ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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.

Two similar realizations with identical data

Example

Let $A = [\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}], B = [\begin{array}{c} 1 \\ 0 \end{array}], C = [\begin{array}{cc} 1 & 1 \end{array}], x_{0} = [\begin{array}{c} 1 \\ 1 \end{array}], T = [\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}], T^{- 1} = [\begin{array}{cc} 1 & - 1 \\ 0 & 1 \end{array}] .$ The transformed realization has $\hat{A} = [\begin{array}{cc} 0 & - 1 \\ 0 & 1 \end{array}], \hat{B} = [\begin{array}{c} 1 \\ 0 \end{array}], \hat{C} = [\begin{array}{cc} 1 & 2 \end{array}], {\hat{x}}_{0} = [\begin{array}{c} 0 \\ 1 \end{array}] .$ The two state-space descriptions differ, but for every common input sequence their exact output sequences agree at every sample.

Example references

Discrete state and output equations

Input-output invariance under state similarity

Dahleh, Dahleh, and Verghese, Lectures on Dynamic Systems and Control, §25.3

Lall, State-Space Realization, §9.2

Facts & Assumptions

Given: The displayed matrices, the common initial/input experiment, and the corresponding transformed matrices and initial state.

[F1]

Discrete linear input-state-output experiments: states obey $x_{k + 1} = A x_{k} + B u_{k}$ and outputs obey $y_{k} = C x_{k}$ .

[F2]

State similarity preserves input-output data: if $T$ is invertible and the transformed matrices and initial state are $T^{- 1} A T$ , $T^{- 1} B$ , $C T$ , and $T^{- 1} x_{0}$ , then common inputs give identical outputs at every sample.

Verification

1.1givenalgebra

Direct multiplication gives $T T^{- 1} = T^{- 1} T = I_{2}$ , so $T$ is invertible. Multiplying the displayed matrices gives $T^{- 1} A T = \hat{A}$ , $T^{- 1} B = \hat{B}$ , $C T = \hat{C}$ , and $T^{- 1} x_{0} = {\hat{x}}_{0}$ . Also $\hat{A} \neq A$ and $\hat{C} \neq C$ , so these are distinct coordinate descriptions.

2.1F1F2step 1.1

The hypotheses of [F2] hold by step 1.1, so every finite common-input experiment has the same output sequence in both descriptions.

3.1step 1.1step 2.1∎

Steps 1.1 and 2.1 exhibit distinct realization matrices and initial coordinates with identical data, as claimed.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Mohammed Dahleh, Munther A. Dahleh, and George Verghese, Lectures on Dynamic Systems and Control, Chapter 25: Minimal State-Space Realization (standard reference, not scraped)
Sanjay Lall, Engr207c: Operator Theory and Linear Systems, State-Space Realization lecture notes (standard reference, not scraped)