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.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
variable
{π : Type*} [NontriviallyNormedField π]
{F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
{R : Type*} [Semi... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 40 | 46 | theorem iteratedDerivWithin_const_add (hn : 0 < n) (c : F) :
iteratedDerivWithin n (fun z => c + f z) s x = iteratedDerivWithin n f s x := by |
obtain β¨n, rflβ© := n.exists_eq_succ_of_ne_zero hn.ne'
rw [iteratedDerivWithin_succ' h hx, iteratedDerivWithin_succ' h hx]
refine iteratedDerivWithin_congr h ?_ hx
intro y hy
exact derivWithin_const_add (h.uniqueDiffWithinAt hy) _
| 5 | 148.413159 | 2 | 1.2 | 10 | 1,290 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
variable
{π : Type*} [NontriviallyNormedField π]
{F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
{R : Type*} [Semi... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 48 | 56 | theorem iteratedDerivWithin_const_neg (hn : 0 < n) (c : F) :
iteratedDerivWithin n (fun z => c - f z) s x = iteratedDerivWithin n (fun z => -f z) s x := by |
obtain β¨n, rflβ© := n.exists_eq_succ_of_ne_zero hn.ne'
rw [iteratedDerivWithin_succ' h hx, iteratedDerivWithin_succ' h hx]
refine iteratedDerivWithin_congr h ?_ hx
intro y hy
have : UniqueDiffWithinAt π s y := h.uniqueDiffWithinAt hy
rw [derivWithin.neg this]
exact derivWithin_const_sub this _
| 7 | 1,096.633158 | 2 | 1.2 | 10 | 1,290 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
variable
{π : Type*} [NontriviallyNormedField π]
{F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
{R : Type*} [Semi... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 58 | 62 | theorem iteratedDerivWithin_const_smul (c : R) (hf : ContDiffOn π n f s) :
iteratedDerivWithin n (c β’ f) s x = c β’ iteratedDerivWithin n f s x := by |
simp_rw [iteratedDerivWithin]
rw [iteratedFDerivWithin_const_smul_apply hf h hx]
simp only [ContinuousMultilinearMap.smul_apply]
| 3 | 20.085537 | 1 | 1.2 | 10 | 1,290 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
variable
{π : Type*} [NontriviallyNormedField π]
{F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
{R : Type*} [Semi... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 64 | 66 | theorem iteratedDerivWithin_const_mul (c : π) {f : π β π} (hf : ContDiffOn π n f s) :
iteratedDerivWithin n (fun z => c * f z) s x = c * iteratedDerivWithin n f s x := by |
simpa using iteratedDerivWithin_const_smul (F := π) hx h c hf
| 1 | 2.718282 | 0 | 1.2 | 10 | 1,290 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
variable
{π : Type*} [NontriviallyNormedField π]
{F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
{R : Type*} [Semi... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 69 | 72 | theorem iteratedDerivWithin_neg :
iteratedDerivWithin n (-f) s x = -iteratedDerivWithin n f s x := by |
rw [iteratedDerivWithin, iteratedDerivWithin, iteratedFDerivWithin_neg_apply h hx,
ContinuousMultilinearMap.neg_apply]
| 2 | 7.389056 | 1 | 1.2 | 10 | 1,290 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
variable
{π : Type*} [NontriviallyNormedField π]
{F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
{R : Type*} [Semi... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 79 | 83 | theorem iteratedDerivWithin_sub (hf : ContDiffOn π n f s) (hg : ContDiffOn π n g s) :
iteratedDerivWithin n (f - g) s x =
iteratedDerivWithin n f s x - iteratedDerivWithin n g s x := by |
rw [sub_eq_add_neg, sub_eq_add_neg, Pi.neg_def, iteratedDerivWithin_add hx h hf hg.neg,
iteratedDerivWithin_neg' hx h]
| 2 | 7.389056 | 1 | 1.2 | 10 | 1,290 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
variable
{π : Type*} [NontriviallyNormedField π]
{F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
{R : Type*} [Semi... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 85 | 100 | theorem iteratedDeriv_const_smul {n : β} {f : π β F} (h : ContDiff π n f) (c : π) :
iteratedDeriv n (fun x => f (c * x)) = fun x => c ^ n β’ iteratedDeriv n f (c * x) := by |
induction n with
| zero => simp
| succ n ih =>
funext x
have hβ : DifferentiableAt π (iteratedDeriv n f) (c * x) :=
h.differentiable_iteratedDeriv n (Nat.cast_lt.mpr n.lt_succ_self) |>.differentiableAt
have hβ : DifferentiableAt π (fun x => iteratedDeriv n f (c * x)) x := by
rw [β Funct... | 14 | 1,202,604.284165 | 2 | 1.2 | 10 | 1,290 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
variable
{π : Type*} [NontriviallyNormedField π]
{F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
{R : Type*} [Semi... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 102 | 104 | theorem iteratedDeriv_const_mul {n : β} {f : π β π} (h : ContDiff π n f) (c : π) :
iteratedDeriv n (fun x => f (c * x)) = fun x => c ^ n * iteratedDeriv n f (c * x) := by |
simpa only [smul_eq_mul] using iteratedDeriv_const_smul h c
| 1 | 2.718282 | 0 | 1.2 | 10 | 1,290 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ΞΉ : Type*}
namespace Finset
section Sigma
variable {Ξ± : ΞΉ β Type*} {Ξ² : Type*} (s sβ sβ : Finset ΞΉ) (... | Mathlib/Data/Finset/Sigma.lean | 60 | 60 | theorem sigma_nonempty : (s.sigma t).Nonempty β β i β s, (t i).Nonempty := by | simp [Finset.Nonempty]
| 1 | 2.718282 | 0 | 1.214286 | 14 | 1,292 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ΞΉ : Type*}
namespace Finset
section Sigma
variable {Ξ± : ΞΉ β Type*} {Ξ² : Type*} (s sβ sβ : Finset ΞΉ) (... | Mathlib/Data/Finset/Sigma.lean | 64 | 65 | theorem sigma_eq_empty : s.sigma t = β
β β i β s, t i = β
:= by |
simp only [β not_nonempty_iff_eq_empty, sigma_nonempty, not_exists, not_and]
| 1 | 2.718282 | 0 | 1.214286 | 14 | 1,292 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ΞΉ : Type*}
namespace Finset
section Sigma
variable {Ξ± : ΞΉ β Type*} {Ξ² : Type*} (s sβ sβ : Finset ΞΉ) (... | Mathlib/Data/Finset/Sigma.lean | 75 | 81 | theorem pairwiseDisjoint_map_sigmaMk :
(s : Set ΞΉ).PairwiseDisjoint fun i => (t i).map (Embedding.sigmaMk i) := by |
intro i _ j _ hij
rw [Function.onFun, disjoint_left]
simp_rw [mem_map, Function.Embedding.sigmaMk_apply]
rintro _ β¨y, _, rflβ© β¨z, _, hz'β©
exact hij (congr_arg Sigma.fst hz'.symm)
| 5 | 148.413159 | 2 | 1.214286 | 14 | 1,292 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ΞΉ : Type*}
namespace Finset
section Sigma
variable {Ξ± : ΞΉ β Type*} {Ξ² : Type*} (s sβ sβ : Finset ΞΉ) (... | Mathlib/Data/Finset/Sigma.lean | 91 | 94 | theorem sigma_eq_biUnion [DecidableEq (Ξ£i, Ξ± i)] (s : Finset ΞΉ) (t : β i, Finset (Ξ± i)) :
s.sigma t = s.biUnion fun i => (t i).map <| Embedding.sigmaMk i := by |
ext β¨x, yβ©
simp [and_left_comm]
| 2 | 7.389056 | 1 | 1.214286 | 14 | 1,292 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ΞΉ : Type*}
namespace Finset
section Sigma
variable {Ξ± : ΞΉ β Type*} {Ξ² : Type*} (s sβ sβ : Finset ΞΉ) (... | Mathlib/Data/Finset/Sigma.lean | 99 | 104 | theorem sup_sigma [SemilatticeSup Ξ²] [OrderBot Ξ²] :
(s.sigma t).sup f = s.sup fun i => (t i).sup fun b => f β¨i, bβ© := by |
simp only [le_antisymm_iff, Finset.sup_le_iff, mem_sigma, and_imp, Sigma.forall]
exact
β¨fun i a hi ha => (le_sup hi).trans' <| le_sup (f := fun a => f β¨i, aβ©) ha, fun i hi a ha =>
le_sup <| mem_sigma.2 β¨hi, haβ©β©
| 4 | 54.59815 | 2 | 1.214286 | 14 | 1,292 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ΞΉ : Type*}
namespace Finset
section Sigma
variable {Ξ± : ΞΉ β Type*} {Ξ² : Type*} (s sβ sβ : Finset ΞΉ) (... | Mathlib/Data/Finset/Sigma.lean | 112 | 114 | theorem _root_.biSup_finsetSigma [CompleteLattice Ξ²] (s : Finset ΞΉ) (t : β i, Finset (Ξ± i))
(f : Sigma Ξ± β Ξ²) : β¨ ij β s.sigma t, f ij = β¨ (i β s) (j β t i), f β¨i, jβ© := by |
simp_rw [β Finset.iSup_coe, Finset.coe_sigma, biSup_sigma]
| 1 | 2.718282 | 0 | 1.214286 | 14 | 1,292 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ΞΉ : Type*}
namespace Finset
section SigmaLift
variable {Ξ± Ξ² Ξ³ : ΞΉ β Type*} [DecidableEq ΞΉ]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 156 | 173 | theorem mem_sigmaLift (f : β β¦iβ¦, Ξ± i β Ξ² i β Finset (Ξ³ i)) (a : Sigma Ξ±) (b : Sigma Ξ²)
(x : Sigma Ξ³) :
x β sigmaLift f a b β β (ha : a.1 = x.1) (hb : b.1 = x.1), x.2 β f (ha βΈ a.2) (hb βΈ b.2) := by |
obtain β¨β¨i, aβ©, j, bβ© := a, b
obtain rfl | h := Decidable.eq_or_ne i j
Β· constructor
Β· simp_rw [sigmaLift]
simp only [dite_eq_ite, ite_true, mem_map, Embedding.sigmaMk_apply, forall_exists_index,
and_imp]
rintro x hx rfl
exact β¨rfl, rfl, hxβ©
Β· rintro β¨β¨β©, β¨β©, hxβ©
rw [sigma... | 15 | 3,269,017.372472 | 2 | 1.214286 | 14 | 1,292 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ΞΉ : Type*}
namespace Finset
section SigmaLift
variable {Ξ± Ξ² Ξ³ : ΞΉ β Type*} [DecidableEq ΞΉ]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 176 | 181 | theorem mk_mem_sigmaLift (f : β β¦iβ¦, Ξ± i β Ξ² i β Finset (Ξ³ i)) (i : ΞΉ) (a : Ξ± i) (b : Ξ² i)
(x : Ξ³ i) : (β¨i, xβ© : Sigma Ξ³) β sigmaLift f β¨i, aβ© β¨i, bβ© β x β f a b := by |
rw [sigmaLift, dif_pos rfl, mem_map]
refine β¨?_, fun hx => β¨_, hx, rflβ©β©
rintro β¨x, hx, _, rflβ©
exact hx
| 4 | 54.59815 | 2 | 1.214286 | 14 | 1,292 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ΞΉ : Type*}
namespace Finset
section SigmaLift
variable {Ξ± Ξ² Ξ³ : ΞΉ β Type*} [DecidableEq ΞΉ]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 184 | 187 | theorem not_mem_sigmaLift_of_ne_left (f : β β¦iβ¦, Ξ± i β Ξ² i β Finset (Ξ³ i)) (a : Sigma Ξ±)
(b : Sigma Ξ²) (x : Sigma Ξ³) (h : a.1 β x.1) : x β sigmaLift f a b := by |
rw [mem_sigmaLift]
exact fun H => h H.fst
| 2 | 7.389056 | 1 | 1.214286 | 14 | 1,292 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ΞΉ : Type*}
namespace Finset
section SigmaLift
variable {Ξ± Ξ² Ξ³ : ΞΉ β Type*} [DecidableEq ΞΉ]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 190 | 193 | theorem not_mem_sigmaLift_of_ne_right (f : β β¦iβ¦, Ξ± i β Ξ² i β Finset (Ξ³ i)) {a : Sigma Ξ±}
(b : Sigma Ξ²) {x : Sigma Ξ³} (h : b.1 β x.1) : x β sigmaLift f a b := by |
rw [mem_sigmaLift]
exact fun H => h H.snd.fst
| 2 | 7.389056 | 1 | 1.214286 | 14 | 1,292 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ΞΉ : Type*}
namespace Finset
section SigmaLift
variable {Ξ± Ξ² Ξ³ : ΞΉ β Type*} [DecidableEq ΞΉ]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 198 | 201 | theorem sigmaLift_nonempty :
(sigmaLift f a b).Nonempty β β h : a.1 = b.1, (f (h βΈ a.2) b.2).Nonempty := by |
simp_rw [nonempty_iff_ne_empty, sigmaLift]
split_ifs with h <;> simp [h]
| 2 | 7.389056 | 1 | 1.214286 | 14 | 1,292 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ΞΉ : Type*}
namespace Finset
section SigmaLift
variable {Ξ± Ξ² Ξ³ : ΞΉ β Type*} [DecidableEq ΞΉ]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 204 | 208 | theorem sigmaLift_eq_empty : sigmaLift f a b = β
β β h : a.1 = b.1, f (h βΈ a.2) b.2 = β
:= by |
simp_rw [sigmaLift]
split_ifs with h
Β· simp [h, forall_prop_of_true h]
Β· simp [h, forall_prop_of_false h]
| 4 | 54.59815 | 2 | 1.214286 | 14 | 1,292 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ΞΉ : Type*}
namespace Finset
section SigmaLift
variable {Ξ± Ξ² Ξ³ : ΞΉ β Type*} [DecidableEq ΞΉ]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 211 | 216 | theorem sigmaLift_mono (h : β β¦iβ¦ β¦a : Ξ± iβ¦ β¦b : Ξ² iβ¦, f a b β g a b) (a : Ξ£i, Ξ± i) (b : Ξ£i, Ξ² i) :
sigmaLift f a b β sigmaLift g a b := by |
rintro x hx
rw [mem_sigmaLift] at hx β’
obtain β¨ha, hb, hxβ© := hx
exact β¨ha, hb, h hxβ©
| 4 | 54.59815 | 2 | 1.214286 | 14 | 1,292 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ΞΉ : Type*}
namespace Finset
section SigmaLift
variable {Ξ± Ξ² Ξ³ : ΞΉ β Type*} [DecidableEq ΞΉ]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 221 | 224 | theorem card_sigmaLift :
(sigmaLift f a b).card = dite (a.1 = b.1) (fun h => (f (h βΈ a.2) b.2).card) fun _ => 0 := by |
simp_rw [sigmaLift]
split_ifs with h <;> simp [h]
| 2 | 7.389056 | 1 | 1.214286 | 14 | 1,292 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 52 | 54 | theorem lineMap_mono_left (ha : a β€ a') (hr : r β€ 1) : lineMap a b r β€ lineMap a' b r := by |
simp only [lineMap_apply_module]
exact add_le_add_right (smul_le_smul_of_nonneg_left ha (sub_nonneg.2 hr)) _
| 2 | 7.389056 | 1 | 1.222222 | 9 | 1,293 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 57 | 59 | theorem lineMap_strict_mono_left (ha : a < a') (hr : r < 1) : lineMap a b r < lineMap a' b r := by |
simp only [lineMap_apply_module]
exact add_lt_add_right (smul_lt_smul_of_pos_left ha (sub_pos.2 hr)) _
| 2 | 7.389056 | 1 | 1.222222 | 9 | 1,293 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 62 | 64 | theorem lineMap_mono_right (hb : b β€ b') (hr : 0 β€ r) : lineMap a b r β€ lineMap a b' r := by |
simp only [lineMap_apply_module]
exact add_le_add_left (smul_le_smul_of_nonneg_left hb hr) _
| 2 | 7.389056 | 1 | 1.222222 | 9 | 1,293 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 67 | 69 | theorem lineMap_strict_mono_right (hb : b < b') (hr : 0 < r) : lineMap a b r < lineMap a b' r := by |
simp only [lineMap_apply_module]
exact add_lt_add_left (smul_lt_smul_of_pos_left hb hr) _
| 2 | 7.389056 | 1 | 1.222222 | 9 | 1,293 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 77 | 80 | theorem lineMap_strict_mono_endpoints (ha : a < a') (hb : b < b') (hβ : 0 β€ r) (hβ : r β€ 1) :
lineMap a b r < lineMap a' b' r := by |
rcases hβ.eq_or_lt with (rfl | hβ); Β· simpa
exact (lineMap_mono_left ha.le hβ).trans_lt (lineMap_strict_mono_right hb hβ)
| 2 | 7.389056 | 1 | 1.222222 | 9 | 1,293 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 83 | 86 | theorem lineMap_lt_lineMap_iff_of_lt (h : r < r') : lineMap a b r < lineMap a b r' β a < b := by |
simp only [lineMap_apply_module]
rw [β lt_sub_iff_add_lt, add_sub_assoc, β sub_lt_iff_lt_add', β sub_smul, β sub_smul,
sub_sub_sub_cancel_left, smul_lt_smul_iff_of_pos_left (sub_pos.2 h)]
| 3 | 20.085537 | 1 | 1.222222 | 9 | 1,293 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 127 | 130 | theorem lineMap_le_lineMap_iff_of_lt (h : r < r') : lineMap a b r β€ lineMap a b r' β a β€ b := by |
simp only [lineMap_apply_module]
rw [β le_sub_iff_add_le, add_sub_assoc, β sub_le_iff_le_add', β sub_smul, β sub_smul,
sub_sub_sub_cancel_left, smul_le_smul_iff_of_pos_left (sub_pos.2 h)]
| 3 | 20.085537 | 1 | 1.222222 | 9 | 1,293 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 206 | 213 | theorem map_le_lineMap_iff_slope_le_slope_left (h : 0 < r * (b - a)) :
f c β€ lineMap (f a) (f b) r β slope f a c β€ slope f a b := by |
rw [lineMap_apply, lineMap_apply, slope, slope, vsub_eq_sub, vsub_eq_sub, vsub_eq_sub,
vadd_eq_add, vadd_eq_add, smul_eq_mul, add_sub_cancel_right, smul_sub, smul_sub, smul_sub,
sub_le_iff_le_add, mul_inv_rev, mul_smul, mul_smul, β smul_sub, β smul_sub, β smul_add,
smul_smul, β mul_inv_rev, inv_smul_le_i... | 6 | 403.428793 | 2 | 1.222222 | 9 | 1,293 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 240 | 248 | theorem map_le_lineMap_iff_slope_le_slope_right (h : 0 < (1 - r) * (b - a)) :
f c β€ lineMap (f a) (f b) r β slope f a b β€ slope f c b := by |
rw [β lineMap_apply_one_sub, β lineMap_apply_one_sub _ _ r]
revert h; generalize 1 - r = r'; clear! r; intro h
simp_rw [lineMap_apply, slope, vsub_eq_sub, vadd_eq_add, smul_eq_mul]
rw [sub_add_eq_sub_sub_swap, sub_self, zero_sub, neg_mul_eq_mul_neg, neg_sub,
le_inv_smul_iff_of_pos h, smul_smul, mul_inv_can... | 7 | 1,096.633158 | 2 | 1.222222 | 9 | 1,293 |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 77 | 85 | theorem IsClosable.leIsClosable {f g : E ββ.[R] F} (hf : f.IsClosable) (hfg : g β€ f) :
g.IsClosable := by |
cases' hf with f' hf
have : g.graph.topologicalClosure β€ f'.graph := by
rw [β hf]
exact Submodule.topologicalClosure_mono (le_graph_of_le hfg)
use g.graph.topologicalClosure.toLinearPMap
rw [Submodule.toLinearPMap_graph_eq]
exact fun _ hx hx' => f'.graph_fst_eq_zero_snd (this hx) hx'
| 7 | 1,096.633158 | 2 | 1.222222 | 9 | 1,294 |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 89 | 92 | theorem IsClosable.existsUnique {f : E ββ.[R] F} (hf : f.IsClosable) :
β! f' : E ββ.[R] F, f.graph.topologicalClosure = f'.graph := by |
refine exists_unique_of_exists_of_unique hf fun _ _ hyβ hyβ => eq_of_eq_graph ?_
rw [β hyβ, β hyβ]
| 2 | 7.389056 | 1 | 1.222222 | 9 | 1,294 |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 103 | 104 | theorem closure_def {f : E ββ.[R] F} (hf : f.IsClosable) : f.closure = hf.choose := by |
simp [closure, hf]
| 1 | 2.718282 | 0 | 1.222222 | 9 | 1,294 |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 107 | 107 | theorem closure_def' {f : E ββ.[R] F} (hf : Β¬f.IsClosable) : f.closure = f := by | simp [closure, hf]
| 1 | 2.718282 | 0 | 1.222222 | 9 | 1,294 |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 112 | 115 | theorem IsClosable.graph_closure_eq_closure_graph {f : E ββ.[R] F} (hf : f.IsClosable) :
f.graph.topologicalClosure = f.closure.graph := by |
rw [closure_def hf]
exact hf.choose_spec
| 2 | 7.389056 | 1 | 1.222222 | 9 | 1,294 |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 119 | 124 | theorem le_closure (f : E ββ.[R] F) : f β€ f.closure := by |
by_cases hf : f.IsClosable
Β· refine le_of_le_graph ?_
rw [β hf.graph_closure_eq_closure_graph]
exact (graph f).le_topologicalClosure
rw [closure_def' hf]
| 5 | 148.413159 | 2 | 1.222222 | 9 | 1,294 |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 127 | 132 | theorem IsClosable.closure_mono {f g : E ββ.[R] F} (hg : g.IsClosable) (h : f β€ g) :
f.closure β€ g.closure := by |
refine le_of_le_graph ?_
rw [β (hg.leIsClosable h).graph_closure_eq_closure_graph]
rw [β hg.graph_closure_eq_closure_graph]
exact Submodule.topologicalClosure_mono (le_graph_of_le h)
| 4 | 54.59815 | 2 | 1.222222 | 9 | 1,294 |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 136 | 138 | theorem IsClosable.closure_isClosed {f : E ββ.[R] F} (hf : f.IsClosable) : f.closure.IsClosed := by |
rw [IsClosed, β hf.graph_closure_eq_closure_graph]
exact f.graph.isClosed_topologicalClosure
| 2 | 7.389056 | 1 | 1.222222 | 9 | 1,294 |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 169 | 179 | theorem closureHasCore (f : E ββ.[R] F) : f.closure.HasCore f.domain := by |
refine β¨f.le_closure.1, ?_β©
congr
ext x y hxy
Β· simp only [domRestrict_domain, Submodule.mem_inf, and_iff_left_iff_imp]
intro hx
exact f.le_closure.1 hx
let z : f.closure.domain := β¨y.1, f.le_closure.1 y.2β©
have hyz : (y : E) = z := by simp
rw [f.le_closure.2 hyz]
exact domRestrict_apply (hxy.t... | 10 | 22,026.465795 | 2 | 1.222222 | 9 | 1,294 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 105 | 105 | theorem areaForm_to_volumeForm (x y : E) : Ο x y = o.volumeForm ![x, y] := by | simp [areaForm]
| 1 | 2.718282 | 0 | 1.222222 | 9 | 1,295 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 109 | 113 | theorem areaForm_apply_self (x : E) : Ο x x = 0 := by |
rw [areaForm_to_volumeForm]
refine o.volumeForm.map_eq_zero_of_eq ![x, x] ?_ (?_ : (0 : Fin 2) β 1)
Β· simp
Β· norm_num
| 4 | 54.59815 | 2 | 1.222222 | 9 | 1,295 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 116 | 121 | theorem areaForm_swap (x y : E) : Ο x y = -Ο y x := by |
simp only [areaForm_to_volumeForm]
convert o.volumeForm.map_swap ![y, x] (_ : (0 : Fin 2) β 1)
Β· ext i
fin_cases i <;> rfl
Β· norm_num
| 5 | 148.413159 | 2 | 1.222222 | 9 | 1,295 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 125 | 127 | theorem areaForm_neg_orientation : (-o).areaForm = -o.areaForm := by |
ext x y
simp [areaForm_to_volumeForm]
| 2 | 7.389056 | 1 | 1.222222 | 9 | 1,295 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 142 | 143 | theorem abs_areaForm_le (x y : E) : |Ο x y| β€ βxβ * βyβ := by |
simpa [areaForm_to_volumeForm, Fin.prod_univ_succ] using o.abs_volumeForm_apply_le ![x, y]
| 1 | 2.718282 | 0 | 1.222222 | 9 | 1,295 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 146 | 147 | theorem areaForm_le (x y : E) : Ο x y β€ βxβ * βyβ := by |
simpa [areaForm_to_volumeForm, Fin.prod_univ_succ] using o.volumeForm_apply_le ![x, y]
| 1 | 2.718282 | 0 | 1.222222 | 9 | 1,295 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 150 | 158 | theorem abs_areaForm_of_orthogonal {x y : E} (h : βͺx, yβ« = 0) : |Ο x y| = βxβ * βyβ := by |
rw [o.areaForm_to_volumeForm, o.abs_volumeForm_apply_of_pairwise_orthogonal]
Β· simp [Fin.prod_univ_succ]
intro i j hij
fin_cases i <;> fin_cases j
Β· simp_all
Β· simpa using h
Β· simpa [real_inner_comm] using h
Β· simp_all
| 8 | 2,980.957987 | 2 | 1.222222 | 9 | 1,295 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 161 | 168 | theorem areaForm_map {F : Type*} [NormedAddCommGroup F] [InnerProductSpace β F]
[hF : Fact (finrank β F = 2)] (Ο : E ββα΅’[β] F) (x y : F) :
(Orientation.map (Fin 2) Ο.toLinearEquiv o).areaForm x y =
o.areaForm (Ο.symm x) (Ο.symm y) := by |
have : Ο.symm β ![x, y] = ![Ο.symm x, Ο.symm y] := by
ext i
fin_cases i <;> rfl
simp [areaForm_to_volumeForm, volumeForm_map, this]
| 4 | 54.59815 | 2 | 1.222222 | 9 | 1,295 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 172 | 180 | theorem areaForm_comp_linearIsometryEquiv (Ο : E ββα΅’[β] E)
(hΟ : 0 < LinearMap.det (Ο.toLinearEquiv : E ββ[β] E)) (x y : E) :
o.areaForm (Ο x) (Ο y) = o.areaForm x y := by |
convert o.areaForm_map Ο (Ο x) (Ο y)
Β· symm
rwa [β o.map_eq_iff_det_pos Ο.toLinearEquiv] at hΟ
rw [@Fact.out (finrank β E = 2), Fintype.card_fin]
Β· simp
Β· simp
| 6 | 403.428793 | 2 | 1.222222 | 9 | 1,295 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 106 | 106 | theorem condexp_of_not_le (hm_not : Β¬m β€ m0) : ΞΌ[f|m] = 0 := by | rw [condexp, dif_neg hm_not]
| 1 | 2.718282 | 0 | 1.222222 | 9 | 1,296 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 109 | 110 | theorem condexp_of_not_sigmaFinite (hm : m β€ m0) (hΞΌm_not : Β¬SigmaFinite (ΞΌ.trim hm)) :
ΞΌ[f|m] = 0 := by | rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hΞΌm_not
| 1 | 2.718282 | 0 | 1.222222 | 9 | 1,296 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 113 | 123 | theorem condexp_of_sigmaFinite (hm : m β€ m0) [hΞΌm : SigmaFinite (ΞΌ.trim hm)] :
ΞΌ[f|m] =
if Integrable f ΞΌ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm ΞΌ f)
else 0 := by |
rw [condexp, dif_pos hm]
simp only [hΞΌm, Ne, true_and_iff]
by_cases hf : Integrable f ΞΌ
Β· rw [dif_pos hf, if_pos hf]
Β· rw [dif_neg hf, if_neg hf]
| 5 | 148.413159 | 2 | 1.222222 | 9 | 1,296 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 126 | 128 | theorem condexp_of_stronglyMeasurable (hm : m β€ m0) [hΞΌm : SigmaFinite (ΞΌ.trim hm)] {f : Ξ± β F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f ΞΌ) : ΞΌ[f|m] = f := by |
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
| 1 | 2.718282 | 0 | 1.222222 | 9 | 1,296 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 136 | 148 | theorem condexp_ae_eq_condexpL1 (hm : m β€ m0) [hΞΌm : SigmaFinite (ΞΌ.trim hm)] (f : Ξ± β F') :
ΞΌ[f|m] =α΅[ΞΌ] condexpL1 hm ΞΌ f := by |
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f ΞΌ
Β· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
Β· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
Β· rw [if_neg hfm]
exact (AEStronglyMeasurable'... | 11 | 59,874.141715 | 2 | 1.222222 | 9 | 1,296 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 152 | 155 | theorem condexp_ae_eq_condexpL1CLM (hm : m β€ m0) [SigmaFinite (ΞΌ.trim hm)] (hf : Integrable f ΞΌ) :
ΞΌ[f|m] =α΅[ΞΌ] condexpL1CLM F' hm ΞΌ (hf.toL1 f) := by |
refine (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => ?_)
rw [condexpL1_eq hf]
| 2 | 7.389056 | 1 | 1.222222 | 9 | 1,296 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 159 | 165 | theorem condexp_undef (hf : Β¬Integrable f ΞΌ) : ΞΌ[f|m] = 0 := by |
by_cases hm : m β€ m0
swap; Β· rw [condexp_of_not_le hm]
by_cases hΞΌm : SigmaFinite (ΞΌ.trim hm)
swap; Β· rw [condexp_of_not_sigmaFinite hm hΞΌm]
haveI : SigmaFinite (ΞΌ.trim hm) := hΞΌm
rw [condexp_of_sigmaFinite, if_neg hf]
| 6 | 403.428793 | 2 | 1.222222 | 9 | 1,296 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 169 | 176 | theorem condexp_zero : ΞΌ[(0 : Ξ± β F')|m] = 0 := by |
by_cases hm : m β€ m0
swap; Β· rw [condexp_of_not_le hm]
by_cases hΞΌm : SigmaFinite (ΞΌ.trim hm)
swap; Β· rw [condexp_of_not_sigmaFinite hm hΞΌm]
haveI : SigmaFinite (ΞΌ.trim hm) := hΞΌm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
| 7 | 1,096.633158 | 2 | 1.222222 | 9 | 1,296 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 179 | 189 | theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (ΞΌ[f|m]) := by |
by_cases hm : m β€ m0
swap; Β· rw [condexp_of_not_le hm]; exact stronglyMeasurable_zero
by_cases hΞΌm : SigmaFinite (ΞΌ.trim hm)
swap; Β· rw [condexp_of_not_sigmaFinite hm hΞΌm]; exact stronglyMeasurable_zero
haveI : SigmaFinite (ΞΌ.trim hm) := hΞΌm
rw [condexp_of_sigmaFinite hm]
split_ifs with hfi hfm
Β· exact... | 10 | 22,026.465795 | 2 | 1.222222 | 9 | 1,296 |
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.matrix.adjugate from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matr... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 82 | 85 | theorem cramer_is_linear : IsLinearMap Ξ± (cramerMap A) := by |
constructor <;> intros <;> ext i
Β· apply (cramerMap_is_linear A i).1
Β· apply (cramerMap_is_linear A i).2
| 3 | 20.085537 | 1 | 1.222222 | 9 | 1,297 |
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.matrix.adjugate from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matr... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 102 | 103 | theorem cramer_transpose_apply (i : n) : cramer Aα΅ b i = (A.updateRow i b).det := by |
rw [cramer_apply, updateColumn_transpose, det_transpose]
| 1 | 2.718282 | 0 | 1.222222 | 9 | 1,297 |
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.matrix.adjugate from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matr... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 106 | 116 | theorem cramer_transpose_row_self (i : n) : Aα΅.cramer (A i) = Pi.single i A.det := by |
ext j
rw [cramer_apply, Pi.single_apply]
split_ifs with h
Β· -- i = j: this entry should be `A.det`
subst h
simp only [updateColumn_transpose, det_transpose, updateRow_eq_self]
Β· -- i β j: this entry should be 0
rw [updateColumn_transpose, det_transpose]
apply det_zero_of_row_eq h
rw [upda... | 10 | 22,026.465795 | 2 | 1.222222 | 9 | 1,297 |
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.matrix.adjugate from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matr... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 119 | 122 | theorem cramer_row_self (i : n) (h : β j, b j = A j i) : A.cramer b = Pi.single i A.det := by |
rw [β transpose_transpose A, det_transpose]
convert cramer_transpose_row_self Aα΅ i
exact funext h
| 3 | 20.085537 | 1 | 1.222222 | 9 | 1,297 |
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.matrix.adjugate from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matr... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 126 | 132 | theorem cramer_one : cramer (1 : Matrix n n Ξ±) = 1 := by |
-- Porting note: was `ext i j`
refine LinearMap.pi_ext' (fun (i : n) => LinearMap.ext_ring (funext (fun (j : n) => ?_)))
convert congr_fun (cramer_row_self (1 : Matrix n n Ξ±) (Pi.single i 1) i _) j
Β· simp
Β· intro j
rw [Matrix.one_eq_pi_single, Pi.single_comm]
| 6 | 403.428793 | 2 | 1.222222 | 9 | 1,297 |
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.matrix.adjugate from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matr... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 141 | 142 | theorem cramer_subsingleton_apply [Subsingleton n] (A : Matrix n n Ξ±) (b : n β Ξ±) (i : n) :
cramer A b i = b i := by | rw [cramer_apply, det_eq_elem_of_subsingleton _ i, updateColumn_self]
| 1 | 2.718282 | 0 | 1.222222 | 9 | 1,297 |
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.matrix.adjugate from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matr... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 145 | 150 | theorem cramer_zero [Nontrivial n] : cramer (0 : Matrix n n Ξ±) = 0 := by |
ext i j
obtain β¨j', hj'β© : β j', j' β j := exists_ne j
apply det_eq_zero_of_column_eq_zero j'
intro j''
simp [updateColumn_ne hj']
| 5 | 148.413159 | 2 | 1.222222 | 9 | 1,297 |
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.matrix.adjugate from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matr... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 160 | 170 | theorem sum_cramer_apply {Ξ²} (s : Finset Ξ²) (f : n β Ξ² β Ξ±) (i : n) :
(β x β s, cramer A (fun j => f j x) i) = cramer A (fun j : n => β x β s, f j x) i :=
calc
(β x β s, cramer A (fun j => f j x) i) = (β x β s, cramer A fun j => f j x) i :=
(Finset.sum_apply i s _).symm
_ = cramer A (fun j : n => β ... |
rw [sum_cramer, cramer_apply, cramer_apply]
simp only [updateColumn]
congr with j
congr
apply Finset.sum_apply
| 5 | 148.413159 | 2 | 1.222222 | 9 | 1,297 |
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.matrix.adjugate from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matr... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 173 | 177 | theorem cramer_submatrix_equiv (A : Matrix m m Ξ±) (e : n β m) (b : n β Ξ±) :
cramer (A.submatrix e e) b = cramer A (b β e.symm) β e := by |
ext i
simp_rw [Function.comp_apply, cramer_apply, updateColumn_submatrix_equiv,
det_submatrix_equiv_self e, Function.comp]
| 3 | 20.085537 | 1 | 1.222222 | 9 | 1,297 |
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.specific_asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Asymptotics
open Topology
sectio... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 28 | 33 | theorem Filter.IsBoundedUnder.isLittleO_sub_self_inv {π E : Type*} [NormedField π] [Norm E] {a : π}
{f : π β E} (h : IsBoundedUnder (Β· β€ Β·) (π[β ] a) (norm β f)) :
f =o[π[β ] a] fun x => (x - a)β»ΒΉ := by |
refine (h.isBigO_const (one_ne_zero' β)).trans_isLittleO (isLittleO_const_left.2 <| Or.inr ?_)
simp only [(Β· β Β·), norm_inv]
exact (tendsto_norm_sub_self_punctured_nhds a).inv_tendsto_zero
| 3 | 20.085537 | 1 | 1.25 | 8 | 1,299 |
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.specific_asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Asymptotics
open Topology
sectio... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 42 | 46 | theorem pow_div_pow_eventuallyEq_atTop {p q : β} :
(fun x : π => x ^ p / x ^ q) =αΆ [atTop] fun x => x ^ ((p : β€) - q) := by |
apply (eventually_gt_atTop (0 : π)).mono fun x hx => _
intro x hx
simp [zpow_subβ hx.ne']
| 3 | 20.085537 | 1 | 1.25 | 8 | 1,299 |
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.specific_asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Asymptotics
open Topology
sectio... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 49 | 53 | theorem pow_div_pow_eventuallyEq_atBot {p q : β} :
(fun x : π => x ^ p / x ^ q) =αΆ [atBot] fun x => x ^ ((p : β€) - q) := by |
apply (eventually_lt_atBot (0 : π)).mono fun x hx => _
intro x hx
simp [zpow_subβ hx.ne]
| 3 | 20.085537 | 1 | 1.25 | 8 | 1,299 |
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.specific_asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Asymptotics
open Topology
sectio... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 56 | 60 | theorem tendsto_pow_div_pow_atTop_atTop {p q : β} (hpq : q < p) :
Tendsto (fun x : π => x ^ p / x ^ q) atTop atTop := by |
rw [tendsto_congr' pow_div_pow_eventuallyEq_atTop]
apply tendsto_zpow_atTop_atTop
omega
| 3 | 20.085537 | 1 | 1.25 | 8 | 1,299 |
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.specific_asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Asymptotics
open Topology
sectio... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 63 | 67 | theorem tendsto_pow_div_pow_atTop_zero [TopologicalSpace π] [OrderTopology π] {p q : β}
(hpq : p < q) : Tendsto (fun x : π => x ^ p / x ^ q) atTop (π 0) := by |
rw [tendsto_congr' pow_div_pow_eventuallyEq_atTop]
apply tendsto_zpow_atTop_zero
omega
| 3 | 20.085537 | 1 | 1.25 | 8 | 1,299 |
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.specific_asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Asymptotics
open Topology
sectio... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 98 | 128 | theorem Asymptotics.IsLittleO.sum_range {Ξ± : Type*} [NormedAddCommGroup Ξ±] {f : β β Ξ±} {g : β β β}
(h : f =o[atTop] g) (hg : 0 β€ g) (h'g : Tendsto (fun n => β i β range n, g i) atTop atTop) :
(fun n => β i β range n, f i) =o[atTop] fun n => β i β range n, g i := by |
have A : β i, βg iβ = g i := fun i => Real.norm_of_nonneg (hg i)
have B : β n, ββ i β range n, g iβ = β i β range n, g i := fun n => by
rwa [Real.norm_eq_abs, abs_sum_of_nonneg']
apply isLittleO_iff.2 fun Ξ΅ Ξ΅pos => _
intro Ξ΅ Ξ΅pos
obtain β¨N, hNβ© : β N : β, β b : β, N β€ b β βf bβ β€ Ξ΅ / 2 * g b := by
si... | 28 | 1,446,257,064,291.475 | 2 | 1.25 | 8 | 1,299 |
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.specific_asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Asymptotics
open Topology
sectio... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 131 | 136 | theorem Asymptotics.isLittleO_sum_range_of_tendsto_zero {Ξ± : Type*} [NormedAddCommGroup Ξ±]
{f : β β Ξ±} (h : Tendsto f atTop (π 0)) :
(fun n => β i β range n, f i) =o[atTop] fun n => (n : β) := by |
have := ((isLittleO_one_iff β).2 h).sum_range fun i => zero_le_one
simp only [sum_const, card_range, Nat.smul_one_eq_cast] at this
exact this tendsto_natCast_atTop_atTop
| 3 | 20.085537 | 1 | 1.25 | 8 | 1,299 |
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.specific_asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Asymptotics
open Topology
sectio... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 140 | 152 | theorem Filter.Tendsto.cesaro_smul {E : Type*} [NormedAddCommGroup E] [NormedSpace β E] {u : β β E}
{l : E} (h : Tendsto u atTop (π l)) :
Tendsto (fun n : β => (nβ»ΒΉ : β) β’ β i β range n, u i) atTop (π l) := by |
rw [β tendsto_sub_nhds_zero_iff, β isLittleO_one_iff β]
have := Asymptotics.isLittleO_sum_range_of_tendsto_zero (tendsto_sub_nhds_zero_iff.2 h)
apply ((isBigO_refl (fun n : β => (n : β)β»ΒΉ) atTop).smul_isLittleO this).congr' _ _
Β· filter_upwards [Ici_mem_atTop 1] with n npos
have nposβ : (0 : β) < n := Nat.... | 10 | 22,026.465795 | 2 | 1.25 | 8 | 1,299 |
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.RingTheory.Ideal.Quotient
#align_import number_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open I... | Mathlib/NumberTheory/Multiplicity.lean | 39 | 43 | theorem dvd_geom_sumβ_iff_of_dvd_sub {x y p : R} (h : p β£ x - y) :
(p β£ β i β range n, x ^ i * y ^ (n - 1 - i)) β p β£ n * y ^ (n - 1) := by |
rw [β mem_span_singleton, β Ideal.Quotient.eq] at h
simp only [β mem_span_singleton, β eq_zero_iff_mem, RingHom.map_geom_sumβ, h, geom_sumβ_self,
_root_.map_mul, map_pow, map_natCast]
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,300 |
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.RingTheory.Ideal.Quotient
#align_import number_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open I... | Mathlib/NumberTheory/Multiplicity.lean | 46 | 48 | theorem dvd_geom_sumβ_iff_of_dvd_sub' {x y p : R} (h : p β£ x - y) :
(p β£ β i β range n, x ^ i * y ^ (n - 1 - i)) β p β£ n * x ^ (n - 1) := by |
rw [geom_sumβ_comm, dvd_geom_sumβ_iff_of_dvd_sub]; simpa using h.neg_right
| 1 | 2.718282 | 0 | 1.25 | 4 | 1,300 |
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.RingTheory.Ideal.Quotient
#align_import number_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open I... | Mathlib/NumberTheory/Multiplicity.lean | 56 | 71 | theorem sq_dvd_add_pow_sub_sub (p x : R) (n : β) :
p ^ 2 β£ (x + p) ^ n - x ^ (n - 1) * p * n - x ^ n := by |
cases' n with n n
Β· simp only [pow_zero, Nat.cast_zero, sub_zero, sub_self, dvd_zero, Nat.zero_eq, mul_zero]
Β· simp only [Nat.succ_sub_succ_eq_sub, tsub_zero, Nat.cast_succ, add_pow, Finset.sum_range_succ,
Nat.choose_self, Nat.succ_sub _, tsub_self, pow_one, Nat.choose_succ_self_right, pow_zero,
mul_... | 14 | 1,202,604.284165 | 2 | 1.25 | 4 | 1,300 |
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.RingTheory.Ideal.Quotient
#align_import number_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open I... | Mathlib/NumberTheory/Multiplicity.lean | 82 | 146 | theorem odd_sq_dvd_geom_sumβ_sub (hp : Odd p) :
(p : R) ^ 2 β£ (β i β range p, (a + p * b) ^ i * a ^ (p - 1 - i)) - p * a ^ (p - 1) := by |
have h1 : β (i : β),
(p : R) ^ 2 β£ (a + βp * b) ^ i - (a ^ (i - 1) * (βp * b) * i + a ^ i) := by
intro i
calc
βp ^ 2 β£ (βp * b) ^ 2 := by simp only [mul_pow, dvd_mul_right]
_ β£ (a + βp * b) ^ i - (a ^ (i - 1) * (βp * b) * βi + a ^ i) := by
simp only [sq_dvd_add_pow_sub_sub (βp * b) ... | 63 | 2,293,783,159,469,610,000,000,000,000 | 2 | 1.25 | 4 | 1,300 |
import Mathlib.Analysis.Calculus.FDeriv.Basic
#align_import analysis.calculus.fderiv.restrict_scalars from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal Filter Asymptotics ENNReal
noncom... | Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean | 92 | 95 | theorem HasFDerivWithinAt.of_restrictScalars {g' : E βL[π] F} (h : HasFDerivWithinAt f g' s x)
(H : f'.restrictScalars π = g') : HasFDerivWithinAt f f' s x := by |
rw [β H] at h
exact .of_isLittleO h.1
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,301 |
import Mathlib.Analysis.Calculus.FDeriv.Basic
#align_import analysis.calculus.fderiv.restrict_scalars from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal Filter Asymptotics ENNReal
noncom... | Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean | 99 | 102 | theorem hasFDerivAt_of_restrictScalars {g' : E βL[π] F} (h : HasFDerivAt f g' x)
(H : f'.restrictScalars π = g') : HasFDerivAt f f' x := by |
rw [β H] at h
exact .of_isLittleO h.1
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,301 |
import Mathlib.Analysis.Calculus.FDeriv.Basic
#align_import analysis.calculus.fderiv.restrict_scalars from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal Filter Asymptotics ENNReal
noncom... | Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean | 110 | 117 | theorem differentiableWithinAt_iff_restrictScalars (hf : DifferentiableWithinAt π f s x)
(hs : UniqueDiffWithinAt π s x) : DifferentiableWithinAt π' f s x β
β g' : E βL[π'] F, g'.restrictScalars π = fderivWithin π f s x := by |
constructor
Β· rintro β¨g', hg'β©
exact β¨g', hs.eq (hg'.restrictScalars π) hf.hasFDerivWithinAtβ©
Β· rintro β¨f', hf'β©
exact β¨f', hf.hasFDerivWithinAt.of_restrictScalars π hf'β©
| 5 | 148.413159 | 2 | 1.25 | 4 | 1,301 |
import Mathlib.Analysis.Calculus.FDeriv.Basic
#align_import analysis.calculus.fderiv.restrict_scalars from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal Filter Asymptotics ENNReal
noncom... | Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean | 120 | 124 | theorem differentiableAt_iff_restrictScalars (hf : DifferentiableAt π f x) :
DifferentiableAt π' f x β β g' : E βL[π'] F, g'.restrictScalars π = fderiv π f x := by |
rw [β differentiableWithinAt_univ, β fderivWithin_univ]
exact
differentiableWithinAt_iff_restrictScalars π hf.differentiableWithinAt uniqueDiffWithinAt_univ
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,301 |
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {Ξ± Ξ² R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 76 | 79 | theorem IsDiag.map [Zero Ξ±] [Zero Ξ²] {A : Matrix n n Ξ±} (ha : A.IsDiag) {f : Ξ± β Ξ²} (hf : f 0 = 0) :
(A.map f).IsDiag := by |
intro i j h
simp [ha h, hf]
| 2 | 7.389056 | 1 | 1.25 | 8 | 1,302 |
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {Ξ± Ξ² R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 82 | 84 | theorem IsDiag.neg [AddGroup Ξ±] {A : Matrix n n Ξ±} (ha : A.IsDiag) : (-A).IsDiag := by |
intro i j h
simp [ha h]
| 2 | 7.389056 | 1 | 1.25 | 8 | 1,302 |
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {Ξ± Ξ² R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 92 | 95 | theorem IsDiag.add [AddZeroClass Ξ±] {A B : Matrix n n Ξ±} (ha : A.IsDiag) (hb : B.IsDiag) :
(A + B).IsDiag := by |
intro i j h
simp [ha h, hb h]
| 2 | 7.389056 | 1 | 1.25 | 8 | 1,302 |
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {Ξ± Ξ² R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 98 | 101 | theorem IsDiag.sub [AddGroup Ξ±] {A B : Matrix n n Ξ±} (ha : A.IsDiag) (hb : B.IsDiag) :
(A - B).IsDiag := by |
intro i j h
simp [ha h, hb h]
| 2 | 7.389056 | 1 | 1.25 | 8 | 1,302 |
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {Ξ± Ξ² R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 104 | 107 | theorem IsDiag.smul [Monoid R] [AddMonoid Ξ±] [DistribMulAction R Ξ±] (k : R) {A : Matrix n n Ξ±}
(ha : A.IsDiag) : (k β’ A).IsDiag := by |
intro i j h
simp [ha h]
| 2 | 7.389056 | 1 | 1.25 | 8 | 1,302 |
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {Ξ± Ξ² R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 143 | 149 | theorem IsDiag.kronecker [MulZeroClass Ξ±] {A : Matrix m m Ξ±} {B : Matrix n n Ξ±} (hA : A.IsDiag)
(hB : B.IsDiag) : (A ββ B).IsDiag := by |
rintro β¨a, bβ© β¨c, dβ© h
simp only [Prod.mk.inj_iff, Ne, not_and_or] at h
cases' h with hac hbd
Β· simp [hA hac]
Β· simp [hB hbd]
| 5 | 148.413159 | 2 | 1.25 | 8 | 1,302 |
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {Ξ± Ξ² R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 152 | 155 | theorem IsDiag.isSymm [Zero Ξ±] {A : Matrix n n Ξ±} (h : A.IsDiag) : A.IsSymm := by |
ext i j
by_cases g : i = j; Β· rw [g, transpose_apply]
simp [h g, h (Ne.symm g)]
| 3 | 20.085537 | 1 | 1.25 | 8 | 1,302 |
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {Ξ± Ξ² R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 159 | 165 | theorem IsDiag.fromBlocks [Zero Ξ±] {A : Matrix m m Ξ±} {D : Matrix n n Ξ±} (ha : A.IsDiag)
(hd : D.IsDiag) : (A.fromBlocks 0 0 D).IsDiag := by |
rintro (i | i) (j | j) hij
Β· exact ha (ne_of_apply_ne _ hij)
Β· rfl
Β· rfl
Β· exact hd (ne_of_apply_ne _ hij)
| 5 | 148.413159 | 2 | 1.25 | 8 | 1,302 |
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finset.Pointwise
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import algebra.group.unique_prods from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
@[to_additive
"Let `G` be a Type with addition, let `A B : Finset G` ... | Mathlib/Algebra/Group/UniqueProds.lean | 67 | 68 | theorem of_subsingleton [Subsingleton G] : UniqueMul A B a0 b0 := by |
simp [UniqueMul, eq_iff_true_of_subsingleton]
| 1 | 2.718282 | 0 | 1.25 | 4 | 1,303 |
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finset.Pointwise
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import algebra.group.unique_prods from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
@[to_additive
"Let `G` be a Type with addition, let `A B : Finset G` ... | Mathlib/Algebra/Group/UniqueProds.lean | 71 | 75 | theorem of_card_le_one (hA : A.Nonempty) (hB : B.Nonempty) (hA1 : A.card β€ 1) (hB1 : B.card β€ 1) :
β a β A, β b β B, UniqueMul A B a b := by |
rw [Finset.card_le_one_iff] at hA1 hB1
obtain β¨a, haβ© := hA; obtain β¨b, hbβ© := hB
exact β¨a, ha, b, hb, fun _ _ ha' hb' _ β¦ β¨hA1 ha' ha, hB1 hb' hbβ©β©
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,303 |
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finset.Pointwise
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import algebra.group.unique_prods from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
@[to_additive
"Let `G` be a Type with addition, let `A B : Finset G` ... | Mathlib/Algebra/Group/UniqueProds.lean | 95 | 101 | theorem set_subsingleton (h : UniqueMul A B a0 b0) :
Set.Subsingleton { ab : G Γ G | ab.1 β A β§ ab.2 β B β§ ab.1 * ab.2 = a0 * b0 } := by |
rintro β¨x1, y1β© (hx : x1 β A β§ y1 β B β§ x1 * y1 = a0 * b0) β¨x2, y2β©
(hy : x2 β A β§ y2 β B β§ x2 * y2 = a0 * b0)
rcases h hx.1 hx.2.1 hx.2.2 with β¨rfl, rflβ©
rcases h hy.1 hy.2.1 hy.2.2 with β¨rfl, rflβ©
rfl
| 5 | 148.413159 | 2 | 1.25 | 4 | 1,303 |
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finset.Pointwise
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import algebra.group.unique_prods from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
@[to_additive
"Let `G` be a Type with addition, let `A B : Finset G` ... | Mathlib/Algebra/Group/UniqueProds.lean | 121 | 129 | theorem iff_card_le_one [DecidableEq G] (ha0 : a0 β A) (hb0 : b0 β B) :
UniqueMul A B a0 b0 β ((A ΓΛ’ B).filter (fun p β¦ p.1 * p.2 = a0 * b0)).card β€ 1 := by |
simp_rw [card_le_one_iff, mem_filter, mem_product]
refine β¨fun h p1 p2 β¨β¨ha1, hb1β©, he1β© β¨β¨ha2, hb2β©, he2β© β¦ ?_, fun h a b ha hb he β¦ ?_β©
Β· have h1 := h ha1 hb1 he1; have h2 := h ha2 hb2 he2
ext
Β· rw [h1.1, h2.1]
Β· rw [h1.2, h2.2]
Β· exact Prod.ext_iff.1 (@h (a, b) (a0, b0) β¨β¨ha, hbβ©, heβ© β¨β¨ha0, hb0... | 7 | 1,096.633158 | 2 | 1.25 | 4 | 1,303 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable s... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 64 | 67 | theorem measurableSet_integrable [SigmaFinite Ξ½] β¦f : Ξ± β Ξ² β Eβ¦
(hf : StronglyMeasurable (uncurry f)) : MeasurableSet {x | Integrable (f x) Ξ½} := by |
simp_rw [Integrable, hf.of_uncurry_left.aestronglyMeasurable, true_and_iff]
exact measurableSet_lt (Measurable.lintegral_prod_right hf.ennnorm) measurable_const
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,304 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable s... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 77 | 122 | theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite Ξ½] β¦f : Ξ± β Ξ² β Eβ¦
(hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => β« y, f x y βΞ½ := by |
by_cases hE : CompleteSpace E; swap; Β· simp [integral, hE, stronglyMeasurable_const]
borelize E
haveI : SeparableSpace (range (uncurry f) βͺ {0} : Set E) :=
hf.separableSpace_range_union_singleton
let s : β β SimpleFunc (Ξ± Γ Ξ²) E :=
SimpleFunc.approxOn _ hf.measurable (range (uncurry f) βͺ {0}) 0 (by sim... | 44 | 12,851,600,114,359,308,000 | 2 | 1.25 | 4 | 1,304 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable s... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 127 | 129 | theorem MeasureTheory.StronglyMeasurable.integral_prod_right' [SigmaFinite Ξ½] β¦f : Ξ± Γ Ξ² β Eβ¦
(hf : StronglyMeasurable f) : StronglyMeasurable fun x => β« y, f (x, y) βΞ½ := by |
rw [β uncurry_curry f] at hf; exact hf.integral_prod_right
| 1 | 2.718282 | 0 | 1.25 | 4 | 1,304 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable s... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 158 | 167 | theorem integrable_measure_prod_mk_left {s : Set (Ξ± Γ Ξ²)} (hs : MeasurableSet s)
(h2s : (ΞΌ.prod Ξ½) s β β) : Integrable (fun x => (Ξ½ (Prod.mk x β»ΒΉ' s)).toReal) ΞΌ := by |
refine β¨(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, ?_β©
simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal toReal_nonneg]
convert h2s.lt_top using 1
-- Porting note: was `simp_rw`
rw [prod_apply hs]
apply lintegral_congr_ae
filter_upwards [ae_measure_lt_top hs h2s] w... | 8 | 2,980.957987 | 2 | 1.25 | 4 | 1,304 |
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.cartan_subalgebra from "leanprover-community/mathlib"@"938fead7abdc0cbbca8eba7a1052865a169dc102"
universe u v w wβ wβ
variable {R : Type u} {L : Type v}
variable [CommRing R] [LieRing L] [LieAlgebra R L] (H : Lie... | Mathlib/Algebra/Lie/CartanSubalgebra.lean | 58 | 61 | theorem normalizer_eq_self_of_isCartanSubalgebra (H : LieSubalgebra R L) [H.IsCartanSubalgebra] :
H.toLieSubmodule.normalizer = H.toLieSubmodule := by |
rw [β LieSubmodule.coe_toSubmodule_eq_iff, coe_normalizer_eq_normalizer,
IsCartanSubalgebra.self_normalizing, coe_toLieSubmodule]
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,305 |
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