WORST_CASE(Omega(n^1),O(n^3)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^3)) + Considered Problem: - Strict TRS: EQ(0(),0()) -> c() EQ(0(),s(z0)) -> c1() EQ(s(z0),0()) -> c2() EQ(s(z0),s(z1)) -> c3(EQ(z0,z1)) IFMIN(false(),cons(z0,cons(z1,z2))) -> c11(MIN(cons(z1,z2))) IFMIN(true(),cons(z0,cons(z1,z2))) -> c10(MIN(cons(z0,z2))) IFREPL(false(),z0,z1,cons(z2,z3)) -> c15(REPLACE(z0,z1,z3)) IFREPL(true(),z0,z1,cons(z2,z3)) -> c14() IFSELSORT(false(),cons(z0,z1)) -> c19(MIN(cons(z0,z1))) IFSELSORT(false(),cons(z0,z1)) -> c20(SELSORT(replace(min(cons(z0,z1)),z0,z1)) ,REPLACE(min(cons(z0,z1)),z0,z1) ,MIN(cons(z0,z1))) IFSELSORT(true(),cons(z0,z1)) -> c18(SELSORT(z1)) LE(0(),z0) -> c4() LE(s(z0),0()) -> c5() LE(s(z0),s(z1)) -> c6(LE(z0,z1)) MIN(cons(z0,cons(z1,z2))) -> c9(IFMIN(le(z0,z1),cons(z0,cons(z1,z2))),LE(z0,z1)) MIN(cons(0(),nil())) -> c7() MIN(cons(s(z0),nil())) -> c8() REPLACE(z0,z1,cons(z2,z3)) -> c13(IFREPL(eq(z0,z2),z0,z1,cons(z2,z3)),EQ(z0,z2)) REPLACE(z0,z1,nil()) -> c12() SELSORT(cons(z0,z1)) -> c17(IFSELSORT(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,EQ(z0,min(cons(z0,z1))) ,MIN(cons(z0,z1))) SELSORT(nil()) -> c16() - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) ifselsort(false(),cons(z0,z1)) -> cons(min(cons(z0,z1)),selsort(replace(min(cons(z0,z1)),z0,z1))) ifselsort(true(),cons(z0,z1)) -> cons(z0,selsort(z1)) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() selsort(cons(z0,z1)) -> ifselsort(eq(z0,min(cons(z0,z1))),cons(z0,z1)) selsort(nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0 ,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ,IFMIN,IFREPL,IFSELSORT,LE,MIN,REPLACE,SELSORT,eq,ifmin ,ifrepl,ifselsort,le,min,replace,selsort} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19 ,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: EQ(0(),0()) -> c() EQ(0(),s(z0)) -> c1() EQ(s(z0),0()) -> c2() EQ(s(z0),s(z1)) -> c3(EQ(z0,z1)) IFMIN(false(),cons(z0,cons(z1,z2))) -> c11(MIN(cons(z1,z2))) IFMIN(true(),cons(z0,cons(z1,z2))) -> c10(MIN(cons(z0,z2))) IFREPL(false(),z0,z1,cons(z2,z3)) -> c15(REPLACE(z0,z1,z3)) IFREPL(true(),z0,z1,cons(z2,z3)) -> c14() IFSELSORT(false(),cons(z0,z1)) -> c19(MIN(cons(z0,z1))) IFSELSORT(false(),cons(z0,z1)) -> c20(SELSORT(replace(min(cons(z0,z1)),z0,z1)) ,REPLACE(min(cons(z0,z1)),z0,z1) ,MIN(cons(z0,z1))) IFSELSORT(true(),cons(z0,z1)) -> c18(SELSORT(z1)) LE(0(),z0) -> c4() LE(s(z0),0()) -> c5() LE(s(z0),s(z1)) -> c6(LE(z0,z1)) MIN(cons(z0,cons(z1,z2))) -> c9(IFMIN(le(z0,z1),cons(z0,cons(z1,z2))),LE(z0,z1)) MIN(cons(0(),nil())) -> c7() MIN(cons(s(z0),nil())) -> c8() REPLACE(z0,z1,cons(z2,z3)) -> c13(IFREPL(eq(z0,z2),z0,z1,cons(z2,z3)),EQ(z0,z2)) REPLACE(z0,z1,nil()) -> c12() SELSORT(cons(z0,z1)) -> c17(IFSELSORT(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,EQ(z0,min(cons(z0,z1))) ,MIN(cons(z0,z1))) SELSORT(nil()) -> c16() - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) ifselsort(false(),cons(z0,z1)) -> cons(min(cons(z0,z1)),selsort(replace(min(cons(z0,z1)),z0,z1))) ifselsort(true(),cons(z0,z1)) -> cons(z0,selsort(z1)) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() selsort(cons(z0,z1)) -> ifselsort(eq(z0,min(cons(z0,z1))),cons(z0,z1)) selsort(nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0 ,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ,IFMIN,IFREPL,IFSELSORT,LE,MIN,REPLACE,SELSORT,eq,ifmin ,ifrepl,ifselsort,le,min,replace,selsort} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19 ,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: EQ(0(),0()) -> c() EQ(0(),s(z0)) -> c1() EQ(s(z0),0()) -> c2() EQ(s(z0),s(z1)) -> c3(EQ(z0,z1)) IFMIN(false(),cons(z0,cons(z1,z2))) -> c11(MIN(cons(z1,z2))) IFMIN(true(),cons(z0,cons(z1,z2))) -> c10(MIN(cons(z0,z2))) IFREPL(false(),z0,z1,cons(z2,z3)) -> c15(REPLACE(z0,z1,z3)) IFREPL(true(),z0,z1,cons(z2,z3)) -> c14() IFSELSORT(false(),cons(z0,z1)) -> c19(MIN(cons(z0,z1))) IFSELSORT(false(),cons(z0,z1)) -> c20(SELSORT(replace(min(cons(z0,z1)),z0,z1)) ,REPLACE(min(cons(z0,z1)),z0,z1) ,MIN(cons(z0,z1))) IFSELSORT(true(),cons(z0,z1)) -> c18(SELSORT(z1)) LE(0(),z0) -> c4() LE(s(z0),0()) -> c5() LE(s(z0),s(z1)) -> c6(LE(z0,z1)) MIN(cons(z0,cons(z1,z2))) -> c9(IFMIN(le(z0,z1),cons(z0,cons(z1,z2))),LE(z0,z1)) MIN(cons(0(),nil())) -> c7() MIN(cons(s(z0),nil())) -> c8() REPLACE(z0,z1,cons(z2,z3)) -> c13(IFREPL(eq(z0,z2),z0,z1,cons(z2,z3)),EQ(z0,z2)) REPLACE(z0,z1,nil()) -> c12() SELSORT(cons(z0,z1)) -> c17(IFSELSORT(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,EQ(z0,min(cons(z0,z1))) ,MIN(cons(z0,z1))) SELSORT(nil()) -> c16() - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) ifselsort(false(),cons(z0,z1)) -> cons(min(cons(z0,z1)),selsort(replace(min(cons(z0,z1)),z0,z1))) ifselsort(true(),cons(z0,z1)) -> cons(z0,selsort(z1)) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() selsort(cons(z0,z1)) -> ifselsort(eq(z0,min(cons(z0,z1))),cons(z0,z1)) selsort(nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0 ,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ,IFMIN,IFREPL,IFSELSORT,LE,MIN,REPLACE,SELSORT,eq,ifmin ,ifrepl,ifselsort,le,min,replace,selsort} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19 ,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: EQ(x,y){x -> s(x),y -> s(y)} = EQ(s(x),s(y)) ->^+ c3(EQ(x,y)) = C[EQ(x,y) = EQ(x,y){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: EQ(0(),0()) -> c() EQ(0(),s(z0)) -> c1() EQ(s(z0),0()) -> c2() EQ(s(z0),s(z1)) -> c3(EQ(z0,z1)) IFMIN(false(),cons(z0,cons(z1,z2))) -> c11(MIN(cons(z1,z2))) IFMIN(true(),cons(z0,cons(z1,z2))) -> c10(MIN(cons(z0,z2))) IFREPL(false(),z0,z1,cons(z2,z3)) -> c15(REPLACE(z0,z1,z3)) IFREPL(true(),z0,z1,cons(z2,z3)) -> c14() IFSELSORT(false(),cons(z0,z1)) -> c19(MIN(cons(z0,z1))) IFSELSORT(false(),cons(z0,z1)) -> c20(SELSORT(replace(min(cons(z0,z1)),z0,z1)) ,REPLACE(min(cons(z0,z1)),z0,z1) ,MIN(cons(z0,z1))) IFSELSORT(true(),cons(z0,z1)) -> c18(SELSORT(z1)) LE(0(),z0) -> c4() LE(s(z0),0()) -> c5() LE(s(z0),s(z1)) -> c6(LE(z0,z1)) MIN(cons(z0,cons(z1,z2))) -> c9(IFMIN(le(z0,z1),cons(z0,cons(z1,z2))),LE(z0,z1)) MIN(cons(0(),nil())) -> c7() MIN(cons(s(z0),nil())) -> c8() REPLACE(z0,z1,cons(z2,z3)) -> c13(IFREPL(eq(z0,z2),z0,z1,cons(z2,z3)),EQ(z0,z2)) REPLACE(z0,z1,nil()) -> c12() SELSORT(cons(z0,z1)) -> c17(IFSELSORT(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,EQ(z0,min(cons(z0,z1))) ,MIN(cons(z0,z1))) SELSORT(nil()) -> c16() - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) ifselsort(false(),cons(z0,z1)) -> cons(min(cons(z0,z1)),selsort(replace(min(cons(z0,z1)),z0,z1))) ifselsort(true(),cons(z0,z1)) -> cons(z0,selsort(z1)) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() selsort(cons(z0,z1)) -> ifselsort(eq(z0,min(cons(z0,z1))),cons(z0,z1)) selsort(nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0 ,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ,IFMIN,IFREPL,IFSELSORT,LE,MIN,REPLACE,SELSORT,eq,ifmin ,ifrepl,ifselsort,le,min,replace,selsort} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19 ,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs EQ#(0(),0()) -> c_1() EQ#(0(),s(z0)) -> c_2() EQ#(s(z0),0()) -> c_3() EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)) IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) IFREPL#(true(),z0,z1,cons(z2,z3)) -> c_8() IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1)) ,replace#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1)) ,REPLACE#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)) LE#(0(),z0) -> c_12() LE#(s(z0),0()) -> c_13() LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1),LE#(z0,z1)) MIN#(cons(0(),nil())) -> c_16() MIN#(cons(s(z0),nil())) -> c_17() REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2),EQ#(z0,z2)) REPLACE#(z0,z1,nil()) -> c_19() SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))) SELSORT#(nil()) -> c_21() Weak DPs eq#(0(),0()) -> c_22() eq#(0(),s(z0)) -> c_23() eq#(s(z0),0()) -> c_24() eq#(s(z0),s(z1)) -> c_25(eq#(z0,z1)) ifmin#(false(),cons(z0,cons(z1,z2))) -> c_26(min#(cons(z1,z2))) ifmin#(true(),cons(z0,cons(z1,z2))) -> c_27(min#(cons(z0,z2))) ifrepl#(false(),z0,z1,cons(z2,z3)) -> c_28(replace#(z0,z1,z3)) ifrepl#(true(),z0,z1,cons(z2,z3)) -> c_29() ifselsort#(false(),cons(z0,z1)) -> c_30(min#(cons(z0,z1)) ,selsort#(replace(min(cons(z0,z1)),z0,z1)) ,replace#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1))) ifselsort#(true(),cons(z0,z1)) -> c_31(selsort#(z1)) le#(0(),z0) -> c_32() le#(s(z0),0()) -> c_33() le#(s(z0),s(z1)) -> c_34(le#(z0,z1)) min#(cons(z0,cons(z1,z2))) -> c_35(ifmin#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1)) min#(cons(0(),nil())) -> c_36() min#(cons(s(z0),nil())) -> c_37() replace#(z0,z1,cons(z2,z3)) -> c_38(ifrepl#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2)) replace#(z0,z1,nil()) -> c_39() selsort#(cons(z0,z1)) -> c_40(ifselsort#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1))) selsort#(nil()) -> c_41() and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: EQ#(0(),0()) -> c_1() EQ#(0(),s(z0)) -> c_2() EQ#(s(z0),0()) -> c_3() EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)) IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) IFREPL#(true(),z0,z1,cons(z2,z3)) -> c_8() IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1)) ,replace#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1)) ,REPLACE#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)) LE#(0(),z0) -> c_12() LE#(s(z0),0()) -> c_13() LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1),LE#(z0,z1)) MIN#(cons(0(),nil())) -> c_16() MIN#(cons(s(z0),nil())) -> c_17() REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2),EQ#(z0,z2)) REPLACE#(z0,z1,nil()) -> c_19() SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))) SELSORT#(nil()) -> c_21() - Weak DPs: eq#(0(),0()) -> c_22() eq#(0(),s(z0)) -> c_23() eq#(s(z0),0()) -> c_24() eq#(s(z0),s(z1)) -> c_25(eq#(z0,z1)) ifmin#(false(),cons(z0,cons(z1,z2))) -> c_26(min#(cons(z1,z2))) ifmin#(true(),cons(z0,cons(z1,z2))) -> c_27(min#(cons(z0,z2))) ifrepl#(false(),z0,z1,cons(z2,z3)) -> c_28(replace#(z0,z1,z3)) ifrepl#(true(),z0,z1,cons(z2,z3)) -> c_29() ifselsort#(false(),cons(z0,z1)) -> c_30(min#(cons(z0,z1)) ,selsort#(replace(min(cons(z0,z1)),z0,z1)) ,replace#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1))) ifselsort#(true(),cons(z0,z1)) -> c_31(selsort#(z1)) le#(0(),z0) -> c_32() le#(s(z0),0()) -> c_33() le#(s(z0),s(z1)) -> c_34(le#(z0,z1)) min#(cons(z0,cons(z1,z2))) -> c_35(ifmin#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1)) min#(cons(0(),nil())) -> c_36() min#(cons(s(z0),nil())) -> c_37() replace#(z0,z1,cons(z2,z3)) -> c_38(ifrepl#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2)) replace#(z0,z1,nil()) -> c_39() selsort#(cons(z0,z1)) -> c_40(ifselsort#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1))) selsort#(nil()) -> c_41() - Weak TRS: EQ(0(),0()) -> c() EQ(0(),s(z0)) -> c1() EQ(s(z0),0()) -> c2() EQ(s(z0),s(z1)) -> c3(EQ(z0,z1)) IFMIN(false(),cons(z0,cons(z1,z2))) -> c11(MIN(cons(z1,z2))) IFMIN(true(),cons(z0,cons(z1,z2))) -> c10(MIN(cons(z0,z2))) IFREPL(false(),z0,z1,cons(z2,z3)) -> c15(REPLACE(z0,z1,z3)) IFREPL(true(),z0,z1,cons(z2,z3)) -> c14() IFSELSORT(false(),cons(z0,z1)) -> c19(MIN(cons(z0,z1))) IFSELSORT(false(),cons(z0,z1)) -> c20(SELSORT(replace(min(cons(z0,z1)),z0,z1)) ,REPLACE(min(cons(z0,z1)),z0,z1) ,MIN(cons(z0,z1))) IFSELSORT(true(),cons(z0,z1)) -> c18(SELSORT(z1)) LE(0(),z0) -> c4() LE(s(z0),0()) -> c5() LE(s(z0),s(z1)) -> c6(LE(z0,z1)) MIN(cons(z0,cons(z1,z2))) -> c9(IFMIN(le(z0,z1),cons(z0,cons(z1,z2))),LE(z0,z1)) MIN(cons(0(),nil())) -> c7() MIN(cons(s(z0),nil())) -> c8() REPLACE(z0,z1,cons(z2,z3)) -> c13(IFREPL(eq(z0,z2),z0,z1,cons(z2,z3)),EQ(z0,z2)) REPLACE(z0,z1,nil()) -> c12() SELSORT(cons(z0,z1)) -> c17(IFSELSORT(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,EQ(z0,min(cons(z0,z1))) ,MIN(cons(z0,z1))) SELSORT(nil()) -> c16() eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) ifselsort(false(),cons(z0,z1)) -> cons(min(cons(z0,z1)),selsort(replace(min(cons(z0,z1)),z0,z1))) ifselsort(true(),cons(z0,z1)) -> cons(z0,selsort(z1)) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() selsort(cons(z0,z1)) -> ifselsort(eq(z0,min(cons(z0,z1))),cons(z0,z1)) selsort(nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/6,c_11/1,c_12/0,c_13/0,c_14/1,c_15/3 ,c_16/0,c_17/0,c_18/3,c_19/0,c_20/6,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,8,12,13,16,17,19,21} by application of Pre({1,2,3,8,12,13,16,17,19,21}) = {4,5,6,7,9,10,11,14,15,18,20}. Here rules are labelled as follows: 1: EQ#(0(),0()) -> c_1() 2: EQ#(0(),s(z0)) -> c_2() 3: EQ#(s(z0),0()) -> c_3() 4: EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)) 5: IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) 6: IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) 7: IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) 8: IFREPL#(true(),z0,z1,cons(z2,z3)) -> c_8() 9: IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) 10: IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1)) ,replace#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1)) ,REPLACE#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))) 11: IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)) 12: LE#(0(),z0) -> c_12() 13: LE#(s(z0),0()) -> c_13() 14: LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)) 15: MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1),LE#(z0,z1)) 16: MIN#(cons(0(),nil())) -> c_16() 17: MIN#(cons(s(z0),nil())) -> c_17() 18: REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2),EQ#(z0,z2)) 19: REPLACE#(z0,z1,nil()) -> c_19() 20: SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))) 21: SELSORT#(nil()) -> c_21() 22: eq#(0(),0()) -> c_22() 23: eq#(0(),s(z0)) -> c_23() 24: eq#(s(z0),0()) -> c_24() 25: eq#(s(z0),s(z1)) -> c_25(eq#(z0,z1)) 26: ifmin#(false(),cons(z0,cons(z1,z2))) -> c_26(min#(cons(z1,z2))) 27: ifmin#(true(),cons(z0,cons(z1,z2))) -> c_27(min#(cons(z0,z2))) 28: ifrepl#(false(),z0,z1,cons(z2,z3)) -> c_28(replace#(z0,z1,z3)) 29: ifrepl#(true(),z0,z1,cons(z2,z3)) -> c_29() 30: ifselsort#(false(),cons(z0,z1)) -> c_30(min#(cons(z0,z1)) ,selsort#(replace(min(cons(z0,z1)),z0,z1)) ,replace#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1))) 31: ifselsort#(true(),cons(z0,z1)) -> c_31(selsort#(z1)) 32: le#(0(),z0) -> c_32() 33: le#(s(z0),0()) -> c_33() 34: le#(s(z0),s(z1)) -> c_34(le#(z0,z1)) 35: min#(cons(z0,cons(z1,z2))) -> c_35(ifmin#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1)) 36: min#(cons(0(),nil())) -> c_36() 37: min#(cons(s(z0),nil())) -> c_37() 38: replace#(z0,z1,cons(z2,z3)) -> c_38(ifrepl#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2)) 39: replace#(z0,z1,nil()) -> c_39() 40: selsort#(cons(z0,z1)) -> c_40(ifselsort#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1))) 41: selsort#(nil()) -> c_41() ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)) IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1)) ,replace#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1)) ,REPLACE#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)) LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1),LE#(z0,z1)) REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2),EQ#(z0,z2)) SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))) - Weak DPs: EQ#(0(),0()) -> c_1() EQ#(0(),s(z0)) -> c_2() EQ#(s(z0),0()) -> c_3() IFREPL#(true(),z0,z1,cons(z2,z3)) -> c_8() LE#(0(),z0) -> c_12() LE#(s(z0),0()) -> c_13() MIN#(cons(0(),nil())) -> c_16() MIN#(cons(s(z0),nil())) -> c_17() REPLACE#(z0,z1,nil()) -> c_19() SELSORT#(nil()) -> c_21() eq#(0(),0()) -> c_22() eq#(0(),s(z0)) -> c_23() eq#(s(z0),0()) -> c_24() eq#(s(z0),s(z1)) -> c_25(eq#(z0,z1)) ifmin#(false(),cons(z0,cons(z1,z2))) -> c_26(min#(cons(z1,z2))) ifmin#(true(),cons(z0,cons(z1,z2))) -> c_27(min#(cons(z0,z2))) ifrepl#(false(),z0,z1,cons(z2,z3)) -> c_28(replace#(z0,z1,z3)) ifrepl#(true(),z0,z1,cons(z2,z3)) -> c_29() ifselsort#(false(),cons(z0,z1)) -> c_30(min#(cons(z0,z1)) ,selsort#(replace(min(cons(z0,z1)),z0,z1)) ,replace#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1))) ifselsort#(true(),cons(z0,z1)) -> c_31(selsort#(z1)) le#(0(),z0) -> c_32() le#(s(z0),0()) -> c_33() le#(s(z0),s(z1)) -> c_34(le#(z0,z1)) min#(cons(z0,cons(z1,z2))) -> c_35(ifmin#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1)) min#(cons(0(),nil())) -> c_36() min#(cons(s(z0),nil())) -> c_37() replace#(z0,z1,cons(z2,z3)) -> c_38(ifrepl#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2)) replace#(z0,z1,nil()) -> c_39() selsort#(cons(z0,z1)) -> c_40(ifselsort#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1))) selsort#(nil()) -> c_41() - Weak TRS: EQ(0(),0()) -> c() EQ(0(),s(z0)) -> c1() EQ(s(z0),0()) -> c2() EQ(s(z0),s(z1)) -> c3(EQ(z0,z1)) IFMIN(false(),cons(z0,cons(z1,z2))) -> c11(MIN(cons(z1,z2))) IFMIN(true(),cons(z0,cons(z1,z2))) -> c10(MIN(cons(z0,z2))) IFREPL(false(),z0,z1,cons(z2,z3)) -> c15(REPLACE(z0,z1,z3)) IFREPL(true(),z0,z1,cons(z2,z3)) -> c14() IFSELSORT(false(),cons(z0,z1)) -> c19(MIN(cons(z0,z1))) IFSELSORT(false(),cons(z0,z1)) -> c20(SELSORT(replace(min(cons(z0,z1)),z0,z1)) ,REPLACE(min(cons(z0,z1)),z0,z1) ,MIN(cons(z0,z1))) IFSELSORT(true(),cons(z0,z1)) -> c18(SELSORT(z1)) LE(0(),z0) -> c4() LE(s(z0),0()) -> c5() LE(s(z0),s(z1)) -> c6(LE(z0,z1)) MIN(cons(z0,cons(z1,z2))) -> c9(IFMIN(le(z0,z1),cons(z0,cons(z1,z2))),LE(z0,z1)) MIN(cons(0(),nil())) -> c7() MIN(cons(s(z0),nil())) -> c8() REPLACE(z0,z1,cons(z2,z3)) -> c13(IFREPL(eq(z0,z2),z0,z1,cons(z2,z3)),EQ(z0,z2)) REPLACE(z0,z1,nil()) -> c12() SELSORT(cons(z0,z1)) -> c17(IFSELSORT(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,EQ(z0,min(cons(z0,z1))) ,MIN(cons(z0,z1))) SELSORT(nil()) -> c16() eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) ifselsort(false(),cons(z0,z1)) -> cons(min(cons(z0,z1)),selsort(replace(min(cons(z0,z1)),z0,z1))) ifselsort(true(),cons(z0,z1)) -> cons(z0,selsort(z1)) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() selsort(cons(z0,z1)) -> ifselsort(eq(z0,min(cons(z0,z1))),cons(z0,z1)) selsort(nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/6,c_11/1,c_12/0,c_13/0,c_14/1,c_15/3 ,c_16/0,c_17/0,c_18/3,c_19/0,c_20/6,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)) -->_1 EQ#(s(z0),0()) -> c_3():14 -->_1 EQ#(0(),s(z0)) -> c_2():13 -->_1 EQ#(0(),0()) -> c_1():12 -->_1 EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)):1 2:S:IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) -->_1 MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1),LE#(z0,z1)):9 -->_1 MIN#(cons(s(z0),nil())) -> c_17():19 -->_1 MIN#(cons(0(),nil())) -> c_16():18 3:S:IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) -->_1 MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1),LE#(z0,z1)):9 -->_1 MIN#(cons(s(z0),nil())) -> c_17():19 -->_1 MIN#(cons(0(),nil())) -> c_16():18 4:S:IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) -->_1 REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2),EQ#(z0,z2)):10 -->_1 REPLACE#(z0,z1,nil()) -> c_19():20 5:S:IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) -->_1 MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1),LE#(z0,z1)):9 -->_1 MIN#(cons(s(z0),nil())) -> c_17():19 -->_1 MIN#(cons(0(),nil())) -> c_16():18 6:S:IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1)) ,replace#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1)) ,REPLACE#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))) -->_2 replace#(z0,z1,cons(z2,z3)) -> c_38(ifrepl#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2)):38 -->_5 min#(cons(z0,cons(z1,z2))) -> c_35(ifmin#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1)):35 -->_3 min#(cons(z0,cons(z1,z2))) -> c_35(ifmin#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1)):35 -->_1 SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))):11 -->_4 REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2),EQ#(z0,z2)):10 -->_6 MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1),LE#(z0,z1)):9 -->_2 replace#(z0,z1,nil()) -> c_39():39 -->_5 min#(cons(s(z0),nil())) -> c_37():37 -->_3 min#(cons(s(z0),nil())) -> c_37():37 -->_5 min#(cons(0(),nil())) -> c_36():36 -->_3 min#(cons(0(),nil())) -> c_36():36 -->_1 SELSORT#(nil()) -> c_21():21 -->_4 REPLACE#(z0,z1,nil()) -> c_19():20 -->_6 MIN#(cons(s(z0),nil())) -> c_17():19 -->_6 MIN#(cons(0(),nil())) -> c_16():18 7:S:IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)) -->_1 SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))):11 -->_1 SELSORT#(nil()) -> c_21():21 8:S:LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)) -->_1 LE#(s(z0),0()) -> c_13():17 -->_1 LE#(0(),z0) -> c_12():16 -->_1 LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)):8 9:S:MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1),LE#(z0,z1)) -->_2 le#(s(z0),s(z1)) -> c_34(le#(z0,z1)):34 -->_2 le#(s(z0),0()) -> c_33():33 -->_2 le#(0(),z0) -> c_32():32 -->_3 LE#(s(z0),0()) -> c_13():17 -->_3 LE#(0(),z0) -> c_12():16 -->_3 LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)):8 -->_1 IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))):3 -->_1 IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))):2 10:S:REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2),EQ#(z0,z2)) -->_2 eq#(s(z0),s(z1)) -> c_25(eq#(z0,z1)):25 -->_2 eq#(s(z0),0()) -> c_24():24 -->_2 eq#(0(),s(z0)) -> c_23():23 -->_2 eq#(0(),0()) -> c_22():22 -->_1 IFREPL#(true(),z0,z1,cons(z2,z3)) -> c_8():15 -->_3 EQ#(s(z0),0()) -> c_3():14 -->_3 EQ#(0(),s(z0)) -> c_2():13 -->_3 EQ#(0(),0()) -> c_1():12 -->_1 IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)):4 -->_3 EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)):1 11:S:SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))) -->_5 min#(cons(z0,cons(z1,z2))) -> c_35(ifmin#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1)):35 -->_3 min#(cons(z0,cons(z1,z2))) -> c_35(ifmin#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1)):35 -->_2 eq#(s(z0),s(z1)) -> c_25(eq#(z0,z1)):25 -->_5 min#(cons(s(z0),nil())) -> c_37():37 -->_3 min#(cons(s(z0),nil())) -> c_37():37 -->_5 min#(cons(0(),nil())) -> c_36():36 -->_3 min#(cons(0(),nil())) -> c_36():36 -->_2 eq#(s(z0),0()) -> c_24():24 -->_2 eq#(0(),s(z0)) -> c_23():23 -->_2 eq#(0(),0()) -> c_22():22 -->_6 MIN#(cons(s(z0),nil())) -> c_17():19 -->_6 MIN#(cons(0(),nil())) -> c_16():18 -->_4 EQ#(s(z0),0()) -> c_3():14 -->_4 EQ#(0(),s(z0)) -> c_2():13 -->_4 EQ#(0(),0()) -> c_1():12 -->_6 MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1),LE#(z0,z1)):9 -->_1 IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)):7 -->_1 IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1)) ,replace#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1)) ,REPLACE#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))):6 -->_1 IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))):5 -->_4 EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)):1 12:W:EQ#(0(),0()) -> c_1() 13:W:EQ#(0(),s(z0)) -> c_2() 14:W:EQ#(s(z0),0()) -> c_3() 15:W:IFREPL#(true(),z0,z1,cons(z2,z3)) -> c_8() 16:W:LE#(0(),z0) -> c_12() 17:W:LE#(s(z0),0()) -> c_13() 18:W:MIN#(cons(0(),nil())) -> c_16() 19:W:MIN#(cons(s(z0),nil())) -> c_17() 20:W:REPLACE#(z0,z1,nil()) -> c_19() 21:W:SELSORT#(nil()) -> c_21() 22:W:eq#(0(),0()) -> c_22() 23:W:eq#(0(),s(z0)) -> c_23() 24:W:eq#(s(z0),0()) -> c_24() 25:W:eq#(s(z0),s(z1)) -> c_25(eq#(z0,z1)) -->_1 eq#(s(z0),s(z1)) -> c_25(eq#(z0,z1)):25 -->_1 eq#(s(z0),0()) -> c_24():24 -->_1 eq#(0(),s(z0)) -> c_23():23 -->_1 eq#(0(),0()) -> c_22():22 26:W:ifmin#(false(),cons(z0,cons(z1,z2))) -> c_26(min#(cons(z1,z2))) -->_1 min#(cons(z0,cons(z1,z2))) -> c_35(ifmin#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1)):35 -->_1 min#(cons(s(z0),nil())) -> c_37():37 -->_1 min#(cons(0(),nil())) -> c_36():36 27:W:ifmin#(true(),cons(z0,cons(z1,z2))) -> c_27(min#(cons(z0,z2))) -->_1 min#(cons(z0,cons(z1,z2))) -> c_35(ifmin#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1)):35 -->_1 min#(cons(s(z0),nil())) -> c_37():37 -->_1 min#(cons(0(),nil())) -> c_36():36 28:W:ifrepl#(false(),z0,z1,cons(z2,z3)) -> c_28(replace#(z0,z1,z3)) -->_1 replace#(z0,z1,cons(z2,z3)) -> c_38(ifrepl#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2)):38 -->_1 replace#(z0,z1,nil()) -> c_39():39 29:W:ifrepl#(true(),z0,z1,cons(z2,z3)) -> c_29() 30:W:ifselsort#(false(),cons(z0,z1)) -> c_30(min#(cons(z0,z1)) ,selsort#(replace(min(cons(z0,z1)),z0,z1)) ,replace#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1))) -->_2 selsort#(cons(z0,z1)) -> c_40(ifselsort#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1))):40 -->_3 replace#(z0,z1,cons(z2,z3)) -> c_38(ifrepl#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2)):38 -->_4 min#(cons(z0,cons(z1,z2))) -> c_35(ifmin#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1)):35 -->_1 min#(cons(z0,cons(z1,z2))) -> c_35(ifmin#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1)):35 -->_2 selsort#(nil()) -> c_41():41 -->_3 replace#(z0,z1,nil()) -> c_39():39 -->_4 min#(cons(s(z0),nil())) -> c_37():37 -->_1 min#(cons(s(z0),nil())) -> c_37():37 -->_4 min#(cons(0(),nil())) -> c_36():36 -->_1 min#(cons(0(),nil())) -> c_36():36 31:W:ifselsort#(true(),cons(z0,z1)) -> c_31(selsort#(z1)) -->_1 selsort#(cons(z0,z1)) -> c_40(ifselsort#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1))):40 -->_1 selsort#(nil()) -> c_41():41 32:W:le#(0(),z0) -> c_32() 33:W:le#(s(z0),0()) -> c_33() 34:W:le#(s(z0),s(z1)) -> c_34(le#(z0,z1)) -->_1 le#(s(z0),s(z1)) -> c_34(le#(z0,z1)):34 -->_1 le#(s(z0),0()) -> c_33():33 -->_1 le#(0(),z0) -> c_32():32 35:W:min#(cons(z0,cons(z1,z2))) -> c_35(ifmin#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1)) -->_2 le#(s(z0),s(z1)) -> c_34(le#(z0,z1)):34 -->_2 le#(s(z0),0()) -> c_33():33 -->_2 le#(0(),z0) -> c_32():32 -->_1 ifmin#(true(),cons(z0,cons(z1,z2))) -> c_27(min#(cons(z0,z2))):27 -->_1 ifmin#(false(),cons(z0,cons(z1,z2))) -> c_26(min#(cons(z1,z2))):26 36:W:min#(cons(0(),nil())) -> c_36() 37:W:min#(cons(s(z0),nil())) -> c_37() 38:W:replace#(z0,z1,cons(z2,z3)) -> c_38(ifrepl#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2)) -->_1 ifrepl#(true(),z0,z1,cons(z2,z3)) -> c_29():29 -->_1 ifrepl#(false(),z0,z1,cons(z2,z3)) -> c_28(replace#(z0,z1,z3)):28 -->_2 eq#(s(z0),s(z1)) -> c_25(eq#(z0,z1)):25 -->_2 eq#(s(z0),0()) -> c_24():24 -->_2 eq#(0(),s(z0)) -> c_23():23 -->_2 eq#(0(),0()) -> c_22():22 39:W:replace#(z0,z1,nil()) -> c_39() 40:W:selsort#(cons(z0,z1)) -> c_40(ifselsort#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1))) -->_3 min#(cons(s(z0),nil())) -> c_37():37 -->_3 min#(cons(0(),nil())) -> c_36():36 -->_3 min#(cons(z0,cons(z1,z2))) -> c_35(ifmin#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1)):35 -->_1 ifselsort#(true(),cons(z0,z1)) -> c_31(selsort#(z1)):31 -->_1 ifselsort#(false(),cons(z0,z1)) -> c_30(min#(cons(z0,z1)) ,selsort#(replace(min(cons(z0,z1)),z0,z1)) ,replace#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1))):30 -->_2 eq#(s(z0),s(z1)) -> c_25(eq#(z0,z1)):25 -->_2 eq#(s(z0),0()) -> c_24():24 -->_2 eq#(0(),s(z0)) -> c_23():23 -->_2 eq#(0(),0()) -> c_22():22 41:W:selsort#(nil()) -> c_41() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 30: ifselsort#(false(),cons(z0,z1)) -> c_30(min#(cons(z0,z1)) ,selsort#(replace(min(cons(z0,z1)),z0,z1)) ,replace#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1))) 40: selsort#(cons(z0,z1)) -> c_40(ifselsort#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1))) 31: ifselsort#(true(),cons(z0,z1)) -> c_31(selsort#(z1)) 41: selsort#(nil()) -> c_41() 21: SELSORT#(nil()) -> c_21() 35: min#(cons(z0,cons(z1,z2))) -> c_35(ifmin#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1)) 27: ifmin#(true(),cons(z0,cons(z1,z2))) -> c_27(min#(cons(z0,z2))) 26: ifmin#(false(),cons(z0,cons(z1,z2))) -> c_26(min#(cons(z1,z2))) 36: min#(cons(0(),nil())) -> c_36() 37: min#(cons(s(z0),nil())) -> c_37() 38: replace#(z0,z1,cons(z2,z3)) -> c_38(ifrepl#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2)) 28: ifrepl#(false(),z0,z1,cons(z2,z3)) -> c_28(replace#(z0,z1,z3)) 39: replace#(z0,z1,nil()) -> c_39() 29: ifrepl#(true(),z0,z1,cons(z2,z3)) -> c_29() 20: REPLACE#(z0,z1,nil()) -> c_19() 15: IFREPL#(true(),z0,z1,cons(z2,z3)) -> c_8() 25: eq#(s(z0),s(z1)) -> c_25(eq#(z0,z1)) 22: eq#(0(),0()) -> c_22() 23: eq#(0(),s(z0)) -> c_23() 24: eq#(s(z0),0()) -> c_24() 18: MIN#(cons(0(),nil())) -> c_16() 19: MIN#(cons(s(z0),nil())) -> c_17() 16: LE#(0(),z0) -> c_12() 17: LE#(s(z0),0()) -> c_13() 34: le#(s(z0),s(z1)) -> c_34(le#(z0,z1)) 32: le#(0(),z0) -> c_32() 33: le#(s(z0),0()) -> c_33() 12: EQ#(0(),0()) -> c_1() 13: EQ#(0(),s(z0)) -> c_2() 14: EQ#(s(z0),0()) -> c_3() ** Step 1.b:4: SimplifyRHS. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)) IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1)) ,replace#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1)) ,REPLACE#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)) LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1),LE#(z0,z1)) REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2),EQ#(z0,z2)) SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))) - Weak TRS: EQ(0(),0()) -> c() EQ(0(),s(z0)) -> c1() EQ(s(z0),0()) -> c2() EQ(s(z0),s(z1)) -> c3(EQ(z0,z1)) IFMIN(false(),cons(z0,cons(z1,z2))) -> c11(MIN(cons(z1,z2))) IFMIN(true(),cons(z0,cons(z1,z2))) -> c10(MIN(cons(z0,z2))) IFREPL(false(),z0,z1,cons(z2,z3)) -> c15(REPLACE(z0,z1,z3)) IFREPL(true(),z0,z1,cons(z2,z3)) -> c14() IFSELSORT(false(),cons(z0,z1)) -> c19(MIN(cons(z0,z1))) IFSELSORT(false(),cons(z0,z1)) -> c20(SELSORT(replace(min(cons(z0,z1)),z0,z1)) ,REPLACE(min(cons(z0,z1)),z0,z1) ,MIN(cons(z0,z1))) IFSELSORT(true(),cons(z0,z1)) -> c18(SELSORT(z1)) LE(0(),z0) -> c4() LE(s(z0),0()) -> c5() LE(s(z0),s(z1)) -> c6(LE(z0,z1)) MIN(cons(z0,cons(z1,z2))) -> c9(IFMIN(le(z0,z1),cons(z0,cons(z1,z2))),LE(z0,z1)) MIN(cons(0(),nil())) -> c7() MIN(cons(s(z0),nil())) -> c8() REPLACE(z0,z1,cons(z2,z3)) -> c13(IFREPL(eq(z0,z2),z0,z1,cons(z2,z3)),EQ(z0,z2)) REPLACE(z0,z1,nil()) -> c12() SELSORT(cons(z0,z1)) -> c17(IFSELSORT(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,EQ(z0,min(cons(z0,z1))) ,MIN(cons(z0,z1))) SELSORT(nil()) -> c16() eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) ifselsort(false(),cons(z0,z1)) -> cons(min(cons(z0,z1)),selsort(replace(min(cons(z0,z1)),z0,z1))) ifselsort(true(),cons(z0,z1)) -> cons(z0,selsort(z1)) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() selsort(cons(z0,z1)) -> ifselsort(eq(z0,min(cons(z0,z1))),cons(z0,z1)) selsort(nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/6,c_11/1,c_12/0,c_13/0,c_14/1,c_15/3 ,c_16/0,c_17/0,c_18/3,c_19/0,c_20/6,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)) -->_1 EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)):1 2:S:IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) -->_1 MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1),LE#(z0,z1)):9 3:S:IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) -->_1 MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1),LE#(z0,z1)):9 4:S:IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) -->_1 REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2),EQ#(z0,z2)):10 5:S:IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) -->_1 MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1),LE#(z0,z1)):9 6:S:IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1)) ,replace#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1)) ,REPLACE#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))) -->_1 SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))):11 -->_4 REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2),EQ#(z0,z2)):10 -->_6 MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1),LE#(z0,z1)):9 7:S:IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)) -->_1 SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))):11 8:S:LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)) -->_1 LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)):8 9:S:MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1),LE#(z0,z1)) -->_3 LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)):8 -->_1 IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))):3 -->_1 IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))):2 10:S:REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),eq#(z0,z2),EQ#(z0,z2)) -->_1 IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)):4 -->_3 EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)):1 11:S:SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,eq#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))) -->_6 MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),le#(z0,z1),LE#(z0,z1)):9 -->_1 IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)):7 -->_1 IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1)) ,replace#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1)) ,REPLACE#(min(cons(z0,z1)),z0,z1) ,min#(cons(z0,z1)) ,MIN#(cons(z0,z1))):6 -->_1 IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))):5 -->_4 EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1)) ,REPLACE#(min(cons(z0,z1)),z0,z1) ,MIN#(cons(z0,z1))) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),LE#(z0,z1)) REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),EQ#(z0,z2)) SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,MIN#(cons(z0,z1))) ** Step 1.b:5: UsableRules. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)) IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1)) ,REPLACE#(min(cons(z0,z1)),z0,z1) ,MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)) LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),LE#(z0,z1)) REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),EQ#(z0,z2)) SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,MIN#(cons(z0,z1))) - Weak TRS: EQ(0(),0()) -> c() EQ(0(),s(z0)) -> c1() EQ(s(z0),0()) -> c2() EQ(s(z0),s(z1)) -> c3(EQ(z0,z1)) IFMIN(false(),cons(z0,cons(z1,z2))) -> c11(MIN(cons(z1,z2))) IFMIN(true(),cons(z0,cons(z1,z2))) -> c10(MIN(cons(z0,z2))) IFREPL(false(),z0,z1,cons(z2,z3)) -> c15(REPLACE(z0,z1,z3)) IFREPL(true(),z0,z1,cons(z2,z3)) -> c14() IFSELSORT(false(),cons(z0,z1)) -> c19(MIN(cons(z0,z1))) IFSELSORT(false(),cons(z0,z1)) -> c20(SELSORT(replace(min(cons(z0,z1)),z0,z1)) ,REPLACE(min(cons(z0,z1)),z0,z1) ,MIN(cons(z0,z1))) IFSELSORT(true(),cons(z0,z1)) -> c18(SELSORT(z1)) LE(0(),z0) -> c4() LE(s(z0),0()) -> c5() LE(s(z0),s(z1)) -> c6(LE(z0,z1)) MIN(cons(z0,cons(z1,z2))) -> c9(IFMIN(le(z0,z1),cons(z0,cons(z1,z2))),LE(z0,z1)) MIN(cons(0(),nil())) -> c7() MIN(cons(s(z0),nil())) -> c8() REPLACE(z0,z1,cons(z2,z3)) -> c13(IFREPL(eq(z0,z2),z0,z1,cons(z2,z3)),EQ(z0,z2)) REPLACE(z0,z1,nil()) -> c12() SELSORT(cons(z0,z1)) -> c17(IFSELSORT(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,EQ(z0,min(cons(z0,z1))) ,MIN(cons(z0,z1))) SELSORT(nil()) -> c16() eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) ifselsort(false(),cons(z0,z1)) -> cons(min(cons(z0,z1)),selsort(replace(min(cons(z0,z1)),z0,z1))) ifselsort(true(),cons(z0,z1)) -> cons(z0,selsort(z1)) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() selsort(cons(z0,z1)) -> ifselsort(eq(z0,min(cons(z0,z1))),cons(z0,z1)) selsort(nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/3,c_11/1,c_12/0,c_13/0,c_14/1,c_15/2 ,c_16/0,c_17/0,c_18/2,c_19/0,c_20/3,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)) IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1)) ,REPLACE#(min(cons(z0,z1)),z0,z1) ,MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)) LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),LE#(z0,z1)) REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),EQ#(z0,z2)) SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,MIN#(cons(z0,z1))) ** Step 1.b:6: DecomposeDG. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)) IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1)) ,REPLACE#(min(cons(z0,z1)),z0,z1) ,MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)) LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),LE#(z0,z1)) REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),EQ#(z0,z2)) SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,MIN#(cons(z0,z1))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/3,c_11/1,c_12/0,c_13/0,c_14/1,c_15/2 ,c_16/0,c_17/0,c_18/2,c_19/0,c_20/3,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,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 IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1)) ,REPLACE#(min(cons(z0,z1)),z0,z1) ,MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)) SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,MIN#(cons(z0,z1))) and a lower component EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)) IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),LE#(z0,z1)) REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),EQ#(z0,z2)) Further, following extension rules are added to the lower component. IFSELSORT#(false(),cons(z0,z1)) -> MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) -> SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(true(),cons(z0,z1)) -> SELSORT#(z1) SELSORT#(cons(z0,z1)) -> EQ#(z0,min(cons(z0,z1))) SELSORT#(cons(z0,z1)) -> IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) -> MIN#(cons(z0,z1)) *** Step 1.b:6.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1)) ,REPLACE#(min(cons(z0,z1)),z0,z1) ,MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)) SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,MIN#(cons(z0,z1))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/3,c_11/1,c_12/0,c_13/0,c_14/1,c_15/2 ,c_16/0,c_17/0,c_18/2,c_19/0,c_20/3,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1)) ,REPLACE#(min(cons(z0,z1)),z0,z1) ,MIN#(cons(z0,z1))) -->_1 SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,MIN#(cons(z0,z1))):3 2:S:IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)) -->_1 SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,MIN#(cons(z0,z1))):3 3:S:SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) ,EQ#(z0,min(cons(z0,z1))) ,MIN#(cons(z0,z1))) -->_1 IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)):2 -->_1 IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1)) ,REPLACE#(min(cons(z0,z1)),z0,z1) ,MIN#(cons(z0,z1))):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1))) SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1))) *** Step 1.b:6.a:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1))) IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)) SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/0,c_14/1,c_15/2 ,c_16/0,c_17/0,c_18/2,c_19/0,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,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_10) = {1}, uargs(c_11) = {1}, uargs(c_20) = {1} Following symbols are considered usable: {ifrepl,replace,EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT#,eq#,ifmin#,ifrepl#,ifselsort#,le# ,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [0] p(EQ) = [0] p(IFMIN) = [0] p(IFREPL) = [0] p(IFSELSORT) = [0] p(LE) = [0] p(MIN) = [0] p(REPLACE) = [0] p(SELSORT) = [1] x1 + [0] p(c) = [0] p(c1) = [0] p(c10) = [0] p(c11) = [0] p(c12) = [0] p(c13) = [0] p(c14) = [0] p(c15) = [1] x1 + [0] p(c16) = [0] p(c17) = [1] x1 + [0] p(c18) = [0] p(c19) = [0] p(c2) = [0] p(c20) = [1] x1 + [0] p(c3) = [0] p(c4) = [0] p(c5) = [0] p(c6) = [0] p(c7) = [0] p(c8) = [0] p(c9) = [1] x2 + [0] p(cons) = [1] x2 + [4] p(eq) = [5] p(false) = [0] p(ifmin) = [1] x2 + [6] p(ifrepl) = [1] x4 + [4] p(ifselsort) = [0] p(le) = [5] x1 + [2] x2 + [0] p(min) = [5] p(nil) = [5] p(replace) = [1] x3 + [4] p(s) = [1] x1 + [1] p(selsort) = [1] x1 + [2] p(true) = [0] p(EQ#) = [4] x1 + [0] p(IFMIN#) = [1] x1 + [2] x2 + [0] p(IFREPL#) = [1] x3 + [2] x4 + [0] p(IFSELSORT#) = [2] x2 + [0] p(LE#) = [2] x1 + [1] x2 + [0] p(MIN#) = [4] x1 + [0] p(REPLACE#) = [2] p(SELSORT#) = [2] x1 + [0] p(eq#) = [0] p(ifmin#) = [0] p(ifrepl#) = [4] x2 + [1] x3 + [0] p(ifselsort#) = [1] x1 + [1] x2 + [0] p(le#) = [4] x1 + [1] x2 + [0] p(min#) = [1] x1 + [1] p(replace#) = [1] x1 + [1] x3 + [0] p(selsort#) = [4] x1 + [2] p(c_1) = [0] p(c_2) = [4] p(c_3) = [2] p(c_4) = [1] x1 + [1] p(c_5) = [1] x1 + [0] p(c_6) = [1] p(c_7) = [1] x1 + [1] p(c_8) = [0] p(c_9) = [2] x1 + [0] p(c_10) = [1] x1 + [0] p(c_11) = [1] x1 + [4] p(c_12) = [4] p(c_13) = [1] p(c_14) = [1] x1 + [2] p(c_15) = [4] x1 + [1] p(c_16) = [4] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [1] x1 + [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [1] p(c_25) = [4] x1 + [4] p(c_26) = [2] p(c_27) = [2] x1 + [1] p(c_28) = [0] p(c_29) = [2] p(c_30) = [4] x2 + [0] p(c_31) = [4] p(c_32) = [0] p(c_33) = [4] p(c_34) = [4] x1 + [0] p(c_35) = [4] x1 + [0] p(c_36) = [0] p(c_37) = [2] p(c_38) = [1] x2 + [2] p(c_39) = [4] p(c_40) = [1] x3 + [4] p(c_41) = [0] Following rules are strictly oriented: IFSELSORT#(true(),cons(z0,z1)) = [2] z1 + [8] > [2] z1 + [4] = c_11(SELSORT#(z1)) Following rules are (at-least) weakly oriented: IFSELSORT#(false(),cons(z0,z1)) = [2] z1 + [8] >= [2] z1 + [8] = c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1))) SELSORT#(cons(z0,z1)) = [2] z1 + [8] >= [2] z1 + [8] = c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1))) ifrepl(false(),z0,z1,cons(z2,z3)) = [1] z3 + [8] >= [1] z3 + [8] = cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) = [1] z3 + [8] >= [1] z3 + [4] = cons(z1,z3) replace(z0,z1,cons(z2,z3)) = [1] z3 + [8] >= [1] z3 + [8] = ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) = [9] >= [5] = nil() *** Step 1.b:6.a:3: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1))) SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1))) - Weak DPs: IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/0,c_14/1,c_15/2 ,c_16/0,c_17/0,c_18/2,c_19/0,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,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_10) = {1}, uargs(c_11) = {1}, uargs(c_20) = {1} Following symbols are considered usable: {ifrepl,replace,EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT#,eq#,ifmin#,ifrepl#,ifselsort#,le# ,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [0] p(EQ) = [1] x2 + [1] p(IFMIN) = [1] x2 + [1] p(IFREPL) = [1] x1 + [1] x3 + [2] x4 + [1] p(IFSELSORT) = [1] x1 + [0] p(LE) = [1] x1 + [2] p(MIN) = [0] p(REPLACE) = [4] x1 + [0] p(SELSORT) = [1] p(c) = [1] p(c1) = [0] p(c10) = [0] p(c11) = [1] x1 + [1] p(c12) = [1] p(c13) = [0] p(c14) = [0] p(c15) = [1] x1 + [1] p(c16) = [1] p(c17) = [2] p(c18) = [0] p(c19) = [1] p(c2) = [4] p(c20) = [1] x2 + [1] x3 + [0] p(c3) = [1] p(c4) = [0] p(c5) = [0] p(c6) = [0] p(c7) = [0] p(c8) = [0] p(c9) = [1] x2 + [1] p(cons) = [1] x2 + [5] p(eq) = [6] p(false) = [0] p(ifmin) = [4] p(ifrepl) = [1] x4 + [0] p(ifselsort) = [1] p(le) = [3] x1 + [3] p(min) = [6] p(nil) = [4] p(replace) = [1] x3 + [0] p(s) = [1] p(selsort) = [0] p(true) = [0] p(EQ#) = [1] x1 + [0] p(IFMIN#) = [2] x2 + [1] p(IFREPL#) = [1] x2 + [4] x4 + [0] p(IFSELSORT#) = [3] x2 + [0] p(LE#) = [1] x1 + [4] x2 + [0] p(MIN#) = [1] x1 + [0] p(REPLACE#) = [4] p(SELSORT#) = [3] x1 + [0] p(eq#) = [4] x1 + [1] x2 + [1] p(ifmin#) = [2] x1 + [0] p(ifrepl#) = [1] x3 + [0] p(ifselsort#) = [1] p(le#) = [0] p(min#) = [4] x1 + [1] p(replace#) = [2] x3 + [1] p(selsort#) = [1] x1 + [4] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] x1 + [1] p(c_5) = [0] p(c_6) = [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] p(c_9) = [1] p(c_10) = [1] x1 + [6] p(c_11) = [1] x1 + [1] p(c_12) = [0] p(c_13) = [1] p(c_14) = [2] x1 + [1] p(c_15) = [1] x1 + [0] p(c_16) = [0] p(c_17) = [1] p(c_18) = [4] x1 + [0] p(c_19) = [0] p(c_20) = [1] x1 + [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [4] p(c_24) = [0] p(c_25) = [4] x1 + [0] p(c_26) = [2] x1 + [1] p(c_27) = [2] x1 + [4] p(c_28) = [1] x1 + [0] p(c_29) = [4] p(c_30) = [2] x1 + [4] x2 + [1] x3 + [4] p(c_31) = [1] x1 + [4] p(c_32) = [0] p(c_33) = [1] p(c_34) = [1] p(c_35) = [1] x1 + [1] x2 + [2] p(c_36) = [1] p(c_37) = [1] p(c_38) = [1] x1 + [1] p(c_39) = [0] p(c_40) = [2] x2 + [1] x3 + [0] p(c_41) = [1] Following rules are strictly oriented: IFSELSORT#(false(),cons(z0,z1)) = [3] z1 + [15] > [3] z1 + [6] = c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1))) Following rules are (at-least) weakly oriented: IFSELSORT#(true(),cons(z0,z1)) = [3] z1 + [15] >= [3] z1 + [1] = c_11(SELSORT#(z1)) SELSORT#(cons(z0,z1)) = [3] z1 + [15] >= [3] z1 + [15] = c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1))) ifrepl(false(),z0,z1,cons(z2,z3)) = [1] z3 + [5] >= [1] z3 + [5] = cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) = [1] z3 + [5] >= [1] z3 + [5] = cons(z1,z3) replace(z0,z1,cons(z2,z3)) = [1] z3 + [5] >= [1] z3 + [5] = ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) = [4] >= [4] = nil() *** Step 1.b:6.a:4: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1))) - Weak DPs: IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1))) IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/0,c_14/1,c_15/2 ,c_16/0,c_17/0,c_18/2,c_19/0,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,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_10) = {1}, uargs(c_11) = {1}, uargs(c_20) = {1} Following symbols are considered usable: {ifrepl,replace,EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT#,eq#,ifmin#,ifrepl#,ifselsort#,le# ,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [0] p(EQ) = [0] p(IFMIN) = [0] p(IFREPL) = [0] p(IFSELSORT) = [0] p(LE) = [0] p(MIN) = [0] p(REPLACE) = [0] p(SELSORT) = [0] p(c) = [0] p(c1) = [0] p(c10) = [1] x1 + [0] p(c11) = [1] x1 + [0] p(c12) = [0] p(c13) = [1] x1 + [1] x2 + [0] p(c14) = [0] p(c15) = [1] x1 + [0] p(c16) = [0] p(c17) = [1] x1 + [1] x2 + [1] x3 + [0] p(c18) = [1] x1 + [0] p(c19) = [1] x1 + [0] p(c2) = [0] p(c20) = [1] x1 + [1] x2 + [1] x3 + [0] p(c3) = [1] x1 + [0] p(c4) = [0] p(c5) = [0] p(c6) = [1] x1 + [0] p(c7) = [0] p(c8) = [0] p(c9) = [1] x1 + [1] x2 + [0] p(cons) = [1] x2 + [4] p(eq) = [7] x1 + [7] p(false) = [0] p(ifmin) = [0] p(ifrepl) = [1] x4 + [0] p(ifselsort) = [0] p(le) = [0] p(min) = [0] p(nil) = [0] p(replace) = [1] x3 + [0] p(s) = [0] p(selsort) = [0] p(true) = [0] p(EQ#) = [0] p(IFMIN#) = [0] p(IFREPL#) = [0] p(IFSELSORT#) = [2] x2 + [0] p(LE#) = [0] p(MIN#) = [0] p(REPLACE#) = [0] p(SELSORT#) = [2] x1 + [2] p(eq#) = [1] x2 + [0] p(ifmin#) = [0] p(ifrepl#) = [1] x2 + [4] x3 + [1] x4 + [0] p(ifselsort#) = [1] x1 + [1] p(le#) = [0] p(min#) = [0] p(replace#) = [2] p(selsort#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [1] p(c_3) = [1] p(c_4) = [0] p(c_5) = [2] x1 + [1] p(c_6) = [4] p(c_7) = [2] x1 + [0] p(c_8) = [0] p(c_9) = [1] p(c_10) = [1] x1 + [0] p(c_11) = [1] x1 + [1] p(c_12) = [4] p(c_13) = [1] p(c_14) = [0] p(c_15) = [1] x1 + [0] p(c_16) = [1] p(c_17) = [1] p(c_18) = [2] p(c_19) = [0] p(c_20) = [1] x1 + [0] p(c_21) = [1] p(c_22) = [0] p(c_23) = [1] p(c_24) = [0] p(c_25) = [1] x1 + [0] p(c_26) = [1] x1 + [0] p(c_27) = [1] x1 + [2] p(c_28) = [2] x1 + [0] p(c_29) = [0] p(c_30) = [1] x1 + [1] x2 + [2] x3 + [1] x4 + [0] p(c_31) = [4] p(c_32) = [1] p(c_33) = [0] p(c_34) = [1] p(c_35) = [4] x2 + [1] p(c_36) = [1] p(c_37) = [0] p(c_38) = [2] x2 + [1] p(c_39) = [0] p(c_40) = [0] p(c_41) = [0] Following rules are strictly oriented: SELSORT#(cons(z0,z1)) = [2] z1 + [10] > [2] z1 + [8] = c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1))) Following rules are (at-least) weakly oriented: IFSELSORT#(false(),cons(z0,z1)) = [2] z1 + [8] >= [2] z1 + [2] = c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1))) IFSELSORT#(true(),cons(z0,z1)) = [2] z1 + [8] >= [2] z1 + [3] = c_11(SELSORT#(z1)) ifrepl(false(),z0,z1,cons(z2,z3)) = [1] z3 + [4] >= [1] z3 + [4] = cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) = [1] z3 + [4] >= [1] z3 + [4] = cons(z1,z3) replace(z0,z1,cons(z2,z3)) = [1] z3 + [4] >= [1] z3 + [4] = ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) = [0] >= [0] = nil() *** Step 1.b:6.a:5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: IFSELSORT#(false(),cons(z0,z1)) -> c_10(SELSORT#(replace(min(cons(z0,z1)),z0,z1))) IFSELSORT#(true(),cons(z0,z1)) -> c_11(SELSORT#(z1)) SELSORT#(cons(z0,z1)) -> c_20(IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/0,c_14/1,c_15/2 ,c_16/0,c_17/0,c_18/2,c_19/0,c_20/1,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:6.b:1: DecomposeDG. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)) IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),LE#(z0,z1)) REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),EQ#(z0,z2)) - Weak DPs: IFSELSORT#(false(),cons(z0,z1)) -> MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) -> SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(true(),cons(z0,z1)) -> SELSORT#(z1) SELSORT#(cons(z0,z1)) -> EQ#(z0,min(cons(z0,z1))) SELSORT#(cons(z0,z1)) -> IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) -> MIN#(cons(z0,z1)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/3,c_11/1,c_12/0,c_13/0,c_14/1,c_15/2 ,c_16/0,c_17/0,c_18/2,c_19/0,c_20/3,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,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 IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) IFSELSORT#(false(),cons(z0,z1)) -> MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) -> SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) -> SELSORT#(z1) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),LE#(z0,z1)) REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),EQ#(z0,z2)) SELSORT#(cons(z0,z1)) -> EQ#(z0,min(cons(z0,z1))) SELSORT#(cons(z0,z1)) -> IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) -> MIN#(cons(z0,z1)) and a lower component EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)) LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)) Further, following extension rules are added to the lower component. IFMIN#(false(),cons(z0,cons(z1,z2))) -> MIN#(cons(z1,z2)) IFMIN#(true(),cons(z0,cons(z1,z2))) -> MIN#(cons(z0,z2)) IFREPL#(false(),z0,z1,cons(z2,z3)) -> REPLACE#(z0,z1,z3) IFSELSORT#(false(),cons(z0,z1)) -> MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) -> SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(true(),cons(z0,z1)) -> SELSORT#(z1) MIN#(cons(z0,cons(z1,z2))) -> IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))) MIN#(cons(z0,cons(z1,z2))) -> LE#(z0,z1) REPLACE#(z0,z1,cons(z2,z3)) -> EQ#(z0,z2) REPLACE#(z0,z1,cons(z2,z3)) -> IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)) SELSORT#(cons(z0,z1)) -> EQ#(z0,min(cons(z0,z1))) SELSORT#(cons(z0,z1)) -> IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) -> MIN#(cons(z0,z1)) **** Step 1.b:6.b:1.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),LE#(z0,z1)) REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),EQ#(z0,z2)) SELSORT#(cons(z0,z1)) -> EQ#(z0,min(cons(z0,z1))) - Weak DPs: IFSELSORT#(false(),cons(z0,z1)) -> MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) -> SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(true(),cons(z0,z1)) -> SELSORT#(z1) SELSORT#(cons(z0,z1)) -> IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) -> MIN#(cons(z0,z1)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/3,c_11/1,c_12/0,c_13/0,c_14/1,c_15/2 ,c_16/0,c_17/0,c_18/2,c_19/0,c_20/3,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) -->_1 MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),LE#(z0,z1)):5 2:S:IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) -->_1 MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),LE#(z0,z1)):5 3:S:IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) -->_1 REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),EQ#(z0,z2)):6 4:S:IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) -->_1 MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),LE#(z0,z1)):5 5:S:MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),LE#(z0,z1)) -->_1 IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))):2 -->_1 IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))):1 6:S:REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),EQ#(z0,z2)) -->_1 IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)):3 7:S:SELSORT#(cons(z0,z1)) -> EQ#(z0,min(cons(z0,z1))) 8:W:IFSELSORT#(false(),cons(z0,z1)) -> MIN#(cons(z0,z1)) -->_1 MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),LE#(z0,z1)):5 9:W:IFSELSORT#(false(),cons(z0,z1)) -> REPLACE#(min(cons(z0,z1)),z0,z1) -->_1 REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)),EQ#(z0,z2)):6 10:W:IFSELSORT#(false(),cons(z0,z1)) -> SELSORT#(replace(min(cons(z0,z1)),z0,z1)) -->_1 SELSORT#(cons(z0,z1)) -> MIN#(cons(z0,z1)):13 -->_1 SELSORT#(cons(z0,z1)) -> IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)):12 -->_1 SELSORT#(cons(z0,z1)) -> EQ#(z0,min(cons(z0,z1))):7 11:W:IFSELSORT#(true(),cons(z0,z1)) -> SELSORT#(z1) -->_1 SELSORT#(cons(z0,z1)) -> MIN#(cons(z0,z1)):13 -->_1 SELSORT#(cons(z0,z1)) -> IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)):12 -->_1 SELSORT#(cons(z0,z1)) -> EQ#(z0,min(cons(z0,z1))):7 12:W:SELSORT#(cons(z0,z1)) -> IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) -->_1 IFSELSORT#(true(),cons(z0,z1)) -> SELSORT#(z1):11 -->_1 IFSELSORT#(false(),cons(z0,z1)) -> SELSORT#(replace(min(cons(z0,z1)),z0,z1)):10 -->_1 IFSELSORT#(false(),cons(z0,z1)) -> REPLACE#(min(cons(z0,z1)),z0,z1):9 -->_1 IFSELSORT#(false(),cons(z0,z1)) -> MIN#(cons(z0,z1)):8 -->_1 IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))):4 13:W:SELSORT#(cons(z0,z1)) -> MIN#(cons(z0,z1)) -->_1 MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))),LE#(z0,z1)):5 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2)))) REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3))) **** Step 1.b:6.b:1.a:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2)))) REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3))) SELSORT#(cons(z0,z1)) -> EQ#(z0,min(cons(z0,z1))) - Weak DPs: IFSELSORT#(false(),cons(z0,z1)) -> MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) -> SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(true(),cons(z0,z1)) -> SELSORT#(z1) SELSORT#(cons(z0,z1)) -> IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) -> MIN#(cons(z0,z1)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/3,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1 ,c_16/0,c_17/0,c_18/1,c_19/0,c_20/3,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,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_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_15) = {1}, uargs(c_18) = {1} Following symbols are considered usable: {ifrepl,le,replace,EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT#,eq#,ifmin#,ifrepl#,ifselsort# ,le#,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [4] p(EQ) = [1] x2 + [0] p(IFMIN) = [2] p(IFREPL) = [1] x1 + [4] p(IFSELSORT) = [1] x2 + [1] p(LE) = [1] x1 + [0] p(MIN) = [1] p(REPLACE) = [1] x2 + [2] x3 + [1] p(SELSORT) = [1] x1 + [1] p(c) = [4] p(c1) = [0] p(c10) = [0] p(c11) = [1] p(c12) = [0] p(c13) = [1] x1 + [1] p(c14) = [2] p(c15) = [1] p(c16) = [1] p(c17) = [4] p(c18) = [1] x1 + [0] p(c19) = [2] p(c2) = [1] p(c20) = [1] x1 + [1] x2 + [2] p(c3) = [1] x1 + [0] p(c4) = [4] p(c5) = [1] p(c6) = [0] p(c7) = [1] p(c8) = [1] p(c9) = [1] x2 + [0] p(cons) = [1] x2 + [4] p(eq) = [0] p(false) = [1] p(ifmin) = [4] x1 + [3] p(ifrepl) = [1] x4 + [0] p(ifselsort) = [4] x2 + [1] p(le) = [1] p(min) = [0] p(nil) = [0] p(replace) = [1] x3 + [0] p(s) = [0] p(selsort) = [1] p(true) = [0] p(EQ#) = [0] p(IFMIN#) = [4] x1 + [1] x2 + [0] p(IFREPL#) = [0] p(IFSELSORT#) = [1] x2 + [6] p(LE#) = [1] p(MIN#) = [1] x1 + [4] p(REPLACE#) = [0] p(SELSORT#) = [1] x1 + [6] p(eq#) = [4] x1 + [4] x2 + [2] p(ifmin#) = [1] p(ifrepl#) = [0] p(ifselsort#) = [2] x1 + [0] p(le#) = [2] x1 + [4] x2 + [0] p(min#) = [0] p(replace#) = [0] p(selsort#) = [1] x1 + [0] p(c_1) = [2] p(c_2) = [1] p(c_3) = [1] p(c_4) = [1] x1 + [1] p(c_5) = [1] x1 + [4] p(c_6) = [1] x1 + [0] p(c_7) = [4] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [2] p(c_10) = [1] x1 + [4] x2 + [1] x3 + [0] p(c_11) = [1] x1 + [1] p(c_12) = [0] p(c_13) = [0] p(c_14) = [2] x1 + [1] p(c_15) = [1] x1 + [0] p(c_16) = [0] p(c_17) = [1] p(c_18) = [4] x1 + [0] p(c_19) = [1] p(c_20) = [1] x1 + [1] x3 + [0] p(c_21) = [0] p(c_22) = [1] p(c_23) = [2] p(c_24) = [0] p(c_25) = [1] x1 + [0] p(c_26) = [1] x1 + [0] p(c_27) = [1] x1 + [0] p(c_28) = [1] p(c_29) = [0] p(c_30) = [1] x1 + [1] x3 + [4] x4 + [1] p(c_31) = [0] p(c_32) = [4] p(c_33) = [1] p(c_34) = [0] p(c_35) = [1] x2 + [2] p(c_36) = [0] p(c_37) = [0] p(c_38) = [4] x1 + [4] x2 + [0] p(c_39) = [2] p(c_40) = [4] x1 + [1] x2 + [0] p(c_41) = [1] Following rules are strictly oriented: SELSORT#(cons(z0,z1)) = [1] z1 + [10] > [0] = EQ#(z0,min(cons(z0,z1))) Following rules are (at-least) weakly oriented: IFMIN#(false(),cons(z0,cons(z1,z2))) = [1] z2 + [12] >= [1] z2 + [12] = c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) = [1] z2 + [8] >= [1] z2 + [8] = c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) = [0] >= [0] = c_7(REPLACE#(z0,z1,z3)) IFSELSORT#(false(),cons(z0,z1)) = [1] z1 + [10] >= [1] z1 + [8] = MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) = [1] z1 + [10] >= [0] = REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) = [1] z1 + [10] >= [1] z1 + [6] = SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(false(),cons(z0,z1)) = [1] z1 + [10] >= [1] z1 + [10] = c_9(MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) = [1] z1 + [10] >= [1] z1 + [6] = SELSORT#(z1) MIN#(cons(z0,cons(z1,z2))) = [1] z2 + [12] >= [1] z2 + [12] = c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2)))) REPLACE#(z0,z1,cons(z2,z3)) = [0] >= [0] = c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3))) SELSORT#(cons(z0,z1)) = [1] z1 + [10] >= [1] z1 + [10] = IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) = [1] z1 + [10] >= [1] z1 + [8] = MIN#(cons(z0,z1)) ifrepl(false(),z0,z1,cons(z2,z3)) = [1] z3 + [4] >= [1] z3 + [4] = cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) = [1] z3 + [4] >= [1] z3 + [4] = cons(z1,z3) le(0(),z0) = [1] >= [0] = true() le(s(z0),0()) = [1] >= [1] = false() le(s(z0),s(z1)) = [1] >= [1] = le(z0,z1) replace(z0,z1,cons(z2,z3)) = [1] z3 + [4] >= [1] z3 + [4] = ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) = [0] >= [0] = nil() **** Step 1.b:6.b:1.a:3: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2)))) REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3))) - Weak DPs: IFSELSORT#(false(),cons(z0,z1)) -> MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) -> SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(true(),cons(z0,z1)) -> SELSORT#(z1) SELSORT#(cons(z0,z1)) -> EQ#(z0,min(cons(z0,z1))) SELSORT#(cons(z0,z1)) -> IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) -> MIN#(cons(z0,z1)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/3,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1 ,c_16/0,c_17/0,c_18/1,c_19/0,c_20/3,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,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_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_15) = {1}, uargs(c_18) = {1} Following symbols are considered usable: {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT#,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} TcT has computed the following interpretation: p(0) = [0] p(EQ) = [0] p(IFMIN) = [0] p(IFREPL) = [4] x1 + [0] p(IFSELSORT) = [2] x2 + [1] p(LE) = [1] x1 + [4] p(MIN) = [1] x1 + [4] p(REPLACE) = [2] x3 + [1] p(SELSORT) = [1] p(c) = [1] p(c1) = [4] p(c10) = [0] p(c11) = [1] x1 + [4] p(c12) = [0] p(c13) = [0] p(c14) = [1] p(c15) = [1] x1 + [1] p(c16) = [4] p(c17) = [1] x2 + [1] p(c18) = [0] p(c19) = [4] p(c2) = [0] p(c20) = [1] p(c3) = [0] p(c4) = [0] p(c5) = [0] p(c6) = [1] p(c7) = [1] p(c8) = [0] p(c9) = [1] p(cons) = [0] p(eq) = [0] p(false) = [0] p(ifmin) = [2] p(ifrepl) = [4] x1 + [1] x2 + [1] x3 + [0] p(ifselsort) = [1] p(le) = [0] p(min) = [2] p(nil) = [1] p(replace) = [0] p(s) = [0] p(selsort) = [1] x1 + [1] p(true) = [0] p(EQ#) = [5] p(IFMIN#) = [0] p(IFREPL#) = [0] p(IFSELSORT#) = [1] x2 + [5] p(LE#) = [2] x1 + [0] p(MIN#) = [0] p(REPLACE#) = [0] p(SELSORT#) = [5] p(eq#) = [1] x1 + [0] p(ifmin#) = [1] x1 + [0] p(ifrepl#) = [4] x1 + [2] x4 + [0] p(ifselsort#) = [4] p(le#) = [4] x1 + [4] x2 + [0] p(min#) = [1] p(replace#) = [1] x1 + [1] x3 + [4] p(selsort#) = [1] x1 + [0] p(c_1) = [1] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [4] x1 + [0] p(c_6) = [4] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] p(c_9) = [1] x1 + [0] p(c_10) = [4] x2 + [1] p(c_11) = [2] p(c_12) = [1] p(c_13) = [0] p(c_14) = [1] p(c_15) = [2] x1 + [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [4] x1 + [0] p(c_19) = [0] p(c_20) = [1] p(c_21) = [1] p(c_22) = [1] p(c_23) = [0] p(c_24) = [4] p(c_25) = [1] p(c_26) = [4] p(c_27) = [1] x1 + [0] p(c_28) = [2] x1 + [0] p(c_29) = [0] p(c_30) = [2] x1 + [4] x2 + [1] x3 + [0] p(c_31) = [4] x1 + [2] p(c_32) = [0] p(c_33) = [4] p(c_34) = [0] p(c_35) = [0] p(c_36) = [4] p(c_37) = [4] p(c_38) = [1] x1 + [1] p(c_39) = [0] p(c_40) = [1] p(c_41) = [0] Following rules are strictly oriented: IFSELSORT#(false(),cons(z0,z1)) = [5] > [0] = c_9(MIN#(cons(z0,z1))) Following rules are (at-least) weakly oriented: IFMIN#(false(),cons(z0,cons(z1,z2))) = [0] >= [0] = c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) = [0] >= [0] = c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) = [0] >= [0] = c_7(REPLACE#(z0,z1,z3)) IFSELSORT#(false(),cons(z0,z1)) = [5] >= [0] = MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) = [5] >= [0] = REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) = [5] >= [5] = SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(true(),cons(z0,z1)) = [5] >= [5] = SELSORT#(z1) MIN#(cons(z0,cons(z1,z2))) = [0] >= [0] = c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2)))) REPLACE#(z0,z1,cons(z2,z3)) = [0] >= [0] = c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3))) SELSORT#(cons(z0,z1)) = [5] >= [5] = EQ#(z0,min(cons(z0,z1))) SELSORT#(cons(z0,z1)) = [5] >= [5] = IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) = [5] >= [0] = MIN#(cons(z0,z1)) **** Step 1.b:6.b:1.a:4: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2)))) REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3))) - Weak DPs: IFSELSORT#(false(),cons(z0,z1)) -> MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) -> SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) -> SELSORT#(z1) SELSORT#(cons(z0,z1)) -> EQ#(z0,min(cons(z0,z1))) SELSORT#(cons(z0,z1)) -> IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) -> MIN#(cons(z0,z1)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/3,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1 ,c_16/0,c_17/0,c_18/1,c_19/0,c_20/3,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,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_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_15) = {1}, uargs(c_18) = {1} Following symbols are considered usable: {ifrepl,replace,EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT#,eq#,ifmin#,ifrepl#,ifselsort#,le# ,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [0] p(EQ) = [1] x1 + [1] x2 + [4] p(IFMIN) = [1] x1 + [0] p(IFREPL) = [1] x2 + [4] x3 + [4] x4 + [0] p(IFSELSORT) = [2] x2 + [0] p(LE) = [4] p(MIN) = [4] p(REPLACE) = [2] x1 + [1] x3 + [0] p(SELSORT) = [1] p(c) = [0] p(c1) = [0] p(c10) = [0] p(c11) = [4] p(c12) = [4] p(c13) = [1] x2 + [1] p(c14) = [1] p(c15) = [1] x1 + [0] p(c16) = [1] p(c17) = [1] p(c18) = [0] p(c19) = [0] p(c2) = [1] p(c20) = [0] p(c3) = [0] p(c4) = [1] p(c5) = [0] p(c6) = [0] p(c7) = [0] p(c8) = [0] p(c9) = [1] x1 + [0] p(cons) = [1] x2 + [1] p(eq) = [0] p(false) = [0] p(ifmin) = [2] x1 + [3] x2 + [3] p(ifrepl) = [1] x4 + [0] p(ifselsort) = [1] x2 + [0] p(le) = [0] p(min) = [0] p(nil) = [4] p(replace) = [1] x3 + [0] p(s) = [0] p(selsort) = [2] x1 + [1] p(true) = [0] p(EQ#) = [2] p(IFMIN#) = [4] x2 + [0] p(IFREPL#) = [0] p(IFSELSORT#) = [4] x2 + [1] p(LE#) = [2] x1 + [1] x2 + [2] p(MIN#) = [4] x1 + [1] p(REPLACE#) = [0] p(SELSORT#) = [4] x1 + [1] p(eq#) = [1] p(ifmin#) = [1] p(ifrepl#) = [1] x1 + [1] p(ifselsort#) = [1] p(le#) = [1] p(min#) = [1] x1 + [2] p(replace#) = [2] x2 + [1] x3 + [4] p(selsort#) = [0] p(c_1) = [1] p(c_2) = [1] p(c_3) = [0] p(c_4) = [4] x1 + [0] p(c_5) = [1] x1 + [1] p(c_6) = [1] x1 + [3] p(c_7) = [4] x1 + [0] p(c_8) = [4] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [0] p(c_11) = [4] x1 + [0] p(c_12) = [0] p(c_13) = [1] p(c_14) = [4] p(c_15) = [1] x1 + [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [4] x1 + [0] p(c_19) = [0] p(c_20) = [1] x3 + [0] p(c_21) = [2] p(c_22) = [1] p(c_23) = [0] p(c_24) = [0] p(c_25) = [1] p(c_26) = [0] p(c_27) = [1] x1 + [0] p(c_28) = [2] p(c_29) = [1] p(c_30) = [1] x3 + [0] p(c_31) = [2] x1 + [0] p(c_32) = [2] p(c_33) = [0] p(c_34) = [4] p(c_35) = [4] x1 + [1] x2 + [1] p(c_36) = [1] p(c_37) = [1] p(c_38) = [0] p(c_39) = [4] p(c_40) = [2] x3 + [2] p(c_41) = [2] Following rules are strictly oriented: IFMIN#(false(),cons(z0,cons(z1,z2))) = [4] z2 + [8] > [4] z2 + [6] = c_5(MIN#(cons(z1,z2))) MIN#(cons(z0,cons(z1,z2))) = [4] z2 + [9] > [4] z2 + [8] = c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2)))) Following rules are (at-least) weakly oriented: IFMIN#(true(),cons(z0,cons(z1,z2))) = [4] z2 + [8] >= [4] z2 + [8] = c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) = [0] >= [0] = c_7(REPLACE#(z0,z1,z3)) IFSELSORT#(false(),cons(z0,z1)) = [4] z1 + [5] >= [4] z1 + [5] = MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) = [4] z1 + [5] >= [0] = REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) = [4] z1 + [5] >= [4] z1 + [1] = SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(false(),cons(z0,z1)) = [4] z1 + [5] >= [4] z1 + [5] = c_9(MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) = [4] z1 + [5] >= [4] z1 + [1] = SELSORT#(z1) REPLACE#(z0,z1,cons(z2,z3)) = [0] >= [0] = c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3))) SELSORT#(cons(z0,z1)) = [4] z1 + [5] >= [2] = EQ#(z0,min(cons(z0,z1))) SELSORT#(cons(z0,z1)) = [4] z1 + [5] >= [4] z1 + [5] = IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) = [4] z1 + [5] >= [4] z1 + [5] = MIN#(cons(z0,z1)) ifrepl(false(),z0,z1,cons(z2,z3)) = [1] z3 + [1] >= [1] z3 + [1] = cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) = [1] z3 + [1] >= [1] z3 + [1] = cons(z1,z3) replace(z0,z1,cons(z2,z3)) = [1] z3 + [1] >= [1] z3 + [1] = ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) = [4] >= [4] = nil() **** Step 1.b:6.b:1.a:5: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3))) - Weak DPs: IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) IFSELSORT#(false(),cons(z0,z1)) -> MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) -> SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) -> SELSORT#(z1) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2)))) SELSORT#(cons(z0,z1)) -> EQ#(z0,min(cons(z0,z1))) SELSORT#(cons(z0,z1)) -> IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) -> MIN#(cons(z0,z1)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/3,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1 ,c_16/0,c_17/0,c_18/1,c_19/0,c_20/3,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,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_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_15) = {1}, uargs(c_18) = {1} Following symbols are considered usable: {ifrepl,le,replace,EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT#,eq#,ifmin#,ifrepl#,ifselsort# ,le#,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [0] p(EQ) = [1] x1 + [1] x2 + [1] p(IFMIN) = [0] p(IFREPL) = [1] x1 + [1] x2 + [1] x3 + [2] x4 + [1] p(IFSELSORT) = [1] x2 + [4] p(LE) = [1] x1 + [2] x2 + [2] p(MIN) = [2] x1 + [4] p(REPLACE) = [1] x2 + [1] p(SELSORT) = [1] x1 + [1] p(c) = [1] p(c1) = [1] p(c10) = [1] x1 + [0] p(c11) = [0] p(c12) = [4] p(c13) = [0] p(c14) = [1] p(c15) = [0] p(c16) = [0] p(c17) = [1] x3 + [4] p(c18) = [0] p(c19) = [0] p(c2) = [4] p(c20) = [1] x2 + [1] x3 + [0] p(c3) = [1] x1 + [4] p(c4) = [0] p(c5) = [0] p(c6) = [0] p(c7) = [2] p(c8) = [1] p(c9) = [1] p(cons) = [1] x2 + [4] p(eq) = [0] p(false) = [2] p(ifmin) = [1] x2 + [2] p(ifrepl) = [1] x4 + [0] p(ifselsort) = [0] p(le) = [3] p(min) = [6] p(nil) = [1] p(replace) = [1] x3 + [0] p(s) = [1] x1 + [0] p(selsort) = [1] p(true) = [3] p(EQ#) = [4] p(IFMIN#) = [2] x1 + [1] x2 + [0] p(IFREPL#) = [0] p(IFSELSORT#) = [1] x2 + [7] p(LE#) = [4] x1 + [0] p(MIN#) = [1] x1 + [6] p(REPLACE#) = [0] p(SELSORT#) = [1] x1 + [7] p(eq#) = [1] x2 + [0] p(ifmin#) = [1] x2 + [0] p(ifrepl#) = [1] x1 + [1] p(ifselsort#) = [1] x1 + [4] p(le#) = [1] x2 + [1] p(min#) = [0] p(replace#) = [4] x2 + [1] x3 + [1] p(selsort#) = [1] p(c_1) = [1] p(c_2) = [2] p(c_3) = [0] p(c_4) = [1] x1 + [1] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [2] p(c_7) = [2] x1 + [0] p(c_8) = [1] p(c_9) = [1] x1 + [1] p(c_10) = [1] x1 + [2] x2 + [2] p(c_11) = [1] p(c_12) = [0] p(c_13) = [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] x1 + [0] p(c_16) = [4] p(c_17) = [0] p(c_18) = [4] x1 + [0] p(c_19) = [0] p(c_20) = [1] x1 + [1] x2 + [4] p(c_21) = [4] p(c_22) = [1] p(c_23) = [0] p(c_24) = [1] p(c_25) = [4] p(c_26) = [2] p(c_27) = [1] p(c_28) = [2] p(c_29) = [1] p(c_30) = [1] x1 + [1] x2 + [1] x3 + [4] x4 + [0] p(c_31) = [1] p(c_32) = [2] p(c_33) = [0] p(c_34) = [0] p(c_35) = [4] x1 + [2] x2 + [4] p(c_36) = [0] p(c_37) = [0] p(c_38) = [1] x1 + [4] p(c_39) = [1] p(c_40) = [1] x1 + [2] x2 + [2] x3 + [0] p(c_41) = [1] Following rules are strictly oriented: IFMIN#(true(),cons(z0,cons(z1,z2))) = [1] z2 + [14] > [1] z2 + [12] = c_6(MIN#(cons(z0,z2))) Following rules are (at-least) weakly oriented: IFMIN#(false(),cons(z0,cons(z1,z2))) = [1] z2 + [12] >= [1] z2 + [10] = c_5(MIN#(cons(z1,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) = [0] >= [0] = c_7(REPLACE#(z0,z1,z3)) IFSELSORT#(false(),cons(z0,z1)) = [1] z1 + [11] >= [1] z1 + [10] = MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) = [1] z1 + [11] >= [0] = REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) = [1] z1 + [11] >= [1] z1 + [7] = SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(false(),cons(z0,z1)) = [1] z1 + [11] >= [1] z1 + [11] = c_9(MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) = [1] z1 + [11] >= [1] z1 + [7] = SELSORT#(z1) MIN#(cons(z0,cons(z1,z2))) = [1] z2 + [14] >= [1] z2 + [14] = c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2)))) REPLACE#(z0,z1,cons(z2,z3)) = [0] >= [0] = c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3))) SELSORT#(cons(z0,z1)) = [1] z1 + [11] >= [4] = EQ#(z0,min(cons(z0,z1))) SELSORT#(cons(z0,z1)) = [1] z1 + [11] >= [1] z1 + [11] = IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) = [1] z1 + [11] >= [1] z1 + [10] = MIN#(cons(z0,z1)) ifrepl(false(),z0,z1,cons(z2,z3)) = [1] z3 + [4] >= [1] z3 + [4] = cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) = [1] z3 + [4] >= [1] z3 + [4] = cons(z1,z3) le(0(),z0) = [3] >= [3] = true() le(s(z0),0()) = [3] >= [2] = false() le(s(z0),s(z1)) = [3] >= [3] = le(z0,z1) replace(z0,z1,cons(z2,z3)) = [1] z3 + [4] >= [1] z3 + [4] = ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) = [1] >= [1] = nil() **** Step 1.b:6.b:1.a:6: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3))) - Weak DPs: IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) IFSELSORT#(false(),cons(z0,z1)) -> MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) -> SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) -> SELSORT#(z1) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2)))) SELSORT#(cons(z0,z1)) -> EQ#(z0,min(cons(z0,z1))) SELSORT#(cons(z0,z1)) -> IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) -> MIN#(cons(z0,z1)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/3,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1 ,c_16/0,c_17/0,c_18/1,c_19/0,c_20/3,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,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_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_15) = {1}, uargs(c_18) = {1} Following symbols are considered usable: {eq,ifrepl,replace,EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT#,eq#,ifmin#,ifrepl#,ifselsort# ,le#,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [0] p(EQ) = [1] p(IFMIN) = [1] x1 + [2] p(IFREPL) = [4] x1 + [1] x3 + [0] p(IFSELSORT) = [0] p(LE) = [1] x1 + [1] x2 + [1] p(MIN) = [2] p(REPLACE) = [2] x1 + [1] x2 + [1] p(SELSORT) = [0] p(c) = [0] p(c1) = [2] p(c10) = [1] p(c11) = [2] p(c12) = [1] p(c13) = [0] p(c14) = [1] p(c15) = [1] p(c16) = [1] p(c17) = [1] x3 + [0] p(c18) = [1] p(c19) = [0] p(c2) = [1] p(c20) = [1] x1 + [1] x2 + [1] x3 + [1] p(c3) = [4] p(c4) = [0] p(c5) = [1] p(c6) = [1] p(c7) = [0] p(c8) = [0] p(c9) = [1] p(cons) = [1] x2 + [1] p(eq) = [2] p(false) = [2] p(ifmin) = [5] p(ifrepl) = [1] x4 + [0] p(ifselsort) = [1] p(le) = [1] x2 + [0] p(min) = [1] x1 + [5] p(nil) = [4] p(replace) = [1] x3 + [0] p(s) = [0] p(selsort) = [1] p(true) = [0] p(EQ#) = [0] p(IFMIN#) = [0] p(IFREPL#) = [2] x1 + [2] x4 + [0] p(IFSELSORT#) = [2] x2 + [3] p(LE#) = [1] p(MIN#) = [0] p(REPLACE#) = [2] x3 + [5] p(SELSORT#) = [2] x1 + [3] p(eq#) = [1] x1 + [1] x2 + [1] p(ifmin#) = [1] p(ifrepl#) = [1] p(ifselsort#) = [4] x2 + [0] p(le#) = [1] x1 + [4] x2 + [0] p(min#) = [2] x1 + [4] p(replace#) = [0] p(selsort#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] p(c_4) = [4] p(c_5) = [1] x1 + [0] p(c_6) = [4] x1 + [0] p(c_7) = [1] x1 + [1] p(c_8) = [1] p(c_9) = [1] x1 + [2] p(c_10) = [2] x2 + [0] p(c_11) = [1] x1 + [1] p(c_12) = [1] p(c_13) = [4] p(c_14) = [1] p(c_15) = [2] x1 + [0] p(c_16) = [1] p(c_17) = [1] p(c_18) = [1] x1 + [0] p(c_19) = [0] p(c_20) = [4] p(c_21) = [4] p(c_22) = [0] p(c_23) = [4] p(c_24) = [0] p(c_25) = [2] x1 + [4] p(c_26) = [2] x1 + [2] p(c_27) = [0] p(c_28) = [1] x1 + [0] p(c_29) = [1] p(c_30) = [4] x2 + [0] p(c_31) = [1] x1 + [1] p(c_32) = [0] p(c_33) = [0] p(c_34) = [0] p(c_35) = [1] x1 + [2] x2 + [1] p(c_36) = [4] p(c_37) = [0] p(c_38) = [2] x1 + [4] p(c_39) = [2] p(c_40) = [1] x3 + [4] p(c_41) = [4] Following rules are strictly oriented: REPLACE#(z0,z1,cons(z2,z3)) = [2] z3 + [7] > [2] z3 + [6] = c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3))) Following rules are (at-least) weakly oriented: IFMIN#(false(),cons(z0,cons(z1,z2))) = [0] >= [0] = c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) = [0] >= [0] = c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) = [2] z3 + [6] >= [2] z3 + [6] = c_7(REPLACE#(z0,z1,z3)) IFSELSORT#(false(),cons(z0,z1)) = [2] z1 + [5] >= [0] = MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) = [2] z1 + [5] >= [2] z1 + [5] = REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) = [2] z1 + [5] >= [2] z1 + [3] = SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(false(),cons(z0,z1)) = [2] z1 + [5] >= [2] = c_9(MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) = [2] z1 + [5] >= [2] z1 + [3] = SELSORT#(z1) MIN#(cons(z0,cons(z1,z2))) = [0] >= [0] = c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2)))) SELSORT#(cons(z0,z1)) = [2] z1 + [5] >= [0] = EQ#(z0,min(cons(z0,z1))) SELSORT#(cons(z0,z1)) = [2] z1 + [5] >= [2] z1 + [5] = IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) = [2] z1 + [5] >= [0] = MIN#(cons(z0,z1)) eq(0(),0()) = [2] >= [0] = true() eq(0(),s(z0)) = [2] >= [2] = false() eq(s(z0),0()) = [2] >= [2] = false() eq(s(z0),s(z1)) = [2] >= [2] = eq(z0,z1) ifrepl(false(),z0,z1,cons(z2,z3)) = [1] z3 + [1] >= [1] z3 + [1] = cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) = [1] z3 + [1] >= [1] z3 + [1] = cons(z1,z3) replace(z0,z1,cons(z2,z3)) = [1] z3 + [1] >= [1] z3 + [1] = ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) = [4] >= [4] = nil() **** Step 1.b:6.b:1.a:7: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) - Weak DPs: IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) IFSELSORT#(false(),cons(z0,z1)) -> MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) -> SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) -> SELSORT#(z1) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2)))) REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3))) SELSORT#(cons(z0,z1)) -> EQ#(z0,min(cons(z0,z1))) SELSORT#(cons(z0,z1)) -> IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) -> MIN#(cons(z0,z1)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/3,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1 ,c_16/0,c_17/0,c_18/1,c_19/0,c_20/3,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,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_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_15) = {1}, uargs(c_18) = {1} Following symbols are considered usable: {eq,ifrepl,replace,EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT#,eq#,ifmin#,ifrepl#,ifselsort# ,le#,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [0] p(EQ) = [0] p(IFMIN) = [0] p(IFREPL) = [0] p(IFSELSORT) = [0] p(LE) = [0] p(MIN) = [0] p(REPLACE) = [0] p(SELSORT) = [0] p(c) = [0] p(c1) = [0] p(c10) = [1] x1 + [0] p(c11) = [1] x1 + [0] p(c12) = [0] p(c13) = [1] x1 + [1] x2 + [0] p(c14) = [0] p(c15) = [1] x1 + [0] p(c16) = [0] p(c17) = [1] x2 + [1] x3 + [0] p(c18) = [1] x1 + [2] p(c19) = [1] x1 + [0] p(c2) = [0] p(c20) = [1] x3 + [0] p(c3) = [4] p(c4) = [0] p(c5) = [1] p(c6) = [0] p(c7) = [0] p(c8) = [1] p(c9) = [1] x2 + [0] p(cons) = [1] x2 + [1] p(eq) = [1] p(false) = [1] p(ifmin) = [4] x1 + [2] p(ifrepl) = [1] x4 + [0] p(ifselsort) = [1] x1 + [0] p(le) = [3] p(min) = [5] p(nil) = [6] p(replace) = [1] x3 + [0] p(s) = [0] p(selsort) = [4] x1 + [0] p(true) = [0] p(EQ#) = [0] p(IFMIN#) = [0] p(IFREPL#) = [4] x1 + [1] x4 + [0] p(IFSELSORT#) = [1] x2 + [4] p(LE#) = [0] p(MIN#) = [0] p(REPLACE#) = [1] x3 + [4] p(SELSORT#) = [1] x1 + [4] p(eq#) = [0] p(ifmin#) = [1] x2 + [0] p(ifrepl#) = [1] x1 + [1] x3 + [0] p(ifselsort#) = [1] x2 + [1] p(le#) = [4] x1 + [0] p(min#) = [4] p(replace#) = [1] x1 + [2] x2 + [2] p(selsort#) = [1] x1 + [1] p(c_1) = [0] p(c_2) = [4] p(c_3) = [0] p(c_4) = [2] p(c_5) = [4] x1 + [0] p(c_6) = [2] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [2] x1 + [1] x2 + [2] p(c_11) = [1] p(c_12) = [2] p(c_13) = [0] p(c_14) = [1] x1 + [2] p(c_15) = [4] x1 + [0] p(c_16) = [1] p(c_17) = [4] p(c_18) = [1] x1 + [0] p(c_19) = [0] p(c_20) = [1] x2 + [4] x3 + [1] p(c_21) = [2] p(c_22) = [1] p(c_23) = [1] p(c_24) = [0] p(c_25) = [4] p(c_26) = [1] p(c_27) = [2] x1 + [0] p(c_28) = [1] x1 + [0] p(c_29) = [1] p(c_30) = [4] x4 + [0] p(c_31) = [1] p(c_32) = [1] p(c_33) = [4] p(c_34) = [1] p(c_35) = [1] x1 + [1] p(c_36) = [2] p(c_37) = [0] p(c_38) = [0] p(c_39) = [2] p(c_40) = [1] x1 + [1] x2 + [4] x3 + [0] p(c_41) = [0] Following rules are strictly oriented: IFREPL#(false(),z0,z1,cons(z2,z3)) = [1] z3 + [5] > [1] z3 + [4] = c_7(REPLACE#(z0,z1,z3)) Following rules are (at-least) weakly oriented: IFMIN#(false(),cons(z0,cons(z1,z2))) = [0] >= [0] = c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) = [0] >= [0] = c_6(MIN#(cons(z0,z2))) IFSELSORT#(false(),cons(z0,z1)) = [1] z1 + [5] >= [0] = MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) = [1] z1 + [5] >= [1] z1 + [4] = REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) = [1] z1 + [5] >= [1] z1 + [4] = SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(false(),cons(z0,z1)) = [1] z1 + [5] >= [0] = c_9(MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) = [1] z1 + [5] >= [1] z1 + [4] = SELSORT#(z1) MIN#(cons(z0,cons(z1,z2))) = [0] >= [0] = c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2)))) REPLACE#(z0,z1,cons(z2,z3)) = [1] z3 + [5] >= [1] z3 + [5] = c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3))) SELSORT#(cons(z0,z1)) = [1] z1 + [5] >= [0] = EQ#(z0,min(cons(z0,z1))) SELSORT#(cons(z0,z1)) = [1] z1 + [5] >= [1] z1 + [5] = IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) = [1] z1 + [5] >= [0] = MIN#(cons(z0,z1)) eq(0(),0()) = [1] >= [0] = true() eq(0(),s(z0)) = [1] >= [1] = false() eq(s(z0),0()) = [1] >= [1] = false() eq(s(z0),s(z1)) = [1] >= [1] = eq(z0,z1) ifrepl(false(),z0,z1,cons(z2,z3)) = [1] z3 + [1] >= [1] z3 + [1] = cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) = [1] z3 + [1] >= [1] z3 + [1] = cons(z1,z3) replace(z0,z1,cons(z2,z3)) = [1] z3 + [1] >= [1] z3 + [1] = ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) = [6] >= [6] = nil() **** Step 1.b:6.b:1.a:8: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: IFMIN#(false(),cons(z0,cons(z1,z2))) -> c_5(MIN#(cons(z1,z2))) IFMIN#(true(),cons(z0,cons(z1,z2))) -> c_6(MIN#(cons(z0,z2))) IFREPL#(false(),z0,z1,cons(z2,z3)) -> c_7(REPLACE#(z0,z1,z3)) IFSELSORT#(false(),cons(z0,z1)) -> MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) -> SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> c_9(MIN#(cons(z0,z1))) IFSELSORT#(true(),cons(z0,z1)) -> SELSORT#(z1) MIN#(cons(z0,cons(z1,z2))) -> c_15(IFMIN#(le(z0,z1),cons(z0,cons(z1,z2)))) REPLACE#(z0,z1,cons(z2,z3)) -> c_18(IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3))) SELSORT#(cons(z0,z1)) -> EQ#(z0,min(cons(z0,z1))) SELSORT#(cons(z0,z1)) -> IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) -> MIN#(cons(z0,z1)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/3,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1 ,c_16/0,c_17/0,c_18/1,c_19/0,c_20/3,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 1.b:6.b:1.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)) LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)) - Weak DPs: IFMIN#(false(),cons(z0,cons(z1,z2))) -> MIN#(cons(z1,z2)) IFMIN#(true(),cons(z0,cons(z1,z2))) -> MIN#(cons(z0,z2)) IFREPL#(false(),z0,z1,cons(z2,z3)) -> REPLACE#(z0,z1,z3) IFSELSORT#(false(),cons(z0,z1)) -> MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) -> SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(true(),cons(z0,z1)) -> SELSORT#(z1) MIN#(cons(z0,cons(z1,z2))) -> IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))) MIN#(cons(z0,cons(z1,z2))) -> LE#(z0,z1) REPLACE#(z0,z1,cons(z2,z3)) -> EQ#(z0,z2) REPLACE#(z0,z1,cons(z2,z3)) -> IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)) SELSORT#(cons(z0,z1)) -> EQ#(z0,min(cons(z0,z1))) SELSORT#(cons(z0,z1)) -> IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) -> MIN#(cons(z0,z1)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/3,c_11/1,c_12/0,c_13/0,c_14/1,c_15/2 ,c_16/0,c_17/0,c_18/2,c_19/0,c_20/3,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_4) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {ifmin,ifrepl,min,replace,EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT#,eq#,ifmin#,ifrepl# ,ifselsort#,le#,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [0] p(EQ) = [0] p(IFMIN) = [4] x1 + [4] x2 + [0] p(IFREPL) = [4] x1 + [1] x3 + [1] x4 + [1] p(IFSELSORT) = [1] x1 + [0] p(LE) = [4] x2 + [0] p(MIN) = [4] x1 + [2] p(REPLACE) = [1] x2 + [2] x3 + [4] p(SELSORT) = [0] p(c) = [0] p(c1) = [1] p(c10) = [1] x1 + [4] p(c11) = [0] p(c12) = [1] p(c13) = [4] p(c14) = [2] p(c15) = [1] x1 + [0] p(c16) = [0] p(c17) = [1] x2 + [1] x3 + [1] p(c18) = [1] x1 + [1] p(c19) = [0] p(c2) = [0] p(c20) = [1] x2 + [4] p(c3) = [1] x1 + [2] p(c4) = [0] p(c5) = [0] p(c6) = [1] x1 + [0] p(c7) = [4] p(c8) = [4] p(c9) = [1] x2 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(eq) = [2] x1 + [0] p(false) = [0] p(ifmin) = [1] x2 + [0] p(ifrepl) = [1] x3 + [1] x4 + [0] p(ifselsort) = [2] x2 + [4] p(le) = [4] p(min) = [1] x1 + [0] p(nil) = [4] p(replace) = [1] x2 + [1] x3 + [0] p(s) = [1] x1 + [3] p(selsort) = [1] p(true) = [0] p(EQ#) = [4] x2 + [1] p(IFMIN#) = [2] x2 + [1] p(IFREPL#) = [4] x4 + [1] p(IFSELSORT#) = [4] x2 + [1] p(LE#) = [2] x1 + [2] x2 + [1] p(MIN#) = [2] x1 + [1] p(REPLACE#) = [4] x3 + [1] p(SELSORT#) = [4] x1 + [1] p(eq#) = [1] x2 + [0] p(ifmin#) = [1] x1 + [4] x2 + [0] p(ifrepl#) = [1] x2 + [1] x4 + [2] p(ifselsort#) = [2] p(le#) = [4] x1 + [4] x2 + [0] p(min#) = [2] x1 + [4] p(replace#) = [1] x2 + [0] p(selsort#) = [2] p(c_1) = [2] p(c_2) = [0] p(c_3) = [1] p(c_4) = [1] x1 + [7] p(c_5) = [1] x1 + [1] p(c_6) = [0] p(c_7) = [1] p(c_8) = [1] p(c_9) = [1] p(c_10) = [2] x2 + [1] x3 + [1] p(c_11) = [1] x1 + [4] p(c_12) = [0] p(c_13) = [0] p(c_14) = [1] x1 + [2] p(c_15) = [1] x2 + [1] p(c_16) = [0] p(c_17) = [1] p(c_18) = [1] p(c_19) = [0] p(c_20) = [4] p(c_21) = [1] p(c_22) = [0] p(c_23) = [1] p(c_24) = [2] p(c_25) = [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [0] p(c_29) = [4] p(c_30) = [1] x1 + [2] x2 + [1] p(c_31) = [1] p(c_32) = [0] p(c_33) = [0] p(c_34) = [0] p(c_35) = [0] p(c_36) = [0] p(c_37) = [0] p(c_38) = [0] p(c_39) = [2] p(c_40) = [1] x2 + [0] p(c_41) = [1] Following rules are strictly oriented: EQ#(s(z0),s(z1)) = [4] z1 + [13] > [4] z1 + [8] = c_4(EQ#(z0,z1)) LE#(s(z0),s(z1)) = [2] z0 + [2] z1 + [13] > [2] z0 + [2] z1 + [3] = c_14(LE#(z0,z1)) Following rules are (at-least) weakly oriented: IFMIN#(false(),cons(z0,cons(z1,z2))) = [2] z0 + [2] z1 + [2] z2 + [1] >= [2] z1 + [2] z2 + [1] = MIN#(cons(z1,z2)) IFMIN#(true(),cons(z0,cons(z1,z2))) = [2] z0 + [2] z1 + [2] z2 + [1] >= [2] z0 + [2] z2 + [1] = MIN#(cons(z0,z2)) IFREPL#(false(),z0,z1,cons(z2,z3)) = [4] z2 + [4] z3 + [1] >= [4] z3 + [1] = REPLACE#(z0,z1,z3) IFSELSORT#(false(),cons(z0,z1)) = [4] z0 + [4] z1 + [1] >= [2] z0 + [2] z1 + [1] = MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) = [4] z0 + [4] z1 + [1] >= [4] z1 + [1] = REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) = [4] z0 + [4] z1 + [1] >= [4] z0 + [4] z1 + [1] = SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(true(),cons(z0,z1)) = [4] z0 + [4] z1 + [1] >= [4] z1 + [1] = SELSORT#(z1) MIN#(cons(z0,cons(z1,z2))) = [2] z0 + [2] z1 + [2] z2 + [1] >= [2] z0 + [2] z1 + [2] z2 + [1] = IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))) MIN#(cons(z0,cons(z1,z2))) = [2] z0 + [2] z1 + [2] z2 + [1] >= [2] z0 + [2] z1 + [1] = LE#(z0,z1) REPLACE#(z0,z1,cons(z2,z3)) = [4] z2 + [4] z3 + [1] >= [4] z2 + [1] = EQ#(z0,z2) REPLACE#(z0,z1,cons(z2,z3)) = [4] z2 + [4] z3 + [1] >= [4] z2 + [4] z3 + [1] = IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)) SELSORT#(cons(z0,z1)) = [4] z0 + [4] z1 + [1] >= [4] z0 + [4] z1 + [1] = EQ#(z0,min(cons(z0,z1))) SELSORT#(cons(z0,z1)) = [4] z0 + [4] z1 + [1] >= [4] z0 + [4] z1 + [1] = IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) = [4] z0 + [4] z1 + [1] >= [2] z0 + [2] z1 + [1] = MIN#(cons(z0,z1)) ifmin(false(),cons(z0,cons(z1,z2))) = [1] z0 + [1] z1 + [1] z2 + [0] >= [1] z1 + [1] z2 + [0] = min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) = [1] z0 + [1] z1 + [1] z2 + [0] >= [1] z0 + [1] z2 + [0] = min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) = [1] z1 + [1] z2 + [1] z3 + [0] >= [1] z1 + [1] z2 + [1] z3 + [0] = cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) = [1] z1 + [1] z2 + [1] z3 + [0] >= [1] z1 + [1] z3 + [0] = cons(z1,z3) min(cons(z0,cons(z1,z2))) = [1] z0 + [1] z1 + [1] z2 + [0] >= [1] z0 + [1] z1 + [1] z2 + [0] = ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) = [4] >= [0] = 0() min(cons(s(z0),nil())) = [1] z0 + [7] >= [1] z0 + [3] = s(z0) replace(z0,z1,cons(z2,z3)) = [1] z1 + [1] z2 + [1] z3 + [0] >= [1] z1 + [1] z2 + [1] z3 + [0] = ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) = [1] z1 + [4] >= [4] = nil() **** Step 1.b:6.b:1.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: EQ#(s(z0),s(z1)) -> c_4(EQ#(z0,z1)) IFMIN#(false(),cons(z0,cons(z1,z2))) -> MIN#(cons(z1,z2)) IFMIN#(true(),cons(z0,cons(z1,z2))) -> MIN#(cons(z0,z2)) IFREPL#(false(),z0,z1,cons(z2,z3)) -> REPLACE#(z0,z1,z3) IFSELSORT#(false(),cons(z0,z1)) -> MIN#(cons(z0,z1)) IFSELSORT#(false(),cons(z0,z1)) -> REPLACE#(min(cons(z0,z1)),z0,z1) IFSELSORT#(false(),cons(z0,z1)) -> SELSORT#(replace(min(cons(z0,z1)),z0,z1)) IFSELSORT#(true(),cons(z0,z1)) -> SELSORT#(z1) LE#(s(z0),s(z1)) -> c_14(LE#(z0,z1)) MIN#(cons(z0,cons(z1,z2))) -> IFMIN#(le(z0,z1),cons(z0,cons(z1,z2))) MIN#(cons(z0,cons(z1,z2))) -> LE#(z0,z1) REPLACE#(z0,z1,cons(z2,z3)) -> EQ#(z0,z2) REPLACE#(z0,z1,cons(z2,z3)) -> IFREPL#(eq(z0,z2),z0,z1,cons(z2,z3)) SELSORT#(cons(z0,z1)) -> EQ#(z0,min(cons(z0,z1))) SELSORT#(cons(z0,z1)) -> IFSELSORT#(eq(z0,min(cons(z0,z1))),cons(z0,z1)) SELSORT#(cons(z0,z1)) -> MIN#(cons(z0,z1)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(z0)) -> false() eq(s(z0),0()) -> false() eq(s(z0),s(z1)) -> eq(z0,z1) ifmin(false(),cons(z0,cons(z1,z2))) -> min(cons(z1,z2)) ifmin(true(),cons(z0,cons(z1,z2))) -> min(cons(z0,z2)) ifrepl(false(),z0,z1,cons(z2,z3)) -> cons(z2,replace(z0,z1,z3)) ifrepl(true(),z0,z1,cons(z2,z3)) -> cons(z1,z3) le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) min(cons(z0,cons(z1,z2))) -> ifmin(le(z0,z1),cons(z0,cons(z1,z2))) min(cons(0(),nil())) -> 0() min(cons(s(z0),nil())) -> s(z0) replace(z0,z1,cons(z2,z3)) -> ifrepl(eq(z0,z2),z0,z1,cons(z2,z3)) replace(z0,z1,nil()) -> nil() - Signature: {EQ/2,IFMIN/2,IFREPL/4,IFSELSORT/2,LE/2,MIN/1,REPLACE/3,SELSORT/1,eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2 ,min/1,replace/3,selsort/1,EQ#/2,IFMIN#/2,IFREPL#/4,IFSELSORT#/2,LE#/2,MIN#/1,REPLACE#/3,SELSORT#/1,eq#/2 ,ifmin#/2,ifrepl#/4,ifselsort#/2,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,c/0,c1/0,c10/1,c11/1,c12/0,c13/2 ,c14/0,c15/1,c16/0,c17/3,c18/1,c19/1,c2/0,c20/3,c3/1,c4/0,c5/0,c6/1,c7/0,c8/0,c9/2,cons/2,false/0,nil/0,s/1 ,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/3,c_11/1,c_12/0,c_13/0,c_14/1,c_15/2 ,c_16/0,c_17/0,c_18/2,c_19/0,c_20/3,c_21/0,c_22/0,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/1,c_29/0,c_30/4 ,c_31/1,c_32/0,c_33/0,c_34/1,c_35/2,c_36/0,c_37/0,c_38/2,c_39/0,c_40/3,c_41/0} - Obligation: innermost runtime complexity wrt. defined symbols {EQ#,IFMIN#,IFREPL#,IFSELSORT#,LE#,MIN#,REPLACE#,SELSORT# ,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15 ,c16,c17,c18,c19,c2,c20,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,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))