21.033/21.033 NO 21.033/21.033 21.033/21.033 Problem 1: 21.033/21.033 21.033/21.033 21.033/21.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 21.033/21.033 Confluence Problem: 21.033/21.033 (VAR vNonEmpty w x y z) 21.033/21.033 (REPLACEMENT-MAP 21.033/21.033 (iadd 1) 21.033/21.033 (imult 1) 21.033/21.033 (0) 21.033/21.033 (fSNonEmpty) 21.033/21.033 (s 1) 21.033/21.033 (tp2 1, 2) 21.033/21.033 ) 21.033/21.033 (RULES 21.033/21.033 iadd(s(z)) -> tp2(s(x),y) | iadd(z) ->* tp2(x,y) 21.033/21.033 iadd(y) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(x,0) 21.033/21.033 imult(s(z)) -> tp2(s(x),s(y)) | iadd(z) ->* tp2(w,y), imult(w) ->* tp2(x,s(y)) 21.033/21.033 ) 21.033/21.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 21.033/21.033 21.033/21.033 21.033/21.033 Problem 1: 21.033/21.033 21.033/21.033 Inlining of Conditions Processor [STERN17]: 21.033/21.033 21.033/21.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 21.033/21.033 Confluence Problem: 21.033/21.033 (VAR vNonEmpty w x y z) 21.033/21.033 (REPLACEMENT-MAP 21.033/21.033 (iadd 1) 21.033/21.033 (imult 1) 21.033/21.033 (0) 21.033/21.033 (fSNonEmpty) 21.033/21.033 (s 1) 21.033/21.033 (tp2 1, 2) 21.033/21.033 ) 21.033/21.033 (RULES 21.033/21.033 iadd(s(z)) -> tp2(s(x),y) | iadd(z) ->* tp2(x,y) 21.033/21.033 iadd(y) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(x,0) 21.033/21.033 imult(s(z)) -> tp2(s(x),s(y)) | iadd(z) ->* tp2(w,y), imult(w) ->* tp2(x,s(y)) 21.033/21.033 ) 21.033/21.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 21.033/21.033 21.033/21.033 21.033/21.033 Problem 1: 21.033/21.033 21.033/21.033 Clean CTRS Processor: 21.033/21.033 21.033/21.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 21.033/21.033 Confluence Problem: 21.033/21.033 (VAR w x y z) 21.033/21.033 (REPLACEMENT-MAP 21.033/21.033 (iadd 1) 21.033/21.033 (imult 1) 21.033/21.033 (0) 21.033/21.033 (fSNonEmpty) 21.033/21.033 (s 1) 21.033/21.033 (tp2 1, 2) 21.033/21.033 ) 21.033/21.033 (RULES 21.033/21.033 iadd(s(z)) -> tp2(s(x),y) | iadd(z) ->* tp2(x,y) 21.033/21.033 iadd(y) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(x,0) 21.033/21.033 imult(s(z)) -> tp2(s(x),s(y)) | iadd(z) ->* tp2(w,y), imult(w) ->* tp2(x,s(y)) 21.033/21.033 ) 21.033/21.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 21.033/21.033 21.033/21.033 CRule InfChecker Info: 21.033/21.033 iadd(s(z)) -> tp2(s(x),y) | iadd(z) ->* tp2(x,y) 21.033/21.033 Rule remains 21.033/21.033 Proof: 21.033/21.033 NO 21.033/21.033 21.033/21.033 Problem 1: 21.033/21.033 21.033/21.033 Infeasibility Problem: 21.033/21.033 [(VAR vNonEmpty w x y z vNonEmpty x y z) 21.033/21.033 (STRATEGY CONTEXTSENSITIVE 21.033/21.033 (iadd 1) 21.033/21.033 (imult 1) 21.033/21.033 (0) 21.033/21.033 (fSNonEmpty) 21.033/21.033 (s 1) 21.033/21.033 (tp2 1 2) 21.033/21.033 ) 21.033/21.033 (RULES 21.033/21.033 iadd(s(z)) -> tp2(s(x),y) | iadd(z) ->* tp2(x,y) 21.033/21.033 iadd(y) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(x,0) 21.033/21.033 imult(s(z)) -> tp2(s(x),s(y)) | iadd(z) ->* tp2(w,y), imult(w) ->* tp2(x,s(y)) 21.033/21.033 )] 21.033/21.033 21.033/21.033 Infeasibility Conditions: 21.033/21.033 iadd(z) ->* tp2(x,y) 21.033/21.033 21.033/21.033 Problem 1: 21.033/21.033 21.033/21.033 Obtaining a proof using Prover9: 21.033/21.033 21.033/21.033 -> Prover9 Output: 21.033/21.033 ============================== Prover9 =============================== 21.033/21.033 Prover9 (64) version 2009-11A, November 2009. 21.033/21.033 Process 10937 was started by ubuntu on ubuntu, 21.033/21.033 Wed Mar 9 09:15:34 2022 21.033/21.033 The command was "./prover9 -f /tmp/prover910928-0.in". 21.033/21.033 ============================== end of head =========================== 21.033/21.033 21.033/21.033 ============================== INPUT ================================= 21.033/21.033 21.033/21.033 % Reading from file /tmp/prover910928-0.in 21.033/21.033 21.033/21.033 assign(max_seconds,20). 21.033/21.033 21.033/21.033 formulas(assumptions). 21.033/21.033 ->_s0(x1,y) -> ->_s0(iadd(x1),iadd(y)) # label(congruence). 21.033/21.033 ->_s0(x1,y) -> ->_s0(imult(x1),imult(y)) # label(congruence). 21.033/21.033 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 21.033/21.033 ->_s0(x1,y) -> ->_s0(tp2(x1,x2),tp2(y,x2)) # label(congruence). 21.033/21.033 ->_s0(x2,y) -> ->_s0(tp2(x1,x2),tp2(x1,y)) # label(congruence). 21.033/21.033 ->*_s0(iadd(x4),tp2(x2,x3)) -> ->_s0(iadd(s(x4)),tp2(s(x2),x3)) # label(replacement). 21.033/21.033 ->_s0(iadd(x3),tp2(0,x3)) # label(replacement). 21.033/21.033 ->_s0(imult(0),tp2(0,x3)) # label(replacement). 21.033/21.033 ->_s0(imult(0),tp2(x2,0)) # label(replacement). 21.033/21.033 ->*_s0(iadd(x4),tp2(x1,x3)) & ->*_s0(imult(x1),tp2(x2,s(x3))) -> ->_s0(imult(s(x4)),tp2(s(x2),s(x3))) # label(replacement). 21.033/21.033 ->*_s0(x,x) # label(reflexivity). 21.033/21.033 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 formulas(goals). 21.033/21.033 (exists x6 exists x7 exists x8 ->*_s0(iadd(x8),tp2(x6,x7))) # label(goal). 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 ============================== end of input ========================== 21.033/21.033 21.033/21.033 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 21.033/21.033 21.033/21.033 % Formulas that are not ordinary clauses: 21.033/21.033 1 ->_s0(x1,y) -> ->_s0(iadd(x1),iadd(y)) # label(congruence) # label(non_clause). [assumption]. 21.033/21.033 2 ->_s0(x1,y) -> ->_s0(imult(x1),imult(y)) # label(congruence) # label(non_clause). [assumption]. 21.033/21.033 3 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 21.033/21.033 4 ->_s0(x1,y) -> ->_s0(tp2(x1,x2),tp2(y,x2)) # label(congruence) # label(non_clause). [assumption]. 21.033/21.033 5 ->_s0(x2,y) -> ->_s0(tp2(x1,x2),tp2(x1,y)) # label(congruence) # label(non_clause). [assumption]. 21.033/21.033 6 ->*_s0(iadd(x4),tp2(x2,x3)) -> ->_s0(iadd(s(x4)),tp2(s(x2),x3)) # label(replacement) # label(non_clause). [assumption]. 21.033/21.033 7 ->*_s0(iadd(x4),tp2(x1,x3)) & ->*_s0(imult(x1),tp2(x2,s(x3))) -> ->_s0(imult(s(x4)),tp2(s(x2),s(x3))) # label(replacement) # label(non_clause). [assumption]. 21.033/21.033 8 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 21.033/21.033 9 (exists x6 exists x7 exists x8 ->*_s0(iadd(x8),tp2(x6,x7))) # label(goal) # label(non_clause) # label(goal). [goal]. 21.033/21.033 21.033/21.033 ============================== end of process non-clausal formulas === 21.033/21.033 21.033/21.033 ============================== PROCESS INITIAL CLAUSES =============== 21.033/21.033 21.033/21.033 % Clauses before input processing: 21.033/21.033 21.033/21.033 formulas(usable). 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 formulas(sos). 21.033/21.033 -->_s0(x,y) | ->_s0(iadd(x),iadd(y)) # label(congruence). [clausify(1)]. 21.033/21.033 -->_s0(x,y) | ->_s0(imult(x),imult(y)) # label(congruence). [clausify(2)]. 21.033/21.033 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(3)]. 21.033/21.033 -->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence). [clausify(4)]. 21.033/21.033 -->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence). [clausify(5)]. 21.033/21.033 -->*_s0(iadd(x),tp2(y,z)) | ->_s0(iadd(s(x)),tp2(s(y),z)) # label(replacement). [clausify(6)]. 21.033/21.033 ->_s0(iadd(x),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 ->_s0(imult(0),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 ->_s0(imult(0),tp2(x,0)) # label(replacement). [assumption]. 21.033/21.033 -->*_s0(iadd(x),tp2(y,z)) | -->*_s0(imult(y),tp2(u,s(z))) | ->_s0(imult(s(x)),tp2(s(u),s(z))) # label(replacement). [clausify(7)]. 21.033/21.033 ->*_s0(x,x) # label(reflexivity). [assumption]. 21.033/21.033 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 21.033/21.033 -->*_s0(iadd(x),tp2(y,z)) # label(goal). [deny(9)]. 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 formulas(demodulators). 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 ============================== PREDICATE ELIMINATION ================= 21.033/21.033 21.033/21.033 No predicates eliminated. 21.033/21.033 21.033/21.033 ============================== end predicate elimination ============= 21.033/21.033 21.033/21.033 Auto_denials: 21.033/21.033 % copying label goal to answer in negative clause 21.033/21.033 21.033/21.033 Term ordering decisions: 21.033/21.033 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 21.033/21.033 Function symbol precedence: function_order([ 0, tp2, s, iadd, imult ]). 21.033/21.033 After inverse_order: (no changes). 21.033/21.033 Unfolding symbols: (none). 21.033/21.033 21.033/21.033 Auto_inference settings: 21.033/21.033 % set(neg_binary_resolution). % (HNE depth_diff=-6) 21.033/21.033 % clear(ordered_res). % (HNE depth_diff=-6) 21.033/21.033 % set(ur_resolution). % (HNE depth_diff=-6) 21.033/21.033 % set(ur_resolution) -> set(pos_ur_resolution). 21.033/21.033 % set(ur_resolution) -> set(neg_ur_resolution). 21.033/21.033 21.033/21.033 Auto_process settings: (no changes). 21.033/21.033 21.033/21.033 kept: 10 -->_s0(x,y) | ->_s0(iadd(x),iadd(y)) # label(congruence). [clausify(1)]. 21.033/21.033 kept: 11 -->_s0(x,y) | ->_s0(imult(x),imult(y)) # label(congruence). [clausify(2)]. 21.033/21.033 kept: 12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(3)]. 21.033/21.033 kept: 13 -->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence). [clausify(4)]. 21.033/21.033 kept: 14 -->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence). [clausify(5)]. 21.033/21.033 kept: 15 -->*_s0(iadd(x),tp2(y,z)) | ->_s0(iadd(s(x)),tp2(s(y),z)) # label(replacement). [clausify(6)]. 21.033/21.033 kept: 16 ->_s0(iadd(x),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 kept: 17 ->_s0(imult(0),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 kept: 18 ->_s0(imult(0),tp2(x,0)) # label(replacement). [assumption]. 21.033/21.033 kept: 19 -->*_s0(iadd(x),tp2(y,z)) | -->*_s0(imult(y),tp2(u,s(z))) | ->_s0(imult(s(x)),tp2(s(u),s(z))) # label(replacement). [clausify(7)]. 21.033/21.033 kept: 20 ->*_s0(x,x) # label(reflexivity). [assumption]. 21.033/21.033 kept: 21 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 21.033/21.033 kept: 22 -->*_s0(iadd(x),tp2(y,z)) # label(goal) # answer(goal). [deny(9)]. 21.033/21.033 21.033/21.033 ============================== end of process initial clauses ======== 21.033/21.033 21.033/21.033 ============================== CLAUSES FOR SEARCH ==================== 21.033/21.033 21.033/21.033 % Clauses after input processing: 21.033/21.033 21.033/21.033 formulas(usable). 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 formulas(sos). 21.033/21.033 10 -->_s0(x,y) | ->_s0(iadd(x),iadd(y)) # label(congruence). [clausify(1)]. 21.033/21.033 11 -->_s0(x,y) | ->_s0(imult(x),imult(y)) # label(congruence). [clausify(2)]. 21.033/21.033 12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(3)]. 21.033/21.033 13 -->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence). [clausify(4)]. 21.033/21.033 14 -->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence). [clausify(5)]. 21.033/21.033 16 ->_s0(iadd(x),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 17 ->_s0(imult(0),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 18 ->_s0(imult(0),tp2(x,0)) # label(replacement). [assumption]. 21.033/21.033 20 ->*_s0(x,x) # label(reflexivity). [assumption]. 21.033/21.033 21 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 21.033/21.033 22 -->*_s0(iadd(x),tp2(y,z)) # label(goal) # answer(goal). [deny(9)]. 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 formulas(demodulators). 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 ============================== end of clauses for search ============= 21.033/21.033 21.033/21.033 ============================== SEARCH ================================ 21.033/21.033 21.033/21.033 % Starting search at 0.00 seconds. 21.033/21.033 21.033/21.033 given #1 (I,wt=8): 10 -->_s0(x,y) | ->_s0(iadd(x),iadd(y)) # label(congruence). [clausify(1)]. 21.033/21.033 21.033/21.033 given #2 (I,wt=8): 11 -->_s0(x,y) | ->_s0(imult(x),imult(y)) # label(congruence). [clausify(2)]. 21.033/21.033 21.033/21.033 given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(3)]. 21.033/21.033 21.033/21.033 given #4 (I,wt=10): 13 -->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence). [clausify(4)]. 21.033/21.033 21.033/21.033 given #5 (I,wt=10): 14 -->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence). [clausify(5)]. 21.033/21.033 21.033/21.033 given #6 (I,wt=6): 16 ->_s0(iadd(x),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 21.033/21.033 given #7 (I,wt=6): 17 ->_s0(imult(0),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 21.033/21.033 given #8 (I,wt=6): 18 ->_s0(imult(0),tp2(x,0)) # label(replacement). [assumption]. 21.033/21.033 21.033/21.033 given #9 (I,wt=3): 20 ->*_s0(x,x) # label(reflexivity). [assumption]. 21.033/21.033 21.033/21.033 given #10 (I,wt=9): 21 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 21.033/21.033 21.033/21.033 ============================== PROOF ================================= 21.033/21.033 21.033/21.033 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 21.033/21.033 % Length of proof is 8. 21.033/21.033 % Level of proof is 3. 21.033/21.033 % Maximum clause weight is 9.000. 21.033/21.033 % Given clauses 10. 21.033/21.033 21.033/21.033 8 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 21.033/21.033 9 (exists x6 exists x7 exists x8 ->*_s0(iadd(x8),tp2(x6,x7))) # label(goal) # label(non_clause) # label(goal). [goal]. 21.033/21.033 16 ->_s0(iadd(x),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 20 ->*_s0(x,x) # label(reflexivity). [assumption]. 21.033/21.033 21 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 21.033/21.033 22 -->*_s0(iadd(x),tp2(y,z)) # label(goal) # answer(goal). [deny(9)]. 21.033/21.033 40 ->*_s0(iadd(x),tp2(0,x)). [ur(21,a,16,a,b,20,a)]. 21.033/21.033 41 $F # answer(goal). [resolve(40,a,22,a)]. 21.033/21.033 21.033/21.033 ============================== end of proof ========================== 21.033/21.033 21.033/21.033 ============================== STATISTICS ============================ 21.033/21.033 21.033/21.033 Given=10. Generated=31. Kept=31. proofs=1. 21.033/21.033 Usable=10. Sos=16. Demods=0. Limbo=2, Disabled=15. Hints=0. 21.033/21.033 Kept_by_rule=0, Deleted_by_rule=0. 21.033/21.033 Forward_subsumed=0. Back_subsumed=2. 21.033/21.033 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 21.033/21.033 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 21.033/21.033 Demod_attempts=0. Demod_rewrites=0. 21.033/21.033 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 21.033/21.033 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=13. 21.033/21.033 Megabytes=0.10. 21.033/21.033 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 21.033/21.033 21.033/21.033 ============================== end of statistics ===================== 21.033/21.033 21.033/21.033 ============================== end of search ========================= 21.033/21.033 21.033/21.033 THEOREM PROVED 21.033/21.033 21.033/21.033 Exiting with 1 proof. 21.033/21.033 21.033/21.033 Process 10937 exit (max_proofs) Wed Mar 9 09:15:34 2022 21.033/21.033 21.033/21.033 21.033/21.033 The problem is feasible. 21.033/21.033 21.033/21.033 21.033/21.033 CRule InfChecker Info: 21.033/21.033 iadd(y) -> tp2(0,y) 21.033/21.033 Rule remains 21.033/21.033 Proof: 21.033/21.033 NO_CONDS 21.033/21.033 21.033/21.033 CRule InfChecker Info: 21.033/21.033 imult(0) -> tp2(0,y) 21.033/21.033 Rule remains 21.033/21.033 Proof: 21.033/21.033 NO_CONDS 21.033/21.033 21.033/21.033 CRule InfChecker Info: 21.033/21.033 imult(0) -> tp2(x,0) 21.033/21.033 Rule remains 21.033/21.033 Proof: 21.033/21.033 NO_CONDS 21.033/21.033 21.033/21.033 CRule InfChecker Info: 21.033/21.033 imult(s(z)) -> tp2(s(x),s(y)) | iadd(z) ->* tp2(w,y), imult(w) ->* tp2(x,s(y)) 21.033/21.033 Rule remains 21.033/21.033 Proof: 21.033/21.033 NO 21.033/21.033 21.033/21.033 Problem 1: 21.033/21.033 21.033/21.033 Infeasibility Problem: 21.033/21.033 [(VAR vNonEmpty w x y z vNonEmpty w x y z) 21.033/21.033 (STRATEGY CONTEXTSENSITIVE 21.033/21.033 (iadd 1) 21.033/21.033 (imult 1) 21.033/21.033 (0) 21.033/21.033 (fSNonEmpty) 21.033/21.033 (s 1) 21.033/21.033 (tp2 1 2) 21.033/21.033 ) 21.033/21.033 (RULES 21.033/21.033 iadd(s(z)) -> tp2(s(x),y) | iadd(z) ->* tp2(x,y) 21.033/21.033 iadd(y) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(x,0) 21.033/21.033 imult(s(z)) -> tp2(s(x),s(y)) | iadd(z) ->* tp2(w,y), imult(w) ->* tp2(x,s(y)) 21.033/21.033 )] 21.033/21.033 21.033/21.033 Infeasibility Conditions: 21.033/21.033 iadd(z) ->* tp2(w,y), imult(w) ->* tp2(x,s(y)) 21.033/21.033 21.033/21.033 Problem 1: 21.033/21.033 21.033/21.033 Obtaining a proof using Prover9: 21.033/21.033 21.033/21.033 -> Prover9 Output: 21.033/21.033 ============================== Prover9 =============================== 21.033/21.033 Prover9 (64) version 2009-11A, November 2009. 21.033/21.033 Process 10959 was started by ubuntu on ubuntu, 21.033/21.033 Wed Mar 9 09:15:34 2022 21.033/21.033 The command was "./prover9 -f /tmp/prover910950-0.in". 21.033/21.033 ============================== end of head =========================== 21.033/21.033 21.033/21.033 ============================== INPUT ================================= 21.033/21.033 21.033/21.033 % Reading from file /tmp/prover910950-0.in 21.033/21.033 21.033/21.033 assign(max_seconds,20). 21.033/21.033 21.033/21.033 formulas(assumptions). 21.033/21.033 ->_s0(x1,y) -> ->_s0(iadd(x1),iadd(y)) # label(congruence). 21.033/21.033 ->_s0(x1,y) -> ->_s0(imult(x1),imult(y)) # label(congruence). 21.033/21.033 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 21.033/21.033 ->_s0(x1,y) -> ->_s0(tp2(x1,x2),tp2(y,x2)) # label(congruence). 21.033/21.033 ->_s0(x2,y) -> ->_s0(tp2(x1,x2),tp2(x1,y)) # label(congruence). 21.033/21.033 ->*_s0(iadd(x4),tp2(x2,x3)) -> ->_s0(iadd(s(x4)),tp2(s(x2),x3)) # label(replacement). 21.033/21.033 ->_s0(iadd(x3),tp2(0,x3)) # label(replacement). 21.033/21.033 ->_s0(imult(0),tp2(0,x3)) # label(replacement). 21.033/21.033 ->_s0(imult(0),tp2(x2,0)) # label(replacement). 21.033/21.033 ->*_s0(iadd(x4),tp2(x1,x3)) & ->*_s0(imult(x1),tp2(x2,s(x3))) -> ->_s0(imult(s(x4)),tp2(s(x2),s(x3))) # label(replacement). 21.033/21.033 ->*_s0(x,x) # label(reflexivity). 21.033/21.033 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 formulas(goals). 21.033/21.033 (exists x6 exists x7 exists x8 exists x9 (->*_s0(iadd(x9),tp2(x6,x8)) & ->*_s0(imult(x6),tp2(x7,s(x8))))) # label(goal). 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 ============================== end of input ========================== 21.033/21.033 21.033/21.033 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 21.033/21.033 21.033/21.033 % Formulas that are not ordinary clauses: 21.033/21.033 1 ->_s0(x1,y) -> ->_s0(iadd(x1),iadd(y)) # label(congruence) # label(non_clause). [assumption]. 21.033/21.033 2 ->_s0(x1,y) -> ->_s0(imult(x1),imult(y)) # label(congruence) # label(non_clause). [assumption]. 21.033/21.033 3 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 21.033/21.033 4 ->_s0(x1,y) -> ->_s0(tp2(x1,x2),tp2(y,x2)) # label(congruence) # label(non_clause). [assumption]. 21.033/21.033 5 ->_s0(x2,y) -> ->_s0(tp2(x1,x2),tp2(x1,y)) # label(congruence) # label(non_clause). [assumption]. 21.033/21.033 6 ->*_s0(iadd(x4),tp2(x2,x3)) -> ->_s0(iadd(s(x4)),tp2(s(x2),x3)) # label(replacement) # label(non_clause). [assumption]. 21.033/21.033 7 ->*_s0(iadd(x4),tp2(x1,x3)) & ->*_s0(imult(x1),tp2(x2,s(x3))) -> ->_s0(imult(s(x4)),tp2(s(x2),s(x3))) # label(replacement) # label(non_clause). [assumption]. 21.033/21.033 8 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 21.033/21.033 9 (exists x6 exists x7 exists x8 exists x9 (->*_s0(iadd(x9),tp2(x6,x8)) & ->*_s0(imult(x6),tp2(x7,s(x8))))) # label(goal) # label(non_clause) # label(goal). [goal]. 21.033/21.033 21.033/21.033 ============================== end of process non-clausal formulas === 21.033/21.033 21.033/21.033 ============================== PROCESS INITIAL CLAUSES =============== 21.033/21.033 21.033/21.033 % Clauses before input processing: 21.033/21.033 21.033/21.033 formulas(usable). 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 formulas(sos). 21.033/21.033 -->_s0(x,y) | ->_s0(iadd(x),iadd(y)) # label(congruence). [clausify(1)]. 21.033/21.033 -->_s0(x,y) | ->_s0(imult(x),imult(y)) # label(congruence). [clausify(2)]. 21.033/21.033 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(3)]. 21.033/21.033 -->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence). [clausify(4)]. 21.033/21.033 -->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence). [clausify(5)]. 21.033/21.033 -->*_s0(iadd(x),tp2(y,z)) | ->_s0(iadd(s(x)),tp2(s(y),z)) # label(replacement). [clausify(6)]. 21.033/21.033 ->_s0(iadd(x),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 ->_s0(imult(0),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 ->_s0(imult(0),tp2(x,0)) # label(replacement). [assumption]. 21.033/21.033 -->*_s0(iadd(x),tp2(y,z)) | -->*_s0(imult(y),tp2(u,s(z))) | ->_s0(imult(s(x)),tp2(s(u),s(z))) # label(replacement). [clausify(7)]. 21.033/21.033 ->*_s0(x,x) # label(reflexivity). [assumption]. 21.033/21.033 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 21.033/21.033 -->*_s0(iadd(x),tp2(y,z)) | -->*_s0(imult(y),tp2(u,s(z))) # label(goal). [deny(9)]. 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 formulas(demodulators). 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 ============================== PREDICATE ELIMINATION ================= 21.033/21.033 21.033/21.033 No predicates eliminated. 21.033/21.033 21.033/21.033 ============================== end predicate elimination ============= 21.033/21.033 21.033/21.033 Auto_denials: 21.033/21.033 % copying label goal to answer in negative clause 21.033/21.033 21.033/21.033 Term ordering decisions: 21.033/21.033 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 21.033/21.033 Function symbol precedence: function_order([ 0, tp2, s, iadd, imult ]). 21.033/21.033 After inverse_order: (no changes). 21.033/21.033 Unfolding symbols: (none). 21.033/21.033 21.033/21.033 Auto_inference settings: 21.033/21.033 % set(neg_binary_resolution). % (HNE depth_diff=-6) 21.033/21.033 % clear(ordered_res). % (HNE depth_diff=-6) 21.033/21.033 % set(ur_resolution). % (HNE depth_diff=-6) 21.033/21.033 % set(ur_resolution) -> set(pos_ur_resolution). 21.033/21.033 % set(ur_resolution) -> set(neg_ur_resolution). 21.033/21.033 21.033/21.033 Auto_process settings: 21.033/21.033 % set(unit_deletion). % (Horn set with negative nonunits) 21.033/21.033 21.033/21.033 kept: 10 -->_s0(x,y) | ->_s0(iadd(x),iadd(y)) # label(congruence). [clausify(1)]. 21.033/21.033 kept: 11 -->_s0(x,y) | ->_s0(imult(x),imult(y)) # label(congruence). [clausify(2)]. 21.033/21.033 kept: 12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(3)]. 21.033/21.033 kept: 13 -->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence). [clausify(4)]. 21.033/21.033 kept: 14 -->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence). [clausify(5)]. 21.033/21.033 kept: 15 -->*_s0(iadd(x),tp2(y,z)) | ->_s0(iadd(s(x)),tp2(s(y),z)) # label(replacement). [clausify(6)]. 21.033/21.033 kept: 16 ->_s0(iadd(x),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 kept: 17 ->_s0(imult(0),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 kept: 18 ->_s0(imult(0),tp2(x,0)) # label(replacement). [assumption]. 21.033/21.033 kept: 19 -->*_s0(iadd(x),tp2(y,z)) | -->*_s0(imult(y),tp2(u,s(z))) | ->_s0(imult(s(x)),tp2(s(u),s(z))) # label(replacement). [clausify(7)]. 21.033/21.033 kept: 20 ->*_s0(x,x) # label(reflexivity). [assumption]. 21.033/21.033 kept: 21 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 21.033/21.033 kept: 22 -->*_s0(iadd(x),tp2(y,z)) | -->*_s0(imult(y),tp2(u,s(z))) # label(goal) # answer(goal). [deny(9)]. 21.033/21.033 21.033/21.033 ============================== end of process initial clauses ======== 21.033/21.033 21.033/21.033 ============================== CLAUSES FOR SEARCH ==================== 21.033/21.033 21.033/21.033 % Clauses after input processing: 21.033/21.033 21.033/21.033 formulas(usable). 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 formulas(sos). 21.033/21.033 10 -->_s0(x,y) | ->_s0(iadd(x),iadd(y)) # label(congruence). [clausify(1)]. 21.033/21.033 11 -->_s0(x,y) | ->_s0(imult(x),imult(y)) # label(congruence). [clausify(2)]. 21.033/21.033 12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(3)]. 21.033/21.033 13 -->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence). [clausify(4)]. 21.033/21.033 14 -->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence). [clausify(5)]. 21.033/21.033 15 -->*_s0(iadd(x),tp2(y,z)) | ->_s0(iadd(s(x)),tp2(s(y),z)) # label(replacement). [clausify(6)]. 21.033/21.033 16 ->_s0(iadd(x),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 17 ->_s0(imult(0),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 18 ->_s0(imult(0),tp2(x,0)) # label(replacement). [assumption]. 21.033/21.033 20 ->*_s0(x,x) # label(reflexivity). [assumption]. 21.033/21.033 21 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 21.033/21.033 22 -->*_s0(iadd(x),tp2(y,z)) | -->*_s0(imult(y),tp2(u,s(z))) # label(goal) # answer(goal). [deny(9)]. 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 formulas(demodulators). 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 ============================== end of clauses for search ============= 21.033/21.033 21.033/21.033 ============================== SEARCH ================================ 21.033/21.033 21.033/21.033 % Starting search at 0.00 seconds. 21.033/21.033 21.033/21.033 given #1 (I,wt=8): 10 -->_s0(x,y) | ->_s0(iadd(x),iadd(y)) # label(congruence). [clausify(1)]. 21.033/21.033 21.033/21.033 given #2 (I,wt=8): 11 -->_s0(x,y) | ->_s0(imult(x),imult(y)) # label(congruence). [clausify(2)]. 21.033/21.033 21.033/21.033 given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(3)]. 21.033/21.033 21.033/21.033 given #4 (I,wt=10): 13 -->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence). [clausify(4)]. 21.033/21.033 21.033/21.033 given #5 (I,wt=10): 14 -->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence). [clausify(5)]. 21.033/21.033 21.033/21.033 given #6 (I,wt=14): 15 -->*_s0(iadd(x),tp2(y,z)) | ->_s0(iadd(s(x)),tp2(s(y),z)) # label(replacement). [clausify(6)]. 21.033/21.033 21.033/21.033 given #7 (I,wt=6): 16 ->_s0(iadd(x),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 21.033/21.033 given #8 (I,wt=6): 17 ->_s0(imult(0),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 21.033/21.033 given #9 (I,wt=6): 18 ->_s0(imult(0),tp2(x,0)) # label(replacement). [assumption]. 21.033/21.033 21.033/21.033 given #10 (I,wt=3): 20 ->*_s0(x,x) # label(reflexivity). [assumption]. 21.033/21.033 21.033/21.033 given #11 (I,wt=9): 21 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 21.033/21.033 21.033/21.033 given #12 (I,wt=13): 22 -->*_s0(iadd(x),tp2(y,z)) | -->*_s0(imult(y),tp2(u,s(z))) # label(goal) # answer(goal). [deny(9)]. 21.033/21.033 21.033/21.033 given #13 (A,wt=10): 23 ->_s0(tp2(x,iadd(y)),tp2(x,tp2(0,y))). [ur(14,a,16,a)]. 21.033/21.033 21.033/21.033 given #14 (F,wt=16): 41 -->*_s0(imult(x),tp2(y,s(z))) | -->_s0(iadd(u),w) | -->*_s0(w,tp2(x,z)) # answer(goal). [resolve(22,a,21,c)]. 21.033/21.033 21.033/21.033 ============================== PROOF ================================= 21.033/21.033 21.033/21.033 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 21.033/21.033 % Length of proof is 11. 21.033/21.033 % Level of proof is 4. 21.033/21.033 % Maximum clause weight is 16.000. 21.033/21.033 % Given clauses 14. 21.033/21.033 21.033/21.033 8 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 21.033/21.033 9 (exists x6 exists x7 exists x8 exists x9 (->*_s0(iadd(x9),tp2(x6,x8)) & ->*_s0(imult(x6),tp2(x7,s(x8))))) # label(goal) # label(non_clause) # label(goal). [goal]. 21.033/21.033 16 ->_s0(iadd(x),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 17 ->_s0(imult(0),tp2(0,x)) # label(replacement). [assumption]. 21.033/21.033 20 ->*_s0(x,x) # label(reflexivity). [assumption]. 21.033/21.033 21 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 21.033/21.033 22 -->*_s0(iadd(x),tp2(y,z)) | -->*_s0(imult(y),tp2(u,s(z))) # label(goal) # answer(goal). [deny(9)]. 21.033/21.033 39 ->*_s0(imult(0),tp2(0,x)). [ur(21,a,17,a,b,20,a)]. 21.033/21.033 41 -->*_s0(imult(x),tp2(y,s(z))) | -->_s0(iadd(u),w) | -->*_s0(w,tp2(x,z)) # answer(goal). [resolve(22,a,21,c)]. 21.033/21.033 54 -->*_s0(imult(0),tp2(x,s(y))) # answer(goal). [ur(41,b,16,a,c,20,a)]. 21.033/21.033 55 $F # answer(goal). [resolve(54,a,39,a)]. 21.033/21.033 21.033/21.033 ============================== end of proof ========================== 21.033/21.033 21.033/21.033 ============================== STATISTICS ============================ 21.033/21.033 21.033/21.033 Given=14. Generated=46. Kept=45. proofs=1. 21.033/21.033 Usable=14. Sos=24. Demods=0. Limbo=5, Disabled=14. Hints=0. 21.033/21.033 Kept_by_rule=0, Deleted_by_rule=0. 21.033/21.033 Forward_subsumed=1. Back_subsumed=1. 21.033/21.033 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 21.033/21.033 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 21.033/21.033 Demod_attempts=0. Demod_rewrites=0. 21.033/21.033 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 21.033/21.033 Nonunit_fsub_feature_tests=13. Nonunit_bsub_feature_tests=17. 21.033/21.033 Megabytes=0.13. 21.033/21.033 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 21.033/21.033 21.033/21.033 ============================== end of statistics ===================== 21.033/21.033 21.033/21.033 ============================== end of search ========================= 21.033/21.033 21.033/21.033 THEOREM PROVED 21.033/21.033 21.033/21.033 Exiting with 1 proof. 21.033/21.033 21.033/21.033 Process 10959 exit (max_proofs) Wed Mar 9 09:15:34 2022 21.033/21.033 21.033/21.033 21.033/21.033 The problem is feasible. 21.033/21.033 21.033/21.033 21.033/21.033 Problem 1: 21.033/21.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 21.033/21.033 Confluence Problem: 21.033/21.033 (VAR w x y z) 21.033/21.033 (REPLACEMENT-MAP 21.033/21.033 (iadd 1) 21.033/21.033 (imult 1) 21.033/21.033 (0) 21.033/21.033 (fSNonEmpty) 21.033/21.033 (s 1) 21.033/21.033 (tp2 1, 2) 21.033/21.033 ) 21.033/21.033 (RULES 21.033/21.033 iadd(s(z)) -> tp2(s(x),y) | iadd(z) ->* tp2(x,y) 21.033/21.033 iadd(y) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(x,0) 21.033/21.033 imult(s(z)) -> tp2(s(x),s(y)) | iadd(z) ->* tp2(w,y), imult(w) ->* tp2(x,s(y)) 21.033/21.033 ) 21.033/21.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 21.033/21.033 21.033/21.033 Critical Pairs Processor: 21.033/21.033 -> Rules: 21.033/21.033 iadd(s(z)) -> tp2(s(x),y) | iadd(z) ->* tp2(x,y) 21.033/21.033 iadd(y) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(x,0) 21.033/21.033 imult(s(z)) -> tp2(s(x),s(y)) | iadd(z) ->* tp2(w,y), imult(w) ->* tp2(x,s(y)) 21.033/21.033 -> Vars: 21.033/21.033 "w", "x", "y", "z" 21.033/21.033 21.033/21.033 -> Rlps: 21.033/21.033 crule: iadd(s(z)) -> tp2(s(x),y) | iadd(z) ->* tp2(x,y), id: 1, possubterms: iadd(s(z))-> [], s(z)-> [1] 21.033/21.033 crule: iadd(y) -> tp2(0,y), id: 2, possubterms: iadd(y)-> [] 21.033/21.033 crule: imult(0) -> tp2(0,y), id: 3, possubterms: imult(0)-> [], 0-> [1] 21.033/21.033 crule: imult(0) -> tp2(x,0), id: 4, possubterms: imult(0)-> [], 0-> [1] 21.033/21.033 crule: imult(s(z)) -> tp2(s(x),s(y)) | iadd(z) ->* tp2(w,y), imult(w) ->* tp2(x,s(y)), id: 5, possubterms: imult(s(z))-> [], s(z)-> [1] 21.033/21.033 21.033/21.033 -> Unifications: 21.033/21.033 R2 unifies with R1 at p: [], l: iadd(y), lp: iadd(y), conds: {iadd(z) ->* tp2(x,y')}, sig: {y -> s(z)}, l': iadd(s(z)), r: tp2(0,y), r': tp2(s(x),y') 21.033/21.033 R4 unifies with R3 at p: [], l: imult(0), lp: imult(0), conds: {}, sig: {}, l': imult(0), r: tp2(x,0), r': tp2(0,y) 21.033/21.033 21.033/21.033 -> Critical pairs info: 21.033/21.033 => Not trivial, Overlay, NW2, N1 21.033/21.033 | iadd(z) ->* tp2(x,y') => Not trivial, Overlay, NW2, N2 21.033/21.033 21.033/21.033 -> Problem conclusions: 21.033/21.033 Left linear, Right linear, Linear 21.033/21.033 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 21.033/21.033 CTRS Type: 4 21.033/21.033 Not deterministic, Maybe strongly deterministic 21.033/21.033 Oriented CTRS, Not properly oriented CTRS, Not join CTRS, Not semiequational CTRS 21.033/21.033 Maybe right-stable CTRS, Overlay CTRS 21.033/21.033 Maybe normal CTRS, Maybe almost normal CTRS 21.033/21.033 Maybe terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 21.033/21.033 Maybe level confluent 21.033/21.033 Maybe confluent 21.033/21.033 21.033/21.033 Problem 1: 21.033/21.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 21.033/21.033 Confluence Problem: 21.033/21.033 (VAR w x y z x y z) 21.033/21.033 (REPLACEMENT-MAP 21.033/21.033 (iadd 1) 21.033/21.033 (imult 1) 21.033/21.033 (0) 21.033/21.033 (fSNonEmpty) 21.033/21.033 (s 1) 21.033/21.033 (tp2 1, 2) 21.033/21.033 ) 21.033/21.033 (RULES 21.033/21.033 iadd(s(z)) -> tp2(s(x),y) | iadd(z) ->* tp2(x,y) 21.033/21.033 iadd(y) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(x,0) 21.033/21.033 imult(s(z)) -> tp2(s(x),s(y)) | iadd(z) ->* tp2(w,y), imult(w) ->* tp2(x,s(y)) 21.033/21.033 ) 21.033/21.033 Critical Pairs: 21.033/21.033 => Not trivial, Overlay, NW2, N1 21.033/21.033 | iadd(z) ->* tp2(x,y') => Not trivial, Overlay, NW2, N2 21.033/21.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 21.033/21.033 21.033/21.033 Conditional Critical Pairs Distributor Processor 21.033/21.033 21.033/21.033 Problem 1: 21.033/21.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 21.033/21.033 Confluence Problem: 21.033/21.033 (VAR w x y z x y z) 21.033/21.033 (REPLACEMENT-MAP 21.033/21.033 (iadd 1) 21.033/21.033 (imult 1) 21.033/21.033 (0) 21.033/21.033 (fSNonEmpty) 21.033/21.033 (s 1) 21.033/21.033 (tp2 1, 2) 21.033/21.033 ) 21.033/21.033 (RULES 21.033/21.033 iadd(s(z)) -> tp2(s(x),y) | iadd(z) ->* tp2(x,y) 21.033/21.033 iadd(y) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(x,0) 21.033/21.033 imult(s(z)) -> tp2(s(x),s(y)) | iadd(z) ->* tp2(w,y), imult(w) ->* tp2(x,s(y)) 21.033/21.033 ) 21.033/21.033 Critical Pairs: 21.033/21.033 => Not trivial, Overlay, NW2, N1 21.033/21.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 21.033/21.033 21.033/21.033 Infeasible Convergence No Conditions CCP Processor: 21.033/21.033 Infeasible convergence? 21.033/21.033 YES 21.033/21.033 21.033/21.033 Problem 1: 21.033/21.033 21.033/21.033 Infeasibility Problem: 21.033/21.033 [(VAR vNonEmpty w x y z vNonEmpty z) 21.033/21.033 (STRATEGY CONTEXTSENSITIVE 21.033/21.033 (iadd 1) 21.033/21.033 (imult 1) 21.033/21.033 (0) 21.033/21.033 (c_x) 21.033/21.033 (c_y) 21.033/21.033 (fSNonEmpty) 21.033/21.033 (s 1) 21.033/21.033 (tp2 1 2) 21.033/21.033 ) 21.033/21.033 (RULES 21.033/21.033 iadd(s(z)) -> tp2(s(x),y) | iadd(z) ->* tp2(x,y) 21.033/21.033 iadd(y) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(0,y) 21.033/21.033 imult(0) -> tp2(x,0) 21.033/21.033 imult(s(z)) -> tp2(s(x),s(y)) | iadd(z) ->* tp2(w,y), imult(w) ->* tp2(x,s(y)) 21.033/21.033 )] 21.033/21.033 21.033/21.033 Infeasibility Conditions: 21.033/21.033 tp2(0,c_y) ->* z, tp2(c_x,0) ->* z 21.033/21.033 21.033/21.033 Problem 1: 21.033/21.033 21.033/21.033 Obtaining a model using Mace4: 21.033/21.033 21.033/21.033 -> Usable Rules: 21.033/21.033 Empty 21.033/21.033 21.033/21.033 -> Mace4 Output: 21.033/21.033 ============================== Mace4 ================================= 21.033/21.033 Mace4 (64) version 2009-11A, November 2009. 21.033/21.033 Process 10987 was started by ubuntu on ubuntu, 21.033/21.033 Wed Mar 9 09:15:34 2022 21.033/21.033 The command was "./mace4 -c -f /tmp/mace410972-2.in". 21.033/21.033 ============================== end of head =========================== 21.033/21.033 21.033/21.033 ============================== INPUT ================================= 21.033/21.033 21.033/21.033 % Reading from file /tmp/mace410972-2.in 21.033/21.033 21.033/21.033 assign(max_seconds,20). 21.033/21.033 21.033/21.033 formulas(assumptions). 21.033/21.033 ->(x1,y) -> ->(iadd(x1),iadd(y)) # label(congruence). 21.033/21.033 ->(x1,y) -> ->(imult(x1),imult(y)) # label(congruence). 21.033/21.033 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence). 21.033/21.033 ->(x1,y) -> ->(tp2(x1,x2),tp2(y,x2)) # label(congruence). 21.033/21.033 ->(x2,y) -> ->(tp2(x1,x2),tp2(x1,y)) # label(congruence). 21.033/21.033 ->*(x,x) # label(reflexivity). 21.033/21.033 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 formulas(goals). 21.033/21.033 (exists x6 (->*(tp2(0,c_y),x6) & ->*(tp2(c_x,0),x6))) # label(goal). 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 ============================== end of input ========================== 21.033/21.033 21.033/21.033 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 21.033/21.033 21.033/21.033 % Formulas that are not ordinary clauses: 21.033/21.033 1 ->(x1,y) -> ->(iadd(x1),iadd(y)) # label(congruence) # label(non_clause). [assumption]. 21.033/21.033 2 ->(x1,y) -> ->(imult(x1),imult(y)) # label(congruence) # label(non_clause). [assumption]. 21.033/21.033 3 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 21.033/21.033 4 ->(x1,y) -> ->(tp2(x1,x2),tp2(y,x2)) # label(congruence) # label(non_clause). [assumption]. 21.033/21.033 5 ->(x2,y) -> ->(tp2(x1,x2),tp2(x1,y)) # label(congruence) # label(non_clause). [assumption]. 21.033/21.033 6 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 21.033/21.033 7 (exists x6 (->*(tp2(0,c_y),x6) & ->*(tp2(c_x,0),x6))) # label(goal) # label(non_clause) # label(goal). [goal]. 21.033/21.033 21.033/21.033 ============================== end of process non-clausal formulas === 21.033/21.033 21.033/21.033 ============================== CLAUSES FOR SEARCH ==================== 21.033/21.033 21.033/21.033 formulas(mace4_clauses). 21.033/21.033 -->(x,y) | ->(iadd(x),iadd(y)) # label(congruence). 21.033/21.033 -->(x,y) | ->(imult(x),imult(y)) # label(congruence). 21.033/21.033 -->(x,y) | ->(s(x),s(y)) # label(congruence). 21.033/21.033 -->(x,y) | ->(tp2(x,z),tp2(y,z)) # label(congruence). 21.033/21.033 -->(x,y) | ->(tp2(z,x),tp2(z,y)) # label(congruence). 21.033/21.033 ->*(x,x) # label(reflexivity). 21.033/21.033 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 21.033/21.033 -->*(tp2(0,c_y),x) | -->*(tp2(c_x,0),x) # label(goal). 21.033/21.033 end_of_list. 21.033/21.033 21.033/21.033 ============================== end of clauses for search ============= 21.033/21.033 21.033/21.033 % There are no natural numbers in the input. 21.033/21.033 21.033/21.033 ============================== DOMAIN SIZE 2 ========================= 21.033/21.033 21.033/21.033 ============================== MODEL ================================= 21.033/21.033 21.033/21.033 interpretation( 2, [number=1, seconds=0], [ 21.033/21.033 21.033/21.033 function(0, [ 0 ]), 21.033/21.033 21.033/21.033 function(c_x, [ 0 ]), 21.033/21.033 21.033/21.033 function(c_y, [ 1 ]), 21.033/21.033 21.033/21.033 function(iadd(_), [ 0, 0 ]), 21.033/21.033 21.033/21.033 function(imult(_), [ 0, 0 ]), 21.033/21.033 21.033/21.033 function(s(_), [ 0, 0 ]), 21.033/21.033 21.033/21.033 function(tp2(_,_), [ 21.033/21.033 0, 1, 21.033/21.033 0, 0 ]), 21.033/21.033 21.033/21.033 relation(->*(_,_), [ 21.033/21.033 1, 0, 21.033/21.033 0, 1 ]), 21.033/21.033 21.033/21.033 relation(->(_,_), [ 21.033/21.033 0, 0, 21.033/21.033 0, 0 ]) 21.033/21.033 ]). 21.033/21.033 21.033/21.033 ============================== end of model ========================== 21.033/21.033 21.033/21.033 ============================== STATISTICS ============================ 21.033/21.033 21.033/21.033 For domain size 2. 21.033/21.033 21.033/21.033 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 21.033/21.033 Ground clauses: seen=40, kept=36. 21.033/21.033 Selections=28, assignments=44, propagations=8, current_models=1. 21.033/21.033 Rewrite_terms=182, rewrite_bools=83, indexes=55. 21.033/21.033 Rules_from_neg_clauses=1, cross_offs=1. 21.033/21.033 21.033/21.033 ============================== end of statistics ===================== 21.033/21.033 21.033/21.033 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 21.033/21.033 21.033/21.033 Exiting with 1 model. 21.033/21.033 21.033/21.033 Process 10987 exit (max_models) Wed Mar 9 09:15:34 2022 21.033/21.033 The process finished Wed Mar 9 09:15:34 2022 21.033/21.033 21.033/21.033 21.033/21.033 Mace4 cooked interpretation: 21.033/21.033 21.033/21.033 21.033/21.033 21.033/21.033 The problem is infeasible. 21.033/21.033 21.033/21.033 21.033/21.033 The problem is not joinable. 21.033/21.033 24.74user 0.68system 0:21.33elapsed 119%CPU (0avgtext+0avgdata 510604maxresident)k 21.033/21.033 8inputs+0outputs (0major+169645minor)pagefaults 0swaps