/bin/sh: line 1: 45366 Quit (core dumped) z3 -T:14.25 -smt2 /tmp/SMTP45290-24 > /tmp/SMTS45290-25 WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: goal(xs,ys) -> revapp(xs,ys) revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) revapp(Nil(),rest) -> rest - Signature: {goal/2,revapp/2} / {Cons/2,Nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {goal,revapp} and constructors {Cons,Nil} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: goal(xs,ys) -> revapp(xs,ys) revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) revapp(Nil(),rest) -> rest - Signature: {goal/2,revapp/2} / {Cons/2,Nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {goal,revapp} and constructors {Cons,Nil} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () *** Step 1.a:1.a:1: Ara. MAYBE + Considered Problem: - Strict TRS: goal(xs,ys) -> revapp(xs,ys) revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) revapp(Nil(),rest) -> rest - Signature: {goal/2,revapp/2} / {Cons/2,Nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {goal,revapp} and constructors {Cons,Nil} + Applied Processor: Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "Cons") :: ["A"(0) x "A"(1)] -(1)-> "A"(1) F (TrsFun "Cons") :: ["A"(0) x "A"(0)] -(0)-> "A"(0) F (TrsFun "Nil") :: [] -(0)-> "A"(1) F (TrsFun "goal") :: ["A"(1) x "A"(0)] -(1)-> "A"(0) F (TrsFun "main") :: ["A"(1) x "A"(0)] -(1)-> "A"(0) F (TrsFun "revapp") :: ["A"(1) x "A"(0)] -(1)-> "A"(0) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: goal(xs,ys) -> revapp(xs,ys) revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) revapp(Nil(),rest) -> rest main(x1,x2) -> revapp(x1,x2) 2. Weak: *** Step 1.a:1.b:1: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: goal(xs,ys) -> revapp(xs,ys) revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) revapp(Nil(),rest) -> rest - Signature: {goal/2,revapp/2} / {Cons/2,Nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {goal,revapp} and constructors {Cons,Nil} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: revapp(y,z){y -> Cons(x,y)} = revapp(Cons(x,y),z) ->^+ revapp(y,Cons(x,z)) = C[revapp(y,Cons(x,z)) = revapp(y,z){z -> Cons(x,z)}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: goal(xs,ys) -> revapp(xs,ys) revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) revapp(Nil(),rest) -> rest - Signature: {goal/2,revapp/2} / {Cons/2,Nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {goal,revapp} and constructors {Cons,Nil} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs goal#(xs,ys) -> c_1(revapp#(xs,ys)) revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))) revapp#(Nil(),rest) -> c_3() Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: goal#(xs,ys) -> c_1(revapp#(xs,ys)) revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))) revapp#(Nil(),rest) -> c_3() - Weak TRS: goal(xs,ys) -> revapp(xs,ys) revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) revapp(Nil(),rest) -> rest - Signature: {goal/2,revapp/2,goal#/2,revapp#/2} / {Cons/2,Nil/0,c_1/1,c_2/1,c_3/0} - Obligation: innermost runtime complexity wrt. defined symbols {goal#,revapp#} and constructors {Cons,Nil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {3} by application of Pre({3}) = {1,2}. Here rules are labelled as follows: 1: goal#(xs,ys) -> c_1(revapp#(xs,ys)) 2: revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))) 3: revapp#(Nil(),rest) -> c_3() ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: goal#(xs,ys) -> c_1(revapp#(xs,ys)) revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))) - Weak DPs: revapp#(Nil(),rest) -> c_3() - Weak TRS: goal(xs,ys) -> revapp(xs,ys) revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) revapp(Nil(),rest) -> rest - Signature: {goal/2,revapp/2,goal#/2,revapp#/2} / {Cons/2,Nil/0,c_1/1,c_2/1,c_3/0} - Obligation: innermost runtime complexity wrt. defined symbols {goal#,revapp#} and constructors {Cons,Nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:goal#(xs,ys) -> c_1(revapp#(xs,ys)) -->_1 revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))):2 -->_1 revapp#(Nil(),rest) -> c_3():3 2:S:revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))) -->_1 revapp#(Nil(),rest) -> c_3():3 -->_1 revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))):2 3:W:revapp#(Nil(),rest) -> c_3() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: revapp#(Nil(),rest) -> c_3() ** Step 1.b:4: RemoveHeads. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: goal#(xs,ys) -> c_1(revapp#(xs,ys)) revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))) - Weak TRS: goal(xs,ys) -> revapp(xs,ys) revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) revapp(Nil(),rest) -> rest - Signature: {goal/2,revapp/2,goal#/2,revapp#/2} / {Cons/2,Nil/0,c_1/1,c_2/1,c_3/0} - Obligation: innermost runtime complexity wrt. defined symbols {goal#,revapp#} and constructors {Cons,Nil} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:goal#(xs,ys) -> c_1(revapp#(xs,ys)) -->_1 revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))):2 2:S:revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))) -->_1 revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))):2 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(1,goal#(xs,ys) -> c_1(revapp#(xs,ys)))] ** Step 1.b:5: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))) - Weak TRS: goal(xs,ys) -> revapp(xs,ys) revapp(Cons(x,xs),rest) -> revapp(xs,Cons(x,rest)) revapp(Nil(),rest) -> rest - Signature: {goal/2,revapp/2,goal#/2,revapp#/2} / {Cons/2,Nil/0,c_1/1,c_2/1,c_3/0} - Obligation: innermost runtime complexity wrt. defined symbols {goal#,revapp#} and constructors {Cons,Nil} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))) ** Step 1.b:6: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))) - Signature: {goal/2,revapp/2,goal#/2,revapp#/2} / {Cons/2,Nil/0,c_1/1,c_2/1,c_3/0} - Obligation: innermost runtime complexity wrt. defined symbols {goal#,revapp#} and constructors {Cons,Nil} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1} Following symbols are considered usable: {goal#,revapp#} TcT has computed the following interpretation: p(Cons) = [1] x2 + [4] p(Nil) = [2] p(goal) = [2] x2 + [1] p(revapp) = [4] x1 + [2] x2 + [2] p(goal#) = [1] x1 + [4] x2 + [0] p(revapp#) = [4] x1 + [4] p(c_1) = [0] p(c_2) = [1] x1 + [15] p(c_3) = [0] Following rules are strictly oriented: revapp#(Cons(x,xs),rest) = [4] xs + [20] > [4] xs + [19] = c_2(revapp#(xs,Cons(x,rest))) Following rules are (at-least) weakly oriented: ** Step 1.b:7: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: revapp#(Cons(x,xs),rest) -> c_2(revapp#(xs,Cons(x,rest))) - Signature: {goal/2,revapp/2,goal#/2,revapp#/2} / {Cons/2,Nil/0,c_1/1,c_2/1,c_3/0} - Obligation: innermost runtime complexity wrt. defined symbols {goal#,revapp#} and constructors {Cons,Nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))