MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fib(0()) -> 0() fib(s(0())) -> s(0()) fib(s(s(x))) -> sp(g(x)) fib(s(s(0()))) -> s(0()) g(0()) -> pair(s(0()),0()) g(s(x)) -> np(g(x)) g(s(0())) -> pair(s(0()),s(0())) np(pair(x,y)) -> pair(+(x,y),x) sp(pair(x,y)) -> +(x,y) - Signature: {+/2,fib/1,g/1,np/1,sp/1} / {0/0,pair/2,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,fib,g,np,sp} and constructors {0,pair,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs +#(x,0()) -> c_1() +#(x,s(y)) -> c_2(+#(x,y)) fib#(0()) -> c_3() fib#(s(0())) -> c_4() fib#(s(s(x))) -> c_5(sp#(g(x)),g#(x)) fib#(s(s(0()))) -> c_6() g#(0()) -> c_7() g#(s(x)) -> c_8(np#(g(x)),g#(x)) g#(s(0())) -> c_9() np#(pair(x,y)) -> c_10(+#(x,y)) sp#(pair(x,y)) -> c_11(+#(x,y)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: +#(x,0()) -> c_1() +#(x,s(y)) -> c_2(+#(x,y)) fib#(0()) -> c_3() fib#(s(0())) -> c_4() fib#(s(s(x))) -> c_5(sp#(g(x)),g#(x)) fib#(s(s(0()))) -> c_6() g#(0()) -> c_7() g#(s(x)) -> c_8(np#(g(x)),g#(x)) g#(s(0())) -> c_9() np#(pair(x,y)) -> c_10(+#(x,y)) sp#(pair(x,y)) -> c_11(+#(x,y)) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fib(0()) -> 0() fib(s(0())) -> s(0()) fib(s(s(x))) -> sp(g(x)) fib(s(s(0()))) -> s(0()) g(0()) -> pair(s(0()),0()) g(s(x)) -> np(g(x)) g(s(0())) -> pair(s(0()),s(0())) np(pair(x,y)) -> pair(+(x,y),x) sp(pair(x,y)) -> +(x,y) - Signature: {+/2,fib/1,g/1,np/1,sp/1,+#/2,fib#/1,g#/1,np#/1,sp#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2,c_6/0 ,c_7/0,c_8/2,c_9/0,c_10/1,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {+#,fib#,g#,np#,sp#} and constructors {0,pair,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) g(0()) -> pair(s(0()),0()) g(s(x)) -> np(g(x)) g(s(0())) -> pair(s(0()),s(0())) np(pair(x,y)) -> pair(+(x,y),x) +#(x,0()) -> c_1() +#(x,s(y)) -> c_2(+#(x,y)) fib#(0()) -> c_3() fib#(s(0())) -> c_4() fib#(s(s(x))) -> c_5(sp#(g(x)),g#(x)) fib#(s(s(0()))) -> c_6() g#(0()) -> c_7() g#(s(x)) -> c_8(np#(g(x)),g#(x)) g#(s(0())) -> c_9() np#(pair(x,y)) -> c_10(+#(x,y)) sp#(pair(x,y)) -> c_11(+#(x,y)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: +#(x,0()) -> c_1() +#(x,s(y)) -> c_2(+#(x,y)) fib#(0()) -> c_3() fib#(s(0())) -> c_4() fib#(s(s(x))) -> c_5(sp#(g(x)),g#(x)) fib#(s(s(0()))) -> c_6() g#(0()) -> c_7() g#(s(x)) -> c_8(np#(g(x)),g#(x)) g#(s(0())) -> c_9() np#(pair(x,y)) -> c_10(+#(x,y)) sp#(pair(x,y)) -> c_11(+#(x,y)) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) g(0()) -> pair(s(0()),0()) g(s(x)) -> np(g(x)) g(s(0())) -> pair(s(0()),s(0())) np(pair(x,y)) -> pair(+(x,y),x) - Signature: {+/2,fib/1,g/1,np/1,sp/1,+#/2,fib#/1,g#/1,np#/1,sp#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2,c_6/0 ,c_7/0,c_8/2,c_9/0,c_10/1,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {+#,fib#,g#,np#,sp#} and constructors {0,pair,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,4,6,7,9} by application of Pre({1,3,4,6,7,9}) = {2,5,8,10,11}. Here rules are labelled as follows: 1: +#(x,0()) -> c_1() 2: +#(x,s(y)) -> c_2(+#(x,y)) 3: fib#(0()) -> c_3() 4: fib#(s(0())) -> c_4() 5: fib#(s(s(x))) -> c_5(sp#(g(x)),g#(x)) 6: fib#(s(s(0()))) -> c_6() 7: g#(0()) -> c_7() 8: g#(s(x)) -> c_8(np#(g(x)),g#(x)) 9: g#(s(0())) -> c_9() 10: np#(pair(x,y)) -> c_10(+#(x,y)) 11: sp#(pair(x,y)) -> c_11(+#(x,y)) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: +#(x,s(y)) -> c_2(+#(x,y)) fib#(s(s(x))) -> c_5(sp#(g(x)),g#(x)) g#(s(x)) -> c_8(np#(g(x)),g#(x)) np#(pair(x,y)) -> c_10(+#(x,y)) sp#(pair(x,y)) -> c_11(+#(x,y)) - Weak DPs: +#(x,0()) -> c_1() fib#(0()) -> c_3() fib#(s(0())) -> c_4() fib#(s(s(0()))) -> c_6() g#(0()) -> c_7() g#(s(0())) -> c_9() - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) g(0()) -> pair(s(0()),0()) g(s(x)) -> np(g(x)) g(s(0())) -> pair(s(0()),s(0())) np(pair(x,y)) -> pair(+(x,y),x) - Signature: {+/2,fib/1,g/1,np/1,sp/1,+#/2,fib#/1,g#/1,np#/1,sp#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2,c_6/0 ,c_7/0,c_8/2,c_9/0,c_10/1,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {+#,fib#,g#,np#,sp#} and constructors {0,pair,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:+#(x,s(y)) -> c_2(+#(x,y)) -->_1 +#(x,0()) -> c_1():6 -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1 2:S:fib#(s(s(x))) -> c_5(sp#(g(x)),g#(x)) -->_1 sp#(pair(x,y)) -> c_11(+#(x,y)):5 -->_2 g#(s(x)) -> c_8(np#(g(x)),g#(x)):3 -->_2 g#(s(0())) -> c_9():11 -->_2 g#(0()) -> c_7():10 3:S:g#(s(x)) -> c_8(np#(g(x)),g#(x)) -->_1 np#(pair(x,y)) -> c_10(+#(x,y)):4 -->_2 g#(s(0())) -> c_9():11 -->_2 g#(0()) -> c_7():10 -->_2 g#(s(x)) -> c_8(np#(g(x)),g#(x)):3 4:S:np#(pair(x,y)) -> c_10(+#(x,y)) -->_1 +#(x,0()) -> c_1():6 -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1 5:S:sp#(pair(x,y)) -> c_11(+#(x,y)) -->_1 +#(x,0()) -> c_1():6 -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1 6:W:+#(x,0()) -> c_1() 7:W:fib#(0()) -> c_3() 8:W:fib#(s(0())) -> c_4() 9:W:fib#(s(s(0()))) -> c_6() 10:W:g#(0()) -> c_7() 11:W:g#(s(0())) -> c_9() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: fib#(s(s(0()))) -> c_6() 8: fib#(s(0())) -> c_4() 7: fib#(0()) -> c_3() 10: g#(0()) -> c_7() 11: g#(s(0())) -> c_9() 6: +#(x,0()) -> c_1() * Step 5: NaturalMI MAYBE + Considered Problem: - Strict DPs: +#(x,s(y)) -> c_2(+#(x,y)) fib#(s(s(x))) -> c_5(sp#(g(x)),g#(x)) g#(s(x)) -> c_8(np#(g(x)),g#(x)) np#(pair(x,y)) -> c_10(+#(x,y)) sp#(pair(x,y)) -> c_11(+#(x,y)) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) g(0()) -> pair(s(0()),0()) g(s(x)) -> np(g(x)) g(s(0())) -> pair(s(0()),s(0())) np(pair(x,y)) -> pair(+(x,y),x) - Signature: {+/2,fib/1,g/1,np/1,sp/1,+#/2,fib#/1,g#/1,np#/1,sp#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2,c_6/0 ,c_7/0,c_8/2,c_9/0,c_10/1,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {+#,fib#,g#,np#,sp#} and constructors {0,pair,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: 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(c_2) = {1}, uargs(c_5) = {1,2}, uargs(c_8) = {1,2}, uargs(c_10) = {1}, uargs(c_11) = {1} Following symbols are considered usable: {+#,fib#,g#,np#,sp#} TcT has computed the following interpretation: p(+) = [0] p(0) = [0] p(fib) = [0] p(g) = [0] p(np) = [0] p(pair) = [15] p(s) = [10] p(sp) = [2] x1 + [8] p(+#) = [0] p(fib#) = [9] p(g#) = [0] p(np#) = [0] p(sp#) = [0] p(c_1) = [0] p(c_2) = [2] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [2] x1 + [1] x2 + [5] p(c_6) = [0] p(c_7) = [0] p(c_8) = [2] x1 + [8] x2 + [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [1] x1 + [0] Following rules are strictly oriented: fib#(s(s(x))) = [9] > [5] = c_5(sp#(g(x)),g#(x)) Following rules are (at-least) weakly oriented: +#(x,s(y)) = [0] >= [0] = c_2(+#(x,y)) g#(s(x)) = [0] >= [0] = c_8(np#(g(x)),g#(x)) np#(pair(x,y)) = [0] >= [0] = c_10(+#(x,y)) sp#(pair(x,y)) = [0] >= [0] = c_11(+#(x,y)) * Step 6: NaturalMI MAYBE + Considered Problem: - Strict DPs: +#(x,s(y)) -> c_2(+#(x,y)) g#(s(x)) -> c_8(np#(g(x)),g#(x)) np#(pair(x,y)) -> c_10(+#(x,y)) sp#(pair(x,y)) -> c_11(+#(x,y)) - Weak DPs: fib#(s(s(x))) -> c_5(sp#(g(x)),g#(x)) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) g(0()) -> pair(s(0()),0()) g(s(x)) -> np(g(x)) g(s(0())) -> pair(s(0()),s(0())) np(pair(x,y)) -> pair(+(x,y),x) - Signature: {+/2,fib/1,g/1,np/1,sp/1,+#/2,fib#/1,g#/1,np#/1,sp#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2,c_6/0 ,c_7/0,c_8/2,c_9/0,c_10/1,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {+#,fib#,g#,np#,sp#} and constructors {0,pair,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: 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(c_2) = {1}, uargs(c_5) = {1,2}, uargs(c_8) = {1,2}, uargs(c_10) = {1}, uargs(c_11) = {1} Following symbols are considered usable: {+#,fib#,g#,np#,sp#} TcT has computed the following interpretation: p(+) = [2] x1 + [2] x2 + [12] p(0) = [0] p(fib) = [1] x1 + [1] p(g) = [4] p(np) = [0] p(pair) = [12] p(s) = [2] p(sp) = [1] x1 + [2] p(+#) = [0] p(fib#) = [2] x1 + [8] p(g#) = [0] p(np#) = [0] p(sp#) = [12] p(c_1) = [1] p(c_2) = [4] x1 + [0] p(c_3) = [0] p(c_4) = [1] p(c_5) = [1] x1 + [1] x2 + [0] p(c_6) = [2] p(c_7) = [0] p(c_8) = [8] x1 + [8] x2 + [0] p(c_9) = [1] p(c_10) = [8] x1 + [0] p(c_11) = [1] x1 + [0] Following rules are strictly oriented: sp#(pair(x,y)) = [12] > [0] = c_11(+#(x,y)) Following rules are (at-least) weakly oriented: +#(x,s(y)) = [0] >= [0] = c_2(+#(x,y)) fib#(s(s(x))) = [12] >= [12] = c_5(sp#(g(x)),g#(x)) g#(s(x)) = [0] >= [0] = c_8(np#(g(x)),g#(x)) np#(pair(x,y)) = [0] >= [0] = c_10(+#(x,y)) * Step 7: Failure MAYBE + Considered Problem: - Strict DPs: +#(x,s(y)) -> c_2(+#(x,y)) g#(s(x)) -> c_8(np#(g(x)),g#(x)) np#(pair(x,y)) -> c_10(+#(x,y)) - Weak DPs: fib#(s(s(x))) -> c_5(sp#(g(x)),g#(x)) sp#(pair(x,y)) -> c_11(+#(x,y)) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) g(0()) -> pair(s(0()),0()) g(s(x)) -> np(g(x)) g(s(0())) -> pair(s(0()),s(0())) np(pair(x,y)) -> pair(+(x,y),x) - Signature: {+/2,fib/1,g/1,np/1,sp/1,+#/2,fib#/1,g#/1,np#/1,sp#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2,c_6/0 ,c_7/0,c_8/2,c_9/0,c_10/1,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {+#,fib#,g#,np#,sp#} and constructors {0,pair,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE