problem([]==>a:pnat=>b:pnat=>c:pnat=>times(plus(b,c),a)=plus(times(b,a),times(c,a))in pnat,
ind_strat(induction(lemma(pnat_primitive)-[(b:pnat)-s(v0)])then[base_case(sym_eval(normalize_term([reduction([1,1,1],[plus1,equ(pnat,left)]),reduction([1,2,1],[times1,equ(pnat,left)]),reduction([2,1],[plus1,equ(pnat,left)])]))then[elementary(intro(new[a])then[intro(new[c])then[identity,wfftacs],wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[1,2,1],[times2,equ(pnat,left)],[])then[wave(direction_out,[1,1,1],[plus2,equ(pnat,left)],[])then[wave(direction_out,[1,1],[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,in,[1],v1)]))))])]),lambda(v0,lambda(v1,lambda(v2,lambda(v3,su(su(su(v6,[v5 of v2],[v6]),[v4 of v0],[v5]),[v3 of v1],[v4]))of lambda(b,p_ind(b,lambda(v0,lambda(v1,su(su(lambda(v2,su(su(v4,[v3 of v1],[v4]),[v2 of v0],[v3]))of lambda(v0,lambda(v1,su(su(lambda(v2,su(su(v4,[v3 of v1],[v4]),[v2 of v0],[v3]))of lambda(v0,lambda(v1,su(su(lambda(v2,su(su(v4,[v3 of v1],[v4]),[v2 of v0],[v3]))of lambda(a,lambda(c,axiom)),[plus1 of times(v1,v0)],[v2]),[term_of(plus1)],[plus1]))),[times1 of v0],[v2]),[term_of(times1)],[times1]))),[plus1 of v1],[v2]),[term_of(plus1)],[plus1]))),[v0,v1,lambda(v2,lambda(v3,su(su(su(lambda(v4,su(su(v6,[v5 of v3],[v6]),[v4 of v2],[v5]))of lambda(v2,lambda(v3,su(su(su(lambda(v4,su(su(v6,[v5 of v3],[v6]),[v4 of v2],[v5]))of lambda(v2,lambda(v3,su(su(su(lambda(v4,su(su(v6,[v5 of v3],[v6]),[v4 of v2],[v5]))of lambda(v2,lambda(v3,su(su(lambda(v4,su(su(v6,[v5 of v3],[v6]),[v4 of v2],[v5]))of _217047,[v4 of v3],[v5]),[v1 of v2],[v4]))),[v4 of v2],[v5]),[times2 of plus(v0,v3)],[v4]),[term_of(times2)],[times2]))),[v4 of v3],[v5]),[plus2 of v0],[v4]),[term_of(plus2)],[plus2]))),[v4 of v2],[v5]),[times2 of v0],[v4]),[term_of(times2)],[times2])))]))))),
[problem([b:pnat,v0:pnat,v1:a:pnat=>c:pnat=>times(plus(v0,c),a)=plus(times(v0,a),times(c,a))in pnat]==>a:pnat=>c:pnat=>plus(plus(times(v0,a),times(c,a)),a)=plus(plus(times(v0,a),a),times(c,a))in pnat,
 generalise(times(c,a),v2:pnat),lambda(v2,lambda(v3,lambda(v4,su(su(su(v7,[v6 of v4],[v7]),[v5 of v3],[v6]),[v2 of times(v4,v3)],[v5]))))of _216830,
 [problem([b:pnat,v0:pnat,v1:a:pnat=>c:pnat=>times(plus(v0,c),a)=plus(times(v0,a),times(c,a))in pnat]==>v2:pnat=>a:pnat=>c:pnat=>plus(plus(times(v0,a),v2),a)=plus(plus(times(v0,a),a),v2)in pnat,
  generalise(times(v0,a),v3:pnat),lambda(v2,lambda(v4,lambda(v5,lambda(v6,su(su(su(su(v9,[v8 of v6],[v9]),[v7 of v5],[v8]),[v3 of v4],[v7]),[v2 of times(v0,v5)],[v3])))))of _216670,
  [problem([b:pnat,v0:pnat,v1:a:pnat=>c:pnat=>times(plus(v0,c),a)=plus(times(v0,a),times(c,a))in pnat]==>v3:pnat=>v2:pnat=>a:pnat=>c:pnat=>plus(plus(v3,v2),a)=plus(plus(v3,a),v2)in pnat,
   ind_strat(induction(lemma(pnat_primitive)-[(v3:pnat)-s(v4)])then[base_case(sym_eval(normalize_term([reduction([1,1,1],[plus1,equ(pnat,left)]),reduction([1,2,1],[plus1,equ(pnat,left)])]))),step_case(ripple(direction_out,wave(direction_out,[1,2,1],[plus2,equ(pnat,left)],[])then[wave(direction_out,[2,1],[plus2,equ(pnat,left)],[])then[wave(direction_out,[1,1,1],[plus2,equ(pnat,left)],[])then[wave(direction_out,[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],v5)]))then elementary(intro(new[v2])then[intro(new[a])then[intro(new[c])then[identity,wfftacs],wfftacs],wfftacs])))])]),lambda(v3,p_ind(v3,lambda(v4,lambda(v2,lambda(v5,su(su(lambda(v6,su(su(su(v9,[v8 of v5],[v9]),[v7 of v2],[v8]),[v6 of v4],[v7]))of lambda(v4,lambda(v2,lambda(v5,su(su(lambda(v6,su(su(su(v9,[v8 of v5],[v9]),[v7 of v2],[v8]),[v6 of v4],[v7]))of _216193,[plus1 of v2],[v6]),[term_of(plus1)],[plus1])))),[plus1 of v4],[v6]),[term_of(plus1)],[plus1])))),[v4,v5,lambda(v6,lambda(v7,lambda(v8,su(su(su(lambda(v2,su(su(su(v11,[v10 of v8],[v11]),[v9 of v7],[v10]),[v2 of v6],[v9]))of lambda(v6,lambda(v7,lambda(v8,su(su(su(lambda(v2,su(su(su(v11,[v10 of v8],[v11]),[v9 of v7],[v10]),[v2 of v6],[v9]))of lambda(v6,lambda(v7,lambda(v8,su(su(su(lambda(v2,su(su(su(v11,[v10 of v8],[v11]),[v9 of v7],[v10]),[v2 of v6],[v9]))of lambda(v6,lambda(v7,lambda(v8,su(su(su(lambda(v2,su(su(su(v11,[v10 of v8],[v11]),[v9 of v7],[v10]),[v2 of v6],[v9]))of lambda(v6,lambda(v7,lambda(v8,su(su(su(lambda(v2,su(su(su(v11,[v10 of v8],[v11]),[v9 of v7],[v10]),[v2 of v6],[v9]))of lambda(v2,lambda(a,lambda(c,axiom))),[v9 of v8],[v10]),[v2 of v7],[v9]),[v5 of v6],[v2])))),[v2 of v7],[v9]),[plus2 of plus(v4,v6)],[v2]),[term_of(plus2)],[plus2])))),[v2 of v6],[v9]),[plus2 of v4],[v2]),[term_of(plus2)],[plus2])))),[v2 of v6],[v9]),[plus2 of plus(v4,v7)],[v2]),[term_of(plus2)],[plus2])))),[v2 of v7],[v9]),[plus2 of v4],[v2]),[term_of(plus2)],[plus2]))))])),
   [problem([b:pnat,v0:pnat,v1:a:pnat=>c:pnat=>times(plus(v0,c),a)=plus(times(v0,a),times(c,a))in pnat,v3:pnat]==>v2:pnat=>a:pnat=>c:pnat=>plus(v2,a)=plus(a,v2)in pnat,
    ind_strat(induction(lemma(pnat_primitive)-[(a:pnat)-s(v4)])then[base_case(sym_eval(normalize_term([reduction([2,1],[plus1,equ(pnat,left)])]))),step_case(ripple(direction_out,wave(direction_out,[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(right,[weak_fertilize(right,in,[1],v5)]))))])]),lambda(v4,lambda(v2,lambda(v5,lambda(v6,su(su(su(v9,[v8 of v5],[v9]),[v7 of v4],[v8]),[v6 of v2],[v7]))of lambda(a,p_ind(a,lambda(v4,lambda(v2,su(su(lambda(v5,su(su(v7,[v6 of v2],[v7]),[v5 of v4],[v6]))of _215440,[plus1 of v4],[v5]),[term_of(plus1)],[plus1]))),[v4,v5,lambda(v6,lambda(v7,su(su(su(lambda(v2,su(su(v9,[v8 of v7],[v9]),[v2 of v6],[v8]))of lambda(v6,lambda(v7,su(su(lambda(v2,su(su(v9,[v8 of v7],[v9]),[v2 of v6],[v8]))of _215333,[v2 of v7],[v8]),[v5 of v6],[v2]))),[v2 of v6],[v8]),[plus2 of v4],[v2]),[term_of(plus2)],[plus2])))]))))),
    [problem([b:pnat,v0:pnat,v1:a:pnat=>c:pnat=>times(plus(v0,c),a)=plus(times(v0,a),times(c,a))in pnat,v3:pnat,a:pnat]==>v2:pnat=>c:pnat=>plus(v2,0)=v2 in pnat,
     ind_strat(induction(lemma(pnat_primitive)-[(v2:pnat)-s(v4)])then[base_case(sym_eval(normalize_term([reduction([1,1],[plus1,equ(pnat,left)])]))then[elementary(intro(new[c])then[identity,wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[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],v5)]))then elementary(intro(new[c])then[identity,wfftacs])))])]),lambda(v2,p_ind(v2,lambda(v4,su(su(lambda(v5,su(v6,[v5 of v4],[v6]))of lambda(c,axiom),[plus1 of 0],[v5]),[term_of(plus1)],[plus1])),[v4,v5,lambda(v6,su(su(su(lambda(v7,su(v8,[v7 of v6],[v8]))of lambda(v6,su(lambda(v7,su(v8,[v7 of v6],[v8]))of lambda(c,axiom),[v5 of v6],[v7])),[v7 of 0],[v8]),[plus2 of v4],[v7]),[term_of(plus2)],[plus2]))])),
     [
     ]) ext _215440,
     problem([b:pnat,v0:pnat,v1:a:pnat=>c:pnat=>times(plus(v0,c),a)=plus(times(v0,a),times(c,a))in pnat,v3:pnat,a:pnat,v4:pnat,v5:v2:pnat=>c:pnat=>plus(v2,v4)=plus(v4,v2)in pnat]==>v2:pnat=>c:pnat=>plus(v2,s(v4))=s(plus(v2,v4))in pnat,
     ind_strat(induction(lemma(pnat_primitive)-[(v2:pnat)-s(v6)])then[base_case(sym_eval(normalize_term([reduction([1,1],[plus1,equ(pnat,left)]),reduction([1,2,1],[plus1,equ(pnat,left)])]))then[elementary(intro(new[c])then[identity,wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[1,1],[plus2,equ(pnat,left)],[])then[wave(direction_out,[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],v7)]))then elementary(intro(new[c])then[identity,wfftacs])))])]),lambda(v2,p_ind(v2,lambda(v6,su(su(lambda(v7,su(v8,[v7 of v6],[v8]))of lambda(v6,su(su(lambda(v7,su(v8,[v7 of v6],[v8]))of lambda(c,axiom),[plus1 of v4],[v7]),[term_of(plus1)],[plus1])),[plus1 of s(v4)],[v7]),[term_of(plus1)],[plus1])),[v6,v7,lambda(v8,su(su(su(lambda(v9,su(v10,[v9 of v8],[v10]))of lambda(v8,su(su(su(lambda(v9,su(v10,[v9 of v8],[v10]))of lambda(v8,su(lambda(v9,su(v10,[v9 of v8],[v10]))of lambda(c,axiom),[v7 of v8],[v9])),[v9 of v4],[v10]),[plus2 of v6],[v9]),[term_of(plus2)],[plus2])),[v9 of s(v4)],[v10]),[plus2 of v6],[v9]),[term_of(plus2)],[plus2]))])),
     [
     ]) ext _215333
    ]) ext _216193
   ]) ext _216670
  ]) ext _216830
 ]) ext _217047
]).
