ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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.

Simultaneous guards need a deterministic priority rule

Example

Consider a one-dimensional transition specification with initial mode $I$ , field $x^{'} = 1$ , initial condition $x (0) = - 1$ , and two outgoing guards $g_{1} (x) = x$ and $g_{2} (x) = x$ , both with positive crossing orientation. Transition $g_{1}$ resets to terminal mode $A$ with $x^{'} = 0$ ; transition $g_{2}$ resets to terminal mode $B$ with $x^{'} = 1$ . Both reset maps set $x$ to $0$ . This data is incomplete under Smooth hybrid ODE modes, guards, resets, and deterministic event priority until a fixed total priority order on the outgoing transitions from $I$ is supplied.

Facts & Assumptions

Given: The oriented-guard and fixed total event-priority conventions of Smooth hybrid ODE modes, guards, resets, and deterministic event priority.

[F1]

Smooth hybrid ODE modes, guards, resets, and deterministic event priority: an event occurs at a zero of an active guard, the reset is applied at that time, and a fixed priority selects among guards vanishing simultaneously.

Verification

1.1givenF1algebra

In mode $I$ , the exact solution is $x (t) = - 1 + t$ , so both guards first vanish at $t = 1$ and have derivative $1 > 0$ there. Thus both transitions are enabled simultaneously.

2.1givenF1step 1.1algebra∎

Completing the specification with priority $g_{1} ≺ g_{2}$ selects transition $g_{1}$ and the continuation in mode $A$ is $x (t) = 0$ for $t \geq 1$ . Completing it instead with priority $g_{2} ≺ g_{1}$ selects transition $g_{2}$ and the continuation in mode $B$ is $x (t) = t - 1$ for $t \geq 1$ . Each completed system has a fixed priority and determines one continuation, but the incomplete specification alone does not determine which completion was intended.

Depends on

Smooth hybrid ODE modes, guards, resets, and deterministic event priority

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Nathan J. Kong, J. Joe Payne, James Zhu, and Aaron M. Johnson, Saltation Matrices: The Essential Tool for Linearizing Hybrid Dynamical Systems (standard reference, not scraped)