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.Quaternion
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Series
#align_import analysis.normed_space.quaternion_exponential from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Quaternion Nat
open... | Mathlib/Analysis/NormedSpace/QuaternionExponential.lean | 39 | 55 | theorem expSeries_even_of_imaginary {q : Quaternion β} (hq : q.re = 0) (n : β) :
expSeries β (Quaternion β) (2 * n) (fun _ => q) =
β((-1 : β) ^ n * βqβ ^ (2 * n) / (2 * n)!) := by |
rw [expSeries_apply_eq]
have hq2 : q ^ 2 = -normSq q := sq_eq_neg_normSq.mpr hq
letI k : β := β(2 * n)!
calc
kβ»ΒΉ β’ q ^ (2 * n) = kβ»ΒΉ β’ (-normSq q) ^ n := by rw [pow_mul, hq2]
_ = kβ»ΒΉ β’ β((-1 : β) ^ n * βqβ ^ (2 * n)) := ?_
_ = β((-1 : β) ^ n * βqβ ^ (2 * n) / k) := ?_
Β· congr 1
rw [neg_pow, n... | 14 | 1,202,604.284165 | 2 | 1.375 | 8 | 1,470 |
import Mathlib.Analysis.Quaternion
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Series
#align_import analysis.normed_space.quaternion_exponential from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Quaternion Nat
open... | Mathlib/Analysis/NormedSpace/QuaternionExponential.lean | 59 | 78 | theorem expSeries_odd_of_imaginary {q : Quaternion β} (hq : q.re = 0) (n : β) :
expSeries β (Quaternion β) (2 * n + 1) (fun _ => q) =
(((-1 : β) ^ n * βqβ ^ (2 * n + 1) / (2 * n + 1)!) / βqβ) β’ q := by |
rw [expSeries_apply_eq]
obtain rfl | hq0 := eq_or_ne q 0
Β· simp
have hq2 : q ^ 2 = -normSq q := sq_eq_neg_normSq.mpr hq
have hqn := norm_ne_zero_iff.mpr hq0
let k : β := β(2 * n + 1)!
calc
kβ»ΒΉ β’ q ^ (2 * n + 1) = kβ»ΒΉ β’ ((-normSq q) ^ n * q) := by rw [pow_succ, pow_mul, hq2]
_ = kβ»ΒΉ β’ ((-1 : β) ^ ... | 17 | 24,154,952.753575 | 2 | 1.375 | 8 | 1,470 |
import Mathlib.Analysis.Quaternion
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Series
#align_import analysis.normed_space.quaternion_exponential from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Quaternion Nat
open... | Mathlib/Analysis/NormedSpace/QuaternionExponential.lean | 82 | 94 | theorem hasSum_expSeries_of_imaginary {q : Quaternion β} (hq : q.re = 0) {c s : β}
(hc : HasSum (fun n => (-1 : β) ^ n * βqβ ^ (2 * n) / (2 * n)!) c)
(hs : HasSum (fun n => (-1 : β) ^ n * βqβ ^ (2 * n + 1) / (2 * n + 1)!) s) :
HasSum (fun n => expSeries β (Quaternion β) n fun _ => q) (βc + (s / βqβ) β’ q) :=... |
replace hc := hasSum_coe.mpr hc
replace hs := (hs.div_const βqβ).smul_const q
refine HasSum.even_add_odd ?_ ?_
Β· convert hc using 1
ext n : 1
rw [expSeries_even_of_imaginary hq]
Β· convert hs using 1
ext n : 1
rw [expSeries_odd_of_imaginary hq]
| 9 | 8,103.083928 | 2 | 1.375 | 8 | 1,470 |
import Mathlib.Analysis.Quaternion
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Series
#align_import analysis.normed_space.quaternion_exponential from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Quaternion Nat
open... | Mathlib/Analysis/NormedSpace/QuaternionExponential.lean | 98 | 103 | theorem exp_of_re_eq_zero (q : Quaternion β) (hq : q.re = 0) :
exp β q = β(Real.cos βqβ) + (Real.sin βqβ / βqβ) β’ q := by |
rw [exp_eq_tsum]
refine HasSum.tsum_eq ?_
simp_rw [β expSeries_apply_eq]
exact hasSum_expSeries_of_imaginary hq (Real.hasSum_cos _) (Real.hasSum_sin _)
| 4 | 54.59815 | 2 | 1.375 | 8 | 1,470 |
import Mathlib.Analysis.Quaternion
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Series
#align_import analysis.normed_space.quaternion_exponential from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Quaternion Nat
open... | Mathlib/Analysis/NormedSpace/QuaternionExponential.lean | 107 | 111 | theorem exp_eq (q : Quaternion β) :
exp β q = exp β q.re β’ (β(Real.cos βq.imβ) + (Real.sin βq.imβ / βq.imβ) β’ q.im) := by |
rw [β exp_of_re_eq_zero q.im q.im_re, β coe_mul_eq_smul, β exp_coe, β exp_add_of_commute,
re_add_im]
exact Algebra.commutes q.re (_ : β[β])
| 3 | 20.085537 | 1 | 1.375 | 8 | 1,470 |
import Mathlib.Analysis.Quaternion
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Series
#align_import analysis.normed_space.quaternion_exponential from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Quaternion Nat
open... | Mathlib/Analysis/NormedSpace/QuaternionExponential.lean | 114 | 114 | theorem re_exp (q : β[β]) : (exp β q).re = exp β q.re * Real.cos βq - q.reβ := by | simp [exp_eq]
| 1 | 2.718282 | 0 | 1.375 | 8 | 1,470 |
import Mathlib.Analysis.Quaternion
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Series
#align_import analysis.normed_space.quaternion_exponential from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Quaternion Nat
open... | Mathlib/Analysis/NormedSpace/QuaternionExponential.lean | 117 | 118 | theorem im_exp (q : β[β]) : (exp β q).im = (exp β q.re * (Real.sin βq.imβ / βq.imβ)) β’ q.im := by |
simp [exp_eq, smul_smul]
| 1 | 2.718282 | 0 | 1.375 | 8 | 1,470 |
import Mathlib.Analysis.Quaternion
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Series
#align_import analysis.normed_space.quaternion_exponential from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Quaternion Nat
open... | Mathlib/Analysis/NormedSpace/QuaternionExponential.lean | 121 | 135 | theorem normSq_exp (q : β[β]) : normSq (exp β q) = exp β q.re ^ 2 :=
calc
normSq (exp β q) =
normSq (exp β q.re β’ (β(Real.cos βq.imβ) + (Real.sin βq.imβ / βq.imβ) β’ q.im)) := by |
rw [exp_eq]
_ = exp β q.re ^ 2 * normSq (β(Real.cos βq.imβ) + (Real.sin βq.imβ / βq.imβ) β’ q.im) := by
rw [normSq_smul]
_ = exp β q.re ^ 2 * (Real.cos βq.imβ ^ 2 + Real.sin βq.imβ ^ 2) := by
congr 1
obtain hv | hv := eq_or_ne βq.imβ 0
Β· simp [hv]
rw [normSq_add, normSq_smul,... | 11 | 59,874.141715 | 2 | 1.375 | 8 | 1,470 |
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.Nat.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Util.AssertExists
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
namespace List
universe u v
variable {Ξ± : Type u} {Ξ² : Type v} (l :... | Mathlib/Data/List/GetD.lean | 38 | 44 | theorem getD_eq_get {n : β} (hn : n < l.length) : l.getD n d = l.get β¨n, hnβ© := by |
induction l generalizing n with
| nil => simp at hn
| cons head tail ih =>
cases n
Β· exact getD_cons_zero
Β· exact ih _
| 6 | 403.428793 | 2 | 1.375 | 8 | 1,471 |
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.Nat.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Util.AssertExists
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
namespace List
universe u v
variable {Ξ± : Type u} {Ξ² : Type v} (l :... | Mathlib/Data/List/GetD.lean | 47 | 53 | theorem getD_map {n : β} (f : Ξ± β Ξ²) : (map f l).getD n (f d) = f (l.getD n d) := by |
induction l generalizing n with
| nil => rfl
| cons head tail ih =>
cases n
Β· rfl
Β· simp [ih]
| 6 | 403.428793 | 2 | 1.375 | 8 | 1,471 |
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.Nat.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Util.AssertExists
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
namespace List
universe u v
variable {Ξ± : Type u} {Ξ² : Type v} (l :... | Mathlib/Data/List/GetD.lean | 57 | 63 | theorem getD_eq_default {n : β} (hn : l.length β€ n) : l.getD n d = d := by |
induction l generalizing n with
| nil => exact getD_nil
| cons head tail ih =>
cases n
Β· simp at hn
Β· exact ih (Nat.le_of_succ_le_succ hn)
| 6 | 403.428793 | 2 | 1.375 | 8 | 1,471 |
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.Nat.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Util.AssertExists
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
namespace List
universe u v
variable {Ξ± : Type u} {Ξ² : Type v} (l :... | Mathlib/Data/List/GetD.lean | 73 | 73 | theorem getD_singleton_default_eq (n : β) : [d].getD n d = d := by | cases n <;> simp
| 1 | 2.718282 | 0 | 1.375 | 8 | 1,471 |
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.Nat.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Util.AssertExists
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
namespace List
universe u v
variable {Ξ± : Type u} {Ξ² : Type v} (l :... | Mathlib/Data/List/GetD.lean | 77 | 80 | theorem getD_replicate_default_eq (r n : β) : (replicate r d).getD n d = d := by |
induction r generalizing n with
| zero => simp
| succ n ih => cases n <;> simp [ih]
| 3 | 20.085537 | 1 | 1.375 | 8 | 1,471 |
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.Nat.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Util.AssertExists
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
namespace List
universe u v
variable {Ξ± : Type u} {Ξ² : Type v} (l :... | Mathlib/Data/List/GetD.lean | 83 | 86 | theorem getD_append (l l' : List Ξ±) (d : Ξ±) (n : β) (h : n < l.length) :
(l ++ l').getD n d = l.getD n d := by |
rw [getD_eq_get _ _ (Nat.lt_of_lt_of_le h (length_append _ _ βΈ Nat.le_add_right _ _)),
get_append _ h, getD_eq_get]
| 2 | 7.389056 | 1 | 1.375 | 8 | 1,471 |
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.Nat.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Util.AssertExists
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
namespace List
universe u v
variable {Ξ± : Type u} {Ξ² : Type v} (l :... | Mathlib/Data/List/GetD.lean | 89 | 99 | theorem getD_append_right (l l' : List Ξ±) (d : Ξ±) (n : β) (h : l.length β€ n) :
(l ++ l').getD n d = l'.getD (n - l.length) d := by |
cases Nat.lt_or_ge n (l ++ l').length with
| inl h' =>
rw [getD_eq_get (l ++ l') d h', get_append_right, getD_eq_get]
Β· rw [length_append] at h'
exact Nat.sub_lt_left_of_lt_add h h'
Β· exact Nat.not_lt_of_le h
| inr h' =>
rw [getD_eq_default _ _ h', getD_eq_default]
rwa [Nat.le_sub_iff_a... | 9 | 8,103.083928 | 2 | 1.375 | 8 | 1,471 |
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.Nat.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Util.AssertExists
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
namespace List
universe u v
variable {Ξ± : Type u} {Ξ² : Type v} (l :... | Mathlib/Data/List/GetD.lean | 102 | 105 | theorem getD_eq_getD_get? (n : β) : l.getD n d = (l.get? n).getD d := by |
cases Nat.lt_or_ge n l.length with
| inl h => rw [getD_eq_get _ _ h, get?_eq_get h, Option.getD_some]
| inr h => rw [getD_eq_default _ _ h, get?_eq_none.mpr h, Option.getD_none]
| 3 | 20.085537 | 1 | 1.375 | 8 | 1,471 |
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
#align_import combinatorics.quiver.symmetric from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
universe v u w v'
namespace Quiver
-- Porting note: no hasNonemptyInstance linter yet
def Symmetrify (V : ... | Mathlib/Combinatorics/Quiver/Symmetric.lean | 61 | 62 | theorem reverse_reverse [h : HasInvolutiveReverse V] {a b : V} (f : a βΆ b) :
reverse (reverse f) = f := by | apply h.inv'
| 1 | 2.718282 | 0 | 1.375 | 8 | 1,472 |
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
#align_import combinatorics.quiver.symmetric from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
universe v u w v'
namespace Quiver
-- Porting note: no hasNonemptyInstance linter yet
def Symmetrify (V : ... | Mathlib/Combinatorics/Quiver/Symmetric.lean | 66 | 72 | theorem reverse_inj [h : HasInvolutiveReverse V] {a b : V}
(f g : a βΆ b) : reverse f = reverse g β f = g := by |
constructor
Β· rintro h
simpa using congr_arg Quiver.reverse h
Β· rintro h
congr
| 5 | 148.413159 | 2 | 1.375 | 8 | 1,472 |
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
#align_import combinatorics.quiver.symmetric from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
universe v u w v'
namespace Quiver
-- Porting note: no hasNonemptyInstance linter yet
def Symmetrify (V : ... | Mathlib/Combinatorics/Quiver/Symmetric.lean | 75 | 77 | theorem eq_reverse_iff [h : HasInvolutiveReverse V] {a b : V} (f : a βΆ b)
(g : b βΆ a) : f = reverse g β reverse f = g := by |
rw [β reverse_inj, reverse_reverse]
| 1 | 2.718282 | 0 | 1.375 | 8 | 1,472 |
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
#align_import combinatorics.quiver.symmetric from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
universe v u w v'
namespace Quiver
-- Porting note: no hasNonemptyInstance linter yet
def Symmetrify (V : ... | Mathlib/Combinatorics/Quiver/Symmetric.lean | 150 | 154 | theorem Path.reverse_comp [HasReverse V] {a b c : V} (p : Path a b) (q : Path b c) :
(p.comp q).reverse = q.reverse.comp p.reverse := by |
induction' q with _ _ _ _ h
Β· simp
Β· simp [h]
| 3 | 20.085537 | 1 | 1.375 | 8 | 1,472 |
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
#align_import combinatorics.quiver.symmetric from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
universe v u w v'
namespace Quiver
-- Porting note: no hasNonemptyInstance linter yet
def Symmetrify (V : ... | Mathlib/Combinatorics/Quiver/Symmetric.lean | 158 | 163 | theorem Path.reverse_reverse [h : HasInvolutiveReverse V] {a b : V} (p : Path a b) :
p.reverse.reverse = p := by |
induction' p with _ _ _ _ h
Β· simp
Β· rw [Path.reverse, Path.reverse_comp, h, Path.reverse_toPath, Quiver.reverse_reverse]
rfl
| 4 | 54.59815 | 2 | 1.375 | 8 | 1,472 |
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
#align_import combinatorics.quiver.symmetric from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
universe v u w v'
namespace Quiver
-- Porting note: no hasNonemptyInstance linter yet
def Symmetrify (V : ... | Mathlib/Combinatorics/Quiver/Symmetric.lean | 188 | 194 | theorem lift_spec [HasReverse V'] (Ο : Prefunctor V V') :
Symmetrify.of.comp (Symmetrify.lift Ο) = Ο := by |
fapply Prefunctor.ext
Β· rintro X
rfl
Β· rintro X Y f
rfl
| 5 | 148.413159 | 2 | 1.375 | 8 | 1,472 |
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
#align_import combinatorics.quiver.symmetric from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
universe v u w v'
namespace Quiver
-- Porting note: no hasNonemptyInstance linter yet
def Symmetrify (V : ... | Mathlib/Combinatorics/Quiver/Symmetric.lean | 197 | 204 | theorem lift_reverse [h : HasInvolutiveReverse V']
(Ο : Prefunctor V V') {X Y : Symmetrify V} (f : X βΆ Y) :
(Symmetrify.lift Ο).map (Quiver.reverse f) = Quiver.reverse ((Symmetrify.lift Ο).map f) := by |
dsimp [Symmetrify.lift]; cases f
Β· simp only
rfl
Β· simp only [reverse_reverse]
rfl
| 5 | 148.413159 | 2 | 1.375 | 8 | 1,472 |
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
#align_import combinatorics.quiver.symmetric from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
universe v u w v'
namespace Quiver
-- Porting note: no hasNonemptyInstance linter yet
def Symmetrify (V : ... | Mathlib/Combinatorics/Quiver/Symmetric.lean | 208 | 219 | theorem lift_unique [HasReverse V'] (Ο : V β₯€q V') (Ξ¦ : Symmetrify V β₯€q V') (hΞ¦ : (of βq Ξ¦) = Ο)
(hΞ¦inv : β {X Y : Symmetrify V} (f : X βΆ Y),
Ξ¦.map (Quiver.reverse f) = Quiver.reverse (Ξ¦.map f)) :
Ξ¦ = Symmetrify.lift Ο := by |
subst_vars
fapply Prefunctor.ext
Β· rintro X
rfl
Β· rintro X Y f
cases f
Β· rfl
· exact hΦinv (Sum.inl _)
| 8 | 2,980.957987 | 2 | 1.375 | 8 | 1,472 |
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.Factors
import Mathlib.GroupTheory.Perm.Closure
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Tactic.NormNum.GCD
#align_import group_theory.perm.cycl... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 57 | 64 | theorem cycleType_eq' {Ο : Perm Ξ±} (s : Finset (Perm Ξ±)) (h1 : β f : Perm Ξ±, f β s β f.IsCycle)
(h2 : (s : Set (Perm Ξ±)).Pairwise Disjoint)
(h0 : s.noncommProd id (h2.imp fun _ _ => Disjoint.commute) = Ο) :
Ο.cycleType = s.1.map (Finset.card β support) := by |
rw [cycleType_def]
congr
rw [cycleFactorsFinset_eq_finset]
exact β¨h1, h2, h0β©
| 4 | 54.59815 | 2 | 1.375 | 8 | 1,473 |
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.Factors
import Mathlib.GroupTheory.Perm.Closure
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Tactic.NormNum.GCD
#align_import group_theory.perm.cycl... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 67 | 75 | theorem cycleType_eq {Ο : Perm Ξ±} (l : List (Perm Ξ±)) (h0 : l.prod = Ο)
(h1 : β Ο : Perm Ξ±, Ο β l β Ο.IsCycle) (h2 : l.Pairwise Disjoint) :
Ο.cycleType = l.map (Finset.card β support) := by |
have hl : l.Nodup := nodup_of_pairwise_disjoint_cycles h1 h2
rw [cycleType_eq' l.toFinset]
Β· simp [List.dedup_eq_self.mpr hl, (Β· β Β·)]
Β· simpa using h1
Β· simpa [hl] using h2
Β· simp [hl, h0]
| 6 | 403.428793 | 2 | 1.375 | 8 | 1,473 |
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.Factors
import Mathlib.GroupTheory.Perm.Closure
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Tactic.NormNum.GCD
#align_import group_theory.perm.cycl... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 79 | 80 | theorem cycleType_eq_zero {Ο : Perm Ξ±} : Ο.cycleType = 0 β Ο = 1 := by |
simp [cycleType_def, cycleFactorsFinset_eq_empty_iff]
| 1 | 2.718282 | 0 | 1.375 | 8 | 1,473 |
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.Factors
import Mathlib.GroupTheory.Perm.Closure
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Tactic.NormNum.GCD
#align_import group_theory.perm.cycl... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 87 | 88 | theorem card_cycleType_eq_zero {Ο : Perm Ξ±} : Multiset.card Ο.cycleType = 0 β Ο = 1 := by |
rw [card_eq_zero, cycleType_eq_zero]
| 1 | 2.718282 | 0 | 1.375 | 8 | 1,473 |
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.Factors
import Mathlib.GroupTheory.Perm.Closure
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Tactic.NormNum.GCD
#align_import group_theory.perm.cycl... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 94 | 98 | theorem two_le_of_mem_cycleType {Ο : Perm Ξ±} {n : β} (h : n β Ο.cycleType) : 2 β€ n := by |
simp only [cycleType_def, β Finset.mem_def, Function.comp_apply, Multiset.mem_map,
mem_cycleFactorsFinset_iff] at h
obtain β¨_, β¨hc, -β©, rflβ© := h
exact hc.two_le_card_support
| 4 | 54.59815 | 2 | 1.375 | 8 | 1,473 |
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.Factors
import Mathlib.GroupTheory.Perm.Closure
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Tactic.NormNum.GCD
#align_import group_theory.perm.cycl... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 110 | 119 | theorem card_cycleType_eq_one {Ο : Perm Ξ±} : Multiset.card Ο.cycleType = 1 β Ο.IsCycle := by |
rw [card_eq_one]
simp_rw [cycleType_def, Multiset.map_eq_singleton, β Finset.singleton_val, Finset.val_inj,
cycleFactorsFinset_eq_singleton_iff]
constructor
Β· rintro β¨_, _, β¨h, -β©, -β©
exact h
Β· intro h
use Ο.support.card, Ο
simp [h]
| 9 | 8,103.083928 | 2 | 1.375 | 8 | 1,473 |
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.Factors
import Mathlib.GroupTheory.Perm.Closure
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Tactic.NormNum.GCD
#align_import group_theory.perm.cycl... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 122 | 126 | theorem Disjoint.cycleType {Ο Ο : Perm Ξ±} (h : Disjoint Ο Ο) :
(Ο * Ο).cycleType = Ο.cycleType + Ο.cycleType := by |
rw [cycleType_def, cycleType_def, cycleType_def, h.cycleFactorsFinset_mul_eq_union, β
Multiset.map_add, Finset.union_val, Multiset.add_eq_union_iff_disjoint.mpr _]
exact Finset.disjoint_val.2 h.disjoint_cycleFactorsFinset
| 3 | 20.085537 | 1 | 1.375 | 8 | 1,473 |
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.Factors
import Mathlib.GroupTheory.Perm.Closure
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Tactic.NormNum.GCD
#align_import group_theory.perm.cycl... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 139 | 144 | theorem cycleType_conj {Ο Ο : Perm Ξ±} : (Ο * Ο * Οβ»ΒΉ).cycleType = Ο.cycleType := by |
induction Ο using cycle_induction_on with
| base_one => simp
| base_cycles Ο hΟ => rw [hΟ.cycleType, hΟ.conj.cycleType, card_support_conj]
| induction_disjoint Ο Ο hd _ hΟ hΟ =>
rw [β conj_mul, hd.cycleType, (hd.conj _).cycleType, hΟ, hΟ]
| 5 | 148.413159 | 2 | 1.375 | 8 | 1,473 |
import Mathlib.Algebra.Polynomial.Degree.Lemmas
open Polynomial
namespace Mathlib.Tactic.ComputeDegree
section recursion_lemmas
variable {R : Type*}
section semiring
variable [Semiring R]
theorem natDegree_C_le (a : R) : natDegree (C a) β€ 0 := (natDegree_C a).le
theorem natDegree_natCast_le (n : β) : natDeg... | Mathlib/Tactic/ComputeDegree.lean | 101 | 103 | theorem coeff_add_of_eq {n : β} {a b : R} {f g : R[X]}
(h_add_left : f.coeff n = a) (h_add_right : g.coeff n = b) :
(f + g).coeff n = a + b := by | subst βΉ_βΊ βΉ_βΊ; apply coeff_add
| 1 | 2.718282 | 0 | 1.4 | 5 | 1,474 |
import Mathlib.Algebra.Polynomial.Degree.Lemmas
open Polynomial
namespace Mathlib.Tactic.ComputeDegree
section recursion_lemmas
variable {R : Type*}
section semiring
variable [Semiring R]
theorem natDegree_C_le (a : R) : natDegree (C a) β€ 0 := (natDegree_C a).le
theorem natDegree_natCast_le (n : β) : natDeg... | Mathlib/Tactic/ComputeDegree.lean | 105 | 115 | theorem coeff_mul_add_of_le_natDegree_of_eq_ite {d df dg : β} {a b : R} {f g : R[X]}
(h_mul_left : natDegree f β€ df) (h_mul_right : natDegree g β€ dg)
(h_mul_left : f.coeff df = a) (h_mul_right : g.coeff dg = b) (ddf : df + dg β€ d) :
(f * g).coeff d = if d = df + dg then a * b else 0 := by |
split_ifs with h
Β· subst h_mul_left h_mul_right h
exact coeff_mul_of_natDegree_le βΉ_βΊ βΉ_βΊ
Β· apply coeff_eq_zero_of_natDegree_lt
apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ddf ?_)
Β· exact natDegree_mul_le_of_le βΉ_βΊ βΉ_βΊ
Β· exact ne_comm.mp h
| 7 | 1,096.633158 | 2 | 1.4 | 5 | 1,474 |
import Mathlib.Algebra.Polynomial.Degree.Lemmas
open Polynomial
namespace Mathlib.Tactic.ComputeDegree
section recursion_lemmas
variable {R : Type*}
section semiring
variable [Semiring R]
theorem natDegree_C_le (a : R) : natDegree (C a) β€ 0 := (natDegree_C a).le
theorem natDegree_natCast_le (n : β) : natDeg... | Mathlib/Tactic/ComputeDegree.lean | 117 | 126 | theorem coeff_pow_of_natDegree_le_of_eq_ite' {m n o : β} {a : R} {p : R[X]}
(h_pow : natDegree p β€ n) (h_exp : m * n β€ o) (h_pow_bas : coeff p n = a) :
coeff (p ^ m) o = if o = m * n then a ^ m else 0 := by |
split_ifs with h
Β· subst h h_pow_bas
exact coeff_pow_of_natDegree_le βΉ_βΊ
Β· apply coeff_eq_zero_of_natDegree_lt
apply lt_of_le_of_lt ?_ (lt_of_le_of_ne βΉ_βΊ ?_)
Β· exact natDegree_pow_le_of_le m βΉ_βΊ
Β· exact Iff.mp ne_comm h
| 7 | 1,096.633158 | 2 | 1.4 | 5 | 1,474 |
import Mathlib.Algebra.Polynomial.Degree.Lemmas
open Polynomial
namespace Mathlib.Tactic.ComputeDegree
section recursion_lemmas
variable {R : Type*}
section semiring
variable [Semiring R]
theorem natDegree_C_le (a : R) : natDegree (C a) β€ 0 := (natDegree_C a).le
theorem natDegree_natCast_le (n : β) : natDeg... | Mathlib/Tactic/ComputeDegree.lean | 150 | 155 | theorem natDegree_eq_of_le_of_coeff_ne_zero' {deg m o : β} {c : R} {p : R[X]}
(h_natDeg_le : natDegree p β€ m) (coeff_eq : coeff p o = c)
(coeff_ne_zero : c β 0) (deg_eq_deg : m = deg) (coeff_eq_deg : o = deg) :
natDegree p = deg := by |
subst coeff_eq deg_eq_deg coeff_eq_deg
exact natDegree_eq_of_le_of_coeff_ne_zero βΉ_βΊ βΉ_βΊ
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,474 |
import Mathlib.Algebra.Polynomial.Degree.Lemmas
open Polynomial
namespace Mathlib.Tactic.ComputeDegree
section recursion_lemmas
variable {R : Type*}
section semiring
variable [Semiring R]
theorem natDegree_C_le (a : R) : natDegree (C a) β€ 0 := (natDegree_C a).le
theorem natDegree_natCast_le (n : β) : natDeg... | Mathlib/Tactic/ComputeDegree.lean | 157 | 165 | theorem degree_eq_of_le_of_coeff_ne_zero' {deg m o : WithBot β} {c : R} {p : R[X]}
(h_deg_le : degree p β€ m) (coeff_eq : coeff p (WithBot.unbot' 0 deg) = c)
(coeff_ne_zero : c β 0) (deg_eq_deg : m = deg) (coeff_eq_deg : o = deg) :
degree p = deg := by |
subst coeff_eq coeff_eq_deg deg_eq_deg
rcases eq_or_ne m β₯ with rfl|hh
Β· exact bot_unique h_deg_le
Β· obtain β¨m, rflβ© := WithBot.ne_bot_iff_exists.mp hh
exact degree_eq_of_le_of_coeff_ne_zero βΉ_βΊ βΉ_βΊ
| 5 | 148.413159 | 2 | 1.4 | 5 | 1,474 |
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
#align_import algebraic_geometry.Gamma_Spec_adjunction from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were u... | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 77 | 79 | theorem not_mem_prime_iff_unit_in_stalk (r : Ξ.obj (op X)) (x : X) :
r β (X.toΞSpecFun x).asIdeal β IsUnit (X.ΞToStalk x r) := by |
erw [LocalRing.mem_maximalIdeal, Classical.not_not]
| 1 | 2.718282 | 0 | 1.4 | 5 | 1,475 |
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
#align_import algebraic_geometry.Gamma_Spec_adjunction from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were u... | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 84 | 87 | theorem toΞSpec_preim_basicOpen_eq (r : Ξ.obj (op X)) :
X.toΞSpecFun β»ΒΉ' (basicOpen r).1 = (X.toRingedSpace.basicOpen r).1 := by |
ext
erw [X.toRingedSpace.mem_top_basicOpen]; apply not_mem_prime_iff_unit_in_stalk
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,475 |
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
#align_import algebraic_geometry.Gamma_Spec_adjunction from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were u... | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 91 | 95 | theorem toΞSpec_continuous : Continuous X.toΞSpecFun := by |
rw [isTopologicalBasis_basic_opens.continuous_iff]
rintro _ β¨r, rflβ©
erw [X.toΞSpec_preim_basicOpen_eq r]
exact (X.toRingedSpace.basicOpen r).2
| 4 | 54.59815 | 2 | 1.4 | 5 | 1,475 |
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
#align_import algebraic_geometry.Gamma_Spec_adjunction from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were u... | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 128 | 134 | theorem isUnit_res_toΞSpecMapBasicOpen : IsUnit (X.toToΞSpecMapBasicOpen r r) := by |
convert
(X.presheaf.map <| (eqToHom <| X.toΞSpecMapBasicOpen_eq r).op).isUnit_map
(X.toRingedSpace.isUnit_res_basicOpen r)
-- Porting note: `rw [comp_apply]` to `erw [comp_apply]`
erw [β comp_apply, β Functor.map_comp]
congr
| 6 | 403.428793 | 2 | 1.4 | 5 | 1,475 |
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
#align_import algebraic_geometry.Gamma_Spec_adjunction from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were u... | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 146 | 160 | theorem toΞSpecCApp_iff
(f :
(structureSheaf <| Ξ.obj <| op X).val.obj (op <| basicOpen r) βΆ
X.presheaf.obj (op <| X.toΞSpecMapBasicOpen r)) :
toOpen _ (basicOpen r) β« f = X.toToΞSpecMapBasicOpen r β f = X.toΞSpecCApp r := by |
-- Porting Note: Type class problem got stuck in `IsLocalization.Away.AwayMap.lift_comp`
-- created instance manually. This replaces the `pick_goal` tactics
have loc_inst := IsLocalization.to_basicOpen (Ξ.obj (op X)) r
rw [β @IsLocalization.Away.AwayMap.lift_comp _ _ _ _ _ _ _ r loc_inst _
(X.isUnit_res_... | 10 | 22,026.465795 | 2 | 1.4 | 5 | 1,475 |
import Batteries.Data.HashMap.Basic
import Batteries.Data.Array.Lemmas
import Batteries.Data.Nat.Lemmas
namespace Batteries.HashMap
namespace Imp
attribute [-simp] Bool.not_eq_true
namespace Buckets
@[ext] protected theorem ext : β {bβ bβ : Buckets Ξ± Ξ²}, bβ.1.data = bβ.1.data β bβ = bβ
| β¨β¨_β©, _β©, β¨β¨_β©, _β©, rfl ... | .lake/packages/batteries/Batteries/Data/HashMap/WF.lean | 23 | 27 | theorem exists_of_update (self : Buckets Ξ± Ξ²) (i d h) :
β lβ lβ, self.1.data = lβ ++ self.1[i] :: lβ β§ List.length lβ = i.toNat β§
(self.update i d h).1.data = lβ ++ d :: lβ := by |
simp only [Array.data_length, Array.ugetElem_eq_getElem, Array.getElem_eq_data_get]
exact List.exists_of_set' h
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,476 |
import Batteries.Data.HashMap.Basic
import Batteries.Data.Array.Lemmas
import Batteries.Data.Nat.Lemmas
namespace Batteries.HashMap
namespace Imp
attribute [-simp] Bool.not_eq_true
namespace Buckets
@[ext] protected theorem ext : β {bβ bβ : Buckets Ξ± Ξ²}, bβ.1.data = bβ.1.data β bβ = bβ
| β¨β¨_β©, _β©, β¨β¨_β©, _β©, rfl ... | .lake/packages/batteries/Batteries/Data/HashMap/WF.lean | 29 | 33 | theorem update_update (self : Buckets Ξ± Ξ²) (i d d' h h') :
(self.update i d h).update i d' h' = self.update i d' h := by |
simp only [update, Array.uset, Array.data_length]
congr 1
rw [Array.set_set]
| 3 | 20.085537 | 1 | 1.4 | 5 | 1,476 |
import Batteries.Data.HashMap.Basic
import Batteries.Data.Array.Lemmas
import Batteries.Data.Nat.Lemmas
namespace Batteries.HashMap
namespace Imp
attribute [-simp] Bool.not_eq_true
namespace Buckets
@[ext] protected theorem ext : β {bβ bβ : Buckets Ξ± Ξ²}, bβ.1.data = bβ.1.data β bβ = bβ
| β¨β¨_β©, _β©, β¨β¨_β©, _β©, rfl ... | .lake/packages/batteries/Batteries/Data/HashMap/WF.lean | 38 | 40 | theorem mk_size (h) : (mk n h : Buckets Ξ± Ξ²).size = 0 := by |
simp only [mk, mkArray, size_eq]; clear h
induction n <;> simp [*]
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,476 |
import Batteries.Data.HashMap.Basic
import Batteries.Data.Array.Lemmas
import Batteries.Data.Nat.Lemmas
namespace Batteries.HashMap
namespace Imp
attribute [-simp] Bool.not_eq_true
namespace Buckets
@[ext] protected theorem ext : β {bβ bβ : Buckets Ξ± Ξ²}, bβ.1.data = bβ.1.data β bβ = bβ
| β¨β¨_β©, _β©, β¨β¨_β©, _β©, rfl ... | .lake/packages/batteries/Batteries/Data/HashMap/WF.lean | 42 | 46 | theorem WF.mk' [BEq Ξ±] [Hashable Ξ±] (h) : (Buckets.mk n h : Buckets Ξ± Ξ²).WF := by |
refine β¨fun _ h => ?_, fun i h => ?_β©
Β· simp only [Buckets.mk, mkArray, List.mem_replicate, ne_eq] at h
simp [h, List.Pairwise.nil]
Β· simp [Buckets.mk, empty', mkArray, Array.getElem_eq_data_get, AssocList.All]
| 4 | 54.59815 | 2 | 1.4 | 5 | 1,476 |
import Batteries.Data.HashMap.Basic
import Batteries.Data.Array.Lemmas
import Batteries.Data.Nat.Lemmas
namespace Batteries.HashMap
namespace Imp
attribute [-simp] Bool.not_eq_true
namespace Buckets
@[ext] protected theorem ext : β {bβ bβ : Buckets Ξ± Ξ²}, bβ.1.data = bβ.1.data β bβ = bβ
| β¨β¨_β©, _β©, β¨β¨_β©, _β©, rfl ... | .lake/packages/batteries/Batteries/Data/HashMap/WF.lean | 48 | 64 | theorem WF.update [BEq Ξ±] [Hashable Ξ±] {buckets : Buckets Ξ± Ξ²} {i d h} (H : buckets.WF)
(hβ : β [PartialEquivBEq Ξ±] [LawfulHashable Ξ±],
(buckets.1[i].toList.Pairwise fun a b => Β¬(a.1 == b.1)) β
d.toList.Pairwise fun a b => Β¬(a.1 == b.1))
(hβ : (buckets.1[i].All fun k _ => ((hash k).toUSize % buckets... |
refine β¨fun l hl => ?_, fun i hi p hp => ?_β©
Β· exact match List.mem_or_eq_of_mem_set hl with
| .inl hl => H.1 _ hl
| .inr rfl => hβ (H.1 _ (Array.getElem_mem_data ..))
Β· revert hp
simp only [Array.getElem_eq_data_get, update_data, List.get_set, Array.data_length, update_size]
split <;> intro hp
... | 10 | 22,026.465795 | 2 | 1.4 | 5 | 1,476 |
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open ... | Mathlib/Algebra/Homology/Homotopy.lean | 51 | 54 | theorem dNext_eq (f : β i j, C.X i βΆ D.X j) {i i' : ΞΉ} (w : c.Rel i i') :
dNext i f = C.d i i' β« f i' i := by |
obtain rfl := c.next_eq' w
rfl
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,477 |
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open ... | Mathlib/Algebra/Homology/Homotopy.lean | 96 | 99 | theorem prevD_eq (f : β i j, C.X i βΆ D.X j) {j j' : ΞΉ} (w : c.Rel j' j) :
prevD j f = f j j' β« D.d j' j := by |
obtain rfl := c.prev_eq' w
rfl
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,477 |
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open ... | Mathlib/Algebra/Homology/Homotopy.lean | 109 | 112 | theorem prevD_comp_right (f : β i j, C.X i βΆ D.X j) (g : D βΆ E) (j : ΞΉ) :
(prevD j fun i j => f i j β« g.f j) = prevD j f β« g.f j := by |
dsimp [prevD]
simp only [assoc, g.comm]
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,477 |
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open ... | Mathlib/Algebra/Homology/Homotopy.lean | 115 | 121 | theorem dNext_nat (C D : ChainComplex V β) (i : β) (f : β i j, C.X i βΆ D.X j) :
dNext i f = C.d i (i - 1) β« f (i - 1) i := by |
dsimp [dNext]
cases i
Β· simp only [shape, ChainComplex.next_nat_zero, ComplexShape.down_Rel, Nat.one_ne_zero,
not_false_iff, zero_comp]
Β· congr <;> simp
| 5 | 148.413159 | 2 | 1.4 | 5 | 1,477 |
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open ... | Mathlib/Algebra/Homology/Homotopy.lean | 124 | 130 | theorem prevD_nat (C D : CochainComplex V β) (i : β) (f : β i j, C.X i βΆ D.X j) :
prevD i f = f i (i - 1) β« D.d (i - 1) i := by |
dsimp [prevD]
cases i
Β· simp only [shape, CochainComplex.prev_nat_zero, ComplexShape.up_Rel, Nat.one_ne_zero,
not_false_iff, comp_zero]
Β· congr <;> simp
| 5 | 148.413159 | 2 | 1.4 | 5 | 1,477 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
noncomputable section
variable {π : Type*} {E F G H : Type*}
open Filter List
open scoped Topol... | Mathlib/Analysis/Analytic/Composition.lean | 106 | 114 | theorem applyComposition_ones (p : FormalMultilinearSeries π E F) (n : β) :
p.applyComposition (Composition.ones n) = fun v i =>
p 1 fun _ => v (Fin.castLE (Composition.length_le _) i) := by |
funext v i
apply p.congr (Composition.ones_blocksFun _ _)
intro j hjn hj1
obtain rfl : j = 0 := by omega
refine congr_arg v ?_
rw [Fin.ext_iff, Fin.coe_castLE, Composition.ones_embedding, Fin.val_mk]
| 6 | 403.428793 | 2 | 1.4 | 5 | 1,478 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
noncomputable section
variable {π : Type*} {E F G H : Type*}
open Filter List
open scoped Topol... | Mathlib/Analysis/Analytic/Composition.lean | 117 | 127 | theorem applyComposition_single (p : FormalMultilinearSeries π E F) {n : β} (hn : 0 < n)
(v : Fin n β E) : p.applyComposition (Composition.single n hn) v = fun _j => p n v := by |
ext j
refine p.congr (by simp) fun i hi1 hi2 => ?_
dsimp
congr 1
convert Composition.single_embedding hn β¨i, hi2β© using 1
cases' j with j_val j_property
have : j_val = 0 := le_bot_iff.1 (Nat.lt_succ_iff.1 j_property)
congr!
simp
| 9 | 8,103.083928 | 2 | 1.4 | 5 | 1,478 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
noncomputable section
variable {π : Type*} {E F G H : Type*}
open Filter List
open scoped Topol... | Mathlib/Analysis/Analytic/Composition.lean | 131 | 134 | theorem removeZero_applyComposition (p : FormalMultilinearSeries π E F) {n : β}
(c : Composition n) : p.removeZero.applyComposition c = p.applyComposition c := by |
ext v i
simp [applyComposition, zero_lt_one.trans_le (c.one_le_blocksFun i), removeZero_of_pos]
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,478 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
noncomputable section
variable {π : Type*} {E F G H : Type*}
open Filter List
open scoped Topol... | Mathlib/Analysis/Analytic/Composition.lean | 140 | 162 | theorem applyComposition_update (p : FormalMultilinearSeries π E F) {n : β} (c : Composition n)
(j : Fin n) (v : Fin n β E) (z : E) :
p.applyComposition c (Function.update v j z) =
Function.update (p.applyComposition c v) (c.index j)
(p (c.blocksFun (c.index j))
(Function.update (v β c.... |
ext k
by_cases h : k = c.index j
Β· rw [h]
let r : Fin (c.blocksFun (c.index j)) β Fin n := c.embedding (c.index j)
simp only [Function.update_same]
change p (c.blocksFun (c.index j)) (Function.update v j z β r) = _
let j' := c.invEmbedding j
suffices B : Function.update v j z β r = Function.u... | 17 | 24,154,952.753575 | 2 | 1.4 | 5 | 1,478 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
noncomputable section
variable {π : Type*} {E F G H : Type*}
open Filter List
open scoped Topol... | Mathlib/Analysis/Analytic/Composition.lean | 166 | 169 | theorem compContinuousLinearMap_applyComposition {n : β} (p : FormalMultilinearSeries π F G)
(f : E βL[π] F) (c : Composition n) (v : Fin n β E) :
(p.compContinuousLinearMap f).applyComposition c v = p.applyComposition c (f β v) := by |
simp (config := {unfoldPartialApp := true}) [applyComposition]; rfl
| 1 | 2.718282 | 0 | 1.4 | 5 | 1,478 |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
namespace List
variable {Ξ± : Type*} [DecidableEq Ξ±]
def nextOr : β (_ : List Ξ±) (_ _ : Ξ±), Ξ±
| [], _, default => default
| [_], _, d... | Mathlib/Data/List/Cycle.lean | 54 | 58 | theorem nextOr_cons_of_ne (xs : List Ξ±) (y x d : Ξ±) (h : x β y) :
nextOr (y :: xs) x d = nextOr xs x d := by |
cases' xs with z zs
Β· rfl
Β· exact if_neg h
| 3 | 20.085537 | 1 | 1.4 | 10 | 1,479 |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
namespace List
variable {Ξ± : Type*} [DecidableEq Ξ±]
def nextOr : β (_ : List Ξ±) (_ _ : Ξ±), Ξ±
| [], _, default => default
| [_], _, d... | Mathlib/Data/List/Cycle.lean | 62 | 73 | theorem nextOr_eq_nextOr_of_mem_of_ne (xs : List Ξ±) (x d d' : Ξ±) (x_mem : x β xs)
(x_ne : x β xs.getLast (ne_nil_of_mem x_mem)) : nextOr xs x d = nextOr xs x d' := by |
induction' xs with y ys IH
Β· cases x_mem
cases' ys with z zs
Β· simp at x_mem x_ne
contradiction
by_cases h : x = y
Β· rw [h, nextOr_self_cons_cons, nextOr_self_cons_cons]
Β· rw [nextOr, nextOr, IH]
Β· simpa [h] using x_mem
Β· simpa using x_ne
| 10 | 22,026.465795 | 2 | 1.4 | 10 | 1,479 |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
namespace List
variable {Ξ± : Type*} [DecidableEq Ξ±]
def nextOr : β (_ : List Ξ±) (_ _ : Ξ±), Ξ±
| [], _, default => default
| [_], _, d... | Mathlib/Data/List/Cycle.lean | 76 | 84 | theorem mem_of_nextOr_ne {xs : List Ξ±} {x d : Ξ±} (h : nextOr xs x d β d) : x β xs := by |
induction' xs with y ys IH
Β· simp at h
cases' ys with z zs
Β· simp at h
Β· by_cases hx : x = y
Β· simp [hx]
Β· rw [nextOr_cons_of_ne _ _ _ _ hx] at h
simpa [hx] using IH h
| 8 | 2,980.957987 | 2 | 1.4 | 10 | 1,479 |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
namespace List
variable {Ξ± : Type*} [DecidableEq Ξ±]
def nextOr : β (_ : List Ξ±) (_ _ : Ξ±), Ξ±
| [], _, default => default
| [_], _, d... | Mathlib/Data/List/Cycle.lean | 87 | 91 | theorem nextOr_concat {xs : List Ξ±} {x : Ξ±} (d : Ξ±) (h : x β xs) : nextOr (xs ++ [x]) x d = d := by |
induction' xs with z zs IH
Β· simp
Β· obtain β¨hz, hzsβ© := not_or.mp (mt mem_cons.2 h)
rw [cons_append, nextOr_cons_of_ne _ _ _ _ hz, IH hzs]
| 4 | 54.59815 | 2 | 1.4 | 10 | 1,479 |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
namespace List
variable {Ξ± : Type*} [DecidableEq Ξ±]
def nextOr : β (_ : List Ξ±) (_ _ : Ξ±), Ξ±
| [], _, default => default
| [_], _, d... | Mathlib/Data/List/Cycle.lean | 94 | 106 | theorem nextOr_mem {xs : List Ξ±} {x d : Ξ±} (hd : d β xs) : nextOr xs x d β xs := by |
revert hd
suffices β xs' : List Ξ±, (β x β xs, x β xs') β d β xs' β nextOr xs x d β xs' by
exact this xs fun _ => id
intro xs' hxs' hd
induction' xs with y ys ih
Β· exact hd
cases' ys with z zs
Β· exact hd
rw [nextOr]
split_ifs with h
Β· exact hxs' _ (mem_cons_of_mem _ (mem_cons_self _ _))
Β· exac... | 12 | 162,754.791419 | 2 | 1.4 | 10 | 1,479 |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
namespace List
variable {Ξ± : Type*} [DecidableEq Ξ±]
def nextOr : β (_ : List Ξ±) (_ _ : Ξ±), Ξ±
| [], _, default => default
| [_], _, d... | Mathlib/Data/List/Cycle.lean | 154 | 155 | theorem next_cons_cons_eq' (y z : Ξ±) (h : x β y :: z :: l) (hx : x = y) :
next (y :: z :: l) x h = z := by | rw [next, nextOr, if_pos hx]
| 1 | 2.718282 | 0 | 1.4 | 10 | 1,479 |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
namespace List
variable {Ξ± : Type*} [DecidableEq Ξ±]
def nextOr : β (_ : List Ξ±) (_ _ : Ξ±), Ξ±
| [], _, default => default
| [_], _, d... | Mathlib/Data/List/Cycle.lean | 163 | 169 | theorem next_ne_head_ne_getLast (h : x β l) (y : Ξ±) (h : x β y :: l) (hy : x β y)
(hx : x β getLast (y :: l) (cons_ne_nil _ _)) :
next (y :: l) x h = next l x (by simpa [hy] using h) := by |
rw [next, next, nextOr_cons_of_ne _ _ _ _ hy, nextOr_eq_nextOr_of_mem_of_ne]
Β· rwa [getLast_cons] at hx
exact ne_nil_of_mem (by assumption)
Β· rwa [getLast_cons] at hx
| 4 | 54.59815 | 2 | 1.4 | 10 | 1,479 |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
namespace List
variable {Ξ± : Type*} [DecidableEq Ξ±]
def nextOr : β (_ : List Ξ±) (_ _ : Ξ±), Ξ±
| [], _, default => default
| [_], _, d... | Mathlib/Data/List/Cycle.lean | 172 | 177 | theorem next_cons_concat (y : Ξ±) (hy : x β y) (hx : x β l)
(h : x β y :: l ++ [x] := mem_append_right _ (mem_singleton_self x)) :
next (y :: l ++ [x]) x h = y := by |
rw [next, nextOr_concat]
Β· rfl
Β· simp [hy, hx]
| 3 | 20.085537 | 1 | 1.4 | 10 | 1,479 |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
open List
def Cycle (Ξ± : Type*) : Type _ :=
Quotient (IsRotated.setoid Ξ±)
#align cycle Cycle
namespace Cycle
variable {Ξ± : Type*}
--... | Mathlib/Data/List/Cycle.lean | 601 | 602 | theorem subsingleton_reverse_iff {s : Cycle Ξ±} : s.reverse.Subsingleton β s.Subsingleton := by |
simp [length_subsingleton_iff]
| 1 | 2.718282 | 0 | 1.4 | 10 | 1,479 |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
open List
def Cycle (Ξ± : Type*) : Type _ :=
Quotient (IsRotated.setoid Ξ±)
#align cycle Cycle
namespace Cycle
variable {Ξ± : Type*}
--... | Mathlib/Data/List/Cycle.lean | 605 | 610 | theorem Subsingleton.congr {s : Cycle Ξ±} (h : Subsingleton s) :
β β¦xβ¦ (_hx : x β s) β¦yβ¦ (_hy : y β s), x = y := by |
induction' s using Quot.inductionOn with l
simp only [length_subsingleton_iff, length_coe, mk_eq_coe, le_iff_lt_or_eq, Nat.lt_add_one_iff,
length_eq_zero, length_eq_one, Nat.not_lt_zero, false_or_iff] at h
rcases h with (rfl | β¨z, rflβ©) <;> simp
| 4 | 54.59815 | 2 | 1.4 | 10 | 1,479 |
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Finset.Pointwise
import Mathlib.Tactic.GCongr
#align_import combinatorics.additive.pluennecke_ruzsa from "leanprover-community/mathlib"@"4aab2abced69a9e579b1e6dc... | Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 45 | 56 | theorem card_div_mul_le_card_div_mul_card_div (A B C : Finset Ξ±) :
(A / C).card * B.card β€ (A / B).card * (B / C).card := by |
rw [β card_product (A / B), β mul_one ((A / B) ΓΛ’ (B / C)).card]
refine card_mul_le_card_mul (fun b ac β¦ ac.1 * ac.2 = b) (fun x hx β¦ ?_)
fun x _ β¦ card_le_one_iff.2 fun hu hv β¦
((mem_bipartiteBelow _).1 hu).2.symm.trans ?_
obtain β¨a, ha, c, hc, rflβ© := mem_div.1 hx
refine card_le_card_of_inj_on (fun... | 10 | 22,026.465795 | 2 | 1.4 | 5 | 1,480 |
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Finset.Pointwise
import Mathlib.Tactic.GCongr
#align_import combinatorics.additive.pluennecke_ruzsa from "leanprover-community/mathlib"@"4aab2abced69a9e579b1e6dc... | Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 63 | 66 | theorem card_div_mul_le_card_mul_mul_card_mul (A B C : Finset Ξ±) :
(A / C).card * B.card β€ (A * B).card * (B * C).card := by |
rw [β div_inv_eq_mul, β card_inv B, β card_inv (B * C), mul_inv, β div_eq_mul_inv]
exact card_div_mul_le_card_div_mul_card_div _ _ _
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,480 |
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Finset.Pointwise
import Mathlib.Tactic.GCongr
#align_import combinatorics.additive.pluennecke_ruzsa from "leanprover-community/mathlib"@"4aab2abced69a9e579b1e6dc... | Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 73 | 76 | theorem card_mul_mul_le_card_div_mul_card_mul (A B C : Finset Ξ±) :
(A * C).card * B.card β€ (A / B).card * (B * C).card := by |
rw [β div_inv_eq_mul, β div_inv_eq_mul B]
exact card_div_mul_le_card_div_mul_card_div _ _ _
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,480 |
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Finset.Pointwise
import Mathlib.Tactic.GCongr
#align_import combinatorics.additive.pluennecke_ruzsa from "leanprover-community/mathlib"@"4aab2abced69a9e579b1e6dc... | Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 83 | 86 | theorem card_mul_mul_le_card_mul_mul_card_div (A B C : Finset Ξ±) :
(A * C).card * B.card β€ (A * B).card * (B / C).card := by |
rw [β div_inv_eq_mul, div_eq_mul_inv B]
exact card_div_mul_le_card_mul_mul_card_mul _ _ _
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,480 |
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Data.Finset.Pointwise
import Mathlib.Tactic.GCongr
#align_import combinatorics.additive.pluennecke_ruzsa from "leanprover-community/mathlib"@"4aab2abced69a9e579b1e6dc... | Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 92 | 118 | theorem mul_pluennecke_petridis (C : Finset Ξ±)
(hA : β A' β A, (A * B).card * A'.card β€ (A' * B).card * A.card) :
(A * B * C).card * A.card β€ (A * B).card * (A * C).card := by |
induction' C using Finset.induction_on with x C _ ih
Β· simp
set A' := A β© (A * C / {x}) with hA'
set C' := insert x C with hC'
have hβ : A' * {x} = A * {x} β© (A * C) := by
rw [hA', inter_mul_singleton, (isUnit_singleton x).div_mul_cancel]
have hβ : A * B * C' = A * B * C βͺ (A * B * {x}) \ (A' * B * {x}... | 24 | 26,489,122,129.84347 | 2 | 1.4 | 5 | 1,480 |
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
noncomputable section
open scoped Classical
open NNReal Topology Filter
local notatio... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 196 | 199 | theorem HasFTaylorSeriesUpToOn.zero_eq' (h : HasFTaylorSeriesUpToOn n f p s) {x : E} (hx : x β s) :
p x 0 = (continuousMultilinearCurryFin0 π E F).symm (f x) := by |
rw [β h.zero_eq x hx]
exact (p x 0).uncurry0_curry0.symm
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,481 |
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
noncomputable section
open scoped Classical
open NNReal Topology Filter
local notatio... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 204 | 208 | theorem HasFTaylorSeriesUpToOn.congr (h : HasFTaylorSeriesUpToOn n f p s)
(hβ : β x β s, fβ x = f x) : HasFTaylorSeriesUpToOn n fβ p s := by |
refine β¨fun x hx => ?_, h.fderivWithin, h.contβ©
rw [hβ x hx]
exact h.zero_eq x hx
| 3 | 20.085537 | 1 | 1.4 | 5 | 1,481 |
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
noncomputable section
open scoped Classical
open NNReal Topology Filter
local notatio... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 223 | 226 | theorem HasFTaylorSeriesUpToOn.continuousOn (h : HasFTaylorSeriesUpToOn n f p s) :
ContinuousOn f s := by |
have := (h.cont 0 bot_le).congr fun x hx => (h.zero_eq' hx).symm
rwa [β (continuousMultilinearCurryFin0 π E F).symm.comp_continuousOn_iff]
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,481 |
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
noncomputable section
open scoped Classical
open NNReal Topology Filter
local notatio... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 229 | 237 | theorem hasFTaylorSeriesUpToOn_zero_iff :
HasFTaylorSeriesUpToOn 0 f p s β ContinuousOn f s β§ β x β s, (p x 0).uncurry0 = f x := by |
refine β¨fun H => β¨H.continuousOn, H.zero_eqβ©, fun H =>
β¨H.2, fun m hm => False.elim (not_le.2 hm bot_le), fun m hm β¦ ?_β©β©
obtain rfl : m = 0 := mod_cast hm.antisymm (zero_le _)
have : EqOn (p Β· 0) ((continuousMultilinearCurryFin0 π E F).symm β f) s := fun x hx β¦
(continuousMultilinearCurryFin0 π E F)... | 7 | 1,096.633158 | 2 | 1.4 | 5 | 1,481 |
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
noncomputable section
open scoped Classical
open NNReal Topology Filter
local notatio... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 240 | 250 | theorem hasFTaylorSeriesUpToOn_top_iff :
HasFTaylorSeriesUpToOn β f p s β β n : β, HasFTaylorSeriesUpToOn n f p s := by |
constructor
Β· intro H n; exact H.of_le le_top
Β· intro H
constructor
Β· exact (H 0).zero_eq
Β· intro m _
apply (H m.succ).fderivWithin m (WithTop.coe_lt_coe.2 (lt_add_one m))
Β· intro m _
apply (H m).cont m le_rfl
| 9 | 8,103.083928 | 2 | 1.4 | 5 | 1,481 |
import Mathlib.Topology.Separation
import Mathlib.Topology.NoetherianSpace
#align_import topology.quasi_separated from "leanprover-community/mathlib"@"5dc6092d09e5e489106865241986f7f2ad28d4c8"
open TopologicalSpace
variable {Ξ± Ξ² : Type*} [TopologicalSpace Ξ±] [TopologicalSpace Ξ²] {f : Ξ± β Ξ²}
def IsQuasiSeparate... | Mathlib/Topology/QuasiSeparated.lean | 53 | 56 | theorem isQuasiSeparated_univ_iff {Ξ± : Type*} [TopologicalSpace Ξ±] :
IsQuasiSeparated (Set.univ : Set Ξ±) β QuasiSeparatedSpace Ξ± := by |
rw [quasiSeparatedSpace_iff]
simp [IsQuasiSeparated]
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,482 |
import Mathlib.Topology.Separation
import Mathlib.Topology.NoetherianSpace
#align_import topology.quasi_separated from "leanprover-community/mathlib"@"5dc6092d09e5e489106865241986f7f2ad28d4c8"
open TopologicalSpace
variable {Ξ± Ξ² : Type*} [TopologicalSpace Ξ±] [TopologicalSpace Ξ²] {f : Ξ± β Ξ²}
def IsQuasiSeparate... | Mathlib/Topology/QuasiSeparated.lean | 64 | 86 | theorem IsQuasiSeparated.image_of_embedding {s : Set Ξ±} (H : IsQuasiSeparated s) (h : Embedding f) :
IsQuasiSeparated (f '' s) := by |
intro U V hU hU' hU'' hV hV' hV''
convert
(H (f β»ΒΉ' U) (f β»ΒΉ' V)
?_ (h.continuous.1 _ hU') ?_ ?_ (h.continuous.1 _ hV') ?_).image h.continuous
Β· symm
rw [β Set.preimage_inter, Set.image_preimage_eq_inter_range, Set.inter_eq_left]
exact Set.inter_subset_left.trans (hU.trans (Set.image_subset_ran... | 21 | 1,318,815,734.483215 | 2 | 1.4 | 5 | 1,482 |
import Mathlib.Topology.Separation
import Mathlib.Topology.NoetherianSpace
#align_import topology.quasi_separated from "leanprover-community/mathlib"@"5dc6092d09e5e489106865241986f7f2ad28d4c8"
open TopologicalSpace
variable {Ξ± Ξ² : Type*} [TopologicalSpace Ξ±] [TopologicalSpace Ξ²] {f : Ξ± β Ξ²}
def IsQuasiSeparate... | Mathlib/Topology/QuasiSeparated.lean | 89 | 96 | theorem OpenEmbedding.isQuasiSeparated_iff (h : OpenEmbedding f) {s : Set Ξ±} :
IsQuasiSeparated s β IsQuasiSeparated (f '' s) := by |
refine β¨fun hs => hs.image_of_embedding h.toEmbedding, ?_β©
intro H U V hU hU' hU'' hV hV' hV''
rw [h.toEmbedding.isCompact_iff, Set.image_inter h.inj]
exact
H (f '' U) (f '' V) (Set.image_subset _ hU) (h.isOpenMap _ hU') (hU''.image h.continuous)
(Set.image_subset _ hV) (h.isOpenMap _ hV') (hV''.imag... | 6 | 403.428793 | 2 | 1.4 | 5 | 1,482 |
import Mathlib.Topology.Separation
import Mathlib.Topology.NoetherianSpace
#align_import topology.quasi_separated from "leanprover-community/mathlib"@"5dc6092d09e5e489106865241986f7f2ad28d4c8"
open TopologicalSpace
variable {Ξ± Ξ² : Type*} [TopologicalSpace Ξ±] [TopologicalSpace Ξ²] {f : Ξ± β Ξ²}
def IsQuasiSeparate... | Mathlib/Topology/QuasiSeparated.lean | 99 | 103 | theorem isQuasiSeparated_iff_quasiSeparatedSpace (s : Set Ξ±) (hs : IsOpen s) :
IsQuasiSeparated s β QuasiSeparatedSpace s := by |
rw [β isQuasiSeparated_univ_iff]
convert (hs.openEmbedding_subtype_val.isQuasiSeparated_iff (s := Set.univ)).symm
simp
| 3 | 20.085537 | 1 | 1.4 | 5 | 1,482 |
import Mathlib.Topology.Separation
import Mathlib.Topology.NoetherianSpace
#align_import topology.quasi_separated from "leanprover-community/mathlib"@"5dc6092d09e5e489106865241986f7f2ad28d4c8"
open TopologicalSpace
variable {Ξ± Ξ² : Type*} [TopologicalSpace Ξ±] [TopologicalSpace Ξ²] {f : Ξ± β Ξ²}
def IsQuasiSeparate... | Mathlib/Topology/QuasiSeparated.lean | 106 | 109 | theorem IsQuasiSeparated.of_subset {s t : Set Ξ±} (ht : IsQuasiSeparated t) (h : s β t) :
IsQuasiSeparated s := by |
intro U V hU hU' hU'' hV hV' hV''
exact ht U V (hU.trans h) hU' hU'' (hV.trans h) hV' hV''
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,482 |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.lemmas from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
open Function Nat
namespace Int
variable {a b : β€} {n : β}
| Mathlib/Data/Int/Order/Lemmas.lean | 28 | 30 | theorem natAbs_eq_iff_mul_self_eq {a b : β€} : a.natAbs = b.natAbs β a * a = b * b := by |
rw [β abs_eq_iff_mul_self_eq, abs_eq_natAbs, abs_eq_natAbs]
exact Int.natCast_inj.symm
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,483 |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.lemmas from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
open Function Nat
namespace Int
variable {a b : β€} {n : β}
theorem natAbs_eq_iff_mul_self_eq {a b : β€} : a.natAbs = b.natAbs β a * a = b * b := by
rw [β a... | Mathlib/Data/Int/Order/Lemmas.lean | 35 | 37 | theorem natAbs_lt_iff_mul_self_lt {a b : β€} : a.natAbs < b.natAbs β a * a < b * b := by |
rw [β abs_lt_iff_mul_self_lt, abs_eq_natAbs, abs_eq_natAbs]
exact Int.ofNat_lt.symm
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,483 |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.lemmas from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
open Function Nat
namespace Int
variable {a b : β€} {n : β}
theorem natAbs_eq_iff_mul_self_eq {a b : β€} : a.natAbs = b.natAbs β a * a = b * b := by
rw [β a... | Mathlib/Data/Int/Order/Lemmas.lean | 40 | 42 | theorem natAbs_le_iff_mul_self_le {a b : β€} : a.natAbs β€ b.natAbs β a * a β€ b * b := by |
rw [β abs_le_iff_mul_self_le, abs_eq_natAbs, abs_eq_natAbs]
exact Int.ofNat_le.symm
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,483 |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.lemmas from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
open Function Nat
namespace Int
variable {a b : β€} {n : β}
theorem natAbs_eq_iff_mul_self_eq {a b : β€} : a.natAbs = b.natAbs β a * a = b * b := by
rw [β a... | Mathlib/Data/Int/Order/Lemmas.lean | 45 | 50 | theorem dvd_div_of_mul_dvd {a b c : β€} (h : a * b β£ c) : b β£ c / a := by |
rcases eq_or_ne a 0 with (rfl | ha)
Β· simp only [Int.ediv_zero, Int.dvd_zero]
rcases h with β¨d, rflβ©
refine β¨d, ?_β©
rw [mul_assoc, Int.mul_ediv_cancel_left _ ha]
| 5 | 148.413159 | 2 | 1.4 | 5 | 1,483 |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.lemmas from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
open Function Nat
namespace Int
variable {a b : β€} {n : β}
theorem natAbs_eq_iff_mul_self_eq {a b : β€} : a.natAbs = b.natAbs β a * a = b * b := by
rw [β a... | Mathlib/Data/Int/Order/Lemmas.lean | 62 | 68 | theorem eq_zero_of_abs_lt_dvd {m x : β€} (h1 : m β£ x) (h2 : |x| < m) : x = 0 := by |
obtain rfl | hm := eq_or_ne m 0
Β· exact Int.zero_dvd.1 h1
rcases h1 with β¨d, rflβ©
apply mul_eq_zero_of_right
rw [β abs_lt_one_iff, β mul_lt_iff_lt_one_right (abs_pos.mpr hm), β abs_mul]
exact lt_of_lt_of_le h2 (le_abs_self m)
| 6 | 403.428793 | 2 | 1.4 | 5 | 1,483 |
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
#align_import data.list.sort from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
open List.Perm
universe u
namespace List
section Sorted
variable {Ξ± : Type u} {r : Ξ± β Ξ± β Prop} {a : Ξ±} {l... | Mathlib/Data/List/Sort.lean | 80 | 85 | theorem Sorted.head!_le [Inhabited Ξ±] [Preorder Ξ±] {a : Ξ±} {l : List Ξ±} (h : Sorted (Β· < Β·) l)
(ha : a β l) : l.head! β€ a := by |
rw [β List.cons_head!_tail (List.ne_nil_of_mem ha)] at h ha
cases ha
Β· exact le_rfl
Β· exact le_of_lt (rel_of_sorted_cons h a (by assumption))
| 4 | 54.59815 | 2 | 1.4 | 5 | 1,484 |
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
#align_import data.list.sort from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
open List.Perm
universe u
namespace List
section Sorted
variable {Ξ± : Type u} {r : Ξ± β Ξ± β Prop} {a : Ξ±} {l... | Mathlib/Data/List/Sort.lean | 87 | 92 | theorem Sorted.le_head! [Inhabited Ξ±] [Preorder Ξ±] {a : Ξ±} {l : List Ξ±} (h : Sorted (Β· > Β·) l)
(ha : a β l) : a β€ l.head! := by |
rw [β List.cons_head!_tail (List.ne_nil_of_mem ha)] at h ha
cases ha
Β· exact le_rfl
Β· exact le_of_lt (rel_of_sorted_cons h a (by assumption))
| 4 | 54.59815 | 2 | 1.4 | 5 | 1,484 |
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
#align_import data.list.sort from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
open List.Perm
universe u
namespace List
section Sorted
variable {Ξ± : Type u} {r : Ξ± β Ξ± β Prop} {a : Ξ±} {l... | Mathlib/Data/List/Sort.lean | 104 | 120 | theorem eq_of_perm_of_sorted [IsAntisymm Ξ± r] {lβ lβ : List Ξ±} (hp : lβ ~ lβ) (hsβ : Sorted r lβ)
(hsβ : Sorted r lβ) : lβ = lβ := by |
induction' hsβ with a lβ hβ hsβ IH generalizing lβ
Β· exact hp.nil_eq
Β· have : a β lβ := hp.subset (mem_cons_self _ _)
rcases append_of_mem this with β¨uβ, vβ, rflβ©
have hp' := (perm_cons a).1 (hp.trans perm_middle)
obtain rfl := IH hp' (hsβ.sublist <| by simp)
change a :: uβ ++ vβ = uβ ++ ([a] ++ ... | 15 | 3,269,017.372472 | 2 | 1.4 | 5 | 1,484 |
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
#align_import data.list.sort from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
open List.Perm
universe u
namespace List
section Sorted
variable {Ξ± : Type u} {r : Ξ± β Ξ± β Prop} {a : Ξ±} {l... | Mathlib/Data/List/Sort.lean | 123 | 126 | theorem sublist_of_subperm_of_sorted [IsAntisymm Ξ± r] {lβ lβ : List Ξ±} (hp : lβ <+~ lβ)
(hsβ : lβ.Sorted r) (hsβ : lβ.Sorted r) : lβ <+ lβ := by |
let β¨_, h, h'β© := hp
rwa [β eq_of_perm_of_sorted h (hsβ.sublist h') hsβ]
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,484 |
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
#align_import data.list.sort from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
open List.Perm
universe u
namespace List
section sort
variable {Ξ± : Type u} (r : Ξ± β Ξ± β Prop) [DecidableRe... | Mathlib/Data/List/Sort.lean | 273 | 275 | theorem orderedInsert_count [DecidableEq Ξ±] (L : List Ξ±) (a b : Ξ±) :
count a (L.orderedInsert r b) = count a L + if a = b then 1 else 0 := by |
rw [(L.perm_orderedInsert r b).count_eq, count_cons]
| 1 | 2.718282 | 0 | 1.4 | 5 | 1,484 |
import Mathlib.Order.Interval.Finset.Fin
#align_import data.fintype.fin from "leanprover-community/mathlib"@"759575657f189ccb424b990164c8b1fa9f55cdfe"
open Finset
open Fintype
namespace Fin
variable {Ξ± Ξ² : Type*} {n : β}
| Mathlib/Data/Fintype/Fin.lean | 25 | 27 | theorem map_valEmbedding_univ : (Finset.univ : Finset (Fin n)).map Fin.valEmbedding = Iio n := by |
ext
simp [orderIsoSubtype.symm.surjective.exists, OrderIso.symm]
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,485 |
import Mathlib.Order.Interval.Finset.Fin
#align_import data.fintype.fin from "leanprover-community/mathlib"@"759575657f189ccb424b990164c8b1fa9f55cdfe"
open Finset
open Fintype
namespace Fin
variable {Ξ± Ξ² : Type*} {n : β}
theorem map_valEmbedding_univ : (Finset.univ : Finset (Fin n)).map Fin.valEmbedding = Iio ... | Mathlib/Data/Fintype/Fin.lean | 41 | 51 | theorem Ioi_succ (i : Fin n) : Ioi i.succ = (Ioi i).map (Fin.succEmb _) := by |
ext i
simp only [mem_filter, mem_Ioi, mem_map, mem_univ, true_and_iff, Function.Embedding.coeFn_mk,
exists_true_left]
constructor
Β· refine cases ?_ ?_ i
Β· rintro β¨β¨β©β©
Β· intro i hi
exact β¨i, succ_lt_succ_iff.mp hi, rflβ©
Β· rintro β¨i, hi, rflβ©
simpa
| 10 | 22,026.465795 | 2 | 1.4 | 5 | 1,485 |
import Mathlib.Order.Interval.Finset.Fin
#align_import data.fintype.fin from "leanprover-community/mathlib"@"759575657f189ccb424b990164c8b1fa9f55cdfe"
open Finset
open Fintype
namespace Fin
variable {Ξ± Ξ² : Type*} {n : β}
theorem map_valEmbedding_univ : (Finset.univ : Finset (Fin n)).map Fin.valEmbedding = Iio ... | Mathlib/Data/Fintype/Fin.lean | 55 | 58 | theorem Iio_castSucc (i : Fin n) : Iio (castSucc i) = (Iio i).map Fin.castSuccEmb := by |
apply Finset.map_injective Fin.valEmbedding
rw [Finset.map_map, Fin.map_valEmbedding_Iio]
exact (Fin.map_valEmbedding_Iio i).symm
| 3 | 20.085537 | 1 | 1.4 | 5 | 1,485 |
import Mathlib.Order.Interval.Finset.Fin
#align_import data.fintype.fin from "leanprover-community/mathlib"@"759575657f189ccb424b990164c8b1fa9f55cdfe"
open Finset
open Fintype
namespace Fin
variable {Ξ± Ξ² : Type*} {n : β}
theorem map_valEmbedding_univ : (Finset.univ : Finset (Fin n)).map Fin.valEmbedding = Iio ... | Mathlib/Data/Fintype/Fin.lean | 61 | 64 | theorem card_filter_univ_succ' (p : Fin (n + 1) β Prop) [DecidablePred p] :
(univ.filter p).card = ite (p 0) 1 0 + (univ.filter (p β Fin.succ)).card := by |
rw [Fin.univ_succ, filter_cons, card_disjUnion, filter_map, card_map]
split_ifs <;> simp
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,485 |
import Mathlib.Order.Interval.Finset.Fin
#align_import data.fintype.fin from "leanprover-community/mathlib"@"759575657f189ccb424b990164c8b1fa9f55cdfe"
open Finset
open Fintype
namespace Fin
variable {Ξ± Ξ² : Type*} {n : β}
theorem map_valEmbedding_univ : (Finset.univ : Finset (Fin n)).map Fin.valEmbedding = Iio ... | Mathlib/Data/Fintype/Fin.lean | 73 | 78 | theorem card_filter_univ_eq_vector_get_eq_count [DecidableEq Ξ±] (a : Ξ±) (v : Vector Ξ± n) :
(univ.filter fun i => a = v.get i).card = v.toList.count a := by |
induction' v with n x xs hxs
Β· simp
Β· simp_rw [card_filter_univ_succ', Vector.get_cons_zero, Vector.toList_cons, Function.comp,
Vector.get_cons_succ, hxs, List.count_cons, add_comm (ite (a = x) 1 0)]
| 4 | 54.59815 | 2 | 1.4 | 5 | 1,485 |
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Localization.NormTrace
#align_import number_theory.number_field.norm from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
open scoped NumberField
open Finset NumberField Algebra FiniteDimensional
namespace RingOfIn... | Mathlib/NumberTheory/NumberField/Norm.lean | 65 | 69 | theorem norm_algebraMap [IsSeparable K L] (x : π K) :
norm K (algebraMap (π K) (π L) x) = x ^ finrank K L := by |
rw [RingOfIntegers.ext_iff, RingOfIntegers.coe_eq_algebraMap,
RingOfIntegers.algebraMap_norm_algebraMap, Algebra.norm_algebraMap,
RingOfIntegers.coe_eq_algebraMap, map_pow]
| 3 | 20.085537 | 1 | 1.4 | 5 | 1,486 |
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Localization.NormTrace
#align_import number_theory.number_field.norm from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
open scoped NumberField
open Finset NumberField Algebra FiniteDimensional
namespace RingOfIn... | Mathlib/NumberTheory/NumberField/Norm.lean | 72 | 85 | theorem isUnit_norm_of_isGalois [IsGalois K L] {x : π L} : IsUnit (norm K x) β IsUnit x := by |
classical
refine β¨fun hx => ?_, IsUnit.map _β©
replace hx : IsUnit (algebraMap (π K) (π L) <| norm K x) := hx.map (algebraMap (π K) <| π L)
refine @isUnit_of_mul_isUnit_right (π L) _
β¨(univ \ {AlgEquiv.refl}).prod fun Ο : L ββ[K] L => Ο x,
prod_mem fun Ο _ => x.2.map (Ο : L β+* L).toIntAlgHomβ© _ ... | 13 | 442,413.392009 | 2 | 1.4 | 5 | 1,486 |
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Localization.NormTrace
#align_import number_theory.number_field.norm from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
open scoped NumberField
open Finset NumberField Algebra FiniteDimensional
namespace RingOfIn... | Mathlib/NumberTheory/NumberField/Norm.lean | 90 | 99 | theorem dvd_norm [IsGalois K L] (x : π L) : x β£ algebraMap (π K) (π L) (norm K x) := by |
classical
have hint :
IsIntegral β€ (β Ο β univ.erase (AlgEquiv.refl : L ββ[K] L), Ο x) :=
IsIntegral.prod _ (fun Ο _ =>
((RingOfIntegers.isIntegral_coe x).map Ο))
refine β¨β¨_, hintβ©, ?_β©
ext
rw [coe_algebraMap_norm K x, norm_eq_prod_automorphisms]
simp [β Finset.mul_prod_erase _ _ (mem_univ Al... | 9 | 8,103.083928 | 2 | 1.4 | 5 | 1,486 |
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