/*  This is a proof plan for theorem:
    prodlwave: []==>w:pnat=>v:pnat=>wl:pnat list=>vl:pnat list=>(prodl(wl)=w in pnat#prodl(vl)=v in pnat)=>prodl(pnatapp(wl,vl))=times(w,v)in pnat
    planner = dplan, clam_version(2.7.0), oyster_version(1.20)

    Time taken to find plan: 8250ms
    Environment:
    []
 */

/* This is the pretty-printed form
normalize(...) then 
  base_case(...) then 
    ind_strat([(wl:pnat list)-v1::v0]) then 
      generalise(prodl(vl),v3:pnat) then 
        generalise(prodl(v0),v4:pnat) then 
          ind_strat([(v1:pnat)-s(v5)]) then 
            generalise(times(v5,v4),v7:pnat) then 
              ind_strat([(v7:pnat)-s(v8)]) then 
                generalise(times(v4,v3),v10:pnat) then 
                  generalise(times(v8,v3),v11:pnat) then 
                    ind_strat([(v11:pnat)-s(v12)]) then 
                      ind_strat([(v10:pnat)-s(v12)]) then 
                        [ind_strat([(v3:pnat)-s(v12)]),
                         ind_strat([(v3:pnat)-s(v14)])
                        ]

*/

proof_plan([]==>w:pnat=>v:pnat=>wl:pnat list=>vl:pnat list=>(prodl(wl)=w in pnat#prodl(vl)=v in pnat)=>prodl(pnatapp(wl,vl))=times(w,v)in pnat,prodlwave,8250,normalize([normal(univ_intro),normal(univ_intro),normal(univ_intro),normal(univ_intro),normal(imply_intro),normal(conjunct_elim(v0,[v1,v2]))])then[base_case(sym_eval(equal(v1,left)then[equal(v2,left)]))then[ind_strat(induction(lemma(list_primitive)-[(wl:pnat list)-v1::v0])then[base_case(sym_eval(normalize_term([reduction([1,1,1],[pnatapp1,equ(pnat list,left)]),reduction([1,2,1],[prodl1,equ(pnat,left)]),reduction([2,1],[times2,equ(pnat,left)]),reduction([1,2,1],[times1,equ(pnat,left)]),reduction([2,1],[plus1,equ(pnat,left)])]))then[elementary(identity)]),step_case(ripple(direction_out,wave(direction_out,[1,2,1],[prodl2,equ(pnat,left)],[])then[wave(direction_out,[1,1,1],[pnatapp2,equ(pnat list,left)],[])then[wave(direction_out,[1,1],[prodl2,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,[2],v2)]))))])])then[generalise(prodl(vl),v3:pnat)then[generalise(prodl(v0),v4:pnat)then[ind_strat(induction(lemma(pnat_primitive)-[(v1:pnat)-s(v5)])then[base_case(sym_eval(normalize_term([reduction([1,1],[times1,equ(pnat,left)]),reduction([1,2,1],[times1,equ(pnat,left)]),reduction([2,1],[times1,equ(pnat,left)])]))then[elementary(intro(new[v4])then[intro(new[v3])then[identity,wfftacs],wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[1,1],[times2,equ(pnat,left)],[])then[wave(direction_out,[1,2,1],[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],v6)]))))])])then[generalise(times(v5,v4),v7:pnat)then[ind_strat(induction(lemma(pnat_primitive)-[(v7:pnat)-s(v8)])then[base_case(sym_eval(normalize_term([reduction([1,1,1],[times1,equ(pnat,left)]),reduction([1,1],[plus1,equ(pnat,left)]),reduction([1,2,1],[plus1,equ(pnat,left)])]))then[elementary(intro(new[v4])then[intro(new[v3])then[identity,wfftacs],wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[1,2,1],[plus2,equ(pnat,left)],[])then[wave(direction_out,[2,1],[times2,equ(pnat,left)],[])then[wave(direction_out,[1,1,1],[times2,equ(pnat,left)],[])]])then[unblock_then_fertilize(weak,unblock_fertilize_lazy([idtac])then fertilize(weak,fertilize_then_ripple(fertilize_left_or_right(right,[weak_fertilize(right,in,[1],v9)]))))])])then[generalise(times(v4,v3),v10:pnat)then[generalise(times(v8,v3),v11:pnat)then[ind_strat(induction(lemma(pnat_primitive)-[(v11:pnat)-s(v12)])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],v13)]))then elementary(intro(new[v10])then[intro(new[v4])then[intro(new[v3])then[identity,wfftacs],wfftacs],wfftacs])))])])then[ind_strat(induction(lemma(pnat_primitive)-[(v10:pnat)-s(v12)])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],v13)]))))])])then[ind_strat(induction(lemma(pnat_primitive)-[(v3:pnat)-s(v12)])then[base_case(sym_eval(normalize_term([reduction([1,1],[plus1,equ(pnat,left)])]))then[elementary(intro(new[v4])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],v13)]))then elementary(intro(new[v4])then[identity,wfftacs])))])]),ind_strat(induction(lemma(pnat_primitive)-[(v3:pnat)-s(v14)])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[v4])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],v15)]))then elementary(intro(new[v4])then[identity,wfftacs])))])])]]]]]]]]]]]],dplan).
