/*  This is a proof plan for theorem:
    notlesstrans3: []==>x:pnat=>y:pnat=>z:pnat=>(leq(z,y)#leq(y,x))=>leq(z,x)
    planner = dplan, clam_version(2.7.0), oyster_version(1.20)

    Time taken to find plan: 2150ms
    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)]) then 
     base_case(...),
   normalize(...) then 
     base_case(...),
   ind_strat([(z:pnat)-s(v3)])
  ]

*/

proof_plan([]==>x:pnat=>y:pnat=>z:pnat=>(leq(z,y)#leq(y,x))=>leq(z,x),notlesstrans3,2150,ind_strat(induction(lemma(pairs)-[(x:pnat)-s(v1),(y:pnat)-s(v0)])then[base_case(sym_eval(normalize_term([reduction([2,1],[leqzero,equiv(left)]),reduction([2],[leqzero,equiv(left)])]))),base_case(sym_eval(normalize_term([reduction([1,1],[leqzero,equiv(left)]),reduction([2,1],[leq1,equ(u(1),left)])]))),step_case(ripple(direction_out,wave(direction_out,[2,1],[leq3,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([1,1],[leqzero,equiv(left)])]))then[elementary(intro(new[v2])then[elim(v2)then hyp(v3),wfftacs])]),base_case(elementary(intro(new[v2])then[identity,wfftacs])),step_case(ripple(direction_out,wave(direction_out,[1,1],[leq3,equ(u(1),left)],[]))then[idtac])])then[base_case(elementary(intro(new[v4])then[elim(v4)then clam_arith(v6:s(v2)=0 in pnat),wfftacs]))],normalize([normal(univ_intro),normal(imply_intro),normal(conjunct_elim(v1,[v2,v3]))])then[base_case(sym_eval(equal(v2,right)then[normalize_term([reduction([],[leq1,equ(u(1),left)])])])then[elementary(hyp(v3))])],ind_strat(induction(lemma(pnat_primitive)-[(z:pnat)-s(v3)])then[base_case(sym_eval(normalize_term([reduction([1,1],[leq1,equ(u(1),left)]),reduction([2],[leq1,equ(u(1),left)])]))then[elementary(intro(new[v3])then[istrue,wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[2],[leq3,equ(u(1),left)],[])then[wave(direction_out,[1,1],[leq3,equ(u(1),left)],[])])then[unblock_then_fertilize(strong,unblock_fertilize_lazy([idtac])then fertilize(strong,v2))])])],dplan).
