/*
 * @(#)$Id: pairs,v 1.1 1994/09/16 09:34:14 dream Exp $
 *
 * $Log: pairs,v $
 * Revision 1.1  1994/09/16 09:34:14  dream
 * Initial revision
 *
 */

problem([]==>phi: (pnat=>pnat=>u(2))=>x:pnat=>y:pnat=> (y:pnat=>phi of 0 of y)=> (x:pnat=>phi of x of 0)=> (x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y))=>phi of x of y,
autotactic(wfftacs),_7380,
[problem([]==>phi: (pnat=>pnat=>u(2))=>x:pnat=>y:pnat=> (y:pnat=>phi of 0 of y)=> (x:pnat=>phi of x of 0)=> (x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y))=>phi of x of y,
 universe(3),_7491,
 [problem([]==>phi: (pnat=>pnat=>u(2))=>x:pnat=>y:pnat=> (y:pnat=>phi of 0 of y)=> (x:pnat=>phi of x of 0)=> (x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y))=>phi of x of y,
  repeat intro,lambda(phi,lambda(x,lambda(y,lambda(v0,lambda(v1,lambda(v2,_7722)))))),
  [problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y)]==>phi of x of y,
   seq(a:acc(pnat#pnat,{pairord})=>phi of fst(a)of snd(a),new[v3]),lambda(v3,_7871)of _7868,
   [problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y)]==>a:acc(pnat#pnat,{pairord})=>phi of fst(a)of snd(a),
    intro,lambda(a,_8004),
    [problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),a:acc(pnat#pnat,{pairord})]==>phi of fst(a)of snd(a),
     elim(a,wo),wo_ind(a,[v5,v3,_8144]),
     [problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),a:acc(pnat#pnat,{pairord}),v5:pnat#pnat,v3:v4:{v4: (pnat#pnat)\{pairord}of v4 of v5}=>phi of fst(v4)of snd(v4)]==>phi of fst(v5)of snd(v5),
      decide(fst(v5)=0 in pnat),pnat_eq(fst(v5),0,su(_8341,[axiom],[v4]),su(_8349,[lambda(~,axiom)],[v4])),
      [problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),a:acc(pnat#pnat,{pairord}),v5:pnat#pnat,v3:v4:{v4: (pnat#pnat)\{pairord}of v4 of v5}=>phi of fst(v4)of snd(v4),v4:fst(v5)=0 in pnat]==>phi of fst(v5)of snd(v5),
       rewrite(v4),_8367,
       [problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),a:acc(pnat#pnat,{pairord}),v5:pnat#pnat,v3:v4:{v4: (pnat#pnat)\{pairord}of v4 of v5}=>phi of fst(v4)of snd(v4),v4:fst(v5)=0 in pnat]==>phi of 0 of snd(v5),
        elim_on(v0,[snd(v5)])then hyp(v6),su(v6,[v0 of snd(v5)],[v6]),
        [
        ]) ext _8367
       ]) ext _8341,
       problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),a:acc(pnat#pnat,{pairord}),v5:pnat#pnat,v3:v4:{v4: (pnat#pnat)\{pairord}of v4 of v5}=>phi of fst(v4)of snd(v4),v4:fst(v5)=0 in pnat=>void]==>phi of fst(v5)of snd(v5),
       decide(snd(v5)=0 in pnat),pnat_eq(snd(v5),0,su(_8975,[axiom],[v6]),su(_8983,[lambda(~,axiom)],[v6])),
       [problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),a:acc(pnat#pnat,{pairord}),v5:pnat#pnat,v3:v4:{v4: (pnat#pnat)\{pairord}of v4 of v5}=>phi of fst(v4)of snd(v4),v4:fst(v5)=0 in pnat=>void,v6:snd(v5)=0 in pnat]==>phi of fst(v5)of snd(v5),
        rewrite(v6),_9001,
        [problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),a:acc(pnat#pnat,{pairord}),v5:pnat#pnat,v3:v4:{v4: (pnat#pnat)\{pairord}of v4 of v5}=>phi of fst(v4)of snd(v4),v4:fst(v5)=0 in pnat=>void,v6:snd(v5)=0 in pnat]==>phi of fst(v5)of 0,
         strong_fertilize(v1),su(v7,[v1 of fst(v5)],[v7]),
         [
         ]) ext _9001
        ]) ext _8975,
        problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),a:acc(pnat#pnat,{pairord}),v5:pnat#pnat,v3:v4:{v4: (pnat#pnat)\{pairord}of v4 of v5}=>phi of fst(v4)of snd(v4),v4:fst(v5)=0 in pnat=>void,v6:snd(v5)=0 in pnat=>void]==>phi of fst(v5)of snd(v5),
        elim(v5),spread(v5,[v7,v8,_9638]),
        [problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),a:acc(pnat#pnat,{pairord}),v5:pnat#pnat,v3:v4:{v4: (pnat#pnat)\{pairord}of v4 of v5}=>phi of fst(v4)of snd(v4),v4:fst(v5)=0 in pnat=>void,v6:snd(v5)=0 in pnat=>void,v7:pnat,v8:pnat,v9:v5=v7&v8 in (pnat#pnat)]==>phi of fst(v7&v8)of snd(v7&v8),
         simplify,_9648,
         [problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),a:acc(pnat#pnat,{pairord}),v5:pnat#pnat,v3:v4:{v4: (pnat#pnat)\{pairord}of v4 of v5}=>phi of fst(v4)of snd(v4),v4:fst(v5)=0 in pnat=>void,v6:snd(v5)=0 in pnat=>void,v7:pnat,v8:pnat,v9:v5=v7&v8 in (pnat#pnat)]==>phi of v7 of v8,
          elim(v3,on(pred(v7)&pred(v8))),su(_10127,[v3 of (pred(v7)&pred(v8))],[v10]),
          [problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),a:acc(pnat#pnat,{pairord}),v5:pnat#pnat,v3:v4:{v4: (pnat#pnat)\{pairord}of v4 of v5}=>phi of fst(v4)of snd(v4),v4:fst(v5)=0 in pnat=>void,v6:snd(v5)=0 in pnat=>void,v7:pnat,v8:pnat,v9:v5=v7&v8 in (pnat#pnat),v10:phi of fst(pred(v7)&pred(v8))of snd(pred(v7)&pred(v8))]==>phi of v7 of v8,
           elim_on(v2,[pred(v7),pred(v8)]),su(su(_10421,[v11 of pred(v8)],[v12]),[v2 of pred(v7)],[v11]),
           [problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),a:acc(pnat#pnat,{pairord}),v5:pnat#pnat,v3:v4:{v4: (pnat#pnat)\{pairord}of v4 of v5}=>phi of fst(v4)of snd(v4),v4:fst(v5)=0 in pnat=>void,v6:snd(v5)=0 in pnat=>void,v7:pnat,v8:pnat,v9:v5=v7&v8 in (pnat#pnat),v10:phi of fst(pred(v7)&pred(v8))of snd(pred(v7)&pred(v8)),v12:phi of pred(v7)of pred(v8)=>phi of s(pred(v7))of s(pred(v8))]==>phi of v7 of v8,
            elim(v12),su(_10738,[v12 of _10745],[v11]),
            [problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),a:acc(pnat#pnat,{pairord}),v5:pnat#pnat,v3:v4:{v4: (pnat#pnat)\{pairord}of v4 of v5}=>phi of fst(v4)of snd(v4),v4:fst(v5)=0 in pnat=>void,v6:snd(v5)=0 in pnat=>void,v7:pnat,v8:pnat,v9:v5=v7&v8 in (pnat#pnat),v10:phi of fst(pred(v7)&pred(v8))of snd(pred(v7)&pred(v8)),v12:phi of pred(v7)of pred(v8)=>phi of s(pred(v7))of s(pred(v8))]==>phi of pred(v7)of pred(v8),
             compute(hyp(v10),[[simplify]]),_10756,
             [problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),a:acc(pnat#pnat,{pairord}),v5:pnat#pnat,v3:v4:{v4: (pnat#pnat)\{pairord}of v4 of v5}=>phi of fst(v4)of snd(v4),v4:fst(v5)=0 in pnat=>void,v6:snd(v5)=0 in pnat=>void,v7:pnat,v8:pnat,v9:v5=v7&v8 in (pnat#pnat),v10:phi of p_ind(v7,0,[a,~,a])of p_ind(v8,0,[a,~,a]),v12:phi of pred(v7)of pred(v8)=>phi of s(pred(v7))of s(pred(v8))]==>phi of pred(v7)of pred(v8),
              simplify,_11062,
              [problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),a:acc(pnat#pnat,{pairord}),v5:pnat#pnat,v3:v4:{v4: (pnat#pnat)\{pairord}of v4 of v5}=>phi of fst(v4)of snd(v4),v4:fst(v5)=0 in pnat=>void,v6:snd(v5)=0 in pnat=>void,v7:pnat,v8:pnat,v9:v5=v7&v8 in (pnat#pnat),v10:phi of p_ind(v7,0,[a,~,a])of p_ind(v8,0,[a,~,a]),v12:phi of pred(v7)of pred(v8)=>phi of s(pred(v7))of s(pred(v8))]==>phi of p_ind(v7,0,[a,~,a])of p_ind(v8,0,[a,~,a]),
               intro,v10,
               [
               ]) ext _11062
              ]) ext _10756
             ]) ext _10745,
             problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),a:acc(pnat#pnat,{pairord}),v5:pnat#pnat,v3:v4:{v4: (pnat#pnat)\{pairord}of v4 of v5}=>phi of fst(v4)of snd(v4),v4:fst(v5)=0 in pnat=>void,v6:snd(v5)=0 in pnat=>void,v7:pnat,v8:pnat,v9:v5=v7&v8 in (pnat#pnat),v10:phi of fst(pred(v7)&pred(v8))of snd(pred(v7)&pred(v8)),v12:phi of pred(v7)of pred(v8)=>phi of s(pred(v7))of s(pred(v8)),v11:phi of s(pred(v7))of s(pred(v8))]==>phi of v7 of v8,
             intro(explicit(v11)),v11,
             [
             ]) ext _10738
            ]) ext _10421
           ]) ext _10127
          ]) ext _9648
         ]) ext _9638
        ]) ext _8983
       ]) ext _8349
      ]) ext _8144
     ]) ext _8004
    ]) ext _7868,
    problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),v3:a:acc(pnat#pnat,{pairord})=>phi of fst(a)of snd(a)]==>phi of x of y,
    elim(v3,on(x&y)),su(_12134,[v3 of (x&y)],[v4]),
    [problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),v3:a:acc(pnat#pnat,{pairord})=>phi of fst(a)of snd(a),v4:phi of fst(x&y)of snd(x&y)]==>phi of x of y,
     compute(hyp(v4),[[simplify]]),_12155,
     [problem([phi:pnat=>pnat=>u(2),x:pnat,y:pnat,v0:y:pnat=>phi of 0 of y,v1:x:pnat=>phi of x of 0,v2:x:pnat=>y:pnat=>phi of x of y=>phi of s(x)of s(y),v3:a:acc(pnat#pnat,{pairord})=>phi of fst(a)of snd(a),v4:phi of x of y]==>phi of x of y,
      intro,v4,
      [
      ]) ext _12155
     ]) ext _12134
    ]) ext _7871
   ]) ext _7722
  ]) ext _7491
 ]) ext _7380
]).
