Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
|---|---|---|---|---|---|---|---|---|---|---|---|
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 129 | 131 | theorem neg_one_lt_goldConj : -1 < Ο := by |
rw [neg_lt, β inv_gold]
exact inv_lt_one one_lt_gold
| 2 | 7.389056 | 1 | 0.894737 | 19 | 776 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 140 | 146 | theorem gold_irrational : Irrational Ο := by |
have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
have := this.rat_add 1
have := this.rat_mul (show (0.5 : β) β 0 by norm_num)
convert this
norm_num
field_simp
| 6 | 403.428793 | 2 | 0.894737 | 19 | 776 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 150 | 156 | theorem goldConj_irrational : Irrational Ο := by |
have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
have := this.rat_sub 1
have := this.rat_mul (show (0.5 : β) β 0 by norm_num)
convert this
norm_num
field_simp
| 6 | 403.428793 | 2 | 0.894737 | 19 | 776 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 178 | 181 | theorem fibRec_charPoly_eq {Ξ² : Type*} [CommRing Ξ²] :
fibRec.charPoly = X ^ 2 - (X + (1 : Ξ²[X])) := by |
rw [fibRec, LinearRecurrence.charPoly]
simp [Finset.sum_fin_eq_sum_range, Finset.sum_range_succ', β smul_X_eq_monomial]
| 2 | 7.389056 | 1 | 0.894737 | 19 | 776 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 187 | 192 | theorem fib_isSol_fibRec : fibRec.IsSolution (fun x => x.fib : β β Ξ±) := by |
rw [fibRec]
intro n
simp only
rw [Nat.fib_add_two, add_comm]
simp [Finset.sum_fin_eq_sum_range, Finset.sum_range_succ']
| 5 | 148.413159 | 2 | 0.894737 | 19 | 776 |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe vβ vβ vβ uβ uβ uβ
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 52 | 56 | theorem eqToHom_trans {X Y Z : C} (p : X = Y) (q : Y = Z) :
eqToHom p β« eqToHom q = eqToHom (p.trans q) := by |
cases p
cases q
simp
| 3 | 20.085537 | 1 | 0.9 | 10 | 777 |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe vβ vβ vβ uβ uβ uβ
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 77 | 80 | theorem eqToHom_naturality {f g : Ξ² β C} (z : β b, f b βΆ g b) {j j' : Ξ²} (w : j = j') :
z j β« eqToHom (by simp [w]) = eqToHom (by simp [w]) β« z j' := by |
cases w
simp
| 2 | 7.389056 | 1 | 0.9 | 10 | 777 |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe vβ vβ vβ uβ uβ uβ
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 86 | 89 | theorem eqToHom_iso_hom_naturality {f g : Ξ² β C} (z : β b, f b β
g b) {j j' : Ξ²} (w : j = j') :
(z j).hom β« eqToHom (by simp [w]) = eqToHom (by simp [w]) β« (z j').hom := by |
cases w
simp
| 2 | 7.389056 | 1 | 0.9 | 10 | 777 |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe vβ vβ vβ uβ uβ uβ
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 95 | 98 | theorem eqToHom_iso_inv_naturality {f g : Ξ² β C} (z : β b, f b β
g b) {j j' : Ξ²} (w : j = j') :
(z j).inv β« eqToHom (by simp [w]) = eqToHom (by simp [w]) β« (z j').inv := by |
cases w
simp
| 2 | 7.389056 | 1 | 0.9 | 10 | 777 |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe vβ vβ vβ uβ uβ uβ
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 104 | 107 | theorem congrArg_cast_hom_left {X Y Z : C} (p : X = Y) (q : Y βΆ Z) :
cast (congrArg (fun W : C => W βΆ Z) p.symm) q = eqToHom p β« q := by |
cases p
simp
| 2 | 7.389056 | 1 | 0.9 | 10 | 777 |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe vβ vβ vβ uβ uβ uβ
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 116 | 119 | theorem congrArg_mpr_hom_left {X Y Z : C} (p : X = Y) (q : Y βΆ Z) :
(congrArg (fun W : C => W βΆ Z) p).mpr q = eqToHom p β« q := by |
cases p
simp
| 2 | 7.389056 | 1 | 0.9 | 10 | 777 |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe vβ vβ vβ uβ uβ uβ
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 126 | 129 | theorem congrArg_cast_hom_right {X Y Z : C} (p : X βΆ Y) (q : Z = Y) :
cast (congrArg (fun W : C => X βΆ W) q.symm) p = p β« eqToHom q.symm := by |
cases q
simp
| 2 | 7.389056 | 1 | 0.9 | 10 | 777 |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe vβ vβ vβ uβ uβ uβ
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 138 | 141 | theorem congrArg_mpr_hom_right {X Y Z : C} (p : X βΆ Y) (q : Z = Y) :
(congrArg (fun W : C => X βΆ W) q).mpr p = p β« eqToHom q.symm := by |
cases q
simp
| 2 | 7.389056 | 1 | 0.9 | 10 | 777 |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe vβ vβ vβ uβ uβ uβ
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 169 | 170 | theorem eqToIso_trans {X Y Z : C} (p : X = Y) (q : Y = Z) :
eqToIso p βͺβ« eqToIso q = eqToIso (p.trans q) := by | ext; simp
| 1 | 2.718282 | 0 | 0.9 | 10 | 777 |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe vβ vβ vβ uβ uβ uβ
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 174 | 176 | theorem eqToHom_op {X Y : C} (h : X = Y) : (eqToHom h).op = eqToHom (congr_arg op h.symm) := by |
cases h
rfl
| 2 | 7.389056 | 1 | 0.9 | 10 | 777 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 71 | 73 | theorem vars_def [DecidableEq Ο] (p : MvPolynomial Ο R) : p.vars = p.degrees.toFinset := by |
rw [vars]
convert rfl
| 2 | 7.389056 | 1 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 77 | 78 | theorem vars_0 : (0 : MvPolynomial Ο R).vars = β
:= by |
classical rw [vars_def, degrees_zero, Multiset.toFinset_zero]
| 1 | 2.718282 | 0 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 82 | 83 | theorem vars_monomial (h : r β 0) : (monomial s r).vars = s.support := by |
classical rw [vars_def, degrees_monomial_eq _ _ h, Finsupp.toFinset_toMultiset]
| 1 | 2.718282 | 0 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 87 | 88 | theorem vars_C : (C r : MvPolynomial Ο R).vars = β
:= by |
classical rw [vars_def, degrees_C, Multiset.toFinset_zero]
| 1 | 2.718282 | 0 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 93 | 94 | theorem vars_X [Nontrivial R] : (X n : MvPolynomial Ο R).vars = {n} := by |
rw [X, vars_monomial (one_ne_zero' R), Finsupp.support_single_ne_zero _ (one_ne_zero' β)]
| 1 | 2.718282 | 0 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 98 | 99 | theorem mem_vars (i : Ο) : i β p.vars β β d β p.support, i β d.support := by |
classical simp only [vars_def, Multiset.mem_toFinset, mem_degrees, mem_support_iff, exists_prop]
| 1 | 2.718282 | 0 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 102 | 105 | theorem mem_support_not_mem_vars_zero {f : MvPolynomial Ο R} {x : Ο ββ β} (H : x β f.support)
{v : Ο} (h : v β vars f) : x v = 0 := by |
contrapose! h
exact (mem_vars v).mpr β¨x, H, Finsupp.mem_support_iff.mpr hβ©
| 2 | 7.389056 | 1 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 108 | 112 | theorem vars_add_subset [DecidableEq Ο] (p q : MvPolynomial Ο R) :
(p + q).vars β p.vars βͺ q.vars := by |
intro x hx
simp only [vars_def, Finset.mem_union, Multiset.mem_toFinset] at hx β’
simpa using Multiset.mem_of_le (degrees_add _ _) hx
| 3 | 20.085537 | 1 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 115 | 119 | theorem vars_add_of_disjoint [DecidableEq Ο] (h : Disjoint p.vars q.vars) :
(p + q).vars = p.vars βͺ q.vars := by |
refine (vars_add_subset p q).antisymm fun x hx => ?_
simp only [vars_def, Multiset.disjoint_toFinset] at h hx β’
rwa [degrees_add_of_disjoint h, Multiset.toFinset_union]
| 3 | 20.085537 | 1 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 124 | 126 | theorem vars_mul [DecidableEq Ο] (Ο Ο : MvPolynomial Ο R) : (Ο * Ο).vars β Ο.vars βͺ Ο.vars := by |
simp_rw [vars_def, β Multiset.toFinset_add, Multiset.toFinset_subset]
exact Multiset.subset_of_le (degrees_mul Ο Ο)
| 2 | 7.389056 | 1 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 134 | 140 | theorem vars_pow (Ο : MvPolynomial Ο R) (n : β) : (Ο ^ n).vars β Ο.vars := by |
classical
induction' n with n ih
Β· simp
Β· rw [pow_succ']
apply Finset.Subset.trans (vars_mul _ _)
exact Finset.union_subset (Finset.Subset.refl _) ih
| 6 | 403.428793 | 2 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 146 | 154 | theorem vars_prod {ΞΉ : Type*} [DecidableEq Ο] {s : Finset ΞΉ} (f : ΞΉ β MvPolynomial Ο R) :
(β i β s, f i).vars β s.biUnion fun i => (f i).vars := by |
classical
induction s using Finset.induction_on with
| empty => simp
| insert hs hsub =>
simp only [hs, Finset.biUnion_insert, Finset.prod_insert, not_false_iff]
apply Finset.Subset.trans (vars_mul _ _)
exact Finset.union_subset_union (Finset.Subset.refl _) hsub
| 7 | 1,096.633158 | 2 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 161 | 168 | theorem vars_C_mul (a : A) (ha : a β 0) (Ο : MvPolynomial Ο A) :
(C a * Ο : MvPolynomial Ο A).vars = Ο.vars := by |
ext1 i
simp only [mem_vars, exists_prop, mem_support_iff]
apply exists_congr
intro d
apply and_congr _ Iff.rfl
rw [coeff_C_mul, mul_ne_zero_iff, eq_true ha, true_and_iff]
| 6 | 403.428793 | 2 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 180 | 189 | theorem vars_sum_subset [DecidableEq Ο] :
(β i β t, Ο i).vars β Finset.biUnion t fun i => (Ο i).vars := by |
classical
induction t using Finset.induction_on with
| empty => simp
| insert has hsum =>
rw [Finset.biUnion_insert, Finset.sum_insert has]
refine Finset.Subset.trans
(vars_add_subset _ _) (Finset.union_subset_union (Finset.Subset.refl _) ?_)
assumption
| 8 | 2,980.957987 | 2 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 192 | 207 | theorem vars_sum_of_disjoint [DecidableEq Ο] (h : Pairwise <| (Disjoint on fun i => (Ο i).vars)) :
(β i β t, Ο i).vars = Finset.biUnion t fun i => (Ο i).vars := by |
classical
induction t using Finset.induction_on with
| empty => simp
| insert has hsum =>
rw [Finset.biUnion_insert, Finset.sum_insert has, vars_add_of_disjoint, hsum]
unfold Pairwise onFun at h
rw [hsum]
simp only [Finset.disjoint_iff_ne] at h β’
intro v hv v2 hv2
rw [Finset.mem_biUnion... | 14 | 1,202,604.284165 | 2 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 217 | 217 | theorem vars_map : (map f p).vars β p.vars := by | classical simp [vars_def, degrees_map]
| 1 | 2.718282 | 0 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 222 | 223 | theorem vars_map_of_injective (hf : Injective f) : (map f p).vars = p.vars := by |
simp [vars, degrees_map_of_injective _ hf]
| 1 | 2.718282 | 0 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 226 | 228 | theorem vars_monomial_single (i : Ο) {e : β} {r : R} (he : e β 0) (hr : r β 0) :
(monomial (Finsupp.single i e) r).vars = {i} := by |
rw [vars_monomial hr, Finsupp.support_single_ne_zero _ he]
| 1 | 2.718282 | 0 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 231 | 234 | theorem vars_eq_support_biUnion_support [DecidableEq Ο] :
p.vars = p.support.biUnion Finsupp.support := by |
ext i
rw [mem_vars, Finset.mem_biUnion]
| 2 | 7.389056 | 1 | 0.9 | 20 | 778 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 248 | 274 | theorem evalβHom_eq_constantCoeff_of_vars (f : R β+* S) {g : Ο β S} {p : MvPolynomial Ο R}
(hp : β i β p.vars, g i = 0) : evalβHom f g p = f (constantCoeff p) := by |
conv_lhs => rw [p.as_sum]
simp only [map_sum, evalβHom_monomial]
by_cases h0 : constantCoeff p = 0
on_goal 1 =>
rw [h0, f.map_zero, Finset.sum_eq_zero]
intro d hd
on_goal 2 =>
rw [Finset.sum_eq_single (0 : Ο ββ β)]
Β· rw [Finsupp.prod_zero_index, mul_one]
rfl
on_goal 1 => intro d hd ... | 25 | 72,004,899,337.38586 | 2 | 0.9 | 20 | 778 |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 137 | 141 | theorem repr_symm_single : b.repr.symm (Finsupp.single i c) = c β’ b i :=
calc
b.repr.symm (Finsupp.single i c) = b.repr.symm (c β’ Finsupp.single i (1 : R)) := by |
{ rw [Finsupp.smul_single', mul_one] }
_ = c β’ b i := by rw [LinearEquiv.map_smul, repr_symm_single_one]
| 2 | 7.389056 | 1 | 0.9 | 10 | 779 |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 149 | 150 | theorem repr_self_apply (j) [Decidable (i = j)] : b.repr (b i) j = if i = j then 1 else 0 := by |
rw [repr_self, Finsupp.single_apply]
| 1 | 2.718282 | 0 | 0.9 | 10 | 779 |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 154 | 158 | theorem repr_symm_apply (v) : b.repr.symm v = Finsupp.total ΞΉ M R b v :=
calc
b.repr.symm v = b.repr.symm (v.sum Finsupp.single) := by | simp
_ = v.sum fun i vi => b.repr.symm (Finsupp.single i vi) := map_finsupp_sum ..
_ = Finsupp.total ΞΉ M R b v := by simp only [repr_symm_single, Finsupp.total_apply]
| 3 | 20.085537 | 1 | 0.9 | 10 | 779 |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 167 | 169 | theorem repr_total (v) : b.repr (Finsupp.total _ _ _ b v) = v := by |
rw [β b.coe_repr_symm]
exact b.repr.apply_symm_apply v
| 2 | 7.389056 | 1 | 0.9 | 10 | 779 |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 173 | 175 | theorem total_repr : Finsupp.total _ _ _ b (b.repr x) = x := by |
rw [β b.coe_repr_symm]
exact b.repr.symm_apply_apply x
| 2 | 7.389056 | 1 | 0.9 | 10 | 779 |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 178 | 179 | theorem repr_range : LinearMap.range (b.repr : M ββ[R] ΞΉ ββ R) = Finsupp.supported R R univ := by |
rw [LinearEquiv.range, Finsupp.supported_univ]
| 1 | 2.718282 | 0 | 0.9 | 10 | 779 |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 186 | 189 | theorem repr_support_subset_of_mem_span (s : Set ΞΉ) {m : M}
(hm : m β span R (b '' s)) : β(b.repr m).support β s := by |
rcases (Finsupp.mem_span_image_iff_total _).1 hm with β¨l, hl, rflβ©
rwa [repr_total, β Finsupp.mem_supported R l]
| 2 | 7.389056 | 1 | 0.9 | 10 | 779 |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 197 | 199 | theorem self_mem_span_image [Nontrivial R] {i : ΞΉ} {s : Set ΞΉ} :
b i β span R (b '' s) β i β s := by |
simp [mem_span_image, Finsupp.support_single_ne_zero]
| 1 | 2.718282 | 0 | 0.9 | 10 | 779 |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 231 | 236 | theorem coe_sumCoords_eq_finsum : (b.sumCoords : M β R) = fun m => βαΆ i, b.coord i m := by |
ext m
simp only [Basis.sumCoords, Basis.coord, Finsupp.lapply_apply, LinearMap.id_coe,
LinearEquiv.coe_coe, Function.comp_apply, Finsupp.coe_lsum, LinearMap.coe_comp,
finsum_eq_sum _ (b.repr m).finite_support, Finsupp.sum, Finset.finite_toSet_toFinset, id,
Finsupp.fun_support_eq]
| 5 | 148.413159 | 2 | 0.9 | 10 | 779 |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 240 | 246 | theorem coe_sumCoords_of_fintype [Fintype ΞΉ] : (b.sumCoords : M β R) = β i, b.coord i := by |
ext m
-- Porting note: - `eq_self_iff_true`
-- + `comp_apply` `LinearMap.coeFn_sum`
simp only [sumCoords, Finsupp.sum_fintype, LinearMap.id_coe, LinearEquiv.coe_coe, coord_apply,
id, Fintype.sum_apply, imp_true_iff, Finsupp.coe_lsum, LinearMap.coe_comp, comp_apply,
LinearMap.coeFn_sum]
| 6 | 403.428793 | 2 | 0.9 | 10 | 779 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 61 | 62 | theorem eq_zero_of_lt {n Ξ½ : β} (h : n < Ξ½) : bernsteinPolynomial R n Ξ½ = 0 := by |
simp [bernsteinPolynomial, Nat.choose_eq_zero_of_lt h]
| 1 | 2.718282 | 0 | 0.9 | 10 | 780 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 70 | 71 | theorem map (f : R β+* S) (n Ξ½ : β) :
(bernsteinPolynomial R n Ξ½).map f = bernsteinPolynomial S n Ξ½ := by | simp [bernsteinPolynomial]
| 1 | 2.718282 | 0 | 0.9 | 10 | 780 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 76 | 78 | theorem flip (n Ξ½ : β) (h : Ξ½ β€ n) :
(bernsteinPolynomial R n Ξ½).comp (1 - X) = bernsteinPolynomial R n (n - Ξ½) := by |
simp [bernsteinPolynomial, h, tsub_tsub_assoc, mul_right_comm]
| 1 | 2.718282 | 0 | 0.9 | 10 | 780 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 81 | 83 | theorem flip' (n Ξ½ : β) (h : Ξ½ β€ n) :
bernsteinPolynomial R n Ξ½ = (bernsteinPolynomial R n (n - Ξ½)).comp (1 - X) := by |
simp [β flip _ _ _ h, Polynomial.comp_assoc]
| 1 | 2.718282 | 0 | 0.9 | 10 | 780 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 86 | 90 | theorem eval_at_0 (n Ξ½ : β) : (bernsteinPolynomial R n Ξ½).eval 0 = if Ξ½ = 0 then 1 else 0 := by |
rw [bernsteinPolynomial]
split_ifs with h
Β· subst h; simp
Β· simp [zero_pow h]
| 4 | 54.59815 | 2 | 0.9 | 10 | 780 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 93 | 99 | theorem eval_at_1 (n Ξ½ : β) : (bernsteinPolynomial R n Ξ½).eval 1 = if Ξ½ = n then 1 else 0 := by |
rw [bernsteinPolynomial]
split_ifs with h
Β· subst h; simp
Β· obtain hΞ½n | hnΞ½ := Ne.lt_or_lt h
Β· simp [zero_pow $ Nat.sub_ne_zero_of_lt hΞ½n]
Β· simp [Nat.choose_eq_zero_of_lt hnΞ½]
| 6 | 403.428793 | 2 | 0.9 | 10 | 780 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 102 | 131 | theorem derivative_succ_aux (n Ξ½ : β) :
Polynomial.derivative (bernsteinPolynomial R (n + 1) (Ξ½ + 1)) =
(n + 1) * (bernsteinPolynomial R n Ξ½ - bernsteinPolynomial R n (Ξ½ + 1)) := by |
rw [bernsteinPolynomial]
suffices ((n + 1).choose (Ξ½ + 1) : R[X]) * ((β(Ξ½ + 1 : β) : R[X]) * X ^ Ξ½) * (1 - X) ^ (n - Ξ½) -
((n + 1).choose (Ξ½ + 1) : R[X]) * X ^ (Ξ½ + 1) * ((β(n - Ξ½) : R[X]) * (1 - X) ^ (n - Ξ½ - 1)) =
(β(n + 1) : R[X]) * ((n.choose Ξ½ : R[X]) * X ^ Ξ½ * (1 - X) ^ (n - Ξ½) -
(n.choos... | 27 | 532,048,240,601.79865 | 2 | 0.9 | 10 | 780 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 134 | 138 | theorem derivative_succ (n Ξ½ : β) : Polynomial.derivative (bernsteinPolynomial R n (Ξ½ + 1)) =
n * (bernsteinPolynomial R (n - 1) Ξ½ - bernsteinPolynomial R (n - 1) (Ξ½ + 1)) := by |
cases n
Β· simp [bernsteinPolynomial]
Β· rw [Nat.cast_succ]; apply derivative_succ_aux
| 3 | 20.085537 | 1 | 0.9 | 10 | 780 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 141 | 143 | theorem derivative_zero (n : β) :
Polynomial.derivative (bernsteinPolynomial R n 0) = -n * bernsteinPolynomial R (n - 1) 0 := by |
simp [bernsteinPolynomial, Polynomial.derivative_pow]
| 1 | 2.718282 | 0 | 0.9 | 10 | 780 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 146 | 161 | theorem iterate_derivative_at_0_eq_zero_of_lt (n : β) {Ξ½ k : β} :
k < Ξ½ β (Polynomial.derivative^[k] (bernsteinPolynomial R n Ξ½)).eval 0 = 0 := by |
cases' Ξ½ with Ξ½
Β· rintro β¨β©
Β· rw [Nat.lt_succ_iff]
induction' k with k ih generalizing n Ξ½
Β· simp [eval_at_0]
Β· simp only [derivative_succ, Int.natCast_eq_zero, mul_eq_zero, Function.comp_apply,
Function.iterate_succ, Polynomial.iterate_derivative_sub,
Polynomial.iterate_derivative_na... | 14 | 1,202,604.284165 | 2 | 0.9 | 10 | 780 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PowerBasis
#align_import field_theory.separable from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
... | Mathlib/FieldTheory/Separable.lean | 52 | 54 | theorem not_separable_zero [Nontrivial R] : Β¬Separable (0 : R[X]) := by |
rintro β¨x, y, hβ©
simp only [derivative_zero, mul_zero, add_zero, zero_ne_one] at h
| 2 | 7.389056 | 1 | 0.9 | 10 | 781 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PowerBasis
#align_import field_theory.separable from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
... | Mathlib/FieldTheory/Separable.lean | 66 | 67 | theorem separable_of_subsingleton [Subsingleton R] (f : R[X]) : f.Separable := by |
simp [Separable, IsCoprime, eq_iff_true_of_subsingleton]
| 1 | 2.718282 | 0 | 0.9 | 10 | 781 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PowerBasis
#align_import field_theory.separable from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
... | Mathlib/FieldTheory/Separable.lean | 70 | 72 | theorem separable_X_add_C (a : R) : (X + C a).Separable := by |
rw [separable_def, derivative_add, derivative_X, derivative_C, add_zero]
exact isCoprime_one_right
| 2 | 7.389056 | 1 | 0.9 | 10 | 781 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PowerBasis
#align_import field_theory.separable from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
... | Mathlib/FieldTheory/Separable.lean | 76 | 78 | theorem separable_X : (X : R[X]).Separable := by |
rw [separable_def, derivative_X]
exact isCoprime_one_right
| 2 | 7.389056 | 1 | 0.9 | 10 | 781 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PowerBasis
#align_import field_theory.separable from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
... | Mathlib/FieldTheory/Separable.lean | 82 | 83 | theorem separable_C (r : R) : (C r).Separable β IsUnit r := by |
rw [separable_def, derivative_C, isCoprime_zero_right, isUnit_C]
| 1 | 2.718282 | 0 | 0.9 | 10 | 781 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PowerBasis
#align_import field_theory.separable from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
... | Mathlib/FieldTheory/Separable.lean | 87 | 89 | theorem Separable.of_mul_left {f g : R[X]} (h : (f * g).Separable) : f.Separable := by |
have := h.of_mul_left_left; rw [derivative_mul] at this
exact IsCoprime.of_mul_right_left (IsCoprime.of_add_mul_left_right this)
| 2 | 7.389056 | 1 | 0.9 | 10 | 781 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PowerBasis
#align_import field_theory.separable from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
... | Mathlib/FieldTheory/Separable.lean | 92 | 94 | theorem Separable.of_mul_right {f g : R[X]} (h : (f * g).Separable) : g.Separable := by |
rw [mul_comm] at h
exact h.of_mul_left
| 2 | 7.389056 | 1 | 0.9 | 10 | 781 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PowerBasis
#align_import field_theory.separable from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
... | Mathlib/FieldTheory/Separable.lean | 97 | 99 | theorem Separable.of_dvd {f g : R[X]} (hf : f.Separable) (hfg : g β£ f) : g.Separable := by |
rcases hfg with β¨f', rflβ©
exact Separable.of_mul_left hf
| 2 | 7.389056 | 1 | 0.9 | 10 | 781 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PowerBasis
#align_import field_theory.separable from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
... | Mathlib/FieldTheory/Separable.lean | 112 | 114 | theorem Separable.isCoprime {f g : R[X]} (h : (f * g).Separable) : IsCoprime f g := by |
have := h.of_mul_left_left; rw [derivative_mul] at this
exact IsCoprime.of_mul_right_right (IsCoprime.of_add_mul_left_right this)
| 2 | 7.389056 | 1 | 0.9 | 10 | 781 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PowerBasis
#align_import field_theory.separable from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
... | Mathlib/FieldTheory/Separable.lean | 138 | 149 | theorem _root_.Associated.separable {f g : R[X]}
(ha : Associated f g) (h : f.Separable) : g.Separable := by |
obtain β¨β¨u, v, h1, h2β©, haβ© := ha
obtain β¨a, b, hβ© := h
refine β¨a * v + b * derivative v, b * v, ?_β©
replace h := congr($h * $(h1))
have h3 := congr(derivative $(h1))
simp only [β ha, derivative_mul, derivative_one] at h3 β’
calc
_ = (a * f + b * derivative f) * (u * v)
+ (b * f) * (derivative u... | 10 | 22,026.465795 | 2 | 0.9 | 10 | 781 |
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {Ξ± Ξ² : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 28 | 28 | theorem join_singleton (l : List Ξ±) : [l].join = l := by | rw [join, join, append_nil]
| 1 | 2.718282 | 0 | 0.9 | 10 | 782 |
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {Ξ± Ξ² : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 38 | 41 | theorem join_append (Lβ Lβ : List (List Ξ±)) : join (Lβ ++ Lβ) = join Lβ ++ join Lβ := by |
induction Lβ
Β· rfl
Β· simp [*]
| 3 | 20.085537 | 1 | 0.9 | 10 | 782 |
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {Ξ± Ξ² : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 44 | 44 | theorem join_concat (L : List (List Ξ±)) (l : List Ξ±) : join (L.concat l) = join L ++ l := by | simp
| 1 | 2.718282 | 0 | 0.9 | 10 | 782 |
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {Ξ± Ξ² : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 60 | 62 | theorem join_filter_ne_nil [DecidablePred fun l : List Ξ± => l β []] {L : List (List Ξ±)} :
join (L.filter fun l => l β []) = L.join := by |
simp [join_filter_not_isEmpty, β isEmpty_iff_eq_nil]
| 1 | 2.718282 | 0 | 0.9 | 10 | 782 |
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {Ξ± Ξ² : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 65 | 66 | theorem join_join (l : List (List (List Ξ±))) : l.join.join = (l.map join).join := by |
induction l <;> simp [*]
| 1 | 2.718282 | 0 | 0.9 | 10 | 782 |
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {Ξ± Ξ² : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 105 | 109 | theorem take_sum_join' (L : List (List Ξ±)) (i : β) :
L.join.take (Nat.sum ((L.map length).take i)) = (L.take i).join := by |
induction L generalizing i
Β· simp
Β· cases i <;> simp [take_append, *]
| 3 | 20.085537 | 1 | 0.9 | 10 | 782 |
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {Ξ± Ξ² : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 115 | 119 | theorem drop_sum_join' (L : List (List Ξ±)) (i : β) :
L.join.drop (Nat.sum ((L.map length).take i)) = (L.drop i).join := by |
induction L generalizing i
Β· simp
Β· cases i <;> simp [drop_append, *]
| 3 | 20.085537 | 1 | 0.9 | 10 | 782 |
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {Ξ± Ξ² : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 123 | 129 | theorem drop_take_succ_eq_cons_get (L : List Ξ±) (i : Fin L.length) :
(L.take (i + 1)).drop i = [get L i] := by |
induction' L with head tail ih
Β· exact (Nat.not_succ_le_zero i i.isLt).elim
rcases i with β¨_ | i, hiβ©
Β· simp
Β· simpa using ih β¨i, Nat.lt_of_succ_lt_succ hiβ©
| 5 | 148.413159 | 2 | 0.9 | 10 | 782 |
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {Ξ± Ξ² : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 135 | 145 | theorem drop_take_succ_eq_cons_nthLe (L : List Ξ±) {i : β} (hi : i < L.length) :
(L.take (i + 1)).drop i = [nthLe L i hi] := by |
induction' L with head tail generalizing i
Β· simp only [length] at hi
exact (Nat.not_succ_le_zero i hi).elim
cases' i with i hi
Β· simp
rfl
have : i < tail.length := by simpa using hi
simp [*]
rfl
| 9 | 8,103.083928 | 2 | 0.9 | 10 | 782 |
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {Ξ± Ξ² : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 153 | 159 | theorem drop_take_succ_join_eq_get' (L : List (List Ξ±)) (i : Fin L.length) :
(L.join.take (Nat.sum ((L.map length).take (i + 1)))).drop (Nat.sum ((L.map length).take i)) =
get L i := by |
have : (L.map length).take i = ((L.take (i + 1)).map length).take i := by
simp [map_take, take_take, Nat.min_eq_left]
simp only [this, length_map, take_sum_join', drop_sum_join', drop_take_succ_eq_cons_get,
join, append_nil]
| 4 | 54.59815 | 2 | 0.9 | 10 | 782 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 40 | 43 | theorem inv_norm_smul_mem_closed_unit_ball (x : E) :
βxββ»ΒΉ β’ x β closedBall (0 : E) 1 := by |
simp only [mem_closedBall_zero_iff, norm_smul, norm_inv, norm_norm, β div_eq_inv_mul,
div_self_le_one]
| 2 | 7.389056 | 1 | 0.9 | 10 | 783 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 46 | 47 | theorem norm_smul_of_nonneg {t : β} (ht : 0 β€ t) (x : E) : βt β’ xβ = t * βxβ := by |
rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg ht]
| 1 | 2.718282 | 0 | 0.9 | 10 | 783 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 50 | 59 | theorem dist_smul_add_one_sub_smul_le {r : β} {x y : E} (h : r β Icc 0 1) :
dist (r β’ x + (1 - r) β’ y) x β€ dist y x :=
calc
dist (r β’ x + (1 - r) β’ y) x = β1 - rβ * βx - yβ := by |
simp_rw [dist_eq_norm', β norm_smul, sub_smul, one_smul, smul_sub, β sub_sub, β sub_add,
sub_right_comm]
_ = (1 - r) * dist y x := by
rw [Real.norm_eq_abs, abs_eq_self.mpr (sub_nonneg.mpr h.2), dist_eq_norm']
_ β€ (1 - 0) * dist y x := by gcongr; exact h.1
_ = dist y x := by rw [sub_zero... | 6 | 403.428793 | 2 | 0.9 | 10 | 783 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 61 | 73 | theorem closure_ball (x : E) {r : β} (hr : r β 0) : closure (ball x r) = closedBall x r := by |
refine Subset.antisymm closure_ball_subset_closedBall fun y hy => ?_
have : ContinuousWithinAt (fun c : β => c β’ (y - x) + x) (Ico 0 1) 1 :=
((continuous_id.smul continuous_const).add continuous_const).continuousWithinAt
convert this.mem_closure _ _
Β· rw [one_smul, sub_add_cancel]
Β· simp [closure_Ico zer... | 12 | 162,754.791419 | 2 | 0.9 | 10 | 783 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 76 | 78 | theorem frontier_ball (x : E) {r : β} (hr : r β 0) :
frontier (ball x r) = sphere x r := by |
rw [frontier, closure_ball x hr, isOpen_ball.interior_eq, closedBall_diff_ball]
| 1 | 2.718282 | 0 | 0.9 | 10 | 783 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 81 | 98 | theorem interior_closedBall (x : E) {r : β} (hr : r β 0) :
interior (closedBall x r) = ball x r := by |
cases' hr.lt_or_lt with hr hr
Β· rw [closedBall_eq_empty.2 hr, ball_eq_empty.2 hr.le, interior_empty]
refine Subset.antisymm ?_ ball_subset_interior_closedBall
intro y hy
rcases (mem_closedBall.1 <| interior_subset hy).lt_or_eq with (hr | rfl)
Β· exact hr
set f : β β E := fun c : β => c β’ (y - x) + x
suf... | 16 | 8,886,110.520508 | 2 | 0.9 | 10 | 783 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 101 | 103 | theorem frontier_closedBall (x : E) {r : β} (hr : r β 0) :
frontier (closedBall x r) = sphere x r := by |
rw [frontier, closure_closedBall, interior_closedBall x hr, closedBall_diff_ball]
| 1 | 2.718282 | 0 | 0.9 | 10 | 783 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 106 | 107 | theorem interior_sphere (x : E) {r : β} (hr : r β 0) : interior (sphere x r) = β
:= by |
rw [β frontier_closedBall x hr, interior_frontier isClosed_ball]
| 1 | 2.718282 | 0 | 0.9 | 10 | 783 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 110 | 111 | theorem frontier_sphere (x : E) {r : β} (hr : r β 0) : frontier (sphere x r) = sphere x r := by |
rw [isClosed_sphere.frontier_eq, interior_sphere x hr, diff_empty]
| 1 | 2.718282 | 0 | 0.9 | 10 | 783 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 124 | 128 | theorem exists_norm_eq {c : β} (hc : 0 β€ c) : β x : E, βxβ = c := by |
rcases exists_ne (0 : E) with β¨x, hxβ©
rw [β norm_ne_zero_iff] at hx
use c β’ βxββ»ΒΉ β’ x
simp [norm_smul, Real.norm_of_nonneg hc, abs_of_nonneg hc, inv_mul_cancel hx]
| 4 | 54.59815 | 2 | 0.9 | 10 | 783 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.simple_graph.inc_matrix from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Finset Matrix SimpleGraph Sym2
open Matrix
namespace SimpleGraph... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 69 | 72 | theorem incMatrix_apply' [Zero R] [One R] [DecidableEq Ξ±] [DecidableRel G.Adj] {a : Ξ±}
{e : Sym2 Ξ±} : G.incMatrix R a e = if e β G.incidenceSet a then 1 else 0 := by |
unfold incMatrix Set.indicator
convert rfl
| 2 | 7.389056 | 1 | 0.9 | 10 | 784 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.simple_graph.inc_matrix from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Finset Matrix SimpleGraph Sym2
open Matrix
namespace SimpleGraph... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 79 | 82 | theorem incMatrix_apply_mul_incMatrix_apply : G.incMatrix R a e * G.incMatrix R b e =
(G.incidenceSet a β© G.incidenceSet b).indicator 1 e := by |
classical simp only [incMatrix, Set.indicator_apply, ite_zero_mul_ite_zero, Pi.one_apply, mul_one,
Set.mem_inter_iff]
| 2 | 7.389056 | 1 | 0.9 | 10 | 784 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.simple_graph.inc_matrix from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Finset Matrix SimpleGraph Sym2
open Matrix
namespace SimpleGraph... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 85 | 89 | theorem incMatrix_apply_mul_incMatrix_apply_of_not_adj (hab : a β b) (h : Β¬G.Adj a b) :
G.incMatrix R a e * G.incMatrix R b e = 0 := by |
rw [incMatrix_apply_mul_incMatrix_apply, Set.indicator_of_not_mem]
rw [G.incidenceSet_inter_incidenceSet_of_not_adj h hab]
exact Set.not_mem_empty e
| 3 | 20.085537 | 1 | 0.9 | 10 | 784 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.simple_graph.inc_matrix from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Finset Matrix SimpleGraph Sym2
open Matrix
namespace SimpleGraph... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 92 | 93 | theorem incMatrix_of_not_mem_incidenceSet (h : e β G.incidenceSet a) : G.incMatrix R a e = 0 := by |
rw [incMatrix_apply, Set.indicator_of_not_mem h]
| 1 | 2.718282 | 0 | 0.9 | 10 | 784 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.simple_graph.inc_matrix from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Finset Matrix SimpleGraph Sym2
open Matrix
namespace SimpleGraph... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 96 | 97 | theorem incMatrix_of_mem_incidenceSet (h : e β G.incidenceSet a) : G.incMatrix R a e = 1 := by |
rw [incMatrix_apply, Set.indicator_of_mem h, Pi.one_apply]
| 1 | 2.718282 | 0 | 0.9 | 10 | 784 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.simple_graph.inc_matrix from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Finset Matrix SimpleGraph Sym2
open Matrix
namespace SimpleGraph... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 102 | 103 | theorem incMatrix_apply_eq_zero_iff : G.incMatrix R a e = 0 β e β G.incidenceSet a := by |
simp only [incMatrix_apply, Set.indicator_apply_eq_zero, Pi.one_apply, one_ne_zero]
| 1 | 2.718282 | 0 | 0.9 | 10 | 784 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.simple_graph.inc_matrix from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Finset Matrix SimpleGraph Sym2
open Matrix
namespace SimpleGraph... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 106 | 112 | theorem incMatrix_apply_eq_one_iff : G.incMatrix R a e = 1 β e β G.incidenceSet a := by |
-- Porting note: was `convert one_ne_zero.ite_eq_left_iff; infer_instance`
unfold incMatrix Set.indicator
simp only [Pi.one_apply]
apply Iff.intro <;> intro h
Β· split at h <;> simp_all only [zero_ne_one]
Β· simp_all only [ite_true]
| 6 | 403.428793 | 2 | 0.9 | 10 | 784 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.simple_graph.inc_matrix from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Finset Matrix SimpleGraph Sym2
open Matrix
namespace SimpleGraph... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 121 | 123 | theorem sum_incMatrix_apply [Fintype (neighborSet G a)] :
β e, G.incMatrix R a e = G.degree a := by |
classical simp [incMatrix_apply', sum_boole, Set.filter_mem_univ_eq_toFinset]
| 1 | 2.718282 | 0 | 0.9 | 10 | 784 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.simple_graph.inc_matrix from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Finset Matrix SimpleGraph Sym2
open Matrix
namespace SimpleGraph... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 126 | 131 | theorem incMatrix_mul_transpose_diag [Fintype (neighborSet G a)] :
(G.incMatrix R * (G.incMatrix R)α΅) a a = G.degree a := by |
classical
rw [β sum_incMatrix_apply]
simp only [mul_apply, incMatrix_apply', transpose_apply, mul_ite, mul_one, mul_zero]
simp_all only [ite_true, sum_boole]
| 4 | 54.59815 | 2 | 0.9 | 10 | 784 |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.simple_graph.inc_matrix from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Finset Matrix SimpleGraph Sym2
open Matrix
namespace SimpleGraph... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 134 | 144 | theorem sum_incMatrix_apply_of_mem_edgeSet [Fintype Ξ±] :
e β G.edgeSet β β a, G.incMatrix R a e = 2 := by |
classical
refine e.ind ?_
intro a b h
rw [mem_edgeSet] at h
rw [β Nat.cast_two, β card_pair h.ne]
simp only [incMatrix_apply', sum_boole, mk'_mem_incidenceSet_iff, h, true_and_iff]
congr 2
ext e
simp only [mem_filter, mem_univ, true_and_iff, mem_insert, mem_singleton]
| 9 | 8,103.083928 | 2 | 0.9 | 10 | 784 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 64 | 65 | theorem cantorFunctionAux_true (h : f n = true) : cantorFunctionAux c f n = c ^ n := by |
simp [cantorFunctionAux, h]
| 1 | 2.718282 | 0 | 0.909091 | 11 | 786 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 69 | 70 | theorem cantorFunctionAux_false (h : f n = false) : cantorFunctionAux c f n = 0 := by |
simp [cantorFunctionAux, h]
| 1 | 2.718282 | 0 | 0.909091 | 11 | 786 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 73 | 75 | theorem cantorFunctionAux_nonneg (h : 0 β€ c) : 0 β€ cantorFunctionAux c f n := by |
cases h' : f n <;> simp [h']
apply pow_nonneg h
| 2 | 7.389056 | 1 | 0.909091 | 11 | 786 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 78 | 79 | theorem cantorFunctionAux_eq (h : f n = g n) :
cantorFunctionAux c f n = cantorFunctionAux c g n := by | simp [cantorFunctionAux, h]
| 1 | 2.718282 | 0 | 0.909091 | 11 | 786 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 82 | 83 | theorem cantorFunctionAux_zero (f : β β Bool) : cantorFunctionAux c f 0 = cond (f 0) 1 0 := by |
cases h : f 0 <;> simp [h]
| 1 | 2.718282 | 0 | 0.909091 | 11 | 786 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.