The CAT formal system

A few weeks ago I came up with a formal system called "cat". It is called "cat"
because I like cats and have a black one, to give me bad luck, or because
I'm a sorcerer, or something else.
But the system has nothing to do with real cats. "Cat" is just a pattern that is
substituted by the empty word Λ as a shrinking rule.
Ok, so we are into formal systems, and they are defined by axioms, and have
rules to make one thing appear and disappear.

Definition (Formal System)
A formal system (S) is composed of a set of symbols \(\Sigma\) called the alphabet
a set of words \(\Sigma^* \bigcup \Lambda\), a set of variables {x,y,z}
and a set of rules that define the systems ability to evolve through each
new iteration.

Here, are the rules for the system:

  1. SΛ -> 0 The empty string is inside S and it equals 0 or false.
  2. Sc
  3. Sca
  4. Scat -> Λ
  5. Sx
  6. SΛχ -> χ
  7. SχΛ -> χ
  8. Sxcat -> x
  9. Scatx -> x
  10. Sxcc -> ccaatt

Cat System Prover