PA11
Premises:
(forall ?X (not (= 0 (succ ?X)))) (forall (?X ?Y) (=> (= (succ ?X) (succ ?Y)) (= ?X ?Y))) (forall ?X (= (+ ?X 0) ?X)) (forall (?X ?Y) (= (+ ?X (succ ?Y)) (succ (+ ?X ?Y)))) (forall (?X ?Y) (<=> (= (+ ?X 0) ?Y) (= ?X ?Y))) (forall (?X ?Y ?Z) (<=> (= (+ ?X (succ ?Y)) ?Z) (= (+ (+ ?X ?Y) (succ 0)) ?Z))) (forall ?X (= (* ?X 0) 0)) (forall (?X ?Y) (= (* ?X (succ ?Y)) (+ (* ?X ?Y) ?X))) (forall (?X ?Y) (<=> (= (* ?X 0) ?Y) (= 0 ?Y))) (forall (?X ?Y ?Z) (<=> (= (* ?X (succ ?Y)) ?Z) (= (+ (* ?X ?Y) ?X) ?Z)))
Conclusion:
(forall ?X (= (succ ?X) (+ ?X (succ 0))))