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.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 150 | 153 | theorem coeff_monomial_ne {m n : σ →₀ ℕ} (h : m ≠ n) (a : R) : coeff R m (monomial R n a) = 0 := by |
classical
rw [monomial_def]
exact LinearMap.stdBasis_ne R (fun _ ↦ R) _ _ h a
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,333 |
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 w₁
namespace LieAlgebra
variable (R : Type u) (L : Type v) [CommRing R] [LieR... | Mathlib/Algebra/Lie/Character.lean | 44 | 45 | theorem lieCharacter_apply_lie (χ : LieCharacter R L) (x y : L) : χ ⁅x, y⁆ = 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.Deprecated.Group
#align_import deprecated.ring from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
universe u v w
variable {α : Type u}
structure IsSemiringHom {α : Type u} {β : Type v} [Semiring α] [Semiring β] (f : α → β) : Prop where
map_zero : f 0 = 0
map... | Mathlib/Deprecated/Ring.lean | 54 | 54 | theorem id : IsSemiringHom (@id α) := by | constructor <;> intros <;> rfl
| 1 | 2.718282 | 0 | 0.5 | 8 | 496 |
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.multivariate.basic from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
universe u v
open MvFunctor
@[pp_with_univ]
structure MvPFunctor (n : ℕ) where
A : Type u
... | Mathlib/Data/PFunctor/Multivariate/Basic.lean | 142 | 144 | theorem comp.get_map (f : α ⟹ β) (x : comp P Q α) :
comp.get (f <$$> x) = (fun i (x : Q i α) => f <$$> x) <$$> comp.get x := by |
rfl
| 1 | 2.718282 | 0 | 0.857143 | 7 | 749 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 99 | 100 | theorem interior_setOf_im_le (a : ℝ) : interior { z : ℂ | z.im ≤ a } = { z | z.im < a } := by |
simpa only [interior_Iic] using interior_preimage_im (Iic a)
| 1 | 2.718282 | 0 | 0 | 10 | 135 |
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped NNReal Matrix
namespace Matrix
variable {R l m n α β : Type*} [Fintype l] [Fintyp... | Mathlib/Analysis/Matrix.lean | 116 | 118 | theorem nnnorm_map_eq (A : Matrix m n α) (f : α → β) (hf : ∀ a, ‖f a‖₊ = ‖a‖₊) :
‖A.map f‖₊ = ‖A‖₊ := by |
simp only [nnnorm_def, Pi.nnnorm_def, Matrix.map_apply, hf]
| 1 | 2.718282 | 0 | 0.533333 | 15 | 509 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 394 | 401 | theorem mulVec_surjective_iff_exists_right_inverse [Finite m] [Fintype n] {A : Matrix m n R} :
Function.Surjective A.mulVec ↔ ∃ B : Matrix n m R, A * B = 1 := by |
cases nonempty_fintype m
refine ⟨fun h ↦ ?_, fun ⟨B, hBA⟩ y ↦ ⟨B *ᵥ y, by simp [hBA]⟩⟩
choose cols hcols using (h <| Pi.single · 1)
refine ⟨(Matrix.of cols)ᵀ, Matrix.ext fun i j ↦ ?_⟩
rw [one_eq_pi_single, Pi.single_comm, ← hcols j]
rfl
| 6 | 403.428793 | 2 | 1 | 9 | 1,111 |
import Mathlib.Control.Monad.Basic
import Mathlib.Control.Monad.Writer
import Mathlib.Init.Control.Lawful
#align_import control.monad.cont from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
universe u v w u₀ u₁ v₀ v₁
structure MonadCont.Label (α : Type w) (m : Type u → Type v) (β : Typ... | Mathlib/Control/Monad/Cont.lean | 193 | 194 | theorem WriterT.goto_mkLabel {α β ω : Type _} [EmptyCollection ω] (x : Label (α × ω) m β) (i : α) :
goto (WriterT.mkLabel x) i = monadLift (goto x (i, ∅)) := by | cases x; rfl
| 1 | 2.718282 | 0 | 0.333333 | 3 | 361 |
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 96 | 97 | theorem ne_of_oangle_ne_zero {x y : V} (h : o.oangle x y ≠ 0) : x ≠ y := by |
rintro rfl; simp at h
| 1 | 2.718282 | 0 | 0.571429 | 7 | 521 |
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
open Nat
namespace List
section countP
variable (p q : α → Bool)
@[simp] theorem countP_nil : countP p [] = 0 := rfl
protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 := by
induction l generalizing n with
| nil... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 75 | 76 | theorem countP_pos : 0 < countP p l ↔ ∃ a ∈ l, p a := by |
simp only [countP_eq_length_filter, length_pos_iff_exists_mem, mem_filter, exists_prop]
| 1 | 2.718282 | 0 | 0.8 | 10 | 703 |
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
#align_import algebra.order.group.min_max from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
section
variable {α : Type*} [Group α] [LinearOrder α] [CovariantClass α α (· * ·) (· ≤ ·)]
-- TODO... | Mathlib/Algebra/Order/Group/MinMax.lean | 57 | 58 | theorem min_div_div_right' (a b c : α) : min (a / c) (b / c) = min a b / c := by |
simpa only [div_eq_mul_inv] using min_mul_mul_right a b c⁻¹
| 1 | 2.718282 | 0 | 0.571429 | 7 | 524 |
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u
variable {α : Type u}
open Nat Function
namespace List
theorem rotate... | Mathlib/Data/List/Rotate.lean | 103 | 108 | theorem rotate'_mod (l : List α) (n : ℕ) : l.rotate' (n % l.length) = l.rotate' n :=
calc
l.rotate' (n % l.length) =
(l.rotate' (n % l.length)).rotate' ((l.rotate' (n % l.length)).length * (n / l.length)) :=
by rw [rotate'_length_mul]
_ = l.rotate' n := by | rw [rotate'_rotate', length_rotate', Nat.mod_add_div]
| 1 | 2.718282 | 0 | 0.153846 | 13 | 257 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
open Topology InnerProductSpace Set
noncomputable section
variable {𝕜 F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F]
variabl... | Mathlib/Analysis/Calculus/Gradient/Basic.lean | 110 | 111 | theorem gradient_eq_zero_of_not_differentiableAt (h : ¬DifferentiableAt 𝕜 f x) : ∇ f x = 0 := by |
rw [gradient, fderiv_zero_of_not_differentiableAt h, map_zero]
| 1 | 2.718282 | 0 | 0.538462 | 13 | 511 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {α β γ δ : Type*} {f : α → β → γ} {a : Option α} {b : Option β} {c : Option γ}
def map₂ (f : α → β → γ) (a : Option α) ... | Mathlib/Data/Option/NAry.lean | 95 | 96 | theorem map₂_map_left (f : γ → β → δ) (g : α → γ) :
map₂ f (a.map g) b = map₂ (fun a b => f (g a) b) a b := by | cases a <;> rfl
| 1 | 2.718282 | 0 | 0 | 14 | 191 |
import Mathlib.Algebra.Ring.Semiconj
import Mathlib.Algebra.Ring.Units
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Data.Bracket
#align_import algebra.ring.commute from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u v w x
variable {α : Type u} {β : Type v} {γ : T... | Mathlib/Algebra/Ring/Commute.lean | 72 | 74 | theorem mul_self_sub_mul_self_eq [NonUnitalNonAssocRing R] {a b : R} (h : Commute a b) :
a * a - b * b = (a + b) * (a - b) := by |
rw [add_mul, mul_sub, mul_sub, h.eq, sub_add_sub_cancel]
| 1 | 2.718282 | 0 | 0.333333 | 3 | 342 |
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-community/mathlib"@"4e24c4bfcff371c71f7ba22050308aa17815626c"
variable {α β : Type*} [LinearOrder α]
open Function
namespace Set
def projIci (a x : α) : Ici a := ⟨max a x,... | Mathlib/Order/Interval/Set/ProjIcc.lean | 113 | 113 | theorem projIci_of_mem (hx : x ∈ Ici a) : projIci a x = ⟨x, hx⟩ := by | simpa [projIci]
| 1 | 2.718282 | 0 | 0.083333 | 12 | 241 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RCLike
open scoped ComplexConjugate
variable {𝕜 E F G : Type... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 85 | 87 | theorem adjointAux_inner_right (A : E →L[𝕜] F) (x : E) (y : F) :
⟪x, adjointAux A y⟫ = ⟪A x, y⟫ := by |
rw [← inner_conj_symm, adjointAux_inner_left, inner_conj_symm]
| 1 | 2.718282 | 0 | 0.875 | 8 | 763 |
import Batteries.Tactic.Alias
import Batteries.Data.Nat.Basic
namespace Nat
@[simp] theorem recAux_zero {motive : Nat → Sort _} (zero : motive 0)
(succ : ∀ n, motive n → motive (n+1)) :
Nat.recAux zero succ 0 = zero := rfl
theorem recAux_succ {motive : Nat → Sort _} (zero : motive 0)
(succ : ∀ n, mo... | .lake/packages/batteries/Batteries/Data/Nat/Lemmas.lean | 81 | 86 | theorem recDiag_succ_zero {motive : Nat → Nat → Sort _} (zero_zero : motive 0 0)
(zero_succ : ∀ n, motive 0 n → motive 0 (n+1)) (succ_zero : ∀ m, motive m 0 → motive (m+1) 0)
(succ_succ : ∀ m n, motive m n → motive (m+1) (n+1)) (m) :
Nat.recDiag zero_zero zero_succ succ_zero succ_succ (m+1) 0
= succ_z... |
simp [Nat.recDiag]; cases m <;> rfl
| 1 | 2.718282 | 0 | 0 | 3 | 30 |
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Basic
import Mathlib.Order.CompleteLatticeIntervals
#align_import order.sup_indep from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
variable {α β ι ι' :... | Mathlib/Order/SupIndep.lean | 92 | 96 | theorem SupIndep.le_sup_iff (hs : s.SupIndep f) (hts : t ⊆ s) (hi : i ∈ s) (hf : ∀ i, f i ≠ ⊥) :
f i ≤ t.sup f ↔ i ∈ t := by |
refine ⟨fun h => ?_, le_sup⟩
by_contra hit
exact hf i (disjoint_self.1 <| (hs hts hi hit).mono_right h)
| 3 | 20.085537 | 1 | 1.6 | 5 | 1,742 |
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.RingTheory.Ideal.Quotient
#align_import topology.algebra.ring.ideal from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
section CommRing
variable {R : Type*} [TopologicalSpace R] [CommRing R] (N : Ideal R)
open Ideal.Quotient
... | Mathlib/Topology/Algebra/Ring/Ideal.lean | 61 | 65 | theorem QuotientRing.isOpenMap_coe : IsOpenMap (mk N) := by |
intro s s_op
change IsOpen (mk N ⁻¹' (mk N '' s))
rw [quotient_ring_saturate]
exact isOpen_iUnion fun ⟨n, _⟩ => isOpenMap_add_left n s s_op
| 4 | 54.59815 | 2 | 2 | 1 | 2,490 |
import Mathlib.Data.Set.Lattice
import Mathlib.Data.Set.Pairwise.Basic
#align_import data.set.pairwise.lattice from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Function Set Order
variable {α β γ ι ι' : Type*} {κ : Sort*} {r p q : α → α → Prop}
section Pairwise
variable {f g : ... | Mathlib/Data/Set/Pairwise/Lattice.lean | 39 | 41 | theorem pairwise_sUnion {r : α → α → Prop} {s : Set (Set α)} (h : DirectedOn (· ⊆ ·) s) :
(⋃₀ s).Pairwise r ↔ ∀ a ∈ s, Set.Pairwise a r := by |
rw [sUnion_eq_iUnion, pairwise_iUnion h.directed_val, SetCoe.forall]
| 1 | 2.718282 | 0 | 1.666667 | 6 | 1,759 |
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#align_import ring_theory.ideal.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u v w
variable {α : Type u} {β : Type v}
open ... | Mathlib/RingTheory/Ideal/Basic.lean | 167 | 168 | theorem isCompactElement_top : CompleteLattice.IsCompactElement (⊤ : Ideal α) := by |
simpa only [← span_singleton_one] using Submodule.singleton_span_isCompactElement 1
| 1 | 2.718282 | 0 | 1 | 3 | 963 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 112 | 112 | theorem laverage_zero_measure (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂(0 : Measure α) = 0 := by | simp [laverage]
| 1 | 2.718282 | 0 | 0.347826 | 23 | 374 |
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.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open scoped Classical Topology
open Filter Asymptotics Set
variable {𝕜... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 119 | 121 | theorem iteratedDerivWithin_one {x : 𝕜} (h : UniqueDiffWithinAt 𝕜 s x) :
iteratedDerivWithin 1 f s x = derivWithin f s x := by |
simp only [iteratedDerivWithin, iteratedFDerivWithin_one_apply h]; rfl
| 1 | 2.718282 | 0 | 0.727273 | 11 | 649 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 73 | 74 | theorem midpoint_comm (x y : P) : midpoint R x y = midpoint R y x := by |
rw [midpoint, ← lineMap_apply_one_sub, one_sub_invOf_two, midpoint]
| 1 | 2.718282 | 0 | 0.444444 | 9 | 412 |
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
universe v v₁ v₂ u u₁ u₂
variable {U : Type*} [Quiver.{u + 1} U]
namespace Quiver
def Hom.cast {u v u' v... | Mathlib/Combinatorics/Quiver/Cast.lean | 57 | 60 | theorem Hom.cast_heq {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) :
HEq (e.cast hu hv) e := by |
subst_vars
rfl
| 2 | 7.389056 | 1 | 1 | 12 | 1,049 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Topology.Algebra.Order.Group
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.floor from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Filter Function Int Set Topology
variable {α β γ : Type*} [LinearOrdere... | Mathlib/Topology/Algebra/Order/Floor.lean | 108 | 110 | theorem tendsto_ceil_right_pure_add_one (n : ℤ) :
Tendsto (ceil : α → ℤ) (𝓝[>] n) (pure (n + 1)) := by |
simpa only [floor_intCast] using tendsto_ceil_right_pure_floor_add_one (n : α)
| 1 | 2.718282 | 0 | 0.5 | 6 | 492 |
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.Lint.Basic
#align_import logic.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
#align_import init.ite_simp from "leanprover-communit... | Mathlib/Logic/Basic.lean | 595 | 598 | theorem eqRec_heq' {α : Sort*} {a' : α} {motive : (a : α) → a' = a → Sort*}
(p : motive a' (rfl : a' = a')) {a : α} (t : a' = a) :
HEq (@Eq.rec α a' motive p a t) p := by |
subst t; rfl
| 1 | 2.718282 | 0 | 0 | 8 | 105 |
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Set.Lattice
#align_import data.nat.pairing from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
assert_not_exists MonoidWithZero
open Prod Decidable Function
namespace Nat
-- Porting note: no pp_nodot
--@[pp_nodot]
def pair (a b : ... | Mathlib/Data/Nat/Pairing.lean | 114 | 114 | theorem left_le_pair (a b : ℕ) : a ≤ pair a b := by | simpa using unpair_left_le (pair a b)
| 1 | 2.718282 | 0 | 1.2 | 10 | 1,270 |
import Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
noncomputable section
open scoped Manifold
open Bundle Set Topology
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
(I : ModelWithCorners 𝕜 E H) {M : Type*} [To... | Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean | 146 | 153 | theorem tangentMap_chart {p q : TangentBundle I M} (h : q.1 ∈ (chartAt H p.1).source) :
tangentMap I I (chartAt H p.1) q =
(TotalSpace.toProd _ _).symm
((chartAt (ModelProd H E) p : TangentBundle I M → ModelProd H E) q) := by |
dsimp [tangentMap]
rw [MDifferentiableAt.mfderiv]
· rfl
· exact mdifferentiableAt_atlas _ (chart_mem_atlas _ _) h
| 4 | 54.59815 | 2 | 2 | 6 | 2,362 |
import Mathlib.CategoryTheory.Sites.Spaces
import Mathlib.Topology.Sheaves.Sheaf
import Mathlib.CategoryTheory.Sites.DenseSubsite
#align_import topology.sheaves.sheaf_condition.sites from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomputable section
set_option linter.uppercaseLe... | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | 90 | 94 | theorem covering_presieve_eq_self {Y : Opens X} (R : Presieve Y) :
presieveOfCoveringAux (coveringOfPresieve Y R) Y = R := by |
funext Z
ext f
exact ⟨fun ⟨⟨_, f', h⟩, rfl⟩ => by rwa [Subsingleton.elim f f'], fun h => ⟨⟨Z, f, h⟩, rfl⟩⟩
| 3 | 20.085537 | 1 | 1.6 | 5 | 1,729 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Data.Complex.Determinant
#align_import analysis.complex.operator_norm from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open ContinuousLinearMap
namespace Complex
@[simp... | Mathlib/Analysis/Complex/OperatorNorm.lean | 37 | 41 | theorem reCLM_norm : ‖reCLM‖ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
calc
1 = ‖reCLM 1‖ := by | simp
_ ≤ ‖reCLM‖ := unit_le_opNorm _ _ (by simp)
| 2 | 7.389056 | 1 | 1 | 2 | 1,141 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
noncomputable section
open scoped Classical
open ... | Mathlib/Geometry/Euclidean/Basic.lean | 112 | 117 | theorem dist_smul_vadd_sq (r : ℝ) (v : V) (p₁ p₂ : P) :
dist (r • v +ᵥ p₁) p₂ * dist (r • v +ᵥ p₁) p₂ =
⟪v, v⟫ * r * r + 2 * ⟪v, p₁ -ᵥ p₂⟫ * r + ⟪p₁ -ᵥ p₂, p₁ -ᵥ p₂⟫ := by |
rw [dist_eq_norm_vsub V _ p₂, ← real_inner_self_eq_norm_mul_norm, vadd_vsub_assoc,
real_inner_add_add_self, real_inner_smul_left, real_inner_smul_left, real_inner_smul_right]
ring
| 3 | 20.085537 | 1 | 1.4 | 5 | 1,493 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Tactic.Ring
#align_import data.nat.hyperoperation from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
def hyperoperation : ℕ → ℕ → ℕ → ℕ
| 0, _, k => k + 1
| 1, m, 0 => m
| 2, _, 0 => 0
| _ + 3, _, 0 => 1
| n + 1, m, k + 1 ... | Mathlib/Data/Nat/Hyperoperation.lean | 53 | 55 | theorem hyperoperation_recursion (n m k : ℕ) :
hyperoperation (n + 1) m (k + 1) = hyperoperation n m (hyperoperation (n + 1) m k) := by |
rw [hyperoperation]
| 1 | 2.718282 | 0 | 1.444444 | 9 | 1,532 |
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.birthday from "leanprover-community/mathlib"@"a347076985674932c0e91da09b9961ed0a79508c"
universe u
open Ordinal
namespace SetTheory
open scoped NaturalOps PGame
namespace PGame
noncomputable def b... | Mathlib/SetTheory/Game/Birthday.lean | 107 | 107 | theorem birthday_one : birthday 1 = 1 := by | rw [birthday_def]; simp
| 1 | 2.718282 | 0 | 0.4 | 10 | 387 |
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.MeasureTheory.Function.LpSpace
#align_import measure_theory.function.lp_order from "leanprover-community/mathlib"@"5dc275ec639221ca4d5f56938eb966f6ad9bc89f"
set_option linter.uppercaseLean3 false
open TopologicalSpace MeasureTheory
open scoped ENNReal
... | Mathlib/MeasureTheory/Function/LpOrder.lean | 45 | 50 | theorem coeFn_nonneg (f : Lp E p μ) : 0 ≤ᵐ[μ] f ↔ 0 ≤ f := by |
rw [← coeFn_le]
have h0 := Lp.coeFn_zero E p μ
constructor <;> intro h <;> filter_upwards [h, h0] with _ _ h2
· rwa [h2]
· rwa [← h2]
| 5 | 148.413159 | 2 | 1 | 2 | 974 |
import Mathlib.Analysis.LocallyConvex.Basic
#align_import analysis.locally_convex.balanced_core_hull from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Pointwise Topology Filter
variable {𝕜 E ι : Type*}
section balancedHull
section SeminormedRing
variable [SeminormedRing ... | Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean | 108 | 109 | theorem mem_balancedHull_iff : x ∈ balancedHull 𝕜 s ↔ ∃ r : 𝕜, ‖r‖ ≤ 1 ∧ x ∈ r • s := by |
simp [balancedHull]
| 1 | 2.718282 | 0 | 0.75 | 4 | 666 |
import Mathlib.Topology.ContinuousFunction.Basic
#align_import topology.compact_open from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter TopologicalSpace
open scoped Topology
namespace ContinuousMap
section CompactOpen
variable {α X Y Z T : Type*}
variable [Topologica... | Mathlib/Topology/CompactOpen.lean | 354 | 354 | theorem image_coev {y : Y} (s : Set X) : coev X Y y '' s = {y} ×ˢ s := by | simp
| 1 | 2.718282 | 0 | 1.4 | 5 | 1,504 |
import Mathlib.Order.Filter.Bases
#align_import order.filter.pi from "leanprover-community/mathlib"@"ce64cd319bb6b3e82f31c2d38e79080d377be451"
open Set Function
open scoped Classical
open Filter
namespace Filter
variable {ι : Type*} {α : ι → Type*} {f f₁ f₂ : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)}
... | Mathlib/Order/Filter/Pi.lean | 229 | 230 | theorem mem_coprodᵢ_iff {s : Set (∀ i, α i)} :
s ∈ Filter.coprodᵢ f ↔ ∀ i : ι, ∃ t₁ ∈ f i, eval i ⁻¹' t₁ ⊆ s := by | simp [Filter.coprodᵢ]
| 1 | 2.718282 | 0 | 0.666667 | 12 | 565 |
import Mathlib.Analysis.SpecialFunctions.JapaneseBracket
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.special_functions.improper_inte... | Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | 57 | 58 | theorem integral_exp_neg_Ioi_zero : (∫ x : ℝ in Ioi 0, exp (-x)) = 1 := by |
simpa only [neg_zero, exp_zero] using integral_exp_neg_Ioi 0
| 1 | 2.718282 | 0 | 1.5 | 8 | 1,667 |
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Data.List.OfFn
import Mathlib.Data.Set.Pointwise.Basic
#align_import data.set.pointwise.list_of_fn from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Set
variable {F α β γ : Type*}
variable [Monoid α] {s t : Set α}... | Mathlib/Data/Set/Pointwise/ListOfFn.lean | 52 | 54 | theorem mem_pow {a : α} {n : ℕ} :
a ∈ s ^ n ↔ ∃ f : Fin n → s, (List.ofFn fun i ↦ (f i : α)).prod = a := by |
rw [← mem_prod_list_ofFn, List.ofFn_const, List.prod_replicate]
| 1 | 2.718282 | 0 | 1.333333 | 3 | 1,443 |
import Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence
import Mathlib.Algebra.ContinuedFractions.TerminatedStable
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
#align_import algebra.continued_fractions.convergents_equiv from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40a... | Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean | 114 | 117 | theorem squashSeq_nth_of_not_terminated {gp_n gp_succ_n : Pair K} (s_nth_eq : s.get? n = some gp_n)
(s_succ_nth_eq : s.get? (n + 1) = some gp_succ_n) :
(squashSeq s n).get? n = some ⟨gp_n.a, gp_n.b + gp_succ_n.a / gp_succ_n.b⟩ := by |
simp [*, squashSeq]
| 1 | 2.718282 | 0 | 1.4 | 5 | 1,492 |
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Init.Algebra.Classes
#align_import algebra.group.commute from "leanprover-community/mathlib"@"05101c3df9d9cfe9430edc205860c79b6d660102"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
variable {G M S : Type*}
@[to_additive "Two elements... | Mathlib/Algebra/Group/Commute/Defs.lean | 262 | 263 | theorem mul_inv_cancel_assoc (h : Commute a b) : a * (b * a⁻¹) = b := by |
rw [← mul_assoc, h.mul_inv_cancel]
| 1 | 2.718282 | 0 | 0 | 1 | 168 |
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.supported from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
universe u v w
namespace MvPolynomial
variable {σ τ : Type*} {R : Type u} {S : Type v} {r : R} {e : ℕ} {n m : σ}
section CommSemiring
variable... | Mathlib/Algebra/MvPolynomial/Supported.lean | 117 | 118 | theorem X_mem_supported [Nontrivial R] {i : σ} : X i ∈ supported R s ↔ i ∈ s := by |
simp [mem_supported]
| 1 | 2.718282 | 0 | 0.888889 | 9 | 773 |
import Mathlib.Algebra.Polynomial.Monic
#align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"47adfab39a11a072db552f47594bf8ed2cf8a722"
open Finset
open Multiset
open Polynomial
universe u w
variable {R : Type u} {ι : Type w}
namespace Polynomial
variable (s : Finset ι)
sectio... | Mathlib/Algebra/Polynomial/BigOperators.lean | 263 | 265 | theorem prod_X_sub_C_nextCoeff {s : Finset ι} (f : ι → R) :
nextCoeff (∏ i ∈ s, (X - C (f i))) = -∑ i ∈ s, f i := by |
simpa using multiset_prod_X_sub_C_nextCoeff (s.1.map f)
| 1 | 2.718282 | 0 | 1.142857 | 7 | 1,216 |
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
assert_not_exists MonoidWithZero
open Multiset
variable {α β γ : Type*}
namespace Finset
section Prod
variable {s s' : Finset α} {t t' : Finset β} {a : α} {b : β}
... | Mathlib/Data/Finset/Prod.lean | 76 | 78 | theorem product_image_fst [DecidableEq α] (ht : t.Nonempty) : (s ×ˢ t).image Prod.fst = s := by |
ext i
simp [mem_image, ht.exists_mem]
| 2 | 7.389056 | 1 | 0.666667 | 3 | 616 |
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Order.Interval.Finset.Nat
#align_import data.polynomial.inductions from "leanprover-community/mathlib"@"57e09a1296bf... | Mathlib/Algebra/Polynomial/Inductions.lean | 45 | 46 | theorem coeff_divX : (divX p).coeff n = p.coeff (n + 1) := by |
rw [add_comm]; cases p; rfl
| 1 | 2.718282 | 0 | 1.142857 | 7 | 1,211 |
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 50 | 52 | theorem equalizerSubobject_arrow :
(equalizerSubobjectIso f g).hom ≫ equalizer.ι f g = (equalizerSubobject f g).arrow := by |
simp [equalizerSubobjectIso]
| 1 | 2.718282 | 0 | 0.263158 | 19 | 308 |
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 120 | 121 | theorem card_fintypeIco : Fintype.card (Set.Ico a b) = b - a := by |
rw [Fintype.card_ofFinset, card_Ico]
| 1 | 2.718282 | 0 | 0.125 | 8 | 253 |
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.GCDMonoid.Nat
#align_import ring_theory.int.basic from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
namespace Int
theorem gcd_eq_one_iff_coprime {a b : ℤ} : Int.gcd a b ... | Mathlib/RingTheory/Int/Basic.lean | 54 | 56 | theorem gcd_ne_one_iff_gcd_mul_right_ne_one {a : ℤ} {m n : ℕ} :
a.gcd (m * n) ≠ 1 ↔ a.gcd m ≠ 1 ∨ a.gcd n ≠ 1 := by |
simp only [gcd_eq_one_iff_coprime, ← not_and_or, not_iff_not, IsCoprime.mul_right_iff]
| 1 | 2.718282 | 0 | 1.153846 | 13 | 1,227 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
#align_import analysis.convex.hull from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set
open Pointwise
variable {𝕜 E F : Type*}
section convexHull
section OrderedSemiring
variable [OrderedSemiring 𝕜]
secti... | Mathlib/Analysis/Convex/Hull.lean | 56 | 57 | theorem convexHull_eq_iInter : convexHull 𝕜 s = ⋂ (t : Set E) (_ : s ⊆ t) (_ : Convex 𝕜 t), t := by |
simp [convexHull, iInter_subtype, iInter_and]
| 1 | 2.718282 | 0 | 1.166667 | 6 | 1,236 |
import Mathlib.Data.PFunctor.Multivariate.Basic
#align_import data.qpf.multivariate.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u
open MvFunctor
class MvQPF {n : ℕ} (F : TypeVec.{u} n → Type*) [MvFunctor F] where
P : MvPFunctor.{u} n
abs : ∀ {α}, P α → F α
... | Mathlib/Data/QPF/Multivariate/Basic.lean | 180 | 181 | theorem supp_eq {α : TypeVec n} {i} (x : F α) :
supp x i = { u | ∀ a f, abs ⟨a, f⟩ = x → u ∈ f i '' univ } := by | ext; apply mem_supp
| 1 | 2.718282 | 0 | 1.666667 | 6 | 1,775 |
import Mathlib.Algebra.Group.Fin
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.matrix.circulant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
variable {α β m n R : Type*}
namespace Matrix
open Function
open Matrix
def circulant [Sub n] (v : n → α)... | Mathlib/LinearAlgebra/Matrix/Circulant.lean | 81 | 82 | theorem transpose_circulant [AddGroup n] (v : n → α) :
(circulant v)ᵀ = circulant fun i => v (-i) := by | ext; simp
| 1 | 2.718282 | 0 | 1 | 6 | 895 |
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 108 | 110 | theorem intervalIntegrable_iff_integrableOn_Ico_of_le [NoAtoms μ] (hab : a ≤ b) :
IntervalIntegrable f μ a b ↔ IntegrableOn f (Ico a b) μ := by |
rw [intervalIntegrable_iff_integrableOn_Icc_of_le hab, integrableOn_Icc_iff_integrableOn_Ico]
| 1 | 2.718282 | 0 | 0.3 | 10 | 319 |
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.GroupTheory.GroupAction.Units
#align_import data.int.absolute_value from "leanprover-community/mathlib"@"9aba7801eeecebb61f58a5763c2b6dd1b47dc6ef"
variable {R S : Type*} [Ring R] [Linea... | Mathlib/Data/Int/AbsoluteValue.lean | 28 | 29 | theorem AbsoluteValue.map_units_int (abv : AbsoluteValue ℤ S) (x : ℤˣ) : abv x = 1 := by |
rcases Int.units_eq_one_or x with (rfl | rfl) <;> simp
| 1 | 2.718282 | 0 | 0 | 3 | 76 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 69 | 70 | theorem apply_ne_one_iff [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i j : V) :
¬A i j = 1 ↔ A i j = 0 := by | obtain h | h := h.zero_or_one i j <;> simp [h]
| 1 | 2.718282 | 0 | 0.285714 | 7 | 315 |
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Integral.PeakFunction
#align_import analysis.special_functions.trigonometric.euler_sine_prod from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open scoped Real Topology
open Real Set Filter intervalIntegra... | Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean | 39 | 46 | theorem antideriv_cos_comp_const_mul (hz : z ≠ 0) (x : ℝ) :
HasDerivAt (fun y : ℝ => Complex.sin (2 * z * y) / (2 * z)) (Complex.cos (2 * z * x)) x := by |
have a : HasDerivAt (fun y : ℂ => y * (2 * z)) _ x := hasDerivAt_mul_const _
have b : HasDerivAt (fun y : ℂ => Complex.sin (y * (2 * z))) _ x :=
HasDerivAt.comp (x : ℂ) (Complex.hasDerivAt_sin (x * (2 * z))) a
have c := b.comp_ofReal.div_const (2 * z)
field_simp at c; simp only [fun y => mul_comm y (2 * z)... | 6 | 403.428793 | 2 | 2 | 5 | 2,257 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Deprecated.Subring
#align_import deprecated.subfield from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
variable {F : Type*} [Field F] (S : Set F)
structure IsSubfield extends IsSubring S : Prop where
inv_mem : ∀ {x : F}, x ∈ S → x⁻... | Mathlib/Deprecated/Subfield.lean | 93 | 99 | theorem closure.isSubmonoid : IsSubmonoid (closure S) :=
{ mul_mem := by |
rintro _ _ ⟨p, hp, q, hq, hq0, rfl⟩ ⟨r, hr, s, hs, hs0, rfl⟩
exact ⟨p * r, IsSubmonoid.mul_mem Ring.closure.isSubring.toIsSubmonoid hp hr, q * s,
IsSubmonoid.mul_mem Ring.closure.isSubring.toIsSubmonoid hq hs,
(div_mul_div_comm _ _ _ _).symm⟩
one_mem := ring_closure_subset <| IsSubmonoi... | 5 | 148.413159 | 2 | 1.5 | 6 | 1,579 |
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-community/mathlib"@"4e24c4bfcff371c71f7ba22050308aa17815626c"
variable {α β : Type*} [LinearOrder α]
open Function
namespace Set
def projIci (a x : α) : Ici a := ⟨max a x,... | Mathlib/Order/Interval/Set/ProjIcc.lean | 102 | 102 | theorem projIic_eq_self : projIic b x = ⟨b, le_rfl⟩ ↔ b ≤ x := by | simp [projIic, Subtype.ext_iff]
| 1 | 2.718282 | 0 | 0.083333 | 12 | 241 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
| Mathlib/NumberTheory/Pell.lean | 83 | 85 | theorem is_pell_solution_iff_mem_unitary {d : ℤ} {a : ℤ√d} :
a.re ^ 2 - d * a.im ^ 2 = 1 ↔ a ∈ unitary (ℤ√d) := by |
rw [← norm_eq_one_iff_mem_unitary, norm_def, sq, sq, ← mul_assoc]
| 1 | 2.718282 | 0 | 1 | 7 | 981 |
import Mathlib.Data.Finset.Pointwise
#align_import combinatorics.additive.e_transform from "leanprover-community/mathlib"@"207c92594599a06e7c134f8d00a030a83e6c7259"
open MulOpposite
open Pointwise
variable {α : Type*} [DecidableEq α]
namespace Finset
section CommGroup
variable [CommGroup α] (e : α) (x : F... | Mathlib/Combinatorics/Additive/ETransform.lean | 58 | 61 | theorem mulDysonETransform.subset :
(mulDysonETransform e x).1 * (mulDysonETransform e x).2 ⊆ x.1 * x.2 := by |
refine union_mul_inter_subset_union.trans (union_subset Subset.rfl ?_)
rw [mul_smul_comm, smul_mul_assoc, inv_smul_smul, mul_comm]
| 2 | 7.389056 | 1 | 0.875 | 8 | 758 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
#align_import algebra.category.Module.monoidal.symmetric from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
suppress_compilation
universe v w x u
open CategoryTheory MonoidalC... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean | 49 | 52 | theorem braiding_naturality_right (X : ModuleCat R) {Y Z : ModuleCat R} (f : Y ⟶ Z) :
X ◁ f ≫ (braiding X Z).hom = (braiding X Y).hom ≫ f ▷ X := by |
simp_rw [← id_tensorHom]
apply braiding_naturality
| 2 | 7.389056 | 1 | 1.2 | 5 | 1,276 |
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.LinearAlgebra.Basis
#align_import analysis.normed_space.linear_isometry from "leanprover-community/mathlib"@"4601791ea62fea875b488dafc4e6dede19e8363f"
open Function Set
variable {R R₂ R₃ R₄ E E₂ E₃ E₄ F 𝓕 : Ty... | Mathlib/Analysis/NormedSpace/LinearIsometry.lean | 170 | 172 | theorem coe_injective : @Injective (E →ₛₗᵢ[σ₁₂] E₂) (E → E₂) (fun f => f) := by |
rintro ⟨_⟩ ⟨_⟩
simp
| 2 | 7.389056 | 1 | 1 | 1 | 979 |
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α : Type*} {N : α → Type*}
namespace DFinsupp
variable [DecidableEq α]
section NHasZero
variable [∀ a, DecidableEq (N a)] [∀ a, Zero (N a)] (f g : Π₀... | 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.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.GroupTheory.GroupAction.SubMulAction
open scoped BigOperators Pointwise
namespace MulAction
section SMul
variable (G : Type*) {X : Type*} [SMul G X]
-- Change termin... | Mathlib/GroupTheory/GroupAction/Blocks.lean | 95 | 97 | theorem IsFixedBlock.isBlock {B : Set X} (hfB : IsFixedBlock G B) :
IsBlock G B := by |
simp [IsBlock.def, hfB _]
| 1 | 2.718282 | 0 | 0.5 | 8 | 486 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section equality
variable {s s₁... | Mathlib/Data/Set/Function.lean | 190 | 191 | theorem eqOn_univ (f₁ f₂ : α → β) : EqOn f₁ f₂ univ ↔ f₁ = f₂ := by |
simp [EqOn, funext_iff]
| 1 | 2.718282 | 0 | 0.8 | 10 | 704 |
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open scoped Classical MeasureTheory NNReal ... | Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | 59 | 60 | theorem withDensityᵥ_apply (hf : Integrable f μ) {s : Set α} (hs : MeasurableSet s) :
μ.withDensityᵥ f s = ∫ x in s, f x ∂μ := by | rw [withDensityᵥ, dif_pos hf]; exact dif_pos hs
| 1 | 2.718282 | 0 | 1.142857 | 7 | 1,217 |
import Mathlib.Geometry.Euclidean.Sphere.Basic
#align_import geometry.euclidean.sphere.second_inter from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RealInnerProductSpace
namespace EuclideanGeometry
variable {V : Type*} {P : Type*} [NormedAddCommGroup V]... | Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | 103 | 108 | theorem Sphere.secondInter_smul (s : Sphere P) (p : P) (v : V) {r : ℝ} (hr : r ≠ 0) :
s.secondInter p (r • v) = s.secondInter p v := by |
simp_rw [Sphere.secondInter, real_inner_smul_left, inner_smul_right, smul_smul,
div_mul_eq_div_div]
rw [mul_comm, ← mul_div_assoc, ← mul_div_assoc, mul_div_cancel_left₀ _ hr, mul_comm, mul_assoc,
mul_div_cancel_left₀ _ hr, mul_comm]
| 4 | 54.59815 | 2 | 1.25 | 8 | 1,314 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 497 | 499 | theorem toSemiring_injective :
Function.Injective (@toSemiring R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 1 | 2.718282 | 0 | 0.4 | 10 | 397 |
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 {𝕜 : Type*} [LinearOrderedField 𝕜] : (P.energy G : 𝕜) =
(∑ uv ∈ P.parts.offDiag, (G.edgeDensity uv.1 uv.2 : 𝕜) ^ 2) / (P.parts.card : 𝕜) ^ 2 := by |
rw [energy]; norm_cast
| 1 | 2.718282 | 0 | 0.666667 | 3 | 575 |
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.Localization.Integral
import Mathlib.RingTheory.IntegrallyClosed
#align_import ring_theory.polynomial.gauss_lemma from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
open... | Mathlib/RingTheory/Polynomial/GaussLemma.lean | 115 | 121 | theorem IsPrimitive.isUnit_iff_isUnit_map_of_injective : IsUnit f ↔ IsUnit (map φ f) := by |
refine ⟨(mapRingHom φ).isUnit_map, fun h => ?_⟩
rcases isUnit_iff.1 h with ⟨_, ⟨u, rfl⟩, hu⟩
have hdeg := degree_C u.ne_zero
rw [hu, degree_map_eq_of_injective hinj] at hdeg
rw [eq_C_of_degree_eq_zero hdeg] at hf ⊢
exact isUnit_C.mpr (isPrimitive_iff_isUnit_of_C_dvd.mp hf (f.coeff 0) dvd_rfl)
| 6 | 403.428793 | 2 | 2 | 4 | 2,151 |
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
namespace Nat
def dist (n m : ℕ) :=
n - m + (m - n)
#align nat.dist Nat.dist
-- Should be aligned to `Nat.dist.eq_def`, but that is generated on demand and isn't pr... | Mathlib/Data/Nat/Dist.lean | 57 | 57 | theorem dist_tri_right (n m : ℕ) : m ≤ n + dist n m := by | rw [add_comm]; apply dist_tri_left
| 1 | 2.718282 | 0 | 0.266667 | 15 | 309 |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.units from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Int
theorem isUnit_iff_abs_eq {x : ℤ} : IsUnit x ↔ abs x = 1 := by
rw [isUnit_iff_natAbs_eq, abs_eq_natAbs, ← Int.ofNat_one, natCast_inj]
#align int.... | Mathlib/Data/Int/Order/Units.lean | 21 | 21 | theorem isUnit_sq {a : ℤ} (ha : IsUnit a) : a ^ 2 = 1 := by | rw [sq, isUnit_mul_self ha]
| 1 | 2.718282 | 0 | 0.222222 | 9 | 285 |
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 158 | 160 | theorem kernelSubobjectMap_arrow (sq : Arrow.mk f ⟶ Arrow.mk f') :
kernelSubobjectMap sq ≫ (kernelSubobject f').arrow = (kernelSubobject f).arrow ≫ sq.left := by |
simp [kernelSubobjectMap]
| 1 | 2.718282 | 0 | 0.263158 | 19 | 308 |
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.GroupTheory.GroupAction.SubMulAction
open scoped BigOperators Pointwise
namespace MulAction
section SMul
variable (G : Type*) {X : Type*} [SMul G X]
-- Change termin... | Mathlib/GroupTheory/GroupAction/Blocks.lean | 85 | 87 | theorem IsBlock.def {B : Set X} :
IsBlock G B ↔ ∀ g g' : G, g • B = g' • B ∨ Disjoint (g • B) (g' • B) := by |
apply Set.pairwiseDisjoint_range_iff
| 1 | 2.718282 | 0 | 0.5 | 8 | 486 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 450 | 451 | theorem mul_one_div (x y : G) : x * (1 / y) = x / y := by |
rw [div_eq_mul_inv, one_mul, div_eq_mul_inv]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 367 |
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.modeq from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
namespace AddCommGroup
variable {α : Type*}
section AddCommGroup
variable [AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}
... | Mathlib/Algebra/ModEq.lean | 311 | 312 | theorem modEq_iff_int_modEq {a b z : ℤ} : a ≡ b [PMOD z] ↔ a ≡ b [ZMOD z] := by |
simp [ModEq, dvd_iff_exists_eq_mul_left, Int.modEq_iff_dvd]
| 1 | 2.718282 | 0 | 0 | 3 | 72 |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe v₁ v₂ v₃ u₁ u₂ u₃
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 169 | 170 | theorem eqToIso_trans {X Y Z : C} (p : X = Y) (q : Y = Z) :
eqToIso p ≪≫ eqToIso q = eqToIso (p.trans q) := by | ext; simp
| 1 | 2.718282 | 0 | 0.9 | 10 | 777 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ι : Sort u} {α : Type v} {β : Type w}
open Set
open OrderDual (toDual)
namespace Set
section LinearOrder
variable [LinearOrder α] ... | Mathlib/Order/Interval/Set/Disjoint.lean | 182 | 184 | theorem biUnion_Ico_eq_Iio_self_iff {p : ι → Prop} {f : ∀ i, p i → α} {a : α} :
⋃ (i) (hi : p i), Ico (f i hi) a = Iio a ↔ ∀ x < a, ∃ i hi, f i hi ≤ x := by |
simp [← Ici_inter_Iio, ← iUnion_inter, subset_def]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 364 |
import Mathlib.Data.ZMod.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Tactic.IntervalCases
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.GroupTheory.SpecificGroups.Cyclic
#align_import group_theory.specific_groups.quaternion from "leanprover-community/mathlib"@"879155bff5af618b9062cbb2915... | Mathlib/GroupTheory/SpecificGroups/Quaternion.lean | 196 | 196 | theorem xa_sq (i : ZMod (2 * n)) : xa i ^ 2 = a n := by | simp [sq]
| 1 | 2.718282 | 0 | 1.333333 | 6 | 1,410 |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 315 | 317 | theorem finsuppTensorFinsuppRid_apply_apply (f : ι →₀ M) (g : κ →₀ R) (a : ι) (b : κ) :
finsuppTensorFinsuppRid R M ι κ (f ⊗ₜ[R] g) (a, b) = g b • f a := by |
simp [finsuppTensorFinsuppRid]
| 1 | 2.718282 | 0 | 0.75 | 8 | 652 |
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Set Filter Topology
variable {X : Type*}
def OnePoint (X : Type*) :=
Option X
#ali... | Mathlib/Topology/Compactification/OnePoint.lean | 152 | 153 | theorem not_mem_range_coe_iff {x : OnePoint X} : x ∉ range some ↔ x = ∞ := by |
rw [← mem_compl_iff, compl_range_coe, mem_singleton_iff]
| 1 | 2.718282 | 0 | 0.25 | 4 | 289 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.add from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Topology Filter ENNReal
open Filter Asymptotics Set
variable... | Mathlib/Analysis/Calculus/Deriv/Add.lean | 97 | 99 | theorem derivWithin_add_const (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F) :
derivWithin (fun y => f y + c) s x = derivWithin f s x := by |
simp only [derivWithin, fderivWithin_add_const hxs]
| 1 | 2.718282 | 0 | 0 | 8 | 26 |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 146 | 147 | theorem degree_eq_iff_natDegree_eq {p : R[X]} {n : ℕ} (hp : p ≠ 0) :
p.degree = n ↔ p.natDegree = n := by | rw [degree_eq_natDegree hp]; exact WithBot.coe_eq_coe
| 1 | 2.718282 | 0 | 0.625 | 8 | 546 |
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Tactic.TypeStar
#align_import logic.nontrivial from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
variable {α : Type*} {β : Type*}
open scoped Classical
class Nontrivial (α : Type*) : Prop where
exists_pair_n... | Mathlib/Logic/Nontrivial/Defs.lean | 83 | 84 | theorem not_nontrivial_iff_subsingleton : ¬Nontrivial α ↔ Subsingleton α := by |
simp only [nontrivial_iff, subsingleton_iff, not_exists, Classical.not_not]
| 1 | 2.718282 | 0 | 0.5 | 2 | 494 |
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.ZornAtoms
#align_import order.filter.ultrafilter from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
universe u v
variable {α : Type u} {β : Type v} {γ : Type*}
open Set Filter Function
open scoped Classical
open Filter
inst... | Mathlib/Order/Filter/Ultrafilter.lean | 115 | 116 | theorem disjoint_iff_not_le {f : Ultrafilter α} {g : Filter α} : Disjoint (↑f) g ↔ ¬↑f ≤ g := by |
rw [← inf_neBot_iff, neBot_iff, Ne, not_not, disjoint_iff]
| 1 | 2.718282 | 0 | 0 | 2 | 56 |
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Data.Finset.Basic
import Mathlib.Order.Interval.Finset.Defs
open Function
namespace Finset
class HasAntidiagonal (A : Type*) [AddMonoid A] where
antidiagonal : A → Finset (A × A)
mem_antidiagonal {n} {a} : a ∈ antidiagonal n ↔ a.fst + a.snd = n
exp... | Mathlib/Data/Finset/Antidiagonal.lean | 80 | 82 | theorem swap_mem_antidiagonal [AddCommMonoid A] [HasAntidiagonal A] {n : A} {xy : A × A}:
xy.swap ∈ antidiagonal n ↔ xy ∈ antidiagonal n := by |
simp [add_comm]
| 1 | 2.718282 | 0 | 1.142857 | 7 | 1,213 |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.tower from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
variable (R A B : Type*) {σ : Type*}
namespace MvPolynomial
section CommSemiring
variable [CommSemiring R] ... | Mathlib/RingTheory/MvPolynomial/Tower.lean | 62 | 65 | theorem aeval_algebraMap_eq_zero_iff_of_injective {x : σ → A} {p : MvPolynomial σ R}
(h : Function.Injective (algebraMap A B)) :
aeval (algebraMap A B ∘ x) p = 0 ↔ aeval x p = 0 := by |
rw [aeval_algebraMap_apply, ← (algebraMap A B).map_zero, h.eq_iff]
| 1 | 2.718282 | 0 | 0.75 | 4 | 671 |
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheory.MeasurableSpace.Basic
open Set
namespace MeasureTheory
variable {ι : Type _} {α : ι → Type _}
section cylinder
def cylinder (s : Finset ι) (S : Set (∀ i : s, α... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 231 | 235 | theorem cylinder_eq_cylinder_union [DecidableEq ι] (I : Finset ι) (S : Set (∀ i : I, α i))
(J : Finset ι) :
cylinder I S =
cylinder (I ∪ J) ((fun f ↦ fun j : I ↦ f ⟨j, Finset.mem_union_left J j.prop⟩) ⁻¹' S) := by |
ext1 f; simp only [mem_cylinder, mem_preimage]
| 1 | 2.718282 | 0 | 0.6875 | 16 | 636 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.Sort
#align_import data.polynomial.basic from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
set_option linter.uppercaseLean3 false
noncomputable section
structure ... | Mathlib/Algebra/Polynomial/Basic.lean | 178 | 180 | theorem ofFinsupp_sub {R : Type u} [Ring R] {a b} : (⟨a - b⟩ : R[X]) = ⟨a⟩ - ⟨b⟩ := by |
rw [sub_eq_add_neg, ofFinsupp_add, ofFinsupp_neg]
rfl
| 2 | 7.389056 | 1 | 1 | 3 | 933 |
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Ty... | Mathlib/Algebra/Polynomial/Roots.lean | 136 | 139 | theorem finite_setOf_isRoot {p : R[X]} (hp : p ≠ 0) : Set.Finite { x | IsRoot p x } := by |
classical
simpa only [← Finset.setOf_mem, Multiset.mem_toFinset, mem_roots hp]
using p.roots.toFinset.finite_toSet
| 3 | 20.085537 | 1 | 1.285714 | 7 | 1,352 |
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.IsomorphismClasses
import Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects
#align_import category_theory.limits.shapes.zero_morphisms from "leanpr... | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 145 | 147 | theorem zero_of_epi_comp {X Y Z : C} (f : X ⟶ Y) {g : Y ⟶ Z} [Epi f] (h : f ≫ g = 0) : g = 0 := by |
rw [← comp_zero, cancel_epi] at h
exact h
| 2 | 7.389056 | 1 | 1 | 4 | 930 |
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.hadamard from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
variable {α β γ m n : Type*}
variable {R : Type*}
namespace Matrix
open Matrix
def hadamard [Mul α] (A : Matrix m n α) (B : Matrix m n α) : Matrix m n α :... | Mathlib/Data/Matrix/Hadamard.lean | 121 | 123 | theorem one_hadamard : (1 : Matrix n n α) ⊙ M = diagonal fun i => M i i := by |
ext i j
by_cases h : i = j <;> simp [h]
| 2 | 7.389056 | 1 | 1 | 3 | 846 |
import Mathlib.Algebra.Module.Submodule.Ker
open Function Submodule
namespace LinearMap
variable {R N M : Type*} [Semiring R] [AddCommMonoid N] [Module R N]
[AddCommMonoid M] [Module R M] (f i : N →ₗ[R] M)
def iterateMapComap (n : ℕ) := (fun K : Submodule R N ↦ (K.map i).comap f)^[n]
theorem iterateMapComap... | Mathlib/Algebra/Module/Submodule/IterateMapComap.lean | 88 | 92 | theorem ker_le_of_iterateMapComap_eq_succ (K : Submodule R N)
(m : ℕ) (heq : f.iterateMapComap i m K = f.iterateMapComap i (m + 1) K)
(hf : Surjective f) (hi : Injective i) : LinearMap.ker f ≤ K := by |
rw [show K = _ from f.iterateMapComap_eq_succ i K m heq hf hi 0]
exact f.ker_le_comap
| 2 | 7.389056 | 1 | 1.666667 | 3 | 1,828 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
noncomputable section
open Filter Finset Function Encodable
open scoped Topology
variable {M : Type*} [CommMonoid M] [TopologicalSpace M] {m m' : M}
variable {G : Type*} [CommGroup G] {g g' : G}
-- don't declare [Topologic... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 290 | 292 | theorem multipliable_iff_nat_tprod_vanishing {f : ℕ → G} : Multipliable f ↔
∀ e ∈ 𝓝 1, ∃ N : ℕ, ∀ t ⊆ {n | N ≤ n}, (∏' n : t, f n) ∈ e := by |
rw [multipliable_iff_cauchySeq_finset, cauchySeq_finset_iff_nat_tprod_vanishing]
| 1 | 2.718282 | 0 | 1.125 | 8 | 1,202 |
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.ByContra
import Mathlib.Util.Delaborators
#align_import data.set.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
... | Mathlib/Data/Set/Basic.lean | 224 | 225 | theorem forall_in_swap {p : α → β → Prop} : (∀ a ∈ s, ∀ (b), p a b) ↔ ∀ (b), ∀ a ∈ s, p a b := by |
tauto
| 1 | 2.718282 | 0 | 0 | 1 | 115 |
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Finsupp.Basic
import Mathlib.LinearAlgebra.Finsupp
#align_import algebra.monoid_algebra.basic from "leanprover-community/mathlib"@... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 249 | 251 | theorem liftNC_one {g_hom : Type*} [FunLike g_hom G R] [OneHomClass g_hom G R]
(f : k →+* R) (g : g_hom) :
liftNC (f : k →+ R) g 1 = 1 := by | simp [one_def]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 591 |
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-community/mathlib"@"4e24c4bfcff371c71f7ba22050308aa17815626c"
variable {α β : Type*} [LinearOrder α]
open Function
namespace Set
def projIci (a x : α) : Ici a := ⟨max a x,... | Mathlib/Order/Interval/Set/ProjIcc.lean | 132 | 134 | theorem projIcc_val (x : Icc a b) : projIcc a b h x = x := by |
cases x
apply projIcc_of_mem
| 2 | 7.389056 | 1 | 0.083333 | 12 | 241 |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.GroupTheory.GroupAction.Ring
#align_import ring_theory.localization.basic from "leanprover-community/mathlib"@"b69c9a770ecf37eb21... | Mathlib/RingTheory/Localization/Basic.lean | 202 | 204 | theorem sec_spec' (z : S) :
algebraMap R S (IsLocalization.sec M z).1 = algebraMap R S (IsLocalization.sec M z).2 * z := by |
rw [mul_comm, sec_spec]
| 1 | 2.718282 | 0 | 1 | 4 | 1,001 |
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