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:

- 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. - A
**lemma**is simultaneously a contention for premises below it and a premise for a contention above it. - 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. - A theorem

### Symbolic analysis and the mathematic dictionary

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

- Symbols are axioms to be started. Some of them are grounded, but some are non-ground: fuzzy, vague or mental models without exact information.
- 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.
- 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.
- 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.

### Some links:

- What is the difference between theorem.. and corollary by David Richeson.
- Definitions for axioms etc at Wiki: http://users.dickinson.edu/~richesod/
- Ideal of Science: https://symbolicanalysis.wordpress.com/2009/10/22/ideal-of-science/
- Transformation process for reverse engineering: https://symbolicanalysis.wordpress.com/2009/09/09/reverse-engineering/ (the figure describes the symbolic analysis transformation process as successive automata: A1-A7).

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