/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),O(n^3)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^3)) + Considered Problem: - Strict TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus,if_mod,le,minus,mod} 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: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus,if_mod,le,minus,mod} and constructors {0,false,s ,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus,if_mod,le,minus,mod} and constructors {0,false,s ,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: le(x,y){x -> s(x),y -> s(y)} = le(s(x),s(y)) ->^+ le(x,y) = C[le(x,y) = le(x,y){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus,if_mod,le,minus,mod} and constructors {0,false,s ,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) if_minus#(true(),s(x),y) -> c_2() if_mod#(false(),s(x),s(y)) -> c_3() if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y)) le#(0(),y) -> c_5() le#(s(x),0()) -> c_6() le#(s(x),s(y)) -> c_7(le#(x,y)) minus#(0(),y) -> c_8() minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) mod#(0(),y) -> c_10() mod#(s(x),0()) -> c_11() mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) if_minus#(true(),s(x),y) -> c_2() if_mod#(false(),s(x),s(y)) -> c_3() if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y)) le#(0(),y) -> c_5() le#(s(x),0()) -> c_6() le#(s(x),s(y)) -> c_7(le#(x,y)) minus#(0(),y) -> c_8() minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) mod#(0(),y) -> c_10() mod#(s(x),0()) -> c_11() mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1 ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,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 {2,3,5,6,8,10,11} by application of Pre({2,3,5,6,8,10,11}) = {1,4,7,9,12}. Here rules are labelled as follows: 1: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) 2: if_minus#(true(),s(x),y) -> c_2() 3: if_mod#(false(),s(x),s(y)) -> c_3() 4: if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y)) 5: le#(0(),y) -> c_5() 6: le#(s(x),0()) -> c_6() 7: le#(s(x),s(y)) -> c_7(le#(x,y)) 8: minus#(0(),y) -> c_8() 9: minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) 10: mod#(0(),y) -> c_10() 11: mod#(s(x),0()) -> c_11() 12: mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y)) le#(s(x),s(y)) -> c_7(le#(x,y)) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) - Weak DPs: if_minus#(true(),s(x),y) -> c_2() if_mod#(false(),s(x),s(y)) -> c_3() le#(0(),y) -> c_5() le#(s(x),0()) -> c_6() minus#(0(),y) -> c_8() mod#(0(),y) -> c_10() mod#(s(x),0()) -> c_11() - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1 ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0 ,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):4 -->_1 minus#(0(),y) -> c_8():10 2:S:if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y)) -->_1 mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):5 -->_2 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):4 -->_1 mod#(0(),y) -> c_10():11 -->_2 minus#(0(),y) -> c_8():10 3:S:le#(s(x),s(y)) -> c_7(le#(x,y)) -->_1 le#(s(x),0()) -> c_6():9 -->_1 le#(0(),y) -> c_5():8 -->_1 le#(s(x),s(y)) -> c_7(le#(x,y)):3 4:S:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) -->_2 le#(s(x),0()) -> c_6():9 -->_1 if_minus#(true(),s(x),y) -> c_2():6 -->_2 le#(s(x),s(y)) -> c_7(le#(x,y)):3 -->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1 5:S:mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) -->_2 le#(s(x),0()) -> c_6():9 -->_2 le#(0(),y) -> c_5():8 -->_1 if_mod#(false(),s(x),s(y)) -> c_3():7 -->_2 le#(s(x),s(y)) -> c_7(le#(x,y)):3 -->_1 if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y)):2 6:W:if_minus#(true(),s(x),y) -> c_2() 7:W:if_mod#(false(),s(x),s(y)) -> c_3() 8:W:le#(0(),y) -> c_5() 9:W:le#(s(x),0()) -> c_6() 10:W:minus#(0(),y) -> c_8() 11:W:mod#(0(),y) -> c_10() 12:W:mod#(s(x),0()) -> c_11() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 12: mod#(s(x),0()) -> c_11() 11: mod#(0(),y) -> c_10() 7: if_mod#(false(),s(x),s(y)) -> c_3() 10: minus#(0(),y) -> c_8() 8: le#(0(),y) -> c_5() 6: if_minus#(true(),s(x),y) -> c_2() 9: le#(s(x),0()) -> c_6() ** Step 1.b:4: UsableRules. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y)) le#(s(x),s(y)) -> c_7(le#(x,y)) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() if_mod(false(),s(x),s(y)) -> s(x) if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) mod(0(),y) -> 0() mod(s(x),0()) -> 0() mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y)) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1 ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0 ,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y)) le#(s(x),s(y)) -> c_7(le#(x,y)) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) ** Step 1.b:5: DecomposeDG. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y)) le#(s(x),s(y)) -> c_7(le#(x,y)) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1 ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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 if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y)) mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) and a lower component if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) le#(s(x),s(y)) -> c_7(le#(x,y)) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) Further, following extension rules are added to the lower component. if_mod#(true(),s(x),s(y)) -> minus#(x,y) if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)) mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)) mod#(s(x),s(y)) -> le#(y,x) *** Step 1.b:5.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y)) mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1 ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0 ,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y)) -->_1 mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):2 2:S:mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) -->_1 if_mod#(true(),s(x),s(y)) -> c_4(mod#(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: if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y))) mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y))) *** Step 1.b:5.a:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y))) mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y))) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1 ,true/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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_4) = {1}, uargs(c_12) = {1} Following symbols are considered usable: {if_minus,minus,if_minus#,if_mod#,le#,minus#,mod#} TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(if_minus) = [1] x2 + [0] p(if_mod) = [1] x1 + [2] x3 + [1] p(le) = [0] p(minus) = [1] x1 + [0] p(mod) = [1] x1 + [1] x2 + [1] p(s) = [1] x1 + [4] p(true) = [0] p(if_minus#) = [1] x1 + [1] p(if_mod#) = [1] x2 + [2] x3 + [4] p(le#) = [0] p(minus#) = [2] x1 + [0] p(mod#) = [1] x1 + [2] x2 + [4] p(c_1) = [1] x1 + [2] p(c_2) = [8] p(c_3) = [0] p(c_4) = [1] x1 + [1] p(c_5) = [2] p(c_6) = [0] p(c_7) = [4] x1 + [0] p(c_8) = [1] p(c_9) = [2] x1 + [4] p(c_10) = [1] p(c_11) = [0] p(c_12) = [1] x1 + [0] Following rules are strictly oriented: if_mod#(true(),s(x),s(y)) = [1] x + [2] y + [16] > [1] x + [2] y + [13] = c_4(mod#(minus(x,y),s(y))) Following rules are (at-least) weakly oriented: mod#(s(x),s(y)) = [1] x + [2] y + [16] >= [1] x + [2] y + [16] = c_12(if_mod#(le(y,x),s(x),s(y))) if_minus(false(),s(x),y) = [1] x + [4] >= [1] x + [4] = s(minus(x,y)) if_minus(true(),s(x),y) = [1] x + [4] >= [0] = 0() minus(0(),y) = [0] >= [0] = 0() minus(s(x),y) = [1] x + [4] >= [1] x + [4] = if_minus(le(s(x),y),s(x),y) *** Step 1.b:5.a:3: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y))) - Weak DPs: if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y))) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1 ,true/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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(if_minus) = {1}, uargs(s) = {1}, uargs(if_mod#) = {1}, uargs(mod#) = {1}, uargs(c_4) = {1}, uargs(c_12) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(if_minus) = [1] x1 + [1] x2 + [1] p(if_mod) = [1] x1 + [0] p(le) = [0] p(minus) = [1] x1 + [1] p(mod) = [1] p(s) = [1] x1 + [2] p(true) = [0] p(if_minus#) = [2] x1 + [1] x2 + [0] p(if_mod#) = [1] x1 + [1] x2 + [6] p(le#) = [4] x2 + [1] p(minus#) = [1] x2 + [2] p(mod#) = [1] x1 + [7] p(c_1) = [0] p(c_2) = [1] p(c_3) = [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x1 + [2] x2 + [2] p(c_10) = [0] p(c_11) = [0] p(c_12) = [1] x1 + [0] Following rules are strictly oriented: mod#(s(x),s(y)) = [1] x + [9] > [1] x + [8] = c_12(if_mod#(le(y,x),s(x),s(y))) Following rules are (at-least) weakly oriented: if_mod#(true(),s(x),s(y)) = [1] x + [8] >= [1] x + [8] = c_4(mod#(minus(x,y),s(y))) if_minus(false(),s(x),y) = [1] x + [3] >= [1] x + [3] = s(minus(x,y)) if_minus(true(),s(x),y) = [1] x + [3] >= [0] = 0() le(0(),y) = [0] >= [0] = true() le(s(x),0()) = [0] >= [0] = false() le(s(x),s(y)) = [0] >= [0] = le(x,y) minus(0(),y) = [1] >= [0] = 0() minus(s(x),y) = [1] x + [3] >= [1] x + [3] = if_minus(le(s(x),y),s(x),y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:5.a:4: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y))) mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y))) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1 ,true/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0 ,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:5.b:1: DecomposeDG. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) le#(s(x),s(y)) -> c_7(le#(x,y)) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) - Weak DPs: if_mod#(true(),s(x),s(y)) -> minus#(x,y) if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)) mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)) mod#(s(x),s(y)) -> le#(y,x) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1 ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) if_mod#(true(),s(x),s(y)) -> minus#(x,y) if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)) mod#(s(x),s(y)) -> le#(y,x) and a lower component le#(s(x),s(y)) -> c_7(le#(x,y)) Further, following extension rules are added to the lower component. if_minus#(false(),s(x),y) -> minus#(x,y) if_mod#(true(),s(x),s(y)) -> minus#(x,y) if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)) minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y) minus#(s(x),y) -> le#(s(x),y) mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)) mod#(s(x),s(y)) -> le#(y,x) **** Step 1.b:5.b:1.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) mod#(s(x),s(y)) -> le#(y,x) - Weak DPs: if_mod#(true(),s(x),s(y)) -> minus#(x,y) if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)) mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1 ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0 ,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2 2:S:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) -->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1 3:S:mod#(s(x),s(y)) -> le#(y,x) 4:W:if_mod#(true(),s(x),s(y)) -> minus#(x,y) -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2 5:W:if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)) -->_1 mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)):6 -->_1 mod#(s(x),s(y)) -> le#(y,x):3 6:W:mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)) -->_1 if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)):5 -->_1 if_mod#(true(),s(x),s(y)) -> minus#(x,y):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y)) **** Step 1.b:5.b:1.a:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y)) mod#(s(x),s(y)) -> le#(y,x) - Weak DPs: if_mod#(true(),s(x),s(y)) -> minus#(x,y) if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)) mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1 ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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_1) = {1}, uargs(c_9) = {1} Following symbols are considered usable: {if_minus,minus,if_minus#,if_mod#,le#,minus#,mod#} TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(if_minus) = [1] x2 + [0] p(if_mod) = [1] x3 + [1] p(le) = [0] p(minus) = [1] x1 + [0] p(mod) = [2] x2 + [1] p(s) = [1] x1 + [0] p(true) = [2] p(if_minus#) = [0] p(if_mod#) = [1] x2 + [5] x3 + [8] p(le#) = [2] x1 + [3] p(minus#) = [0] p(mod#) = [1] x1 + [5] x2 + [8] p(c_1) = [8] x1 + [0] p(c_2) = [1] p(c_3) = [8] p(c_4) = [8] x1 + [1] x2 + [4] p(c_5) = [1] p(c_6) = [0] p(c_7) = [1] p(c_8) = [8] p(c_9) = [8] x1 + [0] p(c_10) = [1] p(c_11) = [0] p(c_12) = [2] x2 + [0] Following rules are strictly oriented: mod#(s(x),s(y)) = [1] x + [5] y + [8] > [2] y + [3] = le#(y,x) Following rules are (at-least) weakly oriented: if_minus#(false(),s(x),y) = [0] >= [0] = c_1(minus#(x,y)) if_mod#(true(),s(x),s(y)) = [1] x + [5] y + [8] >= [0] = minus#(x,y) if_mod#(true(),s(x),s(y)) = [1] x + [5] y + [8] >= [1] x + [5] y + [8] = mod#(minus(x,y),s(y)) minus#(s(x),y) = [0] >= [0] = c_9(if_minus#(le(s(x),y),s(x),y)) mod#(s(x),s(y)) = [1] x + [5] y + [8] >= [1] x + [5] y + [8] = if_mod#(le(y,x),s(x),s(y)) if_minus(false(),s(x),y) = [1] x + [0] >= [1] x + [0] = s(minus(x,y)) if_minus(true(),s(x),y) = [1] x + [0] >= [0] = 0() minus(0(),y) = [0] >= [0] = 0() minus(s(x),y) = [1] x + [0] >= [1] x + [0] = if_minus(le(s(x),y),s(x),y) **** Step 1.b:5.b:1.a:3: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y)) - Weak DPs: if_mod#(true(),s(x),s(y)) -> minus#(x,y) if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)) mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)) mod#(s(x),s(y)) -> le#(y,x) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1 ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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(if_minus) = {1}, uargs(s) = {1}, uargs(if_minus#) = {1}, uargs(if_mod#) = {1}, uargs(mod#) = {1}, uargs(c_1) = {1}, uargs(c_9) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(false) = [0] p(if_minus) = [1] x1 + [1] x2 + [0] p(if_mod) = [1] x1 + [1] x2 + [2] x3 + [0] p(le) = [0] p(minus) = [1] x1 + [0] p(mod) = [1] x1 + [1] p(s) = [1] x1 + [1] p(true) = [0] p(if_minus#) = [1] x1 + [1] x2 + [0] p(if_mod#) = [1] x1 + [1] x2 + [5] p(le#) = [0] p(minus#) = [1] x1 + [6] p(mod#) = [1] x1 + [6] p(c_1) = [1] x1 + [0] p(c_2) = [1] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x1 + [1] p(c_10) = [0] p(c_11) = [1] p(c_12) = [0] Following rules are strictly oriented: minus#(s(x),y) = [1] x + [7] > [1] x + [2] = c_9(if_minus#(le(s(x),y),s(x),y)) Following rules are (at-least) weakly oriented: if_minus#(false(),s(x),y) = [1] x + [1] >= [1] x + [6] = c_1(minus#(x,y)) if_mod#(true(),s(x),s(y)) = [1] x + [6] >= [1] x + [6] = minus#(x,y) if_mod#(true(),s(x),s(y)) = [1] x + [6] >= [1] x + [6] = mod#(minus(x,y),s(y)) mod#(s(x),s(y)) = [1] x + [7] >= [1] x + [6] = if_mod#(le(y,x),s(x),s(y)) mod#(s(x),s(y)) = [1] x + [7] >= [0] = le#(y,x) if_minus(false(),s(x),y) = [1] x + [1] >= [1] x + [1] = s(minus(x,y)) if_minus(true(),s(x),y) = [1] x + [1] >= [1] = 0() le(0(),y) = [0] >= [0] = true() le(s(x),0()) = [0] >= [0] = false() le(s(x),s(y)) = [0] >= [0] = le(x,y) minus(0(),y) = [1] >= [1] = 0() minus(s(x),y) = [1] x + [1] >= [1] x + [1] = if_minus(le(s(x),y),s(x),y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. **** Step 1.b:5.b:1.a:4: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) - Weak DPs: if_mod#(true(),s(x),s(y)) -> minus#(x,y) if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y)) mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)) mod#(s(x),s(y)) -> le#(y,x) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1 ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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(if_minus) = {1}, uargs(s) = {1}, uargs(if_minus#) = {1}, uargs(if_mod#) = {1}, uargs(mod#) = {1}, uargs(c_1) = {1}, uargs(c_9) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(if_minus) = [1] x1 + [1] x2 + [0] p(if_mod) = [4] x1 + [1] x2 + [0] p(le) = [0] p(minus) = [1] x1 + [0] p(mod) = [1] x1 + [2] x2 + [0] p(s) = [1] x1 + [2] p(true) = [0] p(if_minus#) = [1] x1 + [1] x2 + [1] p(if_mod#) = [1] x1 + [1] x2 + [1] p(le#) = [1] x2 + [3] p(minus#) = [1] x1 + [1] p(mod#) = [1] x1 + [1] p(c_1) = [1] x1 + [1] p(c_2) = [4] p(c_3) = [1] p(c_4) = [1] x1 + [1] x2 + [1] p(c_5) = [0] p(c_6) = [4] p(c_7) = [4] x1 + [2] p(c_8) = [1] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] p(c_12) = [4] x1 + [1] x2 + [0] Following rules are strictly oriented: if_minus#(false(),s(x),y) = [1] x + [3] > [1] x + [2] = c_1(minus#(x,y)) Following rules are (at-least) weakly oriented: if_mod#(true(),s(x),s(y)) = [1] x + [3] >= [1] x + [1] = minus#(x,y) if_mod#(true(),s(x),s(y)) = [1] x + [3] >= [1] x + [1] = mod#(minus(x,y),s(y)) minus#(s(x),y) = [1] x + [3] >= [1] x + [3] = c_9(if_minus#(le(s(x),y),s(x),y)) mod#(s(x),s(y)) = [1] x + [3] >= [1] x + [3] = if_mod#(le(y,x),s(x),s(y)) mod#(s(x),s(y)) = [1] x + [3] >= [1] x + [3] = le#(y,x) if_minus(false(),s(x),y) = [1] x + [2] >= [1] x + [2] = s(minus(x,y)) if_minus(true(),s(x),y) = [1] x + [2] >= [0] = 0() le(0(),y) = [0] >= [0] = true() le(s(x),0()) = [0] >= [0] = false() le(s(x),s(y)) = [0] >= [0] = le(x,y) minus(0(),y) = [0] >= [0] = 0() minus(s(x),y) = [1] x + [2] >= [1] x + [2] = if_minus(le(s(x),y),s(x),y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. **** Step 1.b:5.b:1.a:5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) if_mod#(true(),s(x),s(y)) -> minus#(x,y) if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y)) mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)) mod#(s(x),s(y)) -> le#(y,x) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1 ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0 ,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 1.b:5.b:1.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: le#(s(x),s(y)) -> c_7(le#(x,y)) - Weak DPs: if_minus#(false(),s(x),y) -> minus#(x,y) if_mod#(true(),s(x),s(y)) -> minus#(x,y) if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)) minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y) minus#(s(x),y) -> le#(s(x),y) mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)) mod#(s(x),s(y)) -> le#(y,x) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1 ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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_7) = {1} Following symbols are considered usable: {if_minus,minus,if_minus#,if_mod#,le#,minus#,mod#} TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(if_minus) = [1] x2 + [8] p(if_mod) = [1] x1 + [4] x3 + [1] p(le) = [0] p(minus) = [1] x1 + [8] p(mod) = [1] x1 + [2] p(s) = [1] x1 + [8] p(true) = [0] p(if_minus#) = [1] x2 + [14] p(if_mod#) = [1] x2 + [1] x3 + [1] p(le#) = [1] x1 + [0] p(minus#) = [1] x1 + [14] p(mod#) = [1] x1 + [1] x2 + [1] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] p(c_4) = [1] x1 + [2] x2 + [1] p(c_5) = [1] p(c_6) = [1] p(c_7) = [1] x1 + [0] p(c_8) = [1] p(c_9) = [2] x2 + [1] p(c_10) = [0] p(c_11) = [1] p(c_12) = [1] x1 + [2] x2 + [1] Following rules are strictly oriented: le#(s(x),s(y)) = [1] x + [8] > [1] x + [0] = c_7(le#(x,y)) Following rules are (at-least) weakly oriented: if_minus#(false(),s(x),y) = [1] x + [22] >= [1] x + [14] = minus#(x,y) if_mod#(true(),s(x),s(y)) = [1] x + [1] y + [17] >= [1] x + [14] = minus#(x,y) if_mod#(true(),s(x),s(y)) = [1] x + [1] y + [17] >= [1] x + [1] y + [17] = mod#(minus(x,y),s(y)) minus#(s(x),y) = [1] x + [22] >= [1] x + [22] = if_minus#(le(s(x),y),s(x),y) minus#(s(x),y) = [1] x + [22] >= [1] x + [8] = le#(s(x),y) mod#(s(x),s(y)) = [1] x + [1] y + [17] >= [1] x + [1] y + [17] = if_mod#(le(y,x),s(x),s(y)) mod#(s(x),s(y)) = [1] x + [1] y + [17] >= [1] y + [0] = le#(y,x) if_minus(false(),s(x),y) = [1] x + [16] >= [1] x + [16] = s(minus(x,y)) if_minus(true(),s(x),y) = [1] x + [16] >= [0] = 0() minus(0(),y) = [8] >= [0] = 0() minus(s(x),y) = [1] x + [16] >= [1] x + [16] = if_minus(le(s(x),y),s(x),y) **** Step 1.b:5.b:1.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: if_minus#(false(),s(x),y) -> minus#(x,y) if_mod#(true(),s(x),s(y)) -> minus#(x,y) if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)) le#(s(x),s(y)) -> c_7(le#(x,y)) minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y) minus#(s(x),y) -> le#(s(x),y) mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)) mod#(s(x),s(y)) -> le#(y,x) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) - Signature: {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1 ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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^3))