LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Termination of the finite-budget guarded forward-Euler loop

Statement

For every valid input to Finite-budget positive-tube forward Euler loop with finite budget $N$ , the loop returns after at most $N$ candidate evaluations. Its accepted-step count is between $0$ and $N$ , and every returned nodal state lies in the specified compact tube $C$ .

Facts & Assumptions

Given: The algorithm inputs $U^{0}$ , $Δ t > 0$ , a finite integer budget $N \geq 0$ , and a compact tube $C$ as specified in the algorithm definition.

[F1]

Finite-budget positive-tube forward Euler loop: each candidate is either rejected with immediate return of the last accepted vector, or accepted after a nodewise test; after $N$ acceptances, the algorithm returns completed.

Proof

1.1F1given

If $N = 0$ , branch 4 of [F1] returns the input immediately, with zero evaluations and zero accepted stages.

1.2F1given

Suppose $N > 0$ . At each visited stage $n < N$ , branch 3 returns immediately if the candidate fails, so no rejected stage repeats; the algorithm reports $n$ accepted steps because $U^{n}$ follows exactly $n$ earlier acceptances. Otherwise branch 2 accepts one candidate, so the next stored iterate follows one more acceptance. If all $N$ candidates are accepted, branch 4 returns completed with count $N$ . Thus the loop has at most $N$ evaluations and the accepted count lies in $[0, N]$ .

2.1F1step 1.2given∎

The initial nodal vector is in $C$ by the algorithm input. Each later vector becomes the stored state only after every node passes membership in $C$ ; branch 3 returns the preceding stored vector. Therefore every returned nodal state lies in $C$ .

Depends on

Finite-budget positive-tube forward Euler loop

Used by

Conditional fixed-grid forward-Euler error bound Theorem

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

R. Eymard, T. Gallouët, and R. Herbin, Finite Volume Methods, January 2019 updated author text, complete manuscript (standard reference, not scraped)
Eitan Tadmor, Entropy Stable Schemes, Chapter 18, Handbook of Numerical Methods for Hyperbolic Problems: Basic and Fundamental Issues, 2016, complete chapter (standard reference, not scraped)