MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: gcd(x,0()) -> x gcd(0(),s(y)) -> s(y) gcd(s(x),s(y)) -> gcd(mod(s(x),s(y)),mod(s(y),s(x))) if(false(),x,y) -> mod(minus(x,y),y) if(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),x) -> 0() minus(s(x),0()) -> s(x) minus(s(x),s(y)) -> minus(x,y) mod(x,0()) -> 0() mod(x,s(y)) -> if(lt(x,s(y)),x,s(y)) - Signature: {gcd/2,if/3,lt/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if,lt,minus,mod} and constructors {0,false,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs gcd#(x,0()) -> c_1() gcd#(0(),s(y)) -> c_2() gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x))) if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y)) if#(true(),x,y) -> c_5() lt#(x,0()) -> c_6() lt#(0(),s(x)) -> c_7() lt#(s(x),s(y)) -> c_8(lt#(x,y)) minus#(0(),x) -> c_9() minus#(s(x),0()) -> c_10() minus#(s(x),s(y)) -> c_11(minus#(x,y)) mod#(x,0()) -> c_12() mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: gcd#(x,0()) -> c_1() gcd#(0(),s(y)) -> c_2() gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x))) if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y)) if#(true(),x,y) -> c_5() lt#(x,0()) -> c_6() lt#(0(),s(x)) -> c_7() lt#(s(x),s(y)) -> c_8(lt#(x,y)) minus#(0(),x) -> c_9() minus#(s(x),0()) -> c_10() minus#(s(x),s(y)) -> c_11(minus#(x,y)) mod#(x,0()) -> c_12() mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) - Weak TRS: gcd(x,0()) -> x gcd(0(),s(y)) -> s(y) gcd(s(x),s(y)) -> gcd(mod(s(x),s(y)),mod(s(y),s(x))) if(false(),x,y) -> mod(minus(x,y),y) if(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),x) -> 0() minus(s(x),0()) -> s(x) minus(s(x),s(y)) -> minus(x,y) mod(x,0()) -> 0() mod(x,s(y)) -> if(lt(x,s(y)),x,s(y)) - Signature: {gcd/2,if/3,lt/2,minus/2,mod/2,gcd#/2,if#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0 ,c_3/3,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if#,lt#,minus#,mod#} and constructors {0,false,s ,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: if(false(),x,y) -> mod(minus(x,y),y) if(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),x) -> 0() minus(s(x),0()) -> s(x) minus(s(x),s(y)) -> minus(x,y) mod(x,0()) -> 0() mod(x,s(y)) -> if(lt(x,s(y)),x,s(y)) gcd#(x,0()) -> c_1() gcd#(0(),s(y)) -> c_2() gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x))) if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y)) if#(true(),x,y) -> c_5() lt#(x,0()) -> c_6() lt#(0(),s(x)) -> c_7() lt#(s(x),s(y)) -> c_8(lt#(x,y)) minus#(0(),x) -> c_9() minus#(s(x),0()) -> c_10() minus#(s(x),s(y)) -> c_11(minus#(x,y)) mod#(x,0()) -> c_12() mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: gcd#(x,0()) -> c_1() gcd#(0(),s(y)) -> c_2() gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x))) if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y)) if#(true(),x,y) -> c_5() lt#(x,0()) -> c_6() lt#(0(),s(x)) -> c_7() lt#(s(x),s(y)) -> c_8(lt#(x,y)) minus#(0(),x) -> c_9() minus#(s(x),0()) -> c_10() minus#(s(x),s(y)) -> c_11(minus#(x,y)) mod#(x,0()) -> c_12() mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) - Weak TRS: if(false(),x,y) -> mod(minus(x,y),y) if(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),x) -> 0() minus(s(x),0()) -> s(x) minus(s(x),s(y)) -> minus(x,y) mod(x,0()) -> 0() mod(x,s(y)) -> if(lt(x,s(y)),x,s(y)) - Signature: {gcd/2,if/3,lt/2,minus/2,mod/2,gcd#/2,if#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0 ,c_3/3,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if#,lt#,minus#,mod#} and constructors {0,false,s ,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,5,6,7,9,10,12} by application of Pre({1,2,5,6,7,9,10,12}) = {3,4,8,11,13}. Here rules are labelled as follows: 1: gcd#(x,0()) -> c_1() 2: gcd#(0(),s(y)) -> c_2() 3: gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x))) 4: if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y)) 5: if#(true(),x,y) -> c_5() 6: lt#(x,0()) -> c_6() 7: lt#(0(),s(x)) -> c_7() 8: lt#(s(x),s(y)) -> c_8(lt#(x,y)) 9: minus#(0(),x) -> c_9() 10: minus#(s(x),0()) -> c_10() 11: minus#(s(x),s(y)) -> c_11(minus#(x,y)) 12: mod#(x,0()) -> c_12() 13: mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x))) if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y)) lt#(s(x),s(y)) -> c_8(lt#(x,y)) minus#(s(x),s(y)) -> c_11(minus#(x,y)) mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) - Weak DPs: gcd#(x,0()) -> c_1() gcd#(0(),s(y)) -> c_2() if#(true(),x,y) -> c_5() lt#(x,0()) -> c_6() lt#(0(),s(x)) -> c_7() minus#(0(),x) -> c_9() minus#(s(x),0()) -> c_10() mod#(x,0()) -> c_12() - Weak TRS: if(false(),x,y) -> mod(minus(x,y),y) if(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),x) -> 0() minus(s(x),0()) -> s(x) minus(s(x),s(y)) -> minus(x,y) mod(x,0()) -> 0() mod(x,s(y)) -> if(lt(x,s(y)),x,s(y)) - Signature: {gcd/2,if/3,lt/2,minus/2,mod/2,gcd#/2,if#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0 ,c_3/3,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if#,lt#,minus#,mod#} and constructors {0,false,s ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x))) -->_3 mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y))):5 -->_2 mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y))):5 -->_1 gcd#(0(),s(y)) -> c_2():7 -->_1 gcd#(x,0()) -> c_1():6 -->_1 gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x))):1 2:S:if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y)) -->_1 mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y))):5 -->_2 minus#(s(x),s(y)) -> c_11(minus#(x,y)):4 -->_1 mod#(x,0()) -> c_12():13 -->_2 minus#(s(x),0()) -> c_10():12 -->_2 minus#(0(),x) -> c_9():11 3:S:lt#(s(x),s(y)) -> c_8(lt#(x,y)) -->_1 lt#(0(),s(x)) -> c_7():10 -->_1 lt#(x,0()) -> c_6():9 -->_1 lt#(s(x),s(y)) -> c_8(lt#(x,y)):3 4:S:minus#(s(x),s(y)) -> c_11(minus#(x,y)) -->_1 minus#(s(x),0()) -> c_10():12 -->_1 minus#(0(),x) -> c_9():11 -->_1 minus#(s(x),s(y)) -> c_11(minus#(x,y)):4 5:S:mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) -->_2 lt#(0(),s(x)) -> c_7():10 -->_1 if#(true(),x,y) -> c_5():8 -->_2 lt#(s(x),s(y)) -> c_8(lt#(x,y)):3 -->_1 if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y)):2 6:W:gcd#(x,0()) -> c_1() 7:W:gcd#(0(),s(y)) -> c_2() 8:W:if#(true(),x,y) -> c_5() 9:W:lt#(x,0()) -> c_6() 10:W:lt#(0(),s(x)) -> c_7() 11:W:minus#(0(),x) -> c_9() 12:W:minus#(s(x),0()) -> c_10() 13:W:mod#(x,0()) -> c_12() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: gcd#(x,0()) -> c_1() 7: gcd#(0(),s(y)) -> c_2() 13: mod#(x,0()) -> c_12() 11: minus#(0(),x) -> c_9() 12: minus#(s(x),0()) -> c_10() 9: lt#(x,0()) -> c_6() 8: if#(true(),x,y) -> c_5() 10: lt#(0(),s(x)) -> c_7() * Step 5: WeightGap MAYBE + Considered Problem: - Strict DPs: gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x))) if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y)) lt#(s(x),s(y)) -> c_8(lt#(x,y)) minus#(s(x),s(y)) -> c_11(minus#(x,y)) mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) - Weak TRS: if(false(),x,y) -> mod(minus(x,y),y) if(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),x) -> 0() minus(s(x),0()) -> s(x) minus(s(x),s(y)) -> minus(x,y) mod(x,0()) -> 0() mod(x,s(y)) -> if(lt(x,s(y)),x,s(y)) - Signature: {gcd/2,if/3,lt/2,minus/2,mod/2,gcd#/2,if#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0 ,c_3/3,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if#,lt#,minus#,mod#} and constructors {0,false,s ,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, 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 (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(if) = {1}, uargs(mod) = {1}, uargs(gcd#) = {1,2}, uargs(if#) = {1}, uargs(mod#) = {1}, uargs(c_3) = {1,2,3}, uargs(c_4) = {1,2}, uargs(c_8) = {1}, uargs(c_11) = {1}, uargs(c_13) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(gcd) = [0] p(if) = [1] x1 + [1] x2 + [3] x3 + [0] p(lt) = [0] p(minus) = [0] p(mod) = [1] x1 + [0] p(s) = [0] p(true) = [0] p(gcd#) = [1] x1 + [1] x2 + [0] p(if#) = [1] x1 + [1] x3 + [0] p(lt#) = [0] p(minus#) = [1] p(mod#) = [1] x1 + [1] x2 + [6] p(c_1) = [0] p(c_2) = [2] p(c_3) = [1] x1 + [1] x2 + [1] x3 + [2] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [1] p(c_6) = [1] p(c_7) = [1] p(c_8) = [1] x1 + [2] p(c_9) = [1] p(c_10) = [0] p(c_11) = [1] x1 + [2] p(c_12) = [4] p(c_13) = [1] x1 + [1] x2 + [4] Following rules are strictly oriented: mod#(x,s(y)) = [1] x + [6] > [4] = c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) Following rules are (at-least) weakly oriented: gcd#(s(x),s(y)) = [0] >= [14] = c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x))) if#(false(),x,y) = [1] y + [0] >= [1] y + [7] = c_4(mod#(minus(x,y),y),minus#(x,y)) lt#(s(x),s(y)) = [0] >= [2] = c_8(lt#(x,y)) minus#(s(x),s(y)) = [1] >= [3] = c_11(minus#(x,y)) if(false(),x,y) = [1] x + [3] y + [0] >= [0] = mod(minus(x,y),y) if(true(),x,y) = [1] x + [3] y + [0] >= [1] x + [0] = x lt(x,0()) = [0] >= [0] = false() lt(0(),s(x)) = [0] >= [0] = true() lt(s(x),s(y)) = [0] >= [0] = lt(x,y) minus(0(),x) = [0] >= [0] = 0() minus(s(x),0()) = [0] >= [0] = s(x) minus(s(x),s(y)) = [0] >= [0] = minus(x,y) mod(x,0()) = [1] x + [0] >= [0] = 0() mod(x,s(y)) = [1] x + [0] >= [1] x + [0] = if(lt(x,s(y)),x,s(y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: Failure MAYBE + Considered Problem: - Strict DPs: gcd#(s(x),s(y)) -> c_3(gcd#(mod(s(x),s(y)),mod(s(y),s(x))),mod#(s(x),s(y)),mod#(s(y),s(x))) if#(false(),x,y) -> c_4(mod#(minus(x,y),y),minus#(x,y)) lt#(s(x),s(y)) -> c_8(lt#(x,y)) minus#(s(x),s(y)) -> c_11(minus#(x,y)) - Weak DPs: mod#(x,s(y)) -> c_13(if#(lt(x,s(y)),x,s(y)),lt#(x,s(y))) - Weak TRS: if(false(),x,y) -> mod(minus(x,y),y) if(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(x)) -> true() lt(s(x),s(y)) -> lt(x,y) minus(0(),x) -> 0() minus(s(x),0()) -> s(x) minus(s(x),s(y)) -> minus(x,y) mod(x,0()) -> 0() mod(x,s(y)) -> if(lt(x,s(y)),x,s(y)) - Signature: {gcd/2,if/3,lt/2,minus/2,mod/2,gcd#/2,if#/3,lt#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0 ,c_3/3,c_4/2,c_5/0,c_6/0,c_7/0,c_8/1,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if#,lt#,minus#,mod#} and constructors {0,false,s ,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE