WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: *'(z0,0()) -> c2() *'(z0,s(z1)) -> c3(*'(z0,z1)) -'(z0,0()) -> c() -'(s(z0),s(z1)) -> c1(-'(z0,z1)) F(z0,0(),z1) -> c15() F(z0,s(z1),z2) -> c16(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)),ODD(s(z1))) F(z0,s(z1),z2) -> c17(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(z0,z1,*(z0,z2)) ,*'(z0,z2)) F(z0,s(z1),z2) -> c18(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(*(z0,z0),half(s(z1)),z2) ,*'(z0,z0)) F(z0,s(z1),z2) -> c19(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(*(z0,z0),half(s(z1)),z2) ,HALF(s(z1))) HALF(0()) -> c11() HALF(s(0())) -> c12() HALF(s(s(z0))) -> c13(HALF(z0)) IF(false(),z0,z1) -> c5() IF(false(),z0,z1) -> c7() IF(true(),z0,z1) -> c4() IF(true(),z0,z1) -> c6() ODD(0()) -> c8() ODD(s(0())) -> c9() ODD(s(s(z0))) -> c10(ODD(z0)) POW(z0,z1) -> c14(F(z0,z1,s(0()))) - Weak TRS: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) -(z0,0()) -> z0 -(s(z0),s(z1)) -> -(z0,z1) f(z0,0(),z1) -> z1 f(z0,s(z1),z2) -> if(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) if(false(),z0,z1) -> z1 if(false(),z0,z1) -> false() if(true(),z0,z1) -> z0 if(true(),z0,z1) -> true() odd(0()) -> false() odd(s(0())) -> true() odd(s(s(z0))) -> odd(z0) pow(z0,z1) -> f(z0,z1,s(0())) - Signature: {*/2,*'/2,-/2,-'/2,F/3,HALF/1,IF/3,ODD/1,POW/2,f/3,half/1,if/3,odd/1,pow/2} / {+/2,0/0,c/0,c1/1,c10/1,c11/0 ,c12/0,c13/1,c14/1,c15/0,c16/2,c17/3,c18/3,c19/3,c2/0,c3/1,c4/0,c5/0,c6/0,c7/0,c8/0,c9/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {*,*',-,-',F,HALF,IF,ODD,POW,f,half,if,odd ,pow} and constructors {+,0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c3,c4,c5,c6,c7,c8,c9,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: *'(z0,0()) -> c2() *'(z0,s(z1)) -> c3(*'(z0,z1)) -'(z0,0()) -> c() -'(s(z0),s(z1)) -> c1(-'(z0,z1)) F(z0,0(),z1) -> c15() F(z0,s(z1),z2) -> c16(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)),ODD(s(z1))) F(z0,s(z1),z2) -> c17(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(z0,z1,*(z0,z2)) ,*'(z0,z2)) F(z0,s(z1),z2) -> c18(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(*(z0,z0),half(s(z1)),z2) ,*'(z0,z0)) F(z0,s(z1),z2) -> c19(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(*(z0,z0),half(s(z1)),z2) ,HALF(s(z1))) HALF(0()) -> c11() HALF(s(0())) -> c12() HALF(s(s(z0))) -> c13(HALF(z0)) IF(false(),z0,z1) -> c5() IF(false(),z0,z1) -> c7() IF(true(),z0,z1) -> c4() IF(true(),z0,z1) -> c6() ODD(0()) -> c8() ODD(s(0())) -> c9() ODD(s(s(z0))) -> c10(ODD(z0)) POW(z0,z1) -> c14(F(z0,z1,s(0()))) - Weak TRS: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) -(z0,0()) -> z0 -(s(z0),s(z1)) -> -(z0,z1) f(z0,0(),z1) -> z1 f(z0,s(z1),z2) -> if(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) if(false(),z0,z1) -> z1 if(false(),z0,z1) -> false() if(true(),z0,z1) -> z0 if(true(),z0,z1) -> true() odd(0()) -> false() odd(s(0())) -> true() odd(s(s(z0))) -> odd(z0) pow(z0,z1) -> f(z0,z1,s(0())) - Signature: {*/2,*'/2,-/2,-'/2,F/3,HALF/1,IF/3,ODD/1,POW/2,f/3,half/1,if/3,odd/1,pow/2} / {+/2,0/0,c/0,c1/1,c10/1,c11/0 ,c12/0,c13/1,c14/1,c15/0,c16/2,c17/3,c18/3,c19/3,c2/0,c3/1,c4/0,c5/0,c6/0,c7/0,c8/0,c9/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {*,*',-,-',F,HALF,IF,ODD,POW,f,half,if,odd ,pow} and constructors {+,0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c3,c4,c5,c6,c7,c8,c9,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: *'(z0,0()) -> c2() *'(z0,s(z1)) -> c3(*'(z0,z1)) -'(z0,0()) -> c() -'(s(z0),s(z1)) -> c1(-'(z0,z1)) F(z0,0(),z1) -> c15() F(z0,s(z1),z2) -> c16(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)),ODD(s(z1))) F(z0,s(z1),z2) -> c17(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(z0,z1,*(z0,z2)) ,*'(z0,z2)) F(z0,s(z1),z2) -> c18(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(*(z0,z0),half(s(z1)),z2) ,*'(z0,z0)) F(z0,s(z1),z2) -> c19(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(*(z0,z0),half(s(z1)),z2) ,HALF(s(z1))) HALF(0()) -> c11() HALF(s(0())) -> c12() HALF(s(s(z0))) -> c13(HALF(z0)) IF(false(),z0,z1) -> c5() IF(false(),z0,z1) -> c7() IF(true(),z0,z1) -> c4() IF(true(),z0,z1) -> c6() ODD(0()) -> c8() ODD(s(0())) -> c9() ODD(s(s(z0))) -> c10(ODD(z0)) POW(z0,z1) -> c14(F(z0,z1,s(0()))) - Weak TRS: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) -(z0,0()) -> z0 -(s(z0),s(z1)) -> -(z0,z1) f(z0,0(),z1) -> z1 f(z0,s(z1),z2) -> if(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) if(false(),z0,z1) -> z1 if(false(),z0,z1) -> false() if(true(),z0,z1) -> z0 if(true(),z0,z1) -> true() odd(0()) -> false() odd(s(0())) -> true() odd(s(s(z0))) -> odd(z0) pow(z0,z1) -> f(z0,z1,s(0())) - Signature: {*/2,*'/2,-/2,-'/2,F/3,HALF/1,IF/3,ODD/1,POW/2,f/3,half/1,if/3,odd/1,pow/2} / {+/2,0/0,c/0,c1/1,c10/1,c11/0 ,c12/0,c13/1,c14/1,c15/0,c16/2,c17/3,c18/3,c19/3,c2/0,c3/1,c4/0,c5/0,c6/0,c7/0,c8/0,c9/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {*,*',-,-',F,HALF,IF,ODD,POW,f,half,if,odd ,pow} and constructors {+,0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c3,c4,c5,c6,c7,c8,c9,false,s ,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: *'(x,y){y -> s(y)} = *'(x,s(y)) ->^+ c3(*'(x,y)) = C[*'(x,y) = *'(x,y){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: *'(z0,0()) -> c2() *'(z0,s(z1)) -> c3(*'(z0,z1)) -'(z0,0()) -> c() -'(s(z0),s(z1)) -> c1(-'(z0,z1)) F(z0,0(),z1) -> c15() F(z0,s(z1),z2) -> c16(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)),ODD(s(z1))) F(z0,s(z1),z2) -> c17(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(z0,z1,*(z0,z2)) ,*'(z0,z2)) F(z0,s(z1),z2) -> c18(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(*(z0,z0),half(s(z1)),z2) ,*'(z0,z0)) F(z0,s(z1),z2) -> c19(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(*(z0,z0),half(s(z1)),z2) ,HALF(s(z1))) HALF(0()) -> c11() HALF(s(0())) -> c12() HALF(s(s(z0))) -> c13(HALF(z0)) IF(false(),z0,z1) -> c5() IF(false(),z0,z1) -> c7() IF(true(),z0,z1) -> c4() IF(true(),z0,z1) -> c6() ODD(0()) -> c8() ODD(s(0())) -> c9() ODD(s(s(z0))) -> c10(ODD(z0)) POW(z0,z1) -> c14(F(z0,z1,s(0()))) - Weak TRS: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) -(z0,0()) -> z0 -(s(z0),s(z1)) -> -(z0,z1) f(z0,0(),z1) -> z1 f(z0,s(z1),z2) -> if(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) if(false(),z0,z1) -> z1 if(false(),z0,z1) -> false() if(true(),z0,z1) -> z0 if(true(),z0,z1) -> true() odd(0()) -> false() odd(s(0())) -> true() odd(s(s(z0))) -> odd(z0) pow(z0,z1) -> f(z0,z1,s(0())) - Signature: {*/2,*'/2,-/2,-'/2,F/3,HALF/1,IF/3,ODD/1,POW/2,f/3,half/1,if/3,odd/1,pow/2} / {+/2,0/0,c/0,c1/1,c10/1,c11/0 ,c12/0,c13/1,c14/1,c15/0,c16/2,c17/3,c18/3,c19/3,c2/0,c3/1,c4/0,c5/0,c6/0,c7/0,c8/0,c9/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {*,*',-,-',F,HALF,IF,ODD,POW,f,half,if,odd ,pow} and constructors {+,0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c3,c4,c5,c6,c7,c8,c9,false,s ,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs *'#(z0,0()) -> c_1() *'#(z0,s(z1)) -> c_2(*'#(z0,z1)) -'#(z0,0()) -> c_3() -'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)) F#(z0,0(),z1) -> c_5() F#(z0,s(z1),z2) -> c_6(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,ODD#(s(z1))) F#(z0,s(z1),z2) -> c_7(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,*'#(z0,z2)) F#(z0,s(z1),z2) -> c_8(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,*'#(z0,z0)) F#(z0,s(z1),z2) -> c_9(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,HALF#(s(z1))) HALF#(0()) -> c_10() HALF#(s(0())) -> c_11() HALF#(s(s(z0))) -> c_12(HALF#(z0)) IF#(false(),z0,z1) -> c_13() IF#(false(),z0,z1) -> c_14() IF#(true(),z0,z1) -> c_15() IF#(true(),z0,z1) -> c_16() ODD#(0()) -> c_17() ODD#(s(0())) -> c_18() ODD#(s(s(z0))) -> c_19(ODD#(z0)) POW#(z0,z1) -> c_20(F#(z0,z1,s(0()))) Weak DPs *#(z0,0()) -> c_21() *#(z0,s(z1)) -> c_22(*#(z0,z1)) -#(z0,0()) -> c_23() -#(s(z0),s(z1)) -> c_24(-#(z0,z1)) f#(z0,0(),z1) -> c_25() f#(z0,s(z1),z2) -> c_26(if#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1))) half#(0()) -> c_27() half#(s(0())) -> c_28() half#(s(s(z0))) -> c_29(half#(z0)) if#(false(),z0,z1) -> c_30() if#(false(),z0,z1) -> c_31() if#(true(),z0,z1) -> c_32() if#(true(),z0,z1) -> c_33() odd#(0()) -> c_34() odd#(s(0())) -> c_35() odd#(s(s(z0))) -> c_36(odd#(z0)) pow#(z0,z1) -> c_37(f#(z0,z1,s(0()))) and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: *'#(z0,0()) -> c_1() *'#(z0,s(z1)) -> c_2(*'#(z0,z1)) -'#(z0,0()) -> c_3() -'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)) F#(z0,0(),z1) -> c_5() F#(z0,s(z1),z2) -> c_6(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,ODD#(s(z1))) F#(z0,s(z1),z2) -> c_7(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,*'#(z0,z2)) F#(z0,s(z1),z2) -> c_8(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,*'#(z0,z0)) F#(z0,s(z1),z2) -> c_9(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,HALF#(s(z1))) HALF#(0()) -> c_10() HALF#(s(0())) -> c_11() HALF#(s(s(z0))) -> c_12(HALF#(z0)) IF#(false(),z0,z1) -> c_13() IF#(false(),z0,z1) -> c_14() IF#(true(),z0,z1) -> c_15() IF#(true(),z0,z1) -> c_16() ODD#(0()) -> c_17() ODD#(s(0())) -> c_18() ODD#(s(s(z0))) -> c_19(ODD#(z0)) POW#(z0,z1) -> c_20(F#(z0,z1,s(0()))) - Weak DPs: *#(z0,0()) -> c_21() *#(z0,s(z1)) -> c_22(*#(z0,z1)) -#(z0,0()) -> c_23() -#(s(z0),s(z1)) -> c_24(-#(z0,z1)) f#(z0,0(),z1) -> c_25() f#(z0,s(z1),z2) -> c_26(if#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1))) half#(0()) -> c_27() half#(s(0())) -> c_28() half#(s(s(z0))) -> c_29(half#(z0)) if#(false(),z0,z1) -> c_30() if#(false(),z0,z1) -> c_31() if#(true(),z0,z1) -> c_32() if#(true(),z0,z1) -> c_33() odd#(0()) -> c_34() odd#(s(0())) -> c_35() odd#(s(s(z0))) -> c_36(odd#(z0)) pow#(z0,z1) -> c_37(f#(z0,z1,s(0()))) - Weak TRS: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) *'(z0,0()) -> c2() *'(z0,s(z1)) -> c3(*'(z0,z1)) -(z0,0()) -> z0 -(s(z0),s(z1)) -> -(z0,z1) -'(z0,0()) -> c() -'(s(z0),s(z1)) -> c1(-'(z0,z1)) F(z0,0(),z1) -> c15() F(z0,s(z1),z2) -> c16(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)),ODD(s(z1))) F(z0,s(z1),z2) -> c17(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(z0,z1,*(z0,z2)) ,*'(z0,z2)) F(z0,s(z1),z2) -> c18(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(*(z0,z0),half(s(z1)),z2) ,*'(z0,z0)) F(z0,s(z1),z2) -> c19(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(*(z0,z0),half(s(z1)),z2) ,HALF(s(z1))) HALF(0()) -> c11() HALF(s(0())) -> c12() HALF(s(s(z0))) -> c13(HALF(z0)) IF(false(),z0,z1) -> c5() IF(false(),z0,z1) -> c7() IF(true(),z0,z1) -> c4() IF(true(),z0,z1) -> c6() ODD(0()) -> c8() ODD(s(0())) -> c9() ODD(s(s(z0))) -> c10(ODD(z0)) POW(z0,z1) -> c14(F(z0,z1,s(0()))) f(z0,0(),z1) -> z1 f(z0,s(z1),z2) -> if(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) if(false(),z0,z1) -> z1 if(false(),z0,z1) -> false() if(true(),z0,z1) -> z0 if(true(),z0,z1) -> true() odd(0()) -> false() odd(s(0())) -> true() odd(s(s(z0))) -> odd(z0) pow(z0,z1) -> f(z0,z1,s(0())) - Signature: {*/2,*'/2,-/2,-'/2,F/3,HALF/1,IF/3,ODD/1,POW/2,f/3,half/1,if/3,odd/1,pow/2,*#/2,*'#/2,-#/2,-'#/2,F#/3 ,HALF#/1,IF#/3,ODD#/1,POW#/2,f#/3,half#/1,if#/3,odd#/1,pow#/2} / {+/2,0/0,c/0,c1/1,c10/1,c11/0,c12/0,c13/1 ,c14/1,c15/0,c16/2,c17/3,c18/3,c19/3,c2/0,c3/1,c4/0,c5/0,c6/0,c7/0,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1 ,c_3/0,c_4/1,c_5/0,c_6/8,c_7/10,c_8/11,c_9/11,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0 ,c_19/1,c_20/1,c_21/0,c_22/1,c_23/0,c_24/1,c_25/0,c_26/7,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0 ,c_34/0,c_35/0,c_36/1,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,*'#,-#,-'#,F#,HALF#,IF#,ODD#,POW#,f#,half#,if#,odd# ,pow#} and constructors {+,0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c3,c4,c5,c6,c7,c8,c9,false,s ,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,5,10,11,13,14,15,16,17,18} by application of Pre({1,3,5,10,11,13,14,15,16,17,18}) = {2,4,6,7,8,9,12,19,20}. Here rules are labelled as follows: 1: *'#(z0,0()) -> c_1() 2: *'#(z0,s(z1)) -> c_2(*'#(z0,z1)) 3: -'#(z0,0()) -> c_3() 4: -'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)) 5: F#(z0,0(),z1) -> c_5() 6: F#(z0,s(z1),z2) -> c_6(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,ODD#(s(z1))) 7: F#(z0,s(z1),z2) -> c_7(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,*'#(z0,z2)) 8: F#(z0,s(z1),z2) -> c_8(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,*'#(z0,z0)) 9: F#(z0,s(z1),z2) -> c_9(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,HALF#(s(z1))) 10: HALF#(0()) -> c_10() 11: HALF#(s(0())) -> c_11() 12: HALF#(s(s(z0))) -> c_12(HALF#(z0)) 13: IF#(false(),z0,z1) -> c_13() 14: IF#(false(),z0,z1) -> c_14() 15: IF#(true(),z0,z1) -> c_15() 16: IF#(true(),z0,z1) -> c_16() 17: ODD#(0()) -> c_17() 18: ODD#(s(0())) -> c_18() 19: ODD#(s(s(z0))) -> c_19(ODD#(z0)) 20: POW#(z0,z1) -> c_20(F#(z0,z1,s(0()))) 21: *#(z0,0()) -> c_21() 22: *#(z0,s(z1)) -> c_22(*#(z0,z1)) 23: -#(z0,0()) -> c_23() 24: -#(s(z0),s(z1)) -> c_24(-#(z0,z1)) 25: f#(z0,0(),z1) -> c_25() 26: f#(z0,s(z1),z2) -> c_26(if#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1))) 27: half#(0()) -> c_27() 28: half#(s(0())) -> c_28() 29: half#(s(s(z0))) -> c_29(half#(z0)) 30: if#(false(),z0,z1) -> c_30() 31: if#(false(),z0,z1) -> c_31() 32: if#(true(),z0,z1) -> c_32() 33: if#(true(),z0,z1) -> c_33() 34: odd#(0()) -> c_34() 35: odd#(s(0())) -> c_35() 36: odd#(s(s(z0))) -> c_36(odd#(z0)) 37: pow#(z0,z1) -> c_37(f#(z0,z1,s(0()))) ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: *'#(z0,s(z1)) -> c_2(*'#(z0,z1)) -'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)) F#(z0,s(z1),z2) -> c_6(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,ODD#(s(z1))) F#(z0,s(z1),z2) -> c_7(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,*'#(z0,z2)) F#(z0,s(z1),z2) -> c_8(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,*'#(z0,z0)) F#(z0,s(z1),z2) -> c_9(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,HALF#(s(z1))) HALF#(s(s(z0))) -> c_12(HALF#(z0)) ODD#(s(s(z0))) -> c_19(ODD#(z0)) POW#(z0,z1) -> c_20(F#(z0,z1,s(0()))) - Weak DPs: *#(z0,0()) -> c_21() *#(z0,s(z1)) -> c_22(*#(z0,z1)) *'#(z0,0()) -> c_1() -#(z0,0()) -> c_23() -#(s(z0),s(z1)) -> c_24(-#(z0,z1)) -'#(z0,0()) -> c_3() F#(z0,0(),z1) -> c_5() HALF#(0()) -> c_10() HALF#(s(0())) -> c_11() IF#(false(),z0,z1) -> c_13() IF#(false(),z0,z1) -> c_14() IF#(true(),z0,z1) -> c_15() IF#(true(),z0,z1) -> c_16() ODD#(0()) -> c_17() ODD#(s(0())) -> c_18() f#(z0,0(),z1) -> c_25() f#(z0,s(z1),z2) -> c_26(if#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1))) half#(0()) -> c_27() half#(s(0())) -> c_28() half#(s(s(z0))) -> c_29(half#(z0)) if#(false(),z0,z1) -> c_30() if#(false(),z0,z1) -> c_31() if#(true(),z0,z1) -> c_32() if#(true(),z0,z1) -> c_33() odd#(0()) -> c_34() odd#(s(0())) -> c_35() odd#(s(s(z0))) -> c_36(odd#(z0)) pow#(z0,z1) -> c_37(f#(z0,z1,s(0()))) - Weak TRS: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) *'(z0,0()) -> c2() *'(z0,s(z1)) -> c3(*'(z0,z1)) -(z0,0()) -> z0 -(s(z0),s(z1)) -> -(z0,z1) -'(z0,0()) -> c() -'(s(z0),s(z1)) -> c1(-'(z0,z1)) F(z0,0(),z1) -> c15() F(z0,s(z1),z2) -> c16(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)),ODD(s(z1))) F(z0,s(z1),z2) -> c17(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(z0,z1,*(z0,z2)) ,*'(z0,z2)) F(z0,s(z1),z2) -> c18(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(*(z0,z0),half(s(z1)),z2) ,*'(z0,z0)) F(z0,s(z1),z2) -> c19(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(*(z0,z0),half(s(z1)),z2) ,HALF(s(z1))) HALF(0()) -> c11() HALF(s(0())) -> c12() HALF(s(s(z0))) -> c13(HALF(z0)) IF(false(),z0,z1) -> c5() IF(false(),z0,z1) -> c7() IF(true(),z0,z1) -> c4() IF(true(),z0,z1) -> c6() ODD(0()) -> c8() ODD(s(0())) -> c9() ODD(s(s(z0))) -> c10(ODD(z0)) POW(z0,z1) -> c14(F(z0,z1,s(0()))) f(z0,0(),z1) -> z1 f(z0,s(z1),z2) -> if(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) if(false(),z0,z1) -> z1 if(false(),z0,z1) -> false() if(true(),z0,z1) -> z0 if(true(),z0,z1) -> true() odd(0()) -> false() odd(s(0())) -> true() odd(s(s(z0))) -> odd(z0) pow(z0,z1) -> f(z0,z1,s(0())) - Signature: {*/2,*'/2,-/2,-'/2,F/3,HALF/1,IF/3,ODD/1,POW/2,f/3,half/1,if/3,odd/1,pow/2,*#/2,*'#/2,-#/2,-'#/2,F#/3 ,HALF#/1,IF#/3,ODD#/1,POW#/2,f#/3,half#/1,if#/3,odd#/1,pow#/2} / {+/2,0/0,c/0,c1/1,c10/1,c11/0,c12/0,c13/1 ,c14/1,c15/0,c16/2,c17/3,c18/3,c19/3,c2/0,c3/1,c4/0,c5/0,c6/0,c7/0,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1 ,c_3/0,c_4/1,c_5/0,c_6/8,c_7/10,c_8/11,c_9/11,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0 ,c_19/1,c_20/1,c_21/0,c_22/1,c_23/0,c_24/1,c_25/0,c_26/7,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0 ,c_34/0,c_35/0,c_36/1,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,*'#,-#,-'#,F#,HALF#,IF#,ODD#,POW#,f#,half#,if#,odd# ,pow#} and constructors {+,0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c3,c4,c5,c6,c7,c8,c9,false,s ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:*'#(z0,s(z1)) -> c_2(*'#(z0,z1)) -->_1 *'#(z0,0()) -> c_1():12 -->_1 *'#(z0,s(z1)) -> c_2(*'#(z0,z1)):1 2:S:-'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)) -->_1 -'#(z0,0()) -> c_3():15 -->_1 -'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)):2 3:S:F#(z0,s(z1),z2) -> c_6(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,ODD#(s(z1))) -->_2 odd#(s(s(z0))) -> c_36(odd#(z0)):36 -->_7 half#(s(s(z0))) -> c_29(half#(z0)):29 -->_5 f#(z0,s(z1),z2) -> c_26(if#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1))):26 -->_3 f#(z0,s(z1),z2) -> c_26(if#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1))):26 -->_6 *#(z0,s(z1)) -> c_22(*#(z0,z1)):11 -->_4 *#(z0,s(z1)) -> c_22(*#(z0,z1)):11 -->_8 ODD#(s(s(z0))) -> c_19(ODD#(z0)):8 -->_2 odd#(s(0())) -> c_35():35 -->_7 half#(s(0())) -> c_28():28 -->_5 f#(z0,0(),z1) -> c_25():25 -->_3 f#(z0,0(),z1) -> c_25():25 -->_8 ODD#(s(0())) -> c_18():24 -->_1 IF#(true(),z0,z1) -> c_16():22 -->_1 IF#(true(),z0,z1) -> c_15():21 -->_1 IF#(false(),z0,z1) -> c_14():20 -->_1 IF#(false(),z0,z1) -> c_13():19 -->_6 *#(z0,0()) -> c_21():10 -->_4 *#(z0,0()) -> c_21():10 4:S:F#(z0,s(z1),z2) -> c_7(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,*'#(z0,z2)) -->_2 odd#(s(s(z0))) -> c_36(odd#(z0)):36 -->_7 half#(s(s(z0))) -> c_29(half#(z0)):29 -->_5 f#(z0,s(z1),z2) -> c_26(if#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1))):26 -->_3 f#(z0,s(z1),z2) -> c_26(if#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1))):26 -->_9 *#(z0,s(z1)) -> c_22(*#(z0,z1)):11 -->_6 *#(z0,s(z1)) -> c_22(*#(z0,z1)):11 -->_4 *#(z0,s(z1)) -> c_22(*#(z0,z1)):11 -->_8 F#(z0,s(z1),z2) -> c_9(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,HALF#(s(z1))):6 -->_8 F#(z0,s(z1),z2) -> c_8(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,*'#(z0,z0)):5 -->_2 odd#(s(0())) -> c_35():35 -->_7 half#(s(0())) -> c_28():28 -->_5 f#(z0,0(),z1) -> c_25():25 -->_3 f#(z0,0(),z1) -> c_25():25 -->_1 IF#(true(),z0,z1) -> c_16():22 -->_1 IF#(true(),z0,z1) -> c_15():21 -->_1 IF#(false(),z0,z1) -> c_14():20 -->_1 IF#(false(),z0,z1) -> c_13():19 -->_8 F#(z0,0(),z1) -> c_5():16 -->_10 *'#(z0,0()) -> c_1():12 -->_9 *#(z0,0()) -> c_21():10 -->_6 *#(z0,0()) -> c_21():10 -->_4 *#(z0,0()) -> c_21():10 -->_8 F#(z0,s(z1),z2) -> c_7(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,*'#(z0,z2)):4 -->_8 F#(z0,s(z1),z2) -> c_6(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,ODD#(s(z1))):3 -->_10 *'#(z0,s(z1)) -> c_2(*'#(z0,z1)):1 5:S:F#(z0,s(z1),z2) -> c_8(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,*'#(z0,z0)) -->_2 odd#(s(s(z0))) -> c_36(odd#(z0)):36 -->_10 half#(s(s(z0))) -> c_29(half#(z0)):29 -->_7 half#(s(s(z0))) -> c_29(half#(z0)):29 -->_5 f#(z0,s(z1),z2) -> c_26(if#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1))):26 -->_3 f#(z0,s(z1),z2) -> c_26(if#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1))):26 -->_9 *#(z0,s(z1)) -> c_22(*#(z0,z1)):11 -->_6 *#(z0,s(z1)) -> c_22(*#(z0,z1)):11 -->_4 *#(z0,s(z1)) -> c_22(*#(z0,z1)):11 -->_8 F#(z0,s(z1),z2) -> c_9(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,HALF#(s(z1))):6 -->_2 odd#(s(0())) -> c_35():35 -->_10 half#(s(0())) -> c_28():28 -->_7 half#(s(0())) -> c_28():28 -->_5 f#(z0,0(),z1) -> c_25():25 -->_3 f#(z0,0(),z1) -> c_25():25 -->_1 IF#(true(),z0,z1) -> c_16():22 -->_1 IF#(true(),z0,z1) -> c_15():21 -->_1 IF#(false(),z0,z1) -> c_14():20 -->_1 IF#(false(),z0,z1) -> c_13():19 -->_8 F#(z0,0(),z1) -> c_5():16 -->_11 *'#(z0,0()) -> c_1():12 -->_9 *#(z0,0()) -> c_21():10 -->_6 *#(z0,0()) -> c_21():10 -->_4 *#(z0,0()) -> c_21():10 -->_8 F#(z0,s(z1),z2) -> c_8(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,*'#(z0,z0)):5 -->_8 F#(z0,s(z1),z2) -> c_7(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,*'#(z0,z2)):4 -->_8 F#(z0,s(z1),z2) -> c_6(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,ODD#(s(z1))):3 -->_11 *'#(z0,s(z1)) -> c_2(*'#(z0,z1)):1 6:S:F#(z0,s(z1),z2) -> c_9(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,HALF#(s(z1))) -->_2 odd#(s(s(z0))) -> c_36(odd#(z0)):36 -->_10 half#(s(s(z0))) -> c_29(half#(z0)):29 -->_7 half#(s(s(z0))) -> c_29(half#(z0)):29 -->_5 f#(z0,s(z1),z2) -> c_26(if#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1))):26 -->_3 f#(z0,s(z1),z2) -> c_26(if#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1))):26 -->_9 *#(z0,s(z1)) -> c_22(*#(z0,z1)):11 -->_6 *#(z0,s(z1)) -> c_22(*#(z0,z1)):11 -->_4 *#(z0,s(z1)) -> c_22(*#(z0,z1)):11 -->_11 HALF#(s(s(z0))) -> c_12(HALF#(z0)):7 -->_2 odd#(s(0())) -> c_35():35 -->_10 half#(s(0())) -> c_28():28 -->_7 half#(s(0())) -> c_28():28 -->_5 f#(z0,0(),z1) -> c_25():25 -->_3 f#(z0,0(),z1) -> c_25():25 -->_1 IF#(true(),z0,z1) -> c_16():22 -->_1 IF#(true(),z0,z1) -> c_15():21 -->_1 IF#(false(),z0,z1) -> c_14():20 -->_1 IF#(false(),z0,z1) -> c_13():19 -->_11 HALF#(s(0())) -> c_11():18 -->_8 F#(z0,0(),z1) -> c_5():16 -->_9 *#(z0,0()) -> c_21():10 -->_6 *#(z0,0()) -> c_21():10 -->_4 *#(z0,0()) -> c_21():10 -->_8 F#(z0,s(z1),z2) -> c_9(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,HALF#(s(z1))):6 -->_8 F#(z0,s(z1),z2) -> c_8(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,*'#(z0,z0)):5 -->_8 F#(z0,s(z1),z2) -> c_7(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,*'#(z0,z2)):4 -->_8 F#(z0,s(z1),z2) -> c_6(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,ODD#(s(z1))):3 7:S:HALF#(s(s(z0))) -> c_12(HALF#(z0)) -->_1 HALF#(s(0())) -> c_11():18 -->_1 HALF#(0()) -> c_10():17 -->_1 HALF#(s(s(z0))) -> c_12(HALF#(z0)):7 8:S:ODD#(s(s(z0))) -> c_19(ODD#(z0)) -->_1 ODD#(s(0())) -> c_18():24 -->_1 ODD#(0()) -> c_17():23 -->_1 ODD#(s(s(z0))) -> c_19(ODD#(z0)):8 9:S:POW#(z0,z1) -> c_20(F#(z0,z1,s(0()))) -->_1 F#(z0,0(),z1) -> c_5():16 -->_1 F#(z0,s(z1),z2) -> c_9(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,HALF#(s(z1))):6 -->_1 F#(z0,s(z1),z2) -> c_8(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,*'#(z0,z0)):5 -->_1 F#(z0,s(z1),z2) -> c_7(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,*'#(z0,z2)):4 -->_1 F#(z0,s(z1),z2) -> c_6(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,ODD#(s(z1))):3 10:W:*#(z0,0()) -> c_21() 11:W:*#(z0,s(z1)) -> c_22(*#(z0,z1)) -->_1 *#(z0,s(z1)) -> c_22(*#(z0,z1)):11 -->_1 *#(z0,0()) -> c_21():10 12:W:*'#(z0,0()) -> c_1() 13:W:-#(z0,0()) -> c_23() 14:W:-#(s(z0),s(z1)) -> c_24(-#(z0,z1)) -->_1 -#(s(z0),s(z1)) -> c_24(-#(z0,z1)):14 -->_1 -#(z0,0()) -> c_23():13 15:W:-'#(z0,0()) -> c_3() 16:W:F#(z0,0(),z1) -> c_5() 17:W:HALF#(0()) -> c_10() 18:W:HALF#(s(0())) -> c_11() 19:W:IF#(false(),z0,z1) -> c_13() 20:W:IF#(false(),z0,z1) -> c_14() 21:W:IF#(true(),z0,z1) -> c_15() 22:W:IF#(true(),z0,z1) -> c_16() 23:W:ODD#(0()) -> c_17() 24:W:ODD#(s(0())) -> c_18() 25:W:f#(z0,0(),z1) -> c_25() 26:W:f#(z0,s(z1),z2) -> c_26(if#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1))) -->_2 odd#(s(s(z0))) -> c_36(odd#(z0)):36 -->_7 half#(s(s(z0))) -> c_29(half#(z0)):29 -->_2 odd#(s(0())) -> c_35():35 -->_1 if#(true(),z0,z1) -> c_33():33 -->_1 if#(true(),z0,z1) -> c_32():32 -->_1 if#(false(),z0,z1) -> c_31():31 -->_1 if#(false(),z0,z1) -> c_30():30 -->_7 half#(s(0())) -> c_28():28 -->_5 f#(z0,s(z1),z2) -> c_26(if#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1))):26 -->_3 f#(z0,s(z1),z2) -> c_26(if#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1))):26 -->_5 f#(z0,0(),z1) -> c_25():25 -->_3 f#(z0,0(),z1) -> c_25():25 -->_6 *#(z0,s(z1)) -> c_22(*#(z0,z1)):11 -->_4 *#(z0,s(z1)) -> c_22(*#(z0,z1)):11 -->_6 *#(z0,0()) -> c_21():10 -->_4 *#(z0,0()) -> c_21():10 27:W:half#(0()) -> c_27() 28:W:half#(s(0())) -> c_28() 29:W:half#(s(s(z0))) -> c_29(half#(z0)) -->_1 half#(s(s(z0))) -> c_29(half#(z0)):29 -->_1 half#(s(0())) -> c_28():28 -->_1 half#(0()) -> c_27():27 30:W:if#(false(),z0,z1) -> c_30() 31:W:if#(false(),z0,z1) -> c_31() 32:W:if#(true(),z0,z1) -> c_32() 33:W:if#(true(),z0,z1) -> c_33() 34:W:odd#(0()) -> c_34() 35:W:odd#(s(0())) -> c_35() 36:W:odd#(s(s(z0))) -> c_36(odd#(z0)) -->_1 odd#(s(s(z0))) -> c_36(odd#(z0)):36 -->_1 odd#(s(0())) -> c_35():35 -->_1 odd#(0()) -> c_34():34 37:W:pow#(z0,z1) -> c_37(f#(z0,z1,s(0()))) -->_1 f#(z0,s(z1),z2) -> c_26(if#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1))):26 -->_1 f#(z0,0(),z1) -> c_25():25 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 37: pow#(z0,z1) -> c_37(f#(z0,z1,s(0()))) 14: -#(s(z0),s(z1)) -> c_24(-#(z0,z1)) 13: -#(z0,0()) -> c_23() 16: F#(z0,0(),z1) -> c_5() 17: HALF#(0()) -> c_10() 18: HALF#(s(0())) -> c_11() 19: IF#(false(),z0,z1) -> c_13() 20: IF#(false(),z0,z1) -> c_14() 21: IF#(true(),z0,z1) -> c_15() 22: IF#(true(),z0,z1) -> c_16() 23: ODD#(0()) -> c_17() 24: ODD#(s(0())) -> c_18() 26: f#(z0,s(z1),z2) -> c_26(if#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1))) 11: *#(z0,s(z1)) -> c_22(*#(z0,z1)) 10: *#(z0,0()) -> c_21() 25: f#(z0,0(),z1) -> c_25() 30: if#(false(),z0,z1) -> c_30() 31: if#(false(),z0,z1) -> c_31() 32: if#(true(),z0,z1) -> c_32() 33: if#(true(),z0,z1) -> c_33() 29: half#(s(s(z0))) -> c_29(half#(z0)) 27: half#(0()) -> c_27() 28: half#(s(0())) -> c_28() 36: odd#(s(s(z0))) -> c_36(odd#(z0)) 34: odd#(0()) -> c_34() 35: odd#(s(0())) -> c_35() 15: -'#(z0,0()) -> c_3() 12: *'#(z0,0()) -> c_1() ** Step 1.b:4: SimplifyRHS. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: *'#(z0,s(z1)) -> c_2(*'#(z0,z1)) -'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)) F#(z0,s(z1),z2) -> c_6(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,ODD#(s(z1))) F#(z0,s(z1),z2) -> c_7(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,*'#(z0,z2)) F#(z0,s(z1),z2) -> c_8(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,*'#(z0,z0)) F#(z0,s(z1),z2) -> c_9(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,HALF#(s(z1))) HALF#(s(s(z0))) -> c_12(HALF#(z0)) ODD#(s(s(z0))) -> c_19(ODD#(z0)) POW#(z0,z1) -> c_20(F#(z0,z1,s(0()))) - Weak TRS: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) *'(z0,0()) -> c2() *'(z0,s(z1)) -> c3(*'(z0,z1)) -(z0,0()) -> z0 -(s(z0),s(z1)) -> -(z0,z1) -'(z0,0()) -> c() -'(s(z0),s(z1)) -> c1(-'(z0,z1)) F(z0,0(),z1) -> c15() F(z0,s(z1),z2) -> c16(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)),ODD(s(z1))) F(z0,s(z1),z2) -> c17(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(z0,z1,*(z0,z2)) ,*'(z0,z2)) F(z0,s(z1),z2) -> c18(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(*(z0,z0),half(s(z1)),z2) ,*'(z0,z0)) F(z0,s(z1),z2) -> c19(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(*(z0,z0),half(s(z1)),z2) ,HALF(s(z1))) HALF(0()) -> c11() HALF(s(0())) -> c12() HALF(s(s(z0))) -> c13(HALF(z0)) IF(false(),z0,z1) -> c5() IF(false(),z0,z1) -> c7() IF(true(),z0,z1) -> c4() IF(true(),z0,z1) -> c6() ODD(0()) -> c8() ODD(s(0())) -> c9() ODD(s(s(z0))) -> c10(ODD(z0)) POW(z0,z1) -> c14(F(z0,z1,s(0()))) f(z0,0(),z1) -> z1 f(z0,s(z1),z2) -> if(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) if(false(),z0,z1) -> z1 if(false(),z0,z1) -> false() if(true(),z0,z1) -> z0 if(true(),z0,z1) -> true() odd(0()) -> false() odd(s(0())) -> true() odd(s(s(z0))) -> odd(z0) pow(z0,z1) -> f(z0,z1,s(0())) - Signature: {*/2,*'/2,-/2,-'/2,F/3,HALF/1,IF/3,ODD/1,POW/2,f/3,half/1,if/3,odd/1,pow/2,*#/2,*'#/2,-#/2,-'#/2,F#/3 ,HALF#/1,IF#/3,ODD#/1,POW#/2,f#/3,half#/1,if#/3,odd#/1,pow#/2} / {+/2,0/0,c/0,c1/1,c10/1,c11/0,c12/0,c13/1 ,c14/1,c15/0,c16/2,c17/3,c18/3,c19/3,c2/0,c3/1,c4/0,c5/0,c6/0,c7/0,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1 ,c_3/0,c_4/1,c_5/0,c_6/8,c_7/10,c_8/11,c_9/11,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0 ,c_19/1,c_20/1,c_21/0,c_22/1,c_23/0,c_24/1,c_25/0,c_26/7,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0 ,c_34/0,c_35/0,c_36/1,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,*'#,-#,-'#,F#,HALF#,IF#,ODD#,POW#,f#,half#,if#,odd# ,pow#} and constructors {+,0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c3,c4,c5,c6,c7,c8,c9,false,s ,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:*'#(z0,s(z1)) -> c_2(*'#(z0,z1)) -->_1 *'#(z0,s(z1)) -> c_2(*'#(z0,z1)):1 2:S:-'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)) -->_1 -'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)):2 3:S:F#(z0,s(z1),z2) -> c_6(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,ODD#(s(z1))) -->_8 ODD#(s(s(z0))) -> c_19(ODD#(z0)):8 4:S:F#(z0,s(z1),z2) -> c_7(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,*'#(z0,z2)) -->_8 F#(z0,s(z1),z2) -> c_9(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,HALF#(s(z1))):6 -->_8 F#(z0,s(z1),z2) -> c_8(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,*'#(z0,z0)):5 -->_8 F#(z0,s(z1),z2) -> c_7(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,*'#(z0,z2)):4 -->_8 F#(z0,s(z1),z2) -> c_6(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,ODD#(s(z1))):3 -->_10 *'#(z0,s(z1)) -> c_2(*'#(z0,z1)):1 5:S:F#(z0,s(z1),z2) -> c_8(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,*'#(z0,z0)) -->_8 F#(z0,s(z1),z2) -> c_9(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,HALF#(s(z1))):6 -->_8 F#(z0,s(z1),z2) -> c_8(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,*'#(z0,z0)):5 -->_8 F#(z0,s(z1),z2) -> c_7(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,*'#(z0,z2)):4 -->_8 F#(z0,s(z1),z2) -> c_6(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,ODD#(s(z1))):3 -->_11 *'#(z0,s(z1)) -> c_2(*'#(z0,z1)):1 6:S:F#(z0,s(z1),z2) -> c_9(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,HALF#(s(z1))) -->_11 HALF#(s(s(z0))) -> c_12(HALF#(z0)):7 -->_8 F#(z0,s(z1),z2) -> c_9(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,HALF#(s(z1))):6 -->_8 F#(z0,s(z1),z2) -> c_8(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,*'#(z0,z0)):5 -->_8 F#(z0,s(z1),z2) -> c_7(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,*'#(z0,z2)):4 -->_8 F#(z0,s(z1),z2) -> c_6(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,ODD#(s(z1))):3 7:S:HALF#(s(s(z0))) -> c_12(HALF#(z0)) -->_1 HALF#(s(s(z0))) -> c_12(HALF#(z0)):7 8:S:ODD#(s(s(z0))) -> c_19(ODD#(z0)) -->_1 ODD#(s(s(z0))) -> c_19(ODD#(z0)):8 9:S:POW#(z0,z1) -> c_20(F#(z0,z1,s(0()))) -->_1 F#(z0,s(z1),z2) -> c_9(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,HALF#(s(z1))):6 -->_1 F#(z0,s(z1),z2) -> c_8(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,*'#(z0,z0)):5 -->_1 F#(z0,s(z1),z2) -> c_7(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,F#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,*'#(z0,z2)):4 -->_1 F#(z0,s(z1),z2) -> c_6(IF#(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,odd#(s(z1)) ,f#(z0,z1,*(z0,z2)) ,*#(z0,z2) ,f#(*(z0,z0),half(s(z1)),z2) ,*#(z0,z0) ,half#(s(z1)) ,ODD#(s(z1))):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: F#(z0,s(z1),z2) -> c_6(ODD#(s(z1))) F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2)),*'#(z0,z2)) F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2),*'#(z0,z0)) F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2),HALF#(s(z1))) ** Step 1.b:5: UsableRules. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: *'#(z0,s(z1)) -> c_2(*'#(z0,z1)) -'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)) F#(z0,s(z1),z2) -> c_6(ODD#(s(z1))) F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2)),*'#(z0,z2)) F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2),*'#(z0,z0)) F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2),HALF#(s(z1))) HALF#(s(s(z0))) -> c_12(HALF#(z0)) ODD#(s(s(z0))) -> c_19(ODD#(z0)) POW#(z0,z1) -> c_20(F#(z0,z1,s(0()))) - Weak TRS: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) *'(z0,0()) -> c2() *'(z0,s(z1)) -> c3(*'(z0,z1)) -(z0,0()) -> z0 -(s(z0),s(z1)) -> -(z0,z1) -'(z0,0()) -> c() -'(s(z0),s(z1)) -> c1(-'(z0,z1)) F(z0,0(),z1) -> c15() F(z0,s(z1),z2) -> c16(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)),ODD(s(z1))) F(z0,s(z1),z2) -> c17(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(z0,z1,*(z0,z2)) ,*'(z0,z2)) F(z0,s(z1),z2) -> c18(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(*(z0,z0),half(s(z1)),z2) ,*'(z0,z0)) F(z0,s(z1),z2) -> c19(IF(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) ,F(*(z0,z0),half(s(z1)),z2) ,HALF(s(z1))) HALF(0()) -> c11() HALF(s(0())) -> c12() HALF(s(s(z0))) -> c13(HALF(z0)) IF(false(),z0,z1) -> c5() IF(false(),z0,z1) -> c7() IF(true(),z0,z1) -> c4() IF(true(),z0,z1) -> c6() ODD(0()) -> c8() ODD(s(0())) -> c9() ODD(s(s(z0))) -> c10(ODD(z0)) POW(z0,z1) -> c14(F(z0,z1,s(0()))) f(z0,0(),z1) -> z1 f(z0,s(z1),z2) -> if(odd(s(z1)),f(z0,z1,*(z0,z2)),f(*(z0,z0),half(s(z1)),z2)) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) if(false(),z0,z1) -> z1 if(false(),z0,z1) -> false() if(true(),z0,z1) -> z0 if(true(),z0,z1) -> true() odd(0()) -> false() odd(s(0())) -> true() odd(s(s(z0))) -> odd(z0) pow(z0,z1) -> f(z0,z1,s(0())) - Signature: {*/2,*'/2,-/2,-'/2,F/3,HALF/1,IF/3,ODD/1,POW/2,f/3,half/1,if/3,odd/1,pow/2,*#/2,*'#/2,-#/2,-'#/2,F#/3 ,HALF#/1,IF#/3,ODD#/1,POW#/2,f#/3,half#/1,if#/3,odd#/1,pow#/2} / {+/2,0/0,c/0,c1/1,c10/1,c11/0,c12/0,c13/1 ,c14/1,c15/0,c16/2,c17/3,c18/3,c19/3,c2/0,c3/1,c4/0,c5/0,c6/0,c7/0,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1 ,c_3/0,c_4/1,c_5/0,c_6/1,c_7/2,c_8/2,c_9/2,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0 ,c_19/1,c_20/1,c_21/0,c_22/1,c_23/0,c_24/1,c_25/0,c_26/7,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0 ,c_34/0,c_35/0,c_36/1,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,*'#,-#,-'#,F#,HALF#,IF#,ODD#,POW#,f#,half#,if#,odd# ,pow#} and constructors {+,0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c3,c4,c5,c6,c7,c8,c9,false,s ,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) *'#(z0,s(z1)) -> c_2(*'#(z0,z1)) -'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)) F#(z0,s(z1),z2) -> c_6(ODD#(s(z1))) F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2)),*'#(z0,z2)) F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2),*'#(z0,z0)) F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2),HALF#(s(z1))) HALF#(s(s(z0))) -> c_12(HALF#(z0)) ODD#(s(s(z0))) -> c_19(ODD#(z0)) POW#(z0,z1) -> c_20(F#(z0,z1,s(0()))) ** Step 1.b:6: RemoveHeads. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: *'#(z0,s(z1)) -> c_2(*'#(z0,z1)) -'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)) F#(z0,s(z1),z2) -> c_6(ODD#(s(z1))) F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2)),*'#(z0,z2)) F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2),*'#(z0,z0)) F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2),HALF#(s(z1))) HALF#(s(s(z0))) -> c_12(HALF#(z0)) ODD#(s(s(z0))) -> c_19(ODD#(z0)) POW#(z0,z1) -> c_20(F#(z0,z1,s(0()))) - Weak TRS: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) - Signature: {*/2,*'/2,-/2,-'/2,F/3,HALF/1,IF/3,ODD/1,POW/2,f/3,half/1,if/3,odd/1,pow/2,*#/2,*'#/2,-#/2,-'#/2,F#/3 ,HALF#/1,IF#/3,ODD#/1,POW#/2,f#/3,half#/1,if#/3,odd#/1,pow#/2} / {+/2,0/0,c/0,c1/1,c10/1,c11/0,c12/0,c13/1 ,c14/1,c15/0,c16/2,c17/3,c18/3,c19/3,c2/0,c3/1,c4/0,c5/0,c6/0,c7/0,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1 ,c_3/0,c_4/1,c_5/0,c_6/1,c_7/2,c_8/2,c_9/2,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0 ,c_19/1,c_20/1,c_21/0,c_22/1,c_23/0,c_24/1,c_25/0,c_26/7,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0 ,c_34/0,c_35/0,c_36/1,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,*'#,-#,-'#,F#,HALF#,IF#,ODD#,POW#,f#,half#,if#,odd# ,pow#} and constructors {+,0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c3,c4,c5,c6,c7,c8,c9,false,s ,true} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:*'#(z0,s(z1)) -> c_2(*'#(z0,z1)) -->_1 *'#(z0,s(z1)) -> c_2(*'#(z0,z1)):1 2:S:-'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)) -->_1 -'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)):2 3:S:F#(z0,s(z1),z2) -> c_6(ODD#(s(z1))) -->_1 ODD#(s(s(z0))) -> c_19(ODD#(z0)):8 4:S:F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2)),*'#(z0,z2)) -->_1 F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2),HALF#(s(z1))):6 -->_1 F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2),*'#(z0,z0)):5 -->_1 F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2)),*'#(z0,z2)):4 -->_1 F#(z0,s(z1),z2) -> c_6(ODD#(s(z1))):3 -->_2 *'#(z0,s(z1)) -> c_2(*'#(z0,z1)):1 5:S:F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2),*'#(z0,z0)) -->_1 F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2),HALF#(s(z1))):6 -->_1 F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2),*'#(z0,z0)):5 -->_1 F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2)),*'#(z0,z2)):4 -->_1 F#(z0,s(z1),z2) -> c_6(ODD#(s(z1))):3 -->_2 *'#(z0,s(z1)) -> c_2(*'#(z0,z1)):1 6:S:F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2),HALF#(s(z1))) -->_2 HALF#(s(s(z0))) -> c_12(HALF#(z0)):7 -->_1 F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2),HALF#(s(z1))):6 -->_1 F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2),*'#(z0,z0)):5 -->_1 F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2)),*'#(z0,z2)):4 -->_1 F#(z0,s(z1),z2) -> c_6(ODD#(s(z1))):3 7:S:HALF#(s(s(z0))) -> c_12(HALF#(z0)) -->_1 HALF#(s(s(z0))) -> c_12(HALF#(z0)):7 8:S:ODD#(s(s(z0))) -> c_19(ODD#(z0)) -->_1 ODD#(s(s(z0))) -> c_19(ODD#(z0)):8 9:S:POW#(z0,z1) -> c_20(F#(z0,z1,s(0()))) -->_1 F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2),HALF#(s(z1))):6 -->_1 F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2),*'#(z0,z0)):5 -->_1 F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2)),*'#(z0,z2)):4 -->_1 F#(z0,s(z1),z2) -> c_6(ODD#(s(z1))):3 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). [(9,POW#(z0,z1) -> c_20(F#(z0,z1,s(0()))))] ** Step 1.b:7: DecomposeDG. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: *'#(z0,s(z1)) -> c_2(*'#(z0,z1)) -'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)) F#(z0,s(z1),z2) -> c_6(ODD#(s(z1))) F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2)),*'#(z0,z2)) F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2),*'#(z0,z0)) F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2),HALF#(s(z1))) HALF#(s(s(z0))) -> c_12(HALF#(z0)) ODD#(s(s(z0))) -> c_19(ODD#(z0)) - Weak TRS: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) - Signature: {*/2,*'/2,-/2,-'/2,F/3,HALF/1,IF/3,ODD/1,POW/2,f/3,half/1,if/3,odd/1,pow/2,*#/2,*'#/2,-#/2,-'#/2,F#/3 ,HALF#/1,IF#/3,ODD#/1,POW#/2,f#/3,half#/1,if#/3,odd#/1,pow#/2} / {+/2,0/0,c/0,c1/1,c10/1,c11/0,c12/0,c13/1 ,c14/1,c15/0,c16/2,c17/3,c18/3,c19/3,c2/0,c3/1,c4/0,c5/0,c6/0,c7/0,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1 ,c_3/0,c_4/1,c_5/0,c_6/1,c_7/2,c_8/2,c_9/2,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0 ,c_19/1,c_20/1,c_21/0,c_22/1,c_23/0,c_24/1,c_25/0,c_26/7,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0 ,c_34/0,c_35/0,c_36/1,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,*'#,-#,-'#,F#,HALF#,IF#,ODD#,POW#,f#,half#,if#,odd# ,pow#} and constructors {+,0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c3,c4,c5,c6,c7,c8,c9,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 -'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)) F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2)),*'#(z0,z2)) F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2),*'#(z0,z0)) F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2),HALF#(s(z1))) and a lower component *'#(z0,s(z1)) -> c_2(*'#(z0,z1)) F#(z0,s(z1),z2) -> c_6(ODD#(s(z1))) HALF#(s(s(z0))) -> c_12(HALF#(z0)) ODD#(s(s(z0))) -> c_19(ODD#(z0)) Further, following extension rules are added to the lower component. -'#(s(z0),s(z1)) -> -'#(z0,z1) F#(z0,s(z1),z2) -> *'#(z0,z0) F#(z0,s(z1),z2) -> *'#(z0,z2) F#(z0,s(z1),z2) -> F#(z0,z1,*(z0,z2)) F#(z0,s(z1),z2) -> F#(*(z0,z0),half(s(z1)),z2) F#(z0,s(z1),z2) -> HALF#(s(z1)) *** Step 1.b:7.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: -'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)) F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2)),*'#(z0,z2)) F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2),*'#(z0,z0)) F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2),HALF#(s(z1))) - Weak TRS: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) - Signature: {*/2,*'/2,-/2,-'/2,F/3,HALF/1,IF/3,ODD/1,POW/2,f/3,half/1,if/3,odd/1,pow/2,*#/2,*'#/2,-#/2,-'#/2,F#/3 ,HALF#/1,IF#/3,ODD#/1,POW#/2,f#/3,half#/1,if#/3,odd#/1,pow#/2} / {+/2,0/0,c/0,c1/1,c10/1,c11/0,c12/0,c13/1 ,c14/1,c15/0,c16/2,c17/3,c18/3,c19/3,c2/0,c3/1,c4/0,c5/0,c6/0,c7/0,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1 ,c_3/0,c_4/1,c_5/0,c_6/1,c_7/2,c_8/2,c_9/2,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0 ,c_19/1,c_20/1,c_21/0,c_22/1,c_23/0,c_24/1,c_25/0,c_26/7,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0 ,c_34/0,c_35/0,c_36/1,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,*'#,-#,-'#,F#,HALF#,IF#,ODD#,POW#,f#,half#,if#,odd# ,pow#} and constructors {+,0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c3,c4,c5,c6,c7,c8,c9,false,s ,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:-'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)) -->_1 -'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)):1 2:S:F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2)),*'#(z0,z2)) -->_1 F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2),HALF#(s(z1))):4 -->_1 F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2),*'#(z0,z0)):3 -->_1 F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2)),*'#(z0,z2)):2 3:S:F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2),*'#(z0,z0)) -->_1 F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2),HALF#(s(z1))):4 -->_1 F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2),*'#(z0,z0)):3 -->_1 F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2)),*'#(z0,z2)):2 4:S:F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2),HALF#(s(z1))) -->_1 F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2),HALF#(s(z1))):4 -->_1 F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2),*'#(z0,z0)):3 -->_1 F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2)),*'#(z0,z2)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2))) F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2)) F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2)) *** Step 1.b:7.a:2: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: -'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)) F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2))) F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2)) F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2)) - Weak TRS: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) - Signature: {*/2,*'/2,-/2,-'/2,F/3,HALF/1,IF/3,ODD/1,POW/2,f/3,half/1,if/3,odd/1,pow/2,*#/2,*'#/2,-#/2,-'#/2,F#/3 ,HALF#/1,IF#/3,ODD#/1,POW#/2,f#/3,half#/1,if#/3,odd#/1,pow#/2} / {+/2,0/0,c/0,c1/1,c10/1,c11/0,c12/0,c13/1 ,c14/1,c15/0,c16/2,c17/3,c18/3,c19/3,c2/0,c3/1,c4/0,c5/0,c6/0,c7/0,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1 ,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1,c_8/1,c_9/1,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0 ,c_19/1,c_20/1,c_21/0,c_22/1,c_23/0,c_24/1,c_25/0,c_26/7,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0 ,c_34/0,c_35/0,c_36/1,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,*'#,-#,-'#,F#,HALF#,IF#,ODD#,POW#,f#,half#,if#,odd# ,pow#} and constructors {+,0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c3,c4,c5,c6,c7,c8,c9,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(+) = {1}, uargs(s) = {1}, uargs(F#) = {1,2,3}, uargs(c_4) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(*) = [1] x2 + [0] p(*') = [1] x2 + [0] p(+) = [1] x1 + [2] p(-) = [1] x2 + [1] p(-') = [1] x2 + [1] p(0) = [6] p(F) = [2] x2 + [0] p(HALF) = [2] p(IF) = [2] x2 + [1] p(ODD) = [2] x1 + [0] p(POW) = [4] x1 + [4] x2 + [4] p(c) = [2] p(c1) = [1] p(c10) = [1] x1 + [1] p(c11) = [1] p(c12) = [0] p(c13) = [0] p(c14) = [1] x1 + [1] p(c15) = [0] p(c16) = [1] x1 + [1] x2 + [0] p(c17) = [1] x2 + [1] x3 + [0] p(c18) = [1] x1 + [1] x2 + [1] x3 + [0] p(c19) = [1] x1 + [1] x2 + [1] x3 + [0] p(c2) = [0] p(c3) = [1] x1 + [0] p(c4) = [0] p(c5) = [0] p(c6) = [0] p(c7) = [0] p(c8) = [0] p(c9) = [0] p(f) = [0] p(false) = [0] p(half) = [1] x1 + [2] p(if) = [0] p(odd) = [0] p(pow) = [0] p(s) = [1] x1 + [2] p(true) = [0] p(*#) = [0] p(*'#) = [0] p(-#) = [0] p(-'#) = [5] p(F#) = [1] x1 + [1] x2 + [1] x3 + [0] p(HALF#) = [0] p(IF#) = [0] p(ODD#) = [0] p(POW#) = [0] p(f#) = [0] p(half#) = [0] p(if#) = [0] p(odd#) = [0] p(pow#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] x1 + [2] p(c_5) = [1] p(c_6) = [4] x1 + [1] p(c_7) = [1] x1 + [1] p(c_8) = [1] x1 + [3] p(c_9) = [1] x1 + [1] p(c_10) = [1] p(c_11) = [2] p(c_12) = [1] p(c_13) = [1] p(c_14) = [1] p(c_15) = [1] p(c_16) = [2] p(c_17) = [0] p(c_18) = [1] p(c_19) = [1] x1 + [1] p(c_20) = [1] x1 + [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [0] p(c_26) = [2] x1 + [1] x3 + [1] x4 + [2] x5 + [1] x7 + [4] p(c_27) = [4] p(c_28) = [1] p(c_29) = [2] x1 + [2] p(c_30) = [1] p(c_31) = [0] p(c_32) = [0] p(c_33) = [0] p(c_34) = [0] p(c_35) = [2] p(c_36) = [2] x1 + [0] p(c_37) = [1] x1 + [1] Following rules are strictly oriented: F#(z0,s(z1),z2) = [1] z0 + [1] z1 + [1] z2 + [2] > [1] z0 + [1] z1 + [1] z2 + [1] = c_7(F#(z0,z1,*(z0,z2))) Following rules are (at-least) weakly oriented: -'#(s(z0),s(z1)) = [5] >= [7] = c_4(-'#(z0,z1)) F#(z0,s(z1),z2) = [1] z0 + [1] z1 + [1] z2 + [2] >= [1] z0 + [1] z1 + [1] z2 + [7] = c_8(F#(*(z0,z0),half(s(z1)),z2)) F#(z0,s(z1),z2) = [1] z0 + [1] z1 + [1] z2 + [2] >= [1] z0 + [1] z1 + [1] z2 + [5] = c_9(F#(*(z0,z0),half(s(z1)),z2)) *(z0,0()) = [6] >= [6] = 0() *(z0,s(z1)) = [1] z1 + [2] >= [1] z1 + [2] = +(*(z0,z1),z0) half(0()) = [8] >= [6] = 0() half(s(0())) = [10] >= [6] = 0() half(s(s(z0))) = [1] z0 + [6] >= [1] z0 + [4] = s(half(z0)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:7.a:3: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: -'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)) F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2)) F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2)) - Weak DPs: F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2))) - Weak TRS: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) - Signature: {*/2,*'/2,-/2,-'/2,F/3,HALF/1,IF/3,ODD/1,POW/2,f/3,half/1,if/3,odd/1,pow/2,*#/2,*'#/2,-#/2,-'#/2,F#/3 ,HALF#/1,IF#/3,ODD#/1,POW#/2,f#/3,half#/1,if#/3,odd#/1,pow#/2} / {+/2,0/0,c/0,c1/1,c10/1,c11/0,c12/0,c13/1 ,c14/1,c15/0,c16/2,c17/3,c18/3,c19/3,c2/0,c3/1,c4/0,c5/0,c6/0,c7/0,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1 ,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1,c_8/1,c_9/1,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0 ,c_19/1,c_20/1,c_21/0,c_22/1,c_23/0,c_24/1,c_25/0,c_26/7,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0 ,c_34/0,c_35/0,c_36/1,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,*'#,-#,-'#,F#,HALF#,IF#,ODD#,POW#,f#,half#,if#,odd# ,pow#} and constructors {+,0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c3,c4,c5,c6,c7,c8,c9,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(+) = {1}, uargs(s) = {1}, uargs(F#) = {1,2,3}, uargs(c_4) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(*) = [0] p(*') = [1] x2 + [1] p(+) = [1] x1 + [0] p(-) = [1] x1 + [1] p(-') = [2] x1 + [4] x2 + [1] p(0) = [0] p(F) = [1] p(HALF) = [0] p(IF) = [1] x1 + [1] p(ODD) = [0] p(POW) = [4] x1 + [1] x2 + [0] p(c) = [2] p(c1) = [0] p(c10) = [1] p(c11) = [0] p(c12) = [0] p(c13) = [1] x1 + [0] p(c14) = [1] x1 + [0] p(c15) = [0] p(c16) = [1] x1 + [1] x2 + [0] p(c17) = [1] x1 + [1] x2 + [1] x3 + [0] p(c18) = [1] x1 + [1] x2 + [1] x3 + [0] p(c19) = [1] x1 + [1] x2 + [1] x3 + [0] p(c2) = [0] p(c3) = [1] x1 + [0] p(c4) = [0] p(c5) = [0] p(c6) = [0] p(c7) = [0] p(c8) = [0] p(c9) = [0] p(f) = [0] p(false) = [0] p(half) = [1] x1 + [0] p(if) = [0] p(odd) = [0] p(pow) = [0] p(s) = [1] x1 + [4] p(true) = [0] p(*#) = [0] p(*'#) = [0] p(-#) = [0] p(-'#) = [1] x1 + [0] p(F#) = [1] x1 + [1] x2 + [1] x3 + [0] p(HALF#) = [0] p(IF#) = [0] p(ODD#) = [0] p(POW#) = [2] x2 + [0] p(f#) = [4] x1 + [4] x3 + [4] p(half#) = [2] p(if#) = [2] x1 + [4] x2 + [1] x3 + [0] p(odd#) = [0] p(pow#) = [2] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [2] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [1] x1 + [1] p(c_8) = [1] x1 + [5] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [1] x1 + [4] p(c_13) = [0] p(c_14) = [1] p(c_15) = [4] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [1] x1 + [2] p(c_20) = [1] x1 + [4] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [1] x1 + [1] p(c_25) = [4] p(c_26) = [1] x1 + [2] x2 + [4] x3 + [2] x4 + [1] x5 + [1] x6 + [1] x7 + [2] p(c_27) = [0] p(c_28) = [0] p(c_29) = [4] x1 + [1] p(c_30) = [4] p(c_31) = [1] p(c_32) = [0] p(c_33) = [1] p(c_34) = [0] p(c_35) = [4] p(c_36) = [1] x1 + [0] p(c_37) = [4] x1 + [0] Following rules are strictly oriented: -'#(s(z0),s(z1)) = [1] z0 + [4] > [1] z0 + [0] = c_4(-'#(z0,z1)) Following rules are (at-least) weakly oriented: F#(z0,s(z1),z2) = [1] z0 + [1] z1 + [1] z2 + [4] >= [1] z0 + [1] z1 + [1] = c_7(F#(z0,z1,*(z0,z2))) F#(z0,s(z1),z2) = [1] z0 + [1] z1 + [1] z2 + [4] >= [1] z1 + [1] z2 + [9] = c_8(F#(*(z0,z0),half(s(z1)),z2)) F#(z0,s(z1),z2) = [1] z0 + [1] z1 + [1] z2 + [4] >= [1] z1 + [1] z2 + [4] = c_9(F#(*(z0,z0),half(s(z1)),z2)) *(z0,0()) = [0] >= [0] = 0() *(z0,s(z1)) = [0] >= [0] = +(*(z0,z1),z0) half(0()) = [0] >= [0] = 0() half(s(0())) = [4] >= [0] = 0() half(s(s(z0))) = [1] z0 + [8] >= [1] z0 + [4] = s(half(z0)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:7.a:4: Ara. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2)) F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2)) - Weak DPs: -'#(s(z0),s(z1)) -> c_4(-'#(z0,z1)) F#(z0,s(z1),z2) -> c_7(F#(z0,z1,*(z0,z2))) - Weak TRS: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) - Signature: {*/2,*'/2,-/2,-'/2,F/3,HALF/1,IF/3,ODD/1,POW/2,f/3,half/1,if/3,odd/1,pow/2,*#/2,*'#/2,-#/2,-'#/2,F#/3 ,HALF#/1,IF#/3,ODD#/1,POW#/2,f#/3,half#/1,if#/3,odd#/1,pow#/2} / {+/2,0/0,c/0,c1/1,c10/1,c11/0,c12/0,c13/1 ,c14/1,c15/0,c16/2,c17/3,c18/3,c19/3,c2/0,c3/1,c4/0,c5/0,c6/0,c7/0,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1 ,c_3/0,c_4/1,c_5/0,c_6/1,c_7/1,c_8/1,c_9/1,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0 ,c_19/1,c_20/1,c_21/0,c_22/1,c_23/0,c_24/1,c_25/0,c_26/7,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0 ,c_34/0,c_35/0,c_36/1,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,*'#,-#,-'#,F#,HALF#,IF#,ODD#,POW#,f#,half#,if#,odd# ,pow#} and constructors {+,0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c3,c4,c5,c6,c7,c8,c9,false,s ,true} + Applied Processor: Ara {minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "*") :: ["A"(0) x "A"(0)] -(0)-> "A"(0) F (TrsFun "+") :: ["A"(0) x "A"(0)] -(0)-> "A"(0) F (TrsFun "0") :: [] -(0)-> "A"(0) F (TrsFun "0") :: [] -(0)-> "A"(1) F (TrsFun "0") :: [] -(0)-> "A"(2) F (TrsFun "half") :: ["A"(1)] -(0)-> "A"(2) F (TrsFun "s") :: ["A"(2)] -(2)-> "A"(2) F (TrsFun "s") :: ["A"(0)] -(0)-> "A"(0) F (TrsFun "s") :: ["A"(1)] -(1)-> "A"(1) F (DpFun "-'") :: ["A"(0) x "A"(0)] -(0)-> "A"(0) F (DpFun "F") :: ["A"(0) x "A"(2) x "A"(0)] -(0)-> "A"(0) F (ComFun 4) :: ["A"(0)] -(0)-> "A"(0) F (ComFun 7) :: ["A"(0)] -(0)-> "A"(0) F (ComFun 8) :: ["A"(0)] -(0)-> "A"(0) F (ComFun 9) :: ["A"(0)] -(0)-> "A"(0) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: F#(z0,s(z1),z2) -> c_8(F#(*(z0,z0),half(s(z1)),z2)) F#(z0,s(z1),z2) -> c_9(F#(*(z0,z0),half(s(z1)),z2)) 2. Weak: *** Step 1.b:7.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: *'#(z0,s(z1)) -> c_2(*'#(z0,z1)) F#(z0,s(z1),z2) -> c_6(ODD#(s(z1))) HALF#(s(s(z0))) -> c_12(HALF#(z0)) ODD#(s(s(z0))) -> c_19(ODD#(z0)) - Weak DPs: -'#(s(z0),s(z1)) -> -'#(z0,z1) F#(z0,s(z1),z2) -> *'#(z0,z0) F#(z0,s(z1),z2) -> *'#(z0,z2) F#(z0,s(z1),z2) -> F#(z0,z1,*(z0,z2)) F#(z0,s(z1),z2) -> F#(*(z0,z0),half(s(z1)),z2) F#(z0,s(z1),z2) -> HALF#(s(z1)) - Weak TRS: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) - Signature: {*/2,*'/2,-/2,-'/2,F/3,HALF/1,IF/3,ODD/1,POW/2,f/3,half/1,if/3,odd/1,pow/2,*#/2,*'#/2,-#/2,-'#/2,F#/3 ,HALF#/1,IF#/3,ODD#/1,POW#/2,f#/3,half#/1,if#/3,odd#/1,pow#/2} / {+/2,0/0,c/0,c1/1,c10/1,c11/0,c12/0,c13/1 ,c14/1,c15/0,c16/2,c17/3,c18/3,c19/3,c2/0,c3/1,c4/0,c5/0,c6/0,c7/0,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1 ,c_3/0,c_4/1,c_5/0,c_6/1,c_7/2,c_8/2,c_9/2,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0 ,c_19/1,c_20/1,c_21/0,c_22/1,c_23/0,c_24/1,c_25/0,c_26/7,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0 ,c_34/0,c_35/0,c_36/1,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,*'#,-#,-'#,F#,HALF#,IF#,ODD#,POW#,f#,half#,if#,odd# ,pow#} and constructors {+,0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c3,c4,c5,c6,c7,c8,c9,false,s ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:*'#(z0,s(z1)) -> c_2(*'#(z0,z1)) -->_1 *'#(z0,s(z1)) -> c_2(*'#(z0,z1)):1 2:S:F#(z0,s(z1),z2) -> c_6(ODD#(s(z1))) -->_1 ODD#(s(s(z0))) -> c_19(ODD#(z0)):4 3:S:HALF#(s(s(z0))) -> c_12(HALF#(z0)) -->_1 HALF#(s(s(z0))) -> c_12(HALF#(z0)):3 4:S:ODD#(s(s(z0))) -> c_19(ODD#(z0)) -->_1 ODD#(s(s(z0))) -> c_19(ODD#(z0)):4 5:W:-'#(s(z0),s(z1)) -> -'#(z0,z1) -->_1 -'#(s(z0),s(z1)) -> -'#(z0,z1):5 6:W:F#(z0,s(z1),z2) -> *'#(z0,z0) -->_1 *'#(z0,s(z1)) -> c_2(*'#(z0,z1)):1 7:W:F#(z0,s(z1),z2) -> *'#(z0,z2) -->_1 *'#(z0,s(z1)) -> c_2(*'#(z0,z1)):1 8:W:F#(z0,s(z1),z2) -> F#(z0,z1,*(z0,z2)) -->_1 F#(z0,s(z1),z2) -> HALF#(s(z1)):10 -->_1 F#(z0,s(z1),z2) -> F#(*(z0,z0),half(s(z1)),z2):9 -->_1 F#(z0,s(z1),z2) -> F#(z0,z1,*(z0,z2)):8 -->_1 F#(z0,s(z1),z2) -> *'#(z0,z2):7 -->_1 F#(z0,s(z1),z2) -> *'#(z0,z0):6 -->_1 F#(z0,s(z1),z2) -> c_6(ODD#(s(z1))):2 9:W:F#(z0,s(z1),z2) -> F#(*(z0,z0),half(s(z1)),z2) -->_1 F#(z0,s(z1),z2) -> HALF#(s(z1)):10 -->_1 F#(z0,s(z1),z2) -> F#(*(z0,z0),half(s(z1)),z2):9 -->_1 F#(z0,s(z1),z2) -> F#(z0,z1,*(z0,z2)):8 -->_1 F#(z0,s(z1),z2) -> *'#(z0,z2):7 -->_1 F#(z0,s(z1),z2) -> *'#(z0,z0):6 -->_1 F#(z0,s(z1),z2) -> c_6(ODD#(s(z1))):2 10:W:F#(z0,s(z1),z2) -> HALF#(s(z1)) -->_1 HALF#(s(s(z0))) -> c_12(HALF#(z0)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: -'#(s(z0),s(z1)) -> -'#(z0,z1) *** Step 1.b:7.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: *'#(z0,s(z1)) -> c_2(*'#(z0,z1)) F#(z0,s(z1),z2) -> c_6(ODD#(s(z1))) HALF#(s(s(z0))) -> c_12(HALF#(z0)) ODD#(s(s(z0))) -> c_19(ODD#(z0)) - Weak DPs: F#(z0,s(z1),z2) -> *'#(z0,z0) F#(z0,s(z1),z2) -> *'#(z0,z2) F#(z0,s(z1),z2) -> F#(z0,z1,*(z0,z2)) F#(z0,s(z1),z2) -> F#(*(z0,z0),half(s(z1)),z2) F#(z0,s(z1),z2) -> HALF#(s(z1)) - Weak TRS: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) - Signature: {*/2,*'/2,-/2,-'/2,F/3,HALF/1,IF/3,ODD/1,POW/2,f/3,half/1,if/3,odd/1,pow/2,*#/2,*'#/2,-#/2,-'#/2,F#/3 ,HALF#/1,IF#/3,ODD#/1,POW#/2,f#/3,half#/1,if#/3,odd#/1,pow#/2} / {+/2,0/0,c/0,c1/1,c10/1,c11/0,c12/0,c13/1 ,c14/1,c15/0,c16/2,c17/3,c18/3,c19/3,c2/0,c3/1,c4/0,c5/0,c6/0,c7/0,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1 ,c_3/0,c_4/1,c_5/0,c_6/1,c_7/2,c_8/2,c_9/2,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0 ,c_19/1,c_20/1,c_21/0,c_22/1,c_23/0,c_24/1,c_25/0,c_26/7,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0 ,c_34/0,c_35/0,c_36/1,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,*'#,-#,-'#,F#,HALF#,IF#,ODD#,POW#,f#,half#,if#,odd# ,pow#} and constructors {+,0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c3,c4,c5,c6,c7,c8,c9,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_6) = {1}, uargs(c_12) = {1}, uargs(c_19) = {1} Following symbols are considered usable: {*,half,*#,*'#,-#,-'#,F#,HALF#,IF#,ODD#,POW#,f#,half#,if#,odd#,pow#} TcT has computed the following interpretation: p(*) = [0] p(*') = [0] p(+) = [1] x1 + [0] p(-) = [0] p(-') = [0] p(0) = [0] p(F) = [0] p(HALF) = [0] p(IF) = [0] p(ODD) = [0] p(POW) = [0] p(c) = [0] p(c1) = [1] x1 + [0] p(c10) = [1] x1 + [0] p(c11) = [0] p(c12) = [0] p(c13) = [0] p(c14) = [0] p(c15) = [2] p(c16) = [1] p(c17) = [0] p(c18) = [1] p(c19) = [8] p(c2) = [2] p(c3) = [2] p(c4) = [2] p(c5) = [0] p(c6) = [1] p(c7) = [4] p(c8) = [0] p(c9) = [0] p(f) = [1] x1 + [4] x2 + [1] p(false) = [1] p(half) = [1] x1 + [0] p(if) = [1] p(odd) = [1] x1 + [8] p(pow) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [1] p(true) = [2] p(*#) = [1] x1 + [4] p(*'#) = [4] x1 + [0] p(-#) = [2] x1 + [1] p(-'#) = [1] x2 + [2] p(F#) = [4] x1 + [8] x2 + [8] x3 + [2] p(HALF#) = [2] x1 + [8] p(IF#) = [1] x1 + [1] x2 + [1] p(ODD#) = [8] x1 + [0] p(POW#) = [1] x2 + [8] p(f#) = [1] x1 + [1] x2 + [4] p(half#) = [1] p(if#) = [1] x1 + [8] p(odd#) = [2] p(pow#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] p(c_4) = [2] p(c_5) = [4] p(c_6) = [1] x1 + [2] p(c_7) = [2] x2 + [4] p(c_8) = [4] x1 + [2] x2 + [1] p(c_9) = [1] x1 + [1] p(c_10) = [4] p(c_11) = [2] p(c_12) = [1] x1 + [2] p(c_13) = [0] p(c_14) = [8] p(c_15) = [0] p(c_16) = [0] p(c_17) = [1] p(c_18) = [1] p(c_19) = [1] x1 + [8] p(c_20) = [4] x1 + [0] p(c_21) = [4] p(c_22) = [1] x1 + [2] p(c_23) = [0] p(c_24) = [1] p(c_25) = [2] p(c_26) = [8] x2 + [8] x7 + [0] p(c_27) = [8] p(c_28) = [0] p(c_29) = [1] x1 + [2] p(c_30) = [1] p(c_31) = [4] p(c_32) = [2] p(c_33) = [0] p(c_34) = [1] p(c_35) = [2] p(c_36) = [0] p(c_37) = [8] Following rules are strictly oriented: HALF#(s(s(z0))) = [2] z0 + [12] > [2] z0 + [10] = c_12(HALF#(z0)) ODD#(s(s(z0))) = [8] z0 + [16] > [8] z0 + [8] = c_19(ODD#(z0)) Following rules are (at-least) weakly oriented: *'#(z0,s(z1)) = [4] z0 + [0] >= [4] z0 + [0] = c_2(*'#(z0,z1)) F#(z0,s(z1),z2) = [4] z0 + [8] z1 + [8] z2 + [10] >= [4] z0 + [0] = *'#(z0,z0) F#(z0,s(z1),z2) = [4] z0 + [8] z1 + [8] z2 + [10] >= [4] z0 + [0] = *'#(z0,z2) F#(z0,s(z1),z2) = [4] z0 + [8] z1 + [8] z2 + [10] >= [4] z0 + [8] z1 + [2] = F#(z0,z1,*(z0,z2)) F#(z0,s(z1),z2) = [4] z0 + [8] z1 + [8] z2 + [10] >= [8] z1 + [8] z2 + [10] = F#(*(z0,z0),half(s(z1)),z2) F#(z0,s(z1),z2) = [4] z0 + [8] z1 + [8] z2 + [10] >= [2] z1 + [10] = HALF#(s(z1)) F#(z0,s(z1),z2) = [4] z0 + [8] z1 + [8] z2 + [10] >= [8] z1 + [10] = c_6(ODD#(s(z1))) *(z0,0()) = [0] >= [0] = 0() *(z0,s(z1)) = [0] >= [0] = +(*(z0,z1),z0) half(0()) = [0] >= [0] = 0() half(s(0())) = [1] >= [0] = 0() half(s(s(z0))) = [1] z0 + [2] >= [1] z0 + [1] = s(half(z0)) *** Step 1.b:7.b:3: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: *'#(z0,s(z1)) -> c_2(*'#(z0,z1)) F#(z0,s(z1),z2) -> c_6(ODD#(s(z1))) - Weak DPs: F#(z0,s(z1),z2) -> *'#(z0,z0) F#(z0,s(z1),z2) -> *'#(z0,z2) F#(z0,s(z1),z2) -> F#(z0,z1,*(z0,z2)) F#(z0,s(z1),z2) -> F#(*(z0,z0),half(s(z1)),z2) F#(z0,s(z1),z2) -> HALF#(s(z1)) HALF#(s(s(z0))) -> c_12(HALF#(z0)) ODD#(s(s(z0))) -> c_19(ODD#(z0)) - Weak TRS: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) - Signature: {*/2,*'/2,-/2,-'/2,F/3,HALF/1,IF/3,ODD/1,POW/2,f/3,half/1,if/3,odd/1,pow/2,*#/2,*'#/2,-#/2,-'#/2,F#/3 ,HALF#/1,IF#/3,ODD#/1,POW#/2,f#/3,half#/1,if#/3,odd#/1,pow#/2} / {+/2,0/0,c/0,c1/1,c10/1,c11/0,c12/0,c13/1 ,c14/1,c15/0,c16/2,c17/3,c18/3,c19/3,c2/0,c3/1,c4/0,c5/0,c6/0,c7/0,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1 ,c_3/0,c_4/1,c_5/0,c_6/1,c_7/2,c_8/2,c_9/2,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0 ,c_19/1,c_20/1,c_21/0,c_22/1,c_23/0,c_24/1,c_25/0,c_26/7,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0 ,c_34/0,c_35/0,c_36/1,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,*'#,-#,-'#,F#,HALF#,IF#,ODD#,POW#,f#,half#,if#,odd# ,pow#} and constructors {+,0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c3,c4,c5,c6,c7,c8,c9,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_6) = {1}, uargs(c_12) = {1}, uargs(c_19) = {1} Following symbols are considered usable: {*,*#,*'#,-#,-'#,F#,HALF#,IF#,ODD#,POW#,f#,half#,if#,odd#,pow#} TcT has computed the following interpretation: p(*) = [1] x2 + [0] p(*') = [0] p(+) = [1] x1 + [0] p(-) = [0] p(-') = [0] p(0) = [0] p(F) = [0] p(HALF) = [0] p(IF) = [0] p(ODD) = [0] p(POW) = [0] p(c) = [0] p(c1) = [1] x1 + [0] p(c10) = [1] x1 + [0] p(c11) = [0] p(c12) = [0] p(c13) = [1] x1 + [0] p(c14) = [1] x1 + [0] p(c15) = [0] p(c16) = [1] x1 + [1] x2 + [0] p(c17) = [1] x1 + [1] x2 + [1] x3 + [0] p(c18) = [1] x1 + [1] x2 + [1] x3 + [0] p(c19) = [1] x1 + [1] x2 + [1] x3 + [0] p(c2) = [0] p(c3) = [1] x1 + [0] p(c4) = [0] p(c5) = [0] p(c6) = [0] p(c7) = [0] p(c8) = [0] p(c9) = [0] p(f) = [0] p(false) = [0] p(half) = [0] p(if) = [0] p(odd) = [0] p(pow) = [0] p(s) = [1] x1 + [8] p(true) = [0] p(*#) = [0] p(*'#) = [2] x2 + [0] p(-#) = [0] p(-'#) = [1] x2 + [0] p(F#) = [8] x1 + [2] x3 + [1] p(HALF#) = [0] p(IF#) = [0] p(ODD#) = [0] p(POW#) = [0] p(f#) = [0] p(half#) = [0] p(if#) = [0] p(odd#) = [0] p(pow#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [8] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [1] x1 + [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [4] x1 + [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [1] p(c_29) = [8] p(c_30) = [1] p(c_31) = [1] p(c_32) = [1] p(c_33) = [0] p(c_34) = [1] p(c_35) = [2] p(c_36) = [1] x1 + [0] p(c_37) = [1] x1 + [1] Following rules are strictly oriented: *'#(z0,s(z1)) = [2] z1 + [16] > [2] z1 + [8] = c_2(*'#(z0,z1)) F#(z0,s(z1),z2) = [8] z0 + [2] z2 + [1] > [0] = c_6(ODD#(s(z1))) Following rules are (at-least) weakly oriented: F#(z0,s(z1),z2) = [8] z0 + [2] z2 + [1] >= [2] z0 + [0] = *'#(z0,z0) F#(z0,s(z1),z2) = [8] z0 + [2] z2 + [1] >= [2] z2 + [0] = *'#(z0,z2) F#(z0,s(z1),z2) = [8] z0 + [2] z2 + [1] >= [8] z0 + [2] z2 + [1] = F#(z0,z1,*(z0,z2)) F#(z0,s(z1),z2) = [8] z0 + [2] z2 + [1] >= [8] z0 + [2] z2 + [1] = F#(*(z0,z0),half(s(z1)),z2) F#(z0,s(z1),z2) = [8] z0 + [2] z2 + [1] >= [0] = HALF#(s(z1)) HALF#(s(s(z0))) = [0] >= [0] = c_12(HALF#(z0)) ODD#(s(s(z0))) = [0] >= [0] = c_19(ODD#(z0)) *(z0,0()) = [0] >= [0] = 0() *(z0,s(z1)) = [1] z1 + [8] >= [1] z1 + [0] = +(*(z0,z1),z0) *** Step 1.b:7.b:4: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: *'#(z0,s(z1)) -> c_2(*'#(z0,z1)) F#(z0,s(z1),z2) -> *'#(z0,z0) F#(z0,s(z1),z2) -> *'#(z0,z2) F#(z0,s(z1),z2) -> F#(z0,z1,*(z0,z2)) F#(z0,s(z1),z2) -> F#(*(z0,z0),half(s(z1)),z2) F#(z0,s(z1),z2) -> HALF#(s(z1)) F#(z0,s(z1),z2) -> c_6(ODD#(s(z1))) HALF#(s(s(z0))) -> c_12(HALF#(z0)) ODD#(s(s(z0))) -> c_19(ODD#(z0)) - Weak TRS: *(z0,0()) -> 0() *(z0,s(z1)) -> +(*(z0,z1),z0) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) - Signature: {*/2,*'/2,-/2,-'/2,F/3,HALF/1,IF/3,ODD/1,POW/2,f/3,half/1,if/3,odd/1,pow/2,*#/2,*'#/2,-#/2,-'#/2,F#/3 ,HALF#/1,IF#/3,ODD#/1,POW#/2,f#/3,half#/1,if#/3,odd#/1,pow#/2} / {+/2,0/0,c/0,c1/1,c10/1,c11/0,c12/0,c13/1 ,c14/1,c15/0,c16/2,c17/3,c18/3,c19/3,c2/0,c3/1,c4/0,c5/0,c6/0,c7/0,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/1 ,c_3/0,c_4/1,c_5/0,c_6/1,c_7/2,c_8/2,c_9/2,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0 ,c_19/1,c_20/1,c_21/0,c_22/1,c_23/0,c_24/1,c_25/0,c_26/7,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0 ,c_34/0,c_35/0,c_36/1,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,*'#,-#,-'#,F#,HALF#,IF#,ODD#,POW#,f#,half#,if#,odd# ,pow#} and constructors {+,0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c3,c4,c5,c6,c7,c8,c9,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))