/*  This is a proof plan for theorem:
    disttwo: []==>a:pnat=>b:pnat=>c:pnat=>times(plus(b,c),a)=plus(times(b,a),times(c,a))in pnat
    planner = dplan, clam_version(2.7.0), oyster_version(1.20)

    Time taken to find plan: 4870ms
    Environment:
    []
 */

/* This is the pretty-printed form
ind_strat([(b:pnat)-s(v0)]) then 
  generalise(times(c,a),v2:pnat) then 
    generalise(times(v0,a),v3:pnat) then 
      ind_strat([(v3:pnat)-s(v4)]) then 
        ind_strat([(a:pnat)-s(v4)]) then 
          [ind_strat([(v2:pnat)-s(v4)]),
           ind_strat([(v2:pnat)-s(v6)])
          ]

*/

proof_plan([]==>a:pnat=>b:pnat=>c:pnat=>times(plus(b,c),a)=plus(times(b,a),times(c,a))in pnat,disttwo,4870,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)]))))])])then[generalise(times(c,a),v2:pnat)then[generalise(times(v0,a),v3:pnat)then[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])))])])then[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)]))))])])then[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])))])]),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])))])])]]]]],dplan).
