/*  This is a proof plan for theorem:
    exptimes: []==>x:pnat=>m:pnat=>n:pnat=>exp(x,times(n,m))=exp(exp(x,n),m)in pnat
    planner = dplan, clam_version(2.7.0), oyster_version(1.20)

    Time taken to find plan: 18070ms
    Environment:
    []
 */

/* This is the pretty-printed form
ind_strat([(m:pnat)-plus(v1,v0)]) then 
  [ind_strat([(n:pnat)-s(v0)]) then 
     base_case(...) then 
       generalise(exp(x,times(v0,0)),v2:pnat) then 
         ind_strat([(v2:pnat)-s(v3)]) then 
           generalise(times(v3,s(0)),v5:pnat) then 
             ind_strat([(v5:pnat)-plus(v7,v6)]) then 
               ind_strat([(v7:pnat)-s(v10)]),
   ind_strat([(n:pnat)-s(v0)]) then 
     generalise(exp(x,v0),v2:pnat) then 
       ind_strat([(v2:pnat)-plus(v4,v3)]) then 
         [ind_strat([(x:pnat)-s(v3)]) then 
            base_case(...),
          generalise(times(x,s(0)),v3:pnat) then 
            ind_strat([(v3:pnat)-s(v4)]) then 
              ind_strat([(v4:pnat)-s(v6)])
         ]
  ]

*/

proof_plan([]==>x:pnat=>m:pnat=>n:pnat=>exp(x,times(n,m))=exp(exp(x,n),m)in pnat,exptimes,18070,ind_strat(induction(lemma(plusind)-[(m:pnat)-plus(v1,v0)])then[base_case(sym_eval(normalize_term([reduction([2,1],[exp1,equ(pnat,left)])]))),base_case(sym_eval(normalize_term([reduction([2,1],[exp2,equ(pnat,left)]),reduction([2,2,1],[exp1,equ(pnat,left)])]))),step_case(ripple(direction_out,wave(direction_out,[2,1],[expplus,equ(pnat,left)],[])then[wave(direction_out,[2,1,1],[dist,equ(pnat,left)],[])then[wave(direction_out,[1,1],[expplus,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],v2),weak_fertilize(right,in,[2],v3)]))then elementary(intro(new[x])then[intro(new[n])then[identity,wfftacs],wfftacs])))])])then[ind_strat(induction(lemma(pnat_primitive)-[(n:pnat)-s(v0)])then[base_case(sym_eval(normalize_term([reduction([2,1,1],[times1,equ(pnat,left)]),reduction([1,1],[exp1,equ(pnat,left)])]))then[elementary(intro(new[x])then[identity,wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[2,1,1],[times2,equ(pnat,left)],[])then[wave(direction_out,[1,1],[expplus,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)]))))])])then[base_case(sym_eval(normalize_term([reduction([2,1,1],[exp1,equ(pnat,left)])])))then[generalise(exp(x,times(v0,0)),v2:pnat)then[ind_strat(induction(lemma(pnat_primitive)-[(v2:pnat)-s(v3)])then[base_case(sym_eval(normalize_term([reduction([1,1],[times1,equ(pnat,left)])]))then[elementary(intro(new[x])then[identity,wfftacs])]),step_case(ripple(direction_out,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(right,[weak_fertilize(right,in,[1],v4)]))))])])then[generalise(times(v3,s(0)),v5:pnat)then[ind_strat(induction(lemma(plusind)-[(v5:pnat)-plus(v7,v6)])then[base_case(sym_eval(normalize_term([reduction([1,1],[plus1,equ(pnat,left)])]))then[elementary(intro(new[x])then[identity,wfftacs])]),base_case(sym_eval(normalize_term([reduction([1,1],[plus2,equ(pnat,left)]),reduction([1,1,1],[plus1,equ(pnat,left)])]))then[elementary(intro(new[x])then[identity,wfftacs])]),step_case(ripple(direction_out,unblock_then_wave(direction_out,unblock_lazy([unblock(meta_ripple,unused,unused),idtac])then wave(direction_out,[2,1],[plus2,equ(pnat,right)],[])))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],v8)]))))])])then[ind_strat(induction(lemma(pnat_primitive)-[(v7:pnat)-s(v10)])then[base_case(sym_eval(normalize_term([reduction([1,1,1],[plus1,equ(pnat,left)]),reduction([1,2,1],[plus1,equ(pnat,left)]),reduction([2,1],[plus2,equ(pnat,left)]),reduction([1,2,1],[plus1,equ(pnat,left)])]))then[elementary(hyp(v9))]),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],v11)]))then elementary(intro(new[x])then[identity,wfftacs])))])])]]]]]],ind_strat(induction(lemma(pnat_primitive)-[(n:pnat)-s(v0)])then[base_case(sym_eval(normalize_term([reduction([2,1,1],[times1,equ(pnat,left)]),reduction([1,1],[exp1,equ(pnat,left)]),reduction([1,2,1],[exp1,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(intro(new[x])then[identity,wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[1,2,1],[exp2,equ(pnat,left)],[])then[wave(direction_out,[2,1,1],[times2,equ(pnat,left)],[])then[wave(direction_out,[1,1],[expplus,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)])then ripple_and_cancel([unblock(wave_front,[2,1,1],[exp2,equ(pnat,left)]),unblock(wave_front,[2,2,1,1],[exp1,equ(pnat,left)])]))))])])then[generalise(exp(x,v0),v2:pnat)then[ind_strat(induction(lemma(plusind)-[(v2:pnat)-plus(v4,v3)])then[base_case(sym_eval(normalize_term([reduction([1,1,1],[times1,equ(pnat,left)]),reduction([1,1],[times1,equ(pnat,left)])]))),base_case(sym_eval(normalize_term([reduction([1,1,1],[times2,equ(pnat,left)]),reduction([1,1,1,1],[times1,equ(pnat,left)]),reduction([1,1,1],[plus1,equ(pnat,left)]),reduction([1,1],[times2,equ(pnat,left)]),reduction([1,1,1],[times1,equ(pnat,left)]),reduction([1,1],[plus1,equ(pnat,left)])]))),step_case(ripple(direction_out,wave(direction_out,[1,2,1],[dist,equ(pnat,left)],[])then[wave(direction_out,[2,1],[disttwo,equ(pnat,left)],[])then[wave(direction_out,[1,1,1],[disttwo,equ(pnat,left)],[])then[wave(direction_out,[1,1],[disttwo,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),weak_fertilize(right,in,[2],v6)]))then elementary(intro(new[x])then[identity,wfftacs])))])])then[ind_strat(induction(lemma(pnat_primitive)-[(x:pnat)-s(v3)])then[base_case(sym_eval(normalize_term([reduction([1,2,1],[times1,equ(pnat,left)]),reduction([2,1],[times1,equ(pnat,left)])]))then[elementary(identity)]),step_case(ripple(direction_out,wave(direction_out,[1,2,1],[times2,equ(pnat,left)],[])then[wave(direction_out,[2,1],[disttwo,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 ripple_and_cancel([unblock(wave_front,[2,2,1],[times1,equ(pnat,left)])]))))])])then[base_case(sym_eval(normalize_term([reduction([2,1],[plus1,equ(pnat,left)])]))then[elementary(identity)])],generalise(times(x,s(0)),v3:pnat)then[ind_strat(induction(lemma(pnat_primitive)-[(v3:pnat)-s(v4)])then[base_case(sym_eval(normalize_term([reduction([2,1],[times1,equ(pnat,left)])]))then[elementary(intro(new[x])then[identity,wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[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(right,[weak_fertilize(right,in,[1],v5)]))))])])then[ind_strat(induction(lemma(pnat_primitive)-[(v4:pnat)-s(v6)])then[base_case(sym_eval(normalize_term([reduction([2,1],[plus1,equ(pnat,left)])]))then[elementary(intro(new[x])then[identity,wfftacs])]),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],v7)]))then elementary(intro(new[x])then[identity,wfftacs])))])])]]]]]],dplan).
