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

    Time taken to find plan: 6010ms
    Environment:
    []
 */

/* This is the pretty-printed form
ind_strat([(x:pnat)-s(v0)]) then 
  [ind_strat([(y:pnat)-s(v0)]) then 
     base_case(...),
   ind_strat([(y:pnat)-s(v2)]) then 
     generalise(times(v2,v0),v4:pnat) then 
       ind_strat([(v4:pnat)-s(v5)]) then 
         ind_strat([(v0:pnat)-s(v5)]) then 
           [ind_strat([(v2:pnat)-s(v5)]),
            generalise(s(v5),v7:pnat) then 
              ind_strat([(v2:pnat)-s(v8)])
           ]
  ]

*/

proof_plan([]==>x:pnat=>y:pnat=>times(x,y)=times(y,x)in pnat,comm,6010,ind_strat(induction(lemma(pnat_primitive)-[(x:pnat)-s(v0)])then[base_case(sym_eval(normalize_term([reduction([1,1],[times1,equ(pnat,left)])]))),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(left,[weak_fertilize(left,in,[1],v1)]))))])])then[ind_strat(induction(lemma(pnat_primitive)-[(y:pnat)-s(v0)])then[base_case(sym_eval(normalize_term([reduction([2,1],[times1,equ(pnat,left)])]))then[elementary(identity)]),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],v1)]))))])])then[base_case(sym_eval(normalize_term([reduction([2,1],[plus1,equ(pnat,left)])]))then[elementary(identity)])],ind_strat(induction(lemma(pnat_primitive)-[(y:pnat)-s(v2)])then[base_case(sym_eval(normalize_term([reduction([1,1,1],[times1,equ(pnat,left)]),reduction([1,1],[plus1,equ(pnat,left)]),reduction([2,1],[times1,equ(pnat,left)])]))then[elementary(identity)]),step_case(ripple(direction_out,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],v3)]))))])])then[generalise(times(v2,v0),v4:pnat)then[ind_strat(induction(lemma(pnat_primitive)-[(v4:pnat)-s(v5)])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],v6)]))then elementary(identity)))])])then[ind_strat(induction(lemma(pnat_primitive)-[(v0:pnat)-s(v5)])then[base_case(sym_eval(normalize_term([reduction([1,1],[plus1,equ(pnat,left)])]))),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(left,[weak_fertilize(left,in,[1],v6)]))))])])then[ind_strat(induction(lemma(pnat_primitive)-[(v2:pnat)-s(v5)])then[base_case(sym_eval(normalize_term([reduction([2,1],[plus1,equ(pnat,left)])]))then[elementary(identity)]),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],v6)]))then elementary(identity)))])]),generalise(s(v5),v7:pnat)then[ind_strat(induction(lemma(pnat_primitive)-[(v2:pnat)-s(v8)])then[base_case(sym_eval(normalize_term([reduction([1,1,1],[plus1,equ(pnat,left)]),reduction([2,1],[plus1,equ(pnat,left)])]))then[elementary(intro(new[v7])then[identity,wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[2,1],[plus2,equ(pnat,left)],[])then[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],v9)]))then elementary(intro(new[v7])then[identity,wfftacs])))])])]]]]]],dplan).
