/*  This is a proof plan for theorem:
    memunion1: []==>x:int=>a:int list=>b:int list=>member(x,a)=>member(x,union(a,b))
    planner = dplan, clam_version(2.7.0), oyster_version(1.20)

    Time taken to find plan: 2230ms
    Environment:
    []
 */

/* This is the pretty-printed form
ind_strat([(a:int list)-v1::v0]) then 
  [base_case(...),
   base_case(...) then 
     ind_strat([(v0:int list)-v5::v4]) then 
       base_case(...)
  ]

*/

proof_plan([]==>x:int=>a:int list=>b:int list=>member(x,a)=>member(x,union(a,b)),memunion1,2230,ind_strat(induction(lemma(list_primitive)-[(a:int list)-v1::v0])then[base_case(sym_eval(normalize_term([reduction([1],[member1,equ(u(1),left)]),reduction([2,2],[union1,equ(int list,left)])]))then[elementary(intro(new[x])then[intro(new[b])then[intro(new[v0])then[elim(v0),wfftacs],wfftacs],wfftacs])]),step_case(ripple(direction_out,casesplit(disjunction([member(v1,b)=>void,member(v1,b)]))then[wave(direction_out,[2,2],[union4,equ(int list,left)],[])then[casesplit(disjunction([x=v1 in int=>void,x=v1 in int]))then[wave(direction_out,[2],[member3,equ(u(1),left)],[])then[wave(direction_out,[1],[member3,equ(u(1),left)],[])],wave(direction_out,[2],[member2,complementary,equ(u(1),left)],[])]],wave(direction_out,[2,2],[union3,equ(int list,left)],[])then[casesplit(disjunction([x=v1 in int=>void,x=v1 in int]))then[wave(direction_out,[1],[member3,equ(u(1),left)],[]),wave(direction_out,[1],[member2,complementary,equ(u(1),left)],[])]]])then[unblock_then_fertilize(strong,unblock_fertilize_lazy([idtac])then fertilize(strong,v2)),idtac,unblock_then_fertilize(strong,unblock_fertilize_lazy([idtac])then fertilize(strong,v2)),idtac])])then[base_case(elementary(intro(new[v5])then[istrue,wfftacs])),base_case(sym_eval(equal(v4,right)))then[ind_strat(induction(lemma(list_primitive)-[(v0:int list)-v5::v4])then[base_case(sym_eval(normalize_term([reduction([2,2],[union1,equ(int list,left)])]))then[elementary(intro(new[v4])then[hyp(v3),wfftacs])]),step_case(ripple(direction_out,casesplit(disjunction([member(v5,b)=>void,member(v5,b)]))then[wave(direction_out,[2,2],[union4,equ(int list,left)],[])then[casesplit(disjunction([v1=v5 in int=>void,v1=v5 in int]))then[wave(direction_out,[2],[member3,equ(u(1),left)],[]),wave(direction_out,[2],[member2,complementary,equ(u(1),left)],[])]],wave(direction_out,[2,2],[union3,equ(int list,left)],[])])then[unblock_then_fertilize(strong,unblock_fertilize_lazy([idtac])then fertilize(strong,v6)),idtac,unblock_then_fertilize(strong,unblock_fertilize_lazy([idtac])then fertilize(strong,v6))])])then[base_case(elementary(intro(new[v9])then[hyp(v9),wfftacs]))]]],dplan).
