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.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 65 | 65 | theorem oangle_self_right (pβ pβ : P) : β‘ pβ pβ pβ = 0 := by | simp [oangle]
| 1 | 2.718282 | 0 | 0.333333 | 6 | 356 |
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 75 | 76 | theorem left_ne_of_oangle_ne_zero {pβ pβ pβ : P} (h : β‘ pβ pβ pβ β 0) : pβ β pβ := by |
rw [β @vsub_ne_zero V]; exact o.left_ne_zero_of_oangle_ne_zero h
| 1 | 2.718282 | 0 | 0.333333 | 6 | 356 |
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 80 | 81 | theorem right_ne_of_oangle_ne_zero {pβ pβ pβ : P} (h : β‘ pβ pβ pβ β 0) : pβ β pβ := by |
rw [β @vsub_ne_zero V]; exact o.right_ne_zero_of_oangle_ne_zero h
| 1 | 2.718282 | 0 | 0.333333 | 6 | 356 |
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 85 | 86 | theorem left_ne_right_of_oangle_ne_zero {pβ pβ pβ : P} (h : β‘ pβ pβ pβ β 0) : pβ β pβ := by |
rw [β (vsub_left_injective pβ).ne_iff]; exact o.ne_of_oangle_ne_zero h
| 1 | 2.718282 | 0 | 0.333333 | 6 | 356 |
import Mathlib.Algebra.Group.Hom.End
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.GroupTheory.GroupAction.Units
#align_import algebra.module.basic from "leanprover-community/mathlib"@"30413fc89f202a090a54d78e540963ed3de0056e"
assert_n... | Mathlib/Algebra/Module/Defs.lean | 97 | 98 | theorem Convex.combo_self {a b : R} (h : a + b = 1) (x : M) : a β’ x + b β’ x = x := by |
rw [β add_smul, h, one_smul]
| 1 | 2.718282 | 0 | 0.333333 | 3 | 357 |
import Mathlib.Algebra.Group.Hom.End
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.GroupTheory.GroupAction.Units
#align_import algebra.module.basic from "leanprover-community/mathlib"@"30413fc89f202a090a54d78e540963ed3de0056e"
assert_n... | Mathlib/Algebra/Module/Defs.lean | 104 | 104 | theorem two_smul : (2 : R) β’ x = x + x := by | rw [β one_add_one_eq_two, add_smul, one_smul]
| 1 | 2.718282 | 0 | 0.333333 | 3 | 357 |
import Mathlib.Algebra.Group.Hom.End
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.GroupTheory.GroupAction.Units
#align_import algebra.module.basic from "leanprover-community/mathlib"@"30413fc89f202a090a54d78e540963ed3de0056e"
assert_n... | Mathlib/Algebra/Module/Defs.lean | 241 | 245 | theorem Module.ext' {R : Type*} [Semiring R] {M : Type*} [AddCommMonoid M] (P Q : Module R M)
(w : β (r : R) (m : M), (haveI := P; r β’ m) = (haveI := Q; r β’ m)) :
P = Q := by |
ext
exact w _ _
| 2 | 7.389056 | 1 | 0.333333 | 3 | 357 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Ring.Opposite
import Mathlib.Tactic.Abel
#align_import algebra.geom_sum fro... | Mathlib/Algebra/GeomSum.lean | 46 | 48 | theorem geom_sum_succ {x : Ξ±} {n : β} :
β i β range (n + 1), x ^ i = (x * β i β range n, x ^ i) + 1 := by |
simp only [mul_sum, β pow_succ', sum_range_succ', pow_zero]
| 1 | 2.718282 | 0 | 0.333333 | 6 | 358 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Ring.Opposite
import Mathlib.Tactic.Abel
#align_import algebra.geom_sum fro... | Mathlib/Algebra/GeomSum.lean | 60 | 60 | theorem geom_sum_one (x : Ξ±) : β i β range 1, x ^ i = 1 := by | simp [geom_sum_succ']
| 1 | 2.718282 | 0 | 0.333333 | 6 | 358 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Ring.Opposite
import Mathlib.Tactic.Abel
#align_import algebra.geom_sum fro... | Mathlib/Algebra/GeomSum.lean | 64 | 64 | theorem geom_sum_two {x : Ξ±} : β i β range 2, x ^ i = x + 1 := by | simp [geom_sum_succ']
| 1 | 2.718282 | 0 | 0.333333 | 6 | 358 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Ring.Opposite
import Mathlib.Tactic.Abel
#align_import algebra.geom_sum fro... | Mathlib/Algebra/GeomSum.lean | 76 | 76 | theorem one_geom_sum (n : β) : β i β range n, (1 : Ξ±) ^ i = n := by | simp
| 1 | 2.718282 | 0 | 0.333333 | 6 | 358 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Ring.Opposite
import Mathlib.Tactic.Abel
#align_import algebra.geom_sum fro... | Mathlib/Algebra/GeomSum.lean | 81 | 82 | theorem op_geom_sum (x : Ξ±) (n : β) : op (β i β range n, x ^ i) = β i β range n, op x ^ i := by |
simp
| 1 | 2.718282 | 0 | 0.333333 | 6 | 358 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Ring.Opposite
import Mathlib.Tactic.Abel
#align_import algebra.geom_sum fro... | Mathlib/Algebra/GeomSum.lean | 87 | 94 | theorem op_geom_sumβ (x y : Ξ±) (n : β) : β i β range n, op y ^ (n - 1 - i) * op x ^ i =
β i β range n, op y ^ i * op x ^ (n - 1 - i) := by |
rw [β sum_range_reflect]
refine sum_congr rfl fun j j_in => ?_
rw [mem_range, Nat.lt_iff_add_one_le] at j_in
congr
apply tsub_tsub_cancel_of_le
exact le_tsub_of_add_le_right j_in
| 6 | 403.428793 | 2 | 0.333333 | 6 | 358 |
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Slope
noncomputable section
open scoped Topology Filter ENNReal NNReal
open Filter Asymptotics Set
variable {π : Type*} [NontriviallyNormedFiel... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 147 | 150 | theorem hasLineDerivAt_iff_isLittleO_nhds_zero :
HasLineDerivAt π f f' x v β
(fun t : π => f (x + t β’ v) - f x - t β’ f') =o[π 0] fun t => t := by |
simp only [HasLineDerivAt, hasDerivAt_iff_isLittleO_nhds_zero, zero_add, zero_smul, add_zero]
| 1 | 2.718282 | 0 | 0.333333 | 3 | 359 |
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Slope
noncomputable section
open scoped Topology Filter ENNReal NNReal
open Filter Asymptotics Set
variable {π : Type*} [NontriviallyNormedFiel... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 160 | 163 | theorem lineDifferentiableWithinAt_univ :
LineDifferentiableWithinAt π f univ x v β LineDifferentiableAt π f x v := by |
simp only [LineDifferentiableWithinAt, LineDifferentiableAt, preimage_univ,
differentiableWithinAt_univ]
| 2 | 7.389056 | 1 | 0.333333 | 3 | 359 |
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Slope
noncomputable section
open scoped Topology Filter ENNReal NNReal
open Filter Asymptotics Set
variable {π : Type*} [NontriviallyNormedFiel... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 170 | 171 | theorem lineDerivWithin_univ : lineDerivWithin π f univ x v = lineDeriv π f x v := by |
simp [lineDerivWithin, lineDeriv]
| 1 | 2.718282 | 0 | 0.333333 | 3 | 359 |
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
va... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 135 | 137 | theorem coe_smul (r : ββ₯0) : (r : β) β’ j = r β’ j := by |
-- Porting note: replaced `show`
rw [real_smul_def, if_pos (NNReal.coe_nonneg r), Real.toNNReal_coe]
| 2 | 7.389056 | 1 | 0.333333 | 6 | 360 |
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
va... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 148 | 150 | theorem real_smul_posPart_nonneg (r : β) (hr : 0 β€ r) :
(r β’ j).posPart = r.toNNReal β’ j.posPart := by |
rw [real_smul_def, β smul_posPart, if_pos hr]
| 1 | 2.718282 | 0 | 0.333333 | 6 | 360 |
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
va... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 153 | 155 | theorem real_smul_negPart_nonneg (r : β) (hr : 0 β€ r) :
(r β’ j).negPart = r.toNNReal β’ j.negPart := by |
rw [real_smul_def, β smul_negPart, if_pos hr]
| 1 | 2.718282 | 0 | 0.333333 | 6 | 360 |
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
va... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 158 | 160 | theorem real_smul_posPart_neg (r : β) (hr : r < 0) :
(r β’ j).posPart = (-r).toNNReal β’ j.negPart := by |
rw [real_smul_def, β smul_negPart, if_neg (not_le.2 hr), neg_posPart]
| 1 | 2.718282 | 0 | 0.333333 | 6 | 360 |
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
va... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 163 | 165 | theorem real_smul_negPart_neg (r : β) (hr : r < 0) :
(r β’ j).negPart = (-r).toNNReal β’ j.posPart := by |
rw [real_smul_def, β smul_posPart, if_neg (not_le.2 hr), neg_negPart]
| 1 | 2.718282 | 0 | 0.333333 | 6 | 360 |
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
va... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 242 | 248 | theorem toJordanDecomposition_spec (s : SignedMeasure Ξ±) :
β (i : Set Ξ±) (hiβ : MeasurableSet i) (hiβ : 0 β€[i] s) (hiβ : s β€[iαΆ] 0),
s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hiβ hiβ β§
s.toJordanDecomposition.negPart = s.toMeasureOfLEZero iαΆ hiβ.compl hiβ := by |
set i := s.exists_compl_positive_negative.choose
obtain β¨hiβ, hiβ, hiββ© := s.exists_compl_positive_negative.choose_spec
exact β¨i, hiβ, hiβ, hiβ, rfl, rflβ©
| 3 | 20.085537 | 1 | 0.333333 | 6 | 360 |
import Mathlib.Control.Monad.Basic
import Mathlib.Control.Monad.Writer
import Mathlib.Init.Control.Lawful
#align_import control.monad.cont from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
universe u v w uβ uβ vβ vβ
structure MonadCont.Label (Ξ± : Type w) (m : Type u β Type v) (Ξ² : Typ... | Mathlib/Control/Monad/Cont.lean | 101 | 105 | theorem monadLift_bind [Monad m] [LawfulMonad m] {Ξ± Ξ²} (x : m Ξ±) (f : Ξ± β m Ξ²) :
(monadLift (x >>= f) : ContT r m Ξ²) = monadLift x >>= monadLift β f := by |
ext
simp only [monadLift, MonadLift.monadLift, (Β· β Β·), (Β· >>= Β·), bind_assoc, id, run,
ContT.monadLift]
| 3 | 20.085537 | 1 | 0.333333 | 3 | 361 |
import Mathlib.Control.Monad.Basic
import Mathlib.Control.Monad.Writer
import Mathlib.Init.Control.Lawful
#align_import control.monad.cont from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
universe u v w uβ uβ vβ vβ
structure MonadCont.Label (Ξ± : Type w) (m : Type u β Type v) (Ξ² : Typ... | Mathlib/Control/Monad/Cont.lean | 128 | 130 | theorem ExceptT.goto_mkLabel {Ξ± Ξ² Ξ΅ : Type _} (x : Label (Except.{u, u} Ξ΅ Ξ±) m Ξ²) (i : Ξ±) :
goto (ExceptT.mkLabel x) i = ExceptT.mk (Except.ok <$> goto x (Except.ok i)) := by |
cases x; rfl
| 1 | 2.718282 | 0 | 0.333333 | 3 | 361 |
import Mathlib.Control.Monad.Basic
import Mathlib.Control.Monad.Writer
import Mathlib.Init.Control.Lawful
#align_import control.monad.cont from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
universe u v w uβ uβ vβ vβ
structure MonadCont.Label (Ξ± : Type w) (m : Type u β Type v) (Ξ² : Typ... | Mathlib/Control/Monad/Cont.lean | 193 | 194 | theorem WriterT.goto_mkLabel {Ξ± Ξ² Ο : Type _} [EmptyCollection Ο] (x : Label (Ξ± Γ Ο) m Ξ²) (i : Ξ±) :
goto (WriterT.mkLabel x) i = monadLift (goto x (i, β
)) := by | cases x; rfl
| 1 | 2.718282 | 0 | 0.333333 | 3 | 361 |
import Mathlib.Data.Multiset.Nodup
import Mathlib.Data.List.NatAntidiagonal
#align_import data.multiset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
namespace Nat
def antidiagonal (n : β) : Multiset (β Γ β) :=
List.Nat.antidiagonal n
#align... | Mathlib/Data/Multiset/NatAntidiagonal.lean | 36 | 37 | theorem mem_antidiagonal {n : β} {x : β Γ β} : x β antidiagonal n β x.1 + x.2 = n := by |
rw [antidiagonal, mem_coe, List.Nat.mem_antidiagonal]
| 1 | 2.718282 | 0 | 0.333333 | 6 | 362 |
import Mathlib.Data.Multiset.Nodup
import Mathlib.Data.List.NatAntidiagonal
#align_import data.multiset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
namespace Nat
def antidiagonal (n : β) : Multiset (β Γ β) :=
List.Nat.antidiagonal n
#align... | Mathlib/Data/Multiset/NatAntidiagonal.lean | 42 | 43 | theorem card_antidiagonal (n : β) : card (antidiagonal n) = n + 1 := by |
rw [antidiagonal, coe_card, List.Nat.length_antidiagonal]
| 1 | 2.718282 | 0 | 0.333333 | 6 | 362 |
import Mathlib.Data.Multiset.Nodup
import Mathlib.Data.List.NatAntidiagonal
#align_import data.multiset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
namespace Nat
def antidiagonal (n : β) : Multiset (β Γ β) :=
List.Nat.antidiagonal n
#align... | Mathlib/Data/Multiset/NatAntidiagonal.lean | 59 | 61 | theorem antidiagonal_succ {n : β} :
antidiagonal (n + 1) = (0, n + 1) ::β (antidiagonal n).map (Prod.map Nat.succ id) := by |
simp only [antidiagonal, List.Nat.antidiagonal_succ, map_coe, cons_coe]
| 1 | 2.718282 | 0 | 0.333333 | 6 | 362 |
import Mathlib.Data.Multiset.Nodup
import Mathlib.Data.List.NatAntidiagonal
#align_import data.multiset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
namespace Nat
def antidiagonal (n : β) : Multiset (β Γ β) :=
List.Nat.antidiagonal n
#align... | Mathlib/Data/Multiset/NatAntidiagonal.lean | 64 | 67 | theorem antidiagonal_succ' {n : β} :
antidiagonal (n + 1) = (n + 1, 0) ::β (antidiagonal n).map (Prod.map id Nat.succ) := by |
rw [antidiagonal, List.Nat.antidiagonal_succ', β coe_add, add_comm, antidiagonal, map_coe,
coe_add, List.singleton_append, cons_coe]
| 2 | 7.389056 | 1 | 0.333333 | 6 | 362 |
import Mathlib.Data.Multiset.Nodup
import Mathlib.Data.List.NatAntidiagonal
#align_import data.multiset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
namespace Nat
def antidiagonal (n : β) : Multiset (β Γ β) :=
List.Nat.antidiagonal n
#align... | Mathlib/Data/Multiset/NatAntidiagonal.lean | 70 | 74 | theorem antidiagonal_succ_succ' {n : β} :
antidiagonal (n + 2) =
(0, n + 2) ::β (n + 2, 0) ::β (antidiagonal n).map (Prod.map Nat.succ Nat.succ) := by |
rw [antidiagonal_succ, antidiagonal_succ', map_cons, map_map, Prod.map_apply]
rfl
| 2 | 7.389056 | 1 | 0.333333 | 6 | 362 |
import Mathlib.Data.Multiset.Nodup
import Mathlib.Data.List.NatAntidiagonal
#align_import data.multiset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
namespace Nat
def antidiagonal (n : β) : Multiset (β Γ β) :=
List.Nat.antidiagonal n
#align... | Mathlib/Data/Multiset/NatAntidiagonal.lean | 77 | 78 | theorem map_swap_antidiagonal {n : β} : (antidiagonal n).map Prod.swap = antidiagonal n := by |
rw [antidiagonal, map_coe, List.Nat.map_swap_antidiagonal, coe_reverse]
| 1 | 2.718282 | 0 | 0.333333 | 6 | 362 |
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
universe u v w
namespace Ideal
open Set
variabl... | Mathlib/RingTheory/Ideal/Quotient.lean | 129 | 130 | theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 β x β£ y := by |
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
| 1 | 2.718282 | 0 | 0.333333 | 3 | 363 |
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
universe u v w
namespace Ideal
open Set
variabl... | Mathlib/RingTheory/Ideal/Quotient.lean | 137 | 138 | theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y β x - y β I := by |
rw [β eq_zero_iff_mem, map_sub, sub_eq_zero]
| 1 | 2.718282 | 0 | 0.333333 | 3 | 363 |
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
universe u v w
namespace Ideal
open Set
variabl... | Mathlib/RingTheory/Ideal/Quotient.lean | 152 | 154 | theorem subsingleton_iff {I : Ideal R} : Subsingleton (R β§Έ I) β I = β€ := by |
rw [eq_top_iff_one, β subsingleton_iff_zero_eq_one, eq_comm, β (mk I).map_one,
Quotient.eq_zero_iff_mem]
| 2 | 7.389056 | 1 | 0.333333 | 3 | 363 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section Preorder
variable [Preorder Ξ±] {a b c... | Mathlib/Order/Interval/Set/Disjoint.lean | 60 | 61 | theorem Ici_disjoint_Iic : Disjoint (Ici a) (Iic b) β Β¬a β€ b := by |
rw [Set.disjoint_iff_inter_eq_empty, Ici_inter_Iic, Icc_eq_empty_iff]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 364 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section Preorder
variable [Preorder Ξ±] {a b c... | Mathlib/Order/Interval/Set/Disjoint.lean | 87 | 88 | theorem iUnion_Icc_right (a : Ξ±) : β b, Icc a b = Ici a := by |
simp only [β Ici_inter_Iic, β inter_iUnion, iUnion_Iic, inter_univ]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 364 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section Preorder
variable [Preorder Ξ±] {a b c... | Mathlib/Order/Interval/Set/Disjoint.lean | 92 | 93 | theorem iUnion_Ioc_right (a : Ξ±) : β b, Ioc a b = Ioi a := by |
simp only [β Ioi_inter_Iic, β inter_iUnion, iUnion_Iic, inter_univ]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 364 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section Preorder
variable [Preorder Ξ±] {a b c... | Mathlib/Order/Interval/Set/Disjoint.lean | 97 | 98 | theorem iUnion_Icc_left (b : Ξ±) : β a, Icc a b = Iic b := by |
simp only [β Ici_inter_Iic, β iUnion_inter, iUnion_Ici, univ_inter]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 364 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section Preorder
variable [Preorder Ξ±] {a b c... | Mathlib/Order/Interval/Set/Disjoint.lean | 102 | 103 | theorem iUnion_Ico_left (b : Ξ±) : β a, Ico a b = Iio b := by |
simp only [β Ici_inter_Iio, β iUnion_inter, iUnion_Ici, univ_inter]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 364 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section Preorder
variable [Preorder Ξ±] {a b c... | Mathlib/Order/Interval/Set/Disjoint.lean | 117 | 118 | theorem iUnion_Ico_right [NoMaxOrder Ξ±] (a : Ξ±) : β b, Ico a b = Ici a := by |
simp only [β Ici_inter_Iio, β inter_iUnion, iUnion_Iio, inter_univ]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 364 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section Preorder
variable [Preorder Ξ±] {a b c... | Mathlib/Order/Interval/Set/Disjoint.lean | 122 | 123 | theorem iUnion_Ioo_right [NoMaxOrder Ξ±] (a : Ξ±) : β b, Ioo a b = Ioi a := by |
simp only [β Ioi_inter_Iio, β inter_iUnion, iUnion_Iio, inter_univ]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 364 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section Preorder
variable [Preorder Ξ±] {a b c... | Mathlib/Order/Interval/Set/Disjoint.lean | 127 | 128 | theorem iUnion_Ioc_left [NoMinOrder Ξ±] (b : Ξ±) : β a, Ioc a b = Iic b := by |
simp only [β Ioi_inter_Iic, β iUnion_inter, iUnion_Ioi, univ_inter]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 364 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section Preorder
variable [Preorder Ξ±] {a b c... | Mathlib/Order/Interval/Set/Disjoint.lean | 132 | 133 | theorem iUnion_Ioo_left [NoMinOrder Ξ±] (b : Ξ±) : β a, Ioo a b = Iio b := by |
simp only [β Ioi_inter_Iio, β iUnion_inter, iUnion_Ioi, univ_inter]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 364 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section LinearOrder
variable [LinearOrder Ξ±] ... | Mathlib/Order/Interval/Set/Disjoint.lean | 143 | 145 | theorem Ico_disjoint_Ico : Disjoint (Ico aβ aβ) (Ico bβ bβ) β min aβ bβ β€ max aβ bβ := by |
simp_rw [Set.disjoint_iff_inter_eq_empty, Ico_inter_Ico, Ico_eq_empty_iff, inf_eq_min, sup_eq_max,
not_lt]
| 2 | 7.389056 | 1 | 0.333333 | 18 | 364 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section LinearOrder
variable [LinearOrder Ξ±] ... | Mathlib/Order/Interval/Set/Disjoint.lean | 149 | 151 | theorem Ioc_disjoint_Ioc : Disjoint (Ioc aβ aβ) (Ioc bβ bβ) β min aβ bβ β€ max aβ bβ := by |
have h : _ β min (toDual aβ) (toDual bβ) β€ max (toDual aβ) (toDual bβ) := Ico_disjoint_Ico
simpa only [dual_Ico] using h
| 2 | 7.389056 | 1 | 0.333333 | 18 | 364 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section LinearOrder
variable [LinearOrder Ξ±] ... | Mathlib/Order/Interval/Set/Disjoint.lean | 155 | 158 | theorem Ioo_disjoint_Ioo [DenselyOrdered Ξ±] :
Disjoint (Set.Ioo aβ aβ) (Set.Ioo bβ bβ) β min aβ bβ β€ max aβ bβ := by |
simp_rw [Set.disjoint_iff_inter_eq_empty, Ioo_inter_Ioo, Ioo_eq_empty_iff, inf_eq_min, sup_eq_max,
not_lt]
| 2 | 7.389056 | 1 | 0.333333 | 18 | 364 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section LinearOrder
variable [LinearOrder Ξ±] ... | Mathlib/Order/Interval/Set/Disjoint.lean | 162 | 166 | theorem eq_of_Ico_disjoint {xβ xβ yβ yβ : Ξ±} (h : Disjoint (Ico xβ xβ) (Ico yβ yβ)) (hx : xβ < xβ)
(h2 : xβ β Ico yβ yβ) : yβ = xβ := by |
rw [Ico_disjoint_Ico, min_eq_left (le_of_lt h2.2), le_max_iff] at h
apply le_antisymm h2.1
exact h.elim (fun h => absurd hx (not_lt_of_le h)) id
| 3 | 20.085537 | 1 | 0.333333 | 18 | 364 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section LinearOrder
variable [LinearOrder Ξ±] ... | Mathlib/Order/Interval/Set/Disjoint.lean | 170 | 172 | theorem iUnion_Ico_eq_Iio_self_iff {f : ΞΉ β Ξ±} {a : Ξ±} :
β i, Ico (f i) a = Iio a β β x < a, β i, f i β€ x := by |
simp [β Ici_inter_Iio, β iUnion_inter, subset_def]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 364 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section LinearOrder
variable [LinearOrder Ξ±] ... | Mathlib/Order/Interval/Set/Disjoint.lean | 176 | 178 | theorem iUnion_Ioc_eq_Ioi_self_iff {f : ΞΉ β Ξ±} {a : Ξ±} :
β i, Ioc a (f i) = Ioi a β β x, a < x β β i, x β€ f i := by |
simp [β Ioi_inter_Iic, β inter_iUnion, subset_def]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 364 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section LinearOrder
variable [LinearOrder Ξ±] ... | Mathlib/Order/Interval/Set/Disjoint.lean | 182 | 184 | theorem biUnion_Ico_eq_Iio_self_iff {p : ΞΉ β Prop} {f : β i, p i β Ξ±} {a : Ξ±} :
β (i) (hi : p i), Ico (f i hi) a = Iio a β β x < a, β i hi, f i hi β€ x := by |
simp [β Ici_inter_Iio, β iUnion_inter, subset_def]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 364 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section LinearOrder
variable [LinearOrder Ξ±] ... | Mathlib/Order/Interval/Set/Disjoint.lean | 188 | 190 | theorem biUnion_Ioc_eq_Ioi_self_iff {p : ΞΉ β Prop} {f : β i, p i β Ξ±} {a : Ξ±} :
β (i) (hi : p i), Ioc a (f i hi) = Ioi a β β x, a < x β β i hi, x β€ f i hi := by |
simp [β Ioi_inter_Iic, β inter_iUnion, subset_def]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 364 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section UnionIxx
variable [LinearOrder Ξ±] {s ... | Mathlib/Order/Interval/Set/Disjoint.lean | 201 | 205 | theorem IsGLB.biUnion_Ioi_eq (h : IsGLB s a) : β x β s, Ioi x = Ioi a := by |
refine (iUnionβ_subset fun x hx => ?_).antisymm fun x hx => ?_
Β· exact Ioi_subset_Ioi (h.1 hx)
Β· rcases h.exists_between hx with β¨y, hys, _, hyxβ©
exact mem_biUnion hys hyx
| 4 | 54.59815 | 2 | 0.333333 | 18 | 364 |
import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
#align_import computability.NFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Set
open Computability
universe u v
-- Porting note: Required as `NFA` is used in mathlib3
set_option linter.uppercaseLean3 fa... | Mathlib/Computability/NFA.lean | 53 | 54 | theorem mem_stepSet (s : Ο) (S : Set Ο) (a : Ξ±) : s β M.stepSet S a β β t β S, s β M.step t a := by |
simp [stepSet]
| 1 | 2.718282 | 0 | 0.333333 | 6 | 365 |
import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
#align_import computability.NFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Set
open Computability
universe u v
-- Porting note: Required as `NFA` is used in mathlib3
set_option linter.uppercaseLean3 fa... | Mathlib/Computability/NFA.lean | 58 | 58 | theorem stepSet_empty (a : Ξ±) : M.stepSet β
a = β
:= by | simp [stepSet]
| 1 | 2.718282 | 0 | 0.333333 | 6 | 365 |
import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
#align_import computability.NFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Set
open Computability
universe u v
-- Porting note: Required as `NFA` is used in mathlib3
set_option linter.uppercaseLean3 fa... | Mathlib/Computability/NFA.lean | 78 | 80 | theorem evalFrom_append_singleton (S : Set Ο) (x : List Ξ±) (a : Ξ±) :
M.evalFrom S (x ++ [a]) = M.stepSet (M.evalFrom S x) a := by |
simp only [evalFrom, List.foldl_append, List.foldl_cons, List.foldl_nil]
| 1 | 2.718282 | 0 | 0.333333 | 6 | 365 |
import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
#align_import computability.NFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Set
open Computability
universe u v
-- Porting note: Required as `NFA` is used in mathlib3
set_option linter.uppercaseLean3 fa... | Mathlib/Computability/NFA.lean | 108 | 109 | theorem mem_accepts {x : List Ξ±} : x β M.accepts β β S β M.accept, S β M.evalFrom M.start x := by |
rfl
| 1 | 2.718282 | 0 | 0.333333 | 6 | 365 |
import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
#align_import computability.NFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Set
open Computability
universe u v
-- Porting note: Required as `NFA` is used in mathlib3
set_option linter.uppercaseLean3 fa... | Mathlib/Computability/NFA.lean | 120 | 123 | theorem toDFA_correct : M.toDFA.accepts = M.accepts := by |
ext x
rw [mem_accepts, DFA.mem_accepts]
constructor <;> Β· exact fun β¨w, h2, h3β© => β¨w, h3, h2β©
| 3 | 20.085537 | 1 | 0.333333 | 6 | 365 |
import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
#align_import computability.NFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Set
open Computability
universe u v
-- Porting note: Required as `NFA` is used in mathlib3
set_option linter.uppercaseLean3 fa... | Mathlib/Computability/NFA.lean | 126 | 132 | theorem pumping_lemma [Fintype Ο] {x : List Ξ±} (hx : x β M.accepts)
(hlen : Fintype.card (Set Ο) β€ List.length x) :
β a b c,
x = a ++ b ++ c β§
a.length + b.length β€ Fintype.card (Set Ο) β§ b β [] β§ {a} * {b}β * {c} β€ M.accepts := by |
rw [β toDFA_correct] at hx β’
exact M.toDFA.pumping_lemma hx hlen
| 2 | 7.389056 | 1 | 0.333333 | 6 | 365 |
import Lean.Elab.Tactic.Location
import Mathlib.Logic.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Tactic.Conv
import Mathlib.Init.Set
import Lean.Elab.Tactic.Location
set_option autoImplicit true
namespace Mathlib.Tactic.PushNeg
open Lean Meta Elab.Tactic Parser.Tactic
variable (p q : Prop) (s : Ξ± β Prop)... | Mathlib/Tactic/PushNeg.lean | 39 | 42 | theorem not_nonempty_eq (s : Set Ξ³) : (Β¬ s.Nonempty) = (s = β
) := by |
have A : β (x : Ξ³), Β¬(x β (β
: Set Ξ³)) := fun x β¦ id
simp only [Set.Nonempty, not_exists, eq_iff_iff]
exact β¨fun h β¦ Set.ext (fun x β¦ by simp only [h x, false_iff, A]), fun h β¦ by rwa [h]β©
| 3 | 20.085537 | 1 | 0.333333 | 3 | 366 |
import Lean.Elab.Tactic.Location
import Mathlib.Logic.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Tactic.Conv
import Mathlib.Init.Set
import Lean.Elab.Tactic.Location
set_option autoImplicit true
namespace Mathlib.Tactic.PushNeg
open Lean Meta Elab.Tactic Parser.Tactic
variable (p q : Prop) (s : Ξ± β Prop)... | Mathlib/Tactic/PushNeg.lean | 44 | 45 | theorem ne_empty_eq_nonempty (s : Set Ξ³) : (s β β
) = s.Nonempty := by |
rw [ne_eq, β not_nonempty_eq s, not_not]
| 1 | 2.718282 | 0 | 0.333333 | 3 | 366 |
import Lean.Elab.Tactic.Location
import Mathlib.Logic.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Tactic.Conv
import Mathlib.Init.Set
import Lean.Elab.Tactic.Location
set_option autoImplicit true
namespace Mathlib.Tactic.PushNeg
open Lean Meta Elab.Tactic Parser.Tactic
variable (p q : Prop) (s : Ξ± β Prop)... | Mathlib/Tactic/PushNeg.lean | 47 | 48 | theorem empty_ne_eq_nonempty (s : Set Ξ³) : (β
β s) = s.Nonempty := by |
rw [ne_comm, ne_empty_eq_nonempty]
| 1 | 2.718282 | 0 | 0.333333 | 3 | 366 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 117 | 119 | theorem comp_mul_left (x y : Ξ±) : (x * Β·) β (y * Β·) = (x * y * Β·) := by |
ext z
simp [mul_assoc]
| 2 | 7.389056 | 1 | 0.333333 | 18 | 367 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 128 | 130 | theorem comp_mul_right (x y : Ξ±) : (Β· * x) β (Β· * y) = (Β· * (y * x)) := by |
ext z
simp [mul_assoc]
| 2 | 7.389056 | 1 | 0.333333 | 18 | 367 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 146 | 148 | theorem ite_mul_one {P : Prop} [Decidable P] {a b : M} :
ite P (a * b) 1 = ite P a 1 * ite P b 1 := by |
by_cases h:P <;> simp [h]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 367 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 153 | 155 | theorem ite_one_mul {P : Prop} [Decidable P] {a b : M} :
ite P 1 (a * b) = ite P 1 a * ite P 1 b := by |
by_cases h:P <;> simp [h]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 367 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 160 | 161 | theorem eq_one_iff_eq_one_of_mul_eq_one {a b : M} (h : a * b = 1) : a = 1 β b = 1 := by |
constructor <;> (rintro rfl; simpa using h)
| 1 | 2.718282 | 0 | 0.333333 | 18 | 367 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 196 | 197 | theorem mul_mul_mul_comm (a b c d : G) : a * b * (c * d) = a * c * (b * d) := by |
simp only [mul_left_comm, mul_assoc]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 367 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 202 | 203 | theorem mul_rotate (a b c : G) : a * b * c = b * c * a := by |
simp only [mul_left_comm, mul_comm]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 367 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 208 | 209 | theorem mul_rotate' (a b c : G) : a * (b * c) = b * (c * a) := by |
simp only [mul_left_comm, mul_comm]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 367 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 245 | 245 | theorem bit1_zero [One M] : bit1 (0 : M) = 1 := by | rw [bit1, bit0_zero, zero_add]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 367 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 323 | 325 | theorem mul_right_eq_self : a * b = a β b = 1 := calc
a * b = a β a * b = a * 1 := by | rw [mul_one]
_ β b = 1 := mul_left_cancel_iff
| 2 | 7.389056 | 1 | 0.333333 | 18 | 367 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 352 | 354 | theorem mul_left_eq_self : a * b = b β a = 1 := calc
a * b = b β a * b = 1 * b := by | rw [one_mul]
_ β a = 1 := mul_right_cancel_iff
| 2 | 7.389056 | 1 | 0.333333 | 18 | 367 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 445 | 445 | theorem inv_eq_one_div (x : G) : xβ»ΒΉ = 1 / x := by | rw [div_eq_mul_inv, one_mul]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 367 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 450 | 451 | theorem mul_one_div (x y : G) : x * (1 / y) = x / y := by |
rw [div_eq_mul_inv, one_mul, div_eq_mul_inv]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 367 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 456 | 457 | theorem mul_div_assoc (a b c : G) : a * b / c = a * (b / c) := by |
rw [div_eq_mul_inv, div_eq_mul_inv, mul_assoc _ _ _]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 367 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 474 | 474 | theorem mul_div (a b c : G) : a * (b / c) = a * b / c := by | simp only [mul_assoc, div_eq_mul_inv]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 367 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 479 | 479 | theorem div_eq_mul_one_div (a b : G) : a / b = a * (1 / b) := by | rw [div_eq_mul_inv, one_div]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 367 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 490 | 490 | theorem div_one (a : G) : a / 1 = a := by | simp [div_eq_mul_inv]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 367 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 1,426 | 1,432 | theorem multiplicative_of_isTotal (p : Ξ± β Prop) (hswap : β {a b}, p a β p b β f a b * f b a = 1)
(hmul : β {a b c}, r a b β r b c β p a β p b β p c β f a c = f a b * f b c) {a b c : Ξ±}
(pa : p a) (pb : p b) (pc : p c) : f a c = f a b * f b c := by |
apply multiplicative_of_symmetric_of_isTotal (fun a b => p a β§ p b) r f fun _ _ => And.symm
Β· simp_rw [and_imp]; exact @hswap
Β· exact fun rab rbc pab _pbc pac => hmul rab rbc pab.1 pab.2 pac.2
exacts [β¨pa, pbβ©, β¨pb, pcβ©, β¨pa, pcβ©]
| 4 | 54.59815 | 2 | 0.333333 | 18 | 367 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[Mu... | Mathlib/Algebra/Polynomial/Smeval.lean | 54 | 54 | theorem smeval_eq_sum : p.smeval x = p.sum (smul_pow x) := by | rw [smeval_def]
| 1 | 2.718282 | 0 | 0.333333 | 9 | 368 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[Mu... | Mathlib/Algebra/Polynomial/Smeval.lean | 57 | 58 | theorem smeval_C : (C r).smeval x = r β’ x ^ 0 := by |
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_C_index]
| 1 | 2.718282 | 0 | 0.333333 | 9 | 368 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[Mu... | Mathlib/Algebra/Polynomial/Smeval.lean | 61 | 63 | theorem smeval_monomial (n : β) :
(monomial n r).smeval x = r β’ x ^ n := by |
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_monomial_index]
| 1 | 2.718282 | 0 | 0.333333 | 9 | 368 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[Mu... | Mathlib/Algebra/Polynomial/Smeval.lean | 65 | 67 | theorem eval_eq_smeval : p.eval r = p.smeval r := by |
rw [eval_eq_sum, smeval_eq_sum]
rfl
| 2 | 7.389056 | 1 | 0.333333 | 9 | 368 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[Mu... | Mathlib/Algebra/Polynomial/Smeval.lean | 69 | 74 | theorem evalβ_eq_smeval (R : Type*) [Semiring R] {S : Type*} [Semiring S] (f : R β+* S) (p : R[X])
(x: S) : letI : Module R S := RingHom.toModule f
p.evalβ f x = p.smeval x := by |
letI : Module R S := RingHom.toModule f
rw [smeval_eq_sum, evalβ_eq_sum]
rfl
| 3 | 20.085537 | 1 | 0.333333 | 9 | 368 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[Mu... | Mathlib/Algebra/Polynomial/Smeval.lean | 79 | 80 | theorem smeval_zero : (0 : R[X]).smeval x = 0 := by |
simp only [smeval_eq_sum, smul_pow, sum_zero_index]
| 1 | 2.718282 | 0 | 0.333333 | 9 | 368 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[Mu... | Mathlib/Algebra/Polynomial/Smeval.lean | 83 | 85 | theorem smeval_one : (1 : R[X]).smeval x = 1 β’ x ^ 0 := by |
rw [β C_1, smeval_C]
simp only [Nat.cast_one, one_smul]
| 2 | 7.389056 | 1 | 0.333333 | 9 | 368 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[Mu... | Mathlib/Algebra/Polynomial/Smeval.lean | 88 | 90 | theorem smeval_X :
(X : R[X]).smeval x = x ^ 1 := by |
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_X_index, one_smul]
| 1 | 2.718282 | 0 | 0.333333 | 9 | 368 |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[Mu... | Mathlib/Algebra/Polynomial/Smeval.lean | 93 | 95 | theorem smeval_X_pow {n : β} :
(X ^ n : R[X]).smeval x = x ^ n := by |
simp only [smeval_eq_sum, smul_pow, X_pow_eq_monomial, zero_smul, sum_monomial_index, one_smul]
| 1 | 2.718282 | 0 | 0.333333 | 9 | 368 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 108 | 108 | theorem laverage_zero : β¨β» _x, (0 : ββ₯0β) βΞΌ = 0 := by | rw [laverage, lintegral_zero]
| 1 | 2.718282 | 0 | 0.347826 | 23 | 374 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 112 | 112 | theorem laverage_zero_measure (f : Ξ± β ββ₯0β) : β¨β» x, f x β(0 : Measure Ξ±) = 0 := by | simp [laverage]
| 1 | 2.718282 | 0 | 0.347826 | 23 | 374 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 118 | 119 | theorem laverage_eq (f : Ξ± β ββ₯0β) : β¨β» x, f x βΞΌ = (β«β» x, f x βΞΌ) / ΞΌ univ := by |
rw [laverage_eq', lintegral_smul_measure, ENNReal.div_eq_inv_mul]
| 1 | 2.718282 | 0 | 0.347826 | 23 | 374 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 122 | 123 | theorem laverage_eq_lintegral [IsProbabilityMeasure ΞΌ] (f : Ξ± β ββ₯0β) :
β¨β» x, f x βΞΌ = β«β» x, f x βΞΌ := by | rw [laverage, measure_univ, inv_one, one_smul]
| 1 | 2.718282 | 0 | 0.347826 | 23 | 374 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 127 | 131 | theorem measure_mul_laverage [IsFiniteMeasure ΞΌ] (f : Ξ± β ββ₯0β) :
ΞΌ univ * β¨β» x, f x βΞΌ = β«β» x, f x βΞΌ := by |
rcases eq_or_ne ΞΌ 0 with hΞΌ | hΞΌ
Β· rw [hΞΌ, lintegral_zero_measure, laverage_zero_measure, mul_zero]
Β· rw [laverage_eq, ENNReal.mul_div_cancel' (measure_univ_ne_zero.2 hΞΌ) (measure_ne_top _ _)]
| 3 | 20.085537 | 1 | 0.347826 | 23 | 374 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 134 | 135 | theorem setLaverage_eq (f : Ξ± β ββ₯0β) (s : Set Ξ±) :
β¨β» x in s, f x βΞΌ = (β«β» x in s, f x βΞΌ) / ΞΌ s := by | rw [laverage_eq, restrict_apply_univ]
| 1 | 2.718282 | 0 | 0.347826 | 23 | 374 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 138 | 140 | theorem setLaverage_eq' (f : Ξ± β ββ₯0β) (s : Set Ξ±) :
β¨β» x in s, f x βΞΌ = β«β» x, f x β(ΞΌ s)β»ΒΉ β’ ΞΌ.restrict s := by |
simp only [laverage_eq', restrict_apply_univ]
| 1 | 2.718282 | 0 | 0.347826 | 23 | 374 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 145 | 146 | theorem laverage_congr {f g : Ξ± β ββ₯0β} (h : f =α΅[ΞΌ] g) : β¨β» x, f x βΞΌ = β¨β» x, g x βΞΌ := by |
simp only [laverage_eq, lintegral_congr_ae h]
| 1 | 2.718282 | 0 | 0.347826 | 23 | 374 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 149 | 150 | theorem setLaverage_congr (h : s =α΅[ΞΌ] t) : β¨β» x in s, f x βΞΌ = β¨β» x in t, f x βΞΌ := by |
simp only [setLaverage_eq, set_lintegral_congr h, measure_congr h]
| 1 | 2.718282 | 0 | 0.347826 | 23 | 374 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 153 | 155 | theorem setLaverage_congr_fun (hs : MeasurableSet s) (h : βα΅ x βΞΌ, x β s β f x = g x) :
β¨β» x in s, f x βΞΌ = β¨β» x in s, g x βΞΌ := by |
simp only [laverage_eq, set_lintegral_congr_fun hs h]
| 1 | 2.718282 | 0 | 0.347826 | 23 | 374 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 158 | 162 | theorem laverage_lt_top (hf : β«β» x, f x βΞΌ β β) : β¨β» x, f x βΞΌ < β := by |
obtain rfl | hΞΌ := eq_or_ne ΞΌ 0
Β· simp
Β· rw [laverage_eq]
exact div_lt_top hf (measure_univ_ne_zero.2 hΞΌ)
| 4 | 54.59815 | 2 | 0.347826 | 23 | 374 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 169 | 180 | theorem laverage_add_measure :
β¨β» x, f x β(ΞΌ + Ξ½) =
ΞΌ univ / (ΞΌ univ + Ξ½ univ) * β¨β» x, f x βΞΌ + Ξ½ univ / (ΞΌ univ + Ξ½ univ) * β¨β» x, f x βΞ½ := by |
by_cases hΞΌ : IsFiniteMeasure ΞΌ; swap
Β· rw [not_isFiniteMeasure_iff] at hΞΌ
simp [laverage_eq, hΞΌ]
by_cases hΞ½ : IsFiniteMeasure Ξ½; swap
Β· rw [not_isFiniteMeasure_iff] at hΞ½
simp [laverage_eq, hΞ½]
haveI := hΞΌ; haveI := hΞ½
simp only [β ENNReal.mul_div_right_comm, measure_mul_laverage, β ENNReal.add_d... | 9 | 8,103.083928 | 2 | 0.347826 | 23 | 374 |
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