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 = 8] active(f(mark(c),g(mark(c)),mark(g(active(b))))) -> active(f(mark(c),g(mark(c)),mark(g(active(b))))) Let l be the left-hand side and r be the right-hand side of this rule. Let p = epsilon, theta1 = {} and theta2 = {}. We have r|p = active(f(mark(c),g(mark(c)),mark(g(active(b))))) and theta2(theta1(l)) = theta1(r|p). Hence, the term theta1(l) = active(f(mark(c),g(mark(c)),mark(g(active(b))))) 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 = [active^#(f(_0,g(_0),_1)) -> mark^#(f(_1,_1,_1)), mark^#(g(_0)) -> active^#(g(mark(_0))), mark^#(f(_0,_1,_2)) -> active^#(f(_0,_1,_2)), mark^#(g(_0)) -> mark^#(_0)] TRS = {active(f(_0,g(_0),_1)) -> mark(f(_1,_1,_1)), active(g(b)) -> mark(c), active(b) -> mark(c), mark(f(_0,_1,_2)) -> active(f(_0,_1,_2)), mark(g(_0)) -> active(g(mark(_0))), mark(b) -> active(b), mark(c) -> active(c), f(mark(_0),_1,_2) -> f(_0,_1,_2), f(_0,mark(_1),_2) -> f(_0,_1,_2), f(_0,_1,mark(_2)) -> f(_0,_1,_2), f(active(_0),_1,_2) -> f(_0,_1,_2), f(_0,active(_1),_2) -> f(_0,_1,_2), f(_0,_1,active(_2)) -> f(_0,_1,_2), g(mark(_0)) -> g(_0), g(active(_0)) -> g(_0)} ## Trying with homeomorphic embeddings... Failed! ## Trying with polynomial interpretations... Too many coefficients (17)! 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, 4 unfolded rules generated. # Iteration 1: nontermination not detected, 8 unfolded rules generated. # Iteration 2: nontermination not detected, 17 unfolded rules generated. # Iteration 3: nontermination not detected, 50 unfolded rules generated. # Iteration 4: nontermination not detected, 74 unfolded rules generated. # Iteration 5: nontermination not detected, 34 unfolded rules generated. # Iteration 6: nontermination not detected, 40 unfolded rules generated. # Iteration 7: nontermination not detected, 38 unfolded rules generated. # Iteration 8: nontermination not detected, 32 unfolded rules generated. # Iteration 9: nontermination not detected, 12 unfolded rules generated. # Iteration 10: nontermination not detected, 0 unfolded rule generated. Nontermination not detected! # max_depth=3, unfold_variables=true: # Iteration 0: nontermination not detected, 4 unfolded rules generated. # Iteration 1: nontermination not detected, 8 unfolded rules generated. # Iteration 2: nontermination not detected, 17 unfolded rules generated. # Iteration 3: nontermination not detected, 47 unfolded rules generated. # Iteration 4: nontermination not detected, 67 unfolded rules generated. # Iteration 5: nontermination not detected, 32 unfolded rules generated. # Iteration 6: nontermination not detected, 63 unfolded rules generated. # Iteration 7: nontermination not detected, 73 unfolded rules generated. # Iteration 8: nontermination not detected, 70 unfolded rules generated. # Iteration 9: nontermination not detected, 30 unfolded rules generated. # Iteration 10: nontermination not detected, 0 unfolded rule generated. Nontermination not detected! # max_depth=4, unfold_variables=false: # Iteration 0: nontermination not detected, 4 unfolded rules generated. # Iteration 1: nontermination not detected, 8 unfolded rules generated. # Iteration 2: nontermination not detected, 22 unfolded rules generated. # Iteration 3: nontermination not detected, 127 unfolded rules generated. # Iteration 4: nontermination not detected, 596 unfolded rules generated. # Iteration 5: nontermination not detected, 2188 unfolded rules generated. # Iteration 6: nontermination not detected, 5797 unfolded rules generated. # Iteration 7: nontermination not detected, 8072 unfolded rules generated. # Iteration 8: nontermination not detected, 2934 unfolded rules generated. # Iteration 9: nontermination not detected, 5058 unfolded rules generated. # Iteration 10: nontermination not detected, 6486 unfolded rules generated. # Iteration 11: nontermination not detected, 5850 unfolded rules generated. # Iteration 12: nontermination not detected, 2446 unfolded rules generated. # Iteration 13: nontermination not detected, 180 unfolded rules generated. # Iteration 14: nontermination not detected, 12 unfolded rules generated. # Iteration 15: nontermination not detected, 0 unfolded rule generated. Nontermination not detected! # max_depth=4, unfold_variables=true: # Iteration 0: nontermination not detected, 4 unfolded rules generated. # Iteration 1: nontermination not detected, 8 unfolded rules generated. # Iteration 2: nontermination not detected, 22 unfolded rules generated. # Iteration 3: nontermination not detected, 134 unfolded rules generated. # Iteration 4: nontermination not detected, 637 unfolded rules generated. # Iteration 5: nontermination not detected, 2311 unfolded rules generated. # Iteration 6: nontermination not detected, 5970 unfolded rules generated. # Iteration 7: nontermination not detected, 8381 unfolded rules generated. # Iteration 8: nontermination not detected, 3860 unfolded rules generated. # Iteration 9: nontermination not detected, 8544 unfolded rules generated. # Iteration 10: nontermination not detected, 17136 unfolded rules generated. # Iteration 11: nontermination not detected, 29280 unfolded rules generated. # Iteration 12: nontermination not detected, 40680 unfolded rules generated. # Iteration 13: nontermination not detected, 42840 unfolded rules generated. # Iteration 14: nontermination not detected, 30240 unfolded rules generated. # Iteration 15: nontermination not detected, 10080 unfolded rules generated. # Iteration 16: nontermination not detected, 0 unfolded rule generated. Nontermination not detected! # max_depth=5, unfold_variables=false: # Iteration 0: nontermination not detected, 4 unfolded rules generated. # Iteration 1: nontermination not detected, 8 unfolded rules generated. # Iteration 2: nontermination not detected, 22 unfolded rules generated. # Iteration 3: nontermination not detected, 139 unfolded rules generated. # Iteration 4: nontermination not detected, 821 unfolded rules generated. # Iteration 5: nontermination not detected, 4468 unfolded rules generated. # Iteration 6: nontermination not detected, 22001 unfolded rules generated. # Iteration 7: nontermination not detected, 94347 unfolded rules generated. # Iteration 8: nontermination detected, 4546 unfolded rules generated. Here is the successful unfolding. Let IR be the TRS under analysis. L0 = active^#(f(_0,g(_0),_1)) -> mark^#(f(_1,_1,_1)) [trans] is in U_IR^0. D = mark^#(f(_0,_1,_2)) -> active^#(f(_0,_1,_2)) is a dependency pair of IR. We build a composed triple from L0 and D. ==> L1 = active^#(f(_0,g(_0),_1)) -> active^#(f(_1,_1,_1)) [trans] is in U_IR^1. We build a unit triple from L1. ==> L2 = active^#(f(_0,g(_0),_1)) -> active^#(f(_1,_1,_1)) [unit] is in U_IR^2. Let p2 = [0]. We unfold the rule of L2 forwards at position p2 with the rule f(_0,mark(_1),_2) -> f(_0,_1,_2). ==> L3 = active^#(f(_0,g(_0),mark(_1))) -> active^#(f(mark(_1),_1,mark(_1))) [unit] is in U_IR^3. Let p3 = [0, 0]. We unfold the rule of L3 forwards at position p3 with the rule mark(g(_0)) -> active(g(mark(_0))). ==> L4 = active^#(f(_0,g(_0),mark(g(_1)))) -> active^#(f(active(g(mark(_1))),g(_1),mark(g(_1)))) [unit] is in U_IR^4. Let p4 = [0, 0, 0]. We unfold the rule of L4 forwards at position p4 with the rule g(mark(_0)) -> g(_0). ==> L5 = active^#(f(_0,g(_0),mark(g(_1)))) -> active^#(f(active(g(_1)),g(_1),mark(g(_1)))) [unit] is in U_IR^5. Let p5 = [0, 0, 0]. We unfold the rule of L5 forwards at position p5 with the rule g(active(_0)) -> g(_0). ==> L6 = active^#(f(_0,g(_0),mark(g(active(_1))))) -> active^#(f(active(g(_1)),g(active(_1)),mark(g(active(_1))))) [unit] is in U_IR^6. Let p6 = [0, 0]. We unfold the rule of L6 forwards at position p6 with the rule active(g(b)) -> mark(c). ==> L7 = active^#(f(mark(c),g(mark(c)),mark(g(active(b))))) -> active^#(f(mark(c),g(active(b)),mark(g(active(b))))) [unit] is in U_IR^7. Let p7 = [0, 1, 0]. We unfold the rule of L7 forwards at position p7 with the rule active(b) -> mark(c). ==> L8 = active^#(f(mark(c),g(mark(c)),mark(g(active(b))))) -> active^#(f(mark(c),g(mark(c)),mark(g(active(b))))) [unit] is in U_IR^8. This DP problem is infinite. ** END proof description ** Proof stopped at iteration 8 Number of unfolded rules generated by this proof = 366983 Number of unfolded rules generated by all the parallel proofs = 4137535