problem([]==>x:pnat=>y:pnat=>(less(0,x)#less(0,y))=>less(0,times(x,y)),
ind_strat(induction(lemma(pnat_primitive)-[(x:pnat)-s(v0)])then[base_case(sym_eval(normalize_term([reduction([1,1],[less1,equ(u(1),left)]),reduction([2,2],[times1,equ(pnat,left)]),reduction([2],[less1,equ(u(1),left)])]))then[elementary(intro(new[y])then[intro(new[v0])then[elim(v0)then hyp(v1),wfftacs],wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[2,2],[times2,equ(pnat,left)],[]))then[idtac])]),lambda(x,p_ind(x,lambda(v0,su(su(lambda(v1,su(v2,[v1 of v0],[v2]))of lambda(v0,su(su(lambda(v1,su(v2,[v1 of v0],[v2]))of lambda(v0,su(su(lambda(v1,su(v2,[v1 of v0],[v2]))of lambda(y,lambda(v0,spread(v0,[v1,v2,v1]))),[less1 of 0],[v1]),[term_of(less1)],[less1])),[times1 of v0],[v1]),[term_of(times1)],[times1])),[less1 of 0],[v1]),[term_of(less1)],[less1])),[v0,v1,lambda(v2,su(su(su(lambda(v3,su(v4,[v3 of v2],[v4]))of _2167,[v3 of v2],[v4]),[times2 of v0],[v3]),[term_of(times2)],[times2]))])),
[problem([x:pnat,v0:pnat,v1:y:pnat=>(less(0,v0)#less(0,y))=>less(0,times(v0,y))]==>y:pnat=>(less(0,s(v0))#less(0,y))=>less(0,plus(times(v0,y),y)),
 sym_eval(normalize_term([reduction([1,1],[less2,equ(u(1),left)])])),lambda(v2,su(su(lambda(v3,su(v4,[v3 of v2],[v4]))of _2016,[less2 of v0],[v3]),[term_of(less2)],[less2])),
 [problem([x:pnat,v0:pnat,v1:y:pnat=>(less(0,v0)#less(0,y))=>less(0,times(v0,y))]==>y:pnat=>({true}#less(0,y))=>less(0,plus(times(v0,y),y)),
  ind_strat(induction(lemma(pnat_primitive)-[(v0:pnat)-s(v2)])then[base_case(sym_eval(normalize_term([reduction([1,2,2],[times1,equ(pnat,left)]),reduction([2,2],[plus1,equ(pnat,left)])]))then[elementary(intro(new[y])then[intro(new[v2])then[elim(v2)then hyp(v4),wfftacs],wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[1,2,2],[times2,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,=>,[],v3)]))))])]),p_ind(v0,lambda(v2,su(su(lambda(v3,su(v4,[v3 of v2],[v4]))of lambda(v2,su(su(lambda(v3,su(v4,[v3 of v2],[v4]))of lambda(y,lambda(v2,spread(v2,[v3,v4,v4]))),[plus1 of v2],[v3]),[term_of(plus1)],[plus1])),[times1 of v2],[v3]),[term_of(times1)],[times1])),[v2,v3,lambda(v4,su(su(su(lambda(v5,su(v6,[v5 of v4],[v6]))of lambda(v4,lambda(v5,su(lambda(v7,spread(v7,[v8,v9,su(su(su(su(su(su(su(v13,[v12 of v6],[v13]),[v11 of lambda(v12,su(su(su(v15,[v5 of v14],[v15]),[v13 of v9],[v14]),[v12 of v8],[v13]))],[v12]),[v14 of less(0,plus(plus(times(v2,v4),v4),v4))],[v11]),[v13 of less(0,plus(times(v2,v4),v4))],[v14]),[v12 of less(0,v4)],[v13]),[v11 of{true}],[v12]),[term_of(ipc_dp_imp_e2)],[v11])])),[v3 of v4],[v6]))of(lambda(v5,su(v6,[v5 of v4],[v6]))of _1440)),[v5 of v4],[v6]),[times2 of v2],[v5]),[term_of(times2)],[times2]))]),
  [problem([x:pnat,v0:pnat,v1:y:pnat=>(less(0,v0)#less(0,y))=>less(0,times(v0,y)),v2:pnat,v3:y:pnat=>({true}#less(0,y))=>less(0,plus(times(v2,y),y))]==>y:pnat=>less(0,plus(times(v2,y),y))=>less(0,plus(plus(times(v2,y),y),y)),
   generalise(plus(times(v2,y),y),v4:pnat),lambda(v4,lambda(v5,su(su(v7,[v6 of v5],[v7]),[v4 of plus(times(v2,v5),v5)],[v6])))of _1273,
   [problem([x:pnat,v0:pnat,v1:y:pnat=>(less(0,v0)#less(0,y))=>less(0,times(v0,y)),v2:pnat,v3:y:pnat=>({true}#less(0,y))=>less(0,plus(times(v2,y),y))]==>v4:pnat=>y:pnat=>less(0,v4)=>less(0,plus(v4,y)),
    ind_strat(induction(lemma(pnat_primitive)-[(v4:pnat)-s(v5)])then[base_case(sym_eval(normalize_term([reduction([1],[less1,equ(u(1),left)]),reduction([2,2],[plus1,equ(pnat,left)])]))then[elementary(intro(new[y])then[intro(new[v5])then[elim(v5),wfftacs],wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[2,2],[plus2,equ(pnat,left)],[]))then[idtac])]),lambda(v4,p_ind(v4,lambda(v5,su(su(lambda(v6,su(v7,[v6 of v5],[v7]))of lambda(v5,su(su(lambda(v6,su(v7,[v6 of v5],[v7]))of lambda(y,lambda(v5,any(v5))),[plus1 of v5],[v6]),[term_of(plus1)],[plus1])),[less1 of 0],[v6]),[term_of(less1)],[less1])),[v5,v6,lambda(v7,su(su(su(lambda(v8,su(v9,[v8 of v7],[v9]))of _881,[v8 of v7],[v9]),[plus2 of v5],[v8]),[term_of(plus2)],[plus2]))])),
    [problem([x:pnat,v0:pnat,v1:y:pnat=>(less(0,v0)#less(0,y))=>less(0,times(v0,y)),v2:pnat,v3:y:pnat=>({true}#less(0,y))=>less(0,plus(times(v2,y),y)),v4:pnat,v5:pnat,v6:y:pnat=>less(0,v5)=>less(0,plus(v5,y))]==>y:pnat=>less(0,s(v5))=>less(0,s(plus(v5,y))),
     sym_eval(normalize_term([reduction([1],[less2,equ(u(1),left)]),reduction([2],[less2,equ(u(1),left)])]))then[elementary(intro(new[y])then[intro(new[v7])then[hyp(v7),wfftacs],wfftacs])],lambda(v7,su(su(lambda(v8,su(v9,[v8 of v7],[v9]))of lambda(v7,su(su(lambda(v8,su(v9,[v8 of v7],[v9]))of lambda(y,lambda(v7,v7)),[less2 of plus(v5,v7)],[v8]),[term_of(less2)],[less2])),[less2 of v5],[v8]),[term_of(less2)],[less2])),
     [
     ]) ext _881
    ]) ext _1273
   ]) ext _1440
  ]) ext _2016
 ]) ext _2167
]).
