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.GroupTheory.OrderOfElement
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Quotient
#align_import algebra.char_p.quotient from "leanprover-community/mathlib"@"85e3c05a94b27c84dc6f234cf88326d5e0096ec3"
universe u v
| Mathlib/Algebra/CharP/Quotient.lean | 60 | 66 | theorem Ideal.Quotient.index_eq_zero {R : Type*} [CommRing R] (I : Ideal R) :
(βI.toAddSubgroup.index : R β§Έ I) = 0 := by |
rw [AddSubgroup.index, Nat.card_eq]
split_ifs with hq; swap
Β· simp
letI : Fintype (R β§Έ I) := @Fintype.ofFinite _ hq
exact Nat.cast_card_eq_zero (R β§Έ I)
| 5 | 148.413159 | 2 | 1 | 3 | 970 |
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Topology.PartialHomeomorph
#align_import analysis.calculus.inverse from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Function Set Filter Metric
open scoped Topolo... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean | 76 | 77 | theorem approximatesLinearOn_empty (f : E β F) (f' : E βL[π] F) (c : ββ₯0) :
ApproximatesLinearOn f f' β
c := by | simp [ApproximatesLinearOn]
| 1 | 2.718282 | 0 | 1 | 3 | 971 |
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Topology.PartialHomeomorph
#align_import analysis.calculus.inverse from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Function Set Filter Metric
open scoped Topolo... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean | 101 | 105 | theorem approximatesLinearOn_iff_lipschitzOnWith {f : E β F} {f' : E βL[π] F} {s : Set E}
{c : ββ₯0} : ApproximatesLinearOn f f' s c β LipschitzOnWith c (f - βf') s := by |
have : β x y, f x - f y - f' (x - y) = (f - f') x - (f - f') y := fun x y β¦ by
simp only [map_sub, Pi.sub_apply]; abel
simp only [this, lipschitzOnWith_iff_norm_sub_le, ApproximatesLinearOn]
| 3 | 20.085537 | 1 | 1 | 3 | 971 |
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Topology.PartialHomeomorph
#align_import analysis.calculus.inverse from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Function Set Filter Metric
open scoped Topolo... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean | 148 | 280 | theorem surjOn_closedBall_of_nonlinearRightInverse (hf : ApproximatesLinearOn f f' s c)
(f'symm : f'.NonlinearRightInverse) {Ξ΅ : β} {b : E} (Ξ΅0 : 0 β€ Ξ΅) (hΞ΅ : closedBall b Ξ΅ β s) :
SurjOn f (closedBall b Ξ΅) (closedBall (f b) (((f'symm.nnnorm : β)β»ΒΉ - c) * Ξ΅)) := by |
intro y hy
rcases le_or_lt (f'symm.nnnorm : β)β»ΒΉ c with hc | hc
Β· refine β¨b, by simp [Ξ΅0], ?_β©
have : dist y (f b) β€ 0 :=
(mem_closedBall.1 hy).trans (mul_nonpos_of_nonpos_of_nonneg (by linarith) Ξ΅0)
simp only [dist_le_zero] at this
rw [this]
have If' : (0 : β) < f'symm.nnnorm := by rw [β inv... | 128 | 38,877,084,059,945,950,000,000,000,000,000,000,000,000,000,000,000,000,000 | 2 | 1 | 3 | 971 |
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community/mathlib"@"2f3994e1b117b1e1da49bcfb67334f33460c3ce4"
open Nat
namespace Nat
def choose : β β β β β
| _, 0 => 1
| 0, _ + 1 => 0
| n + 1, k + 1 => choose n k + choose n ... | Mathlib/Data/Nat/Choose/Basic.lean | 54 | 54 | theorem choose_zero_right (n : β) : choose n 0 = 1 := by | cases n <;> rfl
| 1 | 2.718282 | 0 | 1 | 5 | 972 |
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community/mathlib"@"2f3994e1b117b1e1da49bcfb67334f33460c3ce4"
open Nat
namespace Nat
def choose : β β β β β
| _, 0 => 1
| 0, _ + 1 => 0
| n + 1, k + 1 => choose n k + choose n ... | Mathlib/Data/Nat/Choose/Basic.lean | 79 | 80 | theorem choose_self (n : β) : choose n n = 1 := by |
induction n <;> simp [*, choose, choose_eq_zero_of_lt (lt_succ_self _)]
| 1 | 2.718282 | 0 | 1 | 5 | 972 |
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community/mathlib"@"2f3994e1b117b1e1da49bcfb67334f33460c3ce4"
open Nat
namespace Nat
def choose : β β β β β
| _, 0 => 1
| 0, _ + 1 => 0
| n + 1, k + 1 => choose n k + choose n ... | Mathlib/Data/Nat/Choose/Basic.lean | 93 | 95 | theorem triangle_succ (n : β) : (n + 1) * (n + 1 - 1) / 2 = n * (n - 1) / 2 + n := by |
rw [β add_mul_div_left, Nat.mul_comm 2 n, β Nat.mul_add, Nat.add_sub_cancel, Nat.mul_comm]
cases n <;> rfl; apply zero_lt_succ
| 2 | 7.389056 | 1 | 1 | 5 | 972 |
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community/mathlib"@"2f3994e1b117b1e1da49bcfb67334f33460c3ce4"
open Nat
namespace Nat
def choose : β β β β β
| _, 0 => 1
| 0, _ + 1 => 0
| n + 1, k + 1 => choose n k + choose n ... | Mathlib/Data/Nat/Choose/Basic.lean | 99 | 103 | theorem choose_two_right (n : β) : choose n 2 = n * (n - 1) / 2 := by |
induction' n with n ih
Β· simp
Β· rw [triangle_succ n, choose, ih]
simp [Nat.add_comm]
| 4 | 54.59815 | 2 | 1 | 5 | 972 |
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community/mathlib"@"2f3994e1b117b1e1da49bcfb67334f33460c3ce4"
open Nat
namespace Nat
def choose : β β β β β
| _, 0 => 1
| 0, _ + 1 => 0
| n + 1, k + 1 => choose n k + choose n ... | Mathlib/Data/Nat/Choose/Basic.lean | 125 | 142 | theorem choose_mul_factorial_mul_factorial : β {n k}, k β€ n β choose n k * k ! * (n - k)! = n !
| 0, _, hk => by simp [Nat.eq_zero_of_le_zero hk]
| n + 1, 0, _ => by simp
| n + 1, succ k, hk => by
rcases lt_or_eq_of_le hk with hkβ | hkβ
Β· have h : choose n k * k.succ ! * (n - k)! = (k + 1) * n ! := by |
rw [β choose_mul_factorial_mul_factorial (le_of_succ_le_succ hk)]
simp [factorial_succ, Nat.mul_comm, Nat.mul_left_comm, Nat.mul_assoc]
have hβ : (n - k)! = (n - k) * (n - k.succ)! := by
rw [β succ_sub_succ, succ_sub (le_of_lt_succ hkβ), factorial_succ]
have hβ : choose n (succ k) *... | 12 | 162,754.791419 | 2 | 1 | 5 | 972 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.floor from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Set
section FloorRing
variable {Ξ± R : Type*} [MeasurableSpace Ξ±] [LinearOrderedRing R] [FloorRing R] [TopologicalSpace R]
... | Mathlib/MeasureTheory/Function/Floor.lean | 47 | 50 | theorem measurable_fract [BorelSpace R] : Measurable (Int.fract : R β R) := by |
intro s hs
rw [Int.preimage_fract]
exact MeasurableSet.iUnion fun z => measurable_id.sub_const _ (hs.inter measurableSet_Ico)
| 3 | 20.085537 | 1 | 1 | 2 | 973 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.floor from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Set
section FloorRing
variable {Ξ± R : Type*} [MeasurableSpace Ξ±] [LinearOrderedRing R] [FloorRing R] [TopologicalSpace R]
... | Mathlib/MeasureTheory/Function/Floor.lean | 59 | 62 | theorem MeasurableSet.image_fract [BorelSpace R] {s : Set R} (hs : MeasurableSet s) :
MeasurableSet (Int.fract '' s) := by |
simp only [Int.image_fract, sub_eq_add_neg, image_add_right']
exact MeasurableSet.iUnion fun m => (measurable_add_const _ hs).inter measurableSet_Ico
| 2 | 7.389056 | 1 | 1 | 2 | 973 |
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.MeasureTheory.Function.LpSpace
#align_import measure_theory.function.lp_order from "leanprover-community/mathlib"@"5dc275ec639221ca4d5f56938eb966f6ad9bc89f"
set_option linter.uppercaseLean3 false
open TopologicalSpace MeasureTheory
open scoped ENNReal
... | Mathlib/MeasureTheory/Function/LpOrder.lean | 41 | 42 | theorem coeFn_le (f g : Lp E p ΞΌ) : f β€α΅[ΞΌ] g β f β€ g := by |
rw [β Subtype.coe_le_coe, β AEEqFun.coeFn_le]
| 1 | 2.718282 | 0 | 1 | 2 | 974 |
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.MeasureTheory.Function.LpSpace
#align_import measure_theory.function.lp_order from "leanprover-community/mathlib"@"5dc275ec639221ca4d5f56938eb966f6ad9bc89f"
set_option linter.uppercaseLean3 false
open TopologicalSpace MeasureTheory
open scoped ENNReal
... | Mathlib/MeasureTheory/Function/LpOrder.lean | 45 | 50 | theorem coeFn_nonneg (f : Lp E p ΞΌ) : 0 β€α΅[ΞΌ] f β 0 β€ f := by |
rw [β coeFn_le]
have h0 := Lp.coeFn_zero E p ΞΌ
constructor <;> intro h <;> filter_upwards [h, h0] with _ _ h2
Β· rwa [h2]
Β· rwa [β h2]
| 5 | 148.413159 | 2 | 1 | 2 | 974 |
import Mathlib.Computability.Halting
#align_import computability.reduce from "leanprover-community/mathlib"@"d13b3a4a392ea7273dfa4727dbd1892e26cfd518"
universe u v w
open Function
def ManyOneReducible {Ξ± Ξ²} [Primcodable Ξ±] [Primcodable Ξ²] (p : Ξ± β Prop) (q : Ξ² β Prop) :=
β f, Computable f β§ β a, p a β q (f a... | Mathlib/Computability/Reduce.lean | 111 | 113 | theorem OneOneReducible.of_equiv_symm {Ξ± Ξ²} [Primcodable Ξ±] [Primcodable Ξ²] {e : Ξ± β Ξ²}
(q : Ξ² β Prop) (h : Computable e.symm) : q β€β (q β e) := by |
convert OneOneReducible.of_equiv _ h; funext; simp
| 1 | 2.718282 | 0 | 1 | 2 | 975 |
import Mathlib.Computability.Halting
#align_import computability.reduce from "leanprover-community/mathlib"@"d13b3a4a392ea7273dfa4727dbd1892e26cfd518"
universe u v w
open Function
def ManyOneReducible {Ξ± Ξ²} [Primcodable Ξ±] [Primcodable Ξ²] (p : Ξ± β Prop) (q : Ξ² β Prop) :=
β f, Computable f β§ β a, p a β q (f a... | Mathlib/Computability/Reduce.lean | 131 | 136 | theorem computable_of_manyOneReducible {p : Ξ± β Prop} {q : Ξ² β Prop} (hβ : p β€β q)
(hβ : ComputablePred q) : ComputablePred p := by |
rcases hβ with β¨f, c, hfβ©
rw [show p = fun a => q (f a) from Set.ext hf]
rcases computable_iff.1 hβ with β¨g, hg, rflβ©
exact β¨by infer_instance, by simpa using hg.comp cβ©
| 4 | 54.59815 | 2 | 1 | 2 | 975 |
import Mathlib.Data.Set.Image
#align_import order.directed from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
open Function
universe u v w
variable {Ξ± : Type u} {Ξ² : Type v} {ΞΉ : Sort w} (r r' s : Ξ± β Ξ± β Prop)
local infixl:50 " βΌ " => r
def Directed (f : ΞΉ β Ξ±) :=
β x y, β z, ... | Mathlib/Order/Directed.lean | 58 | 60 | theorem directedOn_iff_directed {s} : @DirectedOn Ξ± r s β Directed r (Subtype.val : s β Ξ±) := by |
simp only [DirectedOn, Directed, Subtype.exists, exists_and_left, exists_prop, Subtype.forall]
exact forallβ_congr fun x _ => by simp [And.comm, and_assoc]
| 2 | 7.389056 | 1 | 1 | 4 | 976 |
import Mathlib.Data.Set.Image
#align_import order.directed from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
open Function
universe u v w
variable {Ξ± : Type u} {Ξ² : Type v} {ΞΉ : Sort w} (r r' s : Ξ± β Ξ± β Prop)
local infixl:50 " βΌ " => r
def Directed (f : ΞΉ β Ξ±) :=
β x y, β z, ... | Mathlib/Order/Directed.lean | 66 | 67 | theorem directedOn_range {f : ΞΉ β Ξ±} : Directed r f β DirectedOn r (Set.range f) := by |
simp_rw [Directed, DirectedOn, Set.forall_mem_range, Set.exists_range_iff]
| 1 | 2.718282 | 0 | 1 | 4 | 976 |
import Mathlib.Data.Set.Image
#align_import order.directed from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
open Function
universe u v w
variable {Ξ± : Type u} {Ξ² : Type v} {ΞΉ : Sort w} (r r' s : Ξ± β Ξ± β Prop)
local infixl:50 " βΌ " => r
def Directed (f : ΞΉ β Ξ±) :=
β x y, β z, ... | Mathlib/Order/Directed.lean | 77 | 80 | theorem directedOn_image {s : Set Ξ²} {f : Ξ² β Ξ±} :
DirectedOn r (f '' s) β DirectedOn (f β»ΒΉ'o r) s := by |
simp only [DirectedOn, Set.mem_image, exists_exists_and_eq_and, forall_exists_index, and_imp,
forall_apply_eq_imp_iffβ, Order.Preimage]
| 2 | 7.389056 | 1 | 1 | 4 | 976 |
import Mathlib.Data.Set.Image
#align_import order.directed from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
open Function
universe u v w
variable {Ξ± : Type u} {Ξ² : Type v} {ΞΉ : Sort w} (r r' s : Ξ± β Ξ± β Prop)
local infixl:50 " βΌ " => r
def Directed (f : ΞΉ β Ξ±) :=
β x y, β z, ... | Mathlib/Order/Directed.lean | 116 | 128 | theorem Directed.extend_bot [Preorder Ξ±] [OrderBot Ξ±] {e : ΞΉ β Ξ²} {f : ΞΉ β Ξ±}
(hf : Directed (Β· β€ Β·) f) (he : Function.Injective e) :
Directed (Β· β€ Β·) (Function.extend e f β₯) := by |
intro a b
rcases (em (β i, e i = a)).symm with (ha | β¨i, rflβ©)
Β· use b
simp [Function.extend_apply' _ _ _ ha]
rcases (em (β i, e i = b)).symm with (hb | β¨j, rflβ©)
Β· use e i
simp [Function.extend_apply' _ _ _ hb]
rcases hf i j with β¨k, hi, hjβ©
use e k
simp only [he.extend_apply, *, true_and_iff]... | 10 | 22,026.465795 | 2 | 1 | 4 | 976 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.RCLike.Basic
open Set Algebra Filter
open scoped Topology
variable (π : Type*) [RCLike π]
| Mathlib/Analysis/SpecificLimits/RCLike.lean | 19 | 22 | theorem RCLike.tendsto_inverse_atTop_nhds_zero_nat :
Tendsto (fun n : β => (n : π)β»ΒΉ) atTop (π 0) := by |
convert tendsto_algebraMap_inverse_atTop_nhds_zero_nat π
simp
| 2 | 7.389056 | 1 | 1 | 1 | 977 |
import Mathlib.Algebra.Homology.ImageToKernel
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.GradedObject
#align_import algebra.homology.homology from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open CategoryTheory CategoryTheory.Limits... | Mathlib/Algebra/Homology/Homology.lean | 68 | 71 | theorem cycles_eq_top {i} (h : Β¬c.Rel i (c.next i)) : C.cycles' i = β€ := by |
rw [eq_top_iff]
apply le_kernelSubobject
rw [C.dFrom_eq_zero h, comp_zero]
| 3 | 20.085537 | 1 | 1 | 2 | 978 |
import Mathlib.Algebra.Homology.ImageToKernel
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.GradedObject
#align_import algebra.homology.homology from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open CategoryTheory CategoryTheory.Limits... | Mathlib/Algebra/Homology/Homology.lean | 97 | 100 | theorem boundaries_eq_bot [HasZeroObject V] {j} (h : Β¬c.Rel (c.prev j) j) : C.boundaries j = β₯ := by |
rw [eq_bot_iff]
refine imageSubobject_le _ 0 ?_
rw [C.dTo_eq_zero h, zero_comp]
| 3 | 20.085537 | 1 | 1 | 2 | 978 |
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.LinearAlgebra.Basis
#align_import analysis.normed_space.linear_isometry from "leanprover-community/mathlib"@"4601791ea62fea875b488dafc4e6dede19e8363f"
open Function Set
variable {R Rβ Rβ Rβ E Eβ Eβ Eβ F π : Ty... | Mathlib/Analysis/NormedSpace/LinearIsometry.lean | 170 | 172 | theorem coe_injective : @Injective (E βββα΅’[Οββ] Eβ) (E β Eβ) (fun f => f) := by |
rintro β¨_β© β¨_β©
simp
| 2 | 7.389056 | 1 | 1 | 1 | 979 |
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.Order.CauSeq.Completion
#align_import data.real.basic from "leanprover-community/mathlib"@"cb42593171ba005beaaf4549fcfe0dece9ada4c9"
assert_not_exists Finset
assert_not_exists Module
assert_not_exists Submonoid
assert_not_exists FloorRing
structure Real w... | Mathlib/Data/Real/Basic.lean | 130 | 132 | theorem ofCauchy_sub (a b) : (β¨a - bβ© : β) = β¨aβ© - β¨bβ© := by |
rw [sub_eq_add_neg, ofCauchy_add, ofCauchy_neg]
rfl
| 2 | 7.389056 | 1 | 1 | 1 | 980 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
| Mathlib/NumberTheory/Pell.lean | 83 | 85 | theorem is_pell_solution_iff_mem_unitary {d : β€} {a : β€βd} :
a.re ^ 2 - d * a.im ^ 2 = 1 β a β unitary (β€βd) := by |
rw [β norm_eq_one_iff_mem_unitary, norm_def, sq, sq, β mul_assoc]
| 1 | 2.718282 | 0 | 1 | 7 | 981 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 133 | 133 | theorem prop_x (a : Solutionβ d) : a.x ^ 2 = 1 + d * a.y ^ 2 := by | rw [β a.prop]; ring
| 1 | 2.718282 | 0 | 1 | 7 | 981 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 137 | 137 | theorem prop_y (a : Solutionβ d) : d * a.y ^ 2 = a.x ^ 2 - 1 := by | rw [β a.prop]; ring
| 1 | 2.718282 | 0 | 1 | 7 | 981 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 178 | 180 | theorem x_mul (a b : Solutionβ d) : (a * b).x = a.x * b.x + d * (a.y * b.y) := by |
rw [β mul_assoc]
rfl
| 2 | 7.389056 | 1 | 1 | 7 | 981 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 209 | 214 | theorem eq_zero_of_d_neg (hβ : d < 0) (a : Solutionβ d) : a.x = 0 β¨ a.y = 0 := by |
have h := a.prop
contrapose! h
have h1 := sq_pos_of_ne_zero h.1
have h2 := sq_pos_of_ne_zero h.2
nlinarith
| 5 | 148.413159 | 2 | 1 | 7 | 981 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 218 | 222 | theorem x_ne_zero (hβ : 0 β€ d) (a : Solutionβ d) : a.x β 0 := by |
intro hx
have h : 0 β€ d * a.y ^ 2 := mul_nonneg hβ (sq_nonneg _)
rw [a.prop_y, hx, sq, zero_mul, zero_sub] at h
exact not_le.mpr (neg_one_lt_zero : (-1 : β€) < 0) h
| 4 | 54.59815 | 2 | 1 | 7 | 981 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 367 | 434 | theorem exists_of_not_isSquare (hβ : 0 < d) (hd : Β¬IsSquare d) :
β x y : β€, x ^ 2 - d * y ^ 2 = 1 β§ y β 0 := by |
let ΞΎ : β := βd
have hΞΎ : Irrational ΞΎ := by
refine irrational_nrt_of_notint_nrt 2 d (sq_sqrt <| Int.cast_nonneg.mpr hβ.le) ?_ two_pos
rintro β¨x, hxβ©
refine hd β¨x, @Int.cast_injective β _ _ d (x * x) ?_β©
rw [β sq_sqrt <| Int.cast_nonneg.mpr hβ.le, Int.cast_mul, β hx, sq]
obtain β¨M, hMββ© := exists... | 66 | 46,071,866,343,312,910,000,000,000,000 | 2 | 1 | 7 | 981 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
#align_import measure_theory.function.conditional_expectation.condexp_L2 from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 106 | 110 | theorem snorm_condexpL2_le (hm : m β€ m0) (f : Ξ± ββ[ΞΌ] E) :
snorm (F := E) (condexpL2 E π hm f) 2 ΞΌ β€ snorm f 2 ΞΌ := by |
rw [lpMeas_coe, β ENNReal.toReal_le_toReal (Lp.snorm_ne_top _) (Lp.snorm_ne_top _), β
Lp.norm_def, β Lp.norm_def, Submodule.norm_coe]
exact norm_condexpL2_le hm f
| 3 | 20.085537 | 1 | 1 | 2 | 982 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
#align_import measure_theory.function.conditional_expectation.condexp_L2 from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 113 | 117 | theorem norm_condexpL2_coe_le (hm : m β€ m0) (f : Ξ± ββ[ΞΌ] E) :
β(condexpL2 E π hm f : Ξ± ββ[ΞΌ] E)β β€ βfβ := by |
rw [Lp.norm_def, Lp.norm_def, β lpMeas_coe]
refine (ENNReal.toReal_le_toReal ?_ (Lp.snorm_ne_top _)).mpr (snorm_condexpL2_le hm f)
exact Lp.snorm_ne_top _
| 3 | 20.085537 | 1 | 1 | 2 | 982 |
import Mathlib.Algebra.Group.Invertible.Basic
import Mathlib.Algebra.GroupWithZero.Units.Basic
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
assert_not_exists DenselyOrdered
universe u
variable {Ξ± : Type u}
| Mathlib/Algebra/GroupWithZero/Invertible.lean | 23 | 28 | theorem nonzero_of_invertible [MulZeroOneClass Ξ±] (a : Ξ±) [Nontrivial Ξ±] [Invertible a] : a β 0 :=
fun ha =>
zero_ne_one <|
calc
0 = β
a * a := by | simp [ha]
_ = 1 := invOf_mul_self a
| 2 | 7.389056 | 1 | 1 | 1 | 983 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed.group.quotient from "leanprover-community/mathlib"@"2196ab363eb097c008... | Mathlib/Analysis/Normed/Group/Quotient.lean | 113 | 115 | theorem QuotientAddGroup.norm_eq_infDist {S : AddSubgroup M} (x : M β§Έ S) :
βxβ = infDist 0 { m : M | (m : M β§Έ S) = x } := by |
simp only [AddSubgroup.quotient_norm_eq, infDist_eq_iInf, sInf_image', dist_zero_left]
| 1 | 2.718282 | 0 | 1 | 8 | 984 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed.group.quotient from "leanprover-community/mathlib"@"2196ab363eb097c008... | Mathlib/Analysis/Normed/Group/Quotient.lean | 119 | 125 | theorem QuotientAddGroup.norm_mk {S : AddSubgroup M} (x : M) :
β(x : M β§Έ S)β = infDist x S := by |
rw [norm_eq_infDist, β infDist_image (IsometryEquiv.subLeft x).isometry,
IsometryEquiv.subLeft_apply, sub_zero, β IsometryEquiv.preimage_symm]
congr 1 with y
simp only [mem_preimage, IsometryEquiv.subLeft_symm_apply, mem_setOf_eq, QuotientAddGroup.eq,
neg_add, neg_neg, neg_add_cancel_right, SetLike.mem_c... | 5 | 148.413159 | 2 | 1 | 8 | 984 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed.group.quotient from "leanprover-community/mathlib"@"2196ab363eb097c008... | Mathlib/Analysis/Normed/Group/Quotient.lean | 141 | 144 | theorem quotient_norm_neg {S : AddSubgroup M} (x : M β§Έ S) : β-xβ = βxβ := by |
simp only [AddSubgroup.quotient_norm_eq]
congr 1 with r
constructor <;> { rintro β¨m, hm, rflβ©; use -m; simpa [neg_eq_iff_eq_neg] using hm }
| 3 | 20.085537 | 1 | 1 | 8 | 984 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed.group.quotient from "leanprover-community/mathlib"@"2196ab363eb097c008... | Mathlib/Analysis/Normed/Group/Quotient.lean | 147 | 148 | theorem quotient_norm_sub_rev {S : AddSubgroup M} (x y : M β§Έ S) : βx - yβ = βy - xβ := by |
rw [β neg_sub, quotient_norm_neg]
| 1 | 2.718282 | 0 | 1 | 8 | 984 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed.group.quotient from "leanprover-community/mathlib"@"2196ab363eb097c008... | Mathlib/Analysis/Normed/Group/Quotient.lean | 162 | 166 | theorem quotient_norm_mk_eq (S : AddSubgroup M) (m : M) :
βmk' S mβ = sInf ((βm + Β·β) '' S) := by |
rw [mk'_apply, norm_mk, sInf_image', β infDist_image isometry_neg, image_neg,
neg_coe_set (H := S), infDist_eq_iInf]
simp only [dist_eq_norm', sub_neg_eq_add, add_comm]
| 3 | 20.085537 | 1 | 1 | 8 | 984 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed.group.quotient from "leanprover-community/mathlib"@"2196ab363eb097c008... | Mathlib/Analysis/Normed/Group/Quotient.lean | 181 | 184 | theorem quotient_norm_eq_zero_iff (S : AddSubgroup M) (m : M) :
βmk' S mβ = 0 β m β closure (S : Set M) := by |
rw [mk'_apply, norm_mk, β mem_closure_iff_infDist_zero]
exact β¨0, S.zero_memβ©
| 2 | 7.389056 | 1 | 1 | 8 | 984 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed.group.quotient from "leanprover-community/mathlib"@"2196ab363eb097c008... | Mathlib/Analysis/Normed/Group/Quotient.lean | 187 | 190 | theorem QuotientAddGroup.norm_lt_iff {S : AddSubgroup M} {x : M β§Έ S} {r : β} :
βxβ < r β β m : M, βm = x β§ βmβ < r := by |
rw [isGLB_lt_iff (isGLB_quotient_norm _), exists_mem_image]
rfl
| 2 | 7.389056 | 1 | 1 | 8 | 984 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed.group.quotient from "leanprover-community/mathlib"@"2196ab363eb097c008... | Mathlib/Analysis/Normed/Group/Quotient.lean | 200 | 206 | theorem norm_mk_lt' (S : AddSubgroup M) (m : M) {Ξ΅ : β} (hΞ΅ : 0 < Ξ΅) :
β s β S, βm + sβ < βmk' S mβ + Ξ΅ := by |
obtain β¨n : M, hn : mk' S n = mk' S m, hn' : βnβ < βmk' S mβ + Ξ΅β© :=
norm_mk_lt (QuotientAddGroup.mk' S m) hΞ΅
erw [eq_comm, QuotientAddGroup.eq] at hn
use -m + n, hn
rwa [add_neg_cancel_left]
| 5 | 148.413159 | 2 | 1 | 8 | 984 |
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
open Function Set
variable ... | Mathlib/RingTheory/Nilpotent/Lemmas.lean | 25 | 29 | theorem RingHom.ker_isRadical_iff_reduced_of_surjective {S F} [CommSemiring R] [CommRing S]
[FunLike F R S] [RingHomClass F R S] {f : F} (hf : Function.Surjective f) :
(RingHom.ker f).IsRadical β IsReduced S := by |
simp_rw [isReduced_iff, hf.forall, IsNilpotent, β map_pow, β RingHom.mem_ker]
rfl
| 2 | 7.389056 | 1 | 1 | 3 | 985 |
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
open Function Set
variable ... | Mathlib/RingTheory/Nilpotent/Lemmas.lean | 32 | 35 | theorem isRadical_iff_span_singleton [CommSemiring R] :
IsRadical y β (Ideal.span ({y} : Set R)).IsRadical := by |
simp_rw [IsRadical, β Ideal.mem_span_singleton]
exact forall_swap.trans (forall_congr' fun r => exists_imp.symm)
| 2 | 7.389056 | 1 | 1 | 3 | 985 |
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
open Function Set
variable ... | Mathlib/RingTheory/Nilpotent/Lemmas.lean | 123 | 126 | theorem IsNilpotent.mapQ (hnp : IsNilpotent f) : IsNilpotent (p.mapQ p f hp) := by |
obtain β¨k, hkβ© := hnp
use k
simp [β p.mapQ_pow, hk]
| 3 | 20.085537 | 1 | 1 | 3 | 985 |
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
import Mathlib.Order.Filter.Ker
#align_import order.filter.bases from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
set_option autoImplicit true
open Set Filter
open scoped Classical
ope... | Mathlib/Order/Filter/Bases.lean | 268 | 270 | theorem HasBasis.eq_of_same_basis (hl : l.HasBasis p s) (hl' : l'.HasBasis p s) : l = l' := by |
ext t
rw [hl.mem_iff, hl'.mem_iff]
| 2 | 7.389056 | 1 | 1 | 1 | 986 |
import Mathlib.CategoryTheory.Iso
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.comma from "leanprover-community/mathlib"@"8a318021995877a44630c898d0b2bc376fceef3b"
namespace CategoryTheory
open Category
-- declare the `v`'s first; see `Catego... | Mathlib/CategoryTheory/Comma/Basic.lean | 166 | 169 | theorem eqToHom_left (X Y : Comma L R) (H : X = Y) :
CommaMorphism.left (eqToHom H) = eqToHom (by cases H; rfl) := by |
cases H
rfl
| 2 | 7.389056 | 1 | 1 | 2 | 987 |
import Mathlib.CategoryTheory.Iso
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.comma from "leanprover-community/mathlib"@"8a318021995877a44630c898d0b2bc376fceef3b"
namespace CategoryTheory
open Category
-- declare the `v`'s first; see `Catego... | Mathlib/CategoryTheory/Comma/Basic.lean | 173 | 176 | theorem eqToHom_right (X Y : Comma L R) (H : X = Y) :
CommaMorphism.right (eqToHom H) = eqToHom (by cases H; rfl) := by |
cases H
rfl
| 2 | 7.389056 | 1 | 1 | 2 | 987 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Algebra.Group.ULift
#align_import topology.algebra.monoid from "leanprover-community/mathli... | Mathlib/Topology/Algebra/Monoid.lean | 150 | 152 | theorem le_nhds_mul (a b : M) : π a * π b β€ π (a * b) := by |
rw [β mapβ_mul, β map_uncurry_prod, β nhds_prod_eq]
exact continuous_mul.tendsto _
| 2 | 7.389056 | 1 | 1 | 1 | 988 |
import Mathlib.CategoryTheory.Sites.Canonical
#align_import category_theory.sites.types from "leanprover-community/mathlib"@"9f9015c645d85695581237cc761981036be8bd37"
universe u
namespace CategoryTheory
--open scoped CategoryTheory.Type -- Porting note: unknown namespace
def typesGrothendieckTopology : Grothe... | Mathlib/CategoryTheory/Sites/Types.lean | 102 | 105 | theorem eval_typesGlue {S hs Ξ±} (f) : eval.{u} S Ξ± (typesGlue S hs Ξ± f) = f := by |
funext x
apply (IsSheafFor.valid_glue _ _ _ <| β¨PUnit.unit, fun _ => Subsingleton.elim _ _β©).trans
convert FunctorToTypes.map_id_apply S _
| 3 | 20.085537 | 1 | 1 | 3 | 989 |
import Mathlib.CategoryTheory.Sites.Canonical
#align_import category_theory.sites.types from "leanprover-community/mathlib"@"9f9015c645d85695581237cc761981036be8bd37"
universe u
namespace CategoryTheory
--open scoped CategoryTheory.Type -- Porting note: unknown namespace
def typesGrothendieckTopology : Grothe... | Mathlib/CategoryTheory/Sites/Types.lean | 108 | 117 | theorem typesGlue_eval {S hs Ξ±} (s) : typesGlue.{u} S hs Ξ± (eval S Ξ± s) = s := by |
apply (hs.isSheafFor _ _ (generate_discretePresieve_mem Ξ±)).isSeparatedFor.ext
intro Ξ² f hf
apply (IsSheafFor.valid_glue _ _ _ hf).trans
apply (FunctorToTypes.map_comp_apply _ _ _ _).symm.trans
rw [β op_comp]
--congr 2 -- Porting note: This tactic didn't work. Find an alternative.
suffices ((βΎfun _ β¦ PUn... | 9 | 8,103.083928 | 2 | 1 | 3 | 989 |
import Mathlib.CategoryTheory.Sites.Canonical
#align_import category_theory.sites.types from "leanprover-community/mathlib"@"9f9015c645d85695581237cc761981036be8bd37"
universe u
namespace CategoryTheory
--open scoped CategoryTheory.Type -- Porting note: unknown namespace
def typesGrothendieckTopology : Grothe... | Mathlib/CategoryTheory/Sites/Types.lean | 130 | 132 | theorem eval_map (S : Type uα΅α΅ β₯€ Type u) (Ξ± Ξ²) (f : Ξ² βΆ Ξ±) (s x) :
eval S Ξ² (S.map f.op s) x = eval S Ξ± s (f x) := by |
simp_rw [eval, β FunctorToTypes.map_comp_apply, β op_comp]; rfl
| 1 | 2.718282 | 0 | 1 | 3 | 989 |
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
variable {π E : Type*}
namespace LinearMap
variable (π)
section Seminormed
variable [NormedDivisionRing π] [SeminormedAddCommGroup E] [Module π E] [BoundedSMu... | Mathlib/Analysis/NormedSpace/Span.lean | 36 | 39 | theorem toSpanSingleton_homothety (x : E) (c : π) :
βLinearMap.toSpanSingleton π E x cβ = βxβ * βcβ := by |
rw [mul_comm]
exact norm_smul _ _
| 2 | 7.389056 | 1 | 1 | 1 | 990 |
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ΞΉ : Type*} {G : ΞΉ β Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [β i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 88 | 93 | theorem of_comp_eq_base (i : ΞΉ) : (of i).comp (Ο i) = (base Ο) := by |
ext x
apply (Con.eq _).2
refine ConGen.Rel.of _ _ ?_
simp only [MonoidHom.comp_apply, Set.mem_iUnion, Set.mem_range]
exact β¨_, _, rfl, rflβ©
| 5 | 148.413159 | 2 | 1 | 6 | 991 |
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ΞΉ : Type*} {G : ΞΉ β Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [β i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 96 | 97 | theorem of_apply_eq_base (i : ΞΉ) (x : H) : of i (Ο i x) = base Ο x := by |
rw [β MonoidHom.comp_apply, of_comp_eq_base]
| 1 | 2.718282 | 0 | 1 | 6 | 991 |
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ΞΉ : Type*} {G : ΞΉ β Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [β i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 111 | 116 | theorem lift_of (f : β i, G i β* K) (k : H β* K)
(hf : β i, (f i).comp (Ο i) = k)
{i : ΞΉ} (g : G i) : (lift f k hf) (of i g : PushoutI Ο) = f i g := by |
delta PushoutI lift of
simp only [MonoidHom.coe_comp, Con.coe_mk', comp_apply, Con.lift_coe,
lift_apply_inl, CoprodI.lift_of]
| 3 | 20.085537 | 1 | 1 | 6 | 991 |
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ΞΉ : Type*} {G : ΞΉ β Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [β i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 119 | 123 | theorem lift_base (f : β i, G i β* K) (k : H β* K)
(hf : β i, (f i).comp (Ο i) = k)
(g : H) : (lift f k hf) (base Ο g : PushoutI Ο) = k g := by |
delta PushoutI lift base
simp only [MonoidHom.coe_comp, Con.coe_mk', comp_apply, Con.lift_coe, lift_apply_inr]
| 2 | 7.389056 | 1 | 1 | 6 | 991 |
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ΞΉ : Type*} {G : ΞΉ β Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [β i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 163 | 165 | theorem ofCoprodI_of (i : ΞΉ) (g : G i) :
(ofCoprodI (CoprodI.of g) : PushoutI Ο) = of i g := by |
simp [ofCoprodI]
| 1 | 2.718282 | 0 | 1 | 6 | 991 |
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ΞΉ : Type*} {G : ΞΉ β Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [β i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 167 | 184 | theorem induction_on {motive : PushoutI Ο β Prop}
(x : PushoutI Ο)
(of : β (i : ΞΉ) (g : G i), motive (of i g))
(base : β h, motive (base Ο h))
(mul : β x y, motive x β motive y β motive (x * y)) : motive x := by |
delta PushoutI PushoutI.of PushoutI.base at *
induction x using Con.induction_on with
| H x =>
induction x using Coprod.induction_on with
| inl g =>
induction g using CoprodI.induction_on with
| h_of i g => exact of i g
| h_mul x y ihx ihy =>
rw [map_mul]
exact mul _ _ i... | 13 | 442,413.392009 | 2 | 1 | 6 | 991 |
import Mathlib.ModelTheory.Basic
#align_import model_theory.language_map from "leanprover-community/mathlib"@"b3951c65c6e797ff162ae8b69eab0063bcfb3d73"
universe u v u' v' w w'
namespace FirstOrder
set_option linter.uppercaseLean3 false
namespace Language
open Structure Cardinal
open Cardinal
variable (L : L... | Mathlib/ModelTheory/LanguageMap.lean | 153 | 155 | theorem id_comp (F : L βα΄Έ L') : LHom.id L' βα΄Έ F = F := by |
cases F
rfl
| 2 | 7.389056 | 1 | 1 | 2 | 992 |
import Mathlib.ModelTheory.Basic
#align_import model_theory.language_map from "leanprover-community/mathlib"@"b3951c65c6e797ff162ae8b69eab0063bcfb3d73"
universe u v u' v' w w'
namespace FirstOrder
set_option linter.uppercaseLean3 false
namespace Language
open Structure Cardinal
open Cardinal
variable (L : L... | Mathlib/ModelTheory/LanguageMap.lean | 159 | 161 | theorem comp_id (F : L βα΄Έ L') : F βα΄Έ LHom.id L = F := by |
cases F
rfl
| 2 | 7.389056 | 1 | 1 | 2 | 992 |
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.NormNum.GCD
namespace Tactic
namespace NormNum
open Qq Lean Elab.Tactic Mathlib.Meta.NormNum
| Mathlib/Tactic/NormNum/IsCoprime.lean | 23 | 26 | theorem int_not_isCoprime_helper (x y : β€) (d : β) (hd : Int.gcd x y = d)
(h : Nat.beq d 1 = false) : Β¬ IsCoprime x y := by |
rw [Int.isCoprime_iff_gcd_eq_one, hd]
exact Nat.ne_of_beq_eq_false h
| 2 | 7.389056 | 1 | 1 | 1 | 993 |
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {π E : Type*}
variable [NormedField π]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 84 | 88 | theorem smul_ball {c : π} (hc : c β 0) (x : E) (r : β) : c β’ ball x r = ball (c β’ x) (βcβ * r) := by |
ext y
rw [mem_smul_set_iff_inv_smul_memβ hc]
conv_lhs => rw [β inv_smul_smulβ hc x]
simp [β div_eq_inv_mul, div_lt_iff (norm_pos_iff.2 hc), mul_comm _ r, dist_smulβ]
| 4 | 54.59815 | 2 | 1 | 8 | 994 |
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {π E : Type*}
variable [NormedField π]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 91 | 92 | theorem smul_unitBall {c : π} (hc : c β 0) : c β’ ball (0 : E) (1 : β) = ball (0 : E) βcβ := by |
rw [_root_.smul_ball hc, smul_zero, mul_one]
| 1 | 2.718282 | 0 | 1 | 8 | 994 |
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {π E : Type*}
variable [NormedField π]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 95 | 101 | theorem smul_sphere' {c : π} (hc : c β 0) (x : E) (r : β) :
c β’ sphere x r = sphere (c β’ x) (βcβ * r) := by |
ext y
rw [mem_smul_set_iff_inv_smul_memβ hc]
conv_lhs => rw [β inv_smul_smulβ hc x]
simp only [mem_sphere, dist_smulβ, norm_inv, β div_eq_inv_mul, div_eq_iff (norm_pos_iff.2 hc).ne',
mul_comm r]
| 5 | 148.413159 | 2 | 1 | 8 | 994 |
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {π E : Type*}
variable [NormedField π]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 104 | 106 | theorem smul_closedBall' {c : π} (hc : c β 0) (x : E) (r : β) :
c β’ closedBall x r = closedBall (c β’ x) (βcβ * r) := by |
simp only [β ball_union_sphere, Set.smul_set_union, _root_.smul_ball hc, smul_sphere' hc]
| 1 | 2.718282 | 0 | 1 | 8 | 994 |
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {π E : Type*}
variable [NormedField π]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 109 | 115 | theorem set_smul_sphere_zero {s : Set π} (hs : 0 β s) (r : β) :
s β’ sphere (0 : E) r = (βΒ·β) β»ΒΉ' ((βΒ·β * r) '' s) :=
calc
s β’ sphere (0 : E) r = β c β s, c β’ sphere (0 : E) r := iUnion_smul_left_image.symm
_ = β c β s, sphere (0 : E) (βcβ * r) := iUnionβ_congr fun c hc β¦ by
rw [smul_sphere' (ne_of_... | ext; simp [eq_comm]
| 1 | 2.718282 | 0 | 1 | 8 | 994 |
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {π E : Type*}
variable [NormedField π]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 125 | 143 | theorem eventually_singleton_add_smul_subset {x : E} {s : Set E} (hs : Bornology.IsBounded s)
{u : Set E} (hu : u β π x) : βαΆ r in π (0 : π), {x} + r β’ s β u := by |
obtain β¨Ξ΅, Ξ΅pos, hΞ΅β© : β Ξ΅ : β, 0 < Ξ΅ β§ closedBall x Ξ΅ β u := nhds_basis_closedBall.mem_iff.1 hu
obtain β¨R, Rpos, hRβ© : β R : β, 0 < R β§ s β closedBall 0 R := hs.subset_closedBall_lt 0 0
have : Metric.closedBall (0 : π) (Ξ΅ / R) β π (0 : π) := closedBall_mem_nhds _ (div_pos Ξ΅pos Rpos)
filter_upwards [this] w... | 17 | 24,154,952.753575 | 2 | 1 | 8 | 994 |
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {π E : Type*}
variable [NormedField π]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 150 | 151 | theorem smul_unitBall_of_pos {r : β} (hr : 0 < r) : r β’ ball (0 : E) 1 = ball (0 : E) r := by |
rw [smul_unitBall hr.ne', Real.norm_of_nonneg hr.le]
| 1 | 2.718282 | 0 | 1 | 8 | 994 |
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {π E : Type*}
variable [NormedField π]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 162 | 167 | theorem exists_dist_eq (x z : E) {a b : β} (ha : 0 β€ a) (hb : 0 β€ b) (hab : a + b = 1) :
β y, dist x y = b * dist x z β§ dist y z = a * dist x z := by |
use a β’ x + b β’ z
nth_rw 1 [β one_smul β x]
nth_rw 4 [β one_smul β z]
simp [dist_eq_norm, β hab, add_smul, β smul_sub, norm_smul_of_nonneg, ha, hb]
| 4 | 54.59815 | 2 | 1 | 8 | 994 |
import Mathlib.RingTheory.UniqueFactorizationDomain
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.away.basic from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
section CommSemiring
variable {R : Type*} [CommSemiring R] (M : Submonoid R) {S : Type*... | Mathlib/RingTheory/Localization/Away/Basic.lean | 58 | 61 | theorem mul_invSelf : algebraMap R S x * invSelf x = 1 := by |
convert IsLocalization.mk'_mul_mk'_eq_one (M := Submonoid.powers x) (S := S) _ 1
symm
apply IsLocalization.mk'_one
| 3 | 20.085537 | 1 | 1 | 1 | 995 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.UnitaryGroup
#align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
set_... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 114 | 116 | theorem EuclideanSpace.norm_eq {π : Type*} [RCLike π] {n : Type*} [Fintype n]
(x : EuclideanSpace π n) : βxβ = β(β i, βx iβ ^ 2) := by |
simpa only [Real.coe_sqrt, NNReal.coe_sum] using congr_arg ((β) : ββ₯0 β β) x.nnnorm_eq
| 1 | 2.718282 | 0 | 1 | 4 | 996 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.UnitaryGroup
#align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
set_... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 134 | 138 | theorem EuclideanSpace.ball_zero_eq {n : Type*} [Fintype n] (r : β) (hr : 0 β€ r) :
Metric.ball (0 : EuclideanSpace β n) r = {x | β i, x i ^ 2 < r ^ 2} := by |
ext x
have : (0 : β) β€ β i, x i ^ 2 := Finset.sum_nonneg fun _ _ => sq_nonneg _
simp_rw [mem_setOf, mem_ball_zero_iff, norm_eq, norm_eq_abs, sq_abs, sqrt_lt this hr]
| 3 | 20.085537 | 1 | 1 | 4 | 996 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.UnitaryGroup
#align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
set_... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 140 | 143 | theorem EuclideanSpace.closedBall_zero_eq {n : Type*} [Fintype n] (r : β) (hr : 0 β€ r) :
Metric.closedBall (0 : EuclideanSpace β n) r = {x | β i, x i ^ 2 β€ r ^ 2} := by |
ext
simp_rw [mem_setOf, mem_closedBall_zero_iff, norm_eq, norm_eq_abs, sq_abs, sqrt_le_left hr]
| 2 | 7.389056 | 1 | 1 | 4 | 996 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.UnitaryGroup
#align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
set_... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 145 | 150 | theorem EuclideanSpace.sphere_zero_eq {n : Type*} [Fintype n] (r : β) (hr : 0 β€ r) :
Metric.sphere (0 : EuclideanSpace β n) r = {x | β i, x i ^ 2 = r ^ 2} := by |
ext x
have : (0 : β) β€ β i, x i ^ 2 := Finset.sum_nonneg fun _ _ => sq_nonneg _
simp_rw [mem_setOf, mem_sphere_zero_iff_norm, norm_eq, norm_eq_abs, sq_abs,
Real.sqrt_eq_iff_sq_eq this hr, eq_comm]
| 4 | 54.59815 | 2 | 1 | 4 | 996 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 52 | 56 | theorem hasDerivWithinAt_of_bilinear
(hu : HasDerivWithinAt u u' s x) (hv : HasDerivWithinAt v v' s x) :
HasDerivWithinAt (fun x β¦ B (u x) (v x)) (B (u x) v' + B u' (v x)) s x := by |
simpa using (B.hasFDerivWithinAt_of_bilinear
hu.hasFDerivWithinAt hv.hasFDerivWithinAt).hasDerivWithinAt
| 2 | 7.389056 | 1 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 58 | 60 | theorem hasDerivAt_of_bilinear (hu : HasDerivAt u u' x) (hv : HasDerivAt v v' x) :
HasDerivAt (fun x β¦ B (u x) (v x)) (B (u x) v' + B u' (v x)) x := by |
simpa using (B.hasFDerivAt_of_bilinear hu.hasFDerivAt hv.hasFDerivAt).hasDerivAt
| 1 | 2.718282 | 0 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 62 | 65 | theorem hasStrictDerivAt_of_bilinear (hu : HasStrictDerivAt u u' x) (hv : HasStrictDerivAt v v' x) :
HasStrictDerivAt (fun x β¦ B (u x) (v x)) (B (u x) v' + B u' (v x)) x := by |
simpa using
(B.hasStrictFDerivAt_of_bilinear hu.hasStrictFDerivAt hv.hasStrictFDerivAt).hasStrictDerivAt
| 2 | 7.389056 | 1 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 87 | 89 | theorem HasDerivWithinAt.smul (hc : HasDerivWithinAt c c' s x) (hf : HasDerivWithinAt f f' s x) :
HasDerivWithinAt (fun y => c y β’ f y) (c x β’ f' + c' β’ f x) s x := by |
simpa using (HasFDerivWithinAt.smul hc hf).hasDerivWithinAt
| 1 | 2.718282 | 0 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 92 | 95 | theorem HasDerivAt.smul (hc : HasDerivAt c c' x) (hf : HasDerivAt f f' x) :
HasDerivAt (fun y => c y β’ f y) (c x β’ f' + c' β’ f x) x := by |
rw [β hasDerivWithinAt_univ] at *
exact hc.smul hf
| 2 | 7.389056 | 1 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 114 | 117 | theorem HasStrictDerivAt.smul_const (hc : HasStrictDerivAt c c' x) (f : F) :
HasStrictDerivAt (fun y => c y β’ f) (c' β’ f) x := by |
have := hc.smul (hasStrictDerivAt_const x f)
rwa [smul_zero, zero_add] at this
| 2 | 7.389056 | 1 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 120 | 123 | theorem HasDerivWithinAt.smul_const (hc : HasDerivWithinAt c c' s x) (f : F) :
HasDerivWithinAt (fun y => c y β’ f) (c' β’ f) s x := by |
have := hc.smul (hasDerivWithinAt_const x s f)
rwa [smul_zero, zero_add] at this
| 2 | 7.389056 | 1 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 126 | 129 | theorem HasDerivAt.smul_const (hc : HasDerivAt c c' x) (f : F) :
HasDerivAt (fun y => c y β’ f) (c' β’ f) x := by |
rw [β hasDerivWithinAt_univ] at *
exact hc.smul_const f
| 2 | 7.389056 | 1 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 206 | 212 | theorem HasDerivWithinAt.mul (hc : HasDerivWithinAt c c' s x) (hd : HasDerivWithinAt d d' s x) :
HasDerivWithinAt (fun y => c y * d y) (c' * d x + c x * d') s x := by |
have := (HasFDerivWithinAt.mul' hc hd).hasDerivWithinAt
rwa [ContinuousLinearMap.add_apply, ContinuousLinearMap.smul_apply,
ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.smulRight_apply,
ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply, one_smul, one_smul,
add_comm] at this... | 5 | 148.413159 | 2 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 215 | 218 | theorem HasDerivAt.mul (hc : HasDerivAt c c' x) (hd : HasDerivAt d d' x) :
HasDerivAt (fun y => c y * d y) (c' * d x + c x * d') x := by |
rw [β hasDerivWithinAt_univ] at *
exact hc.mul hd
| 2 | 7.389056 | 1 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 221 | 227 | theorem HasStrictDerivAt.mul (hc : HasStrictDerivAt c c' x) (hd : HasStrictDerivAt d d' x) :
HasStrictDerivAt (fun y => c y * d y) (c' * d x + c x * d') x := by |
have := (HasStrictFDerivAt.mul' hc hd).hasStrictDerivAt
rwa [ContinuousLinearMap.add_apply, ContinuousLinearMap.smul_apply,
ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.smulRight_apply,
ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply, one_smul, one_smul,
add_comm] at this... | 5 | 148.413159 | 2 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 242 | 245 | theorem HasDerivWithinAt.mul_const (hc : HasDerivWithinAt c c' s x) (d : πΈ) :
HasDerivWithinAt (fun y => c y * d) (c' * d) s x := by |
convert hc.mul (hasDerivWithinAt_const x s d) using 1
rw [mul_zero, add_zero]
| 2 | 7.389056 | 1 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 248 | 251 | theorem HasDerivAt.mul_const (hc : HasDerivAt c c' x) (d : πΈ) :
HasDerivAt (fun y => c y * d) (c' * d) x := by |
rw [β hasDerivWithinAt_univ] at *
exact hc.mul_const d
| 2 | 7.389056 | 1 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 254 | 255 | theorem hasDerivAt_mul_const (c : π) : HasDerivAt (fun x => x * c) c x := by |
simpa only [one_mul] using (hasDerivAt_id' x).mul_const c
| 1 | 2.718282 | 0 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 258 | 261 | theorem HasStrictDerivAt.mul_const (hc : HasStrictDerivAt c c' x) (d : πΈ) :
HasStrictDerivAt (fun y => c y * d) (c' * d) x := by |
convert hc.mul (hasStrictDerivAt_const x d) using 1
rw [mul_zero, add_zero]
| 2 | 7.389056 | 1 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 274 | 281 | theorem deriv_mul_const_field (v : π') : deriv (fun y => u y * v) x = deriv u x * v := by |
by_cases hu : DifferentiableAt π u x
Β· exact deriv_mul_const hu v
Β· rw [deriv_zero_of_not_differentiableAt hu, zero_mul]
rcases eq_or_ne v 0 with (rfl | hd)
Β· simp only [mul_zero, deriv_const]
Β· refine deriv_zero_of_not_differentiableAt (mt (fun H => ?_) hu)
simpa only [mul_inv_cancel_rightβ h... | 7 | 1,096.633158 | 2 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 336 | 339 | theorem HasDerivAt.finset_prod (hf : β i β u, HasDerivAt (f i) (f' i) x) :
HasDerivAt (β i β u, f i Β·) (β i β u, (β j β u.erase i, f j x) β’ f' i) x := by |
simpa [ContinuousLinearMap.sum_apply, ContinuousLinearMap.smul_apply] using
(HasFDerivAt.finset_prod (fun i hi β¦ (hf i hi).hasFDerivAt)).hasDerivAt
| 2 | 7.389056 | 1 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 341 | 344 | theorem HasDerivWithinAt.finset_prod (hf : β i β u, HasDerivWithinAt (f i) (f' i) s x) :
HasDerivWithinAt (β i β u, f i Β·) (β i β u, (β j β u.erase i, f j x) β’ f' i) s x := by |
simpa [ContinuousLinearMap.sum_apply, ContinuousLinearMap.smul_apply] using
(HasFDerivWithinAt.finset_prod (fun i hi β¦ (hf i hi).hasFDerivWithinAt)).hasDerivWithinAt
| 2 | 7.389056 | 1 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 346 | 349 | theorem HasStrictDerivAt.finset_prod (hf : β i β u, HasStrictDerivAt (f i) (f' i) x) :
HasStrictDerivAt (β i β u, f i Β·) (β i β u, (β j β u.erase i, f j x) β’ f' i) x := by |
simpa [ContinuousLinearMap.sum_apply, ContinuousLinearMap.smul_apply] using
(HasStrictFDerivAt.finset_prod (fun i hi β¦ (hf i hi).hasStrictFDerivAt)).hasStrictDerivAt
| 2 | 7.389056 | 1 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 447 | 451 | theorem HasStrictDerivAt.clm_comp (hc : HasStrictDerivAt c c' x) (hd : HasStrictDerivAt d d' x) :
HasStrictDerivAt (fun y => (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') x := by |
have := (hc.hasStrictFDerivAt.clm_comp hd.hasStrictFDerivAt).hasStrictDerivAt
rwa [add_apply, comp_apply, comp_apply, smulRight_apply, smulRight_apply, one_apply, one_smul,
one_smul, add_comm] at this
| 3 | 20.085537 | 1 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 454 | 459 | theorem HasDerivWithinAt.clm_comp (hc : HasDerivWithinAt c c' s x)
(hd : HasDerivWithinAt d d' s x) :
HasDerivWithinAt (fun y => (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') s x := by |
have := (hc.hasFDerivWithinAt.clm_comp hd.hasFDerivWithinAt).hasDerivWithinAt
rwa [add_apply, comp_apply, comp_apply, smulRight_apply, smulRight_apply, one_apply, one_smul,
one_smul, add_comm] at this
| 3 | 20.085537 | 1 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 462 | 465 | theorem HasDerivAt.clm_comp (hc : HasDerivAt c c' x) (hd : HasDerivAt d d' x) :
HasDerivAt (fun y => (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') x := by |
rw [β hasDerivWithinAt_univ] at *
exact hc.clm_comp hd
| 2 | 7.389056 | 1 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 480 | 484 | theorem HasStrictDerivAt.clm_apply (hc : HasStrictDerivAt c c' x) (hu : HasStrictDerivAt u u' x) :
HasStrictDerivAt (fun y => (c y) (u y)) (c' (u x) + c x u') x := by |
have := (hc.hasStrictFDerivAt.clm_apply hu.hasStrictFDerivAt).hasStrictDerivAt
rwa [add_apply, comp_apply, flip_apply, smulRight_apply, smulRight_apply, one_apply, one_smul,
one_smul, add_comm] at this
| 3 | 20.085537 | 1 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 487 | 492 | theorem HasDerivWithinAt.clm_apply (hc : HasDerivWithinAt c c' s x)
(hu : HasDerivWithinAt u u' s x) :
HasDerivWithinAt (fun y => (c y) (u y)) (c' (u x) + c x u') s x := by |
have := (hc.hasFDerivWithinAt.clm_apply hu.hasFDerivWithinAt).hasDerivWithinAt
rwa [add_apply, comp_apply, flip_apply, smulRight_apply, smulRight_apply, one_apply, one_smul,
one_smul, add_comm] at this
| 3 | 20.085537 | 1 | 1 | 25 | 997 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 495 | 499 | theorem HasDerivAt.clm_apply (hc : HasDerivAt c c' x) (hu : HasDerivAt u u' x) :
HasDerivAt (fun y => (c y) (u y)) (c' (u x) + c x u') x := by |
have := (hc.hasFDerivAt.clm_apply hu.hasFDerivAt).hasDerivAt
rwa [add_apply, comp_apply, flip_apply, smulRight_apply, smulRight_apply, one_apply, one_smul,
one_smul, add_comm] at this
| 3 | 20.085537 | 1 | 1 | 25 | 997 |
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