0.00/0.04 YES 0.00/0.04 0.00/0.04 Problem 1: 0.00/0.04 0.00/0.04 0.00/0.04 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.00/0.04 Confluence Problem: 0.00/0.04 (VAR vNonEmpty x) 0.00/0.04 (REPLACEMENT-MAP 0.00/0.04 (a) 0.00/0.04 (f 1) 0.00/0.04 (b) 0.00/0.04 (fSNonEmpty) 0.00/0.04 ) 0.00/0.04 (RULES 0.00/0.04 a -> b 0.00/0.04 f(x) -> x | x ->* b 0.00/0.04 ) 0.00/0.04 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.00/0.04 0.00/0.04 0.00/0.04 Problem 1: 0.00/0.04 0.00/0.04 Problem 1: 0.00/0.04 0.00/0.04 Clean CTRS Procedure: 0.00/0.04 0.00/0.04 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.00/0.04 Confluence Problem: 0.00/0.04 (VAR x) 0.00/0.04 (REPLACEMENT-MAP 0.00/0.04 (a) 0.00/0.04 (f 1) 0.00/0.04 (b) 0.00/0.04 (fSNonEmpty) 0.00/0.04 ) 0.00/0.04 (RULES 0.00/0.04 a -> b 0.00/0.04 f(x) -> x | x ->* b 0.00/0.04 ) 0.00/0.04 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.00/0.04 0.00/0.04 CRule InfChecker Info: 0.00/0.04 a -> b 0.00/0.04 Rule remains 0.00/0.04 Proof: 0.00/0.04 NO_CONDS 0.00/0.04 0.00/0.04 CRule InfChecker Info: 0.00/0.04 f(x) -> x | x ->* b 0.00/0.04 Rule remains 0.00/0.04 Proof: 0.00/0.04 NO 0.00/0.04 0.00/0.04 Problem 1: 0.00/0.04 0.00/0.04 Infeasibility Problem: 0.00/0.04 [(VAR vNonEmpty x vNonEmpty x) 0.00/0.04 (STRATEGY CONTEXTSENSITIVE 0.00/0.04 (a) 0.00/0.04 (f 1) 0.00/0.04 (b) 0.00/0.04 (fSNonEmpty) 0.00/0.04 ) 0.00/0.04 (RULES 0.00/0.04 a -> b 0.00/0.04 f(x) -> x | x ->* b 0.00/0.04 )] 0.00/0.04 0.00/0.04 Infeasibility Conditions: 0.00/0.04 x ->* b 0.00/0.04 0.00/0.04 Problem 1: 0.00/0.04 0.00/0.04 Obtaining a proof using Prover9: 0.00/0.04 0.00/0.04 -> Prover9 Output: 0.00/0.04 ============================== Prover9 =============================== 0.00/0.04 Prover9 (64) version 2009-11A, November 2009. 0.00/0.04 Process 50855 was started by sandbox2 on n085.star.cs.uiowa.edu, 0.00/0.04 Wed Jul 13 08:28:00 2022 0.00/0.04 The command was "./prover9 -f /tmp/prover950844.in". 0.00/0.04 ============================== end of head =========================== 0.00/0.04 0.00/0.04 ============================== INPUT ================================= 0.00/0.04 0.00/0.04 % Reading from file /tmp/prover950844.in 0.00/0.04 0.00/0.04 assign(max_seconds,20). 0.00/0.04 0.00/0.04 formulas(assumptions). 0.00/0.04 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 0.00/0.04 ->_s0(a,b) # label(replacement). 0.00/0.04 ->*_s0(x1,b) -> ->_s0(f(x1),x1) # label(replacement). 0.00/0.04 ->*_s0(x,x) # label(reflexivity). 0.00/0.04 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 0.00/0.04 end_of_list. 0.00/0.04 0.00/0.04 formulas(goals). 0.00/0.04 (exists x3 ->*_s0(x3,b)) # label(goal). 0.00/0.04 end_of_list. 0.00/0.04 0.00/0.04 ============================== end of input ========================== 0.00/0.04 0.00/0.04 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.00/0.04 0.00/0.04 % Formulas that are not ordinary clauses: 0.00/0.04 1 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 0.00/0.04 2 ->*_s0(x1,b) -> ->_s0(f(x1),x1) # label(replacement) # label(non_clause). [assumption]. 0.00/0.04 3 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 0.00/0.04 4 (exists x3 ->*_s0(x3,b)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.00/0.04 0.00/0.04 ============================== end of process non-clausal formulas === 0.00/0.04 0.00/0.04 ============================== PROCESS INITIAL CLAUSES =============== 0.00/0.04 0.00/0.04 % Clauses before input processing: 0.00/0.04 0.00/0.04 formulas(usable). 0.00/0.04 end_of_list. 0.00/0.04 0.00/0.04 formulas(sos). 0.00/0.04 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 0.00/0.04 ->_s0(a,b) # label(replacement). [assumption]. 0.00/0.04 -->*_s0(x,b) | ->_s0(f(x),x) # label(replacement). [clausify(2)]. 0.00/0.04 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.00/0.04 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(3)]. 0.00/0.04 -->*_s0(x,b) # label(goal). [deny(4)]. 0.00/0.04 end_of_list. 0.00/0.04 0.00/0.04 formulas(demodulators). 0.00/0.04 end_of_list. 0.00/0.04 0.00/0.04 ============================== PREDICATE ELIMINATION ================= 0.00/0.04 0.00/0.04 No predicates eliminated. 0.00/0.04 0.00/0.04 ============================== end predicate elimination ============= 0.00/0.04 0.00/0.04 Auto_denials: 0.00/0.04 % copying label goal to answer in negative clause 0.00/0.04 0.00/0.04 Term ordering decisions: 0.00/0.04 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 0.00/0.04 Function symbol precedence: function_order([ b, a, f ]). 0.00/0.04 After inverse_order: (no changes). 0.00/0.04 Unfolding symbols: (none). 0.00/0.04 0.00/0.04 Auto_inference settings: 0.00/0.04 % set(neg_binary_resolution). % (HNE depth_diff=-2) 0.00/0.04 % clear(ordered_res). % (HNE depth_diff=-2) 0.00/0.04 % set(ur_resolution). % (HNE depth_diff=-2) 0.00/0.04 % set(ur_resolution) -> set(pos_ur_resolution). 0.00/0.04 % set(ur_resolution) -> set(neg_ur_resolution). 0.00/0.04 0.00/0.04 Auto_process settings: (no changes). 0.00/0.04 0.00/0.04 kept: 5 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 0.00/0.04 kept: 6 ->_s0(a,b) # label(replacement). [assumption]. 0.00/0.04 kept: 7 -->*_s0(x,b) | ->_s0(f(x),x) # label(replacement). [clausify(2)]. 0.00/0.04 kept: 8 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.00/0.04 kept: 9 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(3)]. 0.00/0.04 kept: 10 -->*_s0(x,b) # label(goal) # answer(goal). [deny(4)]. 0.00/0.04 0.00/0.04 ============================== PROOF ================================= 0.00/0.04 0.00/0.04 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 0.00/0.04 % Length of proof is 4. 0.00/0.04 % Level of proof is 2. 0.00/0.04 % Maximum clause weight is 3.000. 0.00/0.04 % Given clauses 0. 0.00/0.04 0.00/0.04 4 (exists x3 ->*_s0(x3,b)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.00/0.04 8 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.00/0.04 10 -->*_s0(x,b) # label(goal) # answer(goal). [deny(4)]. 0.00/0.04 11 \$F # answer(goal). [resolve(10,a,8,a)]. 0.00/0.04 0.00/0.04 ============================== end of proof ========================== 0.00/0.04 0.00/0.04 ============================== STATISTICS ============================ 0.00/0.04 0.00/0.04 Given=0. Generated=6. Kept=6. proofs=1. 0.00/0.04 Usable=0. Sos=0. Demods=0. Limbo=5, Disabled=6. Hints=0. 0.00/0.04 Kept_by_rule=0, Deleted_by_rule=0. 0.00/0.04 Forward_subsumed=0. Back_subsumed=0. 0.00/0.04 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 0.00/0.04 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 0.00/0.04 Demod_attempts=0. Demod_rewrites=0. 0.00/0.04 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 0.00/0.04 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0. 0.00/0.04 Megabytes=0.03. 0.00/0.04 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 0.00/0.04 0.00/0.04 ============================== end of statistics ===================== 0.00/0.04 0.00/0.04 ============================== end of search ========================= 0.00/0.04 0.00/0.04 THEOREM PROVED 0.00/0.04 0.00/0.04 Exiting with 1 proof. 0.00/0.04 0.00/0.04 Process 50855 exit (max_proofs) Wed Jul 13 08:28:00 2022 0.00/0.04 0.00/0.04 0.00/0.04 The problem is feasible. 0.00/0.04 0.00/0.04 0.00/0.04 Problem 1: 0.00/0.04 0.00/0.04 Problem 1: 0.00/0.04 0.00/0.04 U_conf Transform CTRS Procedure: 0.00/0.04 0.00/0.04 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.00/0.04 Confluence Problem: 0.00/0.04 (VAR x) 0.00/0.04 (REPLACEMENT-MAP 0.00/0.04 (a) 0.00/0.04 (f 1) 0.00/0.04 (b) 0.00/0.04 (fSNonEmpty) 0.00/0.04 ) 0.00/0.04 (RULES 0.00/0.04 a -> b 0.00/0.04 f(x) -> x | x ->* b 0.00/0.04 ) 0.00/0.04 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.00/0.04 Resulting U_conf(R): 0.00/0.04 (VAR x) 0.00/0.04 (STRATEGY CONTEXTSENSITIVE 0.00/0.04 (a) 0.00/0.04 (f 1) 0.00/0.04 (b) 0.00/0.04 (fSNonEmpty) 0.00/0.04 (U1 1 2) 0.00/0.04 ) 0.00/0.04 (RULES 0.00/0.04 a -> b 0.00/0.04 f(x) -> U1(x,x) 0.00/0.04 U1(b,x) -> x 0.00/0.04 ) 0.00/0.04 UAlphas (debugging): 0.00/0.04 [{f(x) | x}] 0.00/0.04 0.00/0.04 Problem 1: 0.00/0.04 0.00/0.04 0.00/0.04 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.00/0.04 Confluence Problem: 0.00/0.04 (VAR x) 0.00/0.04 (REPLACEMENT-MAP 0.00/0.04 (a) 0.00/0.04 (f 1) 0.00/0.04 (b) 0.00/0.04 (fSNonEmpty) 0.00/0.04 (U1 1, 2) 0.00/0.04 ) 0.00/0.04 (RULES 0.00/0.04 a -> b 0.00/0.04 f(x) -> U1(x,x) 0.00/0.04 U1(b,x) -> x 0.00/0.04 ) 0.00/0.04 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.00/0.04 0.00/0.04 0.00/0.04 Problem 1: 0.00/0.04 0.00/0.04 Problem 1: 0.00/0.04 Not CS-TRS Procedure: 0.00/0.04 R is not a CS-TRS 0.00/0.04 0.00/0.04 Problem 1: 0.00/0.04 Linearity Procedure: 0.00/0.04 Linear? 0.00/0.04 NO 0.00/0.04 0.00/0.04 Problem 1: 0.00/0.04 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.00/0.04 Confluence Problem: 0.00/0.04 (VAR x) 0.00/0.04 (REPLACEMENT-MAP 0.00/0.04 (a) 0.00/0.04 (f 1) 0.00/0.04 (b) 0.00/0.04 (fSNonEmpty) 0.00/0.04 (U1 1, 2) 0.00/0.04 ) 0.00/0.04 (RULES 0.00/0.04 a -> b 0.00/0.04 f(x) -> U1(x,x) 0.00/0.04 U1(b,x) -> x 0.00/0.04 ) 0.00/0.04 Linear:NO 0.00/0.04 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.00/0.04 0.00/0.04 Huet Levy Procedure: 0.00/0.04 -> Rules: 0.00/0.04 a -> b 0.00/0.04 f(x) -> U1(x,x) 0.00/0.04 U1(b,x) -> x 0.00/0.04 -> Vars: 0.00/0.04 x, x 0.00/0.04 0.00/0.04 -> Rlps: 0.00/0.04 (rule: a -> b, id: 1, possubterms: a->[]) 0.00/0.04 (rule: f(x) -> U1(x,x), id: 2, possubterms: f(x)->[]) 0.00/0.04 (rule: U1(b,x) -> x, id: 3, possubterms: U1(b,x)->[], b->[1]) 0.00/0.04 0.00/0.04 -> Unifications: 0.00/0.04 0.00/0.04 0.00/0.04 -> Critical pairs info: 0.00/0.04 0.00/0.04 0.00/0.04 -> Problem conclusions: 0.00/0.04 Left linear, Not right linear, Not linear 0.00/0.04 Weakly orthogonal, Almost orthogonal, Orthogonal 0.00/0.04 Huet-Levy confluent, Not Newman confluent 0.00/0.04 R is a TRS 0.00/0.04 0.00/0.04 0.00/0.04 The problem is confluent. 0.00/10.11 EOF