/bin/sh: line 1: 24562 Quit (core dumped) z3 -T:4.6850000000000005 -smt2 /tmp/SMTP24411-27 > /tmp/SMTS24411-28 /bin/sh: line 1: 24573 Quit (core dumped) z3 -T:7.6 -smt2 /tmp/SMTP24411-76 > /tmp/SMTS24411-78 /bin/sh: line 1: 24567 Quit (core dumped) z3 -T:2.811 -smt2 /tmp/SMTP24411-37 > /tmp/SMTS24411-38 WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () *** Step 1.a:1.a:1: Ara. MAYBE + Considered Problem: - Strict TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "0") :: [] -(0)-> "A"(0) F (TrsFun "0") :: [] -(0)-> "A"(1) F (TrsFun "div") :: ["A"(0) x "A"(0)] -(1)-> "A"(0) F (TrsFun "false") :: [] -(0)-> "A"(0) F (TrsFun "geq") :: ["A"(0) x "A"(0)] -(1)-> "A"(0) F (TrsFun "if") :: ["A"(0) x "A"(0) x "A"(0)] -(1)-> "A"(0) F (TrsFun "main") :: ["A"(1) x "A"(0)] -(1)-> "A"(0) F (TrsFun "minus") :: ["A"(1) x "A"(0)] -(1)-> "A"(0) F (TrsFun "s") :: ["A"(0)] -(0)-> "A"(0) F (TrsFun "s") :: ["A"(1)] -(1)-> "A"(1) F (TrsFun "true") :: [] -(0)-> "A"(0) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) main(x1,x2) -> minus(x1,x2) 2. Weak: *** Step 1.a:1.b:1: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: geq(x,y){x -> s(x),y -> s(y)} = geq(s(x),s(y)) ->^+ geq(x,y) = C[geq(x,y) = geq(x,y){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs div#(0(),s(Y)) -> c_1() div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) ,geq#(X,Y) ,div#(minus(X,Y),s(Y)) ,minus#(X,Y)) geq#(X,0()) -> c_3() geq#(0(),s(Y)) -> c_4() geq#(s(X),s(Y)) -> c_5(geq#(X,Y)) if#(false(),X,Y) -> c_6() if#(true(),X,Y) -> c_7() minus#(0(),Y) -> c_8() minus#(s(X),s(Y)) -> c_9(minus#(X,Y)) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(0(),s(Y)) -> c_1() div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) ,geq#(X,Y) ,div#(minus(X,Y),s(Y)) ,minus#(X,Y)) geq#(X,0()) -> c_3() geq#(0(),s(Y)) -> c_4() geq#(s(X),s(Y)) -> c_5(geq#(X,Y)) if#(false(),X,Y) -> c_6() if#(true(),X,Y) -> c_7() minus#(0(),Y) -> c_8() minus#(s(X),s(Y)) -> c_9(minus#(X,Y)) - Weak TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/4,c_3/0,c_4/0 ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,4,6,7,8} by application of Pre({1,3,4,6,7,8}) = {2,5,9}. Here rules are labelled as follows: 1: div#(0(),s(Y)) -> c_1() 2: div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) ,geq#(X,Y) ,div#(minus(X,Y),s(Y)) ,minus#(X,Y)) 3: geq#(X,0()) -> c_3() 4: geq#(0(),s(Y)) -> c_4() 5: geq#(s(X),s(Y)) -> c_5(geq#(X,Y)) 6: if#(false(),X,Y) -> c_6() 7: if#(true(),X,Y) -> c_7() 8: minus#(0(),Y) -> c_8() 9: minus#(s(X),s(Y)) -> c_9(minus#(X,Y)) ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) ,geq#(X,Y) ,div#(minus(X,Y),s(Y)) ,minus#(X,Y)) geq#(s(X),s(Y)) -> c_5(geq#(X,Y)) minus#(s(X),s(Y)) -> c_9(minus#(X,Y)) - Weak DPs: div#(0(),s(Y)) -> c_1() geq#(X,0()) -> c_3() geq#(0(),s(Y)) -> c_4() if#(false(),X,Y) -> c_6() if#(true(),X,Y) -> c_7() minus#(0(),Y) -> c_8() - Weak TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/4,c_3/0,c_4/0 ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) ,geq#(X,Y) ,div#(minus(X,Y),s(Y)) ,minus#(X,Y)) -->_4 minus#(s(X),s(Y)) -> c_9(minus#(X,Y)):3 -->_2 geq#(s(X),s(Y)) -> c_5(geq#(X,Y)):2 -->_4 minus#(0(),Y) -> c_8():9 -->_1 if#(true(),X,Y) -> c_7():8 -->_1 if#(false(),X,Y) -> c_6():7 -->_2 geq#(0(),s(Y)) -> c_4():6 -->_2 geq#(X,0()) -> c_3():5 -->_3 div#(0(),s(Y)) -> c_1():4 -->_3 div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) ,geq#(X,Y) ,div#(minus(X,Y),s(Y)) ,minus#(X,Y)):1 2:S:geq#(s(X),s(Y)) -> c_5(geq#(X,Y)) -->_1 geq#(0(),s(Y)) -> c_4():6 -->_1 geq#(X,0()) -> c_3():5 -->_1 geq#(s(X),s(Y)) -> c_5(geq#(X,Y)):2 3:S:minus#(s(X),s(Y)) -> c_9(minus#(X,Y)) -->_1 minus#(0(),Y) -> c_8():9 -->_1 minus#(s(X),s(Y)) -> c_9(minus#(X,Y)):3 4:W:div#(0(),s(Y)) -> c_1() 5:W:geq#(X,0()) -> c_3() 6:W:geq#(0(),s(Y)) -> c_4() 7:W:if#(false(),X,Y) -> c_6() 8:W:if#(true(),X,Y) -> c_7() 9:W:minus#(0(),Y) -> c_8() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: div#(0(),s(Y)) -> c_1() 7: if#(false(),X,Y) -> c_6() 8: if#(true(),X,Y) -> c_7() 5: geq#(X,0()) -> c_3() 6: geq#(0(),s(Y)) -> c_4() 9: minus#(0(),Y) -> c_8() ** Step 1.b:4: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) ,geq#(X,Y) ,div#(minus(X,Y),s(Y)) ,minus#(X,Y)) geq#(s(X),s(Y)) -> c_5(geq#(X,Y)) minus#(s(X),s(Y)) -> c_9(minus#(X,Y)) - Weak TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/4,c_3/0,c_4/0 ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) ,geq#(X,Y) ,div#(minus(X,Y),s(Y)) ,minus#(X,Y)) -->_4 minus#(s(X),s(Y)) -> c_9(minus#(X,Y)):3 -->_2 geq#(s(X),s(Y)) -> c_5(geq#(X,Y)):2 -->_3 div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) ,geq#(X,Y) ,div#(minus(X,Y),s(Y)) ,minus#(X,Y)):1 2:S:geq#(s(X),s(Y)) -> c_5(geq#(X,Y)) -->_1 geq#(s(X),s(Y)) -> c_5(geq#(X,Y)):2 3:S:minus#(s(X),s(Y)) -> c_9(minus#(X,Y)) -->_1 minus#(s(X),s(Y)) -> c_9(minus#(X,Y)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y)) ** Step 1.b:5: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y)) geq#(s(X),s(Y)) -> c_5(geq#(X,Y)) minus#(s(X),s(Y)) -> c_9(minus#(X,Y)) - Weak TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()) geq(X,0()) -> true() geq(0(),s(Y)) -> false() geq(s(X),s(Y)) -> geq(X,Y) if(false(),X,Y) -> Y if(true(),X,Y) -> X minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/3,c_3/0,c_4/0 ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y)) geq#(s(X),s(Y)) -> c_5(geq#(X,Y)) minus#(s(X),s(Y)) -> c_9(minus#(X,Y)) ** Step 1.b:6: DecomposeDG. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y)) geq#(s(X),s(Y)) -> c_5(geq#(X,Y)) minus#(s(X),s(Y)) -> c_9(minus#(X,Y)) - Weak TRS: minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/3,c_3/0,c_4/0 ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y)) and a lower component geq#(s(X),s(Y)) -> c_5(geq#(X,Y)) minus#(s(X),s(Y)) -> c_9(minus#(X,Y)) Further, following extension rules are added to the lower component. div#(s(X),s(Y)) -> div#(minus(X,Y),s(Y)) div#(s(X),s(Y)) -> geq#(X,Y) div#(s(X),s(Y)) -> minus#(X,Y) *** Step 1.b:6.a:1: SimplifyRHS. WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y)) - Weak TRS: minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/3,c_3/0,c_4/0 ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y)) -->_2 div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) *** Step 1.b:6.a:2: NaturalMI. WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) - Weak TRS: minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true} + 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: {minus,div#,geq#,if#,minus#} TcT has computed the following interpretation: p(0) = [0] p(div) = [0] p(false) = [8] p(geq) = [1] p(if) = [1] p(minus) = [0] p(s) = [5] p(true) = [1] p(div#) = [1] x1 + [0] p(geq#) = [0] p(if#) = [0] p(minus#) = [0] p(c_1) = [0] p(c_2) = [2] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] Following rules are strictly oriented: div#(s(X),s(Y)) = [5] > [0] = c_2(div#(minus(X,Y),s(Y))) Following rules are (at-least) weakly oriented: minus(0(),Y) = [0] >= [0] = 0() minus(s(X),s(Y)) = [0] >= [0] = minus(X,Y) *** Step 1.b:6.a:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y))) - Weak TRS: minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:6.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: geq#(s(X),s(Y)) -> c_5(geq#(X,Y)) minus#(s(X),s(Y)) -> c_9(minus#(X,Y)) - Weak DPs: div#(s(X),s(Y)) -> div#(minus(X,Y),s(Y)) div#(s(X),s(Y)) -> geq#(X,Y) div#(s(X),s(Y)) -> minus#(X,Y) - Weak TRS: minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/3,c_3/0,c_4/0 ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true} + 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_5) = {1}, uargs(c_9) = {1} Following symbols are considered usable: {minus,div#,geq#,if#,minus#} TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [2] p(false) = [1] p(geq) = [1] x1 + [2] p(if) = [0] p(minus) = [1] x1 + [0] p(s) = [1] x1 + [2] p(true) = [0] p(div#) = [2] x1 + [5] p(geq#) = [1] x1 + [0] p(if#) = [2] x2 + [1] x3 + [8] p(minus#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [1] x3 + [2] p(c_3) = [1] p(c_4) = [1] p(c_5) = [1] x1 + [0] p(c_6) = [8] p(c_7) = [1] p(c_8) = [1] p(c_9) = [8] x1 + [0] Following rules are strictly oriented: geq#(s(X),s(Y)) = [1] X + [2] > [1] X + [0] = c_5(geq#(X,Y)) Following rules are (at-least) weakly oriented: div#(s(X),s(Y)) = [2] X + [9] >= [2] X + [5] = div#(minus(X,Y),s(Y)) div#(s(X),s(Y)) = [2] X + [9] >= [1] X + [0] = geq#(X,Y) div#(s(X),s(Y)) = [2] X + [9] >= [0] = minus#(X,Y) minus#(s(X),s(Y)) = [0] >= [0] = c_9(minus#(X,Y)) minus(0(),Y) = [0] >= [0] = 0() minus(s(X),s(Y)) = [1] X + [2] >= [1] X + [0] = minus(X,Y) *** Step 1.b:6.b:2: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: minus#(s(X),s(Y)) -> c_9(minus#(X,Y)) - Weak DPs: div#(s(X),s(Y)) -> div#(minus(X,Y),s(Y)) div#(s(X),s(Y)) -> geq#(X,Y) div#(s(X),s(Y)) -> minus#(X,Y) geq#(s(X),s(Y)) -> c_5(geq#(X,Y)) - Weak TRS: minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/3,c_3/0,c_4/0 ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(div#) = {1}, uargs(c_5) = {1}, uargs(c_9) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [2] x1 + [1] x2 + [1] p(false) = [1] p(geq) = [2] x2 + [0] p(if) = [1] x1 + [2] x2 + [0] p(minus) = [0] p(s) = [1] x1 + [2] p(true) = [1] p(div#) = [1] x1 + [4] x2 + [2] p(geq#) = [2] x2 + [4] p(if#) = [1] x1 + [1] x3 + [2] p(minus#) = [4] x2 + [4] p(c_1) = [0] p(c_2) = [2] x1 + [1] x2 + [1] x3 + [4] p(c_3) = [2] p(c_4) = [0] p(c_5) = [1] x1 + [1] p(c_6) = [1] p(c_7) = [0] p(c_8) = [2] p(c_9) = [1] x1 + [1] Following rules are strictly oriented: minus#(s(X),s(Y)) = [4] Y + [12] > [4] Y + [5] = c_9(minus#(X,Y)) Following rules are (at-least) weakly oriented: div#(s(X),s(Y)) = [1] X + [4] Y + [12] >= [4] Y + [10] = div#(minus(X,Y),s(Y)) div#(s(X),s(Y)) = [1] X + [4] Y + [12] >= [2] Y + [4] = geq#(X,Y) div#(s(X),s(Y)) = [1] X + [4] Y + [12] >= [4] Y + [4] = minus#(X,Y) geq#(s(X),s(Y)) = [2] Y + [8] >= [2] Y + [5] = c_5(geq#(X,Y)) minus(0(),Y) = [0] >= [0] = 0() minus(s(X),s(Y)) = [0] >= [0] = minus(X,Y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:6.b:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: div#(s(X),s(Y)) -> div#(minus(X,Y),s(Y)) div#(s(X),s(Y)) -> geq#(X,Y) div#(s(X),s(Y)) -> minus#(X,Y) geq#(s(X),s(Y)) -> c_5(geq#(X,Y)) minus#(s(X),s(Y)) -> c_9(minus#(X,Y)) - Weak TRS: minus(0(),Y) -> 0() minus(s(X),s(Y)) -> minus(X,Y) - Signature: {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/3,c_3/0,c_4/0 ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))