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.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
namespace Nat
def dist (n m : ℕ) :=
n - m + (m - n)
#align nat.dist Nat.dist
-- Should be aligned to `Nat.dist.eq_def`, but that is generated on demand and isn't pr... | Mathlib/Data/Nat/Dist.lean | 85 | 89 | theorem dist_eq_intro {n m k l : ℕ} (h : n + m = k + l) : dist n k = dist l m :=
calc
dist n k = dist (n + m) (k + m) := by | rw [dist_add_add_right]
_ = dist (k + l) (k + m) := by rw [h]
_ = dist l m := by rw [dist_add_add_left]
| 3 | 20.085537 | 1 | 0.266667 | 15 | 309 |
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
namespace Nat
def dist (n m : ℕ) :=
n - m + (m - n)
#align nat.dist Nat.dist
-- Should be aligned to `Nat.dist.eq_def`, but that is generated on demand and isn't pr... | Mathlib/Data/Nat/Dist.lean | 92 | 96 | theorem dist.triangle_inequality (n m k : ℕ) : dist n k ≤ dist n m + dist m k := by |
have : dist n m + dist m k = n - m + (m - k) + (k - m + (m - n)) := by
simp [dist, add_comm, add_left_comm, add_assoc]
rw [this, dist]
exact add_le_add tsub_le_tsub_add_tsub tsub_le_tsub_add_tsub
| 4 | 54.59815 | 2 | 0.266667 | 15 | 309 |
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
namespace Nat
def dist (n m : ℕ) :=
n - m + (m - n)
#align nat.dist Nat.dist
-- Should be aligned to `Nat.dist.eq_def`, but that is generated on demand and isn't pr... | Mathlib/Data/Nat/Dist.lean | 99 | 100 | theorem dist_mul_right (n k m : ℕ) : dist (n * k) (m * k) = dist n m * k := by |
rw [dist, dist, right_distrib, tsub_mul n, tsub_mul m]
| 1 | 2.718282 | 0 | 0.266667 | 15 | 309 |
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
namespace Nat
def dist (n m : ℕ) :=
n - m + (m - n)
#align nat.dist Nat.dist
-- Should be aligned to `Nat.dist.eq_def`, but that is generated on demand and isn't pr... | Mathlib/Data/Nat/Dist.lean | 103 | 104 | theorem dist_mul_left (k n m : ℕ) : dist (k * n) (k * m) = k * dist n m := by |
rw [mul_comm k n, mul_comm k m, dist_mul_right, mul_comm]
| 1 | 2.718282 | 0 | 0.266667 | 15 | 309 |
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
namespace Nat
def dist (n m : ℕ) :=
n - m + (m - n)
#align nat.dist Nat.dist
-- Should be aligned to `Nat.dist.eq_def`, but that is generated on demand and isn't pr... | Mathlib/Data/Nat/Dist.lean | 112 | 113 | theorem dist_succ_succ {i j : Nat} : dist (succ i) (succ j) = dist i j := by |
simp [dist, succ_sub_succ]
| 1 | 2.718282 | 0 | 0.266667 | 15 | 309 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
| Mathlib/Order/Interval/Set/OrderIso.lean | 24 | 26 | theorem preimage_Iic (e : α ≃o β) (b : β) : e ⁻¹' Iic b = Iic (e.symm b) := by |
ext x
simp [← e.le_iff_le]
| 2 | 7.389056 | 1 | 0.285714 | 14 | 310 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 30 | 32 | theorem preimage_Ici (e : α ≃o β) (b : β) : e ⁻¹' Ici b = Ici (e.symm b) := by |
ext x
simp [← e.le_iff_le]
| 2 | 7.389056 | 1 | 0.285714 | 14 | 310 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 36 | 38 | theorem preimage_Iio (e : α ≃o β) (b : β) : e ⁻¹' Iio b = Iio (e.symm b) := by |
ext x
simp [← e.lt_iff_lt]
| 2 | 7.389056 | 1 | 0.285714 | 14 | 310 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 42 | 44 | theorem preimage_Ioi (e : α ≃o β) (b : β) : e ⁻¹' Ioi b = Ioi (e.symm b) := by |
ext x
simp [← e.lt_iff_lt]
| 2 | 7.389056 | 1 | 0.285714 | 14 | 310 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 48 | 49 | theorem preimage_Icc (e : α ≃o β) (a b : β) : e ⁻¹' Icc a b = Icc (e.symm a) (e.symm b) := by |
simp [← Ici_inter_Iic]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 310 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 53 | 54 | theorem preimage_Ico (e : α ≃o β) (a b : β) : e ⁻¹' Ico a b = Ico (e.symm a) (e.symm b) := by |
simp [← Ici_inter_Iio]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 310 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 58 | 59 | theorem preimage_Ioc (e : α ≃o β) (a b : β) : e ⁻¹' Ioc a b = Ioc (e.symm a) (e.symm b) := by |
simp [← Ioi_inter_Iic]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 310 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 63 | 64 | theorem preimage_Ioo (e : α ≃o β) (a b : β) : e ⁻¹' Ioo a b = Ioo (e.symm a) (e.symm b) := by |
simp [← Ioi_inter_Iio]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 310 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 68 | 69 | theorem image_Iic (e : α ≃o β) (a : α) : e '' Iic a = Iic (e a) := by |
rw [e.image_eq_preimage, e.symm.preimage_Iic, e.symm_symm]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 310 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 78 | 79 | theorem image_Iio (e : α ≃o β) (a : α) : e '' Iio a = Iio (e a) := by |
rw [e.image_eq_preimage, e.symm.preimage_Iio, e.symm_symm]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 310 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 88 | 89 | theorem image_Ioo (e : α ≃o β) (a b : α) : e '' Ioo a b = Ioo (e a) (e b) := by |
rw [e.image_eq_preimage, e.symm.preimage_Ioo, e.symm_symm]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 310 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 93 | 94 | theorem image_Ioc (e : α ≃o β) (a b : α) : e '' Ioc a b = Ioc (e a) (e b) := by |
rw [e.image_eq_preimage, e.symm.preimage_Ioc, e.symm_symm]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 310 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 98 | 99 | theorem image_Ico (e : α ≃o β) (a b : α) : e '' Ico a b = Ico (e a) (e b) := by |
rw [e.image_eq_preimage, e.symm.preimage_Ico, e.symm_symm]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 310 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 103 | 104 | theorem image_Icc (e : α ≃o β) (a b : α) : e '' Icc a b = Icc (e a) (e b) := by |
rw [e.image_eq_preimage, e.symm.preimage_Icc, e.symm_symm]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 310 |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 48 | 49 | theorem expand_eq_sum {f : R[X]} : expand R p f = f.sum fun e a => C a * (X ^ p) ^ e := by |
simp [expand, eval₂]
| 1 | 2.718282 | 0 | 0.285714 | 7 | 311 |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 65 | 66 | theorem expand_monomial (r : R) : expand R p (monomial q r) = monomial (q * p) r := by |
simp_rw [← smul_X_eq_monomial, AlgHom.map_smul, AlgHom.map_pow, expand_X, mul_comm, pow_mul]
| 1 | 2.718282 | 0 | 0.285714 | 7 | 311 |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 80 | 80 | theorem expand_zero (f : R[X]) : expand R 0 f = C (eval 1 f) := by | simp [expand]
| 1 | 2.718282 | 0 | 0.285714 | 7 | 311 |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 95 | 97 | theorem derivative_expand (f : R[X]) : Polynomial.derivative (expand R p f) =
expand R p (Polynomial.derivative f) * (p * (X ^ (p - 1) : R[X])) := by |
rw [coe_expand, derivative_eval₂_C, derivative_pow, C_eq_natCast, derivative_X, mul_one]
| 1 | 2.718282 | 0 | 0.285714 | 7 | 311 |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 100 | 117 | theorem coeff_expand {p : ℕ} (hp : 0 < p) (f : R[X]) (n : ℕ) :
(expand R p f).coeff n = if p ∣ n then f.coeff (n / p) else 0 := by |
simp only [expand_eq_sum]
simp_rw [coeff_sum, ← pow_mul, C_mul_X_pow_eq_monomial, coeff_monomial, sum]
split_ifs with h
· rw [Finset.sum_eq_single (n / p), Nat.mul_div_cancel' h, if_pos rfl]
· intro b _ hb2
rw [if_neg]
intro hb3
apply hb2
rw [← hb3, Nat.mul_div_cancel_left b hp]
... | 16 | 8,886,110.520508 | 2 | 0.285714 | 7 | 311 |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 121 | 123 | theorem coeff_expand_mul {p : ℕ} (hp : 0 < p) (f : R[X]) (n : ℕ) :
(expand R p f).coeff (n * p) = f.coeff n := by |
rw [coeff_expand hp, if_pos (dvd_mul_left _ _), Nat.mul_div_cancel _ hp]
| 1 | 2.718282 | 0 | 0.285714 | 7 | 311 |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 127 | 128 | theorem coeff_expand_mul' {p : ℕ} (hp : 0 < p) (f : R[X]) (n : ℕ) :
(expand R p f).coeff (p * n) = f.coeff n := by | rw [mul_comm, coeff_expand_mul hp]
| 1 | 2.718282 | 0 | 0.285714 | 7 | 311 |
import Mathlib.Data.Set.Lattice
import Mathlib.Init.Set
import Mathlib.Control.Basic
import Mathlib.Lean.Expr.ExtraRecognizers
#align_import data.set.functor from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u
open Function
namespace Set
variable {α β : Type u} {s : Set α} ... | Mathlib/Data/Set/Functor.lean | 65 | 68 | theorem image2_def {α β γ : Type u} (f : α → β → γ) (s : Set α) (t : Set β) :
image2 f s t = f <$> s <*> t := by |
ext
simp
| 2 | 7.389056 | 1 | 0.285714 | 7 | 312 |
import Mathlib.Data.Set.Lattice
import Mathlib.Init.Set
import Mathlib.Control.Basic
import Mathlib.Lean.Expr.ExtraRecognizers
#align_import data.set.functor from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u
open Function
namespace Set
variable {α β : Type u} {s : Set α} ... | Mathlib/Data/Set/Functor.lean | 93 | 94 | theorem coe_subset : (γ : Set α) ⊆ β := by |
intro _ ⟨_, ⟨⟨⟨_, ha⟩, rfl⟩, _, ⟨_, rfl⟩, _⟩⟩; convert ha
| 1 | 2.718282 | 0 | 0.285714 | 7 | 312 |
import Mathlib.Data.Set.Lattice
import Mathlib.Init.Set
import Mathlib.Control.Basic
import Mathlib.Lean.Expr.ExtraRecognizers
#align_import data.set.functor from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u
open Function
namespace Set
variable {α β : Type u} {s : Set α} ... | Mathlib/Data/Set/Functor.lean | 96 | 97 | theorem mem_of_mem_coe {a : α} (ha : a ∈ (γ : Set α)) : ⟨a, coe_subset ha⟩ ∈ γ := by |
rcases ha with ⟨_, ⟨_, rfl⟩, _, ⟨ha, rfl⟩, _⟩; convert ha
| 1 | 2.718282 | 0 | 0.285714 | 7 | 312 |
import Mathlib.Data.Set.Lattice
import Mathlib.Init.Set
import Mathlib.Control.Basic
import Mathlib.Lean.Expr.ExtraRecognizers
#align_import data.set.functor from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u
open Function
namespace Set
variable {α β : Type u} {s : Set α} ... | Mathlib/Data/Set/Functor.lean | 135 | 139 | theorem coe_eq_image_val (t : Set s) :
@Lean.Internal.coeM Set s α _ Set.monad t = (t : Set α) := by |
change ⋃ (x ∈ t), {x.1} = _
ext
simp
| 3 | 20.085537 | 1 | 0.285714 | 7 | 312 |
import Mathlib.Data.Set.Lattice
import Mathlib.Init.Set
import Mathlib.Control.Basic
import Mathlib.Lean.Expr.ExtraRecognizers
#align_import data.set.functor from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u
open Function
namespace Set
variable {α β : Type u} {s : Set α} ... | Mathlib/Data/Set/Functor.lean | 146 | 147 | theorem image_val_subset : (γ : Set α) ⊆ β := by |
rintro _ ⟨⟨_, ha⟩, _, rfl⟩; exact ha
| 1 | 2.718282 | 0 | 0.285714 | 7 | 312 |
import Mathlib.Data.Set.Lattice
import Mathlib.Init.Set
import Mathlib.Control.Basic
import Mathlib.Lean.Expr.ExtraRecognizers
#align_import data.set.functor from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u
open Function
namespace Set
variable {α β : Type u} {s : Set α} ... | Mathlib/Data/Set/Functor.lean | 149 | 150 | theorem mem_of_mem_image_val (ha : a ∈ (γ : Set α)) : ⟨a, image_val_subset ha⟩ ∈ γ := by |
rcases ha with ⟨_, ha, rfl⟩; exact ha
| 1 | 2.718282 | 0 | 0.285714 | 7 | 312 |
import Mathlib.Data.Set.Lattice
import Mathlib.Init.Set
import Mathlib.Control.Basic
import Mathlib.Lean.Expr.ExtraRecognizers
#align_import data.set.functor from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u
open Function
namespace Set
variable {α β : Type u} {s : Set α} ... | Mathlib/Data/Set/Functor.lean | 155 | 156 | theorem image_image_val_eq_restrict_image {δ : Type*} {f : α → δ} : f '' γ = β.restrict f '' γ := by |
ext; simp
| 1 | 2.718282 | 0 | 0.285714 | 7 | 312 |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Sup
-- can be defined with just `[Bot α]` where some lemmas hold without... | Mathlib/Data/Multiset/Lattice.lean | 79 | 80 | theorem sup_ndunion (s₁ s₂ : Multiset α) : (ndunion s₁ s₂).sup = s₁.sup ⊔ s₂.sup := by |
rw [← sup_dedup, dedup_ext.2, sup_dedup, sup_add]; simp
| 1 | 2.718282 | 0 | 0.285714 | 7 | 313 |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Sup
-- can be defined with just `[Bot α]` where some lemmas hold without... | Mathlib/Data/Multiset/Lattice.lean | 84 | 85 | theorem sup_union (s₁ s₂ : Multiset α) : (s₁ ∪ s₂).sup = s₁.sup ⊔ s₂.sup := by |
rw [← sup_dedup, dedup_ext.2, sup_dedup, sup_add]; simp
| 1 | 2.718282 | 0 | 0.285714 | 7 | 313 |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Sup
-- can be defined with just `[Bot α]` where some lemmas hold without... | Mathlib/Data/Multiset/Lattice.lean | 89 | 90 | theorem sup_ndinsert (a : α) (s : Multiset α) : (ndinsert a s).sup = a ⊔ s.sup := by |
rw [← sup_dedup, dedup_ext.2, sup_dedup, sup_cons]; simp
| 1 | 2.718282 | 0 | 0.285714 | 7 | 313 |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Sup
-- can be defined with just `[Bot α]` where some lemmas hold without... | Mathlib/Data/Multiset/Lattice.lean | 93 | 99 | theorem nodup_sup_iff {α : Type*} [DecidableEq α] {m : Multiset (Multiset α)} :
m.sup.Nodup ↔ ∀ a : Multiset α, a ∈ m → a.Nodup := by |
-- Porting note: this was originally `apply m.induction_on`, which failed due to
-- `failed to elaborate eliminator, expected type is not available`
induction' m using Multiset.induction_on with _ _ h
· simp
· simp [h]
| 5 | 148.413159 | 2 | 0.285714 | 7 | 313 |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Inf
-- can be defined with just `[Top α]` where some lemmas hold with... | Mathlib/Data/Multiset/Lattice.lean | 163 | 164 | theorem inf_ndunion (s₁ s₂ : Multiset α) : (ndunion s₁ s₂).inf = s₁.inf ⊓ s₂.inf := by |
rw [← inf_dedup, dedup_ext.2, inf_dedup, inf_add]; simp
| 1 | 2.718282 | 0 | 0.285714 | 7 | 313 |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Inf
-- can be defined with just `[Top α]` where some lemmas hold with... | Mathlib/Data/Multiset/Lattice.lean | 168 | 169 | theorem inf_union (s₁ s₂ : Multiset α) : (s₁ ∪ s₂).inf = s₁.inf ⊓ s₂.inf := by |
rw [← inf_dedup, dedup_ext.2, inf_dedup, inf_add]; simp
| 1 | 2.718282 | 0 | 0.285714 | 7 | 313 |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Inf
-- can be defined with just `[Top α]` where some lemmas hold with... | Mathlib/Data/Multiset/Lattice.lean | 173 | 174 | theorem inf_ndinsert (a : α) (s : Multiset α) : (ndinsert a s).inf = a ⊓ s.inf := by |
rw [← inf_dedup, dedup_ext.2, inf_dedup, inf_cons]; simp
| 1 | 2.718282 | 0 | 0.285714 | 7 | 313 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 140 | 140 | theorem zero'_eval : zero'.eval = fun v => pure (0 :: v) := by | simp [eval]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 314 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 143 | 143 | theorem succ_eval : succ.eval = fun v => pure [v.headI.succ] := by | simp [eval]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 314 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 146 | 146 | theorem tail_eval : tail.eval = fun v => pure v.tail := by | simp [eval]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 314 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 149 | 152 | theorem cons_eval (f fs) : (cons f fs).eval = fun v => do {
let n ← Code.eval f v
let ns ← Code.eval fs v
pure (n.headI :: ns) } := by | simp [eval]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 314 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 155 | 155 | theorem comp_eval (f g) : (comp f g).eval = fun v => g.eval v >>= f.eval := by | simp [eval]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 314 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 158 | 160 | theorem case_eval (f g) :
(case f g).eval = fun v => v.headI.rec (f.eval v.tail) fun y _ => g.eval (y::v.tail) := by |
simp [eval]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 314 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 163 | 166 | theorem fix_eval (f) : (fix f).eval =
PFun.fix fun v => (f.eval v).map fun v =>
if v.headI = 0 then Sum.inl v.tail else Sum.inr v.tail := by |
simp [eval]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 314 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 174 | 174 | theorem nil_eval (v) : nil.eval v = pure [] := by | simp [nil]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 314 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 183 | 183 | theorem id_eval (v) : id.eval v = pure v := by | simp [id]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 314 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 192 | 192 | theorem head_eval (v) : head.eval v = pure [v.headI] := by | simp [head]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 314 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 201 | 201 | theorem zero_eval (v) : zero.eval v = pure [0] := by | simp [zero]
| 1 | 2.718282 | 0 | 0.285714 | 14 | 314 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 211 | 212 | theorem pred_eval (v) : pred.eval v = pure [v.headI.pred] := by |
simp [pred]; cases v.headI <;> simp
| 1 | 2.718282 | 0 | 0.285714 | 14 | 314 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 264 | 282 | theorem exists_code.comp {m n} {f : Vector ℕ n →. ℕ} {g : Fin n → Vector ℕ m →. ℕ}
(hf : ∃ c : Code, ∀ v : Vector ℕ n, c.eval v.1 = pure <$> f v)
(hg : ∀ i, ∃ c : Code, ∀ v : Vector ℕ m, c.eval v.1 = pure <$> g i v) :
∃ c : Code, ∀ v : Vector ℕ m, c.eval v.1 = pure <$> ((Vector.mOfFn fun i => g i v) >>= f) ... |
rsuffices ⟨cg, hg⟩ :
∃ c : Code, ∀ v : Vector ℕ m, c.eval v.1 = Subtype.val <$> Vector.mOfFn fun i => g i v
· obtain ⟨cf, hf⟩ := hf
exact
⟨cf.comp cg, fun v => by
simp [hg, hf, map_bind, seq_bind_eq, Function.comp]
rfl⟩
clear hf f; induction' n with n IH
· exact ⟨nil, fun v => by ... | 15 | 3,269,017.372472 | 2 | 0.285714 | 14 | 314 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 550 | 554 | theorem Cont.then_eval {k k' : Cont} {v} : (k.then k').eval v = k.eval v >>= k'.eval := by |
induction' k with _ _ _ _ _ _ _ _ _ k_ih _ _ k_ih generalizing v <;>
simp only [Cont.eval, Cont.then, bind_assoc, pure_bind, *]
· simp only [← k_ih]
· split_ifs <;> [rfl; simp only [← k_ih, bind_assoc]]
| 4 | 54.59815 | 2 | 0.285714 | 14 | 314 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 64 | 65 | theorem apply_diag_ne [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i : V) :
¬A i i = 1 := by | simp [h.apply_diag i]
| 1 | 2.718282 | 0 | 0.285714 | 7 | 315 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 69 | 70 | theorem apply_ne_one_iff [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i j : V) :
¬A i j = 1 ↔ A i j = 0 := by | obtain h | h := h.zero_or_one i j <;> simp [h]
| 1 | 2.718282 | 0 | 0.285714 | 7 | 315 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 74 | 75 | theorem apply_ne_zero_iff [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i j : V) :
¬A i j = 0 ↔ A i j = 1 := by | rw [← apply_ne_one_iff h, Classical.not_not]
| 1 | 2.718282 | 0 | 0.285714 | 7 | 315 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 105 | 105 | theorem compl_apply_diag [Zero α] [One α] (i : V) : A.compl i i = 0 := by | simp [compl]
| 1 | 2.718282 | 0 | 0.285714 | 7 | 315 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 109 | 111 | theorem compl_apply [Zero α] [One α] (i j : V) : A.compl i j = 0 ∨ A.compl i j = 1 := by |
unfold compl
split_ifs <;> simp
| 2 | 7.389056 | 1 | 0.285714 | 7 | 315 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 115 | 117 | theorem isSymm_compl [Zero α] [One α] (h : A.IsSymm) : A.compl.IsSymm := by |
ext
simp [compl, h.apply, eq_comm]
| 2 | 7.389056 | 1 | 0.285714 | 7 | 315 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 121 | 122 | theorem isAdjMatrix_compl [Zero α] [One α] (h : A.IsSymm) : IsAdjMatrix A.compl :=
{ symm := by | simp [h] }
| 1 | 2.718282 | 0 | 0.285714 | 7 | 315 |
import Batteries.Data.Sum.Basic
import Batteries.Logic
open Function
namespace Sum
@[simp] protected theorem «forall» {p : α ⊕ β → Prop} :
(∀ x, p x) ↔ (∀ a, p (inl a)) ∧ ∀ b, p (inr b) :=
⟨fun h => ⟨fun _ => h _, fun _ => h _⟩, fun ⟨h₁, h₂⟩ => Sum.rec h₁ h₂⟩
@[simp] protected theorem «exists» {p : α ⊕ β ... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 33 | 38 | theorem forall_sum {γ : α ⊕ β → Sort _} (p : (∀ ab, γ ab) → Prop) :
(∀ fab, p fab) ↔ (∀ fa fb, p (Sum.rec fa fb)) := by |
refine ⟨fun h fa fb => h _, fun h fab => ?_⟩
have h1 : fab = Sum.rec (fun a => fab (Sum.inl a)) (fun b => fab (Sum.inr b)) := by
ext ab; cases ab <;> rfl
rw [h1]; exact h _ _
| 4 | 54.59815 | 2 | 0.285714 | 7 | 316 |
import Batteries.Data.Sum.Basic
import Batteries.Logic
open Function
namespace Sum
@[simp] protected theorem «forall» {p : α ⊕ β → Prop} :
(∀ x, p x) ↔ (∀ a, p (inl a)) ∧ ∀ b, p (inr b) :=
⟨fun h => ⟨fun _ => h _, fun _ => h _⟩, fun ⟨h₁, h₂⟩ => Sum.rec h₁ h₂⟩
@[simp] protected theorem «exists» {p : α ⊕ β ... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 75 | 75 | theorem not_isLeft {x : α ⊕ β} : ¬x.isLeft ↔ x.isRight := by | simp
| 1 | 2.718282 | 0 | 0.285714 | 7 | 316 |
import Batteries.Data.Sum.Basic
import Batteries.Logic
open Function
namespace Sum
@[simp] protected theorem «forall» {p : α ⊕ β → Prop} :
(∀ x, p x) ↔ (∀ a, p (inl a)) ∧ ∀ b, p (inr b) :=
⟨fun h => ⟨fun _ => h _, fun _ => h _⟩, fun ⟨h₁, h₂⟩ => Sum.rec h₁ h₂⟩
@[simp] protected theorem «exists» {p : α ⊕ β ... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 81 | 81 | theorem not_isRight {x : α ⊕ β} : ¬x.isRight ↔ x.isLeft := by | simp
| 1 | 2.718282 | 0 | 0.285714 | 7 | 316 |
import Batteries.Data.Sum.Basic
import Batteries.Logic
open Function
namespace Sum
@[simp] protected theorem «forall» {p : α ⊕ β → Prop} :
(∀ x, p x) ↔ (∀ a, p (inl a)) ∧ ∀ b, p (inr b) :=
⟨fun h => ⟨fun _ => h _, fun _ => h _⟩, fun ⟨h₁, h₂⟩ => Sum.rec h₁ h₂⟩
@[simp] protected theorem «exists» {p : α ⊕ β ... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 83 | 83 | theorem isLeft_iff : x.isLeft ↔ ∃ y, x = Sum.inl y := by | cases x <;> simp
| 1 | 2.718282 | 0 | 0.285714 | 7 | 316 |
import Batteries.Data.Sum.Basic
import Batteries.Logic
open Function
namespace Sum
@[simp] protected theorem «forall» {p : α ⊕ β → Prop} :
(∀ x, p x) ↔ (∀ a, p (inl a)) ∧ ∀ b, p (inr b) :=
⟨fun h => ⟨fun _ => h _, fun _ => h _⟩, fun ⟨h₁, h₂⟩ => Sum.rec h₁ h₂⟩
@[simp] protected theorem «exists» {p : α ⊕ β ... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 85 | 85 | theorem isRight_iff : x.isRight ↔ ∃ y, x = Sum.inr y := by | cases x <;> simp
| 1 | 2.718282 | 0 | 0.285714 | 7 | 316 |
import Batteries.Data.Sum.Basic
import Batteries.Logic
open Function
namespace Sum
@[simp] protected theorem «forall» {p : α ⊕ β → Prop} :
(∀ x, p x) ↔ (∀ a, p (inl a)) ∧ ∀ b, p (inr b) :=
⟨fun h => ⟨fun _ => h _, fun _ => h _⟩, fun ⟨h₁, h₂⟩ => Sum.rec h₁ h₂⟩
@[simp] protected theorem «exists» {p : α ⊕ β ... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 116 | 118 | theorem elim_eq_iff {u u' : α → γ} {v v' : β → γ} :
Sum.elim u v = Sum.elim u' v' ↔ u = u' ∧ v = v' := by |
simp [funext_iff]
| 1 | 2.718282 | 0 | 0.285714 | 7 | 316 |
import Batteries.Data.Sum.Basic
import Batteries.Logic
open Function
namespace Sum
@[simp] protected theorem «forall» {p : α ⊕ β → Prop} :
(∀ x, p x) ↔ (∀ a, p (inl a)) ∧ ∀ b, p (inr b) :=
⟨fun h => ⟨fun _ => h _, fun _ => h _⟩, fun ⟨h₁, h₂⟩ => Sum.rec h₁ h₂⟩
@[simp] protected theorem «exists» {p : α ⊕ β ... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 134 | 136 | theorem elim_map {f₁ : α → β} {f₂ : β → ε} {g₁ : γ → δ} {g₂ : δ → ε} {x} :
Sum.elim f₂ g₂ (Sum.map f₁ g₁ x) = Sum.elim (f₂ ∘ f₁) (g₂ ∘ g₁) x := by |
cases x <;> rfl
| 1 | 2.718282 | 0 | 0.285714 | 7 | 316 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 57 | 73 | theorem coeff_derivative (p : R[X]) (n : ℕ) :
coeff (derivative p) n = coeff p (n + 1) * (n + 1) := by |
rw [derivative_apply]
simp only [coeff_X_pow, coeff_sum, coeff_C_mul]
rw [sum, Finset.sum_eq_single (n + 1)]
· simp only [Nat.add_succ_sub_one, add_zero, mul_one, if_true, eq_self_iff_true]; norm_cast
· intro b
cases b
· intros
rw [Nat.cast_zero, mul_zero, zero_mul]
· intro _ H
rw [Na... | 15 | 3,269,017.372472 | 2 | 0.3 | 10 | 317 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 86 | 89 | theorem derivative_monomial (a : R) (n : ℕ) :
derivative (monomial n a) = monomial (n - 1) (a * n) := by |
rw [derivative_apply, sum_monomial_index, C_mul_X_pow_eq_monomial]
simp
| 2 | 7.389056 | 1 | 0.3 | 10 | 317 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 92 | 93 | theorem derivative_C_mul_X (a : R) : derivative (C a * X) = C a := by |
simp [C_mul_X_eq_monomial, derivative_monomial, Nat.cast_one, mul_one]
| 1 | 2.718282 | 0 | 0.3 | 10 | 317 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 97 | 99 | theorem derivative_C_mul_X_pow (a : R) (n : ℕ) :
derivative (C a * X ^ n) = C (a * n) * X ^ (n - 1) := by |
rw [C_mul_X_pow_eq_monomial, C_mul_X_pow_eq_monomial, derivative_monomial]
| 1 | 2.718282 | 0 | 0.3 | 10 | 317 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 103 | 104 | theorem derivative_C_mul_X_sq (a : R) : derivative (C a * X ^ 2) = C (a * 2) * X := by |
rw [derivative_C_mul_X_pow, Nat.cast_two, pow_one]
| 1 | 2.718282 | 0 | 0.3 | 10 | 317 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 109 | 110 | theorem derivative_X_pow (n : ℕ) : derivative (X ^ n : R[X]) = C (n : R) * X ^ (n - 1) := by |
convert derivative_C_mul_X_pow (1 : R) n <;> simp
| 1 | 2.718282 | 0 | 0.3 | 10 | 317 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 115 | 116 | theorem derivative_X_sq : derivative (X ^ 2 : R[X]) = C 2 * X := by |
rw [derivative_X_pow, Nat.cast_two, pow_one]
| 1 | 2.718282 | 0 | 0.3 | 10 | 317 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 121 | 121 | theorem derivative_C {a : R} : derivative (C a) = 0 := by | simp [derivative_apply]
| 1 | 2.718282 | 0 | 0.3 | 10 | 317 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 125 | 126 | theorem derivative_of_natDegree_zero {p : R[X]} (hp : p.natDegree = 0) : derivative p = 0 := by |
rw [eq_C_of_natDegree_eq_zero hp, derivative_C]
| 1 | 2.718282 | 0 | 0.3 | 10 | 317 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 149 | 150 | theorem derivative_X_add_C (c : R) : derivative (X + C c) = 1 := by |
rw [derivative_add, derivative_X, derivative_C, add_zero]
| 1 | 2.718282 | 0 | 0.3 | 10 | 317 |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section Union
#align list.nil_union List.nil_unio... | Mathlib/Data/List/Lattice.lean | 109 | 110 | theorem forall_mem_union : (∀ x ∈ l₁ ∪ l₂, p x) ↔ (∀ x ∈ l₁, p x) ∧ ∀ x ∈ l₂, p x := by |
simp only [mem_union_iff, or_imp, forall_and]
| 1 | 2.718282 | 0 | 0.3 | 10 | 318 |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section Inter
@[simp]
theorem inter_nil (l : L... | Mathlib/Data/List/Lattice.lean | 134 | 135 | theorem inter_cons_of_mem (l₁ : List α) (h : a ∈ l₂) : (a :: l₁) ∩ l₂ = a :: l₁ ∩ l₂ := by |
simp [Inter.inter, List.inter, h]
| 1 | 2.718282 | 0 | 0.3 | 10 | 318 |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section Inter
@[simp]
theorem inter_nil (l : L... | Mathlib/Data/List/Lattice.lean | 139 | 140 | theorem inter_cons_of_not_mem (l₁ : List α) (h : a ∉ l₂) : (a :: l₁) ∩ l₂ = l₁ ∩ l₂ := by |
simp [Inter.inter, List.inter, h]
| 1 | 2.718282 | 0 | 0.3 | 10 | 318 |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section Inter
@[simp]
theorem inter_nil (l : L... | Mathlib/Data/List/Lattice.lean | 147 | 147 | theorem mem_of_mem_inter_right (h : a ∈ l₁ ∩ l₂) : a ∈ l₂ := by | simpa using of_mem_filter h
| 1 | 2.718282 | 0 | 0.3 | 10 | 318 |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section Inter
@[simp]
theorem inter_nil (l : L... | Mathlib/Data/List/Lattice.lean | 167 | 169 | theorem inter_eq_nil_iff_disjoint : l₁ ∩ l₂ = [] ↔ Disjoint l₁ l₂ := by |
simp only [eq_nil_iff_forall_not_mem, mem_inter_iff, not_and]
rfl
| 2 | 7.389056 | 1 | 0.3 | 10 | 318 |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section Inter
@[simp]
theorem inter_nil (l : L... | Mathlib/Data/List/Lattice.lean | 183 | 184 | theorem inter_reverse {xs ys : List α} : xs.inter ys.reverse = xs.inter ys := by |
simp only [List.inter, elem_eq_mem, mem_reverse]
| 1 | 2.718282 | 0 | 0.3 | 10 | 318 |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section BagInter
@[simp]
| Mathlib/Data/List/Lattice.lean | 195 | 195 | theorem nil_bagInter (l : List α) : [].bagInter l = [] := by | cases l <;> rfl
| 1 | 2.718282 | 0 | 0.3 | 10 | 318 |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section BagInter
@[simp]
theorem nil_bagInt... | Mathlib/Data/List/Lattice.lean | 199 | 199 | theorem bagInter_nil (l : List α) : l.bagInter [] = [] := by | cases l <;> rfl
| 1 | 2.718282 | 0 | 0.3 | 10 | 318 |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section BagInter
@[simp]
theorem nil_bagInt... | Mathlib/Data/List/Lattice.lean | 203 | 207 | theorem cons_bagInter_of_pos (l₁ : List α) (h : a ∈ l₂) :
(a :: l₁).bagInter l₂ = a :: l₁.bagInter (l₂.erase a) := by |
cases l₂
· exact if_pos h
· simp only [List.bagInter, if_pos (elem_eq_true_of_mem h)]
| 3 | 20.085537 | 1 | 0.3 | 10 | 318 |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section BagInter
@[simp]
theorem nil_bagInt... | Mathlib/Data/List/Lattice.lean | 211 | 214 | theorem cons_bagInter_of_neg (l₁ : List α) (h : a ∉ l₂) :
(a :: l₁).bagInter l₂ = l₁.bagInter l₂ := by |
cases l₂; · simp only [bagInter_nil]
simp only [erase_of_not_mem h, List.bagInter, if_neg (mt mem_of_elem_eq_true h)]
| 2 | 7.389056 | 1 | 0.3 | 10 | 318 |
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 83 | 84 | theorem intervalIntegrable_iff : IntervalIntegrable f μ a b ↔ IntegrableOn f (Ι a b) μ := by |
rw [uIoc_eq_union, integrableOn_union, IntervalIntegrable]
| 1 | 2.718282 | 0 | 0.3 | 10 | 319 |
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 93 | 95 | theorem intervalIntegrable_iff_integrableOn_Ioc_of_le (hab : a ≤ b) :
IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioc a b) μ := by |
rw [intervalIntegrable_iff, uIoc_of_le hab]
| 1 | 2.718282 | 0 | 0.3 | 10 | 319 |
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 98 | 100 | theorem intervalIntegrable_iff' [NoAtoms μ] :
IntervalIntegrable f μ a b ↔ IntegrableOn f (uIcc a b) μ := by |
rw [intervalIntegrable_iff, ← Icc_min_max, uIoc, integrableOn_Icc_iff_integrableOn_Ioc]
| 1 | 2.718282 | 0 | 0.3 | 10 | 319 |
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 103 | 105 | theorem intervalIntegrable_iff_integrableOn_Icc_of_le {f : ℝ → E} {a b : ℝ} (hab : a ≤ b)
{μ : Measure ℝ} [NoAtoms μ] : IntervalIntegrable f μ a b ↔ IntegrableOn f (Icc a b) μ := by |
rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hab, integrableOn_Icc_iff_integrableOn_Ioc]
| 1 | 2.718282 | 0 | 0.3 | 10 | 319 |
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 108 | 110 | theorem intervalIntegrable_iff_integrableOn_Ico_of_le [NoAtoms μ] (hab : a ≤ b) :
IntervalIntegrable f μ a b ↔ IntegrableOn f (Ico a b) μ := by |
rw [intervalIntegrable_iff_integrableOn_Icc_of_le hab, integrableOn_Icc_iff_integrableOn_Ico]
| 1 | 2.718282 | 0 | 0.3 | 10 | 319 |
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 112 | 114 | theorem intervalIntegrable_iff_integrableOn_Ioo_of_le [NoAtoms μ] (hab : a ≤ b) :
IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioo a b) μ := by |
rw [intervalIntegrable_iff_integrableOn_Icc_of_le hab, integrableOn_Icc_iff_integrableOn_Ioo]
| 1 | 2.718282 | 0 | 0.3 | 10 | 319 |
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 129 | 131 | theorem intervalIntegrable_const_iff {c : E} :
IntervalIntegrable (fun _ => c) μ a b ↔ c = 0 ∨ μ (Ι a b) < ∞ := by |
simp only [intervalIntegrable_iff, integrableOn_const]
| 1 | 2.718282 | 0 | 0.3 | 10 | 319 |
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 161 | 161 | theorem refl : IntervalIntegrable f μ a a := by | constructor <;> simp
| 1 | 2.718282 | 0 | 0.3 | 10 | 319 |
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 171 | 178 | theorem trans_iterate_Ico {a : ℕ → ℝ} {m n : ℕ} (hmn : m ≤ n)
(hint : ∀ k ∈ Ico m n, IntervalIntegrable f μ (a k) (a <| k + 1)) :
IntervalIntegrable f μ (a m) (a n) := by |
revert hint
refine Nat.le_induction ?_ ?_ n hmn
· simp
· intro p hp IH h
exact (IH fun k hk => h k (Ico_subset_Ico_right p.le_succ hk)).trans (h p (by simp [hp]))
| 5 | 148.413159 | 2 | 0.3 | 10 | 319 |
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 401 | 404 | theorem MonotoneOn.intervalIntegrable {u : ℝ → E} {a b : ℝ} (hu : MonotoneOn u (uIcc a b)) :
IntervalIntegrable u μ a b := by |
rw [intervalIntegrable_iff]
exact (hu.integrableOn_isCompact isCompact_uIcc).mono_set Ioc_subset_Icc_self
| 2 | 7.389056 | 1 | 0.3 | 10 | 319 |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 66 | 67 | theorem lintegral_nnnorm_eq_lintegral_edist (f : α → β) :
∫⁻ a, ‖f a‖₊ ∂μ = ∫⁻ a, edist (f a) 0 ∂μ := by | simp only [edist_eq_coe_nnnorm]
| 1 | 2.718282 | 0 | 0.3 | 10 | 320 |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 70 | 72 | theorem lintegral_norm_eq_lintegral_edist (f : α → β) :
∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ = ∫⁻ a, edist (f a) 0 ∂μ := by |
simp only [ofReal_norm_eq_coe_nnnorm, edist_eq_coe_nnnorm]
| 1 | 2.718282 | 0 | 0.3 | 10 | 320 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.