Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
|---|---|---|---|---|---|---|---|---|---|---|---|
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical
open NNReal ENNReal MeasureTheory
namespace Measur... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 146 | 178 | theorem hasSum_of_disjoint_iUnion [Countable β] {f : β → Set α} (hf₁ : ∀ i, MeasurableSet (f i))
(hf₂ : Pairwise (Disjoint on f)) : HasSum (fun i => v (f i)) (v (⋃ i, f i)) := by |
cases nonempty_encodable β
set g := fun i : ℕ => ⋃ (b : β) (_ : b ∈ Encodable.decode₂ β i), f b with hg
have hg₁ : ∀ i, MeasurableSet (g i) :=
fun _ => MeasurableSet.iUnion fun b => MeasurableSet.iUnion fun _ => hf₁ b
have hg₂ : Pairwise (Disjoint on g) := Encodable.iUnion_decode₂_disjoint_on hf₂
have :=... | 31 | 29,048,849,665,247.426 | 2 | 1.333333 | 3 | 1,442 |
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Data.List.OfFn
import Mathlib.Data.Set.Pointwise.Basic
#align_import data.set.pointwise.list_of_fn from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Set
variable {F α β γ : Type*}
variable [Monoid α] {s t : Set α}... | Mathlib/Data/Set/Pointwise/ListOfFn.lean | 26 | 31 | theorem mem_prod_list_ofFn {a : α} {s : Fin n → Set α} :
a ∈ (List.ofFn s).prod ↔ ∃ f : ∀ i : Fin n, s i, (List.ofFn fun i ↦ (f i : α)).prod = a := by |
induction' n with n ih generalizing a
· simp_rw [List.ofFn_zero, List.prod_nil, Fin.exists_fin_zero_pi, eq_comm, Set.mem_one]
· simp_rw [List.ofFn_succ, List.prod_cons, Fin.exists_fin_succ_pi, Fin.cons_zero, Fin.cons_succ,
mem_mul, @ih, exists_exists_eq_and, SetCoe.exists, exists_prop]
| 4 | 54.59815 | 2 | 1.333333 | 3 | 1,443 |
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Data.List.OfFn
import Mathlib.Data.Set.Pointwise.Basic
#align_import data.set.pointwise.list_of_fn from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Set
variable {F α β γ : Type*}
variable [Monoid α] {s t : Set α}... | Mathlib/Data/Set/Pointwise/ListOfFn.lean | 36 | 47 | theorem mem_list_prod {l : List (Set α)} {a : α} :
a ∈ l.prod ↔
∃ l' : List (Σs : Set α, ↥s),
List.prod (l'.map fun x ↦ (Sigma.snd x : α)) = a ∧ l'.map Sigma.fst = l := by |
induction' l using List.ofFnRec with n f
simp only [mem_prod_list_ofFn, List.exists_iff_exists_tuple, List.map_ofFn, Function.comp,
List.ofFn_inj', Sigma.mk.inj_iff, and_left_comm, exists_and_left, exists_eq_left, heq_eq_eq]
constructor
· rintro ⟨fi, rfl⟩
exact ⟨fun i ↦ ⟨_, fi i⟩, rfl, rfl⟩
· rintro ... | 8 | 2,980.957987 | 2 | 1.333333 | 3 | 1,443 |
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Data.List.OfFn
import Mathlib.Data.Set.Pointwise.Basic
#align_import data.set.pointwise.list_of_fn from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Set
variable {F α β γ : Type*}
variable [Monoid α] {s t : Set α}... | Mathlib/Data/Set/Pointwise/ListOfFn.lean | 52 | 54 | theorem mem_pow {a : α} {n : ℕ} :
a ∈ s ^ n ↔ ∃ f : Fin n → s, (List.ofFn fun i ↦ (f i : α)).prod = a := by |
rw [← mem_prod_list_ofFn, List.ofFn_const, List.prod_replicate]
| 1 | 2.718282 | 0 | 1.333333 | 3 | 1,443 |
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 | 39 | 39 | theorem op_mem_op {s : Set α} {a : α} : op a ∈ s.op ↔ a ∈ s := by | rfl
| 1 | 2.718282 | 0 | 1.333333 | 6 | 1,444 |
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 | 48 | 48 | theorem unop_mem_unop {s : Set αᵒᵖ} {a : αᵒᵖ} : unop a ∈ s.unop ↔ a ∈ s := by | rfl
| 1 | 2.718282 | 0 | 1.333333 | 6 | 1,444 |
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 | 76 | 80 | theorem singleton_op (x : α) : ({x} : Set α).op = {op x} := by |
ext
constructor
· apply unop_injective
· apply op_injective
| 4 | 54.59815 | 2 | 1.333333 | 6 | 1,444 |
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Defs
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
variable {α : Type*}
open Opposite
namespace Set
protected def op (s : Set α) : Set αᵒᵖ :=
unop ⁻¹' s
#align set.op Set.op
protected def u... | Mathlib/Data/Set/Opposite.lean | 84 | 88 | theorem singleton_unop (x : αᵒᵖ) : ({x} : Set αᵒᵖ).unop = {unop x} := by |
ext
constructor
· apply op_injective
· apply unop_injective
| 4 | 54.59815 | 2 | 1.333333 | 6 | 1,444 |
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 | 92 | 96 | theorem singleton_op_unop (x : α) : ({op x} : Set αᵒᵖ).unop = {x} := by |
ext
constructor
· apply op_injective
· apply unop_injective
| 4 | 54.59815 | 2 | 1.333333 | 6 | 1,444 |
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.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
#align_import analysis.normed_space.conformal_linear_map from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open Function LinearIsometry ContinuousLinearMap
def IsConf... | Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean | 62 | 65 | theorem IsConformalMap.smul (hf : IsConformalMap f) {c : R} (hc : c ≠ 0) :
IsConformalMap (c • f) := by |
rcases hf with ⟨c', hc', li, rfl⟩
exact ⟨c * c', mul_ne_zero hc hc', li, smul_smul _ _ _⟩
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,445 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
#align_import analysis.normed_space.conformal_linear_map from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open Function LinearIsometry ContinuousLinearMap
def IsConf... | Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean | 84 | 89 | theorem comp (hg : IsConformalMap g) (hf : IsConformalMap f) : IsConformalMap (g.comp f) := by |
rcases hf with ⟨cf, hcf, lif, rfl⟩
rcases hg with ⟨cg, hcg, lig, rfl⟩
refine ⟨cg * cf, mul_ne_zero hcg hcf, lig.comp lif, ?_⟩
rw [smul_comp, comp_smul, mul_smul]
rfl
| 5 | 148.413159 | 2 | 1.333333 | 3 | 1,445 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
#align_import analysis.normed_space.conformal_linear_map from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open Function LinearIsometry ContinuousLinearMap
def IsConf... | Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean | 97 | 100 | theorem ne_zero [Nontrivial M'] {f' : M' →L[R] N} (hf' : IsConformalMap f') : f' ≠ 0 := by |
rintro rfl
rcases exists_ne (0 : M') with ⟨a, ha⟩
exact ha (hf'.injective rfl)
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,445 |
import Mathlib.NumberTheory.SmoothNumbers
import Mathlib.Analysis.PSeries
open Set Nat
open scoped Topology
-- This needs `Mathlib.Analysis.RCLike.Basic`, so we put it here
-- instead of in `Mathlib.NumberTheory.SmoothNumbers`.
lemma Nat.roughNumbersUpTo_card_le' (N k : ℕ) :
(roughNumbersUpTo N k).card ≤
... | Mathlib/NumberTheory/SumPrimeReciprocals.lean | 64 | 79 | theorem not_summable_one_div_on_primes :
¬ Summable (indicator {p | p.Prime} (fun n : ℕ ↦ (1 : ℝ) / n)) := by |
intro h
obtain ⟨k, hk⟩ := h.nat_tsum_vanishing (Iio_mem_nhds one_half_pos : Iio (1 / 2 : ℝ) ∈ 𝓝 0)
specialize hk ({p | Nat.Prime p} ∩ {p | k ≤ p}) inter_subset_right
rw [tsum_subtype, indicator_indicator, inter_eq_left.mpr fun n hn ↦ hn.1, mem_Iio] at hk
have h' : Summable (indicator ({p | Nat.Prime p} ∩ {p... | 14 | 1,202,604.284165 | 2 | 1.333333 | 3 | 1,446 |
import Mathlib.NumberTheory.SmoothNumbers
import Mathlib.Analysis.PSeries
open Set Nat
open scoped Topology
-- This needs `Mathlib.Analysis.RCLike.Basic`, so we put it here
-- instead of in `Mathlib.NumberTheory.SmoothNumbers`.
lemma Nat.roughNumbersUpTo_card_le' (N k : ℕ) :
(roughNumbersUpTo N k).card ≤
... | Mathlib/NumberTheory/SumPrimeReciprocals.lean | 82 | 83 | theorem Nat.Primes.not_summable_one_div : ¬ Summable (fun p : Nat.Primes ↦ (1 / p : ℝ)) := by |
convert summable_subtype_iff_indicator.mp.mt not_summable_one_div_on_primes
| 1 | 2.718282 | 0 | 1.333333 | 3 | 1,446 |
import Mathlib.NumberTheory.SmoothNumbers
import Mathlib.Analysis.PSeries
open Set Nat
open scoped Topology
-- This needs `Mathlib.Analysis.RCLike.Basic`, so we put it here
-- instead of in `Mathlib.NumberTheory.SmoothNumbers`.
lemma Nat.roughNumbersUpTo_card_le' (N k : ℕ) :
(roughNumbersUpTo N k).card ≤
... | Mathlib/NumberTheory/SumPrimeReciprocals.lean | 86 | 97 | theorem Nat.Primes.summable_rpow {r : ℝ} :
Summable (fun p : Nat.Primes ↦ (p : ℝ) ^ r) ↔ r < -1 := by |
by_cases h : r < -1
· -- case `r < -1`
simp only [h, iff_true]
exact (Real.summable_nat_rpow.mpr h).subtype _
· -- case `-1 ≤ r`
simp only [h, iff_false]
refine fun H ↦ Nat.Primes.not_summable_one_div <| H.of_nonneg_of_le (fun _ ↦ by positivity) ?_
intro p
rw [one_div, ← Real.rpow_neg_one... | 10 | 22,026.465795 | 2 | 1.333333 | 3 | 1,446 |
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Directed
import Mathlib.Order.Hom.Set
#align_import order.antichain from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open Function Set
section General
variable {α β : Type*} {r r₁ r₂ : α →... | Mathlib/Order/Antichain.lean | 89 | 92 | theorem image (hs : IsAntichain r s) (f : α → β) (h : ∀ ⦃a b⦄, r' (f a) (f b) → r a b) :
IsAntichain r' (f '' s) := by |
rintro _ ⟨b, hb, rfl⟩ _ ⟨c, hc, rfl⟩ hbc hr
exact hs hb hc (ne_of_apply_ne _ hbc) (h hr)
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,447 |
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Directed
import Mathlib.Order.Hom.Set
#align_import order.antichain from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open Function Set
section General
variable {α β : Type*} {r r₁ r₂ : α →... | Mathlib/Order/Antichain.lean | 120 | 124 | theorem image_relEmbedding (hs : IsAntichain r s) (φ : r ↪r r') : IsAntichain r' (φ '' s) := by |
intro b hb b' hb' h₁ h₂
rw [Set.mem_image] at hb hb'
obtain ⟨⟨a, has, rfl⟩, ⟨a', has', rfl⟩⟩ := hb, hb'
exact hs has has' (fun haa' => h₁ (by rw [haa'])) (φ.map_rel_iff.mp h₂)
| 4 | 54.59815 | 2 | 1.333333 | 3 | 1,447 |
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Directed
import Mathlib.Order.Hom.Set
#align_import order.antichain from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open Function Set
section General
variable {α β : Type*} {r r₁ r₂ : α →... | Mathlib/Order/Antichain.lean | 313 | 317 | theorem image (hs : IsStrongAntichain r s) {f : α → β} (hf : Surjective f)
(h : ∀ a b, r' (f a) (f b) → r a b) : IsStrongAntichain r' (f '' s) := by |
rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ hab c
obtain ⟨c, rfl⟩ := hf c
exact (hs ha hb (ne_of_apply_ne _ hab) _).imp (mt <| h _ _) (mt <| h _ _)
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,447 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {α : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 55 | 59 | theorem self_subset_generateMeasurableRec (s : Set (Set α)) (i : ω₁) :
s ⊆ generateMeasurableRec s i := by |
unfold generateMeasurableRec
apply_rules [subset_union_of_subset_left]
exact subset_rfl
| 3 | 20.085537 | 1 | 1.333333 | 6 | 1,448 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {α : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 62 | 65 | theorem empty_mem_generateMeasurableRec (s : Set (Set α)) (i : ω₁) :
∅ ∈ generateMeasurableRec s i := by |
unfold generateMeasurableRec
exact mem_union_left _ (mem_union_left _ (mem_union_right _ (mem_singleton ∅)))
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,448 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {α : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 68 | 71 | theorem compl_mem_generateMeasurableRec {s : Set (Set α)} {i j : ω₁} (h : j < i) {t : Set α}
(ht : t ∈ generateMeasurableRec s j) : tᶜ ∈ generateMeasurableRec s i := by |
unfold generateMeasurableRec
exact mem_union_left _ (mem_union_right _ ⟨t, mem_iUnion.2 ⟨⟨j, h⟩, ht⟩, rfl⟩)
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,448 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {α : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 74 | 78 | theorem iUnion_mem_generateMeasurableRec {s : Set (Set α)} {i : ω₁} {f : ℕ → Set α}
(hf : ∀ n, ∃ j < i, f n ∈ generateMeasurableRec s j) :
(⋃ n, f n) ∈ generateMeasurableRec s i := by |
unfold generateMeasurableRec
exact mem_union_right _ ⟨fun n => ⟨f n, let ⟨j, hj, hf⟩ := hf n; mem_iUnion.2 ⟨⟨j, hj⟩, hf⟩⟩, rfl⟩
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,448 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {α : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 91 | 113 | theorem cardinal_generateMeasurableRec_le (s : Set (Set α)) (i : ω₁) :
#(generateMeasurableRec s i) ≤ max #s 2 ^ aleph0.{u} := by |
apply (aleph 1).ord.out.wo.wf.induction i
intro i IH
have A := aleph0_le_aleph 1
have B : aleph 1 ≤ max #s 2 ^ aleph0.{u} :=
aleph_one_le_continuum.trans (power_le_power_right (le_max_right _ _))
have C : ℵ₀ ≤ max #s 2 ^ aleph0.{u} := A.trans B
have J : #(⋃ j : Iio i, generateMeasurableRec s j.1) ≤ max... | 21 | 1,318,815,734.483215 | 2 | 1.333333 | 6 | 1,448 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {α : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 117 | 151 | theorem generateMeasurable_eq_rec (s : Set (Set α)) :
{ t | GenerateMeasurable s t } =
⋃ (i : (Quotient.out (aleph 1).ord).α), generateMeasurableRec s i := by |
ext t; refine ⟨fun ht => ?_, fun ht => ?_⟩
· inhabit ω₁
induction' ht with u hu u _ IH f _ IH
· exact mem_iUnion.2 ⟨default, self_subset_generateMeasurableRec s _ hu⟩
· exact mem_iUnion.2 ⟨default, empty_mem_generateMeasurableRec s _⟩
· rcases mem_iUnion.1 IH with ⟨i, hi⟩
obtain ⟨j, hj⟩ := ex... | 32 | 78,962,960,182,680.7 | 2 | 1.333333 | 6 | 1,448 |
import Mathlib.Data.List.Basic
#align_import data.list.palindrome from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
variable {α β : Type*}
namespace List
inductive Palindrome : List α → Prop
| nil : Palindrome []
| singleton : ∀ x, Palindrome [x]
| cons_concat : ∀ (x) {l}, Pa... | Mathlib/Data/List/Palindrome.lean | 50 | 52 | theorem reverse_eq {l : List α} (p : Palindrome l) : reverse l = l := by |
induction p <;> try (exact rfl)
simpa
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,449 |
import Mathlib.Data.List.Basic
#align_import data.list.palindrome from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
variable {α β : Type*}
namespace List
inductive Palindrome : List α → Prop
| nil : Palindrome []
| singleton : ∀ x, Palindrome [x]
| cons_concat : ∀ (x) {l}, Pa... | Mathlib/Data/List/Palindrome.lean | 55 | 61 | theorem of_reverse_eq {l : List α} : reverse l = l → Palindrome l := by |
refine bidirectionalRecOn l (fun _ => Palindrome.nil) (fun a _ => Palindrome.singleton a) ?_
intro x l y hp hr
rw [reverse_cons, reverse_append] at hr
rw [head_eq_of_cons_eq hr]
have : Palindrome l := hp (append_inj_left' (tail_eq_of_cons_eq hr) rfl)
exact Palindrome.cons_concat x this
| 6 | 403.428793 | 2 | 1.333333 | 3 | 1,449 |
import Mathlib.Data.List.Basic
#align_import data.list.palindrome from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
variable {α β : Type*}
namespace List
inductive Palindrome : List α → Prop
| nil : Palindrome []
| singleton : ∀ x, Palindrome [x]
| cons_concat : ∀ (x) {l}, Pa... | Mathlib/Data/List/Palindrome.lean | 68 | 70 | theorem append_reverse (l : List α) : Palindrome (l ++ reverse l) := by |
apply of_reverse_eq
rw [reverse_append, reverse_reverse]
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,449 |
import Mathlib.Geometry.Manifold.VectorBundle.Tangent
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Classical Topology Manifold
open Set ChartedSpace
section DerivativesDefinitions
variable {𝕜 : Type*} ... | Mathlib/Geometry/Manifold/MFDeriv/Defs.lean | 134 | 177 | theorem differentiable_within_at_localInvariantProp :
(contDiffGroupoid ⊤ I).LocalInvariantProp (contDiffGroupoid ⊤ I')
(DifferentiableWithinAtProp I I') :=
{ is_local := by |
intro s x u f u_open xu
have : I.symm ⁻¹' (s ∩ u) ∩ Set.range I = I.symm ⁻¹' s ∩ Set.range I ∩ I.symm ⁻¹' u := by
simp only [Set.inter_right_comm, Set.preimage_inter]
rw [DifferentiableWithinAtProp, DifferentiableWithinAtProp, this]
symm
apply differentiableWithinAt_inter
ha... | 40 | 235,385,266,837,019,970 | 2 | 1.333333 | 3 | 1,450 |
import Mathlib.Geometry.Manifold.VectorBundle.Tangent
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Classical Topology Manifold
open Set ChartedSpace
section DerivativesDefinitions
variable {𝕜 : Type*} ... | Mathlib/Geometry/Manifold/MFDeriv/Defs.lean | 203 | 207 | theorem mdifferentiableWithinAt_iff' (f : M → M') (s : Set M) (x : M) :
MDifferentiableWithinAt I I' f s x ↔ ContinuousWithinAt f s x ∧
DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f)
((extChartAt I x).symm ⁻¹' s ∩ range I) ((extChartAt I x) x) := by |
rw [MDifferentiableWithinAt, liftPropWithinAt_iff']; rfl
| 1 | 2.718282 | 0 | 1.333333 | 3 | 1,450 |
import Mathlib.Geometry.Manifold.VectorBundle.Tangent
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Classical Topology Manifold
open Set ChartedSpace
section DerivativesDefinitions
variable {𝕜 : Type*} ... | Mathlib/Geometry/Manifold/MFDeriv/Defs.lean | 239 | 246 | theorem mdifferentiableAt_iff (f : M → M') (x : M) :
MDifferentiableAt I I' f x ↔ ContinuousAt f x ∧
DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f) (range I) ((extChartAt I x) x) := by |
rw [MDifferentiableAt, liftPropAt_iff]
congrm _ ∧ ?_
simp [DifferentiableWithinAtProp, Set.univ_inter]
-- Porting note: `rfl` wasn't needed
rfl
| 5 | 148.413159 | 2 | 1.333333 | 3 | 1,450 |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 79 | 82 | theorem IsPiSystem.singleton {α} (S : Set α) : IsPiSystem ({S} : Set (Set α)) := by |
intro s h_s t h_t _
rw [Set.mem_singleton_iff.1 h_s, Set.mem_singleton_iff.1 h_t, Set.inter_self,
Set.mem_singleton_iff]
| 3 | 20.085537 | 1 | 1.333333 | 9 | 1,451 |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 85 | 92 | theorem IsPiSystem.insert_empty {α} {S : Set (Set α)} (h_pi : IsPiSystem S) :
IsPiSystem (insert ∅ S) := by |
intro s hs t ht hst
cases' hs with hs hs
· simp [hs]
· cases' ht with ht ht
· simp [ht]
· exact Set.mem_insert_of_mem _ (h_pi s hs t ht hst)
| 6 | 403.428793 | 2 | 1.333333 | 9 | 1,451 |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 95 | 102 | theorem IsPiSystem.insert_univ {α} {S : Set (Set α)} (h_pi : IsPiSystem S) :
IsPiSystem (insert Set.univ S) := by |
intro s hs t ht hst
cases' hs with hs hs
· cases' ht with ht ht <;> simp [hs, ht]
· cases' ht with ht ht
· simp [hs, ht]
· exact Set.mem_insert_of_mem _ (h_pi s hs t ht hst)
| 6 | 403.428793 | 2 | 1.333333 | 9 | 1,451 |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 105 | 109 | theorem IsPiSystem.comap {α β} {S : Set (Set β)} (h_pi : IsPiSystem S) (f : α → β) :
IsPiSystem { s : Set α | ∃ t ∈ S, f ⁻¹' t = s } := by |
rintro _ ⟨s, hs_mem, rfl⟩ _ ⟨t, ht_mem, rfl⟩ hst
rw [← Set.preimage_inter] at hst ⊢
exact ⟨s ∩ t, h_pi s hs_mem t ht_mem (nonempty_of_nonempty_preimage hst), rfl⟩
| 3 | 20.085537 | 1 | 1.333333 | 9 | 1,451 |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 112 | 120 | theorem isPiSystem_iUnion_of_directed_le {α ι} (p : ι → Set (Set α))
(hp_pi : ∀ n, IsPiSystem (p n)) (hp_directed : Directed (· ≤ ·) p) :
IsPiSystem (⋃ n, p n) := by |
intro t1 ht1 t2 ht2 h
rw [Set.mem_iUnion] at ht1 ht2 ⊢
cases' ht1 with n ht1
cases' ht2 with m ht2
obtain ⟨k, hpnk, hpmk⟩ : ∃ k, p n ≤ p k ∧ p m ≤ p k := hp_directed n m
exact ⟨k, hp_pi k t1 (hpnk ht1) t2 (hpmk ht2) h⟩
| 6 | 403.428793 | 2 | 1.333333 | 9 | 1,451 |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 132 | 134 | theorem isPiSystem_image_Iio (s : Set α) : IsPiSystem (Iio '' s) := by |
rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ -
exact ⟨a ⊓ b, inf_ind a b ha hb, Iio_inter_Iio.symm⟩
| 2 | 7.389056 | 1 | 1.333333 | 9 | 1,451 |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 149 | 151 | theorem isPiSystem_image_Iic (s : Set α) : IsPiSystem (Iic '' s) := by |
rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ -
exact ⟨a ⊓ b, inf_ind a b ha hb, Iic_inter_Iic.symm⟩
| 2 | 7.389056 | 1 | 1.333333 | 9 | 1,451 |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 164 | 170 | theorem isPiSystem_Ixx_mem {Ixx : α → α → Set α} {p : α → α → Prop}
(Hne : ∀ {a b}, (Ixx a b).Nonempty → p a b)
(Hi : ∀ {a₁ b₁ a₂ b₂}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂)) (s t : Set α) :
IsPiSystem { S | ∃ᵉ (l ∈ s) (u ∈ t), p l u ∧ Ixx l u = S } := by |
rintro _ ⟨l₁, hls₁, u₁, hut₁, _, rfl⟩ _ ⟨l₂, hls₂, u₂, hut₂, _, rfl⟩
simp only [Hi]
exact fun H => ⟨l₁ ⊔ l₂, sup_ind l₁ l₂ hls₁ hls₂, u₁ ⊓ u₂, inf_ind u₁ u₂ hut₁ hut₂, Hne H, rfl⟩
| 3 | 20.085537 | 1 | 1.333333 | 9 | 1,451 |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 256 | 261 | theorem generatePiSystem_measurableSet {α} [M : MeasurableSpace α] {S : Set (Set α)}
(h_meas_S : ∀ s ∈ S, MeasurableSet s) (t : Set α) (h_in_pi : t ∈ generatePiSystem S) :
MeasurableSet t := by |
induction' h_in_pi with s h_s s u _ _ _ h_s h_u
· apply h_meas_S _ h_s
· apply MeasurableSet.inter h_s h_u
| 3 | 20.085537 | 1 | 1.333333 | 9 | 1,451 |
import Mathlib.Data.TypeMax
import Mathlib.Logic.UnivLE
import Mathlib.CategoryTheory.Limits.Shapes.Images
#align_import category_theory.limits.types from "leanprover-community/mathlib"@"4aa2a2e17940311e47007f087c9df229e7f12942"
open CategoryTheory CategoryTheory.Limits
universe v u w
namespace CategoryTheory.L... | Mathlib/CategoryTheory/Limits/Types.lean | 52 | 60 | theorem isLimit_iff (c : Cone F) :
Nonempty (IsLimit c) ↔ ∀ s ∈ F.sections, ∃! x : c.pt, ∀ j, c.π.app j x = s j := by |
refine ⟨fun ⟨t⟩ s hs ↦ ?_, fun h ↦ ⟨?_⟩⟩
· let cs := coneOfSection hs
exact ⟨t.lift cs ⟨⟩, fun j ↦ congr_fun (t.fac cs j) ⟨⟩,
fun x hx ↦ congr_fun (t.uniq cs (fun _ ↦ x) fun j ↦ funext fun _ ↦ hx j) ⟨⟩⟩
· choose x hx using fun c y ↦ h _ (sectionOfCone c y).2
exact ⟨x, fun c j ↦ funext fun y ↦ (hx c... | 7 | 1,096.633158 | 2 | 1.333333 | 3 | 1,452 |
import Mathlib.Data.TypeMax
import Mathlib.Logic.UnivLE
import Mathlib.CategoryTheory.Limits.Shapes.Images
#align_import category_theory.limits.types from "leanprover-community/mathlib"@"4aa2a2e17940311e47007f087c9df229e7f12942"
open CategoryTheory CategoryTheory.Limits
universe v u w
namespace CategoryTheory.L... | Mathlib/CategoryTheory/Limits/Types.lean | 62 | 65 | theorem isLimit_iff_bijective_sectionOfCone (c : Cone F) :
Nonempty (IsLimit c) ↔ (Types.sectionOfCone c).Bijective := by |
simp_rw [isLimit_iff, Function.bijective_iff_existsUnique, Subtype.forall, F.sections_ext_iff,
sectionOfCone]
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,452 |
import Mathlib.Data.TypeMax
import Mathlib.Logic.UnivLE
import Mathlib.CategoryTheory.Limits.Shapes.Images
#align_import category_theory.limits.types from "leanprover-community/mathlib"@"4aa2a2e17940311e47007f087c9df229e7f12942"
open CategoryTheory CategoryTheory.Limits
universe v u w
namespace CategoryTheory.L... | Mathlib/CategoryTheory/Limits/Types.lean | 83 | 87 | theorem isLimitEquivSections_symm_apply {c : Cone F} (t : IsLimit c)
(x : F.sections) (j : J) :
c.π.app j ((isLimitEquivSections t).symm x) = (x : ∀ j, F.obj j) j := by |
conv_rhs => rw [← (isLimitEquivSections t).right_inv x]
rfl
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,452 |
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.Algebra.Ring.Subring.Units
#align_import linear_algebra.ray from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
noncomputable section
... | Mathlib/LinearAlgebra/Ray.lean | 61 | 63 | theorem of_subsingleton [Subsingleton M] (x y : M) : SameRay R x y := by |
rw [Subsingleton.elim x 0]
exact zero_left _
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,453 |
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.Algebra.Ring.Subring.Units
#align_import linear_algebra.ray from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
noncomputable section
... | Mathlib/LinearAlgebra/Ray.lean | 74 | 76 | theorem refl (x : M) : SameRay R x x := by |
nontriviality R
exact Or.inr (Or.inr <| ⟨1, 1, zero_lt_one, zero_lt_one, rfl⟩)
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,453 |
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.Algebra.Ring.Subring.Units
#align_import linear_algebra.ray from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
noncomputable section
... | Mathlib/LinearAlgebra/Ray.lean | 102 | 111 | theorem trans (hxy : SameRay R x y) (hyz : SameRay R y z) (hy : y = 0 → x = 0 ∨ z = 0) :
SameRay R x z := by |
rcases eq_or_ne x 0 with (rfl | hx); · exact zero_left z
rcases eq_or_ne z 0 with (rfl | hz); · exact zero_right x
rcases eq_or_ne y 0 with (rfl | hy);
· exact (hy rfl).elim (fun h => (hx h).elim) fun h => (hz h).elim
rcases hxy.exists_pos hx hy with ⟨r₁, r₂, hr₁, hr₂, h₁⟩
rcases hyz.exists_pos hy hz with ... | 8 | 2,980.957987 | 2 | 1.333333 | 3 | 1,453 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.FractionRing
import M... | Mathlib/RingTheory/Localization/Integral.lean | 55 | 58 | theorem coeffIntegerNormalization_of_not_mem_support (p : S[X]) (i : ℕ) (h : coeff p i = 0) :
coeffIntegerNormalization M p i = 0 := by |
simp only [coeffIntegerNormalization, h, mem_support_iff, eq_self_iff_true, not_true, Ne,
dif_neg, not_false_iff]
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,454 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.FractionRing
import M... | Mathlib/RingTheory/Localization/Integral.lean | 61 | 64 | theorem coeffIntegerNormalization_mem_support (p : S[X]) (i : ℕ)
(h : coeffIntegerNormalization M p i ≠ 0) : i ∈ p.support := by |
contrapose h
rw [Ne, Classical.not_not, coeffIntegerNormalization, dif_neg h]
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,454 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.FractionRing
import M... | Mathlib/RingTheory/Localization/Integral.lean | 74 | 77 | theorem integerNormalization_coeff (p : S[X]) (i : ℕ) :
(integerNormalization M p).coeff i = coeffIntegerNormalization M p i := by |
simp (config := { contextual := true }) [integerNormalization, coeff_monomial,
coeffIntegerNormalization_of_not_mem_support]
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,454 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.FractionRing
import M... | Mathlib/RingTheory/Localization/Integral.lean | 80 | 90 | theorem integerNormalization_spec (p : S[X]) :
∃ b : M, ∀ i, algebraMap R S ((integerNormalization M p).coeff i) = (b : R) • p.coeff i := by |
use Classical.choose (exist_integer_multiples_of_finset M (p.support.image p.coeff))
intro i
rw [integerNormalization_coeff, coeffIntegerNormalization]
split_ifs with hi
· exact
Classical.choose_spec
(Classical.choose_spec (exist_integer_multiples_of_finset M (p.support.image p.coeff))
... | 9 | 8,103.083928 | 2 | 1.333333 | 6 | 1,454 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.FractionRing
import M... | Mathlib/RingTheory/Localization/Integral.lean | 112 | 115 | theorem integerNormalization_aeval_eq_zero [Algebra R R'] [Algebra S R'] [IsScalarTower R S R']
(p : S[X]) {x : R'} (hx : aeval x p = 0) : aeval x (integerNormalization M p) = 0 := by |
rw [aeval_def, IsScalarTower.algebraMap_eq R S R',
integerNormalization_eval₂_eq_zero _ (algebraMap _ _) _ hx]
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,454 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.FractionRing
import M... | Mathlib/RingTheory/Localization/Integral.lean | 185 | 201 | theorem RingHom.isIntegralElem_localization_at_leadingCoeff {R S : Type*} [CommRing R] [CommRing S]
(f : R →+* S) (x : S) (p : R[X]) (hf : p.eval₂ f x = 0) (M : Submonoid R)
(hM : p.leadingCoeff ∈ M) {Rₘ Sₘ : Type*} [CommRing Rₘ] [CommRing Sₘ] [Algebra R Rₘ]
[IsLocalization M Rₘ] [Algebra S Sₘ] [IsLocalizat... |
by_cases triv : (1 : Rₘ) = 0
· exact ⟨0, ⟨_root_.trans leadingCoeff_zero triv.symm, eval₂_zero _ _⟩⟩
haveI : Nontrivial Rₘ := nontrivial_of_ne 1 0 triv
obtain ⟨b, hb⟩ := isUnit_iff_exists_inv.mp (map_units Rₘ ⟨p.leadingCoeff, hM⟩)
refine ⟨p.map (algebraMap R Rₘ) * C b, ⟨?_, ?_⟩⟩
· refine monic_mul_C_of_lea... | 12 | 162,754.791419 | 2 | 1.333333 | 6 | 1,454 |
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Trace
#align_import ring_theory.complex from "leanprover-community/mathlib"@"9015c511549dc77a0f8d6eba021d8ac4bba20c82"
open Complex
| Mathlib/RingTheory/Complex.lean | 17 | 28 | theorem Algebra.leftMulMatrix_complex (z : ℂ) :
Algebra.leftMulMatrix Complex.basisOneI z = !![z.re, -z.im; z.im, z.re] := by |
ext i j
rw [Algebra.leftMulMatrix_eq_repr_mul, Complex.coe_basisOneI_repr, Complex.coe_basisOneI, mul_re,
mul_im, Matrix.of_apply]
fin_cases j
· simp only [Fin.mk_zero, Matrix.cons_val_zero, one_re, mul_one, one_im, mul_zero, sub_zero,
zero_add]
fin_cases i <;> rfl
· simp only [Fin.mk_one, Matr... | 10 | 22,026.465795 | 2 | 1.333333 | 3 | 1,455 |
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Trace
#align_import ring_theory.complex from "leanprover-community/mathlib"@"9015c511549dc77a0f8d6eba021d8ac4bba20c82"
open Complex
theorem Algebra.leftMulMatrix_complex (z : ℂ) :
Algebra.leftMulMatrix Complex.basisOn... | Mathlib/RingTheory/Complex.lean | 31 | 34 | theorem Algebra.trace_complex_apply (z : ℂ) : Algebra.trace ℝ ℂ z = 2 * z.re := by |
rw [Algebra.trace_eq_matrix_trace Complex.basisOneI, Algebra.leftMulMatrix_complex,
Matrix.trace_fin_two]
exact (two_mul _).symm
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,455 |
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Trace
#align_import ring_theory.complex from "leanprover-community/mathlib"@"9015c511549dc77a0f8d6eba021d8ac4bba20c82"
open Complex
theorem Algebra.leftMulMatrix_complex (z : ℂ) :
Algebra.leftMulMatrix Complex.basisOn... | Mathlib/RingTheory/Complex.lean | 37 | 40 | theorem Algebra.norm_complex_apply (z : ℂ) : Algebra.norm ℝ z = Complex.normSq z := by |
rw [Algebra.norm_eq_matrix_det Complex.basisOneI, Algebra.leftMulMatrix_complex,
Matrix.det_fin_two, normSq_apply]
simp
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,455 |
import Mathlib.Data.Set.Finite
import Mathlib.Order.Partition.Finpartition
#align_import data.setoid.partition from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
namespace Setoid
variable {α : Type*}
theorem eq_of_mem_eqv_class {c : Set (Set α)} (H : ∀ a, ∃! b ∈ c, a ∈ b) {x b b'}
... | Mathlib/Data/Setoid/Partition.lean | 67 | 71 | theorem classes_ker_subset_fiber_set {β : Type*} (f : α → β) :
(Setoid.ker f).classes ⊆ Set.range fun y => { x | f x = y } := by |
rintro s ⟨x, rfl⟩
rw [Set.mem_range]
exact ⟨f x, rfl⟩
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,456 |
import Mathlib.Data.Set.Finite
import Mathlib.Order.Partition.Finpartition
#align_import data.setoid.partition from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
namespace Setoid
variable {α : Type*}
theorem eq_of_mem_eqv_class {c : Set (Set α)} (H : ∀ a, ∃! b ∈ c, a ∈ b) {x b b'}
... | Mathlib/Data/Setoid/Partition.lean | 78 | 81 | theorem card_classes_ker_le {α β : Type*} [Fintype β] (f : α → β)
[Fintype (Setoid.ker f).classes] : Fintype.card (Setoid.ker f).classes ≤ Fintype.card β := by |
classical exact
le_trans (Set.card_le_card (classes_ker_subset_fiber_set f)) (Fintype.card_range_le _)
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,456 |
import Mathlib.Data.Set.Finite
import Mathlib.Order.Partition.Finpartition
#align_import data.setoid.partition from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
namespace Setoid
variable {α : Type*}
theorem eq_of_mem_eqv_class {c : Set (Set α)} (H : ∀ a, ∃! b ∈ c, a ∈ b) {x b b'}
... | Mathlib/Data/Setoid/Partition.lean | 122 | 130 | theorem eq_eqv_class_of_mem {c : Set (Set α)} (H : ∀ a, ∃! b ∈ c, a ∈ b) {s y}
(hs : s ∈ c) (hy : y ∈ s) : s = { x | (mkClasses c H).Rel x y } := by |
ext x
constructor
· intro hx _s' hs' hx'
rwa [eq_of_mem_eqv_class H hs' hx' hs hx]
· intro hx
obtain ⟨b', ⟨hc, hb'⟩, _⟩ := H x
rwa [eq_of_mem_eqv_class H hs hy hc (hx b' hc hb')]
| 7 | 1,096.633158 | 2 | 1.333333 | 3 | 1,456 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Topology.MetricSpace.CauSeqFilter
#align_import analysis.special_functions.exponential from "leanprover-community/mathlib"@"e1a18cad9cd462973d760af7de36b05776b8811c"
open Filter RCLike ContinuousMultili... | Mathlib/Analysis/SpecialFunctions/Exponential.lean | 67 | 72 | theorem hasStrictFDerivAt_exp_zero_of_radius_pos (h : 0 < (expSeries 𝕂 𝔸).radius) :
HasStrictFDerivAt (exp 𝕂) (1 : 𝔸 →L[𝕂] 𝔸) 0 := by |
convert (hasFPowerSeriesAt_exp_zero_of_radius_pos h).hasStrictFDerivAt
ext x
change x = expSeries 𝕂 𝔸 1 fun _ => x
simp [expSeries_apply_eq, Nat.factorial]
| 4 | 54.59815 | 2 | 1.333333 | 3 | 1,457 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Topology.MetricSpace.CauSeqFilter
#align_import analysis.special_functions.exponential from "leanprover-community/mathlib"@"e1a18cad9cd462973d760af7de36b05776b8811c"
open Filter RCLike ContinuousMultili... | Mathlib/Analysis/SpecialFunctions/Exponential.lean | 220 | 224 | theorem Complex.exp_eq_exp_ℂ : Complex.exp = NormedSpace.exp ℂ := by |
refine funext fun x => ?_
rw [Complex.exp, exp_eq_tsum_div]
have : CauSeq.IsComplete ℂ norm := Complex.instIsComplete
exact tendsto_nhds_unique x.exp'.tendsto_limit (expSeries_div_summable ℝ x).hasSum.tendsto_sum_nat
| 4 | 54.59815 | 2 | 1.333333 | 3 | 1,457 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Topology.MetricSpace.CauSeqFilter
#align_import analysis.special_functions.exponential from "leanprover-community/mathlib"@"e1a18cad9cd462973d760af7de36b05776b8811c"
open Filter RCLike ContinuousMultili... | Mathlib/Analysis/SpecialFunctions/Exponential.lean | 227 | 228 | theorem Real.exp_eq_exp_ℝ : Real.exp = NormedSpace.exp ℝ := by |
ext x; exact mod_cast congr_fun Complex.exp_eq_exp_ℂ x
| 1 | 2.718282 | 0 | 1.333333 | 3 | 1,457 |
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Nat.Lattice
#align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
open Function Set
variable {R ... | Mathlib/RingTheory/Nilpotent/Defs.lean | 64 | 68 | theorem IsNilpotent.of_pow [MonoidWithZero R] {x : R} {m : ℕ}
(h : IsNilpotent (x ^ m)) : IsNilpotent x := by |
obtain ⟨n, h⟩ := h
use m*n
rw [← h, pow_mul x m n]
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,458 |
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Nat.Lattice
#align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
open Function Set
variable {R ... | Mathlib/RingTheory/Nilpotent/Defs.lean | 81 | 85 | theorem IsNilpotent.map [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type*}
[FunLike F R S] [MonoidWithZeroHomClass F R S] (hr : IsNilpotent r) (f : F) :
IsNilpotent (f r) := by |
use hr.choose
rw [← map_pow, hr.choose_spec, map_zero]
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,458 |
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Nat.Lattice
#align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
open Function Set
variable {R ... | Mathlib/RingTheory/Nilpotent/Defs.lean | 197 | 205 | theorem isReduced_of_injective [MonoidWithZero R] [MonoidWithZero S] {F : Type*}
[FunLike F R S] [MonoidWithZeroHomClass F R S]
(f : F) (hf : Function.Injective f) [IsReduced S] :
IsReduced R := by |
constructor
intro x hx
apply hf
rw [map_zero]
exact (hx.map f).eq_zero
| 5 | 148.413159 | 2 | 1.333333 | 3 | 1,458 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Multiset.Lattice
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
import Mathlib.Order.Nat
#align_import... | Mathlib/Data/Finset/Lattice.lean | 82 | 87 | theorem sup_sup : s.sup (f ⊔ g) = s.sup f ⊔ s.sup g := by |
induction s using Finset.cons_induction with
| empty => rw [sup_empty, sup_empty, sup_empty, bot_sup_eq]
| cons _ _ _ ih =>
rw [sup_cons, sup_cons, sup_cons, ih]
exact sup_sup_sup_comm _ _ _ _
| 5 | 148.413159 | 2 | 1.333333 | 3 | 1,459 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Multiset.Lattice
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
import Mathlib.Order.Nat
#align_import... | Mathlib/Data/Finset/Lattice.lean | 90 | 93 | theorem sup_congr {f g : β → α} (hs : s₁ = s₂) (hfg : ∀ a ∈ s₂, f a = g a) :
s₁.sup f = s₂.sup g := by |
subst hs
exact Finset.fold_congr hfg
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,459 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Multiset.Lattice
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
import Mathlib.Order.Nat
#align_import... | Mathlib/Data/Finset/Lattice.lean | 140 | 143 | theorem sup_bot (s : Finset β) : (s.sup fun _ => ⊥) = (⊥ : α) := by |
obtain rfl | hs := s.eq_empty_or_nonempty
· exact sup_empty
· exact sup_const hs _
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,459 |
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.Order.Filter.AtTopBot
import Mathlib.RingTheory.Artinian
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mat... | Mathlib/Algebra/Lie/Nilpotent.lean | 485 | 490 | theorem ucs_mono (k : ℕ) (h : N₁ ≤ N₂) : N₁.ucs k ≤ N₂.ucs k := by |
induction' k with k ih
· simpa
simp only [ucs_succ]
-- Porting note: `mono` makes no progress
apply monotone_normalizer ih
| 5 | 148.413159 | 2 | 1.333333 | 3 | 1,460 |
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.Order.Filter.AtTopBot
import Mathlib.RingTheory.Artinian
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mat... | Mathlib/Algebra/Lie/Nilpotent.lean | 493 | 496 | theorem ucs_eq_self_of_normalizer_eq_self (h : N₁.normalizer = N₁) (k : ℕ) : N₁.ucs k = N₁ := by |
induction' k with k ih
· simp
· rwa [ucs_succ, ih]
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,460 |
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.Order.Filter.AtTopBot
import Mathlib.RingTheory.Artinian
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mat... | Mathlib/Algebra/Lie/Nilpotent.lean | 504 | 508 | theorem ucs_le_of_normalizer_eq_self (h : N₁.normalizer = N₁) (k : ℕ) :
(⊥ : LieSubmodule R L M).ucs k ≤ N₁ := by |
rw [← ucs_eq_self_of_normalizer_eq_self h k]
mono
simp
| 3 | 20.085537 | 1 | 1.333333 | 3 | 1,460 |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
| Mathlib/Data/Real/Sign.lean | 36 | 36 | theorem sign_of_neg {r : ℝ} (hr : r < 0) : sign r = -1 := by | rw [sign, if_pos hr]
| 1 | 2.718282 | 0 | 1.363636 | 11 | 1,467 |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 39 | 39 | theorem sign_of_pos {r : ℝ} (hr : 0 < r) : sign r = 1 := by | rw [sign, if_pos hr, if_neg hr.not_lt]
| 1 | 2.718282 | 0 | 1.363636 | 11 | 1,467 |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 43 | 43 | theorem sign_zero : sign 0 = 0 := by | rw [sign, if_neg (lt_irrefl _), if_neg (lt_irrefl _)]
| 1 | 2.718282 | 0 | 1.363636 | 11 | 1,467 |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 51 | 55 | theorem sign_apply_eq (r : ℝ) : sign r = -1 ∨ sign r = 0 ∨ sign r = 1 := by |
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ)
· exact Or.inl <| sign_of_neg hn
· exact Or.inr <| Or.inl <| sign_zero
· exact Or.inr <| Or.inr <| sign_of_pos hp
| 4 | 54.59815 | 2 | 1.363636 | 11 | 1,467 |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 64 | 71 | theorem sign_eq_zero_iff {r : ℝ} : sign r = 0 ↔ r = 0 := by |
refine ⟨fun h => ?_, fun h => h.symm ▸ sign_zero⟩
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ)
· rw [sign_of_neg hn, neg_eq_zero] at h
exact (one_ne_zero h).elim
· rfl
· rw [sign_of_pos hp] at h
exact (one_ne_zero h).elim
| 7 | 1,096.633158 | 2 | 1.363636 | 11 | 1,467 |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 74 | 79 | theorem sign_intCast (z : ℤ) : sign (z : ℝ) = ↑(Int.sign z) := by |
obtain hn | rfl | hp := lt_trichotomy z (0 : ℤ)
· rw [sign_of_neg (Int.cast_lt_zero.mpr hn), Int.sign_eq_neg_one_of_neg hn, Int.cast_neg,
Int.cast_one]
· rw [Int.cast_zero, sign_zero, Int.sign_zero, Int.cast_zero]
· rw [sign_of_pos (Int.cast_pos.mpr hp), Int.sign_eq_one_of_pos hp, Int.cast_one]
| 5 | 148.413159 | 2 | 1.363636 | 11 | 1,467 |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 85 | 89 | theorem sign_neg {r : ℝ} : sign (-r) = -sign r := by |
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ)
· rw [sign_of_neg hn, sign_of_pos (neg_pos.mpr hn), neg_neg]
· rw [sign_zero, neg_zero, sign_zero]
· rw [sign_of_pos hp, sign_of_neg (neg_lt_zero.mpr hp)]
| 4 | 54.59815 | 2 | 1.363636 | 11 | 1,467 |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 92 | 98 | theorem sign_mul_nonneg (r : ℝ) : 0 ≤ sign r * r := by |
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ)
· rw [sign_of_neg hn]
exact mul_nonneg_of_nonpos_of_nonpos (by norm_num) hn.le
· rw [mul_zero]
· rw [sign_of_pos hp, one_mul]
exact hp.le
| 6 | 403.428793 | 2 | 1.363636 | 11 | 1,467 |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 101 | 104 | theorem sign_mul_pos_of_ne_zero (r : ℝ) (hr : r ≠ 0) : 0 < sign r * r := by |
refine lt_of_le_of_ne (sign_mul_nonneg r) fun h => hr ?_
have hs0 := (zero_eq_mul.mp h).resolve_right hr
exact sign_eq_zero_iff.mp hs0
| 3 | 20.085537 | 1 | 1.363636 | 11 | 1,467 |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 108 | 115 | theorem inv_sign (r : ℝ) : (sign r)⁻¹ = sign r := by |
obtain hn | hz | hp := sign_apply_eq r
· rw [hn]
norm_num
· rw [hz]
exact inv_zero
· rw [hp]
exact inv_one
| 7 | 1,096.633158 | 2 | 1.363636 | 11 | 1,467 |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 119 | 123 | theorem sign_inv (r : ℝ) : sign r⁻¹ = sign r := by |
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ)
· rw [sign_of_neg hn, sign_of_neg (inv_lt_zero.mpr hn)]
· rw [sign_zero, inv_zero, sign_zero]
· rw [sign_of_pos hp, sign_of_pos (inv_pos.mpr hp)]
| 4 | 54.59815 | 2 | 1.363636 | 11 | 1,467 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 71 | 73 | theorem exists_reduced_word (w : W) : ∃ ω, ω.length = ℓ w ∧ w = π ω := by |
have := Nat.find_spec (cs.exists_word_with_prod w)
tauto
| 2 | 7.389056 | 1 | 1.363636 | 11 | 1,468 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 81 | 88 | theorem length_eq_zero_iff {w : W} : ℓ w = 0 ↔ w = 1 := by |
constructor
· intro h
rcases cs.exists_reduced_word w with ⟨ω, hω, rfl⟩
have : ω = [] := eq_nil_of_length_eq_zero (hω.trans h)
rw [this, wordProd_nil]
· rintro rfl
exact cs.length_one
| 7 | 1,096.633158 | 2 | 1.363636 | 11 | 1,468 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 91 | 98 | theorem length_inv (w : W) : ℓ (w⁻¹) = ℓ w := by |
apply Nat.le_antisymm
· rcases cs.exists_reduced_word w with ⟨ω, hω, rfl⟩
have := cs.length_wordProd_le (List.reverse ω)
rwa [wordProd_reverse, length_reverse, hω] at this
· rcases cs.exists_reduced_word w⁻¹ with ⟨ω, hω, h'ω⟩
have := cs.length_wordProd_le (List.reverse ω)
rwa [wordProd_reverse, l... | 7 | 1,096.633158 | 2 | 1.363636 | 11 | 1,468 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 100 | 105 | theorem length_mul_le (w₁ w₂ : W) :
ℓ (w₁ * w₂) ≤ ℓ w₁ + ℓ w₂ := by |
rcases cs.exists_reduced_word w₁ with ⟨ω₁, hω₁, rfl⟩
rcases cs.exists_reduced_word w₂ with ⟨ω₂, hω₂, rfl⟩
have := cs.length_wordProd_le (ω₁ ++ ω₂)
simpa [hω₁, hω₂, wordProd_append] using this
| 4 | 54.59815 | 2 | 1.363636 | 11 | 1,468 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 107 | 109 | theorem length_mul_ge_length_sub_length (w₁ w₂ : W) :
ℓ w₁ - ℓ w₂ ≤ ℓ (w₁ * w₂) := by |
simpa [Nat.sub_le_of_le_add] using cs.length_mul_le (w₁ * w₂) w₂⁻¹
| 1 | 2.718282 | 0 | 1.363636 | 11 | 1,468 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 111 | 113 | theorem length_mul_ge_length_sub_length' (w₁ w₂ : W) :
ℓ w₂ - ℓ w₁ ≤ ℓ (w₁ * w₂) := by |
simpa [Nat.sub_le_of_le_add, add_comm] using cs.length_mul_le w₁⁻¹ (w₁ * w₂)
| 1 | 2.718282 | 0 | 1.363636 | 11 | 1,468 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 131 | 135 | theorem lengthParity_eq_ofAdd_length (w : W) :
cs.lengthParity w = Multiplicative.ofAdd (↑(ℓ w)) := by |
rcases cs.exists_reduced_word w with ⟨ω, hω, rfl⟩
rw [← hω, wordProd, map_list_prod, List.map_map, lengthParity_comp_simple, map_const',
prod_replicate, ← ofAdd_nsmul, nsmul_one]
| 3 | 20.085537 | 1 | 1.363636 | 11 | 1,468 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 137 | 139 | theorem length_mul_mod_two (w₁ w₂ : W) : ℓ (w₁ * w₂) % 2 = (ℓ w₁ + ℓ w₂) % 2 := by |
rw [← ZMod.natCast_eq_natCast_iff', Nat.cast_add]
simpa only [lengthParity_eq_ofAdd_length, ofAdd_add] using map_mul cs.lengthParity w₁ w₂
| 2 | 7.389056 | 1 | 1.363636 | 11 | 1,468 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 142 | 150 | theorem length_simple (i : B) : ℓ (s i) = 1 := by |
apply Nat.le_antisymm
· simpa using cs.length_wordProd_le [i]
· by_contra! length_lt_one
have : cs.lengthParity (s i) = Multiplicative.ofAdd 0 := by
rw [lengthParity_eq_ofAdd_length, Nat.lt_one_iff.mp length_lt_one, Nat.cast_zero]
have : Multiplicative.ofAdd (0 : ZMod 2) = Multiplicative.ofAdd 1 :=... | 8 | 2,980.957987 | 2 | 1.363636 | 11 | 1,468 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 152 | 159 | theorem length_eq_one_iff {w : W} : ℓ w = 1 ↔ ∃ i : B, w = s i := by |
constructor
· intro h
rcases cs.exists_reduced_word w with ⟨ω, hω, rfl⟩
rcases List.length_eq_one.mp (hω.trans h) with ⟨i, rfl⟩
exact ⟨i, cs.wordProd_singleton i⟩
· rintro ⟨i, rfl⟩
exact cs.length_simple i
| 7 | 1,096.633158 | 2 | 1.363636 | 11 | 1,468 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 161 | 169 | theorem length_mul_simple_ne (w : W) (i : B) : ℓ (w * s i) ≠ ℓ w := by |
intro eq
have length_mod_two := cs.length_mul_mod_two w (s i)
rw [eq, length_simple] at length_mod_two
rcases Nat.mod_two_eq_zero_or_one (ℓ w) with even | odd
· rw [even, Nat.succ_mod_two_eq_one_iff.mpr even] at length_mod_two
contradiction
· rw [odd, Nat.succ_mod_two_eq_zero_iff.mpr odd] at length_mod... | 8 | 2,980.957987 | 2 | 1.363636 | 11 | 1,468 |
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin
import Mathlib.Order.Partition.Finpartition
#align_import order.partition.equipartition from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
open Finset Fintype
namespace Finpartition
variable {α : Type*} [DecidableEq α] ... | Mathlib/Order/Partition/Equipartition.lean | 38 | 42 | theorem isEquipartition_iff_card_parts_eq_average :
P.IsEquipartition ↔
∀ a : Finset α,
a ∈ P.parts → a.card = s.card / P.parts.card ∨ a.card = s.card / P.parts.card + 1 := by |
simp_rw [IsEquipartition, Finset.equitableOn_iff, P.sum_card_parts]
| 1 | 2.718282 | 0 | 1.375 | 8 | 1,469 |
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin
import Mathlib.Order.Partition.Finpartition
#align_import order.partition.equipartition from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
open Finset Fintype
namespace Finpartition
variable {α : Type*} [DecidableEq α] ... | Mathlib/Order/Partition/Equipartition.lean | 61 | 66 | theorem IsEquipartition.card_part_eq_average_iff (hP : P.IsEquipartition) (ht : t ∈ P.parts) :
t.card = s.card / P.parts.card ↔ t.card ≠ s.card / P.parts.card + 1 := by |
have a := hP.card_parts_eq_average ht
have b : ¬(t.card = s.card / P.parts.card ∧ t.card = s.card / P.parts.card + 1) := by
by_contra h; exact absurd (h.1 ▸ h.2) (lt_add_one _).ne
tauto
| 4 | 54.59815 | 2 | 1.375 | 8 | 1,469 |
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin
import Mathlib.Order.Partition.Finpartition
#align_import order.partition.equipartition from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
open Finset Fintype
namespace Finpartition
variable {α : Type*} [DecidableEq α] ... | Mathlib/Order/Partition/Equipartition.lean | 68 | 71 | theorem IsEquipartition.average_le_card_part (hP : P.IsEquipartition) (ht : t ∈ P.parts) :
s.card / P.parts.card ≤ t.card := by |
rw [← P.sum_card_parts]
exact Finset.EquitableOn.le hP ht
| 2 | 7.389056 | 1 | 1.375 | 8 | 1,469 |
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin
import Mathlib.Order.Partition.Finpartition
#align_import order.partition.equipartition from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
open Finset Fintype
namespace Finpartition
variable {α : Type*} [DecidableEq α] ... | Mathlib/Order/Partition/Equipartition.lean | 74 | 77 | theorem IsEquipartition.card_part_le_average_add_one (hP : P.IsEquipartition) (ht : t ∈ P.parts) :
t.card ≤ s.card / P.parts.card + 1 := by |
rw [← P.sum_card_parts]
exact Finset.EquitableOn.le_add_one hP ht
| 2 | 7.389056 | 1 | 1.375 | 8 | 1,469 |
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin
import Mathlib.Order.Partition.Finpartition
#align_import order.partition.equipartition from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
open Finset Fintype
namespace Finpartition
variable {α : Type*} [DecidableEq α] ... | Mathlib/Order/Partition/Equipartition.lean | 80 | 85 | theorem IsEquipartition.filter_ne_average_add_one_eq_average (hP : P.IsEquipartition) :
P.parts.filter (fun p ↦ ¬p.card = s.card / P.parts.card + 1) =
P.parts.filter (fun p ↦ p.card = s.card / P.parts.card) := by |
ext p
simp only [mem_filter, and_congr_right_iff]
exact fun hp ↦ (hP.card_part_eq_average_iff hp).symm
| 3 | 20.085537 | 1 | 1.375 | 8 | 1,469 |
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin
import Mathlib.Order.Partition.Finpartition
#align_import order.partition.equipartition from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
open Finset Fintype
namespace Finpartition
variable {α : Type*} [DecidableEq α] ... | Mathlib/Order/Partition/Equipartition.lean | 89 | 100 | theorem IsEquipartition.card_large_parts_eq_mod (hP : P.IsEquipartition) :
(P.parts.filter fun p ↦ p.card = s.card / P.parts.card + 1).card = s.card % P.parts.card := by |
have z := P.sum_card_parts
rw [← sum_filter_add_sum_filter_not (s := P.parts)
(p := fun x ↦ x.card = s.card / P.parts.card + 1),
hP.filter_ne_average_add_one_eq_average,
sum_const_nat (m := s.card / P.parts.card + 1) (by simp),
sum_const_nat (m := s.card / P.parts.card) (by simp),
← hP.filter... | 10 | 22,026.465795 | 2 | 1.375 | 8 | 1,469 |
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin
import Mathlib.Order.Partition.Finpartition
#align_import order.partition.equipartition from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
open Finset Fintype
namespace Finpartition
variable {α : Type*} [DecidableEq α] ... | Mathlib/Order/Partition/Equipartition.lean | 104 | 110 | theorem IsEquipartition.card_small_parts_eq_mod (hP : P.IsEquipartition) :
(P.parts.filter fun p ↦ p.card = s.card / P.parts.card).card =
P.parts.card - s.card % P.parts.card := by |
conv_rhs =>
arg 1
rw [← filter_card_add_filter_neg_card_eq_card (p := fun p ↦ p.card = s.card / P.parts.card + 1)]
rw [hP.card_large_parts_eq_mod, add_tsub_cancel_left, hP.filter_ne_average_add_one_eq_average]
| 4 | 54.59815 | 2 | 1.375 | 8 | 1,469 |
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin
import Mathlib.Order.Partition.Finpartition
#align_import order.partition.equipartition from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
open Finset Fintype
namespace Finpartition
variable {α : Type*} [DecidableEq α] ... | Mathlib/Order/Partition/Equipartition.lean | 114 | 134 | theorem IsEquipartition.exists_partsEquiv (hP : P.IsEquipartition) :
∃ f : P.parts ≃ Fin P.parts.card,
∀ t, t.1.card = s.card / P.parts.card + 1 ↔ f t < s.card % P.parts.card := by |
let el := (P.parts.filter fun p ↦ p.card = s.card / P.parts.card + 1).equivFin
let es := (P.parts.filter fun p ↦ p.card = s.card / P.parts.card).equivFin
simp_rw [mem_filter, hP.card_large_parts_eq_mod] at el
simp_rw [mem_filter, hP.card_small_parts_eq_mod] at es
let sneg : { x // x ∈ P.parts ∧ ¬x.card = s.c... | 18 | 65,659,969.137331 | 2 | 1.375 | 8 | 1,469 |
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