problem([]==>x:pnat=>y:pnat=>z:pnat=>(greater(x,y)#greater(y,z))=>greater(x,z),
ind_strat(induction(lemma(pairs)-[(x:pnat)-s(v1),(y:pnat)-s(v0)])then[base_case(sym_eval(normalize_term([reduction([1,1],[greater1,equ(u(1),left)]),reduction([2],[greater1,equ(u(1),left)])]))then[elementary(intro(new[z])then[intro(new[v1])then[elim(v1)then hyp(v2),wfftacs],wfftacs])]),base_case(sym_eval(normalize_term([reduction([1,1],[greaterzero,equiv(left)]),reduction([2,1],[greater1,equ(u(1),left)])]))then[elementary(intro(new[z])then[intro(new[v1])then[elim(v1)then elim(v3),wfftacs],wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[1,1],[greater3,equ(u(1),left)],[]))then[idtac])]),lambda(v0,lambda(v1,lambda(v2,lambda(v3,su(su(su(v6,[v5 of v2],[v6]),[v4 of v0],[v5]),[v3 of v1],[v4]))of lambda(v0,lambda(v1,lambda(v2,lambda(v3,su(su(su(v6,[v5 of v2],[v6]),[v4 of v0],[v5]),[v3 of v1],[v4]))of lambda(x,lambda(y,su(lambda(v3,su(su(su(su(su(su(v10,[v9 of lambda(v1,lambda(v0,lambda(v2,lambda(v3,su(su(su(lambda(v4,su(v5,[v4 of v3],[v5]))of _121534,[v4 of v0],[v5]),[greater3 of v1],[v4]),[term_of(greater3)],[greater3])))))],[v10]),[v8 of lambda(v0,lambda(v1,su(su(spread(v2,[v3,v4,lambda(v6,lambda(v7,su(v8,[v6 of spread(v7,[v8,v9,su(v11,[v3 of v8],[v11])&v9])],[v8])))of(lambda(v2,su(v3,[v2 of v1],[v3]))of lambda(v1,su(su(lambda(v2,su(v3,[v2 of v1],[v3]))of lambda(z,lambda(v1,spread(v1,[v2,v3,any(v3)]))),[greater1 of v1],[v2]),[term_of(greater1)],[greater1])))]),[greaterzero of v0],[v2]),[term_of(greaterzero)],[greaterzero])))],[v9]),[v7 of lambda(v0,lambda(v1,su(su(lambda(v2,su(v3,[v2 of v1],[v3]))of lambda(v1,su(su(lambda(v2,su(v3,[v2 of v1],[v3]))of lambda(z,lambda(v1,spread(v1,[v2,v3,v2]))),[greater1 of v1],[v2]),[term_of(greater1)],[greater1])),[greater1 of v0],[v2]),[term_of(greater1)],[greater1])))],[v8]),[v6 of y],[v7]),[v5 of x],[v6]),[v3 of lambda(x,lambda(y,z:pnat=>(greater(x,y)#greater(y,z))=>greater(x,z)))],[v5]))of v4,[term_of(pairs)],[v4]))))))))),
[problem([x:pnat,y:pnat,v1:pnat,v0:pnat,v2:z:pnat=>(greater(v1,v0)#greater(v0,z))=>greater(v1,z)]==>z:pnat=>(greater(v1,v0)#greater(s(v0),z))=>greater(s(v1),z),
 ind_strat(induction(lemma(pnat_primitive)-[(z:pnat)-s(v3)])then[base_case(sym_eval(normalize_term([reduction([2,1],[greaterzero,equiv(left)]),reduction([2],[greaterzero,equiv(left)])]))then[elementary(intro(new[v3])then[intro(new[v4])then[clam_arith(v4:s(v1)=0 in pnat),wfftacs],wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[2],[greater3,equ(u(1),left)],[])then[wave(direction_out,[2,1],[greater3,equ(u(1),left)],[])])then[unblock_then_fertilize(strong,unblock_fertilize_lazy([idtac])then fertilize(strong,v2))])]),lambda(z,p_ind(z,su(su(spread(v3,[v4,v5,lambda(v7,lambda(v8,su(v9,[v7 of spread(v8,[v9,v10,v9&su(v12,[v4 of v10],[v12])])],[v9])))of su(su(spread(v3,[v4,v5,lambda(v7,lambda(v8,spread(v8,[v9,v10,spread(v3,[v12,v13,su(su(su(su(su(su(su(v17,[v16 of v13],[v17]),[v15 of(lambda(v16,v16)&lambda(v16,lambda(v17,su(su(su(su(su(su(su(v20,[v19 of v5],[v20]),[v18 of(lambda(v19,su(su(v21,[v20 of v17],[v21]),[v12 of v19],[v20]))&lambda(v19,lambda(v20,su(su(su(su(su(su(su(v23,[v22 of v7],[v23]),[v21 of lambda(v22,su(su(su(su(su(su(su(su(v26,[v25 of v23],[v26]),[v24 of(lambda(v25,su(v26,[v25 of v20],[v26]))&lambda(v25,v10))],[v25]),[v27 of void],[v24]),[v26 of(s(v0)=0 in pnat)],[v27]),[v25 of(s(v1)=0 in pnat=>void)],[v26]),[v24 of void],[v25]),[term_of(ipc_dp_imp_e4)],[v24]),[v22 of v9],[v23]))],[v22]),[v24 of void],[v21]),[v23 of(s(v1)=0 in pnat=>void)],[v24]),[v22 of(s(v0)=0 in pnat=>void)],[v23]),[v21 of greater(v1,v0)],[v22]),[term_of(ipc_dp_imp_e2)],[v21]))))],[v19]),[v21 of void],[v18]),[v20 of(s(v1)=0 in pnat)],[v21]),[v19 of greater(s(v1),0)],[v20]),[v18 of void],[v19]),[term_of(ipc_dp_imp_e4)],[v18]))))],[v16]),[v18 of void],[v15]),[v17 of(s(v1)=0 in pnat)],[v18]),[v16 of greater(s(v1),0)],[v17]),[v15 of greater(s(v1),0)],[v16]),[term_of(ipc_dp_imp_e4)],[v15])])])))of lambda(v3,lambda(v4,su(su(su(any(v7),[v6 of v4],[v7]),[v5 of v1],[v6]),[term_of(arith1)],[v5])))]),[greaterzero of s(v1)],[v3]),[term_of(greaterzero)],[greaterzero])]),[greaterzero of s(v0)],[v3]),[term_of(greaterzero)],[greaterzero]),[v3,v4,su(su(su(su(su(su(su(v5,[v2 of v3],[v5]),[v5 of v3],[v6]),[greater3 of v0],[v5]),[term_of(greater3)],[greater3]),[v5 of v3],[v6]),[greater3 of v1],[v5]),[term_of(greater3)],[greater3])])),
 [
 ]) ext _121534
]).
