In mathematics and logic a traditional proof is done by starting from one or more axioms. Clauses are used as steps (bridges) into higher-order decisions. Lemmas and corollarys are many times useful in order to mark the route to the final decision, which is usually a proof or a theorem.

Below the most important concepts regarding how to derive logical decisions:

1. An axiom (or postulate) is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths.
2. A lemma is simultaneously a contention for premises below it and a premise for a contention above it.
3. A corollary is a statement which follows readily from a previous statement. In mathematics a corollary typically follows a theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective. Proposition B is a corollary of proposition A if B can readily be deduced from A, but the meaning of readily varies depending upon the author and context.
4. A theorem is a statement which has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms

### Symbolic analysis and the mathematic dictionary

In Symbolic Analysis we use the concepts above in the following way:

1. Symbols are axioms to be started. Some of them are grounded, but some are non-ground: fuzzy, vague or mental models without exact information.
2. Those symbols that are ground and captured from source code – have a certain semantics. A lemma is that for any symbol having a semantic notation, a symbolic notation can be created, because each grammar term is independent of other grammar terms by default.  Therefore, each symbol can be simulated separately.
3. The corollary is that each source language symbol is an automaton, which can be programmed in a tool as a state machine having the same principal logic as Turing machine.
4. The final decision from the steps 1, 2 and 3 above is that by simulating source code it is possible to mimic original program execution step by step. This makes a foundation for interactive program proof.