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.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ββ₯0β} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 130 | 130 | theorem not_lt_zero : Β¬a < 0 := by | simp
| 1 | 2.718282 | 0 | 0.666667 | 12 | 570 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ββ₯0β} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 177 | 178 | theorem lt_add_right (ha : a β β) (hb : b β 0) : a < a + b := by |
rwa [β pos_iff_ne_zero, β ENNReal.add_lt_add_iff_left ha, add_zero] at hb
| 1 | 2.718282 | 0 | 0.666667 | 12 | 570 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ββ₯0β} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 193 | 197 | theorem toNNReal_add {rβ rβ : ββ₯0β} (hβ : rβ β β) (hβ : rβ β β) :
(rβ + rβ).toNNReal = rβ.toNNReal + rβ.toNNReal := by |
lift rβ to ββ₯0 using hβ
lift rβ to ββ₯0 using hβ
rfl
| 3 | 20.085537 | 1 | 0.666667 | 12 | 570 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ββ₯0β} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 200 | 200 | theorem not_lt_top {x : ββ₯0β} : Β¬x < β β x = β := by | rw [lt_top_iff_ne_top, Classical.not_not]
| 1 | 2.718282 | 0 | 0.666667 | 12 | 570 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ββ₯0β} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 203 | 203 | theorem add_ne_top : a + b β β β a β β β§ b β β := by | simpa only [lt_top_iff_ne_top] using add_lt_top
| 1 | 2.718282 | 0 | 0.666667 | 12 | 570 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ββ₯0β} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 206 | 206 | theorem mul_top' : a * β = if a = 0 then 0 else β := by | convert WithTop.mul_top' a
| 1 | 2.718282 | 0 | 0.666667 | 12 | 570 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ββ₯0β} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 212 | 212 | theorem top_mul' : β * a = if a = 0 then 0 else β := by | convert WithTop.top_mul' a
| 1 | 2.718282 | 0 | 0.666667 | 12 | 570 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ββ₯0β} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 235 | 235 | theorem mul_ne_top : a β β β b β β β a * b β β := by | simpa only [lt_top_iff_ne_top] using mul_lt_top
| 1 | 2.718282 | 0 | 0.666667 | 12 | 570 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ββ₯0β} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 246 | 252 | theorem mul_lt_top_iff {a b : ββ₯0β} : a * b < β β a < β β§ b < β β¨ a = 0 β¨ b = 0 := by |
constructor
Β· intro h
rw [β or_assoc, or_iff_not_imp_right, or_iff_not_imp_right]
intro hb ha
exact β¨lt_top_of_mul_ne_top_left h.ne hb, lt_top_of_mul_ne_top_right h.ne haβ©
Β· rintro (β¨ha, hbβ© | rfl | rfl) <;> [exact mul_lt_top ha.ne hb.ne; simp; simp]
| 6 | 403.428793 | 2 | 0.666667 | 12 | 570 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ββ₯0β} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 255 | 258 | theorem mul_self_lt_top_iff {a : ββ₯0β} : a * a < β€ β a < β€ := by |
rw [ENNReal.mul_lt_top_iff, and_self, or_self, or_iff_left_iff_imp]
rintro rfl
exact zero_lt_top
| 3 | 20.085537 | 1 | 0.666667 | 12 | 570 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Deprecated.Group
#align_import deprecated.submonoid from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {M : Type*} [Monoid M] {s : Set M}
variable {A : Type*} [AddMonoi... | Mathlib/Deprecated/Submonoid.lean | 232 | 237 | theorem list_prod_mem (hs : IsSubmonoid s) : β {l : List M}, (β x β l, x β s) β l.prod β s
| [], _ => hs.one_mem
| a :: l, h =>
suffices a * l.prod β s by simpa
have : a β s β§ β x β l, x β s := by | simpa using h
hs.mul_mem this.1 (list_prod_mem hs this.2)
| 2 | 7.389056 | 1 | 0.666667 | 3 | 571 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Deprecated.Group
#align_import deprecated.submonoid from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {M : Type*} [Monoid M] {s : Set M}
variable {A : Type*} [AddMonoi... | Mathlib/Deprecated/Submonoid.lean | 246 | 250 | theorem multiset_prod_mem {M} [CommMonoid M] {s : Set M} (hs : IsSubmonoid s) (m : Multiset M) :
(β a β m, a β s) β m.prod β s := by |
refine Quotient.inductionOn m fun l hl => ?_
rw [Multiset.quot_mk_to_coe, Multiset.prod_coe]
exact list_prod_mem hs hl
| 3 | 20.085537 | 1 | 0.666667 | 3 | 571 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Deprecated.Group
#align_import deprecated.submonoid from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {M : Type*} [Monoid M] {s : Set M}
variable {A : Type*} [AddMonoi... | Mathlib/Deprecated/Submonoid.lean | 426 | 427 | theorem Submonoid.isSubmonoid (S : Submonoid M) : IsSubmonoid (S : Set M) := by |
exact β¨S.2, S.1.2β©
| 1 | 2.718282 | 0 | 0.666667 | 3 | 571 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 87 | 87 | theorem transvection_zero : transvection i j (0 : R) = 1 := by | simp [transvection]
| 1 | 2.718282 | 0 | 0.666667 | 12 | 572 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 94 | 108 | theorem updateRow_eq_transvection [Finite n] (c : R) :
updateRow (1 : Matrix n n R) i ((1 : Matrix n n R) i + c β’ (1 : Matrix n n R) j) =
transvection i j c := by |
cases nonempty_fintype n
ext a b
by_cases ha : i = a
Β· by_cases hb : j = b
Β· simp only [updateRow_self, transvection, ha, hb, Pi.add_apply, StdBasisMatrix.apply_same,
one_apply_eq, Pi.smul_apply, mul_one, Algebra.id.smul_eq_mul, add_apply]
Β· simp only [updateRow_self, transvection, ha, hb, StdB... | 12 | 162,754.791419 | 2 | 0.666667 | 12 | 572 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 113 | 116 | theorem transvection_mul_transvection_same (h : i β j) (c d : R) :
transvection i j c * transvection i j d = transvection i j (c + d) := by |
simp [transvection, Matrix.add_mul, Matrix.mul_add, h, h.symm, add_smul, add_assoc,
stdBasisMatrix_add]
| 2 | 7.389056 | 1 | 0.666667 | 12 | 572 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 120 | 121 | theorem transvection_mul_apply_same (b : n) (c : R) (M : Matrix n n R) :
(transvection i j c * M) i b = M i b + c * M j b := by | simp [transvection, Matrix.add_mul]
| 1 | 2.718282 | 0 | 0.666667 | 12 | 572 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 125 | 127 | theorem mul_transvection_apply_same (a : n) (c : R) (M : Matrix n n R) :
(M * transvection i j c) a j = M a j + c * M a i := by |
simp [transvection, Matrix.mul_add, mul_comm]
| 1 | 2.718282 | 0 | 0.666667 | 12 | 572 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 131 | 132 | theorem transvection_mul_apply_of_ne (a b : n) (ha : a β i) (c : R) (M : Matrix n n R) :
(transvection i j c * M) a b = M a b := by | simp [transvection, Matrix.add_mul, ha]
| 1 | 2.718282 | 0 | 0.666667 | 12 | 572 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 136 | 137 | theorem mul_transvection_apply_of_ne (a b : n) (hb : b β j) (c : R) (M : Matrix n n R) :
(M * transvection i j c) a b = M a b := by | simp [transvection, Matrix.mul_add, hb]
| 1 | 2.718282 | 0 | 0.666667 | 12 | 572 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 141 | 142 | theorem det_transvection_of_ne (h : i β j) (c : R) : det (transvection i j c) = 1 := by |
rw [β updateRow_eq_transvection i j, det_updateRow_add_smul_self _ h, det_one]
| 1 | 2.718282 | 0 | 0.666667 | 12 | 572 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 184 | 188 | theorem det_toMatrix_prod [Fintype n] (L : List (TransvectionStruct n π)) :
det (L.map toMatrix).prod = 1 := by |
induction' L with t L IH
Β· simp
Β· simp [IH]
| 3 | 20.085537 | 1 | 0.666667 | 12 | 572 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 205 | 207 | theorem inv_mul (t : TransvectionStruct n R) : t.inv.toMatrix * t.toMatrix = 1 := by |
rcases t with β¨_, _, t_hijβ©
simp [toMatrix, transvection_mul_transvection_same, t_hij]
| 2 | 7.389056 | 1 | 0.666667 | 12 | 572 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 210 | 212 | theorem mul_inv (t : TransvectionStruct n R) : t.toMatrix * t.inv.toMatrix = 1 := by |
rcases t with β¨_, _, t_hijβ©
simp [toMatrix, transvection_mul_transvection_same, t_hij]
| 2 | 7.389056 | 1 | 0.666667 | 12 | 572 |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 371 | 380 | theorem listTransvecCol_mul_last_row_drop (i : Sum (Fin r) Unit) {k : β} (hk : k β€ r) :
(((listTransvecCol M).drop k).prod * M) (inr unit) i = M (inr unit) i := by |
-- Porting note: `apply` didn't work anymore, because of the implicit arguments
refine Nat.decreasingInduction' ?_ hk ?_
Β· intro n hn _ IH
have hn' : n < (listTransvecCol M).length := by simpa [listTransvecCol] using hn
rw [List.drop_eq_get_cons hn']
simpa [listTransvecCol, Matrix.mul_assoc]
Β· simp... | 8 | 2,980.957987 | 2 | 0.666667 | 12 | 572 |
import Mathlib.NumberTheory.NumberField.Embeddings
#align_import number_theory.number_field.units from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
open scoped NumberField
noncomputable section
open NumberField Units
section Rat
| Mathlib/NumberTheory/NumberField/Units/Basic.lean | 40 | 43 | theorem Rat.RingOfIntegers.isUnit_iff {x : π β} : IsUnit x β (x : β) = 1 β¨ (x : β) = -1 := by |
simp_rw [(isUnit_map_iff (Rat.ringOfIntegersEquiv : π β β+* β€) x).symm, Int.isUnit_iff,
RingEquiv.coe_toRingHom, RingEquiv.map_eq_one_iff, RingEquiv.map_eq_neg_one_iff, β
Subtype.coe_injective.eq_iff]; rfl
| 3 | 20.085537 | 1 | 0.666667 | 3 | 573 |
import Mathlib.NumberTheory.NumberField.Embeddings
#align_import number_theory.number_field.units from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
open scoped NumberField
noncomputable section
open NumberField Units
variable (K : Type*) [Field K]
namespace NumberField.Units
secti... | Mathlib/NumberTheory/NumberField/Units/Basic.lean | 78 | 79 | theorem coe_pow (x : (π K)Λ£) (n : β) : ((x ^ n : (π K)Λ£) : K) = (x : K) ^ n := by |
rw [β map_pow, β val_pow_eq_pow_val]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 573 |
import Mathlib.NumberTheory.NumberField.Embeddings
#align_import number_theory.number_field.units from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
open scoped NumberField
noncomputable section
open NumberField Units
variable (K : Type*) [Field K]
namespace NumberField.Units
secti... | Mathlib/NumberTheory/NumberField/Units/Basic.lean | 81 | 83 | theorem coe_zpow (x : (π K)Λ£) (n : β€) : (β(x ^ n) : K) = (x : K) ^ n := by |
change ((Units.coeHom K).comp (map (algebraMap (π K) K))) (x ^ n) = _
exact map_zpow _ x n
| 2 | 7.389056 | 1 | 0.666667 | 3 | 573 |
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {Ξ± : Type*} {N : Ξ± β Type*}
namespace DFinsupp
variable [DecidableEq Ξ±]
section NHasZero
variable [β a, DecidableEq (N a)] [β a, Zero (N a)] (f g : Ξ β... | Mathlib/Data/DFinsupp/NeLocus.lean | 41 | 43 | theorem mem_neLocus {f g : Ξ β a, N a} {a : Ξ±} : a β f.neLocus g β f a β g a := by |
simpa only [neLocus, Finset.mem_filter, Finset.mem_union, mem_support_iff,
and_iff_right_iff_imp] using Ne.ne_or_ne _
| 2 | 7.389056 | 1 | 0.666667 | 3 | 574 |
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {Ξ± : Type*} {N : Ξ± β Type*}
namespace DFinsupp
variable [DecidableEq Ξ±]
section NHasZero
variable [β a, DecidableEq (N a)] [β a, Zero (N a)] (f g : Ξ β... | Mathlib/Data/DFinsupp/NeLocus.lean | 67 | 68 | theorem neLocus_comm : f.neLocus g = g.neLocus f := by |
simp_rw [neLocus, Finset.union_comm, ne_comm]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 574 |
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {Ξ± : Type*} {N : Ξ± β Type*}
namespace DFinsupp
variable [DecidableEq Ξ±]
section NHasZero
variable [β a, DecidableEq (N a)] [β a, Zero (N a)] (f g : Ξ β... | Mathlib/Data/DFinsupp/NeLocus.lean | 72 | 74 | theorem neLocus_zero_right : f.neLocus 0 = f.support := by |
ext
rw [mem_neLocus, mem_support_iff, coe_zero, Pi.zero_apply]
| 2 | 7.389056 | 1 | 0.666667 | 3 | 574 |
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Rat.BigOperators
#align_import combinatorics.simple_graph.regularity.energy from "leanprover-community/mathlib"@"bf7ef0... | Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean | 42 | 43 | theorem energy_nonneg : 0 β€ P.energy G := by |
exact div_nonneg (Finset.sum_nonneg fun _ _ => sq_nonneg _) <| sq_nonneg _
| 1 | 2.718282 | 0 | 0.666667 | 3 | 575 |
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Rat.BigOperators
#align_import combinatorics.simple_graph.regularity.energy from "leanprover-community/mathlib"@"bf7ef0... | Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean | 46 | 57 | theorem energy_le_one : P.energy G β€ 1 :=
div_le_of_nonneg_of_le_mul (sq_nonneg _) zero_le_one <|
calc
β uv β P.parts.offDiag, G.edgeDensity uv.1 uv.2 ^ 2 β€ P.parts.offDiag.card β’ (1 : β) :=
sum_le_card_nsmul _ _ 1 fun uv _ =>
(sq_le_one_iff <| G.edgeDensity_nonneg _ _).2 <| G.edgeDensity_... |
rw [offDiag_card, one_mul]
norm_cast
rw [sq]
exact tsub_le_self
| 4 | 54.59815 | 2 | 0.666667 | 3 | 575 |
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Rat.BigOperators
#align_import combinatorics.simple_graph.regularity.energy from "leanprover-community/mathlib"@"bf7ef0... | Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean | 61 | 63 | theorem coe_energy {π : Type*} [LinearOrderedField π] : (P.energy G : π) =
(β uv β P.parts.offDiag, (G.edgeDensity uv.1 uv.2 : π) ^ 2) / (P.parts.card : π) ^ 2 := by |
rw [energy]; norm_cast
| 1 | 2.718282 | 0 | 0.666667 | 3 | 575 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.Solvable
import Mathlib.LinearAlgebra.Dual
#align_import algebra.lie.character from "leanprover-community/mathlib"@"132328c4dd48da87adca5d408ca54f315282b719"
universe u v w wβ
namespace LieAlgebra
variable (R : Type u) (L : Type v) [CommRing R] [LieR... | Mathlib/Algebra/Lie/Character.lean | 44 | 45 | theorem lieCharacter_apply_lie (Ο : LieCharacter R L) (x y : L) : Ο β
x, yβ = 0 := by |
rw [LieHom.map_lie, LieRing.of_associative_ring_bracket, mul_comm, sub_self]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 576 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.Solvable
import Mathlib.LinearAlgebra.Dual
#align_import algebra.lie.character from "leanprover-community/mathlib"@"132328c4dd48da87adca5d408ca54f315282b719"
universe u v w wβ
namespace LieAlgebra
variable (R : Type u) (L : Type v) [CommRing R] [LieR... | Mathlib/Algebra/Lie/Character.lean | 49 | 50 | theorem lieCharacter_apply_lie' (Ο : LieCharacter R L) (x y : L) : β
Ο x, Ο yβ = 0 := by |
rw [LieRing.of_associative_ring_bracket, mul_comm, sub_self]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 576 |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.Solvable
import Mathlib.LinearAlgebra.Dual
#align_import algebra.lie.character from "leanprover-community/mathlib"@"132328c4dd48da87adca5d408ca54f315282b719"
universe u v w wβ
namespace LieAlgebra
variable (R : Type u) (L : Type v) [CommRing R] [LieR... | Mathlib/Algebra/Lie/Character.lean | 52 | 60 | theorem lieCharacter_apply_of_mem_derived (Ο : LieCharacter R L) {x : L}
(h : x β derivedSeries R L 1) : Ο x = 0 := by |
rw [derivedSeries_def, derivedSeriesOfIdeal_succ, derivedSeriesOfIdeal_zero, β
LieSubmodule.mem_coeSubmodule, LieSubmodule.lieIdeal_oper_eq_linear_span] at h
refine Submodule.span_induction h ?_ ?_ ?_ ?_
Β· rintro y β¨β¨z, hzβ©, β¨β¨w, hwβ©, rflβ©β©; apply lieCharacter_apply_lie
Β· exact Ο.map_zero
Β· intro y z hy ... | 7 | 1,096.633158 | 2 | 0.666667 | 3 | 576 |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.Group
#align_import topology.algebra.order.field from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpace Function
open scoped Pointwise Top... | Mathlib/Topology/Algebra/Order/Field.lean | 30 | 51 | theorem TopologicalRing.of_norm {R π : Type*} [NonUnitalNonAssocRing R] [LinearOrderedField π]
[TopologicalSpace R] [TopologicalAddGroup R] (norm : R β π)
(norm_nonneg : β x, 0 β€ norm x) (norm_mul_le : β x y, norm (x * y) β€ norm x * norm y)
(nhds_basis : (π (0 : R)).HasBasis ((0 : π) < Β·) (fun Ξ΅ β¦ { x ... |
have h0 : β f : R β R, β c β₯ (0 : π), (β x, norm (f x) β€ c * norm x) β
Tendsto f (π 0) (π 0) := by
refine fun f c c0 hf β¦ (nhds_basis.tendsto_iff nhds_basis).2 fun Ξ΅ Ξ΅0 β¦ ?_
rcases exists_pos_mul_lt Ξ΅0 c with β¨Ξ΄, Ξ΄0, hΞ΄β©
refine β¨Ξ΄, Ξ΄0, fun x hx β¦ (hf _).trans_lt ?_β©
exact (mul_le_mul_of_nonn... | 17 | 24,154,952.753575 | 2 | 0.666667 | 9 | 577 |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.Group
#align_import topology.algebra.order.field from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpace Function
open scoped Pointwise Top... | Mathlib/Topology/Algebra/Order/Field.lean | 63 | 67 | theorem Filter.Tendsto.atTop_mul {C : π} (hC : 0 < C) (hf : Tendsto f l atTop)
(hg : Tendsto g l (π C)) : Tendsto (fun x => f x * g x) l atTop := by |
refine tendsto_atTop_mono' _ ?_ (hf.atTop_mul_const (half_pos hC))
filter_upwards [hg.eventually (lt_mem_nhds (half_lt_self hC)), hf.eventually_ge_atTop 0]
with x hg hf using mul_le_mul_of_nonneg_left hg.le hf
| 3 | 20.085537 | 1 | 0.666667 | 9 | 577 |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.Group
#align_import topology.algebra.order.field from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpace Function
open scoped Pointwise Top... | Mathlib/Topology/Algebra/Order/Field.lean | 72 | 74 | theorem Filter.Tendsto.mul_atTop {C : π} (hC : 0 < C) (hf : Tendsto f l (π C))
(hg : Tendsto g l atTop) : Tendsto (fun x => f x * g x) l atTop := by |
simpa only [mul_comm] using hg.atTop_mul hC hf
| 1 | 2.718282 | 0 | 0.666667 | 9 | 577 |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.Group
#align_import topology.algebra.order.field from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpace Function
open scoped Pointwise Top... | Mathlib/Topology/Algebra/Order/Field.lean | 79 | 82 | theorem Filter.Tendsto.atTop_mul_neg {C : π} (hC : C < 0) (hf : Tendsto f l atTop)
(hg : Tendsto g l (π C)) : Tendsto (fun x => f x * g x) l atBot := by |
have := hf.atTop_mul (neg_pos.2 hC) hg.neg
simpa only [(Β· β Β·), neg_mul_eq_mul_neg, neg_neg] using tendsto_neg_atTop_atBot.comp this
| 2 | 7.389056 | 1 | 0.666667 | 9 | 577 |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.Group
#align_import topology.algebra.order.field from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpace Function
open scoped Pointwise Top... | Mathlib/Topology/Algebra/Order/Field.lean | 87 | 89 | theorem Filter.Tendsto.neg_mul_atTop {C : π} (hC : C < 0) (hf : Tendsto f l (π C))
(hg : Tendsto g l atTop) : Tendsto (fun x => f x * g x) l atBot := by |
simpa only [mul_comm] using hg.atTop_mul_neg hC hf
| 1 | 2.718282 | 0 | 0.666667 | 9 | 577 |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.Group
#align_import topology.algebra.order.field from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpace Function
open scoped Pointwise Top... | Mathlib/Topology/Algebra/Order/Field.lean | 94 | 97 | theorem Filter.Tendsto.atBot_mul {C : π} (hC : 0 < C) (hf : Tendsto f l atBot)
(hg : Tendsto g l (π C)) : Tendsto (fun x => f x * g x) l atBot := by |
have := (tendsto_neg_atBot_atTop.comp hf).atTop_mul hC hg
simpa [(Β· β Β·)] using tendsto_neg_atTop_atBot.comp this
| 2 | 7.389056 | 1 | 0.666667 | 9 | 577 |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.Group
#align_import topology.algebra.order.field from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpace Function
open scoped Pointwise Top... | Mathlib/Topology/Algebra/Order/Field.lean | 102 | 105 | theorem Filter.Tendsto.atBot_mul_neg {C : π} (hC : C < 0) (hf : Tendsto f l atBot)
(hg : Tendsto g l (π C)) : Tendsto (fun x => f x * g x) l atTop := by |
have := (tendsto_neg_atBot_atTop.comp hf).atTop_mul_neg hC hg
simpa [(Β· β Β·)] using tendsto_neg_atBot_atTop.comp this
| 2 | 7.389056 | 1 | 0.666667 | 9 | 577 |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.Group
#align_import topology.algebra.order.field from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpace Function
open scoped Pointwise Top... | Mathlib/Topology/Algebra/Order/Field.lean | 110 | 112 | theorem Filter.Tendsto.mul_atBot {C : π} (hC : 0 < C) (hf : Tendsto f l (π C))
(hg : Tendsto g l atBot) : Tendsto (fun x => f x * g x) l atBot := by |
simpa only [mul_comm] using hg.atBot_mul hC hf
| 1 | 2.718282 | 0 | 0.666667 | 9 | 577 |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.Group
#align_import topology.algebra.order.field from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpace Function
open scoped Pointwise Top... | Mathlib/Topology/Algebra/Order/Field.lean | 117 | 119 | theorem Filter.Tendsto.neg_mul_atBot {C : π} (hC : C < 0) (hf : Tendsto f l (π C))
(hg : Tendsto g l atBot) : Tendsto (fun x => f x * g x) l atTop := by |
simpa only [mul_comm] using hg.atBot_mul_neg hC hf
| 1 | 2.718282 | 0 | 0.666667 | 9 | 577 |
import Mathlib.Data.Set.Basic
#align_import data.bundle from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open Function Set
namespace Bundle
variable {B F : Type*} (E : B β Type*)
@[ext]
structure TotalSpace (F : Type*) (E : B β Type*) where
proj : B
snd : E proj
#align bund... | Mathlib/Data/Bundle.lean | 69 | 70 | theorem TotalSpace.mk_cast {x x' : B} (h : x = x') (b : E x) :
.mk' F x' (cast (congr_arg E h) b) = TotalSpace.mk x b := by | subst h; rfl
| 1 | 2.718282 | 0 | 0.666667 | 3 | 578 |
import Mathlib.Data.Set.Basic
#align_import data.bundle from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open Function Set
namespace Bundle
variable {B F : Type*} (E : B β Type*)
@[ext]
structure TotalSpace (F : Type*) (E : B β Type*) where
proj : B
snd : E proj
#align bund... | Mathlib/Data/Bundle.lean | 74 | 75 | theorem TotalSpace.mk_inj {b : B} {y y' : E b} : mk' F b y = mk' F b y' β y = y' := by |
simp [TotalSpace.ext_iff]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 578 |
import Mathlib.Data.Set.Basic
#align_import data.bundle from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open Function Set
namespace Bundle
variable {B F : Type*} (E : B β Type*)
@[ext]
structure TotalSpace (F : Type*) (E : B β Type*) where
proj : B
snd : E proj
#align bund... | Mathlib/Data/Bundle.lean | 95 | 100 | theorem TotalSpace.range_mk (b : B) : range ((β) : E b β TotalSpace F E) = Ο F E β»ΒΉ' {b} := by |
apply Subset.antisymm
Β· rintro _ β¨x, rflβ©
rfl
Β· rintro β¨_, xβ© rfl
exact β¨x, rflβ©
| 5 | 148.413159 | 2 | 0.666667 | 3 | 578 |
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Preserves.Finite
namespace CompHaus
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
universe u w
open Categor... | Mathlib/Topology/Category/CompHaus/Limits.lean | 131 | 134 | theorem pullback_fst_eq :
CompHaus.pullback.fst f g = (pullbackIsoPullback f g).hom β« Limits.pullback.fst := by |
dsimp [pullbackIsoPullback]
simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_Ο]
| 2 | 7.389056 | 1 | 0.666667 | 3 | 579 |
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Preserves.Finite
namespace CompHaus
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
universe u w
open Categor... | Mathlib/Topology/Category/CompHaus/Limits.lean | 136 | 139 | theorem pullback_snd_eq :
CompHaus.pullback.snd f g = (pullbackIsoPullback f g).hom β« Limits.pullback.snd := by |
dsimp [pullbackIsoPullback]
simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_Ο]
| 2 | 7.389056 | 1 | 0.666667 | 3 | 579 |
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Preserves.Finite
namespace CompHaus
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
universe u w
open Categor... | Mathlib/Topology/Category/CompHaus/Limits.lean | 205 | 207 | theorem Sigma.ΞΉ_comp_toFiniteCoproduct (a : Ξ±) :
(Limits.Sigma.ΞΉ X a) β« (coproductIsoCoproduct X).inv = finiteCoproduct.ΞΉ X a := by |
simp [coproductIsoCoproduct]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 579 |
import Mathlib.Control.EquivFunctor
import Mathlib.Data.Option.Basic
import Mathlib.Data.Subtype
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Cases
#align_import logic.equiv.option from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u
namespace Equiv
open Option
vari... | Mathlib/Logic/Equiv/Option.lean | 89 | 91 | theorem removeNone_aux_some {x : Ξ±} (h : β x', e (some x) = some x') :
some (removeNone_aux e x) = e (some x) := by |
simp [removeNone_aux, Option.isSome_iff_exists.mpr h]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 580 |
import Mathlib.Control.EquivFunctor
import Mathlib.Data.Option.Basic
import Mathlib.Data.Subtype
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Cases
#align_import logic.equiv.option from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u
namespace Equiv
open Option
vari... | Mathlib/Logic/Equiv/Option.lean | 95 | 97 | theorem removeNone_aux_none {x : Ξ±} (h : e (some x) = none) :
some (removeNone_aux e x) = e none := by |
simp [removeNone_aux, Option.not_isSome_iff_eq_none.mpr h]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 580 |
import Mathlib.Control.EquivFunctor
import Mathlib.Data.Option.Basic
import Mathlib.Data.Subtype
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Cases
#align_import logic.equiv.option from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u
namespace Equiv
open Option
vari... | Mathlib/Logic/Equiv/Option.lean | 148 | 156 | theorem some_removeNone_iff {x : Ξ±} : some (removeNone e x) = e none β e.symm none = some x := by |
cases' h : e (some x) with a
Β· rw [removeNone_none _ h]
simpa using (congr_arg e.symm h).symm
Β· rw [removeNone_some _ β¨a, hβ©]
have h1 := congr_arg e.symm h
rw [symm_apply_apply] at h1
simp only [false_iff_iff, apply_eq_iff_eq]
simp [h1, apply_eq_iff_eq]
| 8 | 2,980.957987 | 2 | 0.666667 | 3 | 580 |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {Ξ± Ξ² Ξ³ : Type*} {ΞΉ : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 32 | 32 | theorem lift_top (g : Set Ξ± β Filter Ξ²) : (β€ : Filter Ξ±).lift g = g univ := by | simp [Filter.lift]
| 1 | 2.718282 | 0 | 0.666667 | 6 | 581 |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {Ξ± Ξ² Ξ³ : Type*} {ΞΉ : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 45 | 53 | theorem HasBasis.mem_lift_iff {ΞΉ} {p : ΞΉ β Prop} {s : ΞΉ β Set Ξ±} {f : Filter Ξ±}
(hf : f.HasBasis p s) {Ξ² : ΞΉ β Type*} {pg : β i, Ξ² i β Prop} {sg : β i, Ξ² i β Set Ξ³}
{g : Set Ξ± β Filter Ξ³} (hg : β i, (g <| s i).HasBasis (pg i) (sg i)) (gm : Monotone g)
{s : Set Ξ³} : s β f.lift g β β i, p i β§ β x, pg i x β§ sg... |
refine (mem_biInf_of_directed ?_ β¨univ, univ_sets _β©).trans ?_
Β· intro tβ htβ tβ htβ
exact β¨tβ β© tβ, inter_mem htβ htβ, gm inter_subset_left, gm inter_subset_rightβ©
Β· simp only [β (hg _).mem_iff]
exact hf.exists_iff fun tβ tβ ht H => gm ht H
| 5 | 148.413159 | 2 | 0.666667 | 6 | 581 |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {Ξ± Ξ² Ξ³ : Type*} {ΞΉ : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 65 | 70 | theorem HasBasis.lift {ΞΉ} {p : ΞΉ β Prop} {s : ΞΉ β Set Ξ±} {f : Filter Ξ±} (hf : f.HasBasis p s)
{Ξ² : ΞΉ β Type*} {pg : β i, Ξ² i β Prop} {sg : β i, Ξ² i β Set Ξ³} {g : Set Ξ± β Filter Ξ³}
(hg : β i, (g (s i)).HasBasis (pg i) (sg i)) (gm : Monotone g) :
(f.lift g).HasBasis (fun i : Ξ£i, Ξ² i => p i.1 β§ pg i.1 i.2) fun... |
refine β¨fun t => (hf.mem_lift_iff hg gm).trans ?_β©
simp [Sigma.exists, and_assoc, exists_and_left]
| 2 | 7.389056 | 1 | 0.666667 | 6 | 581 |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {Ξ± Ξ² Ξ³ : Type*} {ΞΉ : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 78 | 81 | theorem sInter_lift_sets (hg : Monotone g) :
ββ { s | s β f.lift g } = β s β f, ββ { t | t β g s } := by |
simp only [sInter_eq_biInter, mem_setOf_eq, Filter.mem_sets, mem_lift_sets hg, iInter_exists,
iInter_and, @iInter_comm _ (Set Ξ²)]
| 2 | 7.389056 | 1 | 0.666667 | 6 | 581 |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {Ξ± Ξ² Ξ³ : Type*} {ΞΉ : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 106 | 108 | theorem tendsto_lift {m : Ξ³ β Ξ²} {l : Filter Ξ³} :
Tendsto m l (f.lift g) β β s β f, Tendsto m l (g s) := by |
simp only [Filter.lift, tendsto_iInf]
| 1 | 2.718282 | 0 | 0.666667 | 6 | 581 |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {Ξ± Ξ² Ξ³ : Type*} {ΞΉ : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 117 | 118 | theorem comap_lift_eq {m : Ξ³ β Ξ²} : comap m (f.lift g) = f.lift (comap m β g) := by |
simp only [Filter.lift, comap_iInf]; rfl
| 1 | 2.718282 | 0 | 0.666667 | 6 | 581 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Control.Functor
#align_import control.applicative from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u v w
section Lemmas
open Function
variable {F : Type u β Type v}
variable [Applicative F] [LawfulApplicative F]
variable {Ξ± ... | Mathlib/Control/Applicative.lean | 31 | 33 | theorem Applicative.map_seq_map (f : Ξ± β Ξ² β Ξ³) (g : Ο β Ξ²) (x : F Ξ±) (y : F Ο) :
f <$> x <*> g <$> y = ((Β· β g) β f) <$> x <*> y := by |
simp [flip, functor_norm]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 582 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Control.Functor
#align_import control.applicative from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u v w
section Lemmas
open Function
variable {F : Type u β Type v}
variable [Applicative F] [LawfulApplicative F]
variable {Ξ± ... | Mathlib/Control/Applicative.lean | 36 | 37 | theorem Applicative.pure_seq_eq_map' (f : Ξ± β Ξ²) : ((pure f : F (Ξ± β Ξ²)) <*> Β·) = (f <$> Β·) := by |
ext; simp [functor_norm]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 582 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Control.Functor
#align_import control.applicative from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u v w
section Lemmas
open Function
variable {F : Type u β Type v}
variable [Applicative F] [LawfulApplicative F]
variable {Ξ± ... | Mathlib/Control/Applicative.lean | 40 | 63 | theorem Applicative.ext {F} :
β {A1 : Applicative F} {A2 : Applicative F} [@LawfulApplicative F A1] [@LawfulApplicative F A2],
(β {Ξ± : Type u} (x : Ξ±), @Pure.pure _ A1.toPure _ x = @Pure.pure _ A2.toPure _ x) β
(β {Ξ± Ξ² : Type u} (f : F (Ξ± β Ξ²)) (x : F Ξ±),
@Seq.seq _ A1.toSeq _ _ f (fun _ => x)... |
funext Ξ± x
apply H1
obtain rfl : @s1 = @s2 := by
funext Ξ± Ξ² f x
exact H2 f (x Unit.unit)
obtain β¨seqLeft_eq1, seqRight_eq1, pure_seq1, -β© := L1
obtain β¨seqLeft_eq2, seqRight_eq2, pure_seq2, -β© := L2
obtain rfl : F1 = F2 := by
apply Functor.ext
intros
exact (pur... | 14 | 1,202,604.284165 | 2 | 0.666667 | 3 | 582 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R Ξ± Ξ² Ξ΄ Ξ³ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] [MeasurableSpace Ξ³]
variable {ΞΌ ΞΌβ ΞΌβ ΞΌβ Ξ½ Ξ½' Ξ½... | Mathlib/MeasureTheory/Measure/Restrict.lean | 56 | 59 | theorem restrict_toOuterMeasure_eq_toOuterMeasure_restrict (h : MeasurableSet s) :
(ΞΌ.restrict s).toOuterMeasure = OuterMeasure.restrict s ΞΌ.toOuterMeasure := by |
simp_rw [restrict, restrictβ, liftLinear, LinearMap.coe_mk, AddHom.coe_mk,
toMeasure_toOuterMeasure, OuterMeasure.restrict_trim h, ΞΌ.trimmed]
| 2 | 7.389056 | 1 | 0.666667 | 6 | 583 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R Ξ± Ξ² Ξ΄ Ξ³ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] [MeasurableSpace Ξ³]
variable {ΞΌ ΞΌβ ΞΌβ ΞΌβ Ξ½ Ξ½' Ξ½... | Mathlib/MeasureTheory/Measure/Restrict.lean | 62 | 64 | theorem restrict_applyβ (ht : NullMeasurableSet t (ΞΌ.restrict s)) : ΞΌ.restrict s t = ΞΌ (t β© s) := by |
rw [β restrictβ_apply, restrictβ, liftLinear_applyβ _ ht, OuterMeasure.restrict_apply,
coe_toOuterMeasure]
| 2 | 7.389056 | 1 | 0.666667 | 6 | 583 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R Ξ± Ξ² Ξ΄ Ξ³ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] [MeasurableSpace Ξ³]
variable {ΞΌ ΞΌβ ΞΌβ ΞΌβ Ξ½ Ξ½' Ξ½... | Mathlib/MeasureTheory/Measure/Restrict.lean | 104 | 107 | theorem restrict_apply' (hs : MeasurableSet s) : ΞΌ.restrict s t = ΞΌ (t β© s) := by |
rw [β toOuterMeasure_apply,
Measure.restrict_toOuterMeasure_eq_toOuterMeasure_restrict hs,
OuterMeasure.restrict_apply s t _, toOuterMeasure_apply]
| 3 | 20.085537 | 1 | 0.666667 | 6 | 583 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R Ξ± Ξ² Ξ΄ Ξ³ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] [MeasurableSpace Ξ³]
variable {ΞΌ ΞΌβ ΞΌβ ΞΌβ Ξ½ Ξ½' Ξ½... | Mathlib/MeasureTheory/Measure/Restrict.lean | 110 | 113 | theorem restrict_applyβ' (hs : NullMeasurableSet s ΞΌ) : ΞΌ.restrict s t = ΞΌ (t β© s) := by |
rw [β restrict_congr_set hs.toMeasurable_ae_eq,
restrict_apply' (measurableSet_toMeasurable _ _),
measure_congr ((ae_eq_refl t).inter hs.toMeasurable_ae_eq)]
| 3 | 20.085537 | 1 | 0.666667 | 6 | 583 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R Ξ± Ξ² Ξ΄ Ξ³ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] [MeasurableSpace Ξ³]
variable {ΞΌ ΞΌβ ΞΌβ ΞΌβ Ξ½ Ξ½' Ξ½... | Mathlib/MeasureTheory/Measure/Restrict.lean | 124 | 130 | theorem restrict_eq_self (h : s β t) : ΞΌ.restrict t s = ΞΌ s :=
(le_iff'.1 restrict_le_self s).antisymm <|
calc
ΞΌ s β€ ΞΌ (toMeasurable (ΞΌ.restrict t) s β© t) :=
measure_mono (subset_inter (subset_toMeasurable _ _) h)
_ = ΞΌ.restrict t s := by |
rw [β restrict_apply (measurableSet_toMeasurable _ _), measure_toMeasurable]
| 1 | 2.718282 | 0 | 0.666667 | 6 | 583 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R Ξ± Ξ² Ξ΄ Ξ³ ΞΉ : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace Ξ±} [MeasurableSpace Ξ²] [MeasurableSpace Ξ³]
variable {ΞΌ ΞΌβ ΞΌβ ΞΌβ Ξ½ Ξ½' Ξ½... | Mathlib/MeasureTheory/Measure/Restrict.lean | 140 | 141 | theorem restrict_apply_univ (s : Set Ξ±) : ΞΌ.restrict s univ = ΞΌ s := by |
rw [restrict_apply MeasurableSet.univ, Set.univ_inter]
| 1 | 2.718282 | 0 | 0.666667 | 6 | 583 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Basis
#align_import linear_algebra.determinant from "lea... | Mathlib/LinearAlgebra/Determinant.lean | 77 | 78 | theorem det_comm [DecidableEq n] (M N : Matrix n n A) : det (M * N) = det (N * M) := by |
rw [det_mul, det_mul, mul_comm]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 584 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Basis
#align_import linear_algebra.determinant from "lea... | Mathlib/LinearAlgebra/Determinant.lean | 83 | 90 | theorem det_comm' [DecidableEq m] [DecidableEq n] {M : Matrix n m A} {N : Matrix m n A}
{M' : Matrix m n A} (hMM' : M * M' = 1) (hM'M : M' * M = 1) : det (M * N) = det (N * M) := by |
nontriviality A
-- Although `m` and `n` are different a priori, we will show they have the same cardinality.
-- This turns the problem into one for square matrices, which is easy.
let e := indexEquivOfInv hMM' hM'M
rw [β det_submatrix_equiv_self e, β submatrix_mul_equiv _ _ _ (Equiv.refl n) _, det_comm,
... | 6 | 403.428793 | 2 | 0.666667 | 3 | 584 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Basis
#align_import linear_algebra.determinant from "lea... | Mathlib/LinearAlgebra/Determinant.lean | 96 | 99 | theorem det_conj_of_mul_eq_one [DecidableEq m] [DecidableEq n] {M : Matrix m n A}
{M' : Matrix n m A} {N : Matrix n n A} (hMM' : M * M' = 1) (hM'M : M' * M = 1) :
det (M * N * M') = det N := by |
rw [β det_comm' hM'M hMM', β Matrix.mul_assoc, hM'M, Matrix.one_mul]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 584 |
import Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.equalizers from "leanprover-community/mathlib"@"4698e35ca56a0d4fa53aa5639c3364e0a77f4eba"
noncomputable section
universe w vβ vβ uβ uβ
open Cate... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Equalizers.lean | 104 | 108 | theorem PreservesEqualizer.iso_inv_ΞΉ :
(PreservesEqualizer.iso G f g).inv β« G.map (equalizer.ΞΉ f g) =
equalizer.ΞΉ (G.map f) (G.map g) := by |
rw [β Iso.cancel_iso_hom_left (PreservesEqualizer.iso G f g), β Category.assoc, Iso.hom_inv_id]
simp
| 2 | 7.389056 | 1 | 0.666667 | 3 | 585 |
import Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.equalizers from "leanprover-community/mathlib"@"4698e35ca56a0d4fa53aa5639c3364e0a77f4eba"
noncomputable section
universe w vβ vβ uβ uβ
open Cate... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Equalizers.lean | 207 | 211 | theorem map_Ο_preserves_coequalizer_inv :
G.map (coequalizer.Ο f g) β« (PreservesCoequalizer.iso G f g).inv =
coequalizer.Ο (G.map f) (G.map g) := by |
rw [β ΞΉ_comp_coequalizerComparison_assoc, β PreservesCoequalizer.iso_hom, Iso.hom_inv_id,
comp_id]
| 2 | 7.389056 | 1 | 0.666667 | 3 | 585 |
import Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.equalizers from "leanprover-community/mathlib"@"4698e35ca56a0d4fa53aa5639c3364e0a77f4eba"
noncomputable section
universe w vβ vβ uβ uβ
open Cate... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Equalizers.lean | 215 | 218 | theorem map_Ο_preserves_coequalizer_inv_desc {W : D} (k : G.obj Y βΆ W)
(wk : G.map f β« k = G.map g β« k) : G.map (coequalizer.Ο f g) β«
(PreservesCoequalizer.iso G f g).inv β« coequalizer.desc k wk = k := by |
rw [β Category.assoc, map_Ο_preserves_coequalizer_inv, coequalizer.Ο_desc]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 585 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
#align_import algebra.big_operators.nat_antidiagonal from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
variable {M N : Type*} [CommMonoid M] [AddCommMonoid N]
namespace Finset
namespace Nat
| Mathlib/Algebra/BigOperators/NatAntidiagonal.lean | 23 | 26 | theorem prod_antidiagonal_succ {n : β} {f : β Γ β β M} :
(β p β antidiagonal (n + 1), f p)
= f (0, n + 1) * β p β antidiagonal n, f (p.1 + 1, p.2) := by |
rw [antidiagonal_succ, prod_cons, prod_map]; rfl
| 1 | 2.718282 | 0 | 0.666667 | 3 | 586 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
#align_import algebra.big_operators.nat_antidiagonal from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
variable {M N : Type*} [CommMonoid M] [AddCommMonoid N]
namespace Finset
namespace Nat
t... | Mathlib/Algebra/BigOperators/NatAntidiagonal.lean | 35 | 38 | theorem prod_antidiagonal_swap {n : β} {f : β Γ β β M} :
β p β antidiagonal n, f p.swap = β p β antidiagonal n, f p := by |
conv_lhs => rw [β map_swap_antidiagonal, Finset.prod_map]
rfl
| 2 | 7.389056 | 1 | 0.666667 | 3 | 586 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
#align_import algebra.big_operators.nat_antidiagonal from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
variable {M N : Type*} [CommMonoid M] [AddCommMonoid N]
namespace Finset
namespace Nat
t... | Mathlib/Algebra/BigOperators/NatAntidiagonal.lean | 42 | 45 | theorem prod_antidiagonal_succ' {n : β} {f : β Γ β β M} : (β p β antidiagonal (n + 1), f p) =
f (n + 1, 0) * β p β antidiagonal n, f (p.1, p.2 + 1) := by |
rw [β prod_antidiagonal_swap, prod_antidiagonal_succ, β prod_antidiagonal_swap]
rfl
| 2 | 7.389056 | 1 | 0.666667 | 3 | 586 |
import Mathlib.Algebra.Homology.Single
#align_import algebra.homology.augment from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open CategoryTheory Limits HomologicalComplex
universe v u
variable {V : Type u} [Category.{v} V]
namespace ChainComplex
@[simps]... | Mathlib/Algebra/Homology/Augment.lean | 92 | 94 | theorem augment_d_succ_succ (C : ChainComplex V β) {X : V} (f : C.X 0 βΆ X) (w : C.d 1 0 β« f = 0)
(i j : β) : (augment C f w).d (i + 1) (j + 1) = C.d i j := by |
cases i <;> rfl
| 1 | 2.718282 | 0 | 0.666667 | 3 | 587 |
import Mathlib.Algebra.Homology.Single
#align_import algebra.homology.augment from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open CategoryTheory Limits HomologicalComplex
universe v u
variable {V : Type u} [Category.{v} V]
namespace ChainComplex
@[simps]... | Mathlib/Algebra/Homology/Augment.lean | 132 | 134 | theorem chainComplex_d_succ_succ_zero (C : ChainComplex V β) (i : β) : C.d (i + 2) 0 = 0 := by |
rw [C.shape]
exact i.succ_succ_ne_one.symm
| 2 | 7.389056 | 1 | 0.666667 | 3 | 587 |
import Mathlib.Algebra.Homology.Single
#align_import algebra.homology.augment from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open CategoryTheory Limits HomologicalComplex
universe v u
variable {V : Type u} [Category.{v} V]
namespace CochainComplex
@[simp... | Mathlib/Algebra/Homology/Augment.lean | 325 | 328 | theorem cochainComplex_d_succ_succ_zero (C : CochainComplex V β) (i : β) : C.d 0 (i + 2) = 0 := by |
rw [C.shape]
simp only [ComplexShape.up_Rel, zero_add]
exact (Nat.one_lt_succ_succ _).ne
| 3 | 20.085537 | 1 | 0.666667 | 3 | 587 |
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Antisymmetrization
#align_import order.cover from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Set OrderDual
variable {Ξ± Ξ² : Type*}
section WeaklyCovers
section Preorder
variable [Preorder Ξ±] [Preorder Ξ²] {a ... | Mathlib/Order/Cover.lean | 96 | 97 | theorem not_wcovBy_iff (h : a β€ b) : Β¬a β©Ώ b β β c, a < c β§ c < b := by |
simp_rw [WCovBy, h, true_and_iff, not_forall, exists_prop, not_not]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 588 |
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Antisymmetrization
#align_import order.cover from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Set OrderDual
variable {Ξ± Ξ² : Type*}
section WeaklyCovers
section Preorder
variable [Preorder Ξ±] [Preorder Ξ²] {a ... | Mathlib/Order/Cover.lean | 122 | 126 | theorem WCovBy.image (f : Ξ± βͺo Ξ²) (hab : a β©Ώ b) (h : (range f).OrdConnected) : f a β©Ώ f b := by |
refine β¨f.monotone hab.le, fun c ha hb => ?_β©
obtain β¨c, rflβ© := h.out (mem_range_self _) (mem_range_self _) β¨ha.le, hb.leβ©
rw [f.lt_iff_lt] at ha hb
exact hab.2 ha hb
| 4 | 54.59815 | 2 | 0.666667 | 3 | 588 |
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Antisymmetrization
#align_import order.cover from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Set OrderDual
variable {Ξ± Ξ² : Type*}
section WeaklyCovers
section LT
variable [LT Ξ±] {a b : Ξ±}
def CovBy (a b :... | Mathlib/Order/Cover.lean | 233 | 234 | theorem not_covBy_iff (h : a < b) : Β¬a β b β β c, a < c β§ c < b := by |
simp_rw [CovBy, h, true_and_iff, not_forall, exists_prop, not_not]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 588 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 131 | 133 | theorem mod_add_div' (m k : R) : m % k + m / k * k = m := by |
rw [mul_comm]
exact mod_add_div _ _
| 2 | 7.389056 | 1 | 0.666667 | 6 | 589 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 136 | 138 | theorem div_add_mod' (m k : R) : m / k * k + m % k = m := by |
rw [mul_comm]
exact div_add_mod _ _
| 2 | 7.389056 | 1 | 0.666667 | 6 | 589 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 141 | 144 | theorem mod_eq_sub_mul_div {R : Type*} [EuclideanDomain R] (a b : R) : a % b = a - b * (a / b) :=
calc
a % b = b * (a / b) + a % b - b * (a / b) := (add_sub_cancel_left _ _).symm
_ = a - b * (a / b) := by | rw [div_add_mod]
| 1 | 2.718282 | 0 | 0.666667 | 6 | 589 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 151 | 153 | theorem mul_right_not_lt {a : R} (b) (h : a β 0) : Β¬a * b βΊ b := by |
rw [mul_comm]
exact mul_left_not_lt b h
| 2 | 7.389056 | 1 | 0.666667 | 6 | 589 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 157 | 157 | theorem mod_zero (a : R) : a % 0 = a := by | simpa only [zero_mul, zero_add] using div_add_mod a 0
| 1 | 2.718282 | 0 | 0.666667 | 6 | 589 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 209 | 211 | theorem gcd_zero_left (a : R) : gcd 0 a = a := by |
rw [gcd]
exact if_pos rfl
| 2 | 7.389056 | 1 | 0.666667 | 6 | 589 |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {Ξ± Ξ² Ξ³ : Type*} {ΞΉ ΞΉ' : Sort*}
section Preimage
variable {f : Ξ± β Ξ²} {g : Ξ² β Ξ³... | Mathlib/Data/Set/Image.lean | 53 | 55 | theorem preimage_congr {f g : Ξ± β Ξ²} {s : Set Ξ²} (h : β x : Ξ±, f x = g x) : f β»ΒΉ' s = g β»ΒΉ' s := by |
congr with x
simp [h]
| 2 | 7.389056 | 1 | 0.666667 | 15 | 590 |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {Ξ± Ξ² Ξ³ : Type*} {ΞΉ ΞΉ' : Sort*}
section Preimage
variable {f : Ξ± β Ξ²} {g : Ξ² β Ξ³... | Mathlib/Data/Set/Image.lean | 133 | 136 | theorem preimage_const (b : Ξ²) (s : Set Ξ²) [Decidable (b β s)] :
(fun _ : Ξ± => b) β»ΒΉ' s = if b β s then univ else β
:= by |
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
| 2 | 7.389056 | 1 | 0.666667 | 15 | 590 |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {Ξ± Ξ² Ξ³ : Type*} {ΞΉ ΞΉ' : Sort*}
section Preimage
variable {f : Ξ± β Ξ²} {g : Ξ² β Ξ³... | Mathlib/Data/Set/Image.lean | 157 | 159 | theorem preimage_iterate_eq {f : Ξ± β Ξ±} {n : β} : Set.preimage f^[n] = (Set.preimage f)^[n] := by |
induction' n with n ih; Β· simp
rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
| 2 | 7.389056 | 1 | 0.666667 | 15 | 590 |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {Ξ± Ξ² Ξ³ : Type*} {ΞΉ ΞΉ' : Sort*}
section Image
variable {f : Ξ± β Ξ²} {s t : Set... | Mathlib/Data/Set/Image.lean | 223 | 224 | theorem forall_mem_image {f : Ξ± β Ξ²} {s : Set Ξ±} {p : Ξ² β Prop} :
(β y β f '' s, p y) β β β¦xβ¦, x β s β p (f x) := by | simp
| 1 | 2.718282 | 0 | 0.666667 | 15 | 590 |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {Ξ± Ξ² Ξ³ : Type*} {ΞΉ ΞΉ' : Sort*}
section Image
variable {f : Ξ± β Ξ²} {s t : Set... | Mathlib/Data/Set/Image.lean | 227 | 228 | theorem exists_mem_image {f : Ξ± β Ξ²} {s : Set Ξ±} {p : Ξ² β Prop} :
(β y β f '' s, p y) β β x β s, p (f x) := by | simp
| 1 | 2.718282 | 0 | 0.666667 | 15 | 590 |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {Ξ± Ξ² Ξ³ : Type*} {ΞΉ ΞΉ' : Sort*}
section Image
variable {f : Ξ± β Ξ²} {s t : Set... | Mathlib/Data/Set/Image.lean | 249 | 251 | theorem image_congr {f g : Ξ± β Ξ²} {s : Set Ξ±} (h : β a β s, f a = g a) : f '' s = g '' s := by |
ext x
exact exists_congr fun a β¦ and_congr_right fun ha β¦ by rw [h a ha]
| 2 | 7.389056 | 1 | 0.666667 | 15 | 590 |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {Ξ± Ξ² Ξ³ : Type*} {ΞΉ ΞΉ' : Sort*}
section Image
variable {f : Ξ± β Ξ²} {s t : Set... | Mathlib/Data/Set/Image.lean | 263 | 263 | theorem image_comp (f : Ξ² β Ξ³) (g : Ξ± β Ξ²) (a : Set Ξ±) : f β g '' a = f '' (g '' a) := by | aesop
| 1 | 2.718282 | 0 | 0.666667 | 15 | 590 |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {Ξ± Ξ² Ξ³ : Type*} {ΞΉ ΞΉ' : Sort*}
section Image
variable {f : Ξ± β Ξ²} {s t : Set... | Mathlib/Data/Set/Image.lean | 266 | 266 | theorem image_comp_eq {g : Ξ² β Ξ³} : image (g β f) = image g β image f := by | ext; simp
| 1 | 2.718282 | 0 | 0.666667 | 15 | 590 |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {Ξ± Ξ² Ξ³ : Type*} {ΞΉ ΞΉ' : Sort*}
section Image
variable {f : Ξ± β Ξ²} {s t : Set... | Mathlib/Data/Set/Image.lean | 273 | 275 | theorem image_comm {Ξ²'} {f : Ξ² β Ξ³} {g : Ξ± β Ξ²} {f' : Ξ± β Ξ²'} {g' : Ξ²' β Ξ³}
(h_comm : β a, f (g a) = g' (f' a)) : (s.image g).image f = (s.image f').image g' := by |
simp_rw [image_image, h_comm]
| 1 | 2.718282 | 0 | 0.666667 | 15 | 590 |
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