20.025/20.025 YES 20.025/20.025 20.025/20.025 Problem 1: 20.025/20.025 20.025/20.025 20.025/20.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.025/20.025 Confluence Problem: 20.025/20.025 (VAR vNonEmpty:S x':S x'':S x:S y':S y:S y'':S) 20.025/20.025 (STRATEGY CONTEXTSENSITIVE 20.025/20.025 (f 1 2) 20.025/20.025 (fSNonEmpty) 20.025/20.025 (g 1) 20.025/20.025 (h 1) 20.025/20.025 ) 20.025/20.025 (RULES 20.025/20.025 f(x':S,x'':S) -> g(x:S) | x':S ->* x:S, x'':S ->* x:S 20.025/20.025 f(y':S,h(y'':S)) -> g(y:S) | y':S ->* y:S, y'':S ->* y:S 20.025/20.025 ) 20.025/20.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.025/20.025 20.025/20.025 20.025/20.025 Problem 1: 20.025/20.025 20.025/20.025 Inlining of Conditions Processor [STERN17]: 20.025/20.025 20.025/20.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.025/20.025 Confluence Problem: 20.025/20.025 (VAR vNonEmpty:S x':S x'':S x:S y':S y:S y'':S) 20.025/20.025 (STRATEGY CONTEXTSENSITIVE 20.025/20.025 (f 1 2) 20.025/20.025 (fSNonEmpty) 20.025/20.025 (g 1) 20.025/20.025 (h 1) 20.025/20.025 ) 20.025/20.025 (RULES 20.025/20.025 f(x':S,x'':S) -> g(x:S) | x':S ->* x:S, x'':S ->* x:S 20.025/20.025 f(y':S,h(y'':S)) -> g(y:S) | y':S ->* y:S, y'':S ->* y:S 20.025/20.025 ) 20.025/20.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.025/20.025 20.025/20.025 20.025/20.025 Problem 1: 20.025/20.025 20.025/20.025 Clean CTRS Processor: 20.025/20.025 20.025/20.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.025/20.025 Confluence Problem: 20.025/20.025 (VAR x':S x'':S x:S y':S y:S y'':S) 20.025/20.025 (STRATEGY CONTEXTSENSITIVE 20.025/20.025 (f 1 2) 20.025/20.025 (fSNonEmpty) 20.025/20.025 (g 1) 20.025/20.025 (h 1) 20.025/20.025 ) 20.025/20.025 (RULES 20.025/20.025 f(x':S,x'':S) -> g(x:S) | x':S ->* x:S, x'':S ->* x:S 20.025/20.025 f(y':S,h(y'':S)) -> g(y:S) | y':S ->* y:S, y'':S ->* y:S 20.025/20.025 ) 20.025/20.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.025/20.025 20.025/20.025 CRule InfChecker Info: 20.025/20.025 f(x':S,x'':S) -> g(x:S) | x':S ->* x:S, x'':S ->* x:S 20.025/20.025 Rule remains 20.025/20.025 Proof: 20.025/20.025 NO 20.025/20.025 20.025/20.025 Problem 1: 20.025/20.025 20.025/20.025 Infeasibility Problem: 20.025/20.025 [(VAR vNonEmpty:S vNonEmpty:S:S x':S:S x'':S:S x:S:S y':S:S y:S:S y'':S:S) 20.025/20.025 (STRATEGY CONTEXTSENSITIVE 20.025/20.025 (f 1 2) 20.025/20.025 (fSNonEmpty) 20.025/20.025 (g 1) 20.025/20.025 (h 1) 20.025/20.025 ) 20.025/20.025 (RULES 20.025/20.025 f(x':S:S,x'':S:S) -> g(x:S:S) | x':S:S ->* x:S:S, x'':S:S ->* x:S:S 20.025/20.025 f(y':S:S,h(y'':S:S)) -> g(y:S:S) | y':S:S ->* y:S:S, y'':S:S ->* y:S:S 20.025/20.025 )] 20.025/20.025 20.025/20.025 Infeasibility Conditions: 20.025/20.025 x':S:S ->* x:S:S, x'':S:S ->* x:S:S 20.025/20.025 20.025/20.025 Problem 1: 20.025/20.025 20.025/20.025 Obtaining a proof using Prover9: 20.025/20.025 20.025/20.025 -> Prover9 Output: 20.025/20.025 ============================== Prover9 =============================== 20.025/20.025 Prover9 (64) version 2009-11A, November 2009. 20.025/20.025 Process 106414 was started by ubuntu on ubuntu, 20.025/20.025 Fri Jul 16 14:54:52 2021 20.025/20.025 The command was "./prover9 -f /tmp/prover9106405-0.in". 20.025/20.025 ============================== end of head =========================== 20.025/20.025 20.025/20.025 ============================== INPUT ================================= 20.025/20.025 20.025/20.025 % Reading from file /tmp/prover9106405-0.in 20.025/20.025 20.025/20.025 assign(max_seconds,20). 20.025/20.025 20.025/20.025 formulas(assumptions). 20.025/20.025 ->*_s0(x,x) # label(reflexivity). 20.025/20.025 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.025/20.025 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 20.025/20.025 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 20.025/20.025 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 20.025/20.025 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence). 20.025/20.025 ->*_s0(x2,x4) & ->*_s0(x3,x4) -> ->_s0(f(x2,x3),g(x4)) # label(replacement). 20.025/20.025 ->*_s0(x5,x6) & ->*_s0(x7,x6) -> ->_s0(f(x5,h(x7)),g(x6)) # label(replacement). 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 formulas(goals). 20.025/20.025 (exists x2 exists x3 exists x4 (->*_s0(x2,x4) & ->*_s0(x3,x4))) # label(goal). 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 ============================== end of input ========================== 20.025/20.025 20.025/20.025 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.025/20.025 20.025/20.025 % Formulas that are not ordinary clauses: 20.025/20.025 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.025/20.025 2 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.025/20.025 3 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.025/20.025 4 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 20.025/20.025 5 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 20.025/20.025 6 ->*_s0(x2,x4) & ->*_s0(x3,x4) -> ->_s0(f(x2,x3),g(x4)) # label(replacement) # label(non_clause). [assumption]. 20.025/20.025 7 ->*_s0(x5,x6) & ->*_s0(x7,x6) -> ->_s0(f(x5,h(x7)),g(x6)) # label(replacement) # label(non_clause). [assumption]. 20.025/20.025 8 (exists x2 exists x3 exists x4 (->*_s0(x2,x4) & ->*_s0(x3,x4))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.025/20.025 20.025/20.025 ============================== end of process non-clausal formulas === 20.025/20.025 20.025/20.025 ============================== PROCESS INITIAL CLAUSES =============== 20.025/20.025 20.025/20.025 % Clauses before input processing: 20.025/20.025 20.025/20.025 formulas(usable). 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 formulas(sos). 20.025/20.025 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.025/20.025 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 20.025/20.025 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 20.025/20.025 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 20.025/20.025 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 20.025/20.025 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 20.025/20.025 -->*_s0(x,y) | -->*_s0(z,y) | ->_s0(f(x,z),g(y)) # label(replacement). [clausify(6)]. 20.025/20.025 -->*_s0(x,y) | -->*_s0(z,y) | ->_s0(f(x,h(z)),g(y)) # label(replacement). [clausify(7)]. 20.025/20.025 -->*_s0(x,y) | -->*_s0(z,y) # label(goal). [deny(8)]. 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 formulas(demodulators). 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 ============================== PREDICATE ELIMINATION ================= 20.025/20.025 20.025/20.025 No predicates eliminated. 20.025/20.025 20.025/20.025 ============================== end predicate elimination ============= 20.025/20.025 20.025/20.025 Auto_denials: 20.025/20.025 % copying label goal to answer in negative clause 20.025/20.025 20.025/20.025 Term ordering decisions: 20.025/20.025 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 20.025/20.025 Function symbol precedence: function_order([ f, g, h ]). 20.025/20.025 After inverse_order: (no changes). 20.025/20.025 Unfolding symbols: (none). 20.025/20.025 20.025/20.025 Auto_inference settings: 20.025/20.025 % set(neg_binary_resolution). % (HNE depth_diff=-7) 20.025/20.025 % clear(ordered_res). % (HNE depth_diff=-7) 20.025/20.025 % set(ur_resolution). % (HNE depth_diff=-7) 20.025/20.025 % set(ur_resolution) -> set(pos_ur_resolution). 20.025/20.025 % set(ur_resolution) -> set(neg_ur_resolution). 20.025/20.025 20.025/20.025 Auto_process settings: 20.025/20.025 % set(unit_deletion). % (Horn set with negative nonunits) 20.025/20.025 20.025/20.025 kept: 9 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.025/20.025 kept: 10 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 20.025/20.025 kept: 11 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 20.025/20.025 kept: 12 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 20.025/20.025 kept: 13 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 20.025/20.025 kept: 14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 20.025/20.025 kept: 15 -->*_s0(x,y) | -->*_s0(z,y) | ->_s0(f(x,z),g(y)) # label(replacement). [clausify(6)]. 20.025/20.025 kept: 16 -->*_s0(x,y) | -->*_s0(z,y) | ->_s0(f(x,h(z)),g(y)) # label(replacement). [clausify(7)]. 20.025/20.025 kept: 17 -->*_s0(x,y) | -->*_s0(z,y) # label(goal) # answer(goal). [deny(8)]. 20.025/20.025 20.025/20.025 ============================== end of process initial clauses ======== 20.025/20.025 20.025/20.025 ============================== CLAUSES FOR SEARCH ==================== 20.025/20.025 20.025/20.025 % Clauses after input processing: 20.025/20.025 20.025/20.025 formulas(usable). 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 formulas(sos). 20.025/20.025 9 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.025/20.025 10 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 20.025/20.025 11 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 20.025/20.025 12 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 20.025/20.025 13 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 20.025/20.025 14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 20.025/20.025 17 -->*_s0(x,y) | -->*_s0(z,y) # label(goal) # answer(goal). [deny(8)]. 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 formulas(demodulators). 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 ============================== end of clauses for search ============= 20.025/20.025 20.025/20.025 ============================== SEARCH ================================ 20.025/20.025 20.025/20.025 % Starting search at 0.01 seconds. 20.025/20.025 20.025/20.025 given #1 (I,wt=3): 9 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.025/20.025 20.025/20.025 given #2 (I,wt=9): 10 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 20.025/20.025 20.025/20.025 given #3 (I,wt=10): 11 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 20.025/20.025 20.025/20.025 given #4 (I,wt=10): 12 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 20.025/20.025 20.025/20.025 given #5 (I,wt=8): 13 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 20.025/20.025 20.025/20.025 given #6 (I,wt=8): 14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 20.025/20.025 20.025/20.025 given #7 (I,wt=6): 17 -->*_s0(x,y) | -->*_s0(z,y) # label(goal) # answer(goal). [deny(8)]. 20.025/20.025 20.025/20.025 ============================== PROOF ================================= 20.025/20.025 20.025/20.025 % Proof 1 at 0.01 (+ 0.00) seconds: goal. 20.025/20.025 % Length of proof is 5. 20.025/20.025 % Level of proof is 3. 20.025/20.025 % Maximum clause weight is 6.000. 20.025/20.025 % Given clauses 7. 20.025/20.025 20.025/20.025 8 (exists x2 exists x3 exists x4 (->*_s0(x2,x4) & ->*_s0(x3,x4))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.025/20.025 9 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.025/20.025 17 -->*_s0(x,y) | -->*_s0(z,y) # label(goal) # answer(goal). [deny(8)]. 20.025/20.025 18 -->*_s0(x,y) # answer(goal). [resolve(17,a,9,a)]. 20.025/20.025 19 $F # answer(goal). [resolve(18,a,9,a)]. 20.025/20.025 20.025/20.025 ============================== end of proof ========================== 20.025/20.025 20.025/20.025 ============================== STATISTICS ============================ 20.025/20.025 20.025/20.025 Given=7. Generated=11. Kept=10. proofs=1. 20.025/20.025 Usable=7. Sos=0. Demods=0. Limbo=0, Disabled=11. Hints=0. 20.025/20.025 Kept_by_rule=0, Deleted_by_rule=0. 20.025/20.025 Forward_subsumed=1. Back_subsumed=2. 20.025/20.025 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.025/20.025 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.025/20.025 Demod_attempts=0. Demod_rewrites=0. 20.025/20.025 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.025/20.025 Nonunit_fsub_feature_tests=3. Nonunit_bsub_feature_tests=14. 20.025/20.025 Megabytes=0.06. 20.025/20.025 User_CPU=0.01, System_CPU=0.00, Wall_clock=0. 20.025/20.025 20.025/20.025 ============================== end of statistics ===================== 20.025/20.025 20.025/20.025 ============================== end of search ========================= 20.025/20.025 20.025/20.025 THEOREM PROVED 20.025/20.025 20.025/20.025 Exiting with 1 proof. 20.025/20.025 20.025/20.025 Process 106414 exit (max_proofs) Fri Jul 16 14:54:52 2021 20.025/20.025 20.025/20.025 20.025/20.025 The problem is feasible. 20.025/20.025 20.025/20.025 20.025/20.025 CRule InfChecker Info: 20.025/20.025 f(y':S,h(y'':S)) -> g(y:S) | y':S ->* y:S, y'':S ->* y:S 20.025/20.025 Rule remains 20.025/20.025 Proof: 20.025/20.025 NO 20.025/20.025 20.025/20.025 Problem 1: 20.025/20.025 20.025/20.025 Infeasibility Problem: 20.025/20.025 [(VAR vNonEmpty:S vNonEmpty:S:S x':S:S x'':S:S x:S:S y':S:S y:S:S y'':S:S) 20.025/20.025 (STRATEGY CONTEXTSENSITIVE 20.025/20.025 (f 1 2) 20.025/20.025 (fSNonEmpty) 20.025/20.025 (g 1) 20.025/20.025 (h 1) 20.025/20.025 ) 20.025/20.025 (RULES 20.025/20.025 f(x':S:S,x'':S:S) -> g(x:S:S) | x':S:S ->* x:S:S, x'':S:S ->* x:S:S 20.025/20.025 f(y':S:S,h(y'':S:S)) -> g(y:S:S) | y':S:S ->* y:S:S, y'':S:S ->* y:S:S 20.025/20.025 )] 20.025/20.025 20.025/20.025 Infeasibility Conditions: 20.025/20.025 y':S:S ->* y:S:S, y'':S:S ->* y:S:S 20.025/20.025 20.025/20.025 Problem 1: 20.025/20.025 20.025/20.025 Obtaining a proof using Prover9: 20.025/20.025 20.025/20.025 -> Prover9 Output: 20.025/20.025 ============================== Prover9 =============================== 20.025/20.025 Prover9 (64) version 2009-11A, November 2009. 20.025/20.025 Process 106436 was started by ubuntu on ubuntu, 20.025/20.025 Fri Jul 16 14:54:52 2021 20.025/20.025 The command was "./prover9 -f /tmp/prover9106427-0.in". 20.025/20.025 ============================== end of head =========================== 20.025/20.025 20.025/20.025 ============================== INPUT ================================= 20.025/20.025 20.025/20.025 % Reading from file /tmp/prover9106427-0.in 20.025/20.025 20.025/20.025 assign(max_seconds,20). 20.025/20.025 20.025/20.025 formulas(assumptions). 20.025/20.025 ->*_s0(x,x) # label(reflexivity). 20.025/20.025 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.025/20.025 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 20.025/20.025 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 20.025/20.025 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 20.025/20.025 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence). 20.025/20.025 ->*_s0(x2,x4) & ->*_s0(x3,x4) -> ->_s0(f(x2,x3),g(x4)) # label(replacement). 20.025/20.025 ->*_s0(x5,x6) & ->*_s0(x7,x6) -> ->_s0(f(x5,h(x7)),g(x6)) # label(replacement). 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 formulas(goals). 20.025/20.025 (exists x5 exists x6 exists x7 (->*_s0(x5,x6) & ->*_s0(x7,x6))) # label(goal). 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 ============================== end of input ========================== 20.025/20.025 20.025/20.025 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.025/20.025 20.025/20.025 % Formulas that are not ordinary clauses: 20.025/20.025 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.025/20.025 2 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.025/20.025 3 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.025/20.025 4 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 20.025/20.025 5 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 20.025/20.025 6 ->*_s0(x2,x4) & ->*_s0(x3,x4) -> ->_s0(f(x2,x3),g(x4)) # label(replacement) # label(non_clause). [assumption]. 20.025/20.025 7 ->*_s0(x5,x6) & ->*_s0(x7,x6) -> ->_s0(f(x5,h(x7)),g(x6)) # label(replacement) # label(non_clause). [assumption]. 20.025/20.025 8 (exists x5 exists x6 exists x7 (->*_s0(x5,x6) & ->*_s0(x7,x6))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.025/20.025 20.025/20.025 ============================== end of process non-clausal formulas === 20.025/20.025 20.025/20.025 ============================== PROCESS INITIAL CLAUSES =============== 20.025/20.025 20.025/20.025 % Clauses before input processing: 20.025/20.025 20.025/20.025 formulas(usable). 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 formulas(sos). 20.025/20.025 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.025/20.025 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 20.025/20.025 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 20.025/20.025 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 20.025/20.025 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 20.025/20.025 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 20.025/20.025 -->*_s0(x,y) | -->*_s0(z,y) | ->_s0(f(x,z),g(y)) # label(replacement). [clausify(6)]. 20.025/20.025 -->*_s0(x,y) | -->*_s0(z,y) | ->_s0(f(x,h(z)),g(y)) # label(replacement). [clausify(7)]. 20.025/20.025 -->*_s0(x,y) | -->*_s0(z,y) # label(goal). [deny(8)]. 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 formulas(demodulators). 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 ============================== PREDICATE ELIMINATION ================= 20.025/20.025 20.025/20.025 No predicates eliminated. 20.025/20.025 20.025/20.025 ============================== end predicate elimination ============= 20.025/20.025 20.025/20.025 Auto_denials: 20.025/20.025 % copying label goal to answer in negative clause 20.025/20.025 20.025/20.025 Term ordering decisions: 20.025/20.025 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 20.025/20.025 Function symbol precedence: function_order([ f, g, h ]). 20.025/20.025 After inverse_order: (no changes). 20.025/20.025 Unfolding symbols: (none). 20.025/20.025 20.025/20.025 Auto_inference settings: 20.025/20.025 % set(neg_binary_resolution). % (HNE depth_diff=-7) 20.025/20.025 % clear(ordered_res). % (HNE depth_diff=-7) 20.025/20.025 % set(ur_resolution). % (HNE depth_diff=-7) 20.025/20.025 % set(ur_resolution) -> set(pos_ur_resolution). 20.025/20.025 % set(ur_resolution) -> set(neg_ur_resolution). 20.025/20.025 20.025/20.025 Auto_process settings: 20.025/20.025 % set(unit_deletion). % (Horn set with negative nonunits) 20.025/20.025 20.025/20.025 kept: 9 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.025/20.025 kept: 10 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 20.025/20.025 kept: 11 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 20.025/20.025 kept: 12 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 20.025/20.025 kept: 13 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 20.025/20.025 kept: 14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 20.025/20.025 kept: 15 -->*_s0(x,y) | -->*_s0(z,y) | ->_s0(f(x,z),g(y)) # label(replacement). [clausify(6)]. 20.025/20.025 kept: 16 -->*_s0(x,y) | -->*_s0(z,y) | ->_s0(f(x,h(z)),g(y)) # label(replacement). [clausify(7)]. 20.025/20.025 kept: 17 -->*_s0(x,y) | -->*_s0(z,y) # label(goal) # answer(goal). [deny(8)]. 20.025/20.025 20.025/20.025 ============================== end of process initial clauses ======== 20.025/20.025 20.025/20.025 ============================== CLAUSES FOR SEARCH ==================== 20.025/20.025 20.025/20.025 % Clauses after input processing: 20.025/20.025 20.025/20.025 formulas(usable). 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 formulas(sos). 20.025/20.025 9 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.025/20.025 10 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 20.025/20.025 11 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 20.025/20.025 12 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 20.025/20.025 13 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 20.025/20.025 14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 20.025/20.025 17 -->*_s0(x,y) | -->*_s0(z,y) # label(goal) # answer(goal). [deny(8)]. 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 formulas(demodulators). 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 ============================== end of clauses for search ============= 20.025/20.025 20.025/20.025 ============================== SEARCH ================================ 20.025/20.025 20.025/20.025 % Starting search at 0.00 seconds. 20.025/20.025 20.025/20.025 given #1 (I,wt=3): 9 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.025/20.025 20.025/20.025 given #2 (I,wt=9): 10 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 20.025/20.025 20.025/20.025 given #3 (I,wt=10): 11 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 20.025/20.025 20.025/20.025 given #4 (I,wt=10): 12 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 20.025/20.025 20.025/20.025 given #5 (I,wt=8): 13 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 20.025/20.025 20.025/20.025 given #6 (I,wt=8): 14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 20.025/20.025 20.025/20.025 given #7 (I,wt=6): 17 -->*_s0(x,y) | -->*_s0(z,y) # label(goal) # answer(goal). [deny(8)]. 20.025/20.025 20.025/20.025 ============================== PROOF ================================= 20.025/20.025 20.025/20.025 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.025/20.025 % Length of proof is 5. 20.025/20.025 % Level of proof is 3. 20.025/20.025 % Maximum clause weight is 6.000. 20.025/20.025 % Given clauses 7. 20.025/20.025 20.025/20.025 8 (exists x5 exists x6 exists x7 (->*_s0(x5,x6) & ->*_s0(x7,x6))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.025/20.025 9 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.025/20.025 17 -->*_s0(x,y) | -->*_s0(z,y) # label(goal) # answer(goal). [deny(8)]. 20.025/20.025 18 -->*_s0(x,y) # answer(goal). [resolve(17,a,9,a)]. 20.025/20.025 19 $F # answer(goal). [resolve(18,a,9,a)]. 20.025/20.025 20.025/20.025 ============================== end of proof ========================== 20.025/20.025 20.025/20.025 ============================== STATISTICS ============================ 20.025/20.025 20.025/20.025 Given=7. Generated=11. Kept=10. proofs=1. 20.025/20.025 Usable=7. Sos=0. Demods=0. Limbo=0, Disabled=11. Hints=0. 20.025/20.025 Kept_by_rule=0, Deleted_by_rule=0. 20.025/20.025 Forward_subsumed=1. Back_subsumed=2. 20.025/20.025 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.025/20.025 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.025/20.025 Demod_attempts=0. Demod_rewrites=0. 20.025/20.025 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.025/20.025 Nonunit_fsub_feature_tests=3. Nonunit_bsub_feature_tests=14. 20.025/20.025 Megabytes=0.06. 20.025/20.025 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.025/20.025 20.025/20.025 ============================== end of statistics ===================== 20.025/20.025 20.025/20.025 ============================== end of search ========================= 20.025/20.025 20.025/20.025 THEOREM PROVED 20.025/20.025 20.025/20.025 Exiting with 1 proof. 20.025/20.025 20.025/20.025 Process 106436 exit (max_proofs) Fri Jul 16 14:54:52 2021 20.025/20.025 20.025/20.025 20.025/20.025 The problem is feasible. 20.025/20.025 20.025/20.025 20.025/20.025 Problem 1: 20.025/20.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.025/20.025 Confluence Problem: 20.025/20.025 (VAR x':S x'':S x:S y':S y:S y'':S) 20.025/20.025 (STRATEGY CONTEXTSENSITIVE 20.025/20.025 (f 1 2) 20.025/20.025 (fSNonEmpty) 20.025/20.025 (g 1) 20.025/20.025 (h 1) 20.025/20.025 ) 20.025/20.025 (RULES 20.025/20.025 f(x':S,x'':S) -> g(x:S) | x':S ->* x:S, x'':S ->* x:S 20.025/20.025 f(y':S,h(y'':S)) -> g(y:S) | y':S ->* y:S, y'':S ->* y:S 20.025/20.025 ) 20.025/20.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.025/20.025 20.025/20.025 Critical Pairs Processor: 20.025/20.025 -> Rules: 20.025/20.025 f(x':S,x'':S) -> g(x:S) | x':S ->* x:S, x'':S ->* x:S 20.025/20.025 f(y':S,h(y'':S)) -> g(y:S) | y':S ->* y:S, y'':S ->* y:S 20.025/20.025 -> Vars: 20.025/20.025 "x'", "x''", "x", "y'", "y", "y''" 20.025/20.025 -> FVars: 20.025/20.025 "x7", "x8", "x9", "x10", "x11", "x12" 20.025/20.025 -> PVars: 20.025/20.025 "x'": ["x7"], "x''": ["x8"], "x": ["x9"], "y'": ["x10"], "y": ["x11"], "y''": ["x12"] 20.025/20.025 20.025/20.025 -> Rlps: 20.025/20.025 crule: f(x7:S,x8:S) -> g(x9:S) | x7:S ->* x9:S, x8:S ->* x9:S, id: 1, possubterms: f(x7:S,x8:S)-> [] 20.025/20.025 crule: f(x10:S,h(x12:S)) -> g(x11:S) | x10:S ->* x11:S, x12:S ->* x11:S, id: 2, possubterms: f(x10:S,h(x12:S))-> [], h(x12:S)-> [2] 20.025/20.025 20.025/20.025 -> Unifications: 20.025/20.025 R2 unifies with R1 at p: [], l: f(x10:S,h(x12:S)), lp: f(x10:S,h(x12:S)), conds: {x10:S ->* x11:S, x12:S ->* x11:S, x':S ->* x:S, x'':S ->* x:S}, sig: {x'':S -> h(x12:S),x10:S -> x':S}, l': f(x':S,x'':S), r: g(x11:S), r': g(x:S) 20.025/20.025 20.025/20.025 -> Critical pairs info: 20.025/20.025 | x':S ->* y:S, y'':S ->* y:S, x':S ->* x:S, h(y'':S) ->* x:S => Not trivial, Overlay, N1 20.025/20.025 20.025/20.025 -> Problem conclusions: 20.025/20.025 Left linear, Right linear, Linear 20.025/20.025 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.025/20.025 CTRS Type: 3 20.025/20.025 Deterministic, Strongly deterministic 20.025/20.025 Oriented CTRS, Properly oriented CTRS, Not join CTRS 20.025/20.025 Maybe right-stable CTRS, Overlay CTRS 20.025/20.025 Maybe normal CTRS, Maybe almost normal CTRS 20.025/20.025 Maybe terminating CTRS, Maybe joinable CCPs 20.025/20.025 Maybe level confluent 20.025/20.025 Maybe confluent 20.025/20.025 20.025/20.025 Problem 1: 20.025/20.025 Clean Infeasible CCPs Processor: 20.025/20.025 Num of CPIs: 1 20.025/20.025 Timeout: 30 20.025/20.025 Timeout for each infeasibility problem: 30 s 20.025/20.025 | x':S ->* y:S, y'':S ->* y:S, x':S ->* x:S, h(y'':S) ->* x:S => Not trivial, Overlay, N1 20.025/20.025 (PROBLEM INFEASIBILITY) 20.025/20.025 (VAR x1 x11 x y1 y y11) 20.025/20.025 (RULES 20.025/20.025 f(x1,x11) -> g(x) | x1 ->* x, x11 ->* x 20.025/20.025 f(y1,h(y11)) -> g(y) | y1 ->* y, y11 ->* y) 20.025/20.025 (VAR x9 x10 x11 x12) 20.025/20.025 (CONDITION x9 ->* x11, x12 ->* x11, x9 ->* x10, h(x12) ->* x10) 20.025/20.025 20.025/20.025 Proof: 20.025/20.025 YES 20.025/20.025 20.025/20.025 Problem 1: 20.025/20.025 20.025/20.025 Infeasibility Problem: 20.025/20.025 [(VAR vNonEmpty:S x1:S x11:S x:S y1:S y:S y11:S x9:S x10:S x12:S) 20.025/20.025 (STRATEGY CONTEXTSENSITIVE 20.025/20.025 (f 1 2) 20.025/20.025 (fSNonEmpty) 20.025/20.025 (g 1) 20.025/20.025 (h 1) 20.025/20.025 ) 20.025/20.025 (RULES 20.025/20.025 f(x1:S,x11:S) -> g(x:S) | x1:S ->* x:S, x11:S ->* x:S 20.025/20.025 f(y1:S,h(y11:S)) -> g(y:S) | y1:S ->* y:S, y11:S ->* y:S 20.025/20.025 )] 20.025/20.025 20.025/20.025 Infeasibility Conditions: 20.025/20.025 x9:S ->* x11:S, x12:S ->* x11:S, x9:S ->* x10:S, h(x12:S) ->* x10:S 20.025/20.025 20.025/20.025 Problem 1: 20.025/20.025 20.025/20.025 Obtaining a model using Mace4: 20.025/20.025 20.025/20.025 -> Usable Rules: 20.025/20.025 f(x1:S,x11:S) -> g(x:S) | x1:S ->* x:S, x11:S ->* x:S 20.025/20.025 f(y1:S,h(y11:S)) -> g(y:S) | y1:S ->* y:S, y11:S ->* y:S 20.025/20.025 20.025/20.025 -> Mace4 Output: 20.025/20.025 ============================== Mace4 ================================= 20.025/20.025 Mace4 (64) version 2009-11A, November 2009. 20.025/20.025 Process 106470 was started by ubuntu on ubuntu, 20.025/20.025 Fri Jul 16 14:54:52 2021 20.025/20.025 The command was "./mace4 -c -f /tmp/mace4106453-2.in". 20.025/20.025 ============================== end of head =========================== 20.025/20.025 20.025/20.025 ============================== INPUT ================================= 20.025/20.025 20.025/20.025 % Reading from file /tmp/mace4106453-2.in 20.025/20.025 20.025/20.025 assign(max_seconds,10). 20.025/20.025 20.025/20.025 formulas(assumptions). 20.025/20.025 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence). 20.025/20.025 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence). 20.025/20.025 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence). 20.025/20.025 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence). 20.025/20.025 ->*(x1,x3) & ->*(x2,x3) -> ->(f(x1,x2),g(x3)) # label(replacement). 20.025/20.025 ->*(x4,x5) & ->*(x6,x5) -> ->(f(x4,h(x6)),g(x5)) # label(replacement). 20.025/20.025 ->*(x,x) # label(reflexivity). 20.025/20.025 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 formulas(goals). 20.025/20.025 (exists x2 exists x7 exists x8 exists x9 (->*(x7,x2) & ->*(x9,x2) & ->*(x7,x8) & ->*(h(x9),x8))) # label(goal). 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 ============================== end of input ========================== 20.025/20.025 20.025/20.025 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.025/20.025 20.025/20.025 % Formulas that are not ordinary clauses: 20.025/20.025 1 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.025/20.025 2 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.025/20.025 3 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 20.025/20.025 4 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 20.025/20.025 5 ->*(x1,x3) & ->*(x2,x3) -> ->(f(x1,x2),g(x3)) # label(replacement) # label(non_clause). [assumption]. 20.025/20.025 6 ->*(x4,x5) & ->*(x6,x5) -> ->(f(x4,h(x6)),g(x5)) # label(replacement) # label(non_clause). [assumption]. 20.025/20.025 7 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.025/20.025 8 (exists x2 exists x7 exists x8 exists x9 (->*(x7,x2) & ->*(x9,x2) & ->*(x7,x8) & ->*(h(x9),x8))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.025/20.025 20.025/20.025 ============================== end of process non-clausal formulas === 20.025/20.025 20.025/20.025 ============================== CLAUSES FOR SEARCH ==================== 20.025/20.025 20.025/20.025 formulas(mace4_clauses). 20.025/20.025 -->(x,y) | ->(f(x,z),f(y,z)) # label(congruence). 20.025/20.025 -->(x,y) | ->(f(z,x),f(z,y)) # label(congruence). 20.025/20.025 -->(x,y) | ->(g(x),g(y)) # label(congruence). 20.025/20.025 -->(x,y) | ->(h(x),h(y)) # label(congruence). 20.025/20.025 -->*(x,y) | -->*(z,y) | ->(f(x,z),g(y)) # label(replacement). 20.025/20.025 -->*(x,y) | -->*(z,y) | ->(f(x,h(z)),g(y)) # label(replacement). 20.025/20.025 ->*(x,x) # label(reflexivity). 20.025/20.025 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 20.025/20.025 -->*(x,y) | -->*(z,y) | -->*(x,u) | -->*(h(z),u) # label(goal). 20.025/20.025 end_of_list. 20.025/20.025 20.025/20.025 ============================== end of clauses for search ============= 20.025/20.025 20.025/20.025 % There are no natural numbers in the input. 20.025/20.025 20.025/20.025 ============================== DOMAIN SIZE 2 ========================= 20.025/20.025 20.025/20.025 ============================== MODEL ================================= 20.025/20.025 20.025/20.025 interpretation( 2, [number=1, seconds=0], [ 20.025/20.025 20.025/20.025 function(g(_), [ 0, 0 ]), 20.025/20.025 20.025/20.025 function(h(_), [ 1, 0 ]), 20.025/20.025 20.025/20.025 function(f(_,_), [ 20.025/20.025 0, 0, 20.025/20.025 0, 0 ]), 20.025/20.025 20.025/20.025 relation(->*(_,_), [ 20.025/20.025 1, 0, 20.025/20.025 0, 1 ]), 20.025/20.025 20.025/20.025 relation(->(_,_), [ 20.025/20.025 1, 0, 20.025/20.025 0, 1 ]) 20.025/20.025 ]). 20.025/20.025 20.025/20.025 ============================== end of model ========================== 20.025/20.025 20.025/20.025 ============================== STATISTICS ============================ 20.025/20.025 20.025/20.025 For domain size 2. 20.025/20.025 20.025/20.025 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 20.025/20.025 Ground clauses: seen=66, kept=49. 20.025/20.025 Selections=2, assignments=2, propagations=14, current_models=1. 20.025/20.025 Rewrite_terms=90, rewrite_bools=73, indexes=14. 20.025/20.025 Rules_from_neg_clauses=6, cross_offs=6. 20.025/20.025 20.025/20.025 ============================== end of statistics ===================== 20.025/20.025 20.025/20.025 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.025/20.025 20.025/20.025 Exiting with 1 model. 20.025/20.025 20.025/20.025 Process 106470 exit (max_models) Fri Jul 16 14:54:52 2021 20.025/20.025 The process finished Fri Jul 16 14:54:52 2021 20.025/20.025 20.025/20.025 20.025/20.025 Mace4 cooked interpretation: 20.025/20.025 20.025/20.025 20.025/20.025 20.025/20.025 The problem is infeasible. 20.025/20.025 20.025/20.025 20.025/20.025 20.025/20.025 20.025/20.025 -> Problem conclusions: 20.025/20.025 Left linear, Right linear, Linear 20.025/20.025 Weakly orthogonal, Almost orthogonal, Orthogonal 20.025/20.025 CTRS Type: 3 20.025/20.025 Deterministic, Strongly deterministic 20.025/20.025 Oriented CTRS, Properly oriented CTRS, Not join CTRS 20.025/20.025 Maybe right-stable CTRS, Overlay CTRS 20.025/20.025 Maybe normal CTRS, Maybe almost normal CTRS 20.025/20.025 Maybe terminating CTRS, Joinable CCPs 20.025/20.025 Maybe level confluent 20.025/20.025 Maybe confluent 20.025/20.025 20.025/20.025 Resulting CCPs: 20.025/20.025 No CCPs left 20.025/20.025 20.025/20.025 Problem 1: 20.025/20.025 Operation Termination Processor: 20.025/20.025 Operational terminating? 20.025/20.025 YES 20.025/20.025 20.025/20.025 Problem 1: 20.025/20.025 20.025/20.025 (VAR vu95NonEmpty:S xu39u58S:S xu39u39u58S:S xu58S:S yu39u58S:S yu58S:S yu39u39u58S:S) 20.025/20.025 (RULES 20.025/20.025 f(xu39u58S:S,xu39u39u58S:S) -> g(xu58S:S) | xu39u58S:S ->* xu58S:S, xu39u39u58S:S ->* xu58S:S 20.025/20.025 f(yu39u58S:S,h(yu39u39u58S:S)) -> g(yu58S:S) | yu39u58S:S ->* yu58S:S, yu39u39u58S:S ->* yu58S:S 20.025/20.025 ) 20.025/20.025 20.025/20.025 Problem 1: 20.025/20.025 Valid CTRS Processor: 20.025/20.025 -> Rules: 20.025/20.025 f(xu39u58S:S,xu39u39u58S:S) -> g(xu58S:S) | xu39u58S:S ->* xu58S:S, xu39u39u58S:S ->* xu58S:S 20.025/20.025 f(yu39u58S:S,h(yu39u39u58S:S)) -> g(yu58S:S) | yu39u58S:S ->* yu58S:S, yu39u39u58S:S ->* yu58S:S 20.025/20.025 -> The system is a deterministic 3-CTRS. 20.025/20.025 20.025/20.025 Problem 1: 20.025/20.025 20.025/20.025 2D Dependency Pair Processor: 20.025/20.025 20.025/20.025 Conditional Termination Problem 1: 20.025/20.025 -> Pairs: 20.025/20.025 Empty 20.025/20.025 -> QPairs: 20.025/20.025 Empty 20.025/20.025 -> Rules: 20.025/20.025 f(xu39u58S:S,xu39u39u58S:S) -> g(xu58S:S) | xu39u58S:S ->* xu58S:S, xu39u39u58S:S ->* xu58S:S 20.025/20.025 f(yu39u58S:S,h(yu39u39u58S:S)) -> g(yu58S:S) | yu39u58S:S ->* yu58S:S, yu39u39u58S:S ->* yu58S:S 20.025/20.025 20.025/20.025 Conditional Termination Problem 2: 20.025/20.025 -> Pairs: 20.025/20.025 Empty 20.025/20.025 -> QPairs: 20.025/20.025 Empty 20.025/20.025 -> Rules: 20.025/20.025 f(xu39u58S:S,xu39u39u58S:S) -> g(xu58S:S) | xu39u58S:S ->* xu58S:S, xu39u39u58S:S ->* xu58S:S 20.025/20.025 f(yu39u58S:S,h(yu39u39u58S:S)) -> g(yu58S:S) | yu39u58S:S ->* yu58S:S, yu39u39u58S:S ->* yu58S:S 20.025/20.025 20.025/20.025 20.025/20.025 The problem is decomposed in 2 subproblems. 20.025/20.025 20.025/20.025 Problem 1.1: 20.025/20.025 20.025/20.025 SCC Processor: 20.025/20.025 -> Pairs: 20.025/20.025 Empty 20.025/20.025 -> QPairs: 20.025/20.025 Empty 20.025/20.025 -> Rules: 20.025/20.025 f(xu39u58S:S,xu39u39u58S:S) -> g(xu58S:S) | xu39u58S:S ->* xu58S:S, xu39u39u58S:S ->* xu58S:S 20.025/20.025 f(yu39u58S:S,h(yu39u39u58S:S)) -> g(yu58S:S) | yu39u58S:S ->* yu58S:S, yu39u39u58S:S ->* yu58S:S 20.025/20.025 ->Strongly Connected Components: 20.025/20.025 There is no strongly connected component 20.025/20.025 20.025/20.025 The problem is finite. 20.025/20.025 20.025/20.025 Problem 1.2: 20.025/20.025 20.025/20.025 SCC Processor: 20.025/20.025 -> Pairs: 20.025/20.025 Empty 20.025/20.025 -> QPairs: 20.025/20.025 Empty 20.025/20.025 -> Rules: 20.025/20.025 f(xu39u58S:S,xu39u39u58S:S) -> g(xu58S:S) | xu39u58S:S ->* xu58S:S, xu39u39u58S:S ->* xu58S:S 20.025/20.025 f(yu39u58S:S,h(yu39u39u58S:S)) -> g(yu58S:S) | yu39u58S:S ->* yu58S:S, yu39u39u58S:S ->* yu58S:S 20.025/20.025 ->Strongly Connected Components: 20.025/20.025 There is no strongly connected component 20.025/20.025 20.025/20.025 The problem is finite. 20.025/20.025 20.025/20.025 20.025/20.025 -> Problem conclusions: 20.025/20.025 Left linear, Right linear, Linear 20.025/20.025 Weakly orthogonal, Almost orthogonal, Orthogonal 20.025/20.025 CTRS Type: 3 20.025/20.025 Deterministic, Strongly deterministic 20.025/20.025 Oriented CTRS, Properly oriented CTRS, Not join CTRS 20.025/20.025 Maybe right-stable CTRS, Overlay CTRS 20.025/20.025 Maybe normal CTRS, Maybe almost normal CTRS 20.025/20.025 Terminating CTRS, Joinable CCPs 20.025/20.025 Maybe level confluent 20.025/20.025 Confluent 20.025/20.025 20.025/20.025 The problem is joinable. 20.025/20.025 45.00user 1.76system 0:20.25elapsed 230%CPU (0avgtext+0avgdata 72488maxresident)k 20.025/20.025 8inputs+0outputs (0major+65620minor)pagefaults 0swaps