/*  This is a proof plan for theorem:
    notlesstrans: []==>x:pnat=>y:pnat=>z:pnat=>leq(z,y)=>less(x,z)=>less(x,y)
    planner = dplan, clam_version(2.7.0), oyster_version(1.20)

    Time taken to find plan: 2610ms
    Environment:
    []
 */

/* This is the pretty-printed form
ind_strat([(x:pnat)-s(v1),(y:pnat)-s(v0)]) then 
  [ind_strat([(v0:pnat)-s(v2),(z:pnat)-s(v1)]),
   ind_strat([(v0:pnat)-s(v2),(z:pnat)-s(v1)]),
   ind_strat([(z:pnat)-s(v3)])
  ]

*/

proof_plan([]==>x:pnat=>y:pnat=>z:pnat=>leq(z,y)=>less(x,z)=>less(x,y),notlesstrans,2610,ind_strat(induction(lemma(pairs)-[(x:pnat)-s(v1),(y:pnat)-s(v0)])then[base_case(sym_eval(normalize_term([reduction([1,2],[lesszero,equiv(left)]),reduction([2,2],[lesszero,equiv(left)])]))),base_case(sym_eval(normalize_term([reduction([1],[leqzero,equiv(left)]),reduction([2,2],[less1,equ(u(1),left)])]))),step_case(ripple(direction_out,wave(direction_out,[2,2],[less3,equ(u(1),left)],[]))then[idtac])])then[ind_strat(induction(lemma(pairs)-[(v0:pnat)-s(v2),(z:pnat)-s(v1)])then[base_case(sym_eval(normalize_term([reduction([1],[leqzero,equiv(left)])]))then[elementary(intro(new[v2])then[intro(new[v3])then[intro(new[v4])then[elim(v3)then[hyp(v2),hyp(v5)],wfftacs],wfftacs],wfftacs])]),base_case(elementary(intro(new[v2])then[intro(new[v3])then[intro(new[v4])then[elim(v3)then[identity,hyp(v5)],wfftacs],wfftacs],wfftacs])),step_case(ripple(direction_out,wave(direction_out,[1],[leq3,equ(u(1),left)],[]))then[unblock_then_fertilize(weak,unblock_fertilize_lazy([idtac])then fertilize(weak,fertilize_then_ripple(fertilize_left_or_right(left,[weak_fertilize(left,=>,[],v3)]))then elementary(intro(new[v4])then[intro(new[v5])then[intro(new[v6])then[clam_arith(v6:s(v2)=0 in pnat),wfftacs],wfftacs],wfftacs])))])]),ind_strat(induction(lemma(pairs)-[(v0:pnat)-s(v2),(z:pnat)-s(v1)])then[base_case(sym_eval(normalize_term([reduction([1,2],[lesszero,equiv(left)])]))then[elementary(intro(new[v2])then[intro(new[v3])then[elim(v3)then[hyp(v2),hyp(v4)],wfftacs],wfftacs])]),base_case(sym_eval(normalize_term([reduction([1,2],[less1,equ(u(1),left)])]))then[elementary(intro(new[v2])then[intro(new[v3])then[hyp(v3),wfftacs],wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[1,2],[less3,equ(u(1),left)],[]))then[unblock_then_fertilize(weak,unblock_fertilize_lazy([idtac])then fertilize(weak,fertilize_then_ripple(fertilize_left_or_right(right,[weak_fertilize(right,=>,[],v3)]))then elementary(intro(new[v4])then[clam_arith(v4:s(v1)=0 in pnat),wfftacs])))])]),ind_strat(induction(lemma(pnat_primitive)-[(z:pnat)-s(v3)])then[base_case(sym_eval(normalize_term([reduction([1],[leq1,equ(u(1),left)]),reduction([1,2],[less1,equ(u(1),left)])]))then[elementary(intro(new[v3])then[intro(new[v4])then[elim(v4),wfftacs],wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[1],[leq3,equ(u(1),left)],[])then[wave(direction_out,[1,2],[less3,equ(u(1),left)],[])])then[unblock_then_fertilize(strong,unblock_fertilize_lazy([idtac])then fertilize(strong,v2))])])],dplan).
