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• Introduction-to-the-theorem-prover

Example-induction-scheme-on-several-variables

Induction on several variables

See logic-knowledge-taken-for-granted-inductive-proof for an explanation of what we mean by the induction suggested by a recursive function or a term.

Induction on Several Variables To (p n x) for all n and all x, prove each of the following:

Base Case 1: 
(implies (endp x) 
         (p n x)) 

Base Case 2: 
(implies (and (not (endp x)) 
              (zp n)) 
         (p n x)) 

Induction Step: 
(implies (and (not (endp x)) 
              (not (zp n)) 
              (p (- n 1) (cdr x))) 
         (p n x)) 

A function that suggests this induction is

(defun nth (n x)
  (if (endp x)
      nil
      (if (zp n)
          (car x)
          (nth (- n 1) (cdr x))))).