0.00/0.18 YES 0.00/0.18 0.00/0.18 Problem 1: 0.00/0.18 0.00/0.18 (VAR v_NonEmpty:S X:S Y:S) 0.00/0.18 (RULES 0.00/0.18 +(X:S,0) -> X:S 0.00/0.18 +(X:S,s(Y:S)) -> s(+(X:S,Y:S)) 0.00/0.18 f(0,s(0),X:S) -> f(X:S,+(X:S,X:S),X:S) 0.00/0.18 g(X:S,Y:S) -> X:S 0.00/0.18 g(X:S,Y:S) -> Y:S 0.00/0.18 ) 0.00/0.18 (STRATEGY INNERMOST) 0.00/0.18 0.00/0.18 Problem 1: 0.00/0.18 0.00/0.18 Dependency Pairs Processor: 0.00/0.18 -> Pairs: 0.00/0.18 +#(X:S,s(Y:S)) -> +#(X:S,Y:S) 0.00/0.18 F(0,s(0),X:S) -> +#(X:S,X:S) 0.00/0.18 F(0,s(0),X:S) -> F(X:S,+(X:S,X:S),X:S) 0.00/0.18 -> Rules: 0.00/0.18 +(X:S,0) -> X:S 0.00/0.18 +(X:S,s(Y:S)) -> s(+(X:S,Y:S)) 0.00/0.18 f(0,s(0),X:S) -> f(X:S,+(X:S,X:S),X:S) 0.00/0.18 g(X:S,Y:S) -> X:S 0.00/0.18 g(X:S,Y:S) -> Y:S 0.00/0.18 0.00/0.18 Problem 1: 0.00/0.18 0.00/0.18 SCC Processor: 0.00/0.18 -> Pairs: 0.00/0.18 +#(X:S,s(Y:S)) -> +#(X:S,Y:S) 0.00/0.18 F(0,s(0),X:S) -> +#(X:S,X:S) 0.00/0.18 F(0,s(0),X:S) -> F(X:S,+(X:S,X:S),X:S) 0.00/0.18 -> Rules: 0.00/0.18 +(X:S,0) -> X:S 0.00/0.18 +(X:S,s(Y:S)) -> s(+(X:S,Y:S)) 0.00/0.18 f(0,s(0),X:S) -> f(X:S,+(X:S,X:S),X:S) 0.00/0.18 g(X:S,Y:S) -> X:S 0.00/0.18 g(X:S,Y:S) -> Y:S 0.00/0.18 ->Strongly Connected Components: 0.00/0.18 ->->Cycle: 0.00/0.18 ->->-> Pairs: 0.00/0.18 +#(X:S,s(Y:S)) -> +#(X:S,Y:S) 0.00/0.18 ->->-> Rules: 0.00/0.18 +(X:S,0) -> X:S 0.00/0.18 +(X:S,s(Y:S)) -> s(+(X:S,Y:S)) 0.00/0.18 f(0,s(0),X:S) -> f(X:S,+(X:S,X:S),X:S) 0.00/0.18 g(X:S,Y:S) -> X:S 0.00/0.18 g(X:S,Y:S) -> Y:S 0.00/0.18 ->->Cycle: 0.00/0.18 ->->-> Pairs: 0.00/0.18 F(0,s(0),X:S) -> F(X:S,+(X:S,X:S),X:S) 0.00/0.18 ->->-> Rules: 0.00/0.18 +(X:S,0) -> X:S 0.00/0.18 +(X:S,s(Y:S)) -> s(+(X:S,Y:S)) 0.00/0.18 f(0,s(0),X:S) -> f(X:S,+(X:S,X:S),X:S) 0.00/0.18 g(X:S,Y:S) -> X:S 0.00/0.18 g(X:S,Y:S) -> Y:S 0.00/0.18 0.00/0.18 0.00/0.18 The problem is decomposed in 2 subproblems. 0.00/0.18 0.00/0.18 Problem 1.1: 0.00/0.18 0.00/0.18 Subterm Processor: 0.00/0.18 -> Pairs: 0.00/0.18 +#(X:S,s(Y:S)) -> +#(X:S,Y:S) 0.00/0.18 -> Rules: 0.00/0.18 +(X:S,0) -> X:S 0.00/0.18 +(X:S,s(Y:S)) -> s(+(X:S,Y:S)) 0.00/0.18 f(0,s(0),X:S) -> f(X:S,+(X:S,X:S),X:S) 0.00/0.18 g(X:S,Y:S) -> X:S 0.00/0.18 g(X:S,Y:S) -> Y:S 0.00/0.18 ->Projection: 0.00/0.18 pi(+#) = 2 0.00/0.18 0.00/0.18 Problem 1.1: 0.00/0.18 0.00/0.18 SCC Processor: 0.00/0.18 -> Pairs: 0.00/0.18 Empty 0.00/0.18 -> Rules: 0.00/0.18 +(X:S,0) -> X:S 0.00/0.18 +(X:S,s(Y:S)) -> s(+(X:S,Y:S)) 0.00/0.18 f(0,s(0),X:S) -> f(X:S,+(X:S,X:S),X:S) 0.00/0.18 g(X:S,Y:S) -> X:S 0.00/0.18 g(X:S,Y:S) -> Y:S 0.00/0.18 ->Strongly Connected Components: 0.00/0.18 There is no strongly connected component 0.00/0.18 0.00/0.18 The problem is finite. 0.00/0.18 0.00/0.18 Problem 1.2: 0.00/0.18 0.00/0.18 Reduction Pair Processor: 0.00/0.18 -> Pairs: 0.00/0.18 F(0,s(0),X:S) -> F(X:S,+(X:S,X:S),X:S) 0.00/0.18 -> Rules: 0.00/0.18 +(X:S,0) -> X:S 0.00/0.18 +(X:S,s(Y:S)) -> s(+(X:S,Y:S)) 0.00/0.18 f(0,s(0),X:S) -> f(X:S,+(X:S,X:S),X:S) 0.00/0.18 g(X:S,Y:S) -> X:S 0.00/0.18 g(X:S,Y:S) -> Y:S 0.00/0.18 -> Usable rules: 0.00/0.18 +(X:S,0) -> X:S 0.00/0.18 +(X:S,s(Y:S)) -> s(+(X:S,Y:S)) 0.00/0.18 ->Mace4 Output: 0.00/0.18 ============================== Mace4 ================================= 0.00/0.18 Mace4 (64) version 2009-11A, November 2009. 0.00/0.18 Process 20061 was started by sandbox on n063.star.cs.uiowa.edu, 0.00/0.18 Thu Mar 28 22:14:43 2019 0.00/0.18 The command was "./mace4 -c -f /tmp/mace4336465782861021530.in". 0.00/0.18 ============================== end of head =========================== 0.00/0.18 0.00/0.18 ============================== INPUT ================================= 0.00/0.18 0.00/0.18 % Reading from file /tmp/mace4336465782861021530.in 0.00/0.18 0.00/0.18 assign(max_seconds,20). 0.00/0.18 0.00/0.18 formulas(assumptions). 0.00/0.18 gtrsim_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility). 0.00/0.18 succeq_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility). 0.00/0.18 gtrsim_s0(x,y) & succeq_s0(y,z) -> gtrsim_s0(x,z) # label(compatibility). 0.00/0.18 arrow_s0(x1,y) -> arrow_s0(f2(x1,x2),f2(y,x2)) # label(congruence). 0.00/0.18 arrow_s0(x2,y) -> arrow_s0(f2(x1,x2),f2(x1,y)) # label(congruence). 0.00/0.18 arrow_s0(x1,y) -> arrow_s0(f3(x1,x2,x3),f3(y,x2,x3)) # label(congruence). 0.00/0.18 arrow_s0(x2,y) -> arrow_s0(f3(x1,x2,x3),f3(x1,y,x3)) # label(congruence). 0.00/0.18 arrow_s0(x3,y) -> arrow_s0(f3(x1,x2,x3),f3(x1,x2,y)) # label(congruence). 0.00/0.18 arrow_s0(x1,y) -> arrow_s0(f4(x1,x2),f4(y,x2)) # label(congruence). 0.00/0.18 arrow_s0(x2,y) -> arrow_s0(f4(x1,x2),f4(x1,y)) # label(congruence). 0.00/0.18 arrow_s0(x1,y) -> arrow_s0(f7(x1),f7(y)) # label(congruence). 0.00/0.18 arrow_s0(x1,y) -> arrow_s0(f9(x1,x2),f9(y,x2)) # label(congruence). 0.00/0.18 arrow_s0(x2,y) -> arrow_s0(f9(x1,x2),f9(x1,y)) # label(congruence). 0.00/0.18 arrow_s0(x1,y) -> arrow_s0(f10(x1,x2,x3),f10(y,x2,x3)) # label(congruence). 0.00/0.18 arrow_s0(x2,y) -> arrow_s0(f10(x1,x2,x3),f10(x1,y,x3)) # label(congruence). 0.00/0.18 arrow_s0(x3,y) -> arrow_s0(f10(x1,x2,x3),f10(x1,x2,y)) # label(congruence). 0.00/0.18 arrow_s0(f2(x1,f5),x1) # label(replacement). 0.00/0.18 arrow_s0(f2(x1,f7(x2)),f7(f2(x1,x2))) # label(replacement). 0.00/0.18 arrow_s0(x,y) -> gtrsim_s0(x,y) # label(inclusion). 0.00/0.18 sqsupset_s0(f10(f5,f7(f5),x1),f10(x1,f2(x1,x1),x1)) # label(replacement). 0.00/0.18 sqsupset_s0(x,y) -> sqsupsetStar_s0(x,y) # label(inclusion). 0.00/0.18 sqsupset_s0(x,y) & sqsupsetStar_s0(y,z) -> sqsupsetStar_s0(x,z) # label(compatibility). 0.00/0.18 end_of_list. 0.00/0.18 0.00/0.18 formulas(goals). 0.00/0.18 (exists x sqsupsetStar_s0(x,x)) # label(wellfoundedness). 0.00/0.18 end_of_list. 0.00/0.18 0.00/0.18 ============================== end of input ========================== 0.00/0.18 0.00/0.18 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.00/0.18 0.00/0.18 % Formulas that are not ordinary clauses: 0.00/0.18 1 gtrsim_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility) # label(non_clause). [assumption]. 0.00/0.18 2 succeq_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility) # label(non_clause). [assumption]. 0.00/0.18 3 gtrsim_s0(x,y) & succeq_s0(y,z) -> gtrsim_s0(x,z) # label(compatibility) # label(non_clause). [assumption]. 0.00/0.18 4 arrow_s0(x1,y) -> arrow_s0(f2(x1,x2),f2(y,x2)) # label(congruence) # label(non_clause). [assumption]. 0.00/0.18 5 arrow_s0(x2,y) -> arrow_s0(f2(x1,x2),f2(x1,y)) # label(congruence) # label(non_clause). [assumption]. 0.00/0.18 6 arrow_s0(x1,y) -> arrow_s0(f3(x1,x2,x3),f3(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 0.00/0.18 7 arrow_s0(x2,y) -> arrow_s0(f3(x1,x2,x3),f3(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 0.00/0.18 8 arrow_s0(x3,y) -> arrow_s0(f3(x1,x2,x3),f3(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 0.00/0.18 9 arrow_s0(x1,y) -> arrow_s0(f4(x1,x2),f4(y,x2)) # label(congruence) # label(non_clause). [assumption]. 0.00/0.18 10 arrow_s0(x2,y) -> arrow_s0(f4(x1,x2),f4(x1,y)) # label(congruence) # label(non_clause). [assumption]. 0.00/0.18 11 arrow_s0(x1,y) -> arrow_s0(f7(x1),f7(y)) # label(congruence) # label(non_clause). [assumption]. 0.00/0.18 12 arrow_s0(x1,y) -> arrow_s0(f9(x1,x2),f9(y,x2)) # label(congruence) # label(non_clause). [assumption]. 0.00/0.18 13 arrow_s0(x2,y) -> arrow_s0(f9(x1,x2),f9(x1,y)) # label(congruence) # label(non_clause). [assumption]. 0.00/0.18 14 arrow_s0(x1,y) -> arrow_s0(f10(x1,x2,x3),f10(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 0.00/0.18 15 arrow_s0(x2,y) -> arrow_s0(f10(x1,x2,x3),f10(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 0.00/0.18 16 arrow_s0(x3,y) -> arrow_s0(f10(x1,x2,x3),f10(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 0.00/0.18 17 arrow_s0(x,y) -> gtrsim_s0(x,y) # label(inclusion) # label(non_clause). [assumption]. 0.00/0.18 18 sqsupset_s0(x,y) -> sqsupsetStar_s0(x,y) # label(inclusion) # label(non_clause). [assumption]. 0.00/0.18 19 sqsupset_s0(x,y) & sqsupsetStar_s0(y,z) -> sqsupsetStar_s0(x,z) # label(compatibility) # label(non_clause). [assumption]. 0.00/0.18 20 (exists x sqsupsetStar_s0(x,x)) # label(wellfoundedness) # label(non_clause) # label(goal). [goal]. 0.00/0.18 0.00/0.18 ============================== end of process non-clausal formulas === 0.00/0.18 0.00/0.18 ============================== CLAUSES FOR SEARCH ==================== 0.00/0.18 0.00/0.18 formulas(mace4_clauses). 0.00/0.18 -gtrsim_s0(x,y) | -sqsupset_s0(y,z) | sqsupset_s0(x,z) # label(compatibility). 0.00/0.18 -succeq_s0(x,y) | -sqsupset_s0(y,z) | sqsupset_s0(x,z) # label(compatibility). 0.00/0.18 -gtrsim_s0(x,y) | -succeq_s0(y,z) | gtrsim_s0(x,z) # label(compatibility). 0.00/0.18 -arrow_s0(x,y) | arrow_s0(f2(x,z),f2(y,z)) # label(congruence). 0.00/0.18 -arrow_s0(x,y) | arrow_s0(f2(z,x),f2(z,y)) # label(congruence). 0.00/0.18 -arrow_s0(x,y) | arrow_s0(f3(x,z,u),f3(y,z,u)) # label(congruence). 0.00/0.18 -arrow_s0(x,y) | arrow_s0(f3(z,x,u),f3(z,y,u)) # label(congruence). 0.00/0.18 -arrow_s0(x,y) | arrow_s0(f3(z,u,x),f3(z,u,y)) # label(congruence). 0.00/0.18 -arrow_s0(x,y) | arrow_s0(f4(x,z),f4(y,z)) # label(congruence). 0.00/0.18 -arrow_s0(x,y) | arrow_s0(f4(z,x),f4(z,y)) # label(congruence). 0.00/0.18 -arrow_s0(x,y) | arrow_s0(f7(x),f7(y)) # label(congruence). 0.00/0.18 -arrow_s0(x,y) | arrow_s0(f9(x,z),f9(y,z)) # label(congruence). 0.00/0.18 -arrow_s0(x,y) | arrow_s0(f9(z,x),f9(z,y)) # label(congruence). 0.00/0.18 -arrow_s0(x,y) | arrow_s0(f10(x,z,u),f10(y,z,u)) # label(congruence). 0.00/0.18 -arrow_s0(x,y) | arrow_s0(f10(z,x,u),f10(z,y,u)) # label(congruence). 0.00/0.18 -arrow_s0(x,y) | arrow_s0(f10(z,u,x),f10(z,u,y)) # label(congruence). 0.00/0.18 arrow_s0(f2(x,f5),x) # label(replacement). 0.00/0.18 arrow_s0(f2(x,f7(y)),f7(f2(x,y))) # label(replacement). 0.00/0.18 -arrow_s0(x,y) | gtrsim_s0(x,y) # label(inclusion). 0.00/0.18 sqsupset_s0(f10(f5,f7(f5),x),f10(x,f2(x,x),x)) # label(replacement). 0.00/0.18 -sqsupset_s0(x,y) | sqsupsetStar_s0(x,y) # label(inclusion). 0.00/0.18 -sqsupset_s0(x,y) | -sqsupsetStar_s0(y,z) | sqsupsetStar_s0(x,z) # label(compatibility). 0.00/0.18 -sqsupsetStar_s0(x,x) # label(wellfoundedness). 0.00/0.18 end_of_list. 0.00/0.18 0.00/0.18 ============================== end of clauses for search ============= 0.00/0.18 0.00/0.18 % There are no natural numbers in the input. 0.00/0.18 0.00/0.18 ============================== DOMAIN SIZE 2 ========================= 0.00/0.18 0.00/0.18 ============================== MODEL ================================= 0.00/0.18 0.00/0.18 interpretation( 2, [number=1, seconds=0], [ 0.00/0.18 0.00/0.18 function(f5, [ 0 ]), 0.00/0.18 0.00/0.18 function(f7(_), [ 1, 0 ]), 0.00/0.18 0.00/0.18 function(f2(_,_), [ 0.00/0.18 0, 1, 0.00/0.18 1, 0 ]), 0.00/0.18 0.00/0.18 function(f4(_,_), [ 0.00/0.18 0, 0, 0.00/0.18 0, 0 ]), 0.00/0.18 0.00/0.18 function(f9(_,_), [ 0.00/0.18 0, 0, 0.00/0.18 0, 0 ]), 0.00/0.18 0.00/0.18 function(f10(_,_,_), [ 0.00/0.18 0, 0, 0.00/0.18 1, 1, 0.00/0.18 0, 0, 0.00/0.18 0, 0 ]), 0.00/0.18 0.00/0.18 function(f3(_,_,_), [ 0.00/0.18 0, 0, 0.00/0.18 0, 0, 0.00/0.18 0, 0, 0.00/0.18 0, 0 ]), 0.00/0.18 0.00/0.18 relation(arrow_s0(_,_), [ 0.00/0.18 1, 0, 0.00/0.18 0, 1 ]), 0.00/0.18 0.00/0.18 relation(gtrsim_s0(_,_), [ 0.00/0.18 1, 0, 0.00/0.18 0, 1 ]), 0.00/0.18 0.00/0.18 relation(sqsupsetStar_s0(_,_), [ 0.00/0.18 0, 0, 0.00/0.18 1, 0 ]), 0.00/0.18 0.00/0.18 relation(sqsupset_s0(_,_), [ 0.00/0.18 0, 0, 0.00/0.18 1, 0 ]), 0.00/0.18 0.00/0.18 relation(succeq_s0(_,_), [ 0.00/0.18 0, 0, 0.00/0.18 0, 0 ]) 0.00/0.18 ]). 0.00/0.18 0.00/0.18 ============================== end of model ========================== 0.00/0.18 0.00/0.18 ============================== STATISTICS ============================ 0.00/0.18 0.00/0.18 For domain size 2. 0.00/0.18 0.00/0.18 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 0.00/0.18 Ground clauses: seen=198, kept=198. 0.00/0.18 Selections=44, assignments=61, propagations=52, current_models=1. 0.00/0.18 Rewrite_terms=642, rewrite_bools=688, indexes=89. 0.00/0.18 Rules_from_neg_clauses=14, cross_offs=14. 0.00/0.18 0.00/0.18 ============================== end of statistics ===================== 0.00/0.18 0.00/0.18 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 0.00/0.18 0.00/0.18 Exiting with 1 model. 0.00/0.18 0.00/0.18 Process 20061 exit (max_models) Thu Mar 28 22:14:43 2019 0.00/0.18 The process finished Thu Mar 28 22:14:43 2019 0.00/0.18 0.00/0.18 0.00/0.18 Mace4 cooked interpretation: 0.00/0.18 0.00/0.18 % number = 1 0.00/0.18 % seconds = 0 0.00/0.18 0.00/0.18 % Interpretation of size 2 0.00/0.18 0.00/0.18 f5 = 0. 0.00/0.18 0.00/0.18 f7(0) = 1. 0.00/0.18 f7(1) = 0. 0.00/0.18 0.00/0.18 f2(0,0) = 0. 0.00/0.18 f2(0,1) = 1. 0.00/0.18 f2(1,0) = 1. 0.00/0.18 f2(1,1) = 0. 0.00/0.18 0.00/0.18 f4(0,0) = 0. 0.00/0.18 f4(0,1) = 0. 0.00/0.18 f4(1,0) = 0. 0.00/0.18 f4(1,1) = 0. 0.00/0.18 0.00/0.18 f9(0,0) = 0. 0.00/0.18 f9(0,1) = 0. 0.00/0.18 f9(1,0) = 0. 0.00/0.18 f9(1,1) = 0. 0.00/0.18 0.00/0.18 f10(0,0,0) = 0. 0.00/0.18 f10(0,0,1) = 0. 0.00/0.18 f10(0,1,0) = 1. 0.00/0.18 f10(0,1,1) = 1. 0.00/0.18 f10(1,0,0) = 0. 0.00/0.18 f10(1,0,1) = 0. 0.00/0.18 f10(1,1,0) = 0. 0.00/0.18 f10(1,1,1) = 0. 0.00/0.18 0.00/0.18 f3(0,0,0) = 0. 0.00/0.18 f3(0,0,1) = 0. 0.00/0.18 f3(0,1,0) = 0. 0.00/0.18 f3(0,1,1) = 0. 0.00/0.18 f3(1,0,0) = 0. 0.00/0.18 f3(1,0,1) = 0. 0.00/0.18 f3(1,1,0) = 0. 0.00/0.18 f3(1,1,1) = 0. 0.00/0.18 0.00/0.18 arrow_s0(0,0). 0.00/0.18 - arrow_s0(0,1). 0.00/0.18 - arrow_s0(1,0). 0.00/0.18 arrow_s0(1,1). 0.00/0.18 0.00/0.18 gtrsim_s0(0,0). 0.00/0.18 - gtrsim_s0(0,1). 0.00/0.18 - gtrsim_s0(1,0). 0.00/0.18 gtrsim_s0(1,1). 0.00/0.18 0.00/0.18 - sqsupsetStar_s0(0,0). 0.00/0.18 - sqsupsetStar_s0(0,1). 0.00/0.18 sqsupsetStar_s0(1,0). 0.00/0.18 - sqsupsetStar_s0(1,1). 0.00/0.18 0.00/0.18 - sqsupset_s0(0,0). 0.00/0.18 - sqsupset_s0(0,1). 0.00/0.18 sqsupset_s0(1,0). 0.00/0.18 - sqsupset_s0(1,1). 0.00/0.18 0.00/0.18 - succeq_s0(0,0). 0.00/0.18 - succeq_s0(0,1). 0.00/0.18 - succeq_s0(1,0). 0.00/0.18 - succeq_s0(1,1). 0.00/0.18 0.00/0.18 0.00/0.18 Problem 1.2: 0.00/0.18 0.00/0.18 SCC Processor: 0.00/0.18 -> Pairs: 0.00/0.18 Empty 0.00/0.18 -> Rules: 0.00/0.18 +(X:S,0) -> X:S 0.00/0.18 +(X:S,s(Y:S)) -> s(+(X:S,Y:S)) 0.00/0.18 f(0,s(0),X:S) -> f(X:S,+(X:S,X:S),X:S) 0.00/0.18 g(X:S,Y:S) -> X:S 0.00/0.18 g(X:S,Y:S) -> Y:S 0.00/0.18 ->Strongly Connected Components: 0.00/0.18 There is no strongly connected component 0.00/0.18 0.00/0.18 The problem is finite. 0.00/0.18 EOF