NO Prover = TRS(tech=GUIDED_UNF_TRIPLES, nb_unfoldings=unlimited, unfold_variables=false, max_nb_coefficients=12, max_nb_unfolded_rules=-1, strategy=LEFTMOST_NE) ** BEGIN proof argument ** The following rule was generated while unfolding the analyzed TRS: [iteration = 12] eq(n__s(n__length(cons(_0,n__inf(_1)))),n__s(n__length(cons(_2,n__inf(_3))))) -> eq(n__s(n__length(cons(_1,n__inf(n__s(_1))))),n__s(n__length(cons(_3,n__inf(s(_3)))))) Let l be the left-hand side and r be the right-hand side of this rule. Let p = epsilon, theta1 = {} and theta2 = {_0->_1, _1->n__s(_1), _3->s(_3), _2->_3}. We have r|p = eq(n__s(n__length(cons(_1,n__inf(n__s(_1))))),n__s(n__length(cons(_3,n__inf(s(_3)))))) and theta2(theta1(l)) = theta1(r|p). Hence, the term theta1(l) = eq(n__s(n__length(cons(_0,n__inf(_1)))),n__s(n__length(cons(_2,n__inf(_3))))) loops w.r.t. the analyzed TRS. ** END proof argument ** ** BEGIN proof description ** ## Searching for a generalized rewrite rule (a rule whose right-hand side contains a variable that does not occur in the left-hand side)... No generalized rewrite rule found! ## Applying the DP framework... ## Round 1: ## DP problem: Dependency pairs = [eq^#(n__s(_0),n__s(_1)) -> eq^#(activate(_0),activate(_1))] TRS = {eq(n__0,n__0) -> true, eq(n__s(_0),n__s(_1)) -> eq(activate(_0),activate(_1)), eq(_0,_1) -> false, inf(_0) -> cons(_0,n__inf(s(_0))), take(0,_0) -> nil, take(s(_0),cons(_1,_2)) -> cons(activate(_1),n__take(activate(_0),activate(_2))), length(nil) -> 0, length(cons(_0,_1)) -> s(n__length(activate(_1))), 0 -> n__0, s(_0) -> n__s(_0), inf(_0) -> n__inf(_0), take(_0,_1) -> n__take(_0,_1), length(_0) -> n__length(_0), activate(n__0) -> 0, activate(n__s(_0)) -> s(_0), activate(n__inf(_0)) -> inf(_0), activate(n__take(_0,_1)) -> take(_0,_1), activate(n__length(_0)) -> length(_0), activate(_0) -> _0} ## Trying with homeomorphic embeddings... Failed! ## Trying with polynomial interpretations... Too many coefficients (36)! Aborting! ## Trying with lexicographic path orders... Failed! ## Trying to prove nontermination by unfolding the dependency pairs with the rules of the TRS # max_depth=3, unfold_variables=false: # Iteration 0: nontermination not detected, 1 unfolded rule generated. # Iteration 1: nontermination not detected, 1 unfolded rule generated. # Iteration 2: nontermination not detected, 3 unfolded rules generated. # Iteration 3: nontermination not detected, 6 unfolded rules generated. # Iteration 4: nontermination not detected, 3 unfolded rules generated. # Iteration 5: nontermination not detected, 0 unfolded rule generated. Nontermination not detected! # max_depth=3, unfold_variables=true: # Iteration 0: nontermination not detected, 1 unfolded rule generated. # Iteration 1: nontermination not detected, 1 unfolded rule generated. # Iteration 2: nontermination not detected, 3 unfolded rules generated. # Iteration 3: nontermination not detected, 6 unfolded rules generated. # Iteration 4: nontermination not detected, 3 unfolded rules generated. # Iteration 5: nontermination not detected, 0 unfolded rule generated. Nontermination not detected! # max_depth=4, unfold_variables=false: # Iteration 0: nontermination not detected, 1 unfolded rule generated. # Iteration 1: nontermination not detected, 1 unfolded rule generated. # Iteration 2: nontermination not detected, 3 unfolded rules generated. # Iteration 3: nontermination not detected, 4 unfolded rules generated. # Iteration 4: nontermination not detected, 6 unfolded rules generated. # Iteration 5: nontermination not detected, 7 unfolded rules generated. # Iteration 6: nontermination not detected, 11 unfolded rules generated. # Iteration 7: nontermination not detected, 5 unfolded rules generated. # Iteration 8: nontermination not detected, 9 unfolded rules generated. # Iteration 9: nontermination not detected, 5 unfolded rules generated. # Iteration 10: nontermination not detected, 0 unfolded rule generated. Nontermination not detected! # max_depth=4, unfold_variables=true: # Iteration 0: nontermination not detected, 1 unfolded rule generated. # Iteration 1: nontermination not detected, 1 unfolded rule generated. # Iteration 2: nontermination not detected, 3 unfolded rules generated. # Iteration 3: nontermination not detected, 4 unfolded rules generated. # Iteration 4: nontermination not detected, 6 unfolded rules generated. # Iteration 5: nontermination not detected, 7 unfolded rules generated. # Iteration 6: nontermination not detected, 11 unfolded rules generated. # Iteration 7: nontermination not detected, 5 unfolded rules generated. # Iteration 8: nontermination not detected, 9 unfolded rules generated. # Iteration 9: nontermination not detected, 5 unfolded rules generated. # Iteration 10: nontermination not detected, 0 unfolded rule generated. Nontermination not detected! # max_depth=5, unfold_variables=false: # Iteration 0: nontermination not detected, 1 unfolded rule generated. # Iteration 1: nontermination not detected, 1 unfolded rule generated. # Iteration 2: nontermination not detected, 3 unfolded rules generated. # Iteration 3: nontermination not detected, 4 unfolded rules generated. # Iteration 4: nontermination not detected, 10 unfolded rules generated. # Iteration 5: nontermination not detected, 21 unfolded rules generated. # Iteration 6: nontermination not detected, 53 unfolded rules generated. # Iteration 7: nontermination not detected, 37 unfolded rules generated. # Iteration 8: nontermination not detected, 87 unfolded rules generated. # Iteration 9: nontermination not detected, 159 unfolded rules generated. # Iteration 10: nontermination not detected, 25 unfolded rules generated. # Iteration 11: nontermination not detected, 1 unfolded rule generated. # Iteration 12: nontermination not detected, 0 unfolded rule generated. Nontermination not detected! # max_depth=5, unfold_variables=true: # Iteration 0: nontermination not detected, 1 unfolded rule generated. # Iteration 1: nontermination not detected, 1 unfolded rule generated. # Iteration 2: nontermination not detected, 3 unfolded rules generated. # Iteration 3: nontermination not detected, 4 unfolded rules generated. # Iteration 4: nontermination not detected, 10 unfolded rules generated. # Iteration 5: nontermination not detected, 21 unfolded rules generated. # Iteration 6: nontermination not detected, 53 unfolded rules generated. # Iteration 7: nontermination not detected, 37 unfolded rules generated. # Iteration 8: nontermination not detected, 87 unfolded rules generated. # Iteration 9: nontermination not detected, 159 unfolded rules generated. # Iteration 10: nontermination not detected, 25 unfolded rules generated. # Iteration 11: nontermination not detected, 1 unfolded rule generated. # Iteration 12: nontermination not detected, 0 unfolded rule generated. Nontermination not detected! # max_depth=6, unfold_variables=false: # Iteration 0: nontermination not detected, 1 unfolded rule generated. # Iteration 1: nontermination not detected, 1 unfolded rule generated. # Iteration 2: nontermination not detected, 3 unfolded rules generated. # Iteration 3: nontermination not detected, 4 unfolded rules generated. # Iteration 4: nontermination not detected, 10 unfolded rules generated. # Iteration 5: nontermination not detected, 24 unfolded rules generated. # Iteration 6: nontermination not detected, 60 unfolded rules generated. # Iteration 7: nontermination not detected, 103 unfolded rules generated. # Iteration 8: nontermination not detected, 281 unfolded rules generated. # Iteration 9: nontermination not detected, 586 unfolded rules generated. # Iteration 10: nontermination not detected, 979 unfolded rules generated. # Iteration 11: nontermination not detected, 1590 unfolded rules generated. # Iteration 12: nontermination detected, 218 unfolded rules generated. Here is the successful unfolding. Let IR be the TRS under analysis. L0 = eq^#(n__s(_0),n__s(_1)) -> eq^#(activate(_0),activate(_1)) [trans] is in U_IR^0. We build a unit triple from L0. ==> L1 = eq^#(n__s(_0),n__s(_1)) -> eq^#(activate(_0),activate(_1)) [unit] is in U_IR^1. Let p1 = [0]. We unfold the rule of L1 forwards at position p1 with the rule activate(n__length(_0)) -> length(_0). ==> L2 = eq^#(n__s(n__length(_0)),n__s(_1)) -> eq^#(length(_0),activate(_1)) [unit] is in U_IR^2. Let p2 = [0]. We unfold the rule of L2 forwards at position p2 with the rule length(cons(_0,_1)) -> s(n__length(activate(_1))). ==> L3 = eq^#(n__s(n__length(cons(_0,_1))),n__s(_2)) -> eq^#(s(n__length(activate(_1))),activate(_2)) [unit] is in U_IR^3. Let p3 = [0]. We unfold the rule of L3 forwards at position p3 with the rule s(_0) -> n__s(_0). ==> L4 = eq^#(n__s(n__length(cons(_0,_1))),n__s(_2)) -> eq^#(n__s(n__length(activate(_1))),activate(_2)) [unit] is in U_IR^4. Let p4 = [0, 0, 0]. We unfold the rule of L4 forwards at position p4 with the rule activate(n__inf(_0)) -> inf(_0). ==> L5 = eq^#(n__s(n__length(cons(_0,n__inf(_1)))),n__s(_2)) -> eq^#(n__s(n__length(inf(_1))),activate(_2)) [unit] is in U_IR^5. Let p5 = [0, 0, 0]. We unfold the rule of L5 forwards at position p5 with the rule inf(_0) -> cons(_0,n__inf(s(_0))). ==> L6 = eq^#(n__s(n__length(cons(_0,n__inf(_1)))),n__s(_2)) -> eq^#(n__s(n__length(cons(_1,n__inf(s(_1))))),activate(_2)) [unit] is in U_IR^6. Let p6 = [0, 0, 0, 1, 0]. We unfold the rule of L6 forwards at position p6 with the rule s(_0) -> n__s(_0). ==> L7 = eq^#(n__s(n__length(cons(_0,n__inf(_1)))),n__s(_2)) -> eq^#(n__s(n__length(cons(_1,n__inf(n__s(_1))))),activate(_2)) [unit] is in U_IR^7. Let p7 = [1]. We unfold the rule of L7 forwards at position p7 with the rule activate(n__length(_0)) -> length(_0). ==> L8 = eq^#(n__s(n__length(cons(_0,n__inf(_1)))),n__s(n__length(_2))) -> eq^#(n__s(n__length(cons(_1,n__inf(n__s(_1))))),length(_2)) [unit] is in U_IR^8. Let p8 = [1]. We unfold the rule of L8 forwards at position p8 with the rule length(cons(_0,_1)) -> s(n__length(activate(_1))). ==> L9 = eq^#(n__s(n__length(cons(_0,n__inf(_1)))),n__s(n__length(cons(_2,_3)))) -> eq^#(n__s(n__length(cons(_1,n__inf(n__s(_1))))),s(n__length(activate(_3)))) [unit] is in U_IR^9. Let p9 = [1]. We unfold the rule of L9 forwards at position p9 with the rule s(_0) -> n__s(_0). ==> L10 = eq^#(n__s(n__length(cons(_0,n__inf(_1)))),n__s(n__length(cons(_2,_3)))) -> eq^#(n__s(n__length(cons(_1,n__inf(n__s(_1))))),n__s(n__length(activate(_3)))) [unit] is in U_IR^10. Let p10 = [1, 0, 0]. We unfold the rule of L10 forwards at position p10 with the rule activate(n__inf(_0)) -> inf(_0). ==> L11 = eq^#(n__s(n__length(cons(_0,n__inf(_1)))),n__s(n__length(cons(_2,n__inf(_3))))) -> eq^#(n__s(n__length(cons(_1,n__inf(n__s(_1))))),n__s(n__length(inf(_3)))) [unit] is in U_IR^11. Let p11 = [1, 0, 0]. We unfold the rule of L11 forwards at position p11 with the rule inf(_0) -> cons(_0,n__inf(s(_0))). ==> L12 = eq^#(n__s(n__length(cons(_0,n__inf(_1)))),n__s(n__length(cons(_2,n__inf(_3))))) -> eq^#(n__s(n__length(cons(_1,n__inf(n__s(_1))))),n__s(n__length(cons(_3,n__inf(s(_3)))))) [unit] is in U_IR^12. This DP problem is infinite. ** END proof description ** Proof stopped at iteration 12 Number of unfolded rules generated by this proof = 4796 Number of unfolded rules generated by all the parallel proofs = 59831