YES Problem 1: (VAR v_NonEmpty:S x:S y:S z:S) (RULES f(x:S,y:S,z:S) -> g(x:S,y:S,z:S) g(0,1,x:S) -> f(x:S,x:S,x:S) ) Problem 1: Innermost Equivalent Processor: -> Rules: f(x:S,y:S,z:S) -> g(x:S,y:S,z:S) g(0,1,x:S) -> f(x:S,x:S,x:S) -> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination. Problem 1: Dependency Pairs Processor: -> Pairs: F(x:S,y:S,z:S) -> G(x:S,y:S,z:S) G(0,1,x:S) -> F(x:S,x:S,x:S) -> Rules: f(x:S,y:S,z:S) -> g(x:S,y:S,z:S) g(0,1,x:S) -> f(x:S,x:S,x:S) Problem 1: SCC Processor: -> Pairs: F(x:S,y:S,z:S) -> G(x:S,y:S,z:S) G(0,1,x:S) -> F(x:S,x:S,x:S) -> Rules: f(x:S,y:S,z:S) -> g(x:S,y:S,z:S) g(0,1,x:S) -> f(x:S,x:S,x:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: F(x:S,y:S,z:S) -> G(x:S,y:S,z:S) G(0,1,x:S) -> F(x:S,x:S,x:S) ->->-> Rules: f(x:S,y:S,z:S) -> g(x:S,y:S,z:S) g(0,1,x:S) -> f(x:S,x:S,x:S) Problem 1: Reduction Pair Processor: -> Pairs: F(x:S,y:S,z:S) -> G(x:S,y:S,z:S) G(0,1,x:S) -> F(x:S,x:S,x:S) -> Rules: f(x:S,y:S,z:S) -> g(x:S,y:S,z:S) g(0,1,x:S) -> f(x:S,x:S,x:S) -> Usable rules: Empty ->Mace4 Output: ============================== Mace4 ================================= Mace4 (64) version 2009-11A, November 2009. Process 53783 was started by sandbox2 on n182.star.cs.uiowa.edu, Mon Jun 22 01:11:23 2020 The command was "./mace4 -c -f /tmp/mace41497983152038664370.in". ============================== end of head =========================== ============================== INPUT ================================= % Reading from file /tmp/mace41497983152038664370.in assign(max_seconds,20). formulas(assumptions). gtrsim_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility). succeq_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility). gtrsim_s0(x,y) & succeq_s0(y,z) -> gtrsim_s0(x,z) # label(compatibility). arrow_s0(x1,y) -> arrow_s0(f2(x1,x2,x3),f2(y,x2,x3)) # label(congruence). arrow_s0(x2,y) -> arrow_s0(f2(x1,x2,x3),f2(x1,y,x3)) # label(congruence). arrow_s0(x3,y) -> arrow_s0(f2(x1,x2,x3),f2(x1,x2,y)) # label(congruence). arrow_s0(x1,y) -> arrow_s0(f3(x1,x2,x3),f3(y,x2,x3)) # label(congruence). arrow_s0(x2,y) -> arrow_s0(f3(x1,x2,x3),f3(x1,y,x3)) # label(congruence). arrow_s0(x3,y) -> arrow_s0(f3(x1,x2,x3),f3(x1,x2,y)) # label(congruence). arrow_s0(x1,y) -> arrow_s0(f8(x1,x2,x3),f8(y,x2,x3)) # label(congruence). arrow_s0(x2,y) -> arrow_s0(f8(x1,x2,x3),f8(x1,y,x3)) # label(congruence). arrow_s0(x3,y) -> arrow_s0(f8(x1,x2,x3),f8(x1,x2,y)) # label(congruence). arrow_s0(x1,y) -> arrow_s0(f9(x1,x2,x3),f9(y,x2,x3)) # label(congruence). arrow_s0(x2,y) -> arrow_s0(f9(x1,x2,x3),f9(x1,y,x3)) # label(congruence). arrow_s0(x3,y) -> arrow_s0(f9(x1,x2,x3),f9(x1,x2,y)) # label(congruence). arrow_s0(x,y) -> gtrsim_s0(x,y) # label(inclusion). sqsupset_s0(f8(x1,x2,x3),f9(x1,x2,x3)) # label(replacement). succeq_s0(f9(f4,f5,x1),f8(x1,x1,x1)) # label(replacement). sqsupset_s0(x,y) -> sqsupsetStar_s0(x,y) # label(inclusion). sqsupset_s0(x,y) & sqsupsetStar_s0(y,z) -> sqsupsetStar_s0(x,z) # label(compatibility). end_of_list. formulas(goals). (exists x sqsupsetStar_s0(x,x)) # label(wellfoundedness). end_of_list. ============================== end of input ========================== ============================== PROCESS NON-CLAUSAL FORMULAS ========== % Formulas that are not ordinary clauses: 1 gtrsim_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility) # label(non_clause). [assumption]. 2 succeq_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility) # label(non_clause). [assumption]. 3 gtrsim_s0(x,y) & succeq_s0(y,z) -> gtrsim_s0(x,z) # label(compatibility) # label(non_clause). [assumption]. 4 arrow_s0(x1,y) -> arrow_s0(f2(x1,x2,x3),f2(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 5 arrow_s0(x2,y) -> arrow_s0(f2(x1,x2,x3),f2(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 6 arrow_s0(x3,y) -> arrow_s0(f2(x1,x2,x3),f2(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 7 arrow_s0(x1,y) -> arrow_s0(f3(x1,x2,x3),f3(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 8 arrow_s0(x2,y) -> arrow_s0(f3(x1,x2,x3),f3(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 9 arrow_s0(x3,y) -> arrow_s0(f3(x1,x2,x3),f3(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 10 arrow_s0(x1,y) -> arrow_s0(f8(x1,x2,x3),f8(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 11 arrow_s0(x2,y) -> arrow_s0(f8(x1,x2,x3),f8(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 12 arrow_s0(x3,y) -> arrow_s0(f8(x1,x2,x3),f8(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 13 arrow_s0(x1,y) -> arrow_s0(f9(x1,x2,x3),f9(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 14 arrow_s0(x2,y) -> arrow_s0(f9(x1,x2,x3),f9(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 15 arrow_s0(x3,y) -> arrow_s0(f9(x1,x2,x3),f9(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 16 arrow_s0(x,y) -> gtrsim_s0(x,y) # label(inclusion) # label(non_clause). [assumption]. 17 sqsupset_s0(x,y) -> sqsupsetStar_s0(x,y) # label(inclusion) # label(non_clause). [assumption]. 18 sqsupset_s0(x,y) & sqsupsetStar_s0(y,z) -> sqsupsetStar_s0(x,z) # label(compatibility) # label(non_clause). [assumption]. 19 (exists x sqsupsetStar_s0(x,x)) # label(wellfoundedness) # label(non_clause) # label(goal). [goal]. ============================== end of process non-clausal formulas === ============================== CLAUSES FOR SEARCH ==================== formulas(mace4_clauses). -gtrsim_s0(x,y) | -sqsupset_s0(y,z) | sqsupset_s0(x,z) # label(compatibility). -succeq_s0(x,y) | -sqsupset_s0(y,z) | sqsupset_s0(x,z) # label(compatibility). -gtrsim_s0(x,y) | -succeq_s0(y,z) | gtrsim_s0(x,z) # label(compatibility). -arrow_s0(x,y) | arrow_s0(f2(x,z,u),f2(y,z,u)) # label(congruence). -arrow_s0(x,y) | arrow_s0(f2(z,x,u),f2(z,y,u)) # label(congruence). -arrow_s0(x,y) | arrow_s0(f2(z,u,x),f2(z,u,y)) # label(congruence). -arrow_s0(x,y) | arrow_s0(f3(x,z,u),f3(y,z,u)) # label(congruence). -arrow_s0(x,y) | arrow_s0(f3(z,x,u),f3(z,y,u)) # label(congruence). -arrow_s0(x,y) | arrow_s0(f3(z,u,x),f3(z,u,y)) # label(congruence). -arrow_s0(x,y) | arrow_s0(f8(x,z,u),f8(y,z,u)) # label(congruence). -arrow_s0(x,y) | arrow_s0(f8(z,x,u),f8(z,y,u)) # label(congruence). -arrow_s0(x,y) | arrow_s0(f8(z,u,x),f8(z,u,y)) # label(congruence). -arrow_s0(x,y) | arrow_s0(f9(x,z,u),f9(y,z,u)) # label(congruence). -arrow_s0(x,y) | arrow_s0(f9(z,x,u),f9(z,y,u)) # label(congruence). -arrow_s0(x,y) | arrow_s0(f9(z,u,x),f9(z,u,y)) # label(congruence). -arrow_s0(x,y) | gtrsim_s0(x,y) # label(inclusion). sqsupset_s0(f8(x,y,z),f9(x,y,z)) # label(replacement). succeq_s0(f9(f4,f5,x),f8(x,x,x)) # label(replacement). -sqsupset_s0(x,y) | sqsupsetStar_s0(x,y) # label(inclusion). -sqsupset_s0(x,y) | -sqsupsetStar_s0(y,z) | sqsupsetStar_s0(x,z) # label(compatibility). -sqsupsetStar_s0(x,x) # label(wellfoundedness). end_of_list. ============================== end of clauses for search ============= % There are no natural numbers in the input. ============================== DOMAIN SIZE 2 ========================= ============================== STATISTICS ============================ For domain size 2. Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). Ground clauses: seen=244, kept=244. Selections=16, assignments=31, propagations=140, current_models=0. Rewrite_terms=806, rewrite_bools=1008, indexes=232. Rules_from_neg_clauses=32, cross_offs=32. ============================== end of statistics ===================== ============================== DOMAIN SIZE 3 ========================= ============================== MODEL ================================= interpretation( 3, [number=1, seconds=0], [ function(f4, [ 0 ]), function(f5, [ 1 ]), function(f2(_,_,_), [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]), function(f3(_,_,_), [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]), function(f8(_,_,_), [ 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]), function(f9(_,_,_), [ 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]), relation(arrow_s0(_,_), [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ]), relation(gtrsim_s0(_,_), [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ]), relation(sqsupsetStar_s0(_,_), [ 0, 1, 0, 0, 0, 0, 1, 1, 0 ]), relation(sqsupset_s0(_,_), [ 0, 1, 0, 0, 0, 0, 1, 0, 0 ]), relation(succeq_s0(_,_), [ 1, 0, 0, 0, 0, 0, 0, 0, 0 ]) ]). ============================== end of model ========================== ============================== STATISTICS ============================ For domain size 3. Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). Ground clauses: seen=1131, kept=1131. Selections=119, assignments=165, propagations=188, current_models=1. Rewrite_terms=3124, rewrite_bools=1421, indexes=253. Rules_from_neg_clauses=32, cross_offs=83. ============================== end of statistics ===================== User_CPU=0.00, System_CPU=0.01, Wall_clock=0. Exiting with 1 model. Process 53783 exit (max_models) Mon Jun 22 01:11:23 2020 The process finished Mon Jun 22 01:11:23 2020 Mace4 cooked interpretation: % number = 1 % seconds = 0 % Interpretation of size 3 f4 = 0. f5 = 1. f2(0,0,0) = 0. f2(0,0,1) = 0. f2(0,0,2) = 0. f2(0,1,0) = 0. f2(0,1,1) = 0. f2(0,1,2) = 0. f2(0,2,0) = 0. f2(0,2,1) = 0. f2(0,2,2) = 0. f2(1,0,0) = 0. f2(1,0,1) = 0. f2(1,0,2) = 0. f2(1,1,0) = 0. f2(1,1,1) = 0. f2(1,1,2) = 0. f2(1,2,0) = 0. f2(1,2,1) = 0. f2(1,2,2) = 0. f2(2,0,0) = 0. f2(2,0,1) = 0. f2(2,0,2) = 0. f2(2,1,0) = 0. f2(2,1,1) = 0. f2(2,1,2) = 0. f2(2,2,0) = 0. f2(2,2,1) = 0. f2(2,2,2) = 0. f3(0,0,0) = 0. f3(0,0,1) = 0. f3(0,0,2) = 0. f3(0,1,0) = 0. f3(0,1,1) = 0. f3(0,1,2) = 0. f3(0,2,0) = 0. f3(0,2,1) = 0. f3(0,2,2) = 0. f3(1,0,0) = 0. f3(1,0,1) = 0. f3(1,0,2) = 0. f3(1,1,0) = 0. f3(1,1,1) = 0. f3(1,1,2) = 0. f3(1,2,0) = 0. f3(1,2,1) = 0. f3(1,2,2) = 0. f3(2,0,0) = 0. f3(2,0,1) = 0. f3(2,0,2) = 0. f3(2,1,0) = 0. f3(2,1,1) = 0. f3(2,1,2) = 0. f3(2,2,0) = 0. f3(2,2,1) = 0. f3(2,2,2) = 0. f8(0,0,0) = 0. f8(0,0,1) = 0. f8(0,0,2) = 0. f8(0,1,0) = 2. f8(0,1,1) = 2. f8(0,1,2) = 2. f8(0,2,0) = 0. f8(0,2,1) = 0. f8(0,2,2) = 0. f8(1,0,0) = 0. f8(1,0,1) = 0. f8(1,0,2) = 0. f8(1,1,0) = 0. f8(1,1,1) = 0. f8(1,1,2) = 0. f8(1,2,0) = 0. f8(1,2,1) = 0. f8(1,2,2) = 0. f8(2,0,0) = 0. f8(2,0,1) = 0. f8(2,0,2) = 0. f8(2,1,0) = 0. f8(2,1,1) = 0. f8(2,1,2) = 0. f8(2,2,0) = 0. f8(2,2,1) = 0. f8(2,2,2) = 0. f9(0,0,0) = 1. f9(0,0,1) = 1. f9(0,0,2) = 1. f9(0,1,0) = 0. f9(0,1,1) = 0. f9(0,1,2) = 0. f9(0,2,0) = 1. f9(0,2,1) = 1. f9(0,2,2) = 1. f9(1,0,0) = 1. f9(1,0,1) = 1. f9(1,0,2) = 1. f9(1,1,0) = 1. f9(1,1,1) = 1. f9(1,1,2) = 1. f9(1,2,0) = 1. f9(1,2,1) = 1. f9(1,2,2) = 1. f9(2,0,0) = 1. f9(2,0,1) = 1. f9(2,0,2) = 1. f9(2,1,0) = 1. f9(2,1,1) = 1. f9(2,1,2) = 1. f9(2,2,0) = 1. f9(2,2,1) = 1. f9(2,2,2) = 1. - arrow_s0(0,0). - arrow_s0(0,1). - arrow_s0(0,2). - arrow_s0(1,0). - arrow_s0(1,1). - arrow_s0(1,2). - arrow_s0(2,0). - arrow_s0(2,1). - arrow_s0(2,2). - gtrsim_s0(0,0). - gtrsim_s0(0,1). - gtrsim_s0(0,2). - gtrsim_s0(1,0). - gtrsim_s0(1,1). - gtrsim_s0(1,2). - gtrsim_s0(2,0). - gtrsim_s0(2,1). - gtrsim_s0(2,2). - sqsupsetStar_s0(0,0). sqsupsetStar_s0(0,1). - sqsupsetStar_s0(0,2). - sqsupsetStar_s0(1,0). - sqsupsetStar_s0(1,1). - sqsupsetStar_s0(1,2). sqsupsetStar_s0(2,0). sqsupsetStar_s0(2,1). - sqsupsetStar_s0(2,2). - sqsupset_s0(0,0). sqsupset_s0(0,1). - sqsupset_s0(0,2). - sqsupset_s0(1,0). - sqsupset_s0(1,1). - sqsupset_s0(1,2). sqsupset_s0(2,0). - sqsupset_s0(2,1). - sqsupset_s0(2,2). succeq_s0(0,0). - succeq_s0(0,1). - succeq_s0(0,2). - succeq_s0(1,0). - succeq_s0(1,1). - succeq_s0(1,2). - succeq_s0(2,0). - succeq_s0(2,1). - succeq_s0(2,2). Problem 1: SCC Processor: -> Pairs: G(0,1,x:S) -> F(x:S,x:S,x:S) -> Rules: f(x:S,y:S,z:S) -> g(x:S,y:S,z:S) g(0,1,x:S) -> f(x:S,x:S,x:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite.