20.017/20.017 YES 20.017/20.017 20.017/20.017 Problem 1: 20.017/20.017 20.017/20.017 20.017/20.017 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.017/20.017 Confluence Problem: 20.017/20.017 (VAR vNonEmpty x y xs ys) 20.017/20.017 (REPLACEMENT-MAP 20.017/20.017 (isnoc 1) 20.017/20.017 (cons 1, 2) 20.017/20.017 (fSNonEmpty) 20.017/20.017 (nil) 20.017/20.017 (tp2 1, 2) 20.017/20.017 ) 20.017/20.017 (RULES 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y) 20.017/20.017 ) 20.017/20.017 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.017/20.017 20.017/20.017 20.017/20.017 Problem 1: 20.017/20.017 20.017/20.017 Inlining of Conditions Processor [STERN17]: 20.017/20.017 20.017/20.017 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.017/20.017 Confluence Problem: 20.017/20.017 (VAR vNonEmpty x y xs ys) 20.017/20.017 (REPLACEMENT-MAP 20.017/20.017 (isnoc 1) 20.017/20.017 (cons 1, 2) 20.017/20.017 (fSNonEmpty) 20.017/20.017 (nil) 20.017/20.017 (tp2 1, 2) 20.017/20.017 ) 20.017/20.017 (RULES 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y) 20.017/20.017 ) 20.017/20.017 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.017/20.017 20.017/20.017 20.017/20.017 Problem 1: 20.017/20.017 20.017/20.017 Clean CTRS Processor: 20.017/20.017 20.017/20.017 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.017/20.017 Confluence Problem: 20.017/20.017 (VAR x y xs ys) 20.017/20.017 (REPLACEMENT-MAP 20.017/20.017 (isnoc 1) 20.017/20.017 (cons 1, 2) 20.017/20.017 (fSNonEmpty) 20.017/20.017 (nil) 20.017/20.017 (tp2 1, 2) 20.017/20.017 ) 20.017/20.017 (RULES 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y) 20.017/20.017 ) 20.017/20.017 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.017/20.017 20.017/20.017 CRule InfChecker Info: 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 Rule remains 20.017/20.017 Proof: 20.017/20.017 NO 20.017/20.017 20.017/20.017 Problem 1: 20.017/20.017 20.017/20.017 Infeasibility Problem: 20.017/20.017 [(VAR vNonEmpty x y xs ys vNonEmpty y xs ys) 20.017/20.017 (STRATEGY CONTEXTSENSITIVE 20.017/20.017 (isnoc 1) 20.017/20.017 (cons 1 2) 20.017/20.017 (fSNonEmpty) 20.017/20.017 (nil) 20.017/20.017 (tp2 1 2) 20.017/20.017 ) 20.017/20.017 (RULES 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y) 20.017/20.017 )] 20.017/20.017 20.017/20.017 Infeasibility Conditions: 20.017/20.017 isnoc(ys) ->* tp2(xs,y) 20.017/20.017 20.017/20.017 Problem 1: 20.017/20.017 20.017/20.017 Obtaining a proof using Prover9: 20.017/20.017 20.017/20.017 -> Prover9 Output: 20.017/20.017 ============================== Prover9 =============================== 20.017/20.017 Prover9 (64) version 2009-11A, November 2009. 20.017/20.017 Process 12730 was started by ubuntu on ubuntu, 20.017/20.017 Wed Mar 9 09:20:47 2022 20.017/20.017 The command was "./prover9 -f /tmp/prover912721-0.in". 20.017/20.017 ============================== end of head =========================== 20.017/20.017 20.017/20.017 ============================== INPUT ================================= 20.017/20.017 20.017/20.017 % Reading from file /tmp/prover912721-0.in 20.017/20.017 20.017/20.017 assign(max_seconds,20). 20.017/20.017 20.017/20.017 formulas(assumptions). 20.017/20.017 ->_s0(x1,y) -> ->_s0(isnoc(x1),isnoc(y)) # label(congruence). 20.017/20.017 ->_s0(x1,y) -> ->_s0(cons(x1,x2),cons(y,x2)) # label(congruence). 20.017/20.017 ->_s0(x2,y) -> ->_s0(cons(x1,x2),cons(x1,y)) # label(congruence). 20.017/20.017 ->_s0(x1,y) -> ->_s0(tp2(x1,x2),tp2(y,x2)) # label(congruence). 20.017/20.017 ->_s0(x2,y) -> ->_s0(tp2(x1,x2),tp2(x1,y)) # label(congruence). 20.017/20.017 ->*_s0(isnoc(x4),tp2(x3,x2)) -> ->_s0(isnoc(cons(x1,x4)),tp2(cons(x1,x3),x2)) # label(replacement). 20.017/20.017 ->_s0(isnoc(cons(x2,nil)),tp2(nil,x2)) # label(replacement). 20.017/20.017 ->*_s0(x,x) # label(reflexivity). 20.017/20.017 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.017/20.017 end_of_list. 20.017/20.017 20.017/20.017 formulas(goals). 20.017/20.017 (exists x6 exists x7 exists x8 ->*_s0(isnoc(x8),tp2(x7,x6))) # label(goal). 20.017/20.017 end_of_list. 20.017/20.017 20.017/20.017 ============================== end of input ========================== 20.017/20.017 20.017/20.017 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.017/20.017 20.017/20.017 % Formulas that are not ordinary clauses: 20.017/20.017 1 ->_s0(x1,y) -> ->_s0(isnoc(x1),isnoc(y)) # label(congruence) # label(non_clause). [assumption]. 20.017/20.017 2 ->_s0(x1,y) -> ->_s0(cons(x1,x2),cons(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.017/20.017 3 ->_s0(x2,y) -> ->_s0(cons(x1,x2),cons(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.017/20.017 4 ->_s0(x1,y) -> ->_s0(tp2(x1,x2),tp2(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.017/20.017 5 ->_s0(x2,y) -> ->_s0(tp2(x1,x2),tp2(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.017/20.017 6 ->*_s0(isnoc(x4),tp2(x3,x2)) -> ->_s0(isnoc(cons(x1,x4)),tp2(cons(x1,x3),x2)) # label(replacement) # label(non_clause). [assumption]. 20.017/20.017 7 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.017/20.017 8 (exists x6 exists x7 exists x8 ->*_s0(isnoc(x8),tp2(x7,x6))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.017/20.017 20.017/20.017 ============================== end of process non-clausal formulas === 20.017/20.017 20.017/20.017 ============================== PROCESS INITIAL CLAUSES =============== 20.017/20.017 20.017/20.017 % Clauses before input processing: 20.017/20.017 20.017/20.017 formulas(usable). 20.017/20.017 end_of_list. 20.017/20.017 20.017/20.017 formulas(sos). 20.017/20.017 -->_s0(x,y) | ->_s0(isnoc(x),isnoc(y)) # label(congruence). [clausify(1)]. 20.017/20.017 -->_s0(x,y) | ->_s0(cons(x,z),cons(y,z)) # label(congruence). [clausify(2)]. 20.017/20.017 -->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence). [clausify(3)]. 20.017/20.017 -->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence). [clausify(4)]. 20.017/20.017 -->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence). [clausify(5)]. 20.017/20.017 -->*_s0(isnoc(x),tp2(y,z)) | ->_s0(isnoc(cons(u,x)),tp2(cons(u,y),z)) # label(replacement). [clausify(6)]. 20.017/20.017 ->_s0(isnoc(cons(x,nil)),tp2(nil,x)) # label(replacement). [assumption]. 20.017/20.017 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.017/20.017 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(7)]. 20.017/20.017 -->*_s0(isnoc(x),tp2(y,z)) # label(goal). [deny(8)]. 20.017/20.017 end_of_list. 20.017/20.017 20.017/20.017 formulas(demodulators). 20.017/20.017 end_of_list. 20.017/20.017 20.017/20.017 ============================== PREDICATE ELIMINATION ================= 20.017/20.017 20.017/20.017 No predicates eliminated. 20.017/20.017 20.017/20.017 ============================== end predicate elimination ============= 20.017/20.017 20.017/20.017 Auto_denials: 20.017/20.017 % copying label goal to answer in negative clause 20.017/20.017 20.017/20.017 Term ordering decisions: 20.017/20.017 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 20.017/20.017 Function symbol precedence: function_order([ nil, cons, tp2, isnoc ]). 20.017/20.017 After inverse_order: (no changes). 20.017/20.017 Unfolding symbols: (none). 20.017/20.017 20.017/20.017 Auto_inference settings: 20.017/20.017 % set(neg_binary_resolution). % (HNE depth_diff=-6) 20.017/20.017 % clear(ordered_res). % (HNE depth_diff=-6) 20.017/20.017 % set(ur_resolution). % (HNE depth_diff=-6) 20.017/20.017 % set(ur_resolution) -> set(pos_ur_resolution). 20.017/20.017 % set(ur_resolution) -> set(neg_ur_resolution). 20.017/20.017 20.017/20.017 Auto_process settings: (no changes). 20.017/20.017 20.017/20.017 kept: 9 -->_s0(x,y) | ->_s0(isnoc(x),isnoc(y)) # label(congruence). [clausify(1)]. 20.017/20.017 kept: 10 -->_s0(x,y) | ->_s0(cons(x,z),cons(y,z)) # label(congruence). [clausify(2)]. 20.017/20.017 kept: 11 -->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence). [clausify(3)]. 20.017/20.017 kept: 12 -->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence). [clausify(4)]. 20.017/20.017 kept: 13 -->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence). [clausify(5)]. 20.017/20.017 kept: 14 -->*_s0(isnoc(x),tp2(y,z)) | ->_s0(isnoc(cons(u,x)),tp2(cons(u,y),z)) # label(replacement). [clausify(6)]. 20.017/20.017 kept: 15 ->_s0(isnoc(cons(x,nil)),tp2(nil,x)) # label(replacement). [assumption]. 20.017/20.017 kept: 16 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.017/20.017 kept: 17 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(7)]. 20.017/20.017 kept: 18 -->*_s0(isnoc(x),tp2(y,z)) # label(goal) # answer(goal). [deny(8)]. 20.017/20.017 20.017/20.017 ============================== end of process initial clauses ======== 20.017/20.017 20.017/20.017 ============================== CLAUSES FOR SEARCH ==================== 20.017/20.017 20.017/20.017 % Clauses after input processing: 20.017/20.017 20.017/20.017 formulas(usable). 20.017/20.017 end_of_list. 20.017/20.017 20.017/20.017 formulas(sos). 20.017/20.017 9 -->_s0(x,y) | ->_s0(isnoc(x),isnoc(y)) # label(congruence). [clausify(1)]. 20.017/20.017 10 -->_s0(x,y) | ->_s0(cons(x,z),cons(y,z)) # label(congruence). [clausify(2)]. 20.017/20.017 11 -->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence). [clausify(3)]. 20.017/20.017 12 -->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence). [clausify(4)]. 20.017/20.017 13 -->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence). [clausify(5)]. 20.017/20.017 15 ->_s0(isnoc(cons(x,nil)),tp2(nil,x)) # label(replacement). [assumption]. 20.017/20.017 16 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.017/20.017 17 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(7)]. 20.017/20.017 18 -->*_s0(isnoc(x),tp2(y,z)) # label(goal) # answer(goal). [deny(8)]. 20.017/20.017 end_of_list. 20.017/20.017 20.017/20.017 formulas(demodulators). 20.017/20.017 end_of_list. 20.017/20.017 20.017/20.017 ============================== end of clauses for search ============= 20.017/20.017 20.017/20.017 ============================== SEARCH ================================ 20.017/20.017 20.017/20.017 % Starting search at 0.00 seconds. 20.017/20.017 20.017/20.017 given #1 (I,wt=8): 9 -->_s0(x,y) | ->_s0(isnoc(x),isnoc(y)) # label(congruence). [clausify(1)]. 20.017/20.017 20.017/20.017 given #2 (I,wt=10): 10 -->_s0(x,y) | ->_s0(cons(x,z),cons(y,z)) # label(congruence). [clausify(2)]. 20.017/20.017 20.017/20.017 given #3 (I,wt=10): 11 -->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence). [clausify(3)]. 20.017/20.017 20.017/20.017 given #4 (I,wt=10): 12 -->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence). [clausify(4)]. 20.017/20.017 20.017/20.017 given #5 (I,wt=10): 13 -->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence). [clausify(5)]. 20.017/20.017 20.017/20.017 given #6 (I,wt=8): 15 ->_s0(isnoc(cons(x,nil)),tp2(nil,x)) # label(replacement). [assumption]. 20.017/20.017 20.017/20.017 given #7 (I,wt=3): 16 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.017/20.017 20.017/20.017 given #8 (I,wt=9): 17 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(7)]. 20.017/20.017 20.017/20.017 ============================== PROOF ================================= 20.017/20.017 20.017/20.017 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.017/20.017 % Length of proof is 8. 20.017/20.017 % Level of proof is 3. 20.017/20.017 % Maximum clause weight is 9.000. 20.017/20.017 % Given clauses 8. 20.017/20.017 20.017/20.017 7 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.017/20.017 8 (exists x6 exists x7 exists x8 ->*_s0(isnoc(x8),tp2(x7,x6))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.017/20.017 15 ->_s0(isnoc(cons(x,nil)),tp2(nil,x)) # label(replacement). [assumption]. 20.017/20.017 16 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.017/20.017 17 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(7)]. 20.017/20.017 18 -->*_s0(isnoc(x),tp2(y,z)) # label(goal) # answer(goal). [deny(8)]. 20.017/20.017 24 ->*_s0(isnoc(cons(x,nil)),tp2(nil,x)). [ur(17,a,15,a,b,16,a)]. 20.017/20.017 25 $F # answer(goal). [resolve(24,a,18,a)]. 20.017/20.017 20.017/20.017 ============================== end of proof ========================== 20.017/20.017 20.017/20.017 ============================== STATISTICS ============================ 20.017/20.017 20.017/20.017 Given=8. Generated=16. Kept=16. proofs=1. 20.017/20.017 Usable=8. Sos=6. Demods=0. Limbo=0, Disabled=11. Hints=0. 20.017/20.017 Kept_by_rule=0, Deleted_by_rule=0. 20.017/20.017 Forward_subsumed=0. Back_subsumed=1. 20.017/20.017 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.017/20.017 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.017/20.017 Demod_attempts=0. Demod_rewrites=0. 20.017/20.017 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.017/20.017 Nonunit_fsub_feature_tests=2. Nonunit_bsub_feature_tests=13. 20.017/20.017 Megabytes=0.08. 20.017/20.017 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.017/20.017 20.017/20.017 ============================== end of statistics ===================== 20.017/20.017 20.017/20.017 ============================== end of search ========================= 20.017/20.017 20.017/20.017 THEOREM PROVED 20.017/20.017 20.017/20.017 Exiting with 1 proof. 20.017/20.017 20.017/20.017 Process 12730 exit (max_proofs) Wed Mar 9 09:20:47 2022 20.017/20.017 20.017/20.017 20.017/20.017 The problem is feasible. 20.017/20.017 20.017/20.017 20.017/20.017 CRule InfChecker Info: 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y) 20.017/20.017 Rule remains 20.017/20.017 Proof: 20.017/20.017 NO_CONDS 20.017/20.017 20.017/20.017 Problem 1: 20.017/20.017 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.017/20.017 Confluence Problem: 20.017/20.017 (VAR x y xs ys) 20.017/20.017 (REPLACEMENT-MAP 20.017/20.017 (isnoc 1) 20.017/20.017 (cons 1, 2) 20.017/20.017 (fSNonEmpty) 20.017/20.017 (nil) 20.017/20.017 (tp2 1, 2) 20.017/20.017 ) 20.017/20.017 (RULES 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y) 20.017/20.017 ) 20.017/20.017 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.017/20.017 20.017/20.017 Critical Pairs Processor: 20.017/20.017 -> Rules: 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y) 20.017/20.017 -> Vars: 20.017/20.017 "x", "y", "xs", "ys" 20.017/20.017 20.017/20.017 -> Rlps: 20.017/20.017 crule: isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y), id: 1, possubterms: isnoc(cons(x,ys))-> [], cons(x,ys)-> [1] 20.017/20.017 crule: isnoc(cons(y,nil)) -> tp2(nil,y), id: 2, possubterms: isnoc(cons(y,nil))-> [], cons(y,nil)-> [1], nil-> [1,2] 20.017/20.017 20.017/20.017 -> Unifications: 20.017/20.017 R2 unifies with R1 at p: [], l: isnoc(cons(y,nil)), lp: isnoc(cons(y,nil)), conds: {isnoc(ys) ->* tp2(xs,y')}, sig: {y -> x,ys -> nil}, l': isnoc(cons(x,ys)), r: tp2(nil,y), r': tp2(cons(x,xs),y') 20.017/20.017 20.017/20.017 -> Critical pairs info: 20.017/20.017 | isnoc(nil) ->* tp2(xs,y') => Not trivial, Overlay, NW2, N1 20.017/20.017 20.017/20.017 -> Problem conclusions: 20.017/20.017 Left linear, Right linear, Linear 20.017/20.017 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.017/20.017 CTRS Type: 3 20.017/20.017 Deterministic, Strongly deterministic 20.017/20.017 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.017/20.017 Maybe right-stable CTRS, Overlay CTRS 20.017/20.017 Maybe normal CTRS, Maybe almost normal CTRS 20.017/20.017 Maybe terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 20.017/20.017 Maybe level confluent 20.017/20.017 Maybe confluent 20.017/20.017 20.017/20.017 Problem 1: 20.017/20.017 Clean Infeasible CCPs Processor: 20.017/20.017 Num of CPIs: 1 20.017/20.017 Timeout: 60 20.017/20.017 Timeout for each infeasibility problem: 60 s 20.017/20.017 | isnoc(nil) ->* tp2(xs,y') => Not trivial, Overlay, NW2, N1 20.017/20.017 (CONDITIONTYPE ORIENTED) 20.017/20.017 (VAR x y xs ys x y1 xs) 20.017/20.017 (RULES 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y)) 20.017/20.017 (VAR x13 x14) 20.017/20.017 (CONDITION isnoc(nil) ->* tp2(x14,x13)) 20.017/20.017 20.017/20.017 Proof: 20.017/20.017 YES 20.017/20.017 20.017/20.017 Problem 1: 20.017/20.017 20.017/20.017 Infeasibility Problem: 20.017/20.017 [(VAR vNonEmpty x y xs ys y1 vNonEmpty x13 x14) 20.017/20.017 (STRATEGY CONTEXTSENSITIVE 20.017/20.017 (isnoc 1) 20.017/20.017 (cons 1 2) 20.017/20.017 (fSNonEmpty) 20.017/20.017 (nil) 20.017/20.017 (tp2 1 2) 20.017/20.017 ) 20.017/20.017 (RULES 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y) 20.017/20.017 )] 20.017/20.017 20.017/20.017 Infeasibility Conditions: 20.017/20.017 isnoc(nil) ->* tp2(x14,x13) 20.017/20.017 20.017/20.017 Problem 1: 20.017/20.017 20.017/20.017 Obtaining a model using Mace4: 20.017/20.017 20.017/20.017 -> Usable Rules: 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y) 20.017/20.017 20.017/20.017 -> Mace4 Output: 20.017/20.017 ============================== Mace4 ================================= 20.017/20.017 Mace4 (64) version 2009-11A, November 2009. 20.017/20.017 Process 12766 was started by ubuntu on ubuntu, 20.017/20.017 Wed Mar 9 09:20:47 2022 20.017/20.017 The command was "./mace4 -c -f /tmp/mace412747-2.in". 20.017/20.017 ============================== end of head =========================== 20.017/20.017 20.017/20.017 ============================== INPUT ================================= 20.017/20.017 20.017/20.017 % Reading from file /tmp/mace412747-2.in 20.017/20.017 20.017/20.017 assign(max_seconds,20). 20.017/20.017 20.017/20.017 formulas(assumptions). 20.017/20.017 ->(x1,y) -> ->(isnoc(x1),isnoc(y)) # label(congruence). 20.017/20.017 ->(x1,y) -> ->(cons(x1,x2),cons(y,x2)) # label(congruence). 20.017/20.017 ->(x2,y) -> ->(cons(x1,x2),cons(x1,y)) # label(congruence). 20.017/20.017 ->(x1,y) -> ->(tp2(x1,x2),tp2(y,x2)) # label(congruence). 20.017/20.017 ->(x2,y) -> ->(tp2(x1,x2),tp2(x1,y)) # label(congruence). 20.017/20.017 ->*(isnoc(x4),tp2(x3,x2)) -> ->(isnoc(cons(x1,x4)),tp2(cons(x1,x3),x2)) # label(replacement). 20.017/20.017 ->(isnoc(cons(x2,nil)),tp2(nil,x2)) # label(replacement). 20.017/20.017 ->*(x,x) # label(reflexivity). 20.017/20.017 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 20.017/20.017 end_of_list. 20.017/20.017 20.017/20.017 formulas(goals). 20.017/20.017 (exists x7 exists x8 ->*(isnoc(nil),tp2(x8,x7))) # label(goal). 20.017/20.017 end_of_list. 20.017/20.017 20.017/20.017 ============================== end of input ========================== 20.017/20.017 20.017/20.017 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.017/20.017 20.017/20.017 % Formulas that are not ordinary clauses: 20.017/20.017 1 ->(x1,y) -> ->(isnoc(x1),isnoc(y)) # label(congruence) # label(non_clause). [assumption]. 20.017/20.017 2 ->(x1,y) -> ->(cons(x1,x2),cons(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.017/20.017 3 ->(x2,y) -> ->(cons(x1,x2),cons(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.017/20.017 4 ->(x1,y) -> ->(tp2(x1,x2),tp2(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.017/20.017 5 ->(x2,y) -> ->(tp2(x1,x2),tp2(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.017/20.017 6 ->*(isnoc(x4),tp2(x3,x2)) -> ->(isnoc(cons(x1,x4)),tp2(cons(x1,x3),x2)) # label(replacement) # label(non_clause). [assumption]. 20.017/20.017 7 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.017/20.017 8 (exists x7 exists x8 ->*(isnoc(nil),tp2(x8,x7))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.017/20.017 20.017/20.017 ============================== end of process non-clausal formulas === 20.017/20.017 20.017/20.017 ============================== CLAUSES FOR SEARCH ==================== 20.017/20.017 20.017/20.017 formulas(mace4_clauses). 20.017/20.017 -->(x,y) | ->(isnoc(x),isnoc(y)) # label(congruence). 20.017/20.017 -->(x,y) | ->(cons(x,z),cons(y,z)) # label(congruence). 20.017/20.017 -->(x,y) | ->(cons(z,x),cons(z,y)) # label(congruence). 20.017/20.017 -->(x,y) | ->(tp2(x,z),tp2(y,z)) # label(congruence). 20.017/20.017 -->(x,y) | ->(tp2(z,x),tp2(z,y)) # label(congruence). 20.017/20.017 -->*(isnoc(x),tp2(y,z)) | ->(isnoc(cons(u,x)),tp2(cons(u,y),z)) # label(replacement). 20.017/20.017 ->(isnoc(cons(x,nil)),tp2(nil,x)) # label(replacement). 20.017/20.017 ->*(x,x) # label(reflexivity). 20.017/20.017 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 20.017/20.017 -->*(isnoc(nil),tp2(x,y)) # label(goal). 20.017/20.017 end_of_list. 20.017/20.017 20.017/20.017 ============================== end of clauses for search ============= 20.017/20.017 20.017/20.017 % There are no natural numbers in the input. 20.017/20.017 20.017/20.017 ============================== DOMAIN SIZE 2 ========================= 20.017/20.017 20.017/20.017 ============================== MODEL ================================= 20.017/20.017 20.017/20.017 interpretation( 2, [number=1, seconds=0], [ 20.017/20.017 20.017/20.017 function(nil, [ 0 ]), 20.017/20.017 20.017/20.017 function(isnoc(_), [ 0, 1 ]), 20.017/20.017 20.017/20.017 function(cons(_,_), [ 20.017/20.017 1, 1, 20.017/20.017 1, 1 ]), 20.017/20.017 20.017/20.017 function(tp2(_,_), [ 20.017/20.017 1, 1, 20.017/20.017 1, 1 ]), 20.017/20.017 20.017/20.017 relation(->*(_,_), [ 20.017/20.017 1, 0, 20.017/20.017 0, 1 ]), 20.017/20.017 20.017/20.017 relation(->(_,_), [ 20.017/20.017 0, 0, 20.017/20.017 0, 1 ]) 20.017/20.017 ]). 20.017/20.017 20.017/20.017 ============================== end of model ========================== 20.017/20.017 20.017/20.017 ============================== STATISTICS ============================ 20.017/20.017 20.017/20.017 For domain size 2. 20.017/20.017 20.017/20.017 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 20.017/20.017 Ground clauses: seen=68, kept=64. 20.017/20.017 Selections=8, assignments=12, propagations=11, current_models=1. 20.017/20.017 Rewrite_terms=202, rewrite_bools=87, indexes=37. 20.017/20.017 Rules_from_neg_clauses=5, cross_offs=5. 20.017/20.017 20.017/20.017 ============================== end of statistics ===================== 20.017/20.017 20.017/20.017 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.017/20.017 20.017/20.017 Exiting with 1 model. 20.017/20.017 20.017/20.017 Process 12766 exit (max_models) Wed Mar 9 09:20:47 2022 20.017/20.017 The process finished Wed Mar 9 09:20:47 2022 20.017/20.017 20.017/20.017 20.017/20.017 Mace4 cooked interpretation: 20.017/20.017 20.017/20.017 20.017/20.017 20.017/20.017 The problem is infeasible. 20.017/20.017 20.017/20.017 20.017/20.017 20.017/20.017 20.017/20.017 -> Problem conclusions: 20.017/20.017 Left linear, Right linear, Linear 20.017/20.017 Weakly orthogonal, Almost orthogonal, Orthogonal 20.017/20.017 CTRS Type: 3 20.017/20.017 Deterministic, Strongly deterministic 20.017/20.017 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.017/20.017 Maybe right-stable CTRS, Overlay CTRS 20.017/20.017 Maybe normal CTRS, Maybe almost normal CTRS 20.017/20.017 Maybe terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.017/20.017 Maybe level confluent 20.017/20.017 Maybe confluent 20.017/20.017 20.017/20.017 Resulting CCPs: 20.017/20.017 No CCPs left 20.017/20.017 20.017/20.017 Problem 1: 20.017/20.017 Operation Termination Processor: 20.017/20.017 Operational terminating? 20.017/20.017 YES 20.017/20.017 20.017/20.017 Problem 1: 20.017/20.017 20.017/20.017 (VAR vu95NonEmpty x y xs ys yu39) 20.017/20.017 (RULES 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y) 20.017/20.017 ) 20.017/20.017 20.017/20.017 Problem 1: 20.017/20.017 Valid CTRS Processor: 20.017/20.017 -> Rules: 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y) 20.017/20.017 -> The system is a deterministic 3-CTRS. 20.017/20.017 20.017/20.017 Problem 1: 20.017/20.017 20.017/20.017 2D Dependency Pair Processor: 20.017/20.017 20.017/20.017 Conditional Termination Problem 1: 20.017/20.017 -> Pairs: 20.017/20.017 Empty 20.017/20.017 -> QPairs: 20.017/20.017 Empty 20.017/20.017 -> Rules: 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y) 20.017/20.017 20.017/20.017 Conditional Termination Problem 2: 20.017/20.017 -> Pairs: 20.017/20.017 ISNOC(cons(x,ys)) -> ISNOC(ys) 20.017/20.017 -> QPairs: 20.017/20.017 Empty 20.017/20.017 -> Rules: 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y) 20.017/20.017 20.017/20.017 20.017/20.017 The problem is decomposed in 2 subproblems. 20.017/20.017 20.017/20.017 Problem 1.1: 20.017/20.017 20.017/20.017 SCC Processor: 20.017/20.017 -> Pairs: 20.017/20.017 Empty 20.017/20.017 -> QPairs: 20.017/20.017 Empty 20.017/20.017 -> Rules: 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y) 20.017/20.017 ->Strongly Connected Components: 20.017/20.017 There is no strongly connected component 20.017/20.017 20.017/20.017 The problem is finite. 20.017/20.017 20.017/20.017 Problem 1.2: 20.017/20.017 20.017/20.017 SCC Processor: 20.017/20.017 -> Pairs: 20.017/20.017 ISNOC(cons(x,ys)) -> ISNOC(ys) 20.017/20.017 -> QPairs: 20.017/20.017 Empty 20.017/20.017 -> Rules: 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y) 20.017/20.017 ->Strongly Connected Components: 20.017/20.017 ->->Cycle: 20.017/20.017 ->->-> Pairs: 20.017/20.017 ISNOC(cons(x,ys)) -> ISNOC(ys) 20.017/20.017 -> QPairs: 20.017/20.017 Empty 20.017/20.017 ->->-> Rules: 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y) 20.017/20.017 20.017/20.017 Problem 1.2: 20.017/20.017 20.017/20.017 Subterm Processor: 20.017/20.017 -> Pairs: 20.017/20.017 ISNOC(cons(x,ys)) -> ISNOC(ys) 20.017/20.017 -> QPairs: 20.017/20.017 Empty 20.017/20.017 -> Rules: 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y) 20.017/20.017 ->Projection: 20.017/20.017 pi(ISNOC) = 1 20.017/20.017 20.017/20.017 Problem 1.2: 20.017/20.017 20.017/20.017 SCC Processor: 20.017/20.017 -> Pairs: 20.017/20.017 Empty 20.017/20.017 -> QPairs: 20.017/20.017 Empty 20.017/20.017 -> Rules: 20.017/20.017 isnoc(cons(x,ys)) -> tp2(cons(x,xs),y) | isnoc(ys) ->* tp2(xs,y) 20.017/20.017 isnoc(cons(y,nil)) -> tp2(nil,y) 20.017/20.017 ->Strongly Connected Components: 20.017/20.017 There is no strongly connected component 20.017/20.017 20.017/20.017 The problem is finite. 20.017/20.017 20.017/20.017 20.017/20.017 -> Problem conclusions: 20.017/20.017 Left linear, Right linear, Linear 20.017/20.017 Weakly orthogonal, Almost orthogonal, Orthogonal 20.017/20.017 CTRS Type: 3 20.017/20.017 Deterministic, Strongly deterministic 20.017/20.017 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.017/20.017 Maybe right-stable CTRS, Overlay CTRS 20.017/20.017 Maybe normal CTRS, Maybe almost normal CTRS 20.017/20.017 Terminating CTRS, Operational terminating CTRS, Joinable CCPs 20.017/20.017 Maybe level confluent 20.017/20.017 Confluent 20.017/20.017 20.017/20.017 The problem is joinable. 20.017/20.017 39.58user 1.09system 0:20.17elapsed 201%CPU (0avgtext+0avgdata 50844maxresident)k 20.017/20.017 8inputs+0outputs (0major+45189minor)pagefaults 0swaps