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import Mathlib.Data.PFunctor.Univariate.M #align_import data.qpf.univariate.basic from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7" universe u class QPF (F : Type u → Type u) [Functor F] where P : PFunctor.{u} abs : ∀ {α}, P α → F α repr : ∀ {α}, F α → P α abs_repr : ∀ {α} (...
Mathlib/Data/QPF/Univariate/Basic.lean
117
131
theorem liftp_iff' {α : Type u} (p : α → Prop) (x : F α) : Liftp p x ↔ ∃ u : q.P α, abs u = x ∧ ∀ i, p (u.snd i) := by
constructor · rintro ⟨y, hy⟩ cases' h : repr y with a f use ⟨a, fun i => (f i).val⟩ dsimp constructor · rw [← hy, ← abs_repr y, h, ← abs_map] rfl intro i apply (f i).property rintro ⟨⟨a, f⟩, h₀, h₁⟩; dsimp at * use abs ⟨a, fun i => ⟨f i, h₁ i⟩⟩ rw [← abs_map, ← h₀]; rfl
13
442,413.392009
2
1.571429
7
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import Mathlib.CategoryTheory.EqToHom import Mathlib.CategoryTheory.Quotient import Mathlib.Combinatorics.Quiver.Path #align_import category_theory.path_category from "leanprover-community/mathlib"@"c6dd521ebdce53bb372c527569dd7c25de53a08b" universe v₁ v₂ u₁ u₂ namespace CategoryTheory section def Paths (V : ...
Mathlib/CategoryTheory/PathCategory.lean
93
100
theorem lift_spec {C} [Category C] (φ : V ⥤q C) : of ⋙q (lift φ).toPrefunctor = φ := by
fapply Prefunctor.ext · rintro X rfl · rintro X Y f rcases φ with ⟨φo, φm⟩ dsimp [lift, Quiver.Hom.toPath] simp only [Category.id_comp]
7
1,096.633158
2
1.75
4
1,866
import Mathlib.CategoryTheory.Abelian.Basic #align_import category_theory.idempotents.basic from "leanprover-community/mathlib"@"3a061790136d13594ec10c7c90d202335ac5d854" open CategoryTheory open CategoryTheory.Category open CategoryTheory.Limits open CategoryTheory.Preadditive open Opposite namespace Catego...
Mathlib/CategoryTheory/Idempotents/Basic.lean
130
140
theorem split_imp_of_iso {X X' : C} (φ : X ≅ X') (p : X ⟶ X) (p' : X' ⟶ X') (hpp' : p ≫ φ.hom = φ.hom ≫ p') (h : ∃ (Y : C) (i : Y ⟶ X) (e : X ⟶ Y), i ≫ e = 𝟙 Y ∧ e ≫ i = p) : ∃ (Y' : C) (i' : Y' ⟶ X') (e' : X' ⟶ Y'), i' ≫ e' = 𝟙 Y' ∧ e' ≫ i' = p' := by
rcases h with ⟨Y, i, e, ⟨h₁, h₂⟩⟩ use Y, i ≫ φ.hom, φ.inv ≫ e constructor · slice_lhs 2 3 => rw [φ.hom_inv_id] rw [id_comp, h₁] · slice_lhs 2 3 => rw [h₂] rw [hpp', ← assoc, φ.inv_hom_id, id_comp]
7
1,096.633158
2
1.6
5
1,735
import Mathlib.Analysis.Convex.Topology import Mathlib.LinearAlgebra.Dimension.DivisionRing import Mathlib.Topology.Algebra.Module.Cardinality open Convex Set Metric section TopologicalVectorSpace variable {E : Type*} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E]
Mathlib/Analysis/NormedSpace/Connected.lean
34
103
theorem Set.Countable.isPathConnected_compl_of_one_lt_rank (h : 1 < Module.rank ℝ E) {s : Set E} (hs : s.Countable) : IsPathConnected sᶜ := by
have : Nontrivial E := (rank_pos_iff_nontrivial (R := ℝ)).1 (zero_lt_one.trans h) -- the set `sᶜ` is dense, therefore nonempty. Pick `a ∈ sᶜ`. We have to show that any -- `b ∈ sᶜ` can be joined to `a`. obtain ⟨a, ha⟩ : sᶜ.Nonempty := (hs.dense_compl ℝ).nonempty refine ⟨a, ha, ?_⟩ intro b hb rcases eq_or_...
67
125,236,317,084,221,370,000,000,000,000
2
2
1
2,081
import Mathlib.CategoryTheory.Abelian.Opposite import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels import Mathlib.CategoryTheory.Preadditive.LeftExact import Mathlib.CategoryTheory.Adjunction.Limits import Mathlib.Algebra.Homology.Exact import Mathli...
Mathlib/CategoryTheory/Abelian/Exact.lean
84
93
theorem exact_iff' {cg : KernelFork g} (hg : IsLimit cg) {cf : CokernelCofork f} (hf : IsColimit cf) : Exact f g ↔ f ≫ g = 0 ∧ cg.ι ≫ cf.π = 0 := by
constructor · intro h exact ⟨h.1, fork_ι_comp_cofork_π f g h cg cf⟩ · rw [exact_iff] refine fun h => ⟨h.1, ?_⟩ apply zero_of_epi_comp (IsLimit.conePointUniqueUpToIso hg (limit.isLimit _)).hom apply zero_of_comp_mono (IsColimit.coconePointUniqueUpToIso (colimit.isColimit _) hf).hom simp [h.2]
8
2,980.957987
2
1.8
5
1,886
import Mathlib.Data.List.Cycle import Mathlib.GroupTheory.Perm.Cycle.Type import Mathlib.GroupTheory.Perm.List #align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a" open Equiv Equiv.Perm List variable {α : Type*} namespace Cycle variable [...
Mathlib/GroupTheory/Perm/Cycle/Concrete.lean
149
156
theorem formPerm_subsingleton (s : Cycle α) (h : Subsingleton s) : formPerm s h.nodup = 1 := by
induction' s using Quot.inductionOn with s simp only [formPerm_coe, mk_eq_coe] simp only [length_subsingleton_iff, length_coe, mk_eq_coe] at h cases' s with hd tl · simp · simp only [length_eq_zero, add_le_iff_nonpos_left, List.length, nonpos_iff_eq_zero] at h simp [h]
7
1,096.633158
2
1
18
1,030
import Mathlib.ModelTheory.Ultraproducts import Mathlib.ModelTheory.Bundled import Mathlib.ModelTheory.Skolem #align_import model_theory.satisfiability from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728" set_option linter.uppercaseLean3 false universe u v w w' open Cardinal CategoryTh...
Mathlib/ModelTheory/Satisfiability.lean
138
154
theorem isSatisfiable_union_distinctConstantsTheory_of_card_le (T : L.Theory) (s : Set α) (M : Type w') [Nonempty M] [L.Structure M] [M ⊨ T] (h : Cardinal.lift.{w'} #s ≤ Cardinal.lift.{w} #M) : ((L.lhomWithConstants α).onTheory T ∪ L.distinctConstantsTheory s).IsSatisfiable := by
haveI : Inhabited M := Classical.inhabited_of_nonempty inferInstance rw [Cardinal.lift_mk_le'] at h letI : (constantsOn α).Structure M := constantsOn.structure (Function.extend (↑) h.some default) have : M ⊨ (L.lhomWithConstants α).onTheory T ∪ L.distinctConstantsTheory s := by refine ((LHom.onTheory_model...
13
442,413.392009
2
2
5
2,364
import Mathlib.Topology.Constructions import Mathlib.Topology.ContinuousOn #align_import topology.bases from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4" open Set Filter Function Topology noncomputable section namespace TopologicalSpace universe u variable {α : Type u} {β : Type*} ...
Mathlib/Topology/Bases.lean
93
103
theorem IsTopologicalBasis.diff_empty {s : Set (Set α)} (h : IsTopologicalBasis s) : IsTopologicalBasis (s \ {∅}) := by
refine ⟨?_, by rw [sUnion_diff_singleton_empty, h.sUnion_eq], ?_⟩ · rintro t₁ ⟨h₁, -⟩ t₂ ⟨h₂, -⟩ x hx obtain ⟨t₃, h₃, hs⟩ := h.exists_subset_inter _ h₁ _ h₂ x hx exact ⟨t₃, ⟨h₃, Nonempty.ne_empty ⟨x, hs.1⟩⟩, hs⟩ · rw [h.eq_generateFrom] refine le_antisymm (generateFrom_anti diff_subset) (le_generateF...
9
8,103.083928
2
2
5
2,210
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp #align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" noncomputable section open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise open Set Filter TopologicalSpace ENNR...
Mathlib/MeasureTheory/Integral/SetToL1.lean
130
135
theorem smul_measure (c : ℝ≥0∞) (hc_ne_zero : c ≠ 0) (hT : FinMeasAdditive μ T) : FinMeasAdditive (c • μ) T := by
refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top] simp only [hc_ne_zero, true_and_iff, Ne, not_false_iff] exact Or.inl hμs
4
54.59815
2
1.8
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1,900
import Mathlib.MeasureTheory.Measure.Restrict #align_import measure_theory.measure.mutually_singular from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570" open Set open MeasureTheory NNReal ENNReal namespace MeasureTheory namespace Measure variable {α : Type*} {m0 : MeasurableSpace α}...
Mathlib/MeasureTheory/Measure/MutuallySingular.lean
114
120
theorem sum_left {ι : Type*} [Countable ι] {μ : ι → Measure α} : sum μ ⟂ₘ ν ↔ ∀ i, μ i ⟂ₘ ν := by
refine ⟨fun h i => h.mono (le_sum _ _) le_rfl, fun H => ?_⟩ choose s hsm hsμ hsν using H refine ⟨⋂ i, s i, MeasurableSet.iInter hsm, ?_, ?_⟩ · rw [sum_apply _ (MeasurableSet.iInter hsm), ENNReal.tsum_eq_zero] exact fun i => measure_mono_null (iInter_subset _ _) (hsμ i) · rwa [compl_iInter, measure_iUnion...
6
403.428793
2
1
3
913
import Mathlib.Algebra.Polynomial.BigOperators import Mathlib.Algebra.Polynomial.Derivative import Mathlib.Data.Nat.Choose.Cast import Mathlib.Data.Nat.Choose.Vandermonde import Mathlib.Tactic.FieldSimp #align_import data.polynomial.hasse_deriv from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358...
Mathlib/Algebra/Polynomial/HasseDeriv.lean
143
161
theorem factorial_smul_hasseDeriv : ⇑(k ! • @hasseDeriv R _ k) = (@derivative R _)^[k] := by
induction' k with k ih · rw [hasseDeriv_zero, factorial_zero, iterate_zero, one_smul, LinearMap.id_coe] ext f n : 2 rw [iterate_succ_apply', ← ih] simp only [LinearMap.smul_apply, coeff_smul, LinearMap.map_smul_of_tower, coeff_derivative, hasseDeriv_coeff, ← @choose_symm_add _ k] simp only [nsmul_eq_mu...
18
65,659,969.137331
2
1.2
10
1,278
import Mathlib.Data.Nat.Prime import Mathlib.Tactic.NormNum.Basic #align_import data.nat.prime_norm_num from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1" open Nat Qq Lean Meta namespace Mathlib.Meta.NormNum theorem not_prime_mul_of_ble (a b n : ℕ) (h : a * b = n) (h₁ : a.ble 1 = fals...
Mathlib/Tactic/NormNum/Prime.lean
90
95
theorem minFacHelper_3 {n k k' : ℕ} (e : k + 2 = k') (nk : (n % k).beq 0 = false) (h : MinFacHelper n k) : MinFacHelper n k' := by
refine minFacHelper_1 e h λ h2 ↦ ?_ have nk := Nat.ne_of_beq_eq_false nk rw [← Nat.dvd_iff_mod_eq_zero, ← h2] at nk exact nk <| minFac_dvd n
4
54.59815
2
1.8
5
1,898
import Mathlib.Data.Rat.Cast.Defs import Mathlib.Algebra.Field.Basic #align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441" namespace Rat variable {α : Type*} [DivisionRing α] -- Porting note: rewrote proof @[simp] theorem cast_inv_nat (n : ℕ) : ((n⁻¹ : ℚ) : α...
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 [Nat.cast_pos] using q.den_pos simp only [Int.cast_natCast, Nat.cast_inj] at hn hd ...
6
403.428793
2
1.333333
6
1,436
import Mathlib.LinearAlgebra.Dimension.Free import Mathlib.Algebra.Homology.ShortComplex.ModuleCat open CategoryTheory namespace ModuleCat variable {ι ι' R : Type*} [Ring R] {S : ShortComplex (ModuleCat R)} (hS : S.Exact) (hS' : S.ShortExact) {v : ι → S.X₁} open CategoryTheory Submodule Set section LinearInde...
Mathlib/Algebra/Category/ModuleCat/Free.lean
72
78
theorem linearIndependent_shortExact {w : ι' → S.X₃} (hw : LinearIndependent R w) : LinearIndependent R (Sum.elim (S.f ∘ v) (S.g.toFun.invFun ∘ w)) := by
apply linearIndependent_leftExact hS'.exact hv _ hS'.mono_f rfl dsimp convert hw ext apply Function.rightInverse_invFun ((epi_iff_surjective _).mp hS'.epi_g)
5
148.413159
2
2
5
2,042
import Mathlib.Data.Nat.Defs import Mathlib.Order.Interval.Set.Basic import Mathlib.Tactic.Monotonicity.Attr #align_import data.nat.log from "leanprover-community/mathlib"@"3e00d81bdcbf77c8188bbd18f5524ddc3ed8cac6" namespace Nat --@[pp_nodot] porting note: unknown attribute def log (b : ℕ) : ℕ → ℕ | n => i...
Mathlib/Data/Nat/Log.lean
89
101
theorem pow_le_iff_le_log {b : ℕ} (hb : 1 < b) {x y : ℕ} (hy : y ≠ 0) : b ^ x ≤ y ↔ x ≤ log b y := by
induction' y using Nat.strong_induction_on with y ih generalizing x cases x with | zero => dsimp; omega | succ x => rw [log]; split_ifs with h · have b_pos : 0 < b := lt_of_succ_lt hb rw [Nat.add_le_add_iff_right, ← ih (y / b) (div_lt_self (Nat.pos_iff_ne_zero.2 hy) hb) (Nat.div_pos h.1 b...
11
59,874.141715
2
1
6
1,106
import Mathlib.Algebra.Polynomial.Inductions import Mathlib.Algebra.Polynomial.Monic import Mathlib.RingTheory.Multiplicity import Mathlib.RingTheory.Ideal.Maps #align_import data.polynomial.div from "leanprover-community/mathlib"@"e1e7190efdcefc925cb36f257a8362ef22944204" noncomputable section open Polynomial ...
Mathlib/Algebra/Polynomial/Div.lean
166
173
theorem natDegree_modByMonic_lt (p : R[X]) {q : R[X]} (hmq : Monic q) (hq : q ≠ 1) : natDegree (p %ₘ q) < q.natDegree := by
by_cases hpq : p %ₘ q = 0 · rw [hpq, natDegree_zero, Nat.pos_iff_ne_zero] contrapose! hq exact eq_one_of_monic_natDegree_zero hmq hq · haveI := Nontrivial.of_polynomial_ne hpq exact natDegree_lt_natDegree hpq (degree_modByMonic_lt p hmq)
6
403.428793
2
2
4
2,144
import Mathlib.Combinatorics.SimpleGraph.Connectivity namespace SimpleGraph universe u v variable {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} namespace Subgraph protected structure Preconnected (H : G.Subgraph) : Prop where protected coe : H.coe.Preconnected instance {H : G.Subgraph}...
Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean
64
69
theorem singletonSubgraph_connected {v : V} : (G.singletonSubgraph v).Connected := by
refine ⟨⟨?_⟩⟩ rintro ⟨a, ha⟩ ⟨b, hb⟩ simp only [singletonSubgraph_verts, Set.mem_singleton_iff] at ha hb subst_vars rfl
5
148.413159
2
2
2
2,427
import Mathlib.Topology.Algebra.Module.WeakDual import Mathlib.Algebra.Algebra.Spectrum import Mathlib.Topology.ContinuousFunction.Algebra import Mathlib.Data.Set.Lattice #align_import topology.algebra.module.character_space from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64" namespace ...
Mathlib/Topology/Algebra/Module/CharacterSpace.lean
128
134
theorem union_zero_isClosed [T2Space 𝕜] [ContinuousMul 𝕜] : IsClosed (characterSpace 𝕜 A ∪ {0}) := by
simp only [union_zero, Set.setOf_forall] exact isClosed_iInter fun x => isClosed_iInter fun y => isClosed_eq (eval_continuous _) <| (eval_continuous _).mul (eval_continuous _)
5
148.413159
2
2
1
2,022
import Mathlib.Algebra.Polynomial.Mirror import Mathlib.Analysis.Complex.Polynomial #align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836" namespace Polynomial open scoped Polynomial open Finset section Semiring variable {R : Type*} [Semirin...
Mathlib/Algebra/Polynomial/UnitTrinomial.lean
67
78
theorem trinomial_natDegree (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) : (trinomial k m n u v w).natDegree = n := by
refine natDegree_eq_of_degree_eq_some ((Finset.sup_le fun i h => ?_).antisymm <| le_degree_of_ne_zero <| by rwa [trinomial_leading_coeff' hkm hmn]) replace h := support_trinomial' k m n u v w h rw [mem_insert, mem_insert, mem_singleton] at h rcases h with (rfl | rfl | rfl) · exact WithBot.c...
10
22,026.465795
2
1.111111
9
1,193
import Mathlib.Algebra.Homology.Homotopy import Mathlib.Algebra.Category.ModuleCat.Abelian import Mathlib.Algebra.Category.ModuleCat.Subobject import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory #align_import algebra.homology.Module from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225...
Mathlib/Algebra/Homology/ModuleCat.lean
72
79
theorem cycles'Map_toCycles' (f : C ⟶ D) {i : ι} (x : LinearMap.ker (C.dFrom i)) : (cycles'Map f i) (toCycles' x) = toCycles' ⟨f.f i x.1, by -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644 rw [LinearMap.mem_ker]; erw [Hom.comm_from_apply, x.2, map_zero]⟩ := by
ext -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644 erw [cycles'Map_arrow_apply, toKernelSubobject_arrow, toKernelSubobject_arrow] rfl
4
54.59815
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import Mathlib.Algebra.Module.Card import Mathlib.SetTheory.Cardinal.CountableCover import Mathlib.SetTheory.Cardinal.Continuum import Mathlib.Analysis.SpecificLimits.Normed import Mathlib.Topology.MetricSpace.Perfect universe u v open Filter Pointwise Set Function Cardinal open scoped Cardinal Topology theorem c...
Mathlib/Topology/Algebra/Module/Cardinality.lean
97
106
theorem cardinal_eq_of_mem_nhds {E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E] {s : Set E} {x : E} (hs : s ∈ 𝓝 x) : #s = #E := by
let g := Homeomorph.addLeft x let t := g ⁻¹' s have : t ∈ 𝓝 0 := g.continuous.continuousAt.preimage_mem_nhds (by simpa [g] using hs) have A : #t = #E := cardinal_eq_of_mem_nhds_zero 𝕜 this have B : #t = #s := Cardinal.mk_subtype_of_equiv s g.toEquiv rwa [B] at A
6
403.428793
2
1.2
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1,284
import Mathlib.RingTheory.IntegralClosure import Mathlib.RingTheory.Localization.Integral #align_import ring_theory.integrally_closed from "leanprover-community/mathlib"@"d35b4ff446f1421bd551fafa4b8efd98ac3ac408" open scoped nonZeroDivisors Polynomial open Polynomial abbrev IsIntegrallyClosedIn (R A : Type*) [...
Mathlib/RingTheory/IntegrallyClosed.lean
110
120
theorem isIntegrallyClosedIn_iff {R A : Type*} [CommRing R] [CommRing A] [Algebra R A] : IsIntegrallyClosedIn R A ↔ Function.Injective (algebraMap R A) ∧ ∀ {x : A}, IsIntegral R x → ∃ y, algebraMap R A y = x := by
constructor · rintro ⟨_, cl⟩ aesop · rintro ⟨inj, cl⟩ refine ⟨inj, by aesop, ?_⟩ rintro ⟨y, rfl⟩ apply isIntegral_algebraMap
7
1,096.633158
2
1.75
4
1,859
import Mathlib.Analysis.LocallyConvex.Bounded import Mathlib.Topology.Algebra.Module.StrongTopology #align_import analysis.normed_space.compact_operator from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9" open Function Set Filter Bornology Metric Pointwise Topology def IsCompactOperat...
Mathlib/Analysis/NormedSpace/CompactOperator.lean
336
365
theorem IsCompactOperator.continuous {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator f) : Continuous f := by
letI : UniformSpace M₂ := TopologicalAddGroup.toUniformSpace _ haveI : UniformAddGroup M₂ := comm_topologicalAddGroup_is_uniform -- Since `f` is linear, we only need to show that it is continuous at zero. -- Let `U` be a neighborhood of `0` in `M₂`. refine continuous_of_continuousAt_zero f fun U hU => ?_ r...
28
1,446,257,064,291.475
2
1.6
5
1,713
import Mathlib.Topology.Separation import Mathlib.Topology.UniformSpace.Basic import Mathlib.Topology.UniformSpace.Cauchy #align_import topology.uniform_space.uniform_convergence from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9" noncomputable section open Topology Uniformity Filter S...
Mathlib/Topology/UniformSpace/UniformConvergence.lean
166
171
theorem TendstoUniformlyOnFilter.tendsto_at (h : TendstoUniformlyOnFilter F f p p') (hx : 𝓟 {x} ≤ p') : Tendsto (fun n => F n x) p <| 𝓝 (f x) := by
refine Uniform.tendsto_nhds_right.mpr fun u hu => mem_map.mpr ?_ filter_upwards [(h u hu).curry] intro i h simpa using h.filter_mono hx
4
54.59815
2
0.571429
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import Mathlib.Algebra.Category.GroupCat.Basic import Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects #align_import algebra.category.Group.zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a" open CategoryTheory open CategoryTheory.Limits universe u namespace GroupCat @[to_addi...
Mathlib/Algebra/Category/GroupCat/Zero.lean
28
34
theorem isZero_of_subsingleton (G : GroupCat) [Subsingleton G] : IsZero G := by
refine ⟨fun X => ⟨⟨⟨1⟩, fun f => ?_⟩⟩, fun X => ⟨⟨⟨1⟩, fun f => ?_⟩⟩⟩ · ext x have : x = 1 := Subsingleton.elim _ _ rw [this, map_one, map_one] · ext apply Subsingleton.elim
6
403.428793
2
2
2
2,492
import Mathlib.Order.Interval.Finset.Nat import Mathlib.Data.PNat.Defs #align_import data.pnat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29" open Finset Function PNat namespace PNat variable (a b : ℕ+) instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ+ := Subtype....
Mathlib/Data/PNat/Interval.lean
67
72
theorem card_Icc : (Icc a b).card = b + 1 - a := by
rw [← Nat.card_Icc] -- Porting note: I had to change this to `erw` *and* provide the proof, yuck. -- https://github.com/leanprover-community/mathlib4/issues/5164 erw [← Finset.map_subtype_embedding_Icc _ a b (fun c x _ hx _ hc _ => hc.trans_le hx)] rw [card_map]
5
148.413159
2
1
8
948
import Mathlib.Data.Complex.Exponential import Mathlib.Analysis.SpecialFunctions.Log.Deriv #align_import data.complex.exponential_bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973" namespace Real open IsAbsoluteValue Finset CauSeq Complex theorem exp_one_near_10 : |exp 1 - 224...
Mathlib/Data/Complex/ExponentialBounds.lean
28
33
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
apply exp_approx_start iterate 21 refine exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) ?_ norm_num1 refine exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) ?_ rw [_root_.abs_one, abs_of_pos] <;> norm_num1
5
148.413159
2
2
5
2,495
import Mathlib.Analysis.Calculus.ContDiff.Basic import Mathlib.Analysis.Calculus.Deriv.Linear import Mathlib.Analysis.Complex.Conformal import Mathlib.Analysis.Calculus.Conformal.NormedSpace #align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" se...
Mathlib/Analysis/Complex/RealDeriv.lean
84
89
theorem ContDiffAt.real_of_complex {n : ℕ∞} (h : ContDiffAt ℂ n e z) : ContDiffAt ℝ n (fun x : ℝ => (e x).re) z := by
have A : ContDiffAt ℝ n ((↑) : ℝ → ℂ) z := ofRealCLM.contDiff.contDiffAt have B : ContDiffAt ℝ n e z := h.restrict_scalars ℝ have C : ContDiffAt ℝ n re (e z) := reCLM.contDiff.contDiffAt exact C.comp z (B.comp z A)
4
54.59815
2
1
11
1,076
import Mathlib.Algebra.Group.Basic import Mathlib.Algebra.Order.Monoid.Canonical.Defs import Mathlib.Data.Set.Function import Mathlib.Order.Interval.Set.Basic #align_import data.set.intervals.monoid from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9" namespace Set variable {M : Type*} ...
Mathlib/Algebra/Order/Interval/Set/Monoid.lean
58
62
theorem Ioc_add_bij : BijOn (· + d) (Ioc a b) (Ioc (a + d) (b + d)) := by
rw [← Ioi_inter_Iic, ← Ioi_inter_Iic] exact (Ioi_add_bij a d).inter_mapsTo (fun x hx => add_le_add_right hx _) fun x hx => le_of_add_le_add_right hx.2
4
54.59815
2
1.090909
11
1,187
import Mathlib.Algebra.Group.ConjFinite import Mathlib.GroupTheory.Abelianization import Mathlib.GroupTheory.GroupAction.ConjAct import Mathlib.GroupTheory.GroupAction.Quotient import Mathlib.GroupTheory.Index import Mathlib.GroupTheory.SpecificGroups.Dihedral import Mathlib.Tactic.FieldSimp import Mathlib.Tactic.Line...
Mathlib/GroupTheory/CommutingProbability.lean
54
60
theorem commProb_pi {α : Type*} (i : α → Type*) [Fintype α] [∀ a, Mul (i a)] : commProb (∀ a, i a) = ∏ a, commProb (i a) := by
simp_rw [commProb_def, Finset.prod_div_distrib, Finset.prod_pow, ← Nat.cast_prod, ← Nat.card_pi, Commute, SemiconjBy, Function.funext_iff] congr 2 exact Nat.card_congr ⟨fun x a => ⟨⟨x.1.1 a, x.1.2 a⟩, x.2 a⟩, fun x => ⟨⟨fun a => (x a).1.1, fun a => (x a).1.2⟩, fun a => (x a).2⟩, fun x => rfl, fun x => rf...
5
148.413159
2
1.625
8
1,749
import Mathlib.Algebra.Polynomial.AlgebraMap import Mathlib.Algebra.Polynomial.Degree.Lemmas import Mathlib.Algebra.Polynomial.HasseDeriv #align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a" noncomputable section namespace Polynomial open Polynomial...
Mathlib/Algebra/Polynomial/Taylor.lean
130
134
theorem taylor_injective {R} [CommRing R] (r : R) : Function.Injective (taylor r) := by
intro f g h apply_fun taylor (-r) at h simpa only [taylor_apply, comp_assoc, add_comp, X_comp, C_comp, C_neg, neg_add_cancel_right, comp_X] using h
4
54.59815
2
0.466667
15
415
import Mathlib.Tactic.FinCases import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Finsupp import Mathlib.Algebra.Field.IsField #align_import ring_theory.ideal.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988" universe u v w variable {α : Type u} {β : Type v} open ...
Mathlib/RingTheory/Ideal/Basic.lean
106
110
theorem unit_mul_mem_iff_mem {x y : α} (hy : IsUnit y) : y * x ∈ I ↔ x ∈ I := by
refine ⟨fun h => ?_, fun h => I.mul_mem_left y h⟩ obtain ⟨y', hy'⟩ := hy.exists_left_inv have := I.mul_mem_left y' h rwa [← mul_assoc, hy', one_mul] at this
4
54.59815
2
1
3
963
import Mathlib.Algebra.Polynomial.Degree.Definitions import Mathlib.Data.ENat.Basic #align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836" noncomputable section open Function Polynomial Finsupp Finset open scoped Polynomial namespace ...
Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean
158
164
theorem natTrailingDegree_le_of_ne_zero (h : coeff p n ≠ 0) : natTrailingDegree p ≤ n := by
have : WithTop.some (natTrailingDegree p) = Nat.cast (natTrailingDegree p) := rfl rw [← WithTop.coe_le_coe, this, ← trailingDegree_eq_natTrailingDegree] · exact trailingDegree_le_of_ne_zero h · intro h subst h exact h rfl
6
403.428793
2
1.666667
6
1,754
import Mathlib.Data.Opposite import Mathlib.Data.Set.Defs #align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e" variable {α : Type*} open Opposite namespace Set protected def op (s : Set α) : Set αᵒᵖ := unop ⁻¹' s #align set.op Set.op protected def u...
Mathlib/Data/Set/Opposite.lean
100
104
theorem singleton_unop_op (x : αᵒᵖ) : ({unop x} : Set α).op = {x} := by
ext constructor · apply unop_injective · apply op_injective
4
54.59815
2
1.333333
6
1,444
import Mathlib.LinearAlgebra.FreeModule.PID import Mathlib.LinearAlgebra.FreeModule.Finite.Basic import Mathlib.LinearAlgebra.BilinearForm.DualLattice import Mathlib.RingTheory.DedekindDomain.Basic import Mathlib.RingTheory.Localization.Module import Mathlib.RingTheory.Trace #align_import ring_theory.dedekind_domain....
Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean
119
138
theorem exists_integral_multiples (s : Finset L) : ∃ y ≠ (0 : A), ∀ x ∈ s, IsIntegral A (y • x) := by
haveI := Classical.decEq L refine s.induction ?_ ?_ · use 1, one_ne_zero rintro x ⟨⟩ · rintro x s hx ⟨y, hy, hs⟩ have := exists_integral_multiple ((IsFractionRing.isAlgebraic_iff A K L).mpr (.of_finite _ x)) ((injective_iff_map_eq_zero (algebraMap A L)).mp ?_) · rcases this with ⟨x', y'...
18
65,659,969.137331
2
1.8
5
1,881
import Mathlib.Analysis.SpecialFunctions.Pow.Real import Mathlib.LinearAlgebra.FreeModule.PID import Mathlib.LinearAlgebra.Matrix.AbsoluteValue import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue import Mathlib.RingTheory.ClassGroup import Mathlib.RingTheory.DedekindDomain.IntegralClosure import Mathlib.Ri...
Mathlib/NumberTheory/ClassNumber/Finite.lean
119
135
theorem exists_min (I : (Ideal S)⁰) : ∃ b ∈ (I : Ideal S), b ≠ 0 ∧ ∀ c ∈ (I : Ideal S), abv (Algebra.norm R c) < abv (Algebra.norm R b) → c = (0 : S) := by
obtain ⟨_, ⟨b, b_mem, b_ne_zero, rfl⟩, min⟩ := @Int.exists_least_of_bdd (fun a => ∃ b ∈ (I : Ideal S), b ≠ (0 : S) ∧ abv (Algebra.norm R b) = a) (by use 0 rintro _ ⟨b, _, _, rfl⟩ apply abv.nonneg) (by obtain ⟨b, b_mem, b_ne_zero⟩ := (I : Ideal S).ne_bot_iff.mp (nonZeroDivisors.c...
13
442,413.392009
2
2
4
2,456
import Mathlib.CategoryTheory.Limits.Types import Mathlib.CategoryTheory.IsConnected import Mathlib.CategoryTheory.Limits.Final import Mathlib.CategoryTheory.Conj universe w v u namespace CategoryTheory.Limits.Types variable (C : Type u) [Category.{v} C] def constPUnitFunctor : C ⥤ Type w := (Functor.const C).o...
Mathlib/CategoryTheory/Limits/IsConnected.lean
97
104
theorem isConnected_iff_colimit_constPUnitFunctor_iso_pUnit [HasColimit (constPUnitFunctor.{w} C)] : IsConnected C ↔ Nonempty (colimit (constPUnitFunctor.{w} C) ≅ PUnit) := by
refine ⟨fun _ => ⟨colimitConstPUnitIsoPUnit.{w} C⟩, fun ⟨h⟩ => ?_⟩ have : Nonempty C := nonempty_of_nonempty_colimit <| Nonempty.map h.inv inferInstance refine zigzag_isConnected <| fun c d => ?_ refine zigzag_of_eqvGen_quot_rel _ (constPUnitFunctor C) ⟨c, PUnit.unit⟩ ⟨d, PUnit.unit⟩ ?_ exact colimit_eq <| h...
5
148.413159
2
2
4
2,211
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms import Mathlib.CategoryTheory.Limits.Shapes.Kernels import Mathlib.CategoryTheory.Abelian.Basic import Mathlib.CategoryTheory.Subobject.Lattice import Mathlib.Order.Atoms #align_import category_theory.simple from "leanprover-community/mathlib"@"4ed0bcaef698011...
Mathlib/CategoryTheory/Simple.lean
170
175
theorem cokernel_zero_of_nonzero_to_simple {X Y : C} [Simple Y] {f : X ⟶ Y} (w : f ≠ 0) : cokernel.π f = 0 := by
classical by_contra h haveI := isIso_of_epi_of_nonzero h exact w (eq_zero_of_mono_cokernel f)
4
54.59815
2
1.5
8
1,594
import Mathlib.RingTheory.Localization.Module import Mathlib.RingTheory.Norm import Mathlib.RingTheory.Discriminant #align_import ring_theory.localization.norm from "leanprover-community/mathlib"@"2e59a6de168f95d16b16d217b808a36290398c0a" open scoped nonZeroDivisors variable (R : Type*) {S : Type*} [CommRing R] ...
Mathlib/RingTheory/Localization/NormTrace.lean
83
92
theorem Algebra.trace_localization [Module.Free R S] [Module.Finite R S] (a : S) : Algebra.trace Rₘ Sₘ (algebraMap S Sₘ a) = algebraMap R Rₘ (Algebra.trace R S a) := by
cases subsingleton_or_nontrivial R · haveI : Subsingleton Rₘ := Module.subsingleton R Rₘ simp [eq_iff_true_of_subsingleton] let b := Module.Free.chooseBasis R S letI := Classical.decEq (Module.Free.ChooseBasisIndex R S) rw [Algebra.trace_eq_matrix_trace (b.localizationLocalization Rₘ M Sₘ), Algebra.t...
8
2,980.957987
2
1.6
5
1,731
import Mathlib.Topology.Algebra.Algebra import Mathlib.Analysis.InnerProductSpace.Basic #align_import analysis.inner_product_space.of_norm from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb" open RCLike open scoped ComplexConjugate variable {𝕜 : Type*} [RCLike 𝕜] (E : Type*) [Normed...
Mathlib/Analysis/InnerProductSpace/OfNorm.lean
127
136
theorem inner_.norm_sq (x : E) : ‖x‖ ^ 2 = re (inner_ 𝕜 x x) := by
simp only [inner_] have h₁ : RCLike.normSq (4 : 𝕜) = 16 := by have : ((4 : ℝ) : 𝕜) = (4 : 𝕜) := by norm_cast rw [← this, normSq_eq_def', RCLike.norm_of_nonneg (by norm_num : (0 : ℝ) ≤ 4)] norm_num have h₂ : ‖x + x‖ = 2 * ‖x‖ := by rw [← two_smul 𝕜, norm_smul, RCLike.norm_two] simp only [h₁, h₂,...
9
8,103.083928
2
1.75
4
1,861
import Mathlib.Algebra.MvPolynomial.Basic import Mathlib.RingTheory.Polynomial.Basic import Mathlib.RingTheory.PrincipalIdealDomain #align_import ring_theory.adjoin.fg from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23" universe u v w open Subsemiring Ring Submodule open Pointwise na...
Mathlib/RingTheory/Adjoin/FG.lean
170
179
theorem induction_on_adjoin [IsNoetherian R A] (P : Subalgebra R A → Prop) (base : P ⊥) (ih : ∀ (S : Subalgebra R A) (x : A), P S → P (Algebra.adjoin R (insert x S))) (S : Subalgebra R A) : P S := by
classical obtain ⟨t, rfl⟩ := S.fg_of_noetherian refine Finset.induction_on t ?_ ?_ · simpa using base intro x t _ h rw [Finset.coe_insert] simpa only [Algebra.adjoin_insert_adjoin] using ih _ x h
7
1,096.633158
2
2
3
2,099
import Mathlib.Algebra.Order.BigOperators.Group.Finset import Mathlib.Data.Nat.Factors import Mathlib.Order.Interval.Finset.Nat #align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" open scoped Classical open Finset namespace Nat variable (n : ℕ) d...
Mathlib/NumberTheory/Divisors.lean
147
151
theorem divisor_le {m : ℕ} : n ∈ divisors m → n ≤ m := by
cases' m with m · simp · simp only [mem_divisors, Nat.succ_ne_zero m, and_true_iff, Ne, not_false_iff] exact Nat.le_of_dvd (Nat.succ_pos m)
4
54.59815
2
1
11
1,070
import Mathlib.GroupTheory.Perm.Cycle.Basic #align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" open Equiv Function Finset variable {ι α β : Type*} namespace Equiv.Perm section Generation variable [Finite β] open Subgroup theorem closure...
Mathlib/GroupTheory/Perm/Closure.lean
111
122
theorem closure_prime_cycle_swap {σ τ : Perm α} (h0 : (Fintype.card α).Prime) (h1 : IsCycle σ) (h2 : σ.support = Finset.univ) (h3 : IsSwap τ) : closure ({σ, τ} : Set (Perm α)) = ⊤ := by
obtain ⟨x, y, h4, h5⟩ := h3 obtain ⟨i, hi⟩ := h1.exists_pow_eq (mem_support.mp ((Finset.ext_iff.mp h2 x).mpr (Finset.mem_univ x))) (mem_support.mp ((Finset.ext_iff.mp h2 y).mpr (Finset.mem_univ y))) rw [h5, ← hi] refine closure_cycle_coprime_swap (Nat.Coprime.symm (h0.coprime_iff_not_dvd.mpr fun ...
10
22,026.465795
2
2
4
2,314
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition import Mathlib.LinearAlgebra.FreeModule.Basic #align_import linear_algebra.free_module.pid from "leanprover-community/mathlib"@"d87199d51218d36a0a42c66c82d147b5a7ff87b3" universe u v section IsDomain variable {ι : Type*} {R : Type*} [CommRing R] [IsDoma...
Mathlib/LinearAlgebra/FreeModule/PID.lean
93
98
theorem dvd_generator_iff {I : Ideal R} [I.IsPrincipal] {x : R} (hx : x ∈ I) : x ∣ generator I ↔ I = Ideal.span {x} := by
conv_rhs => rw [← span_singleton_generator I] rw [Ideal.submodule_span_eq, Ideal.span_singleton_eq_span_singleton, ← dvd_dvd_iff_associated, ← mem_iff_generator_dvd] exact ⟨fun h ↦ ⟨hx, h⟩, fun h ↦ h.2⟩
4
54.59815
2
2
3
2,115
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic import Mathlib.NumberTheory.GaussSum #align_import number_theory.legendre_symbol.quadratic_char.gauss_sum from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9" section SpecialValues open ZMod MulChar variable {F : Type*} ...
Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean
130
143
theorem FiniteField.isSquare_odd_prime_iff (hF : ringChar F ≠ 2) {p : ℕ} [Fact p.Prime] (hp : p ≠ 2) : IsSquare (p : F) ↔ quadraticChar (ZMod p) (χ₄ (Fintype.card F) * Fintype.card F) ≠ -1 := by
classical by_cases hFp : ringChar F = p · rw [show (p : F) = 0 by rw [← hFp]; exact ringChar.Nat.cast_ringChar] simp only [isSquare_zero, Ne, true_iff_iff, map_mul] obtain ⟨n, _, hc⟩ := FiniteField.card F (ringChar F) have hchar : ringChar F = ringChar (ZMod p) := by rw [hFp]; exact (ringChar_zmod_n ...
11
59,874.141715
2
1.833333
6
1,917
import Mathlib.Analysis.LocallyConvex.Bounded import Mathlib.Topology.Algebra.Module.Multilinear.Basic open Bornology Filter Set Function open scoped Topology namespace Bornology.IsVonNBounded variable {ι 𝕜 F : Type*} {E : ι → Type*} [NormedField 𝕜] [∀ i, AddCommGroup (E i)] [∀ i, Module 𝕜 (E i)] [∀ i, Topol...
Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean
44
83
theorem image_multilinear' [Nonempty ι] {s : Set (∀ i, E i)} (hs : IsVonNBounded 𝕜 s) (f : ContinuousMultilinearMap 𝕜 E F) : IsVonNBounded 𝕜 (f '' s) := fun V hV ↦ by classical if h₁ : ∀ c : 𝕜, ‖c‖ ≤ 1 then exact absorbs_iff_norm.2 ⟨2, fun c hc ↦ by linarith [h₁ c]⟩ else let _ : NontriviallyNormed...
have hfV : f ⁻¹' V ∈ 𝓝 0 := (map_continuous f).tendsto' _ _ f.map_zero hV rwa [nhds_pi, Filter.mem_pi, exists_finite_iff_finset] at hfV have : ∀ i, ∃ c : 𝕜, c ≠ 0 ∧ ∀ c' : 𝕜, ‖c'‖ ≤ ‖c‖ → ∀ x ∈ s, c' • x i ∈ t i := fun i ↦ by rw [isVonNBounded_pi_iff] at hs have := (hs i).tendsto_smallSe...
31
29,048,849,665,247.426
2
2
2
2,153
import Mathlib.Logic.Function.Iterate import Mathlib.Order.Monotone.Basic #align_import order.iterate from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f" open Function open Function (Commute) namespace Monotone variable {α : Type*} [Preorder α] {f : α → α} {x y : ℕ → α} theorem se...
Mathlib/Order/Iterate.lean
51
60
theorem seq_pos_lt_seq_of_lt_of_le (hf : Monotone f) {n : ℕ} (hn : 0 < n) (h₀ : x 0 ≤ y 0) (hx : ∀ k < n, x (k + 1) < f (x k)) (hy : ∀ k < n, f (y k) ≤ y (k + 1)) : x n < y n := by
induction' n with n ihn · exact hn.false.elim suffices x n ≤ y n from (hx n n.lt_succ_self).trans_le ((hf this).trans <| hy n n.lt_succ_self) cases n with | zero => exact h₀ | succ n => refine (ihn n.zero_lt_succ (fun k hk => hx _ ?_) fun k hk => hy _ ?_).le <;> exact hk.trans n.succ.lt_succ_self
8
2,980.957987
2
1.666667
3
1,800
import Mathlib.Algebra.CharP.Invertible import Mathlib.Algebra.MvPolynomial.Variables import Mathlib.Algebra.MvPolynomial.CommRing import Mathlib.Algebra.MvPolynomial.Expand import Mathlib.Data.Fintype.BigOperators import Mathlib.Data.ZMod.Basic #align_import ring_theory.witt_vector.witt_polynomial from "leanprover-c...
Mathlib/RingTheory/WittVector/WittPolynomial.lean
170
181
theorem wittPolynomial_vars [CharZero R] (n : ℕ) : (wittPolynomial p R n).vars = range (n + 1) := by
have : ∀ i, (monomial (Finsupp.single i (p ^ (n - i))) ((p : R) ^ i)).vars = {i} := by intro i refine vars_monomial_single i (pow_ne_zero _ hp.1) ?_ rw [← Nat.cast_pow, Nat.cast_ne_zero] exact pow_ne_zero i hp.1 rw [wittPolynomial, vars_sum_of_disjoint] · simp only [this, biUnion_singleton_eq_sel...
11
59,874.141715
2
1.181818
11
1,247
import Mathlib.Algebra.MvPolynomial.Basic import Mathlib.Data.Fintype.Card import Mathlib.RingTheory.Algebraic #align_import field_theory.ax_grothendieck from "leanprover-community/mathlib"@"4e529b03dd62b7b7d13806c3fb974d9d4848910e" noncomputable section open MvPolynomial Finset Function
Mathlib/FieldTheory/AxGrothendieck.lean
33
66
theorem ax_grothendieck_of_locally_finite {ι K R : Type*} [Field K] [Finite K] [CommRing R] [Finite ι] [Algebra K R] [Algebra.IsAlgebraic K R] (ps : ι → MvPolynomial ι R) (hinj : Injective fun v i => MvPolynomial.eval v (ps i)) : Surjective fun v i => MvPolynomial.eval v (ps i) := by
classical intro v cases nonempty_fintype ι /- `s` is the set of all coefficients of the polynomial, as well as all of the coordinates of `v`, the point I am trying to find the preimage of. -/ let s : Finset R := (Finset.biUnion (univ : Finset ι) fun i => (ps i).support.image fun x => coef...
30
10,686,474,581,524.463
2
2
1
2,155
import Mathlib.Analysis.Fourier.Inversion open Real Complex Set MeasureTheory variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] open scoped FourierTransform private theorem rexp_neg_deriv_aux : ∀ x ∈ univ, HasDerivWithinAt (rexp ∘ Neg.neg) (-rexp (-x)) univ x := fun x _ ↦ mul_neg_one (rexp (-x)...
Mathlib/Analysis/MellinInversion.lean
89
121
theorem mellin_inversion (σ : ℝ) (f : ℝ → E) {x : ℝ} (hx : 0 < x) (hf : MellinConvergent f σ) (hFf : VerticalIntegrable (mellin f) σ) (hfx : ContinuousAt f x) : mellinInv σ (mellin f) x = f x := by
let g := fun (u : ℝ) => Real.exp (-σ * u) • f (Real.exp (-u)) replace hf : Integrable g := by rw [MellinConvergent, ← rexp_neg_image_aux, integrableOn_image_iff_integrableOn_abs_deriv_smul MeasurableSet.univ rexp_neg_deriv_aux rexp_neg_injOn_aux] at hf replace hf : Integrable fun (x : ℝ) ↦ cexp (-↑σ ...
30
10,686,474,581,524.463
2
2
3
1,966
import Mathlib.Data.List.Chain import Mathlib.Data.List.Enum import Mathlib.Data.List.Nodup import Mathlib.Data.List.Pairwise import Mathlib.Data.List.Zip #align_import data.list.range from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213" set_option autoImplicit true universe u open Nat...
Mathlib/Data/List/Range.lean
104
112
theorem chain'_range_succ (r : ℕ → ℕ → Prop) (n : ℕ) : Chain' r (range n.succ) ↔ ∀ m < n, r m m.succ := by
rw [range_succ] induction' n with n hn · simp · rw [range_succ] simp only [append_assoc, singleton_append, chain'_append_cons_cons, chain'_singleton, and_true_iff] rw [hn, forall_lt_succ]
7
1,096.633158
2
0.571429
7
516
import Mathlib.Topology.ExtendFrom import Mathlib.Topology.Order.DenselyOrdered #align_import topology.algebra.order.extend_from from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977" set_option autoImplicit true open Filter Set TopologicalSpace open scoped Classical open Topology
Mathlib/Topology/Order/ExtendFrom.lean
23
33
theorem continuousOn_Icc_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α] [OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {la lb : β} (hab : a ≠ b) (hf : ContinuousOn f (Ioo a b)) (ha : Tendsto f (𝓝[>] a) (𝓝 la)) (hb : Tendsto f (𝓝[<] b) (𝓝 lb)) : Cont...
apply continuousOn_extendFrom · rw [closure_Ioo hab] · intro x x_in rcases eq_endpoints_or_mem_Ioo_of_mem_Icc x_in with (rfl | rfl | h) · exact ⟨la, ha.mono_left <| nhdsWithin_mono _ Ioo_subset_Ioi_self⟩ · exact ⟨lb, hb.mono_left <| nhdsWithin_mono _ Ioo_subset_Iio_self⟩ · exact ⟨f x, hf x h⟩
7
1,096.633158
2
1.8
5
1,887
import Mathlib.Data.Finset.Fold import Mathlib.Algebra.GCDMonoid.Multiset #align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853" #align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d" variab...
Mathlib/Algebra/GCDMonoid/Finset.lean
77
82
theorem lcm_insert [DecidableEq β] {b : β} : (insert b s : Finset β).lcm f = GCDMonoid.lcm (f b) (s.lcm f) := by
by_cases h : b ∈ s · rw [insert_eq_of_mem h, (lcm_eq_right_iff (f b) (s.lcm f) (Multiset.normalize_lcm (s.1.map f))).2 (dvd_lcm h)] apply fold_insert h
4
54.59815
2
1
13
925
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv import Mathlib.Analysis.Calculus.FDeriv.Add import Mathlib.Analysis.Calculus.FDeriv.Prod import Mathlib.Analysis.NormedSpace.Complemented #align_import analysis.calculus.implicit from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a21598...
Mathlib/Analysis/Calculus/Implicit.lean
201
214
theorem implicitFunction_hasStrictFDerivAt (g'inv : G →L[𝕜] E) (hg'inv : φ.rightDeriv.comp g'inv = ContinuousLinearMap.id 𝕜 G) (hg'invf : φ.leftDeriv.comp g'inv = 0) : HasStrictFDerivAt (φ.implicitFunction (φ.leftFun φ.pt)) g'inv (φ.rightFun φ.pt) := by
have := φ.hasStrictFDerivAt.to_localInverse simp only [prodFun] at this convert this.comp (φ.rightFun φ.pt) ((hasStrictFDerivAt_const _ _).prod (hasStrictFDerivAt_id _)) -- Porting note: added parentheses to help `simp` simp only [ContinuousLinearMap.ext_iff, (ContinuousLinearMap.comp_apply)] at hg'inv hg'in...
10
22,026.465795
2
2
1
2,392
import Mathlib.RingTheory.FiniteType import Mathlib.RingTheory.Localization.AtPrime import Mathlib.RingTheory.Localization.Away.Basic import Mathlib.RingTheory.Localization.Integer import Mathlib.RingTheory.Localization.Submodule import Mathlib.RingTheory.Nilpotent.Lemmas import Mathlib.RingTheory.RingHomProperties im...
Mathlib/RingTheory/LocalProperties.lean
290
300
theorem eq_zero_of_localization (r : R) (h : ∀ (J : Ideal R) (hJ : J.IsMaximal), algebraMap R (Localization.AtPrime J) r = 0) : r = 0 := by
rw [← Ideal.span_singleton_eq_bot] apply ideal_eq_bot_of_localization intro J hJ delta IsLocalization.coeSubmodule erw [Submodule.map_span, Submodule.span_eq_bot] rintro _ ⟨_, h', rfl⟩ cases Set.mem_singleton_iff.mpr h' exact h J hJ
8
2,980.957987
2
1.833333
6
1,920
import Mathlib.Algebra.Group.Subgroup.Basic import Mathlib.Data.Fintype.Basic import Mathlib.Data.List.Sublists import Mathlib.Data.List.InsertNth #align_import group_theory.free_group from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6" open Relation universe u v w variable {α : Type u...
Mathlib/GroupTheory/FreeGroup/Basic.lean
151
155
theorem not_step_nil : ¬Step [] L := by
generalize h' : [] = L' intro h cases' h with L₁ L₂ simp [List.nil_eq_append] at h'
4
54.59815
2
1.333333
3
1,423
import Mathlib.Algebra.Algebra.Quasispectrum import Mathlib.FieldTheory.IsAlgClosed.Spectrum import Mathlib.Analysis.Complex.Liouville import Mathlib.Analysis.Complex.Polynomial import Mathlib.Analysis.Analytic.RadiusLiminf import Mathlib.Topology.Algebra.Module.CharacterSpace import Mathlib.Analysis.NormedSpace.Expon...
Mathlib/Analysis/NormedSpace/Spectrum.lean
104
113
theorem mem_resolventSet_of_norm_lt_mul {a : A} {k : 𝕜} (h : ‖a‖ * ‖(1 : A)‖ < ‖k‖) : k ∈ ρ a := by
rw [resolventSet, Set.mem_setOf_eq, Algebra.algebraMap_eq_smul_one] nontriviality A have hk : k ≠ 0 := ne_zero_of_norm_ne_zero ((mul_nonneg (norm_nonneg _) (norm_nonneg _)).trans_lt h).ne' letI ku := Units.map ↑ₐ.toMonoidHom (Units.mk0 k hk) rw [← inv_inv ‖(1 : A)‖, mul_inv_lt_iff (inv_pos.2 <| norm_...
9
8,103.083928
2
1
4
1,004
import Mathlib.Analysis.Calculus.FDeriv.Equiv import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn #align_import analysis.calculus.inverse from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" open Function Set Filter Metric open scoped Topology Classical NNReal n...
Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.lean
74
83
theorem approximates_deriv_on_nhds {f : E → F} {f' : E →L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f f' a) {c : ℝ≥0} (hc : Subsingleton E ∨ 0 < c) : ∃ s ∈ 𝓝 a, ApproximatesLinearOn f f' s c := by
cases' hc with hE hc · refine ⟨univ, IsOpen.mem_nhds isOpen_univ trivial, fun x _ y _ => ?_⟩ simp [@Subsingleton.elim E hE x y] have := hf.def hc rw [nhds_prod_eq, Filter.Eventually, mem_prod_same_iff] at this rcases this with ⟨s, has, hs⟩ exact ⟨s, has, fun x hx y hy => hs (mk_mem_prod hx hy)⟩
7
1,096.633158
2
2
3
2,275
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace suppress_compilation set_option linter.uppercaseLean3 false open Metric open scoped Classical NNReal Topology Uniformity variable {𝕜 E : Type*} [NontriviallyNormedField 𝕜] section SemiNormed variable [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] ...
Mathlib/Analysis/NormedSpace/OperatorNorm/Mul.lean
226
231
theorem norm_toSpanSingleton (x : E) : ‖toSpanSingleton 𝕜 x‖ = ‖x‖ := by
refine opNorm_eq_of_bounds (norm_nonneg _) (fun x => ?_) fun N _ h => ?_ · rw [toSpanSingleton_apply, norm_smul, mul_comm] · specialize h 1 rw [toSpanSingleton_apply, norm_smul, mul_comm] at h exact (mul_le_mul_right (by simp)).mp h
5
148.413159
2
1.5
2
1,587
import Mathlib.NumberTheory.LegendreSymbol.Basic import Mathlib.Analysis.Normed.Field.Basic #align_import number_theory.legendre_symbol.gauss_eisenstein_lemmas from "leanprover-community/mathlib"@"8818fdefc78642a7e6afcd20be5c184f3c7d9699" open Finset Nat open scoped Nat section GaussEisenstein namespace ZMod ...
Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean
30
60
theorem Ico_map_valMinAbs_natAbs_eq_Ico_map_id (p : ℕ) [hp : Fact p.Prime] (a : ZMod p) (hap : a ≠ 0) : ((Ico 1 (p / 2).succ).1.map fun (x : ℕ) => (a * x).valMinAbs.natAbs) = (Ico 1 (p / 2).succ).1.map fun a => a := by
have he : ∀ {x}, x ∈ Ico 1 (p / 2).succ → x ≠ 0 ∧ x ≤ p / 2 := by simp (config := { contextual := true }) [Nat.lt_succ_iff, Nat.succ_le_iff, pos_iff_ne_zero] have hep : ∀ {x}, x ∈ Ico 1 (p / 2).succ → x < p := fun hx => lt_of_le_of_lt (he hx).2 (Nat.div_lt_self hp.1.pos (by decide)) have hpe : ∀ {x}, x ∈...
28
1,446,257,064,291.475
2
2
1
2,438
import Mathlib.Algebra.Star.Basic import Mathlib.Algebra.Star.Pointwise import Mathlib.Algebra.Group.Centralizer variable {R : Type*} [Mul R] [StarMul R] {a : R} {s : Set R}
Mathlib/Algebra/Star/Center.lean
14
34
theorem Set.star_mem_center (ha : a ∈ Set.center R) : star a ∈ Set.center R where comm := by
simpa only [star_mul, star_star] using fun g => congr_arg star (((Set.mem_center_iff R).mp ha).comm <| star g).symm left_assoc b c := calc star a * (b * c) = star a * (star (star b) * star (star c)) := by rw [star_star, star_star] _ = star a * star (star c * star b) := by rw [star_mul] _ = star ((sta...
20
485,165,195.40979
2
2
1
2,109
import Mathlib.Algebra.Order.Monoid.Canonical.Defs import Mathlib.Data.List.Infix import Mathlib.Data.List.MinMax import Mathlib.Data.List.EditDistance.Defs set_option autoImplicit true variable {C : Levenshtein.Cost α β δ} [CanonicallyLinearOrderedAddCommMonoid δ] theorem suffixLevenshtein_minimum_le_levenshtein...
Mathlib/Data/List/EditDistance/Bounds.lean
81
87
theorem suffixLevenshtein_minimum_le_levenshtein_append (xs ys₁ ys₂) : (suffixLevenshtein C xs ys₂).1.minimum ≤ levenshtein C xs (ys₁ ++ ys₂) := by
cases ys₁ with | nil => exact List.minimum_le_of_mem' (List.get_mem _ _ _) | cons y ys₁ => exact (le_suffixLevenshtein_append_minimum _ _ _).trans (suffixLevenshtein_minimum_le_levenshtein_cons _ _ _)
5
148.413159
2
1.5
6
1,674
import Mathlib.Analysis.SpecialFunctions.Pow.Real import Mathlib.LinearAlgebra.FreeModule.PID import Mathlib.LinearAlgebra.Matrix.AbsoluteValue import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue import Mathlib.RingTheory.ClassGroup import Mathlib.RingTheory.DedekindDomain.IntegralClosure import Mathlib.Ri...
Mathlib/NumberTheory/ClassNumber/Finite.lean
58
71
theorem normBound_pos : 0 < normBound abv bS := by
obtain ⟨i, j, k, hijk⟩ : ∃ i j k, Algebra.leftMulMatrix bS (bS i) j k ≠ 0 := by by_contra! h obtain ⟨i⟩ := bS.index_nonempty apply bS.ne_zero i apply (injective_iff_map_eq_zero (Algebra.leftMulMatrix bS)).mp (Algebra.leftMulMatrix_injective bS) ext j k simp [h, DMatrix.zero_apply] sim...
13
442,413.392009
2
2
4
2,456
import Mathlib.Algebra.Group.Basic import Mathlib.Algebra.Order.Monoid.Canonical.Defs import Mathlib.Data.Set.Function import Mathlib.Order.Interval.Set.Basic #align_import data.set.intervals.monoid from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9" namespace Set variable {M : Type*} ...
Mathlib/Algebra/Order/Interval/Set/Monoid.lean
27
32
theorem Ici_add_bij : BijOn (· + d) (Ici a) (Ici (a + d)) := by
refine ⟨fun x h => add_le_add_right (mem_Ici.mp h) _, (add_left_injective d).injOn, fun _ h => ?_⟩ obtain ⟨c, rfl⟩ := exists_add_of_le (mem_Ici.mp h) rw [mem_Ici, add_right_comm, add_le_add_iff_right] at h exact ⟨a + c, h, by rw [add_right_comm]⟩
5
148.413159
2
1.090909
11
1,187
import Batteries.Data.List.Lemmas import Batteries.Data.Array.Basic import Batteries.Tactic.SeqFocus import Batteries.Util.ProofWanted namespace Array
.lake/packages/batteries/Batteries/Data/Array/Lemmas.lean
14
29
theorem forIn_eq_data_forIn [Monad m] (as : Array α) (b : β) (f : α → β → m (ForInStep β)) : forIn as b f = forIn as.data b f := by
let rec loop : ∀ {i h b j}, j + i = as.size → Array.forIn.loop as f i h b = forIn (as.data.drop j) b f | 0, _, _, _, rfl => by rw [List.drop_length]; rfl | i+1, _, _, j, ij => by simp only [forIn.loop, Nat.add] have j_eq : j = size as - 1 - i := by simp [← ij, ← Nat.add_assoc] have : ...
13
442,413.392009
2
1.5
6
1,680
import Mathlib.NumberTheory.Padics.PadicIntegers import Mathlib.RingTheory.ZMod #align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950" noncomputable section open scoped Classical open Nat LocalRing Padic namespace PadicInt variable {p : ℕ} [h...
Mathlib/NumberTheory/Padics/RingHoms.lean
586
597
theorem limNthHom_spec (r : R) : ∀ ε : ℝ, 0 < ε → ∃ N : ℕ, ∀ n ≥ N, ‖limNthHom f_compat r - nthHom f r n‖ < ε := by
intro ε hε obtain ⟨ε', hε'0, hε'⟩ : ∃ v : ℚ, (0 : ℝ) < v ∧ ↑v < ε := exists_rat_btwn hε norm_cast at hε'0 obtain ⟨N, hN⟩ := padicNormE.defn (nthHomSeq f_compat r) hε'0 use N intro n hn apply _root_.lt_trans _ hε' change (padicNormE _ : ℝ) < _ norm_cast exact hN _ hn
10
22,026.465795
2
1.833333
12
1,916
import Mathlib.MeasureTheory.MeasurableSpace.Basic import Mathlib.Data.Set.MemPartition import Mathlib.Order.Filter.CountableSeparatingOn open Set MeasureTheory namespace MeasurableSpace variable {α β : Type*} class CountablyGenerated (α : Type*) [m : MeasurableSpace α] : Prop where isCountablyGenerated : ∃ b...
Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean
96
101
theorem CountablyGenerated.comap [m : MeasurableSpace β] [h : CountablyGenerated β] (f : α → β) : @CountablyGenerated α (.comap f m) := by
rcases h with ⟨⟨b, hbc, rfl⟩⟩ rw [comap_generateFrom] letI := generateFrom (preimage f '' b) exact ⟨_, hbc.image _, rfl⟩
4
54.59815
2
1.5
4
1,686
import Mathlib.Analysis.Calculus.ContDiff.Basic import Mathlib.Analysis.Calculus.UniformLimitsDeriv import Mathlib.Topology.Algebra.InfiniteSum.Module import Mathlib.Analysis.NormedSpace.FunctionSeries #align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982...
Mathlib/Analysis/Calculus/SmoothSeries.lean
72
84
theorem hasFDerivAt_tsum_of_isPreconnected (hu : Summable u) (hs : IsOpen s) (h's : IsPreconnected s) (hf : ∀ n x, x ∈ s → HasFDerivAt (f n) (f' n x) x) (hf' : ∀ n x, x ∈ s → ‖f' n x‖ ≤ u n) (hx₀ : x₀ ∈ s) (hf0 : Summable fun n => f n x₀) (hx : x ∈ s) : HasFDerivAt (fun y => ∑' n, f n y) (∑' n, f' n x) x :=...
classical have A : ∀ x : E, x ∈ s → Tendsto (fun t : Finset α => ∑ n ∈ t, f n x) atTop (𝓝 (∑' n, f n x)) := by intro y hy apply Summable.hasSum exact summable_of_summable_hasFDerivAt_of_isPreconnected hu hs h's hf hf' hx₀ hf0 hy refine hasFDerivAt_of_tendstoUniformlyOn hs (tendstoUni...
9
8,103.083928
2
1.6
5
1,721
import Mathlib.Algebra.GeomSum import Mathlib.Algebra.Polynomial.Roots import Mathlib.GroupTheory.SpecificGroups.Cyclic #align_import ring_theory.integral_domain from "leanprover-community/mathlib"@"6e70e0d419bf686784937d64ed4bfde866ff229e" section open Finset Polynomial Function Nat variable {R : Type*} {G : Ty...
Mathlib/RingTheory/IntegralDomain.lean
200
254
theorem sum_hom_units_eq_zero (f : G →* R) (hf : f ≠ 1) : ∑ g : G, f g = 0 := by
classical obtain ⟨x, hx⟩ : ∃ x : MonoidHom.range f.toHomUnits, ∀ y : MonoidHom.range f.toHomUnits, y ∈ Submonoid.powers x := IsCyclic.exists_monoid_generator have hx1 : x ≠ 1 := by rintro rfl apply hf ext g rw [MonoidHom.one_apply] cases' hx ⟨f.toHomUnits g, g, rfl...
53
104,137,594,330,290,870,000,000
2
2
6
2,481
import Mathlib.Algebra.Module.Card import Mathlib.SetTheory.Cardinal.CountableCover import Mathlib.SetTheory.Cardinal.Continuum import Mathlib.Analysis.SpecificLimits.Normed import Mathlib.Topology.MetricSpace.Perfect universe u v open Filter Pointwise Set Function Cardinal open scoped Cardinal Topology
Mathlib/Topology/Algebra/Module/Cardinality.lean
29
45
theorem continuum_le_cardinal_of_nontriviallyNormedField (𝕜 : Type*) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] : 𝔠 ≤ #𝕜 := by
suffices ∃ f : (ℕ → Bool) → 𝕜, range f ⊆ univ ∧ Continuous f ∧ Injective f by rcases this with ⟨f, -, -, f_inj⟩ simpa using lift_mk_le_lift_mk_of_injective f_inj apply Perfect.exists_nat_bool_injection _ univ_nonempty refine ⟨isClosed_univ, preperfect_iff_nhds.2 (fun x _ U hU ↦ ?_)⟩ rcases NormedField...
15
3,269,017.372472
2
1.2
5
1,284
import Mathlib.Algebra.BigOperators.WithTop import Mathlib.Algebra.GroupWithZero.Divisibility import Mathlib.Data.ENNReal.Basic #align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520" open Set NNReal ENNReal namespace ENNReal variable {a b c d : ℝ≥0∞} {r p q...
Mathlib/Data/ENNReal/Operations.lean
33
41
theorem mul_lt_mul (ac : a < c) (bd : b < d) : a * b < c * d := by
rcases lt_iff_exists_nnreal_btwn.1 ac with ⟨a', aa', a'c⟩ lift a to ℝ≥0 using ne_top_of_lt aa' rcases lt_iff_exists_nnreal_btwn.1 bd with ⟨b', bb', b'd⟩ lift b to ℝ≥0 using ne_top_of_lt bb' norm_cast at * calc ↑(a * b) < ↑(a' * b') := coe_lt_coe.2 (mul_lt_mul₀ aa' bb') _ ≤ c * d := mul_le_mul' a'c....
8
2,980.957987
2
0.666667
12
570
import Mathlib.Topology.Sheaves.Presheaf import Mathlib.Topology.Sheaves.Stalks import Mathlib.CategoryTheory.Limits.Preserves.Filtered import Mathlib.CategoryTheory.Sites.LocallySurjective #align_import topology.sheaves.locally_surjective from "leanprover-community/mathlib"@"fb7698eb37544cbb66292b68b40e54d001f8d1a9"...
Mathlib/Topology/Sheaves/LocallySurjective.lean
78
118
theorem locally_surjective_iff_surjective_on_stalks (T : ℱ ⟶ 𝒢) : IsLocallySurjective T ↔ ∀ x : X, Function.Surjective ((stalkFunctor C x).map T) := by
constructor <;> intro hT · /- human proof: Let g ∈ Γₛₜ 𝒢 x be a germ. Represent it on an open set U ⊆ X as ⟨t, U⟩. By local surjectivity, pass to a smaller open set V on which there exists s ∈ Γ_ ℱ V mapping to t |_ V. Then the germ of s maps to g -/ -- Let g ∈ Γₛₜ 𝒢 x be a ge...
39
86,593,400,423,993,740
2
2
1
2,242
import Mathlib.Data.List.Lattice import Mathlib.Data.List.Range import Mathlib.Data.Bool.Basic #align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213" open Nat namespace List def Ico (n m : ℕ) : List ℕ := range' n (m - n) #align list.Ico List.Ico names...
Mathlib/Data/List/Intervals.lean
95
100
theorem append_consecutive {n m l : ℕ} (hnm : n ≤ m) (hml : m ≤ l) : Ico n m ++ Ico m l = Ico n l := by
dsimp only [Ico] convert range'_append n (m-n) (l-m) 1 using 2 · rw [Nat.one_mul, Nat.add_sub_cancel' hnm] · rw [Nat.sub_add_sub_cancel hml hnm]
4
54.59815
2
0.9375
16
794
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset import Mathlib.Data.Finite.Card import Mathlib.GroupTheory.Finiteness import Mathlib.GroupTheory.GroupAction.Quotient #align_import group_theory.index from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988" namespace Subgroup open Ca...
Mathlib/GroupTheory/Index.lean
62
76
theorem index_comap_of_surjective {G' : Type*} [Group G'] {f : G' →* G} (hf : Function.Surjective f) : (H.comap f).index = H.index := by
letI := QuotientGroup.leftRel H letI := QuotientGroup.leftRel (H.comap f) have key : ∀ x y : G', Setoid.r x y ↔ Setoid.r (f x) (f y) := by simp only [QuotientGroup.leftRel_apply] exact fun x y => iff_of_eq (congr_arg (· ∈ H) (by rw [f.map_mul, f.map_inv])) refine Cardinal.toNat_congr (Equiv.ofBijective...
13
442,413.392009
2
0.666667
6
611
import Mathlib.Algebra.Algebra.Subalgebra.Directed import Mathlib.FieldTheory.IntermediateField import Mathlib.FieldTheory.Separable import Mathlib.FieldTheory.SplittingField.IsSplittingField import Mathlib.RingTheory.TensorProduct.Basic #align_import field_theory.adjoin from "leanprover-community/mathlib"@"df76f4335...
Mathlib/FieldTheory/Adjoin.lean
62
67
theorem mem_adjoin_simple_iff {α : E} (x : E) : x ∈ adjoin F {α} ↔ ∃ r s : F[X], x = aeval α r / aeval α s := by
simp only [adjoin, mem_mk, Subring.mem_toSubsemiring, Subfield.mem_toSubring, Subfield.mem_closure_iff, ← Algebra.adjoin_eq_ring_closure, Subalgebra.mem_toSubring, Algebra.adjoin_singleton_eq_range_aeval, AlgHom.mem_range, exists_exists_eq_and] tauto
4
54.59815
2
2
2
1,932
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix import Mathlib.Topology.Algebra.Module.Simple import Mathlib.Topology.Algebra.Module.Determinant import Mathlib.RingTheory.Ideal.LocalRing #align_import topology.algebra.module.finite_dimension from "leanprove...
Mathlib/Topology/Algebra/Module/FiniteDimension.lean
77
127
theorem unique_topology_of_t2 {t : TopologicalSpace 𝕜} (h₁ : @TopologicalAddGroup 𝕜 t _) (h₂ : @ContinuousSMul 𝕜 𝕜 _ hnorm.toUniformSpace.toTopologicalSpace t) (h₃ : @T2Space 𝕜 t) : t = hnorm.toUniformSpace.toTopologicalSpace := by
-- Let `𝓣₀` denote the topology on `𝕜` induced by the norm, and `𝓣` be any T2 vector -- topology on `𝕜`. To show that `𝓣₀ = 𝓣`, it suffices to show that they have the same -- neighborhoods of 0. refine TopologicalAddGroup.ext h₁ inferInstance (le_antisymm ?_ ?_) · -- To show `𝓣 ≤ 𝓣₀`, we have to show...
48
701,673,591,209,763,100,000
2
2
2
2,137
import Mathlib.Algebra.MonoidAlgebra.Support import Mathlib.Algebra.Polynomial.Basic import Mathlib.Algebra.Regular.Basic import Mathlib.Data.Nat.Choose.Sum #align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c" set_option linter.uppercaseLean3 false no...
Mathlib/Algebra/Polynomial/Coeff.lean
60
65
theorem support_smul [SMulZeroClass S R] (r : S) (p : R[X]) : support (r • p) ⊆ support p := by
intro i hi simp? [mem_support_iff] at hi ⊢ says simp only [mem_support_iff, coeff_smul, ne_eq] at hi ⊢ contrapose! hi simp [hi]
4
54.59815
2
1.125
8
1,203
import Mathlib.Algebra.BigOperators.NatAntidiagonal import Mathlib.Algebra.Polynomial.RingDivision #align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2" namespace Polynomial open Polynomial section Semiring variable {R : Type*} [Semiring R] (p q : R...
Mathlib/Algebra/Polynomial/Mirror.lean
66
72
theorem mirror_natDegree : p.mirror.natDegree = p.natDegree := by
by_cases hp : p = 0 · rw [hp, mirror_zero] nontriviality R rw [mirror, natDegree_mul', reverse_natDegree, natDegree_X_pow, tsub_add_cancel_of_le p.natTrailingDegree_le_natDegree] rwa [leadingCoeff_X_pow, mul_one, reverse_leadingCoeff, Ne, trailingCoeff_eq_zero]
6
403.428793
2
1.571429
7
1,704
import Mathlib.MeasureTheory.SetSemiring open MeasurableSpace Set namespace MeasureTheory variable {α : Type*} {𝒜 : Set (Set α)} {s t : Set α} structure IsSetAlgebra (𝒜 : Set (Set α)) : Prop where empty_mem : ∅ ∈ 𝒜 compl_mem : ∀ ⦃s⦄, s ∈ 𝒜 → sᶜ ∈ 𝒜 union_mem : ∀ ⦃s t⦄, s ∈ 𝒜 → t ∈ 𝒜 → s ∪ t ∈ 𝒜 ...
Mathlib/MeasureTheory/SetAlgebra.lean
122
134
theorem generateFrom_generateSetAlgebra_eq : generateFrom (generateSetAlgebra 𝒜) = generateFrom 𝒜 := by
refine le_antisymm (fun s ms ↦ ?_) (generateFrom_mono self_subset_generateSetAlgebra) refine @generateFrom_induction _ _ (generateSetAlgebra 𝒜) (fun t ht ↦ ?_) (@MeasurableSet.empty _ (generateFrom 𝒜)) (fun t ↦ MeasurableSet.compl) (fun f hf ↦ MeasurableSet.iUnion hf) s ms induction ht with |...
11
59,874.141715
2
2
4
1,933
import Mathlib.Data.Finset.Image import Mathlib.Data.Multiset.Pi #align_import data.finset.pi from "leanprover-community/mathlib"@"b2c89893177f66a48daf993b7ba5ef7cddeff8c9" namespace Finset open Multiset section Pi variable {α : Type*} def Pi.empty (β : α → Sort*) (a : α) (h : a ∈ (∅ : Finset α)) : β a :=...
Mathlib/Data/Finset/Pi.lean
96
112
theorem pi_insert [∀ a, DecidableEq (β a)] {s : Finset α} {t : ∀ a : α, Finset (β a)} {a : α} (ha : a ∉ s) : pi (insert a s) t = (t a).biUnion fun b => (pi s t).image (Pi.cons s a b) := by
apply eq_of_veq rw [← (pi (insert a s) t).2.dedup] refine (fun s' (h : s' = a ::ₘ s.1) => (?_ : dedup (Multiset.pi s' fun a => (t a).1) = dedup ((t a).1.bind fun b => dedup <| (Multiset.pi s.1 fun a : α => (t a).val).map fun f a' h...
15
3,269,017.372472
2
1.666667
3
1,804
import Mathlib.SetTheory.Ordinal.Arithmetic namespace OrdinalApprox universe u variable {α : Type u} variable [CompleteLattice α] (f : α →o α) (x : α) open Function fixedPoints Cardinal Order OrderHom set_option linter.unusedVariables false in def lfpApprox (a : Ordinal.{u}) : α := sSup ({ f (lfpApprox b) | ...
Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean
92
112
theorem lfpApprox_add_one (h : x ≤ f x) (a : Ordinal) : lfpApprox f x (a+1) = f (lfpApprox f x a) := by
apply le_antisymm · conv => left; unfold lfpApprox apply sSup_le simp only [Ordinal.add_one_eq_succ, lt_succ_iff, exists_prop, Set.union_singleton, Set.mem_insert_iff, Set.mem_setOf_eq, forall_eq_or_imp, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂] apply And.intro · apply le_...
19
178,482,300.963187
2
1.8
5
1,896
import Mathlib.RingTheory.Ideal.Cotangent import Mathlib.RingTheory.QuotientNilpotent import Mathlib.RingTheory.TensorProduct.Basic import Mathlib.RingTheory.FinitePresentation import Mathlib.RingTheory.Localization.Away.Basic import Mathlib.RingTheory.Localization.Away.AdjoinRoot #align_import ring_theory.etale from ...
Mathlib/RingTheory/Smooth/Basic.lean
121
131
theorem liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) : g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644 erw [← FormallySmooth.mk_lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom...
8
2,980.957987
2
2
4
2,341
import Mathlib.MeasureTheory.Covering.DensityTheorem import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar #align_import measure_theory.covering.one_dim from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844" open Set MeasureTheory IsUnifLocDoublingMeasure Filter open scoped Topology names...
Mathlib/MeasureTheory/Covering/OneDim.lean
51
59
theorem tendsto_Icc_vitaliFamily_left (x : ℝ) : Tendsto (fun y => Icc y x) (𝓝[<] x) ((vitaliFamily (volume : Measure ℝ) 1).filterAt x) := by
refine (VitaliFamily.tendsto_filterAt_iff _).2 ⟨?_, ?_⟩ · filter_upwards [self_mem_nhdsWithin] with y hy using Icc_mem_vitaliFamily_at_left hy · intro ε εpos have : x ∈ Ioc (x - ε) x := ⟨by linarith, le_refl _⟩ filter_upwards [Icc_mem_nhdsWithin_Iio this] with y hy rw [closedBall_eq_Icc] exact Ic...
7
1,096.633158
2
1.5
4
1,590
import Mathlib.Geometry.Manifold.MFDeriv.Atlas noncomputable section open scoped Manifold open Set section UniqueMDiff variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type*} [Topolog...
Mathlib/Geometry/Manifold/MFDeriv/UniqueDifferential.lean
98
107
theorem UniqueMDiffOn.uniqueDiffOn_inter_preimage (hs : UniqueMDiffOn I s) (x : M) (y : M') {f : M → M'} (hf : ContinuousOn f s) : UniqueDiffOn 𝕜 ((extChartAt I x).target ∩ (extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source)) := haveI : UniqueMDiffOn I (s ∩ f ⁻¹' (extChartAt I' y).source) :=...
intro z hz apply (hs z hz.1).inter' apply (hf z hz.1).preimage_mem_nhdsWithin exact (isOpen_extChartAt_source I' y).mem_nhds hz.2 this.uniqueDiffOn_target_inter _
5
148.413159
2
2
4
2,009
import Mathlib.MeasureTheory.Integral.SetIntegral #align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520" open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function open scoped Topology ENNReal Convex variable...
Mathlib/MeasureTheory/Integral/Average.lean
158
162
theorem laverage_lt_top (hf : ∫⁻ x, f x ∂μ ≠ ∞) : ⨍⁻ x, f x ∂μ < ∞ := by
obtain rfl | hμ := eq_or_ne μ 0 · simp · rw [laverage_eq] exact div_lt_top hf (measure_univ_ne_zero.2 hμ)
4
54.59815
2
0.347826
23
374
import Mathlib.Analysis.SpecialFunctions.ExpDeriv #align_import analysis.ODE.gronwall from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics Fil...
Mathlib/Analysis/ODE/Gronwall.lean
113
132
theorem le_gronwallBound_of_liminf_deriv_right_le {f f' : ℝ → ℝ} {δ K ε : ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, (z - x)⁻¹ * (f z - f x) < r) (ha : f a ≤ δ) (bound : ∀ x ∈ Ico a b, f' x ≤ K * f x + ε) : ∀ x ∈ Icc a b, f x ≤ gronwallBound δ K ε...
have H : ∀ x ∈ Icc a b, ∀ ε' ∈ Ioi ε, f x ≤ gronwallBound δ K ε' (x - a) := by intro x hx ε' hε' apply image_le_of_liminf_slope_right_lt_deriv_boundary hf hf' · rwa [sub_self, gronwallBound_x0] · exact fun x => hasDerivAt_gronwallBound_shift δ K ε' x a · intro x hx hfB rw [← hfB] appl...
15
3,269,017.372472
2
1.428571
7
1,514
import Mathlib.MeasureTheory.Integral.FundThmCalculus import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv import Mathlib.Analysis.SpecialFunctions.NonIntegrable import Mathlib.Analysis.SpecialFunctions.Pow.Deriv #align_import analysis.special_functions.integrals from "leanprover-community/mathlib"@"011...
Mathlib/Analysis/SpecialFunctions/Integrals.lean
120
164
theorem intervalIntegrable_cpow {r : ℂ} (h : 0 ≤ r.re ∨ (0 : ℝ) ∉ [[a, b]]) : IntervalIntegrable (fun x : ℝ => (x : ℂ) ^ r) μ a b := by
by_cases h2 : (0 : ℝ) ∉ [[a, b]] · -- Easy case #1: 0 ∉ [a, b] -- use continuity. refine (ContinuousAt.continuousOn fun x hx => ?_).intervalIntegrable exact Complex.continuousAt_ofReal_cpow_const _ _ (Or.inr <| ne_of_mem_of_not_mem hx h2) rw [eq_false h2, or_false_iff] at h rcases lt_or_eq_of_le h with...
43
4,727,839,468,229,346,000
2
2
2
2,258
import Mathlib.Algebra.Order.Ring.Abs import Mathlib.Data.Set.Pointwise.Iterate import Mathlib.Dynamics.Ergodic.Ergodic import Mathlib.MeasureTheory.Covering.DensityTheorem import Mathlib.MeasureTheory.Group.AddCircle import Mathlib.MeasureTheory.Measure.Haar.Unique #align_import dynamics.ergodic.add_circle from "lea...
Mathlib/Dynamics/Ergodic/AddCircle.lean
45
101
theorem ae_empty_or_univ_of_forall_vadd_ae_eq_self {s : Set <| AddCircle T} (hs : NullMeasurableSet s volume) {ι : Type*} {l : Filter ι} [l.NeBot] {u : ι → AddCircle T} (hu₁ : ∀ i, (u i +ᵥ s : Set _) =ᵐ[volume] s) (hu₂ : Tendsto (addOrderOf ∘ u) l atTop) : s =ᵐ[volume] (∅ : Set <| AddCircle T) ∨ s =ᵐ[volume...
/- Sketch of proof: Assume `T = 1` for simplicity and let `μ` be the Haar measure. We may assume `s` has positive measure since otherwise there is nothing to prove. In this case, by Lebesgue's density theorem, there exists a point `d` of positive density. Let `Iⱼ` be the sequence of closed balls about `d...
53
104,137,594,330,290,870,000,000
2
2
2
2,345
import Mathlib.Algebra.Field.Basic import Mathlib.Algebra.Order.Group.Basic import Mathlib.Algebra.Order.Ring.Basic import Mathlib.RingTheory.Int.Basic import Mathlib.Tactic.Ring import Mathlib.Tactic.FieldSimp import Mathlib.Data.Int.NatPrime import Mathlib.Data.ZMod.Basic #align_import number_theory.pythagorean_tri...
Mathlib/NumberTheory/PythagoreanTriples.lean
87
93
theorem mul_iff (k : ℤ) (hk : k ≠ 0) : PythagoreanTriple (k * x) (k * y) (k * z) ↔ PythagoreanTriple x y z := by
refine ⟨?_, fun h => h.mul k⟩ simp only [PythagoreanTriple] intro h rw [← mul_left_inj' (mul_ne_zero hk hk)] convert h using 1 <;> ring
5
148.413159
2
1.2
10
1,251
import Mathlib.CategoryTheory.EffectiveEpi.Basic import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks import Mathlib.Tactic.ApplyFun namespace CategoryTheory open Limits variable {C : Type*} [Category C] noncomputable def effectiveEpiStructIsColimitDescOfEffectiveEpiFamily {B : C} {α : Type*} (X : α → C) (c...
Mathlib/CategoryTheory/EffectiveEpi/Coproduct.lean
61
93
theorem effectiveEpiFamilyStructOfEffectiveEpiDesc_aux {B : C} {α : Type*} {X : α → C} {π : (a : α) → X a ⟶ B} [HasCoproduct X] [∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)] [∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a ↦ pullback g (Sigma.ι X a)] [∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.des...
apply_fun ((Sigma.desc fun a ↦ pullback.fst (f := g₁) (g := (Sigma.ι X a))) ≫ ·) using (fun a b ↦ (cancel_epi _).mp) ext a simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_pt, Cofan.mk_ι_app] rw [← Category.assoc, pullback.condition] simp only [Category.assoc, colimit.ι_desc, Cofan.mk_pt, ...
24
26,489,122,129.84347
2
2
1
1,990
import Mathlib.RingTheory.Adjoin.FG #align_import ring_theory.adjoin.tower from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a" open Pointwise universe u v w u₁ variable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) namespace Algebra
Mathlib/RingTheory/Adjoin/Tower.lean
30
46
theorem adjoin_restrictScalars (C D E : Type*) [CommSemiring C] [CommSemiring D] [CommSemiring E] [Algebra C D] [Algebra C E] [Algebra D E] [IsScalarTower C D E] (S : Set E) : (Algebra.adjoin D S).restrictScalars C = (Algebra.adjoin ((⊤ : Subalgebra C D).map (IsScalarTower.toAlgHom C D E)) S).restrictScal...
suffices Set.range (algebraMap D E) = Set.range (algebraMap ((⊤ : Subalgebra C D).map (IsScalarTower.toAlgHom C D E)) E) by ext x change x ∈ Subsemiring.closure (_ ∪ S) ↔ x ∈ Subsemiring.closure (_ ∪ S) rw [this] ext x constructor · rintro ⟨y, hy⟩ exact ⟨⟨algebraMap D E y, ⟨y, ⟨Algebr...
12
162,754.791419
2
2
3
2,229
import Mathlib.Algebra.Group.Indicator import Mathlib.Algebra.Module.Defs import Mathlib.Algebra.Order.Field.Rat import Mathlib.GroupTheory.GroupAction.Group import Mathlib.GroupTheory.GroupAction.Pi #align_import algebra.module.basic from "leanprover-community/mathlib"@"30413fc89f202a090a54d78e540963ed3de0056e" o...
Mathlib/Algebra/Module/Basic.lean
49
55
theorem map_inv_intCast_smul [AddCommGroup M] [AddCommGroup M₂] {F : Type*} [FunLike F M M₂] [AddMonoidHomClass F M M₂] (f : F) (R S : Type*) [DivisionRing R] [DivisionRing S] [Module R M] [Module S M₂] (z : ℤ) (x : M) : f ((z⁻¹ : R) • x) = (z⁻¹ : S) • f x := by
obtain ⟨n, rfl | rfl⟩ := z.eq_nat_or_neg · rw [Int.cast_natCast, Int.cast_natCast, map_inv_natCast_smul _ R S] · simp_rw [Int.cast_neg, Int.cast_natCast, inv_neg, neg_smul, map_neg, map_inv_natCast_smul _ R S]
4
54.59815
2
1.666667
3
1,767
import Mathlib.NumberTheory.Padics.PadicIntegers import Mathlib.RingTheory.ZMod #align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950" noncomputable section open scoped Classical open Nat LocalRing Padic namespace PadicInt variable {p : ℕ} [h...
Mathlib/NumberTheory/Padics/RingHoms.lean
514
525
theorem isCauSeq_nthHom (r : R) : IsCauSeq (padicNorm p) fun n => nthHom f r n := by
intro ε hε obtain ⟨k, hk⟩ : ∃ k : ℕ, (p : ℚ) ^ (-((k : ℕ) : ℤ)) < ε := exists_pow_neg_lt_rat p hε use k intro j hj refine lt_of_le_of_lt ?_ hk -- Need to do beta reduction first, as `norm_cast` doesn't. -- Added to adapt to leanprover/lean4#2734. beta_reduce norm_cast rw [← padicNorm.dvd_iff_norm_l...
11
59,874.141715
2
1.833333
12
1,916
import Mathlib.Data.Finset.Basic variable {ι : Sort _} {π : ι → Sort _} {x : ∀ i, π i} [DecidableEq ι] namespace Function def updateFinset (x : ∀ i, π i) (s : Finset ι) (y : ∀ i : ↥s, π i) (i : ι) : π i := if hi : i ∈ s then y ⟨i, hi⟩ else x i open Finset Equiv theorem updateFinset_def {s : Finset ι} {y} : ...
Mathlib/Data/Finset/Update.lean
35
41
theorem updateFinset_singleton {i y} : updateFinset x {i} y = Function.update x i (y ⟨i, mem_singleton_self i⟩) := by
congr with j by_cases hj : j = i · cases hj simp only [dif_pos, Finset.mem_singleton, update_same, updateFinset] · simp [hj, updateFinset]
5
148.413159
2
2
3
2,441
import Mathlib.RingTheory.DiscreteValuationRing.Basic import Mathlib.RingTheory.MvPowerSeries.Inverse import Mathlib.RingTheory.PowerSeries.Basic import Mathlib.RingTheory.PowerSeries.Order #align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60" ...
Mathlib/RingTheory/PowerSeries/Inverse.lean
54
81
theorem coeff_inv_aux (n : ℕ) (a : R) (φ : R⟦X⟧) : coeff R n (inv.aux a φ) = if n = 0 then a else -a * ∑ x ∈ antidiagonal n, if x.2 < n then coeff R x.1 φ * coeff R x.2 (inv.aux a φ) else 0 := by
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644 erw [coeff, inv.aux, MvPowerSeries.coeff_inv_aux] simp only [Finsupp.single_eq_zero] split_ifs; · rfl congr 1 symm apply Finset.sum_nbij' (fun (a, b) ↦ (single () a, single () b)) fun (f, g) ↦ (f (), g ()) · aesop · aesop · ...
21
1,318,815,734.483215
2
1
2
1,105
import Mathlib.Algebra.Group.Subgroup.Basic import Mathlib.Data.Fintype.Card import Mathlib.GroupTheory.Nilpotent import Mathlib.Order.Radical def frattini (G : Type*) [Group G] : Subgroup G := Order.radical (Subgroup G) variable {G H : Type*} [Group G] [Group H] {φ : G →* H} (hφ : Function.Surjective φ) lemma...
Mathlib/GroupTheory/Frattini.lean
59
74
theorem frattini_nilpotent [Finite G] : Group.IsNilpotent (frattini G) := by
-- We use the characterisation of nilpotency in terms of all Sylow subgroups being normal. have q := (isNilpotent_of_finite_tfae (G := frattini G)).out 0 3 rw [q]; clear q -- Consider each prime `p` and Sylow `p`-subgroup `P` of `frattini G`. intro p p_prime P -- The Frattini argument shows that the normal...
15
3,269,017.372472
2
2
1
2,381
import Mathlib.Algebra.Order.Group.Abs import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax #align_import algebra.order.group.min_max from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1" section variable {α : Type*} [Group α] [LinearOrder α] [CovariantClass α α (· * ·) (· ≤ ·)] -- TODO...
Mathlib/Algebra/Order/Group/MinMax.lean
96
100
theorem abs_max_sub_max_le_max (a b c d : α) : |max a b - max c d| ≤ max |a - c| |b - d| := by
refine abs_sub_le_iff.2 ⟨?_, ?_⟩ · exact (max_sub_max_le_max _ _ _ _).trans (max_le_max (le_abs_self _) (le_abs_self _)) · rw [abs_sub_comm a c, abs_sub_comm b d] exact (max_sub_max_le_max _ _ _ _).trans (max_le_max (le_abs_self _) (le_abs_self _))
4
54.59815
2
0.571429
7
524
import Mathlib.Algebra.Homology.ExactSequence import Mathlib.CategoryTheory.Abelian.Refinements #align_import category_theory.abelian.diagram_lemmas.four from "leanprover-community/mathlib"@"d34cbcf6c94953e965448c933cd9cc485115ebbd" namespace CategoryTheory open Category Limits Preadditive namespace Abelian va...
Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean
95
120
theorem epi_of_epi_of_epi_of_mono' (hR₁ : (mk₂ (R₁.map' 1 2) (R₁.map' 2 3)).Exact) (hR₂ : (mk₂ (R₂.map' 0 1) (R₂.map' 1 2)).Exact) (hR₂' : R₂.map' 1 3 = 0) (h₀ : Epi (app' φ 0)) (h₂ : Epi (app' φ 2)) (h₃ : Mono (app' φ 3)) : Epi (app' φ 1) := by
rw [epi_iff_surjective_up_to_refinements] intro A g₁ obtain ⟨A₁, π₁, _, f₂, h₁⟩ := surjective_up_to_refinements_of_epi (app' φ 2 _) (g₁ ≫ R₂.map' 1 2) have h₂ : f₂ ≫ R₁.map' 2 3 = 0 := by rw [← cancel_mono (app' φ 3 _), assoc, zero_comp, NatTrans.naturality, ← reassoc_of% h₁, ← R₂.map'_comp 1 2 3...
21
1,318,815,734.483215
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import Mathlib.Data.ZMod.Quotient #align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f" open Set open scoped Pointwise namespace Subgroup variable {G : Type*} [Group G] (H K : Subgroup G) (S T : Set G) @[to_additive "`S` and `T` are complements if ...
Mathlib/GroupTheory/Complement.lean
133
139
theorem isComplement_singleton_right {g : G} : IsComplement S {g} ↔ S = univ := by
refine ⟨fun h => top_le_iff.mp fun x _ => ?_, fun h => h ▸ isComplement_univ_singleton⟩ obtain ⟨y, hy⟩ := h.2 (x * g) conv_rhs at hy => rw [← show y.2.1 = g from y.2.2] rw [← mul_right_cancel hy] exact y.1.2
6
403.428793
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import Mathlib.Algebra.Algebra.Operations import Mathlib.Data.Fintype.Lattice import Mathlib.RingTheory.Coprime.Lemmas #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" assert_not_exists Basis -- See `RingTheory.Ideal.Basis` assert_not_exists ...
Mathlib/RingTheory/Ideal/Operations.lean
82
96
theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by
rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx ...
13
442,413.392009
2
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