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 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.Ordinal.Family
/-! # Ordinal exponential
In this file we define the power function and the lo... | Mathlib/SetTheory/Ordinal/Exponential.lean | 385 | 394 | theorem lt_log_of_lt_opow {b x c : Ordinal} (hc : c ≠ 0) : x < b ^ c → log b x < c :=
lt_imp_lt_of_le_imp_le <| opow_le_of_le_log hc
theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o := by | rwa [← succ_le_iff, succ_zero, ← opow_le_iff_le_log hb ho, opow_one]
theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0 := by
rcases eq_or_ne o 0 with (rfl | ho)
· exact log_zero_right b
rcases le_or_lt b 1 with hb | hb |
/-
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.FieldTheory.Extension
import Mathlib.FieldTheory.Normal.Defs
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
/-!
# Algebraica... | Mathlib/FieldTheory/IsAlgClosed/Basic.lean | 89 | 96 | theorem exists_pow_nat_eq [IsAlgClosed k] (x : k) {n : ℕ} (hn : 0 < n) : ∃ z, z ^ n = x := by | have : degree (X ^ n - C x) ≠ 0 := by
rw [degree_X_pow_sub_C hn x]
exact ne_of_gt (WithBot.coe_lt_coe.2 hn)
obtain ⟨z, hz⟩ := exists_root (X ^ n - C x) this
use z
simp only [eval_C, eval_X, eval_pow, eval_sub, IsRoot.def] at hz
exact sub_eq_zero.1 hz |
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Basic Translation Lemmas Between Functions Defined for Continued... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 35 | 35 | theorem terminatedAt_iff_s_none : g.TerminatedAt n ↔ g.s.get? n = none := by | rfl |
/-
Copyright (c) 2024 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.ContinuousMap.CocompactMap
import Mathlib.Topology.MetricSpace.Bounded
/-!
# Cocompact maps in normed g... | Mathlib/Analysis/Normed/Group/CocompactMap.lean | 30 | 40 | theorem CocompactMapClass.norm_le [ProperSpace F] [FunLike 𝓕 E F] [CocompactMapClass 𝓕 E F]
(ε : ℝ) : ∃ r : ℝ, ∀ x : E, r < ‖x‖ → ε < ‖f x‖ := by | have h := cocompact_tendsto f
rw [tendsto_def] at h
specialize h (Metric.closedBall 0 ε)ᶜ (mem_cocompact_of_closedBall_compl_subset 0 ⟨ε, rfl.subset⟩)
rcases closedBall_compl_subset_of_mem_cocompact h 0 with ⟨r, hr⟩
use r
intro x hx
suffices x ∈ f⁻¹' (Metric.closedBall 0 ε)ᶜ by aesop
apply hr
simp [hx] |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Mario Carneiro, Sean Leather
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finite sets in `Option α`
In this file we define
* `Option.t... | Mathlib/Data/Finset/Option.lean | 70 | 72 | theorem some_mem_insertNone {s : Finset α} {a : α} : some a ∈ insertNone s ↔ a ∈ s := by | simp
lemma none_mem_insertNone {s : Finset α} : none ∈ insertNone s := by simp |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Constructions.Polish.Basic
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.Probability.Martingale.Upcrossing
/-!
# Mar... | Mathlib/Probability/Martingale/Convergence.lean | 186 | 190 | theorem Submartingale.exists_ae_tendsto_of_bdd [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hbdd : ∀ n, eLpNorm (f n) 1 μ ≤ R) : ∀ᵐ ω ∂μ, ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) := by | filter_upwards [hf.upcrossings_ae_lt_top hbdd, ae_bdd_liminf_atTop_of_eLpNorm_bdd one_ne_zero
(fun n => (hf.stronglyMeasurable n).measurable.mono (ℱ.le n) le_rfl) hbdd] with ω h₁ h₂
exact tendsto_of_uncrossing_lt_top h₂ h₁ |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 217 | 221 | theorem oangle_rotation_self_left {x : V} (hx : x ≠ 0) (θ : Real.Angle) :
o.oangle (o.rotation θ x) x = -θ := by | simp [hx]
/-- A vector has an angle of `θ` from the rotation of that vector by `θ`. -/
@[simp] |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Topology.Alge... | Mathlib/Topology/Algebra/Algebra.lean | 47 | 51 | theorem continuous_algebraMap_iff_smul [ContinuousMul A] :
Continuous (algebraMap R A) ↔ Continuous fun p : R × A => p.1 • p.2 := by | refine ⟨fun h => ?_, fun h => have : ContinuousSMul R A := ⟨h⟩; continuous_algebraMap _ _⟩
simp only [Algebra.smul_def]
exact (h.comp continuous_fst).mul continuous_snd |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Ri... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 241 | 250 | theorem U_eval_neg_one (n : ℤ) : (U R n).eval (-1) = n.negOnePow * (n + 1) := by | induction n using Polynomial.Chebyshev.induct with
| zero => simp
| one => simp; norm_num
| add_two n ih1 ih2 =>
simp only [U_add_two, eval_sub, eval_mul, eval_ofNat, eval_X, mul_neg, mul_one, ih1,
Int.cast_add, Int.cast_natCast, Int.cast_one, neg_mul, ih2, Int.cast_ofNat, Int.negOnePow_add,
Int.n... |
/-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.EffectiveEpi.Preserves
import Mathlib.CategoryTheory.EffectiveEpi.Coproduct
import Mathlib.CategoryTheory.Extensive
import Mathlib.C... | Mathlib/CategoryTheory/EffectiveEpi/Extensive.lean | 24 | 29 | theorem effectiveEpi_desc_iff_effectiveEpiFamily {α : Type} [Finite α]
{B : C} (X : α → C) (π : (a : α) → X a ⟶ B) :
EffectiveEpi (Sigma.desc π) ↔ EffectiveEpiFamily X π := by | exact ⟨fun h ↦ ⟨⟨@effectiveEpiFamilyStructOfEffectiveEpiDesc _ _ _ _ X π _ h _ _ (fun g ↦
(FinitaryPreExtensive.sigma_desc_iso (fun a ↦ Sigma.ι X a) g inferInstance).epi_of_iso)⟩⟩,
fun _ ↦ inferInstance⟩ |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finsets in product types
This file defines finset constru... | Mathlib/Data/Finset/Prod.lean | 76 | 78 | theorem product_image_snd [DecidableEq β] (ht : s.Nonempty) : (s ×ˢ t).image Prod.snd = t := by | ext i
simp [mem_image, ht.exists_mem] |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 482 | 486 | theorem exists_degree_lt [Fintype σ] (f : MvPolynomial σ R) (n : ℕ)
(h : f.totalDegree < n * Fintype.card σ) {d : σ →₀ ℕ} (hd : d ∈ f.support) : ∃ i, d i < n := by | contrapose! h
calc
n * Fintype.card σ = ∑ _s : σ, n := by |
/-
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.CharP.Frobenius
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Polynomial.Basic
/... | Mathlib/Algebra/Polynomial/Expand.lean | 201 | 204 | theorem contract_one {f : R[X]} : contract 1 f = f :=
ext fun n ↦ by rw [coeff_contract one_ne_zero, mul_one]
@[simp] theorem contract_C (r : R) : contract p (C r) = C r := by | simp [contract] |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Batteries.WF
import Mathlib.Data.Part
import Mathlib.Data.Rel
import Mathlib.Tactic.GeneralizeProofs
/-!
# Partial functions
This... | Mathlib/Data/PFun.lean | 585 | 586 | theorem prodMap_id_id : (PFun.id α).prodMap (PFun.id β) = PFun.id _ := by | aesop |
/-
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.Module.Submodule.Ker
/-!
# Iterate maps and comaps of submodules
Some preliminary work for establishing the strong rank condition for noetherian rings.
Giv... | Mathlib/Algebra/Module/Submodule/IterateMapComap.lean | 45 | 56 | theorem iterateMapComap_le_succ (K : Submodule R N) (h : K.map f ≤ K.map i) (n : ℕ) :
f.iterateMapComap i n K ≤ f.iterateMapComap i (n + 1) K := by | nth_rw 2 [iterateMapComap]
rw [iterate_succ', Function.comp_apply, ← iterateMapComap, ← map_le_iff_le_comap]
induction n with
| zero => exact h
| succ n ih =>
simp_rw [iterateMapComap, iterate_succ', Function.comp_apply]
calc
_ ≤ (f.iterateMapComap i n K).map i := map_comap_le _ _
_ ≤ (((f.i... |
/-
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.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | Mathlib/MeasureTheory/Measure/Restrict.lean | 568 | 569 | theorem ae_restrict_iff'₀ {p : α → Prop} (hs : NullMeasurableSet s μ) :
(∀ᵐ x ∂μ.restrict s, p x) ↔ ∀ᵐ x ∂μ, x ∈ s → p x := by | |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Tactic.Positivity
/-!
# Inverse of analytic functions
We ... | Mathlib/Analysis/Analytic/Inverse.lean | 182 | 183 | theorem rightInv_coeff_one (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E) :
p.rightInv i x 1 = (continuousMultilinearCurryFin1 𝕜 F E).symm i.symm := by | rw [rightInv] |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.List.Lattice
import Mathlib.Data.Bool.Basic
import Mathlib.Order.Lattice
/-!
# Intervals in ℕ
This file defines intervals of naturals. `List.Ico m n... | Mathlib/Data/List/Intervals.lean | 162 | 166 | theorem filter_le_of_le {n m l : ℕ} (hnl : n ≤ l) :
((Ico n m).filter fun x => l ≤ x) = Ico l m := by | rcases le_total l m with hlm | hml
· rw [← append_consecutive hnl hlm, filter_append, filter_le_of_top_le (le_refl l),
filter_le_of_le_bot (le_refl l), nil_append] |
/-
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.Data.Set.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | Mathlib/Topology/Connected/Basic.lean | 511 | 520 | theorem isConnected_connectedComponentIn_iff {x : α} {F : Set α} :
IsConnected (connectedComponentIn F x) ↔ x ∈ F := by | simp_rw [← connectedComponentIn_nonempty_iff, IsConnected, isPreconnected_connectedComponentIn,
and_true]
theorem IsPreconnected.subset_connectedComponent {x : α} {s : Set α} (H1 : IsPreconnected s)
(H2 : x ∈ s) : s ⊆ connectedComponent x := fun _z hz => mem_sUnion_of_mem hz ⟨H1, H2⟩
theorem IsPreconnected.su... |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.SetTheory.Cardinal.ENat
/-!
# Finite Cardinality Funct... | Mathlib/SetTheory/Cardinal/Finite.lean | 207 | 208 | theorem card_sum [Finite α] [Finite β] : Nat.card (α ⊕ β) = Nat.card α + Nat.card β := by | have := Fintype.ofFinite α |
/-
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
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Zify
/-!
# The length function, reduced word... | Mathlib/GroupTheory/Coxeter/Length.lean | 281 | 289 | theorem isLeftDescent_inv_iff {w : W} {i : B} :
cs.IsLeftDescent w⁻¹ i ↔ cs.IsRightDescent w i := by | unfold IsLeftDescent IsRightDescent
nth_rw 1 [← length_inv]
simp
theorem isRightDescent_inv_iff {w : W} {i : B} :
cs.IsRightDescent w⁻¹ i ↔ cs.IsLeftDescent w i := by
simpa using (cs.isLeftDescent_inv_iff (w := w⁻¹)).symm |
/-
Copyright (c) 2020 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.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
/-!
# Reverse of a univariate polynomial
The main definition is `r... | Mathlib/Algebra/Polynomial/Reverse.lean | 272 | 277 | theorem natTrailingDegree_reverse (f : R[X]) : f.reverse.natTrailingDegree = 0 := by | rw [natTrailingDegree_eq_zero, reverse_eq_zero, coeff_zero_reverse, leadingCoeff_ne_zero]
exact eq_or_ne _ _
theorem reverse_trailingCoeff (f : R[X]) : f.reverse.trailingCoeff = f.leadingCoeff := by
rw [trailingCoeff, natTrailingDegree_reverse, coeff_zero_reverse] |
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
/-!
# Termination of a hydra game
This file deals with the following version... | Mathlib/Logic/Hydra.lean | 104 | 107 | theorem not_cutExpand_zero [IsIrrefl α r] (s) : ¬CutExpand r s 0 := by | classical
rw [cutExpand_iff]
rintro ⟨_, _, _, ⟨⟩, _⟩ |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finsets in product types
This file defines finset constru... | Mathlib/Data/Finset/Prod.lean | 342 | 345 | theorem product_sdiff_offDiag : s ×ˢ s \ s.offDiag = s.diag := by | rw [← diag_union_offDiag, union_sdiff_self, sdiff_eq_self_of_disjoint (disjoint_diag_offDiag _)]
theorem diag_inter : (s ∩ t).diag = s.diag ∩ t.diag := |
/-
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.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 700 | 704 | theorem dist_div_sin_oangle_left_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
(h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : dist p₃ p₂ / Real.Angle.sin (∡ p₃ p₁ p₂) = dist p₁ p₃ := by | have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm, Real.Angle.sin_coe, dist_comm p₁ p₃,
dist_div_sin_angle_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two 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.MeasureTheory.Integral.IntegrableOn
/-!
# Locally integrable functions
A function is called *locally integrable* (`MeasureTheory.LocallyIntegrabl... | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 597 | 609 | theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg' : ContinuousOn g' K)
(hA : MeasurableSet A) (hK : IsCompact K) (hAK : A ⊆ K) :
IntegrableOn (fun x => g x * g' x) A μ := by | rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩
rw [IntegrableOn, ← memLp_one_iff_integrable] at hg ⊢
have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g x‖ := by
filter_upwards [ae_restrict_mem hA] with x hx
refine (norm_mul_le _ _).trans ?_
rw [mul_comm]
gcongr
exact hC x (hAK h... |
/-
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 | 467 | 481 | theorem card_edgeFinset_induce_support :
#(G.induce G.support).edgeFinset = #G.edgeFinset :=
card_edgeFinset_induce_of_support_subset subset_rfl
theorem map_neighborFinset_induce (v : s) :
((G.induce s).neighborFinset v).map (.subtype s)
= G.neighborFinset v ∩ s.toFinset := by | ext; simp [Set.mem_def]
theorem map_neighborFinset_induce_of_neighborSet_subset {v : s} (h : G.neighborSet v ⊆ s) :
((G.induce s).neighborFinset v).map (.subtype s) = G.neighborFinset v := by
rwa [← Set.toFinset_subset_toFinset, ← neighborFinset_def, ← inter_eq_left,
← map_neighborFinset_induce v] at h
/-- ... |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.IsSheafFor
import Mathlib.CategoryTheory.Limits.Types.Shapes
import Mathlib.Tactic.ApplyFun
/-!
# The equalizer diagram sheaf conditi... | Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean | 320 | 323 | theorem compatible_iff (x : FirstObj P X) : (Arrows.Compatible P π ((Types.productIso _).hom x)) ↔
firstMap P X π x = secondMap P X π x := by | rw [Arrows.pullbackCompatible_iff]
constructor |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yaël Dillies
-/
import Mathlib.Order.Interval.Set.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Tactic.Contrapose
import Mathl... | Mathlib/Data/Nat/Log.lean | 42 | 44 | theorem log_eq_zero_iff {b n : ℕ} : log b n = 0 ↔ n < b ∨ b ≤ 1 := by | rw [log, dite_eq_right_iff]
simp only [Nat.add_eq_zero_iff, Nat.one_ne_zero, and_false, imp_false, not_and_or, not_le, not_lt] |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Yury Kudryashov
-/
import Mathlib.Topology.Order.Basic
/-!
# Bounded monotone sequences converge
In this file we prove a few theorems of the form “if the range of a... | Mathlib/Topology/Order/MonotoneConvergence.lean | 103 | 104 | theorem tendsto_atTop_isGLB (h_anti : Antitone f) (ha : IsGLB (Set.range f) a) :
Tendsto f atTop (𝓝 a) := by | convert tendsto_atBot_isLUB h_anti.dual ha.dual using 1 |
/-
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, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.Module.Submodule.... | Mathlib/Algebra/Module/Submodule/Ker.lean | 187 | 187 | theorem _root_.LinearMapClass.ker_eq_bot : ker f = ⊥ ↔ Injective f := by | |
/-
Copyright (c) 2019 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.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Mul
import ... | Mathlib/Analysis/Calculus/MeanValue.lean | 599 | 604 | theorem _root_.IsOpen.exists_is_const_of_fderiv_eq_zero
(hs : IsOpen s) (hs' : IsPreconnected s) (hf : DifferentiableOn 𝕜 f s)
(hf' : s.EqOn (fderiv 𝕜 f) 0) : ∃ a, ∀ x ∈ s, f x = a := by | obtain (rfl|⟨y, hy⟩) := s.eq_empty_or_nonempty
· exact ⟨0, by simp⟩
· refine ⟨f y, fun x hx ↦ ?_⟩ |
/-
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.LocallyConvex.Bounded
import Mathlib.Analysis.Seminorm
import Mathlib.Data.Real.Sqrt
import Mathlib.Topology.Algebra.Equicontinuit... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 655 | 697 | theorem _root_.WithSeminorms.uniformEquicontinuous_iff_bddAbove_and_continuous_iSup {κ : Type*}
{q : SeminormFamily 𝕜₂ F ι'} [UniformSpace E] [IsUniformAddGroup E] [u : UniformSpace F]
[IsUniformAddGroup F] (hq : WithSeminorms q) [ContinuousSMul 𝕜 E]
(f : κ → E →ₛₗ[σ₁₂] F) :
UniformEquicontinuous ((↑)... | constructor
rw [hp.topology_eq_withSeminorms]
clear hp t
refine le_antisymm ?_ ?_ <;>
rw [← continuous_id_iff_le] <;>
refine continuous_from_bounded (.mk (topology := _) rfl) (.mk (topology := _) rfl)
LinearMap.id (by assumption)
protected theorem finset_sups {p : SeminormFamily 𝕜 E ι} [TopologicalSpace... |
/-
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.Polynomial.Eval.Degree
import Mathlib.Algebra.Prime.Lemmas
/-!
# Theory of degrees of polynomials
S... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 76 | 81 | theorem natDegree_le_iff_coeff_eq_zero : p.natDegree ≤ n ↔ ∀ N : ℕ, n < N → p.coeff N = 0 := by | simp_rw [natDegree_le_iff_degree_le, degree_le_iff_coeff_zero, Nat.cast_lt]
theorem natDegree_add_le_iff_left {n : ℕ} (p q : R[X]) (qn : q.natDegree ≤ n) :
(p + q).natDegree ≤ n ↔ p.natDegree ≤ n := by
refine ⟨fun h => ?_, fun h => natDegree_add_le_of_degree_le h qn⟩ |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 478 | 491 | theorem oangle_add_cyc3_neg_left {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
o.oangle (-x) y + o.oangle (-y) z + o.oangle (-z) x = π := by | rw [o.oangle_neg_left hx hy, o.oangle_neg_left hy hz, o.oangle_neg_left hz hx,
show o.oangle x y + π + (o.oangle y z + π) + (o.oangle z x + π) =
o.oangle x y + o.oangle y z + o.oangle z x + (π + π + π : Real.Angle) by abel,
o.oangle_add_cyc3 hx hy hz, Real.Angle.coe_pi_add_coe_pi, zero_add, zero_add]
/--... |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 598 | 600 | theorem sin_toReal (θ : Angle) : Real.sin θ.toReal = sin θ := by | conv_rhs => rw [← coe_toReal θ, sin_coe] |
/-
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.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 820 | 824 | theorem foldl_concat
(f : β → α → β) (b : β) (x : α) (xs : List α) :
List.foldl f b (xs ++ [x]) = f (List.foldl f b xs) x := by | simp only [List.foldl_append, List.foldl] |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Batteries.WF
import Mathlib.Data.Part
import Mathlib.Data.Rel
import Mathlib.Tactic.GeneralizeProofs
/-!
# Partial functions
This... | Mathlib/Data/PFun.lean | 514 | 516 | theorem Part.bind_comp (f : β →. γ) (g : α →. β) (a : Part α) :
a.bind (f.comp g) = (a.bind g).bind f := by | ext c |
/-
Copyright (c) 2017 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.PFunctor.Univariate.Basic
/-!
# M-types
M types are potentially infinite tree-like structures. They are defined
as the greatest fixpoint of a polynomi... | Mathlib/Data/PFunctor/Univariate/M.lean | 568 | 569 | theorem eq_of_bisim [Nonempty (M F)] (bisim : IsBisimulation R) : ∀ s₁ s₂, R s₁ s₂ → s₁ = s₂ := by | inhabit M F |
/-
Copyright (c) 2018 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topolo... | Mathlib/Topology/MetricSpace/Lipschitz.lean | 191 | 194 | theorem max_const (hf : LipschitzWith Kf f) (a : ℝ) : LipschitzWith Kf fun x => max (f x) a := by | simpa only [max_eq_left (zero_le Kf)] using hf.max (LipschitzWith.const a)
theorem const_max (hf : LipschitzWith Kf f) (a : ℝ) : LipschitzWith Kf fun x => max a (f x) := by |
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Basic
/-!
# Monad Operations for Probability Mass Functions
This file constructs two operations on `PMF` ... | Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 239 | 241 | theorem pure_bindOnSupport (a : α) (f : ∀ (a' : α) (_ : a' ∈ (pure a).support), PMF β) :
(pure a).bindOnSupport f = f a ((mem_support_pure_iff a a).mpr rfl) := by | refine PMF.ext fun b => ?_ |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence
import Mathlib.Algebra.ContinuedFractions.TerminatedStable
import Mathlib.Tactic.FieldSimp
import ... | Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean | 204 | 213 | theorem squashGCF_nth_of_lt {m : ℕ} (m_lt_n : m < n) :
(squashGCF g (n + 1)).s.get? m = g.s.get? m := by | simp only [squashGCF, squashSeq_nth_of_lt m_lt_n, Nat.add_eq, add_zero]
/-- `convs'` returns the same value for a gcf and the corresponding squashed gcf at the
squashed position. -/
theorem succ_nth_conv'_eq_squashGCF_nth_conv' :
g.convs' (n + 1) = (squashGCF g n).convs' n := by
cases n with
| zero => |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Eval.SMul
/-!
# Scalar-multiple polynomial ev... | Mathlib/Algebra/Polynomial/Smeval.lean | 111 | 114 | theorem smeval_natCast (n : ℕ) : (n : R[X]).smeval x = n • x ^ 0 := by | induction n with
| zero => simp only [smeval_zero, Nat.cast_zero, zero_smul]
| succ n ih => rw [n.cast_succ, smeval_add, ih, smeval_one, ← add_nsmul] |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.Degree.Domain
import Mathlib.Algebra.Ring.NonZeroDivisors
import Mathlib.RingTheory.Localization.FractionRing
/-!
# The field of rational... | Mathlib/FieldTheory/RatFunc/Defs.lean | 168 | 171 | theorem mk_eq_mk {p q p' q' : K[X]} (hq : q ≠ 0) (hq' : q' ≠ 0) :
RatFunc.mk p q = RatFunc.mk p' q' ↔ p * q' = p' * q := by | rw [mk_def_of_ne _ hq, mk_def_of_ne _ hq', ofFractionRing_injective.eq_iff,
IsLocalization.mk'_eq_iff_eq', |
/-
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
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWi... | Mathlib/Data/Nat/GCD/Basic.lean | 75 | 76 | theorem lcm_mul_left {m n k : ℕ} : (m * n).lcm (m * k) = m * n.lcm k := by | apply dvd_antisymm |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
import Mathlib.SetTheory.Nimber.Basic
/-!
# Nim and the Sprague-Grundy theore... | Mathlib/SetTheory/Game/Nim.lean | 255 | 256 | theorem grundyValue_le_of_forall_moveLeft {G : PGame} {o : Nimber}
(h : ∀ i, grundyValue (G.moveLeft i) ≠ o) : G.grundyValue ≤ o := by | |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Tactic.Peel
import... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 155 | 159 | theorem norm_nonneg (f : PadicSeq p) : 0 ≤ f.norm := by | classical exact if hf : f ≈ 0 then by simp [hf, norm] else by simp [norm, hf, padicNorm.nonneg]
/-- An auxiliary lemma for manipulating sequence indices. -/
theorem lift_index_left_left {f : PadicSeq p} (hf : ¬f ≈ 0) (v2 v3 : ℕ) : |
/-
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.Order.Group.Finset
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 275 | 283 | theorem monic_map_iff [Semiring S] [Nontrivial S] {f : R →+* S} {p : R[X]} :
(p.map f).Monic ↔ p.Monic := by | rw [Monic, leadingCoeff_map, ← f.map_one, Function.Injective.eq_iff f.injective, Monic]
end DivisionRing
section Field
variable [Field R] {p q : R[X]} |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Nikolas Kuhn
-/
import Mathlib.CategoryTheory.Sites.Coherent.CoherentSheaves
/-!
# Description of the covering sieves of the coherent topology
This file characterises the cov... | Mathlib/CategoryTheory/Sites/Coherent/CoherentTopology.lean | 44 | 68 | theorem EffectiveEpiFamily.transitive_of_finite {α : Type} [Finite α] {Y : α → C}
(π : (a : α) → (Y a ⟶ X)) (h : EffectiveEpiFamily Y π) {β : α → Type} [∀ (a : α), Finite (β a)]
{Y_n : (a : α) → β a → C} (π_n : (a : α) → (b : β a) → (Y_n a b ⟶ Y a))
(H : ∀ a, EffectiveEpiFamily (Y_n a) (π_n a)) :
Effect... | rw [← Sieve.effectiveEpimorphic_family]
suffices h₂ : (Sieve.generate (Presieve.ofArrows (fun (⟨a, b⟩ : Σ _, β _) => Y_n a b)
(fun ⟨a,b⟩ => π_n a b ≫ π a))) ∈ (coherentTopology C) X by
change Nonempty _
rw [← Sieve.forallYonedaIsSheaf_iff_colimit]
exact fun W => coherentTopology.isSheaf_yoneda_obj... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.InfiniteSum.Defs
imp... | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 491 | 506 | theorem tprod_range {g : γ → β} (f : β → α) (hg : Injective g) :
∏' x : Set.range g, f x = ∏' x, f (g x) := by | rw [← Set.image_univ, tprod_image f hg.injOn]
simp_rw [← comp_apply (g := g), tprod_univ (f ∘ g)]
/-- If `f b = 1` for all `b ∈ t`, then the product of `f a` with `a ∈ s` is the same as the
product of `f a` with `a ∈ s ∖ t`. -/
@[to_additive "If `f b = 0` for all `b ∈ t`, then the sum of `f a` with `a ∈ s` is the sa... |
/-
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.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 535 | 537 | theorem oangle_left_eq_arccos_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
∡ p₃ p₁ p₂ = Real.arccos (dist p₁ p₂ / dist p₁ p₃) := by | have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two] |
/-
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.Ordinal.Family
/-! # Ordinal exponential
In this file we define the power function and the lo... | Mathlib/SetTheory/Ordinal/Exponential.lean | 147 | 162 | theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c := by | by_cases a0 : a = 0
-- Porting note: `le_refl` is required.
· subst a
by_cases c0 : c = 0
· subst c
simp only [opow_zero, le_refl]
· simp only [zero_opow c0, Ordinal.zero_le]
· induction c using limitRecOn with
| zero => simp only [opow_zero, le_refl]
| succ c IH =>
simpa only [opo... |
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearG... | Mathlib/NumberTheory/Modular.lean | 381 | 388 | theorem three_lt_four_mul_im_sq_of_mem_fdo (h : z ∈ 𝒟ᵒ) : 3 < 4 * z.im ^ 2 := by | have : 1 < z.re * z.re + z.im * z.im := by simpa [Complex.normSq_apply] using h.1
have := h.2
cases abs_cases z.re <;> nlinarith
/-- non-strict variant of `ModularGroup.three_le_four_mul_im_sq_of_mem_fdo` -/
theorem three_le_four_mul_im_sq_of_mem_fd {τ : ℍ} (h : τ ∈ 𝒟) : 3 ≤ 4 * τ.im ^ 2 := by
have : 1 ≤ τ.re *... |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Eric Wieser
-/
import Mathlib.Data.ENNReal.Holder
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
import Mathl... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 88 | 92 | theorem eLpNorm'_le_eLpNorm'_of_exponent_le {p q : ℝ} (hp0_lt : 0 < p)
(hpq : p ≤ q) (μ : Measure α) [IsProbabilityMeasure μ] (hf : AEStronglyMeasurable f μ) :
eLpNorm' f p μ ≤ eLpNorm' f q μ := by | have h_le_μ := eLpNorm'_le_eLpNorm'_mul_rpow_measure_univ hp0_lt hpq hf
rwa [measure_univ, ENNReal.one_rpow, mul_one] at h_le_μ |
/-
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.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.Prime.Basic
import Ma... | Mathlib/Data/Nat/Factors.lean | 197 | 201 | theorem perm_primeFactorsList_mul_of_coprime {a b : ℕ} (hab : Coprime a b) :
(a * b).primeFactorsList ~ a.primeFactorsList ++ b.primeFactorsList := by | rcases a.eq_zero_or_pos with (rfl | ha)
· simp [(coprime_zero_left _).mp hab]
rcases b.eq_zero_or_pos with (rfl | hb) |
/-
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 | 257 | 260 | theorem inv_strictAntiOn : StrictAntiOn (fun x : α => x⁻¹) (Set.Ioi 0) := fun _ hx _ hy xy =>
(inv_lt_inv₀ hy hx).2 xy
theorem inv_pow_le_inv_pow_of_le (a1 : 1 ≤ a) {m n : ℕ} (mn : m ≤ n) : (a ^ n)⁻¹ ≤ (a ^ m)⁻¹ := by | |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity.WalkCounting
import Math... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 69 | 70 | theorem apply_ne_zero_iff [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i j : V) :
¬A i j = 0 ↔ A i j = 1 := by | rw [← apply_ne_one_iff h, Classical.not_not] |
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | Mathlib/Algebra/Polynomial/Roots.lean | 460 | 462 | theorem aroots_X_pow [CommRing S] [IsDomain S] [Algebra T S] (n : ℕ) :
(X ^ n : T[X]).aroots S = n • ({0} : Multiset S) := by | rw [aroots_pow, aroots_X] |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity.WalkCounting
import Math... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 105 | 105 | theorem isSymm_compl [Zero α] [One α] (h : A.IsSymm) : A.compl.IsSymm := by | |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Algebra.MvPolynomial.Eval
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mat... | Mathlib/RingTheory/Adjoin/FG.lean | 40 | 80 | theorem fg_trans (h1 : (adjoin R s).toSubmodule.FG) (h2 : (adjoin (adjoin R s) t).toSubmodule.FG) :
(adjoin R (s ∪ t)).toSubmodule.FG := by | rcases fg_def.1 h1 with ⟨p, hp, hp'⟩
rcases fg_def.1 h2 with ⟨q, hq, hq'⟩
refine fg_def.2 ⟨p * q, hp.mul hq, le_antisymm ?_ ?_⟩
· rw [span_le, Set.mul_subset_iff]
intro x hx y hy
change x * y ∈ adjoin R (s ∪ t)
refine Subalgebra.mul_mem _ ?_ ?_
· have : x ∈ Subalgebra.toSubmodule (adjoin R s) := b... |
/-
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.NumberTheory.ModularForms.JacobiTheta.TwoVariable
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
/-! # Jacobi's theta function
This file define... | Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean | 30 | 31 | theorem jacobiTheta_two_add (τ : ℂ) : jacobiTheta (2 + τ) = jacobiTheta τ := by | simp_rw [jacobiTheta_eq_jacobiTheta₂, add_comm, jacobiTheta₂_add_right] |
/-
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.Analysis.InnerProductSpace.Subspace
import Mathlib.LinearAlgebra.SesquilinearForm
/-!
# Orthogonal complements of su... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 103 | 107 | theorem orthogonal_eq_inter : Kᗮ = ⨅ v : K, LinearMap.ker (innerSL 𝕜 (v : E)) := by | apply le_antisymm
· rw [le_iInf_iff]
rintro ⟨v, hv⟩ w hw
simpa using hw _ hv |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 455 | 457 | theorem totalDegree_multiset_prod (s : Multiset (MvPolynomial σ R)) :
s.prod.totalDegree ≤ (s.map MvPolynomial.totalDegree).sum := by | refine Quotient.inductionOn s fun l => ?_ |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.Submodule.Basic
/-!
# Decompositions of additive monoids, groups, and modules into dire... | Mathlib/Algebra/DirectSum/Decomposition.lean | 140 | 142 | theorem degree_eq_of_mem_mem {x : M} {i j : ι} (hxi : x ∈ ℳ i) (hxj : x ∈ ℳ j) (hx : x ≠ 0) :
i = j := by | contrapose! hx; rw [← decompose_of_mem_same ℳ hxj, decompose_of_mem_ne ℳ hxi hx] |
/-
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 | 179 | 186 | theorem mem_powers_of_prime_card {p : ℕ} [hp : Fact p.Prime]
(h : Nat.card G = p) {g g' : G} (hg : g ≠ 1) : g' ∈ Submonoid.powers g := by | have : Finite G := Nat.finite_of_card_ne_zero (h ▸ hp.1.ne_zero)
rw [mem_powers_iff_mem_zpowers]
exact mem_zpowers_of_prime_card h hg
@[to_additive]
theorem powers_eq_top_of_prime_card {p : ℕ} |
/-
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.MeasureTheory.Group.Measure
/-!
# Lebesgue Integration on Groups
We develop properties of integrals with a group as domain.
This file contains pr... | Mathlib/MeasureTheory/Group/LIntegral.lean | 34 | 37 | theorem lintegral_inv_eq_self [IsInvInvariant μ] (f : G → ℝ≥0∞) :
∫⁻ x, f x⁻¹ ∂μ = ∫⁻ x, f x ∂μ := by | simpa using (lintegral_map_equiv f (μ := μ) <| MeasurableEquiv.inv G).symm |
/-
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.Polynomial.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... | Mathlib/Algebra/Polynomial/Derivative.lean | 460 | 464 | theorem derivative_sq (p : R[X]) : derivative (p ^ 2) = C 2 * p * derivative p := by | rw [derivative_pow_succ, Nat.cast_one, one_add_one_eq_two, pow_one]
theorem pow_sub_one_dvd_derivative_of_pow_dvd {p q : R[X]} {n : ℕ}
(dvd : q ^ n ∣ p) : q ^ (n - 1) ∣ derivative p := by |
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... | Mathlib/Data/Vector/MapLemmas.lean | 27 | 35 | theorem mapAccumr_mapAccumr :
mapAccumr f₁ (mapAccumr f₂ xs s₂).snd s₁
= let m := (mapAccumr (fun x s =>
let r₂ := f₂ x s.snd
let r₁ := f₁ r₂.snd s.fst
((r₁.fst, r₂.fst), r₁.snd)
) xs (s₁, s₂))
(m.fst.fst, m.snd) := by | induction xs using Vector.revInductionOn generalizing s₁ s₂ <;> simp_all |
/-
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.Data.Set.BooleanAlgebra
import Mathlib.Data.Set.Piecewise
import Mathlib.Order.Interval.Set.Basic
import Mathli... | Mathlib/Order/Interval/Set/Pi.lean | 112 | 118 | theorem image_update_Icc (f : ∀ i, α i) (i : ι) (a b : α i) :
update f i '' Icc a b = Icc (update f i a) (update f i b) := by | ext x
rw [← Set.pi_univ_Icc]
refine ⟨?_, fun h => ⟨x i, ?_, ?_⟩⟩
· rintro ⟨c, hc, rfl⟩
simpa [update_le_update_iff] |
/-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Algebra.Pointwise.Stabilizer
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.GroupTh... | Mathlib/GroupTheory/GroupAction/Blocks.lean | 121 | 124 | theorem IsTrivialBlock.image {φ : M → N} {f : α →ₑ[φ] β}
(hf : Function.Surjective f) {B : Set α} (hB : IsTrivialBlock B) :
IsTrivialBlock (f '' B) := by | obtain hB | hB := hB |
/-
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.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 507 | 516 | theorem _root_.Directed.measure_iInter [Countable ι] {s : ι → Set α}
(h : ∀ i, NullMeasurableSet (s i) μ) (hd : Directed (· ⊇ ·) s) (hfin : ∃ i, μ (s i) ≠ ∞) :
μ (⋂ i, s i) = ⨅ i, μ (s i) := by | rcases hfin with ⟨k, hk⟩
have : ∀ t ⊆ s k, μ t ≠ ∞ := fun t ht => ne_top_of_le_ne_top hk (measure_mono ht)
rw [← ENNReal.sub_sub_cancel hk (iInf_le (fun i => μ (s i)) k), ENNReal.sub_iInf, ←
ENNReal.sub_sub_cancel hk (measure_mono (iInter_subset _ k)), ←
measure_diff (iInter_subset _ k) (.iInter h) (this _ ... |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Action.Center
import Mathlib.GroupTheory.ClassEquation
import Mathlib.RingTheory.Polynomial.Cyclotomic.Eval
/-!
... | Mathlib/RingTheory/LittleWedderburn.lean | 167 | 172 | theorem Finite.isDomain_to_isField (D : Type*) [Finite D] [Ring D] [IsDomain D] : IsField D := by | classical
cases nonempty_fintype D
let _ := Fintype.divisionRingOfIsDomain D
let _ := littleWedderburn D
exact Field.toIsField D |
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... | Mathlib/Data/Vector/MapLemmas.lean | 50 | 52 | theorem map_map (f₁ : β → γ) (f₂ : α → β) :
map f₁ (map f₂ xs) = map (fun x => f₁ <| f₂ x) xs := by | induction xs <;> simp_all |
/-
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.Separation.Regular
import Mathlib.Topology.UniformSpace.Basic
/-!
# Hausdorff... | Mathlib/Topology/UniformSpace/Separation.lean | 202 | 216 | theorem comap_map_mk_uniformity : comap (Prod.map mk mk) (map (Prod.map mk mk) (𝓤 α)) = 𝓤 α := by | refine le_antisymm ?_ le_comap_map
refine ((((𝓤 α).basis_sets.map _).comap _).le_basis_iff uniformity_hasBasis_open).2 fun U hU ↦ ?_
refine ⟨U, hU.1, fun (x₁, x₂) ⟨(y₁, y₂), hyU, hxy⟩ ↦ ?_⟩
simp only [Prod.map, Prod.ext_iff, mk_eq_mk] at hxy
exact ((hxy.1.prod hxy.2).mem_open_iff hU.2).1 hyU
instance instUnif... |
/-
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.Polynomial.Splits
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.Ro... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 355 | 357 | theorem cyclotomic_prime (R : Type*) [Ring R] (p : ℕ) [hp : Fact p.Prime] :
cyclotomic p R = ∑ i ∈ Finset.range p, X ^ i := by | suffices cyclotomic p ℤ = ∑ i ∈ range p, X ^ i by |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 600 | 601 | theorem eventually_const {f : Filter α} [t : NeBot f] {p : Prop} : (∀ᶠ _ in f, p) ↔ p := by | by_cases h : p <;> simp [h, t.ne] |
/-
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.Logic.Basic
import Mathlib.Tactic.Positivity.Basic
/-!
# Algebraic order homomorphism classes
This file defines hom classes for common properties at the ... | Mathlib/Algebra/Order/Hom/Basic.lean | 131 | 133 | theorem le_map_div_mul_map_div [Group α] [Mul β] [LE β] [SubmultiplicativeHomClass F α β]
(f : F) (a b c : α) : f (a / c) ≤ f (a / b) * f (b / c) := by | simpa only [div_mul_div_cancel] using map_mul_le_mul f (a / b) (b / c) |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 292 | 302 | theorem sin_eq_iff_eq_or_add_eq_pi {θ ψ : Angle} : sin θ = sin ψ ↔ θ = ψ ∨ θ + ψ = π := by | induction ψ using Real.Angle.induction_on
exact sin_eq_real_sin_iff_eq_or_add_eq_pi
@[simp]
theorem sin_zero : sin (0 : Angle) = 0 := by rw [← coe_zero, sin_coe, Real.sin_zero]
theorem sin_coe_pi : sin (π : Angle) = 0 := by rw [sin_coe, Real.sin_pi]
theorem sin_eq_zero_iff {θ : Angle} : sin θ = 0 ↔ θ = 0 ∨ θ = π :... |
/-
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, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.GroupWithZero.... | Mathlib/Algebra/BigOperators/Associated.lean | 29 | 36 | theorem exists_mem_multiset_dvd (hp : Prime p) {s : Multiset α} : p ∣ s.prod → ∃ a ∈ s, p ∣ a :=
Multiset.induction_on s (fun h => (hp.not_dvd_one h).elim) fun a s ih h =>
have : p ∣ a * s.prod := by | simpa using h
match hp.dvd_or_dvd this with
| Or.inl h => ⟨a, Multiset.mem_cons_self a s, h⟩
| Or.inr h =>
let ⟨a, has, h⟩ := ih h
⟨a, Multiset.mem_cons_of_mem has, h⟩ |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | Mathlib/Order/LiminfLimsup.lean | 835 | 836 | theorem frequently_lt_of_liminf_lt {b : β}
(hu : f.IsCoboundedUnder (· ≥ ·) u := by | isBoundedDefault) |
/-
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 | 675 | 678 | theorem ncard_image_le (hs : s.Finite := by | toFinite_tac) : (f '' s).ncard ≤ s.ncard := by
to_encard_tac; rw [hs.cast_ncard_eq, (hs.image _).cast_ncard_eq]; apply encard_image_le
theorem ncard_image_of_injOn (H : Set.InjOn f s) : (f '' s).ncard = s.ncard := |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | Mathlib/Data/Nat/Factorization/Basic.lean | 544 | 554 | theorem eq_iff_prime_padicValNat_eq (a b : ℕ) (ha : a ≠ 0) (hb : b ≠ 0) :
a = b ↔ ∀ p : ℕ, p.Prime → padicValNat p a = padicValNat p b := by | constructor
· rintro rfl
simp
· intro h
refine eq_of_factorization_eq ha hb fun p => ?_
by_cases pp : p.Prime
· simp [factorization_def, pp, h p pp]
· simp [factorization_eq_zero_of_non_prime, pp] |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Basic
/-!
# Compact subsets of products as limits in `Profinite`
This file exhibits a compact subset `C` of a product... | Mathlib/Topology/Category/Profinite/Product.lean | 68 | 75 | theorem eq_of_forall_π_app_eq (a b : C)
(h : ∀ (J : Finset ι), π_app C (· ∈ J) a = π_app C (· ∈ J) b) : a = b := by | ext i
specialize h ({i} : Finset ι)
rw [Subtype.ext_iff] at h
simp only [π_app, ContinuousMap.precomp, ContinuousMap.coe_mk,
Set.MapsTo.val_restrict_apply] at h
exact congr_fun h ⟨i, Finset.mem_singleton.mpr rfl⟩ |
/-
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, Benjamin Davidson
-/
import Mathlib.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,146 | 1,146 | theorem tan_pi_div_two_sub (x : ℂ) : tan (π / 2 - x) = (tan x)⁻¹ := by | |
/-
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.Lemmas
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Topology.Algebra.Monoid
/-!
# Topological group with zero
I... | Mathlib/Topology/Algebra/GroupWithZero.lean | 69 | 71 | theorem Continuous.div_const (hf : Continuous f) (y : G₀) : Continuous fun x => f x / y := by | simpa only [div_eq_mul_inv] using hf.mul continuous_const |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Opposites
/-!
# Morphisms from equations between objects.
When working categorically, sometimes one encounters an equation `h ... | Mathlib/CategoryTheory/EqToHom.lean | 320 | 321 | theorem eqToHom_map (F : C ⥤ D) {X Y : C} (p : X = Y) :
F.map (eqToHom p) = eqToHom (congr_arg F.obj p) := by | cases p; simp |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.TakeWhile
import Mathlib.Order.Fin.Basic
/-!
# So... | Mathlib/Data/List/Sort.lean | 474 | 499 | theorem insertionSort_cons {a : α} {l : List α} (h : ∀ b ∈ l, r a b) :
insertionSort r (a :: l) = a :: insertionSort r l := by | rw [insertionSort]
cases hi : insertionSort r l with
| nil => rfl
| cons b m =>
rw [orderedInsert_of_le]
apply h b <| (mem_insertionSort r).1 _
rw [hi]
exact mem_cons_self
theorem map_insertionSort (f : α → β) (l : List α) (hl : ∀ a ∈ l, ∀ b ∈ l, a ≼ b ↔ f a ≼ f b) :
(l.insertionSort r).map f... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Anne Baanen
-/
import Mathlib.Tactic.Ring.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Tactic.Conv
import Mathlib.Util.Qq
/-!
# `ring_nf` tactic
A tactic which ... | Mathlib/Tactic/Ring/RingNF.lean | 120 | 120 | theorem nat_rawCast_1 : (Nat.rawCast 1 : R) = 1 := by | simp |
/-
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 | 553 | 554 | theorem sdiff_triangle (a b c : α) : a \ c ≤ a \ b ⊔ b \ c := by | rw [sdiff_le_iff, sup_left_comm, ← sdiff_le_iff] |
/-
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.Topology.MetricSpace.HausdorffDistance
/-!
# Topological study of spaces `Π (n : ℕ), E n`
When `E n` are topological spaces, the space `Π (n : ... | Mathlib/Topology/MetricSpace/PiNat.lean | 468 | 474 | theorem firstDiff_lt_shortestPrefixDiff {s : Set (∀ n, E n)} (hs : IsClosed s) {x y : ∀ n, E n}
(hx : x ∉ s) (hy : y ∈ s) : firstDiff x y < shortestPrefixDiff x s := by | have A := exists_disjoint_cylinder hs hx
rw [shortestPrefixDiff, dif_pos A]
classical
have B := Nat.find_spec A
contrapose! B |
/-
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.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Algebra.Module.LinearMap.Prod
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Analysis... | Mathlib/Analysis/Convex/Star.lean | 76 | 81 | theorem starConvex_iff_segment_subset : StarConvex 𝕜 x s ↔ ∀ ⦃y⦄, y ∈ s → [x -[𝕜] y] ⊆ s := by | constructor
· rintro h y hy z ⟨a, b, ha, hb, hab, rfl⟩
exact h hy ha hb hab
· rintro h y hy a b ha hb hab
exact h hy ⟨a, b, ha, hb, hab, rfl⟩ |
/-
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.Probability.Independence.Basic
import Mathlib.Probability.Independence.Conditional
/-!
# Kolmogorov's 0-1 law
Let `s : ι → MeasurableSpace Ω` be an indep... | Mathlib/Probability/Independence/ZeroOne.lean | 130 | 141 | theorem Kernel.indep_iSup_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
(hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) :
Indep (⨆ a, ⨆ n ∈ ns a, s n) (limsup s f) κ μα := by | rcases eq_or_ne μα 0 with rfl | hμ
· simp
obtain ⟨η, η_eq, hη⟩ : ∃ (η : Kernel α Ω), κ =ᵐ[μα] η ∧ IsMarkovKernel η :=
exists_ae_eq_isMarkovKernel h_indep.ae_isProbabilityMeasure hμ
replace h_indep := h_indep.congr η_eq
apply Indep.congr (Filter.EventuallyEq.symm η_eq)
apply indep_iSup_of_directed_le
· e... |
/-
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 | 153 | 154 | theorem toPoly_eq_zero_iff (P : Cubic R) : P.toPoly = 0 ↔ P = 0 := by | rw [← zero, toPoly_injective] |
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.Grou... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 326 | 333 | theorem add_mul_div_right (x y z : R) (h1 : y ≠ 0) (h2 : y ∣ x) : (x + z * y) / y = x / y + z := by | rw [mul_comm z y]
exact add_mul_div_left _ _ _ h1 h2
theorem sub_mul_div_left (x y z : R) (h1 : y ≠ 0) (h2 : y ∣ x) : (x - y * z) / y = x / y - z := by
rw [eq_comm]
apply eq_div_of_mul_eq_right h1
rw [mul_sub, EuclideanDomain.mul_div_cancel' h1 h2] |
/-
Copyright (c) 2022 Yaël Dillies, 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.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... | Mathlib/Combinatorics/SimpleGraph/Clique.lean | 114 | 116 | theorem isClique_map_iff {f : α ↪ β} {t : Set β} :
(G.map f).IsClique t ↔ t.Subsingleton ∨ ∃ (s : Set α), G.IsClique s ∧ f '' s = t := by | obtain (ht | ht) := t.subsingleton_or_nontrivial |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
/-!
# The ideal class group
This file defines the ideal class group `ClassGroup R` of fractional ideals of `R`
inside its f... | Mathlib/RingTheory/ClassGroup.lean | 221 | 229 | theorem FractionalIdeal.coe_mk0 [IsDedekindDomain R] (I : (Ideal R)⁰) :
(FractionalIdeal.mk0 K I : FractionalIdeal R⁰ K) = I := rfl
theorem FractionalIdeal.canonicalEquiv_mk0 [IsDedekindDomain R] (K' : Type*) [Field K']
[Algebra R K'] [IsFractionRing R K'] (I : (Ideal R)⁰) :
FractionalIdeal.canonicalEquiv ... | simp only [FractionalIdeal.coe_mk0, FractionalIdeal.canonicalEquiv_coeIdeal] |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 75 | 76 | theorem volume_Ico {a b : ℝ} : volume (Ico a b) = ofReal (b - a) := by | simp [volume_val] |
/-
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 | 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) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Minor.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most o... | Mathlib/Data/Matroid/Constructions.lean | 177 | 179 | theorem ground_indep_iff_eq_freeOn : M.Indep M.E ↔ M = freeOn M.E := by | simp [eq_freeOn_iff] |
/-
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.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Ordering.Lemmas
import Mathlib.Data.PNat.Basic
import Mathlib.SetTheory.Ordinal.Principal
import Ma... | Mathlib/SetTheory/Ordinal/Notation.lean | 225 | 226 | theorem NF.zero_of_zero {e n a} (h : NF (ONote.oadd e n a)) (e0 : e = 0) : a = 0 := by | simpa [e0, NFBelow_zero] using h.snd' |
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