YES Problem 1: (VAR v_NonEmpty:S fun:S x:S xs:S y:S) (RULES app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(app(f,0),app(s,0)),x:S) -> app(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,x:S),y:S) -> x:S app(app(g,x:S),y:S) -> y:S app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(app(plus,x:S),app(s,y:S)) -> app(s,app(app(plus,x:S),y:S)) app(app(plus,x:S),0) -> x:S ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(cons,x:S),app(app(filter,fun:S),xs:S)) APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) APP(app(app(f,0),app(s,0)),x:S) -> APP(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) APP(app(app(f,0),app(s,0)),x:S) -> APP(app(f,x:S),app(app(plus,x:S),x:S)) APP(app(app(f,0),app(s,0)),x:S) -> APP(app(plus,x:S),x:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(filter2,app(fun:S,x:S)),fun:S),x:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(filter2,app(fun:S,x:S)),fun:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(filter2,app(fun:S,x:S)) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(map,fun:S),xs:S) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(cons,app(fun:S,x:S)) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) APP(app(plus,x:S),app(s,y:S)) -> APP(app(plus,x:S),y:S) APP(app(plus,x:S),app(s,y:S)) -> APP(s,app(app(plus,x:S),y:S)) -> Rules: app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(app(f,0),app(s,0)),x:S) -> app(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,x:S),y:S) -> x:S app(app(g,x:S),y:S) -> y:S app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(app(plus,x:S),app(s,y:S)) -> app(s,app(app(plus,x:S),y:S)) app(app(plus,x:S),0) -> x:S Problem 1: SCC Processor: -> Pairs: APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(cons,x:S),app(app(filter,fun:S),xs:S)) APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) APP(app(app(f,0),app(s,0)),x:S) -> APP(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) APP(app(app(f,0),app(s,0)),x:S) -> APP(app(f,x:S),app(app(plus,x:S),x:S)) APP(app(app(f,0),app(s,0)),x:S) -> APP(app(plus,x:S),x:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(filter2,app(fun:S,x:S)),fun:S),x:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(filter2,app(fun:S,x:S)),fun:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(filter2,app(fun:S,x:S)) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(map,fun:S),xs:S) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(cons,app(fun:S,x:S)) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) APP(app(plus,x:S),app(s,y:S)) -> APP(app(plus,x:S),y:S) APP(app(plus,x:S),app(s,y:S)) -> APP(s,app(app(plus,x:S),y:S)) -> Rules: app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(app(f,0),app(s,0)),x:S) -> app(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,x:S),y:S) -> x:S app(app(g,x:S),y:S) -> y:S app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(app(plus,x:S),app(s,y:S)) -> app(s,app(app(plus,x:S),y:S)) app(app(plus,x:S),0) -> x:S ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: APP(app(plus,x:S),app(s,y:S)) -> APP(app(plus,x:S),y:S) ->->-> Rules: app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(app(f,0),app(s,0)),x:S) -> app(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,x:S),y:S) -> x:S app(app(g,x:S),y:S) -> y:S app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(app(plus,x:S),app(s,y:S)) -> app(s,app(app(plus,x:S),y:S)) app(app(plus,x:S),0) -> x:S ->->Cycle: ->->-> Pairs: APP(app(app(f,0),app(s,0)),x:S) -> APP(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) ->->-> Rules: app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(app(f,0),app(s,0)),x:S) -> app(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,x:S),y:S) -> x:S app(app(g,x:S),y:S) -> y:S app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(app(plus,x:S),app(s,y:S)) -> app(s,app(app(plus,x:S),y:S)) app(app(plus,x:S),0) -> x:S ->->Cycle: ->->-> Pairs: APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(map,fun:S),xs:S) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) ->->-> Rules: app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(app(f,0),app(s,0)),x:S) -> app(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,x:S),y:S) -> x:S app(app(g,x:S),y:S) -> y:S app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(app(plus,x:S),app(s,y:S)) -> app(s,app(app(plus,x:S),y:S)) app(app(plus,x:S),0) -> x:S The problem is decomposed in 3 subproblems. Problem 1.1: Subterm Processor: -> Pairs: APP(app(plus,x:S),app(s,y:S)) -> APP(app(plus,x:S),y:S) -> Rules: app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(app(f,0),app(s,0)),x:S) -> app(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,x:S),y:S) -> x:S app(app(g,x:S),y:S) -> y:S app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(app(plus,x:S),app(s,y:S)) -> app(s,app(app(plus,x:S),y:S)) app(app(plus,x:S),0) -> x:S ->Projection: pi(APP) = 2 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(app(f,0),app(s,0)),x:S) -> app(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,x:S),y:S) -> x:S app(app(g,x:S),y:S) -> y:S app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(app(plus,x:S),app(s,y:S)) -> app(s,app(app(plus,x:S),y:S)) app(app(plus,x:S),0) -> x:S ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Instantiation Processor: -> Pairs: APP(app(app(f,0),app(s,0)),x:S) -> APP(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) -> Rules: app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(app(f,0),app(s,0)),x:S) -> app(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,x:S),y:S) -> x:S app(app(g,x:S),y:S) -> y:S app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(app(plus,x:S),app(s,y:S)) -> app(s,app(app(plus,x:S),y:S)) app(app(plus,x:S),0) -> x:S ->Instantiated Pairs: ->->Original Pair: APP(app(app(f,0),app(s,0)),x:S) -> APP(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) ->-> Instantiated pairs: APP(app(app(f,0),app(s,0)),0) -> APP(app(app(f,0),app(app(plus,0),0)),0) Problem 1.2: SCC Processor: -> Pairs: APP(app(app(f,0),app(s,0)),0) -> APP(app(app(f,0),app(app(plus,0),0)),0) -> Rules: app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(app(f,0),app(s,0)),x:S) -> app(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,x:S),y:S) -> x:S app(app(g,x:S),y:S) -> y:S app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(app(plus,x:S),app(s,y:S)) -> app(s,app(app(plus,x:S),y:S)) app(app(plus,x:S),0) -> x:S ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: APP(app(app(f,0),app(s,0)),0) -> APP(app(app(f,0),app(app(plus,0),0)),0) ->->-> Rules: app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(app(f,0),app(s,0)),x:S) -> app(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,x:S),y:S) -> x:S app(app(g,x:S),y:S) -> y:S app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(app(plus,x:S),app(s,y:S)) -> app(s,app(app(plus,x:S),y:S)) app(app(plus,x:S),0) -> x:S Problem 1.2: Reduction Pair Processor: -> Pairs: APP(app(app(f,0),app(s,0)),0) -> APP(app(app(f,0),app(app(plus,0),0)),0) -> Rules: app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(app(f,0),app(s,0)),x:S) -> app(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,x:S),y:S) -> x:S app(app(g,x:S),y:S) -> y:S app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(app(plus,x:S),app(s,y:S)) -> app(s,app(app(plus,x:S),y:S)) app(app(plus,x:S),0) -> x:S -> Usable rules: app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(app(f,0),app(s,0)),x:S) -> app(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,x:S),y:S) -> x:S app(app(g,x:S),y:S) -> y:S app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(app(plus,x:S),app(s,y:S)) -> app(s,app(app(plus,x:S),y:S)) app(app(plus,x:S),0) -> x:S ->Mace4 Output: ============================== Mace4 ================================= Mace4 (64) version 2009-11A, November 2009. Process 1611 was started by sandbox on n147.star.cs.uiowa.edu, Sun Jun 21 22:37:59 2020 The command was "./mace4 -c -f /tmp/mace41911759956749241873.in". ============================== end of head =========================== ============================== INPUT ================================= % Reading from file /tmp/mace41911759956749241873.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),f2(y,x2)) # label(congruence). arrow_s0(x2,y) -> arrow_s0(f2(x1,x2),f2(x1,y)) # label(congruence). arrow_s0(x1,y) -> arrow_s0(f17(x1,x2),f17(y,x2)) # label(congruence). arrow_s0(x2,y) -> arrow_s0(f17(x1,x2),f17(x1,y)) # label(congruence). arrow_s0(f2(f2(f2(f2(f9,f7),x1),x2),x3),f2(f2(f8,x1),x3)) # label(replacement). arrow_s0(f2(f2(f2(f2(f9,f15),x1),x2),x3),f2(f2(f4,x2),f2(f2(f8,x1),x3))) # label(replacement). arrow_s0(f2(f2(f2(f5,f3),f2(f14,f3)),x2),f2(f2(f2(f5,x2),f2(f2(f13,x2),x2)),x2)) # label(replacement). arrow_s0(f2(f2(f8,x1),f2(f2(f4,x2),x3)),f2(f2(f2(f2(f9,f2(x1,x2)),x1),x2),x3)) # label(replacement). arrow_s0(f2(f2(f8,x1),f12),f12) # label(replacement). arrow_s0(f2(f2(f10,x2),x4),x2) # label(replacement). arrow_s0(f2(f2(f10,x2),x4),x4) # label(replacement). arrow_s0(f2(f2(f11,x1),f2(f2(f4,x2),x3)),f2(f2(f4,f2(x1,x2)),f2(f2(f11,x1),x3))) # label(replacement). arrow_s0(f2(f2(f11,x1),f12),f12) # label(replacement). arrow_s0(f2(f2(f13,x2),f2(f14,x4)),f2(f14,f2(f2(f13,x2),x4))) # label(replacement). arrow_s0(f2(f2(f13,x2),f3),x2) # label(replacement). arrow_s0(x,y) -> gtrsim_s0(x,y) # label(inclusion). sqsupset_s0(f17(f2(f2(f5,f3),f2(f14,f3)),f3),f17(f2(f2(f5,f3),f2(f2(f13,f3),f3)),f3)) # 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),f2(y,x2)) # label(congruence) # label(non_clause). [assumption]. 5 arrow_s0(x2,y) -> arrow_s0(f2(x1,x2),f2(x1,y)) # label(congruence) # label(non_clause). [assumption]. 6 arrow_s0(x1,y) -> arrow_s0(f17(x1,x2),f17(y,x2)) # label(congruence) # label(non_clause). [assumption]. 7 arrow_s0(x2,y) -> arrow_s0(f17(x1,x2),f17(x1,y)) # label(congruence) # label(non_clause). [assumption]. 8 arrow_s0(x,y) -> gtrsim_s0(x,y) # label(inclusion) # label(non_clause). [assumption]. 9 sqsupset_s0(x,y) -> sqsupsetStar_s0(x,y) # label(inclusion) # label(non_clause). [assumption]. 10 sqsupset_s0(x,y) & sqsupsetStar_s0(y,z) -> sqsupsetStar_s0(x,z) # label(compatibility) # label(non_clause). [assumption]. 11 (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),f2(y,z)) # label(congruence). -arrow_s0(x,y) | arrow_s0(f2(z,x),f2(z,y)) # label(congruence). -arrow_s0(x,y) | arrow_s0(f17(x,z),f17(y,z)) # label(congruence). -arrow_s0(x,y) | arrow_s0(f17(z,x),f17(z,y)) # label(congruence). arrow_s0(f2(f2(f2(f2(f9,f7),x),y),z),f2(f2(f8,x),z)) # label(replacement). arrow_s0(f2(f2(f2(f2(f9,f15),x),y),z),f2(f2(f4,y),f2(f2(f8,x),z))) # label(replacement). arrow_s0(f2(f2(f2(f5,f3),f2(f14,f3)),x),f2(f2(f2(f5,x),f2(f2(f13,x),x)),x)) # label(replacement). arrow_s0(f2(f2(f8,x),f2(f2(f4,y),z)),f2(f2(f2(f2(f9,f2(x,y)),x),y),z)) # label(replacement). arrow_s0(f2(f2(f8,x),f12),f12) # label(replacement). arrow_s0(f2(f2(f10,x),y),x) # label(replacement). arrow_s0(f2(f2(f10,x),y),y) # label(replacement). arrow_s0(f2(f2(f11,x),f2(f2(f4,y),z)),f2(f2(f4,f2(x,y)),f2(f2(f11,x),z))) # label(replacement). arrow_s0(f2(f2(f11,x),f12),f12) # label(replacement). arrow_s0(f2(f2(f13,x),f2(f14,y)),f2(f14,f2(f2(f13,x),y))) # label(replacement). arrow_s0(f2(f2(f13,x),f3),x) # label(replacement). -arrow_s0(x,y) | gtrsim_s0(x,y) # label(inclusion). sqsupset_s0(f17(f2(f2(f5,f3),f2(f14,f3)),f3),f17(f2(f2(f5,f3),f2(f2(f13,f3),f3)),f3)) # 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 ========================= ============================== MODEL ================================= interpretation( 2, [number=1, seconds=0], [ function(f10, [ 0 ]), function(f11, [ 0 ]), function(f12, [ 0 ]), function(f13, [ 0 ]), function(f14, [ 1 ]), function(f15, [ 0 ]), function(f3, [ 0 ]), function(f4, [ 0 ]), function(f5, [ 0 ]), function(f7, [ 0 ]), function(f8, [ 0 ]), function(f9, [ 0 ]), function(f17(_,_), [ 0, 0, 1, 1 ]), function(f2(_,_), [ 0, 1, 1, 1 ]), relation(arrow_s0(_,_), [ 1, 0, 1, 1 ]), relation(gtrsim_s0(_,_), [ 1, 0, 1, 1 ]), relation(sqsupsetStar_s0(_,_), [ 0, 0, 1, 0 ]), relation(sqsupset_s0(_,_), [ 0, 0, 1, 0 ]), relation(succeq_s0(_,_), [ 0, 0, 0, 0 ]) ]). ============================== end of model ========================== ============================== STATISTICS ============================ For domain size 2. Current CPU time: 0.00 seconds (total CPU time: 0.68 seconds). Ground clauses: seen=127, kept=127. Selections=48032, assignments=96047, propagations=97780, current_models=1. Rewrite_terms=5884534, rewrite_bools=1706923, indexes=681434. Rules_from_neg_clauses=769, cross_offs=769. ============================== end of statistics ===================== User_CPU=0.68, System_CPU=0.04, Wall_clock=0. Exiting with 1 model. Process 1611 exit (max_models) Sun Jun 21 22:37:59 2020 The process finished Sun Jun 21 22:37:59 2020 Mace4 cooked interpretation: % number = 1 % seconds = 0 % Interpretation of size 2 f10 = 0. f11 = 0. f12 = 0. f13 = 0. f14 = 1. f15 = 0. f3 = 0. f4 = 0. f5 = 0. f7 = 0. f8 = 0. f9 = 0. f17(0,0) = 0. f17(0,1) = 0. f17(1,0) = 1. f17(1,1) = 1. f2(0,0) = 0. f2(0,1) = 1. f2(1,0) = 1. f2(1,1) = 1. arrow_s0(0,0). - arrow_s0(0,1). arrow_s0(1,0). arrow_s0(1,1). gtrsim_s0(0,0). - gtrsim_s0(0,1). gtrsim_s0(1,0). gtrsim_s0(1,1). - sqsupsetStar_s0(0,0). - sqsupsetStar_s0(0,1). sqsupsetStar_s0(1,0). - sqsupsetStar_s0(1,1). - sqsupset_s0(0,0). - sqsupset_s0(0,1). sqsupset_s0(1,0). - sqsupset_s0(1,1). - succeq_s0(0,0). - succeq_s0(0,1). - succeq_s0(1,0). - succeq_s0(1,1). Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(app(f,0),app(s,0)),x:S) -> app(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,x:S),y:S) -> x:S app(app(g,x:S),y:S) -> y:S app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(app(plus,x:S),app(s,y:S)) -> app(s,app(app(plus,x:S),y:S)) app(app(plus,x:S),0) -> x:S ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(map,fun:S),xs:S) APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) -> Rules: app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(app(f,0),app(s,0)),x:S) -> app(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,x:S),y:S) -> x:S app(app(g,x:S),y:S) -> y:S app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(app(plus,x:S),app(s,y:S)) -> app(s,app(app(plus,x:S),y:S)) app(app(plus,x:S),0) -> x:S ->Projection: pi(APP) = 2 Problem 1.3: SCC Processor: -> Pairs: APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) -> Rules: app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) app(app(app(f,0),app(s,0)),x:S) -> app(app(app(f,x:S),app(app(plus,x:S),x:S)),x:S) app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) app(app(filter,fun:S),nil) -> nil app(app(g,x:S),y:S) -> x:S app(app(g,x:S),y:S) -> y:S app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) app(app(map,fun:S),nil) -> nil app(app(plus,x:S),app(s,y:S)) -> app(s,app(app(plus,x:S),y:S)) app(app(plus,x:S),0) -> x:S ->Strongly Connected Components: There is no strongly connected component The problem is finite.