| [ |
| { |
| "id": "O1", |
| "full_name": "Lattice", |
| "file": "Mathlib/Order/Lattice.lean:342", |
| "difficulty": "Easy", |
| "kind": "class", |
| "q1a_lean": "a poset with `⊔` and `⊓`", |
| "q1b_latex": "a poset with binary $\\sqcup$ and $\\sqcap$", |
| "q1c_natural": "a partial order in which any two elements have both a join and a meet", |
| "q2_slogan": "a poset with binary joins and meets", |
| "q3_nickname": "lattice", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "O2", |
| "full_name": "GaloisConnection", |
| "file": "Mathlib/Order/GaloisConnection/Defs.lean:41", |
| "difficulty": "Easy", |
| "kind": "def", |
| "q1a_lean": "a pair `f g` with `f a ≤ b ↔ a ≤ g b`", |
| "q1b_latex": "$f(a) \\le b \\iff a \\le g(b)$ for $f, g$ between preorders", |
| "q1c_natural": "two monotone maps related by an adjoint condition", |
| "q2_slogan": "an adjunction between two preorders", |
| "q3_nickname": "Galois connection", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "O3", |
| "full_name": "Nucleus", |
| "file": "Mathlib/Order/Nucleus.lean:34", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "an `InfHom` `j` with `x ≤ j x` and `j (j x) ≤ j x`", |
| "q1b_latex": "$j : L \\to L$ inflationary, idempotent, $j(x \\sqcap y) = j(x) \\sqcap j(y)$", |
| "q1c_natural": "an inflationary, idempotent self-map that also preserves meets", |
| "q2_slogan": "", |
| "q3_nickname": "nucleus", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "O4", |
| "full_name": "Order.IsNormal", |
| "file": "Mathlib/Order/IsNormal.lean:34", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "strictly monotone `f` on well-orders, with `f a = sup (f '' Iio a)` at successor-limits", |
| "q1b_latex": "$f$ strictly increasing with $f(a) = \\sup_{b<a} f(b)$ at each successor-limit $a$", |
| "q1c_natural": "a strictly increasing function between well-orders that is continuous at every limit", |
| "q2_slogan": "strictly increasing and limit-preserving on well-orders", |
| "q3_nickname": "normal function", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "O5", |
| "full_name": "iSup_iInf_le_iInf_iSup", |
| "file": "Mathlib/Order/CompleteLattice/Basic.lean:287", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`⨆ i, ⨅ j, f i j ≤ ⨅ j, ⨆ i, f i j`", |
| "q1b_latex": "$\\sup_i \\inf_j f(i, j) \\le \\inf_j \\sup_i f(i, j)$", |
| "q1c_natural": "the sup of infs is always at most the inf of sups", |
| "q2_slogan": "sup of infs bounded by inf of sups", |
| "q3_nickname": "max-min inequality", |
| "q4_special_case": "$\\max(\\min(a,b), \\min(c,d)) \\le \\min(\\max(a,c), \\max(b,d))$" |
| }, |
| { |
| "id": "O6", |
| "full_name": "GaloisConnection.l_iSup", |
| "file": "Mathlib/Order/GaloisConnection/Basic.lean:116", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "for adjoint `f`, `f (⨆ i, x i) = ⨆ i, f (x i)`", |
| "q1b_latex": "$f(\\sup_i x_i) = \\sup_i f(x_i)$ for a left adjoint $f$", |
| "q1c_natural": "a left adjoint commutes with arbitrary joins", |
| "q2_slogan": "left adjoints preserve suprema", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "O7", |
| "full_name": "OrderHom.map_lfp", |
| "file": "Mathlib/Order/FixedPoints.lean:71", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`f (lfp f) = lfp f`", |
| "q1b_latex": "$f(\\mathrm{lfp}\\,f) = \\mathrm{lfp}\\,f$", |
| "q1c_natural": "", |
| "q2_slogan": "", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "O8", |
| "full_name": "Filter.instCardinalInterFilter_cocardinal", |
| "file": "Mathlib/Order/Filter/Cocardinal.lean:50", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "the cocardinal filter is closed under `<c`-many intersections", |
| "q1b_latex": "$\\bigcap_{i < \\kappa'} A_i \\in \\mathrm{cocard}_\\kappa$ for $\\kappa' < \\kappa$, $\\kappa$ regular", |
| "q1c_natural": "the filter of complements of small sets is closed under intersections of fewer than κ many sets", |
| "q2_slogan": "cocardinal filter has the cardinal-intersection property", |
| "q3_nickname": "", |
| "q4_special_case": "the cofinite filter is closed under finite intersections" |
| }, |
| { |
| "id": "T1", |
| "full_name": "MetricSpace", |
| "file": "Mathlib/Topology/MetricSpace/Defs.lean:57", |
| "difficulty": "Easy", |
| "kind": "class", |
| "q1a_lean": "a type with `dist : α → α → ℝ`, `dist x y = 0 ↔ x = y`, symmetric, triangle inequality", |
| "q1b_latex": "$X$ with $d : X \\times X \\to \\mathbb{R}$, $d(x,y)=0 \\iff x=y$, symmetric, $\\triangle$-inequality", |
| "q1c_natural": "a set with a distance satisfying positivity, symmetry, and the triangle inequality", |
| "q2_slogan": "a space with a distance function", |
| "q3_nickname": "metric space", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "T2", |
| "full_name": "IsCompact", |
| "file": "Mathlib/Topology/Defs/Filter.lean:276", |
| "difficulty": "Easy", |
| "kind": "def", |
| "q1a_lean": "a set `s` such that every nontrivial filter `≤ 𝓟 s` clusters at some point of `s`", |
| "q1b_latex": "$s$ such that every nontrivial filter $\\mathcal{F} \\le \\mathcal{P}(s)$ has a cluster point in $s$", |
| "q1c_natural": "a set in which every filter clusters somewhere inside it", |
| "q2_slogan": "every filter on a compact set has a cluster point", |
| "q3_nickname": "compact set", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "T3", |
| "full_name": "AlexandrovDiscrete", |
| "file": "Mathlib/Topology/AlexandrovDiscrete.lean:38", |
| "difficulty": "Hard", |
| "kind": "class", |
| "q1a_lean": "a `TopologicalSpace` where `⋂₀ S` is open for any family `S` of opens", |
| "q1b_latex": "$\\bigcap_{i \\in I} U_i$ open for arbitrary index set $I$", |
| "q1c_natural": "a topological space where intersections of arbitrary open families remain open", |
| "q2_slogan": "arbitrary intersections of opens stay open", |
| "q3_nickname": "Alexandrov-discrete space", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "T4", |
| "full_name": "FiberBundle", |
| "file": "Mathlib/Topology/FiberBundle/Basic.lean:186", |
| "difficulty": "Hard", |
| "kind": "class", |
| "q1a_lean": "a space `E → B` with a trivialization making it locally `≅ U × F`", |
| "q1b_latex": "a space $E$ over $B$ such that locally $E \\cong U \\times F$", |
| "q1c_natural": "a space over a base that locally looks like a direct product with a fixed fiber", |
| "q2_slogan": "a space that locally looks like a product with a fixed fiber", |
| "q3_nickname": "fiber bundle", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "T5", |
| "full_name": "isCompact_iff_finite_subcover", |
| "file": "Mathlib/Topology/Compactness/Compact.lean:368", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`IsCompact s` iff every open cover of `s` has a finite subcover", |
| "q1b_latex": "$K$ compact $\\iff \\forall \\mathcal{U}$ open cover, $\\exists \\mathcal{V} \\subseteq \\mathcal{U}$ finite, $K \\subseteq \\bigcup \\mathcal{V}$", |
| "q1c_natural": "a set is compact iff every open cover admits a finite subcover", |
| "q2_slogan": "compact = finite subcover property", |
| "q3_nickname": "finite subcover characterization", |
| "q4_special_case": "$[0,1] \\subseteq \\mathbb{R}$ has a finite subcover for every open cover" |
| }, |
| { |
| "id": "T6", |
| "full_name": "continuous_iff_isClosed", |
| "file": "Mathlib/Topology/Continuous.lean:182", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`Continuous f ↔ ∀ s, IsClosed s → IsClosed (f ⁻¹' s)`", |
| "q1b_latex": "$f$ continuous $\\iff \\forall C$ closed, $f^{-1}(C)$ closed", |
| "q1c_natural": "a function is continuous iff the preimage of every closed set is closed", |
| "q2_slogan": "continuity = closed preimages of closed sets", |
| "q3_nickname": "", |
| "q4_special_case": "$f : \\mathbb{R} \\to \\mathbb{R}$ continuous iff $f^{-1}([a,b])$ closed for every closed interval" |
| }, |
| { |
| "id": "T7", |
| "full_name": "exists_continuous_zero_one_of_isClosed", |
| "file": "Mathlib/Topology/UrysohnsLemma.lean:321", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "for disjoint closed `s t` in a `NormalSpace`, exists continuous `f` with `f = 0` on `s`, `f = 1` on `t`", |
| "q1b_latex": "$\\exists f \\in C(X, [0,1]),\\ f|_S = 0,\\ f|_T = 1$ for $S, T$ disjoint closed in normal $X$", |
| "q1c_natural": "in a normal space, two disjoint closed sets can be separated by a continuous function valued 0 on one and 1 on the other", |
| "q2_slogan": "disjoint closed sets in a normal space are separated by a continuous function", |
| "q3_nickname": "Urysohn's lemma", |
| "q4_special_case": "in $\\mathbb{R}$, $[0,1]$ and $[2,3]$ are separated by a continuous ramp" |
| }, |
| { |
| "id": "T8", |
| "full_name": "T4Space.of_paracompactSpace_t2Space", |
| "file": "Mathlib/Topology/Compactness/Paracompact.lean:300", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "paracompact + Hausdorff ⇒ T4", |
| "q1b_latex": "paracompact $+\\ T_2 \\Rightarrow T_4$", |
| "q1c_natural": "every paracompact Hausdorff space is normal", |
| "q2_slogan": "paracompact Hausdorff spaces are normal", |
| "q3_nickname": "Dieudonné's theorem", |
| "q4_special_case": "every metric space is normal" |
| }, |
| { |
| "id": "A1", |
| "full_name": "LieRing", |
| "file": "Mathlib/Algebra/Lie/Basic.lean", |
| "difficulty": "Easy", |
| "kind": "class", |
| "q1a_lean": "`AddCommGroup` with bilinear `⁅·,·⁆`, `⁅x,x⁆ = 0`, and Jacobi", |
| "q1b_latex": "an additive group with a bracket $[\\cdot,\\cdot]$ that is bilinear, alternating, and satisfies $[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0$", |
| "q1c_natural": "an additive group equipped with a bracket that is bilinear, alternating, and satisfies the Jacobi identity", |
| "q2_slogan": "abelian group with a Jacobi-style bracket", |
| "q3_nickname": "Lie ring", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "A2", |
| "full_name": "Subalgebra", |
| "file": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", |
| "difficulty": "Easy", |
| "kind": "structure", |
| "q1a_lean": "a `Subsemiring` of `A` containing the image of `algebraMap R A`", |
| "q1b_latex": "$S \\le A$ closed under $+, \\cdot, 0, 1$ with $\\mathrm{algebraMap}(R) \\subseteq S$", |
| "q1c_natural": "a sub(semi)ring containing the image of the algebra map", |
| "q2_slogan": "subring closed under the scalar multiplication", |
| "q3_nickname": "subalgebra", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "A3", |
| "full_name": "GCDMonoid", |
| "file": "Mathlib/Algebra/GCDMonoid/Basic.lean:246", |
| "difficulty": "Hard", |
| "kind": "class", |
| "q1a_lean": "`IsCancelMulZero` with `gcd`, `lcm` and the standard divisibility axioms", |
| "q1b_latex": "comm monoid with $\\gcd, \\mathrm{lcm}$ such that $\\gcd(a,b) \\mid a$, $\\gcd(a,b) \\mid b$, $\\gcd(a,b) \\cdot \\mathrm{lcm}(a,b) \\sim ab$", |
| "q1c_natural": "a cancel-zero comm monoid equipped with gcd and lcm", |
| "q2_slogan": "comm monoid with gcd and lcm", |
| "q3_nickname": "GCD monoid", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "A4", |
| "full_name": "CategoryTheory.ShortComplex.HomologyData", |
| "file": "Mathlib/Algebra/Homology/ShortComplex/Homology.lean:50", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "a `LeftHomologyData` and `RightHomologyData` with an iso between their `H` and a compatibility square", |
| "q1b_latex": "$H_{\\mathrm{left}} \\xrightarrow{\\sim} H_{\\mathrm{right}}$ on a short complex with compatible square", |
| "q1c_natural": "a left and right homology data of a short complex together with an iso between them", |
| "q2_slogan": "paired left/right homology data of a short complex", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "A5", |
| "full_name": "Convex.combo_self", |
| "file": "Mathlib/Algebra/Module/Defs.lean:76", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`a + b = 1 → a • x + b • x = x`", |
| "q1b_latex": "$a + b = 1 \\Rightarrow a x + b x = x$", |
| "q1c_natural": "a convex combination of a point with itself equals the point", |
| "q2_slogan": "", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "A6", |
| "full_name": "normalize_associated", |
| "file": "Mathlib/Algebra/GCDMonoid/Basic.lean:108", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`Associated (normalize x) x`", |
| "q1b_latex": "$\\mathrm{normalize}(x) \\sim x$", |
| "q1c_natural": "the normalized form of an element is associated to the element itself", |
| "q2_slogan": "normalize stays in the same associativity class", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "A7", |
| "full_name": "GCDMonoid.gcd_mul_lcm", |
| "file": "Mathlib/Algebra/GCDMonoid/Basic.lean:303", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`Associated (gcd a b * lcm a b) (a * b)`", |
| "q1b_latex": "$\\gcd(a,b) \\cdot \\mathrm{lcm}(a,b) \\sim a \\cdot b$", |
| "q1c_natural": "the product of gcd and lcm equals the original product up to a unit", |
| "q2_slogan": "gcd × lcm equals the product up to a unit", |
| "q3_nickname": "gcd-lcm identity", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "A8", |
| "full_name": "LieAlgebra.ofAssociativeAlgebra", |
| "file": "Mathlib/Algebra/Lie/OfAssociative.lean", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "every `Algebra R A` is a `LieAlgebra R A` via the commutator", |
| "q1b_latex": "any associative $R$-algebra $A$ becomes a Lie algebra under $[x, y] := xy - yx$", |
| "q1c_natural": "every associative algebra carries a canonical Lie algebra structure", |
| "q2_slogan": "associative algebras are Lie algebras under the commutator bracket", |
| "q3_nickname": "commutator Lie algebra", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "AG1", |
| "full_name": "AlgebraicGeometry.Scheme", |
| "file": "Mathlib/AlgebraicGeometry/Scheme.lean:42", |
| "difficulty": "Easy", |
| "kind": "structure", |
| "q1a_lean": "a `LocallyRingedSpace` that is locally `Spec R` near every point", |
| "q1b_latex": "a locally ringed space $(X, \\mathcal{O}_X)$ in which every point has a neighborhood isomorphic to $\\mathrm{Spec}\\,R$ for some commutative ring $R$", |
| "q1c_natural": "a locally ringed space that is locally isomorphic to the spectrum of a commutative ring", |
| "q2_slogan": "a locally ringed space modeled on Spec of rings", |
| "q3_nickname": "scheme", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "AG2", |
| "full_name": "AlgebraicGeometry.IsAffine", |
| "file": "Mathlib/AlgebraicGeometry/AffineScheme.lean:59", |
| "difficulty": "Easy", |
| "kind": "class", |
| "q1a_lean": "", |
| "q1b_latex": "$X \\cong \\mathrm{Spec}\\,\\Gamma(X, \\mathcal{O}_X)$", |
| "q1c_natural": "a scheme that is isomorphic to Spec A", |
| "q2_slogan": "a scheme that is the spectrum of its ring of functions", |
| "q3_nickname": "affine scheme", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "AG3", |
| "full_name": "AlgebraicGeometry.ValuativeCommSq", |
| "file": "Mathlib/AlgebraicGeometry/ValuativeCriterion.lean:53", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "a comm square `Spec K → X` and `Spec R → Y` with `R` a DVR and `K` its fraction field", |
| "q1b_latex": "commutative square $\\mathrm{Spec}\\,K \\to X,\\,\\mathrm{Spec}\\,R \\to Y$ over $R$ a DVR with fraction field $K$", |
| "q1c_natural": "a commutative square over a valuation ring and its fraction field used to test properties of a morphism", |
| "q2_slogan": "test square for valuative criteria of morphisms", |
| "q3_nickname": "valuative criteria diagram", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "AG4", |
| "full_name": "AlgebraicGeometry.Scheme.GlueData", |
| "file": "Mathlib/AlgebraicGeometry/Gluing.lean:90", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "a categorical `GlueData` for `Scheme` whose `f i j` are open immersions", |
| "q1b_latex": "glue data on $\\mathrm{Scheme}$ with each $f_{ij}$ an open immersion", |
| "q1c_natural": "gluing data for schemes via open immersions on overlap pieces", |
| "q2_slogan": "gluing data for schemes", |
| "q3_nickname": "scheme glue data", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "AG5", |
| "full_name": "AlgebraicGeometry.IsAffineOpen.basicOpen_basicOpen_is_basicOpen", |
| "file": "Mathlib/AlgebraicGeometry/AffineScheme.lean", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`IsAffineOpen U → IsAffineOpen (U.basicOpen f)`", |
| "q1b_latex": "$D(f) \\subseteq U$ with $U$ affine open $\\Rightarrow D(f)$ affine open", |
| "q1c_natural": "a basic open inside an affine open is itself affine", |
| "q2_slogan": "basic opens of affine opens are affine", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "AG6", |
| "full_name": "AlgebraicGeometry.affine_isReduced_iff", |
| "file": "Mathlib/AlgebraicGeometry/Properties.lean:111", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`IsReduced (Spec R) ↔ IsReduced R`", |
| "q1b_latex": "$\\mathrm{Spec}\\,R$ reduced $\\iff R$ reduced", |
| "q1c_natural": "an affine scheme is reduced iff its ring of functions has no nilpotents", |
| "q2_slogan": "affine scheme reduced iff ring is reduced", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "AG7", |
| "full_name": "AlgebraicGeometry.IsAffine.of_isIso", |
| "file": "Mathlib/AlgebraicGeometry/AffineScheme.lean:122", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`f : X ⟶ Y` an iso, `Y` affine `⇒` `X` affine", |
| "q1b_latex": "$X \\cong Y$ as schemes and $Y$ affine $\\Rightarrow X$ affine", |
| "q1c_natural": "a scheme isomorphic to an affine scheme is itself affine", |
| "q2_slogan": "affineness transfers along scheme isomorphisms", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "AG8", |
| "full_name": "AlgebraicGeometry.IsOpenImmersion.comp", |
| "file": "Mathlib/AlgebraicGeometry/OpenImmersion.lean:40", |
| "difficulty": "Easy", |
| "kind": "instance", |
| "q1a_lean": "`[IsOpenImmersion f] [IsOpenImmersion g] : IsOpenImmersion (f ≫ g)`", |
| "q1b_latex": "$f, g$ open immersions $\\Rightarrow$ $g \\circ f$ open immersion", |
| "q1c_natural": "composition of two open immersions is an open immersion", |
| "q2_slogan": "open immersions are closed under composition", |
| "q3_nickname": "", |
| "q4_special_case": "a chain $U \\hookrightarrow V \\hookrightarrow X$ of open subscheme inclusions composes to an open immersion" |
| }, |
| { |
| "id": "AT1", |
| "full_name": "CategoryTheory.FundamentalGroupoid", |
| "file": "Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean:237", |
| "difficulty": "Easy", |
| "kind": "structure", |
| "q1a_lean": "a wrapper around `X` carrying a `Groupoid` instance via paths up to homotopy", |
| "q1b_latex": "$\\Pi_1(X)$", |
| "q1c_natural": "the carrier type of the fundamental groupoid of a topological space", |
| "q2_slogan": "carrier of the fundamental groupoid", |
| "q3_nickname": "fundamental groupoid", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "AT2", |
| "full_name": "SSet.Quasicategory", |
| "file": "Mathlib/AlgebraicTopology/Quasicategory/Basic.lean:39", |
| "difficulty": "Easy", |
| "kind": "class", |
| "q1a_lean": "a simplicial set in which every inner horn `Λⁿᵢ → S` extends to a `Δⁿ → S`", |
| "q1b_latex": "a simplicial set $S$ such that every inner horn $\\Lambda^n_i \\to S$ ($0 < i < n$) extends to a simplex $\\Delta^n \\to S$", |
| "q1c_natural": "a simplicial set in which every inner horn extends to a full simplex", |
| "q2_slogan": "a simplicial set with all inner horns fillable", |
| "q3_nickname": "quasicategory", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "AT3", |
| "full_name": "HomotopicalAlgebra.ModelCategory", |
| "file": "Mathlib/AlgebraicTopology/ModelCategory/Basic.lean:49", |
| "difficulty": "Hard", |
| "kind": "class", |
| "q1a_lean": "a category equipped with `cofibrations`, `fibrations`, `weakEquivalences` satisfying CM1–CM5", |
| "q1b_latex": "a category $C$ with classes of cofibrations, fibrations, weak equivalences satisfying Quillen\\'s axioms", |
| "q1c_natural": "a category equipped with three classes of morphisms satisfying the Quillen axioms", |
| "q2_slogan": "category equipped to do homotopy theory à la Quillen", |
| "q3_nickname": "Quillen model category", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "AT4", |
| "full_name": "SimplicialObject.Augmented.ExtraDegeneracy", |
| "file": "Mathlib/AlgebraicTopology/ExtraDegeneracy.lean:64", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "`s' : X.right ⟶ X.left.obj 0`, `s n : X.left.obj n ⟶ X.left.obj (n+1)` satisfying simplicial identities with face/degeneracy maps", |
| "q1b_latex": "extra degeneracy maps $s\\', s_n$ on an augmented simplicial object compatible with $\\delta_i$ and $\\sigma_i$", |
| "q1c_natural": "additional degeneracy maps on an augmented simplicial object satisfying compatibility with face and degeneracy maps", |
| "q2_slogan": "extra degeneracy on an augmented simplicial object", |
| "q3_nickname": "extra degeneracy", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "AT5", |
| "full_name": "CategoryTheory.SimplicialObject.δ_comp_δ", |
| "file": "Mathlib/AlgebraicTopology/SimplicialObject/Basic.lean:115", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`δ j.succ ≫ δ i = δ i.castSucc ≫ δ j` for `i ≤ j`", |
| "q1b_latex": "$\\delta_{j+1} \\circ \\delta_i = \\delta_i \\circ \\delta_j$ for $i \\le j$", |
| "q1c_natural": "face maps in a simplicial object commute with appropriate index shifting", |
| "q2_slogan": "first simplicial identity (faces commute)", |
| "q3_nickname": "simplicial identity", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "AT6", |
| "full_name": "CategoryTheory.Arrow.AugmentedCechNerve.ExtraDegeneracy.s_comp_π_succ", |
| "file": "Mathlib/AlgebraicTopology/ExtraDegeneracy.lean:270", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`s f S n ≫ π_(i+1) = π_i` in the augmented Čech nerve", |
| "q1b_latex": "$s_n \\circ \\pi_{i+1} = \\pi_i$ in the augmented Čech nerve", |
| "q1c_natural": "composing the extra-degeneracy map with the next projection gives the previous projection", |
| "q2_slogan": "extra degeneracy and projection match up via index shift", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "AT7", |
| "full_name": "AlgebraicTopology.NormalizedMooreComplex.d_squared", |
| "file": "Mathlib/AlgebraicTopology/MooreComplex.lean:99", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`d (n+1) ≫ d n = 0` in the normalized Moore complex", |
| "q1b_latex": "$d_{n+1} \\circ d_n = 0$ in the normalized Moore complex of a simplicial object", |
| "q1c_natural": "consecutive differentials in the normalized Moore complex compose to zero", |
| "q2_slogan": "d squared is zero in the normalized Moore complex", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "AT8", |
| "full_name": "SSet.instQuasicategoryOfKanComplex", |
| "file": "Mathlib/AlgebraicTopology/Quasicategory/Basic.lean:58", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "`[KanComplex S] : Quasicategory S`", |
| "q1b_latex": "$S$ Kan complex $\\Rightarrow$ $S$ quasicategory", |
| "q1c_natural": "a Kan complex is in particular a quasicategory", |
| "q2_slogan": "Kan complexes are quasicategories", |
| "q3_nickname": "", |
| "q4_special_case": "the singular complex of a topological space is a Kan complex hence a quasicategory" |
| }, |
| { |
| "id": "An1", |
| "full_name": "UniformConvexSpace", |
| "file": "Mathlib/Analysis/Convex/Uniform.lean:44", |
| "difficulty": "Easy", |
| "kind": "class", |
| "q1a_lean": "`SeminormedAddCommGroup` where `‖x‖=‖y‖=1, ‖x-y‖≥ε ⇒ ‖x+y‖ ≤ 2-δ(ε)`", |
| "q1b_latex": "$|x|=|y|=1, |x-y|\\ge\\varepsilon \\Rightarrow |\\tfrac{x+y}{2}| \\le 1-\\delta(\\varepsilon)$", |
| "q1c_natural": "a normed space where the triangle inequality is uniformly strict for far-apart unit vectors", |
| "q2_slogan": "uniformly strict triangle inequality on the unit sphere", |
| "q3_nickname": "uniformly convex space", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "An2", |
| "full_name": "BoxIntegral.Box", |
| "file": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean:76", |
| "difficulty": "Easy", |
| "kind": "structure", |
| "q1a_lean": "`lower, upper : ι → ℝ` with `lower i < upper i`", |
| "q1b_latex": "$\\prod_i (l_i, u_i]$", |
| "q1c_natural": "a half-open rectangular product of intervals indexed by ι", |
| "q2_slogan": "rectangular box in ℝ^ι", |
| "q3_nickname": "rectangular box", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "An3", |
| "full_name": "CompletelyPositiveMap", |
| "file": "Mathlib/Analysis/CStarAlgebra/CompletelyPositiveMap.lean:50", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "an `A₁ →ₗ[ℂ] A₂` such that for every `k`, the lifted map `M_k(A₁) → M_k(A₂)` preserves positivity", |
| "q1b_latex": "linear $\\varphi : A_1 \\to A_2$ such that $\\mathrm{id}_{M_k} \\otimes \\varphi$ preserves positivity for every $k$", |
| "q1c_natural": "a linear map between C*-algebras whose all matrix amplifications stay positive", |
| "q2_slogan": "linear map remaining positive after every matrix amplification", |
| "q3_nickname": "completely positive map (CP map)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "An4", |
| "full_name": "BoxIntegral.Prepartition", |
| "file": "Mathlib/Analysis/BoxIntegral/Partition/Basic.lean:54", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "a `Finset` of sub-boxes `J ≤ I` that are pairwise disjoint", |
| "q1b_latex": "a finite collection of pairwise disjoint sub-boxes contained in $I$", |
| "q1c_natural": "a finite collection of pairwise disjoint sub-boxes of a given box", |
| "q2_slogan": "prepartition by disjoint sub-boxes", |
| "q3_nickname": "prepartition", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "An5", |
| "full_name": "inner_mul_inner_self_le", |
| "file": "Mathlib/Analysis/InnerProductSpace/Basic.lean:281", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`‖⟪x, y⟫‖ * ‖⟪y, x⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫`", |
| "q1b_latex": "$|\\langle x, y\\rangle| \\cdot |\\langle y, x\\rangle| \\le \\mathrm{Re}\\langle x, x\\rangle \\cdot \\mathrm{Re}\\langle y, y\\rangle$", |
| "q1c_natural": "the inner product magnitude is bounded by the product of the squared norms", |
| "q2_slogan": "Cauchy–Schwarz inequality", |
| "q3_nickname": "Cauchy–Schwarz", |
| "q4_special_case": "$\\left(\\sum_i x_i y_i\\right)^2 \\le \\left(\\sum_i x_i^2\\right) \\left(\\sum_i y_i^2\\right)$" |
| }, |
| { |
| "id": "An6", |
| "full_name": "Convex.inter", |
| "file": "Mathlib/Analysis/Convex/Basic.lean:90", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`Convex 𝕜 s → Convex 𝕜 t → Convex 𝕜 (s ∩ t)`", |
| "q1b_latex": "$S, T$ convex $\\Rightarrow S \\cap T$ convex", |
| "q1c_natural": "the intersection of two convex sets is convex", |
| "q2_slogan": "convex sets are closed under intersection", |
| "q3_nickname": "", |
| "q4_special_case": "intersection of two half-planes in $\\mathbb{R}^2$ is convex" |
| }, |
| { |
| "id": "An7", |
| "full_name": "image_norm_le_of_norm_deriv_right_le_deriv_boundary", |
| "file": "Mathlib/Analysis/Calculus/MeanValue.lean", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`‖f' x‖ ≤ B' x` and `‖f a‖ ≤ B a` `⇒` `‖f x‖ ≤ B x` on `[a,b]`", |
| "q1b_latex": "$|f\\'(x)| \\le \\varphi\\'(x)$ on $[a,b]$ and $|f(a)| \\le \\varphi(a) \\Rightarrow |f(x)| \\le \\varphi(x)$ for all $x \\in [a,b]$", |
| "q1c_natural": "if the norm of the derivative of f is bounded by the derivative of φ and f at the start is bounded by φ at the start, then f stays bounded by φ", |
| "q2_slogan": "derivative-norm bound transfers to function bound", |
| "q3_nickname": "mean-value inequality (Banach-space form)", |
| "q4_special_case": "for $f : [0,1] \\to \\mathbb{R}^n$, $|f(1) - f(0)| \\le \\sup_x |f\\'(x)|$" |
| }, |
| { |
| "id": "An8", |
| "full_name": "ContinuousLinearMap.instCStarRingId", |
| "file": "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "`instance : CStarRing (E →L[𝕜] E)` via the operator adjoint", |
| "q1b_latex": "$|T^*T| = |T|^2$ for bounded operators on a Hilbert space", |
| "q1c_natural": "bounded operators on a Hilbert space form a C*-algebra under the operator adjoint as star", |
| "q2_slogan": "bounded operators on a Hilbert space form a C*-algebra", |
| "q3_nickname": "$B(H)$ is a $C^*$-algebra", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "C1", |
| "full_name": "CategoryTheory.Functor", |
| "file": "Mathlib/CategoryTheory/Functor/Basic.lean:40", |
| "difficulty": "Easy", |
| "kind": "structure", |
| "q1a_lean": "a map between categories with `obj : C → D` and `map : (X ⟶ Y) → (F X ⟶ F Y)` preserving `𝟙` and `≫`", |
| "q1b_latex": "$F : C \\to D$ with $F(\\mathrm{id}_X) = \\mathrm{id}_{F X}$ and $F(g \\circ f) = F g \\circ F f$", |
| "q1c_natural": "a map between categories that preserves identities and composition", |
| "q2_slogan": "a structure-preserving map between categories", |
| "q3_nickname": "functor", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "C2", |
| "full_name": "CategoryTheory.NatTrans", |
| "file": "Mathlib/CategoryTheory/NatTrans.lean:52", |
| "difficulty": "Easy", |
| "kind": "structure", |
| "q1a_lean": "`app (X : C) : F.obj X ⟶ G.obj X` with `naturality (f : X ⟶ Y)`", |
| "q1b_latex": "family $\\alpha_X : F(X) \\to G(X)$ satisfying $\\alpha_Y \\circ F(f) = G(f) \\circ \\alpha_X$", |
| "q1c_natural": "a family of morphisms between two functors that commutes with morphism action", |
| "q2_slogan": "morphism between functors", |
| "q3_nickname": "natural transformation", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "C3", |
| "full_name": "CategoryTheory.Bicategory", |
| "file": "Mathlib/CategoryTheory/Bicategory/Basic.lean:61", |
| "difficulty": "Hard", |
| "kind": "class", |
| "q1a_lean": "a `CategoryStruct` with `whiskerLeft`, `whiskerRight`, `associator`, `leftUnitor`, `rightUnitor` satisfying pentagon and triangle", |
| "q1b_latex": "category enriched in categories with weak associator/unitor satisfying pentagon and triangle laws", |
| "q1c_natural": "a category whose composition is associative only up to coherent 2-isomorphism", |
| "q2_slogan": "weak 2-category with coherent associator and unitors", |
| "q3_nickname": "bicategory", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "C4", |
| "full_name": "CategoryTheory.GrothendieckTopology", |
| "file": "Mathlib/CategoryTheory/Sites/Grothendieck.lean:76", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "`sieves : (X : C) → Set (Sieve X)` satisfying maximality, pullback stability, transitivity", |
| "q1b_latex": "assignment $X \\mapsto J(X) \\subseteq \\mathrm{Sieve}(X)$ closed under maximality, pullback, transitivity", |
| "q1c_natural": "an assignment of covering sieves to each object satisfying maximality, pullback stability, and transitivity", |
| "q2_slogan": "axiomatic notion of \"open cover\" in a category", |
| "q3_nickname": "Grothendieck topology", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "C5", |
| "full_name": "CategoryTheory.Limits.limit.lift_π", |
| "file": "Mathlib/CategoryTheory/Limits/HasLimits.lean", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`lift F c ≫ π F j = c.π.app j`", |
| "q1b_latex": "$\\mathrm{lift}(c) \\circ \\pi_j = c.\\pi_j$ for any cone $c$ over $F$", |
| "q1c_natural": "composing the universal lift with each limit projection recovers the original cone", |
| "q2_slogan": "universal lift composes correctly with the projections", |
| "q3_nickname": "universal property of limit", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "C6", |
| "full_name": "CategoryTheory.NatTrans.vcomp_app", |
| "file": "Mathlib/CategoryTheory/NatTrans.lean:107", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`(α.vcomp β).app X = α.app X ≫ β.app X`", |
| "q1b_latex": "$(\\beta \\circ \\alpha)_X = \\beta_X \\circ \\alpha_X$", |
| "q1c_natural": "vertical composition of natural transformations is computed component-wise", |
| "q2_slogan": "vertical composition is component-wise", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "C7", |
| "full_name": "CategoryTheory.Adjunction.homEquiv_naturality_left", |
| "file": "Mathlib/CategoryTheory/Adjunction/Basic.lean:227", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "for `F ⊣ G`, `Φ (F.map f ≫ g) = f ≫ Φ g`", |
| "q1b_latex": "$\\Phi(F(f) \\circ g) = f \\circ \\Phi(g)$ for the hom-equivalence of an adjunction $F \\dashv G$", |
| "q1c_natural": "the hom-set bijection of an adjunction is natural in the source object", |
| "q2_slogan": "adjunction hom-set bijection is natural in the source", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "C8", |
| "full_name": "CategoryTheory.Pretriangulated.triangleCategory", |
| "file": "Mathlib/CategoryTheory/Triangulated/Basic.lean:146", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "`instance : Category (Triangle C)`", |
| "q1b_latex": "triangles in a pretriangulated category form a category", |
| "q1c_natural": "distinguished triangles in a pretriangulated category form their own category", |
| "q2_slogan": "triangles form a category", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cb1", |
| "full_name": "SimpleGraph", |
| "file": "Mathlib/Combinatorics/SimpleGraph/Basic.lean:92", |
| "difficulty": "Easy", |
| "kind": "structure", |
| "q1a_lean": "`Adj : V → V → Prop` symmetric and irreflexive", |
| "q1b_latex": "$G = (V, E)$ with $E$ symmetric and irreflexive on $V$", |
| "q1c_natural": "a vertex set with a symmetric, irreflexive adjacency relation", |
| "q2_slogan": "irreflexive symmetric adjacency on vertices", |
| "q3_nickname": "simple graph", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cb2", |
| "full_name": "YoungDiagram", |
| "file": "Mathlib/Combinatorics/Young/YoungDiagram.lean:66", |
| "difficulty": "Easy", |
| "kind": "structure", |
| "q1a_lean": "a `Finset (ℕ × ℕ)` that is a lower set", |
| "q1b_latex": "a finite lower set in $\\mathbb{N} \\times \\mathbb{N}$", |
| "q1c_natural": "a finite collection of cells forming a lower set in the integer grid", |
| "q2_slogan": "finite lower set of cells in the integer grid", |
| "q3_nickname": "Young diagram", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cb3", |
| "full_name": "Matroid", |
| "file": "Mathlib/Combinatorics/Matroid/Basic.lean:192", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "ground set `E`, base predicate `IsBase`, indep predicate `Indep` satisfying exchange and maximality", |
| "q1b_latex": "a ground set $E$ with families $\\mathcal{B}, \\mathcal{I}$ of bases and independents satisfying exchange and maximality", |
| "q1c_natural": "a ground set with families of bases and independent sets satisfying exchange and maximality", |
| "q2_slogan": "combinatorial abstraction of linear independence", |
| "q3_nickname": "matroid", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cb4", |
| "full_name": "Combinatorics.Line", |
| "file": "Mathlib/Combinatorics/HalesJewett.lean:175", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "`idxFun : ι → Option α` with at least one `none` (the free coordinate)", |
| "q1b_latex": "$\\ell : \\alpha \\to (\\iota \\to \\alpha)$ where some coordinates vary together and others are fixed", |
| "q1c_natural": "a parametric family of points in a hypercube where some coordinates vary together and others are fixed", |
| "q2_slogan": "non-trivial axis-parallel line in a hypercube", |
| "q3_nickname": "combinatorial line", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cb5", |
| "full_name": "SimpleGraph.adj_comm", |
| "file": "Mathlib/Combinatorics/SimpleGraph/Basic.lean:166", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`G.Adj u v ↔ G.Adj v u`", |
| "q1b_latex": "$u \\sim v \\iff v \\sim u$", |
| "q1c_natural": "adjacency in a simple graph is symmetric", |
| "q2_slogan": "edges of a simple graph are undirected", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cb6", |
| "full_name": "Finset.all_card_le_biUnion_card_iff_existsInjective'", |
| "file": "Mathlib/Combinatorics/Hall/Finite.lean:243", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`(∀ s : Finset ι, #s ≤ #(s.biUnion t)) ↔ ∃ f : ι → α, Injective f ∧ ∀ x, f x ∈ t x`", |
| "q1b_latex": "$\\forall S, |S| \\le \\left|\\bigcup_{i \\in S} t(i)\\right| \\iff \\exists f$ injective, $f(i) \\in t(i)$", |
| "q1c_natural": "a finite system has a system of distinct representatives iff every subfamily covers enough elements", |
| "q2_slogan": "Hall's condition characterizes existence of an SDR", |
| "q3_nickname": "Hall's marriage theorem", |
| "q4_special_case": "$n$ men and $n$ women have a perfect matching iff every $k$-subset of men collectively likes at least $k$ women" |
| }, |
| { |
| "id": "Cb7", |
| "full_name": "Finset.exists_lt_card_fiber_of_nsmul_lt_card_of_maps_to", |
| "file": "Mathlib/Combinatorics/Pigeonhole.lean:220", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`|t| • b < |s| → ∃ y ∈ t, b < |{x ∈ s | f x = y}|`", |
| "q1b_latex": "$|t| \\cdot b < |s| \\Rightarrow \\exists y \\in t,\\ b < |\\{x \\in s : f(x) = y\\}|$", |
| "q1c_natural": "if more than $b \\cdot |t|$ elements of $s$ map into $t$, some fiber has more than $b$ elements", |
| "q2_slogan": "more pigeons than holes×threshold ⇒ some hole exceeds threshold", |
| "q3_nickname": "pigeonhole principle (fiber form)", |
| "q4_special_case": "distributing 11 balls into 3 boxes gives some box with at least 4 balls" |
| }, |
| { |
| "id": "Cb8", |
| "full_name": "instFintypeSimpleGraphOfDecidableEq", |
| "file": "Mathlib/Combinatorics/SimpleGraph/Basic.lean:114", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "`[Fintype V] [DecidableEq V] : Fintype (SimpleGraph V)`", |
| "q1b_latex": "$V$ finite with decidable equality $\\Rightarrow$ $\\mathrm{SimpleGraph}\\,V$ is finite", |
| "q1c_natural": "when V is finite with decidable equality, the type of simple graphs on V is finite", |
| "q2_slogan": "simple graphs on a finite vertex set are finite", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cp1", |
| "full_name": "DFA", |
| "file": "Mathlib/Computability/DFA.lean:58", |
| "difficulty": "Easy", |
| "kind": "structure", |
| "q1a_lean": "`step : σ → α → σ`, `start : σ`, `accept : Set σ`", |
| "q1b_latex": "$(Q, \\Sigma, \\delta, q_0, F)$", |
| "q1c_natural": "a finite-state automaton with deterministic transition, start state, and accept set", |
| "q2_slogan": "deterministic finite automaton", |
| "q3_nickname": "DFA (deterministic finite automaton)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cp2", |
| "full_name": "RegularExpression", |
| "file": "Mathlib/Computability/RegularExpressions.lean:47", |
| "difficulty": "Easy", |
| "kind": "inductive", |
| "q1a_lean": "6 constructors: `zero`, `epsilon`, `char`, `plus`, `comp`, `star`", |
| "q1b_latex": "$0,\\ \\varepsilon,\\ a,\\ r+s,\\ r\\cdot s,\\ r^*$", |
| "q1c_natural": "term language built from zero, epsilon, character, sum, concat, and Kleene star", |
| "q2_slogan": "term language for regular languages", |
| "q3_nickname": "regular expression", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cp3", |
| "full_name": "Nat.Partrec.Code", |
| "file": "Mathlib/Computability/PartrecCode.lean:76", |
| "difficulty": "Hard", |
| "kind": "inductive", |
| "q1a_lean": "8 constructors `zero, succ, left, right, pair, comp, prec, rfind'`", |
| "q1b_latex": "inductive Gödel codes for partial recursive functions on $\\mathbb{N}$", |
| "q1c_natural": "inductive codes parameterizing all partial recursive functions on naturals", |
| "q2_slogan": "Gödel codes for partial recursive functions", |
| "q3_nickname": "Gödel numbering of partial recursive functions", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cp4", |
| "full_name": "Nat.RecursiveIn", |
| "file": "Mathlib/Computability/RecursiveIn.lean:42", |
| "difficulty": "Hard", |
| "kind": "inductive", |
| "q1a_lean": "oracle-relativized partial recursive functions, with `oracle (g : ℕ →. ℕ) : g ∈ O → RecursiveIn O g`", |
| "q1b_latex": "inductive type for $f$ recursive in oracle $O$", |
| "q1c_natural": "inductive predicate for partial functions computable using a given oracle", |
| "q2_slogan": "partial recursive with access to an oracle", |
| "q3_nickname": "oracle computability", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cp5", |
| "full_name": "DFA.evalFrom_append_singleton", |
| "file": "Mathlib/Computability/DFA.lean:91", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`M.evalFrom s (x ++ [a]) = M.step (M.evalFrom s x) a`", |
| "q1b_latex": "$\\mathrm{eval}_s(x \\cdot a) = \\delta(\\mathrm{eval}_s(x), a)$", |
| "q1c_natural": "appending one symbol to a DFA run takes one extra transition", |
| "q2_slogan": "DFA appending one symbol = one transition step", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cp6", |
| "full_name": "DFA.mem_acceptsFrom", |
| "file": "Mathlib/Computability/DFA.lean:120", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`x ∈ M.acceptsFrom s ↔ M.evalFrom s x ∈ M.accept`", |
| "q1b_latex": "$x \\in \\mathrm{acceptsFrom}(s) \\iff \\mathrm{eval}_s(x) \\in F$", |
| "q1c_natural": "a string is accepted iff its run ends in an accepting state", |
| "q2_slogan": "DFA acceptance = ending in an accept state", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cp7", |
| "full_name": "Nat.Partrec.Code.computable_recOn", |
| "file": "Mathlib/Computability/PartrecCode.lean:366", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`Computable (fun a => Code.recOn (c a) ...)` given each branch is computable", |
| "q1b_latex": "structural recursion on partial recursive codes is itself computable", |
| "q1c_natural": "recursing on a Code with computable branches yields a computable function", |
| "q2_slogan": "recursion on Code is a computable operation", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cp8", |
| "full_name": "Primcodable.list", |
| "file": "Mathlib/Computability/Primrec/List.lean", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "`[Primcodable α] : Primcodable (List α)`", |
| "q1b_latex": "$\\alpha$ Primcodable $\\Rightarrow$ $\\mathrm{List}\\,\\alpha$ Primcodable", |
| "q1c_natural": "lists over a Primcodable base type are themselves Primcodable", |
| "q2_slogan": "List preserves Primcodable", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cd1", |
| "full_name": "Condensed", |
| "file": "Mathlib/Condensed/Basic.lean:45", |
| "difficulty": "Easy", |
| "kind": "def", |
| "q1a_lean": "`Sheaf (coherentTopology CompHaus) C`", |
| "q1b_latex": "$\\mathrm{Sheaf}(\\mathrm{CompHaus},\\,J_{\\mathrm{coh}};\\,C)$", |
| "q1c_natural": "sheaves on the coherent site of compact Hausdorff spaces valued in a category", |
| "q2_slogan": "sheaves on the coherent CompHaus site", |
| "q3_nickname": "condensed object", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cd2", |
| "full_name": "CondensedSet", |
| "file": "Mathlib/Condensed/Basic.lean:55", |
| "difficulty": "Easy", |
| "kind": "def", |
| "q1a_lean": "`Condensed.{u} (Type (u+1))`", |
| "q1b_latex": "$\\mathrm{Condensed}(\\mathbf{Set})$", |
| "q1c_natural": "condensed objects valued in sets", |
| "q2_slogan": "condensed sets = condensed objects in Set", |
| "q3_nickname": "condensed set", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cd3", |
| "full_name": "CondensedMod.IsSolid", |
| "file": "Mathlib/Condensed/Solid.lean:84", |
| "difficulty": "Hard", |
| "kind": "class", |
| "q1a_lean": "a condensed module `A` such that `IsIso ((yoneda.obj A).map (profiniteSolidification ...).op)`", |
| "q1b_latex": "condensed $R$-module $A$ on which the solidification map induces a Yoneda iso", |
| "q1c_natural": "a condensed module fixed (up to Yoneda iso) by the solidification functor", |
| "q2_slogan": "solid = solidification map is a Yoneda iso", |
| "q3_nickname": "solid module", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cd4", |
| "full_name": "Condensed.profiniteSolid", |
| "file": "Mathlib/Condensed/Solid.lean:49", |
| "difficulty": "Hard", |
| "kind": "def", |
| "q1a_lean": "`Profinite ⥤ CondensedMod R` sending `S` to `R[S]^solid`", |
| "q1b_latex": "$S \\mapsto R[S]^{\\mathrm{solid}}$ as a functor $\\mathrm{Profinite} \\to \\mathrm{CondensedMod}\\,R$", |
| "q1c_natural": "the functor sending a profinite set to its solid free R-module", |
| "q2_slogan": "profinite-to-solid free module functor", |
| "q3_nickname": "profinite solidification", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cd5", |
| "full_name": "Condensed.hasExactColimitsOfShape", |
| "file": "Mathlib/Condensed/AB.lean:37", |
| "difficulty": "Easy", |
| "kind": "lemma", |
| "q1a_lean": "`HasExactColimitsOfShape J (Condensed A)` given exactness on coefficients", |
| "q1b_latex": "$J$-shaped colimits in $\\mathrm{Condensed}\\,A$ exist and are exact when colimits in $A$ are", |
| "q1c_natural": "filtered colimits in the condensed category exist and remain exact", |
| "q2_slogan": "condensed inherits exact colimits from coefficient category", |
| "q3_nickname": "AB5 for condensed", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cd6", |
| "full_name": "Condensed.instAB4CondensedMod", |
| "file": "Mathlib/Condensed/AB.lean:64", |
| "difficulty": "Easy", |
| "kind": "instance", |
| "q1a_lean": "`instance : AB4 (CondensedMod R)`", |
| "q1b_latex": "$\\mathrm{CondensedMod}\\,R$ satisfies Grothendieck axiom AB4", |
| "q1c_natural": "condensed modules satisfy Grothendieck AB4 (infinite direct sums are exact)", |
| "q2_slogan": "CondensedMod satisfies AB4", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cd7", |
| "full_name": "Condensed.StoneanCompHaus.equivalence", |
| "file": "Mathlib/Condensed/Equivalence.lean:50", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`Sheaf (coherentTopology Stonean) A ≌ Condensed A`", |
| "q1b_latex": "$\\mathrm{Sh}(J_{\\mathrm{coh}}, \\mathrm{Stonean}; A) \\simeq \\mathrm{Condensed}\\,A$", |
| "q1c_natural": "sheaves on Stonean spaces with the coherent topology are equivalent to condensed objects", |
| "q2_slogan": "condensed = sheaves on Stonean = sheaves on CompHaus", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Cd8", |
| "full_name": "Condensed.discreteUnderlyingAdj", |
| "file": "Mathlib/Condensed/Discrete/Basic.lean", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`discrete C ⊣ underlying C`", |
| "q1b_latex": "$\\mathrm{discrete} \\dashv \\mathrm{underlying}$", |
| "q1c_natural": "the discrete-object functor is left adjoint to the underlying functor", |
| "q2_slogan": "discrete is left adjoint to underlying", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "D1", |
| "full_name": "Sym2", |
| "file": "Mathlib/Data/Sym/Sym2.lean", |
| "difficulty": "Easy", |
| "kind": "def", |
| "q1a_lean": "`Quot (Sym2.Rel α)`", |
| "q1b_latex": "$\\mathrm{Sym}^2 \\alpha = (\\alpha \\times \\alpha) / \\sim$", |
| "q1c_natural": "type of unordered pairs of elements (symmetric square)", |
| "q2_slogan": "type of unordered pairs", |
| "q3_nickname": "unordered pair (symmetric square)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "D2", |
| "full_name": "DFinsupp", |
| "file": "Mathlib/Data/DFinsupp/Defs.lean", |
| "difficulty": "Easy", |
| "kind": "structure", |
| "q1a_lean": "`Π₀ i, β i` — dependent function with finite support", |
| "q1b_latex": "$\\Pi_0\\,i,\\,\\beta(i)$", |
| "q1c_natural": "a dependent function over a family of types with finite support", |
| "q2_slogan": "dependent finitely-supported function", |
| "q3_nickname": "dependent finsupp", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "D3", |
| "full_name": "Bracket", |
| "file": "Mathlib/Data/Bracket.lean:37", |
| "difficulty": "Hard", |
| "kind": "class", |
| "q1a_lean": "`class Bracket L M where bracket : L → M → M`", |
| "q1b_latex": "typeclass providing a binary operation $L \\to M \\to M$ written `⁅·, ·⁆`", |
| "q1c_natural": "typeclass providing a generic bracket operation", |
| "q2_slogan": "generic bracket notation", |
| "q3_nickname": "bracket notation", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "D4", |
| "full_name": "Bundle.TotalSpace", |
| "file": "Mathlib/Data/Bundle.lean:56", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "`proj : B`, `snd : E proj`", |
| "q1b_latex": "$\\Sigma\\,b : B,\\,E(b)$", |
| "q1c_natural": "a dependent pair of base point and fiber element representing the total space of a bundle", |
| "q2_slogan": "total space of a bundle as dependent pairs", |
| "q3_nickname": "total space (of a bundle)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "D5", |
| "full_name": "List.singleton_injective", |
| "file": "Mathlib/Data/List/Basic.lean:58", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`Function.Injective (fun a : α => [a])`", |
| "q1b_latex": "$a \\mapsto [a]$ is injective", |
| "q1c_natural": "mapping an element to its singleton list is injective", |
| "q2_slogan": "singleton-list constructor is injective", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "D6", |
| "full_name": "Finset.erase_singleton", |
| "file": "Mathlib/Data/Finset/Basic.lean:129", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`({a} : Finset α).erase a = ∅`", |
| "q1b_latex": "$\\{a\\} \\setminus \\{a\\} = \\emptyset$ as Finset", |
| "q1c_natural": "removing the unique element from a singleton finset gives the empty set", |
| "q2_slogan": "erasing the only element of a singleton finset", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "D7", |
| "full_name": "Holor.sum_unitVec_mul_slice", |
| "file": "Mathlib/Data/Holor.lean", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`∑ i, (unitVec d i).mul (x.slice i ⋯) = x`", |
| "q1b_latex": "$T = \\sum_i e_i \\otimes T_i$ where $T_i$ is the $i$-th slice of $T$", |
| "q1c_natural": "a holor reconstructs from the sum of unit vectors tensored with its slices along the first axis", |
| "q2_slogan": "holor decomposes via unit vectors and slices", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "D8", |
| "full_name": "ENNReal.instCommSemiring", |
| "file": "Mathlib/Data/ENNReal/Basic.lean", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "`instance : CommSemiring ℝ≥0∞`", |
| "q1b_latex": "$\\overline{\\mathbb{R}}_{\\ge 0}$ is a commutative semiring", |
| "q1c_natural": "extended nonneg reals form a commutative semiring with appropriate ∞ conventions", |
| "q2_slogan": "extended nonneg reals form a comm semiring", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Dy1", |
| "full_name": "Flow", |
| "file": "Mathlib/Dynamics/Flow.lean:94", |
| "difficulty": "Easy", |
| "kind": "structure", |
| "q1a_lean": "`toFun : τ → α → α` continuous, `toFun 0 x = x`, `toFun (s+t) x = toFun s (toFun t x)`", |
| "q1b_latex": "continuous $\\Phi : \\tau \\times \\alpha \\to \\alpha$ with $\\Phi(0, x) = x$ and $\\Phi(s+t, x) = \\Phi(s, \\Phi(t, x))$", |
| "q1c_natural": "a continuous additive-monoid action of a time monoid on a topological space", |
| "q2_slogan": "continuous time-evolution of a space", |
| "q3_nickname": "flow (continuous dynamical system)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Dy2", |
| "full_name": "Function.fixedPoints", |
| "file": "Mathlib/Dynamics/FixedPoints/Defs.lean:118", |
| "difficulty": "Easy", |
| "kind": "def", |
| "q1a_lean": "`{ x | f x = x }`", |
| "q1b_latex": "$\\mathrm{Fix}(f) = \\{x : f(x) = x\\}$", |
| "q1c_natural": "set of points fixed by a function", |
| "q2_slogan": "set of fixed points of a function", |
| "q3_nickname": "fixed point set", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Dy3", |
| "full_name": "MulAction.IsMinimal", |
| "file": "Mathlib/Dynamics/Minimal.lean:40", |
| "difficulty": "Hard", |
| "kind": "class", |
| "q1a_lean": "`∀ x, Dense (orbit M x)`", |
| "q1b_latex": "$\\forall x \\in \\alpha,\\ \\overline{Mx} = \\alpha$", |
| "q1c_natural": "a group action all of whose orbits are dense", |
| "q2_slogan": "every orbit is dense", |
| "q3_nickname": "minimal action", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Dy4", |
| "full_name": "MulAction.IsTopologicallyTransitive", |
| "file": "Mathlib/Dynamics/Transitive.lean:40", |
| "difficulty": "Hard", |
| "kind": "class", |
| "q1a_lean": "`∀ U V open nonempty, ∃ m, (m • U ∩ V).Nonempty`", |
| "q1b_latex": "$\\forall U, V$ open nonempty, $\\exists m \\in M,\\ (m \\cdot U) \\cap V \\ne \\emptyset$", |
| "q1c_natural": "for any two nonempty opens, some group element moves the first to overlap the second", |
| "q2_slogan": "action mixes nonempty open sets", |
| "q3_nickname": "topological transitivity", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Dy5", |
| "full_name": "omegaLimit_union", |
| "file": "Mathlib/Dynamics/OmegaLimit.lean:153", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`omegaLimit f ϕ (s₁ ∪ s₂) = omegaLimit f ϕ s₁ ∪ omegaLimit f ϕ s₂`", |
| "q1b_latex": "$\\omega(s_1 \\cup s_2) = \\omega(s_1) \\cup \\omega(s_2)$", |
| "q1c_natural": "the ω-limit of a union is the union of the ω-limits", |
| "q2_slogan": "ω-limit distributes over union", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Dy6", |
| "full_name": "Function.IsFixedPt.comp", |
| "file": "Mathlib/Dynamics/FixedPoints/Basic.lean", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`IsFixedPt f x → IsFixedPt g x → IsFixedPt (f ∘ g) x`", |
| "q1b_latex": "$f(x)=x \\wedge g(x)=x \\Rightarrow (f \\circ g)(x) = x$", |
| "q1c_natural": "a common fixed point of f and g is a fixed point of f∘g", |
| "q2_slogan": "common fixed point survives composition", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Dy7", |
| "full_name": "Ergodic.of_mem_extremePoints_measure_univ_eq", |
| "file": "Mathlib/Dynamics/Ergodic/Extreme.lean:32", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`μ ∈ extremePoints {ν | MeasurePreserving f ν ν ∧ ν univ = c} → Ergodic f μ`", |
| "q1b_latex": "$\\mu$ extreme in $\\{\\nu : \\mathrm{MeasurePreserving}(f, \\nu, \\nu),\\ \\nu(\\Omega) = c\\} \\Rightarrow$ $\\mu$ ergodic", |
| "q1c_natural": "an extreme point of the convex set of invariant measures with fixed total mass is ergodic", |
| "q2_slogan": "extreme points of the invariant-measure simplex are ergodic", |
| "q3_nickname": "extreme = ergodic", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Dy8", |
| "full_name": "MulAction.isMinimal_of_pretransitive", |
| "file": "Mathlib/Dynamics/Minimal.lean:58", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "`[IsPretransitive M α] : IsMinimal M α`", |
| "q1b_latex": "pretransitive action $\\Rightarrow$ minimal action", |
| "q1c_natural": "a topologically pretransitive group action is automatically minimal", |
| "q2_slogan": "pretransitive ⇒ minimal", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "F1", |
| "full_name": "Polynomial.Separable", |
| "file": "Mathlib/FieldTheory/Separable.lean:49", |
| "difficulty": "Easy", |
| "kind": "def", |
| "q1a_lean": "`IsCoprime f (Polynomial.derivative f)`", |
| "q1b_latex": "$\\gcd(f, f\\') = 1$", |
| "q1c_natural": "a polynomial coprime with its derivative", |
| "q2_slogan": "polynomial coprime with its derivative", |
| "q3_nickname": "separable polynomial", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "F2", |
| "full_name": "PerfectRing", |
| "file": "Mathlib/FieldTheory/Perfect.lean:46", |
| "difficulty": "Easy", |
| "kind": "class", |
| "q1a_lean": "`Function.Bijective (fun x : R => x ^ p)`", |
| "q1b_latex": "$x \\mapsto x^p$ is a bijection on $R$", |
| "q1c_natural": "a ring of characteristic p where the Frobenius map is a bijection", |
| "q2_slogan": "Frobenius is a bijection", |
| "q3_nickname": "perfect ring", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "F3", |
| "full_name": "IsPRadical", |
| "file": "Mathlib/FieldTheory/IsPerfectClosure.lean:147", |
| "difficulty": "Hard", |
| "kind": "class", |
| "q1a_lean": "`i : K →+* L` with `∀ x, ∃ n y, i y = x ^ p^n` and `ker i ≤ pNilradical K p`", |
| "q1b_latex": "ring map $i : K \\to L$ such that every $x \\in L$ has $x^{p^n} \\in i(K)$ and $\\ker i \\subseteq \\mathrm{pNilradical}(K)$", |
| "q1c_natural": "a ring map where every element of the target has a $p^n$-power in the source", |
| "q2_slogan": "every element of L has a p-power in K", |
| "q3_nickname": "p-radical (purely inseparable)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "F4", |
| "full_name": "IntermediateField.Lifts", |
| "file": "Mathlib/FieldTheory/Extension.lean:35", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "`carrier : IntermediateField F E` with `emb : carrier →ₐ[F] K`", |
| "q1b_latex": "pair $(L, \\phi : L \\to K)$ where $L$ is an intermediate field", |
| "q1c_natural": "a partial intermediate field together with a chosen algebra map into a target", |
| "q2_slogan": "partial extension of an algebra hom", |
| "q3_nickname": "lifts (for extension lifting)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "F5", |
| "full_name": "mem_algebraicClosure_iff", |
| "file": "Mathlib/FieldTheory/AlgebraicClosure.lean:55", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`x ∈ algebraicClosure F E ↔ IsAlgebraic F x`", |
| "q1b_latex": "$x \\in \\overline{F}^E \\iff x$ is algebraic over $F$", |
| "q1c_natural": "an element belongs to the algebraic closure iff it is algebraic over the base", |
| "q2_slogan": "algebraic closure = algebraic elements", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "F6", |
| "full_name": "Polynomial.separable_def", |
| "file": "Mathlib/FieldTheory/Separable.lean", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`Separable f ↔ IsCoprime f (derivative f)`", |
| "q1b_latex": "$\\mathrm{Separable}(f) \\iff \\gcd(f, f\\') = 1$", |
| "q1c_natural": "a polynomial is separable iff it is coprime with its derivative", |
| "q2_slogan": "separability = coprime with derivative", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "F7", |
| "full_name": "IntermediateField.exists_algHom_adjoin_of_splits'", |
| "file": "Mathlib/FieldTheory/Extension.lean:253", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`f : L →ₐ[F] K` extends to `adjoin L S →ₐ[F] K` if elements of S are integral and minpolys split in K", |
| "q1b_latex": "all $s \\in S$ integral over $L$, minpolys $\\mathrm{minpoly}_L(s)$ splitting in $K$ $\\Rightarrow$ alg-hom $L \\to K$ extends to $\\mathrm{adjoin}(L, S) \\to K$", |
| "q1c_natural": "if a polynomial splits in the target, a partial algebra map extends to an adjoin of its roots", |
| "q2_slogan": "alg-hom extends to adjoin when the polynomial splits in the target", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "F8", |
| "full_name": "IsGalois.of_fixedField_normal_subgroup", |
| "file": "Mathlib/FieldTheory/Galois/Basic.lean", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "`[IsGalois K L] (H : Subgroup Gal(L/K)) [H.Normal] : IsGalois K (fixedField H)`", |
| "q1b_latex": "$H \\trianglelefteq \\mathrm{Gal}(L/K) \\Rightarrow L^H / K$ is Galois", |
| "q1c_natural": "the fixed field of a normal subgroup of a Galois group is itself Galois over the base", |
| "q2_slogan": "fixed field of a normal subgroup is Galois", |
| "q3_nickname": "Galois fundamental theorem (one direction)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "G1", |
| "full_name": "ChartedSpace", |
| "file": "Mathlib/Geometry/Manifold/ChartedSpace.lean:138", |
| "difficulty": "Easy", |
| "kind": "class", |
| "q1a_lean": "`atlas : Set (OpenPartialHomeomorph M H)` and `chartAt : M → ...` with chart-source axioms", |
| "q1b_latex": "a topological space $M$ equipped with an atlas of partial homeomorphisms onto a model space $H$", |
| "q1c_natural": "a topological space equipped with an atlas of local homeomorphisms onto a model space", |
| "q2_slogan": "a space locally modeled on H", |
| "q3_nickname": "charted space", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "G2", |
| "full_name": "Polygon", |
| "file": "Mathlib/Geometry/Polygon/Basic.lean:29", |
| "difficulty": "Easy", |
| "kind": "structure", |
| "q1a_lean": "`vertices : Fin n → P`", |
| "q1b_latex": "an indexed sequence of $n$ vertices in $P$", |
| "q1c_natural": "an indexed family of n vertices in a space", |
| "q2_slogan": "indexed family of n vertices", |
| "q3_nickname": "polygon", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "G3", |
| "full_name": "DiffeologicalSpace", |
| "file": "Mathlib/Geometry/Diffeology/Basic.lean:157", |
| "difficulty": "Hard", |
| "kind": "class", |
| "q1a_lean": "`plots (n : ℕ) : Set (𝔼ⁿ → X)` satisfying constant, reparametrization, locality", |
| "q1b_latex": "space with a family of \"smooth\" maps from $\\mathbb{R}^n$ satisfying covering, reparametrization, locality", |
| "q1c_natural": "a space equipped with a family of plots from Euclidean spaces satisfying smoothness axioms", |
| "q2_slogan": "space defined by which Euclidean-domain maps are smooth", |
| "q3_nickname": "diffeological space", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "G4", |
| "full_name": "AlgebraicGeometry.PresheafedSpace", |
| "file": "Mathlib/Geometry/RingedSpace/PresheafedSpace.lean:42", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "`carrier : TopCat`, `presheaf : carrier.Presheaf C`", |
| "q1b_latex": "$(X, \\mathcal{F} : \\mathrm{Open}(X)^{\\mathrm{op}} \\to C)$", |
| "q1c_natural": "a topological space together with a presheaf on it valued in some category", |
| "q2_slogan": "topological space + presheaf", |
| "q3_nickname": "presheafed space", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "G5", |
| "full_name": "chart_source_mem_nhds", |
| "file": "Mathlib/Geometry/Manifold/ChartedSpace.lean:185", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`(chartAt H x).source ∈ 𝓝 x`", |
| "q1b_latex": "$\\mathrm{source}(\\mathrm{chartAt}(x)) \\in \\mathcal{N}(x)$", |
| "q1c_natural": "the source of the chart at x is a neighborhood of x", |
| "q2_slogan": "chart source is a neighborhood of the chart's center", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "G6", |
| "full_name": "StructureGroupoid.trans", |
| "file": "Mathlib/Geometry/Manifold/StructureGroupoid.lean:161", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`e ∈ G → e' ∈ G → e.trans e' ∈ G`", |
| "q1b_latex": "$e, e\\' \\in G \\Rightarrow e \\circ e\\' \\in G$", |
| "q1c_natural": "composition of two members of a structure groupoid is again in the groupoid", |
| "q2_slogan": "structure groupoid closed under composition", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "G7", |
| "full_name": "IsOpen.uniqueMDiffWithinAt", |
| "file": "Mathlib/Geometry/Manifold/MFDeriv/Basic.lean:91", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`IsOpen s → x ∈ s → UniqueMDiffWithinAt I s x`", |
| "q1b_latex": "$U$ open, $x \\in U \\Rightarrow \\mathrm{UniqueMDiffWithinAt}(I, U, x)$", |
| "q1c_natural": "at any point of an open set, the manifold derivative is uniquely determined", |
| "q2_slogan": "derivatives within an open set are uniquely determined", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "G8", |
| "full_name": "AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces", |
| "file": "Mathlib/Geometry/RingedSpace/PresheafedSpace.lean:119", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "`instance : Category (PresheafedSpace C)`", |
| "q1b_latex": "$\\mathrm{PresheafedSpace}\\,C$ is a category", |
| "q1c_natural": "presheafed spaces with appropriate morphisms form a category", |
| "q2_slogan": "presheafed spaces form a category", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Gr1", |
| "full_name": "Sylow", |
| "file": "Mathlib/GroupTheory/Sylow.lean:53", |
| "difficulty": "Easy", |
| "kind": "structure", |
| "q1a_lean": "`extends Subgroup G` (maximal p-subgroup)", |
| "q1b_latex": "maximal p-subgroup of $G$", |
| "q1c_natural": "a maximal p-subgroup of a group", |
| "q2_slogan": "maximal p-subgroup", |
| "q3_nickname": "Sylow subgroup", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Gr2", |
| "full_name": "IsPGroup", |
| "file": "Mathlib/GroupTheory/PGroup.lean:26", |
| "difficulty": "Easy", |
| "kind": "def", |
| "q1a_lean": "`∀ g : G, ∃ k, g ^ p^k = 1`", |
| "q1b_latex": "$\\forall g \\in G,\\ \\exists k \\in \\mathbb{N},\\ g^{p^k} = 1$", |
| "q1c_natural": "a group in which every element has order a power of p", |
| "q2_slogan": "every element has p-power order", |
| "q3_nickname": "p-group", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Gr3", |
| "full_name": "Monoid.CoprodI.Word", |
| "file": "Mathlib/GroupTheory/CoprodI.lean:106", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "`toList : List ((i : ι) × M i)` with `ne_one` and `chain_ne` (no 1's, adjacent letters from different summands)", |
| "q1b_latex": "list of nontrivial elements from indexed monoids with consecutive ones from different factors", |
| "q1c_natural": "a list of nontrivial elements where consecutive entries come from different summands", |
| "q2_slogan": "reduced word in a free product", |
| "q3_nickname": "reduced word", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Gr4", |
| "full_name": "IsSolvable", |
| "file": "Mathlib/GroupTheory/Solvable.lean:104", |
| "difficulty": "Hard", |
| "kind": "class", |
| "q1a_lean": "`∃ n : ℕ, derivedSeries G n = ⊥`", |
| "q1b_latex": "$\\exists n \\in \\mathbb{N},\\ G^{(n)} = \\{e\\}$", |
| "q1c_natural": "a group whose derived series eventually reaches the trivial subgroup", |
| "q2_slogan": "derived series eventually trivial", |
| "q3_nickname": "solvable group", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Gr5", |
| "full_name": "Sylow.nonempty", |
| "file": "Mathlib/GroupTheory/Sylow.lean:177", |
| "difficulty": "Easy", |
| "kind": "instance", |
| "q1a_lean": "`Nonempty (Sylow p G)` for finite `G` and prime `p`", |
| "q1b_latex": "$\\mathrm{Sylow}_p(G) \\ne \\emptyset$ for $G$ finite and $p$ prime", |
| "q1c_natural": "every finite group has at least one Sylow p-subgroup for any prime p", |
| "q2_slogan": "Sylow p-subgroups always exist", |
| "q3_nickname": "first Sylow theorem", |
| "q4_special_case": "$S_4$ has a Sylow 2-subgroup of order $8$" |
| }, |
| { |
| "id": "Gr6", |
| "full_name": "MonoidHom.FixedPointFree.commutatorMap_injective", |
| "file": "Mathlib/GroupTheory/FixedPointFree.lean:42", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`(hφ : FixedPointFree φ) : Function.Injective (commutatorMap φ)`", |
| "q1b_latex": "$\\varphi$ FPF $\\Rightarrow$ $g \\mapsto g^{-1} \\varphi(g)$ injective", |
| "q1c_natural": "a fixed-point-free endomorphism gives an injective commutator map", |
| "q2_slogan": "commutator map of a fixed-point-free auto is injective", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Gr7", |
| "full_name": "Monoid.CoprodI.of_apply", |
| "file": "Mathlib/GroupTheory/CoprodI.lean:123", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`of m = (conGen (Rel M)).mk' (FreeMonoid.of ⟨i, m⟩)`", |
| "q1b_latex": "$\\mathrm{of}(m) = \\overline{\\mathrm{FreeMonoid.of}\\,\\langle i, m\\rangle}$", |
| "q1c_natural": "the canonical inclusion into the free product is the equivalence class of the singleton free-monoid word", |
| "q2_slogan": "free product inclusion = singleton free-monoid word modulo relation", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Gr8", |
| "full_name": "Sylow.exists_subgroup_card_pow_prime", |
| "file": "Mathlib/GroupTheory/Sylow.lean", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`p^n ∣ Nat.card G → ∃ K : Subgroup G, Nat.card K = p^n`", |
| "q1b_latex": "$p^n \\mid |G| \\Rightarrow \\exists H \\le G,\\ |H| = p^n$", |
| "q1c_natural": "whenever a prime power divides the group order, a subgroup of that order exists", |
| "q2_slogan": "subgroup of order $p^n$ exists when $p^n$ divides $|G|$", |
| "q3_nickname": "generalized first Sylow theorem", |
| "q4_special_case": "$|G| = 12$ has subgroups of orders 1, 2, 4" |
| }, |
| { |
| "id": "I1", |
| "full_name": "hammingDist", |
| "file": "Mathlib/InformationTheory/Hamming.lean:41", |
| "difficulty": "Easy", |
| "kind": "def", |
| "q1a_lean": "`#{i | x i ≠ y i}`", |
| "q1b_latex": "$d_H(x, y) = \\#\\{i : x_i \\ne y_i\\}$", |
| "q1c_natural": "number of coordinates where two tuples differ", |
| "q2_slogan": "counting disagreeing coordinates", |
| "q3_nickname": "Hamming distance", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "I2", |
| "full_name": "InformationTheory.UniquelyDecodable", |
| "file": "Mathlib/InformationTheory/Coding/UniquelyDecodable.lean:35", |
| "difficulty": "Easy", |
| "kind": "def", |
| "q1a_lean": "`L₁.flatten = L₂.flatten → L₁ = L₂` for `L₁, L₂ : List (List α)` of code-words", |
| "q1b_latex": "$L_1.\\mathrm{flatten} = L_2.\\mathrm{flatten} \\Rightarrow L_1 = L_2$ for code-word lists", |
| "q1c_natural": "a code where two lists of code-words yielding the same concatenation must be equal", |
| "q2_slogan": "distinct code-word lists never share a concatenation", |
| "q3_nickname": "uniquely decodable code", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "I3", |
| "full_name": "InformationTheory.klFun", |
| "file": "Mathlib/InformationTheory/KullbackLeibler/KLFun.lean", |
| "difficulty": "Hard", |
| "kind": "def", |
| "q1a_lean": "`x * Real.log x + 1 - x`", |
| "q1b_latex": "$\\varphi(x) = x \\log x + 1 - x$", |
| "q1c_natural": "convex kernel function whose Bregman divergence gives KL", |
| "q2_slogan": "kernel function for KL divergence", |
| "q3_nickname": "KL kernel function", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "I4", |
| "full_name": "InformationTheory.klDiv", |
| "file": "Mathlib/InformationTheory/KullbackLeibler/Basic.lean:57", |
| "difficulty": "Hard", |
| "kind": "def", |
| "q1a_lean": "`klDiv μ ν : ℝ≥0∞` (irreducible def via Radon–Nikodym + log integral)", |
| "q1b_latex": "$D_{\\mathrm{KL}}(\\mu | \\nu) = \\int \\log \\tfrac{d\\mu}{d\\nu}\\,d\\mu$", |
| "q1c_natural": "relative entropy of one measure with respect to another", |
| "q2_slogan": "relative entropy of two measures", |
| "q3_nickname": "KL divergence (relative entropy)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "I5", |
| "full_name": "hammingDist_triangle", |
| "file": "Mathlib/InformationTheory/Hamming.lean:58", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`hammingDist x z ≤ hammingDist x y + hammingDist y z`", |
| "q1b_latex": "$d_H(x, z) \\le d_H(x, y) + d_H(y, z)$", |
| "q1c_natural": "Hamming distance satisfies the triangle inequality", |
| "q2_slogan": "Hamming distance triangle inequality", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "I6", |
| "full_name": "Hamming.instPseudoMetricSpace", |
| "file": "Mathlib/InformationTheory/Hamming.lean", |
| "difficulty": "Easy", |
| "kind": "instance", |
| "q1a_lean": "`instance : PseudoMetricSpace (Hamming β)`", |
| "q1b_latex": "$\\mathrm{Hamming}\\,\\beta$ is a pseudo-metric space", |
| "q1c_natural": "the Hamming type carries the pseudo-metric structure from the Hamming distance", |
| "q2_slogan": "Hamming type is a pseudo-metric space", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "I7", |
| "full_name": "InformationTheory.kraft_mcmillan_inequality", |
| "file": "Mathlib/InformationTheory/Coding/KraftMcMillan.lean:147", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`UniquelyDecodable S → ∑ w ∈ S, (1/D)^w.length ≤ 1` where `D = |alphabet|`", |
| "q1b_latex": "$\\sum_{w \\in S} D^{-|w|} \\le 1$ for $S$ uniquely decodable, alphabet size $D$", |
| "q1c_natural": "for any uniquely decodable code, the sum of $D^{-|w|}$ over codewords is at most 1", |
| "q2_slogan": "codeword lengths satisfy Kraft's inequality", |
| "q3_nickname": "Kraft–McMillan inequality", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "I8", |
| "full_name": "InformationTheory.klDiv_eq_zero_iff", |
| "file": "Mathlib/InformationTheory/KullbackLeibler/Basic.lean", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`[IsFiniteMeasure μ] [IsFiniteMeasure ν] : klDiv μ ν = 0 ↔ μ = ν`", |
| "q1b_latex": "$D_{\\mathrm{KL}}(\\mu | \\nu) = 0 \\iff \\mu = \\nu$", |
| "q1c_natural": "KL divergence between two finite measures vanishes iff the measures coincide", |
| "q2_slogan": "KL divergence is zero iff measures agree", |
| "q3_nickname": "converse Gibbs inequality", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "L1", |
| "full_name": "Module.Basis", |
| "file": "Mathlib/LinearAlgebra/Basis/Defs.lean:88", |
| "difficulty": "Easy", |
| "kind": "structure", |
| "q1a_lean": "`M ≃ₗ[R] (ι →₀ R)`", |
| "q1b_latex": "$M \\cong_R \\bigoplus_\\iota R$ via a chosen isomorphism", |
| "q1c_natural": "a linear isomorphism between the module and a free module on an index set", |
| "q2_slogan": "a linear iso with a free module on an index set", |
| "q3_nickname": "basis of a module", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "L2", |
| "full_name": "Module.End.genEigenspace", |
| "file": "Mathlib/LinearAlgebra/Eigenspace/Basic.lean:73", |
| "difficulty": "Easy", |
| "kind": "def", |
| "q1a_lean": "`ℕ∞ →o Submodule R M`, value at `k` is `ker ((f - μ) ^ k)`", |
| "q1b_latex": "$\\ker((f - \\mu \\cdot \\mathrm{id})^k)$ as an increasing chain in $k$", |
| "q1c_natural": "the increasing chain of kernels of powers of $f - \\mu$", |
| "q2_slogan": "increasing chain of generalized eigenspace kernels", |
| "q3_nickname": "generalized eigenspace", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "L3", |
| "full_name": "CliffordAlgebra.Rel", |
| "file": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean:62", |
| "difficulty": "Hard", |
| "kind": "inductive", |
| "q1a_lean": "`inductive Rel ... | of (m : M) : Rel ((ι R m) * (ι R m)) (algebraMap R _ (Q m))`", |
| "q1b_latex": "relation imposing $\\iota(m) \\cdot \\iota(m) = Q(m)$ in the tensor algebra", |
| "q1c_natural": "the relation on the tensor algebra that imposes $v \\cdot v = Q(v) \\cdot 1$", |
| "q2_slogan": "relation defining Clifford algebra from tensor algebra", |
| "q3_nickname": "Clifford relation", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "L4", |
| "full_name": "RootPairing", |
| "file": "Mathlib/LinearAlgebra/RootSystem/Defs.lean", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "`extends M →ₗ[R] N →ₗ[R] R; root, coroot, reflectionPerm` with axioms", |
| "q1b_latex": "perfect pairing $M \\otimes N \\to R$ + indexed roots/coroots + reflection permutations", |
| "q1c_natural": "a perfect pairing on dual modules with families of roots, coroots, and reflection permutations satisfying compatibility", |
| "q2_slogan": "data for a root system: roots, coroots, perfect pairing, reflections", |
| "q3_nickname": "root pairing", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "L5", |
| "full_name": "Module.rank_lt_aleph0", |
| "file": "Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean:490", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`[Module.Finite R M] : Module.rank R M < Cardinal.aleph0`", |
| "q1b_latex": "$M$ finitely generated over $R \\Rightarrow \\mathrm{rank}(M) < \\aleph_0$", |
| "q1c_natural": "a finitely generated module has rank strictly less than ℵ₀", |
| "q2_slogan": "finitely generated module has finite rank", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "L6", |
| "full_name": "Matrix.det_comm", |
| "file": "Mathlib/LinearAlgebra/Determinant.lean:83", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`(M * N).det = (N * M).det`", |
| "q1b_latex": "$\\det(MN) = \\det(NM)$", |
| "q1c_natural": "the determinant is symmetric in the two factors of a product", |
| "q2_slogan": "determinant of a product is symmetric", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "L7", |
| "full_name": "Module.Free.of_det_ne_one", |
| "file": "Mathlib/LinearAlgebra/Determinant.lean:188", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`(hf : LinearMap.det f ≠ 1) : Module.Free R M`", |
| "q1b_latex": "$\\det(f) \\ne 1$ for some endomorphism $f \\Rightarrow$ $R$-module $M$ is free", |
| "q1c_natural": "existence of an endomorphism with non-unit determinant forces the module to be free", |
| "q2_slogan": "nontrivial determinant forces freeness", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "L8", |
| "full_name": "instAlgebraCliffordAlgebra", |
| "file": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "`instance : Algebra R (CliffordAlgebra Q)`", |
| "q1b_latex": "$\\mathrm{CliffordAlgebra}\\,Q$ is an $R$-algebra", |
| "q1c_natural": "the Clifford algebra of a quadratic form carries an R-algebra structure", |
| "q2_slogan": "Clifford algebra is an R-algebra", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Lo1", |
| "full_name": "Equiv", |
| "file": "Mathlib/Logic/Equiv/Defs.lean:69", |
| "difficulty": "Easy", |
| "kind": "structure", |
| "q1a_lean": "`α ≃ β` with `toFun, invFun, left_inv, right_inv`", |
| "q1b_latex": "$\\alpha \\cong \\beta$ with two-sided inverse", |
| "q1c_natural": "a function with a two-sided inverse", |
| "q2_slogan": "bijection bundled with its inverse", |
| "q3_nickname": "isomorphism", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Lo2", |
| "full_name": "Denumerable", |
| "file": "Mathlib/Logic/Denumerable.lean", |
| "difficulty": "Easy", |
| "kind": "class", |
| "q1a_lean": "`class Denumerable extends Encodable; decode_inv (n) : ∃ a ∈ decode n, encode a = n`", |
| "q1b_latex": "a constructive bijection $\\alpha \\cong \\mathbb{N}$", |
| "q1c_natural": "a type with a constructive bijection to the natural numbers", |
| "q2_slogan": "countably infinite type", |
| "q3_nickname": "denumerable (countably infinite type)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Lo3", |
| "full_name": "Encodable", |
| "file": "Mathlib/Logic/Encodable/Basic.lean:52", |
| "difficulty": "Hard", |
| "kind": "class", |
| "q1a_lean": "`encode : α → ℕ`, `decode : ℕ → Option α`, `encodek (a) : decode (encode a) = some a`", |
| "q1b_latex": "$\\mathrm{encode} : \\alpha \\to \\mathbb{N}$, $\\mathrm{decode} : \\mathbb{N} \\to \\mathrm{Option}\\,\\alpha$, with $\\mathrm{decode}(\\mathrm{encode}(a)) = \\mathrm{some}\\,a$", |
| "q1c_natural": "a type with computable encode/decode functions to natural numbers and a round-trip property", |
| "q2_slogan": "type with computable encode/decode to ℕ", |
| "q3_nickname": "encodable (constructively countable)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Lo4", |
| "full_name": "Relation.CutExpand", |
| "file": "Mathlib/Logic/Hydra.lean:58", |
| "difficulty": "Hard", |
| "kind": "def", |
| "q1a_lean": "`∃ t a, (∀ a' ∈ t, r a' a) ∧ s' + {a} = s + t`", |
| "q1b_latex": "$s\\' + \\{a\\} = s + t$ where $\\forall a\\' \\in t,\\ r(a\\', a)$", |
| "q1c_natural": "rewriting move on multisets that cuts one element and adds smaller ones", |
| "q2_slogan": "cut-and-expand move on multisets", |
| "q3_nickname": "Hydra game move (Kirby–Paris)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Lo5", |
| "full_name": "Pairwise.mono", |
| "file": "Mathlib/Logic/Pairwise.lean", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`Pairwise r → (∀ i j, r i j → p i j) → Pairwise p`", |
| "q1b_latex": "$\\mathrm{Pairwise}\\,r \\wedge r \\le p \\Rightarrow \\mathrm{Pairwise}\\,p$", |
| "q1c_natural": "if a relation holds pairwise and implies another, the implied one also holds pairwise", |
| "q2_slogan": "Pairwise is monotone in the relation", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Lo6", |
| "full_name": "Function.cantor_surjective", |
| "file": "Mathlib/Logic/Function/Basic.lean", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`¬ Surjective (f : α → Set α)`", |
| "q1b_latex": "$\\nexists\\, f : \\alpha \\twoheadrightarrow \\mathcal{P}(\\alpha)$", |
| "q1c_natural": "no function from a type to its powerset can be surjective", |
| "q2_slogan": "a type cannot surject onto its power set", |
| "q3_nickname": "Cantor's theorem", |
| "q4_special_case": "no surjection $\\mathbb{N} \\to \\mathcal{P}(\\mathbb{N})$" |
| }, |
| { |
| "id": "Lo7", |
| "full_name": "Encodable.encode_injective", |
| "file": "Mathlib/Logic/Encodable/Basic.lean", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`Function.Injective (encode : α → ℕ)`", |
| "q1b_latex": "$\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ is injective", |
| "q1c_natural": "the encoding function of an encodable type is injective", |
| "q2_slogan": "encode is injective", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "Lo8", |
| "full_name": "Denumerable.sigma", |
| "file": "Mathlib/Logic/Denumerable.lean", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "`[Denumerable α] [∀ a, Denumerable (β a)] : Denumerable (Σ a, β a)`", |
| "q1b_latex": "Sigma type over a denumerable index with denumerable fibers is denumerable", |
| "q1c_natural": "a dependent pair type over a denumerable index with denumerable fibers is denumerable", |
| "q2_slogan": "Sigma preserves denumerability", |
| "q3_nickname": "", |
| "q4_special_case": "$\\Sigma\\,n : \\mathbb{N},\\,\\mathrm{Fin}\\,n$ is denumerable" |
| }, |
| { |
| "id": "M1", |
| "full_name": "MeasureTheory.OuterMeasure", |
| "file": "Mathlib/MeasureTheory/OuterMeasure/Defs.lean:51", |
| "difficulty": "Easy", |
| "kind": "structure", |
| "q1a_lean": "`measureOf : Set α → ℝ≥0∞`, `empty: measureOf ∅ = 0`, `mono`, `iUnion_nat` (countable subadditivity)", |
| "q1b_latex": "$\\mu : \\mathcal{P}(\\alpha) \\to \\overline{\\mathbb{R}}_{\\ge 0}$ with $\\mu(\\emptyset) = 0$, monotone, countably subadditive", |
| "q1c_natural": "a monotone, countably subadditive set function vanishing on the empty set", |
| "q2_slogan": "monotone countably-subadditive set function with $\\mu(\\emptyset) = 0$", |
| "q3_nickname": "outer measure", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "M2", |
| "full_name": "IsPiSystem", |
| "file": "Mathlib/MeasureTheory/PiSystem.lean:72", |
| "difficulty": "Easy", |
| "kind": "def", |
| "q1a_lean": "`∀ s ∈ C, ∀ t ∈ C, (s ∩ t).Nonempty → s ∩ t ∈ C`", |
| "q1b_latex": "$s, t \\in \\mathcal{C},\\ s \\cap t \\ne \\emptyset \\Rightarrow s \\cap t \\in \\mathcal{C}$", |
| "q1c_natural": "a family of sets closed under non-empty pairwise intersection", |
| "q2_slogan": "closed under non-empty pairwise intersection", |
| "q3_nickname": "π-system", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "M3", |
| "full_name": "MeasureTheory.VectorMeasure", |
| "file": "Mathlib/MeasureTheory/VectorMeasure/Basic.lean:61", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "`measureOf' : Set α → M`, σ-additive on countable disjoint families", |
| "q1b_latex": "$\\mu\\bigl(\\bigsqcup_n A_n\\bigr) = \\sum_n \\mu(A_n)$, $\\mu : \\Sigma \\to M$", |
| "q1c_natural": "a σ-additive set function valued in a topological additive monoid", |
| "q2_slogan": "σ-additive set function in a topological group", |
| "q3_nickname": "vector measure", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "M4", |
| "full_name": "MeasureTheory.IsSetSemiring", |
| "file": "Mathlib/MeasureTheory/SetSemiring.lean:65", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "`∅ ∈ C`, closed under intersection, differences split into finite disjoint unions in C", |
| "q1b_latex": "family $\\mathcal{C}$ with $\\emptyset \\in \\mathcal{C}$, $\\mathcal{C}$ closed under intersection, and $s \\setminus t = \\bigsqcup_i I_i$ with $I_i \\in \\mathcal{C}$", |
| "q1c_natural": "a family of sets closed under intersection where set differences split as finite disjoint unions", |
| "q2_slogan": "family closed under intersection with finite-disjoint difference splitting", |
| "q3_nickname": "set semiring", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "M5", |
| "full_name": "MeasureTheory.lintegral_const", |
| "file": "Mathlib/MeasureTheory/Integral/Lebesgue/Basic.lean", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`∫⁻ x, c ∂μ = c * μ Set.univ`", |
| "q1b_latex": "$\\int^- c\\,d\\mu = c \\cdot \\mu(\\Omega)$", |
| "q1c_natural": "integrating a constant gives constant times total mass", |
| "q2_slogan": "integral of a constant", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "M6", |
| "full_name": "MeasureTheory.MemLp.aestronglyMeasurable", |
| "file": "Mathlib/MeasureTheory/Function/LpSeminorm/Defs.lean:119", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`MemLp f p μ → AEStronglyMeasurable f μ`", |
| "q1b_latex": "$f \\in L^p(\\mu) \\Rightarrow f$ is a.e. strongly measurable", |
| "q1c_natural": "a function in Lp is a.e. strongly measurable", |
| "q2_slogan": "Lp membership entails a.e. strong measurability", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "M7", |
| "full_name": "MeasureTheory.tendsto_lintegral_of_dominated_convergence", |
| "file": "Mathlib/MeasureTheory/Integral/Lebesgue/DominatedConvergence.lean:46", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`F n ≤ᵐ bound`, `bound` integrable, `F n → f` a.e. ⇒ `∫⁻ F n → ∫⁻ f`", |
| "q1b_latex": "$|F_n| \\le g$ integrable, $F_n \\to f$ a.e. $\\Rightarrow \\int F_n \\to \\int f$", |
| "q1c_natural": "a sequence dominated by an integrable function and converging a.e. has its integrals converging to the integral of the limit", |
| "q2_slogan": "bounded a.e. convergence transfers to integrals", |
| "q3_nickname": "dominated convergence theorem (DCT)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "M8", |
| "full_name": "MeasureTheory.Measure.instOuterMeasureClass", |
| "file": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean:95", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "`instance : OuterMeasureClass (Measure α) α`", |
| "q1b_latex": "$\\mathrm{Measure}\\,\\alpha$ satisfies the OuterMeasureClass interface", |
| "q1c_natural": "a measure can be regarded as an outer measure via the natural class", |
| "q2_slogan": "Measure carries OuterMeasureClass", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "MT1", |
| "full_name": "FirstOrder.Language", |
| "file": "Mathlib/ModelTheory/Basic.lean:58", |
| "difficulty": "Easy", |
| "kind": "structure", |
| "q1a_lean": "`Functions : ℕ → Type u`, `Relations : ℕ → Type v`", |
| "q1b_latex": "$\\mathcal{L} = (\\{\\mathrm{Func}_n\\}_n, \\{\\mathrm{Rel}_n\\}_n)$", |
| "q1c_natural": "a signature giving function and relation symbols of every arity", |
| "q2_slogan": "signature of function and relation symbols by arity", |
| "q3_nickname": "first-order language (signature)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "MT2", |
| "full_name": "FirstOrder.Language.Structure", |
| "file": "Mathlib/ModelTheory/Basic.lean:159", |
| "difficulty": "Easy", |
| "kind": "class", |
| "q1a_lean": "`funMap : L.Functions n → (Fin n → M) → M`, `RelMap : L.Relations n → (Fin n → M) → Prop`", |
| "q1b_latex": "type with interpretations of all $\\mathcal{L}$-function and relation symbols", |
| "q1c_natural": "a type equipped with interpretations of all symbols in a language", |
| "q2_slogan": "interpretation of L-symbols on a type", |
| "q3_nickname": "L-structure", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "MT3", |
| "full_name": "FirstOrder.Language.ElementaryEmbedding", |
| "file": "Mathlib/ModelTheory/ElementaryMaps.lean:45", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "`toFun : M → N` with `map_formula' (φ) (x) : φ.Realize (f ∘ x) ↔ φ.Realize x`", |
| "q1b_latex": "$f : M \\to N$ such that $M \\models \\varphi(\\bar x) \\iff N \\models \\varphi(f \\bar x)$ for every formula $\\varphi$", |
| "q1c_natural": "a structure-embedding preserving truth of all first-order formulas", |
| "q2_slogan": "embedding preserving truth of every formula", |
| "q3_nickname": "elementary embedding", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "MT4", |
| "full_name": "FirstOrder.Language.IsFraisse", |
| "file": "Mathlib/ModelTheory/Fraisse.lean:118", |
| "difficulty": "Hard", |
| "kind": "class", |
| "q1a_lean": "class with `is_nonempty`, `FG`, `is_essentially_countable`, `hereditary`, `jointEmbedding`, `amalgamation`", |
| "q1b_latex": "class of finitely generated structures satisfying the 6 Fraïssé conditions", |
| "q1c_natural": "a class of finitely generated structures satisfying nonempty, FG, essentially countable, hereditary, JEP, AP", |
| "q2_slogan": "class with the Fraïssé properties", |
| "q3_nickname": "Fraïssé class", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "MT5", |
| "full_name": "FirstOrder.Language.Substructure.ext", |
| "file": "Mathlib/ModelTheory/Substructures.lean:124", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`(∀ x, x ∈ S ↔ x ∈ T) → S = T`", |
| "q1b_latex": "$(\\forall x,\\,x \\in S \\iff x \\in T) \\Rightarrow S = T$", |
| "q1c_natural": "two substructures with the same elements are equal", |
| "q2_slogan": "substructure extensionality", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "MT6", |
| "full_name": "FirstOrder.Language.ClosedUnder.inter", |
| "file": "Mathlib/ModelTheory/Substructures.lean:78", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`ClosedUnder f s → ClosedUnder f t → ClosedUnder f (s ∩ t)`", |
| "q1b_latex": "$\\mathrm{ClosedUnder}\\,f\\,s \\wedge \\mathrm{ClosedUnder}\\,f\\,t \\Rightarrow \\mathrm{ClosedUnder}\\,f\\,(s \\cap t)$", |
| "q1c_natural": "the intersection of two sets closed under a function operation is itself closed", |
| "q2_slogan": "closure under f preserved by intersection", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "MT7", |
| "full_name": "FirstOrder.Language.exists_elementaryEmbedding_card_eq_of_le", |
| "file": "Mathlib/ModelTheory/Satisfiability.lean:199", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`ℵ₀ ≤ κ ≤ #M`, `L.card ≤ κ` `→` `∃ N : Bundled L.Structure, N ↪ₑ M ∧ #N = κ`", |
| "q1b_latex": "$\\aleph_0 \\le \\kappa \\le |M|$, $|\\mathcal{L}| \\le \\kappa \\Rightarrow \\exists N \\hookrightarrow_e M,\\ |N| = \\kappa$", |
| "q1c_natural": "for any cardinal between countable and the size of M, there is an elementary substructure of that cardinality", |
| "q2_slogan": "every infinite structure has elementary substructures of any prescribed smaller infinite cardinality", |
| "q3_nickname": "downward Löwenheim–Skolem", |
| "q4_special_case": "every uncountable model has a countable elementary substructure" |
| }, |
| { |
| "id": "MT8", |
| "full_name": "FirstOrder.Language.Ultraproduct.structure", |
| "file": "Mathlib/ModelTheory/Ultraproducts.lean", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "`instance : L.Structure ((↑u).Product M)`", |
| "q1b_latex": "ultraproduct of $\\mathcal{L}$-structures along $u$ inherits an $\\mathcal{L}$-structure", |
| "q1c_natural": "the ultraproduct of L-structures along an ultrafilter is itself an L-structure", |
| "q2_slogan": "ultraproduct inherits L-structure", |
| "q3_nickname": "structure-side of Łoś's theorem", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "N1", |
| "full_name": "ArithmeticFunction.moebius", |
| "file": "Mathlib/NumberTheory/ArithmeticFunction/Moebius.lean:51", |
| "difficulty": "Easy", |
| "kind": "def", |
| "q1a_lean": "`if Squarefree n then (-1) ^ cardFactors n else 0`", |
| "q1b_latex": "$\\mu(n) = (-1)^{\\omega(n)}$ if $n$ squarefree, else $0$", |
| "q1c_natural": "the arithmetic function used in Möbius inversion", |
| "q2_slogan": "signed-by-squarefree-prime-count function", |
| "q3_nickname": "Möbius μ function", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "N2", |
| "full_name": "Nat.divisors", |
| "file": "Mathlib/NumberTheory/Divisors.lean:51", |
| "difficulty": "Easy", |
| "kind": "def", |
| "q1a_lean": "`{d ∈ Ico 1 (n+1) | d ∣ n}`", |
| "q1b_latex": "$\\{d \\in [1, n+1) : d \\mid n\\}$", |
| "q1c_natural": "the finset of positive divisors of n", |
| "q2_slogan": "positive divisors of a natural number", |
| "q3_nickname": "divisors of n", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "N3", |
| "full_name": "IsCyclotomicExtension", |
| "file": "Mathlib/NumberTheory/Cyclotomic/Basic.lean:76", |
| "difficulty": "Hard", |
| "kind": "class", |
| "q1a_lean": "$B = A\\bigl(\\zeta_n : n \\in S\\bigr)$ generated by primitive n-th roots of unity", |
| "q1b_latex": "a ring extension generated by primitive n-th roots of unity for each n in a set", |
| "q1c_natural": "extension adjoining primitive roots of unity for indices in S", |
| "q2_slogan": "cyclotomic extension", |
| "q3_nickname": "$\\mathbb{Q}(\\zeta_n)/\\mathbb{Q}$", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "N4", |
| "full_name": "LSeries", |
| "file": "Mathlib/NumberTheory/LSeries/Basic.lean", |
| "difficulty": "Hard", |
| "kind": "def", |
| "q1a_lean": "`∑' n, LSeries.term f s n`", |
| "q1b_latex": "$L(f, s) = \\sum_{n \\ge 1} \\frac{f(n)}{n^s}$", |
| "q1c_natural": "the Dirichlet series of f at s (returning 0 when not absolutely convergent)", |
| "q2_slogan": "Dirichlet series of an arithmetic function", |
| "q3_nickname": "L-series (Dirichlet series)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "N5", |
| "full_name": "Nat.sum_divisors_eq_sum_properDivisors_add_self", |
| "file": "Mathlib/NumberTheory/Divisors.lean", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`∑ i ∈ n.divisors, i = ∑ i ∈ n.properDivisors, i + n`", |
| "q1b_latex": "$\\sum_{d \\mid n} d = \\sum_{d \\mid n,\\ d < n} d + n$", |
| "q1c_natural": "the sum of all divisors equals the sum of proper divisors plus n", |
| "q2_slogan": "sum of all divisors = sum of proper divisors + n", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "N6", |
| "full_name": "bernoulli'_zero", |
| "file": "Mathlib/NumberTheory/Bernoulli.lean", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`bernoulli' 0 = 1`", |
| "q1b_latex": "$B'_0 = 1$", |
| "q1c_natural": "the zeroth Bernoulli number is 1", |
| "q2_slogan": "$B'_0 = 1$", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "N7", |
| "full_name": "EulerProduct.eulerProduct_hasProd", |
| "file": "Mathlib/NumberTheory/EulerProduct/Basic.lean:174", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`f` multiplicative, `Summable ‖f ·‖`, `f 0 = 0` ⇒ `HasProd (fun p prime ↦ ∑' e, f (p^e)) (∑' n, f n)`", |
| "q1b_latex": "$f$ multiplicative, $\\sum |f(n)| < \\infty$, $f(0) = 0 \\Rightarrow \\prod_p \\sum_{e \\ge 0} f(p^e) = \\sum_n f(n)$", |
| "q1c_natural": "for an absolutely convergent multiplicative function, its sum equals the Euler product over primes", |
| "q2_slogan": "multiplicative absolutely-convergent series equals its Euler product", |
| "q3_nickname": "Euler product formula", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "N8", |
| "full_name": "NumberField.RingOfIntegers.instIsDedekindDomain", |
| "file": "Mathlib/NumberTheory/NumberField/Basic.lean", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "`[NumberField K] : IsDedekindDomain (RingOfIntegers K)`", |
| "q1b_latex": "$K$ a number field $\\Rightarrow$ $\\mathcal{O}_K$ is a Dedekind domain", |
| "q1c_natural": "the ring of integers of a number field is a Dedekind domain", |
| "q2_slogan": "ring of integers of a number field is Dedekind", |
| "q3_nickname": "", |
| "q4_special_case": "$\\mathbb{Z}[i]$ is Dedekind" |
| }, |
| { |
| "id": "P1", |
| "full_name": "PMF", |
| "file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean:46", |
| "difficulty": "Easy", |
| "kind": "def", |
| "q1a_lean": "`{f : α → ℝ≥0∞ // HasSum f 1}`", |
| "q1b_latex": "$\\{f : \\alpha \\to \\overline{\\mathbb{R}}_{\\ge 0} : \\sum_a f(a) = 1\\}$", |
| "q1c_natural": "a nonneg-extended-real function on a type whose values sum to 1", |
| "q2_slogan": "discrete probability distribution", |
| "q3_nickname": "PMF (probability mass function)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "P2", |
| "full_name": "ProbabilityTheory.cond", |
| "file": "Mathlib/Probability/ConditionalProbability.lean:74", |
| "difficulty": "Easy", |
| "kind": "def", |
| "q1a_lean": "`(μ s)⁻¹ • μ.restrict s`", |
| "q1b_latex": "$\\mu[\\cdot \\mid s] = \\frac{1}{\\mu(s)}\\,\\mu(\\cdot \\cap s)$", |
| "q1c_natural": "the measure restricted to a set and rescaled to total mass 1", |
| "q2_slogan": "conditional measure = restrict and normalize", |
| "q3_nickname": "conditional probability measure", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "P3", |
| "full_name": "ProbabilityTheory.Kernel", |
| "file": "Mathlib/Probability/Kernel/Defs.lean:55", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "`toFun : α → Measure β` with `measurable' : Measurable toFun`", |
| "q1b_latex": "a measurable family $\\kappa : \\alpha \\to \\mathrm{Measure}\\,\\beta$", |
| "q1c_natural": "a measurable family of measures parameterized by a base space", |
| "q2_slogan": "measurable family of measures", |
| "q3_nickname": "Markov kernel (transition kernel)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "P4", |
| "full_name": "MeasureTheory.Martingale", |
| "file": "Mathlib/Probability/Martingale/Basic.lean:53", |
| "difficulty": "Hard", |
| "kind": "def", |
| "q1a_lean": "`StronglyAdapted ℱ f ∧ ∀ i ≤ j, μ[f j | ℱ i] =ᵐ f i`", |
| "q1b_latex": "an adapted process $X$ such that $\\mathbb{E}[X_{n+1} \\mid \\mathcal{F}_n] = X_n$", |
| "q1c_natural": "an adapted process whose conditional expectation given the past equals the present", |
| "q2_slogan": "fair-game stochastic process", |
| "q3_nickname": "martingale", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "P5", |
| "full_name": "ProbabilityTheory.cond_inter_self", |
| "file": "Mathlib/Probability/ConditionalProbability.lean:224", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`μ[s ∩ t | s] = μ[t | s]`", |
| "q1b_latex": "$\\mu[s \\cap t \\mid s] = \\mu[t \\mid s]$", |
| "q1c_natural": "conditioning on s absorbs intersection with s", |
| "q2_slogan": "conditioning absorbs intersection with the conditioning set", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "P6", |
| "full_name": "ProbabilityTheory.cond_isProbabilityMeasure", |
| "file": "Mathlib/Probability/ConditionalProbability.lean:162", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`[IsFiniteMeasure μ] (hcs : μ s ≠ 0) : IsProbabilityMeasure μ[·|s]`", |
| "q1b_latex": "$\\mu$ finite, $\\mu(s) \\ne 0 \\Rightarrow \\mu[\\cdot \\mid s]$ is a probability measure", |
| "q1c_natural": "conditioning a finite measure on a positive-measure set gives a probability measure", |
| "q2_slogan": "conditioning a finite measure yields a probability measure", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "P7", |
| "full_name": "ProbabilityTheory.cond_cond_eq_cond_inter", |
| "file": "Mathlib/Probability/ConditionalProbability.lean:252", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`(μ[·|s])[·|t] = μ[·|s ∩ t]`", |
| "q1b_latex": "$(\\mu[\\cdot \\mid s])[\\cdot \\mid t] = \\mu[\\cdot \\mid s \\cap t]$", |
| "q1c_natural": "conditioning first on s and then on t equals conditioning on the intersection", |
| "q2_slogan": "iterated conditioning = conditioning on intersection", |
| "q3_nickname": "chain rule for conditional probability", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "P8", |
| "full_name": "ProbabilityTheory.PMF.instFunLike", |
| "file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean:51", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "`instance : FunLike (PMF α) α ℝ≥0∞`", |
| "q1b_latex": "a PMF is callable as a function $\\alpha \\to \\overline{\\mathbb{R}}_{\\ge 0}$", |
| "q1c_natural": "a PMF can be coerced to a function returning probability values", |
| "q2_slogan": "PMF callable as a function returning probabilities", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "R1", |
| "full_name": "FDRep.character", |
| "file": "Mathlib/RepresentationTheory/Character.lean:54", |
| "difficulty": "Easy", |
| "kind": "def", |
| "q1a_lean": "`LinearMap.trace k V (V.ρ g)`", |
| "q1b_latex": "$\\chi_V(g) = \\mathrm{tr}(\\rho(g))$", |
| "q1c_natural": "trace of the representation matrix at a group element", |
| "q2_slogan": "trace of a representation matrix", |
| "q3_nickname": "character of a representation", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "R2", |
| "full_name": "Subrepresentation", |
| "file": "Mathlib/RepresentationTheory/Subrepresentation.lean:30", |
| "difficulty": "Easy", |
| "kind": "structure", |
| "q1a_lean": "`toSubmodule : Submodule A W` with `apply_mem_toSubmodule : ∀ g v, v ∈ toSubmodule → ρ g v ∈ toSubmodule`", |
| "q1b_latex": "a submodule $W \\le V$ with $\\rho(g)(W) \\subseteq W$ for all $g$", |
| "q1c_natural": "a submodule that is invariant under the group action", |
| "q2_slogan": "invariant submodule under the action", |
| "q3_nickname": "subrepresentation (invariant subspace)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "R3", |
| "full_name": "Representation.IntertwiningMap", |
| "file": "Mathlib/RepresentationTheory/Intertwining.lean:36", |
| "difficulty": "Hard", |
| "kind": "structure", |
| "q1a_lean": "`extends V →ₗ[A] W` with `isIntertwining' : ∀ g, toLinearMap ∘ ρ g = σ g ∘ toLinearMap`", |
| "q1b_latex": "a linear map $f : V \\to W$ with $f \\circ \\rho(g) = \\sigma(g) \\circ f$", |
| "q1c_natural": "a linear map between representations that commutes with the group action", |
| "q2_slogan": "G-equivariant linear map between representations", |
| "q3_nickname": "intertwining map (G-equivariant map)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "R4", |
| "full_name": "LinearMap.equivariantProjection", |
| "file": "Mathlib/RepresentationTheory/Maschke.lean:114", |
| "difficulty": "Hard", |
| "kind": "def", |
| "q1a_lean": "`Ring.inverse (Fintype.card G : k) • ∑ g, ρ(g⁻¹) ∘ π ∘ ρ(g)`", |
| "q1b_latex": "$\\frac{1}{|G|} \\sum_{g \\in G} \\rho(g^{-1}) \\circ \\pi \\circ \\rho(g)$", |
| "q1c_natural": "averaged G-equivariant retraction obtained from a linear projection by group averaging", |
| "q2_slogan": "averaged equivariant retraction (Maschke averaging)", |
| "q3_nickname": "Maschke averaging", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "R5", |
| "full_name": "FDRep.char_one", |
| "file": "Mathlib/RepresentationTheory/Character.lean", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`V.character 1 = finrank k V`", |
| "q1b_latex": "$\\chi_V(1) = \\dim V$", |
| "q1c_natural": "evaluating a character at the identity element gives the dimension", |
| "q2_slogan": "$\\chi(1) = \\dim V$", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "R6", |
| "full_name": "Representation.invariants_eq_inter", |
| "file": "Mathlib/RepresentationTheory/Invariants.lean", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`ρ.invariants.carrier = ⋂ g, Function.fixedPoints (ρ g)`", |
| "q1b_latex": "$V^G = \\bigcap_g \\mathrm{Fix}(\\rho(g))$", |
| "q1c_natural": "the G-invariants equal the intersection of fixed-point sets over each group element", |
| "q2_slogan": "invariants = intersection of pointwise fixed sets", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "R7", |
| "full_name": "MonoidAlgebra.Submodule.exists_isCompl", |
| "file": "Mathlib/RepresentationTheory/Maschke.lean", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "every `Submodule (k[G]) V` has a `k[G]`-complement when `|G|` is invertible in `k`", |
| "q1b_latex": "every $k[G]$-submodule $W \\le V$ admits an equivariant complement, i.e. $\\exists W'\\!,\\ V = W \\oplus W'$", |
| "q1c_natural": "every subrepresentation of a finite-group representation over a field of suitable characteristic admits an equivariant complement", |
| "q2_slogan": "subrepresentations of finite-group reps are direct summands", |
| "q3_nickname": "Maschke's theorem", |
| "q4_special_case": "every complex representation of a finite group decomposes as a direct sum of irreducibles" |
| }, |
| { |
| "id": "R8", |
| "full_name": "Representation.IntertwiningMap.instLinearMapClass", |
| "file": "Mathlib/RepresentationTheory/Intertwining.lean", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "`instance : LinearMapClass (IntertwiningMap A ρ σ) A V W`", |
| "q1b_latex": "$\\mathrm{IntertwiningMap}\\,A\\,\\rho\\,\\sigma$ belongs to $\\mathrm{LinearMapClass}$", |
| "q1c_natural": "intertwining maps form a class of bundled linear maps", |
| "q2_slogan": "intertwining maps fit the LinearMapClass abstraction", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "RT1", |
| "full_name": "IsNoetherian", |
| "file": "Mathlib/RingTheory/Noetherian/Defs.lean:62", |
| "difficulty": "Easy", |
| "kind": "class", |
| "q1a_lean": "`∀ s : Submodule R M, s.FG`", |
| "q1b_latex": "every submodule $N \\le M$ is finitely generated", |
| "q1c_natural": "a module in which every submodule is finitely generated", |
| "q2_slogan": "every submodule is finitely generated", |
| "q3_nickname": "Noetherian module", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "RT2", |
| "full_name": "Ring.DimensionLEOne", |
| "file": "Mathlib/RingTheory/DedekindDomain/Basic.lean:56", |
| "difficulty": "Easy", |
| "kind": "class", |
| "q1a_lean": "`∀ p : Ideal R, p ≠ ⊥ → p.IsPrime → p.IsMaximal`", |
| "q1b_latex": "$\\forall \\mathfrak{p}$ prime, $\\mathfrak{p} \\ne 0 \\Rightarrow \\mathfrak{p}$ maximal", |
| "q1c_natural": "a ring whose every nonzero prime ideal is maximal", |
| "q2_slogan": "nonzero primes are maximal", |
| "q3_nickname": "Krull dimension ≤ 1", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "RT3", |
| "full_name": "Algebra.FormallyEtale", |
| "file": "Mathlib/RingTheory/Etale/Basic.lean:47", |
| "difficulty": "Hard", |
| "kind": "class", |
| "q1a_lean": "`Subsingleton Ω[A⁄R]` and `Subsingleton (H1Cotangent R A)`", |
| "q1b_latex": "$\\Omega^1_{A/R} = 0$ and $H^1(L_{A/R}) = 0$", |
| "q1c_natural": "an R-algebra with vanishing Kähler differentials and first cotangent cohomology", |
| "q2_slogan": "vanishing differentials and obstructions", |
| "q3_nickname": "formally étale", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "RT4", |
| "full_name": "Coalgebra", |
| "file": "Mathlib/RingTheory/Coalgebra/Basic.lean:81", |
| "difficulty": "Hard", |
| "kind": "class", |
| "q1a_lean": "`extends CoalgebraStruct` with `coassoc`, `rTensor_counit_comp_comul`, `lTensor_counit_comp_comul`", |
| "q1b_latex": "comultiplication $\\Delta : A \\to A \\otimes A$ + counit $\\varepsilon : A \\to R$ with coassociativity and counitality", |
| "q1c_natural": "a module equipped with comultiplication and counit dual to the algebra axioms", |
| "q2_slogan": "dual of an algebra: comultiplication + counit", |
| "q3_nickname": "coalgebra", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "RT5", |
| "full_name": "Ideal.mul_mem_left", |
| "file": "Mathlib/RingTheory/Ideal/Defs.lean:61", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`b ∈ I → a * b ∈ I`", |
| "q1b_latex": "$b \\in I \\Rightarrow a \\cdot b \\in I$", |
| "q1c_natural": "an ideal absorbs left multiplication by ring elements", |
| "q2_slogan": "ideals absorb left multiplication", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "RT6", |
| "full_name": "Ideal.IsPrime.isMaximal", |
| "file": "Mathlib/RingTheory/DedekindDomain/Basic.lean:61", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`[Ring.DimensionLEOne R] {p : Ideal R} (h : p.IsPrime) (hp : p ≠ ⊥) : p.IsMaximal`", |
| "q1b_latex": "in a Krull-dim-1 ring, every nonzero prime ideal is maximal", |
| "q1c_natural": "in a 1-dimensional ring, every nonzero prime ideal is maximal", |
| "q2_slogan": "nonzero primes are maximal in dim-1 rings", |
| "q3_nickname": "", |
| "q4_special_case": "nonzero prime ideals in $\\mathbb{Z}$ are maximal" |
| }, |
| { |
| "id": "RT7", |
| "full_name": "UniqueFactorizationMonoid.factors_unique", |
| "file": "Mathlib/RingTheory/UniqueFactorizationDomain/Basic.lean", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`(hf : ∀ x ∈ f, Irreducible x) (hg : ∀ x ∈ g, Irreducible x) (h : Associated f.prod g.prod) : Multiset.Rel Associated f g`", |
| "q1b_latex": "in a UFM, two irreducible-factorizations of associated products are related elementwise by associated", |
| "q1c_natural": "irreducible factorizations in a UFM are unique up to permutation and associated", |
| "q2_slogan": "irreducible factorizations are unique up to associates", |
| "q3_nickname": "unique factorization in a UFM", |
| "q4_special_case": "integer $60$ factors uniquely as $2^2 \\cdot 3 \\cdot 5$" |
| }, |
| { |
| "id": "RT8", |
| "full_name": "Associates.ufm", |
| "file": "Mathlib/RingTheory/UniqueFactorizationDomain/Basic.lean:69", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "`instance : UniqueFactorizationMonoid (Associates α)`", |
| "q1b_latex": "$\\mathrm{Associates}\\,\\alpha$ inherits UFM from $\\alpha$", |
| "q1c_natural": "the quotient by the associated relation of a UFM is itself a UFM", |
| "q2_slogan": "Associates of a UFM is again a UFM", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "S1", |
| "full_name": "Lists'", |
| "file": "Mathlib/SetTheory/Lists.lean:57", |
| "difficulty": "Easy", |
| "kind": "inductive", |
| "q1a_lean": "`atom : α → Lists' α false`, `nil : Lists' α true`, `cons' : ... → ... → Lists' α true`", |
| "q1b_latex": "inductive type of atoms or pre-pended lists, indexed by an \"is-atom\" boolean", |
| "q1c_natural": "hereditarily finite lists distinguishing atoms from proper lists by a boolean", |
| "q2_slogan": "hereditarily finite lists with atoms", |
| "q3_nickname": "hereditarily finite lists", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "S2", |
| "full_name": "ZFSet", |
| "file": "Mathlib/SetTheory/ZFC/Basic.lean:48", |
| "difficulty": "Easy", |
| "kind": "def", |
| "q1a_lean": "`Quotient PSet.setoid`", |
| "q1b_latex": "$\\mathrm{ZFSet} = \\mathrm{PSet} / \\sim_{\\mathrm{ext}}$", |
| "q1c_natural": "pre-sets quotiented by extensional equivalence", |
| "q2_slogan": "pre-sets modulo extensional equivalence", |
| "q3_nickname": "ZFC sets", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "S3", |
| "full_name": "Ordinal.veblenWith", |
| "file": "Mathlib/SetTheory/Ordinal/Veblen.lean:66", |
| "difficulty": "Hard", |
| "kind": "def", |
| "q1a_lean": "`(f : Ordinal → Ordinal) (o : Ordinal) : Ordinal → Ordinal` defined recursively", |
| "q1b_latex": "$\\varphi_o(f)$: the $o$-th iteration of the Veblen function starting from $f$", |
| "q1c_natural": "the o-th iteration of the Veblen function starting from a base normal function", |
| "q2_slogan": "o-th member of the Veblen hierarchy", |
| "q3_nickname": "Veblen hierarchy", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "S4", |
| "full_name": "ONote", |
| "file": "Mathlib/SetTheory/Ordinal/Notation.lean:40", |
| "difficulty": "Hard", |
| "kind": "inductive", |
| "q1a_lean": "`zero | oadd : ONote → ℕ+ → ONote → ONote`", |
| "q1b_latex": "$\\omega^e \\cdot n + a$ — Cantor normal form below $\\varepsilon_0$", |
| "q1c_natural": "ordinal terms below $\\varepsilon_0$ in Cantor normal form", |
| "q2_slogan": "Cantor normal form notation for ordinals below $\\varepsilon_0$", |
| "q3_nickname": "ordinal notation (Cantor normal form)", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "S5", |
| "full_name": "Ordinal.add_eq_zero_iff", |
| "file": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`a + b = 0 ↔ a = 0 ∧ b = 0`", |
| "q1b_latex": "$a + b = 0 \\iff a = 0 \\wedge b = 0$", |
| "q1c_natural": "the sum of two ordinals is zero iff both are zero", |
| "q2_slogan": "ordinal sum is zero iff both summands zero", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "S6", |
| "full_name": "Cardinal.aleph0_add_aleph0", |
| "file": "Mathlib/SetTheory/Cardinal/Arithmetic.lean", |
| "difficulty": "Easy", |
| "kind": "theorem", |
| "q1a_lean": "`aleph0 + aleph0 = aleph0`", |
| "q1b_latex": "$\\aleph_0 + \\aleph_0 = \\aleph_0$", |
| "q1c_natural": "countable infinity absorbs itself under addition", |
| "q2_slogan": "countable infinity absorbs under addition", |
| "q3_nickname": "", |
| "q4_special_case": "$\\mathbb{N} \\sqcup \\mathbb{N}$ is countable" |
| }, |
| { |
| "id": "S7", |
| "full_name": "Ordinal.inductionOn", |
| "file": "Mathlib/SetTheory/Ordinal/Basic.lean:233", |
| "difficulty": "Hard", |
| "kind": "theorem", |
| "q1a_lean": "`(o : Ordinal) (type : ∀ α r [IsWellOrder α r], motive (Ordinal.type r)) : motive o`", |
| "q1b_latex": "every property of ordinals can be proved by induction on the representing well-order", |
| "q1c_natural": "ordinals can be inducted on via their representing well-orders", |
| "q2_slogan": "ordinal induction on representing well-orders", |
| "q3_nickname": "ordinal induction principle", |
| "q4_special_case": "" |
| }, |
| { |
| "id": "S8", |
| "full_name": "ONote.nf_opow", |
| "file": "Mathlib/SetTheory/Ordinal/Notation.lean", |
| "difficulty": "Hard", |
| "kind": "instance", |
| "q1a_lean": "`(o₁ o₂ : ONote) [o₁.NF] [o₂.NF] : (o₁ ^ o₂).NF`", |
| "q1b_latex": "$o_1, o_2$ in Cantor normal form $\\Rightarrow$ $o_1^{o_2}$ in Cantor normal form", |
| "q1c_natural": "exponentiating two ordinal notations in normal form yields a notation in normal form", |
| "q2_slogan": "Cantor normal form preserved under exponentiation", |
| "q3_nickname": "", |
| "q4_special_case": "" |
| } |
| ] |
