**Completeness condition (for model of principle)**

*It is assumed that the set of diagrams (processes), consistent with a given principle and called the model of this principle, must be the largest convex set possible. It is allowed to have many alternative sets (models) that meet the above condition, in the sense that a given set (model) cannot be enlarged without breaking compliance with the principle or without breaking the convexity requirement.*

The completeness condition allows for the consideration of richer models (sets of diagrams) for a given principle. Thanks to this condition, diagrams (processes) whose compliance with the principle is not obvious can be attached to the model. An attached diagram (process) can be created as a result of adding two trivial diagrams or diagrams already belonging to the model. In addition, a diagram (process) can be attached to the model, the addition of which to existing diagrams results in a trivial diagram or a diagram belonging to the model. This procedure can fork the model into two models or the entire model family. Regardless of the number of models in the family, each model should be considered an alternative independent model. If there is only one model, it will be called a homogeneous model here, and in the literature it is sometimes called a stable model.

Given the above theoretical structure, we can study the formal implications (or lack thereof) between the four elementary principles of thermodynamics (*C*0, *C I*, *K*, *CII*). There are as many as 12 formal implications from Table 2 to be proved or disproved. Proof of the *Z*<sup>1</sup> |→ *Z*<sup>2</sup> implication will be performed double, once by checking the content of the sets of diagrams (models) of both principles {*Z*1}⊂{*Z*2}, two by formal constructive proof (proof by contradiction will not be used). Similarly, disproving the implications will consist, firstly, in checking that the models do not contain themselves {*Z*1} ⊂ {*Z*2}, and secondly, in pointing to a counterexample in the form of a diagram *d*<sup>1</sup> consistent with principle *Z*1, but inconsistent with principle *Z*<sup>2</sup> (see Figure 2b). The negation of the formal implication

will be denoted by the standard negation operator ¬(*Z*<sup>1</sup> |→ *Z*2) or by the crossed arrow *<sup>Z</sup>*<sup>1</sup> <sup>|</sup>-*Z*2.

The proof and rebuttals will be presented in the Discussion section and summarized in the Results section. A synthesis will be given in the Conclusions.
