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 Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
open Nat
namespace List
section countP
variable (p q : α → Bool)
@[simp] theorem countP_nil : countP p [] = 0 := rfl
protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 := by
induction l generalizing n with
| nil... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 44 | 45 | theorem countP_cons (a : α) (l) : countP p (a :: l) = countP p l + if p a then 1 else 0 := by |
by_cases h : p a <;> simp [h]
| 1 | 2.718282 | 0 | 0.8 | 10 | 703 |
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
open Nat
namespace List
section countP
variable (p q : α → Bool)
@[simp] theorem countP_nil : countP p [] = 0 := rfl
protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 := by
induction l generalizing n with
| nil... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 47 | 58 | theorem length_eq_countP_add_countP (l) : length l = countP p l + countP (fun a => ¬p a) l := by |
induction l with
| nil => rfl
| cons x h ih =>
if h : p x then
rw [countP_cons_of_pos _ _ h, countP_cons_of_neg _ _ _, length, ih]
· rw [Nat.add_assoc, Nat.add_comm _ 1, Nat.add_assoc]
· simp only [h, not_true_eq_false, decide_False, not_false_eq_true]
else
rw [countP_cons_of_pos ... | 11 | 59,874.141715 | 2 | 0.8 | 10 | 703 |
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
open Nat
namespace List
section countP
variable (p q : α → Bool)
@[simp] theorem countP_nil : countP p [] = 0 := rfl
protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 := by
induction l generalizing n with
| nil... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 60 | 66 | theorem countP_eq_length_filter (l) : countP p l = length (filter p l) := by |
induction l with
| nil => rfl
| cons x l ih =>
if h : p x
then rw [countP_cons_of_pos p l h, ih, filter_cons_of_pos l h, length]
else rw [countP_cons_of_neg p l h, ih, filter_cons_of_neg l h]
| 6 | 403.428793 | 2 | 0.8 | 10 | 703 |
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
open Nat
namespace List
section countP
variable (p q : α → Bool)
@[simp] theorem countP_nil : countP p [] = 0 := rfl
protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 := by
induction l generalizing n with
| nil... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 68 | 70 | theorem countP_le_length : countP p l ≤ l.length := by |
simp only [countP_eq_length_filter]
apply length_filter_le
| 2 | 7.389056 | 1 | 0.8 | 10 | 703 |
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
open Nat
namespace List
section countP
variable (p q : α → Bool)
@[simp] theorem countP_nil : countP p [] = 0 := rfl
protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 := by
induction l generalizing n with
| nil... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 75 | 76 | theorem countP_pos : 0 < countP p l ↔ ∃ a ∈ l, p a := by |
simp only [countP_eq_length_filter, length_pos_iff_exists_mem, mem_filter, exists_prop]
| 1 | 2.718282 | 0 | 0.8 | 10 | 703 |
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
open Nat
namespace List
section countP
variable (p q : α → Bool)
@[simp] theorem countP_nil : countP p [] = 0 := rfl
protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 := by
induction l generalizing n with
| nil... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 78 | 79 | theorem countP_eq_zero : countP p l = 0 ↔ ∀ a ∈ l, ¬p a := by |
simp only [countP_eq_length_filter, length_eq_zero, filter_eq_nil]
| 1 | 2.718282 | 0 | 0.8 | 10 | 703 |
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
open Nat
namespace List
section countP
variable (p q : α → Bool)
@[simp] theorem countP_nil : countP p [] = 0 := rfl
protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 := by
induction l generalizing n with
| nil... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 81 | 82 | theorem countP_eq_length : countP p l = l.length ↔ ∀ a ∈ l, p a := by |
rw [countP_eq_length_filter, filter_length_eq_length]
| 1 | 2.718282 | 0 | 0.8 | 10 | 703 |
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
open Nat
namespace List
section countP
variable (p q : α → Bool)
@[simp] theorem countP_nil : countP p [] = 0 := rfl
protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 := by
induction l generalizing n with
| nil... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 84 | 86 | theorem Sublist.countP_le (s : l₁ <+ l₂) : countP p l₁ ≤ countP p l₂ := by |
simp only [countP_eq_length_filter]
apply s.filter _ |>.length_le
| 2 | 7.389056 | 1 | 0.8 | 10 | 703 |
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
open Nat
namespace List
section countP
variable (p q : α → Bool)
@[simp] theorem countP_nil : countP p [] = 0 := rfl
protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 := by
induction l generalizing n with
| nil... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 88 | 90 | theorem countP_filter (l : List α) :
countP p (filter q l) = countP (fun a => p a ∧ q a) l := by |
simp only [countP_eq_length_filter, filter_filter]
| 1 | 2.718282 | 0 | 0.8 | 10 | 703 |
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
open Nat
namespace List
section countP
variable (p q : α → Bool)
@[simp] theorem countP_nil : countP p [] = 0 := rfl
protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 := by
induction l generalizing n with
| nil... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 107 | 119 | theorem countP_mono_left (h : ∀ x ∈ l, p x → q x) : countP p l ≤ countP q l := by |
induction l with
| nil => apply Nat.le_refl
| cons a l ihl =>
rw [forall_mem_cons] at h
have ⟨ha, hl⟩ := h
simp [countP_cons]
cases h : p a
. simp
apply Nat.le_trans ?_ (Nat.le_add_right _ _)
apply ihl hl
. simp [ha h]
apply ihl hl
| 12 | 162,754.791419 | 2 | 0.8 | 10 | 703 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section restrict
def restrict (... | Mathlib/Data/Set/Function.lean | 74 | 76 | theorem image_restrict (f : α → β) (s t : Set α) :
s.restrict f '' (Subtype.val ⁻¹' t) = f '' (t ∩ s) := by |
rw [restrict_eq, image_comp, image_preimage_eq_inter_range, Subtype.range_coe]
| 1 | 2.718282 | 0 | 0.8 | 10 | 704 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section restrict
def restrict (... | Mathlib/Data/Set/Function.lean | 117 | 120 | theorem restrict_extend_range (f : α → β) (g : α → γ) (g' : β → γ) :
(range f).restrict (extend f g g') = fun x => g x.coe_prop.choose := by |
classical
exact restrict_dite _ _
| 2 | 7.389056 | 1 | 0.8 | 10 | 704 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section restrict
def restrict (... | Mathlib/Data/Set/Function.lean | 124 | 127 | theorem restrict_extend_compl_range (f : α → β) (g : α → γ) (g' : β → γ) :
(range f)ᶜ.restrict (extend f g g') = g' ∘ Subtype.val := by |
classical
exact restrict_dite_compl _ _
| 2 | 7.389056 | 1 | 0.8 | 10 | 704 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section restrict
def restrict (... | Mathlib/Data/Set/Function.lean | 130 | 136 | theorem range_extend_subset (f : α → β) (g : α → γ) (g' : β → γ) :
range (extend f g g') ⊆ range g ∪ g' '' (range f)ᶜ := by |
classical
rintro _ ⟨y, rfl⟩
rw [extend_def]
split_ifs with h
exacts [Or.inl (mem_range_self _), Or.inr (mem_image_of_mem _ h)]
| 5 | 148.413159 | 2 | 0.8 | 10 | 704 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section restrict
def restrict (... | Mathlib/Data/Set/Function.lean | 139 | 143 | theorem range_extend {f : α → β} (hf : Injective f) (g : α → γ) (g' : β → γ) :
range (extend f g g') = range g ∪ g' '' (range f)ᶜ := by |
refine (range_extend_subset _ _ _).antisymm ?_
rintro z (⟨x, rfl⟩ | ⟨y, hy, rfl⟩)
exacts [⟨f x, hf.extend_apply _ _ _⟩, ⟨y, extend_apply' _ _ _ hy⟩]
| 3 | 20.085537 | 1 | 0.8 | 10 | 704 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section equality
variable {s s₁... | Mathlib/Data/Set/Function.lean | 185 | 186 | theorem eqOn_singleton : Set.EqOn f₁ f₂ {a} ↔ f₁ a = f₂ a := by |
simp [Set.EqOn]
| 1 | 2.718282 | 0 | 0.8 | 10 | 704 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section equality
variable {s s₁... | Mathlib/Data/Set/Function.lean | 190 | 191 | theorem eqOn_univ (f₁ f₂ : α → β) : EqOn f₁ f₂ univ ↔ f₁ = f₂ := by |
simp [EqOn, funext_iff]
| 1 | 2.718282 | 0 | 0.8 | 10 | 704 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section Order
variable {s : Se... | Mathlib/Data/Set/Function.lean | 264 | 267 | theorem _root_.MonotoneOn.congr (h₁ : MonotoneOn f₁ s) (h : s.EqOn f₁ f₂) : MonotoneOn f₂ s := by |
intro a ha b hb hab
rw [← h ha, ← h hb]
exact h₁ ha hb hab
| 3 | 20.085537 | 1 | 0.8 | 10 | 704 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section Order
variable {s : Se... | Mathlib/Data/Set/Function.lean | 274 | 278 | theorem _root_.StrictMonoOn.congr (h₁ : StrictMonoOn f₁ s) (h : s.EqOn f₁ f₂) :
StrictMonoOn f₂ s := by |
intro a ha b hb hab
rw [← h ha, ← h hb]
exact h₁ ha hb hab
| 3 | 20.085537 | 1 | 0.8 | 10 | 704 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
variable {s s₁ s₂ : Set α} {t ... | Mathlib/Data/Set/Function.lean | 360 | 363 | theorem MapsTo.coe_iterate_restrict {f : α → α} (h : MapsTo f s s) (x : s) (k : ℕ) :
h.restrict^[k] x = f^[k] x := by |
induction' k with k ih; · simp
simp only [iterate_succ', comp_apply, val_restrict_apply, ih]
| 2 | 7.389056 | 1 | 0.8 | 10 | 704 |
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Finiteness
#align_import group_theory.abelianization from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
universe u v w
-- Let G be a group.
variable (G : Type u) [Group G]
open Subgroup (... | Mathlib/GroupTheory/Abelianization.lean | 49 | 50 | theorem commutator_eq_closure : commutator G = Subgroup.closure (commutatorSet G) := by |
simp [commutator, Subgroup.commutator_def, commutatorSet]
| 1 | 2.718282 | 0 | 0.8 | 5 | 705 |
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Finiteness
#align_import group_theory.abelianization from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
universe u v w
-- Let G be a group.
variable (G : Type u) [Group G]
open Subgroup (... | Mathlib/GroupTheory/Abelianization.lean | 53 | 54 | theorem commutator_eq_normalClosure : commutator G = Subgroup.normalClosure (commutatorSet G) := by |
simp [commutator, Subgroup.commutator_def', commutatorSet]
| 1 | 2.718282 | 0 | 0.8 | 5 | 705 |
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Finiteness
#align_import group_theory.abelianization from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
universe u v w
-- Let G be a group.
variable (G : Type u) [Group G]
open Subgroup (... | Mathlib/GroupTheory/Abelianization.lean | 65 | 68 | theorem rank_commutator_le_card [Finite (commutatorSet G)] :
Group.rank (commutator G) ≤ Nat.card (commutatorSet G) := by |
rw [Subgroup.rank_congr (commutator_eq_closure G)]
apply Subgroup.rank_closure_finite_le_nat_card
| 2 | 7.389056 | 1 | 0.8 | 5 | 705 |
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Finiteness
#align_import group_theory.abelianization from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
universe u v w
-- Let G be a group.
variable (G : Type u) [Group G]
open Subgroup (... | Mathlib/GroupTheory/Abelianization.lean | 71 | 79 | theorem commutator_centralizer_commutator_le_center :
⁅centralizer (commutator G : Set G), centralizer (commutator G)⁆ ≤ Subgroup.center G := by |
rw [← Subgroup.centralizer_univ, ← Subgroup.coe_top, ←
Subgroup.commutator_eq_bot_iff_le_centralizer]
suffices ⁅⁅⊤, centralizer (commutator G : Set G)⁆, centralizer (commutator G : Set G)⁆ = ⊥ by
refine Subgroup.commutator_commutator_eq_bot_of_rotate ?_ this
rwa [Subgroup.commutator_comm (centralizer (... | 7 | 1,096.633158 | 2 | 0.8 | 5 | 705 |
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Finiteness
#align_import group_theory.abelianization from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
universe u v w
-- Let G be a group.
variable (G : Type u) [Group G]
open Subgroup (... | Mathlib/GroupTheory/Abelianization.lean | 132 | 135 | theorem commutator_subset_ker : commutator G ≤ f.ker := by |
rw [commutator_eq_closure, Subgroup.closure_le]
rintro x ⟨p, q, rfl⟩
simp [MonoidHom.mem_ker, mul_right_comm (f p) (f q), commutatorElement_def]
| 3 | 20.085537 | 1 | 0.8 | 5 | 705 |
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 93 | 99 | theorem braiding_tensor_left (X Y Z : C) :
(β_ (X ⊗ Y) Z).hom =
(α_ X Y Z).hom ≫ X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫
(β_ X Z).hom ▷ Y ≫ (α_ Z X Y).hom := by |
apply (cancel_epi (α_ X Y Z).inv).1
apply (cancel_mono (α_ Z X Y).inv).1
simp [hexagon_reverse]
| 3 | 20.085537 | 1 | 0.8 | 5 | 706 |
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 102 | 108 | theorem braiding_tensor_right (X Y Z : C) :
(β_ X (Y ⊗ Z)).hom =
(α_ X Y Z).inv ≫ (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫
Y ◁ (β_ X Z).hom ≫ (α_ Y Z X).inv := by |
apply (cancel_epi (α_ X Y Z).hom).1
apply (cancel_mono (α_ Y Z X).hom).1
simp [hexagon_forward]
| 3 | 20.085537 | 1 | 0.8 | 5 | 706 |
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 125 | 128 | theorem braiding_naturality {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :
(f ⊗ g) ≫ (braiding Y Y').hom = (braiding X X').hom ≫ (g ⊗ f) := by |
rw [tensorHom_def' f g, tensorHom_def g f]
simp_rw [Category.assoc, braiding_naturality_left, braiding_naturality_right_assoc]
| 2 | 7.389056 | 1 | 0.8 | 5 | 706 |
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 146 | 153 | theorem yang_baxter (X Y Z : C) :
(α_ X Y Z).inv ≫ (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫
Y ◁ (β_ X Z).hom ≫ (α_ Y Z X).inv ≫ (β_ Y Z).hom ▷ X ≫ (α_ Z Y X).hom =
X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫ (β_ X Z).hom ▷ Y ≫
(α_ Z X Y).hom ≫ Z ◁ (β_ X Y).hom := by |
rw [← braiding_tensor_right_assoc X Y Z, ← cancel_mono (α_ Z Y X).inv]
repeat rw [assoc]
rw [Iso.hom_inv_id, comp_id, ← braiding_naturality_right, braiding_tensor_right]
| 3 | 20.085537 | 1 | 0.8 | 5 | 706 |
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 267 | 271 | theorem braiding_leftUnitor_aux₁ (X : C) :
(α_ (𝟙_ C) (𝟙_ C) X).hom ≫
(𝟙_ C ◁ (β_ X (𝟙_ C)).inv) ≫ (α_ _ X _).inv ≫ ((λ_ X).hom ▷ _) =
((λ_ _).hom ▷ X) ≫ (β_ X (𝟙_ C)).inv := by |
coherence
| 1 | 2.718282 | 0 | 0.8 | 5 | 706 |
import Mathlib.Algebra.Group.Commutator
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Bracket
import Mathlib.GroupTheory.Subgroup.Centralizer
import Mathlib.Tactic.Group
#align_import group_theory.commutator from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
variable... | Mathlib/GroupTheory/Commutator.lean | 31 | 32 | theorem commutatorElement_eq_one_iff_mul_comm : ⁅g₁, g₂⁆ = 1 ↔ g₁ * g₂ = g₂ * g₁ := by |
rw [commutatorElement_def, mul_inv_eq_one, mul_inv_eq_iff_eq_mul]
| 1 | 2.718282 | 0 | 0.8 | 5 | 707 |
import Mathlib.Algebra.Group.Commutator
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Bracket
import Mathlib.GroupTheory.Subgroup.Centralizer
import Mathlib.Tactic.Group
#align_import group_theory.commutator from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
variable... | Mathlib/GroupTheory/Commutator.lean | 61 | 62 | theorem commutatorElement_inv : ⁅g₁, g₂⁆⁻¹ = ⁅g₂, g₁⁆ := by |
simp_rw [commutatorElement_def, mul_inv_rev, inv_inv, mul_assoc]
| 1 | 2.718282 | 0 | 0.8 | 5 | 707 |
import Mathlib.Algebra.Group.Commutator
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Bracket
import Mathlib.GroupTheory.Subgroup.Centralizer
import Mathlib.Tactic.Group
#align_import group_theory.commutator from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
variable... | Mathlib/GroupTheory/Commutator.lean | 65 | 66 | theorem map_commutatorElement : (f ⁅g₁, g₂⁆ : G') = ⁅f g₁, f g₂⁆ := by |
simp_rw [commutatorElement_def, map_mul f, map_inv f]
| 1 | 2.718282 | 0 | 0.8 | 5 | 707 |
import Mathlib.Algebra.Group.Commutator
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Bracket
import Mathlib.GroupTheory.Subgroup.Centralizer
import Mathlib.Tactic.Group
#align_import group_theory.commutator from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
variable... | Mathlib/GroupTheory/Commutator.lean | 100 | 104 | theorem commutator_eq_bot_iff_le_centralizer : ⁅H₁, H₂⁆ = ⊥ ↔ H₁ ≤ centralizer H₂ := by |
rw [eq_bot_iff, commutator_le]
refine forall_congr' fun p =>
forall_congr' fun _hp => forall_congr' fun q => forall_congr' fun hq => ?_
rw [mem_bot, commutatorElement_eq_one_iff_mul_comm, eq_comm]
| 4 | 54.59815 | 2 | 0.8 | 5 | 707 |
import Mathlib.Algebra.Group.Commutator
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Bracket
import Mathlib.GroupTheory.Subgroup.Centralizer
import Mathlib.Tactic.Group
#align_import group_theory.commutator from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
variable... | Mathlib/GroupTheory/Commutator.lean | 108 | 116 | theorem commutator_commutator_eq_bot_of_rotate (h1 : ⁅⁅H₂, H₃⁆, H₁⁆ = ⊥) (h2 : ⁅⁅H₃, H₁⁆, H₂⁆ = ⊥) :
⁅⁅H₁, H₂⁆, H₃⁆ = ⊥ := by |
simp_rw [commutator_eq_bot_iff_le_centralizer, commutator_le,
mem_centralizer_iff_commutator_eq_one, ← commutatorElement_def] at h1 h2 ⊢
intro x hx y hy z hz
trans x * z * ⁅y, ⁅z⁻¹, x⁻¹⁆⁆⁻¹ * z⁻¹ * y * ⁅x⁻¹, ⁅y⁻¹, z⁆⁆⁻¹ * y⁻¹ * x⁻¹
· group
· rw [h1 _ (H₂.inv_mem hy) _ hz _ (H₁.inv_mem hx), h2 _ (H₃.inv_m... | 7 | 1,096.633158 | 2 | 0.8 | 5 | 707 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
#align_import probability.martingale.basic from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheor... | Mathlib/Probability/Martingale/Basic.lean | 75 | 79 | theorem martingale_const_fun [OrderBot ι] (ℱ : Filtration ι m0) (μ : Measure Ω) [IsFiniteMeasure μ]
{f : Ω → E} (hf : StronglyMeasurable[ℱ ⊥] f) (hfint : Integrable f μ) :
Martingale (fun _ => f) ℱ μ := by |
refine ⟨fun i => hf.mono <| ℱ.mono bot_le, fun i j _ => ?_⟩
rw [condexp_of_stronglyMeasurable (ℱ.le _) (hf.mono <| ℱ.mono bot_le) hfint]
| 2 | 7.389056 | 1 | 0.8 | 5 | 708 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
#align_import probability.martingale.basic from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheor... | Mathlib/Probability/Martingale/Basic.lean | 109 | 113 | theorem setIntegral_eq [SigmaFiniteFiltration μ ℱ] (hf : Martingale f ℱ μ) {i j : ι} (hij : i ≤ j)
{s : Set Ω} (hs : MeasurableSet[ℱ i] s) : ∫ ω in s, f i ω ∂μ = ∫ ω in s, f j ω ∂μ := by |
rw [← @setIntegral_condexp _ _ _ _ _ (ℱ i) m0 _ _ _ (ℱ.le i) _ (hf.integrable j) hs]
refine setIntegral_congr_ae (ℱ.le i s hs) ?_
filter_upwards [hf.2 i j hij] with _ heq _ using heq.symm
| 3 | 20.085537 | 1 | 0.8 | 5 | 708 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
#align_import probability.martingale.basic from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheor... | Mathlib/Probability/Martingale/Basic.lean | 119 | 121 | theorem add (hf : Martingale f ℱ μ) (hg : Martingale g ℱ μ) : Martingale (f + g) ℱ μ := by |
refine ⟨hf.adapted.add hg.adapted, fun i j hij => ?_⟩
exact (condexp_add (hf.integrable j) (hg.integrable j)).trans ((hf.2 i j hij).add (hg.2 i j hij))
| 2 | 7.389056 | 1 | 0.8 | 5 | 708 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
#align_import probability.martingale.basic from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheor... | Mathlib/Probability/Martingale/Basic.lean | 128 | 129 | theorem sub (hf : Martingale f ℱ μ) (hg : Martingale g ℱ μ) : Martingale (f - g) ℱ μ := by |
rw [sub_eq_add_neg]; exact hf.add hg.neg
| 1 | 2.718282 | 0 | 0.8 | 5 | 708 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
#align_import probability.martingale.basic from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheor... | Mathlib/Probability/Martingale/Basic.lean | 132 | 135 | theorem smul (c : ℝ) (hf : Martingale f ℱ μ) : Martingale (c • f) ℱ μ := by |
refine ⟨hf.adapted.smul c, fun i j hij => ?_⟩
refine (condexp_smul c (f j)).trans ((hf.2 i j hij).mono fun x hx => ?_)
simp only [Pi.smul_apply, hx]
| 3 | 20.085537 | 1 | 0.8 | 5 | 708 |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.PartialHomeomorph
#align_import analysis.asymptotics.asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open ... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 89 | 89 | theorem isBigOWith_iff : IsBigOWith c l f g ↔ ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := by | rw [IsBigOWith_def]
| 1 | 2.718282 | 0 | 0.8 | 5 | 709 |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.PartialHomeomorph
#align_import analysis.asymptotics.asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open ... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 109 | 109 | theorem isBigO_iff_isBigOWith : f =O[l] g ↔ ∃ c : ℝ, IsBigOWith c l f g := by | rw [IsBigO_def]
| 1 | 2.718282 | 0 | 0.8 | 5 | 709 |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.PartialHomeomorph
#align_import analysis.asymptotics.asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open ... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 113 | 114 | theorem isBigO_iff : f =O[l] g ↔ ∃ c : ℝ, ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := by |
simp only [IsBigO_def, IsBigOWith_def]
| 1 | 2.718282 | 0 | 0.8 | 5 | 709 |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.PartialHomeomorph
#align_import analysis.asymptotics.asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open ... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 118 | 132 | theorem isBigO_iff' {g : α → E'''} :
f =O[l] g ↔ ∃ c > 0, ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := by |
refine ⟨fun h => ?mp, fun h => ?mpr⟩
case mp =>
rw [isBigO_iff] at h
obtain ⟨c, hc⟩ := h
refine ⟨max c 1, zero_lt_one.trans_le (le_max_right _ _), ?_⟩
filter_upwards [hc] with x hx
apply hx.trans
gcongr
exact le_max_left _ _
case mpr =>
rw [isBigO_iff]
obtain ⟨c, ⟨_, hc⟩⟩ := h... | 13 | 442,413.392009 | 2 | 0.8 | 5 | 709 |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.PartialHomeomorph
#align_import analysis.asymptotics.asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open ... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 135 | 149 | theorem isBigO_iff'' {g : α → E'''} :
f =O[l] g ↔ ∃ c > 0, ∀ᶠ x in l, c * ‖f x‖ ≤ ‖g x‖ := by |
refine ⟨fun h => ?mp, fun h => ?mpr⟩
case mp =>
rw [isBigO_iff'] at h
obtain ⟨c, ⟨hc_pos, hc⟩⟩ := h
refine ⟨c⁻¹, ⟨by positivity, ?_⟩⟩
filter_upwards [hc] with x hx
rwa [inv_mul_le_iff (by positivity)]
case mpr =>
rw [isBigO_iff']
obtain ⟨c, ⟨hc_pos, hc⟩⟩ := h
refine ⟨c⁻¹, ⟨by posi... | 13 | 442,413.392009 | 2 | 0.8 | 5 | 709 |
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 39 | 41 | theorem tendsto_const_div_atTop_nhds_zero_nat (C : ℝ) :
Tendsto (fun n : ℕ ↦ C / n) atTop (𝓝 0) := by |
simpa only [mul_zero] using tendsto_const_nhds.mul tendsto_inverse_atTop_nhds_zero_nat
| 1 | 2.718282 | 0 | 0.8 | 5 | 710 |
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 51 | 54 | theorem NNReal.tendsto_inverse_atTop_nhds_zero_nat :
Tendsto (fun n : ℕ ↦ (n : ℝ≥0)⁻¹) atTop (𝓝 0) := by |
rw [← NNReal.tendsto_coe]
exact _root_.tendsto_inverse_atTop_nhds_zero_nat
| 2 | 7.389056 | 1 | 0.8 | 5 | 710 |
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 59 | 61 | theorem NNReal.tendsto_const_div_atTop_nhds_zero_nat (C : ℝ≥0) :
Tendsto (fun n : ℕ ↦ C / n) atTop (𝓝 0) := by |
simpa using tendsto_const_nhds.mul NNReal.tendsto_inverse_atTop_nhds_zero_nat
| 1 | 2.718282 | 0 | 0.8 | 5 | 710 |
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 74 | 79 | theorem NNReal.tendsto_algebraMap_inverse_atTop_nhds_zero_nat (𝕜 : Type*) [Semiring 𝕜]
[Algebra ℝ≥0 𝕜] [TopologicalSpace 𝕜] [ContinuousSMul ℝ≥0 𝕜] :
Tendsto (algebraMap ℝ≥0 𝕜 ∘ fun n : ℕ ↦ (n : ℝ≥0)⁻¹) atTop (𝓝 0) := by |
convert (continuous_algebraMap ℝ≥0 𝕜).continuousAt.tendsto.comp
tendsto_inverse_atTop_nhds_zero_nat
rw [map_zero]
| 3 | 20.085537 | 1 | 0.8 | 5 | 710 |
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 97 | 113 | theorem tendsto_natCast_div_add_atTop {𝕜 : Type*} [DivisionRing 𝕜] [TopologicalSpace 𝕜]
[CharZero 𝕜] [Algebra ℝ 𝕜] [ContinuousSMul ℝ 𝕜] [TopologicalDivisionRing 𝕜] (x : 𝕜) :
Tendsto (fun n : ℕ ↦ (n : 𝕜) / (n + x)) atTop (𝓝 1) := by |
convert Tendsto.congr' ((eventually_ne_atTop 0).mp (eventually_of_forall fun n hn ↦ _)) _
· exact fun n : ℕ ↦ 1 / (1 + x / n)
· field_simp [Nat.cast_ne_zero.mpr hn]
· have : 𝓝 (1 : 𝕜) = 𝓝 (1 / (1 + x * (0 : 𝕜))) := by
rw [mul_zero, add_zero, div_one]
rw [this]
refine tendsto_const_nhds.div (t... | 14 | 1,202,604.284165 | 2 | 0.8 | 5 | 710 |
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import ring_theory.simple_module from "leanprover-community/mathlib"@"cce7f68a7eaadadf74c82bbac207... | Mathlib/RingTheory/SimpleModule.lean | 86 | 88 | theorem isSimpleModule_iff_isAtom : IsSimpleModule R m ↔ IsAtom m := by |
rw [← Set.isSimpleOrder_Iic_iff_isAtom]
exact m.mapIic.isSimpleOrder_iff
| 2 | 7.389056 | 1 | 0.8 | 5 | 711 |
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import ring_theory.simple_module from "leanprover-community/mathlib"@"cce7f68a7eaadadf74c82bbac207... | Mathlib/RingTheory/SimpleModule.lean | 91 | 94 | theorem isSimpleModule_iff_isCoatom : IsSimpleModule R (M ⧸ m) ↔ IsCoatom m := by |
rw [← Set.isSimpleOrder_Ici_iff_isCoatom]
apply OrderIso.isSimpleOrder_iff
exact Submodule.comapMkQRelIso m
| 3 | 20.085537 | 1 | 0.8 | 5 | 711 |
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import ring_theory.simple_module from "leanprover-community/mathlib"@"cce7f68a7eaadadf74c82bbac207... | Mathlib/RingTheory/SimpleModule.lean | 97 | 101 | theorem covBy_iff_quot_is_simple {A B : Submodule R M} (hAB : A ≤ B) :
A ⋖ B ↔ IsSimpleModule R (B ⧸ Submodule.comap B.subtype A) := by |
set f : Submodule R B ≃o Set.Iic B := B.mapIic with hf
rw [covBy_iff_coatom_Iic hAB, isSimpleModule_iff_isCoatom, ← OrderIso.isCoatom_iff f, hf]
simp [-OrderIso.isCoatom_iff, Submodule.map_comap_subtype, inf_eq_right.2 hAB]
| 3 | 20.085537 | 1 | 0.8 | 5 | 711 |
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import ring_theory.simple_module from "leanprover-community/mathlib"@"cce7f68a7eaadadf74c82bbac207... | Mathlib/RingTheory/SimpleModule.lean | 125 | 127 | theorem toSpanSingleton_surjective {m : M} (hm : m ≠ 0) :
Function.Surjective (toSpanSingleton R M m) := by |
rw [← range_eq_top, ← span_singleton_eq_range, span_singleton_eq_top R hm]
| 1 | 2.718282 | 0 | 0.8 | 5 | 711 |
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import ring_theory.simple_module from "leanprover-community/mathlib"@"cce7f68a7eaadadf74c82bbac207... | Mathlib/RingTheory/SimpleModule.lean | 129 | 132 | theorem ker_toSpanSingleton_isMaximal {m : M} (hm : m ≠ 0) :
Ideal.IsMaximal (ker (toSpanSingleton R M m)) := by |
rw [Ideal.isMaximal_def, ← isSimpleModule_iff_isCoatom]
exact congr (quotKerEquivOfSurjective _ <| toSpanSingleton_surjective R hm)
| 2 | 7.389056 | 1 | 0.8 | 5 | 711 |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 46 | 49 | theorem Duplicate.mem (h : x ∈+ l) : x ∈ l := by |
induction' h with l' _ y l' _ hm
· exact mem_cons_self _ _
· exact mem_cons_of_mem _ hm
| 3 | 20.085537 | 1 | 0.818182 | 11 | 719 |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 52 | 55 | theorem Duplicate.mem_cons_self (h : x ∈+ x :: l) : x ∈ l := by |
cases' h with _ h _ _ h
· exact h
· exact h.mem
| 3 | 20.085537 | 1 | 0.818182 | 11 | 719 |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 70 | 73 | theorem Duplicate.ne_singleton (h : x ∈+ l) (y : α) : l ≠ [y] := by |
induction' h with l' h z l' h _
· simp [ne_nil_of_mem h]
· simp [ne_nil_of_mem h.mem]
| 3 | 20.085537 | 1 | 0.818182 | 11 | 719 |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 88 | 95 | theorem duplicate_cons_iff {y : α} : x ∈+ y :: l ↔ y = x ∧ x ∈ l ∨ x ∈+ l := by |
refine ⟨fun h => ?_, fun h => ?_⟩
· cases' h with _ hm _ _ hm
· exact Or.inl ⟨rfl, hm⟩
· exact Or.inr hm
· rcases h with (⟨rfl | h⟩ | h)
· simpa
· exact h.cons_duplicate
| 7 | 1,096.633158 | 2 | 0.818182 | 11 | 719 |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 98 | 99 | theorem Duplicate.of_duplicate_cons {y : α} (h : x ∈+ y :: l) (hx : x ≠ y) : x ∈+ l := by |
simpa [duplicate_cons_iff, hx.symm] using h
| 1 | 2.718282 | 0 | 0.818182 | 11 | 719 |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 102 | 103 | theorem duplicate_cons_iff_of_ne {y : α} (hne : x ≠ y) : x ∈+ y :: l ↔ x ∈+ l := by |
simp [duplicate_cons_iff, hne.symm]
| 1 | 2.718282 | 0 | 0.818182 | 11 | 719 |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 106 | 113 | theorem Duplicate.mono_sublist {l' : List α} (hx : x ∈+ l) (h : l <+ l') : x ∈+ l' := by |
induction' h with l₁ l₂ y _ IH l₁ l₂ y h IH
· exact hx
· exact (IH hx).duplicate_cons _
· rw [duplicate_cons_iff] at hx ⊢
rcases hx with (⟨rfl, hx⟩ | hx)
· simp [h.subset hx]
· simp [IH hx]
| 7 | 1,096.633158 | 2 | 0.818182 | 11 | 719 |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 117 | 126 | theorem duplicate_iff_sublist : x ∈+ l ↔ [x, x] <+ l := by |
induction' l with y l IH
· simp
· by_cases hx : x = y
· simp [hx, cons_sublist_cons, singleton_sublist]
· rw [duplicate_cons_iff_of_ne hx, IH]
refine ⟨sublist_cons_of_sublist y, fun h => ?_⟩
cases h
· assumption
· contradiction
| 9 | 8,103.083928 | 2 | 0.818182 | 11 | 719 |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 129 | 130 | theorem nodup_iff_forall_not_duplicate : Nodup l ↔ ∀ x : α, ¬x ∈+ l := by |
simp_rw [nodup_iff_sublist, duplicate_iff_sublist]
| 1 | 2.718282 | 0 | 0.818182 | 11 | 719 |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 133 | 134 | theorem exists_duplicate_iff_not_nodup : (∃ x : α, x ∈+ l) ↔ ¬Nodup l := by |
simp [nodup_iff_forall_not_duplicate]
| 1 | 2.718282 | 0 | 0.818182 | 11 | 719 |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 141 | 142 | theorem duplicate_iff_two_le_count [DecidableEq α] : x ∈+ l ↔ 2 ≤ count x l := by |
simp [duplicate_iff_sublist, le_count_iff_replicate_sublist]
| 1 | 2.718282 | 0 | 0.818182 | 11 | 719 |
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
open Nat
namespace Int
| Mathlib/Data/Int/Lemmas.lean | 26 | 29 | theorem le_natCast_sub (m n : ℕ) : (m - n : ℤ) ≤ ↑(m - n : ℕ) := by |
by_cases h : m ≥ n
· exact le_of_eq (Int.ofNat_sub h).symm
· simp [le_of_not_ge h, ofNat_le]
| 3 | 20.085537 | 1 | 0.818182 | 11 | 720 |
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
open Nat
namespace Int
theorem le_natCast_sub (m n : ℕ) : (m ... | Mathlib/Data/Int/Lemmas.lean | 45 | 47 | theorem natAbs_eq_iff_sq_eq {a b : ℤ} : a.natAbs = b.natAbs ↔ a ^ 2 = b ^ 2 := by |
rw [sq, sq]
exact natAbs_eq_iff_mul_self_eq
| 2 | 7.389056 | 1 | 0.818182 | 11 | 720 |
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
open Nat
namespace Int
theorem le_natCast_sub (m n : ℕ) : (m ... | Mathlib/Data/Int/Lemmas.lean | 50 | 52 | theorem natAbs_lt_iff_sq_lt {a b : ℤ} : a.natAbs < b.natAbs ↔ a ^ 2 < b ^ 2 := by |
rw [sq, sq]
exact natAbs_lt_iff_mul_self_lt
| 2 | 7.389056 | 1 | 0.818182 | 11 | 720 |
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
open Nat
namespace Int
theorem le_natCast_sub (m n : ℕ) : (m ... | Mathlib/Data/Int/Lemmas.lean | 55 | 57 | theorem natAbs_le_iff_sq_le {a b : ℤ} : a.natAbs ≤ b.natAbs ↔ a ^ 2 ≤ b ^ 2 := by |
rw [sq, sq]
exact natAbs_le_iff_mul_self_le
| 2 | 7.389056 | 1 | 0.818182 | 11 | 720 |
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
open Nat
namespace Int
theorem le_natCast_sub (m n : ℕ) : (m ... | Mathlib/Data/Int/Lemmas.lean | 60 | 61 | theorem natAbs_inj_of_nonneg_of_nonneg {a b : ℤ} (ha : 0 ≤ a) (hb : 0 ≤ b) :
natAbs a = natAbs b ↔ a = b := by | rw [← sq_eq_sq ha hb, ← natAbs_eq_iff_sq_eq]
| 1 | 2.718282 | 0 | 0.818182 | 11 | 720 |
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
open Nat
namespace Int
theorem le_natCast_sub (m n : ℕ) : (m ... | Mathlib/Data/Int/Lemmas.lean | 64 | 67 | theorem natAbs_inj_of_nonpos_of_nonpos {a b : ℤ} (ha : a ≤ 0) (hb : b ≤ 0) :
natAbs a = natAbs b ↔ a = b := by |
simpa only [Int.natAbs_neg, neg_inj] using
natAbs_inj_of_nonneg_of_nonneg (neg_nonneg_of_nonpos ha) (neg_nonneg_of_nonpos hb)
| 2 | 7.389056 | 1 | 0.818182 | 11 | 720 |
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
open Nat
namespace Int
theorem le_natCast_sub (m n : ℕ) : (m ... | Mathlib/Data/Int/Lemmas.lean | 70 | 72 | theorem natAbs_inj_of_nonneg_of_nonpos {a b : ℤ} (ha : 0 ≤ a) (hb : b ≤ 0) :
natAbs a = natAbs b ↔ a = -b := by |
simpa only [Int.natAbs_neg] using natAbs_inj_of_nonneg_of_nonneg ha (neg_nonneg_of_nonpos hb)
| 1 | 2.718282 | 0 | 0.818182 | 11 | 720 |
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
open Nat
namespace Int
theorem le_natCast_sub (m n : ℕ) : (m ... | Mathlib/Data/Int/Lemmas.lean | 75 | 77 | theorem natAbs_inj_of_nonpos_of_nonneg {a b : ℤ} (ha : a ≤ 0) (hb : 0 ≤ b) :
natAbs a = natAbs b ↔ -a = b := by |
simpa only [Int.natAbs_neg] using natAbs_inj_of_nonneg_of_nonneg (neg_nonneg_of_nonpos ha) hb
| 1 | 2.718282 | 0 | 0.818182 | 11 | 720 |
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
open Nat
namespace Int
theorem le_natCast_sub (m n : ℕ) : (m ... | Mathlib/Data/Int/Lemmas.lean | 82 | 86 | theorem natAbs_coe_sub_coe_le_of_le {a b n : ℕ} (a_le_n : a ≤ n) (b_le_n : b ≤ n) :
natAbs (a - b : ℤ) ≤ n := by |
rw [← Nat.cast_le (α := ℤ), natCast_natAbs]
exact abs_sub_le_of_nonneg_of_le (ofNat_nonneg a) (ofNat_le.mpr a_le_n)
(ofNat_nonneg b) (ofNat_le.mpr b_le_n)
| 3 | 20.085537 | 1 | 0.818182 | 11 | 720 |
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
open Nat
namespace Int
theorem le_natCast_sub (m n : ℕ) : (m ... | Mathlib/Data/Int/Lemmas.lean | 90 | 94 | theorem natAbs_coe_sub_coe_lt_of_lt {a b n : ℕ} (a_lt_n : a < n) (b_lt_n : b < n) :
natAbs (a - b : ℤ) < n := by |
rw [← Nat.cast_lt (α := ℤ), natCast_natAbs]
exact abs_sub_lt_of_nonneg_of_lt (ofNat_nonneg a) (ofNat_lt.mpr a_lt_n)
(ofNat_nonneg b) (ofNat_lt.mpr b_lt_n)
| 3 | 20.085537 | 1 | 0.818182 | 11 | 720 |
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
open Nat
namespace Int
theorem le_natCast_sub (m n : ℕ) : (m ... | Mathlib/Data/Int/Lemmas.lean | 137 | 143 | theorem div2_bit (b n) : div2 (bit b n) = n := by |
rw [bit_val, div2_val, add_comm, Int.add_mul_ediv_left, (_ : (_ / 2 : ℤ) = 0), zero_add]
cases b
· decide
· show ofNat _ = _
rw [Nat.div_eq_of_lt] <;> simp
· decide
| 6 | 403.428793 | 2 | 0.818182 | 11 | 720 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 39 | 44 | theorem convexOn_norm (hs : Convex ℝ s) : ConvexOn ℝ s norm :=
⟨hs, fun x _ y _ a b ha hb _ =>
calc
‖a • x + b • y‖ ≤ ‖a • x‖ + ‖b • y‖ := norm_add_le _ _
_ = a * ‖x‖ + b * ‖y‖ := by |
rw [norm_smul, norm_smul, Real.norm_of_nonneg ha, Real.norm_of_nonneg hb]⟩
| 1 | 2.718282 | 0 | 0.818182 | 11 | 721 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 53 | 55 | theorem convexOn_dist (z : E) (hs : Convex ℝ s) : ConvexOn ℝ s fun z' => dist z' z := by |
simpa [dist_eq_norm, preimage_preimage] using
(convexOn_norm (hs.translate (-z))).comp_affineMap (AffineMap.id ℝ E - AffineMap.const ℝ E z)
| 2 | 7.389056 | 1 | 0.818182 | 11 | 721 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 62 | 63 | theorem convex_ball (a : E) (r : ℝ) : Convex ℝ (Metric.ball a r) := by |
simpa only [Metric.ball, sep_univ] using (convexOn_univ_dist a).convex_lt r
| 1 | 2.718282 | 0 | 0.818182 | 11 | 721 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 66 | 67 | theorem convex_closedBall (a : E) (r : ℝ) : Convex ℝ (Metric.closedBall a r) := by |
simpa only [Metric.closedBall, sep_univ] using (convexOn_univ_dist a).convex_le r
| 1 | 2.718282 | 0 | 0.818182 | 11 | 721 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 70 | 72 | theorem Convex.thickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (thickening δ s) := by |
rw [← add_ball_zero]
exact hs.add (convex_ball 0 _)
| 2 | 7.389056 | 1 | 0.818182 | 11 | 721 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 75 | 80 | theorem Convex.cthickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (cthickening δ s) := by |
obtain hδ | hδ := le_total 0 δ
· rw [cthickening_eq_iInter_thickening hδ]
exact convex_iInter₂ fun _ _ => hs.thickening _
· rw [cthickening_of_nonpos hδ]
exact hs.closure
| 5 | 148.413159 | 2 | 0.818182 | 11 | 721 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 92 | 97 | theorem convexHull_exists_dist_ge2 {s t : Set E} {x y : E} (hx : x ∈ convexHull ℝ s)
(hy : y ∈ convexHull ℝ t) : ∃ x' ∈ s, ∃ y' ∈ t, dist x y ≤ dist x' y' := by |
rcases convexHull_exists_dist_ge hx y with ⟨x', hx', Hx'⟩
rcases convexHull_exists_dist_ge hy x' with ⟨y', hy', Hy'⟩
use x', hx', y', hy'
exact le_trans Hx' (dist_comm y x' ▸ dist_comm y' x' ▸ Hy')
| 4 | 54.59815 | 2 | 0.818182 | 11 | 721 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 102 | 108 | theorem convexHull_ediam (s : Set E) : EMetric.diam (convexHull ℝ s) = EMetric.diam s := by |
refine (EMetric.diam_le fun x hx y hy => ?_).antisymm (EMetric.diam_mono <| subset_convexHull ℝ s)
rcases convexHull_exists_dist_ge2 hx hy with ⟨x', hx', y', hy', H⟩
rw [edist_dist]
apply le_trans (ENNReal.ofReal_le_ofReal H)
rw [← edist_dist]
exact EMetric.edist_le_diam_of_mem hx' hy'
| 6 | 403.428793 | 2 | 0.818182 | 11 | 721 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 113 | 114 | theorem convexHull_diam (s : Set E) : Metric.diam (convexHull ℝ s) = Metric.diam s := by |
simp only [Metric.diam, convexHull_ediam]
| 1 | 2.718282 | 0 | 0.818182 | 11 | 721 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 119 | 121 | theorem isBounded_convexHull {s : Set E} :
Bornology.IsBounded (convexHull ℝ s) ↔ Bornology.IsBounded s := by |
simp only [Metric.isBounded_iff_ediam_ne_top, convexHull_ediam]
| 1 | 2.718282 | 0 | 0.818182 | 11 | 721 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 133 | 136 | theorem Wbtw.dist_add_dist {x y z : P} (h : Wbtw ℝ x y z) :
dist x y + dist y z = dist x z := by |
obtain ⟨a, ⟨ha₀, ha₁⟩, rfl⟩ := h
simp [abs_of_nonneg, ha₀, ha₁, sub_mul]
| 2 | 7.389056 | 1 | 0.818182 | 11 | 721 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 49 | 50 | theorem normed_sub (x : E) : f.normed μ (c - x) = f.normed μ (c + x) := by |
simp_rw [f.normed_def, f.sub]
| 1 | 2.718282 | 0 | 0.818182 | 11 | 722 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 53 | 54 | theorem normed_neg (f : ContDiffBump (0 : E)) (x : E) : f.normed μ (-x) = f.normed μ x := by |
simp_rw [f.normed_def, f.neg]
| 1 | 2.718282 | 0 | 0.818182 | 11 | 722 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 69 | 72 | theorem integral_pos : 0 < ∫ x, f x ∂μ := by |
refine (integral_pos_iff_support_of_nonneg f.nonneg' f.integrable).mpr ?_
rw [f.support_eq]
exact measure_ball_pos μ c f.rOut_pos
| 3 | 20.085537 | 1 | 0.818182 | 11 | 722 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 75 | 77 | theorem integral_normed : ∫ x, f.normed μ x ∂μ = 1 := by |
simp_rw [ContDiffBump.normed, div_eq_mul_inv, mul_comm (f _), ← smul_eq_mul, integral_smul]
exact inv_mul_cancel f.integral_pos.ne'
| 2 | 7.389056 | 1 | 0.818182 | 11 | 722 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 80 | 82 | theorem support_normed_eq : Function.support (f.normed μ) = Metric.ball c f.rOut := by |
unfold ContDiffBump.normed
rw [support_div, f.support_eq, support_const f.integral_pos.ne', inter_univ]
| 2 | 7.389056 | 1 | 0.818182 | 11 | 722 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 85 | 86 | theorem tsupport_normed_eq : tsupport (f.normed μ) = Metric.closedBall c f.rOut := by |
rw [tsupport, f.support_normed_eq, closure_ball _ f.rOut_pos.ne']
| 1 | 2.718282 | 0 | 0.818182 | 11 | 722 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 89 | 90 | theorem hasCompactSupport_normed : HasCompactSupport (f.normed μ) := by |
simp only [HasCompactSupport, f.tsupport_normed_eq (μ := μ), isCompact_closedBall]
| 1 | 2.718282 | 0 | 0.818182 | 11 | 722 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 93 | 101 | theorem tendsto_support_normed_smallSets {ι} {φ : ι → ContDiffBump c} {l : Filter ι}
(hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)) :
Tendsto (fun i => Function.support fun x => (φ i).normed μ x) l (𝓝 c).smallSets := by |
simp_rw [NormedAddCommGroup.tendsto_nhds_zero, Real.norm_eq_abs,
abs_eq_self.mpr (φ _).rOut_pos.le] at hφ
rw [nhds_basis_ball.smallSets.tendsto_right_iff]
refine fun ε hε ↦ (hφ ε hε).mono fun i hi ↦ ?_
rw [(φ i).support_normed_eq]
exact ball_subset_ball hi.le
| 6 | 403.428793 | 2 | 0.818182 | 11 | 722 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 106 | 108 | theorem integral_normed_smul {X} [NormedAddCommGroup X] [NormedSpace ℝ X]
[CompleteSpace X] (z : X) : ∫ x, f.normed μ x • z ∂μ = z := by |
simp_rw [integral_smul_const, f.integral_normed (μ := μ), one_smul]
| 1 | 2.718282 | 0 | 0.818182 | 11 | 722 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 111 | 115 | theorem measure_closedBall_le_integral : (μ (closedBall c f.rIn)).toReal ≤ ∫ x, f x ∂μ := by | calc
(μ (closedBall c f.rIn)).toReal = ∫ x in closedBall c f.rIn, 1 ∂μ := by simp
_ = ∫ x in closedBall c f.rIn, f x ∂μ := setIntegral_congr measurableSet_closedBall
(fun x hx ↦ (one_of_mem_closedBall f hx).symm)
_ ≤ ∫ x, f x ∂μ := setIntegral_le_integral f.integrable (eventually_of_forall (fun x ↦ f.non... | 5 | 148.413159 | 2 | 0.818182 | 11 | 722 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 117 | 123 | theorem normed_le_div_measure_closedBall_rIn (x : E) :
f.normed μ x ≤ 1 / (μ (closedBall c f.rIn)).toReal := by |
rw [normed_def]
gcongr
· exact ENNReal.toReal_pos (measure_closedBall_pos _ _ f.rIn_pos).ne' measure_closedBall_lt_top.ne
· exact f.le_one
· exact f.measure_closedBall_le_integral μ
| 5 | 148.413159 | 2 | 0.818182 | 11 | 722 |
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.comm_ring from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/CommRing.lean | 96 | 97 | theorem degrees_neg (p : MvPolynomial σ R) : (-p).degrees = p.degrees := by |
rw [degrees, support_neg]; rfl
| 1 | 2.718282 | 0 | 0.833333 | 6 | 723 |
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