/*  This is a proof plan for theorem:
    binom_one: []==>x:pnat=>minus(binom(x,s(0)),x)=0 in pnat
    planner = dplan, clam_version(2.7.0), oyster_version(1.20)

    Time taken to find plan: 5440ms
    Environment:
    []
 */

/* This is the pretty-printed form
ind_strat([(x:pnat)-s(v0)]) then 
  base_case(...) then 
    generalise(binom(v0,s(0)),v2:pnat) then 
      ind_strat([(v0:pnat)-s(v3)]) then 
        ind_strat([(v2:pnat)-s(v3)]) then 
          base_case(...) then 
            ind_strat([(v3:pnat)-s(v5)]) then 
              base_case(...) then 
                ind_strat([(v5:pnat)-s(v7)]) then 
                  generalise(plus(v7,s(0)),v9:pnat) then 
                    base_case(...)

*/

proof_plan([]==>x:pnat=>minus(binom(x,s(0)),x)=0 in pnat,binom_one,5440,ind_strat(induction(lemma(pnat_primitive)-[(x:pnat)-s(v0)])then[base_case(sym_eval(normalize_term([reduction([1,1],[minus1,equ(pnat,left)]),reduction([1,1],[binom2,equ(pnat,left)])]))then[elementary(identity)]),step_case(ripple(direction_out,wave(direction_out,[1,1],[minus2,equ(pnat,left)],[])then[wave(direction_out,[1,1,1,1],[binom3,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([2,1,1,1,1],[binom1,equ(pnat,left)])])))then[generalise(binom(v0,s(0)),v2:pnat)then[ind_strat(induction(lemma(pnat_primitive)-[(v0:pnat)-s(v3)])then[base_case(sym_eval(normalize_term([reduction([1,1,1],[minus1,equ(pnat,left)]),reduction([2,1],[minus1,equ(pnat,left)])]))),step_case(ripple(direction_out,wave(direction_out,[2,1],[minus2,equ(pnat,left)],[])then[wave(direction_out,[1,1,1],[minus2,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 elementary(intro(new[v2])then[identity,wfftacs])))])])then[ind_strat(induction(lemma(pnat_primitive)-[(v2:pnat)-s(v3)])then[base_case(sym_eval(normalize_term([reduction([1,1,1],[plus1,equ(pnat,left)]),reduction([1,1],[pred2,equ(pnat,left)])]))then[elementary(identity)]),step_case(ripple(direction_out,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],v4)]))))])])then[base_case(sym_eval(normalize_term([reduction([1,1],[pred2,equ(pnat,left)])])))then[ind_strat(induction(lemma(pnat_primitive)-[(v3:pnat)-s(v5)])then[base_case(sym_eval(normalize_term([reduction([1,1],[plus1,equ(pnat,left)]),reduction([1,1,2,1],[plus1,equ(pnat,left)]),reduction([1,2,1],[pred2,equ(pnat,left)])]))then[elementary(identity)]),step_case(ripple(direction_out,wave(direction_out,[1,1],[plus2,equ(pnat,left)],[])then[wave(direction_out,[1,1,2,1],[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],v6)]))))])])then[base_case(sym_eval(normalize_term([reduction([1,2,1],[pred2,equ(pnat,left)])])))then[ind_strat(induction(lemma(pnat_primitive)-[(v5:pnat)-s(v7)])then[base_case(sym_eval(normalize_term([reduction([1,1,1,1,1],[plus1,equ(pnat,left)]),reduction([1,1,1,1],[pred2,equ(pnat,left)]),reduction([1,2,1],[plus1,equ(pnat,left)])]))then[elementary(identity)]),step_case(ripple(direction_out,wave(direction_out,[1,2,1],[plus2,equ(pnat,left)],[])then[wave(direction_out,[1,1,1,1,1],[plus2,equ(pnat,left)],[])])then[idtac])])then[generalise(plus(v7,s(0)),v9:pnat)then[base_case(sym_eval(normalize_term([reduction([1,1,1,1],[pred2,equ(pnat,left)])]))then[elementary(intro(new[v9])then[identity,wfftacs])])]]]]]]]]]],dplan).
