/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^2)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: 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(x,0()) -> x minus(s(x),s(y)) -> minus(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_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {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_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(x,0()) -> x minus(s(x),s(y)) -> minus(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_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {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_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(x,0()) -> x minus(s(x),s(y)) -> minus(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_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {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^2)) + Considered Problem: - Strict TRS: 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(x,0()) -> x minus(s(x),s(y)) -> minus(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_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {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_mod#(false(),s(x),s(y)) -> c_1() if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)) le#(0(),y) -> c_3() le#(s(x),0()) -> c_4() le#(s(x),s(y)) -> c_5(le#(x,y)) minus#(x,0()) -> c_6() minus#(s(x),s(y)) -> c_7(minus#(x,y)) mod#(0(),y) -> c_8() mod#(s(x),0()) -> c_9() mod#(s(x),s(y)) -> c_10(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^2)) + Considered Problem: - Strict DPs: if_mod#(false(),s(x),s(y)) -> c_1() if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)) le#(0(),y) -> c_3() le#(s(x),0()) -> c_4() le#(s(x),s(y)) -> c_5(le#(x,y)) minus#(x,0()) -> c_6() minus#(s(x),s(y)) -> c_7(minus#(x,y)) mod#(0(),y) -> c_8() mod#(s(x),0()) -> c_9() mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) - Weak TRS: 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(x,0()) -> x minus(s(x),s(y)) -> minus(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_mod/3,le/2,minus/2,mod/2,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {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 {1,3,4,6,8,9} by application of Pre({1,3,4,6,8,9}) = {2,5,7,10}. Here rules are labelled as follows: 1: if_mod#(false(),s(x),s(y)) -> c_1() 2: if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)) 3: le#(0(),y) -> c_3() 4: le#(s(x),0()) -> c_4() 5: le#(s(x),s(y)) -> c_5(le#(x,y)) 6: minus#(x,0()) -> c_6() 7: minus#(s(x),s(y)) -> c_7(minus#(x,y)) 8: mod#(0(),y) -> c_8() 9: mod#(s(x),0()) -> c_9() 10: mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)) le#(s(x),s(y)) -> c_5(le#(x,y)) minus#(s(x),s(y)) -> c_7(minus#(x,y)) mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) - Weak DPs: if_mod#(false(),s(x),s(y)) -> c_1() le#(0(),y) -> c_3() le#(s(x),0()) -> c_4() minus#(x,0()) -> c_6() mod#(0(),y) -> c_8() mod#(s(x),0()) -> c_9() - Weak TRS: 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(x,0()) -> x minus(s(x),s(y)) -> minus(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_mod/3,le/2,minus/2,mod/2,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#} and constructors {0,false,s ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)) -->_1 mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):4 -->_2 minus#(s(x),s(y)) -> c_7(minus#(x,y)):3 -->_1 mod#(0(),y) -> c_8():9 -->_2 minus#(x,0()) -> c_6():8 2:S:le#(s(x),s(y)) -> c_5(le#(x,y)) -->_1 le#(s(x),0()) -> c_4():7 -->_1 le#(0(),y) -> c_3():6 -->_1 le#(s(x),s(y)) -> c_5(le#(x,y)):2 3:S:minus#(s(x),s(y)) -> c_7(minus#(x,y)) -->_1 minus#(x,0()) -> c_6():8 -->_1 minus#(s(x),s(y)) -> c_7(minus#(x,y)):3 4:S:mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) -->_2 le#(s(x),0()) -> c_4():7 -->_2 le#(0(),y) -> c_3():6 -->_1 if_mod#(false(),s(x),s(y)) -> c_1():5 -->_2 le#(s(x),s(y)) -> c_5(le#(x,y)):2 -->_1 if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)):1 5:W:if_mod#(false(),s(x),s(y)) -> c_1() 6:W:le#(0(),y) -> c_3() 7:W:le#(s(x),0()) -> c_4() 8:W:minus#(x,0()) -> c_6() 9:W:mod#(0(),y) -> c_8() 10:W:mod#(s(x),0()) -> c_9() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: mod#(s(x),0()) -> c_9() 9: mod#(0(),y) -> c_8() 8: minus#(x,0()) -> c_6() 5: if_mod#(false(),s(x),s(y)) -> c_1() 6: le#(0(),y) -> c_3() 7: le#(s(x),0()) -> c_4() ** Step 1.b:4: UsableRules. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)) le#(s(x),s(y)) -> c_5(le#(x,y)) minus#(s(x),s(y)) -> c_7(minus#(x,y)) mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) - Weak TRS: 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(x,0()) -> x minus(s(x),s(y)) -> minus(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_mod/3,le/2,minus/2,mod/2,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#} and constructors {0,false,s ,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)) le#(s(x),s(y)) -> c_5(le#(x,y)) minus#(s(x),s(y)) -> c_7(minus#(x,y)) mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) ** Step 1.b:5: DecomposeDG. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)) le#(s(x),s(y)) -> c_5(le#(x,y)) minus#(s(x),s(y)) -> c_7(minus#(x,y)) mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {if_mod/3,le/2,minus/2,mod/2,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {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_2(mod#(minus(x,y),s(y)),minus#(x,y)) mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) and a lower component le#(s(x),s(y)) -> c_5(le#(x,y)) minus#(s(x),s(y)) -> c_7(minus#(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_2(mod#(minus(x,y),s(y)),minus#(x,y)) mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {if_mod/3,le/2,minus/2,mod/2,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {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_2(mod#(minus(x,y),s(y)),minus#(x,y)) -->_1 mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):2 2:S:mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)) -->_1 if_mod#(true(),s(x),s(y)) -> c_2(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_2(mod#(minus(x,y),s(y))) mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y))) *** Step 1.b:5.a:2: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y))) mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y))) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {if_mod/3,le/2,minus/2,mod/2,if_mod#/3,le#/2,minus#/2,mod#/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_7/1,c_8/0,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {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_mod#) = {1}, uargs(mod#) = {1}, uargs(c_2) = {1}, uargs(c_10) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(if_mod) = [2] x1 + [1] x3 + [0] p(le) = [4] p(minus) = [1] x1 + [4] p(mod) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [8] p(true) = [0] p(if_mod#) = [1] x1 + [1] x2 + [1] x3 + [0] p(le#) = [0] p(minus#) = [0] p(mod#) = [1] x1 + [1] x2 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] p(c_7) = [1] p(c_8) = [8] p(c_9) = [0] p(c_10) = [1] x1 + [2] Following rules are strictly oriented: if_mod#(true(),s(x),s(y)) = [1] x + [1] y + [16] > [1] x + [1] y + [12] = c_2(mod#(minus(x,y),s(y))) Following rules are (at-least) weakly oriented: mod#(s(x),s(y)) = [1] x + [1] y + [16] >= [1] x + [1] y + [22] = c_10(if_mod#(le(y,x),s(x),s(y))) le(0(),y) = [4] >= [0] = true() le(s(x),0()) = [4] >= [0] = false() le(s(x),s(y)) = [4] >= [4] = le(x,y) minus(x,0()) = [1] x + [4] >= [1] x + [0] = x minus(s(x),s(y)) = [1] x + [12] >= [1] x + [4] = minus(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:5.a:3: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y))) - Weak DPs: if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y))) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {if_mod/3,le/2,minus/2,mod/2,if_mod#/3,le#/2,minus#/2,mod#/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_7/1,c_8/0,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {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_2) = {1}, uargs(c_10) = {1} Following symbols are considered usable: {le,minus,if_mod#,le#,minus#,mod#} TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(if_mod) = [1] x1 + [8] x3 + [1] p(le) = [1] p(minus) = [1] x1 + [0] p(mod) = [1] x1 + [2] x2 + [4] p(s) = [1] x1 + [2] p(true) = [1] p(if_mod#) = [4] x1 + [2] x2 + [2] p(le#) = [1] x1 + [1] x2 + [0] p(minus#) = [1] p(mod#) = [2] x1 + [10] p(c_1) = [1] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [2] x1 + [0] p(c_6) = [1] p(c_7) = [0] p(c_8) = [4] p(c_9) = [0] p(c_10) = [1] x1 + [0] Following rules are strictly oriented: mod#(s(x),s(y)) = [2] x + [14] > [2] x + [10] = c_10(if_mod#(le(y,x),s(x),s(y))) Following rules are (at-least) weakly oriented: if_mod#(true(),s(x),s(y)) = [2] x + [10] >= [2] x + [10] = c_2(mod#(minus(x,y),s(y))) le(0(),y) = [1] >= [1] = true() le(s(x),0()) = [1] >= [0] = false() le(s(x),s(y)) = [1] >= [1] = le(x,y) minus(x,0()) = [1] x + [0] >= [1] x + [0] = x minus(s(x),s(y)) = [1] x + [2] >= [1] x + [0] = minus(x,y) *** Step 1.b:5.a:4: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y))) mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y))) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {if_mod/3,le/2,minus/2,mod/2,if_mod#/3,le#/2,minus#/2,mod#/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_7/1,c_8/0,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {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: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: le#(s(x),s(y)) -> c_5(le#(x,y)) minus#(s(x),s(y)) -> c_7(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)) mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)) mod#(s(x),s(y)) -> le#(y,x) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {if_mod/3,le/2,minus/2,mod/2,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {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_mod#) = {1}, uargs(mod#) = {1}, uargs(c_5) = {1}, uargs(c_7) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(false) = [1] p(if_mod) = [2] x3 + [4] p(le) = [1] p(minus) = [1] x1 + [2] p(mod) = [1] x2 + [4] p(s) = [1] x1 + [2] p(true) = [1] p(if_mod#) = [1] x1 + [1] x2 + [1] x3 + [0] p(le#) = [1] x2 + [5] p(minus#) = [1] p(mod#) = [1] x1 + [1] x2 + [1] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [1] x1 + [5] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] Following rules are strictly oriented: le#(s(x),s(y)) = [1] y + [7] > [1] y + [5] = c_5(le#(x,y)) Following rules are (at-least) weakly oriented: if_mod#(true(),s(x),s(y)) = [1] x + [1] y + [5] >= [1] = minus#(x,y) if_mod#(true(),s(x),s(y)) = [1] x + [1] y + [5] >= [1] x + [1] y + [5] = mod#(minus(x,y),s(y)) minus#(s(x),s(y)) = [1] >= [6] = c_7(minus#(x,y)) mod#(s(x),s(y)) = [1] x + [1] y + [5] >= [1] x + [1] y + [5] = if_mod#(le(y,x),s(x),s(y)) mod#(s(x),s(y)) = [1] x + [1] y + [5] >= [1] x + [5] = le#(y,x) le(0(),y) = [1] >= [1] = true() le(s(x),0()) = [1] >= [1] = false() le(s(x),s(y)) = [1] >= [1] = le(x,y) minus(x,0()) = [1] x + [2] >= [1] x + [0] = x minus(s(x),s(y)) = [1] x + [4] >= [1] x + [2] = minus(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:5.b:2: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: minus#(s(x),s(y)) -> c_7(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)) le#(s(x),s(y)) -> c_5(le#(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: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {if_mod/3,le/2,minus/2,mod/2,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {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_mod#) = {1}, uargs(mod#) = {1}, uargs(c_5) = {1}, uargs(c_7) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(if_mod) = [0] p(le) = [0] p(minus) = [1] x1 + [0] p(mod) = [0] p(s) = [1] x1 + [5] p(true) = [0] p(if_mod#) = [1] x1 + [1] x2 + [2] x3 + [0] p(le#) = [1] x2 + [6] p(minus#) = [1] x1 + [1] x2 + [2] p(mod#) = [1] x1 + [2] x2 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] p(c_4) = [1] p(c_5) = [1] x1 + [0] p(c_6) = [2] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] Following rules are strictly oriented: minus#(s(x),s(y)) = [1] x + [1] y + [12] > [1] x + [1] y + [2] = c_7(minus#(x,y)) Following rules are (at-least) weakly oriented: if_mod#(true(),s(x),s(y)) = [1] x + [2] y + [15] >= [1] x + [1] y + [2] = minus#(x,y) if_mod#(true(),s(x),s(y)) = [1] x + [2] y + [15] >= [1] x + [2] y + [10] = mod#(minus(x,y),s(y)) le#(s(x),s(y)) = [1] y + [11] >= [1] y + [6] = c_5(le#(x,y)) mod#(s(x),s(y)) = [1] x + [2] y + [15] >= [1] x + [2] y + [15] = if_mod#(le(y,x),s(x),s(y)) mod#(s(x),s(y)) = [1] x + [2] y + [15] >= [1] x + [6] = le#(y,x) le(0(),y) = [0] >= [0] = true() le(s(x),0()) = [0] >= [0] = false() le(s(x),s(y)) = [0] >= [0] = le(x,y) minus(x,0()) = [1] x + [0] >= [1] x + [0] = x minus(s(x),s(y)) = [1] x + [5] >= [1] x + [0] = minus(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:5.b:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - 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)) le#(s(x),s(y)) -> c_5(le#(x,y)) minus#(s(x),s(y)) -> c_7(minus#(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: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {if_mod/3,le/2,minus/2,mod/2,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {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^2))