WORST_CASE(Omega(n^1),O(n^3)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^3)) + Considered Problem: - Strict TRS: F(0(),z0,0(),z1) -> c10() F(0(),z0,s(z1),z2) -> c11() F(s(z0),0(),z1,z2) -> c12(F(z0,z2,minus(z1,s(z0)),z2),MINUS(z1,s(z0))) F(s(z0),s(z1),z2,z3) -> c13(IF(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE(z0,z1)) F(s(z0),s(z1),z2,z3) -> c14(IF(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F(s(z0),minus(z1,z0),z2,z3) ,MINUS(z1,z0)) F(s(z0),s(z1),z2,z3) -> c15(IF(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),F(z0,z3,z2,z3)) IF(false(),z0,z1) -> c7() IF(true(),z0,z1) -> c6() LE(0(),z0) -> c3() LE(s(z0),0()) -> c4() LE(s(z0),s(z1)) -> c5(LE(z0,z1)) MINUS(0(),z0) -> c() MINUS(s(z0),0()) -> c1() MINUS(s(z0),s(z1)) -> c2(MINUS(z0,z1)) PERFECTP(0()) -> c8() PERFECTP(s(z0)) -> c9(F(z0,s(0()),s(z0),s(z0))) - Weak TRS: f(0(),z0,0(),z1) -> true() f(0(),z0,s(z1),z2) -> false() f(s(z0),0(),z1,z2) -> f(z0,z2,minus(z1,s(z0)),z2) f(s(z0),s(z1),z2,z3) -> if(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) perfectp(0()) -> false() perfectp(s(z0)) -> f(z0,s(0()),s(z0),s(z0)) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2 ,c14/3,c15/2,c2/1,c3/0,c4/0,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {F,IF,LE,MINUS,PERFECTP,f,if,le,minus ,perfectp} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,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: F(0(),z0,0(),z1) -> c10() F(0(),z0,s(z1),z2) -> c11() F(s(z0),0(),z1,z2) -> c12(F(z0,z2,minus(z1,s(z0)),z2),MINUS(z1,s(z0))) F(s(z0),s(z1),z2,z3) -> c13(IF(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE(z0,z1)) F(s(z0),s(z1),z2,z3) -> c14(IF(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F(s(z0),minus(z1,z0),z2,z3) ,MINUS(z1,z0)) F(s(z0),s(z1),z2,z3) -> c15(IF(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),F(z0,z3,z2,z3)) IF(false(),z0,z1) -> c7() IF(true(),z0,z1) -> c6() LE(0(),z0) -> c3() LE(s(z0),0()) -> c4() LE(s(z0),s(z1)) -> c5(LE(z0,z1)) MINUS(0(),z0) -> c() MINUS(s(z0),0()) -> c1() MINUS(s(z0),s(z1)) -> c2(MINUS(z0,z1)) PERFECTP(0()) -> c8() PERFECTP(s(z0)) -> c9(F(z0,s(0()),s(z0),s(z0))) - Weak TRS: f(0(),z0,0(),z1) -> true() f(0(),z0,s(z1),z2) -> false() f(s(z0),0(),z1,z2) -> f(z0,z2,minus(z1,s(z0)),z2) f(s(z0),s(z1),z2,z3) -> if(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) perfectp(0()) -> false() perfectp(s(z0)) -> f(z0,s(0()),s(z0),s(z0)) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2 ,c14/3,c15/2,c2/1,c3/0,c4/0,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {F,IF,LE,MINUS,PERFECTP,f,if,le,minus ,perfectp} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,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: F(0(),z0,0(),z1) -> c10() F(0(),z0,s(z1),z2) -> c11() F(s(z0),0(),z1,z2) -> c12(F(z0,z2,minus(z1,s(z0)),z2),MINUS(z1,s(z0))) F(s(z0),s(z1),z2,z3) -> c13(IF(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE(z0,z1)) F(s(z0),s(z1),z2,z3) -> c14(IF(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F(s(z0),minus(z1,z0),z2,z3) ,MINUS(z1,z0)) F(s(z0),s(z1),z2,z3) -> c15(IF(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),F(z0,z3,z2,z3)) IF(false(),z0,z1) -> c7() IF(true(),z0,z1) -> c6() LE(0(),z0) -> c3() LE(s(z0),0()) -> c4() LE(s(z0),s(z1)) -> c5(LE(z0,z1)) MINUS(0(),z0) -> c() MINUS(s(z0),0()) -> c1() MINUS(s(z0),s(z1)) -> c2(MINUS(z0,z1)) PERFECTP(0()) -> c8() PERFECTP(s(z0)) -> c9(F(z0,s(0()),s(z0),s(z0))) - Weak TRS: f(0(),z0,0(),z1) -> true() f(0(),z0,s(z1),z2) -> false() f(s(z0),0(),z1,z2) -> f(z0,z2,minus(z1,s(z0)),z2) f(s(z0),s(z1),z2,z3) -> if(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) perfectp(0()) -> false() perfectp(s(z0)) -> f(z0,s(0()),s(z0),s(z0)) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2 ,c14/3,c15/2,c2/1,c3/0,c4/0,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {F,IF,LE,MINUS,PERFECTP,f,if,le,minus ,perfectp} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,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: LE(x,y){x -> s(x),y -> s(y)} = LE(s(x),s(y)) ->^+ c5(LE(x,y)) = C[LE(x,y) = LE(x,y){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: F(0(),z0,0(),z1) -> c10() F(0(),z0,s(z1),z2) -> c11() F(s(z0),0(),z1,z2) -> c12(F(z0,z2,minus(z1,s(z0)),z2),MINUS(z1,s(z0))) F(s(z0),s(z1),z2,z3) -> c13(IF(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE(z0,z1)) F(s(z0),s(z1),z2,z3) -> c14(IF(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F(s(z0),minus(z1,z0),z2,z3) ,MINUS(z1,z0)) F(s(z0),s(z1),z2,z3) -> c15(IF(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),F(z0,z3,z2,z3)) IF(false(),z0,z1) -> c7() IF(true(),z0,z1) -> c6() LE(0(),z0) -> c3() LE(s(z0),0()) -> c4() LE(s(z0),s(z1)) -> c5(LE(z0,z1)) MINUS(0(),z0) -> c() MINUS(s(z0),0()) -> c1() MINUS(s(z0),s(z1)) -> c2(MINUS(z0,z1)) PERFECTP(0()) -> c8() PERFECTP(s(z0)) -> c9(F(z0,s(0()),s(z0),s(z0))) - Weak TRS: f(0(),z0,0(),z1) -> true() f(0(),z0,s(z1),z2) -> false() f(s(z0),0(),z1,z2) -> f(z0,z2,minus(z1,s(z0)),z2) f(s(z0),s(z1),z2,z3) -> if(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) perfectp(0()) -> false() perfectp(s(z0)) -> f(z0,s(0()),s(z0),s(z0)) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2 ,c14/3,c15/2,c2/1,c3/0,c4/0,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {F,IF,LE,MINUS,PERFECTP,f,if,le,minus ,perfectp} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs F#(0(),z0,0(),z1) -> c_1() F#(0(),z0,s(z1),z2) -> c_2() F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_4(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE#(z0,z1)) F#(s(z0),s(z1),z2,z3) -> c_5(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(s(z0),minus(z1,z0),z2,z3) ,MINUS#(z1,z0)) F#(s(z0),s(z1),z2,z3) -> c_6(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),F#(z0,z3,z2,z3)) IF#(false(),z0,z1) -> c_7() IF#(true(),z0,z1) -> c_8() LE#(0(),z0) -> c_9() LE#(s(z0),0()) -> c_10() LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) MINUS#(0(),z0) -> c_12() MINUS#(s(z0),0()) -> c_13() MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) PERFECTP#(0()) -> c_15() PERFECTP#(s(z0)) -> c_16(F#(z0,s(0()),s(z0),s(z0))) Weak DPs f#(0(),z0,0(),z1) -> c_17() f#(0(),z0,s(z1),z2) -> c_18() f#(s(z0),0(),z1,z2) -> c_19(f#(z0,z2,minus(z1,s(z0)),z2)) f#(s(z0),s(z1),z2,z3) -> c_20(if#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3))) if#(false(),z0,z1) -> c_21() if#(true(),z0,z1) -> c_22() le#(0(),z0) -> c_23() le#(s(z0),0()) -> c_24() le#(s(z0),s(z1)) -> c_25(le#(z0,z1)) minus#(0(),z0) -> c_26() minus#(s(z0),0()) -> c_27() minus#(s(z0),s(z1)) -> c_28(minus#(z0,z1)) perfectp#(0()) -> c_29() perfectp#(s(z0)) -> c_30(f#(z0,s(0()),s(z0),s(z0))) and mark the set of starting terms. ** Step 1.b:2: UsableRules. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: F#(0(),z0,0(),z1) -> c_1() F#(0(),z0,s(z1),z2) -> c_2() F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_4(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE#(z0,z1)) F#(s(z0),s(z1),z2,z3) -> c_5(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(s(z0),minus(z1,z0),z2,z3) ,MINUS#(z1,z0)) F#(s(z0),s(z1),z2,z3) -> c_6(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),F#(z0,z3,z2,z3)) IF#(false(),z0,z1) -> c_7() IF#(true(),z0,z1) -> c_8() LE#(0(),z0) -> c_9() LE#(s(z0),0()) -> c_10() LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) MINUS#(0(),z0) -> c_12() MINUS#(s(z0),0()) -> c_13() MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) PERFECTP#(0()) -> c_15() PERFECTP#(s(z0)) -> c_16(F#(z0,s(0()),s(z0),s(z0))) - Strict TRS: F(0(),z0,0(),z1) -> c10() F(0(),z0,s(z1),z2) -> c11() F(s(z0),0(),z1,z2) -> c12(F(z0,z2,minus(z1,s(z0)),z2),MINUS(z1,s(z0))) F(s(z0),s(z1),z2,z3) -> c13(IF(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE(z0,z1)) F(s(z0),s(z1),z2,z3) -> c14(IF(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F(s(z0),minus(z1,z0),z2,z3) ,MINUS(z1,z0)) F(s(z0),s(z1),z2,z3) -> c15(IF(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),F(z0,z3,z2,z3)) IF(false(),z0,z1) -> c7() IF(true(),z0,z1) -> c6() LE(0(),z0) -> c3() LE(s(z0),0()) -> c4() LE(s(z0),s(z1)) -> c5(LE(z0,z1)) MINUS(0(),z0) -> c() MINUS(s(z0),0()) -> c1() MINUS(s(z0),s(z1)) -> c2(MINUS(z0,z1)) PERFECTP(0()) -> c8() PERFECTP(s(z0)) -> c9(F(z0,s(0()),s(z0),s(z0))) - Weak DPs: f#(0(),z0,0(),z1) -> c_17() f#(0(),z0,s(z1),z2) -> c_18() f#(s(z0),0(),z1,z2) -> c_19(f#(z0,z2,minus(z1,s(z0)),z2)) f#(s(z0),s(z1),z2,z3) -> c_20(if#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3))) if#(false(),z0,z1) -> c_21() if#(true(),z0,z1) -> c_22() le#(0(),z0) -> c_23() le#(s(z0),0()) -> c_24() le#(s(z0),s(z1)) -> c_25(le#(z0,z1)) minus#(0(),z0) -> c_26() minus#(s(z0),0()) -> c_27() minus#(s(z0),s(z1)) -> c_28(minus#(z0,z1)) perfectp#(0()) -> c_29() perfectp#(s(z0)) -> c_30(f#(z0,s(0()),s(z0),s(z0))) - Weak TRS: f(0(),z0,0(),z1) -> true() f(0(),z0,s(z1),z2) -> false() f(s(z0),0(),z1,z2) -> f(z0,z2,minus(z1,s(z0)),z2) f(s(z0),s(z1),z2,z3) -> if(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) perfectp(0()) -> false() perfectp(s(z0)) -> f(z0,s(0()),s(z0),s(z0)) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/3,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: f(0(),z0,0(),z1) -> true() f(0(),z0,s(z1),z2) -> false() f(s(z0),0(),z1,z2) -> f(z0,z2,minus(z1,s(z0)),z2) f(s(z0),s(z1),z2,z3) -> if(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) F#(0(),z0,0(),z1) -> c_1() F#(0(),z0,s(z1),z2) -> c_2() F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_4(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE#(z0,z1)) F#(s(z0),s(z1),z2,z3) -> c_5(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(s(z0),minus(z1,z0),z2,z3) ,MINUS#(z1,z0)) F#(s(z0),s(z1),z2,z3) -> c_6(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),F#(z0,z3,z2,z3)) IF#(false(),z0,z1) -> c_7() IF#(true(),z0,z1) -> c_8() LE#(0(),z0) -> c_9() LE#(s(z0),0()) -> c_10() LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) MINUS#(0(),z0) -> c_12() MINUS#(s(z0),0()) -> c_13() MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) PERFECTP#(0()) -> c_15() PERFECTP#(s(z0)) -> c_16(F#(z0,s(0()),s(z0),s(z0))) f#(0(),z0,0(),z1) -> c_17() f#(0(),z0,s(z1),z2) -> c_18() f#(s(z0),0(),z1,z2) -> c_19(f#(z0,z2,minus(z1,s(z0)),z2)) f#(s(z0),s(z1),z2,z3) -> c_20(if#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3))) if#(false(),z0,z1) -> c_21() if#(true(),z0,z1) -> c_22() le#(0(),z0) -> c_23() le#(s(z0),0()) -> c_24() le#(s(z0),s(z1)) -> c_25(le#(z0,z1)) minus#(0(),z0) -> c_26() minus#(s(z0),0()) -> c_27() minus#(s(z0),s(z1)) -> c_28(minus#(z0,z1)) perfectp#(0()) -> c_29() perfectp#(s(z0)) -> c_30(f#(z0,s(0()),s(z0),s(z0))) ** Step 1.b:3: PredecessorEstimation. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: F#(0(),z0,0(),z1) -> c_1() F#(0(),z0,s(z1),z2) -> c_2() F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_4(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE#(z0,z1)) F#(s(z0),s(z1),z2,z3) -> c_5(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(s(z0),minus(z1,z0),z2,z3) ,MINUS#(z1,z0)) F#(s(z0),s(z1),z2,z3) -> c_6(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),F#(z0,z3,z2,z3)) IF#(false(),z0,z1) -> c_7() IF#(true(),z0,z1) -> c_8() LE#(0(),z0) -> c_9() LE#(s(z0),0()) -> c_10() LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) MINUS#(0(),z0) -> c_12() MINUS#(s(z0),0()) -> c_13() MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) PERFECTP#(0()) -> c_15() PERFECTP#(s(z0)) -> c_16(F#(z0,s(0()),s(z0),s(z0))) - Weak DPs: f#(0(),z0,0(),z1) -> c_17() f#(0(),z0,s(z1),z2) -> c_18() f#(s(z0),0(),z1,z2) -> c_19(f#(z0,z2,minus(z1,s(z0)),z2)) f#(s(z0),s(z1),z2,z3) -> c_20(if#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3))) if#(false(),z0,z1) -> c_21() if#(true(),z0,z1) -> c_22() le#(0(),z0) -> c_23() le#(s(z0),0()) -> c_24() le#(s(z0),s(z1)) -> c_25(le#(z0,z1)) minus#(0(),z0) -> c_26() minus#(s(z0),0()) -> c_27() minus#(s(z0),s(z1)) -> c_28(minus#(z0,z1)) perfectp#(0()) -> c_29() perfectp#(s(z0)) -> c_30(f#(z0,s(0()),s(z0),s(z0))) - Weak TRS: f(0(),z0,0(),z1) -> true() f(0(),z0,s(z1),z2) -> false() f(s(z0),0(),z1,z2) -> f(z0,z2,minus(z1,s(z0)),z2) f(s(z0),s(z1),z2,z3) -> if(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/3,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,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 {1,2,7,8,9,10,12,13,15} by application of Pre({1,2,7,8,9,10,12,13,15}) = {3,4,5,6,11,14,16}. Here rules are labelled as follows: 1: F#(0(),z0,0(),z1) -> c_1() 2: F#(0(),z0,s(z1),z2) -> c_2() 3: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) 4: F#(s(z0),s(z1),z2,z3) -> c_4(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE#(z0,z1)) 5: F#(s(z0),s(z1),z2,z3) -> c_5(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(s(z0),minus(z1,z0),z2,z3) ,MINUS#(z1,z0)) 6: F#(s(z0),s(z1),z2,z3) -> c_6(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),F#(z0,z3,z2,z3)) 7: IF#(false(),z0,z1) -> c_7() 8: IF#(true(),z0,z1) -> c_8() 9: LE#(0(),z0) -> c_9() 10: LE#(s(z0),0()) -> c_10() 11: LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) 12: MINUS#(0(),z0) -> c_12() 13: MINUS#(s(z0),0()) -> c_13() 14: MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) 15: PERFECTP#(0()) -> c_15() 16: PERFECTP#(s(z0)) -> c_16(F#(z0,s(0()),s(z0),s(z0))) 17: f#(0(),z0,0(),z1) -> c_17() 18: f#(0(),z0,s(z1),z2) -> c_18() 19: f#(s(z0),0(),z1,z2) -> c_19(f#(z0,z2,minus(z1,s(z0)),z2)) 20: f#(s(z0),s(z1),z2,z3) -> c_20(if#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3))) 21: if#(false(),z0,z1) -> c_21() 22: if#(true(),z0,z1) -> c_22() 23: le#(0(),z0) -> c_23() 24: le#(s(z0),0()) -> c_24() 25: le#(s(z0),s(z1)) -> c_25(le#(z0,z1)) 26: minus#(0(),z0) -> c_26() 27: minus#(s(z0),0()) -> c_27() 28: minus#(s(z0),s(z1)) -> c_28(minus#(z0,z1)) 29: perfectp#(0()) -> c_29() 30: perfectp#(s(z0)) -> c_30(f#(z0,s(0()),s(z0),s(z0))) ** Step 1.b:4: RemoveWeakSuffixes. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_4(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE#(z0,z1)) F#(s(z0),s(z1),z2,z3) -> c_5(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(s(z0),minus(z1,z0),z2,z3) ,MINUS#(z1,z0)) F#(s(z0),s(z1),z2,z3) -> c_6(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),F#(z0,z3,z2,z3)) LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) PERFECTP#(s(z0)) -> c_16(F#(z0,s(0()),s(z0),s(z0))) - Weak DPs: F#(0(),z0,0(),z1) -> c_1() F#(0(),z0,s(z1),z2) -> c_2() IF#(false(),z0,z1) -> c_7() IF#(true(),z0,z1) -> c_8() LE#(0(),z0) -> c_9() LE#(s(z0),0()) -> c_10() MINUS#(0(),z0) -> c_12() MINUS#(s(z0),0()) -> c_13() PERFECTP#(0()) -> c_15() f#(0(),z0,0(),z1) -> c_17() f#(0(),z0,s(z1),z2) -> c_18() f#(s(z0),0(),z1,z2) -> c_19(f#(z0,z2,minus(z1,s(z0)),z2)) f#(s(z0),s(z1),z2,z3) -> c_20(if#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3))) if#(false(),z0,z1) -> c_21() if#(true(),z0,z1) -> c_22() le#(0(),z0) -> c_23() le#(s(z0),0()) -> c_24() le#(s(z0),s(z1)) -> c_25(le#(z0,z1)) minus#(0(),z0) -> c_26() minus#(s(z0),0()) -> c_27() minus#(s(z0),s(z1)) -> c_28(minus#(z0,z1)) perfectp#(0()) -> c_29() perfectp#(s(z0)) -> c_30(f#(z0,s(0()),s(z0),s(z0))) - Weak TRS: f(0(),z0,0(),z1) -> true() f(0(),z0,s(z1),z2) -> false() f(s(z0),0(),z1,z2) -> f(z0,z2,minus(z1,s(z0)),z2) f(s(z0),s(z1),z2,z3) -> if(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/3,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) -->_2 MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)):6 -->_1 F#(s(z0),s(z1),z2,z3) -> c_6(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(z0,z3,z2,z3)):4 -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(s(z0),minus(z1,z0),z2,z3) ,MINUS#(z1,z0)):3 -->_1 F#(s(z0),s(z1),z2,z3) -> c_4(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE#(z0,z1)):2 -->_2 MINUS#(0(),z0) -> c_12():14 -->_1 F#(0(),z0,s(z1),z2) -> c_2():9 -->_1 F#(0(),z0,0(),z1) -> c_1():8 -->_1 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))):1 2:S:F#(s(z0),s(z1),z2,z3) -> c_4(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE#(z0,z1)) -->_2 LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)):5 -->_2 LE#(s(z0),0()) -> c_10():13 -->_2 LE#(0(),z0) -> c_9():12 -->_1 IF#(true(),z0,z1) -> c_8():11 -->_1 IF#(false(),z0,z1) -> c_7():10 3:S:F#(s(z0),s(z1),z2,z3) -> c_5(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(s(z0),minus(z1,z0),z2,z3) ,MINUS#(z1,z0)) -->_3 MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)):6 -->_2 F#(s(z0),s(z1),z2,z3) -> c_6(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(z0,z3,z2,z3)):4 -->_3 MINUS#(s(z0),0()) -> c_13():15 -->_3 MINUS#(0(),z0) -> c_12():14 -->_1 IF#(true(),z0,z1) -> c_8():11 -->_1 IF#(false(),z0,z1) -> c_7():10 -->_2 F#(s(z0),s(z1),z2,z3) -> c_5(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(s(z0),minus(z1,z0),z2,z3) ,MINUS#(z1,z0)):3 -->_2 F#(s(z0),s(z1),z2,z3) -> c_4(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE#(z0,z1)):2 -->_2 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))):1 4:S:F#(s(z0),s(z1),z2,z3) -> c_6(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),F#(z0,z3,z2,z3)) -->_1 IF#(true(),z0,z1) -> c_8():11 -->_1 IF#(false(),z0,z1) -> c_7():10 -->_2 F#(0(),z0,s(z1),z2) -> c_2():9 -->_2 F#(0(),z0,0(),z1) -> c_1():8 -->_2 F#(s(z0),s(z1),z2,z3) -> c_6(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(z0,z3,z2,z3)):4 -->_2 F#(s(z0),s(z1),z2,z3) -> c_5(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(s(z0),minus(z1,z0),z2,z3) ,MINUS#(z1,z0)):3 -->_2 F#(s(z0),s(z1),z2,z3) -> c_4(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE#(z0,z1)):2 -->_2 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))):1 5:S:LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) -->_1 LE#(s(z0),0()) -> c_10():13 -->_1 LE#(0(),z0) -> c_9():12 -->_1 LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)):5 6:S:MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) -->_1 MINUS#(s(z0),0()) -> c_13():15 -->_1 MINUS#(0(),z0) -> c_12():14 -->_1 MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)):6 7:S:PERFECTP#(s(z0)) -> c_16(F#(z0,s(0()),s(z0),s(z0))) -->_1 F#(0(),z0,s(z1),z2) -> c_2():9 -->_1 F#(s(z0),s(z1),z2,z3) -> c_6(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(z0,z3,z2,z3)):4 -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(s(z0),minus(z1,z0),z2,z3) ,MINUS#(z1,z0)):3 -->_1 F#(s(z0),s(z1),z2,z3) -> c_4(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE#(z0,z1)):2 8:W:F#(0(),z0,0(),z1) -> c_1() 9:W:F#(0(),z0,s(z1),z2) -> c_2() 10:W:IF#(false(),z0,z1) -> c_7() 11:W:IF#(true(),z0,z1) -> c_8() 12:W:LE#(0(),z0) -> c_9() 13:W:LE#(s(z0),0()) -> c_10() 14:W:MINUS#(0(),z0) -> c_12() 15:W:MINUS#(s(z0),0()) -> c_13() 16:W:PERFECTP#(0()) -> c_15() 17:W:f#(0(),z0,0(),z1) -> c_17() 18:W:f#(0(),z0,s(z1),z2) -> c_18() 19:W:f#(s(z0),0(),z1,z2) -> c_19(f#(z0,z2,minus(z1,s(z0)),z2)) -->_1 f#(s(z0),s(z1),z2,z3) -> c_20(if#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3))):20 -->_1 f#(s(z0),0(),z1,z2) -> c_19(f#(z0,z2,minus(z1,s(z0)),z2)):19 -->_1 f#(0(),z0,s(z1),z2) -> c_18():18 -->_1 f#(0(),z0,0(),z1) -> c_17():17 20:W:f#(s(z0),s(z1),z2,z3) -> c_20(if#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3))) -->_1 if#(true(),z0,z1) -> c_22():22 -->_1 if#(false(),z0,z1) -> c_21():21 21:W:if#(false(),z0,z1) -> c_21() 22:W:if#(true(),z0,z1) -> c_22() 23:W:le#(0(),z0) -> c_23() 24:W:le#(s(z0),0()) -> c_24() 25:W:le#(s(z0),s(z1)) -> c_25(le#(z0,z1)) -->_1 le#(s(z0),s(z1)) -> c_25(le#(z0,z1)):25 -->_1 le#(s(z0),0()) -> c_24():24 -->_1 le#(0(),z0) -> c_23():23 26:W:minus#(0(),z0) -> c_26() 27:W:minus#(s(z0),0()) -> c_27() 28:W:minus#(s(z0),s(z1)) -> c_28(minus#(z0,z1)) -->_1 minus#(s(z0),s(z1)) -> c_28(minus#(z0,z1)):28 -->_1 minus#(s(z0),0()) -> c_27():27 -->_1 minus#(0(),z0) -> c_26():26 29:W:perfectp#(0()) -> c_29() 30:W:perfectp#(s(z0)) -> c_30(f#(z0,s(0()),s(z0),s(z0))) -->_1 f#(s(z0),s(z1),z2,z3) -> c_20(if#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3))):20 -->_1 f#(0(),z0,s(z1),z2) -> c_18():18 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 30: perfectp#(s(z0)) -> c_30(f#(z0,s(0()),s(z0),s(z0))) 29: perfectp#(0()) -> c_29() 28: minus#(s(z0),s(z1)) -> c_28(minus#(z0,z1)) 27: minus#(s(z0),0()) -> c_27() 26: minus#(0(),z0) -> c_26() 25: le#(s(z0),s(z1)) -> c_25(le#(z0,z1)) 24: le#(s(z0),0()) -> c_24() 23: le#(0(),z0) -> c_23() 19: f#(s(z0),0(),z1,z2) -> c_19(f#(z0,z2,minus(z1,s(z0)),z2)) 20: f#(s(z0),s(z1),z2,z3) -> c_20(if#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3))) 21: if#(false(),z0,z1) -> c_21() 22: if#(true(),z0,z1) -> c_22() 18: f#(0(),z0,s(z1),z2) -> c_18() 17: f#(0(),z0,0(),z1) -> c_17() 16: PERFECTP#(0()) -> c_15() 12: LE#(0(),z0) -> c_9() 13: LE#(s(z0),0()) -> c_10() 8: F#(0(),z0,0(),z1) -> c_1() 9: F#(0(),z0,s(z1),z2) -> c_2() 10: IF#(false(),z0,z1) -> c_7() 11: IF#(true(),z0,z1) -> c_8() 14: MINUS#(0(),z0) -> c_12() 15: MINUS#(s(z0),0()) -> c_13() ** Step 1.b:5: SimplifyRHS. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_4(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE#(z0,z1)) F#(s(z0),s(z1),z2,z3) -> c_5(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(s(z0),minus(z1,z0),z2,z3) ,MINUS#(z1,z0)) F#(s(z0),s(z1),z2,z3) -> c_6(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),F#(z0,z3,z2,z3)) LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) PERFECTP#(s(z0)) -> c_16(F#(z0,s(0()),s(z0),s(z0))) - Weak TRS: f(0(),z0,0(),z1) -> true() f(0(),z0,s(z1),z2) -> false() f(s(z0),0(),z1,z2) -> f(z0,z2,minus(z1,s(z0)),z2) f(s(z0),s(z1),z2,z3) -> if(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/2,c_5/3,c_6/2,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) -->_2 MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)):6 -->_1 F#(s(z0),s(z1),z2,z3) -> c_6(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(z0,z3,z2,z3)):4 -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(s(z0),minus(z1,z0),z2,z3) ,MINUS#(z1,z0)):3 -->_1 F#(s(z0),s(z1),z2,z3) -> c_4(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE#(z0,z1)):2 -->_1 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))):1 2:S:F#(s(z0),s(z1),z2,z3) -> c_4(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE#(z0,z1)) -->_2 LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)):5 3:S:F#(s(z0),s(z1),z2,z3) -> c_5(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(s(z0),minus(z1,z0),z2,z3) ,MINUS#(z1,z0)) -->_3 MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)):6 -->_2 F#(s(z0),s(z1),z2,z3) -> c_6(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(z0,z3,z2,z3)):4 -->_2 F#(s(z0),s(z1),z2,z3) -> c_5(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(s(z0),minus(z1,z0),z2,z3) ,MINUS#(z1,z0)):3 -->_2 F#(s(z0),s(z1),z2,z3) -> c_4(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE#(z0,z1)):2 -->_2 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))):1 4:S:F#(s(z0),s(z1),z2,z3) -> c_6(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),F#(z0,z3,z2,z3)) -->_2 F#(s(z0),s(z1),z2,z3) -> c_6(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(z0,z3,z2,z3)):4 -->_2 F#(s(z0),s(z1),z2,z3) -> c_5(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(s(z0),minus(z1,z0),z2,z3) ,MINUS#(z1,z0)):3 -->_2 F#(s(z0),s(z1),z2,z3) -> c_4(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE#(z0,z1)):2 -->_2 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))):1 5:S:LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) -->_1 LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)):5 6:S:MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) -->_1 MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)):6 7:S:PERFECTP#(s(z0)) -> c_16(F#(z0,s(0()),s(z0),s(z0))) -->_1 F#(s(z0),s(z1),z2,z3) -> c_6(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(z0,z3,z2,z3)):4 -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) ,F#(s(z0),minus(z1,z0),z2,z3) ,MINUS#(z1,z0)):3 -->_1 F#(s(z0),s(z1),z2,z3) -> c_4(IF#(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)),LE#(z0,z1)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,z1)) F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) ** Step 1.b:6: UsableRules. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,z1)) F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) PERFECTP#(s(z0)) -> c_16(F#(z0,s(0()),s(z0),s(z0))) - Weak TRS: f(0(),z0,0(),z1) -> true() f(0(),z0,s(z1),z2) -> false() f(s(z0),0(),z1,z2) -> f(z0,z2,minus(z1,s(z0)),z2) f(s(z0),s(z1),z2,z3) -> if(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,z1)) F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) PERFECTP#(s(z0)) -> c_16(F#(z0,s(0()),s(z0),s(z0))) ** Step 1.b:7: RemoveHeads. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,z1)) F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) PERFECTP#(s(z0)) -> c_16(F#(z0,s(0()),s(z0),s(z0))) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) -->_2 MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)):6 -->_1 F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)):4 -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)):3 -->_1 F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,z1)):2 -->_1 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))):1 2:S:F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,z1)) -->_1 LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)):5 3:S:F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) -->_2 MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)):6 -->_1 F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)):4 -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)):3 -->_1 F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,z1)):2 -->_1 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))):1 4:S:F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) -->_1 F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)):4 -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)):3 -->_1 F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,z1)):2 -->_1 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))):1 5:S:LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) -->_1 LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)):5 6:S:MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) -->_1 MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)):6 7:S:PERFECTP#(s(z0)) -> c_16(F#(z0,s(0()),s(z0),s(z0))) -->_1 F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)):4 -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)):3 -->_1 F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,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). [(7,PERFECTP#(s(z0)) -> c_16(F#(z0,s(0()),s(z0),s(z0))))] ** Step 1.b:8: PredecessorEstimationCP. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,z1)) F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) 2: F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,z1)) 4: F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) 5: LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) The strictly oriented rules are moved into the weak component. *** Step 1.b:8.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,z1)) F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,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 first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1,2}, uargs(c_4) = {1}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_11) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus#,perfectp#} TcT has computed the following interpretation: p(0) = [2] p(F) = [8] x1 + [1] x2 + [1] x3 + [1] x4 + [1] p(IF) = [2] x1 + [1] x2 + [1] p(LE) = [1] x1 + [1] x2 + [0] p(MINUS) = [0] p(PERFECTP) = [1] x1 + [8] p(c) = [0] p(c1) = [2] p(c10) = [2] p(c11) = [4] p(c12) = [1] x1 + [8] p(c13) = [1] x1 + [1] x2 + [1] p(c14) = [1] x1 + [2] p(c15) = [1] x2 + [4] p(c2) = [1] x1 + [0] p(c3) = [8] p(c4) = [1] p(c5) = [0] p(c6) = [4] p(c7) = [0] p(c8) = [8] p(c9) = [1] p(f) = [4] x3 + [2] x4 + [0] p(false) = [0] p(if) = [2] p(le) = [8] x2 + [1] p(minus) = [0] p(perfectp) = [2] x1 + [1] p(s) = [1] x1 + [2] p(true) = [4] p(F#) = [8] x1 + [0] p(IF#) = [1] x1 + [0] p(LE#) = [2] x1 + [2] p(MINUS#) = [0] p(PERFECTP#) = [1] x1 + [0] p(f#) = [1] x2 + [4] x3 + [1] p(if#) = [1] x1 + [1] x2 + [2] p(le#) = [4] x1 + [2] x2 + [1] p(minus#) = [0] p(perfectp#) = [1] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] x1 + [8] x2 + [8] p(c_4) = [4] x1 + [0] p(c_5) = [1] x1 + [2] x2 + [0] p(c_6) = [1] x1 + [8] p(c_7) = [0] p(c_8) = [1] p(c_9) = [1] p(c_10) = [8] p(c_11) = [1] x1 + [0] p(c_12) = [1] p(c_13) = [4] p(c_14) = [4] x1 + [0] p(c_15) = [8] p(c_16) = [0] p(c_17) = [2] p(c_18) = [8] p(c_19) = [1] x1 + [1] p(c_20) = [1] x1 + [0] p(c_21) = [0] p(c_22) = [1] p(c_23) = [1] p(c_24) = [2] p(c_25) = [1] p(c_26) = [1] p(c_27) = [2] p(c_28) = [1] x1 + [1] p(c_29) = [4] p(c_30) = [0] Following rules are strictly oriented: F#(s(z0),0(),z1,z2) = [8] z0 + [16] > [8] z0 + [8] = c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) = [8] z0 + [16] > [8] z0 + [8] = c_4(LE#(z0,z1)) F#(s(z0),s(z1),z2,z3) = [8] z0 + [16] > [8] z0 + [8] = c_6(F#(z0,z3,z2,z3)) LE#(s(z0),s(z1)) = [2] z0 + [6] > [2] z0 + [2] = c_11(LE#(z0,z1)) Following rules are (at-least) weakly oriented: F#(s(z0),s(z1),z2,z3) = [8] z0 + [16] >= [8] z0 + [16] = c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) MINUS#(s(z0),s(z1)) = [0] >= [0] = c_14(MINUS#(z0,z1)) *** Step 1.b:8.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) - Weak DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,z1)) F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () *** Step 1.b:8.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) - Weak DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,z1)) F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) -->_1 F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)):5 -->_1 F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,z1)):4 -->_1 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))):3 -->_2 MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)):2 -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)):1 2:S:MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) -->_1 MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)):2 3:W:F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) -->_1 F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)):5 -->_1 F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,z1)):4 -->_1 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))):3 -->_2 MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)):2 -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)):1 4:W:F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,z1)) -->_1 LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)):6 5:W:F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) -->_1 F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)):5 -->_1 F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,z1)):4 -->_1 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))):3 -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)):1 6:W:LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) -->_1 LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)):6 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: F#(s(z0),s(z1),z2,z3) -> c_4(LE#(z0,z1)) 6: LE#(s(z0),s(z1)) -> c_11(LE#(z0,z1)) *** Step 1.b:8.b:2: DecomposeDG. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) - Weak DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) and a lower component MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) Further, following extension rules are added to the lower component. F#(s(z0),0(),z1,z2) -> F#(z0,z2,minus(z1,s(z0)),z2) F#(s(z0),0(),z1,z2) -> MINUS#(z1,s(z0)) F#(s(z0),s(z1),z2,z3) -> F#(z0,z3,z2,z3) F#(s(z0),s(z1),z2,z3) -> F#(s(z0),minus(z1,z0),z2,z3) F#(s(z0),s(z1),z2,z3) -> MINUS#(z1,z0) **** Step 1.b:8.b:2.a:1: PredecessorEstimationCP. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) - Weak DPs: F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) The strictly oriented rules are moved into the weak component. ***** Step 1.b:8.b:2.a:1.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) - Weak DPs: F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,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 first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus#,perfectp#} TcT has computed the following interpretation: p(0) = [0] p(F) = [1] x2 + [2] p(IF) = [1] x1 + [2] x2 + [1] x3 + [1] p(LE) = [1] x1 + [2] x2 + [0] p(MINUS) = [1] p(PERFECTP) = [2] p(c) = [0] p(c1) = [1] p(c10) = [0] p(c11) = [0] p(c12) = [1] x1 + [0] p(c13) = [0] p(c14) = [1] p(c15) = [0] p(c2) = [0] p(c3) = [1] p(c4) = [1] p(c5) = [1] x1 + [2] p(c6) = [2] p(c7) = [2] p(c8) = [0] p(c9) = [0] p(f) = [1] x1 + [1] x4 + [0] p(false) = [0] p(if) = [4] x2 + [1] p(le) = [0] p(minus) = [2] x1 + [8] p(perfectp) = [1] x1 + [1] p(s) = [1] x1 + [1] p(true) = [4] p(F#) = [2] x1 + [2] p(IF#) = [1] x1 + [1] x2 + [1] x3 + [0] p(LE#) = [1] x1 + [1] x2 + [0] p(MINUS#) = [1] p(PERFECTP#) = [1] p(f#) = [1] x1 + [8] x3 + [2] x4 + [0] p(if#) = [2] x1 + [1] x3 + [2] p(le#) = [2] x2 + [0] p(minus#) = [1] x1 + [1] x2 + [2] p(perfectp#) = [8] x1 + [0] p(c_1) = [2] p(c_2) = [1] p(c_3) = [1] x1 + [1] x2 + [0] p(c_4) = [1] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [2] p(c_7) = [2] p(c_8) = [1] p(c_9) = [0] p(c_10) = [4] p(c_11) = [8] x1 + [0] p(c_12) = [2] p(c_13) = [1] p(c_14) = [1] x1 + [0] p(c_15) = [1] p(c_16) = [4] p(c_17) = [0] p(c_18) = [0] p(c_19) = [4] x1 + [1] p(c_20) = [1] x1 + [2] p(c_21) = [2] p(c_22) = [0] p(c_23) = [0] p(c_24) = [1] p(c_25) = [2] p(c_26) = [1] p(c_27) = [1] p(c_28) = [4] p(c_29) = [1] p(c_30) = [2] x1 + [1] Following rules are strictly oriented: F#(s(z0),0(),z1,z2) = [2] z0 + [4] > [2] z0 + [3] = c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) Following rules are (at-least) weakly oriented: F#(s(z0),s(z1),z2,z3) = [2] z0 + [4] >= [2] z0 + [4] = c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) F#(s(z0),s(z1),z2,z3) = [2] z0 + [4] >= [2] z0 + [4] = c_6(F#(z0,z3,z2,z3)) ***** Step 1.b:8.b:2.a:1.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) - Weak DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () ***** Step 1.b:8.b:2.a:1.b:1: PredecessorEstimationCP. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) - Weak DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) The strictly oriented rules are moved into the weak component. ****** Step 1.b:8.b:2.a:1.b:1.a:1: NaturalPI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) - Weak DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {minus,F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus#,perfectp#} TcT has computed the following interpretation: p(0) = 2 p(F) = 0 p(IF) = 0 p(LE) = 0 p(MINUS) = 0 p(PERFECTP) = 0 p(c) = 0 p(c1) = 0 p(c10) = 0 p(c11) = 0 p(c12) = x1 + x2 p(c13) = x1 + x2 p(c14) = x1 + x2 + x3 p(c15) = x2 p(c2) = x1 p(c3) = 0 p(c4) = 0 p(c5) = 0 p(c6) = 0 p(c7) = 0 p(c8) = 0 p(c9) = 0 p(f) = x1*x3 + 2*x1*x4 + x2^2 + x3^2 + x4 p(false) = 1 p(if) = 2 + 2*x1 + 2*x1^2 + x2*x3 + 2*x2^2 + x3 p(le) = x1 + x1*x2 + x1^2 p(minus) = x1 p(perfectp) = 1 + x1 p(s) = 2 + x1 p(true) = 0 p(F#) = x1*x4 + 2*x2 + x3*x4 + 2*x3^2 + x4 p(IF#) = 1 + x2^2 + 2*x3^2 p(LE#) = 2 + 2*x1 + x1^2 p(MINUS#) = 1 p(PERFECTP#) = 2*x1^2 p(f#) = 2 + 2*x1^2 + x2*x3 + x3*x4 + x4 p(if#) = 1 + 2*x2^2 p(le#) = 2 + x2 + 2*x2^2 p(minus#) = 2*x1*x2 + x1^2 + x2 p(perfectp#) = x1^2 p(c_1) = 1 p(c_2) = 2 p(c_3) = x1 + x2 p(c_4) = 2 p(c_5) = x1 p(c_6) = 2 + x1 p(c_7) = 0 p(c_8) = 0 p(c_9) = 0 p(c_10) = 1 p(c_11) = 0 p(c_12) = 0 p(c_13) = 0 p(c_14) = 2 p(c_15) = 0 p(c_16) = x1 p(c_17) = 0 p(c_18) = 2 p(c_19) = 1 p(c_20) = 1 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) = 2 p(c_29) = 2 p(c_30) = 1 Following rules are strictly oriented: F#(s(z0),s(z1),z2,z3) = 4 + z0*z3 + 2*z1 + z2*z3 + 2*z2^2 + 3*z3 > z0*z3 + 2*z1 + z2*z3 + 2*z2^2 + 3*z3 = c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) Following rules are (at-least) weakly oriented: F#(s(z0),0(),z1,z2) = 4 + z0*z2 + z1*z2 + 2*z1^2 + 3*z2 >= 1 + z0*z2 + z1*z2 + 2*z1^2 + 3*z2 = c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) = 4 + z0*z3 + 2*z1 + z2*z3 + 2*z2^2 + 3*z3 >= 2 + z0*z3 + z2*z3 + 2*z2^2 + 3*z3 = c_6(F#(z0,z3,z2,z3)) minus(0(),z0) = 2 >= 2 = 0() minus(s(z0),0()) = 2 + z0 >= 2 + z0 = s(z0) minus(s(z0),s(z1)) = 2 + z0 >= z0 = minus(z0,z1) ****** Step 1.b:8.b:2.a:1.b:1.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () ****** Step 1.b:8.b:2.a:1.b:1.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) -->_1 F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)):3 -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)):2 -->_1 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))):1 2:W:F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) -->_1 F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)):3 -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)):2 -->_1 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))):1 3:W:F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) -->_1 F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)):3 -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)):2 -->_1 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2),MINUS#(z1,s(z0))) 3: F#(s(z0),s(z1),z2,z3) -> c_6(F#(z0,z3,z2,z3)) 2: F#(s(z0),s(z1),z2,z3) -> c_5(F#(s(z0),minus(z1,z0),z2,z3),MINUS#(z1,z0)) ****** Step 1.b:8.b:2.a:1.b:1.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,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). **** Step 1.b:8.b:2.b:1: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) - Weak DPs: F#(s(z0),0(),z1,z2) -> F#(z0,z2,minus(z1,s(z0)),z2) F#(s(z0),0(),z1,z2) -> MINUS#(z1,s(z0)) F#(s(z0),s(z1),z2,z3) -> F#(z0,z3,z2,z3) F#(s(z0),s(z1),z2,z3) -> F#(s(z0),minus(z1,z0),z2,z3) F#(s(z0),s(z1),z2,z3) -> MINUS#(z1,z0) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) The strictly oriented rules are moved into the weak component. ***** Step 1.b:8.b:2.b:1.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) - Weak DPs: F#(s(z0),0(),z1,z2) -> F#(z0,z2,minus(z1,s(z0)),z2) F#(s(z0),0(),z1,z2) -> MINUS#(z1,s(z0)) F#(s(z0),s(z1),z2,z3) -> F#(z0,z3,z2,z3) F#(s(z0),s(z1),z2,z3) -> F#(s(z0),minus(z1,z0),z2,z3) F#(s(z0),s(z1),z2,z3) -> MINUS#(z1,z0) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,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 first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_14) = {1} Following symbols are considered usable: {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus#,perfectp#} TcT has computed the following interpretation: p(0) = [0] p(F) = [0] p(IF) = [0] p(LE) = [0] p(MINUS) = [0] p(PERFECTP) = [0] p(c) = [0] p(c1) = [0] p(c10) = [0] p(c11) = [0] p(c12) = [1] x1 + [1] x2 + [0] p(c13) = [1] x1 + [1] x2 + [0] p(c14) = [1] x1 + [1] x2 + [1] x3 + [0] p(c15) = [1] x1 + [1] x2 + [0] p(c2) = [1] x1 + [0] p(c3) = [0] p(c4) = [0] p(c5) = [1] x1 + [0] p(c6) = [0] p(c7) = [0] p(c8) = [0] p(c9) = [1] x1 + [0] p(f) = [0] p(false) = [0] p(if) = [0] p(le) = [0] p(minus) = [1] x2 + [0] p(perfectp) = [0] p(s) = [1] x1 + [4] p(true) = [0] p(F#) = [4] x1 + [8] p(IF#) = [0] p(LE#) = [0] p(MINUS#) = [2] x2 + [0] p(PERFECTP#) = [0] p(f#) = [0] p(if#) = [1] x3 + [0] p(le#) = [1] x1 + [0] p(minus#) = [0] p(perfectp#) = [2] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [8] x2 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [2] x1 + [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [0] p(c_13) = [2] p(c_14) = [1] x1 + [6] p(c_15) = [0] p(c_16) = [0] p(c_17) = [2] p(c_18) = [1] p(c_19) = [2] p(c_20) = [1] x1 + [2] p(c_21) = [4] p(c_22) = [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [8] x1 + [0] p(c_26) = [2] p(c_27) = [0] p(c_28) = [2] p(c_29) = [0] p(c_30) = [2] Following rules are strictly oriented: MINUS#(s(z0),s(z1)) = [2] z1 + [8] > [2] z1 + [6] = c_14(MINUS#(z0,z1)) Following rules are (at-least) weakly oriented: F#(s(z0),0(),z1,z2) = [4] z0 + [24] >= [4] z0 + [8] = F#(z0,z2,minus(z1,s(z0)),z2) F#(s(z0),0(),z1,z2) = [4] z0 + [24] >= [2] z0 + [8] = MINUS#(z1,s(z0)) F#(s(z0),s(z1),z2,z3) = [4] z0 + [24] >= [4] z0 + [8] = F#(z0,z3,z2,z3) F#(s(z0),s(z1),z2,z3) = [4] z0 + [24] >= [4] z0 + [24] = F#(s(z0),minus(z1,z0),z2,z3) F#(s(z0),s(z1),z2,z3) = [4] z0 + [24] >= [2] z0 + [0] = MINUS#(z1,z0) ***** Step 1.b:8.b:2.b:1.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: F#(s(z0),0(),z1,z2) -> F#(z0,z2,minus(z1,s(z0)),z2) F#(s(z0),0(),z1,z2) -> MINUS#(z1,s(z0)) F#(s(z0),s(z1),z2,z3) -> F#(z0,z3,z2,z3) F#(s(z0),s(z1),z2,z3) -> F#(s(z0),minus(z1,z0),z2,z3) F#(s(z0),s(z1),z2,z3) -> MINUS#(z1,z0) MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () ***** Step 1.b:8.b:2.b:1.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: F#(s(z0),0(),z1,z2) -> F#(z0,z2,minus(z1,s(z0)),z2) F#(s(z0),0(),z1,z2) -> MINUS#(z1,s(z0)) F#(s(z0),s(z1),z2,z3) -> F#(z0,z3,z2,z3) F#(s(z0),s(z1),z2,z3) -> F#(s(z0),minus(z1,z0),z2,z3) F#(s(z0),s(z1),z2,z3) -> MINUS#(z1,z0) MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:F#(s(z0),0(),z1,z2) -> F#(z0,z2,minus(z1,s(z0)),z2) -->_1 F#(s(z0),s(z1),z2,z3) -> MINUS#(z1,z0):5 -->_1 F#(s(z0),s(z1),z2,z3) -> F#(s(z0),minus(z1,z0),z2,z3):4 -->_1 F#(s(z0),s(z1),z2,z3) -> F#(z0,z3,z2,z3):3 -->_1 F#(s(z0),0(),z1,z2) -> MINUS#(z1,s(z0)):2 -->_1 F#(s(z0),0(),z1,z2) -> F#(z0,z2,minus(z1,s(z0)),z2):1 2:W:F#(s(z0),0(),z1,z2) -> MINUS#(z1,s(z0)) -->_1 MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)):6 3:W:F#(s(z0),s(z1),z2,z3) -> F#(z0,z3,z2,z3) -->_1 F#(s(z0),s(z1),z2,z3) -> MINUS#(z1,z0):5 -->_1 F#(s(z0),s(z1),z2,z3) -> F#(s(z0),minus(z1,z0),z2,z3):4 -->_1 F#(s(z0),s(z1),z2,z3) -> F#(z0,z3,z2,z3):3 -->_1 F#(s(z0),0(),z1,z2) -> MINUS#(z1,s(z0)):2 -->_1 F#(s(z0),0(),z1,z2) -> F#(z0,z2,minus(z1,s(z0)),z2):1 4:W:F#(s(z0),s(z1),z2,z3) -> F#(s(z0),minus(z1,z0),z2,z3) -->_1 F#(s(z0),s(z1),z2,z3) -> MINUS#(z1,z0):5 -->_1 F#(s(z0),s(z1),z2,z3) -> F#(s(z0),minus(z1,z0),z2,z3):4 -->_1 F#(s(z0),s(z1),z2,z3) -> F#(z0,z3,z2,z3):3 -->_1 F#(s(z0),0(),z1,z2) -> MINUS#(z1,s(z0)):2 -->_1 F#(s(z0),0(),z1,z2) -> F#(z0,z2,minus(z1,s(z0)),z2):1 5:W:F#(s(z0),s(z1),z2,z3) -> MINUS#(z1,z0) -->_1 MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)):6 6:W:MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) -->_1 MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)):6 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: F#(s(z0),0(),z1,z2) -> F#(z0,z2,minus(z1,s(z0)),z2) 4: F#(s(z0),s(z1),z2,z3) -> F#(s(z0),minus(z1,z0),z2,z3) 3: F#(s(z0),s(z1),z2,z3) -> F#(z0,z3,z2,z3) 2: F#(s(z0),0(),z1,z2) -> MINUS#(z1,s(z0)) 5: F#(s(z0),s(z1),z2,z3) -> MINUS#(z1,z0) 6: MINUS#(s(z0),s(z1)) -> c_14(MINUS#(z0,z1)) ***** Step 1.b:8.b:2.b:1.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: minus(0(),z0) -> 0() minus(s(z0),0()) -> s(z0) minus(s(z0),s(z1)) -> minus(z0,z1) - Signature: {F/4,IF/3,LE/2,MINUS/2,PERFECTP/1,f/4,if/3,le/2,minus/2,perfectp/1,F#/4,IF#/3,LE#/2,MINUS#/2,PERFECTP#/1 ,f#/4,if#/3,le#/2,minus#/2,perfectp#/1} / {0/0,c/0,c1/0,c10/0,c11/0,c12/2,c13/2,c14/3,c15/2,c2/1,c3/0,c4/0 ,c5/1,c6/0,c7/0,c8/0,c9/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0 ,c_11/1,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/0,c_19/1,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1 ,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,IF#,LE#,MINUS#,PERFECTP#,f#,if#,le#,minus# ,perfectp#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,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^3))