/export/starexec/sandbox/solver/bin/starexec_run_tct_rci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: gcd(x,y) -> if1(ge(x,y),x,y) ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) if1(false(),x,y) -> gcd(y,x) if1(true(),x,y) -> if2(gt(y,0()),x,y) if2(false(),x,y) -> x if2(true(),x,y) -> gcd(minus(x,y),y) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,ge,gt,if,if1,if2,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: gcd(x,y) -> if1(ge(x,y),x,y) ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) if1(false(),x,y) -> gcd(y,x) if1(true(),x,y) -> if2(gt(y,0()),x,y) if2(false(),x,y) -> x if2(true(),x,y) -> gcd(minus(x,y),y) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,ge,gt,if,if1,if2,minus} 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: gcd(x,y) -> if1(ge(x,y),x,y) ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) if1(false(),x,y) -> gcd(y,x) if1(true(),x,y) -> if2(gt(y,0()),x,y) if2(false(),x,y) -> x if2(true(),x,y) -> gcd(minus(x,y),y) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,ge,gt,if,if1,if2,minus} and constructors {0,false,s ,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: ge(x,y){x -> s(x),y -> s(y)} = ge(s(x),s(y)) ->^+ ge(x,y) = C[ge(x,y) = ge(x,y){}] ** Step 1.b:1: DependencyPairs. MAYBE + Considered Problem: - Strict TRS: gcd(x,y) -> if1(ge(x,y),x,y) ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) if1(false(),x,y) -> gcd(y,x) if1(true(),x,y) -> if2(gt(y,0()),x,y) if2(false(),x,y) -> x if2(true(),x,y) -> gcd(minus(x,y),y) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,ge,gt,if,if1,if2,minus} and constructors {0,false,s ,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)) ge#(x,0()) -> c_2() ge#(0(),s(x)) -> c_3() ge#(s(x),s(y)) -> c_4(ge#(x,y)) gt#(0(),y) -> c_5() gt#(s(x),0()) -> c_6() gt#(s(x),s(y)) -> c_7(gt#(x,y)) if#(false(),x,y) -> c_8() if#(true(),x,y) -> c_9(minus#(x,y)) if1#(false(),x,y) -> c_10(gcd#(y,x)) if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0())) if2#(false(),x,y) -> c_12() if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y),minus#(x,y)) minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. MAYBE + Considered Problem: - Strict DPs: gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)) ge#(x,0()) -> c_2() ge#(0(),s(x)) -> c_3() ge#(s(x),s(y)) -> c_4(ge#(x,y)) gt#(0(),y) -> c_5() gt#(s(x),0()) -> c_6() gt#(s(x),s(y)) -> c_7(gt#(x,y)) if#(false(),x,y) -> c_8() if#(true(),x,y) -> c_9(minus#(x,y)) if1#(false(),x,y) -> c_10(gcd#(y,x)) if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0())) if2#(false(),x,y) -> c_12() if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y),minus#(x,y)) minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)) - Weak TRS: gcd(x,y) -> if1(ge(x,y),x,y) ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) if1(false(),x,y) -> gcd(y,x) if1(true(),x,y) -> if2(gt(y,0()),x,y) if2(false(),x,y) -> x if2(true(),x,y) -> gcd(minus(x,y),y) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,gcd#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2} / {0/0,false/0 ,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/2,c_12/0,c_13/2,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,ge#,gt#,if#,if1#,if2#,minus#} and constructors {0 ,false,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,3,5,6,8,12} by application of Pre({2,3,5,6,8,12}) = {1,4,7,11,14}. Here rules are labelled as follows: 1: gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)) 2: ge#(x,0()) -> c_2() 3: ge#(0(),s(x)) -> c_3() 4: ge#(s(x),s(y)) -> c_4(ge#(x,y)) 5: gt#(0(),y) -> c_5() 6: gt#(s(x),0()) -> c_6() 7: gt#(s(x),s(y)) -> c_7(gt#(x,y)) 8: if#(false(),x,y) -> c_8() 9: if#(true(),x,y) -> c_9(minus#(x,y)) 10: if1#(false(),x,y) -> c_10(gcd#(y,x)) 11: if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0())) 12: if2#(false(),x,y) -> c_12() 13: if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y),minus#(x,y)) 14: minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)) ** Step 1.b:3: RemoveWeakSuffixes. MAYBE + Considered Problem: - Strict DPs: gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)) ge#(s(x),s(y)) -> c_4(ge#(x,y)) gt#(s(x),s(y)) -> c_7(gt#(x,y)) if#(true(),x,y) -> c_9(minus#(x,y)) if1#(false(),x,y) -> c_10(gcd#(y,x)) if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0())) if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y),minus#(x,y)) minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)) - Weak DPs: ge#(x,0()) -> c_2() ge#(0(),s(x)) -> c_3() gt#(0(),y) -> c_5() gt#(s(x),0()) -> c_6() if#(false(),x,y) -> c_8() if2#(false(),x,y) -> c_12() - Weak TRS: gcd(x,y) -> if1(ge(x,y),x,y) ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) if1(false(),x,y) -> gcd(y,x) if1(true(),x,y) -> if2(gt(y,0()),x,y) if2(false(),x,y) -> x if2(true(),x,y) -> gcd(minus(x,y),y) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,gcd#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2} / {0/0,false/0 ,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/2,c_12/0,c_13/2,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,ge#,gt#,if#,if1#,if2#,minus#} and constructors {0 ,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)) -->_1 if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0())):6 -->_1 if1#(false(),x,y) -> c_10(gcd#(y,x)):5 -->_2 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2 -->_2 ge#(0(),s(x)) -> c_3():10 -->_2 ge#(x,0()) -> c_2():9 2:S:ge#(s(x),s(y)) -> c_4(ge#(x,y)) -->_1 ge#(0(),s(x)) -> c_3():10 -->_1 ge#(x,0()) -> c_2():9 -->_1 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2 3:S:gt#(s(x),s(y)) -> c_7(gt#(x,y)) -->_1 gt#(s(x),0()) -> c_6():12 -->_1 gt#(0(),y) -> c_5():11 -->_1 gt#(s(x),s(y)) -> c_7(gt#(x,y)):3 4:S:if#(true(),x,y) -> c_9(minus#(x,y)) -->_1 minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)):8 5:S:if1#(false(),x,y) -> c_10(gcd#(y,x)) -->_1 gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)):1 6:S:if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0())) -->_1 if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y),minus#(x,y)):7 -->_1 if2#(false(),x,y) -> c_12():14 -->_2 gt#(s(x),0()) -> c_6():12 -->_2 gt#(0(),y) -> c_5():11 7:S:if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y),minus#(x,y)) -->_2 minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)):8 -->_1 gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)):1 8:S:minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)) -->_1 if#(false(),x,y) -> c_8():13 -->_2 gt#(s(x),0()) -> c_6():12 -->_1 if#(true(),x,y) -> c_9(minus#(x,y)):4 -->_2 gt#(s(x),s(y)) -> c_7(gt#(x,y)):3 9:W:ge#(x,0()) -> c_2() 10:W:ge#(0(),s(x)) -> c_3() 11:W:gt#(0(),y) -> c_5() 12:W:gt#(s(x),0()) -> c_6() 13:W:if#(false(),x,y) -> c_8() 14:W:if2#(false(),x,y) -> c_12() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: ge#(x,0()) -> c_2() 10: ge#(0(),s(x)) -> c_3() 14: if2#(false(),x,y) -> c_12() 11: gt#(0(),y) -> c_5() 12: gt#(s(x),0()) -> c_6() 13: if#(false(),x,y) -> c_8() ** Step 1.b:4: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)) ge#(s(x),s(y)) -> c_4(ge#(x,y)) gt#(s(x),s(y)) -> c_7(gt#(x,y)) if#(true(),x,y) -> c_9(minus#(x,y)) if1#(false(),x,y) -> c_10(gcd#(y,x)) if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0())) if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y),minus#(x,y)) minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)) - Weak TRS: gcd(x,y) -> if1(ge(x,y),x,y) ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) if1(false(),x,y) -> gcd(y,x) if1(true(),x,y) -> if2(gt(y,0()),x,y) if2(false(),x,y) -> x if2(true(),x,y) -> gcd(minus(x,y),y) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,gcd#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2} / {0/0,false/0 ,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/2,c_12/0,c_13/2,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,ge#,gt#,if#,if1#,if2#,minus#} and constructors {0 ,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)) -->_1 if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0())):6 -->_1 if1#(false(),x,y) -> c_10(gcd#(y,x)):5 -->_2 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2 2:S:ge#(s(x),s(y)) -> c_4(ge#(x,y)) -->_1 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2 3:S:gt#(s(x),s(y)) -> c_7(gt#(x,y)) -->_1 gt#(s(x),s(y)) -> c_7(gt#(x,y)):3 4:S:if#(true(),x,y) -> c_9(minus#(x,y)) -->_1 minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)):8 5:S:if1#(false(),x,y) -> c_10(gcd#(y,x)) -->_1 gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)):1 6:S:if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y),gt#(y,0())) -->_1 if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y),minus#(x,y)):7 7:S:if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y),minus#(x,y)) -->_2 minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)):8 -->_1 gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)):1 8:S:minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)) -->_1 if#(true(),x,y) -> c_9(minus#(x,y)):4 -->_2 gt#(s(x),s(y)) -> c_7(gt#(x,y)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y)) ** Step 1.b:5: UsableRules. MAYBE + Considered Problem: - Strict DPs: gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)) ge#(s(x),s(y)) -> c_4(ge#(x,y)) gt#(s(x),s(y)) -> c_7(gt#(x,y)) if#(true(),x,y) -> c_9(minus#(x,y)) if1#(false(),x,y) -> c_10(gcd#(y,x)) if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y)) if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y),minus#(x,y)) minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)) - Weak TRS: gcd(x,y) -> if1(ge(x,y),x,y) ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) if1(false(),x,y) -> gcd(y,x) if1(true(),x,y) -> if2(gt(y,0()),x,y) if2(false(),x,y) -> x if2(true(),x,y) -> gcd(minus(x,y),y) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,gcd#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2} / {0/0,false/0 ,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/2,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,ge#,gt#,if#,if1#,if2#,minus#} and constructors {0 ,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) minus(s(x),y) -> if(gt(s(x),y),x,y) gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)) ge#(s(x),s(y)) -> c_4(ge#(x,y)) gt#(s(x),s(y)) -> c_7(gt#(x,y)) if#(true(),x,y) -> c_9(minus#(x,y)) if1#(false(),x,y) -> c_10(gcd#(y,x)) if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y)) if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y),minus#(x,y)) minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)) ** Step 1.b:6: DecomposeDG. MAYBE + Considered Problem: - Strict DPs: gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)) ge#(s(x),s(y)) -> c_4(ge#(x,y)) gt#(s(x),s(y)) -> c_7(gt#(x,y)) if#(true(),x,y) -> c_9(minus#(x,y)) if1#(false(),x,y) -> c_10(gcd#(y,x)) if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y)) if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y),minus#(x,y)) minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,gcd#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2} / {0/0,false/0 ,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/2,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,ge#,gt#,if#,if1#,if2#,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 gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)) if1#(false(),x,y) -> c_10(gcd#(y,x)) if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y)) if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y),minus#(x,y)) and a lower component ge#(s(x),s(y)) -> c_4(ge#(x,y)) gt#(s(x),s(y)) -> c_7(gt#(x,y)) if#(true(),x,y) -> c_9(minus#(x,y)) minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)) Further, following extension rules are added to the lower component. gcd#(x,y) -> ge#(x,y) gcd#(x,y) -> if1#(ge(x,y),x,y) if1#(false(),x,y) -> gcd#(y,x) if1#(true(),x,y) -> if2#(gt(y,0()),x,y) if2#(true(),x,y) -> gcd#(minus(x,y),y) if2#(true(),x,y) -> minus#(x,y) *** Step 1.b:6.a:1: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)) if1#(false(),x,y) -> c_10(gcd#(y,x)) if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y)) if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y),minus#(x,y)) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,gcd#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2} / {0/0,false/0 ,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/2,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,ge#,gt#,if#,if1#,if2#,minus#} and constructors {0 ,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)) -->_1 if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y)):3 -->_1 if1#(false(),x,y) -> c_10(gcd#(y,x)):2 2:S:if1#(false(),x,y) -> c_10(gcd#(y,x)) -->_1 gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)):1 3:S:if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y)) -->_1 if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y),minus#(x,y)):4 4:S:if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y),minus#(x,y)) -->_1 gcd#(x,y) -> c_1(if1#(ge(x,y),x,y),ge#(x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: gcd#(x,y) -> c_1(if1#(ge(x,y),x,y)) if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y)) *** Step 1.b:6.a:2: WeightGap. MAYBE + Considered Problem: - Strict DPs: gcd#(x,y) -> c_1(if1#(ge(x,y),x,y)) if1#(false(),x,y) -> c_10(gcd#(y,x)) if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y)) if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y)) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,gcd#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/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_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,ge#,gt#,if#,if1#,if2#,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(if) = {1}, uargs(s) = {1}, uargs(gcd#) = {1}, uargs(if1#) = {1}, uargs(if2#) = {1}, uargs(c_1) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_13) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [2] p(gcd) = [1] x1 + [0] p(ge) = [3] p(gt) = [2] p(if) = [1] x1 + [0] p(if1) = [1] x1 + [2] x2 + [1] x3 + [2] p(if2) = [1] x2 + [1] p(minus) = [2] p(s) = [1] x1 + [0] p(true) = [2] p(gcd#) = [1] x1 + [1] x2 + [2] p(ge#) = [1] x1 + [2] x2 + [2] p(gt#) = [1] x2 + [0] p(if#) = [1] x2 + [0] p(if1#) = [1] x1 + [1] x2 + [1] x3 + [5] p(if2#) = [1] x1 + [1] x3 + [4] p(minus#) = [4] x1 + [1] x2 + [0] p(c_1) = [1] x1 + [6] p(c_2) = [1] p(c_3) = [4] p(c_4) = [4] p(c_5) = [0] p(c_6) = [2] p(c_7) = [1] p(c_8) = [0] p(c_9) = [1] x1 + [1] p(c_10) = [1] x1 + [0] p(c_11) = [1] x1 + [0] p(c_12) = [0] p(c_13) = [1] x1 + [6] p(c_14) = [4] x2 + [1] Following rules are strictly oriented: if1#(false(),x,y) = [1] x + [1] y + [7] > [1] x + [1] y + [2] = c_10(gcd#(y,x)) if1#(true(),x,y) = [1] x + [1] y + [7] > [1] y + [6] = c_11(if2#(gt(y,0()),x,y)) Following rules are (at-least) weakly oriented: gcd#(x,y) = [1] x + [1] y + [2] >= [1] x + [1] y + [14] = c_1(if1#(ge(x,y),x,y)) if2#(true(),x,y) = [1] y + [6] >= [1] y + [10] = c_13(gcd#(minus(x,y),y)) ge(x,0()) = [3] >= [2] = true() ge(0(),s(x)) = [3] >= [2] = false() ge(s(x),s(y)) = [3] >= [3] = ge(x,y) gt(0(),y) = [2] >= [2] = false() gt(s(x),0()) = [2] >= [2] = true() gt(s(x),s(y)) = [2] >= [2] = gt(x,y) if(false(),x,y) = [2] >= [0] = 0() if(true(),x,y) = [2] >= [2] = s(minus(x,y)) minus(s(x),y) = [2] >= [2] = if(gt(s(x),y),x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:6.a:3: WeightGap. MAYBE + Considered Problem: - Strict DPs: gcd#(x,y) -> c_1(if1#(ge(x,y),x,y)) if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y)) - Weak DPs: if1#(false(),x,y) -> c_10(gcd#(y,x)) if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y)) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,gcd#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/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_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,ge#,gt#,if#,if1#,if2#,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(if) = {1}, uargs(s) = {1}, uargs(gcd#) = {1}, uargs(if1#) = {1}, uargs(if2#) = {1}, uargs(c_1) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_13) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(gcd) = [0] p(ge) = [3] p(gt) = [0] p(if) = [1] x1 + [0] p(if1) = [0] p(if2) = [0] p(minus) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(gcd#) = [1] x1 + [1] x2 + [0] p(ge#) = [0] p(gt#) = [2] x1 + [0] p(if#) = [1] x3 + [0] p(if1#) = [1] x1 + [1] x2 + [1] x3 + [1] p(if2#) = [1] x1 + [1] x2 + [1] x3 + [1] p(minus#) = [0] p(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [2] p(c_5) = [0] p(c_6) = [1] p(c_7) = [1] x1 + [0] p(c_8) = [2] p(c_9) = [1] x1 + [1] p(c_10) = [1] x1 + [1] p(c_11) = [1] x1 + [0] p(c_12) = [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] x2 + [1] Following rules are strictly oriented: if2#(true(),x,y) = [1] x + [1] y + [1] > [1] y + [0] = c_13(gcd#(minus(x,y),y)) Following rules are (at-least) weakly oriented: gcd#(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [4] = c_1(if1#(ge(x,y),x,y)) if1#(false(),x,y) = [1] x + [1] y + [1] >= [1] x + [1] y + [1] = c_10(gcd#(y,x)) if1#(true(),x,y) = [1] x + [1] y + [1] >= [1] x + [1] y + [1] = c_11(if2#(gt(y,0()),x,y)) ge(x,0()) = [3] >= [0] = true() ge(0(),s(x)) = [3] >= [0] = false() ge(s(x),s(y)) = [3] >= [3] = ge(x,y) gt(0(),y) = [0] >= [0] = false() gt(s(x),0()) = [0] >= [0] = true() gt(s(x),s(y)) = [0] >= [0] = gt(x,y) if(false(),x,y) = [0] >= [0] = 0() if(true(),x,y) = [0] >= [0] = s(minus(x,y)) minus(s(x),y) = [0] >= [0] = if(gt(s(x),y),x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:6.a:4: Failure MAYBE + Considered Problem: - Strict DPs: gcd#(x,y) -> c_1(if1#(ge(x,y),x,y)) - Weak DPs: if1#(false(),x,y) -> c_10(gcd#(y,x)) if1#(true(),x,y) -> c_11(if2#(gt(y,0()),x,y)) if2#(true(),x,y) -> c_13(gcd#(minus(x,y),y)) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,gcd#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/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_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,ge#,gt#,if#,if1#,if2#,minus#} and constructors {0 ,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. *** Step 1.b:6.b:1: DecomposeDG. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ge#(s(x),s(y)) -> c_4(ge#(x,y)) gt#(s(x),s(y)) -> c_7(gt#(x,y)) if#(true(),x,y) -> c_9(minus#(x,y)) minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)) - Weak DPs: gcd#(x,y) -> ge#(x,y) gcd#(x,y) -> if1#(ge(x,y),x,y) if1#(false(),x,y) -> gcd#(y,x) if1#(true(),x,y) -> if2#(gt(y,0()),x,y) if2#(true(),x,y) -> gcd#(minus(x,y),y) if2#(true(),x,y) -> minus#(x,y) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,gcd#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2} / {0/0,false/0 ,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/2,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,ge#,gt#,if#,if1#,if2#,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 gcd#(x,y) -> ge#(x,y) gcd#(x,y) -> if1#(ge(x,y),x,y) ge#(s(x),s(y)) -> c_4(ge#(x,y)) if#(true(),x,y) -> c_9(minus#(x,y)) if1#(false(),x,y) -> gcd#(y,x) if1#(true(),x,y) -> if2#(gt(y,0()),x,y) if2#(true(),x,y) -> gcd#(minus(x,y),y) if2#(true(),x,y) -> minus#(x,y) minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)) and a lower component gt#(s(x),s(y)) -> c_7(gt#(x,y)) Further, following extension rules are added to the lower component. gcd#(x,y) -> ge#(x,y) gcd#(x,y) -> if1#(ge(x,y),x,y) ge#(s(x),s(y)) -> ge#(x,y) if#(true(),x,y) -> minus#(x,y) if1#(false(),x,y) -> gcd#(y,x) if1#(true(),x,y) -> if2#(gt(y,0()),x,y) if2#(true(),x,y) -> gcd#(minus(x,y),y) if2#(true(),x,y) -> minus#(x,y) minus#(s(x),y) -> gt#(s(x),y) minus#(s(x),y) -> if#(gt(s(x),y),x,y) **** Step 1.b:6.b:1.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ge#(s(x),s(y)) -> c_4(ge#(x,y)) if#(true(),x,y) -> c_9(minus#(x,y)) minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)) - Weak DPs: gcd#(x,y) -> ge#(x,y) gcd#(x,y) -> if1#(ge(x,y),x,y) if1#(false(),x,y) -> gcd#(y,x) if1#(true(),x,y) -> if2#(gt(y,0()),x,y) if2#(true(),x,y) -> gcd#(minus(x,y),y) if2#(true(),x,y) -> minus#(x,y) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,gcd#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2} / {0/0,false/0 ,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/2,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,ge#,gt#,if#,if1#,if2#,minus#} and constructors {0 ,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:ge#(s(x),s(y)) -> c_4(ge#(x,y)) -->_1 ge#(s(x),s(y)) -> c_4(ge#(x,y)):1 2:S:if#(true(),x,y) -> c_9(minus#(x,y)) -->_1 minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)):3 3:S:minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)) -->_1 if#(true(),x,y) -> c_9(minus#(x,y)):2 4:W:gcd#(x,y) -> ge#(x,y) -->_1 ge#(s(x),s(y)) -> c_4(ge#(x,y)):1 5:W:gcd#(x,y) -> if1#(ge(x,y),x,y) -->_1 if1#(true(),x,y) -> if2#(gt(y,0()),x,y):7 -->_1 if1#(false(),x,y) -> gcd#(y,x):6 6:W:if1#(false(),x,y) -> gcd#(y,x) -->_1 gcd#(x,y) -> if1#(ge(x,y),x,y):5 -->_1 gcd#(x,y) -> ge#(x,y):4 7:W:if1#(true(),x,y) -> if2#(gt(y,0()),x,y) -->_1 if2#(true(),x,y) -> minus#(x,y):9 -->_1 if2#(true(),x,y) -> gcd#(minus(x,y),y):8 8:W:if2#(true(),x,y) -> gcd#(minus(x,y),y) -->_1 gcd#(x,y) -> if1#(ge(x,y),x,y):5 -->_1 gcd#(x,y) -> ge#(x,y):4 9:W:if2#(true(),x,y) -> minus#(x,y) -->_1 minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y),gt#(s(x),y)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y)) **** Step 1.b:6.b:1.a:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ge#(s(x),s(y)) -> c_4(ge#(x,y)) if#(true(),x,y) -> c_9(minus#(x,y)) minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y)) - Weak DPs: gcd#(x,y) -> ge#(x,y) gcd#(x,y) -> if1#(ge(x,y),x,y) if1#(false(),x,y) -> gcd#(y,x) if1#(true(),x,y) -> if2#(gt(y,0()),x,y) if2#(true(),x,y) -> gcd#(minus(x,y),y) if2#(true(),x,y) -> minus#(x,y) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,gcd#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2} / {0/0,false/0 ,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/2,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,ge#,gt#,if#,if1#,if2#,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_4) = {1}, uargs(c_9) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {ge,if,minus,gcd#,ge#,gt#,if#,if1#,if2#,minus#} TcT has computed the following interpretation: p(0) = [8] p(false) = [4] p(gcd) = [2] x1 + [8] p(ge) = [4] p(gt) = [0] p(if) = [1] x2 + [14] p(if1) = [2] x1 + [2] x2 + [1] p(if2) = [1] x1 + [1] x2 + [1] x3 + [0] p(minus) = [1] x1 + [0] p(s) = [1] x1 + [14] p(true) = [4] p(gcd#) = [10] x1 + [10] x2 + [12] p(ge#) = [1] x1 + [0] p(gt#) = [1] x1 + [4] x2 + [0] p(if#) = [0] p(if1#) = [1] x1 + [10] x2 + [10] x3 + [8] p(if2#) = [10] x2 + [10] x3 + [12] p(minus#) = [0] p(c_1) = [8] x2 + [0] p(c_2) = [1] p(c_3) = [0] p(c_4) = [1] x1 + [4] p(c_5) = [1] p(c_6) = [1] p(c_7) = [1] x1 + [2] p(c_8) = [0] p(c_9) = [8] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [1] p(c_12) = [0] p(c_13) = [2] x1 + [1] x2 + [0] p(c_14) = [8] x1 + [0] Following rules are strictly oriented: ge#(s(x),s(y)) = [1] x + [14] > [1] x + [4] = c_4(ge#(x,y)) Following rules are (at-least) weakly oriented: gcd#(x,y) = [10] x + [10] y + [12] >= [1] x + [0] = ge#(x,y) gcd#(x,y) = [10] x + [10] y + [12] >= [10] x + [10] y + [12] = if1#(ge(x,y),x,y) if#(true(),x,y) = [0] >= [0] = c_9(minus#(x,y)) if1#(false(),x,y) = [10] x + [10] y + [12] >= [10] x + [10] y + [12] = gcd#(y,x) if1#(true(),x,y) = [10] x + [10] y + [12] >= [10] x + [10] y + [12] = if2#(gt(y,0()),x,y) if2#(true(),x,y) = [10] x + [10] y + [12] >= [10] x + [10] y + [12] = gcd#(minus(x,y),y) if2#(true(),x,y) = [10] x + [10] y + [12] >= [0] = minus#(x,y) minus#(s(x),y) = [0] >= [0] = c_14(if#(gt(s(x),y),x,y)) ge(x,0()) = [4] >= [4] = true() ge(0(),s(x)) = [4] >= [4] = false() ge(s(x),s(y)) = [4] >= [4] = ge(x,y) if(false(),x,y) = [1] x + [14] >= [8] = 0() if(true(),x,y) = [1] x + [14] >= [1] x + [14] = s(minus(x,y)) minus(s(x),y) = [1] x + [14] >= [1] x + [14] = if(gt(s(x),y),x,y) **** Step 1.b:6.b:1.a:3: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if#(true(),x,y) -> c_9(minus#(x,y)) minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y)) - Weak DPs: gcd#(x,y) -> ge#(x,y) gcd#(x,y) -> if1#(ge(x,y),x,y) ge#(s(x),s(y)) -> c_4(ge#(x,y)) if1#(false(),x,y) -> gcd#(y,x) if1#(true(),x,y) -> if2#(gt(y,0()),x,y) if2#(true(),x,y) -> gcd#(minus(x,y),y) if2#(true(),x,y) -> minus#(x,y) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,gcd#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2} / {0/0,false/0 ,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/2,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,ge#,gt#,if#,if1#,if2#,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(if) = {1}, uargs(s) = {1}, uargs(gcd#) = {1}, uargs(if#) = {1}, uargs(if1#) = {1}, uargs(if2#) = {1}, uargs(c_4) = {1}, uargs(c_9) = {1}, uargs(c_14) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(gcd) = [0] p(ge) = [0] p(gt) = [0] p(if) = [1] x1 + [0] p(if1) = [0] p(if2) = [0] p(minus) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(gcd#) = [1] x1 + [1] x2 + [4] p(ge#) = [1] x1 + [1] x2 + [4] p(gt#) = [0] p(if#) = [1] x1 + [0] p(if1#) = [1] x1 + [1] x2 + [1] x3 + [4] p(if2#) = [1] x1 + [1] x3 + [4] p(minus#) = [4] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [1] x1 + [0] Following rules are strictly oriented: minus#(s(x),y) = [4] > [0] = c_14(if#(gt(s(x),y),x,y)) Following rules are (at-least) weakly oriented: gcd#(x,y) = [1] x + [1] y + [4] >= [1] x + [1] y + [4] = ge#(x,y) gcd#(x,y) = [1] x + [1] y + [4] >= [1] x + [1] y + [4] = if1#(ge(x,y),x,y) ge#(s(x),s(y)) = [1] x + [1] y + [4] >= [1] x + [1] y + [4] = c_4(ge#(x,y)) if#(true(),x,y) = [0] >= [4] = c_9(minus#(x,y)) if1#(false(),x,y) = [1] x + [1] y + [4] >= [1] x + [1] y + [4] = gcd#(y,x) if1#(true(),x,y) = [1] x + [1] y + [4] >= [1] y + [4] = if2#(gt(y,0()),x,y) if2#(true(),x,y) = [1] y + [4] >= [1] y + [4] = gcd#(minus(x,y),y) if2#(true(),x,y) = [1] y + [4] >= [4] = minus#(x,y) ge(x,0()) = [0] >= [0] = true() ge(0(),s(x)) = [0] >= [0] = false() ge(s(x),s(y)) = [0] >= [0] = ge(x,y) gt(0(),y) = [0] >= [0] = false() gt(s(x),0()) = [0] >= [0] = true() gt(s(x),s(y)) = [0] >= [0] = gt(x,y) if(false(),x,y) = [0] >= [0] = 0() if(true(),x,y) = [0] >= [0] = s(minus(x,y)) minus(s(x),y) = [0] >= [0] = if(gt(s(x),y),x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. **** Step 1.b:6.b:1.a:4: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if#(true(),x,y) -> c_9(minus#(x,y)) - Weak DPs: gcd#(x,y) -> ge#(x,y) gcd#(x,y) -> if1#(ge(x,y),x,y) ge#(s(x),s(y)) -> c_4(ge#(x,y)) if1#(false(),x,y) -> gcd#(y,x) if1#(true(),x,y) -> if2#(gt(y,0()),x,y) if2#(true(),x,y) -> gcd#(minus(x,y),y) if2#(true(),x,y) -> minus#(x,y) minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y)) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,gcd#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2} / {0/0,false/0 ,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/2,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,ge#,gt#,if#,if1#,if2#,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(if) = {1}, uargs(s) = {1}, uargs(gcd#) = {1}, uargs(if#) = {1}, uargs(if1#) = {1}, uargs(if2#) = {1}, uargs(c_4) = {1}, uargs(c_9) = {1}, uargs(c_14) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(gcd) = [2] x1 + [0] p(ge) = [0] p(gt) = [0] p(if) = [1] x1 + [1] x2 + [1] p(if1) = [4] x2 + [1] x3 + [1] p(if2) = [1] x1 + [1] x2 + [1] p(minus) = [1] x1 + [0] p(s) = [1] x1 + [1] p(true) = [0] p(gcd#) = [1] x1 + [1] x2 + [0] p(ge#) = [1] x1 + [1] x2 + [0] p(gt#) = [1] x1 + [1] x2 + [1] p(if#) = [1] x1 + [1] x2 + [1] p(if1#) = [1] x1 + [1] x2 + [1] x3 + [0] p(if2#) = [1] x1 + [1] x2 + [1] x3 + [0] p(minus#) = [1] x1 + [0] p(c_1) = [1] x2 + [4] p(c_2) = [1] p(c_3) = [2] p(c_4) = [1] x1 + [0] p(c_5) = [1] p(c_6) = [1] p(c_7) = [4] x1 + [1] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [1] p(c_11) = [1] p(c_12) = [4] p(c_13) = [2] x2 + [2] p(c_14) = [1] x1 + [0] Following rules are strictly oriented: if#(true(),x,y) = [1] x + [1] > [1] x + [0] = c_9(minus#(x,y)) Following rules are (at-least) weakly oriented: gcd#(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = ge#(x,y) gcd#(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = if1#(ge(x,y),x,y) ge#(s(x),s(y)) = [1] x + [1] y + [2] >= [1] x + [1] y + [0] = c_4(ge#(x,y)) if1#(false(),x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = gcd#(y,x) if1#(true(),x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = if2#(gt(y,0()),x,y) if2#(true(),x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = gcd#(minus(x,y),y) if2#(true(),x,y) = [1] x + [1] y + [0] >= [1] x + [0] = minus#(x,y) minus#(s(x),y) = [1] x + [1] >= [1] x + [1] = c_14(if#(gt(s(x),y),x,y)) ge(x,0()) = [0] >= [0] = true() ge(0(),s(x)) = [0] >= [0] = false() ge(s(x),s(y)) = [0] >= [0] = ge(x,y) gt(0(),y) = [0] >= [0] = false() gt(s(x),0()) = [0] >= [0] = true() gt(s(x),s(y)) = [0] >= [0] = gt(x,y) if(false(),x,y) = [1] x + [1] >= [0] = 0() if(true(),x,y) = [1] x + [1] >= [1] x + [1] = s(minus(x,y)) minus(s(x),y) = [1] x + [1] >= [1] x + [1] = if(gt(s(x),y),x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. **** Step 1.b:6.b:1.a:5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: gcd#(x,y) -> ge#(x,y) gcd#(x,y) -> if1#(ge(x,y),x,y) ge#(s(x),s(y)) -> c_4(ge#(x,y)) if#(true(),x,y) -> c_9(minus#(x,y)) if1#(false(),x,y) -> gcd#(y,x) if1#(true(),x,y) -> if2#(gt(y,0()),x,y) if2#(true(),x,y) -> gcd#(minus(x,y),y) if2#(true(),x,y) -> minus#(x,y) minus#(s(x),y) -> c_14(if#(gt(s(x),y),x,y)) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,gcd#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2} / {0/0,false/0 ,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/2,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,ge#,gt#,if#,if1#,if2#,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.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: gt#(s(x),s(y)) -> c_7(gt#(x,y)) - Weak DPs: gcd#(x,y) -> ge#(x,y) gcd#(x,y) -> if1#(ge(x,y),x,y) ge#(s(x),s(y)) -> ge#(x,y) if#(true(),x,y) -> minus#(x,y) if1#(false(),x,y) -> gcd#(y,x) if1#(true(),x,y) -> if2#(gt(y,0()),x,y) if2#(true(),x,y) -> gcd#(minus(x,y),y) if2#(true(),x,y) -> minus#(x,y) minus#(s(x),y) -> gt#(s(x),y) minus#(s(x),y) -> if#(gt(s(x),y),x,y) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,gcd#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2} / {0/0,false/0 ,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/2,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,ge#,gt#,if#,if1#,if2#,minus#} and constructors {0 ,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:gt#(s(x),s(y)) -> c_7(gt#(x,y)) -->_1 gt#(s(x),s(y)) -> c_7(gt#(x,y)):1 2:W:gcd#(x,y) -> ge#(x,y) -->_1 ge#(s(x),s(y)) -> ge#(x,y):4 3:W:gcd#(x,y) -> if1#(ge(x,y),x,y) -->_1 if1#(true(),x,y) -> if2#(gt(y,0()),x,y):7 -->_1 if1#(false(),x,y) -> gcd#(y,x):6 4:W:ge#(s(x),s(y)) -> ge#(x,y) -->_1 ge#(s(x),s(y)) -> ge#(x,y):4 5:W:if#(true(),x,y) -> minus#(x,y) -->_1 minus#(s(x),y) -> if#(gt(s(x),y),x,y):11 -->_1 minus#(s(x),y) -> gt#(s(x),y):10 6:W:if1#(false(),x,y) -> gcd#(y,x) -->_1 gcd#(x,y) -> if1#(ge(x,y),x,y):3 -->_1 gcd#(x,y) -> ge#(x,y):2 7:W:if1#(true(),x,y) -> if2#(gt(y,0()),x,y) -->_1 if2#(true(),x,y) -> minus#(x,y):9 -->_1 if2#(true(),x,y) -> gcd#(minus(x,y),y):8 8:W:if2#(true(),x,y) -> gcd#(minus(x,y),y) -->_1 gcd#(x,y) -> if1#(ge(x,y),x,y):3 -->_1 gcd#(x,y) -> ge#(x,y):2 9:W:if2#(true(),x,y) -> minus#(x,y) -->_1 minus#(s(x),y) -> if#(gt(s(x),y),x,y):11 -->_1 minus#(s(x),y) -> gt#(s(x),y):10 10:W:minus#(s(x),y) -> gt#(s(x),y) -->_1 gt#(s(x),s(y)) -> c_7(gt#(x,y)):1 11:W:minus#(s(x),y) -> if#(gt(s(x),y),x,y) -->_1 if#(true(),x,y) -> minus#(x,y):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: gcd#(x,y) -> ge#(x,y) 4: ge#(s(x),s(y)) -> ge#(x,y) **** Step 1.b:6.b:1.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: gt#(s(x),s(y)) -> c_7(gt#(x,y)) - Weak DPs: gcd#(x,y) -> if1#(ge(x,y),x,y) if#(true(),x,y) -> minus#(x,y) if1#(false(),x,y) -> gcd#(y,x) if1#(true(),x,y) -> if2#(gt(y,0()),x,y) if2#(true(),x,y) -> gcd#(minus(x,y),y) if2#(true(),x,y) -> minus#(x,y) minus#(s(x),y) -> gt#(s(x),y) minus#(s(x),y) -> if#(gt(s(x),y),x,y) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,gcd#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2} / {0/0,false/0 ,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/2,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,ge#,gt#,if#,if1#,if2#,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_7) = {1} Following symbols are considered usable: {if,minus,gcd#,ge#,gt#,if#,if1#,if2#,minus#} TcT has computed the following interpretation: p(0) = [2] p(false) = [2] p(gcd) = [1] p(ge) = [8] x1 + [8] p(gt) = [0] p(if) = [1] x2 + [2] p(if1) = [1] x1 + [1] x3 + [8] p(if2) = [1] x1 + [1] x2 + [1] x3 + [0] p(minus) = [1] x1 + [0] p(s) = [1] x1 + [2] p(true) = [0] p(gcd#) = [8] x1 + [8] x2 + [0] p(ge#) = [2] x1 + [1] x2 + [0] p(gt#) = [8] x1 + [0] p(if#) = [8] x2 + [5] p(if1#) = [8] x2 + [8] x3 + [0] p(if2#) = [8] x2 + [8] x3 + [0] p(minus#) = [8] x1 + [0] p(c_1) = [2] x1 + [1] x2 + [4] p(c_2) = [0] p(c_3) = [4] p(c_4) = [1] p(c_5) = [1] p(c_6) = [0] p(c_7) = [1] x1 + [8] p(c_8) = [1] p(c_9) = [4] x1 + [1] p(c_10) = [2] p(c_11) = [0] p(c_12) = [1] p(c_13) = [1] x1 + [0] p(c_14) = [1] x2 + [1] Following rules are strictly oriented: gt#(s(x),s(y)) = [8] x + [16] > [8] x + [8] = c_7(gt#(x,y)) Following rules are (at-least) weakly oriented: gcd#(x,y) = [8] x + [8] y + [0] >= [8] x + [8] y + [0] = if1#(ge(x,y),x,y) if#(true(),x,y) = [8] x + [5] >= [8] x + [0] = minus#(x,y) if1#(false(),x,y) = [8] x + [8] y + [0] >= [8] x + [8] y + [0] = gcd#(y,x) if1#(true(),x,y) = [8] x + [8] y + [0] >= [8] x + [8] y + [0] = if2#(gt(y,0()),x,y) if2#(true(),x,y) = [8] x + [8] y + [0] >= [8] x + [8] y + [0] = gcd#(minus(x,y),y) if2#(true(),x,y) = [8] x + [8] y + [0] >= [8] x + [0] = minus#(x,y) minus#(s(x),y) = [8] x + [16] >= [8] x + [16] = gt#(s(x),y) minus#(s(x),y) = [8] x + [16] >= [8] x + [5] = if#(gt(s(x),y),x,y) if(false(),x,y) = [1] x + [2] >= [2] = 0() if(true(),x,y) = [1] x + [2] >= [1] x + [2] = s(minus(x,y)) minus(s(x),y) = [1] x + [2] >= [1] x + [2] = if(gt(s(x),y),x,y) **** Step 1.b:6.b:1.b:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: gcd#(x,y) -> if1#(ge(x,y),x,y) gt#(s(x),s(y)) -> c_7(gt#(x,y)) if#(true(),x,y) -> minus#(x,y) if1#(false(),x,y) -> gcd#(y,x) if1#(true(),x,y) -> if2#(gt(y,0()),x,y) if2#(true(),x,y) -> gcd#(minus(x,y),y) if2#(true(),x,y) -> minus#(x,y) minus#(s(x),y) -> gt#(s(x),y) minus#(s(x),y) -> if#(gt(s(x),y),x,y) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) gt(0(),y) -> false() gt(s(x),0()) -> true() gt(s(x),s(y)) -> gt(x,y) if(false(),x,y) -> 0() if(true(),x,y) -> s(minus(x,y)) minus(s(x),y) -> if(gt(s(x),y),x,y) - Signature: {gcd/2,ge/2,gt/2,if/3,if1/3,if2/3,minus/2,gcd#/2,ge#/2,gt#/2,if#/3,if1#/3,if2#/3,minus#/2} / {0/0,false/0 ,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/2,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,ge#,gt#,if#,if1#,if2#,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),?)