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.Data.Int.Interval
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Count
import Mathlib.Data.Rat.Floor
import Mathlib.Order.Interval.Finset.Nat
open Finset Int
namespace Int
variable (a b : β€) {r : β€} (hr : 0 < r)
lemma Ico_filter_dvd_eq : (Ico a b).filter (r β£ Β·) =
(Ico βa / (r : β)β βb... | Mathlib/Data/Int/CardIntervalMod.lean | 71 | 73 | theorem Ioc_filter_modEq_card (v : β€) : ((Ioc a b).filter (Β· β‘ v [ZMOD r])).card =
max (β(b - v) / (r : β)β - β(a - v) / (r : β)β) 0 := by |
simp [Ioc_filter_modEq_eq, Ioc_filter_dvd_eq, toNat_eq_max, hr]
| 1 | 2.718282 | 0 | 0 | 4 | 158 |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ΞΉ : Type*}
| Mathlib/Data/Nat/GCD/BigOperators.lean | 20 | 22 | theorem coprime_list_prod_left_iff {l : List β} {k : β} :
Coprime l.prod k β β n β l, Coprime n k := by |
induction l <;> simp [Nat.coprime_mul_iff_left, *]
| 1 | 2.718282 | 0 | 0 | 8 | 159 |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ΞΉ : Type*}
theorem coprime_list_prod_left_iff {l : List β} {k : β} :
Coprime l.prod k β β n β l, Coprime n k := by
... | Mathlib/Data/Nat/GCD/BigOperators.lean | 24 | 26 | theorem coprime_list_prod_right_iff {k : β} {l : List β} :
Coprime k l.prod β β n β l, Coprime k n := by |
simp_rw [coprime_comm (n := k), coprime_list_prod_left_iff]
| 1 | 2.718282 | 0 | 0 | 8 | 159 |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ΞΉ : Type*}
theorem coprime_list_prod_left_iff {l : List β} {k : β} :
Coprime l.prod k β β n β l, Coprime n k := by
... | Mathlib/Data/Nat/GCD/BigOperators.lean | 28 | 30 | theorem coprime_multiset_prod_left_iff {m : Multiset β} {k : β} :
Coprime m.prod k β β n β m, Coprime n k := by |
induction m using Quotient.inductionOn; simpa using coprime_list_prod_left_iff
| 1 | 2.718282 | 0 | 0 | 8 | 159 |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ΞΉ : Type*}
theorem coprime_list_prod_left_iff {l : List β} {k : β} :
Coprime l.prod k β β n β l, Coprime n k := by
... | Mathlib/Data/Nat/GCD/BigOperators.lean | 32 | 34 | theorem coprime_multiset_prod_right_iff {k : β} {m : Multiset β} :
Coprime k m.prod β β n β m, Coprime k n := by |
induction m using Quotient.inductionOn; simpa using coprime_list_prod_right_iff
| 1 | 2.718282 | 0 | 0 | 8 | 159 |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ΞΉ : Type*}
theorem coprime_list_prod_left_iff {l : List β} {k : β} :
Coprime l.prod k β β n β l, Coprime n k := by
... | Mathlib/Data/Nat/GCD/BigOperators.lean | 36 | 38 | theorem coprime_prod_left_iff {t : Finset ΞΉ} {s : ΞΉ β β} {x : β} :
Coprime (β i β t, s i) x β β i β t, Coprime (s i) x := by |
simpa using coprime_multiset_prod_left_iff (m := t.val.map s)
| 1 | 2.718282 | 0 | 0 | 8 | 159 |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ΞΉ : Type*}
theorem coprime_list_prod_left_iff {l : List β} {k : β} :
Coprime l.prod k β β n β l, Coprime n k := by
... | Mathlib/Data/Nat/GCD/BigOperators.lean | 40 | 42 | theorem coprime_prod_right_iff {x : β} {t : Finset ΞΉ} {s : ΞΉ β β} :
Coprime x (β i β t, s i) β β i β t, Coprime x (s i) := by |
simpa using coprime_multiset_prod_right_iff (m := t.val.map s)
| 1 | 2.718282 | 0 | 0 | 8 | 159 |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ΞΉ : Type*}
theorem coprime_list_prod_left_iff {l : List β} {k : β} :
Coprime l.prod k β β n β l, Coprime n k := by
... | Mathlib/Data/Nat/GCD/BigOperators.lean | 52 | 54 | theorem coprime_fintype_prod_left_iff [Fintype ΞΉ] {s : ΞΉ β β} {x : β} :
Coprime (β i, s i) x β β i, Coprime (s i) x := by |
simp [coprime_prod_left_iff]
| 1 | 2.718282 | 0 | 0 | 8 | 159 |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ΞΉ : Type*}
theorem coprime_list_prod_left_iff {l : List β} {k : β} :
Coprime l.prod k β β n β l, Coprime n k := by
... | Mathlib/Data/Nat/GCD/BigOperators.lean | 56 | 58 | theorem coprime_fintype_prod_right_iff [Fintype ΞΉ] {x : β} {s : ΞΉ β β} :
Coprime x (β i, s i) β β i, Coprime x (s i) := by |
simp [coprime_prod_right_iff]
| 1 | 2.718282 | 0 | 0 | 8 | 159 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.LinearAlgebra.TensorProduct.Graded.Internal
import Mathlib.LinearAlgebra.QuadraticForm.Prod
suppress_compilation
variable {R Mβ Mβ N : Type*}
variable [CommRing R] [AddCommGroup Mβ] [AddCommGroup Mβ] [AddCommGroup N]
variable [Module R Mβ] [Module... | Mathlib/LinearAlgebra/CliffordAlgebra/Prod.lean | 101 | 104 | theorem map_mul_map_eq_neg_of_isOrtho_of_mem_evenOdd_one
(hmβ : mβ β evenOdd Qβ 1) (hmβ : mβ β evenOdd Qβ 1) :
map fβ mβ * map fβ mβ = - map fβ mβ * map fβ mβ := by |
simp [map_mul_map_of_isOrtho_of_mem_evenOdd _ _ hf _ _ hmβ hmβ]
| 1 | 2.718282 | 0 | 0 | 1 | 160 |
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Homeomorph
#align_import topology.algebra.group_with_zero from "leanprover-community/mathlib"@"c10e724be91096453ee3db13862b9fb9a992fef2"
open Topology Filter Function
variable {Ξ± Ξ² Gβ : Type*}
section DivConst... | Mathlib/Topology/Algebra/GroupWithZero.lean | 52 | 54 | theorem Filter.Tendsto.div_const {x : Gβ} (hf : Tendsto f l (π x)) (y : Gβ) :
Tendsto (fun a => f a / y) l (π (x / y)) := by |
simpa only [div_eq_mul_inv] using hf.mul tendsto_const_nhds
| 1 | 2.718282 | 0 | 0 | 3 | 161 |
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Homeomorph
#align_import topology.algebra.group_with_zero from "leanprover-community/mathlib"@"c10e724be91096453ee3db13862b9fb9a992fef2"
open Topology Filter Function
variable {Ξ± Ξ² Gβ : Type*}
section DivConst... | Mathlib/Topology/Algebra/GroupWithZero.lean | 69 | 71 | theorem ContinuousOn.div_const (hf : ContinuousOn f s) (y : Gβ) :
ContinuousOn (fun x => f x / y) s := by |
simpa only [div_eq_mul_inv] using hf.mul continuousOn_const
| 1 | 2.718282 | 0 | 0 | 3 | 161 |
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Homeomorph
#align_import topology.algebra.group_with_zero from "leanprover-community/mathlib"@"c10e724be91096453ee3db13862b9fb9a992fef2"
open Topology Filter Function
variable {Ξ± Ξ² Gβ : Type*}
section DivConst... | Mathlib/Topology/Algebra/GroupWithZero.lean | 75 | 76 | theorem Continuous.div_const (hf : Continuous f) (y : Gβ) : Continuous fun x => f x / y := by |
simpa only [div_eq_mul_inv] using hf.mul continuous_const
| 1 | 2.718282 | 0 | 0 | 3 | 161 |
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.calculus.deriv.basic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical Topology Filter ENNReal ... | Mathlib/Analysis/Calculus/Deriv/Basic.lean | 161 | 162 | theorem hasFDerivAtFilter_iff_hasDerivAtFilter {f' : π βL[π] F} :
HasFDerivAtFilter f f' x L β HasDerivAtFilter f (f' 1) x L := by | simp [HasDerivAtFilter]
| 1 | 2.718282 | 0 | 0 | 2 | 162 |
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.calculus.deriv.basic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical Topology Filter ENNReal ... | Mathlib/Analysis/Calculus/Deriv/Basic.lean | 201 | 203 | theorem hasStrictFDerivAt_iff_hasStrictDerivAt {f' : π βL[π] F} :
HasStrictFDerivAt f f' x β HasStrictDerivAt f (f' 1) x := by |
simp [HasStrictDerivAt, HasStrictFDerivAt]
| 1 | 2.718282 | 0 | 0 | 2 | 162 |
import Mathlib.CategoryTheory.NatTrans
import Mathlib.CategoryTheory.Iso
#align_import category_theory.functor.category from "leanprover-community/mathlib"@"63721b2c3eba6c325ecf8ae8cca27155a4f6306f"
namespace CategoryTheory
-- declare the `v`'s first; see note [CategoryTheory universes].
universe vβ vβ vβ uβ uβ u... | Mathlib/CategoryTheory/Functor/Category.lean | 68 | 68 | theorem congr_app {Ξ± Ξ² : F βΆ G} (h : Ξ± = Ξ²) (X : C) : Ξ±.app X = Ξ².app X := by | rw [h]
| 1 | 2.718282 | 0 | 0 | 4 | 163 |
import Mathlib.CategoryTheory.NatTrans
import Mathlib.CategoryTheory.Iso
#align_import category_theory.functor.category from "leanprover-community/mathlib"@"63721b2c3eba6c325ecf8ae8cca27155a4f6306f"
namespace CategoryTheory
-- declare the `v`'s first; see note [CategoryTheory universes].
universe vβ vβ vβ uβ uβ u... | Mathlib/CategoryTheory/Functor/Category.lean | 121 | 122 | theorem hcomp_id_app {H : D β₯€ E} (Ξ± : F βΆ G) (X : C) : (Ξ± β« π H).app X = H.map (Ξ±.app X) := by |
simp
| 1 | 2.718282 | 0 | 0 | 4 | 163 |
import Mathlib.CategoryTheory.NatTrans
import Mathlib.CategoryTheory.Iso
#align_import category_theory.functor.category from "leanprover-community/mathlib"@"63721b2c3eba6c325ecf8ae8cca27155a4f6306f"
namespace CategoryTheory
-- declare the `v`'s first; see note [CategoryTheory universes].
universe vβ vβ vβ uβ uβ u... | Mathlib/CategoryTheory/Functor/Category.lean | 125 | 125 | theorem id_hcomp_app {H : E β₯€ C} (Ξ± : F βΆ G) (X : E) : (π H β« Ξ±).app X = Ξ±.app _ := by | simp
| 1 | 2.718282 | 0 | 0 | 4 | 163 |
import Mathlib.CategoryTheory.NatTrans
import Mathlib.CategoryTheory.Iso
#align_import category_theory.functor.category from "leanprover-community/mathlib"@"63721b2c3eba6c325ecf8ae8cca27155a4f6306f"
namespace CategoryTheory
-- declare the `v`'s first; see note [CategoryTheory universes].
universe vβ vβ vβ uβ uβ u... | Mathlib/CategoryTheory/Functor/Category.lean | 132 | 134 | theorem exchange {I J K : D β₯€ E} (Ξ± : F βΆ G) (Ξ² : G βΆ H) (Ξ³ : I βΆ J) (Ξ΄ : J βΆ K) :
(Ξ± β« Ξ²) β« (Ξ³ β« Ξ΄) = (Ξ± β« Ξ³) β« Ξ² β« Ξ΄ := by |
aesop_cat
| 1 | 2.718282 | 0 | 0 | 4 | 163 |
import Mathlib.Control.Functor
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6eb4348fb1749c302d8abc303d34a"
universe uβ uβ uβ vβ vβ vβ
open Function
class Bifunctor (F : Type uβ β Type uβ β Type uβ) where
bimap : β {Ξ± Ξ±' Ξ² Ξ²'}, (Ξ± β Ξ±') β (Ξ² β Ξ²'... | Mathlib/Control/Bifunctor.lean | 86 | 87 | theorem comp_fst {Ξ±β Ξ±β Ξ±β Ξ²} (f : Ξ±β β Ξ±β) (f' : Ξ±β β Ξ±β) (x : F Ξ±β Ξ²) :
fst f' (fst f x) = fst (f' β f) x := by | simp [fst, bimap_bimap]
| 1 | 2.718282 | 0 | 0 | 4 | 164 |
import Mathlib.Control.Functor
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6eb4348fb1749c302d8abc303d34a"
universe uβ uβ uβ vβ vβ vβ
open Function
class Bifunctor (F : Type uβ β Type uβ β Type uβ) where
bimap : β {Ξ± Ξ±' Ξ² Ξ²'}, (Ξ± β Ξ±') β (Ξ² β Ξ²'... | Mathlib/Control/Bifunctor.lean | 92 | 93 | theorem fst_snd {Ξ±β Ξ±β Ξ²β Ξ²β} (f : Ξ±β β Ξ±β) (f' : Ξ²β β Ξ²β) (x : F Ξ±β Ξ²β) :
fst f (snd f' x) = bimap f f' x := by | simp [fst, bimap_bimap]
| 1 | 2.718282 | 0 | 0 | 4 | 164 |
import Mathlib.Control.Functor
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6eb4348fb1749c302d8abc303d34a"
universe uβ uβ uβ vβ vβ vβ
open Function
class Bifunctor (F : Type uβ β Type uβ β Type uβ) where
bimap : β {Ξ± Ξ±' Ξ² Ξ²'}, (Ξ± β Ξ±') β (Ξ² β Ξ²'... | Mathlib/Control/Bifunctor.lean | 98 | 99 | theorem snd_fst {Ξ±β Ξ±β Ξ²β Ξ²β} (f : Ξ±β β Ξ±β) (f' : Ξ²β β Ξ²β) (x : F Ξ±β Ξ²β) :
snd f' (fst f x) = bimap f f' x := by | simp [snd, bimap_bimap]
| 1 | 2.718282 | 0 | 0 | 4 | 164 |
import Mathlib.Control.Functor
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6eb4348fb1749c302d8abc303d34a"
universe uβ uβ uβ vβ vβ vβ
open Function
class Bifunctor (F : Type uβ β Type uβ β Type uβ) where
bimap : β {Ξ± Ξ±' Ξ² Ξ²'}, (Ξ± β Ξ±') β (Ξ² β Ξ²'... | Mathlib/Control/Bifunctor.lean | 104 | 105 | theorem comp_snd {Ξ± Ξ²β Ξ²β Ξ²β} (g : Ξ²β β Ξ²β) (g' : Ξ²β β Ξ²β) (x : F Ξ± Ξ²β) :
snd g' (snd g x) = snd (g' β g) x := by | simp [snd, bimap_bimap]
| 1 | 2.718282 | 0 | 0 | 4 | 164 |
import Mathlib.Algebra.TrivSqZeroExt
#align_import algebra.dual_number from "leanprover-community/mathlib"@"b8d2eaa69d69ce8f03179a5cda774fc0cde984e4"
variable {R A B : Type*}
abbrev DualNumber (R : Type*) : Type _ :=
TrivSqZeroExt R R
#align dual_number DualNumber
def DualNumber.eps [Zero R] [One R] : DualN... | Mathlib/Algebra/DualNumber.lean | 96 | 97 | theorem commute_eps_left [Semiring R] (x : DualNumber R) : Commute Ξ΅ x := by |
ext <;> simp
| 1 | 2.718282 | 0 | 0 | 1 | 165 |
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
import Mathlib.Logic.Basic
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Function
universe u v w
namespace Function
section
variable {Ξ± Ξ² Ξ³ : Sort*} {f : Ξ± β Ξ²}
@[reducible, simp] de... | Mathlib/Logic/Function/Basic.lean | 89 | 91 | theorem Injective.beq_eq {Ξ± Ξ² : Type*} [BEq Ξ±] [LawfulBEq Ξ±] [BEq Ξ²] [LawfulBEq Ξ²] {f : Ξ± β Ξ²}
(I : Injective f) {a b : Ξ±} : (f a == f b) = (a == b) := by |
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
| 1 | 2.718282 | 0 | 0 | 2 | 166 |
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
import Mathlib.Logic.Basic
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Function
universe u v w
namespace Function
section
variable {Ξ± Ξ² Ξ³ : Sort*} {f : Ξ± β Ξ²}
@[reducible, simp] de... | Mathlib/Logic/Function/Basic.lean | 109 | 110 | theorem not_injective_iff : Β¬ Injective f β β a b, f a = f b β§ a β b := by |
simp only [Injective, not_forall, exists_prop]
| 1 | 2.718282 | 0 | 0 | 2 | 166 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
#align_import linear_algebra.clifford_algebra.star from "leanprover-community/mathlib"@"4d66277cfec381260ba05c68f9ae6ce2a118031d"
variable {R : Type*} [CommRing R]
variable {M : Type*} [AddCommGroup M] [Module R M]
variable {Q : QuadraticForm R M}
namespac... | Mathlib/LinearAlgebra/CliffordAlgebra/Star.lean | 50 | 50 | theorem star_ΞΉ (m : M) : star (ΞΉ Q m) = -ΞΉ Q m := by | rw [star_def, involute_ΞΉ, map_neg, reverse_ΞΉ]
| 1 | 2.718282 | 0 | 0 | 3 | 167 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
#align_import linear_algebra.clifford_algebra.star from "leanprover-community/mathlib"@"4d66277cfec381260ba05c68f9ae6ce2a118031d"
variable {R : Type*} [CommRing R]
variable {M : Type*} [AddCommGroup M] [Module R M]
variable {Q : QuadraticForm R M}
namespac... | Mathlib/LinearAlgebra/CliffordAlgebra/Star.lean | 57 | 58 | theorem star_smul (r : R) (x : CliffordAlgebra Q) : star (r β’ x) = r β’ star x := by |
rw [star_def, star_def, map_smul, map_smul]
| 1 | 2.718282 | 0 | 0 | 3 | 167 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
#align_import linear_algebra.clifford_algebra.star from "leanprover-community/mathlib"@"4d66277cfec381260ba05c68f9ae6ce2a118031d"
variable {R : Type*} [CommRing R]
variable {M : Type*} [AddCommGroup M] [Module R M]
variable {Q : QuadraticForm R M}
namespac... | Mathlib/LinearAlgebra/CliffordAlgebra/Star.lean | 62 | 64 | theorem star_algebraMap (r : R) :
star (algebraMap R (CliffordAlgebra Q) r) = algebraMap R (CliffordAlgebra Q) r := by |
rw [star_def, involute.commutes, reverse.commutes]
| 1 | 2.718282 | 0 | 0 | 3 | 167 |
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Init.Algebra.Classes
#align_import algebra.group.commute from "leanprover-community/mathlib"@"05101c3df9d9cfe9430edc205860c79b6d660102"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
variable {G M S : Type*}
@[to_additive "Two elements... | Mathlib/Algebra/Group/Commute/Defs.lean | 262 | 263 | theorem mul_inv_cancel_assoc (h : Commute a b) : a * (b * aβ»ΒΉ) = b := by |
rw [β mul_assoc, h.mul_inv_cancel]
| 1 | 2.718282 | 0 | 0 | 1 | 168 |
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.Algebra.DirectSum.Module
#align_import linear_algebra.direct_sum.tensor_product from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
suppress_compilation
universe u vβ vβ wβ wβ' wβ wβ'
section Ring
namespace TensorProduct
... | Mathlib/LinearAlgebra/DirectSum/TensorProduct.lean | 150 | 153 | theorem directSum_lof_tmul_lof (iβ : ΞΉβ) (mβ : Mβ iβ) (iβ : ΞΉβ) (mβ : Mβ iβ) :
TensorProduct.directSum R S Mβ Mβ (DirectSum.lof S ΞΉβ Mβ iβ mβ ββ DirectSum.lof R ΞΉβ Mβ iβ mβ) =
DirectSum.lof S (ΞΉβ Γ ΞΉβ) (fun i => Mβ i.1 β[R] Mβ i.2) (iβ, iβ) (mβ ββ mβ) := by |
simp [TensorProduct.directSum]
| 1 | 2.718282 | 0 | 0 | 1 | 169 |
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.DifferentialObject
#align_import algebra.homology.differential_object from "leanprover-community/mathlib"@"b535c2d5d996acd9b0554b76395d9c920e186f4f"
open CategoryTheory CategoryTheory.Limits
open scoped Classical
noncomputable secti... | Mathlib/Algebra/Homology/DifferentialObject.lean | 53 | 54 | theorem objEqToHom_d {x y : Ξ²} (h : x = y) :
X.objEqToHom h β« X.d y = X.d x β« X.objEqToHom (by cases h; rfl) := by | cases h; dsimp; simp
| 1 | 2.718282 | 0 | 0 | 3 | 170 |
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.DifferentialObject
#align_import algebra.homology.differential_object from "leanprover-community/mathlib"@"b535c2d5d996acd9b0554b76395d9c920e186f4f"
open CategoryTheory CategoryTheory.Limits
open scoped Classical
noncomputable secti... | Mathlib/Algebra/Homology/DifferentialObject.lean | 61 | 62 | theorem eqToHom_f' {X Y : DifferentialObject β€ (GradedObjectWithShift b V)} (f : X βΆ Y) {x y : Ξ²}
(h : x = y) : X.objEqToHom h β« f.f y = f.f x β« Y.objEqToHom h := by | cases h; simp
| 1 | 2.718282 | 0 | 0 | 3 | 170 |
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.DifferentialObject
#align_import algebra.homology.differential_object from "leanprover-community/mathlib"@"b535c2d5d996acd9b0554b76395d9c920e186f4f"
open CategoryTheory CategoryTheory.Limits
open scoped Classical
noncomputable secti... | Mathlib/Algebra/Homology/DifferentialObject.lean | 78 | 79 | theorem d_eqToHom (X : HomologicalComplex V (ComplexShape.up' b)) {x y z : Ξ²} (h : y = z) :
X.d x y β« eqToHom (congr_arg X.X h) = X.d x z := by | cases h; simp
| 1 | 2.718282 | 0 | 0 | 3 | 170 |
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Convert
#align_import control.equiv_functor from "leanprover-community/mathlib"@"d6aae1bcbd04b8de2022b9b83a5b5b10e10c777d"
universe uβ uβ uβ vβ vβ vβ
open Function
class EquivFunctor (f : Type uβ β Type uβ) where
map : β {Ξ± Ξ²}, Ξ± β Ξ² β f Ξ± β f Ξ²
m... | Mathlib/Control/EquivFunctor.lean | 70 | 71 | theorem mapEquiv_refl (Ξ±) : mapEquiv f (Equiv.refl Ξ±) = Equiv.refl (f Ξ±) := by |
simp only [mapEquiv, map_refl', Equiv.refl_symm]; rfl
| 1 | 2.718282 | 0 | 0 | 1 | 171 |
import Mathlib.CategoryTheory.Monoidal.Mon_
#align_import category_theory.monoidal.Mod_ from "leanprover-community/mathlib"@"33085c9739c41428651ac461a323fde9a2688d9b"
universe vβ vβ uβ uβ
open CategoryTheory MonoidalCategory
variable (C : Type uβ) [Category.{vβ} C] [MonoidalCategory.{vβ} C]
variable {C}
struc... | Mathlib/CategoryTheory/Monoidal/Mod_.lean | 37 | 38 | theorem assoc_flip :
(A.X β M.act) β« M.act = (Ξ±_ A.X A.X M.X).inv β« (A.mul β· M.X) β« M.act := by | simp
| 1 | 2.718282 | 0 | 0 | 2 | 172 |
import Mathlib.CategoryTheory.Monoidal.Mon_
#align_import category_theory.monoidal.Mod_ from "leanprover-community/mathlib"@"33085c9739c41428651ac461a323fde9a2688d9b"
universe vβ vβ uβ uβ
open CategoryTheory MonoidalCategory
variable (C : Type uβ) [Category.{vβ} C] [MonoidalCategory.{vβ} C]
variable {C}
struc... | Mathlib/CategoryTheory/Monoidal/Mod_.lean | 81 | 82 | theorem id_hom' (M : Mod_ A) : (π M : M βΆ M).hom = π M.X := by |
rfl
| 1 | 2.718282 | 0 | 0 | 2 | 172 |
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
open Finset Function Sym
universe u
variab... | Mathlib/Data/Sym/Sym2.lean | 69 | 69 | theorem Rel.symm {x y : Ξ± Γ Ξ±} : Rel Ξ± x y β Rel Ξ± y x := by | aesop (rule_sets := [Sym2])
| 1 | 2.718282 | 0 | 0 | 4 | 173 |
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
open Finset Function Sym
universe u
variab... | Mathlib/Data/Sym/Sym2.lean | 73 | 74 | theorem Rel.trans {x y z : Ξ± Γ Ξ±} (a : Rel Ξ± x y) (b : Rel Ξ± y z) : Rel Ξ± x z := by |
aesop (rule_sets := [Sym2])
| 1 | 2.718282 | 0 | 0 | 4 | 173 |
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
open Finset Function Sym
universe u
variab... | Mathlib/Data/Sym/Sym2.lean | 88 | 89 | theorem rel_iff' {p q : Ξ± Γ Ξ±} : Rel Ξ± p q β p = q β¨ p = q.swap := by |
aesop (rule_sets := [Sym2])
| 1 | 2.718282 | 0 | 0 | 4 | 173 |
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
open Finset Function Sym
universe u
variab... | Mathlib/Data/Sym/Sym2.lean | 91 | 92 | theorem rel_iff {x y z w : Ξ±} : Rel Ξ± (x, y) (z, w) β x = z β§ y = w β¨ x = w β§ y = z := by |
simp
| 1 | 2.718282 | 0 | 0 | 4 | 173 |
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.BoundedOrder
#align_import order.succ_pred.limit from "leanprover-community/mathlib"@"1e05171a5e8cf18d98d9cf7b207540acb044acae"
variable {Ξ± : Type*}
namespace Order
open Function Set OrderDual
section LT
variable [LT Ξ±]
def IsSuccLimit (a : Ξ±) : Pr... | Mathlib/Order/SuccPred/Limit.lean | 46 | 47 | theorem not_isSuccLimit_iff_exists_covBy (a : Ξ±) : Β¬IsSuccLimit a β β b, b β a := by |
simp [IsSuccLimit]
| 1 | 2.718282 | 0 | 0 | 1 | 174 |
import Mathlib.Mathport.Rename
#align_import init.meta.well_founded_tactics from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
-- Porting note: meta code used to implement well-founded recursion is not ported
theorem Nat.lt_add_of_zero_lt_left (a b : Nat) (h : 0 < b) : a < a + b :=
show a... | Mathlib/Init/Meta/WellFoundedTactics.lean | 18 | 18 | theorem Nat.zero_lt_one_add (a : Nat) : 0 < 1 + a := by | simp [Nat.one_add]
| 1 | 2.718282 | 0 | 0 | 1 | 175 |
import Batteries.Data.Nat.Gcd
import Mathlib.Init.Data.Nat.Notation
import Mathlib.Mathport.Rename
#align_import init.data.nat.gcd from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
open WellFounded
namespace Nat
#align nat.gcd Nat.gcd
#align nat.gcd_zero_left Nat.gcd_zero_left
#alig... | Mathlib/Init/Data/Nat/GCD.lean | 35 | 36 | theorem gcd_def (x y : β) : gcd x y = if x = 0 then y else gcd (y % x) x := by |
cases x <;> simp [Nat.gcd_succ]
| 1 | 2.718282 | 0 | 0 | 1 | 176 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposites
#align_import linear_algebra.clifford_algebra.conjugation from "leanprover-community/mathlib"@"34020e531ebc4e8aac6d449d9eecbcd1508ea8d0"
variable {R : Type*} [CommRing R]
variable {M : Type*} [AddCommGroup M] [Module R M]... | Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 55 | 56 | theorem involute_comp_involute : involute.comp involute = AlgHom.id R (CliffordAlgebra Q) := by |
ext; simp
| 1 | 2.718282 | 0 | 0 | 2 | 177 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposites
#align_import linear_algebra.clifford_algebra.conjugation from "leanprover-community/mathlib"@"34020e531ebc4e8aac6d449d9eecbcd1508ea8d0"
variable {R : Type*} [CommRing R]
variable {M : Type*} [AddCommGroup M] [Module R M]... | Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 111 | 111 | theorem reverse_ΞΉ (m : M) : reverse (ΞΉ Q m) = ΞΉ Q m := by | simp [reverse]
| 1 | 2.718282 | 0 | 0 | 2 | 177 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Logic.Unique
#align_import algebra.group_with_zero.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
assert_not_exists DenselyOrdered
open scoped Classical
open Function
variable {Ξ± M... | Mathlib/Algebra/GroupWithZero/Basic.lean | 110 | 111 | theorem eq_zero_of_zero_eq_one (h : (0 : Mβ) = 1) (a : Mβ) : a = 0 := by |
rw [β mul_one a, β h, mul_zero]
| 1 | 2.718282 | 0 | 0 | 1 | 178 |
import Mathlib.Topology.IsLocalHomeomorph
import Mathlib.Topology.FiberBundle.Basic
#align_import topology.covering from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open Bundle
variable {E X : Type*} [TopologicalSpace E] [TopologicalSpace X] (f : E β X) (s : Set X)
def IsEvenlyCov... | Mathlib/Topology/Covering.lean | 140 | 141 | theorem isCoveringMap_iff_isCoveringMapOn_univ : IsCoveringMap f β IsCoveringMapOn f Set.univ := by |
simp only [IsCoveringMap, IsCoveringMapOn, Set.mem_univ, forall_true_left]
| 1 | 2.718282 | 0 | 0 | 1 | 179 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Deprecated.Submonoid
#align_import deprecated.subgroup from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Set Function
variable {G : Type*} {H : Type*} {A : Type*} {a aβ aβ b c : G}
section Group
variable [Group G] [Add... | Mathlib/Deprecated/Subgroup.lean | 57 | 58 | theorem IsSubgroup.div_mem {s : Set G} (hs : IsSubgroup s) {x y : G} (hx : x β s) (hy : y β s) :
x / y β s := by | simpa only [div_eq_mul_inv] using hs.mul_mem hx (hs.inv_mem hy)
| 1 | 2.718282 | 0 | 0 | 1 | 180 |
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span
#align_import algebra.algebra.tower from "leanprover-community/mathlib"@"71150516f28d9826c7341f8815b31f7d8770c212"
open Pointwise
universe u v w uβ vβ
variable (R : Type u) (S : Type v) (A : Type w) (B : Type uβ) (M : Type vβ)
namespace IsS... | Mathlib/Algebra/Algebra/Tower.lean | 88 | 90 | theorem algebraMap_smul [SMul R M] [IsScalarTower R A M] (r : R) (x : M) :
algebraMap R A r β’ x = r β’ x := by |
rw [Algebra.algebraMap_eq_smul_one, smul_assoc, one_smul]
| 1 | 2.718282 | 0 | 0 | 4 | 181 |
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span
#align_import algebra.algebra.tower from "leanprover-community/mathlib"@"71150516f28d9826c7341f8815b31f7d8770c212"
open Pointwise
universe u v w uβ vβ
variable (R : Type u) (S : Type v) (A : Type w) (B : Type uβ) (M : Type vβ)
namespace IsS... | Mathlib/Algebra/Algebra/Tower.lean | 94 | 96 | theorem of_algebraMap_smul [SMul R M] (h : β (r : R) (x : M), algebraMap R A r β’ x = r β’ x) :
IsScalarTower R A M where
smul_assoc r a x := by | rw [Algebra.smul_def, mul_smul, h]
| 1 | 2.718282 | 0 | 0 | 4 | 181 |
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span
#align_import algebra.algebra.tower from "leanprover-community/mathlib"@"71150516f28d9826c7341f8815b31f7d8770c212"
open Pointwise
universe u v w uβ vβ
variable (R : Type u) (S : Type v) (A : Type w) (B : Type uβ) (M : Type vβ)
namespace IsS... | Mathlib/Algebra/Algebra/Tower.lean | 130 | 131 | theorem algebraMap_apply (x : R) : algebraMap R A x = algebraMap S A (algebraMap R S x) := by |
rw [algebraMap_eq R S A, RingHom.comp_apply]
| 1 | 2.718282 | 0 | 0 | 4 | 181 |
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span
#align_import algebra.algebra.tower from "leanprover-community/mathlib"@"71150516f28d9826c7341f8815b31f7d8770c212"
open Pointwise
universe u v w uβ vβ
variable (R : Type u) (S : Type v) (A : Type w) (B : Type uβ) (M : Type vβ)
namespace IsS... | Mathlib/Algebra/Algebra/Tower.lean | 162 | 164 | theorem _root_.AlgHom.map_algebraMap (f : A ββ[S] B) (r : R) :
f (algebraMap R A r) = algebraMap R B r := by |
rw [algebraMap_apply R S A r, f.commutes, β algebraMap_apply R S B]
| 1 | 2.718282 | 0 | 0 | 4 | 181 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.Submodule.Basic
#align_import algebra.direct_sum.decomposition from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
variable {ΞΉ R M Ο : Type*}
open DirectSum
namespace DirectSum
section AddCommMonoid
variable [Deci... | Mathlib/Algebra/DirectSum/Decomposition.lean | 127 | 128 | theorem decompose_coe {i : ΞΉ} (x : β³ i) : decompose β³ (x : M) = DirectSum.of _ i x := by |
rw [β decompose_symm_of _, Equiv.apply_symm_apply]
| 1 | 2.718282 | 0 | 0 | 4 | 182 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.Submodule.Basic
#align_import algebra.direct_sum.decomposition from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
variable {ΞΉ R M Ο : Type*}
open DirectSum
namespace DirectSum
section AddCommMonoid
variable [Deci... | Mathlib/Algebra/DirectSum/Decomposition.lean | 136 | 137 | theorem decompose_of_mem_same {x : M} {i : ΞΉ} (hx : x β β³ i) : (decompose β³ x i : M) = x := by |
rw [decompose_of_mem _ hx, DirectSum.of_eq_same, Subtype.coe_mk]
| 1 | 2.718282 | 0 | 0 | 4 | 182 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.Submodule.Basic
#align_import algebra.direct_sum.decomposition from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
variable {ΞΉ R M Ο : Type*}
open DirectSum
namespace DirectSum
section AddCommMonoid
variable [Deci... | Mathlib/Algebra/DirectSum/Decomposition.lean | 140 | 142 | theorem decompose_of_mem_ne {x : M} {i j : ΞΉ} (hx : x β β³ i) (hij : i β j) :
(decompose β³ x j : M) = 0 := by |
rw [decompose_of_mem _ hx, DirectSum.of_eq_of_ne _ _ _ _ hij, ZeroMemClass.coe_zero]
| 1 | 2.718282 | 0 | 0 | 4 | 182 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.Submodule.Basic
#align_import algebra.direct_sum.decomposition from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
variable {ΞΉ R M Ο : Type*}
open DirectSum
namespace DirectSum
section AddCommMonoid
variable [Deci... | Mathlib/Algebra/DirectSum/Decomposition.lean | 145 | 147 | theorem degree_eq_of_mem_mem {x : M} {i j : ΞΉ} (hxi : x β β³ i) (hxj : x β β³ j) (hx : x β 0) :
i = j := by |
contrapose! hx; rw [β decompose_of_mem_same β³ hxj, decompose_of_mem_ne β³ hxi hx]
| 1 | 2.718282 | 0 | 0 | 4 | 182 |
import Mathlib.Algebra.Group.Prod
#align_import data.nat.cast.prod from "leanprover-community/mathlib"@"ee0c179cd3c8a45aa5bffbf1b41d8dbede452865"
assert_not_exists MonoidWithZero
variable {Ξ± Ξ² : Type*}
namespace Prod
variable [AddMonoidWithOne Ξ±] [AddMonoidWithOne Ξ²]
instance instAddMonoidWithOne : AddMonoidWi... | Mathlib/Data/Nat/Cast/Prod.lean | 29 | 29 | theorem fst_natCast (n : β) : (n : Ξ± Γ Ξ²).fst = n := by | induction n <;> simp [*]
| 1 | 2.718282 | 0 | 0 | 2 | 183 |
import Mathlib.Algebra.Group.Prod
#align_import data.nat.cast.prod from "leanprover-community/mathlib"@"ee0c179cd3c8a45aa5bffbf1b41d8dbede452865"
assert_not_exists MonoidWithZero
variable {Ξ± Ξ² : Type*}
namespace Prod
variable [AddMonoidWithOne Ξ±] [AddMonoidWithOne Ξ²]
instance instAddMonoidWithOne : AddMonoidWi... | Mathlib/Data/Nat/Cast/Prod.lean | 39 | 39 | theorem snd_natCast (n : β) : (n : Ξ± Γ Ξ²).snd = n := by | induction n <;> simp [*]
| 1 | 2.718282 | 0 | 0 | 2 | 183 |
import Mathlib.MeasureTheory.Measure.AEMeasurable
#align_import dynamics.ergodic.measure_preserving from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} [MeasurableSpace Ξ±] [MeasurableSpace Ξ²] [MeasurableSpace Ξ³]
[MeasurableSpace Ξ΄]
namespace MeasureTheory
... | Mathlib/Dynamics/Ergodic/MeasurePreserving.lean | 87 | 89 | theorem restrict_image_emb {f : Ξ± β Ξ²} (hf : MeasurePreserving f ΞΌa ΞΌb) (hβ : MeasurableEmbedding f)
(s : Set Ξ±) : MeasurePreserving f (ΞΌa.restrict s) (ΞΌb.restrict (f '' s)) := by |
simpa only [Set.preimage_image_eq _ hβ.injective] using hf.restrict_preimage_emb hβ (f '' s)
| 1 | 2.718282 | 0 | 0 | 2 | 184 |
import Mathlib.MeasureTheory.Measure.AEMeasurable
#align_import dynamics.ergodic.measure_preserving from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} [MeasurableSpace Ξ±] [MeasurableSpace Ξ²] [MeasurableSpace Ξ³]
[MeasurableSpace Ξ΄]
namespace MeasureTheory
... | Mathlib/Dynamics/Ergodic/MeasurePreserving.lean | 92 | 94 | theorem aemeasurable_comp_iff {f : Ξ± β Ξ²} (hf : MeasurePreserving f ΞΌa ΞΌb)
(hβ : MeasurableEmbedding f) {g : Ξ² β Ξ³} : AEMeasurable (g β f) ΞΌa β AEMeasurable g ΞΌb := by |
rw [β hf.map_eq, hβ.aemeasurable_map_iff]
| 1 | 2.718282 | 0 | 0 | 2 | 184 |
import Mathlib.Algebra.Star.Order
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.Order.MonotoneContinuity
#align_import data.real.sqrt from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
open Set Filter
open scoped Filter NNReal Topology
namespace NNReal
variable {x y... | Mathlib/Data/Real/Sqrt.lean | 97 | 98 | theorem sqrt_mul (x y : ββ₯0) : sqrt (x * y) = sqrt x * sqrt y := by |
rw [sqrt_eq_iff_eq_sq, mul_pow, sq_sqrt, sq_sqrt]
| 1 | 2.718282 | 0 | 0 | 1 | 185 |
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 57 | 57 | theorem dichotomy (b : Bool) : b = false β¨ b = true := by | cases b <;> simp
| 1 | 2.718282 | 0 | 0 | 6 | 186 |
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 99 | 99 | theorem or_inl {a b : Bool} (H : a) : a || b := by | simp [H]
| 1 | 2.718282 | 0 | 0 | 6 | 186 |
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 102 | 102 | theorem or_inr {a b : Bool} (H : b) : a || b := by | cases a <;> simp [H]
| 1 | 2.718282 | 0 | 0 | 6 | 186 |
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 109 | 109 | theorem and_elim_left : β {a b : Bool}, a && b β a := by | decide
| 1 | 2.718282 | 0 | 0 | 6 | 186 |
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 112 | 112 | theorem and_intro : β {a b : Bool}, a β b β a && b := by | decide
| 1 | 2.718282 | 0 | 0 | 6 | 186 |
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 115 | 115 | theorem and_elim_right : β {a b : Bool}, a && b β b := by | decide
| 1 | 2.718282 | 0 | 0 | 6 | 186 |
import Mathlib.Geometry.Manifold.ContMDiff.NormedSpace
#align_import geometry.manifold.vector_bundle.fiberwise_linear from "leanprover-community/mathlib"@"be2c24f56783935652cefffb4bfca7e4b25d167e"
noncomputable section
open Set TopologicalSpace
open scoped Manifold Topology
variable {π B F : Type*} [Topolog... | Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean | 74 | 82 | theorem source_trans_partialHomeomorph (hU : IsOpen U)
(hΟ : ContinuousOn (fun x => Ο x : B β F βL[π] F) U)
(h2Ο : ContinuousOn (fun x => (Ο x).symm : B β F βL[π] F) U) (hU' : IsOpen U')
(hΟ' : ContinuousOn (fun x => Ο' x : B β F βL[π] F) U')
(h2Ο' : ContinuousOn (fun x => (Ο' x).symm : B β F βL[π] ... |
dsimp only [FiberwiseLinear.partialHomeomorph]; mfld_set_tac
| 1 | 2.718282 | 0 | 0 | 2 | 187 |
import Mathlib.Geometry.Manifold.ContMDiff.NormedSpace
#align_import geometry.manifold.vector_bundle.fiberwise_linear from "leanprover-community/mathlib"@"be2c24f56783935652cefffb4bfca7e4b25d167e"
noncomputable section
open Set TopologicalSpace
open scoped Manifold Topology
variable {π B F : Type*} [Topolog... | Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean | 87 | 95 | theorem target_trans_partialHomeomorph (hU : IsOpen U)
(hΟ : ContinuousOn (fun x => Ο x : B β F βL[π] F) U)
(h2Ο : ContinuousOn (fun x => (Ο x).symm : B β F βL[π] F) U) (hU' : IsOpen U')
(hΟ' : ContinuousOn (fun x => Ο' x : B β F βL[π] F) U')
(h2Ο' : ContinuousOn (fun x => (Ο' x).symm : B β F βL[π] ... |
dsimp only [FiberwiseLinear.partialHomeomorph]; mfld_set_tac
| 1 | 2.718282 | 0 | 0 | 2 | 187 |
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Ring.Action.Subobjects
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.Submonoid.Centralizer
import Mathlib.RingTheory.NonUnitalSubsem... | Mathlib/Algebra/Ring/Subsemiring/Basic.lean | 39 | 40 | theorem natCast_mem [AddSubmonoidWithOneClass S R] (n : β) : (n : R) β s := by |
induction n <;> simp [zero_mem, add_mem, one_mem, *]
| 1 | 2.718282 | 0 | 0 | 1 | 188 |
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
import Mathlib.Analysis.InnerProductSpace.ConformalLinearMap
#align_import analysis.calculus.conformal.inner_product from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
variable {E F : Type*}
variable [NormedA... | Mathlib/Analysis/Calculus/Conformal/InnerProduct.lean | 29 | 31 | theorem conformalAt_iff' {f : E β F} {x : E} : ConformalAt f x β
β c : β, 0 < c β§ β u v : E, βͺfderiv β f x u, fderiv β f x vβ« = c * βͺu, vβ« := by |
rw [conformalAt_iff_isConformalMap_fderiv, isConformalMap_iff]
| 1 | 2.718282 | 0 | 0 | 2 | 189 |
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
import Mathlib.Analysis.InnerProductSpace.ConformalLinearMap
#align_import analysis.calculus.conformal.inner_product from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
variable {E F : Type*}
variable [NormedA... | Mathlib/Analysis/Calculus/Conformal/InnerProduct.lean | 36 | 38 | theorem conformalAt_iff {f : E β F} {x : E} {f' : E βL[β] F} (h : HasFDerivAt f f' x) :
ConformalAt f x β β c : β, 0 < c β§ β u v : E, βͺf' u, f' vβ« = c * βͺu, vβ« := by |
simp only [conformalAt_iff', h.fderiv]
| 1 | 2.718282 | 0 | 0 | 2 | 189 |
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
#align_import category_theory.limits.preserves.shapes.images from "leanprover-community/mathlib"@"fc78e3c190c72a109699385da6be2725e88df841"
noncomputable section
namespace CategoryTheory
namespace Prese... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Images.lean | 52 | 53 | theorem factorThruImage_comp_hom {X Y : A} (f : X βΆ Y) :
factorThruImage (L.map f) β« (iso L f).hom = L.map (factorThruImage f) := by | simp
| 1 | 2.718282 | 0 | 0 | 3 | 190 |
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
#align_import category_theory.limits.preserves.shapes.images from "leanprover-community/mathlib"@"fc78e3c190c72a109699385da6be2725e88df841"
noncomputable section
namespace CategoryTheory
namespace Prese... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Images.lean | 57 | 58 | theorem hom_comp_map_image_ΞΉ {X Y : A} (f : X βΆ Y) :
(iso L f).hom β« L.map (image.ΞΉ f) = image.ΞΉ (L.map f) := by | rw [iso_hom, image.lift_fac]
| 1 | 2.718282 | 0 | 0 | 3 | 190 |
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
#align_import category_theory.limits.preserves.shapes.images from "leanprover-community/mathlib"@"fc78e3c190c72a109699385da6be2725e88df841"
noncomputable section
namespace CategoryTheory
namespace Prese... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Images.lean | 62 | 63 | theorem inv_comp_image_ΞΉ_map {X Y : A} (f : X βΆ Y) :
(iso L f).inv β« image.ΞΉ (L.map f) = L.map (image.ΞΉ f) := by | simp
| 1 | 2.718282 | 0 | 0 | 3 | 190 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {f : Ξ± β Ξ² β Ξ³} {a : Option Ξ±} {b : Option Ξ²} {c : Option Ξ³}
def mapβ (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) ... | Mathlib/Data/Option/NAry.lean | 46 | 48 | theorem mapβ_def {Ξ± Ξ² Ξ³ : Type u} (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) (b : Option Ξ²) :
mapβ f a b = f <$> a <*> b := by |
cases a <;> rfl
| 1 | 2.718282 | 0 | 0 | 14 | 191 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {f : Ξ± β Ξ² β Ξ³} {a : Option Ξ±} {b : Option Ξ²} {c : Option Ξ³}
def mapβ (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) ... | Mathlib/Data/Option/NAry.lean | 63 | 63 | theorem mapβ_none_right (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) : mapβ f a none = none := by | cases a <;> rfl
| 1 | 2.718282 | 0 | 0 | 14 | 191 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {f : Ξ± β Ξ² β Ξ³} {a : Option Ξ±} {b : Option Ξ²} {c : Option Ξ³}
def mapβ (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) ... | Mathlib/Data/Option/NAry.lean | 73 | 74 | theorem mapβ_coe_right (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) (b : Ξ²) :
mapβ f a b = a.map fun a => f a b := by | cases a <;> rfl
| 1 | 2.718282 | 0 | 0 | 14 | 191 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {f : Ξ± β Ξ² β Ξ³} {a : Option Ξ±} {b : Option Ξ²} {c : Option Ξ³}
def mapβ (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) ... | Mathlib/Data/Option/NAry.lean | 78 | 79 | theorem mem_mapβ_iff {c : Ξ³} : c β mapβ f a b β β a' b', a' β a β§ b' β b β§ f a' b' = c := by |
simp [mapβ, bind_eq_some]
| 1 | 2.718282 | 0 | 0 | 14 | 191 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {f : Ξ± β Ξ² β Ξ³} {a : Option Ξ±} {b : Option Ξ²} {c : Option Ξ³}
def mapβ (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) ... | Mathlib/Data/Option/NAry.lean | 83 | 84 | theorem mapβ_eq_none_iff : mapβ f a b = none β a = none β¨ b = none := by |
cases a <;> cases b <;> simp
| 1 | 2.718282 | 0 | 0 | 14 | 191 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {f : Ξ± β Ξ² β Ξ³} {a : Option Ξ±} {b : Option Ξ²} {c : Option Ξ³}
def mapβ (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) ... | Mathlib/Data/Option/NAry.lean | 87 | 88 | theorem mapβ_swap (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) (b : Option Ξ²) :
mapβ f a b = mapβ (fun a b => f b a) b a := by | cases a <;> cases b <;> rfl
| 1 | 2.718282 | 0 | 0 | 14 | 191 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {f : Ξ± β Ξ² β Ξ³} {a : Option Ξ±} {b : Option Ξ²} {c : Option Ξ³}
def mapβ (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) ... | Mathlib/Data/Option/NAry.lean | 91 | 92 | theorem map_mapβ (f : Ξ± β Ξ² β Ξ³) (g : Ξ³ β Ξ΄) :
(mapβ f a b).map g = mapβ (fun a b => g (f a b)) a b := by | cases a <;> cases b <;> rfl
| 1 | 2.718282 | 0 | 0 | 14 | 191 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {f : Ξ± β Ξ² β Ξ³} {a : Option Ξ±} {b : Option Ξ²} {c : Option Ξ³}
def mapβ (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) ... | Mathlib/Data/Option/NAry.lean | 95 | 96 | theorem mapβ_map_left (f : Ξ³ β Ξ² β Ξ΄) (g : Ξ± β Ξ³) :
mapβ f (a.map g) b = mapβ (fun a b => f (g a) b) a b := by | cases a <;> rfl
| 1 | 2.718282 | 0 | 0 | 14 | 191 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {f : Ξ± β Ξ² β Ξ³} {a : Option Ξ±} {b : Option Ξ²} {c : Option Ξ³}
def mapβ (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) ... | Mathlib/Data/Option/NAry.lean | 99 | 100 | theorem mapβ_map_right (f : Ξ± β Ξ³ β Ξ΄) (g : Ξ² β Ξ³) :
mapβ f a (b.map g) = mapβ (fun a b => f a (g b)) a b := by | cases b <;> rfl
| 1 | 2.718282 | 0 | 0 | 14 | 191 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {f : Ξ± β Ξ² β Ξ³} {a : Option Ξ±} {b : Option Ξ²} {c : Option Ξ³}
def mapβ (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) ... | Mathlib/Data/Option/NAry.lean | 109 | 110 | theorem map_uncurry (f : Ξ± β Ξ² β Ξ³) (x : Option (Ξ± Γ Ξ²)) :
x.map (uncurry f) = mapβ f (x.map Prod.fst) (x.map Prod.snd) := by | cases x <;> rfl
| 1 | 2.718282 | 0 | 0 | 14 | 191 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {f : Ξ± β Ξ² β Ξ³} {a : Option Ξ±} {b : Option Ξ²} {c : Option Ξ³}
def mapβ (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) ... | Mathlib/Data/Option/NAry.lean | 124 | 127 | theorem mapβ_assoc {f : Ξ΄ β Ξ³ β Ξ΅} {g : Ξ± β Ξ² β Ξ΄} {f' : Ξ± β Ξ΅' β Ξ΅} {g' : Ξ² β Ξ³ β Ξ΅'}
(h_assoc : β a b c, f (g a b) c = f' a (g' b c)) :
mapβ f (mapβ g a b) c = mapβ f' a (mapβ g' b c) := by |
cases a <;> cases b <;> cases c <;> simp [h_assoc]
| 1 | 2.718282 | 0 | 0 | 14 | 191 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {f : Ξ± β Ξ² β Ξ³} {a : Option Ξ±} {b : Option Ξ²} {c : Option Ξ³}
def mapβ (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) ... | Mathlib/Data/Option/NAry.lean | 130 | 131 | theorem mapβ_comm {g : Ξ² β Ξ± β Ξ³} (h_comm : β a b, f a b = g b a) : mapβ f a b = mapβ g b a := by |
cases a <;> cases b <;> simp [h_comm]
| 1 | 2.718282 | 0 | 0 | 14 | 191 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {f : Ξ± β Ξ² β Ξ³} {a : Option Ξ±} {b : Option Ξ²} {c : Option Ξ³}
def mapβ (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) ... | Mathlib/Data/Option/NAry.lean | 134 | 137 | theorem mapβ_left_comm {f : Ξ± β Ξ΄ β Ξ΅} {g : Ξ² β Ξ³ β Ξ΄} {f' : Ξ± β Ξ³ β Ξ΄'} {g' : Ξ² β Ξ΄' β Ξ΅}
(h_left_comm : β a b c, f a (g b c) = g' b (f' a c)) :
mapβ f a (mapβ g b c) = mapβ g' b (mapβ f' a c) := by |
cases a <;> cases b <;> cases c <;> simp [h_left_comm]
| 1 | 2.718282 | 0 | 0 | 14 | 191 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {f : Ξ± β Ξ² β Ξ³} {a : Option Ξ±} {b : Option Ξ²} {c : Option Ξ³}
def mapβ (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) ... | Mathlib/Data/Option/NAry.lean | 140 | 143 | theorem mapβ_right_comm {f : Ξ΄ β Ξ³ β Ξ΅} {g : Ξ± β Ξ² β Ξ΄} {f' : Ξ± β Ξ³ β Ξ΄'} {g' : Ξ΄' β Ξ² β Ξ΅}
(h_right_comm : β a b c, f (g a b) c = g' (f' a c) b) :
mapβ f (mapβ g a b) c = mapβ g' (mapβ f' a c) b := by |
cases a <;> cases b <;> cases c <;> simp [h_right_comm]
| 1 | 2.718282 | 0 | 0 | 14 | 191 |
import Mathlib.Data.Bool.Set
import Mathlib.Data.Nat.Set
import Mathlib.Data.Set.Prod
import Mathlib.Data.ULift
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Hom.Set
import Mathlib.Order.SetNotation
#align_import order.complete_lattice from "leanprover-community/mathlib"@"5709b0d8725255e76f47debca6400c07b5c2... | Mathlib/Order/CompleteLattice.lean | 110 | 111 | theorem le_iSup_iff {s : ΞΉ β Ξ±} : a β€ iSup s β β b, (β i, s i β€ b) β a β€ b := by |
simp [iSup, le_sSup_iff, upperBounds]
| 1 | 2.718282 | 0 | 0 | 2 | 192 |
import Mathlib.Data.Bool.Set
import Mathlib.Data.Nat.Set
import Mathlib.Data.Set.Prod
import Mathlib.Data.ULift
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Hom.Set
import Mathlib.Order.SetNotation
#align_import order.complete_lattice from "leanprover-community/mathlib"@"5709b0d8725255e76f47debca6400c07b5c2... | Mathlib/Order/CompleteLattice.lean | 180 | 181 | theorem iInf_le_iff {s : ΞΉ β Ξ±} : iInf s β€ a β β b, (β i, b β€ s i) β b β€ a := by |
simp [iInf, sInf_le_iff, lowerBounds]
| 1 | 2.718282 | 0 | 0 | 2 | 192 |
import Mathlib.GroupTheory.Subgroup.Center
import Mathlib.GroupTheory.Submonoid.Centralizer
#align_import group_theory.subgroup.basic from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
open Function
open Int
variable {G : Type*} [Group G]
namespace Subgroup
variable {H K : Subgroup ... | Mathlib/GroupTheory/Subgroup/Centralizer.lean | 42 | 44 | theorem mem_centralizer_iff_commutator_eq_one {g : G} {s : Set G} :
g β centralizer s β β h β s, h * g * hβ»ΒΉ * gβ»ΒΉ = 1 := by |
simp only [mem_centralizer_iff, mul_inv_eq_iff_eq_mul, one_mul]
| 1 | 2.718282 | 0 | 0 | 1 | 193 |
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.QuotientGroup
#align_import group_theory.torsion from "leanprover-community/mathlib"@"1f4705ccdfe1e557fc54a0ce081a05e33d2e6240"
... | Mathlib/GroupTheory/Torsion.lean | 63 | 64 | theorem not_isTorsion_iff : Β¬IsTorsion G β β g : G, Β¬IsOfFinOrder g := by |
rw [IsTorsion, not_forall]
| 1 | 2.718282 | 0 | 0 | 1 | 194 |
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
def HomRel (C) [Quiver C] :=
β β¦X Y : Cβ¦, (X βΆ Y) β (X βΆ Y) β Prop
#align hom_rel HomRel
-- Porting Note: `deriving I... | Mathlib/CategoryTheory/Quotient.lean | 65 | 66 | theorem CompClosure.of {a b : C} (mβ mβ : a βΆ b) (h : r mβ mβ) : CompClosure r mβ mβ := by |
simpa using CompClosure.intro (π _) mβ mβ (π _) h
| 1 | 2.718282 | 0 | 0 | 1 | 195 |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Data.Rat.Cast.Defs
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {F ΞΉ Ξ± Ξ² : Type*}
namespace Rat
open Rat
section WithDivRing
variable [DivisionRing Ξ±]
@[simp, norm_cast]
th... | Mathlib/Data/Rat/Cast/CharZero.lean | 46 | 46 | theorem cast_eq_zero [CharZero Ξ±] {n : β} : (n : Ξ±) = 0 β n = 0 := by | rw [β cast_zero, cast_inj]
| 1 | 2.718282 | 0 | 0 | 3 | 196 |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Data.Rat.Cast.Defs
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {F ΞΉ Ξ± Ξ² : Type*}
namespace Rat
open Rat
section WithDivRing
variable [DivisionRing Ξ±]
@[simp, norm_cast]
th... | Mathlib/Data/Rat/Cast/CharZero.lean | 78 | 79 | theorem cast_bit1 [CharZero Ξ±] (n : β) : ((bit1 n : β) : Ξ±) = (bit1 n : Ξ±) := by |
rw [bit1, cast_add, cast_one, cast_bit0]; rfl
| 1 | 2.718282 | 0 | 0 | 3 | 196 |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Data.Rat.Cast.Defs
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {F ΞΉ Ξ± Ξ² : Type*}
namespace Rat
open Rat
section WithDivRing
variable [DivisionRing Ξ±]
@[simp, norm_cast]
th... | Mathlib/Data/Rat/Cast/CharZero.lean | 119 | 120 | theorem cast_mk (a b : β€) : (a /. b : Ξ±) = a / b := by |
simp only [divInt_eq_div, cast_div, cast_intCast]
| 1 | 2.718282 | 0 | 0 | 3 | 196 |
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Data.List.Prime
#align_import data.polynomial.splits from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
noncomputable section
open Polynomial
universe u v w
variable {R : Type*} {F... | Mathlib/Algebra/Polynomial/Splits.lean | 124 | 125 | theorem splits_map_iff (j : L β+* F) {f : K[X]} : Splits j (f.map i) β Splits (j.comp i) f := by |
simp [Splits, Polynomial.map_map]
| 1 | 2.718282 | 0 | 0 | 1 | 197 |
import Mathlib.Data.Fin.Fin2
import Mathlib.Data.PFun
import Mathlib.Data.Vector3
import Mathlib.NumberTheory.PellMatiyasevic
#align_import number_theory.dioph from "leanprover-community/mathlib"@"a66d07e27d5b5b8ac1147cacfe353478e5c14002"
open Fin2 Function Nat Sum
local infixr:67 " ::β " => Option.elim'
local ... | Mathlib/NumberTheory/Dioph.lean | 85 | 86 | theorem IsPoly.neg {f : (Ξ± β β) β β€} : IsPoly f β IsPoly (-f) := by |
rw [β zero_sub]; exact (IsPoly.const 0).sub
| 1 | 2.718282 | 0 | 0 | 2 | 198 |
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