Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
|---|---|---|---|---|---|---|---|---|---|---|---|
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.extreme from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14637e8eb963"
open Function Set
open scoped Classical
open Affine
variable {π E F ΞΉ : Type*} {Ο : ΞΉ β Type*}
section SMul
variable (π) [OrderedSemiring π] [AddCommMonoi... | Mathlib/Analysis/Convex/Extreme.lean | 111 | 117 | theorem isExtreme_iInter {ΞΉ : Sort*} [Nonempty ΞΉ] {F : ΞΉ β Set E}
(hAF : β i : ΞΉ, IsExtreme π A (F i)) : IsExtreme π A (β i : ΞΉ, F i) := by |
obtain i := Classical.arbitrary ΞΉ
refine β¨iInter_subset_of_subset i (hAF i).1, fun xβ hxβA xβ hxβA x hxF hx β¦ ?_β©
simp_rw [mem_iInter] at hxF β’
have h := fun i β¦ (hAF i).2 hxβA hxβA (hxF i) hx
exact β¨fun i β¦ (h i).1, fun i β¦ (h i).2β©
| 5 | 148.413159 | 2 | 1.25 | 4 | 1,326 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.extreme from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14637e8eb963"
open Function Set
open scoped Classical
open Affine
variable {π E F ΞΉ : Type*} {Ο : ΞΉ β Type*}
section SMul
variable (π) [OrderedSemiring π] [AddCommMonoi... | Mathlib/Analysis/Convex/Extreme.lean | 120 | 123 | theorem isExtreme_biInter {F : Set (Set E)} (hF : F.Nonempty) (hA : β B β F, IsExtreme π A B) :
IsExtreme π A (β B β F, B) := by |
haveI := hF.to_subtype
simpa only [iInter_subtype] using isExtreme_iInter fun i : F β¦ hA _ i.2
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,326 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.extreme from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14637e8eb963"
open Function Set
open scoped Classical
open Affine
variable {π E F ΞΉ : Type*} {Ο : ΞΉ β Type*}
section SMul
variable (π) [OrderedSemiring π] [AddCommMonoi... | Mathlib/Analysis/Convex/Extreme.lean | 126 | 127 | theorem isExtreme_sInter {F : Set (Set E)} (hF : F.Nonempty) (hAF : β B β F, IsExtreme π A B) :
IsExtreme π A (ββ F) := by | simpa [sInter_eq_biInter] using isExtreme_biInter hF hAF
| 1 | 2.718282 | 0 | 1.25 | 4 | 1,326 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Init.Data.Nat.Lemmas
#align_import data.nat.psub from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
namespace Nat
def ppred : β β Option β
| 0 => none
| n + 1 => some n
#align nat.ppred Nat.ppred
@... | Mathlib/Data/Nat/PSub.lean | 54 | 54 | theorem pred_eq_ppred (n : β) : pred n = (ppred n).getD 0 := by | cases n <;> rfl
| 1 | 2.718282 | 0 | 1.25 | 4 | 1,327 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Init.Data.Nat.Lemmas
#align_import data.nat.psub from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
namespace Nat
def ppred : β β Option β
| 0 => none
| n + 1 => some n
#align nat.ppred Nat.ppred
@... | Mathlib/Data/Nat/PSub.lean | 85 | 93 | theorem psub_eq_none {m n : β} : psub m n = none β m < n := by |
cases s : psub m n <;> simp [eq_comm]
Β· show m < n
refine lt_of_not_ge fun h => ?_
cases' le.dest h with k e
injection s.symm.trans (psub_eq_some.2 <| (add_comm _ _).trans e)
Β· show n β€ m
rw [β psub_eq_some.1 s]
apply Nat.le_add_left
| 8 | 2,980.957987 | 2 | 1.25 | 4 | 1,327 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Init.Data.Nat.Lemmas
#align_import data.nat.psub from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
namespace Nat
def ppred : β β Option β
| 0 => none
| n + 1 => some n
#align nat.ppred Nat.ppred
@... | Mathlib/Data/Nat/PSub.lean | 105 | 109 | theorem psub_add (m n k) :
psub m (n + k) = (do psub (β psub m n) k) := by |
induction k with
| zero => simp only [zero_eq, add_zero, psub_zero, Option.bind_eq_bind, Option.bind_some]
| succ n ih => simp only [ih, add_succ, psub_succ, bind_assoc]
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,327 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Init.Data.Nat.Lemmas
#align_import data.nat.psub from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
namespace Nat
def ppred : β β Option β
| 0 => none
| n + 1 => some n
#align nat.ppred Nat.ppred
@... | Mathlib/Data/Nat/PSub.lean | 118 | 122 | theorem psub'_eq_psub (m n) : psub' m n = psub m n := by |
rw [psub']
split_ifs with h
Β· exact (psub_eq_sub h).symm
Β· exact (psub_eq_none.2 (not_le.1 h)).symm
| 4 | 54.59815 | 2 | 1.25 | 4 | 1,327 |
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
#align_import analysis.convex.simplicial_complex.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Finset Set
variable (π E : Type*) {ΞΉ : Type*} [OrderedRing π] [AddCommGroup E] [Mod... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 86 | 87 | theorem mem_space_iff : x β K.space β β s β K.faces, x β convexHull π (s : Set E) := by |
simp [space]
| 1 | 2.718282 | 0 | 1.25 | 4 | 1,328 |
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
#align_import analysis.convex.simplicial_complex.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Finset Set
variable (π E : Type*) {ΞΉ : Type*} [OrderedRing π] [AddCommGroup E] [Mod... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 91 | 93 | theorem convexHull_subset_space (hs : s β K.faces) : convexHull π βs β K.space := by |
convert subset_biUnion_of_mem hs
rfl
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,328 |
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
#align_import analysis.convex.simplicial_complex.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Finset Set
variable (π E : Type*) {ΞΉ : Type*} [OrderedRing π] [AddCommGroup E] [Mod... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 110 | 119 | theorem disjoint_or_exists_inter_eq_convexHull (hs : s β K.faces) (ht : t β K.faces) :
Disjoint (convexHull π (s : Set E)) (convexHull π βt) β¨
β u β K.faces, convexHull π (s : Set E) β© convexHull π βt = convexHull π βu := by |
classical
by_contra! h
refine h.2 (s β© t) (K.down_closed hs inter_subset_left fun hst => h.1 <|
disjoint_iff_inf_le.mpr <| (K.inter_subset_convexHull hs ht).trans ?_) ?_
Β· rw [β coe_inter, hst, coe_empty, convexHull_empty]
rfl
Β· rw [coe_inter, convexHull_inter_convexHull hs ht]
| 7 | 1,096.633158 | 2 | 1.25 | 4 | 1,328 |
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
#align_import analysis.convex.simplicial_complex.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Finset Set
variable (π E : Type*) {ΞΉ : Type*} [OrderedRing π] [AddCommGroup E] [Mod... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 158 | 162 | theorem vertices_eq : K.vertices = β k β K.faces, (k : Set E) := by |
ext x
refine β¨fun h => mem_biUnion h <| mem_coe.2 <| mem_singleton_self x, fun h => ?_β©
obtain β¨s, hs, hxβ© := mem_iUnionβ.1 h
exact K.down_closed hs (Finset.singleton_subset_iff.2 <| mem_coe.1 hx) (singleton_ne_empty _)
| 4 | 54.59815 | 2 | 1.25 | 4 | 1,328 |
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.congruence_subgroups from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f... | Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 56 | 66 | theorem Gamma_mem (N : β) (Ξ³ : SL(2, β€)) : Ξ³ β Gamma N β ((ββΞ³ 0 0 : β€) : ZMod N) = 1 β§
((ββΞ³ 0 1 : β€) : ZMod N) = 0 β§ ((ββΞ³ 1 0 : β€) : ZMod N) = 0 β§ ((ββΞ³ 1 1 : β€) : ZMod N) = 1 := by |
rw [Gamma_mem']
constructor
Β· intro h
simp [β SL_reduction_mod_hom_val N Ξ³, h]
Β· intro h
ext i j
rw [SL_reduction_mod_hom_val N Ξ³]
fin_cases i <;> fin_cases j <;> simp only [h]
exacts [h.1, h.2.1, h.2.2.1, h.2.2.2]
| 9 | 8,103.083928 | 2 | 1.25 | 4 | 1,329 |
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.congruence_subgroups from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f... | Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 73 | 75 | theorem Gamma_one_top : Gamma 1 = β€ := by |
ext
simp [eq_iff_true_of_subsingleton]
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,329 |
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.congruence_subgroups from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f... | Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 78 | 88 | theorem Gamma_zero_bot : Gamma 0 = β₯ := by |
ext
simp only [Gamma_mem, coe_matrix_coe, Int.coe_castRingHom, map_apply, Int.cast_id,
Subgroup.mem_bot]
constructor
Β· intro h
ext i j
fin_cases i <;> fin_cases j <;> simp only [h]
exacts [h.1, h.2.1, h.2.2.1, h.2.2.2]
Β· intro h
simp [h]
| 10 | 22,026.465795 | 2 | 1.25 | 4 | 1,329 |
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.congruence_subgroups from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f... | Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 125 | 125 | theorem Gamma0_det (N : β) (A : Gamma0 N) : (A.1.1.det : ZMod N) = 1 := by | simp [A.1.property]
| 1 | 2.718282 | 0 | 1.25 | 4 | 1,329 |
import Mathlib.Data.Int.Bitwise
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.matrix.zpow from "leanprover-community/mathlib"@"03fda9112aa6708947da13944a19310684bfdfcb"
open Matrix
namespace Matrix
variable {n' : Type*} [Decidab... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 44 | 47 | theorem inv_pow' (A : M) (n : β) : Aβ»ΒΉ ^ n = (A ^ n)β»ΒΉ := by |
induction' n with n ih
Β· simp
Β· rw [pow_succ A, mul_inv_rev, β ih, β pow_succ']
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,330 |
import Mathlib.Data.Int.Bitwise
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.matrix.zpow from "leanprover-community/mathlib"@"03fda9112aa6708947da13944a19310684bfdfcb"
open Matrix
namespace Matrix
variable {n' : Type*} [Decidab... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 50 | 54 | theorem pow_sub' (A : M) {m n : β} (ha : IsUnit A.det) (h : n β€ m) :
A ^ (m - n) = A ^ m * (A ^ n)β»ΒΉ := by |
rw [β tsub_add_cancel_of_le h, pow_add, Matrix.mul_assoc, mul_nonsing_inv,
tsub_add_cancel_of_le h, Matrix.mul_one]
simpa using ha.pow n
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,330 |
import Mathlib.Data.Int.Bitwise
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.matrix.zpow from "leanprover-community/mathlib"@"03fda9112aa6708947da13944a19310684bfdfcb"
open Matrix
namespace Matrix
variable {n' : Type*} [Decidab... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 57 | 70 | theorem pow_inv_comm' (A : M) (m n : β) : Aβ»ΒΉ ^ m * A ^ n = A ^ n * Aβ»ΒΉ ^ m := by |
induction' n with n IH generalizing m
Β· simp
cases' m with m m
Β· simp
rcases nonsing_inv_cancel_or_zero A with (β¨h, h'β© | h)
Β· calc
Aβ»ΒΉ ^ (m + 1) * A ^ (n + 1) = Aβ»ΒΉ ^ m * (Aβ»ΒΉ * A) * A ^ n := by
simp only [pow_succ Aβ»ΒΉ, pow_succ' A, Matrix.mul_assoc]
_ = A ^ n * Aβ»ΒΉ ^ m := by simp onl... | 13 | 442,413.392009 | 2 | 1.25 | 4 | 1,330 |
import Mathlib.Data.Int.Bitwise
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.matrix.zpow from "leanprover-community/mathlib"@"03fda9112aa6708947da13944a19310684bfdfcb"
open Matrix
namespace Matrix
variable {n' : Type*} [Decidab... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 92 | 95 | theorem zero_zpow_eq (n : β€) : (0 : M) ^ n = if n = 0 then 1 else 0 := by |
split_ifs with h
Β· rw [h, zpow_zero]
Β· rw [zero_zpow _ h]
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,330 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Perm.Fin
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.IntervalCases
#align_import group_theory.specific_groups.alternating from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
-- An example on how to de... | Mathlib/GroupTheory/SpecificGroups/Alternating.lean | 77 | 80 | theorem prod_list_swap_mem_alternatingGroup_iff_even_length {l : List (Perm Ξ±)}
(hl : β g β l, IsSwap g) : l.prod β alternatingGroup Ξ± β Even l.length := by |
rw [mem_alternatingGroup, sign_prod_list_swap hl, neg_one_pow_eq_one_iff_even]
decide
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,331 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Perm.Fin
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.IntervalCases
#align_import group_theory.specific_groups.alternating from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
-- An example on how to de... | Mathlib/GroupTheory/SpecificGroups/Alternating.lean | 89 | 91 | theorem finRotate_bit1_mem_alternatingGroup {n : β} :
finRotate (bit1 n) β alternatingGroup (Fin (bit1 n)) := by |
rw [mem_alternatingGroup, bit1, sign_finRotate, pow_bit0', Int.units_mul_self, one_pow]
| 1 | 2.718282 | 0 | 1.25 | 4 | 1,331 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Perm.Fin
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.IntervalCases
#align_import group_theory.specific_groups.alternating from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
-- An example on how to de... | Mathlib/GroupTheory/SpecificGroups/Alternating.lean | 96 | 101 | theorem two_mul_card_alternatingGroup [Nontrivial Ξ±] :
2 * card (alternatingGroup Ξ±) = card (Perm Ξ±) := by |
let this := (QuotientGroup.quotientKerEquivOfSurjective _ (sign_surjective Ξ±)).toEquiv
rw [β Fintype.card_units_int, β Fintype.card_congr this]
simp only [β Nat.card_eq_fintype_card]
apply (Subgroup.card_eq_card_quotient_mul_card_subgroup _).symm
| 4 | 54.59815 | 2 | 1.25 | 4 | 1,331 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Perm.Fin
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.IntervalCases
#align_import group_theory.specific_groups.alternating from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
-- An example on how to de... | Mathlib/GroupTheory/SpecificGroups/Alternating.lean | 219 | 224 | theorem nontrivial_of_three_le_card (h3 : 3 β€ card Ξ±) : Nontrivial (alternatingGroup Ξ±) := by |
haveI := Fintype.one_lt_card_iff_nontrivial.1 (lt_trans (by decide) h3)
rw [β Fintype.one_lt_card_iff_nontrivial]
refine lt_of_mul_lt_mul_left ?_ (le_of_lt Nat.prime_two.pos)
rw [two_mul_card_alternatingGroup, card_perm, β Nat.succ_le_iff]
exact le_trans h3 (card Ξ±).self_le_factorial
| 5 | 148.413159 | 2 | 1.25 | 4 | 1,331 |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def Οβ : MulChar (ZMod 4) β€... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 48 | 52 | theorem isQuadratic_Οβ : Οβ.IsQuadratic := by |
intro a
-- Porting note (#11043): was `decide!`
fin_cases a
all_goals decide
| 4 | 54.59815 | 2 | 1.25 | 12 | 1,332 |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def Οβ : MulChar (ZMod 4) β€... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 56 | 56 | theorem Οβ_nat_mod_four (n : β) : Οβ n = Οβ (n % 4 : β) := by | rw [β ZMod.natCast_mod n 4]
| 1 | 2.718282 | 0 | 1.25 | 12 | 1,332 |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def Οβ : MulChar (ZMod 4) β€... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 60 | 62 | theorem Οβ_int_mod_four (n : β€) : Οβ n = Οβ (n % 4 : β€) := by |
rw [β ZMod.intCast_mod n 4]
norm_cast
| 2 | 7.389056 | 1 | 1.25 | 12 | 1,332 |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def Οβ : MulChar (ZMod 4) β€... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 66 | 71 | theorem Οβ_int_eq_if_mod_four (n : β€) :
Οβ n = if n % 2 = 0 then 0 else if n % 4 = 1 then 1 else -1 := by |
have help : β m : β€, 0 β€ m β m < 4 β Οβ m = if m % 2 = 0 then 0 else if m = 1 then 1 else -1 := by
decide
rw [β Int.emod_emod_of_dvd n (by decide : (2 : β€) β£ 4), β ZMod.intCast_mod n 4]
exact help (n % 4) (Int.emod_nonneg n (by norm_num)) (Int.emod_lt n (by norm_num))
| 4 | 54.59815 | 2 | 1.25 | 12 | 1,332 |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def Οβ : MulChar (ZMod 4) β€... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 80 | 91 | theorem Οβ_eq_neg_one_pow {n : β} (hn : n % 2 = 1) : Οβ n = (-1) ^ (n / 2) := by |
rw [Οβ_nat_eq_if_mod_four]
simp only [hn, Nat.one_ne_zero, if_false]
conv_rhs => -- Porting note: was `nth_rw`
arg 2; rw [β Nat.div_add_mod n 4]
enter [1, 1, 1]; rw [(by norm_num : 4 = 2 * 2)]
rw [mul_assoc, add_comm, Nat.add_mul_div_left _ _ (by norm_num : 0 < 2), pow_add, pow_mul,
neg_one_sq, one... | 11 | 59,874.141715 | 2 | 1.25 | 12 | 1,332 |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def Οβ : MulChar (ZMod 4) β€... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 95 | 97 | theorem Οβ_nat_one_mod_four {n : β} (hn : n % 4 = 1) : Οβ n = 1 := by |
rw [Οβ_nat_mod_four, hn]
rfl
| 2 | 7.389056 | 1 | 1.25 | 12 | 1,332 |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def Οβ : MulChar (ZMod 4) β€... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 101 | 103 | theorem Οβ_nat_three_mod_four {n : β} (hn : n % 4 = 3) : Οβ n = -1 := by |
rw [Οβ_nat_mod_four, hn]
rfl
| 2 | 7.389056 | 1 | 1.25 | 12 | 1,332 |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def Οβ : MulChar (ZMod 4) β€... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 107 | 109 | theorem Οβ_int_one_mod_four {n : β€} (hn : n % 4 = 1) : Οβ n = 1 := by |
rw [Οβ_int_mod_four, hn]
rfl
| 2 | 7.389056 | 1 | 1.25 | 12 | 1,332 |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def Οβ : MulChar (ZMod 4) β€... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 113 | 115 | theorem Οβ_int_three_mod_four {n : β€} (hn : n % 4 = 3) : Οβ n = -1 := by |
rw [Οβ_int_mod_four, hn]
rfl
| 2 | 7.389056 | 1 | 1.25 | 12 | 1,332 |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def Οβ : MulChar (ZMod 4) β€... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 119 | 121 | theorem neg_one_pow_div_two_of_one_mod_four {n : β} (hn : n % 4 = 1) : (-1 : β€) ^ (n / 2) = 1 := by |
rw [β Οβ_eq_neg_one_pow (Nat.odd_of_mod_four_eq_one hn), β natCast_mod, hn]
rfl
| 2 | 7.389056 | 1 | 1.25 | 12 | 1,332 |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def Οβ : MulChar (ZMod 4) β€... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 125 | 128 | theorem neg_one_pow_div_two_of_three_mod_four {n : β} (hn : n % 4 = 3) :
(-1 : β€) ^ (n / 2) = -1 := by |
rw [β Οβ_eq_neg_one_pow (Nat.odd_of_mod_four_eq_three hn), β natCast_mod, hn]
rfl
| 2 | 7.389056 | 1 | 1.25 | 12 | 1,332 |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def Οβ : MulChar (ZMod 4) β€... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 142 | 146 | theorem isQuadratic_Οβ : Οβ.IsQuadratic := by |
intro a
-- Porting note: was `decide!`
fin_cases a
all_goals decide
| 4 | 54.59815 | 2 | 1.25 | 12 | 1,332 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 127 | 131 | theorem monomial_def [DecidableEq Ο] (n : Ο ββ β) :
(monomial R n) = LinearMap.stdBasis R (fun _ β¦ R) n := by |
rw [monomial]
-- unify the `Decidable` arguments
convert rfl
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,333 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 134 | 140 | theorem coeff_monomial [DecidableEq Ο] (m n : Ο ββ β) (a : R) :
coeff R m (monomial R n a) = if m = n then a else 0 := by |
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [coeff, monomial_def, LinearMap.proj_apply (i := m)]
dsimp only
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [LinearMap.stdBasis_apply, Function.update_apply, Pi.zero_apply]
| 5 | 148.413159 | 2 | 1.25 | 4 | 1,333 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 144 | 147 | theorem coeff_monomial_same (n : Ο ββ β) (a : R) : coeff R n (monomial R n a) = a := by |
classical
rw [monomial_def]
exact LinearMap.stdBasis_same R (fun _ β¦ R) n a
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,333 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 150 | 153 | theorem coeff_monomial_ne {m n : Ο ββ β} (h : m β n) (a : R) : coeff R m (monomial R n a) = 0 := by |
classical
rw [monomial_def]
exact LinearMap.stdBasis_ne R (fun _ β¦ R) _ _ h a
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,333 |
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.restrict from "leanprover-community/mathlib"@"09258fb7f75d741b7eda9fa18d5c869e2135d9f1"
variable {k Vβ Pβ Vβ Pβ : Type*} [Ring k] [AddCommGroup Vβ] [AddCommGroup Vβ] [Module k Vβ]
[Module k Vβ] [AddTorsor Vβ Pβ] [A... | Mathlib/LinearAlgebra/AffineSpace/Restrict.lean | 33 | 36 | theorem AffineSubspace.nonempty_map {E : AffineSubspace k Pβ} [Ene : Nonempty E] {Ο : Pβ βα΅[k] Pβ} :
Nonempty (E.map Ο) := by |
obtain β¨x, hxβ© := id Ene
exact β¨β¨Ο x, AffineSubspace.mem_map.mpr β¨x, hx, rflβ©β©β©
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,334 |
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.restrict from "leanprover-community/mathlib"@"09258fb7f75d741b7eda9fa18d5c869e2135d9f1"
variable {k Vβ Pβ Vβ Pβ : Type*} [Ring k] [AddCommGroup Vβ] [AddCommGroup Vβ] [Module k Vβ]
[Module k Vβ] [AddTorsor Vβ Pβ] [A... | Mathlib/LinearAlgebra/AffineSpace/Restrict.lean | 61 | 64 | theorem AffineMap.restrict.linear_aux {Ο : Pβ βα΅[k] Pβ} {E : AffineSubspace k Pβ}
{F : AffineSubspace k Pβ} (hEF : E.map Ο β€ F) : E.direction β€ F.direction.comap Ο.linear := by |
rw [β Submodule.map_le_iff_le_comap, β AffineSubspace.map_direction]
exact AffineSubspace.direction_le hEF
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,334 |
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.restrict from "leanprover-community/mathlib"@"09258fb7f75d741b7eda9fa18d5c869e2135d9f1"
variable {k Vβ Pβ Vβ Pβ : Type*} [Ring k] [AddCommGroup Vβ] [AddCommGroup Vβ] [Module k Vβ]
[Module k Vβ] [AddTorsor Vβ Pβ] [A... | Mathlib/LinearAlgebra/AffineSpace/Restrict.lean | 73 | 78 | theorem AffineMap.restrict.injective {Ο : Pβ βα΅[k] Pβ} (hΟ : Function.Injective Ο)
{E : AffineSubspace k Pβ} {F : AffineSubspace k Pβ} [Nonempty E] [Nonempty F]
(hEF : E.map Ο β€ F) : Function.Injective (AffineMap.restrict Ο hEF) := by |
intro x y h
simp only [Subtype.ext_iff, Subtype.coe_mk, AffineMap.restrict.coe_apply] at h β’
exact hΟ h
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,334 |
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.restrict from "leanprover-community/mathlib"@"09258fb7f75d741b7eda9fa18d5c869e2135d9f1"
variable {k Vβ Pβ Vβ Pβ : Type*} [Ring k] [AddCommGroup Vβ] [AddCommGroup Vβ] [Module k Vβ]
[Module k Vβ] [AddTorsor Vβ Pβ] [A... | Mathlib/LinearAlgebra/AffineSpace/Restrict.lean | 81 | 87 | theorem AffineMap.restrict.surjective (Ο : Pβ βα΅[k] Pβ) {E : AffineSubspace k Pβ}
{F : AffineSubspace k Pβ} [Nonempty E] [Nonempty F] (h : E.map Ο = F) :
Function.Surjective (AffineMap.restrict Ο (le_of_eq h)) := by |
rintro β¨x, hx : x β Fβ©
rw [β h, AffineSubspace.mem_map] at hx
obtain β¨y, hy, rflβ© := hx
exact β¨β¨y, hyβ©, rflβ©
| 4 | 54.59815 | 2 | 1.25 | 4 | 1,334 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.Sort
#align_import data.pnat.factors from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
-- Porting note: `deriving` contained Inhabited, Canonic... | Mathlib/Data/PNat/Factors.lean | 89 | 91 | theorem coeNat_prime (v : PrimeMultiset) (p : β) (h : p β (v : Multiset β)) : p.Prime := by |
rcases Multiset.mem_map.mp h with β¨β¨_, hp'β©, β¨_, h_eqβ©β©
exact h_eq βΈ hp'
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,335 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.Sort
#align_import data.pnat.factors from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
-- Porting note: `deriving` contained Inhabited, Canonic... | Mathlib/Data/PNat/Factors.lean | 121 | 123 | theorem coePNat_prime (v : PrimeMultiset) (p : β+) (h : p β (v : Multiset β+)) : p.Prime := by |
rcases Multiset.mem_map.mp h with β¨β¨_, hp'β©, β¨_, h_eqβ©β©
exact h_eq βΈ hp'
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,335 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.Sort
#align_import data.pnat.factors from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
-- Porting note: `deriving` contained Inhabited, Canonic... | Mathlib/Data/PNat/Factors.lean | 130 | 133 | theorem coePNat_nat (v : PrimeMultiset) : ((v : Multiset β+) : Multiset β) = (v : Multiset β) := by |
change (v.map (Coe.coe : Nat.Primes β β+)).map Subtype.val = v.map Subtype.val
rw [Multiset.map_map]
congr
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,335 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.Sort
#align_import data.pnat.factors from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
-- Porting note: `deriving` contained Inhabited, Canonic... | Mathlib/Data/PNat/Factors.lean | 141 | 146 | theorem coe_prod (v : PrimeMultiset) : (v.prod : β) = (v : Multiset β).prod := by |
let h : (v.prod : β) = ((v.map Coe.coe).map Coe.coe).prod :=
PNat.coeMonoidHom.map_multiset_prod v.toPNatMultiset
rw [Multiset.map_map] at h
have : (Coe.coe : β+ β β) β (Coe.coe : Nat.Primes β β+) = Coe.coe := funext fun p => rfl
rw [this] at h; exact h
| 5 | 148.413159 | 2 | 1.25 | 4 | 1,335 |
import Mathlib.FieldTheory.SeparableDegree
import Mathlib.FieldTheory.IsSepClosed
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField Field
noncomputable section
universe u v w
variable (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E]
variable (K : Type w) [Field K] [... | Mathlib/FieldTheory/SeparableClosure.lean | 94 | 96 | theorem map_mem_separableClosure_iff (i : E ββ[F] K) {x : E} :
i x β separableClosure F K β x β separableClosure F E := by |
simp_rw [mem_separableClosure_iff, minpoly.algHom_eq i i.injective]
| 1 | 2.718282 | 0 | 1.25 | 4 | 1,336 |
import Mathlib.FieldTheory.SeparableDegree
import Mathlib.FieldTheory.IsSepClosed
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField Field
noncomputable section
universe u v w
variable (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E]
variable (K : Type w) [Field K] [... | Mathlib/FieldTheory/SeparableClosure.lean | 100 | 103 | theorem separableClosure.comap_eq_of_algHom (i : E ββ[F] K) :
(separableClosure F K).comap i = separableClosure F E := by |
ext x
exact map_mem_separableClosure_iff i
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,336 |
import Mathlib.FieldTheory.SeparableDegree
import Mathlib.FieldTheory.IsSepClosed
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField Field
noncomputable section
universe u v w
variable (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E]
variable (K : Type w) [Field K] [... | Mathlib/FieldTheory/SeparableClosure.lean | 115 | 121 | theorem separableClosure.map_eq_of_separableClosure_eq_bot [Algebra E K] [IsScalarTower F E K]
(h : separableClosure E K = β₯) :
(separableClosure F E).map (IsScalarTower.toAlgHom F E K) = separableClosure F K := by |
refine le_antisymm (map_le_of_algHom _) (fun x hx β¦ ?_)
obtain β¨y, rflβ© := mem_bot.1 <| h βΈ mem_separableClosure_iff.2
(mem_separableClosure_iff.1 hx |>.map_minpoly E)
exact β¨y, (map_mem_separableClosure_iff <| IsScalarTower.toAlgHom F E K).mp hx, rflβ©
| 4 | 54.59815 | 2 | 1.25 | 4 | 1,336 |
import Mathlib.FieldTheory.SeparableDegree
import Mathlib.FieldTheory.IsSepClosed
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField Field
noncomputable section
universe u v w
variable (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E]
variable (K : Type w) [Field K] [... | Mathlib/FieldTheory/SeparableClosure.lean | 186 | 192 | theorem IsSepClosed.separableClosure_eq_bot_iff [IsSepClosed E] :
separableClosure F E = β₯ β IsSepClosed F := by |
refine β¨fun h β¦ IsSepClosed.of_exists_root _ fun p _ hirr hsep β¦ ?_,
fun _ β¦ IntermediateField.eq_bot_of_isSepClosed_of_isSeparable _β©
obtain β¨x, hxβ© := IsSepClosed.exists_aeval_eq_zero E p (degree_pos_of_irreducible hirr).ne' hsep
obtain β¨x, rflβ© := h βΈ mem_separableClosure_iff.2 (hsep.of_dvd <| minpoly.dvd... | 5 | 148.413159 | 2 | 1.25 | 4 | 1,336 |
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Category.ModuleCat.Abelian
import Mathlib.Algebra.Category.ModuleCat.Subobject
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import algebra.homology.Module from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225... | Mathlib/Algebra/Homology/ModuleCat.lean | 37 | 49 | theorem homology'_ext {L M N K : ModuleCat.{u} R} {f : L βΆ M} {g : M βΆ N} (w : f β« g = 0)
{h k : homology' f g w βΆ K}
(w :
β x : LinearMap.ker g,
h (cokernel.Ο (imageToKernel _ _ w) (toKernelSubobject x)) =
k (cokernel.Ο (imageToKernel _ _ w) (toKernelSubobject x))) :
h = k := by |
refine Concrete.cokernel_funext fun n => ?_
-- Porting note: as `equiv_rw` was not ported, it was replaced by `Equiv.surjective`
-- Gosh it would be nice if `equiv_rw` could directly use an isomorphism, or an enriched `β`.
obtain β¨n, rflβ© := (kernelSubobjectIso g βͺβ«
ModuleCat.kernelIsoKer g).toLinearEquiv.... | 6 | 403.428793 | 2 | 1.25 | 4 | 1,337 |
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Category.ModuleCat.Abelian
import Mathlib.Algebra.Category.ModuleCat.Subobject
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import algebra.homology.Module from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225... | Mathlib/Algebra/Homology/ModuleCat.lean | 61 | 65 | theorem cycles'_ext {C : HomologicalComplex (ModuleCat.{u} R) c} {i : ΞΉ}
{x y : (C.cycles' i : Type u)}
(w : (C.cycles' i).arrow x = (C.cycles' i).arrow y) : x = y := by |
apply_fun (C.cycles' i).arrow using (ModuleCat.mono_iff_injective _).mp (cycles' C i).arrow_mono
exact w
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,337 |
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Category.ModuleCat.Abelian
import Mathlib.Algebra.Category.ModuleCat.Subobject
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import algebra.homology.Module from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225... | Mathlib/Algebra/Homology/ModuleCat.lean | 72 | 79 | theorem cycles'Map_toCycles' (f : C βΆ D) {i : ΞΉ} (x : LinearMap.ker (C.dFrom i)) :
(cycles'Map f i) (toCycles' x) = toCycles' β¨f.f i x.1, by
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
rw [LinearMap.mem_ker]; erw [Hom.comm_from_apply, x.2, map_zero]β© := by |
ext
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [cycles'Map_arrow_apply, toKernelSubobject_arrow, toKernelSubobject_arrow]
rfl
| 4 | 54.59815 | 2 | 1.25 | 4 | 1,337 |
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Category.ModuleCat.Abelian
import Mathlib.Algebra.Category.ModuleCat.Subobject
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import algebra.homology.Module from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225... | Mathlib/Algebra/Homology/ModuleCat.lean | 91 | 93 | theorem homology'_ext' {M : ModuleCat R} (i : ΞΉ) {h k : C.homology' i βΆ M}
(w : β x : LinearMap.ker (C.dFrom i), h (toHomology' x) = k (toHomology' x)) : h = k := by |
apply homology'_ext _ w
| 1 | 2.718282 | 0 | 1.25 | 4 | 1,337 |
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import linear_algebra.eigenspace.is_alg_closed from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
open Set Function Module FiniteDimensional
variable {K V : Type*} [Field K] [AddCommGro... | Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | 51 | 54 | theorem exists_eigenvalue [IsAlgClosed K] [FiniteDimensional K V] [Nontrivial V] (f : End K V) :
β c : K, f.HasEigenvalue c := by |
simp_rw [hasEigenvalue_iff_mem_spectrum]
exact spectrum.nonempty_of_isAlgClosed_of_finiteDimensional K f
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,338 |
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import linear_algebra.eigenspace.is_alg_closed from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
open Set Function Module FiniteDimensional
variable {K V : Type*} [Field K] [AddCommGro... | Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | 64 | 123 | theorem iSup_genEigenspace_eq_top [IsAlgClosed K] [FiniteDimensional K V] (f : End K V) :
β¨ (ΞΌ : K) (k : β), f.genEigenspace ΞΌ k = β€ := by |
-- We prove the claim by strong induction on the dimension of the vector space.
induction' h_dim : finrank K V using Nat.strong_induction_on with n ih generalizing V
cases' n with n
-- If the vector space is 0-dimensional, the result is trivial.
Β· rw [β top_le_iff]
simp only [Submodule.finrank_eq_zero.1 ... | 58 | 15,455,389,355,901,040,000,000,000 | 2 | 1.25 | 4 | 1,338 |
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import linear_algebra.eigenspace.is_alg_closed from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
open Set Function Module FiniteDimensional
variable {K V : Type*} [Field K] [AddCommGro... | Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | 132 | 192 | theorem inf_iSup_genEigenspace [FiniteDimensional K V] (h : β x β p, f x β p) :
p β β¨ ΞΌ, β¨ k, f.genEigenspace ΞΌ k = β¨ ΞΌ, β¨ k, p β f.genEigenspace ΞΌ k := by |
simp_rw [β (f.genEigenspace _).mono.directed_le.inf_iSup_eq]
refine le_antisymm (fun m hm β¦ ?_)
(le_inf_iff.mpr β¨iSup_le fun ΞΌ β¦ inf_le_left, iSup_mono fun ΞΌ β¦ inf_le_rightβ©)
classical
obtain β¨hmβ : m β p, hmβ : m β β¨ ΞΌ, β¨ k, f.genEigenspace ΞΌ kβ© := hm
obtain β¨m, hmβ, rflβ© := (mem_iSup_iff_exists_finsupp... | 59 | 42,012,104,037,905,144,000,000,000 | 2 | 1.25 | 4 | 1,338 |
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import linear_algebra.eigenspace.is_alg_closed from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
open Set Function Module FiniteDimensional
variable {K V : Type*} [Field K] [AddCommGro... | Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | 194 | 197 | theorem eq_iSup_inf_genEigenspace [FiniteDimensional K V]
(h : β x β p, f x β p) (h' : β¨ ΞΌ, β¨ k, f.genEigenspace ΞΌ k = β€) :
p = β¨ ΞΌ, β¨ k, p β f.genEigenspace ΞΌ k := by |
rw [β inf_iSup_genEigenspace h, h', inf_top_eq]
| 1 | 2.718282 | 0 | 1.25 | 4 | 1,338 |
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.MvPolynomial.Basic
#align_import data.mv_polynomial.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
variable {Ο R : Type*} [CommSemiring R]
namespace MvPolynomial
| Mathlib/Algebra/MvPolynomial/Division.lean | 221 | 240 | theorem monomial_dvd_monomial {r s : R} {i j : Ο ββ β} :
monomial i r β£ monomial j s β (s = 0 β¨ i β€ j) β§ r β£ s := by |
constructor
Β· rintro β¨x, hxβ©
rw [MvPolynomial.ext_iff] at hx
have hj := hx j
have hi := hx i
classical
simp_rw [coeff_monomial, if_pos] at hj hi
simp_rw [coeff_monomial_mul'] at hi hj
split_ifs at hi hj with hi hi
Β· exact β¨Or.inr hi, _, hjβ©
Β· exact β¨Or.inl hj, hj.symm ... | 18 | 65,659,969.137331 | 2 | 1.25 | 4 | 1,339 |
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.MvPolynomial.Basic
#align_import data.mv_polynomial.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
variable {Ο R : Type*} [CommSemiring R]
namespace MvPolynomial
theorem monomial_dvd_monomial {r s : ... | Mathlib/Algebra/MvPolynomial/Division.lean | 244 | 247 | theorem monomial_one_dvd_monomial_one [Nontrivial R] {i j : Ο ββ β} :
monomial i (1 : R) β£ monomial j 1 β i β€ j := by |
rw [monomial_dvd_monomial]
simp_rw [one_ne_zero, false_or_iff, dvd_rfl, and_true_iff]
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,339 |
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.MvPolynomial.Basic
#align_import data.mv_polynomial.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
variable {Ο R : Type*} [CommSemiring R]
namespace MvPolynomial
theorem monomial_dvd_monomial {r s : ... | Mathlib/Algebra/MvPolynomial/Division.lean | 251 | 255 | theorem X_dvd_X [Nontrivial R] {i j : Ο} :
(X i : MvPolynomial Ο R) β£ (X j : MvPolynomial Ο R) β i = j := by |
refine monomial_one_dvd_monomial_one.trans ?_
simp_rw [Finsupp.single_le_iff, Nat.one_le_iff_ne_zero, Finsupp.single_apply_ne_zero,
ne_eq, not_false_eq_true, and_true]
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,339 |
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.MvPolynomial.Basic
#align_import data.mv_polynomial.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
variable {Ο R : Type*} [CommSemiring R]
namespace MvPolynomial
theorem monomial_dvd_monomial {r s : ... | Mathlib/Algebra/MvPolynomial/Division.lean | 260 | 263 | theorem X_dvd_monomial {i : Ο} {j : Ο ββ β} {r : R} :
(X i : MvPolynomial Ο R) β£ monomial j r β r = 0 β¨ j i β 0 := by |
refine monomial_dvd_monomial.trans ?_
simp_rw [one_dvd, and_true_iff, Finsupp.single_le_iff, Nat.one_le_iff_ne_zero]
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,339 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 61 | 63 | theorem sSup_inv (s : Set Ξ±) : sSup sβ»ΒΉ = (sInf s)β»ΒΉ := by |
rw [β image_inv, sSup_image]
exact ((OrderIso.inv Ξ±).map_sInf _).symm
| 2 | 7.389056 | 1 | 1.25 | 16 | 1,340 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 68 | 70 | theorem sInf_inv (s : Set Ξ±) : sInf sβ»ΒΉ = (sSup s)β»ΒΉ := by |
rw [β image_inv, sInf_image]
exact ((OrderIso.inv Ξ±).map_sSup _).symm
| 2 | 7.389056 | 1 | 1.25 | 16 | 1,340 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 89 | 89 | theorem sSup_div : sSup (s / t) = sSup s / sInf t := by | simp_rw [div_eq_mul_inv, sSup_mul, sSup_inv]
| 1 | 2.718282 | 0 | 1.25 | 16 | 1,340 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 94 | 94 | theorem sInf_div : sInf (s / t) = sInf s / sSup t := by | simp_rw [div_eq_mul_inv, sInf_mul, sInf_inv]
| 1 | 2.718282 | 0 | 1.25 | 16 | 1,340 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 130 | 132 | theorem csSup_inv (hsβ : s.Nonempty) (hsβ : BddBelow s) : sSup sβ»ΒΉ = (sInf s)β»ΒΉ := by |
rw [β image_inv]
exact ((OrderIso.inv Ξ±).map_csInf' hsβ hsβ).symm
| 2 | 7.389056 | 1 | 1.25 | 16 | 1,340 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 137 | 139 | theorem csInf_inv (hsβ : s.Nonempty) (hsβ : BddAbove s) : sInf sβ»ΒΉ = (sSup s)β»ΒΉ := by |
rw [β image_inv]
exact ((OrderIso.inv Ξ±).map_csSup' hsβ hsβ).symm
| 2 | 7.389056 | 1 | 1.25 | 16 | 1,340 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 160 | 162 | theorem csSup_div (hsβ : s.Nonempty) (hsβ : BddAbove s) (htβ : t.Nonempty) (htβ : BddBelow t) :
sSup (s / t) = sSup s / sInf t := by |
rw [div_eq_mul_inv, csSup_mul hsβ hsβ htβ.inv htβ.inv, csSup_inv htβ htβ, div_eq_mul_inv]
| 1 | 2.718282 | 0 | 1.25 | 16 | 1,340 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 167 | 169 | theorem csInf_div (hsβ : s.Nonempty) (hsβ : BddBelow s) (htβ : t.Nonempty) (htβ : BddAbove t) :
sInf (s / t) = sInf s / sSup t := by |
rw [div_eq_mul_inv, csInf_mul hsβ hsβ htβ.inv htβ.inv, csInf_inv htβ htβ, div_eq_mul_inv]
| 1 | 2.718282 | 0 | 1.25 | 16 | 1,340 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 183 | 194 | theorem smul_Ioo : r β’ Ioo a b = Ioo (r β’ a) (r β’ b) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Ioo]
constructor
Β· rintro β¨a, β¨a_h_left_left, a_h_left_rightβ©, rflβ©
constructor
Β· exact (mul_lt_mul_left hr).mpr a_h_left_left
Β· exact (mul_lt_mul_left hr).mpr a_h_left_right
Β· rintro β¨a_left, a_rightβ©
use x / r
refine β¨β¨(lt_div_iff' hr).mpr... | 11 | 59,874.141715 | 2 | 1.25 | 16 | 1,340 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 197 | 208 | theorem smul_Icc : r β’ Icc a b = Icc (r β’ a) (r β’ b) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Icc]
constructor
Β· rintro β¨a, β¨a_h_left_left, a_h_left_rightβ©, rflβ©
constructor
Β· exact (mul_le_mul_left hr).mpr a_h_left_left
Β· exact (mul_le_mul_left hr).mpr a_h_left_right
Β· rintro β¨a_left, a_rightβ©
use x / r
refine β¨β¨(le_div_iff' hr).mpr... | 11 | 59,874.141715 | 2 | 1.25 | 16 | 1,340 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 211 | 222 | theorem smul_Ico : r β’ Ico a b = Ico (r β’ a) (r β’ b) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Ico]
constructor
Β· rintro β¨a, β¨a_h_left_left, a_h_left_rightβ©, rflβ©
constructor
Β· exact (mul_le_mul_left hr).mpr a_h_left_left
Β· exact (mul_lt_mul_left hr).mpr a_h_left_right
Β· rintro β¨a_left, a_rightβ©
use x / r
refine β¨β¨(le_div_iff' hr).mpr... | 11 | 59,874.141715 | 2 | 1.25 | 16 | 1,340 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 225 | 236 | theorem smul_Ioc : r β’ Ioc a b = Ioc (r β’ a) (r β’ b) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Ioc]
constructor
Β· rintro β¨a, β¨a_h_left_left, a_h_left_rightβ©, rflβ©
constructor
Β· exact (mul_lt_mul_left hr).mpr a_h_left_left
Β· exact (mul_le_mul_left hr).mpr a_h_left_right
Β· rintro β¨a_left, a_rightβ©
use x / r
refine β¨β¨(lt_div_iff' hr).mpr... | 11 | 59,874.141715 | 2 | 1.25 | 16 | 1,340 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 239 | 249 | theorem smul_Ioi : r β’ Ioi a = Ioi (r β’ a) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Ioi]
constructor
Β· rintro β¨a_w, a_h_left, rflβ©
exact (mul_lt_mul_left hr).mpr a_h_left
Β· rintro h
use x / r
constructor
Β· exact (lt_div_iff' hr).mpr h
Β· exact mul_div_cancelβ _ (ne_of_gt hr)
| 10 | 22,026.465795 | 2 | 1.25 | 16 | 1,340 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 252 | 262 | theorem smul_Iio : r β’ Iio a = Iio (r β’ a) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Iio]
constructor
Β· rintro β¨a_w, a_h_left, rflβ©
exact (mul_lt_mul_left hr).mpr a_h_left
Β· rintro h
use x / r
constructor
Β· exact (div_lt_iff' hr).mpr h
Β· exact mul_div_cancelβ _ (ne_of_gt hr)
| 10 | 22,026.465795 | 2 | 1.25 | 16 | 1,340 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 265 | 275 | theorem smul_Ici : r β’ Ici a = Ici (r β’ a) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Ioi]
constructor
Β· rintro β¨a_w, a_h_left, rflβ©
exact (mul_le_mul_left hr).mpr a_h_left
Β· rintro h
use x / r
constructor
Β· exact (le_div_iff' hr).mpr h
Β· exact mul_div_cancelβ _ (ne_of_gt hr)
| 10 | 22,026.465795 | 2 | 1.25 | 16 | 1,340 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 278 | 288 | theorem smul_Iic : r β’ Iic a = Iic (r β’ a) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Iio]
constructor
Β· rintro β¨a_w, a_h_left, rflβ©
exact (mul_le_mul_left hr).mpr a_h_left
Β· rintro h
use x / r
constructor
Β· exact (div_le_iff' hr).mpr h
Β· exact mul_div_cancelβ _ (ne_of_gt hr)
| 10 | 22,026.465795 | 2 | 1.25 | 16 | 1,340 |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Topology.Instances.ENNReal
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Metric TopologicalSpace Function Filter
open scoped Topology NNReal
variable {Ξ± Ξ² F : Type*} [N... | Mathlib/Analysis/NormedSpace/FunctionSeries.lean | 28 | 39 | theorem tendstoUniformlyOn_tsum {f : Ξ± β Ξ² β F} (hu : Summable u) {s : Set Ξ²}
(hfu : β n x, x β s β βf n xβ β€ u n) :
TendstoUniformlyOn (fun t : Finset Ξ± => fun x => β n β t, f n x) (fun x => β' n, f n x) atTop
s := by |
refine tendstoUniformlyOn_iff.2 fun Ξ΅ Ξ΅pos => ?_
filter_upwards [(tendsto_order.1 (tendsto_tsum_compl_atTop_zero u)).2 _ Ξ΅pos] with t ht x hx
have A : Summable fun n => βf n xβ :=
.of_nonneg_of_le (fun _ β¦ norm_nonneg _) (fun n => hfu n x hx) hu
rw [dist_eq_norm, β sum_add_tsum_subtype_compl A.of_norm t, a... | 8 | 2,980.957987 | 2 | 1.25 | 4 | 1,341 |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Topology.Instances.ENNReal
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Metric TopologicalSpace Function Filter
open scoped Topology NNReal
variable {Ξ± Ξ² F : Type*} [N... | Mathlib/Analysis/NormedSpace/FunctionSeries.lean | 53 | 56 | theorem tendstoUniformly_tsum {f : Ξ± β Ξ² β F} (hu : Summable u) (hfu : β n x, βf n xβ β€ u n) :
TendstoUniformly (fun t : Finset Ξ± => fun x => β n β t, f n x)
(fun x => β' n, f n x) atTop := by |
rw [β tendstoUniformlyOn_univ]; exact tendstoUniformlyOn_tsum hu fun n x _ => hfu n x
| 1 | 2.718282 | 0 | 1.25 | 4 | 1,341 |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Topology.Instances.ENNReal
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Metric TopologicalSpace Function Filter
open scoped Topology NNReal
variable {Ξ± Ξ² F : Type*} [N... | Mathlib/Analysis/NormedSpace/FunctionSeries.lean | 70 | 76 | theorem continuousOn_tsum [TopologicalSpace Ξ²] {f : Ξ± β Ξ² β F} {s : Set Ξ²}
(hf : β i, ContinuousOn (f i) s) (hu : Summable u) (hfu : β n x, x β s β βf n xβ β€ u n) :
ContinuousOn (fun x => β' n, f n x) s := by |
classical
refine (tendstoUniformlyOn_tsum hu hfu).continuousOn (eventually_of_forall ?_)
intro t
exact continuousOn_finset_sum _ fun i _ => hf i
| 4 | 54.59815 | 2 | 1.25 | 4 | 1,341 |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Topology.Instances.ENNReal
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Metric TopologicalSpace Function Filter
open scoped Topology NNReal
variable {Ξ± Ξ² F : Type*} [N... | Mathlib/Analysis/NormedSpace/FunctionSeries.lean | 81 | 84 | theorem continuous_tsum [TopologicalSpace Ξ²] {f : Ξ± β Ξ² β F} (hf : β i, Continuous (f i))
(hu : Summable u) (hfu : β n x, βf n xβ β€ u n) : Continuous fun x => β' n, f n x := by |
simp_rw [continuous_iff_continuousOn_univ] at hf β’
exact continuousOn_tsum hf hu fun n x _ => hfu n x
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,341 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 61 | 67 | theorem ExpChar.eq {p q : β} (hp : ExpChar R p) (hq : ExpChar R q) : p = q := by |
cases' hp with hp _ hp' hp
Β· cases' hq with hq _ hq' hq
exacts [rfl, False.elim (Nat.not_prime_zero (CharP.eq R hq (CharP.ofCharZero R) βΈ hq'))]
Β· cases' hq with hq _ hq' hq
exacts [False.elim (Nat.not_prime_zero (CharP.eq R hp (CharP.ofCharZero R) βΈ hp')),
CharP.eq R hp hq]
| 6 | 403.428793 | 2 | 1.272727 | 11 | 1,345 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 74 | 79 | theorem ringExpChar.eq (q : β) [h : ExpChar R q] : ringExpChar R = q := by |
cases' h with _ _ h _
Β· haveI := CharP.ofCharZero R
rw [ringExpChar, ringChar.eq R 0]; rfl
rw [ringExpChar, ringChar.eq R q]
exact Nat.max_eq_left h.one_lt.le
| 5 | 148.413159 | 2 | 1.272727 | 11 | 1,345 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 82 | 83 | theorem ringExpChar.eq_one (R : Type*) [NonAssocSemiring R] [CharZero R] : ringExpChar R = 1 := by |
rw [ringExpChar, ringChar.eq_zero, max_eq_right zero_le_one]
| 1 | 2.718282 | 0 | 1.272727 | 11 | 1,345 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 86 | 89 | theorem expChar_one_of_char_zero (q : β) [hp : CharP R 0] [hq : ExpChar R q] : q = 1 := by |
cases' hq with q hq_one hq_prime hq_hchar
Β· rfl
Β· exact False.elim <| hq_prime.ne_zero <| hq_hchar.eq R hp
| 3 | 20.085537 | 1 | 1.272727 | 11 | 1,345 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 93 | 97 | theorem char_eq_expChar_iff (p q : β) [hp : CharP R p] [hq : ExpChar R q] : p = q β p.Prime := by |
cases' hq with q hq_one hq_prime hq_hchar
Β· rw [(CharP.eq R hp inferInstance : p = 0)]
decide
Β· exact β¨fun hpq => hpq.symm βΈ hq_prime, fun _ => CharP.eq R hp hq_hcharβ©
| 4 | 54.59815 | 2 | 1.272727 | 11 | 1,345 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 105 | 108 | theorem char_zero_of_expChar_one (p : β) [hp : CharP R p] [hq : ExpChar R 1] : p = 0 := by |
cases hq
Β· exact CharP.eq R hp inferInstance
Β· exact False.elim (CharP.char_ne_one R 1 rfl)
| 3 | 20.085537 | 1 | 1.272727 | 11 | 1,345 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 113 | 116 | theorem charZero_of_expChar_one' [hq : ExpChar R 1] : CharZero R := by |
cases hq
Β· assumption
Β· exact False.elim (CharP.char_ne_one R 1 rfl)
| 3 | 20.085537 | 1 | 1.272727 | 11 | 1,345 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 120 | 125 | theorem expChar_one_iff_char_zero (p q : β) [CharP R p] [ExpChar R q] : q = 1 β p = 0 := by |
constructor
Β· rintro rfl
exact char_zero_of_expChar_one R p
Β· rintro rfl
exact expChar_one_of_char_zero R q
| 5 | 148.413159 | 2 | 1.272727 | 11 | 1,345 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 133 | 136 | theorem char_prime_of_ne_zero {p : β} [hp : CharP R p] (p_ne_zero : p β 0) : Nat.Prime p := by |
cases' CharP.char_is_prime_or_zero R p with h h
Β· exact h
Β· contradiction
| 3 | 20.085537 | 1 | 1.272727 | 11 | 1,345 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 141 | 144 | theorem expChar_is_prime_or_one (q : β) [hq : ExpChar R q] : Nat.Prime q β¨ q = 1 := by |
cases hq with
| zero => exact .inr rfl
| prime hp => exact .inl hp
| 3 | 20.085537 | 1 | 1.272727 | 11 | 1,345 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 148 | 150 | theorem expChar_pos (q : β) [ExpChar R q] : 0 < q := by |
rcases expChar_is_prime_or_one R q with h | rfl
exacts [Nat.Prime.pos h, Nat.one_pos]
| 2 | 7.389056 | 1 | 1.272727 | 11 | 1,345 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
import Mathlib.Tactic.FieldSimp
#align_import number_theory.bernoulli from "leanprover-community/mat... | Mathlib/NumberTheory/Bernoulli.lean | 78 | 80 | theorem bernoulli'_def (n : β) :
bernoulli' n = 1 - β k β range n, n.choose k / (n - k + 1) * bernoulli' k := by |
rw [bernoulli'_def', β Fin.sum_univ_eq_sum_range]
| 1 | 2.718282 | 0 | 1.272727 | 11 | 1,346 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
import Mathlib.Tactic.FieldSimp
#align_import number_theory.bernoulli from "leanprover-community/mat... | Mathlib/NumberTheory/Bernoulli.lean | 83 | 88 | theorem bernoulli'_spec (n : β) :
(β k β range n.succ, (n.choose (n - k) : β) / (n - k + 1) * bernoulli' k) = 1 := by |
rw [sum_range_succ_comm, bernoulli'_def n, tsub_self, choose_zero_right, sub_self, zero_add,
div_one, cast_one, one_mul, sub_add, β sum_sub_distrib, β sub_eq_zero, sub_sub_cancel_left,
neg_eq_zero]
exact Finset.sum_eq_zero (fun x hx => by rw [choose_symm (le_of_lt (mem_range.1 hx)), sub_self])
| 4 | 54.59815 | 2 | 1.272727 | 11 | 1,346 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
import Mathlib.Tactic.FieldSimp
#align_import number_theory.bernoulli from "leanprover-community/mat... | Mathlib/NumberTheory/Bernoulli.lean | 91 | 95 | theorem bernoulli'_spec' (n : β) :
(β k β antidiagonal n, ((k.1 + k.2).choose k.2 : β) / (k.2 + 1) * bernoulli' k.1) = 1 := by |
refine ((sum_antidiagonal_eq_sum_range_succ_mk _ n).trans ?_).trans (bernoulli'_spec n)
refine sum_congr rfl fun x hx => ?_
simp only [add_tsub_cancel_of_le, mem_range_succ_iff.mp hx, cast_sub]
| 3 | 20.085537 | 1 | 1.272727 | 11 | 1,346 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
import Mathlib.Tactic.FieldSimp
#align_import number_theory.bernoulli from "leanprover-community/mat... | Mathlib/NumberTheory/Bernoulli.lean | 104 | 106 | theorem bernoulli'_zero : bernoulli' 0 = 1 := by |
rw [bernoulli'_def]
norm_num
| 2 | 7.389056 | 1 | 1.272727 | 11 | 1,346 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
import Mathlib.Tactic.FieldSimp
#align_import number_theory.bernoulli from "leanprover-community/mat... | Mathlib/NumberTheory/Bernoulli.lean | 110 | 112 | theorem bernoulli'_one : bernoulli' 1 = 1 / 2 := by |
rw [bernoulli'_def]
norm_num
| 2 | 7.389056 | 1 | 1.272727 | 11 | 1,346 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
import Mathlib.Tactic.FieldSimp
#align_import number_theory.bernoulli from "leanprover-community/mat... | Mathlib/NumberTheory/Bernoulli.lean | 116 | 118 | theorem bernoulli'_two : bernoulli' 2 = 1 / 6 := by |
rw [bernoulli'_def]
norm_num [sum_range_succ, sum_range_succ, sum_range_zero]
| 2 | 7.389056 | 1 | 1.272727 | 11 | 1,346 |
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