WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: DIVP(z0,z1) -> c7() PRIME(0()) -> c() PRIME(s(0())) -> c1() PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)),s(z0))) PRIME1(z0,0()) -> c3() PRIME1(z0,s(0())) -> c4() PRIME1(z0,s(s(z1))) -> c5(DIVP(s(s(z1)),z0)) PRIME1(z0,s(s(z1))) -> c6(PRIME1(z0,s(z1))) - Weak TRS: divp(z0,z1) -> =(rem(z0,z1),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(z0))) -> prime1(s(s(z0)),s(z0)) prime1(z0,0()) -> false() prime1(z0,s(0())) -> true() prime1(z0,s(s(z1))) -> and(not(divp(s(s(z1)),z0)),prime1(z0,s(z1))) - Signature: {DIVP/2,PRIME/1,PRIME1/2,divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,c/0,c1/0,c2/1,c3/0,c4/0,c5/1,c6/1,c7/0 ,false/0,not/1,rem/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {DIVP,PRIME,PRIME1,divp,prime,prime1} and constructors {0 ,=,and,c,c1,c2,c3,c4,c5,c6,c7,false,not,rem,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: DIVP(z0,z1) -> c7() PRIME(0()) -> c() PRIME(s(0())) -> c1() PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)),s(z0))) PRIME1(z0,0()) -> c3() PRIME1(z0,s(0())) -> c4() PRIME1(z0,s(s(z1))) -> c5(DIVP(s(s(z1)),z0)) PRIME1(z0,s(s(z1))) -> c6(PRIME1(z0,s(z1))) - Weak TRS: divp(z0,z1) -> =(rem(z0,z1),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(z0))) -> prime1(s(s(z0)),s(z0)) prime1(z0,0()) -> false() prime1(z0,s(0())) -> true() prime1(z0,s(s(z1))) -> and(not(divp(s(s(z1)),z0)),prime1(z0,s(z1))) - Signature: {DIVP/2,PRIME/1,PRIME1/2,divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,c/0,c1/0,c2/1,c3/0,c4/0,c5/1,c6/1,c7/0 ,false/0,not/1,rem/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {DIVP,PRIME,PRIME1,divp,prime,prime1} and constructors {0 ,=,and,c,c1,c2,c3,c4,c5,c6,c7,false,not,rem,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: DIVP(z0,z1) -> c7() PRIME(0()) -> c() PRIME(s(0())) -> c1() PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)),s(z0))) PRIME1(z0,0()) -> c3() PRIME1(z0,s(0())) -> c4() PRIME1(z0,s(s(z1))) -> c5(DIVP(s(s(z1)),z0)) PRIME1(z0,s(s(z1))) -> c6(PRIME1(z0,s(z1))) - Weak TRS: divp(z0,z1) -> =(rem(z0,z1),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(z0))) -> prime1(s(s(z0)),s(z0)) prime1(z0,0()) -> false() prime1(z0,s(0())) -> true() prime1(z0,s(s(z1))) -> and(not(divp(s(s(z1)),z0)),prime1(z0,s(z1))) - Signature: {DIVP/2,PRIME/1,PRIME1/2,divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,c/0,c1/0,c2/1,c3/0,c4/0,c5/1,c6/1,c7/0 ,false/0,not/1,rem/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {DIVP,PRIME,PRIME1,divp,prime,prime1} and constructors {0 ,=,and,c,c1,c2,c3,c4,c5,c6,c7,false,not,rem,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: PRIME1(x,s(y)){y -> s(y)} = PRIME1(x,s(s(y))) ->^+ c6(PRIME1(x,s(y))) = C[PRIME1(x,s(y)) = PRIME1(x,s(y)){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: DIVP(z0,z1) -> c7() PRIME(0()) -> c() PRIME(s(0())) -> c1() PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)),s(z0))) PRIME1(z0,0()) -> c3() PRIME1(z0,s(0())) -> c4() PRIME1(z0,s(s(z1))) -> c5(DIVP(s(s(z1)),z0)) PRIME1(z0,s(s(z1))) -> c6(PRIME1(z0,s(z1))) - Weak TRS: divp(z0,z1) -> =(rem(z0,z1),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(z0))) -> prime1(s(s(z0)),s(z0)) prime1(z0,0()) -> false() prime1(z0,s(0())) -> true() prime1(z0,s(s(z1))) -> and(not(divp(s(s(z1)),z0)),prime1(z0,s(z1))) - Signature: {DIVP/2,PRIME/1,PRIME1/2,divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,c/0,c1/0,c2/1,c3/0,c4/0,c5/1,c6/1,c7/0 ,false/0,not/1,rem/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {DIVP,PRIME,PRIME1,divp,prime,prime1} and constructors {0 ,=,and,c,c1,c2,c3,c4,c5,c6,c7,false,not,rem,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs DIVP#(z0,z1) -> c_1() PRIME#(0()) -> c_2() PRIME#(s(0())) -> c_3() PRIME#(s(s(z0))) -> c_4(PRIME1#(s(s(z0)),s(z0))) PRIME1#(z0,0()) -> c_5() PRIME1#(z0,s(0())) -> c_6() PRIME1#(z0,s(s(z1))) -> c_7(DIVP#(s(s(z1)),z0)) PRIME1#(z0,s(s(z1))) -> c_8(PRIME1#(z0,s(z1))) Weak DPs divp#(z0,z1) -> c_9() prime#(0()) -> c_10() prime#(s(0())) -> c_11() prime#(s(s(z0))) -> c_12(prime1#(s(s(z0)),s(z0))) prime1#(z0,0()) -> c_13() prime1#(z0,s(0())) -> c_14() prime1#(z0,s(s(z1))) -> c_15(divp#(s(s(z1)),z0),prime1#(z0,s(z1))) and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DIVP#(z0,z1) -> c_1() PRIME#(0()) -> c_2() PRIME#(s(0())) -> c_3() PRIME#(s(s(z0))) -> c_4(PRIME1#(s(s(z0)),s(z0))) PRIME1#(z0,0()) -> c_5() PRIME1#(z0,s(0())) -> c_6() PRIME1#(z0,s(s(z1))) -> c_7(DIVP#(s(s(z1)),z0)) PRIME1#(z0,s(s(z1))) -> c_8(PRIME1#(z0,s(z1))) - Weak DPs: divp#(z0,z1) -> c_9() prime#(0()) -> c_10() prime#(s(0())) -> c_11() prime#(s(s(z0))) -> c_12(prime1#(s(s(z0)),s(z0))) prime1#(z0,0()) -> c_13() prime1#(z0,s(0())) -> c_14() prime1#(z0,s(s(z1))) -> c_15(divp#(s(s(z1)),z0),prime1#(z0,s(z1))) - Weak TRS: DIVP(z0,z1) -> c7() PRIME(0()) -> c() PRIME(s(0())) -> c1() PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)),s(z0))) PRIME1(z0,0()) -> c3() PRIME1(z0,s(0())) -> c4() PRIME1(z0,s(s(z1))) -> c5(DIVP(s(s(z1)),z0)) PRIME1(z0,s(s(z1))) -> c6(PRIME1(z0,s(z1))) divp(z0,z1) -> =(rem(z0,z1),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(z0))) -> prime1(s(s(z0)),s(z0)) prime1(z0,0()) -> false() prime1(z0,s(0())) -> true() prime1(z0,s(s(z1))) -> and(not(divp(s(s(z1)),z0)),prime1(z0,s(z1))) - Signature: {DIVP/2,PRIME/1,PRIME1/2,divp/2,prime/1,prime1/2,DIVP#/2,PRIME#/1,PRIME1#/2,divp#/2,prime#/1 ,prime1#/2} / {0/0,=/2,and/2,c/0,c1/0,c2/1,c3/0,c4/0,c5/1,c6/1,c7/0,false/0,not/1,rem/2,s/1,true/0,c_1/0 ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/2} - Obligation: innermost runtime complexity wrt. defined symbols {DIVP#,PRIME#,PRIME1#,divp#,prime# ,prime1#} and constructors {0,=,and,c,c1,c2,c3,c4,c5,c6,c7,false,not,rem,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,5,6} by application of Pre({1,2,3,5,6}) = {4,7,8}. Here rules are labelled as follows: 1: DIVP#(z0,z1) -> c_1() 2: PRIME#(0()) -> c_2() 3: PRIME#(s(0())) -> c_3() 4: PRIME#(s(s(z0))) -> c_4(PRIME1#(s(s(z0)),s(z0))) 5: PRIME1#(z0,0()) -> c_5() 6: PRIME1#(z0,s(0())) -> c_6() 7: PRIME1#(z0,s(s(z1))) -> c_7(DIVP#(s(s(z1)),z0)) 8: PRIME1#(z0,s(s(z1))) -> c_8(PRIME1#(z0,s(z1))) 9: divp#(z0,z1) -> c_9() 10: prime#(0()) -> c_10() 11: prime#(s(0())) -> c_11() 12: prime#(s(s(z0))) -> c_12(prime1#(s(s(z0)),s(z0))) 13: prime1#(z0,0()) -> c_13() 14: prime1#(z0,s(0())) -> c_14() 15: prime1#(z0,s(s(z1))) -> c_15(divp#(s(s(z1)),z0),prime1#(z0,s(z1))) ** Step 1.b:3: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: PRIME#(s(s(z0))) -> c_4(PRIME1#(s(s(z0)),s(z0))) PRIME1#(z0,s(s(z1))) -> c_7(DIVP#(s(s(z1)),z0)) PRIME1#(z0,s(s(z1))) -> c_8(PRIME1#(z0,s(z1))) - Weak DPs: DIVP#(z0,z1) -> c_1() PRIME#(0()) -> c_2() PRIME#(s(0())) -> c_3() PRIME1#(z0,0()) -> c_5() PRIME1#(z0,s(0())) -> c_6() divp#(z0,z1) -> c_9() prime#(0()) -> c_10() prime#(s(0())) -> c_11() prime#(s(s(z0))) -> c_12(prime1#(s(s(z0)),s(z0))) prime1#(z0,0()) -> c_13() prime1#(z0,s(0())) -> c_14() prime1#(z0,s(s(z1))) -> c_15(divp#(s(s(z1)),z0),prime1#(z0,s(z1))) - Weak TRS: DIVP(z0,z1) -> c7() PRIME(0()) -> c() PRIME(s(0())) -> c1() PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)),s(z0))) PRIME1(z0,0()) -> c3() PRIME1(z0,s(0())) -> c4() PRIME1(z0,s(s(z1))) -> c5(DIVP(s(s(z1)),z0)) PRIME1(z0,s(s(z1))) -> c6(PRIME1(z0,s(z1))) divp(z0,z1) -> =(rem(z0,z1),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(z0))) -> prime1(s(s(z0)),s(z0)) prime1(z0,0()) -> false() prime1(z0,s(0())) -> true() prime1(z0,s(s(z1))) -> and(not(divp(s(s(z1)),z0)),prime1(z0,s(z1))) - Signature: {DIVP/2,PRIME/1,PRIME1/2,divp/2,prime/1,prime1/2,DIVP#/2,PRIME#/1,PRIME1#/2,divp#/2,prime#/1 ,prime1#/2} / {0/0,=/2,and/2,c/0,c1/0,c2/1,c3/0,c4/0,c5/1,c6/1,c7/0,false/0,not/1,rem/2,s/1,true/0,c_1/0 ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/2} - Obligation: innermost runtime complexity wrt. defined symbols {DIVP#,PRIME#,PRIME1#,divp#,prime# ,prime1#} and constructors {0,=,and,c,c1,c2,c3,c4,c5,c6,c7,false,not,rem,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2} by application of Pre({2}) = {1,3}. Here rules are labelled as follows: 1: PRIME#(s(s(z0))) -> c_4(PRIME1#(s(s(z0)),s(z0))) 2: PRIME1#(z0,s(s(z1))) -> c_7(DIVP#(s(s(z1)),z0)) 3: PRIME1#(z0,s(s(z1))) -> c_8(PRIME1#(z0,s(z1))) 4: DIVP#(z0,z1) -> c_1() 5: PRIME#(0()) -> c_2() 6: PRIME#(s(0())) -> c_3() 7: PRIME1#(z0,0()) -> c_5() 8: PRIME1#(z0,s(0())) -> c_6() 9: divp#(z0,z1) -> c_9() 10: prime#(0()) -> c_10() 11: prime#(s(0())) -> c_11() 12: prime#(s(s(z0))) -> c_12(prime1#(s(s(z0)),s(z0))) 13: prime1#(z0,0()) -> c_13() 14: prime1#(z0,s(0())) -> c_14() 15: prime1#(z0,s(s(z1))) -> c_15(divp#(s(s(z1)),z0),prime1#(z0,s(z1))) ** Step 1.b:4: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: PRIME#(s(s(z0))) -> c_4(PRIME1#(s(s(z0)),s(z0))) PRIME1#(z0,s(s(z1))) -> c_8(PRIME1#(z0,s(z1))) - Weak DPs: DIVP#(z0,z1) -> c_1() PRIME#(0()) -> c_2() PRIME#(s(0())) -> c_3() PRIME1#(z0,0()) -> c_5() PRIME1#(z0,s(0())) -> c_6() PRIME1#(z0,s(s(z1))) -> c_7(DIVP#(s(s(z1)),z0)) divp#(z0,z1) -> c_9() prime#(0()) -> c_10() prime#(s(0())) -> c_11() prime#(s(s(z0))) -> c_12(prime1#(s(s(z0)),s(z0))) prime1#(z0,0()) -> c_13() prime1#(z0,s(0())) -> c_14() prime1#(z0,s(s(z1))) -> c_15(divp#(s(s(z1)),z0),prime1#(z0,s(z1))) - Weak TRS: DIVP(z0,z1) -> c7() PRIME(0()) -> c() PRIME(s(0())) -> c1() PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)),s(z0))) PRIME1(z0,0()) -> c3() PRIME1(z0,s(0())) -> c4() PRIME1(z0,s(s(z1))) -> c5(DIVP(s(s(z1)),z0)) PRIME1(z0,s(s(z1))) -> c6(PRIME1(z0,s(z1))) divp(z0,z1) -> =(rem(z0,z1),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(z0))) -> prime1(s(s(z0)),s(z0)) prime1(z0,0()) -> false() prime1(z0,s(0())) -> true() prime1(z0,s(s(z1))) -> and(not(divp(s(s(z1)),z0)),prime1(z0,s(z1))) - Signature: {DIVP/2,PRIME/1,PRIME1/2,divp/2,prime/1,prime1/2,DIVP#/2,PRIME#/1,PRIME1#/2,divp#/2,prime#/1 ,prime1#/2} / {0/0,=/2,and/2,c/0,c1/0,c2/1,c3/0,c4/0,c5/1,c6/1,c7/0,false/0,not/1,rem/2,s/1,true/0,c_1/0 ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/2} - Obligation: innermost runtime complexity wrt. defined symbols {DIVP#,PRIME#,PRIME1#,divp#,prime# ,prime1#} and constructors {0,=,and,c,c1,c2,c3,c4,c5,c6,c7,false,not,rem,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:PRIME#(s(s(z0))) -> c_4(PRIME1#(s(s(z0)),s(z0))) -->_1 PRIME1#(z0,s(s(z1))) -> c_7(DIVP#(s(s(z1)),z0)):8 -->_1 PRIME1#(z0,s(s(z1))) -> c_8(PRIME1#(z0,s(z1))):2 -->_1 PRIME1#(z0,s(0())) -> c_6():7 2:S:PRIME1#(z0,s(s(z1))) -> c_8(PRIME1#(z0,s(z1))) -->_1 PRIME1#(z0,s(s(z1))) -> c_7(DIVP#(s(s(z1)),z0)):8 -->_1 PRIME1#(z0,s(0())) -> c_6():7 -->_1 PRIME1#(z0,s(s(z1))) -> c_8(PRIME1#(z0,s(z1))):2 3:W:DIVP#(z0,z1) -> c_1() 4:W:PRIME#(0()) -> c_2() 5:W:PRIME#(s(0())) -> c_3() 6:W:PRIME1#(z0,0()) -> c_5() 7:W:PRIME1#(z0,s(0())) -> c_6() 8:W:PRIME1#(z0,s(s(z1))) -> c_7(DIVP#(s(s(z1)),z0)) -->_1 DIVP#(z0,z1) -> c_1():3 9:W:divp#(z0,z1) -> c_9() 10:W:prime#(0()) -> c_10() 11:W:prime#(s(0())) -> c_11() 12:W:prime#(s(s(z0))) -> c_12(prime1#(s(s(z0)),s(z0))) -->_1 prime1#(z0,s(s(z1))) -> c_15(divp#(s(s(z1)),z0),prime1#(z0,s(z1))):15 -->_1 prime1#(z0,s(0())) -> c_14():14 13:W:prime1#(z0,0()) -> c_13() 14:W:prime1#(z0,s(0())) -> c_14() 15:W:prime1#(z0,s(s(z1))) -> c_15(divp#(s(s(z1)),z0),prime1#(z0,s(z1))) -->_2 prime1#(z0,s(s(z1))) -> c_15(divp#(s(s(z1)),z0),prime1#(z0,s(z1))):15 -->_2 prime1#(z0,s(0())) -> c_14():14 -->_1 divp#(z0,z1) -> c_9():9 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 13: prime1#(z0,0()) -> c_13() 12: prime#(s(s(z0))) -> c_12(prime1#(s(s(z0)),s(z0))) 15: prime1#(z0,s(s(z1))) -> c_15(divp#(s(s(z1)),z0),prime1#(z0,s(z1))) 14: prime1#(z0,s(0())) -> c_14() 11: prime#(s(0())) -> c_11() 10: prime#(0()) -> c_10() 9: divp#(z0,z1) -> c_9() 6: PRIME1#(z0,0()) -> c_5() 5: PRIME#(s(0())) -> c_3() 4: PRIME#(0()) -> c_2() 7: PRIME1#(z0,s(0())) -> c_6() 8: PRIME1#(z0,s(s(z1))) -> c_7(DIVP#(s(s(z1)),z0)) 3: DIVP#(z0,z1) -> c_1() ** Step 1.b:5: RemoveHeads. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: PRIME#(s(s(z0))) -> c_4(PRIME1#(s(s(z0)),s(z0))) PRIME1#(z0,s(s(z1))) -> c_8(PRIME1#(z0,s(z1))) - Weak TRS: DIVP(z0,z1) -> c7() PRIME(0()) -> c() PRIME(s(0())) -> c1() PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)),s(z0))) PRIME1(z0,0()) -> c3() PRIME1(z0,s(0())) -> c4() PRIME1(z0,s(s(z1))) -> c5(DIVP(s(s(z1)),z0)) PRIME1(z0,s(s(z1))) -> c6(PRIME1(z0,s(z1))) divp(z0,z1) -> =(rem(z0,z1),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(z0))) -> prime1(s(s(z0)),s(z0)) prime1(z0,0()) -> false() prime1(z0,s(0())) -> true() prime1(z0,s(s(z1))) -> and(not(divp(s(s(z1)),z0)),prime1(z0,s(z1))) - Signature: {DIVP/2,PRIME/1,PRIME1/2,divp/2,prime/1,prime1/2,DIVP#/2,PRIME#/1,PRIME1#/2,divp#/2,prime#/1 ,prime1#/2} / {0/0,=/2,and/2,c/0,c1/0,c2/1,c3/0,c4/0,c5/1,c6/1,c7/0,false/0,not/1,rem/2,s/1,true/0,c_1/0 ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/2} - Obligation: innermost runtime complexity wrt. defined symbols {DIVP#,PRIME#,PRIME1#,divp#,prime# ,prime1#} and constructors {0,=,and,c,c1,c2,c3,c4,c5,c6,c7,false,not,rem,s,true} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:PRIME#(s(s(z0))) -> c_4(PRIME1#(s(s(z0)),s(z0))) -->_1 PRIME1#(z0,s(s(z1))) -> c_8(PRIME1#(z0,s(z1))):2 2:S:PRIME1#(z0,s(s(z1))) -> c_8(PRIME1#(z0,s(z1))) -->_1 PRIME1#(z0,s(s(z1))) -> c_8(PRIME1#(z0,s(z1))):2 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(1,PRIME#(s(s(z0))) -> c_4(PRIME1#(s(s(z0)),s(z0))))] ** Step 1.b:6: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: PRIME1#(z0,s(s(z1))) -> c_8(PRIME1#(z0,s(z1))) - Weak TRS: DIVP(z0,z1) -> c7() PRIME(0()) -> c() PRIME(s(0())) -> c1() PRIME(s(s(z0))) -> c2(PRIME1(s(s(z0)),s(z0))) PRIME1(z0,0()) -> c3() PRIME1(z0,s(0())) -> c4() PRIME1(z0,s(s(z1))) -> c5(DIVP(s(s(z1)),z0)) PRIME1(z0,s(s(z1))) -> c6(PRIME1(z0,s(z1))) divp(z0,z1) -> =(rem(z0,z1),0()) prime(0()) -> false() prime(s(0())) -> false() prime(s(s(z0))) -> prime1(s(s(z0)),s(z0)) prime1(z0,0()) -> false() prime1(z0,s(0())) -> true() prime1(z0,s(s(z1))) -> and(not(divp(s(s(z1)),z0)),prime1(z0,s(z1))) - Signature: {DIVP/2,PRIME/1,PRIME1/2,divp/2,prime/1,prime1/2,DIVP#/2,PRIME#/1,PRIME1#/2,divp#/2,prime#/1 ,prime1#/2} / {0/0,=/2,and/2,c/0,c1/0,c2/1,c3/0,c4/0,c5/1,c6/1,c7/0,false/0,not/1,rem/2,s/1,true/0,c_1/0 ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/2} - Obligation: innermost runtime complexity wrt. defined symbols {DIVP#,PRIME#,PRIME1#,divp#,prime# ,prime1#} and constructors {0,=,and,c,c1,c2,c3,c4,c5,c6,c7,false,not,rem,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: PRIME1#(z0,s(s(z1))) -> c_8(PRIME1#(z0,s(z1))) ** Step 1.b:7: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: PRIME1#(z0,s(s(z1))) -> c_8(PRIME1#(z0,s(z1))) - Signature: {DIVP/2,PRIME/1,PRIME1/2,divp/2,prime/1,prime1/2,DIVP#/2,PRIME#/1,PRIME1#/2,divp#/2,prime#/1 ,prime1#/2} / {0/0,=/2,and/2,c/0,c1/0,c2/1,c3/0,c4/0,c5/1,c6/1,c7/0,false/0,not/1,rem/2,s/1,true/0,c_1/0 ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/2} - Obligation: innermost runtime complexity wrt. defined symbols {DIVP#,PRIME#,PRIME1#,divp#,prime# ,prime1#} and constructors {0,=,and,c,c1,c2,c3,c4,c5,c6,c7,false,not,rem,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_8) = {1} Following symbols are considered usable: {DIVP#,PRIME#,PRIME1#,divp#,prime#,prime1#} TcT has computed the following interpretation: p(0) = [0] p(=) = [1] x1 + [1] x2 + [0] p(DIVP) = [0] p(PRIME) = [0] p(PRIME1) = [0] p(and) = [1] x1 + [0] p(c) = [0] p(c1) = [0] p(c2) = [0] p(c3) = [0] p(c4) = [2] p(c5) = [1] x1 + [0] p(c6) = [1] p(c7) = [4] p(divp) = [2] x2 + [0] p(false) = [1] p(not) = [8] p(prime) = [4] x1 + [0] p(prime1) = [1] x2 + [0] p(rem) = [1] x2 + [8] p(s) = [1] x1 + [1] p(true) = [0] p(DIVP#) = [2] x1 + [1] x2 + [1] p(PRIME#) = [1] x1 + [1] p(PRIME1#) = [8] x2 + [9] p(divp#) = [2] p(prime#) = [4] x1 + [1] p(prime1#) = [1] p(c_1) = [1] p(c_2) = [1] p(c_3) = [0] p(c_4) = [1] x1 + [1] p(c_5) = [1] p(c_6) = [0] p(c_7) = [1] x1 + [4] p(c_8) = [1] x1 + [7] p(c_9) = [1] p(c_10) = [2] p(c_11) = [8] p(c_12) = [1] x1 + [1] p(c_13) = [0] p(c_14) = [2] p(c_15) = [4] x1 + [2] Following rules are strictly oriented: PRIME1#(z0,s(s(z1))) = [8] z1 + [25] > [8] z1 + [24] = c_8(PRIME1#(z0,s(z1))) Following rules are (at-least) weakly oriented: ** Step 1.b:8: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: PRIME1#(z0,s(s(z1))) -> c_8(PRIME1#(z0,s(z1))) - Signature: {DIVP/2,PRIME/1,PRIME1/2,divp/2,prime/1,prime1/2,DIVP#/2,PRIME#/1,PRIME1#/2,divp#/2,prime#/1 ,prime1#/2} / {0/0,=/2,and/2,c/0,c1/0,c2/1,c3/0,c4/0,c5/1,c6/1,c7/0,false/0,not/1,rem/2,s/1,true/0,c_1/0 ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/2} - Obligation: innermost runtime complexity wrt. defined symbols {DIVP#,PRIME#,PRIME1#,divp#,prime# ,prime1#} and constructors {0,=,and,c,c1,c2,c3,c4,c5,c6,c7,false,not,rem,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))