dynamics/BirkhoffSum/NormedSpace.lean | 75 | 84 | theorem tendsto_birkhoffAverage_apply_sub_birkhoffAverage {f : α → α} {g : α → E} {x : α}
(h : Bornology.IsBounded (range (g <| f^[·] x))) :
Tendsto (fun n ↦ birkhoffAverage 𝕜 f g n (f x) - birkhoffAverage 𝕜 f g n x) atTop (𝓝 0) := by | rcases Metric.isBounded_range_iff.1 h with ⟨C, hC⟩
have : Tendsto (fun n : ℕ ↦ C / n) atTop (𝓝 0) :=
tendsto_const_nhds.div_atTop tendsto_natCast_atTop_atTop
refine squeeze_zero_norm (fun n ↦ ?_) this
rw [← dist_eq_norm, dist_birkhoffAverage_apply_birkhoffAverage]
gcongr
exact hC n 0 |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Finset.Max
import Mathlib.Data.Fintype.Pigeonhole
import Mathlib.Logic.Enco... | Mathlib/Order/SuccPred/LinearLocallyFinite.lean | 103 | 106 | theorem succFn_spec (i : ι) : IsGLB (Set.Ioi i) (succFn i) :=
(exists_glb_Ioi i).choose_spec
theorem le_succFn (i : ι) : i ≤ succFn i := by | |
/-
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.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Normed.Operator.BoundedLinear... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 751 | 762 | theorem measurableSet_of_differentiableWithinAt_Ioi :
MeasurableSet { x | DifferentiableWithinAt ℝ f (Ioi x) x } := by | simpa [differentiableWithinAt_Ioi_iff_Ici] using measurableSet_of_differentiableWithinAt_Ici f
@[measurability, fun_prop]
theorem measurable_derivWithin_Ioi [MeasurableSpace F] [BorelSpace F] :
Measurable fun x => derivWithin f (Ioi x) x := by
simpa [derivWithin_Ioi_eq_Ici] using measurable_derivWithin_Ici f
th... |
/-
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.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
/-!
# Proba... | Mathlib/Probability/Density.lean | 247 | 255 | theorem quasiMeasurePreserving_hasPDF (hg : QuasiMeasurePreserving g μ ν)
(hmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν) : HasPDF (g ∘ X) ℙ ν := by | have hgm : AEMeasurable g (map X ℙ) := hg.aemeasurable.mono_ac HasPDF.absolutelyContinuous
rw [hasPDF_iff, ← AEMeasurable.map_map_of_aemeasurable hgm (HasPDF.aemeasurable X ℙ μ)]
refine ⟨hg.measurable.comp_aemeasurable (HasPDF.aemeasurable _ _ μ), hmap, ?_⟩
exact (HasPDF.absolutelyContinuous.map hg.1).trans hg.2
... |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Violeta Hernández Palacios, Grayson Burton, Floris van Doorn
-/
import Mathlib.Order.Antisymmetrization
import Mathlib.Order.Hom.WithTopBot
import Mathlib.Order.Interval.Se... | Mathlib/Order/Cover.lean | 557 | 562 | theorem wcovBy_iff : x ⩿ y ↔ x.1 ⩿ y.1 ∧ x.2 = y.2 ∨ x.2 ⩿ y.2 ∧ x.1 = y.1 := by | cases x
cases y
exact mk_wcovBy_mk_iff
theorem covBy_iff : x ⋖ y ↔ x.1 ⋖ y.1 ∧ x.2 = y.2 ∨ x.2 ⋖ y.2 ∧ x.1 = y.1 := by |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Stopping times, stopped processes and stopped va... | Mathlib/Probability/Process/Stopping.lean | 526 | 534 | theorem measurableSpace_min (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π) :
(hτ.min hπ).measurableSpace = hτ.measurableSpace ⊓ hπ.measurableSpace := by | refine le_antisymm ?_ ?_
· exact le_inf (measurableSpace_mono _ hτ fun _ => min_le_left _ _)
(measurableSpace_mono _ hπ fun _ => min_le_right _ _)
· intro s
change MeasurableSet[hτ.measurableSpace] s ∧ MeasurableSet[hπ.measurableSpace] s →
MeasurableSet[(hτ.min hπ).measurableSpace] s
simp_rw [Is... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 451 | 455 | theorem ord_cof (hf : IsFundamentalSequence a o f) :
IsFundamentalSequence a a.cof.ord fun i hi => f i (hi.trans_le (by rw [hf.cof_eq])) := by | have H := hf.cof_eq
subst H
exact hf |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Cardinal
import Mathlib.SetTheory.Cardi... | Mathlib/Data/Real/Cardinality.lean | 225 | 243 | theorem mk_Ici_real (a : ℝ) : #(Ici a) = 𝔠 :=
le_antisymm (mk_real ▸ mk_set_le _) (mk_Ioi_real a ▸ mk_le_mk_of_subset Ioi_subset_Ici_self)
/-- The cardinality of the interval (-∞, a). -/
theorem mk_Iio_real (a : ℝ) : #(Iio a) = 𝔠 := by | refine le_antisymm (mk_real ▸ mk_set_le _) ?_
have h2 : (fun x => a + a - x) '' Iio a = Ioi a := by
simp only [image_const_sub_Iio, add_sub_cancel_right]
exact mk_Ioi_real a ▸ h2 ▸ mk_image_le
/-- The cardinality of the interval (-∞, a]. -/
theorem mk_Iic_real (a : ℝ) : #(Iic a) = 𝔠 :=
le_antisymm (mk_real ... |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton
-/
import Mathlib.Topology.Hom.ContinuousEval
import Mathlib.Topology.ContinuousMap.Basic
import Mathlib.Topology.Separation.Regular
/-!
# The compact-open topology
In this ... | Mathlib/Topology/CompactOpen.lean | 274 | 282 | theorem nhds_compactOpen_eq_iInf_nhds_induced (f : C(X, Y)) :
𝓝 f = ⨅ (s) (_ : IsCompact s), (𝓝 (f.restrict s)).comap (ContinuousMap.restrict s) := by | rw [compactOpen_eq_iInf_induced]
simp only [nhds_iInf, nhds_induced]
theorem tendsto_compactOpen_restrict {ι : Type*} {l : Filter ι} {F : ι → C(X, Y)} {f : C(X, Y)}
(hFf : Filter.Tendsto F l (𝓝 f)) (s : Set X) :
Tendsto (fun i => (F i).restrict s) l (𝓝 (f.restrict s)) :=
(continuous_restrict s).continuou... |
/-
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
/-!
# Lists with no duplicates
`List.Nodup` is defined in `Data/List/Basic`. In this file we prove various properties of... | Mathlib/Data/List/Nodup.lean | 330 | 344 | theorem Nodup.union [DecidableEq α] (l₁ : List α) (h : Nodup l₂) : (l₁ ∪ l₂).Nodup := by | induction l₁ generalizing l₂ with
| nil => exact h
| cons a l₁ ih => exact (ih h).insert
theorem Nodup.inter [DecidableEq α] (l₂ : List α) : Nodup l₁ → Nodup (l₁ ∩ l₂) :=
Nodup.filter _
theorem Nodup.diff_eq_filter [BEq α] [LawfulBEq α] :
∀ {l₁ l₂ : List α} (_ : l₁.Nodup), l₁.diff l₂ = l₁.filter (· ∉ l₂)
... |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Order.Filter.Prod
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Filter.Finite
import Mathlib.Order.Filter.Bases.Basic
/... | Mathlib/Order/Filter/Lift.lean | 252 | 254 | theorem comap_lift'_eq {m : γ → β} : comap m (f.lift' h) = f.lift' (preimage m ∘ h) := by | simp only [Filter.lift', comap_lift_eq, comp_def, comap_principal] |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Eric Wieser
-/
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Matrices as a normed space
In this file we provide the fo... | Mathlib/Analysis/Matrix.lean | 573 | 574 | theorem frobenius_nnnorm_diagonal [DecidableEq n] (v : n → α) :
‖diagonal v‖₊ = ‖(WithLp.equiv 2 _).symm v‖₊ := by | |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.Topology.MetricSpace.ThickenedIndicator
/-!
# Spaces where indicators of closed sets have ... | Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean | 56 | 65 | theorem tendsto_lintegral_nn_filter_of_le_const {ι : Type*} {L : Filter ι} [L.IsCountablyGenerated]
(μ : Measure Ω) [IsFiniteMeasure μ] {fs : ι → Ω →ᵇ ℝ≥0} {c : ℝ≥0}
(fs_le_const : ∀ᶠ i in L, ∀ᵐ ω : Ω ∂μ, fs i ω ≤ c) {f : Ω → ℝ≥0}
(fs_lim : ∀ᵐ ω : Ω ∂μ, Tendsto (fun i ↦ fs i ω) L (𝓝 (f ω))) :
Tendsto (... | refine tendsto_lintegral_filter_of_dominated_convergence (fun _ ↦ c)
(Eventually.of_forall fun i ↦ (ENNReal.continuous_coe.comp (fs i).continuous).measurable) ?_
(@lintegral_const_lt_top _ _ μ _ _ (@ENNReal.coe_ne_top c)).ne ?_
· simpa only [Function.comp_apply, ENNReal.coe_le_coe] using fs_le_const
· simpa... |
/-
Copyright (c) 2023 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Lemmas
/-!
# `compute_degree` and `monicity`: tactics for explicit polynomials
This file defines two related tactics: `comp... | Mathlib/Tactic/ComputeDegree.lean | 184 | 185 | theorem coeff_intCast_ite {n : ℕ} {a : ℤ} : (Int.cast a : R[X]).coeff n = ite (n = 0) a 0 := by | simp only [← C_eq_intCast, coeff_C, Int.cast_ite, Int.cast_zero] |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Real
/-!
# Properties of addition, multiplication and subtraction on extended non-negative real numbers
In this file we... | Mathlib/Data/ENNReal/Operations.lean | 212 | 212 | theorem mul_self_lt_top_iff {a : ℝ≥0∞} : a * a < ⊤ ↔ a < ⊤ := by | |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
/-!
# Permutations from a list
A list `l : List α` ... | Mathlib/GroupTheory/Perm/List.lean | 315 | 319 | theorem mem_of_formPerm_ne_self (l : List α) (x : α) (h : formPerm l x ≠ x) : x ∈ l := by | suffices x ∈ { y | formPerm l y ≠ y } by
rw [← mem_toFinset]
exact support_formPerm_le' _ this
simpa using h |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.C... | Mathlib/Topology/Instances/AddCircle.lean | 64 | 79 | theorem continuous_right_toIcoMod : ContinuousWithinAt (toIcoMod hp a) (Ici x) x := by | intro s h
rw [Filter.mem_map, mem_nhdsWithin_iff_exists_mem_nhds_inter]
haveI : Nontrivial 𝕜 := ⟨⟨0, p, hp.ne⟩⟩
simp_rw [mem_nhds_iff_exists_Ioo_subset] at h ⊢
obtain ⟨l, u, hxI, hIs⟩ := h
let d := toIcoDiv hp a x • p
have hd := toIcoMod_mem_Ico hp a x
simp_rw [subset_def, mem_inter_iff]
refine ⟨_, ⟨l ... |
/-
Copyright (c) 2022 Dylan MacKenzie. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dylan MacKenzie
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.Module.Defs
import Mathlib.Tactic.Abel
/-!
# Summation by parts
-/
namespace Finset
variable ... | Mathlib/Algebra/BigOperators/Module.lean | 21 | 57 | theorem sum_Ico_by_parts (hmn : m < n) :
∑ i ∈ Ico m n, f i • g i =
f (n - 1) • G n - f m • G m - ∑ i ∈ Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by | have h₁ : (∑ i ∈ Ico (m + 1) n, f i • G i) = ∑ i ∈ Ico m (n - 1), f (i + 1) • G (i + 1) := by
rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn), ← sum_Ico_add']
simp only [tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,
tsub_eq_zero_iff_le, add_tsub_cancel_right]
have h₂ :
(∑ i ∈ Ico (m... |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.PNat.Basic
/-!
# Primality and GCD on pnat
This file extends the theory of `ℕ+` with ... | Mathlib/Data/PNat/Prime.lean | 257 | 260 | theorem gcd_eq_left {m n : ℕ+} : m ∣ n → m.gcd n = m := by | rw [dvd_iff]
intro h
apply eq |
/-
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.Option.NAry
import Mathlib.Data.Seq.Computation
import Mathlib.Tactic.ApplyFun
import Mathlib.Data.List.Basic
/-!
# Possibly infinite lists
This... | Mathlib/Data/Seq/Seq.lean | 806 | 807 | theorem length_toList (s : Seq α) (h : s.Terminates) : (toList s h).length = length s h := by | rw [toList, length_take_of_le_length] |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
/-!
# Quadratic characters on ℤ/nℤ
This file defines some quadr... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 100 | 102 | theorem χ₄_int_three_mod_four {n : ℤ} (hn : n % 4 = 3) : χ₄ n = -1 := by | rw [χ₄_int_mod_four, hn]
rfl |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury Kudryashov
-/
import Mathlib.MeasureTheory.OuterMeasure.Basic
/-!
# The “almost everywhere” filter of co-null sets.
If `μ` is an outer measure or a measure on `α... | Mathlib/MeasureTheory/OuterMeasure/AE.lean | 231 | 233 | theorem _root_.Set.mulIndicator_ae_eq_one {M : Type*} [One M] {f : α → M} {s : Set α} :
s.mulIndicator f =ᵐ[μ] 1 ↔ μ (s ∩ f.mulSupport) = 0 := by | simp [EventuallyEq, eventually_iff, ae, compl_setOf]; rfl |
/-
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.GroupTheory.MonoidLocalization.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matri... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 327 | 332 | theorem _root_.AlgHom.map_adjugate {R A B : Type*} [CommSemiring R] [CommRing A] [CommRing B]
[Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (M : Matrix n n A) :
f.mapMatrix M.adjugate = Matrix.adjugate (f.mapMatrix M) :=
f.toRingHom.map_adjugate _
theorem det_adjugate (A : Matrix n n α) : (adjugate A).det = A.... | |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
import... | Mathlib/MeasureTheory/Measure/Stieltjes.lean | 484 | 488 | theorem measure_univ {l u : ℝ} (hfl : Tendsto f atBot (𝓝 l)) (hfu : Tendsto f atTop (𝓝 u)) :
f.measure univ = ofReal (u - l) := by | refine tendsto_nhds_unique (tendsto_measure_Iic_atTop _) ?_
simp_rw [measure_Iic f hfl]
exact ENNReal.tendsto_ofReal (Tendsto.sub_const hfu _) |
/-
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.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
/-!
# Compositions
A compositio... | Mathlib/Combinatorics/Enumerative/Composition.lean | 256 | 257 | theorem boundary_zero : c.boundary 0 = 0 := by | simp [boundary, Fin.ext_iff] |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Module.Submodule.Map
import Mathlib.Algebra.Polynomial.Eval.Defs
import Mathlib.RingTheory.Ideal.Quotient.Defs
/-!
# modular equivalence for submodule
-... | Mathlib/LinearAlgebra/SModEq.lean | 90 | 92 | theorem mul {I : Ideal A} {x₁ x₂ y₁ y₂ : A} (hxy₁ : x₁ ≡ y₁ [SMOD I])
(hxy₂ : x₂ ≡ y₂ [SMOD I]) : x₁ * x₂ ≡ y₁ * y₂ [SMOD I] := by | simp only [SModEq.def, Ideal.Quotient.mk_eq_mk, map_mul] at hxy₁ hxy₂ ⊢ |
/-
Copyright (c) 2022 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.Basic
/-!
# Conditional expectation of indicator functions
This file proves some results about the conditi... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean | 63 | 70 | theorem condExp_indicator_aux (hs : MeasurableSet[m] s) (hf : f =ᵐ[μ.restrict sᶜ] 0) :
μ[s.indicator f|m] =ᵐ[μ] s.indicator (μ[f|m]) := by | by_cases hm : m ≤ m0
swap; · simp_rw [condExp_of_not_le hm, Set.indicator_zero']; rfl
have hsf_zero : ∀ g : α → E, g =ᵐ[μ.restrict sᶜ] 0 → s.indicator g =ᵐ[μ] g := fun g =>
indicator_ae_eq_of_restrict_compl_ae_eq_zero (hm _ hs)
refine ((hsf_zero (μ[f|m]) (condExp_ae_eq_restrict_zero hs.compl hf)).trans ?_).sy... |
/-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.RepresentationTheory.FDRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
/-!
# Characters of representations
Th... | Mathlib/RepresentationTheory/Character.lean | 106 | 109 | theorem char_orthonormal (V W : FDRep k G) [Simple V] [Simple W] :
⅟ (Fintype.card G : k) • ∑ g : G, V.character g * W.character g⁻¹ =
if Nonempty (V ≅ W) then ↑1 else ↑0 := by | -- First, we can rewrite the summand `V.character g * W.character g⁻¹` as the character |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... | Mathlib/Order/Interval/Finset/Basic.lean | 220 | 222 | theorem Icc_subset_Ioo_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ < a₁ ∧ b₁ < b₂ := by | rw [← coe_subset, coe_Icc, coe_Ioo, Set.Icc_subset_Ioo_iff h₁] |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... | Mathlib/CategoryTheory/Monoidal/Category.lean | 246 | 250 | theorem comp_whiskerRight {W X Y : C} (f : W ⟶ X) (g : X ⟶ Y) (Z : C) :
(f ≫ g) ▷ Z = f ▷ Z ≫ g ▷ Z := by | simp only [← tensorHom_id, ← tensor_comp, id_comp]
@[reassoc, simp] |
/-
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.Complex.Log
/-! # Power funct... | Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 49 | 51 | theorem cpow_ne_zero_iff_of_exponent_ne_zero {x y : ℂ} (hy : y ≠ 0) :
x ^ y ≠ 0 ↔ x ≠ 0 := by | simp [hy] |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Nonneg.Field
import Mathlib.Data.Rat.Cast.Defs
import Mathlib.Tactic.Positivity.Basic
/-!
# Some exiled lemmas about cas... | Mathlib/Data/Rat/Cast/Lemmas.lean | 55 | 57 | theorem cast_ofScientific {K} [DivisionRing K] (m : ℕ) (s : Bool) (e : ℕ) :
(OfScientific.ofScientific m s e : ℚ) = (OfScientific.ofScientific m s e : K) := by | rw [← NNRat.cast_ofScientific (K := K), ← NNRat.cast_ofScientific, cast_nnratCast] |
/-
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.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | Mathlib/Data/Matroid/Basic.lean | 979 | 985 | theorem IsBasis.exists_isBase (hI : M.IsBasis I X) : ∃ B, M.IsBase B ∧ I = B ∩ X :=
let ⟨B,hB, hIB⟩ := hI.indep.exists_isBase_superset
⟨B, hB, subset_antisymm (subset_inter hIB hI.subset)
(by rw [hI.eq_of_subset_indep (hB.indep.inter_right X) (subset_inter hIB hI.subset)
inter_subset_right])⟩
@[simp] theor... |
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