problem([]==>x:pnat=>minus(binom(x,s(0)),x)=0 in pnat,
ind_strat(induction(lemma(pnat_primitive)-[(x:pnat)-s(v0)])then[base_case(sym_eval(normalize_term([reduction([1,1],[minus1,equ(pnat,left)]),reduction([1,1],[binom2,equ(pnat,left)])]))then[elementary(identity)]),step_case(ripple(direction_out,wave(direction_out,[1,1],[minus2,equ(pnat,left)],[])then[wave(direction_out,[1,1,1,1],[binom3,equ(pnat,left)],[])])then[unblock_then_fertilize(weak,unblock_fertilize_lazy([idtac])then fertilize(weak,fertilize_then_ripple(fertilize_left_or_right(right,[weak_fertilize(right,in,[],v1)]))))])]),lambda(x,p_ind(x,su(su(su(su(axiom,[binom2 of 0],[v0]),[term_of(binom2)],[binom2]),[minus1 of binom(0,s(0))],[v0]),[term_of(minus1)],[minus1]),[v0,v1,su(su(su(su(su(su(_52525,[v2 of 0],[v3]),[binom3 of v0],[v2]),[term_of(binom3)],[binom3]),[v2 of v0],[v3]),[minus2 of binom(s(v0),s(0))],[v2]),[term_of(minus2)],[minus2])])),
[problem([x:pnat,v0:pnat,v1:minus(binom(v0,s(0)),v0)=0 in pnat]==>pred(minus(plus(binom(v0,s(0)),binom(v0,0)),v0))=minus(binom(v0,s(0)),v0)in pnat,
 sym_eval(normalize_term([reduction([2,1,1,1,1],[binom1,equ(pnat,left)])])),su(su(_52372,[binom1 of v0],[v2]),[term_of(binom1)],[binom1]),
 [problem([x:pnat,v0:pnat,v1:minus(binom(v0,s(0)),v0)=0 in pnat]==>pred(minus(plus(binom(v0,s(0)),s(0)),v0))=minus(binom(v0,s(0)),v0)in pnat,
  generalise(binom(v0,s(0)),v2:pnat),lambda(v2,su(v3,[v2 of binom(v0,s(0))],[v3]))of _52275,
  [problem([x:pnat,v0:pnat,v1:minus(binom(v0,s(0)),v0)=0 in pnat]==>v2:pnat=>pred(minus(plus(v2,s(0)),v0))=minus(v2,v0)in pnat,
   ind_strat(induction(lemma(pnat_primitive)-[(v0:pnat)-s(v3)])then[base_case(sym_eval(normalize_term([reduction([1,1,1],[minus1,equ(pnat,left)]),reduction([2,1],[minus1,equ(pnat,left)])]))),step_case(ripple(direction_out,wave(direction_out,[2,1],[minus2,equ(pnat,left)],[])then[wave(direction_out,[1,1,1],[minus2,equ(pnat,left)],[])])then[unblock_then_fertilize(weak,unblock_fertilize_lazy([idtac])then fertilize(weak,fertilize_then_ripple(fertilize_left_or_right(right,[weak_fertilize(right,in,[1],v4)]))then elementary(intro(new[v2])then[identity,wfftacs])))])]),p_ind(v0,lambda(v3,su(su(lambda(v2,su(v4,[v2 of v3],[v4]))of lambda(v3,su(su(lambda(v2,su(v4,[v2 of v3],[v4]))of _51983,[minus1 of v3],[v2]),[term_of(minus1)],[minus1])),[minus1 of plus(v3,s(0))],[v2]),[term_of(minus1)],[minus1])),[v3,v4,lambda(v5,su(su(su(lambda(v2,su(v6,[v2 of v5],[v6]))of lambda(v5,su(su(su(lambda(v2,su(v6,[v2 of v5],[v6]))of lambda(v5,su(lambda(v2,su(v6,[v2 of v5],[v6]))of lambda(v2,axiom),[v4 of v5],[v2])),[v2 of v3],[v6]),[minus2 of plus(v5,s(0))],[v2]),[term_of(minus2)],[minus2])),[v2 of v3],[v6]),[minus2 of v5],[v2]),[term_of(minus2)],[minus2]))]),
   [problem([x:pnat,v0:pnat,v1:minus(binom(v0,s(0)),v0)=0 in pnat]==>v2:pnat=>pred(plus(v2,s(0)))=v2 in pnat,
    ind_strat(induction(lemma(pnat_primitive)-[(v2:pnat)-s(v3)])then[base_case(sym_eval(normalize_term([reduction([1,1,1],[plus1,equ(pnat,left)]),reduction([1,1],[pred2,equ(pnat,left)])]))then[elementary(identity)]),step_case(ripple(direction_out,wave(direction_out,[1,1,1],[plus2,equ(pnat,left)],[]))then[unblock_then_fertilize(weak,unblock_fertilize_lazy([idtac])then fertilize(weak,fertilize_then_ripple(fertilize_left_or_right(right,[weak_fertilize(right,in,[1],v4)]))))])]),lambda(v2,p_ind(v2,su(su(su(su(axiom,[pred2 of 0],[v3]),[term_of(pred2)],[pred2]),[plus1 of s(0)],[v3]),[term_of(plus1)],[plus1]),[v3,v4,su(su(su(_51530,[v5 of s(0)],[v6]),[plus2 of v3],[v5]),[term_of(plus2)],[plus2])])),
    [problem([x:pnat,v0:pnat,v1:minus(binom(v0,s(0)),v0)=0 in pnat,v2:pnat,v3:pnat,v4:pred(plus(v3,s(0)))=v3 in pnat]==>pred(s(plus(v3,s(0))))=s(pred(plus(v3,s(0))))in pnat,
     sym_eval(normalize_term([reduction([1,1],[pred2,equ(pnat,left)])])),su(su(_51386,[pred2 of plus(v3,s(0))],[v5]),[term_of(pred2)],[pred2]),
     [problem([x:pnat,v0:pnat,v1:minus(binom(v0,s(0)),v0)=0 in pnat,v2:pnat,v3:pnat,v4:pred(plus(v3,s(0)))=v3 in pnat]==>plus(v3,s(0))=s(pred(plus(v3,s(0))))in pnat,
      ind_strat(induction(lemma(pnat_primitive)-[(v3:pnat)-s(v5)])then[base_case(sym_eval(normalize_term([reduction([1,1],[plus1,equ(pnat,left)]),reduction([1,1,2,1],[plus1,equ(pnat,left)]),reduction([1,2,1],[pred2,equ(pnat,left)])]))then[elementary(identity)]),step_case(ripple(direction_out,wave(direction_out,[1,1],[plus2,equ(pnat,left)],[])then[wave(direction_out,[1,1,2,1],[plus2,equ(pnat,left)],[])])then[unblock_then_fertilize(weak,unblock_fertilize_lazy([idtac])then fertilize(weak,fertilize_then_ripple(fertilize_left_or_right(left,[weak_fertilize(left,in,[1],v6)]))))])]),p_ind(v3,su(su(su(su(su(su(axiom,[pred2 of 0],[v5]),[term_of(pred2)],[pred2]),[plus1 of s(0)],[v5]),[term_of(plus1)],[plus1]),[plus1 of s(0)],[v5]),[term_of(plus1)],[plus1]),[v5,v6,su(su(su(su(su(su(_51006,[v7 of s(0)],[v8]),[plus2 of v5],[v7]),[term_of(plus2)],[plus2]),[v7 of s(0)],[v8]),[plus2 of v5],[v7]),[term_of(plus2)],[plus2])]),
      [problem([x:pnat,v0:pnat,v1:minus(binom(v0,s(0)),v0)=0 in pnat,v2:pnat,v3:pnat,v4:pred(plus(v3,s(0)))=v3 in pnat,v5:pnat,v6:plus(v5,s(0))=s(pred(plus(v5,s(0))))in pnat]==>s(s(pred(plus(v5,s(0)))))=s(pred(s(plus(v5,s(0)))))in pnat,
       sym_eval(normalize_term([reduction([1,2,1],[pred2,equ(pnat,left)])])),su(su(_50804,[pred2 of plus(v5,s(0))],[v7]),[term_of(pred2)],[pred2]),
       [problem([x:pnat,v0:pnat,v1:minus(binom(v0,s(0)),v0)=0 in pnat,v2:pnat,v3:pnat,v4:pred(plus(v3,s(0)))=v3 in pnat,v5:pnat,v6:plus(v5,s(0))=s(pred(plus(v5,s(0))))in pnat]==>s(s(pred(plus(v5,s(0)))))=s(plus(v5,s(0)))in pnat,
        ind_strat(induction(lemma(pnat_primitive)-[(v5:pnat)-s(v7)])then[base_case(sym_eval(normalize_term([reduction([1,1,1,1,1],[plus1,equ(pnat,left)]),reduction([1,1,1,1],[pred2,equ(pnat,left)]),reduction([1,2,1],[plus1,equ(pnat,left)])]))then[elementary(identity)]),step_case(ripple(direction_out,wave(direction_out,[1,2,1],[plus2,equ(pnat,left)],[])then[wave(direction_out,[1,1,1,1,1],[plus2,equ(pnat,left)],[])])then[idtac])]),p_ind(v5,su(su(su(su(su(su(axiom,[plus1 of s(0)],[v7]),[term_of(plus1)],[plus1]),[pred2 of 0],[v7]),[term_of(pred2)],[pred2]),[plus1 of s(0)],[v7]),[term_of(plus1)],[plus1]),[v7,v8,su(su(su(su(su(su(_50407,[v9 of s(0)],[v10]),[plus2 of v7],[v9]),[term_of(plus2)],[plus2]),[v9 of s(0)],[v10]),[plus2 of v7],[v9]),[term_of(plus2)],[plus2])]),
        [problem([x:pnat,v0:pnat,v1:minus(binom(v0,s(0)),v0)=0 in pnat,v2:pnat,v3:pnat,v4:pred(plus(v3,s(0)))=v3 in pnat,v5:pnat,v6:plus(v5,s(0))=s(pred(plus(v5,s(0))))in pnat,v7:pnat,v8:s(s(pred(plus(v7,s(0)))))=s(plus(v7,s(0)))in pnat]==>s(s(pred(s(plus(v7,s(0))))))=s(s(plus(v7,s(0))))in pnat,
         generalise(plus(v7,s(0)),v9:pnat),lambda(v9,su(v10,[v9 of plus(v7,s(0))],[v10]))of _50188,
         [problem([x:pnat,v0:pnat,v1:minus(binom(v0,s(0)),v0)=0 in pnat,v2:pnat,v3:pnat,v4:pred(plus(v3,s(0)))=v3 in pnat,v5:pnat,v6:plus(v5,s(0))=s(pred(plus(v5,s(0))))in pnat,v7:pnat,v8:s(s(pred(plus(v7,s(0)))))=s(plus(v7,s(0)))in pnat]==>v9:pnat=>s(s(pred(s(v9))))=s(s(v9))in pnat,
          sym_eval(normalize_term([reduction([1,1,1,1],[pred2,equ(pnat,left)])]))then[elementary(intro(new[v9])then[identity,wfftacs])],lambda(v10,su(su(lambda(v9,su(v11,[v9 of v10],[v11]))of lambda(v9,axiom),[pred2 of v10],[v9]),[term_of(pred2)],[pred2])),
          [
          ]) ext _50188
         ]) ext _50407
        ]) ext _50804
       ]) ext _51006
      ]) ext _51386
     ]) ext _51530
    ]) ext _51983
   ]) ext _52275
  ]) ext _52372
 ]) ext _52525
]).
