Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
|---|---|---|---|---|---|---|---|---|---|---|---|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 156 | 161 | theorem fib_coprime_fib_succ (n : β) : Nat.Coprime (fib n) (fib (n + 1)) := by |
induction' n with n ih
Β· simp
Β· rw [fib_add_two]
simp only [coprime_add_self_right]
simp [Coprime, ih.symm]
| 5 | 148.413159 | 2 | 1.181818 | 11 | 1,246 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 165 | 171 | theorem fib_add (m n : β) : fib (m + n + 1) = fib m * fib n + fib (m + 1) * fib (n + 1) := by |
induction' n with n ih generalizing m
Β· simp
Β· specialize ih (m + 1)
rw [add_assoc m 1 n, add_comm 1 n] at ih
simp only [fib_add_two, succ_eq_add_one, ih]
ring
| 6 | 403.428793 | 2 | 1.181818 | 11 | 1,246 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 174 | 179 | theorem fib_two_mul (n : β) : fib (2 * n) = fib n * (2 * fib (n + 1) - fib n) := by |
cases n
Β· simp
Β· rw [two_mul, β add_assoc, fib_add, fib_add_two, two_mul]
simp only [β add_assoc, add_tsub_cancel_right]
ring
| 5 | 148.413159 | 2 | 1.181818 | 11 | 1,246 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 182 | 184 | theorem fib_two_mul_add_one (n : β) : fib (2 * n + 1) = fib (n + 1) ^ 2 + fib n ^ 2 := by |
rw [two_mul, fib_add]
ring
| 2 | 7.389056 | 1 | 1.181818 | 11 | 1,246 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 187 | 194 | theorem fib_two_mul_add_two (n : β) :
fib (2 * n + 2) = fib (n + 1) * (2 * fib n + fib (n + 1)) := by |
rw [fib_add_two, fib_two_mul, fib_two_mul_add_one]
-- Porting note: A bunch of issues similar to [this zulip thread](https://github.com/leanprover-community/mathlib4/pull/1576) with `zify`
have : fib n β€ 2 * fib (n + 1) :=
le_trans fib_le_fib_succ (mul_comm 2 _ βΈ Nat.le_mul_of_pos_right _ two_pos)
zify [th... | 6 | 403.428793 | 2 | 1.181818 | 11 | 1,246 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.ZMod.Basic
#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-c... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 81 | 86 | theorem wittPolynomial_eq_sum_C_mul_X_pow (n : β) :
wittPolynomial p R n = β i β range (n + 1), C ((p : R) ^ i) * X i ^ p ^ (n - i) := by |
apply sum_congr rfl
rintro i -
rw [monomial_eq, Finsupp.prod_single_index]
rw [pow_zero]
| 4 | 54.59815 | 2 | 1.181818 | 11 | 1,247 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.ZMod.Basic
#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-c... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 116 | 119 | theorem map_wittPolynomial (f : R β+* S) (n : β) : map f (W n) = W n := by |
rw [wittPolynomial, map_sum, wittPolynomial]
refine sum_congr rfl fun i _ => ?_
rw [map_monomial, RingHom.map_pow, map_natCast]
| 3 | 20.085537 | 1 | 1.181818 | 11 | 1,247 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.ZMod.Basic
#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-c... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 125 | 132 | theorem constantCoeff_wittPolynomial [hp : Fact p.Prime] (n : β) :
constantCoeff (wittPolynomial p R n) = 0 := by |
simp only [wittPolynomial, map_sum, constantCoeff_monomial]
rw [sum_eq_zero]
rintro i _
rw [if_neg]
rw [Finsupp.single_eq_zero]
exact ne_of_gt (pow_pos hp.1.pos _)
| 6 | 403.428793 | 2 | 1.181818 | 11 | 1,247 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.ZMod.Basic
#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-c... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 136 | 137 | theorem wittPolynomial_zero : wittPolynomial p R 0 = X 0 := by |
simp only [wittPolynomial, X, sum_singleton, range_one, pow_zero, zero_add, tsub_self]
| 1 | 2.718282 | 0 | 1.181818 | 11 | 1,247 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.ZMod.Basic
#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-c... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 141 | 143 | theorem wittPolynomial_one : wittPolynomial p R 1 = C (p : R) * X 1 + X 0 ^ p := by |
simp only [wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ_comm, range_one, sum_singleton,
one_mul, pow_one, C_1, pow_zero, tsub_self, tsub_zero]
| 2 | 7.389056 | 1 | 1.181818 | 11 | 1,247 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.ZMod.Basic
#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-c... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 146 | 148 | theorem aeval_wittPolynomial {A : Type*} [CommRing A] [Algebra R A] (f : β β A) (n : β) :
aeval f (W_ R n) = β i β range (n + 1), (p : A) ^ i * f i ^ p ^ (n - i) := by |
simp [wittPolynomial, AlgHom.map_sum, aeval_monomial, Finsupp.prod_single_index]
| 1 | 2.718282 | 0 | 1.181818 | 11 | 1,247 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.ZMod.Basic
#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-c... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 154 | 163 | theorem wittPolynomial_zmod_self (n : β) :
W_ (ZMod (p ^ (n + 1))) (n + 1) = expand p (W_ (ZMod (p ^ (n + 1))) n) := by |
simp only [wittPolynomial_eq_sum_C_mul_X_pow]
rw [sum_range_succ, β Nat.cast_pow, CharP.cast_eq_zero (ZMod (p ^ (n + 1))) (p ^ (n + 1)), C_0,
zero_mul, add_zero, AlgHom.map_sum, sum_congr rfl]
intro k hk
rw [AlgHom.map_mul, AlgHom.map_pow, expand_X, algHom_C, β pow_mul, β pow_succ']
congr
rw [mem_range... | 8 | 2,980.957987 | 2 | 1.181818 | 11 | 1,247 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.ZMod.Basic
#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-c... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 170 | 181 | theorem wittPolynomial_vars [CharZero R] (n : β) : (wittPolynomial p R n).vars = range (n + 1) := by |
have : β i, (monomial (Finsupp.single i (p ^ (n - i))) ((p : R) ^ i)).vars = {i} := by
intro i
refine vars_monomial_single i (pow_ne_zero _ hp.1) ?_
rw [β Nat.cast_pow, Nat.cast_ne_zero]
exact pow_ne_zero i hp.1
rw [wittPolynomial, vars_sum_of_disjoint]
Β· simp only [this, biUnion_singleton_eq_sel... | 11 | 59,874.141715 | 2 | 1.181818 | 11 | 1,247 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.ZMod.Basic
#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-c... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 184 | 186 | theorem wittPolynomial_vars_subset (n : β) : (wittPolynomial p R n).vars β range (n + 1) := by |
rw [β map_wittPolynomial p (Int.castRingHom R), β wittPolynomial_vars p β€]
apply vars_map
| 2 | 7.389056 | 1 | 1.181818 | 11 | 1,247 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.ZMod.Basic
#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-c... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 211 | 213 | theorem xInTermsOfW_eq [Invertible (p : R)] {n : β} : xInTermsOfW p R n =
(X n - β i β range n, C ((p: R) ^ i) * xInTermsOfW p R i ^ p ^ (n - i)) * C ((β
p : R) ^ n) := by |
rw [xInTermsOfW, β Fin.sum_univ_eq_sum_range]
| 1 | 2.718282 | 0 | 1.181818 | 11 | 1,247 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.ZMod.Basic
#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-c... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 218 | 234 | theorem constantCoeff_xInTermsOfW [hp : Fact p.Prime] [Invertible (p : R)] (n : β) :
constantCoeff (xInTermsOfW p R n) = 0 := by |
apply Nat.strongInductionOn n; clear n
intro n IH
rw [xInTermsOfW_eq, mul_comm, RingHom.map_mul, RingHom.map_sub, map_sum, constantCoeff_C,
constantCoeff_X, zero_sub, mul_neg, neg_eq_zero]
-- Porting note: here, we should be able to do `rw [sum_eq_zero]`, but the goal that
-- is created is not what we ex... | 15 | 3,269,017.372472 | 2 | 1.181818 | 11 | 1,247 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n Ξ± : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 52 | 59 | theorem fromBlocks_eq_of_invertibleββ (A : Matrix m m Ξ±) (B : Matrix m n Ξ±) (C : Matrix l m Ξ±)
(D : Matrix l n Ξ±) [Invertible A] :
fromBlocks A B C D =
fromBlocks 1 0 (C * β
A) 1 * fromBlocks A 0 0 (D - C * β
A * B) *
fromBlocks 1 (β
A * B) 0 1 := by |
simp only [fromBlocks_multiply, Matrix.mul_zero, Matrix.zero_mul, add_zero, zero_add,
Matrix.one_mul, Matrix.mul_one, invOf_mul_self, Matrix.mul_invOf_self_assoc,
Matrix.mul_invOf_mul_self_cancel, Matrix.mul_assoc, add_sub_cancel]
| 3 | 20.085537 | 1 | 1.1875 | 16 | 1,248 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n Ξ± : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 100 | 104 | theorem invOf_fromBlocks_zeroββ_eq (A : Matrix m m Ξ±) (B : Matrix m n Ξ±) (D : Matrix n n Ξ±)
[Invertible A] [Invertible D] [Invertible (fromBlocks A B 0 D)] :
β
(fromBlocks A B 0 D) = fromBlocks (β
A) (-(β
A * B * β
D)) 0 (β
D) := by |
letI := fromBlocksZeroββInvertible A B D
convert (rfl : β
(fromBlocks A B 0 D) = _)
| 2 | 7.389056 | 1 | 1.1875 | 16 | 1,248 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n Ξ± : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 107 | 111 | theorem invOf_fromBlocks_zeroββ_eq (A : Matrix m m Ξ±) (C : Matrix n m Ξ±) (D : Matrix n n Ξ±)
[Invertible A] [Invertible D] [Invertible (fromBlocks A 0 C D)] :
β
(fromBlocks A 0 C D) = fromBlocks (β
A) 0 (-(β
D * C * β
A)) (β
D) := by |
letI := fromBlocksZeroββInvertible A C D
convert (rfl : β
(fromBlocks A 0 C D) = _)
| 2 | 7.389056 | 1 | 1.1875 | 16 | 1,248 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n Ξ± : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 390 | 394 | theorem det_fromBlocksββ (A : Matrix m m Ξ±) (B : Matrix m n Ξ±) (C : Matrix n m Ξ±)
(D : Matrix n n Ξ±) [Invertible A] :
(Matrix.fromBlocks A B C D).det = det A * det (D - C * β
A * B) := by |
rw [fromBlocks_eq_of_invertibleββ (A := A), det_mul, det_mul, det_fromBlocks_zeroββ,
det_fromBlocks_zeroββ, det_fromBlocks_zeroββ, det_one, det_one, one_mul, one_mul, mul_one]
| 2 | 7.389056 | 1 | 1.1875 | 16 | 1,248 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n Ξ± : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 398 | 401 | theorem det_fromBlocks_oneββ (B : Matrix m n Ξ±) (C : Matrix n m Ξ±) (D : Matrix n n Ξ±) :
(Matrix.fromBlocks 1 B C D).det = det (D - C * B) := by |
haveI : Invertible (1 : Matrix m m Ξ±) := invertibleOne
rw [det_fromBlocksββ, invOf_one, Matrix.mul_one, det_one, one_mul]
| 2 | 7.389056 | 1 | 1.1875 | 16 | 1,248 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n Ξ± : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 406 | 413 | theorem det_fromBlocksββ (A : Matrix m m Ξ±) (B : Matrix m n Ξ±) (C : Matrix n m Ξ±)
(D : Matrix n n Ξ±) [Invertible D] :
(Matrix.fromBlocks A B C D).det = det D * det (A - B * β
D * C) := by |
have : fromBlocks A B C D =
(fromBlocks D C B A).submatrix (Equiv.sumComm _ _) (Equiv.sumComm _ _) := by
ext (i j)
cases i <;> cases j <;> rfl
rw [this, det_submatrix_equiv_self, det_fromBlocksββ]
| 5 | 148.413159 | 2 | 1.1875 | 16 | 1,248 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n Ξ± : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 417 | 420 | theorem det_fromBlocks_oneββ (A : Matrix m m Ξ±) (B : Matrix m n Ξ±) (C : Matrix n m Ξ±) :
(Matrix.fromBlocks A B C 1).det = det (A - B * C) := by |
haveI : Invertible (1 : Matrix n n Ξ±) := invertibleOne
rw [det_fromBlocksββ, invOf_one, Matrix.mul_one, det_one, one_mul]
| 2 | 7.389056 | 1 | 1.1875 | 16 | 1,248 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n Ξ± : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 425 | 430 | theorem det_one_add_mul_comm (A : Matrix m n Ξ±) (B : Matrix n m Ξ±) :
det (1 + A * B) = det (1 + B * A) :=
calc
det (1 + A * B) = det (fromBlocks 1 (-A) B 1) := by |
rw [det_fromBlocks_oneββ, Matrix.neg_mul, sub_neg_eq_add]
_ = det (1 + B * A) := by rw [det_fromBlocks_oneββ, Matrix.mul_neg, sub_neg_eq_add]
| 2 | 7.389056 | 1 | 1.1875 | 16 | 1,248 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n Ξ± : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 434 | 435 | theorem det_mul_add_one_comm (A : Matrix m n Ξ±) (B : Matrix n m Ξ±) :
det (A * B + 1) = det (B * A + 1) := by | rw [add_comm, det_one_add_mul_comm, add_comm]
| 1 | 2.718282 | 0 | 1.1875 | 16 | 1,248 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n Ξ± : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 438 | 440 | theorem det_one_sub_mul_comm (A : Matrix m n Ξ±) (B : Matrix n m Ξ±) :
det (1 - A * B) = det (1 - B * A) := by |
rw [sub_eq_add_neg, β Matrix.neg_mul, det_one_add_mul_comm, Matrix.mul_neg, β sub_eq_add_neg]
| 1 | 2.718282 | 0 | 1.1875 | 16 | 1,248 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n Ξ± : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 444 | 446 | theorem det_one_add_col_mul_row (u v : m β Ξ±) : det (1 + col u * row v) = 1 + v β¬α΅₯ u := by |
rw [det_one_add_mul_comm, det_unique, Pi.add_apply, Pi.add_apply, Matrix.one_apply_eq,
Matrix.row_mul_col_apply]
| 2 | 7.389056 | 1 | 1.1875 | 16 | 1,248 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n Ξ± : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 454 | 459 | theorem det_add_col_mul_row {A : Matrix m m Ξ±} (hA : IsUnit A.det) (u v : m β Ξ±) :
(A + col u * row v).det = A.det * (1 + row v * Aβ»ΒΉ * col u).det := by |
nth_rewrite 1 [β Matrix.mul_one A]
rwa [β Matrix.mul_nonsing_inv_cancel_left A (col u * row v),
β Matrix.mul_add, det_mul, β Matrix.mul_assoc, det_one_add_mul_comm,
β Matrix.mul_assoc]
| 4 | 54.59815 | 2 | 1.1875 | 16 | 1,248 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n Ξ± : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 482 | 491 | theorem schur_complement_eqββ [Fintype m] [DecidableEq m] [Fintype n] {A : Matrix m m π}
(B : Matrix m n π) (D : Matrix n n π) (x : m β π) (y : n β π) [Invertible A]
(hA : A.IsHermitian) :
(star (x βα΅₯ y)) α΅₯* (fromBlocks A B Bα΄΄ D) β¬α΅₯ (x βα΅₯ y) =
(star (x + (Aβ»ΒΉ * B) *α΅₯ y)) α΅₯* A β¬α΅₯ (x + (Aβ»ΒΉ * B) *α΅₯... |
simp [Function.star_sum_elim, fromBlocks_mulVec, vecMul_fromBlocks, add_vecMul,
dotProduct_mulVec, vecMul_sub, Matrix.mul_assoc, vecMul_mulVec, hA.eq,
conjTranspose_nonsing_inv, star_mulVec]
abel
| 4 | 54.59815 | 2 | 1.1875 | 16 | 1,248 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n Ξ± : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 494 | 503 | theorem schur_complement_eqββ [Fintype m] [Fintype n] [DecidableEq n] (A : Matrix m m π)
(B : Matrix m n π) {D : Matrix n n π} (x : m β π) (y : n β π) [Invertible D]
(hD : D.IsHermitian) :
(star (x βα΅₯ y)) α΅₯* (fromBlocks A B Bα΄΄ D) β¬α΅₯ (x βα΅₯ y) =
(star ((Dβ»ΒΉ * Bα΄΄) *α΅₯ x + y)) α΅₯* D β¬α΅₯ ((Dβ»ΒΉ * Bα΄΄) *α΅₯ x... |
simp [Function.star_sum_elim, fromBlocks_mulVec, vecMul_fromBlocks, add_vecMul,
dotProduct_mulVec, vecMul_sub, Matrix.mul_assoc, vecMul_mulVec, hD.eq,
conjTranspose_nonsing_inv, star_mulVec]
abel
| 4 | 54.59815 | 2 | 1.1875 | 16 | 1,248 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n Ξ± : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 506 | 519 | theorem IsHermitian.fromBlocksββ [Fintype m] [DecidableEq m] {A : Matrix m m π} (B : Matrix m n π)
(D : Matrix n n π) (hA : A.IsHermitian) :
(Matrix.fromBlocks A B Bα΄΄ D).IsHermitian β (D - Bα΄΄ * Aβ»ΒΉ * B).IsHermitian := by |
have hBAB : (Bα΄΄ * Aβ»ΒΉ * B).IsHermitian := by
apply isHermitian_conjTranspose_mul_mul
apply hA.inv
rw [isHermitian_fromBlocks_iff]
constructor
Β· intro h
apply IsHermitian.sub h.2.2.2 hBAB
Β· intro h
refine β¨hA, rfl, conjTranspose_conjTranspose B, ?_β©
rw [β sub_add_cancel D]
apply IsHerm... | 11 | 59,874.141715 | 2 | 1.1875 | 16 | 1,248 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n Ξ± : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 522 | 527 | theorem IsHermitian.fromBlocksββ [Fintype n] [DecidableEq n] (A : Matrix m m π) (B : Matrix m n π)
{D : Matrix n n π} (hD : D.IsHermitian) :
(Matrix.fromBlocks A B Bα΄΄ D).IsHermitian β (A - B * Dβ»ΒΉ * Bα΄΄).IsHermitian := by |
rw [β isHermitian_submatrix_equiv (Equiv.sumComm n m), Equiv.sumComm_apply,
fromBlocks_submatrix_sum_swap_sum_swap]
convert IsHermitian.fromBlocksββ _ _ hD <;> simp
| 3 | 20.085537 | 1 | 1.1875 | 16 | 1,248 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 61 | 70 | theorem conj_apply {x : ((K β+* β) β β)} (Ο : K β+* β)
(hx : x β Submodule.span β (Set.range (canonicalEmbedding K))) :
conj (x Ο) = x (ComplexEmbedding.conjugate Ο) := by |
refine Submodule.span_induction hx ?_ ?_ (fun _ _ hx hy => ?_) (fun a _ hx => ?_)
Β· rintro _ β¨x, rflβ©
rw [apply_at, apply_at, ComplexEmbedding.conjugate_coe_eq]
Β· rw [Pi.zero_apply, Pi.zero_apply, map_zero]
Β· rw [Pi.add_apply, Pi.add_apply, map_add, hx, hy]
Β· rw [Pi.smul_apply, Complex.real_smul, map_mul... | 7 | 1,096.633158 | 2 | 1.1875 | 16 | 1,249 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 72 | 74 | theorem nnnorm_eq [NumberField K] (x : K) :
βcanonicalEmbedding K xββ = Finset.univ.sup (fun Ο : K β+* β => βΟ xββ) := by |
simp_rw [Pi.nnnorm_def, apply_at]
| 1 | 2.718282 | 0 | 1.1875 | 16 | 1,249 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 76 | 85 | theorem norm_le_iff [NumberField K] (x : K) (r : β) :
βcanonicalEmbedding K xβ β€ r β β Ο : K β+* β, βΟ xβ β€ r := by |
obtain hr | hr := lt_or_le r 0
Β· obtain β¨Οβ© := (inferInstance : Nonempty (K β+* β))
refine iff_of_false ?_ ?_
Β· exact (hr.trans_le (norm_nonneg _)).not_le
Β· exact fun h => hr.not_le (le_trans (norm_nonneg _) (h Ο))
Β· lift r to NNReal using hr
simp_rw [β coe_nnnorm, nnnorm_eq, NNReal.coe_le_coe, F... | 8 | 2,980.957987 | 2 | 1.1875 | 16 | 1,249 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 93 | 105 | theorem integerLattice.inter_ball_finite [NumberField K] (r : β) :
((integerLattice K : Set ((K β+* β) β β)) β© Metric.closedBall 0 r).Finite := by |
obtain hr | _ := lt_or_le r 0
Β· simp [Metric.closedBall_eq_empty.2 hr]
Β· have heq : β x, canonicalEmbedding K x β Metric.closedBall 0 r β
β Ο : K β+* β, βΟ xβ β€ r := by
intro x; rw [β norm_le_iff, mem_closedBall_zero_iff]
convert (Embeddings.finite_of_norm_le K β r).image (canonicalEmbedding K)... | 11 | 59,874.141715 | 2 | 1.1875 | 16 | 1,249 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 139 | 142 | theorem latticeBasis_apply [NumberField K] (i : Free.ChooseBasisIndex β€ (π K)) :
latticeBasis K i = (canonicalEmbedding K) (integralBasis K i) := by |
simp only [latticeBasis, integralBasis_apply, coe_basisOfLinearIndependentOfCardEqFinrank,
Function.comp_apply, Equiv.apply_symm_apply]
| 2 | 7.389056 | 1 | 1.1875 | 16 | 1,249 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 185 | 187 | theorem _root_.NumberField.mixedEmbedding_injective [NumberField K] :
Function.Injective (NumberField.mixedEmbedding K) := by |
exact RingHom.injective _
| 1 | 2.718282 | 0 | 1.1875 | 16 | 1,249 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 259 | 262 | theorem normAtPlace_nonneg (w : InfinitePlace K) (x : E K) :
0 β€ normAtPlace w x := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]
split_ifs <;> exact norm_nonneg _
| 2 | 7.389056 | 1 | 1.1875 | 16 | 1,249 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 264 | 267 | theorem normAtPlace_neg (w : InfinitePlace K) (x : E K) :
normAtPlace w (- x) = normAtPlace w x := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]
split_ifs <;> simp
| 2 | 7.389056 | 1 | 1.1875 | 16 | 1,249 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 269 | 272 | theorem normAtPlace_add_le (w : InfinitePlace K) (x y : E K) :
normAtPlace w (x + y) β€ normAtPlace w x + normAtPlace w y := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]
split_ifs <;> exact norm_add_le _ _
| 2 | 7.389056 | 1 | 1.1875 | 16 | 1,249 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 274 | 279 | theorem normAtPlace_smul (w : InfinitePlace K) (x : E K) (c : β) :
normAtPlace w (c β’ x) = |c| * normAtPlace w x := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]
split_ifs
Β· rw [Prod.smul_fst, Pi.smul_apply, norm_smul, Real.norm_eq_abs]
Β· rw [Prod.smul_snd, Pi.smul_apply, norm_smul, Real.norm_eq_abs, Complex.norm_eq_abs]
| 4 | 54.59815 | 2 | 1.1875 | 16 | 1,249 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 281 | 284 | theorem normAtPlace_real (w : InfinitePlace K) (c : β) :
normAtPlace w ((fun _ β¦ c, fun _ β¦ c) : (E K)) = |c| := by |
rw [show ((fun _ β¦ c, fun _ β¦ c) : (E K)) = c β’ 1 by ext <;> simp, normAtPlace_smul, map_one,
mul_one]
| 2 | 7.389056 | 1 | 1.1875 | 16 | 1,249 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 286 | 288 | theorem normAtPlace_apply_isReal {w : InfinitePlace K} (hw : IsReal w) (x : E K):
normAtPlace w x = βx.1 β¨w, hwβ©β := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, dif_pos]
| 1 | 2.718282 | 0 | 1.1875 | 16 | 1,249 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 290 | 293 | theorem normAtPlace_apply_isComplex {w : InfinitePlace K} (hw : IsComplex w) (x : E K) :
normAtPlace w x = βx.2 β¨w, hwβ©β := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk,
dif_neg (not_isReal_iff_isComplex.mpr hw)]
| 2 | 7.389056 | 1 | 1.1875 | 16 | 1,249 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 296 | 300 | theorem normAtPlace_apply (w : InfinitePlace K) (x : K) :
normAtPlace w (mixedEmbedding K x) = w x := by |
simp_rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, mixedEmbedding,
RingHom.prod_apply, Pi.ringHom_apply, norm_embedding_of_isReal, norm_embedding_eq, dite_eq_ite,
ite_id]
| 3 | 20.085537 | 1 | 1.1875 | 16 | 1,249 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 302 | 308 | theorem normAtPlace_eq_zero {x : E K} :
(β w, normAtPlace w x = 0) β x = 0 := by |
refine β¨fun h β¦ ?_, fun h β¦ ?_β©
Β· ext w
Β· exact norm_eq_zero'.mp (normAtPlace_apply_isReal w.prop _ βΈ h w.1)
Β· exact norm_eq_zero'.mp (normAtPlace_apply_isComplex w.prop _ βΈ h w.1)
Β· simp_rw [h, map_zero, implies_true]
| 5 | 148.413159 | 2 | 1.1875 | 16 | 1,249 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 312 | 323 | theorem nnnorm_eq_sup_normAtPlace (x : E K) :
βxββ = univ.sup fun w β¦ β¨normAtPlace w x, normAtPlace_nonneg w xβ© := by |
rw [show (univ : Finset (InfinitePlace K)) = (univ.image
(fun w : {w : InfinitePlace K // IsReal w} β¦ w.1)) βͺ
(univ.image (fun w : {w : InfinitePlace K // IsComplex w} β¦ w.1))
by ext; simp [isReal_or_isComplex], sup_union, univ.sup_image, univ.sup_image, sup_eq_max,
Prod.nnnorm_def', Pi.nnnorm_def, P... | 10 | 22,026.465795 | 2 | 1.1875 | 16 | 1,249 |
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 83 | 86 | theorem nonempty_monotone_mem {s : Set Ξ±} (hs : s.Nonempty) :
Nonempty { u // Monotone u β§ β i : β, u i β s } := by |
obtain β¨x, hxβ© := hs
exact β¨β¨fun _ => x, fun i j _ => le_rfl, fun _ => hxβ©β©
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,250 |
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 89 | 94 | theorem eq_of_edist_zero_on {f f' : Ξ± β E} {s : Set Ξ±} (h : β β¦xβ¦, x β s β edist (f x) (f' x) = 0) :
eVariationOn f s = eVariationOn f' s := by |
dsimp only [eVariationOn]
congr 1 with p : 1
congr 1 with i : 1
rw [edist_congr_right (h <| p.snd.prop.2 (i + 1)), edist_congr_left (h <| p.snd.prop.2 i)]
| 4 | 54.59815 | 2 | 1.2 | 5 | 1,250 |
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 107 | 124 | theorem sum_le_of_monotoneOn_Icc (f : Ξ± β E) {s : Set Ξ±} {m n : β} {u : β β Ξ±}
(hu : MonotoneOn u (Icc m n)) (us : β i β Icc m n, u i β s) :
(β i β Finset.Ico m n, edist (f (u (i + 1))) (f (u i))) β€ eVariationOn f s := by |
rcases le_total n m with hnm | hmn
Β· simp [Finset.Ico_eq_empty_of_le hnm]
let Ο := projIcc m n hmn
let v i := u (Ο i)
calc
β i β Finset.Ico m n, edist (f (u (i + 1))) (f (u i))
= β i β Finset.Ico m n, edist (f (v (i + 1))) (f (v i)) :=
Finset.sum_congr rfl fun i hi β¦ by
rw [Finset.m... | 15 | 3,269,017.372472 | 2 | 1.2 | 5 | 1,250 |
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 127 | 130 | theorem sum_le_of_monotoneOn_Iic (f : Ξ± β E) {s : Set Ξ±} {n : β} {u : β β Ξ±}
(hu : MonotoneOn u (Iic n)) (us : β i β€ n, u i β s) :
(β i β Finset.range n, edist (f (u (i + 1))) (f (u i))) β€ eVariationOn f s := by |
simpa using sum_le_of_monotoneOn_Icc f (m := 0) (hu.mono Icc_subset_Iic_self) fun i hi β¦ us i hi.2
| 1 | 2.718282 | 0 | 1.2 | 5 | 1,250 |
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 133 | 136 | theorem mono (f : Ξ± β E) {s t : Set Ξ±} (hst : t β s) : eVariationOn f t β€ eVariationOn f s := by |
apply iSup_le _
rintro β¨n, β¨u, hu, utβ©β©
exact sum_le f n hu fun i => hst (ut i)
| 3 | 20.085537 | 1 | 1.2 | 5 | 1,250 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 32 | 34 | theorem sq_ne_two_fin_zmod_four (z : ZMod 4) : z * z β 2 := by |
change Fin 4 at z
fin_cases z <;> decide
| 2 | 7.389056 | 1 | 1.2 | 10 | 1,251 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 37 | 40 | theorem Int.sq_ne_two_mod_four (z : β€) : z * z % 4 β 2 := by |
suffices Β¬z * z % (4 : β) = 2 % (4 : β) by exact this
rw [β ZMod.intCast_eq_intCast_iff']
simpa using sq_ne_two_fin_zmod_four _
| 3 | 20.085537 | 1 | 1.2 | 10 | 1,251 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 54 | 56 | theorem pythagoreanTriple_comm {x y z : β€} : PythagoreanTriple x y z β PythagoreanTriple y x z := by |
delta PythagoreanTriple
rw [add_comm]
| 2 | 7.389056 | 1 | 1.2 | 10 | 1,251 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 60 | 61 | theorem PythagoreanTriple.zero : PythagoreanTriple 0 0 0 := by |
simp only [PythagoreanTriple, zero_mul, zero_add]
| 1 | 2.718282 | 0 | 1.2 | 10 | 1,251 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 73 | 73 | theorem symm : PythagoreanTriple y x z := by | rwa [pythagoreanTriple_comm]
| 1 | 2.718282 | 0 | 1.2 | 10 | 1,251 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 78 | 82 | theorem mul (k : β€) : PythagoreanTriple (k * x) (k * y) (k * z) :=
calc
k * x * (k * x) + k * y * (k * y) = k ^ 2 * (x * x + y * y) := by | ring
_ = k ^ 2 * (z * z) := by rw [h.eq]
_ = k * z * (k * z) := by ring
| 3 | 20.085537 | 1 | 1.2 | 10 | 1,251 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 87 | 93 | theorem mul_iff (k : β€) (hk : k β 0) :
PythagoreanTriple (k * x) (k * y) (k * z) β PythagoreanTriple x y z := by |
refine β¨?_, fun h => h.mul kβ©
simp only [PythagoreanTriple]
intro h
rw [β mul_left_inj' (mul_ne_zero hk hk)]
convert h using 1 <;> ring
| 5 | 148.413159 | 2 | 1.2 | 10 | 1,251 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 120 | 129 | theorem mul_isClassified (k : β€) (hc : h.IsClassified) : (h.mul k).IsClassified := by |
obtain β¨l, m, n, β¨β¨rfl, rflβ© | β¨rfl, rflβ©, coβ©β© := hc
Β· use k * l, m, n
apply And.intro _ co
left
constructor <;> ring
Β· use k * l, m, n
apply And.intro _ co
right
constructor <;> ring
| 9 | 8,103.083928 | 2 | 1.2 | 10 | 1,251 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 132 | 161 | theorem even_odd_of_coprime (hc : Int.gcd x y = 1) :
x % 2 = 0 β§ y % 2 = 1 β¨ x % 2 = 1 β§ y % 2 = 0 := by |
cases' Int.emod_two_eq_zero_or_one x with hx hx <;>
cases' Int.emod_two_eq_zero_or_one y with hy hy
-- x even, y even
Β· exfalso
apply Nat.not_coprime_of_dvd_of_dvd (by decide : 1 < 2) _ _ hc
Β· apply Int.natCast_dvd.1
apply Int.dvd_of_emod_eq_zero hx
Β· apply Int.natCast_dvd.1
apply Int... | 28 | 1,446,257,064,291.475 | 2 | 1.2 | 10 | 1,251 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 164 | 182 | theorem gcd_dvd : (Int.gcd x y : β€) β£ z := by |
by_cases h0 : Int.gcd x y = 0
Β· have hx : x = 0 := by
apply Int.natAbs_eq_zero.mp
apply Nat.eq_zero_of_gcd_eq_zero_left h0
have hy : y = 0 := by
apply Int.natAbs_eq_zero.mp
apply Nat.eq_zero_of_gcd_eq_zero_right h0
have hz : z = 0 := by
simpa only [PythagoreanTriple, hx, hy, a... | 18 | 65,659,969.137331 | 2 | 1.2 | 10 | 1,251 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-community/mathlib"@"de87d5053a9fe5cbde723172c0fb7e27e7436473"
open Function
variable {Ξ± Ξ² : Type*}
section CovariantClassMulLe
variable [LinearOrder Ξ±]
section Mul
variable [Mul Ξ±]
@[to_additive... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 90 | 94 | theorem lt_or_lt_of_mul_lt_mul [CovariantClass Ξ± Ξ± (Β· * Β·) (Β· β€ Β·)]
[CovariantClass Ξ± Ξ± (Function.swap (Β· * Β·)) (Β· β€ Β·)] {aβ aβ bβ bβ : Ξ±} :
aβ * bβ < aβ * bβ β aβ < aβ β¨ bβ < bβ := by |
contrapose!
exact fun h => mul_le_mul' h.1 h.2
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,252 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-community/mathlib"@"de87d5053a9fe5cbde723172c0fb7e27e7436473"
open Function
variable {Ξ± Ξ² : Type*}
section CovariantClassMulLe
variable [LinearOrder Ξ±]
section Mul
variable [Mul Ξ±]
@[to_additive... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 99 | 103 | theorem le_or_lt_of_mul_le_mul [CovariantClass Ξ± Ξ± (Β· * Β·) (Β· β€ Β·)]
[CovariantClass Ξ± Ξ± (Function.swap (Β· * Β·)) (Β· < Β·)] {aβ aβ bβ bβ : Ξ±} :
aβ * bβ β€ aβ * bβ β aβ β€ aβ β¨ bβ < bβ := by |
contrapose!
exact fun h => mul_lt_mul_of_lt_of_le h.1 h.2
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,252 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-community/mathlib"@"de87d5053a9fe5cbde723172c0fb7e27e7436473"
open Function
variable {Ξ± Ξ² : Type*}
section CovariantClassMulLe
variable [LinearOrder Ξ±]
section Mul
variable [Mul Ξ±]
@[to_additive... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 108 | 112 | theorem lt_or_le_of_mul_le_mul [CovariantClass Ξ± Ξ± (Β· * Β·) (Β· < Β·)]
[CovariantClass Ξ± Ξ± (Function.swap (Β· * Β·)) (Β· β€ Β·)] {aβ aβ bβ bβ : Ξ±} :
aβ * bβ β€ aβ * bβ β aβ < aβ β¨ bβ β€ bβ := by |
contrapose!
exact fun h => mul_lt_mul_of_le_of_lt h.1 h.2
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,252 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-community/mathlib"@"de87d5053a9fe5cbde723172c0fb7e27e7436473"
open Function
variable {Ξ± Ξ² : Type*}
section CovariantClassMulLe
variable [LinearOrder Ξ±]
section Mul
variable [Mul Ξ±]
@[to_additive... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 117 | 121 | theorem le_or_le_of_mul_le_mul [CovariantClass Ξ± Ξ± (Β· * Β·) (Β· < Β·)]
[CovariantClass Ξ± Ξ± (Function.swap (Β· * Β·)) (Β· < Β·)] {aβ aβ bβ bβ : Ξ±} :
aβ * bβ β€ aβ * bβ β aβ β€ aβ β¨ bβ β€ bβ := by |
contrapose!
exact fun h => mul_lt_mul_of_lt_of_lt h.1 h.2
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,252 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-community/mathlib"@"de87d5053a9fe5cbde723172c0fb7e27e7436473"
open Function
variable {Ξ± Ξ² : Type*}
section CovariantClassMulLe
variable [LinearOrder Ξ±]
section Mul
variable [Mul Ξ±]
@[to_additive... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 126 | 133 | theorem mul_lt_mul_iff_of_le_of_le [CovariantClass Ξ± Ξ± (Β· * Β·) (Β· β€ Β·)]
[CovariantClass Ξ± Ξ± (Function.swap (Β· * Β·)) (Β· β€ Β·)] [CovariantClass Ξ± Ξ± (Β· * Β·) (Β· < Β·)]
[CovariantClass Ξ± Ξ± (Function.swap (Β· * Β·)) (Β· < Β·)] {aβ aβ bβ bβ : Ξ±} (ha : aβ β€ aβ)
(hb : bβ β€ bβ) : aβ * bβ < aβ * bβ β aβ < aβ β¨ bβ < bβ := by |
refine β¨lt_or_lt_of_mul_lt_mul, fun h => ?_β©
cases' h with ha' hb'
Β· exact mul_lt_mul_of_lt_of_le ha' hb
Β· exact mul_lt_mul_of_le_of_lt ha hb'
| 4 | 54.59815 | 2 | 1.2 | 5 | 1,252 |
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Pointwise
#align_import analysis.normed_space.is_R_or_C from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric
variable {π : Type*} [RCLike π] {E :... | Mathlib/Analysis/NormedSpace/RCLike.lean | 36 | 36 | theorem RCLike.norm_coe_norm {z : E} : β(βzβ : π)β = βzβ := by | simp
| 1 | 2.718282 | 0 | 1.2 | 5 | 1,253 |
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Pointwise
#align_import analysis.normed_space.is_R_or_C from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric
variable {π : Type*} [RCLike π] {E :... | Mathlib/Analysis/NormedSpace/RCLike.lean | 43 | 45 | theorem norm_smul_inv_norm {x : E} (hx : x β 0) : β(βxββ»ΒΉ : π) β’ xβ = 1 := by |
have : βxβ β 0 := by simp [hx]
field_simp [norm_smul]
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,253 |
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Pointwise
#align_import analysis.normed_space.is_R_or_C from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric
variable {π : Type*} [RCLike π] {E :... | Mathlib/Analysis/NormedSpace/RCLike.lean | 49 | 52 | theorem norm_smul_inv_norm' {r : β} (r_nonneg : 0 β€ r) {x : E} (hx : x β 0) :
β((r : π) * (βxβ : π)β»ΒΉ) β’ xβ = r := by |
have : βxβ β 0 := by simp [hx]
field_simp [norm_smul, r_nonneg, rclike_simps]
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,253 |
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Pointwise
#align_import analysis.normed_space.is_R_or_C from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric
variable {π : Type*} [RCLike π] {E :... | Mathlib/Analysis/NormedSpace/RCLike.lean | 55 | 75 | theorem LinearMap.bound_of_sphere_bound {r : β} (r_pos : 0 < r) (c : β) (f : E ββ[π] π)
(h : β z β sphere (0 : E) r, βf zβ β€ c) (z : E) : βf zβ β€ c / r * βzβ := by |
by_cases z_zero : z = 0
Β· rw [z_zero]
simp only [LinearMap.map_zero, norm_zero, mul_zero]
exact le_rfl
set zβ := ((r : π) * (βzβ : π)β»ΒΉ) β’ z with hzβ
have norm_f_zβ : βf zββ β€ c := by
apply h
rw [mem_sphere_zero_iff_norm]
exact norm_smul_inv_norm' r_pos.le z_zero
have r_ne_zero : (r : οΏ½... | 19 | 178,482,300.963187 | 2 | 1.2 | 5 | 1,253 |
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Pointwise
#align_import analysis.normed_space.is_R_or_C from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric
variable {π : Type*} [RCLike π] {E :... | Mathlib/Analysis/NormedSpace/RCLike.lean | 85 | 93 | theorem ContinuousLinearMap.opNorm_bound_of_ball_bound {r : β} (r_pos : 0 < r) (c : β)
(f : E βL[π] π) (h : β z β closedBall (0 : E) r, βf zβ β€ c) : βfβ β€ c / r := by |
apply ContinuousLinearMap.opNorm_le_bound
Β· apply div_nonneg _ r_pos.le
exact
(norm_nonneg _).trans
(h 0 (by simp only [norm_zero, mem_closedBall, dist_zero_left, r_pos.le]))
apply LinearMap.bound_of_ball_bound' r_pos
exact fun z hz => h z hz
| 7 | 1,096.633158 | 2 | 1.2 | 5 | 1,253 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.NormedSpace.Connected
import Mathlib.LinearAlgebra.AffineSpace.ContinuousAffineEquiv
open Set
variable {F : Type*} [AddCommGroup F] [Module β F] [TopologicalSpace F]
def AmpleSet (s : Set F) : Prop :=
β x β s, convexHull β (connectedComponentIn s ... | Mathlib/Analysis/Convex/AmpleSet.lean | 53 | 56 | theorem ampleSet_univ {F : Type*} [NormedAddCommGroup F] [NormedSpace β F] :
AmpleSet (univ : Set F) := by |
intro x _
rw [connectedComponentIn_univ, PreconnectedSpace.connectedComponent_eq_univ, convexHull_univ]
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,254 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.NormedSpace.Connected
import Mathlib.LinearAlgebra.AffineSpace.ContinuousAffineEquiv
open Set
variable {F : Type*} [AddCommGroup F] [Module β F] [TopologicalSpace F]
def AmpleSet (s : Set F) : Prop :=
β x β s, convexHull β (connectedComponentIn s ... | Mathlib/Analysis/Convex/AmpleSet.lean | 65 | 74 | theorem union {s t : Set F} (hs : AmpleSet s) (ht : AmpleSet t) : AmpleSet (s βͺ t) := by |
intro x hx
rcases hx with (h | h) <;>
-- The connected component of `x β s` in `s βͺ t` contains the connected component of `x` in `s`,
-- hence is also full; similarly for `t`.
[have hx := hs x h; have hx := ht x h] <;>
rw [β Set.univ_subset_iff, β hx] <;>
apply convexHull_mono <;>
apply connectedCompo... | 9 | 8,103.083928 | 2 | 1.2 | 5 | 1,254 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.NormedSpace.Connected
import Mathlib.LinearAlgebra.AffineSpace.ContinuousAffineEquiv
open Set
variable {F : Type*} [AddCommGroup F] [Module β F] [TopologicalSpace F]
def AmpleSet (s : Set F) : Prop :=
β x β s, convexHull β (connectedComponentIn s ... | Mathlib/Analysis/Convex/AmpleSet.lean | 79 | 86 | theorem image {s : Set E} (h : AmpleSet s) (L : E βα΅L[β] F) :
AmpleSet (L '' s) := forall_mem_image.mpr fun x hx β¦
calc (convexHull β) (connectedComponentIn (L '' s) (L x))
_ = (convexHull β) (L '' (connectedComponentIn s x)) :=
.symm <| congrArg _ <| L.toHomeomorph.image_connectedComponentIn hx
... | rw [h x hx, image_univ, L.surjective.range_eq]
| 1 | 2.718282 | 0 | 1.2 | 5 | 1,254 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.NormedSpace.Connected
import Mathlib.LinearAlgebra.AffineSpace.ContinuousAffineEquiv
open Set
variable {F : Type*} [AddCommGroup F] [Module β F] [TopologicalSpace F]
def AmpleSet (s : Set F) : Prop :=
β x β s, convexHull β (connectedComponentIn s ... | Mathlib/Analysis/Convex/AmpleSet.lean | 94 | 96 | theorem preimage {s : Set F} (h : AmpleSet s) (L : E βα΅L[β] F) : AmpleSet (L β»ΒΉ' s) := by |
rw [β L.image_symm_eq_preimage]
exact h.image L.symm
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,254 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.NormedSpace.Connected
import Mathlib.LinearAlgebra.AffineSpace.ContinuousAffineEquiv
open Set
variable {F : Type*} [AddCommGroup F] [Module β F] [TopologicalSpace F]
def AmpleSet (s : Set F) : Prop :=
β x β s, convexHull β (connectedComponentIn s ... | Mathlib/Analysis/Convex/AmpleSet.lean | 120 | 132 | theorem of_one_lt_codim [TopologicalAddGroup F] [ContinuousSMul β F] {E : Submodule β F}
(hcodim : 1 < Module.rank β (F β§Έ E)) :
AmpleSet (EαΆ : Set F) := fun x hx β¦ by
rw [E.connectedComponentIn_eq_self_of_one_lt_codim hcodim hx, eq_univ_iff_forall]
intro y
by_cases h : y β E
Β· obtain β¨z, hzβ© : β z, z β ... |
rw [β not_forall, β Submodule.eq_top_iff']
rintro rfl
simp [rank_zero_iff.2 inferInstance] at hcodim
refine segment_subset_convexHull ?_ ?_ (mem_segment_sub_add y z) <;>
simpa [sub_eq_add_neg, Submodule.add_mem_iff_right _ h]
Β· exact subset_convexHull β (EαΆ : Set F) h
| 6 | 403.428793 | 2 | 1.2 | 5 | 1,254 |
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
#align_import measure_theory.measure.finite_measure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 168 | 171 | theorem apply_mono (ΞΌ : FiniteMeasure Ξ©) {sβ sβ : Set Ξ©} (h : sβ β sβ) : ΞΌ sβ β€ ΞΌ sβ := by |
change ((ΞΌ : Measure Ξ©) sβ).toNNReal β€ ((ΞΌ : Measure Ξ©) sβ).toNNReal
have key : (ΞΌ : Measure Ξ©) sβ β€ (ΞΌ : Measure Ξ©) sβ := (ΞΌ : Measure Ξ©).mono h
apply (ENNReal.toNNReal_le_toNNReal (measure_ne_top _ sβ) (measure_ne_top _ sβ)).mpr key
| 3 | 20.085537 | 1 | 1.2 | 5 | 1,255 |
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
#align_import measure_theory.measure.finite_measure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 200 | 204 | theorem mass_zero_iff (ΞΌ : FiniteMeasure Ξ©) : ΞΌ.mass = 0 β ΞΌ = 0 := by |
refine β¨fun ΞΌ_mass => ?_, fun hΞΌ => by simp only [hΞΌ, zero_mass]β©
apply toMeasure_injective
apply Measure.measure_univ_eq_zero.mp
rwa [β ennreal_mass, ENNReal.coe_eq_zero]
| 4 | 54.59815 | 2 | 1.2 | 5 | 1,255 |
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
#align_import measure_theory.measure.finite_measure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 207 | 209 | theorem mass_nonzero_iff (ΞΌ : FiniteMeasure Ξ©) : ΞΌ.mass β 0 β ΞΌ β 0 := by |
rw [not_iff_not]
exact FiniteMeasure.mass_zero_iff ΞΌ
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,255 |
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
#align_import measure_theory.measure.finite_measure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 213 | 217 | theorem eq_of_forall_toMeasure_apply_eq (ΞΌ Ξ½ : FiniteMeasure Ξ©)
(h : β s : Set Ξ©, MeasurableSet s β (ΞΌ : Measure Ξ©) s = (Ξ½ : Measure Ξ©) s) : ΞΌ = Ξ½ := by |
apply Subtype.ext
ext1 s s_mble
exact h s s_mble
| 3 | 20.085537 | 1 | 1.2 | 5 | 1,255 |
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
#align_import measure_theory.measure.finite_measure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 220 | 223 | theorem eq_of_forall_apply_eq (ΞΌ Ξ½ : FiniteMeasure Ξ©)
(h : β s : Set Ξ©, MeasurableSet s β ΞΌ s = Ξ½ s) : ΞΌ = Ξ½ := by |
ext1 s s_mble
simpa [ennreal_coeFn_eq_coeFn_toMeasure] using congr_arg ((β) : ββ₯0 β ββ₯0β) (h s s_mble)
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,255 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import algebra.monoid_algebra.grading from "leanprover-community/mathlib"@"feb99064803fd3108e37c18b0f77d0a8344677a3"
noncomputable sectio... | Mathlib/Algebra/MonoidAlgebra/Grading.lean | 63 | 64 | theorem mem_gradeBy_iff (f : M β ΞΉ) (i : ΞΉ) (a : R[M]) :
a β gradeBy R f i β (a.support : Set M) β f β»ΒΉ' {i} := by | rfl
| 1 | 2.718282 | 0 | 1.2 | 5 | 1,256 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import algebra.monoid_algebra.grading from "leanprover-community/mathlib"@"feb99064803fd3108e37c18b0f77d0a8344677a3"
noncomputable sectio... | Mathlib/Algebra/MonoidAlgebra/Grading.lean | 67 | 69 | theorem mem_grade_iff (m : M) (a : R[M]) : a β grade R m β a.support β {m} := by |
rw [β Finset.coe_subset, Finset.coe_singleton]
rfl
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,256 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import algebra.monoid_algebra.grading from "leanprover-community/mathlib"@"feb99064803fd3108e37c18b0f77d0a8344677a3"
noncomputable sectio... | Mathlib/Algebra/MonoidAlgebra/Grading.lean | 72 | 78 | theorem mem_grade_iff' (m : M) (a : R[M]) :
a β grade R m β a β (LinearMap.range (Finsupp.lsingle m : R ββ[R] M ββ R) :
Submodule R R[M]) := by |
rw [mem_grade_iff, Finsupp.support_subset_singleton']
apply exists_congr
intro r
constructor <;> exact Eq.symm
| 4 | 54.59815 | 2 | 1.2 | 5 | 1,256 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import algebra.monoid_algebra.grading from "leanprover-community/mathlib"@"feb99064803fd3108e37c18b0f77d0a8344677a3"
noncomputable sectio... | Mathlib/Algebra/MonoidAlgebra/Grading.lean | 86 | 89 | theorem single_mem_gradeBy {R} [CommSemiring R] (f : M β ΞΉ) (m : M) (r : R) :
Finsupp.single m r β gradeBy R f (f m) := by |
intro x hx
rw [Finset.mem_singleton.mp (Finsupp.support_single_subset hx)]
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,256 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import algebra.monoid_algebra.grading from "leanprover-community/mathlib"@"feb99064803fd3108e37c18b0f77d0a8344677a3"
noncomputable sectio... | Mathlib/Algebra/MonoidAlgebra/Grading.lean | 140 | 150 | theorem decomposeAux_single (m : M) (r : R) :
decomposeAux f (Finsupp.single m r) =
DirectSum.of (fun i : ΞΉ => gradeBy R f i) (f m)
β¨Finsupp.single m r, single_mem_gradeBy _ _ _β© := by |
refine (lift_single _ _ _).trans ?_
refine (DirectSum.of_smul R _ _ _).symm.trans ?_
apply DirectSum.of_eq_of_gradedMonoid_eq
refine Sigma.subtype_ext rfl ?_
refine (Finsupp.smul_single' _ _ _).trans ?_
rw [mul_one]
rfl
| 7 | 1,096.633158 | 2 | 1.2 | 5 | 1,256 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.LinearAlgebra.AffineSpace.Slope
#align_import analysis.calculus.deriv.slope from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open Topology Filter TopologicalSpace
open Filter Set
secti... | Mathlib/Analysis/Calculus/Deriv/Slope.lean | 51 | 63 | theorem hasDerivAtFilter_iff_tendsto_slope {x : π} {L : Filter π} :
HasDerivAtFilter f f' x L β Tendsto (slope f x) (L β π {x}αΆ) (π f') :=
calc HasDerivAtFilter f f' x L
β Tendsto (fun y β¦ slope f x y - (y - x)β»ΒΉ β’ (y - x) β’ f') L (π 0) := by |
simp only [hasDerivAtFilter_iff_tendsto, β norm_inv, β norm_smul,
β tendsto_zero_iff_norm_tendsto_zero, slope_def_module, smul_sub]
_ β Tendsto (fun y β¦ slope f x y - (y - x)β»ΒΉ β’ (y - x) β’ f') (L β π {x}αΆ) (π 0) :=
.symm <| tendsto_inf_principal_nhds_iff_of_forall_eq <| by simp
_ β Tend... | 9 | 8,103.083928 | 2 | 1.2 | 5 | 1,257 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.LinearAlgebra.AffineSpace.Slope
#align_import analysis.calculus.deriv.slope from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open Topology Filter TopologicalSpace
open Filter Set
secti... | Mathlib/Analysis/Calculus/Deriv/Slope.lean | 66 | 69 | theorem hasDerivWithinAt_iff_tendsto_slope :
HasDerivWithinAt f f' s x β Tendsto (slope f x) (π[s \ {x}] x) (π f') := by |
simp only [HasDerivWithinAt, nhdsWithin, diff_eq, β inf_assoc, inf_principal.symm]
exact hasDerivAtFilter_iff_tendsto_slope
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,257 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.LinearAlgebra.AffineSpace.Slope
#align_import analysis.calculus.deriv.slope from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open Topology Filter TopologicalSpace
open Filter Set
secti... | Mathlib/Analysis/Calculus/Deriv/Slope.lean | 72 | 74 | theorem hasDerivWithinAt_iff_tendsto_slope' (hs : x β s) :
HasDerivWithinAt f f' s x β Tendsto (slope f x) (π[s] x) (π f') := by |
rw [hasDerivWithinAt_iff_tendsto_slope, diff_singleton_eq_self hs]
| 1 | 2.718282 | 0 | 1.2 | 5 | 1,257 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.LinearAlgebra.AffineSpace.Slope
#align_import analysis.calculus.deriv.slope from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open Topology Filter TopologicalSpace
open Filter Set
secti... | Mathlib/Analysis/Calculus/Deriv/Slope.lean | 81 | 85 | theorem hasDerivAt_iff_tendsto_slope_zero :
HasDerivAt f f' x β Tendsto (fun t β¦ tβ»ΒΉ β’ (f (x + t) - f x)) (π[β ] 0) (π f') := by |
have : π[β ] x = Filter.map (fun t β¦ x + t) (π[β ] 0) := by
simp [nhdsWithin, map_add_left_nhds_zero x, Filter.map_inf, add_right_injective x]
simp [hasDerivAt_iff_tendsto_slope, this, slope, Function.comp]
| 3 | 20.085537 | 1 | 1.2 | 5 | 1,257 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.LinearAlgebra.AffineSpace.Slope
#align_import analysis.calculus.deriv.slope from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open Topology Filter TopologicalSpace
open Filter Set
secti... | Mathlib/Analysis/Calculus/Deriv/Slope.lean | 99 | 134 | theorem range_derivWithin_subset_closure_span_image
(f : π β F) {s t : Set π} (h : s β closure (s β© t)) :
range (derivWithin f s) β closure (Submodule.span π (f '' t)) := by |
rintro - β¨x, rflβ©
rcases eq_or_neBot (π[s \ {x}] x) with H|H
Β· simp [derivWithin, fderivWithin, H]
exact subset_closure (zero_mem _)
by_cases H' : DifferentiableWithinAt π f s x; swap
Β· rw [derivWithin_zero_of_not_differentiableWithinAt H']
exact subset_closure (zero_mem _)
have I : (π[(s β© t) \... | 33 | 214,643,579,785,916.06 | 2 | 1.2 | 5 | 1,257 |
import Mathlib.Data.List.Sublists
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {Ξ± : Type*}
-- Porting note (#11215): TODO: Write a more efficient version
def powerset... | Mathlib/Data/Multiset/Powerset.lean | 45 | 46 | theorem powersetAux_perm_powersetAux' {l : List Ξ±} : powersetAux l ~ powersetAux' l := by |
rw [powersetAux_eq_map_coe]; exact (sublists_perm_sublists' _).map _
| 1 | 2.718282 | 0 | 1.2 | 5 | 1,258 |
import Mathlib.Data.List.Sublists
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {Ξ± : Type*}
-- Porting note (#11215): TODO: Write a more efficient version
def powerset... | Mathlib/Data/Multiset/Powerset.lean | 55 | 57 | theorem powersetAux'_cons (a : Ξ±) (l : List Ξ±) :
powersetAux' (a :: l) = powersetAux' l ++ List.map (cons a) (powersetAux' l) := by |
simp only [powersetAux', sublists'_cons, map_append, List.map_map, append_cancel_left_eq]; rfl
| 1 | 2.718282 | 0 | 1.2 | 5 | 1,258 |
import Mathlib.Data.List.Sublists
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {Ξ± : Type*}
-- Porting note (#11215): TODO: Write a more efficient version
def powerset... | Mathlib/Data/Multiset/Powerset.lean | 60 | 70 | theorem powerset_aux'_perm {lβ lβ : List Ξ±} (p : lβ ~ lβ) : powersetAux' lβ ~ powersetAux' lβ := by |
induction' p with a lβ lβ p IH a b l lβ lβ lβ _ _ IHβ IHβ
Β· simp
Β· simp only [powersetAux'_cons]
exact IH.append (IH.map _)
Β· simp only [powersetAux'_cons, map_append, List.map_map, append_assoc]
apply Perm.append_left
rw [β append_assoc, β append_assoc,
(by funext s; simp [cons_swap] : cons ... | 10 | 22,026.465795 | 2 | 1.2 | 5 | 1,258 |
import Mathlib.Data.List.Sublists
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {Ξ± : Type*}
-- Porting note (#11215): TODO: Write a more efficient version
def powerset... | Mathlib/Data/Multiset/Powerset.lean | 125 | 129 | theorem revzip_powersetAux {l : List Ξ±} β¦xβ¦ (h : x β revzip (powersetAux l)) : x.1 + x.2 = βl := by |
rw [revzip, powersetAux_eq_map_coe, β map_reverse, zip_map, β revzip, List.mem_map] at h
simp only [Prod.map_apply, Prod.exists] at h
rcases h with β¨lβ, lβ, h, rfl, rflβ©
exact Quot.sound (revzip_sublists _ _ _ h)
| 4 | 54.59815 | 2 | 1.2 | 5 | 1,258 |
import Mathlib.Data.List.Sublists
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {Ξ± : Type*}
-- Porting note (#11215): TODO: Write a more efficient version
def powerset... | Mathlib/Data/Multiset/Powerset.lean | 132 | 137 | theorem revzip_powersetAux' {l : List Ξ±} β¦xβ¦ (h : x β revzip (powersetAux' l)) :
x.1 + x.2 = βl := by |
rw [revzip, powersetAux', β map_reverse, zip_map, β revzip, List.mem_map] at h
simp only [Prod.map_apply, Prod.exists] at h
rcases h with β¨lβ, lβ, h, rfl, rflβ©
exact Quot.sound (revzip_sublists' _ _ _ h)
| 4 | 54.59815 | 2 | 1.2 | 5 | 1,258 |
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
#align_import set_theory.ordinal.topology from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
noncomputable section
universe u v
open Cardinal Order Topology
namespace Ordina... | Mathlib/SetTheory/Ordinal/Topology.lean | 41 | 53 | theorem isOpen_singleton_iff : IsOpen ({a} : Set Ordinal) β Β¬IsLimit a := by |
refine β¨fun h β¨hβ, hsuccβ© => ?_, fun ha => ?_β©
Β· obtain β¨b, c, hbc, hbc'β© :=
(mem_nhds_iff_exists_Ioo_subset' β¨0, Ordinal.pos_iff_ne_zero.2 hββ© β¨_, lt_succ aβ©).1
(h.mem_nhds rfl)
have hba := hsucc b hbc.1
exact hba.ne (hbc' β¨lt_succ b, hba.trans hbc.2β©)
Β· rcases zero_or_succ_or_limit a with... | 12 | 162,754.791419 | 2 | 1.2 | 5 | 1,259 |
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
#align_import set_theory.ordinal.topology from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
noncomputable section
universe u v
open Cardinal Order Topology
namespace Ordina... | Mathlib/SetTheory/Ordinal/Topology.lean | 60 | 61 | theorem nhds_left'_eq_nhds_ne (a : Ordinal) : π[<] a = π[β ] a := by |
rw [β nhds_left'_sup_nhds_right', nhds_right', sup_bot_eq]
| 1 | 2.718282 | 0 | 1.2 | 5 | 1,259 |
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