Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
|---|---|---|---|---|---|---|---|---|---|---|---|
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.GroupTheory.GroupAction.Units
#align_import data.int.absolute_value from "leanprover-community/mathlib"@"9aba7801eeecebb61f58a5763c2b6dd1b47dc6ef"
variable {R S : Type*} [Ring R] [Linea... | Mathlib/Data/Int/AbsoluteValue.lean | 41 | 42 | theorem AbsoluteValue.map_units_int_smul (abv : AbsoluteValue R S) (x : β€Λ£) (y : R) :
abv (x β’ y) = abv y := by | rcases Int.units_eq_one_or x with (rfl | rfl) <;> simp
| 1 | 2.718282 | 0 | 0 | 3 | 76 |
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Hom.Set
#align_import order.bounds.order_iso from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
set_option autoImplicit true
open Set
namespace OrderIso
variable [Preorder Ξ±] [Preorder Ξ²] (f : Ξ± βo Ξ²)
theorem upperBounds_image {... | Mathlib/Order/Bounds/OrderIso.lean | 41 | 42 | theorem isLUB_image' {s : Set Ξ±} {x : Ξ±} : IsLUB (f '' s) (f x) β IsLUB s x := by |
rw [isLUB_image, f.symm_apply_apply]
| 1 | 2.718282 | 0 | 0 | 3 | 77 |
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Hom.Set
#align_import order.bounds.order_iso from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
set_option autoImplicit true
open Set
namespace OrderIso
variable [Preorder Ξ±] [Preorder Ξ²] (f : Ξ± βo Ξ²)
theorem upperBounds_image {... | Mathlib/Order/Bounds/OrderIso.lean | 55 | 56 | theorem isLUB_preimage {s : Set Ξ²} {x : Ξ±} : IsLUB (f β»ΒΉ' s) x β IsLUB s (f x) := by |
rw [β f.symm_symm, β image_eq_preimage, isLUB_image]
| 1 | 2.718282 | 0 | 0 | 3 | 77 |
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Hom.Set
#align_import order.bounds.order_iso from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
set_option autoImplicit true
open Set
namespace OrderIso
variable [Preorder Ξ±] [Preorder Ξ²] (f : Ξ± βo Ξ²)
theorem upperBounds_image {... | Mathlib/Order/Bounds/OrderIso.lean | 59 | 60 | theorem isLUB_preimage' {s : Set Ξ²} {x : Ξ²} : IsLUB (f β»ΒΉ' s) (f.symm x) β IsLUB s x := by |
rw [isLUB_preimage, f.apply_symm_apply]
| 1 | 2.718282 | 0 | 0 | 3 | 77 |
import Mathlib.Logic.Basic
import Mathlib.Init.ZeroOne
import Mathlib.Init.Order.Defs
#align_import algebra.ne_zero from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
variable {R : Type*} [Zero R]
class NeZero (n : R) : Prop where
out : n β 0
#align ne_zero NeZero
theorem NeZero... | Mathlib/Algebra/NeZero.lean | 45 | 45 | theorem not_neZero {n : R} : Β¬NeZero n β n = 0 := by | simp [neZero_iff]
| 1 | 2.718282 | 0 | 0 | 1 | 78 |
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.sums.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace CategoryTheory
universe vβ uβ
-- morphism levels before object levels. See note [category_theory universes].
open Sum
section
variable (C : Ty... | Mathlib/CategoryTheory/Sums/Basic.lean | 62 | 63 | theorem hom_inl_inr_false {X : C} {Y : D} (f : Sum.inl X βΆ Sum.inr Y) : False := by |
cases f
| 1 | 2.718282 | 0 | 0 | 2 | 79 |
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.sums.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace CategoryTheory
universe vβ uβ
-- morphism levels before object levels. See note [category_theory universes].
open Sum
section
variable (C : Ty... | Mathlib/CategoryTheory/Sums/Basic.lean | 66 | 67 | theorem hom_inr_inl_false {X : C} {Y : D} (f : Sum.inr X βΆ Sum.inl Y) : False := by |
cases f
| 1 | 2.718282 | 0 | 0 | 2 | 79 |
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Algebra.Module.Basic
import Mathlib.Topology.Separation
#align_import topology.support from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
open Function Set Filter Topology
variable {X Ξ± Ξ±' Ξ² Ξ³ Ξ΄ M E R : Type*}
section One
... | Mathlib/Topology/Support.lean | 63 | 64 | theorem mulTSupport_eq_empty_iff {f : X β Ξ±} : mulTSupport f = β
β f = 1 := by |
rw [mulTSupport, closure_empty_iff, mulSupport_eq_empty_iff]
| 1 | 2.718282 | 0 | 0 | 1 | 80 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 74 | 77 | theorem HasDerivAtFilter.scomp (hg : HasDerivAtFilter gβ gβ' (h x) L')
(hh : HasDerivAtFilter h h' x L) (hL : Tendsto h L L') :
HasDerivAtFilter (gβ β h) (h' β’ gβ') x L := by |
simpa using ((hg.restrictScalars π).comp x hh hL).hasDerivAtFilter
| 1 | 2.718282 | 0 | 0 | 14 | 81 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 80 | 83 | theorem HasDerivAtFilter.scomp_of_eq (hg : HasDerivAtFilter gβ gβ' y L')
(hh : HasDerivAtFilter h h' x L) (hy : y = h x) (hL : Tendsto h L L') :
HasDerivAtFilter (gβ β h) (h' β’ gβ') x L := by |
rw [hy] at hg; exact hg.scomp x hh hL
| 1 | 2.718282 | 0 | 0 | 14 | 81 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 90 | 93 | theorem HasDerivWithinAt.scomp_hasDerivAt_of_eq (hg : HasDerivWithinAt gβ gβ' s' y)
(hh : HasDerivAt h h' x) (hs : β x, h x β s') (hy : y = h x) :
HasDerivAt (gβ β h) (h' β’ gβ') x := by |
rw [hy] at hg; exact hg.scomp_hasDerivAt x hh hs
| 1 | 2.718282 | 0 | 0 | 14 | 81 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 101 | 104 | theorem HasDerivWithinAt.scomp_of_eq (hg : HasDerivWithinAt gβ gβ' t' y)
(hh : HasDerivWithinAt h h' s x) (hst : MapsTo h s t') (hy : y = h x) :
HasDerivWithinAt (gβ β h) (h' β’ gβ') s x := by |
rw [hy] at hg; exact hg.scomp x hh hst
| 1 | 2.718282 | 0 | 0 | 14 | 81 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 113 | 116 | theorem HasDerivAt.scomp_of_eq
(hg : HasDerivAt gβ gβ' y) (hh : HasDerivAt h h' x) (hy : y = h x) :
HasDerivAt (gβ β h) (h' β’ gβ') x := by |
rw [hy] at hg; exact hg.scomp x hh
| 1 | 2.718282 | 0 | 0 | 14 | 81 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 118 | 120 | theorem HasStrictDerivAt.scomp (hg : HasStrictDerivAt gβ gβ' (h x)) (hh : HasStrictDerivAt h h' x) :
HasStrictDerivAt (gβ β h) (h' β’ gβ') x := by |
simpa using ((hg.restrictScalars π).comp x hh).hasStrictDerivAt
| 1 | 2.718282 | 0 | 0 | 14 | 81 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 123 | 126 | theorem HasStrictDerivAt.scomp_of_eq
(hg : HasStrictDerivAt gβ gβ' y) (hh : HasStrictDerivAt h h' x) (hy : y = h x) :
HasStrictDerivAt (gβ β h) (h' β’ gβ') x := by |
rw [hy] at hg; exact hg.scomp x hh
| 1 | 2.718282 | 0 | 0 | 14 | 81 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 133 | 136 | theorem HasDerivAt.scomp_hasDerivWithinAt_of_eq (hg : HasDerivAt gβ gβ' y)
(hh : HasDerivWithinAt h h' s x) (hy : y = h x) :
HasDerivWithinAt (gβ β h) (h' β’ gβ') s x := by |
rw [hy] at hg; exact hg.scomp_hasDerivWithinAt x hh
| 1 | 2.718282 | 0 | 0 | 14 | 81 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 357 | 361 | theorem HasFDerivWithinAt.comp_hasDerivWithinAt_of_eq {t : Set F}
(hl : HasFDerivWithinAt l l' t y)
(hf : HasDerivWithinAt f f' s x) (hst : MapsTo f s t) (hy : y = f x) :
HasDerivWithinAt (l β f) (l' f') s x := by |
rw [hy] at hl; exact hl.comp_hasDerivWithinAt x hf hst
| 1 | 2.718282 | 0 | 0 | 14 | 81 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 368 | 371 | theorem HasFDerivAt.comp_hasDerivWithinAt_of_eq (hl : HasFDerivAt l l' y)
(hf : HasDerivWithinAt f f' s x) (hy : y = f x) :
HasDerivWithinAt (l β f) (l' f') s x := by |
rw [hy] at hl; exact hl.comp_hasDerivWithinAt x hf
| 1 | 2.718282 | 0 | 0 | 14 | 81 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 382 | 385 | theorem HasFDerivAt.comp_hasDerivAt_of_eq
(hl : HasFDerivAt l l' y) (hf : HasDerivAt f f' x) (hy : y = f x) :
HasDerivAt (l β f) (l' f') x := by |
rw [hy] at hl; exact hl.comp_hasDerivAt x hf
| 1 | 2.718282 | 0 | 0 | 14 | 81 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 393 | 396 | theorem HasStrictFDerivAt.comp_hasStrictDerivAt_of_eq (hl : HasStrictFDerivAt l l' y)
(hf : HasStrictDerivAt f f' x) (hy : y = f x) :
HasStrictDerivAt (l β f) (l' f') x := by |
rw [hy] at hl; exact hl.comp_hasStrictDerivAt x hf
| 1 | 2.718282 | 0 | 0 | 14 | 81 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 404 | 408 | theorem fderivWithin.comp_derivWithin_of_eq {t : Set F} (hl : DifferentiableWithinAt π l t y)
(hf : DifferentiableWithinAt π f s x) (hs : MapsTo f s t) (hxs : UniqueDiffWithinAt π s x)
(hy : y = f x) :
derivWithin (l β f) s x = (fderivWithin π l t (f x) : F β E) (derivWithin f s x) := by |
rw [hy] at hl; exact fderivWithin.comp_derivWithin x hl hf hs hxs
| 1 | 2.718282 | 0 | 0 | 14 | 81 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 415 | 418 | theorem fderiv.comp_deriv_of_eq (hl : DifferentiableAt π l y) (hf : DifferentiableAt π f x)
(hy : y = f x) :
deriv (l β f) x = (fderiv π l (f x) : F β E) (deriv f x) := by |
rw [hy] at hl; exact fderiv.comp_deriv x hl hf
| 1 | 2.718282 | 0 | 0 | 14 | 81 |
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists AddCommMonoidWithOne
assert_not_exists MonoidWithZero... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 67 | 68 | theorem prod_val [CommMonoid Ξ±] (s : Finset Ξ±) : s.1.prod = s.prod id := by |
rw [Finset.prod, Multiset.map_id]
| 1 | 2.718282 | 0 | 0 | 1 | 82 |
import Mathlib.Algebra.Group.Aut
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Logic.Function.Basic
#align_import group_theory.semidirect_product from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable (N : Type*) (G : Type*) {H : Type*} [Group N] [Group G] [Group H]
... | Mathlib/GroupTheory/SemidirectProduct.lean | 157 | 158 | theorem inl_aut (g : G) (n : N) : (inl (Ο g n) : N β[Ο] G) = inr g * inl n * inr gβ»ΒΉ := by |
ext <;> simp
| 1 | 2.718282 | 0 | 0 | 2 | 83 |
import Mathlib.Algebra.Group.Aut
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Logic.Function.Basic
#align_import group_theory.semidirect_product from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable (N : Type*) (G : Type*) {H : Type*} [Group N] [Group G] [Group H]
... | Mathlib/GroupTheory/SemidirectProduct.lean | 161 | 162 | theorem inl_aut_inv (g : G) (n : N) : (inl ((Ο g)β»ΒΉ n) : N β[Ο] G) = inr gβ»ΒΉ * inl n * inr g := by |
rw [β MonoidHom.map_inv, inl_aut, inv_inv]
| 1 | 2.718282 | 0 | 0 | 2 | 83 |
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Homology.Linear
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
import Mathlib.CategoryTheory.Quotient.Linear
import Mathlib.CategoryTheory.Quotient.Preadditive
#align_import algebra.homology.homotopy_category from "leanprover-community/mathl... | Mathlib/Algebra/Homology/HomotopyCategory.lean | 138 | 139 | theorem quotient_map_out_comp_out {C D E : HomotopyCategory V c} (f : C βΆ D) (g : D βΆ E) :
(quotient V c).map (Quot.out f β« Quot.out g) = f β« g := by | simp
| 1 | 2.718282 | 0 | 0 | 1 | 84 |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical.Basic
import Mathlib.Algebra.Order.Nonneg.Field
import Mathlib.Algebra.Order.Nonneg.Floor
import Mathlib.Data.Real.Pointwise
import Mathlib.Order.ConditionallyCompleteLattice.Group
imp... | Mathlib/Data/Real/NNReal.lean | 125 | 126 | theorem _root_.Real.toNNReal_of_nonneg {r : β} (hr : 0 β€ r) : r.toNNReal = β¨r, hrβ© := by |
simp_rw [Real.toNNReal, max_eq_left hr]
| 1 | 2.718282 | 0 | 0 | 1 | 85 |
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.SetTheory.Game.Ordinal
#align_import set_theory.surreal.basic from "leanprover-community/mathlib"@"8900d545017cd21961daa2a1734bb658ef52c618"
universe u
namespace SetTheory
open scoped PGame
namespace PGame
def Numeric : PGame β Prop
| β¨_, _, L, Rβ© => (... | Mathlib/SetTheory/Surreal/Basic.lean | 71 | 75 | theorem numeric_def {x : PGame} :
Numeric x β
(β i j, x.moveLeft i < x.moveRight j) β§
(β i, Numeric (x.moveLeft i)) β§ β j, Numeric (x.moveRight j) := by |
cases x; rfl
| 1 | 2.718282 | 0 | 0 | 4 | 86 |
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.SetTheory.Game.Ordinal
#align_import set_theory.surreal.basic from "leanprover-community/mathlib"@"8900d545017cd21961daa2a1734bb658ef52c618"
universe u
namespace SetTheory
open scoped PGame
namespace PGame
def Numeric : PGame β Prop
| β¨_, _, L, Rβ© => (... | Mathlib/SetTheory/Surreal/Basic.lean | 85 | 86 | theorem left_lt_right {x : PGame} (o : Numeric x) (i : x.LeftMoves) (j : x.RightMoves) :
x.moveLeft i < x.moveRight j := by | cases x; exact o.1 i j
| 1 | 2.718282 | 0 | 0 | 4 | 86 |
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.SetTheory.Game.Ordinal
#align_import set_theory.surreal.basic from "leanprover-community/mathlib"@"8900d545017cd21961daa2a1734bb658ef52c618"
universe u
namespace SetTheory
open scoped PGame
namespace PGame
def Numeric : PGame β Prop
| β¨_, _, L, Rβ© => (... | Mathlib/SetTheory/Surreal/Basic.lean | 89 | 90 | theorem moveLeft {x : PGame} (o : Numeric x) (i : x.LeftMoves) : Numeric (x.moveLeft i) := by |
cases x; exact o.2.1 i
| 1 | 2.718282 | 0 | 0 | 4 | 86 |
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.SetTheory.Game.Ordinal
#align_import set_theory.surreal.basic from "leanprover-community/mathlib"@"8900d545017cd21961daa2a1734bb658ef52c618"
universe u
namespace SetTheory
open scoped PGame
namespace PGame
def Numeric : PGame β Prop
| β¨_, _, L, Rβ© => (... | Mathlib/SetTheory/Surreal/Basic.lean | 93 | 94 | theorem moveRight {x : PGame} (o : Numeric x) (j : x.RightMoves) : Numeric (x.moveRight j) := by |
cases x; exact o.2.2 j
| 1 | 2.718282 | 0 | 0 | 4 | 86 |
import Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Split
#align_import analysis.box_integral.partition.filter from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Function Filter Metric Finset Bool
open scoped Classical
o... | Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | 276 | 276 | theorem henstock_le_riemann : Henstock β€ Riemann := by | trivial
| 1 | 2.718282 | 0 | 0 | 3 | 87 |
import Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Split
#align_import analysis.box_integral.partition.filter from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Function Filter Metric Finset Bool
open scoped Classical
o... | Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | 280 | 280 | theorem henstock_le_mcShane : Henstock β€ McShane := by | trivial
| 1 | 2.718282 | 0 | 0 | 3 | 87 |
import Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Split
#align_import analysis.box_integral.partition.filter from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Function Filter Metric Finset Bool
open scoped Classical
o... | Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | 347 | 349 | theorem rCond_of_bRiemann_eq_false {ΞΉ} (l : IntegrationParams) (hl : l.bRiemann = false)
{r : (ΞΉ β β) β Ioi (0 : β)} : l.RCond r := by |
simp [RCond, hl]
| 1 | 2.718282 | 0 | 0 | 3 | 87 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Int.Sqrt
#align_import data.rat.sqrt from "leanprover-community/mathlib"@"46a64b5b4268c594af770c44d9e502afc6a515cb"
namespace Rat
-- @[pp_nodot] porting note: unknown attribute
def sqrt... | Mathlib/Data/Rat/Sqrt.lean | 30 | 31 | theorem sqrt_eq (q : β) : Rat.sqrt (q * q) = |q| := by |
rw [sqrt, mul_self_num, mul_self_den, Int.sqrt_eq, Nat.sqrt_eq, abs_def, divInt_ofNat]
| 1 | 2.718282 | 0 | 0 | 1 | 88 |
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Logic.Function.Iterate
#align_import dynamics.flow from "leanprover-community/mathlib"@"717c073262cd9d59b1a1dcda7e8ab570c5b63370"
open Set Function Filter
section Invariant
variable {Ο : Type*} {Ξ± : Type*}
def IsInvariant (Ο : Ο β Ξ± β Ξ±) (s : Set Ξ±) ... | Mathlib/Dynamics/Flow.lean | 49 | 50 | theorem isInvariant_iff_image : IsInvariant Ο s β β t, Ο t '' s β s := by |
simp_rw [IsInvariant, mapsTo']
| 1 | 2.718282 | 0 | 0 | 1 | 89 |
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal
namespace ENNReal
noncomputable section Inv
variable {a b c d : ββ₯0β} {r p q : ββ₯0}
protected theorem div_eq_inv_mul : a / b = bβ»ΒΉ * a := by rw [... | Mathlib/Data/ENNReal/Inv.lean | 68 | 68 | theorem coe_inv_two : ((2β»ΒΉ : ββ₯0) : ββ₯0β) = 2β»ΒΉ := by | rw [coe_inv _root_.two_ne_zero, coe_two]
| 1 | 2.718282 | 0 | 0 | 5 | 90 |
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal
namespace ENNReal
noncomputable section Inv
variable {a b c d : ββ₯0β} {r p q : ββ₯0}
protected theorem div_eq_inv_mul : a / b = bβ»ΒΉ * a := by rw [... | Mathlib/Data/ENNReal/Inv.lean | 72 | 73 | theorem coe_div (hr : r β 0) : (β(p / r) : ββ₯0β) = p / r := by |
rw [div_eq_mul_inv, div_eq_mul_inv, coe_mul, coe_inv hr]
| 1 | 2.718282 | 0 | 0 | 5 | 90 |
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal
namespace ENNReal
noncomputable section Inv
variable {a b c d : ββ₯0β} {r p q : ββ₯0}
protected theorem div_eq_inv_mul : a / b = bβ»ΒΉ * a := by rw [... | Mathlib/Data/ENNReal/Inv.lean | 79 | 79 | theorem div_zero (h : a β 0) : a / 0 = β := by | simp [div_eq_mul_inv, h]
| 1 | 2.718282 | 0 | 0 | 5 | 90 |
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal
namespace ENNReal
noncomputable section Inv
variable {a b c d : ββ₯0β} {r p q : ββ₯0}
protected theorem div_eq_inv_mul : a / b = bβ»ΒΉ * a := by rw [... | Mathlib/Data/ENNReal/Inv.lean | 133 | 133 | theorem inv_ne_top : aβ»ΒΉ β β β a β 0 := by | simp
| 1 | 2.718282 | 0 | 0 | 5 | 90 |
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal
namespace ENNReal
noncomputable section Inv
variable {a b c d : ββ₯0β} {r p q : ββ₯0}
protected theorem div_eq_inv_mul : a / b = bβ»ΒΉ * a := by rw [... | Mathlib/Data/ENNReal/Inv.lean | 137 | 138 | theorem inv_lt_top {x : ββ₯0β} : xβ»ΒΉ < β β 0 < x := by |
simp only [lt_top_iff_ne_top, inv_ne_top, pos_iff_ne_zero]
| 1 | 2.718282 | 0 | 0 | 5 | 90 |
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Multivariate.M
import Mathlib.Data.QPF.Multivariate.Basic
#align_import data.qpf.multivariate.constructions.cofix from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean | 64 | 66 | theorem corecF_eq {Ξ± : TypeVec n} {Ξ² : Type u} (g : Ξ² β F (Ξ±.append1 Ξ²)) (x : Ξ²) :
M.dest q.P (corecF g x) = appendFun id (corecF g) <$$> repr (g x) := by |
rw [corecF, M.dest_corec]
| 1 | 2.718282 | 0 | 0 | 1 | 91 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.LinearAlgebra.Alternating.Basic
#align_import linear_algebra.exterior_algebra.basic from "leanprover-community/mathlib"@"b8d2eaa69d69ce8f03179a5cda774fc0cde984e4"
universe u1 u2 u3 u4 u5
variable (R : Type u1) [CommRing R]
variable (M : Type u2) [... | Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean | 97 | 98 | theorem comp_ΞΉ_sq_zero (g : ExteriorAlgebra R M ββ[R] A) (m : M) : g (ΞΉ R m) * g (ΞΉ R m) = 0 := by |
rw [β AlgHom.map_mul, ΞΉ_sq_zero, AlgHom.map_zero]
| 1 | 2.718282 | 0 | 0 | 1 | 92 |
import Mathlib.Data.PFunctor.Multivariate.Basic
#align_import data.pfunctor.multivariate.W from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u v
namespace MvPFunctor
open TypeVec
open MvFunctor
variable {n : β} (P : MvPFunctor.{u} (n + 1))
inductive WPath : P.last.W β F... | Mathlib/Data/PFunctor/Multivariate/W.lean | 109 | 111 | theorem wPathCasesOn_eta {Ξ± : TypeVec n} {a : P.A} {f : P.last.B a β P.last.W}
(h : P.WPath β¨a, fβ© βΉ Ξ±) : P.wPathCasesOn (P.wPathDestLeft h) (P.wPathDestRight h) = h := by |
ext i x; cases x <;> rfl
| 1 | 2.718282 | 0 | 0 | 2 | 93 |
import Mathlib.Data.PFunctor.Multivariate.Basic
#align_import data.pfunctor.multivariate.W from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u v
namespace MvPFunctor
open TypeVec
open MvFunctor
variable {n : β} (P : MvPFunctor.{u} (n + 1))
inductive WPath : P.last.W β F... | Mathlib/Data/PFunctor/Multivariate/W.lean | 115 | 118 | theorem comp_wPathCasesOn {Ξ± Ξ² : TypeVec n} (h : Ξ± βΉ Ξ²) {a : P.A} {f : P.last.B a β P.last.W}
(g' : P.drop.B a βΉ Ξ±) (g : β j : P.last.B a, P.WPath (f j) βΉ Ξ±) :
h β P.wPathCasesOn g' g = P.wPathCasesOn (h β g') fun i => h β g i := by |
ext i x; cases x <;> rfl
| 1 | 2.718282 | 0 | 0 | 2 | 93 |
import Mathlib.Topology.Category.TopCat.Opens
import Mathlib.Data.Set.Subsingleton
#align_import topology.category.Top.open_nhds from "leanprover-community/mathlib"@"1ec4876214bf9f1ddfbf97ae4b0d777ebd5d6938"
open CategoryTheory TopologicalSpace Opposite
universe u
variable {X Y : TopCat.{u}} (f : X βΆ Y)
namesp... | Mathlib/Topology/Category/TopCat/OpenNhds.lean | 124 | 125 | theorem map_id_obj_unop (x : X) (U : (OpenNhds x)α΅α΅) : (map (π X) x).obj (unop U) = unop U := by |
simp
| 1 | 2.718282 | 0 | 0 | 2 | 94 |
import Mathlib.Topology.Category.TopCat.Opens
import Mathlib.Data.Set.Subsingleton
#align_import topology.category.Top.open_nhds from "leanprover-community/mathlib"@"1ec4876214bf9f1ddfbf97ae4b0d777ebd5d6938"
open CategoryTheory TopologicalSpace Opposite
universe u
variable {X Y : TopCat.{u}} (f : X βΆ Y)
namesp... | Mathlib/Topology/Category/TopCat/OpenNhds.lean | 129 | 129 | theorem op_map_id_obj (x : X) (U : (OpenNhds x)α΅α΅) : (map (π X) x).op.obj U = U := by | simp
| 1 | 2.718282 | 0 | 0 | 2 | 94 |
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.shapes.kernels from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
noncomputable section
universe v vβ u u' uβ
open CategoryTheory
open CategoryTheory.Limits.WalkingParallelPair
namespace... | Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 86 | 87 | theorem KernelFork.condition (s : KernelFork f) : Fork.ΞΉ s β« f = 0 := by |
erw [Fork.condition, HasZeroMorphisms.comp_zero]
| 1 | 2.718282 | 0 | 0 | 2 | 95 |
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.shapes.kernels from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
noncomputable section
universe v vβ u u' uβ
open CategoryTheory
open CategoryTheory.Limits.WalkingParallelPair
namespace... | Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 91 | 92 | theorem KernelFork.app_one (s : KernelFork f) : s.Ο.app one = 0 := by |
simp [Fork.app_one_eq_ΞΉ_comp_right]
| 1 | 2.718282 | 0 | 0 | 2 | 95 |
import Batteries.Tactic.Lint.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Init.Data.Int.Order
set_option autoImplicit true
namespace Linarith
theorem lt_irrefl {Ξ± : Type u} ... | Mathlib/Tactic/Linarith/Lemmas.lean | 27 | 28 | theorem eq_of_eq_of_eq {Ξ±} [OrderedSemiring Ξ±] {a b : Ξ±} (ha : a = 0) (hb : b = 0) : a + b = 0 := by |
simp [*]
| 1 | 2.718282 | 0 | 0 | 6 | 96 |
import Batteries.Tactic.Lint.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Init.Data.Int.Order
set_option autoImplicit true
namespace Linarith
theorem lt_irrefl {Ξ± : Type u} ... | Mathlib/Tactic/Linarith/Lemmas.lean | 30 | 31 | theorem le_of_eq_of_le {Ξ±} [OrderedSemiring Ξ±] {a b : Ξ±} (ha : a = 0) (hb : b β€ 0) : a + b β€ 0 := by |
simp [*]
| 1 | 2.718282 | 0 | 0 | 6 | 96 |
import Batteries.Tactic.Lint.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Init.Data.Int.Order
set_option autoImplicit true
namespace Linarith
theorem lt_irrefl {Ξ± : Type u} ... | Mathlib/Tactic/Linarith/Lemmas.lean | 33 | 34 | theorem lt_of_eq_of_lt {Ξ±} [OrderedSemiring Ξ±] {a b : Ξ±} (ha : a = 0) (hb : b < 0) : a + b < 0 := by |
simp [*]
| 1 | 2.718282 | 0 | 0 | 6 | 96 |
import Batteries.Tactic.Lint.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Init.Data.Int.Order
set_option autoImplicit true
namespace Linarith
theorem lt_irrefl {Ξ± : Type u} ... | Mathlib/Tactic/Linarith/Lemmas.lean | 36 | 37 | theorem le_of_le_of_eq {Ξ±} [OrderedSemiring Ξ±] {a b : Ξ±} (ha : a β€ 0) (hb : b = 0) : a + b β€ 0 := by |
simp [*]
| 1 | 2.718282 | 0 | 0 | 6 | 96 |
import Batteries.Tactic.Lint.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Init.Data.Int.Order
set_option autoImplicit true
namespace Linarith
theorem lt_irrefl {Ξ± : Type u} ... | Mathlib/Tactic/Linarith/Lemmas.lean | 39 | 40 | theorem lt_of_lt_of_eq {Ξ±} [OrderedSemiring Ξ±] {a b : Ξ±} (ha : a < 0) (hb : b = 0) : a + b < 0 := by |
simp [*]
| 1 | 2.718282 | 0 | 0 | 6 | 96 |
import Batteries.Tactic.Lint.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Init.Data.Int.Order
set_option autoImplicit true
namespace Linarith
theorem lt_irrefl {Ξ± : Type u} ... | Mathlib/Tactic/Linarith/Lemmas.lean | 52 | 53 | theorem mul_eq {Ξ±} [OrderedSemiring Ξ±] {a b : Ξ±} (ha : a = 0) (_ : 0 < b) : b * a = 0 := by |
simp [*]
| 1 | 2.718282 | 0 | 0 | 6 | 96 |
import Mathlib.MeasureTheory.Measure.AEMeasurable
#align_import measure_theory.group.arithmetic from "leanprover-community/mathlib"@"a75898643b2d774cced9ae7c0b28c21663b99666"
open MeasureTheory
open scoped Pointwise
universe u v
variable {Ξ± : Type*}
class MeasurableAdd (M : Type*) [MeasurableSpace M] [Add M]... | Mathlib/MeasureTheory/Group/Arithmetic.lean | 188 | 189 | theorem measurable_div_const' {G : Type*} [DivInvMonoid G] [MeasurableSpace G] [MeasurableMul G]
(g : G) : Measurable fun h => h / g := by | simp_rw [div_eq_mul_inv, measurable_mul_const]
| 1 | 2.718282 | 0 | 0 | 1 | 97 |
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Rat.Lemmas
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e... | Mathlib/Data/Rat/Cast/Defs.lean | 120 | 121 | theorem cast_natCast (n : β) : ((n : β) : Ξ±) = n := by |
rw [β Int.cast_natCast, cast_intCast, Int.cast_natCast]
| 1 | 2.718282 | 0 | 0 | 3 | 98 |
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Rat.Lemmas
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e... | Mathlib/Data/Rat/Cast/Defs.lean | 143 | 144 | theorem cast_commute (r : β) (a : Ξ±) : Commute (βr) a := by |
simpa only [cast_def] using (r.1.cast_commute a).div_left (r.2.cast_commute a)
| 1 | 2.718282 | 0 | 0 | 3 | 98 |
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Rat.Lemmas
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e... | Mathlib/Data/Rat/Cast/Defs.lean | 237 | 238 | theorem map_ratCast [DivisionRing Ξ±] [DivisionRing Ξ²] [RingHomClass F Ξ± Ξ²] (f : F) (q : β) :
f q = q := by | rw [cast_def, map_divβ, map_intCast, map_natCast, cast_def]
| 1 | 2.718282 | 0 | 0 | 3 | 98 |
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Ring.Prod
import Mathlib.Algebra.Ring.Subring.Basic
import Mathlib.Topology.Algebra.Group.Basic
#align_import topology.algebra.ring.basic from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpac... | Mathlib/Topology/Algebra/Ring/Basic.lean | 63 | 66 | theorem TopologicalSemiring.continuousNeg_of_mul [TopologicalSpace Ξ±] [NonAssocRing Ξ±]
[ContinuousMul Ξ±] : ContinuousNeg Ξ± where
continuous_neg := by |
simpa using (continuous_const.mul continuous_id : Continuous fun x : Ξ± => -1 * x)
| 1 | 2.718282 | 0 | 0 | 1 | 99 |
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Star.Unitary
import Mathlib.Data.Nat.ModEq
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic.Monotonicity
#align_import number_theory.pell_matiyasevic from "leanprover-community/mathlib"@"795b501869b9f... | Mathlib/NumberTheory/PellMatiyasevic.lean | 151 | 151 | theorem xn_one : xn a1 1 = a := by | simp
| 1 | 2.718282 | 0 | 0 | 2 | 100 |
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Star.Unitary
import Mathlib.Data.Nat.ModEq
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic.Monotonicity
#align_import number_theory.pell_matiyasevic from "leanprover-community/mathlib"@"795b501869b9f... | Mathlib/NumberTheory/PellMatiyasevic.lean | 155 | 155 | theorem yn_one : yn a1 1 = 1 := by | simp
| 1 | 2.718282 | 0 | 0 | 2 | 100 |
import Mathlib.Data.Finset.Option
import Mathlib.Data.PFun
import Mathlib.Data.Part
#align_import data.finset.pimage from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {Ξ± Ξ² : Type*}
namespace Part
def toFinset (o : Part Ξ±) [Decidable o.Dom] : Finset Ξ± :=
o.toOption.toFins... | Mathlib/Data/Finset/PImage.lean | 34 | 35 | theorem mem_toFinset {o : Part Ξ±} [Decidable o.Dom] {x : Ξ±} : x β o.toFinset β x β o := by |
simp [toFinset]
| 1 | 2.718282 | 0 | 0 | 3 | 101 |
import Mathlib.Data.Finset.Option
import Mathlib.Data.PFun
import Mathlib.Data.Part
#align_import data.finset.pimage from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {Ξ± Ξ² : Type*}
namespace Part
def toFinset (o : Part Ξ±) [Decidable o.Dom] : Finset Ξ± :=
o.toOption.toFins... | Mathlib/Data/Finset/PImage.lean | 39 | 40 | theorem toFinset_none [Decidable (none : Part Ξ±).Dom] : none.toFinset = (β
: Finset Ξ±) := by |
simp [toFinset]
| 1 | 2.718282 | 0 | 0 | 3 | 101 |
import Mathlib.Data.Finset.Option
import Mathlib.Data.PFun
import Mathlib.Data.Part
#align_import data.finset.pimage from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {Ξ± Ξ² : Type*}
namespace Part
def toFinset (o : Part Ξ±) [Decidable o.Dom] : Finset Ξ± :=
o.toOption.toFins... | Mathlib/Data/Finset/PImage.lean | 44 | 45 | theorem toFinset_some {a : Ξ±} [Decidable (some a).Dom] : (some a).toFinset = {a} := by |
simp [toFinset]
| 1 | 2.718282 | 0 | 0 | 3 | 101 |
import Mathlib.Algebra.DualNumber
import Mathlib.Analysis.NormedSpace.TrivSqZeroExt
#align_import analysis.normed_space.dual_number from "leanprover-community/mathlib"@"806c0bb86f6128cfa2f702285727518eb5244390"
open NormedSpace -- For `NormedSpace.exp`.
namespace DualNumber
open TrivSqZeroExt
variable (π : Typ... | Mathlib/Analysis/NormedSpace/DualNumber.lean | 38 | 39 | theorem exp_smul_eps (r : R) : exp π (r β’ eps : DualNumber R) = 1 + r β’ eps := by |
rw [eps, β inr_smul, exp_inr]
| 1 | 2.718282 | 0 | 0 | 1 | 102 |
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
#align_import data.is_R_or_C.basic from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
section
local notation "οΏ½... | Mathlib/Analysis/RCLike/Basic.lean | 105 | 106 | theorem real_smul_eq_coe_smul [AddCommGroup E] [Module K E] [Module β E] [IsScalarTower β K E]
(r : β) (x : E) : r β’ x = (r : K) β’ x := by | rw [RCLike.ofReal_alg, smul_one_smul]
| 1 | 2.718282 | 0 | 0 | 3 | 103 |
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
#align_import data.is_R_or_C.basic from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
section
local notation "οΏ½... | Mathlib/Analysis/RCLike/Basic.lean | 162 | 162 | theorem one_re : re (1 : K) = 1 := by | rw [β ofReal_one, ofReal_re]
| 1 | 2.718282 | 0 | 0 | 3 | 103 |
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
#align_import data.is_R_or_C.basic from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
section
local notation "οΏ½... | Mathlib/Analysis/RCLike/Basic.lean | 166 | 166 | theorem one_im : im (1 : K) = 0 := by | rw [β ofReal_one, ofReal_im]
| 1 | 2.718282 | 0 | 0 | 3 | 103 |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
-- Porting note (#11081): cannot automatically derive Fintype, adde... | Mathlib/Data/Sign.lean | 162 | 162 | theorem nonneg_iff {a : SignType} : 0 β€ a β a = 0 β¨ a = 1 := by | cases a <;> decide
| 1 | 2.718282 | 0 | 0 | 6 | 104 |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
-- Porting note (#11081): cannot automatically derive Fintype, adde... | Mathlib/Data/Sign.lean | 165 | 165 | theorem nonneg_iff_ne_neg_one {a : SignType} : 0 β€ a β a β -1 := by | cases a <;> decide
| 1 | 2.718282 | 0 | 0 | 6 | 104 |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
-- Porting note (#11081): cannot automatically derive Fintype, adde... | Mathlib/Data/Sign.lean | 168 | 168 | theorem neg_one_lt_iff {a : SignType} : -1 < a β 0 β€ a := by | cases a <;> decide
| 1 | 2.718282 | 0 | 0 | 6 | 104 |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
-- Porting note (#11081): cannot automatically derive Fintype, adde... | Mathlib/Data/Sign.lean | 171 | 171 | theorem nonpos_iff {a : SignType} : a β€ 0 β a = -1 β¨ a = 0 := by | cases a <;> decide
| 1 | 2.718282 | 0 | 0 | 6 | 104 |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
-- Porting note (#11081): cannot automatically derive Fintype, adde... | Mathlib/Data/Sign.lean | 174 | 174 | theorem nonpos_iff_ne_one {a : SignType} : a β€ 0 β a β 1 := by | cases a <;> decide
| 1 | 2.718282 | 0 | 0 | 6 | 104 |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
-- Porting note (#11081): cannot automatically derive Fintype, adde... | Mathlib/Data/Sign.lean | 177 | 177 | theorem lt_one_iff {a : SignType} : a < 1 β a β€ 0 := by | cases a <;> decide
| 1 | 2.718282 | 0 | 0 | 6 | 104 |
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.Lint.Basic
#align_import logic.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
#align_import init.ite_simp from "leanprover-communit... | Mathlib/Logic/Basic.lean | 59 | 61 | theorem congr_heq {Ξ± Ξ² Ξ³ : Sort _} {f : Ξ± β Ξ³} {g : Ξ² β Ξ³} {x : Ξ±} {y : Ξ²}
(hβ : HEq f g) (hβ : HEq x y) : f x = g y := by |
cases hβ; cases hβ; rfl
| 1 | 2.718282 | 0 | 0 | 8 | 105 |
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.Lint.Basic
#align_import logic.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
#align_import init.ite_simp from "leanprover-communit... | Mathlib/Logic/Basic.lean | 591 | 592 | theorem Eq.rec_eq_cast {Ξ± : Sort _} {P : Ξ± β Sort _} {x y : Ξ±} (h : x = y) (z : P x) :
h βΈ z = cast (congr_arg P h) z := by | induction h; rfl
| 1 | 2.718282 | 0 | 0 | 8 | 105 |
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.Lint.Basic
#align_import logic.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
#align_import init.ite_simp from "leanprover-communit... | Mathlib/Logic/Basic.lean | 595 | 598 | theorem eqRec_heq' {Ξ± : Sort*} {a' : Ξ±} {motive : (a : Ξ±) β a' = a β Sort*}
(p : motive a' (rfl : a' = a')) {a : Ξ±} (t : a' = a) :
HEq (@Eq.rec Ξ± a' motive p a t) p := by |
subst t; rfl
| 1 | 2.718282 | 0 | 0 | 8 | 105 |
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.Lint.Basic
#align_import logic.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
#align_import init.ite_simp from "leanprover-communit... | Mathlib/Logic/Basic.lean | 601 | 602 | theorem rec_heq_of_heq {C : Ξ± β Sort*} {x : C a} {y : Ξ²} (e : a = b) (h : HEq x y) :
HEq (e βΈ x) y := by | subst e; exact h
| 1 | 2.718282 | 0 | 0 | 8 | 105 |
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.Lint.Basic
#align_import logic.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
#align_import init.ite_simp from "leanprover-communit... | Mathlib/Logic/Basic.lean | 606 | 607 | theorem rec_heq_iff_heq {C : Ξ± β Sort*} {x : C a} {y : Ξ²} {e : a = b} :
HEq (e βΈ x) y β HEq x y := by | subst e; rfl
| 1 | 2.718282 | 0 | 0 | 8 | 105 |
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.Lint.Basic
#align_import logic.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
#align_import init.ite_simp from "leanprover-communit... | Mathlib/Logic/Basic.lean | 611 | 612 | theorem heq_rec_iff_heq {C : Ξ± β Sort*} {x : Ξ²} {y : C a} {e : a = b} :
HEq x (e βΈ y) β HEq x y := by | subst e; rfl
| 1 | 2.718282 | 0 | 0 | 8 | 105 |
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.Lint.Basic
#align_import logic.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
#align_import init.ite_simp from "leanprover-communit... | Mathlib/Logic/Basic.lean | 1,092 | 1,093 | theorem bex_eq_left {a : Ξ±} : (β (x : _) (_ : x = a), p x) β p a := by |
simp only [exists_prop, exists_eq_left]
| 1 | 2.718282 | 0 | 0 | 8 | 105 |
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.Lint.Basic
#align_import logic.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
#align_import init.ite_simp from "leanprover-communit... | Mathlib/Logic/Basic.lean | 1,131 | 1,131 | theorem existsβ_imp : (β x h, P x h) β b β β x h, P x h β b := by | simp
| 1 | 2.718282 | 0 | 0 | 8 | 105 |
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot
import Mathlib.Order.Hom.Basic
#align_import data.multiset.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
universe v
... | Mathlib/Data/Multiset/Basic.lean | 157 | 158 | theorem cons_inj_right (a : Ξ±) : β {s t : Multiset Ξ±}, a ::β s = a ::β t β s = t := by |
rintro β¨lββ© β¨lββ©; simp
| 1 | 2.718282 | 0 | 0 | 1 | 106 |
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.Init.Data.Prod
import Mathlib.RingTheory.OreLocalization.Basic
#align_import group_theory.monoid_localization from "leanprover-community/... | Mathlib/GroupTheory/MonoidLocalization.lean | 206 | 207 | theorem r_iff_exists {x y : M Γ S} : r S x y β β c : S, βc * (βy.2 * x.1) = c * (x.2 * y.1) := by |
rw [r_eq_r' S]; rfl
| 1 | 2.718282 | 0 | 0 | 1 | 107 |
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.Util.AddRelatedDecl
import Batteries.Tactic.Lint
set_option autoImplicit true
open Lean Meta Elab Tactic
open Mathlib.Tactic
namespace Tactic.Elementwise
open CategoryTheory
section theorems
theorem forall_congr_forget_Type (Ξ± : Type u) (p : Ξ±... | Mathlib/Tactic/CategoryTheory/Elementwise.lean | 52 | 53 | theorem hom_elementwise [Category C] [ConcreteCategory C]
{X Y : C} {f g : X βΆ Y} (h : f = g) (x : X) : f x = g x := by | rw [h]
| 1 | 2.718282 | 0 | 0 | 1 | 108 |
import Mathlib.Data.PNat.Defs
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Set.Basic
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Positive.Ring
import Mathlib.Order.Hom.Basic
#align_import data.pnat.basic from "leanprover-community/mathlib"@"172bf2812857f5e56938cc148b7a5... | Mathlib/Data/PNat/Basic.lean | 33 | 34 | theorem one_add_natPred (n : β+) : 1 + n.natPred = n := by |
rw [natPred, add_tsub_cancel_iff_le.mpr <| show 1 β€ (n : β) from n.2]
| 1 | 2.718282 | 0 | 0 | 1 | 109 |
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 46 | 46 | theorem mem_map {f : Ξ± β Ξ²} {y : Ξ²} {o : Option Ξ±} : y β o.map f β β x β o, f x = y := by | simp
| 1 | 2.718282 | 0 | 0 | 8 | 110 |
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 53 | 55 | theorem mem_map_of_injective {f : Ξ± β Ξ²} (H : Function.Injective f) {a : Ξ±} {o : Option Ξ±} :
f a β o.map f β a β o := by |
aesop
| 1 | 2.718282 | 0 | 0 | 8 | 110 |
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 57 | 58 | theorem forall_mem_map {f : Ξ± β Ξ²} {o : Option Ξ±} {p : Ξ² β Prop} :
(β y β o.map f, p y) β β x β o, p (f x) := by | simp
| 1 | 2.718282 | 0 | 0 | 8 | 110 |
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 61 | 62 | theorem exists_mem_map {f : Ξ± β Ξ²} {o : Option Ξ±} {p : Ξ² β Prop} :
(β y β o.map f, p y) β β x β o, p (f x) := by | simp
| 1 | 2.718282 | 0 | 0 | 8 | 110 |
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 101 | 103 | theorem bind_eq_some' {x : Option Ξ±} {f : Ξ± β Option Ξ²} {b : Ξ²} :
x.bind f = some b β β a, x = some a β§ f a = some b := by |
cases x <;> simp
| 1 | 2.718282 | 0 | 0 | 8 | 110 |
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 108 | 110 | theorem bind_congr {f g : Ξ± β Option Ξ²} {x : Option Ξ±}
(h : β a β x, f a = g a) : x.bind f = x.bind g := by |
cases x <;> simp only [some_bind, none_bind, mem_def, h]
| 1 | 2.718282 | 0 | 0 | 8 | 110 |
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 151 | 153 | theorem map_comm {fβ : Ξ± β Ξ²} {fβ : Ξ± β Ξ³} {gβ : Ξ² β Ξ΄} {gβ : Ξ³ β Ξ΄} (h : gβ β fβ = gβ β fβ)
(a : Ξ±) :
(Option.map fβ a).map gβ = (Option.map fβ a).map gβ := by | rw [map_map, h, β map_map]
| 1 | 2.718282 | 0 | 0 | 8 | 110 |
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 162 | 163 | theorem pbind_eq_bind (f : Ξ± β Option Ξ²) (x : Option Ξ±) : (x.pbind fun a _ β¦ f a) = x.bind f := by |
cases x <;> simp only [pbind, none_bind', some_bind']
| 1 | 2.718282 | 0 | 0 | 8 | 110 |
import Mathlib.CategoryTheory.Limits.Creates
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.IsConnected
#align_import category_theory.limits.constructions.over.connected from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
universe v u
-- morphism levels before o... | Mathlib/CategoryTheory/Limits/Constructions/Over/Connected.lean | 60 | 62 | theorem raised_cone_lowers_to_original [IsConnected J] {B : C} {F : J β₯€ Over B}
(c : Cone (F β forget B)) :
(forget B).mapCone (raiseCone c) = c := by | aesop_cat
| 1 | 2.718282 | 0 | 0 | 1 | 111 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open Equiv Function Finset
variable {... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 90 | 90 | theorem sameCycle_one : SameCycle 1 x y β x = y := by | simp [SameCycle]
| 1 | 2.718282 | 0 | 0 | 7 | 112 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open Equiv Function Finset
variable {... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 107 | 108 | theorem SameCycle.conj : SameCycle f x y β SameCycle (g * f * gβ»ΒΉ) (g x) (g y) := by |
simp [sameCycle_conj]
| 1 | 2.718282 | 0 | 0 | 7 | 112 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open Equiv Function Finset
variable {... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 132 | 133 | theorem sameCycle_apply_right : SameCycle f x (f y) β SameCycle f x y := by |
rw [sameCycle_comm, sameCycle_apply_left, sameCycle_comm]
| 1 | 2.718282 | 0 | 0 | 7 | 112 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open Equiv Function Finset
variable {... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 137 | 138 | theorem sameCycle_inv_apply_left : SameCycle f (fβ»ΒΉ x) y β SameCycle f x y := by |
rw [β sameCycle_apply_left, apply_inv_self]
| 1 | 2.718282 | 0 | 0 | 7 | 112 |
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