problem([]==>x:pnat=>y:pnat=>z:pnat=>leq(z,y)=>less(x,z)=>less(x,y),
ind_strat(induction(lemma(pairs)-[(x:pnat)-s(v1),(y:pnat)-s(v0)])then[base_case(sym_eval(normalize_term([reduction([1,2],[lesszero,equiv(left)]),reduction([2,2],[lesszero,equiv(left)])]))),base_case(sym_eval(normalize_term([reduction([1],[leqzero,equiv(left)]),reduction([2,2],[less1,equ(u(1),left)])]))),step_case(ripple(direction_out,wave(direction_out,[2,2],[less3,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 _199851,[v4 of v0],[v5]),[less3 of v1],[v4]),[term_of(less3)],[less3])))))],[v10]),[v8 of lambda(v0,lambda(v1,su(su(spread(v2,[v3,v4,lambda(v6,lambda(v7,su(v8,[v6 of su(v8,[v3 of v7],[v8])],[v8])))of(lambda(v2,su(v3,[v2 of v1],[v3]))of lambda(v1,su(su(lambda(v2,su(v3,[v2 of v1],[v3]))of _199725,[less1 of v0],[v2]),[term_of(less1)],[less1])))]),[leqzero of v1],[v2]),[term_of(leqzero)],[leqzero])))],[v9]),[v7 of lambda(v0,lambda(v1,su(su(spread(v2,[v3,v4,lambda(v6,lambda(v7,su(lambda(v9,su(v10,[v8 of su(v10,[v3 of v9],[v10])],[v10])),[v6 of v7],[v8])))of(lambda(v2,su(v3,[v2 of v1],[v3]))of lambda(v1,su(su(spread(v2,[v3,v4,lambda(v6,lambda(v7,su(lambda(v9,su(su(v11,[v4 of v10],[v11]),[v8 of v9],[v10])),[v6 of v7],[v8])))of(lambda(v2,su(v3,[v2 of v1],[v3]))of _199525)]),[lesszero of v0],[v2]),[term_of(lesszero)],[lesszero])))]),[lesszero of v1],[v2]),[term_of(lesszero)],[lesszero])))],[v8]),[v6 of y],[v7]),[v5 of x],[v6]),[v3 of lambda(x,lambda(y,z:pnat=>leq(z,y)=>less(x,z)=>less(x,y)))],[v5]))of v4,[term_of(pairs)],[v4]))))))))),
[problem([x:pnat,y:pnat,v0:pnat]==>z:pnat=>leq(z,v0)=>(z=0 in pnat=>void)=>v0=0 in pnat=>void,
 ind_strat(induction(lemma(pairs)-[(v0:pnat)-s(v2),(z:pnat)-s(v1)])then[base_case(sym_eval(normalize_term([reduction([1],[leqzero,equiv(left)])]))then[elementary(intro(new[v2])then[intro(new[v3])then[intro(new[v4])then[elim(v3)then[hyp(v2),hyp(v5)],wfftacs],wfftacs],wfftacs])]),base_case(elementary(intro(new[v2])then[intro(new[v3])then[intro(new[v4])then[elim(v3)then[identity,hyp(v5)],wfftacs],wfftacs],wfftacs])),step_case(ripple(direction_out,wave(direction_out,[1],[leq3,equ(u(1),left)],[]))then[unblock_then_fertilize(weak,unblock_fertilize_lazy([idtac])then fertilize(weak,fertilize_then_ripple(fertilize_left_or_right(left,[weak_fertilize(left,=>,[],v3)]))then elementary(intro(new[v4])then[intro(new[v5])then[intro(new[v6])then[clam_arith(v6:s(v2)=0 in pnat),wfftacs],wfftacs],wfftacs])))])]),lambda(z,su(lambda(v4,su(su(su(su(su(su(v11,[v10 of lambda(v2,lambda(v1,lambda(v3,su(su(su(lambda(v4,lambda(v5,lambda(v6,lambda(v7,su(su(su(su(su(su(su(su(v11,[v10 of v8],[v11]),[v9 of(lambda(v10,su(su(su(su(su(su(su(v13,[v12 of v4],[v13]),[v11 of(lambda(v12,su(su(su(su(su(su(su(v15,[v14 of v12],[v15]),[v13 of(lambda(v14,su(v15,[v14 of v7],[v15]))&lambda(v14,v6))],[v14]),[v16 of void],[v13]),[v15 of(s(v1)=0 in pnat)],[v16]),[v14 of(s(v2)=0 in pnat=>void)],[v15]),[v13 of void],[v14]),[term_of(ipc_dp_imp_e4)],[v13]))&lambda(v12,lambda(v13,v10)))],[v12]),[v14 of(v2=0 in pnat=>void)],[v11]),[v13 of(v1=0 in pnat=>void)],[v14]),[v12 of((s(v1)=0 in pnat=>void)=>s(v2)=0 in pnat=>void)],[v13]),[v11 of void],[v12]),[term_of(ipc_dp_imp_e4)],[v11]))&lambda(v10,lambda(v11,su(su(su(su(su(su(su(v14,[v13 of v4],[v14]),[v12 of(lambda(v13,su(su(su(su(su(su(su(v16,[v15 of v13],[v16]),[v14 of(lambda(v15,su(v16,[v15 of v7],[v16]))&lambda(v15,v6))],[v15]),[v17 of void],[v14]),[v16 of(s(v1)=0 in pnat)],[v17]),[v15 of(s(v2)=0 in pnat=>void)],[v16]),[v14 of void],[v15]),[term_of(ipc_dp_imp_e4)],[v14]))&lambda(v13,lambda(v14,lambda(v15,su(v16,[v14 of v11],[v16])))))],[v13]),[v15 of(v2=0 in pnat=>void)],[v12]),[v14 of(v1=0 in pnat=>void)],[v15]),[v13 of((s(v1)=0 in pnat=>void)=>s(v2)=0 in pnat=>void)],[v14]),[v12 of void],[v13]),[term_of(ipc_dp_imp_e4)],[v12]))))],[v10]),[v12 of void],[v9]),[v11 of(v1=0 in pnat)],[v12]),[v10 of(v2=0 in pnat=>void)],[v11]),[v9 of void],[v10]),[term_of(ipc_dp_imp_e4)],[v9]),[v3 of v5],[v8])))))of lambda(v4,lambda(v5,lambda(v6,su(su(su(any(v9),[v8 of v6],[v9]),[v7 of v2],[v8]),[term_of(arith1)],[v7])))),[v4 of v2],[v5]),[leq3 of v1],[v4]),[term_of(leq3)],[leq3]))))],[v11]),[v9 of lambda(v1,lambda(v2,lambda(v3,lambda(v4,su(v5,[v3 of axiom],[v5])))))],[v10]),[v8 of lambda(v1,su(su(spread(v2,[v3,v4,lambda(v6,lambda(v7,su(v8,[v6 of su(v8,[v3 of v7],[v8])],[v8])))of lambda(v2,lambda(v3,lambda(v4,su(v5,[v3 of v2],[v5]))))]),[leqzero of v1],[v2]),[term_of(leqzero)],[leqzero]))],[v9]),[v7 of z],[v8]),[v6 of v0],[v7]),[v4 of lambda(v0,lambda(z,leq(z,v0)=>(z=0 in pnat=>void)=>v0=0 in pnat=>void))],[v6]))of v5,[term_of(pairs)],[v5])),
 [
 ]) ext _199525,
 problem([x:pnat,y:pnat,v0:pnat]==>z:pnat=>z=0 in pnat=>less(v0,z)=>void,
 ind_strat(induction(lemma(pairs)-[(v0:pnat)-s(v2),(z:pnat)-s(v1)])then[base_case(sym_eval(normalize_term([reduction([1,2],[lesszero,equiv(left)])]))then[elementary(intro(new[v2])then[intro(new[v3])then[elim(v3)then[hyp(v2),hyp(v4)],wfftacs],wfftacs])]),base_case(sym_eval(normalize_term([reduction([1,2],[less1,equ(u(1),left)])]))then[elementary(intro(new[v2])then[intro(new[v3])then[hyp(v3),wfftacs],wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[1,2],[less3,equ(u(1),left)],[]))then[unblock_then_fertilize(weak,unblock_fertilize_lazy([idtac])then fertilize(weak,fertilize_then_ripple(fertilize_left_or_right(right,[weak_fertilize(right,=>,[],v3)]))then elementary(intro(new[v4])then[clam_arith(v4:s(v1)=0 in pnat),wfftacs])))])]),lambda(z,su(lambda(v4,su(su(su(su(su(su(v11,[v10 of lambda(v2,lambda(v1,lambda(v3,su(su(su(lambda(v4,lambda(v5,lambda(v6,su(su(su(v9,[v8 of v6],[v9]),[v3 of v7],[v8]),[v4 of v5],[v7]))))of lambda(v4,su(su(su(any(v7),[v6 of v4],[v7]),[v5 of v1],[v6]),[term_of(arith1)],[v5])),[v4 of v1],[v5]),[less3 of v2],[v4]),[term_of(less3)],[less3]))))],[v11]),[v9 of lambda(v1,su(su(lambda(v2,lambda(v3,v3)),[less1 of v1],[v2]),[term_of(less1)],[less1]))],[v10]),[v8 of lambda(v1,su(su(spread(v2,[v3,v4,lambda(v6,lambda(v7,su(lambda(v9,su(v10,[v8 of su(v10,[v3 of v9],[v10])],[v10])),[v6 of v7],[v8])))of lambda(v2,lambda(v3,su(v4,[v3 of v2],[v4])))]),[lesszero of v1],[v2]),[term_of(lesszero)],[lesszero]))],[v9]),[v7 of z],[v8]),[v6 of v0],[v7]),[v4 of lambda(v0,lambda(z,z=0 in pnat=>less(v0,z)=>void))],[v6]))of v5,[term_of(pairs)],[v5])),
 [
 ]) ext _199725,
 problem([x:pnat,y:pnat,v1:pnat,v0:pnat,v2:z:pnat=>leq(z,v0)=>less(v1,z)=>less(v1,v0)]==>z:pnat=>leq(z,s(v0))=>less(s(v1),z)=>less(v1,v0),
 ind_strat(induction(lemma(pnat_primitive)-[(z:pnat)-s(v3)])then[base_case(sym_eval(normalize_term([reduction([1],[leq1,equ(u(1),left)]),reduction([1,2],[less1,equ(u(1),left)])]))then[elementary(intro(new[v3])then[intro(new[v4])then[elim(v4),wfftacs],wfftacs])]),step_case(ripple(direction_out,wave(direction_out,[1],[leq3,equ(u(1),left)],[])then[wave(direction_out,[1,2],[less3,equ(u(1),left)],[])])then[unblock_then_fertilize(strong,unblock_fertilize_lazy([idtac])then fertilize(strong,v2))])]),lambda(z,p_ind(z,su(su(su(su(lambda(v3,lambda(v4,any(v4))),[less1 of s(v1)],[v3]),[term_of(less1)],[less1]),[leq1 of s(v0)],[v3]),[term_of(leq1)],[leq1]),[v3,v4,su(su(su(su(su(su(su(v5,[v2 of v3],[v5]),[v5 of v3],[v6]),[less3 of v1],[v5]),[term_of(less3)],[less3]),[v5 of v0],[v6]),[leq3 of v3],[v5]),[term_of(leq3)],[leq3])])),
 [
 ]) ext _199851
]).
