583.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 583

References:

erdosproblems.com/583
[AnBa23] Anto, Nevil and Basavaraju, Manu, Gallai's path decomposition for 2-degenerate graphs. Discrete Math. Theor. Comput. Sci. (2023), Paper No. 16, 11.
[BBB21] A. Blanché, M. Bonamy, and N. Bonichon, Gallai's path decomposition in planar graphs. arXiv:2110.08870 (2021).
[BoPe19] Bonamy, Marthe and Perrett, Thomas J., Gallai's path decomposition conjecture for graphs of small maximum degree. Discrete Math. (2019), 1293--1299.
[CFZ26] Chu, Yanan and Fan, Genghua and Zhou, Chuixiang, Gallai's conjecture and the path number of odd semi-cliques. Discrete Math. (2026), Paper No. 114725, 6.
[Ch78] Chung, F. R. K., On partitions of graphs into trees. Discrete Math. (1978), 23-30.
[DeKo00] Dean, Nathaniel and Kouider, Mekkia, Gallai's conjecture for disconnected graphs. Discrete Math. (2000), 43--54.
[Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.
[Fa02] Fan, Genghua, Subgraph coverings and edge switchings. J. Combin. Theory Ser. B (2002), 54-83.
[Lo68] Lovász, L., On covering of graphs. Theory of Graphs (Proc. Colloq., Tihany, 1966) (1968), 231-236.
[Py96] Pyber, L., Covering the edges of a connected graph by paths. J. Combin. Theory Ser. B (1996), 152-159.

Imports (1)

FormalConjecturesUtil

def Erdos583.IsPathSubgraph A subgraph H of G is a path subgraph if it is the subgraph traced out by a path in G, i.e. a walk with no repeated vertices line 56 source
Source documentation
A subgraph H of G is a path subgraph if it is the subgraph traced out by a path in G, i.e. a walk with no repeated vertices.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, SimpleGraph.Subgraph
Proof uses: Exists, SimpleGraph.Walk, And, SimpleGraph.Walk.IsPath, Eq, SimpleGraph.Walk.toSubgraph
```
def IsPathSubgraph {V : Type*} {G : SimpleGraph V} (H : G.Subgraph) : Prop :=
…
```
theorem Erdos583.erdos_583 Every connected graph on $n$ vertices can be partitioned into at most $\lceil n/2\rceil$ edge-disjoint paths sorry line 66 source
Source documentation
Every connected graph on $n$ vertices can be partitioned into at most $\lceil n/2\rceil$ edge-disjoint paths.
A problem of Erdős and Gallai.

depends on sorry what this means
Statement uses: Fintype, SimpleGraph, SimpleGraph.Connected, Exists, Finset, SimpleGraph.Subgraph, And, Membership.mem, Erdos583.IsPathSubgraph, SimpleGraph.IsDecomposition, LE.le, Nat, Finset.card, Nat.ceil, Rat, HDiv.hDiv, Fintype.card, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
31 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_583 {V : Type*} [Fintype V] (G : SimpleGraph V) (hG : G.Connected) :
    ∃ D : Finset G.Subgraph,
      (∀ H ∈ D, IsPathSubgraph H) ∧
      IsDecomposition G D ∧
      D.card ≤ ⌈(Fintype.card V : ℚ) / 2⌉₊ := by
…
```
Mentions: Erdos583.IsPathSubgraph