By Robert S. Boyer
Not like such a lot texts on common sense and arithmetic, this e-book is ready the way to turn out theorems instead of evidence of particular effects. We supply our solutions to such questions as: - while may still induction be used? - How does one invent a suitable induction argument? - while may still a definition be multiplied?
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Among the theorems that can be derived from the above axioms is the theorem that if X and Y are numeric, then (ADDI X) is equal to (ADDI Y) if and only if X is equal to Y. See Appendix B. We will abbreviate (ZERO) as 0 , and any well-formed nest of ADDl's around a 0 as the decimal numeral expressing the number of ADDI terms in the nest. Thus, 1 is an abbreviation for ( ADDI 0 ) , 2 is an abbreviation for (ADDI (ADDI 0) ) , etc. G. LITERAL ATOMS We want to be able to prove theorems about functions that manipu late symbols.
I. DEFINITIONS We have already defined certain simple functions, such as AND, OR, NOT, and IMPLIES. For example, Definition (AND P Q) ( I F P ( I F Q T F) F). Another simple function we define is ZEROP; it returns T if its argu ment is virtually 0 (in the sense that ADDI and SUBI treat it as 0) and F otherwise: Definition (ZEROP X) (OR (EQUAL X 0) (NOT (NUMBERP X ) ) ) . In general, if the function symbol f is new, if Xi, . . , x n are dis tinct variables, if the term body mentions no symbol as a variable other than these Xi, and if body does not mention f as a function sym bol, then adding the axiom ( f xi .
Xn) ) ) , means to add as an axiom the defining equation: ( f Xi . . xn) = body. We now illustrate an application of the principle of definition. Imagine that LESSP is well founded and that the axioms CAR. LESSP and CDR. LESSP have been added as in Chapter II. The defining equa tion for MC. FLATTEN is added to our theory by the following instantia tion of our principle of definition, f is the function symbol MC . FLATTEN (CDR X) ANS)) (CONS X ANS)); r is LESSP ; m is the function symbol COUNT 1, where ( COUNT 1 X Y ) is 46 / III.
A Computational Logic by Robert S. Boyer
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